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ONote.repr_opow_aux₁ ** e a : ONote Ne : NF e Na : NF a a' : Ordinal.{0} e0 : repr e ≠ 0 h : a' < ω ^ repr e aa : repr a = a' n : ℕ+ ⊢ (ω ^ repr e * ↑↑n + a') ^ ω = (ω ^ repr e) ^ ω ** subst aa ** e a : ONote Ne : NF e Na : NF a e0 : repr e ≠ 0 n : ℕ+ h : repr a < ω ^ repr e ⊢ (ω ^ repr e * ↑↑n + repr a) ^ ω = (ω ^ repr e) ^ ω ** have No := Ne.oadd n (Na.below_of_lt' h) ** e a : ONote Ne : NF e Na : NF a e0 : repr e ≠ 0 n : ℕ+ h : repr a < ω ^ repr e No : NF (oadd e n a) ⊢ (ω ^ repr e * ↑↑n + repr a) ^ ω = (ω ^ repr e) ^ ω ** have := omega_le_oadd e n a ** e a : ONote Ne : NF e Na : NF a e0 : repr e ≠ 0 n : ℕ+ h : repr a < ω ^ repr e No : NF (oadd e n a) this : ω ^ repr e ≤ repr (oadd e n a) ⊢ (ω ^ repr e * ↑↑n + repr a) ^ ω = (ω ^ repr e) ^ ω ** rw [repr] at this ** e a : ONote Ne : NF e Na : NF a e0 : repr e ≠ 0 n : ℕ+ h : repr a < ω ^ repr e No : NF (oadd e n a) this : ω ^ repr e ≤ ω ^ repr e * ↑↑n + repr a ⊢ (ω ^ repr e * ↑↑n + repr a) ^ ω = (ω ^ repr e) ^ ω ** refine' le_antisymm _ (opow_le_opow_left _ this) ** e a : ONote Ne : NF e Na : NF a e0 : repr e ≠ 0 n : ℕ+ h : repr a < ω ^ repr e No : NF (oadd e n a) this : ω ^ repr e ≤ ω ^ repr e * ↑↑n + repr a ⊢ (ω ^ repr e * ↑↑n + repr a) ^ ω ≤ (ω ^ repr e) ^ ω ** apply (opow_le_of_limit ((opow_pos _ omega_pos).trans_le this).ne' omega_isLimit).2 ** e a : ONote Ne : NF e Na : NF a e0 : repr e ≠ 0 n : ℕ+ h : repr a < ω ^ repr e No : NF (oadd e n a) this : ω ^ repr e ≤ ω ^ repr e * ↑↑n + repr a ⊢ ∀ (b' : Ordinal.{0}), b' < ω → (ω ^ repr e * ↑↑n + repr a) ^ b' ≤ (ω ^ repr e) ^ ω ** intro b l ** e a : ONote Ne : NF e Na : NF a e0 : repr e ≠ 0 n : ℕ+ h : repr a < ω ^ repr e No : NF (oadd e n a) this : ω ^ repr e ≤ ω ^ repr e * ↑↑n + repr a b : Ordinal.{0} l : b < ω ⊢ (ω ^ repr e * ↑↑n + repr a) ^ b ≤ (ω ^ repr e) ^ ω ** have := (No.below_of_lt (lt_succ _)).repr_lt ** e a : ONote Ne : NF e Na : NF a e0 : repr e ≠ 0 n : ℕ+ h : repr a < ω ^ repr e No : NF (oadd e n a) this✝ : ω ^ repr e ≤ ω ^ repr e * ↑↑n + repr a b : Ordinal.{0} l : b < ω this : repr (oadd e n a) < ω ^ succ (repr e) ⊢ (ω ^ repr e * ↑↑n + repr a) ^ b ≤ (ω ^ repr e) ^ ω ** rw [repr] at this ** e a : ONote Ne : NF e Na : NF a e0 : repr e ≠ 0 n : ℕ+ h : repr a < ω ^ repr e No : NF (oadd e n a) this✝ : ω ^ repr e ≤ ω ^ repr e * ↑↑n + repr a b : Ordinal.{0} l : b < ω this : ω ^ repr e * ↑↑n + repr a < ω ^ succ (repr e) ⊢ (ω ^ repr e * ↑↑n + repr a) ^ b ≤ (ω ^ repr e) ^ ω ** apply (opow_le_opow_left b <| this.le).trans ** e a : ONote Ne : NF e Na : NF a e0 : repr e ≠ 0 n : ℕ+ h : repr a < ω ^ repr e No : NF (oadd e n a) this✝ : ω ^ repr e ≤ ω ^ repr e * ↑↑n + repr a b : Ordinal.{0} l : b < ω this : ω ^ repr e * ↑↑n + repr a < ω ^ succ (repr e) ⊢ (ω ^ succ (repr e)) ^ b ≤ (ω ^ repr e) ^ ω ** rw [← opow_mul, ← opow_mul] ** e a : ONote Ne : NF e Na : NF a e0 : repr e ≠ 0 n : ℕ+ h : repr a < ω ^ repr e No : NF (oadd e n a) this✝ : ω ^ repr e ≤ ω ^ repr e * ↑↑n + repr a b : Ordinal.{0} l : b < ω this : ω ^ repr e * ↑↑n + repr a < ω ^ succ (repr e) ⊢ ω ^ (succ (repr e) * b) ≤ ω ^ (repr e * ω) ** apply opow_le_opow_right omega_pos ** e a : ONote Ne : NF e Na : NF a e0 : repr e ≠ 0 n : ℕ+ h : repr a < ω ^ repr e No : NF (oadd e n a) this✝ : ω ^ repr e ≤ ω ^ repr e * ↑↑n + repr a b : Ordinal.{0} l : b < ω this : ω ^ repr e * ↑↑n + repr a < ω ^ succ (repr e) ⊢ succ (repr e) * b ≤ repr e * ω ** cases' le_or_lt ω (repr e) with h h ** case inl e a : ONote Ne : NF e Na : NF a e0 : repr e ≠ 0 n : ℕ+ h✝ : repr a < ω ^ repr e No : NF (oadd e n a) this✝ : ω ^ repr e ≤ ω ^ repr e * ↑↑n + repr a b : Ordinal.{0} l : b < ω this : ω ^ repr e * ↑↑n + repr a < ω ^ succ (repr e) h : ω ≤ repr e ⊢ succ (repr e) * b ≤ repr e * ω ** apply (mul_le_mul_left' (le_succ b) _).trans ** case inl e a : ONote Ne : NF e Na : NF a e0 : repr e ≠ 0 n : ℕ+ h✝ : repr a < ω ^ repr e No : NF (oadd e n a) this✝ : ω ^ repr e ≤ ω ^ repr e * ↑↑n + repr a b : Ordinal.{0} l : b < ω this : ω ^ repr e * ↑↑n + repr a < ω ^ succ (repr e) h : ω ≤ repr e ⊢ succ (repr e) * succ b ≤ repr e * ω ** rw [← add_one_eq_succ, add_mul_succ _ (one_add_of_omega_le h), add_one_eq_succ, succ_le_iff,
Ordinal.mul_lt_mul_iff_left (Ordinal.pos_iff_ne_zero.2 e0)] ** case inl e a : ONote Ne : NF e Na : NF a e0 : repr e ≠ 0 n : ℕ+ h✝ : repr a < ω ^ repr e No : NF (oadd e n a) this✝ : ω ^ repr e ≤ ω ^ repr e * ↑↑n + repr a b : Ordinal.{0} l : b < ω this : ω ^ repr e * ↑↑n + repr a < ω ^ succ (repr e) h : ω ≤ repr e ⊢ succ b < ω ** exact omega_isLimit.2 _ l ** case inr e a : ONote Ne : NF e Na : NF a e0 : repr e ≠ 0 n : ℕ+ h✝ : repr a < ω ^ repr e No : NF (oadd e n a) this✝ : ω ^ repr e ≤ ω ^ repr e * ↑↑n + repr a b : Ordinal.{0} l : b < ω this : ω ^ repr e * ↑↑n + repr a < ω ^ succ (repr e) h : repr e < ω ⊢ succ (repr e) * b ≤ repr e * ω ** apply (principal_mul_omega (omega_isLimit.2 _ h) l).le.trans ** case inr e a : ONote Ne : NF e Na : NF a e0 : repr e ≠ 0 n : ℕ+ h✝ : repr a < ω ^ repr e No : NF (oadd e n a) this✝ : ω ^ repr e ≤ ω ^ repr e * ↑↑n + repr a b : Ordinal.{0} l : b < ω this : ω ^ repr e * ↑↑n + repr a < ω ^ succ (repr e) h : repr e < ω ⊢ ω ≤ repr e * ω ** simpa using mul_le_mul_right' (one_le_iff_ne_zero.2 e0) ω ** Qed | |
ONote.repr_opow_aux₂ ** a0 a' : ONote N0 : NF a0 Na' : NF a' m : ℕ d : ω ∣ repr a' e0 : repr a0 ≠ 0 h : repr a' + ↑m < ω ^ repr a0 n : ℕ+ k : ℕ ⊢ let R := repr (opowAux 0 a0 (oadd a0 n a' * ↑m) k m); (k ≠ 0 → R < (ω ^ repr a0) ^ succ ↑k) ∧ (ω ^ repr a0) ^ ↑k * (ω ^ repr a0 * ↑↑n + repr a') + R = (ω ^ repr a0 * ↑↑n + repr a' + ↑m) ^ succ ↑k ** intro R' ** a0 a' : ONote N0 : NF a0 Na' : NF a' m : ℕ d : ω ∣ repr a' e0 : repr a0 ≠ 0 h : repr a' + ↑m < ω ^ repr a0 n : ℕ+ k : ℕ R' : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑m) k m) ⊢ (k ≠ 0 → R' < (ω ^ repr a0) ^ succ ↑k) ∧ (ω ^ repr a0) ^ ↑k * (ω ^ repr a0 * ↑↑n + repr a') + R' = (ω ^ repr a0 * ↑↑n + repr a' + ↑m) ^ succ ↑k ** haveI No : NF (oadd a0 n a') :=
N0.oadd n (Na'.below_of_lt' <| lt_of_le_of_lt (le_add_right _ _) h) ** a0 a' : ONote N0 : NF a0 Na' : NF a' m : ℕ d : ω ∣ repr a' e0 : repr a0 ≠ 0 h : repr a' + ↑m < ω ^ repr a0 n : ℕ+ k : ℕ R' : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑m) k m) No : NF (oadd a0 n a') ⊢ (k ≠ 0 → R' < (ω ^ repr a0) ^ succ ↑k) ∧ (ω ^ repr a0) ^ ↑k * (ω ^ repr a0 * ↑↑n + repr a') + R' = (ω ^ repr a0 * ↑↑n + repr a' + ↑m) ^ succ ↑k ** induction' k with k IH ** case succ a0 a' : ONote N0 : NF a0 Na' : NF a' m : ℕ d : ω ∣ repr a' e0 : repr a0 ≠ 0 h : repr a' + ↑m < ω ^ repr a0 n : ℕ+ k✝ : ℕ R'✝ : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑m) k✝ m) No : NF (oadd a0 n a') k : ℕ IH : let R' := repr (opowAux 0 a0 (oadd a0 n a' * ↑m) k m); (k ≠ 0 → R' < (ω ^ repr a0) ^ succ ↑k) ∧ (ω ^ repr a0) ^ ↑k * (ω ^ repr a0 * ↑↑n + repr a') + R' = (ω ^ repr a0 * ↑↑n + repr a' + ↑m) ^ succ ↑k R' : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑m) (Nat.succ k) m) ⊢ (Nat.succ k ≠ 0 → R' < (ω ^ repr a0) ^ succ ↑(Nat.succ k)) ∧ (ω ^ repr a0) ^ ↑(Nat.succ k) * (ω ^ repr a0 * ↑↑n + repr a') + R' = (ω ^ repr a0 * ↑↑n + repr a' + ↑m) ^ succ ↑(Nat.succ k) ** let R := repr (opowAux 0 a0 (oadd a0 n a' * ofNat m) k m) ** case succ a0 a' : ONote N0 : NF a0 Na' : NF a' m : ℕ d : ω ∣ repr a' e0 : repr a0 ≠ 0 h : repr a' + ↑m < ω ^ repr a0 n : ℕ+ k✝ : ℕ R'✝ : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑m) k✝ m) No : NF (oadd a0 n a') k : ℕ IH : let R' := repr (opowAux 0 a0 (oadd a0 n a' * ↑m) k m); (k ≠ 0 → R' < (ω ^ repr a0) ^ succ ↑k) ∧ (ω ^ repr a0) ^ ↑k * (ω ^ repr a0 * ↑↑n + repr a') + R' = (ω ^ repr a0 * ↑↑n + repr a' + ↑m) ^ succ ↑k R' : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑m) (Nat.succ k) m) R : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑m) k m) ⊢ (Nat.succ k ≠ 0 → R' < (ω ^ repr a0) ^ succ ↑(Nat.succ k)) ∧ (ω ^ repr a0) ^ ↑(Nat.succ k) * (ω ^ repr a0 * ↑↑n + repr a') + R' = (ω ^ repr a0 * ↑↑n + repr a' + ↑m) ^ succ ↑(Nat.succ k) ** let ω0 := ω ^ repr a0 ** case succ a0 a' : ONote N0 : NF a0 Na' : NF a' m : ℕ d : ω ∣ repr a' e0 : repr a0 ≠ 0 h : repr a' + ↑m < ω ^ repr a0 n : ℕ+ k✝ : ℕ R'✝ : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑m) k✝ m) No : NF (oadd a0 n a') k : ℕ IH : let R' := repr (opowAux 0 a0 (oadd a0 n a' * ↑m) k m); (k ≠ 0 → R' < (ω ^ repr a0) ^ succ ↑k) ∧ (ω ^ repr a0) ^ ↑k * (ω ^ repr a0 * ↑↑n + repr a') + R' = (ω ^ repr a0 * ↑↑n + repr a' + ↑m) ^ succ ↑k R' : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑m) (Nat.succ k) m) R : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑m) k m) ω0 : Ordinal.{0} := ω ^ repr a0 ⊢ (Nat.succ k ≠ 0 → R' < (ω ^ repr a0) ^ succ ↑(Nat.succ k)) ∧ (ω ^ repr a0) ^ ↑(Nat.succ k) * (ω ^ repr a0 * ↑↑n + repr a') + R' = (ω ^ repr a0 * ↑↑n + repr a' + ↑m) ^ succ ↑(Nat.succ k) ** let α' := ω0 * n + repr a' ** case succ a0 a' : ONote N0 : NF a0 Na' : NF a' m : ℕ d : ω ∣ repr a' e0 : repr a0 ≠ 0 h : repr a' + ↑m < ω ^ repr a0 n : ℕ+ k✝ : ℕ R'✝ : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑m) k✝ m) No : NF (oadd a0 n a') k : ℕ IH : let R' := repr (opowAux 0 a0 (oadd a0 n a' * ↑m) k m); (k ≠ 0 → R' < (ω ^ repr a0) ^ succ ↑k) ∧ (ω ^ repr a0) ^ ↑k * (ω ^ repr a0 * ↑↑n + repr a') + R' = (ω ^ repr a0 * ↑↑n + repr a' + ↑m) ^ succ ↑k R' : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑m) (Nat.succ k) m) R : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑m) k m) ω0 : Ordinal.{0} := ω ^ repr a0 α' : Ordinal.{0} := ω0 * ↑↑n + repr a' ⊢ (Nat.succ k ≠ 0 → R' < (ω ^ repr a0) ^ succ ↑(Nat.succ k)) ∧ (ω ^ repr a0) ^ ↑(Nat.succ k) * (ω ^ repr a0 * ↑↑n + repr a') + R' = (ω ^ repr a0 * ↑↑n + repr a' + ↑m) ^ succ ↑(Nat.succ k) ** change (k ≠ 0 → R < (ω0 ^ succ ↑k)) ∧ (ω0 ^ k) * α' + R = (α' + m) ^ succ ↑k at IH ** case succ a0 a' : ONote N0 : NF a0 Na' : NF a' m : ℕ d : ω ∣ repr a' e0 : repr a0 ≠ 0 h : repr a' + ↑m < ω ^ repr a0 n : ℕ+ k✝ : ℕ R'✝ : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑m) k✝ m) No : NF (oadd a0 n a') k : ℕ R' : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑m) (Nat.succ k) m) R : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑m) k m) ω0 : Ordinal.{0} := ω ^ repr a0 α' : Ordinal.{0} := ω0 * ↑↑n + repr a' IH : (k ≠ 0 → R < ω0 ^ succ ↑k) ∧ ω0 ^ ↑k * α' + R = (α' + ↑m) ^ succ ↑k RR : R' = ω0 ^ ↑k * (α' * ↑m) + R ⊢ (Nat.succ k ≠ 0 → R' < (ω ^ repr a0) ^ succ ↑(Nat.succ k)) ∧ (ω ^ repr a0) ^ ↑(Nat.succ k) * (ω ^ repr a0 * ↑↑n + repr a') + R' = (ω ^ repr a0 * ↑↑n + repr a' + ↑m) ^ succ ↑(Nat.succ k) ** have α0 : 0 < α' := by simpa [lt_def, repr] using oadd_pos a0 n a' ** case succ a0 a' : ONote N0 : NF a0 Na' : NF a' m : ℕ d : ω ∣ repr a' e0 : repr a0 ≠ 0 h : repr a' + ↑m < ω ^ repr a0 n : ℕ+ k✝ : ℕ R'✝ : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑m) k✝ m) No : NF (oadd a0 n a') k : ℕ R' : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑m) (Nat.succ k) m) R : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑m) k m) ω0 : Ordinal.{0} := ω ^ repr a0 α' : Ordinal.{0} := ω0 * ↑↑n + repr a' IH : (k ≠ 0 → R < ω0 ^ succ ↑k) ∧ ω0 ^ ↑k * α' + R = (α' + ↑m) ^ succ ↑k RR : R' = ω0 ^ ↑k * (α' * ↑m) + R α0 : 0 < α' ⊢ (Nat.succ k ≠ 0 → R' < (ω ^ repr a0) ^ succ ↑(Nat.succ k)) ∧ (ω ^ repr a0) ^ ↑(Nat.succ k) * (ω ^ repr a0 * ↑↑n + repr a') + R' = (ω ^ repr a0 * ↑↑n + repr a' + ↑m) ^ succ ↑(Nat.succ k) ** have ω00 : 0 < (ω0 ^ k) := opow_pos _ (opow_pos _ omega_pos) ** case succ a0 a' : ONote N0 : NF a0 Na' : NF a' m : ℕ d : ω ∣ repr a' e0 : repr a0 ≠ 0 h : repr a' + ↑m < ω ^ repr a0 n : ℕ+ k✝ : ℕ R'✝ : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑m) k✝ m) No : NF (oadd a0 n a') k : ℕ R' : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑m) (Nat.succ k) m) R : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑m) k m) ω0 : Ordinal.{0} := ω ^ repr a0 α' : Ordinal.{0} := ω0 * ↑↑n + repr a' IH : (k ≠ 0 → R < ω0 ^ succ ↑k) ∧ ω0 ^ ↑k * α' + R = (α' + ↑m) ^ succ ↑k RR : R' = ω0 ^ ↑k * (α' * ↑m) + R α0 : 0 < α' ω00 : 0 < ω0 ^ ↑k Rl : R < ω ^ (repr a0 * succ ↑k) ⊢ (Nat.succ k ≠ 0 → R' < (ω ^ repr a0) ^ succ ↑(Nat.succ k)) ∧ (ω ^ repr a0) ^ ↑(Nat.succ k) * (ω ^ repr a0 * ↑↑n + repr a') + R' = (ω ^ repr a0 * ↑↑n + repr a' + ↑m) ^ succ ↑(Nat.succ k) ** refine' ⟨fun _ => _, _⟩ ** case succ.refine'_2 a0 a' : ONote N0 : NF a0 Na' : NF a' m : ℕ d : ω ∣ repr a' e0 : repr a0 ≠ 0 h : repr a' + ↑m < ω ^ repr a0 n : ℕ+ k✝ : ℕ R'✝ : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑m) k✝ m) No : NF (oadd a0 n a') k : ℕ R' : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑m) (Nat.succ k) m) R : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑m) k m) ω0 : Ordinal.{0} := ω ^ repr a0 α' : Ordinal.{0} := ω0 * ↑↑n + repr a' IH : (k ≠ 0 → R < ω0 ^ succ ↑k) ∧ ω0 ^ ↑k * α' + R = (α' + ↑m) ^ succ ↑k RR : R' = ω0 ^ ↑k * (α' * ↑m) + R α0 : 0 < α' ω00 : 0 < ω0 ^ ↑k Rl : R < ω ^ (repr a0 * succ ↑k) ⊢ (ω ^ repr a0) ^ ↑(Nat.succ k) * (ω ^ repr a0 * ↑↑n + repr a') + R' = (ω ^ repr a0 * ↑↑n + repr a' + ↑m) ^ succ ↑(Nat.succ k) ** calc
(ω0 ^ k.succ) * α' + R'
_ = (ω0 ^ succ ↑k) * α' + ((ω0 ^ k) * α' * m + R) := by rw [nat_cast_succ, RR, ← mul_assoc]
_ = ((ω0 ^ k) * α' + R) * α' + ((ω0 ^ k) * α' + R) * m := ?_
_ = (α' + m) ^ succ ↑k.succ := by rw [← mul_add, nat_cast_succ, opow_succ, IH.2] ** case succ.refine'_2 a0 a' : ONote N0 : NF a0 Na' : NF a' m : ℕ d : ω ∣ repr a' e0 : repr a0 ≠ 0 h : repr a' + ↑m < ω ^ repr a0 n : ℕ+ k✝ : ℕ R'✝ : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑m) k✝ m) No : NF (oadd a0 n a') k : ℕ R' : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑m) (Nat.succ k) m) R : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑m) k m) ω0 : Ordinal.{0} := ω ^ repr a0 α' : Ordinal.{0} := ω0 * ↑↑n + repr a' IH : (k ≠ 0 → R < ω0 ^ succ ↑k) ∧ ω0 ^ ↑k * α' + R = (α' + ↑m) ^ succ ↑k RR : R' = ω0 ^ ↑k * (α' * ↑m) + R α0 : 0 < α' ω00 : 0 < ω0 ^ ↑k Rl : R < ω ^ (repr a0 * succ ↑k) ⊢ ω0 ^ succ ↑k * α' + (ω0 ^ ↑k * α' * ↑m + R) = (ω0 ^ ↑k * α' + R) * α' + (ω0 ^ ↑k * α' + R) * ↑m ** congr 1 ** case zero a0 a' : ONote N0 : NF a0 Na' : NF a' m : ℕ d : ω ∣ repr a' e0 : repr a0 ≠ 0 h : repr a' + ↑m < ω ^ repr a0 n : ℕ+ k : ℕ R'✝ : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑m) k m) No : NF (oadd a0 n a') R' : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑m) Nat.zero m) ⊢ (Nat.zero ≠ 0 → R' < (ω ^ repr a0) ^ succ ↑Nat.zero) ∧ (ω ^ repr a0) ^ ↑Nat.zero * (ω ^ repr a0 * ↑↑n + repr a') + R' = (ω ^ repr a0 * ↑↑n + repr a' + ↑m) ^ succ ↑Nat.zero ** cases m <;> simp [opowAux] ** a0 a' : ONote N0 : NF a0 Na' : NF a' m : ℕ d : ω ∣ repr a' e0 : repr a0 ≠ 0 h : repr a' + ↑m < ω ^ repr a0 n : ℕ+ k✝ : ℕ R'✝ : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑m) k✝ m) No : NF (oadd a0 n a') k : ℕ R' : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑m) (Nat.succ k) m) R : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑m) k m) ω0 : Ordinal.{0} := ω ^ repr a0 α' : Ordinal.{0} := ω0 * ↑↑n + repr a' IH : (k ≠ 0 → R < ω0 ^ succ ↑k) ∧ ω0 ^ ↑k * α' + R = (α' + ↑m) ^ succ ↑k ⊢ R' = ω0 ^ ↑k * (α' * ↑m) + R ** by_cases h : m = 0 ** case pos a0 a' : ONote N0 : NF a0 Na' : NF a' m : ℕ d : ω ∣ repr a' e0 : repr a0 ≠ 0 h✝ : repr a' + ↑m < ω ^ repr a0 n : ℕ+ k✝ : ℕ R'✝ : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑m) k✝ m) No : NF (oadd a0 n a') k : ℕ R' : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑m) (Nat.succ k) m) R : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑m) k m) ω0 : Ordinal.{0} := ω ^ repr a0 α' : Ordinal.{0} := ω0 * ↑↑n + repr a' IH : (k ≠ 0 → R < ω0 ^ succ ↑k) ∧ ω0 ^ ↑k * α' + R = (α' + ↑m) ^ succ ↑k h : m = 0 ⊢ R' = ω0 ^ ↑k * (α' * ↑m) + R ** simp only [h, ONote.ofNat, Nat.cast_zero, zero_add, ONote.repr, mul_zero, ONote.opowAux,
add_zero] ** case neg a0 a' : ONote N0 : NF a0 Na' : NF a' m : ℕ d : ω ∣ repr a' e0 : repr a0 ≠ 0 h✝ : repr a' + ↑m < ω ^ repr a0 n : ℕ+ k✝ : ℕ R'✝ : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑m) k✝ m) No : NF (oadd a0 n a') k : ℕ R' : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑m) (Nat.succ k) m) R : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑m) k m) ω0 : Ordinal.{0} := ω ^ repr a0 α' : Ordinal.{0} := ω0 * ↑↑n + repr a' IH : (k ≠ 0 → R < ω0 ^ succ ↑k) ∧ ω0 ^ ↑k * α' + R = (α' + ↑m) ^ succ ↑k h : ¬m = 0 ⊢ R' = ω0 ^ ↑k * (α' * ↑m) + R ** simp only [ONote.repr_scale, ONote.repr, ONote.mulNat_eq_mul, ONote.opowAux, ONote.repr_ofNat,
ONote.repr_mul, ONote.repr_add, Ordinal.opow_mul, ONote.zero_add] ** a0 a' : ONote N0 : NF a0 Na' : NF a' m : ℕ d : ω ∣ repr a' e0 : repr a0 ≠ 0 h : repr a' + ↑m < ω ^ repr a0 n : ℕ+ k✝ : ℕ R'✝ : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑m) k✝ m) No : NF (oadd a0 n a') k : ℕ R' : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑m) (Nat.succ k) m) R : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑m) k m) ω0 : Ordinal.{0} := ω ^ repr a0 α' : Ordinal.{0} := ω0 * ↑↑n + repr a' IH : (k ≠ 0 → R < ω0 ^ succ ↑k) ∧ ω0 ^ ↑k * α' + R = (α' + ↑m) ^ succ ↑k RR : R' = ω0 ^ ↑k * (α' * ↑m) + R ⊢ 0 < α' ** simpa [lt_def, repr] using oadd_pos a0 n a' ** a0 a' : ONote N0 : NF a0 Na' : NF a' m : ℕ d : ω ∣ repr a' e0 : repr a0 ≠ 0 h : repr a' + ↑m < ω ^ repr a0 n : ℕ+ k✝ : ℕ R'✝ : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑m) k✝ m) No : NF (oadd a0 n a') k : ℕ R' : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑m) (Nat.succ k) m) R : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑m) k m) ω0 : Ordinal.{0} := ω ^ repr a0 α' : Ordinal.{0} := ω0 * ↑↑n + repr a' IH : (k ≠ 0 → R < ω0 ^ succ ↑k) ∧ ω0 ^ ↑k * α' + R = (α' + ↑m) ^ succ ↑k RR : R' = ω0 ^ ↑k * (α' * ↑m) + R α0 : 0 < α' ω00 : 0 < ω0 ^ ↑k ⊢ R < ω ^ (repr a0 * succ ↑k) ** by_cases k0 : k = 0 ** case pos a0 a' : ONote N0 : NF a0 Na' : NF a' m : ℕ d : ω ∣ repr a' e0 : repr a0 ≠ 0 h : repr a' + ↑m < ω ^ repr a0 n : ℕ+ k✝ : ℕ R'✝ : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑m) k✝ m) No : NF (oadd a0 n a') k : ℕ R' : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑m) (Nat.succ k) m) R : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑m) k m) ω0 : Ordinal.{0} := ω ^ repr a0 α' : Ordinal.{0} := ω0 * ↑↑n + repr a' IH : (k ≠ 0 → R < ω0 ^ succ ↑k) ∧ ω0 ^ ↑k * α' + R = (α' + ↑m) ^ succ ↑k RR : R' = ω0 ^ ↑k * (α' * ↑m) + R α0 : 0 < α' ω00 : 0 < ω0 ^ ↑k k0 : k = 0 ⊢ R < ω ^ (repr a0 * succ ↑k) ** simp [k0] ** case pos a0 a' : ONote N0 : NF a0 Na' : NF a' m : ℕ d : ω ∣ repr a' e0 : repr a0 ≠ 0 h : repr a' + ↑m < ω ^ repr a0 n : ℕ+ k✝ : ℕ R'✝ : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑m) k✝ m) No : NF (oadd a0 n a') k : ℕ R' : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑m) (Nat.succ k) m) R : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑m) k m) ω0 : Ordinal.{0} := ω ^ repr a0 α' : Ordinal.{0} := ω0 * ↑↑n + repr a' IH : (k ≠ 0 → R < ω0 ^ succ ↑k) ∧ ω0 ^ ↑k * α' + R = (α' + ↑m) ^ succ ↑k RR : R' = ω0 ^ ↑k * (α' * ↑m) + R α0 : 0 < α' ω00 : 0 < ω0 ^ ↑k k0 : k = 0 ⊢ repr (opowAux 0 a0 (oadd a0 n a' * ↑m) 0 m) < ω ^ repr a0 ** refine' lt_of_lt_of_le _ (opow_le_opow_right omega_pos (one_le_iff_ne_zero.2 e0)) ** case pos a0 a' : ONote N0 : NF a0 Na' : NF a' m : ℕ d : ω ∣ repr a' e0 : repr a0 ≠ 0 h : repr a' + ↑m < ω ^ repr a0 n : ℕ+ k✝ : ℕ R'✝ : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑m) k✝ m) No : NF (oadd a0 n a') k : ℕ R' : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑m) (Nat.succ k) m) R : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑m) k m) ω0 : Ordinal.{0} := ω ^ repr a0 α' : Ordinal.{0} := ω0 * ↑↑n + repr a' IH : (k ≠ 0 → R < ω0 ^ succ ↑k) ∧ ω0 ^ ↑k * α' + R = (α' + ↑m) ^ succ ↑k RR : R' = ω0 ^ ↑k * (α' * ↑m) + R α0 : 0 < α' ω00 : 0 < ω0 ^ ↑k k0 : k = 0 ⊢ repr (opowAux 0 a0 (oadd a0 n a' * ↑m) 0 m) < ω ^ 1 ** cases' m with m <;> simp [opowAux, omega_pos] ** case pos.succ a0 a' : ONote N0 : NF a0 Na' : NF a' d : ω ∣ repr a' e0 : repr a0 ≠ 0 n : ℕ+ k✝ : ℕ No : NF (oadd a0 n a') k : ℕ ω0 : Ordinal.{0} := ω ^ repr a0 α' : Ordinal.{0} := ω0 * ↑↑n + repr a' α0 : 0 < α' ω00 : 0 < ω0 ^ ↑k k0 : k = 0 m : ℕ h : repr a' + ↑(Nat.succ m) < ω ^ repr a0 R'✝ : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑(Nat.succ m)) k✝ (Nat.succ m)) R' : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑(Nat.succ m)) (Nat.succ k) (Nat.succ m)) R : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑(Nat.succ m)) k (Nat.succ m)) IH : (k ≠ 0 → R < ω0 ^ succ ↑k) ∧ ω0 ^ ↑k * α' + R = (α' + ↑(Nat.succ m)) ^ succ ↑k RR : R' = ω0 ^ ↑k * (α' * ↑(Nat.succ m)) + R ⊢ succ ↑m < ω ** rw [← add_one_eq_succ, ← Nat.cast_succ] ** case pos.succ a0 a' : ONote N0 : NF a0 Na' : NF a' d : ω ∣ repr a' e0 : repr a0 ≠ 0 n : ℕ+ k✝ : ℕ No : NF (oadd a0 n a') k : ℕ ω0 : Ordinal.{0} := ω ^ repr a0 α' : Ordinal.{0} := ω0 * ↑↑n + repr a' α0 : 0 < α' ω00 : 0 < ω0 ^ ↑k k0 : k = 0 m : ℕ h : repr a' + ↑(Nat.succ m) < ω ^ repr a0 R'✝ : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑(Nat.succ m)) k✝ (Nat.succ m)) R' : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑(Nat.succ m)) (Nat.succ k) (Nat.succ m)) R : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑(Nat.succ m)) k (Nat.succ m)) IH : (k ≠ 0 → R < ω0 ^ succ ↑k) ∧ ω0 ^ ↑k * α' + R = (α' + ↑(Nat.succ m)) ^ succ ↑k RR : R' = ω0 ^ ↑k * (α' * ↑(Nat.succ m)) + R ⊢ ↑(Nat.succ m) < ω ** apply nat_lt_omega ** case neg a0 a' : ONote N0 : NF a0 Na' : NF a' m : ℕ d : ω ∣ repr a' e0 : repr a0 ≠ 0 h : repr a' + ↑m < ω ^ repr a0 n : ℕ+ k✝ : ℕ R'✝ : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑m) k✝ m) No : NF (oadd a0 n a') k : ℕ R' : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑m) (Nat.succ k) m) R : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑m) k m) ω0 : Ordinal.{0} := ω ^ repr a0 α' : Ordinal.{0} := ω0 * ↑↑n + repr a' IH : (k ≠ 0 → R < ω0 ^ succ ↑k) ∧ ω0 ^ ↑k * α' + R = (α' + ↑m) ^ succ ↑k RR : R' = ω0 ^ ↑k * (α' * ↑m) + R α0 : 0 < α' ω00 : 0 < ω0 ^ ↑k k0 : ¬k = 0 ⊢ R < ω ^ (repr a0 * succ ↑k) ** rw [opow_mul] ** case neg a0 a' : ONote N0 : NF a0 Na' : NF a' m : ℕ d : ω ∣ repr a' e0 : repr a0 ≠ 0 h : repr a' + ↑m < ω ^ repr a0 n : ℕ+ k✝ : ℕ R'✝ : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑m) k✝ m) No : NF (oadd a0 n a') k : ℕ R' : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑m) (Nat.succ k) m) R : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑m) k m) ω0 : Ordinal.{0} := ω ^ repr a0 α' : Ordinal.{0} := ω0 * ↑↑n + repr a' IH : (k ≠ 0 → R < ω0 ^ succ ↑k) ∧ ω0 ^ ↑k * α' + R = (α' + ↑m) ^ succ ↑k RR : R' = ω0 ^ ↑k * (α' * ↑m) + R α0 : 0 < α' ω00 : 0 < ω0 ^ ↑k k0 : ¬k = 0 ⊢ R < (ω ^ repr a0) ^ succ ↑k ** exact IH.1 k0 ** case succ.refine'_1 a0 a' : ONote N0 : NF a0 Na' : NF a' m : ℕ d : ω ∣ repr a' e0 : repr a0 ≠ 0 h : repr a' + ↑m < ω ^ repr a0 n : ℕ+ k✝ : ℕ R'✝ : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑m) k✝ m) No : NF (oadd a0 n a') k : ℕ R' : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑m) (Nat.succ k) m) R : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑m) k m) ω0 : Ordinal.{0} := ω ^ repr a0 α' : Ordinal.{0} := ω0 * ↑↑n + repr a' IH : (k ≠ 0 → R < ω0 ^ succ ↑k) ∧ ω0 ^ ↑k * α' + R = (α' + ↑m) ^ succ ↑k RR : R' = ω0 ^ ↑k * (α' * ↑m) + R α0 : 0 < α' ω00 : 0 < ω0 ^ ↑k Rl : R < ω ^ (repr a0 * succ ↑k) x✝ : Nat.succ k ≠ 0 ⊢ R' < (ω ^ repr a0) ^ succ ↑(Nat.succ k) ** rw [RR, ← opow_mul _ _ (succ k.succ)] ** case succ.refine'_1 a0 a' : ONote N0 : NF a0 Na' : NF a' m : ℕ d : ω ∣ repr a' e0 : repr a0 ≠ 0 h : repr a' + ↑m < ω ^ repr a0 n : ℕ+ k✝ : ℕ R'✝ : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑m) k✝ m) No : NF (oadd a0 n a') k : ℕ R' : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑m) (Nat.succ k) m) R : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑m) k m) ω0 : Ordinal.{0} := ω ^ repr a0 α' : Ordinal.{0} := ω0 * ↑↑n + repr a' IH : (k ≠ 0 → R < ω0 ^ succ ↑k) ∧ ω0 ^ ↑k * α' + R = (α' + ↑m) ^ succ ↑k RR : R' = ω0 ^ ↑k * (α' * ↑m) + R α0 : 0 < α' ω00 : 0 < ω0 ^ ↑k Rl : R < ω ^ (repr a0 * succ ↑k) x✝ : Nat.succ k ≠ 0 ⊢ ω0 ^ ↑k * (α' * ↑m) + R < ω ^ (repr a0 * succ ↑(Nat.succ k)) ** have e0 := Ordinal.pos_iff_ne_zero.2 e0 ** case succ.refine'_1 a0 a' : ONote N0 : NF a0 Na' : NF a' m : ℕ d : ω ∣ repr a' e0✝ : repr a0 ≠ 0 h : repr a' + ↑m < ω ^ repr a0 n : ℕ+ k✝ : ℕ R'✝ : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑m) k✝ m) No : NF (oadd a0 n a') k : ℕ R' : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑m) (Nat.succ k) m) R : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑m) k m) ω0 : Ordinal.{0} := ω ^ repr a0 α' : Ordinal.{0} := ω0 * ↑↑n + repr a' IH : (k ≠ 0 → R < ω0 ^ succ ↑k) ∧ ω0 ^ ↑k * α' + R = (α' + ↑m) ^ succ ↑k RR : R' = ω0 ^ ↑k * (α' * ↑m) + R α0 : 0 < α' ω00 : 0 < ω0 ^ ↑k Rl : R < ω ^ (repr a0 * succ ↑k) x✝ : Nat.succ k ≠ 0 e0 : 0 < repr a0 ⊢ ω0 ^ ↑k * (α' * ↑m) + R < ω ^ (repr a0 * succ ↑(Nat.succ k)) ** have rr0 : 0 < repr a0 + repr a0 := lt_of_lt_of_le e0 (le_add_left _ _) ** case succ.refine'_1 a0 a' : ONote N0 : NF a0 Na' : NF a' m : ℕ d : ω ∣ repr a' e0✝ : repr a0 ≠ 0 h : repr a' + ↑m < ω ^ repr a0 n : ℕ+ k✝ : ℕ R'✝ : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑m) k✝ m) No : NF (oadd a0 n a') k : ℕ R' : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑m) (Nat.succ k) m) R : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑m) k m) ω0 : Ordinal.{0} := ω ^ repr a0 α' : Ordinal.{0} := ω0 * ↑↑n + repr a' IH : (k ≠ 0 → R < ω0 ^ succ ↑k) ∧ ω0 ^ ↑k * α' + R = (α' + ↑m) ^ succ ↑k RR : R' = ω0 ^ ↑k * (α' * ↑m) + R α0 : 0 < α' ω00 : 0 < ω0 ^ ↑k Rl : R < ω ^ (repr a0 * succ ↑k) x✝ : Nat.succ k ≠ 0 e0 : 0 < repr a0 rr0 : 0 < repr a0 + repr a0 ⊢ ω0 ^ ↑k * (α' * ↑m) + R < ω ^ (repr a0 * succ ↑(Nat.succ k)) ** apply principal_add_omega_opow ** case succ.refine'_1.a a0 a' : ONote N0 : NF a0 Na' : NF a' m : ℕ d : ω ∣ repr a' e0✝ : repr a0 ≠ 0 h : repr a' + ↑m < ω ^ repr a0 n : ℕ+ k✝ : ℕ R'✝ : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑m) k✝ m) No : NF (oadd a0 n a') k : ℕ R' : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑m) (Nat.succ k) m) R : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑m) k m) ω0 : Ordinal.{0} := ω ^ repr a0 α' : Ordinal.{0} := ω0 * ↑↑n + repr a' IH : (k ≠ 0 → R < ω0 ^ succ ↑k) ∧ ω0 ^ ↑k * α' + R = (α' + ↑m) ^ succ ↑k RR : R' = ω0 ^ ↑k * (α' * ↑m) + R α0 : 0 < α' ω00 : 0 < ω0 ^ ↑k Rl : R < ω ^ (repr a0 * succ ↑k) x✝ : Nat.succ k ≠ 0 e0 : 0 < repr a0 rr0 : 0 < repr a0 + repr a0 ⊢ ω0 ^ ↑k * (α' * ↑m) < ω ^ (repr a0 * succ ↑(Nat.succ k)) ** simp [opow_mul, opow_add, mul_assoc] ** case succ.refine'_1.a a0 a' : ONote N0 : NF a0 Na' : NF a' m : ℕ d : ω ∣ repr a' e0✝ : repr a0 ≠ 0 h : repr a' + ↑m < ω ^ repr a0 n : ℕ+ k✝ : ℕ R'✝ : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑m) k✝ m) No : NF (oadd a0 n a') k : ℕ R' : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑m) (Nat.succ k) m) R : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑m) k m) ω0 : Ordinal.{0} := ω ^ repr a0 α' : Ordinal.{0} := ω0 * ↑↑n + repr a' IH : (k ≠ 0 → R < ω0 ^ succ ↑k) ∧ ω0 ^ ↑k * α' + R = (α' + ↑m) ^ succ ↑k RR : R' = ω0 ^ ↑k * (α' * ↑m) + R α0 : 0 < α' ω00 : 0 < ω0 ^ ↑k Rl : R < ω ^ (repr a0 * succ ↑k) x✝ : Nat.succ k ≠ 0 e0 : 0 < repr a0 rr0 : 0 < repr a0 + repr a0 ⊢ (ω ^ repr a0) ^ ↑k * ((ω ^ repr a0 * ↑↑n + repr a') * ↑m) < (ω ^ repr a0) ^ ↑k * (ω ^ repr a0 * ω ^ repr a0) ** rw [Ordinal.mul_lt_mul_iff_left ω00, ← Ordinal.opow_add] ** case succ.refine'_1.a a0 a' : ONote N0 : NF a0 Na' : NF a' m : ℕ d : ω ∣ repr a' e0✝ : repr a0 ≠ 0 h : repr a' + ↑m < ω ^ repr a0 n : ℕ+ k✝ : ℕ R'✝ : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑m) k✝ m) No : NF (oadd a0 n a') k : ℕ R' : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑m) (Nat.succ k) m) R : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑m) k m) ω0 : Ordinal.{0} := ω ^ repr a0 α' : Ordinal.{0} := ω0 * ↑↑n + repr a' IH : (k ≠ 0 → R < ω0 ^ succ ↑k) ∧ ω0 ^ ↑k * α' + R = (α' + ↑m) ^ succ ↑k RR : R' = ω0 ^ ↑k * (α' * ↑m) + R α0 : 0 < α' ω00 : 0 < ω0 ^ ↑k Rl : R < ω ^ (repr a0 * succ ↑k) x✝ : Nat.succ k ≠ 0 e0 : 0 < repr a0 rr0 : 0 < repr a0 + repr a0 ⊢ (ω ^ repr a0 * ↑↑n + repr a') * ↑m < ω ^ (repr a0 + repr a0) ** have : _ < ω ^ (repr a0 + repr a0) := (No.below_of_lt ?_).repr_lt ** case succ.refine'_1.a.refine_2 a0 a' : ONote N0 : NF a0 Na' : NF a' m : ℕ d : ω ∣ repr a' e0✝ : repr a0 ≠ 0 h : repr a' + ↑m < ω ^ repr a0 n : ℕ+ k✝ : ℕ R'✝ : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑m) k✝ m) No : NF (oadd a0 n a') k : ℕ R' : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑m) (Nat.succ k) m) R : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑m) k m) ω0 : Ordinal.{0} := ω ^ repr a0 α' : Ordinal.{0} := ω0 * ↑↑n + repr a' IH : (k ≠ 0 → R < ω0 ^ succ ↑k) ∧ ω0 ^ ↑k * α' + R = (α' + ↑m) ^ succ ↑k RR : R' = ω0 ^ ↑k * (α' * ↑m) + R α0 : 0 < α' ω00 : 0 < ω0 ^ ↑k Rl : R < ω ^ (repr a0 * succ ↑k) x✝ : Nat.succ k ≠ 0 e0 : 0 < repr a0 rr0 : 0 < repr a0 + repr a0 this : repr (oadd a0 n a') < ω ^ (repr a0 + repr a0) ⊢ (ω ^ repr a0 * ↑↑n + repr a') * ↑m < ω ^ (repr a0 + repr a0) case succ.refine'_1.a.refine_1 a0 a' : ONote N0 : NF a0 Na' : NF a' m : ℕ d : ω ∣ repr a' e0✝ : repr a0 ≠ 0 h : repr a' + ↑m < ω ^ repr a0 n : ℕ+ k✝ : ℕ R'✝ : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑m) k✝ m) No : NF (oadd a0 n a') k : ℕ R' : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑m) (Nat.succ k) m) R : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑m) k m) ω0 : Ordinal.{0} := ω ^ repr a0 α' : Ordinal.{0} := ω0 * ↑↑n + repr a' IH : (k ≠ 0 → R < ω0 ^ succ ↑k) ∧ ω0 ^ ↑k * α' + R = (α' + ↑m) ^ succ ↑k RR : R' = ω0 ^ ↑k * (α' * ↑m) + R α0 : 0 < α' ω00 : 0 < ω0 ^ ↑k Rl : R < ω ^ (repr a0 * succ ↑k) x✝ : Nat.succ k ≠ 0 e0 : 0 < repr a0 rr0 : 0 < repr a0 + repr a0 ⊢ repr a0 < repr a0 + repr a0 ** refine' mul_lt_omega_opow rr0 this (nat_lt_omega _) ** case succ.refine'_1.a.refine_1 a0 a' : ONote N0 : NF a0 Na' : NF a' m : ℕ d : ω ∣ repr a' e0✝ : repr a0 ≠ 0 h : repr a' + ↑m < ω ^ repr a0 n : ℕ+ k✝ : ℕ R'✝ : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑m) k✝ m) No : NF (oadd a0 n a') k : ℕ R' : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑m) (Nat.succ k) m) R : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑m) k m) ω0 : Ordinal.{0} := ω ^ repr a0 α' : Ordinal.{0} := ω0 * ↑↑n + repr a' IH : (k ≠ 0 → R < ω0 ^ succ ↑k) ∧ ω0 ^ ↑k * α' + R = (α' + ↑m) ^ succ ↑k RR : R' = ω0 ^ ↑k * (α' * ↑m) + R α0 : 0 < α' ω00 : 0 < ω0 ^ ↑k Rl : R < ω ^ (repr a0 * succ ↑k) x✝ : Nat.succ k ≠ 0 e0 : 0 < repr a0 rr0 : 0 < repr a0 + repr a0 ⊢ repr a0 < repr a0 + repr a0 ** simpa using (add_lt_add_iff_left (repr a0)).2 e0 ** case succ.refine'_1.a a0 a' : ONote N0 : NF a0 Na' : NF a' m : ℕ d : ω ∣ repr a' e0✝ : repr a0 ≠ 0 h : repr a' + ↑m < ω ^ repr a0 n : ℕ+ k✝ : ℕ R'✝ : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑m) k✝ m) No : NF (oadd a0 n a') k : ℕ R' : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑m) (Nat.succ k) m) R : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑m) k m) ω0 : Ordinal.{0} := ω ^ repr a0 α' : Ordinal.{0} := ω0 * ↑↑n + repr a' IH : (k ≠ 0 → R < ω0 ^ succ ↑k) ∧ ω0 ^ ↑k * α' + R = (α' + ↑m) ^ succ ↑k RR : R' = ω0 ^ ↑k * (α' * ↑m) + R α0 : 0 < α' ω00 : 0 < ω0 ^ ↑k Rl : R < ω ^ (repr a0 * succ ↑k) x✝ : Nat.succ k ≠ 0 e0 : 0 < repr a0 rr0 : 0 < repr a0 + repr a0 ⊢ R < ω ^ (repr a0 * succ ↑(Nat.succ k)) ** refine'
lt_of_lt_of_le Rl
(opow_le_opow_right omega_pos <|
mul_le_mul_left' (succ_le_succ_iff.2 (nat_cast_le.2 (le_of_lt k.lt_succ_self))) _) ** a0 a' : ONote N0 : NF a0 Na' : NF a' m : ℕ d : ω ∣ repr a' e0 : repr a0 ≠ 0 h : repr a' + ↑m < ω ^ repr a0 n : ℕ+ k✝ : ℕ R'✝ : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑m) k✝ m) No : NF (oadd a0 n a') k : ℕ R' : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑m) (Nat.succ k) m) R : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑m) k m) ω0 : Ordinal.{0} := ω ^ repr a0 α' : Ordinal.{0} := ω0 * ↑↑n + repr a' IH : (k ≠ 0 → R < ω0 ^ succ ↑k) ∧ ω0 ^ ↑k * α' + R = (α' + ↑m) ^ succ ↑k RR : R' = ω0 ^ ↑k * (α' * ↑m) + R α0 : 0 < α' ω00 : 0 < ω0 ^ ↑k Rl : R < ω ^ (repr a0 * succ ↑k) ⊢ ω0 ^ ↑(Nat.succ k) * α' + R' = ω0 ^ succ ↑k * α' + (ω0 ^ ↑k * α' * ↑m + R) ** rw [nat_cast_succ, RR, ← mul_assoc] ** a0 a' : ONote N0 : NF a0 Na' : NF a' m : ℕ d : ω ∣ repr a' e0 : repr a0 ≠ 0 h : repr a' + ↑m < ω ^ repr a0 n : ℕ+ k✝ : ℕ R'✝ : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑m) k✝ m) No : NF (oadd a0 n a') k : ℕ R' : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑m) (Nat.succ k) m) R : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑m) k m) ω0 : Ordinal.{0} := ω ^ repr a0 α' : Ordinal.{0} := ω0 * ↑↑n + repr a' IH : (k ≠ 0 → R < ω0 ^ succ ↑k) ∧ ω0 ^ ↑k * α' + R = (α' + ↑m) ^ succ ↑k RR : R' = ω0 ^ ↑k * (α' * ↑m) + R α0 : 0 < α' ω00 : 0 < ω0 ^ ↑k Rl : R < ω ^ (repr a0 * succ ↑k) ⊢ (ω0 ^ ↑k * α' + R) * α' + (ω0 ^ ↑k * α' + R) * ↑m = (α' + ↑m) ^ succ ↑(Nat.succ k) ** rw [← mul_add, nat_cast_succ, opow_succ, IH.2] ** case succ.refine'_2.e_a a0 a' : ONote N0 : NF a0 Na' : NF a' m : ℕ d : ω ∣ repr a' e0 : repr a0 ≠ 0 h : repr a' + ↑m < ω ^ repr a0 n : ℕ+ k✝ : ℕ R'✝ : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑m) k✝ m) No : NF (oadd a0 n a') k : ℕ R' : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑m) (Nat.succ k) m) R : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑m) k m) ω0 : Ordinal.{0} := ω ^ repr a0 α' : Ordinal.{0} := ω0 * ↑↑n + repr a' IH : (k ≠ 0 → R < ω0 ^ succ ↑k) ∧ ω0 ^ ↑k * α' + R = (α' + ↑m) ^ succ ↑k RR : R' = ω0 ^ ↑k * (α' * ↑m) + R α0 : 0 < α' ω00 : 0 < ω0 ^ ↑k Rl : R < ω ^ (repr a0 * succ ↑k) ⊢ ω0 ^ succ ↑k * α' = (ω0 ^ ↑k * α' + R) * α' ** have αd : ω ∣ α' :=
dvd_add (dvd_mul_of_dvd_left (by simpa using opow_dvd_opow ω (one_le_iff_ne_zero.2 e0)) _) d ** case succ.refine'_2.e_a a0 a' : ONote N0 : NF a0 Na' : NF a' m : ℕ d : ω ∣ repr a' e0 : repr a0 ≠ 0 h : repr a' + ↑m < ω ^ repr a0 n : ℕ+ k✝ : ℕ R'✝ : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑m) k✝ m) No : NF (oadd a0 n a') k : ℕ R' : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑m) (Nat.succ k) m) R : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑m) k m) ω0 : Ordinal.{0} := ω ^ repr a0 α' : Ordinal.{0} := ω0 * ↑↑n + repr a' IH : (k ≠ 0 → R < ω0 ^ succ ↑k) ∧ ω0 ^ ↑k * α' + R = (α' + ↑m) ^ succ ↑k RR : R' = ω0 ^ ↑k * (α' * ↑m) + R α0 : 0 < α' ω00 : 0 < ω0 ^ ↑k Rl : R < ω ^ (repr a0 * succ ↑k) αd : ω ∣ α' ⊢ ω0 ^ succ ↑k * α' = (ω0 ^ ↑k * α' + R) * α' ** rw [mul_add (ω0^k), add_assoc, ← mul_assoc, ← opow_succ,
add_mul_limit _ (isLimit_iff_omega_dvd.2 ⟨ne_of_gt α0, αd⟩), mul_assoc,
@mul_omega_dvd n (nat_cast_pos.2 n.pos) (nat_lt_omega _) _ αd] ** a0 a' : ONote N0 : NF a0 Na' : NF a' m : ℕ d : ω ∣ repr a' e0 : repr a0 ≠ 0 h : repr a' + ↑m < ω ^ repr a0 n : ℕ+ k✝ : ℕ R'✝ : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑m) k✝ m) No : NF (oadd a0 n a') k : ℕ R' : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑m) (Nat.succ k) m) R : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑m) k m) ω0 : Ordinal.{0} := ω ^ repr a0 α' : Ordinal.{0} := ω0 * ↑↑n + repr a' IH : (k ≠ 0 → R < ω0 ^ succ ↑k) ∧ ω0 ^ ↑k * α' + R = (α' + ↑m) ^ succ ↑k RR : R' = ω0 ^ ↑k * (α' * ↑m) + R α0 : 0 < α' ω00 : 0 < ω0 ^ ↑k Rl : R < ω ^ (repr a0 * succ ↑k) αd : ω ∣ α' ⊢ ω0 ^ ↑k * repr a' + R + ω0 ^ succ ↑k * ↑↑n = ω0 ^ succ ↑k * ↑↑n ** apply @add_absorp _ (repr a0 * succ ↑k) ** a0 a' : ONote N0 : NF a0 Na' : NF a' m : ℕ d : ω ∣ repr a' e0 : repr a0 ≠ 0 h : repr a' + ↑m < ω ^ repr a0 n : ℕ+ k✝ : ℕ R'✝ : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑m) k✝ m) No : NF (oadd a0 n a') k : ℕ R' : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑m) (Nat.succ k) m) R : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑m) k m) ω0 : Ordinal.{0} := ω ^ repr a0 α' : Ordinal.{0} := ω0 * ↑↑n + repr a' IH : (k ≠ 0 → R < ω0 ^ succ ↑k) ∧ ω0 ^ ↑k * α' + R = (α' + ↑m) ^ succ ↑k RR : R' = ω0 ^ ↑k * (α' * ↑m) + R α0 : 0 < α' ω00 : 0 < ω0 ^ ↑k Rl : R < ω ^ (repr a0 * succ ↑k) ⊢ ω ∣ ω0 ** simpa using opow_dvd_opow ω (one_le_iff_ne_zero.2 e0) ** case h₁ a0 a' : ONote N0 : NF a0 Na' : NF a' m : ℕ d : ω ∣ repr a' e0 : repr a0 ≠ 0 h : repr a' + ↑m < ω ^ repr a0 n : ℕ+ k✝ : ℕ R'✝ : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑m) k✝ m) No : NF (oadd a0 n a') k : ℕ R' : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑m) (Nat.succ k) m) R : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑m) k m) ω0 : Ordinal.{0} := ω ^ repr a0 α' : Ordinal.{0} := ω0 * ↑↑n + repr a' IH : (k ≠ 0 → R < ω0 ^ succ ↑k) ∧ ω0 ^ ↑k * α' + R = (α' + ↑m) ^ succ ↑k RR : R' = ω0 ^ ↑k * (α' * ↑m) + R α0 : 0 < α' ω00 : 0 < ω0 ^ ↑k Rl : R < ω ^ (repr a0 * succ ↑k) αd : ω ∣ α' ⊢ ω0 ^ ↑k * repr a' + R < ω ^ (repr a0 * succ ↑k) ** refine' principal_add_omega_opow _ _ Rl ** case h₁ a0 a' : ONote N0 : NF a0 Na' : NF a' m : ℕ d : ω ∣ repr a' e0 : repr a0 ≠ 0 h : repr a' + ↑m < ω ^ repr a0 n : ℕ+ k✝ : ℕ R'✝ : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑m) k✝ m) No : NF (oadd a0 n a') k : ℕ R' : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑m) (Nat.succ k) m) R : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑m) k m) ω0 : Ordinal.{0} := ω ^ repr a0 α' : Ordinal.{0} := ω0 * ↑↑n + repr a' IH : (k ≠ 0 → R < ω0 ^ succ ↑k) ∧ ω0 ^ ↑k * α' + R = (α' + ↑m) ^ succ ↑k RR : R' = ω0 ^ ↑k * (α' * ↑m) + R α0 : 0 < α' ω00 : 0 < ω0 ^ ↑k Rl : R < ω ^ (repr a0 * succ ↑k) αd : ω ∣ α' ⊢ ω0 ^ ↑k * repr a' < ω ^ (repr a0 * succ ↑k) ** rw [opow_mul, opow_succ, Ordinal.mul_lt_mul_iff_left ω00] ** case h₁ a0 a' : ONote N0 : NF a0 Na' : NF a' m : ℕ d : ω ∣ repr a' e0 : repr a0 ≠ 0 h : repr a' + ↑m < ω ^ repr a0 n : ℕ+ k✝ : ℕ R'✝ : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑m) k✝ m) No : NF (oadd a0 n a') k : ℕ R' : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑m) (Nat.succ k) m) R : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑m) k m) ω0 : Ordinal.{0} := ω ^ repr a0 α' : Ordinal.{0} := ω0 * ↑↑n + repr a' IH : (k ≠ 0 → R < ω0 ^ succ ↑k) ∧ ω0 ^ ↑k * α' + R = (α' + ↑m) ^ succ ↑k RR : R' = ω0 ^ ↑k * (α' * ↑m) + R α0 : 0 < α' ω00 : 0 < ω0 ^ ↑k Rl : R < ω ^ (repr a0 * succ ↑k) αd : ω ∣ α' ⊢ repr a' < ω ^ repr a0 ** exact No.snd'.repr_lt ** case h₂ a0 a' : ONote N0 : NF a0 Na' : NF a' m : ℕ d : ω ∣ repr a' e0 : repr a0 ≠ 0 h : repr a' + ↑m < ω ^ repr a0 n : ℕ+ k✝ : ℕ R'✝ : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑m) k✝ m) No : NF (oadd a0 n a') k : ℕ R' : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑m) (Nat.succ k) m) R : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑m) k m) ω0 : Ordinal.{0} := ω ^ repr a0 α' : Ordinal.{0} := ω0 * ↑↑n + repr a' IH : (k ≠ 0 → R < ω0 ^ succ ↑k) ∧ ω0 ^ ↑k * α' + R = (α' + ↑m) ^ succ ↑k RR : R' = ω0 ^ ↑k * (α' * ↑m) + R α0 : 0 < α' ω00 : 0 < ω0 ^ ↑k Rl : R < ω ^ (repr a0 * succ ↑k) αd : ω ∣ α' ⊢ ω ^ (repr a0 * succ ↑k) ≤ ω0 ^ succ ↑k * ↑↑n ** have := mul_le_mul_left' (one_le_iff_pos.2 <| nat_cast_pos.2 n.pos) (ω0^succ k) ** case h₂ a0 a' : ONote N0 : NF a0 Na' : NF a' m : ℕ d : ω ∣ repr a' e0 : repr a0 ≠ 0 h : repr a' + ↑m < ω ^ repr a0 n : ℕ+ k✝ : ℕ R'✝ : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑m) k✝ m) No : NF (oadd a0 n a') k : ℕ R' : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑m) (Nat.succ k) m) R : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑m) k m) ω0 : Ordinal.{0} := ω ^ repr a0 α' : Ordinal.{0} := ω0 * ↑↑n + repr a' IH : (k ≠ 0 → R < ω0 ^ succ ↑k) ∧ ω0 ^ ↑k * α' + R = (α' + ↑m) ^ succ ↑k RR : R' = ω0 ^ ↑k * (α' * ↑m) + R α0 : 0 < α' ω00 : 0 < ω0 ^ ↑k Rl : R < ω ^ (repr a0 * succ ↑k) αd : ω ∣ α' this : ω0 ^ ↑(succ k) * 1 ≤ ω0 ^ ↑(succ k) * ↑↑n ⊢ ω ^ (repr a0 * succ ↑k) ≤ ω0 ^ succ ↑k * ↑↑n ** rw [opow_mul] ** case h₂ a0 a' : ONote N0 : NF a0 Na' : NF a' m : ℕ d : ω ∣ repr a' e0 : repr a0 ≠ 0 h : repr a' + ↑m < ω ^ repr a0 n : ℕ+ k✝ : ℕ R'✝ : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑m) k✝ m) No : NF (oadd a0 n a') k : ℕ R' : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑m) (Nat.succ k) m) R : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑m) k m) ω0 : Ordinal.{0} := ω ^ repr a0 α' : Ordinal.{0} := ω0 * ↑↑n + repr a' IH : (k ≠ 0 → R < ω0 ^ succ ↑k) ∧ ω0 ^ ↑k * α' + R = (α' + ↑m) ^ succ ↑k RR : R' = ω0 ^ ↑k * (α' * ↑m) + R α0 : 0 < α' ω00 : 0 < ω0 ^ ↑k Rl : R < ω ^ (repr a0 * succ ↑k) αd : ω ∣ α' this : ω0 ^ ↑(succ k) * 1 ≤ ω0 ^ ↑(succ k) * ↑↑n ⊢ (ω ^ repr a0) ^ succ ↑k ≤ ω0 ^ succ ↑k * ↑↑n ** simpa [-opow_succ] ** case succ.refine'_2.e_a a0 a' : ONote N0 : NF a0 Na' : NF a' m : ℕ d : ω ∣ repr a' e0 : repr a0 ≠ 0 h : repr a' + ↑m < ω ^ repr a0 n : ℕ+ k✝ : ℕ R'✝ : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑m) k✝ m) No : NF (oadd a0 n a') k : ℕ R' : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑m) (Nat.succ k) m) R : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑m) k m) ω0 : Ordinal.{0} := ω ^ repr a0 α' : Ordinal.{0} := ω0 * ↑↑n + repr a' IH : (k ≠ 0 → R < ω0 ^ succ ↑k) ∧ ω0 ^ ↑k * α' + R = (α' + ↑m) ^ succ ↑k RR : R' = ω0 ^ ↑k * (α' * ↑m) + R α0 : 0 < α' ω00 : 0 < ω0 ^ ↑k Rl : R < ω ^ (repr a0 * succ ↑k) ⊢ ω0 ^ ↑k * α' * ↑m + R = (ω0 ^ ↑k * α' + R) * ↑m ** cases m ** case succ.refine'_2.e_a.zero a0 a' : ONote N0 : NF a0 Na' : NF a' d : ω ∣ repr a' e0 : repr a0 ≠ 0 n : ℕ+ k✝ : ℕ No : NF (oadd a0 n a') k : ℕ ω0 : Ordinal.{0} := ω ^ repr a0 α' : Ordinal.{0} := ω0 * ↑↑n + repr a' α0 : 0 < α' ω00 : 0 < ω0 ^ ↑k h : repr a' + ↑Nat.zero < ω ^ repr a0 R'✝ : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑Nat.zero) k✝ Nat.zero) R' : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑Nat.zero) (Nat.succ k) Nat.zero) R : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑Nat.zero) k Nat.zero) IH : (k ≠ 0 → R < ω0 ^ succ ↑k) ∧ ω0 ^ ↑k * α' + R = (α' + ↑Nat.zero) ^ succ ↑k RR : R' = ω0 ^ ↑k * (α' * ↑Nat.zero) + R Rl : R < ω ^ (repr a0 * succ ↑k) ⊢ ω0 ^ ↑k * α' * ↑Nat.zero + R = (ω0 ^ ↑k * α' + R) * ↑Nat.zero ** have : R = 0 := by cases k <;> simp [opowAux] ** case succ.refine'_2.e_a.zero a0 a' : ONote N0 : NF a0 Na' : NF a' d : ω ∣ repr a' e0 : repr a0 ≠ 0 n : ℕ+ k✝ : ℕ No : NF (oadd a0 n a') k : ℕ ω0 : Ordinal.{0} := ω ^ repr a0 α' : Ordinal.{0} := ω0 * ↑↑n + repr a' α0 : 0 < α' ω00 : 0 < ω0 ^ ↑k h : repr a' + ↑Nat.zero < ω ^ repr a0 R'✝ : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑Nat.zero) k✝ Nat.zero) R' : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑Nat.zero) (Nat.succ k) Nat.zero) R : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑Nat.zero) k Nat.zero) IH : (k ≠ 0 → R < ω0 ^ succ ↑k) ∧ ω0 ^ ↑k * α' + R = (α' + ↑Nat.zero) ^ succ ↑k RR : R' = ω0 ^ ↑k * (α' * ↑Nat.zero) + R Rl : R < ω ^ (repr a0 * succ ↑k) this : R = 0 ⊢ ω0 ^ ↑k * α' * ↑Nat.zero + R = (ω0 ^ ↑k * α' + R) * ↑Nat.zero ** simp [this] ** a0 a' : ONote N0 : NF a0 Na' : NF a' d : ω ∣ repr a' e0 : repr a0 ≠ 0 n : ℕ+ k✝ : ℕ No : NF (oadd a0 n a') k : ℕ ω0 : Ordinal.{0} := ω ^ repr a0 α' : Ordinal.{0} := ω0 * ↑↑n + repr a' α0 : 0 < α' ω00 : 0 < ω0 ^ ↑k h : repr a' + ↑Nat.zero < ω ^ repr a0 R'✝ : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑Nat.zero) k✝ Nat.zero) R' : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑Nat.zero) (Nat.succ k) Nat.zero) R : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑Nat.zero) k Nat.zero) IH : (k ≠ 0 → R < ω0 ^ succ ↑k) ∧ ω0 ^ ↑k * α' + R = (α' + ↑Nat.zero) ^ succ ↑k RR : R' = ω0 ^ ↑k * (α' * ↑Nat.zero) + R Rl : R < ω ^ (repr a0 * succ ↑k) ⊢ R = 0 ** cases k <;> simp [opowAux] ** case succ.refine'_2.e_a.succ a0 a' : ONote N0 : NF a0 Na' : NF a' d : ω ∣ repr a' e0 : repr a0 ≠ 0 n : ℕ+ k✝ : ℕ No : NF (oadd a0 n a') k : ℕ ω0 : Ordinal.{0} := ω ^ repr a0 α' : Ordinal.{0} := ω0 * ↑↑n + repr a' α0 : 0 < α' ω00 : 0 < ω0 ^ ↑k n✝ : ℕ h : repr a' + ↑(Nat.succ n✝) < ω ^ repr a0 R'✝ : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑(Nat.succ n✝)) k✝ (Nat.succ n✝)) R' : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑(Nat.succ n✝)) (Nat.succ k) (Nat.succ n✝)) R : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑(Nat.succ n✝)) k (Nat.succ n✝)) IH : (k ≠ 0 → R < ω0 ^ succ ↑k) ∧ ω0 ^ ↑k * α' + R = (α' + ↑(Nat.succ n✝)) ^ succ ↑k RR : R' = ω0 ^ ↑k * (α' * ↑(Nat.succ n✝)) + R Rl : R < ω ^ (repr a0 * succ ↑k) ⊢ ω0 ^ ↑k * α' * ↑(Nat.succ n✝) + R = (ω0 ^ ↑k * α' + R) * ↑(Nat.succ n✝) ** rw [nat_cast_succ, add_mul_succ] ** case succ.refine'_2.e_a.succ.ba a0 a' : ONote N0 : NF a0 Na' : NF a' d : ω ∣ repr a' e0 : repr a0 ≠ 0 n : ℕ+ k✝ : ℕ No : NF (oadd a0 n a') k : ℕ ω0 : Ordinal.{0} := ω ^ repr a0 α' : Ordinal.{0} := ω0 * ↑↑n + repr a' α0 : 0 < α' ω00 : 0 < ω0 ^ ↑k n✝ : ℕ h : repr a' + ↑(Nat.succ n✝) < ω ^ repr a0 R'✝ : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑(Nat.succ n✝)) k✝ (Nat.succ n✝)) R' : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑(Nat.succ n✝)) (Nat.succ k) (Nat.succ n✝)) R : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑(Nat.succ n✝)) k (Nat.succ n✝)) IH : (k ≠ 0 → R < ω0 ^ succ ↑k) ∧ ω0 ^ ↑k * α' + R = (α' + ↑(Nat.succ n✝)) ^ succ ↑k RR : R' = ω0 ^ ↑k * (α' * ↑(Nat.succ n✝)) + R Rl : R < ω ^ (repr a0 * succ ↑k) ⊢ R + ω0 ^ ↑k * α' = ω0 ^ ↑k * α' ** apply add_absorp Rl ** case succ.refine'_2.e_a.succ.ba a0 a' : ONote N0 : NF a0 Na' : NF a' d : ω ∣ repr a' e0 : repr a0 ≠ 0 n : ℕ+ k✝ : ℕ No : NF (oadd a0 n a') k : ℕ ω0 : Ordinal.{0} := ω ^ repr a0 α' : Ordinal.{0} := ω0 * ↑↑n + repr a' α0 : 0 < α' ω00 : 0 < ω0 ^ ↑k n✝ : ℕ h : repr a' + ↑(Nat.succ n✝) < ω ^ repr a0 R'✝ : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑(Nat.succ n✝)) k✝ (Nat.succ n✝)) R' : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑(Nat.succ n✝)) (Nat.succ k) (Nat.succ n✝)) R : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑(Nat.succ n✝)) k (Nat.succ n✝)) IH : (k ≠ 0 → R < ω0 ^ succ ↑k) ∧ ω0 ^ ↑k * α' + R = (α' + ↑(Nat.succ n✝)) ^ succ ↑k RR : R' = ω0 ^ ↑k * (α' * ↑(Nat.succ n✝)) + R Rl : R < ω ^ (repr a0 * succ ↑k) ⊢ ω ^ (repr a0 * succ ↑k) ≤ ω0 ^ ↑k * α' ** rw [opow_mul, opow_succ] ** case succ.refine'_2.e_a.succ.ba a0 a' : ONote N0 : NF a0 Na' : NF a' d : ω ∣ repr a' e0 : repr a0 ≠ 0 n : ℕ+ k✝ : ℕ No : NF (oadd a0 n a') k : ℕ ω0 : Ordinal.{0} := ω ^ repr a0 α' : Ordinal.{0} := ω0 * ↑↑n + repr a' α0 : 0 < α' ω00 : 0 < ω0 ^ ↑k n✝ : ℕ h : repr a' + ↑(Nat.succ n✝) < ω ^ repr a0 R'✝ : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑(Nat.succ n✝)) k✝ (Nat.succ n✝)) R' : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑(Nat.succ n✝)) (Nat.succ k) (Nat.succ n✝)) R : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑(Nat.succ n✝)) k (Nat.succ n✝)) IH : (k ≠ 0 → R < ω0 ^ succ ↑k) ∧ ω0 ^ ↑k * α' + R = (α' + ↑(Nat.succ n✝)) ^ succ ↑k RR : R' = ω0 ^ ↑k * (α' * ↑(Nat.succ n✝)) + R Rl : R < ω ^ (repr a0 * succ ↑k) ⊢ (ω ^ repr a0) ^ ↑k * ω ^ repr a0 ≤ ω0 ^ ↑k * α' ** apply mul_le_mul_left' ** case succ.refine'_2.e_a.succ.ba.bc a0 a' : ONote N0 : NF a0 Na' : NF a' d : ω ∣ repr a' e0 : repr a0 ≠ 0 n : ℕ+ k✝ : ℕ No : NF (oadd a0 n a') k : ℕ ω0 : Ordinal.{0} := ω ^ repr a0 α' : Ordinal.{0} := ω0 * ↑↑n + repr a' α0 : 0 < α' ω00 : 0 < ω0 ^ ↑k n✝ : ℕ h : repr a' + ↑(Nat.succ n✝) < ω ^ repr a0 R'✝ : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑(Nat.succ n✝)) k✝ (Nat.succ n✝)) R' : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑(Nat.succ n✝)) (Nat.succ k) (Nat.succ n✝)) R : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑(Nat.succ n✝)) k (Nat.succ n✝)) IH : (k ≠ 0 → R < ω0 ^ succ ↑k) ∧ ω0 ^ ↑k * α' + R = (α' + ↑(Nat.succ n✝)) ^ succ ↑k RR : R' = ω0 ^ ↑k * (α' * ↑(Nat.succ n✝)) + R Rl : R < ω ^ (repr a0 * succ ↑k) ⊢ ω ^ repr a0 ≤ α' ** simpa [repr] using omega_le_oadd a0 n a' ** Qed | |
ONote.repr_opow ** o₁ o₂ : ONote inst✝¹ : NF o₁ inst✝ : NF o₂ ⊢ repr (o₁ ^ o₂) = repr o₁ ^ repr o₂ ** cases' e₁ : split o₁ with a m ** case mk o₁ o₂ : ONote inst✝¹ : NF o₁ inst✝ : NF o₂ a : ONote m : ℕ e₁ : split o₁ = (a, m) ⊢ repr (o₁ ^ o₂) = repr o₁ ^ repr o₂ ** cases' nf_repr_split e₁ with N₁ r₁ ** case mk.intro o₁ o₂ : ONote inst✝¹ : NF o₁ inst✝ : NF o₂ a : ONote m : ℕ e₁ : split o₁ = (a, m) N₁ : NF a r₁ : repr o₁ = repr a + ↑m ⊢ repr (o₁ ^ o₂) = repr o₁ ^ repr o₂ ** cases' a with a0 n a' ** case mk.intro.zero o₁ o₂ : ONote inst✝¹ : NF o₁ inst✝ : NF o₂ m : ℕ e₁ : split o₁ = (zero, m) N₁ : NF zero r₁ : repr o₁ = repr zero + ↑m ⊢ repr (o₁ ^ o₂) = repr o₁ ^ repr o₂ ** cases' m with m ** case mk.intro.zero.zero o₁ o₂ : ONote inst✝¹ : NF o₁ inst✝ : NF o₂ N₁ : NF zero e₁ : split o₁ = (zero, Nat.zero) r₁ : repr o₁ = repr zero + ↑Nat.zero ⊢ repr (o₁ ^ o₂) = repr o₁ ^ repr o₂ ** by_cases o₂ = 0 <;> simp [opow_def, opowAux2, opow, e₁, h, r₁] ** case neg o₁ o₂ : ONote inst✝¹ : NF o₁ inst✝ : NF o₂ N₁ : NF zero e₁ : split o₁ = (zero, Nat.zero) r₁ : repr o₁ = repr zero + ↑Nat.zero h : ¬o₂ = 0 ⊢ 0 = 0 ^ repr o₂ ** have := mt repr_inj.1 h ** case neg o₁ o₂ : ONote inst✝¹ : NF o₁ inst✝ : NF o₂ N₁ : NF zero e₁ : split o₁ = (zero, Nat.zero) r₁ : repr o₁ = repr zero + ↑Nat.zero h : ¬o₂ = 0 this : ¬repr o₂ = repr 0 ⊢ 0 = 0 ^ repr o₂ ** rw [zero_opow this] ** case mk.intro.zero.succ o₁ o₂ : ONote inst✝¹ : NF o₁ inst✝ : NF o₂ N₁ : NF zero m : ℕ e₁ : split o₁ = (zero, Nat.succ m) r₁ : repr o₁ = repr zero + ↑(Nat.succ m) ⊢ repr (o₁ ^ o₂) = repr o₁ ^ repr o₂ ** cases' e₂ : split' o₂ with b' k ** case mk.intro.zero.succ.mk o₁ o₂ : ONote inst✝¹ : NF o₁ inst✝ : NF o₂ N₁ : NF zero m : ℕ e₁ : split o₁ = (zero, Nat.succ m) r₁ : repr o₁ = repr zero + ↑(Nat.succ m) b' : ONote k : ℕ e₂ : split' o₂ = (b', k) ⊢ repr (o₁ ^ o₂) = repr o₁ ^ repr o₂ ** cases' nf_repr_split' e₂ with _ r₂ ** case mk.intro.zero.succ.mk.intro o₁ o₂ : ONote inst✝¹ : NF o₁ inst✝ : NF o₂ N₁ : NF zero m : ℕ e₁ : split o₁ = (zero, Nat.succ m) r₁ : repr o₁ = repr zero + ↑(Nat.succ m) b' : ONote k : ℕ e₂ : split' o₂ = (b', k) left✝ : NF b' r₂ : repr o₂ = ω * repr b' + ↑k ⊢ repr (o₁ ^ o₂) = repr o₁ ^ repr o₂ ** by_cases m = 0 ** case neg o₁ o₂ : ONote inst✝¹ : NF o₁ inst✝ : NF o₂ N₁ : NF zero m : ℕ e₁ : split o₁ = (zero, Nat.succ m) r₁ : repr o₁ = repr zero + ↑(Nat.succ m) b' : ONote k : ℕ e₂ : split' o₂ = (b', k) left✝ : NF b' r₂ : repr o₂ = ω * repr b' + ↑k h : ¬m = 0 ⊢ repr (o₁ ^ o₂) = repr o₁ ^ repr o₂ ** simp only [opow_def, opowAux2, opow, e₁, h, r₁, e₂, r₂, repr,
opow_zero, Nat.succPNat_coe, Nat.cast_succ, Nat.cast_zero, _root_.zero_add, mul_one,
add_zero, one_opow, npow_eq_pow] ** case neg o₁ o₂ : ONote inst✝¹ : NF o₁ inst✝ : NF o₂ N₁ : NF zero m : ℕ e₁ : split o₁ = (zero, Nat.succ m) r₁ : repr o₁ = repr zero + ↑(Nat.succ m) b' : ONote k : ℕ e₂ : split' o₂ = (b', k) left✝ : NF b' r₂ : repr o₂ = ω * repr b' + ↑k h : ¬m = 0 ⊢ ω ^ repr b' * ↑↑(Nat.succPNat m ^ k) = (↑m + 1) ^ (ω * repr b' + ↑k) ** rw [opow_add, opow_mul, opow_omega, add_one_eq_succ] ** case neg o₁ o₂ : ONote inst✝¹ : NF o₁ inst✝ : NF o₂ N₁ : NF zero m : ℕ e₁ : split o₁ = (zero, Nat.succ m) r₁ : repr o₁ = repr zero + ↑(Nat.succ m) b' : ONote k : ℕ e₂ : split' o₂ = (b', k) left✝ : NF b' r₂ : repr o₂ = ω * repr b' + ↑k h : ¬m = 0 ⊢ ω ^ repr b' * ↑↑(Nat.succPNat m ^ k) = ω ^ repr b' * succ ↑m ^ ↑k case neg.a1 o₁ o₂ : ONote inst✝¹ : NF o₁ inst✝ : NF o₂ N₁ : NF zero m : ℕ e₁ : split o₁ = (zero, Nat.succ m) r₁ : repr o₁ = repr zero + ↑(Nat.succ m) b' : ONote k : ℕ e₂ : split' o₂ = (b', k) left✝ : NF b' r₂ : repr o₂ = ω * repr b' + ↑k h : ¬m = 0 ⊢ 1 < ↑m + 1 case neg.h o₁ o₂ : ONote inst✝¹ : NF o₁ inst✝ : NF o₂ N₁ : NF zero m : ℕ e₁ : split o₁ = (zero, Nat.succ m) r₁ : repr o₁ = repr zero + ↑(Nat.succ m) b' : ONote k : ℕ e₂ : split' o₂ = (b', k) left✝ : NF b' r₂ : repr o₂ = ω * repr b' + ↑k h : ¬m = 0 ⊢ ↑m + 1 < ω ** congr ** case neg.e_a o₁ o₂ : ONote inst✝¹ : NF o₁ inst✝ : NF o₂ N₁ : NF zero m : ℕ e₁ : split o₁ = (zero, Nat.succ m) r₁ : repr o₁ = repr zero + ↑(Nat.succ m) b' : ONote k : ℕ e₂ : split' o₂ = (b', k) left✝ : NF b' r₂ : repr o₂ = ω * repr b' + ↑k h : ¬m = 0 ⊢ ↑↑(Nat.succPNat m ^ k) = succ ↑m ^ ↑k case neg.a1 o₁ o₂ : ONote inst✝¹ : NF o₁ inst✝ : NF o₂ N₁ : NF zero m : ℕ e₁ : split o₁ = (zero, Nat.succ m) r₁ : repr o₁ = repr zero + ↑(Nat.succ m) b' : ONote k : ℕ e₂ : split' o₂ = (b', k) left✝ : NF b' r₂ : repr o₂ = ω * repr b' + ↑k h : ¬m = 0 ⊢ 1 < ↑m + 1 case neg.h o₁ o₂ : ONote inst✝¹ : NF o₁ inst✝ : NF o₂ N₁ : NF zero m : ℕ e₁ : split o₁ = (zero, Nat.succ m) r₁ : repr o₁ = repr zero + ↑(Nat.succ m) b' : ONote k : ℕ e₂ : split' o₂ = (b', k) left✝ : NF b' r₂ : repr o₂ = ω * repr b' + ↑k h : ¬m = 0 ⊢ ↑m + 1 < ω ** conv_lhs =>
simp [HPow.hPow]
simp [Pow.pow, opow, Ordinal.succ_ne_zero] ** case pos o₁ o₂ : ONote inst✝¹ : NF o₁ inst✝ : NF o₂ N₁ : NF zero m : ℕ e₁ : split o₁ = (zero, Nat.succ m) r₁ : repr o₁ = repr zero + ↑(Nat.succ m) b' : ONote k : ℕ e₂ : split' o₂ = (b', k) left✝ : NF b' r₂ : repr o₂ = ω * repr b' + ↑k h : m = 0 ⊢ repr (o₁ ^ o₂) = repr o₁ ^ repr o₂ ** simp [opow_def, opow, e₁, h, r₁, e₂, r₂, -Nat.cast_succ, ← Nat.one_eq_succ_zero] ** case neg.a1 o₁ o₂ : ONote inst✝¹ : NF o₁ inst✝ : NF o₂ N₁ : NF zero m : ℕ e₁ : split o₁ = (zero, Nat.succ m) r₁ : repr o₁ = repr zero + ↑(Nat.succ m) b' : ONote k : ℕ e₂ : split' o₂ = (b', k) left✝ : NF b' r₂ : repr o₂ = ω * repr b' + ↑k h : ¬m = 0 ⊢ 1 < ↑m + 1 ** simpa using nat_cast_lt.2 (Nat.succ_lt_succ <| pos_iff_ne_zero.2 h) ** case neg.h o₁ o₂ : ONote inst✝¹ : NF o₁ inst✝ : NF o₂ N₁ : NF zero m : ℕ e₁ : split o₁ = (zero, Nat.succ m) r₁ : repr o₁ = repr zero + ↑(Nat.succ m) b' : ONote k : ℕ e₂ : split' o₂ = (b', k) left✝ : NF b' r₂ : repr o₂ = ω * repr b' + ↑k h : ¬m = 0 ⊢ ↑m + 1 < ω ** rw [←Nat.cast_succ, lt_omega] ** case neg.h o₁ o₂ : ONote inst✝¹ : NF o₁ inst✝ : NF o₂ N₁ : NF zero m : ℕ e₁ : split o₁ = (zero, Nat.succ m) r₁ : repr o₁ = repr zero + ↑(Nat.succ m) b' : ONote k : ℕ e₂ : split' o₂ = (b', k) left✝ : NF b' r₂ : repr o₂ = ω * repr b' + ↑k h : ¬m = 0 ⊢ ∃ n, ↑(Nat.succ m) = ↑n ** exact ⟨_, rfl⟩ ** case mk.intro.oadd o₁ o₂ : ONote inst✝¹ : NF o₁ inst✝ : NF o₂ m : ℕ a0 : ONote n : ℕ+ a' : ONote e₁ : split o₁ = (oadd a0 n a', m) N₁ : NF (oadd a0 n a') r₁ : repr o₁ = repr (oadd a0 n a') + ↑m ⊢ repr (o₁ ^ o₂) = repr o₁ ^ repr o₂ ** haveI := N₁.fst ** case mk.intro.oadd o₁ o₂ : ONote inst✝¹ : NF o₁ inst✝ : NF o₂ m : ℕ a0 : ONote n : ℕ+ a' : ONote e₁ : split o₁ = (oadd a0 n a', m) N₁ : NF (oadd a0 n a') r₁ : repr o₁ = repr (oadd a0 n a') + ↑m this : NF a0 ⊢ repr (o₁ ^ o₂) = repr o₁ ^ repr o₂ ** haveI := N₁.snd ** case mk.intro.oadd o₁ o₂ : ONote inst✝¹ : NF o₁ inst✝ : NF o₂ m : ℕ a0 : ONote n : ℕ+ a' : ONote e₁ : split o₁ = (oadd a0 n a', m) N₁ : NF (oadd a0 n a') r₁ : repr o₁ = repr (oadd a0 n a') + ↑m this✝ : NF a0 this : NF a' ⊢ repr (o₁ ^ o₂) = repr o₁ ^ repr o₂ ** cases' N₁.of_dvd_omega (split_dvd e₁) with a00 ad ** case mk.intro.oadd.intro o₁ o₂ : ONote inst✝¹ : NF o₁ inst✝ : NF o₂ m : ℕ a0 : ONote n : ℕ+ a' : ONote e₁ : split o₁ = (oadd a0 n a', m) N₁ : NF (oadd a0 n a') r₁ : repr o₁ = repr (oadd a0 n a') + ↑m this✝ : NF a0 this : NF a' a00 : repr a0 ≠ 0 ad : ω ∣ repr a' ⊢ repr (o₁ ^ o₂) = repr o₁ ^ repr o₂ ** have al := split_add_lt e₁ ** case mk.intro.oadd.intro o₁ o₂ : ONote inst✝¹ : NF o₁ inst✝ : NF o₂ m : ℕ a0 : ONote n : ℕ+ a' : ONote e₁ : split o₁ = (oadd a0 n a', m) N₁ : NF (oadd a0 n a') r₁ : repr o₁ = repr (oadd a0 n a') + ↑m this✝ : NF a0 this : NF a' a00 : repr a0 ≠ 0 ad : ω ∣ repr a' al : repr a' + ↑m < ω ^ repr a0 ⊢ repr (o₁ ^ o₂) = repr o₁ ^ repr o₂ ** have aa : repr (a' + ofNat m) = repr a' + m := by
simp only [eq_self_iff_true, ONote.repr_ofNat, ONote.repr_add] ** case mk.intro.oadd.intro o₁ o₂ : ONote inst✝¹ : NF o₁ inst✝ : NF o₂ m : ℕ a0 : ONote n : ℕ+ a' : ONote e₁ : split o₁ = (oadd a0 n a', m) N₁ : NF (oadd a0 n a') r₁ : repr o₁ = repr (oadd a0 n a') + ↑m this✝ : NF a0 this : NF a' a00 : repr a0 ≠ 0 ad : ω ∣ repr a' al : repr a' + ↑m < ω ^ repr a0 aa : repr (a' + ↑m) = repr a' + ↑m ⊢ repr (o₁ ^ o₂) = repr o₁ ^ repr o₂ ** cases' e₂ : split' o₂ with b' k ** case mk.intro.oadd.intro.mk o₁ o₂ : ONote inst✝¹ : NF o₁ inst✝ : NF o₂ m : ℕ a0 : ONote n : ℕ+ a' : ONote e₁ : split o₁ = (oadd a0 n a', m) N₁ : NF (oadd a0 n a') r₁ : repr o₁ = repr (oadd a0 n a') + ↑m this✝ : NF a0 this : NF a' a00 : repr a0 ≠ 0 ad : ω ∣ repr a' al : repr a' + ↑m < ω ^ repr a0 aa : repr (a' + ↑m) = repr a' + ↑m b' : ONote k : ℕ e₂ : split' o₂ = (b', k) ⊢ repr (o₁ ^ o₂) = repr o₁ ^ repr o₂ ** cases' nf_repr_split' e₂ with _ r₂ ** case mk.intro.oadd.intro.mk.intro o₁ o₂ : ONote inst✝¹ : NF o₁ inst✝ : NF o₂ m : ℕ a0 : ONote n : ℕ+ a' : ONote e₁ : split o₁ = (oadd a0 n a', m) N₁ : NF (oadd a0 n a') r₁ : repr o₁ = repr (oadd a0 n a') + ↑m this✝ : NF a0 this : NF a' a00 : repr a0 ≠ 0 ad : ω ∣ repr a' al : repr a' + ↑m < ω ^ repr a0 aa : repr (a' + ↑m) = repr a' + ↑m b' : ONote k : ℕ e₂ : split' o₂ = (b', k) left✝ : NF b' r₂ : repr o₂ = ω * repr b' + ↑k ⊢ repr (o₁ ^ o₂) = repr o₁ ^ repr o₂ ** simp only [opow_def, opow, e₁, r₁, split_eq_scale_split' e₂, opowAux2, repr] ** case mk.intro.oadd.intro.mk.intro o₁ o₂ : ONote inst✝¹ : NF o₁ inst✝ : NF o₂ m : ℕ a0 : ONote n : ℕ+ a' : ONote e₁ : split o₁ = (oadd a0 n a', m) N₁ : NF (oadd a0 n a') r₁ : repr o₁ = repr (oadd a0 n a') + ↑m this✝ : NF a0 this : NF a' a00 : repr a0 ≠ 0 ad : ω ∣ repr a' al : repr a' + ↑m < ω ^ repr a0 aa : repr (a' + ↑m) = repr a' + ↑m b' : ONote k : ℕ e₂ : split' o₂ = (b', k) left✝ : NF b' r₂ : repr o₂ = ω * repr b' + ↑k ⊢ repr (match (scale 1 b', k) with | (b, 0) => oadd (a0 * b) 1 0 | (b, Nat.succ k) => scale (a0 * b + mulNat a0 k) (oadd a0 n a') + opowAux (a0 * b) a0 (mulNat (oadd a0 n a') m) k m) = (ω ^ repr a0 * ↑↑n + repr a' + ↑m) ^ repr o₂ ** cases' k with k <;> skip ** o₁ o₂ : ONote inst✝¹ : NF o₁ inst✝ : NF o₂ m : ℕ a0 : ONote n : ℕ+ a' : ONote e₁ : split o₁ = (oadd a0 n a', m) N₁ : NF (oadd a0 n a') r₁ : repr o₁ = repr (oadd a0 n a') + ↑m this✝ : NF a0 this : NF a' a00 : repr a0 ≠ 0 ad : ω ∣ repr a' al : repr a' + ↑m < ω ^ repr a0 ⊢ repr (a' + ↑m) = repr a' + ↑m ** simp only [eq_self_iff_true, ONote.repr_ofNat, ONote.repr_add] ** case mk.intro.oadd.intro.mk.intro.zero o₁ o₂ : ONote inst✝¹ : NF o₁ inst✝ : NF o₂ m : ℕ a0 : ONote n : ℕ+ a' : ONote e₁ : split o₁ = (oadd a0 n a', m) N₁ : NF (oadd a0 n a') r₁ : repr o₁ = repr (oadd a0 n a') + ↑m this✝ : NF a0 this : NF a' a00 : repr a0 ≠ 0 ad : ω ∣ repr a' al : repr a' + ↑m < ω ^ repr a0 aa : repr (a' + ↑m) = repr a' + ↑m b' : ONote left✝ : NF b' e₂ : split' o₂ = (b', Nat.zero) r₂ : repr o₂ = ω * repr b' + ↑Nat.zero ⊢ repr (match (scale 1 b', Nat.zero) with | (b, 0) => oadd (a0 * b) 1 0 | (b, Nat.succ k) => scale (a0 * b + mulNat a0 k) (oadd a0 n a') + opowAux (a0 * b) a0 (mulNat (oadd a0 n a') m) k m) = (ω ^ repr a0 * ↑↑n + repr a' + ↑m) ^ repr o₂ ** simp [opow, r₂, opow_mul, repr_opow_aux₁ a00 al aa,
add_assoc, split_eq_scale_split' e₂] ** case mk.intro.oadd.intro.mk.intro.succ o₁ o₂ : ONote inst✝¹ : NF o₁ inst✝ : NF o₂ m : ℕ a0 : ONote n : ℕ+ a' : ONote e₁ : split o₁ = (oadd a0 n a', m) N₁ : NF (oadd a0 n a') r₁ : repr o₁ = repr (oadd a0 n a') + ↑m this✝ : NF a0 this : NF a' a00 : repr a0 ≠ 0 ad : ω ∣ repr a' al : repr a' + ↑m < ω ^ repr a0 aa : repr (a' + ↑m) = repr a' + ↑m b' : ONote left✝ : NF b' k : ℕ e₂ : split' o₂ = (b', Nat.succ k) r₂ : repr o₂ = ω * repr b' + ↑(Nat.succ k) ⊢ repr (match (scale 1 b', Nat.succ k) with | (b, 0) => oadd (a0 * b) 1 0 | (b, Nat.succ k) => scale (a0 * b + mulNat a0 k) (oadd a0 n a') + opowAux (a0 * b) a0 (mulNat (oadd a0 n a') m) k m) = (ω ^ repr a0 * ↑↑n + repr a' + ↑m) ^ repr o₂ ** simp [opow, opowAux2, r₂, opow_add, opow_mul, mul_assoc, add_assoc, -repr] ** case mk.intro.oadd.intro.mk.intro.succ o₁ o₂ : ONote inst✝¹ : NF o₁ inst✝ : NF o₂ m : ℕ a0 : ONote n : ℕ+ a' : ONote e₁ : split o₁ = (oadd a0 n a', m) N₁ : NF (oadd a0 n a') r₁ : repr o₁ = repr (oadd a0 n a') + ↑m this✝ : NF a0 this : NF a' a00 : repr a0 ≠ 0 ad : ω ∣ repr a' al : repr a' + ↑m < ω ^ repr a0 aa : repr (a' + ↑m) = repr a' + ↑m b' : ONote left✝ : NF b' k : ℕ e₂ : split' o₂ = (b', Nat.succ k) r₂ : repr o₂ = ω * repr b' + ↑(Nat.succ k) ⊢ ((ω ^ repr a0) ^ ω ^ repr 1) ^ repr b' * ((ω ^ repr a0) ^ ↑k * repr (oadd a0 n a')) + repr (opowAux (a0 * scale 1 b') a0 (oadd a0 n a' * ↑m) k m) = ((ω ^ repr a0 * ↑↑n + (repr a' + ↑m)) ^ ω) ^ repr b' * ((ω ^ repr a0 * ↑↑n + (repr a' + ↑m)) ^ ↑k * (ω ^ repr a0 * ↑↑n + (repr a' + ↑m))) ** simp only [repr, opow_zero, Nat.succPNat_coe, Nat.cast_one, mul_one, add_zero, opow_one] ** case mk.intro.oadd.intro.mk.intro.succ o₁ o₂ : ONote inst✝¹ : NF o₁ inst✝ : NF o₂ m : ℕ a0 : ONote n : ℕ+ a' : ONote e₁ : split o₁ = (oadd a0 n a', m) N₁ : NF (oadd a0 n a') r₁ : repr o₁ = repr (oadd a0 n a') + ↑m this✝ : NF a0 this : NF a' a00 : repr a0 ≠ 0 ad : ω ∣ repr a' al : repr a' + ↑m < ω ^ repr a0 aa : repr (a' + ↑m) = repr a' + ↑m b' : ONote left✝ : NF b' k : ℕ e₂ : split' o₂ = (b', Nat.succ k) r₂ : repr o₂ = ω * repr b' + ↑(Nat.succ k) ⊢ ((ω ^ repr a0) ^ ω) ^ repr b' * ((ω ^ repr a0) ^ ↑k * (ω ^ repr a0 * ↑↑n + repr a')) + repr (opowAux (a0 * scale 1 b') a0 (oadd a0 n a' * ↑m) k m) = ((ω ^ repr a0 * ↑↑n + (repr a' + ↑m)) ^ ω) ^ repr b' * ((ω ^ repr a0 * ↑↑n + (repr a' + ↑m)) ^ ↑k * (ω ^ repr a0 * ↑↑n + (repr a' + ↑m))) ** rw [repr_opow_aux₁ a00 al aa, scale_opowAux] ** case mk.intro.oadd.intro.mk.intro.succ o₁ o₂ : ONote inst✝¹ : NF o₁ inst✝ : NF o₂ m : ℕ a0 : ONote n : ℕ+ a' : ONote e₁ : split o₁ = (oadd a0 n a', m) N₁ : NF (oadd a0 n a') r₁ : repr o₁ = repr (oadd a0 n a') + ↑m this✝ : NF a0 this : NF a' a00 : repr a0 ≠ 0 ad : ω ∣ repr a' al : repr a' + ↑m < ω ^ repr a0 aa : repr (a' + ↑m) = repr a' + ↑m b' : ONote left✝ : NF b' k : ℕ e₂ : split' o₂ = (b', Nat.succ k) r₂ : repr o₂ = ω * repr b' + ↑(Nat.succ k) ⊢ ((ω ^ repr a0) ^ ω) ^ repr b' * ((ω ^ repr a0) ^ ↑k * (ω ^ repr a0 * ↑↑n + repr a')) + ω ^ repr (a0 * scale 1 b') * repr (opowAux 0 a0 (oadd a0 n a' * ↑m) k m) = ((ω ^ repr a0) ^ ω) ^ repr b' * ((ω ^ repr a0 * ↑↑n + (repr a' + ↑m)) ^ ↑k * (ω ^ repr a0 * ↑↑n + (repr a' + ↑m))) ** simp only [repr_mul, repr_scale, repr, opow_zero, Nat.succPNat_coe, Nat.cast_one, mul_one,
add_zero, opow_one, opow_mul] ** case mk.intro.oadd.intro.mk.intro.succ o₁ o₂ : ONote inst✝¹ : NF o₁ inst✝ : NF o₂ m : ℕ a0 : ONote n : ℕ+ a' : ONote e₁ : split o₁ = (oadd a0 n a', m) N₁ : NF (oadd a0 n a') r₁ : repr o₁ = repr (oadd a0 n a') + ↑m this✝ : NF a0 this : NF a' a00 : repr a0 ≠ 0 ad : ω ∣ repr a' al : repr a' + ↑m < ω ^ repr a0 aa : repr (a' + ↑m) = repr a' + ↑m b' : ONote left✝ : NF b' k : ℕ e₂ : split' o₂ = (b', Nat.succ k) r₂ : repr o₂ = ω * repr b' + ↑(Nat.succ k) ⊢ ((ω ^ repr a0) ^ ω) ^ repr b' * ((ω ^ repr a0) ^ ↑k * (ω ^ repr a0 * ↑↑n + repr a')) + ((ω ^ repr a0) ^ ω) ^ repr b' * repr (opowAux 0 a0 (oadd a0 n a' * ↑m) k m) = ((ω ^ repr a0) ^ ω) ^ repr b' * ((ω ^ repr a0 * ↑↑n + (repr a' + ↑m)) ^ ↑k * (ω ^ repr a0 * ↑↑n + (repr a' + ↑m))) ** rw [← mul_add, ← add_assoc ((ω : Ordinal.{0}) ^ repr a0 * (n : ℕ))] ** case mk.intro.oadd.intro.mk.intro.succ o₁ o₂ : ONote inst✝¹ : NF o₁ inst✝ : NF o₂ m : ℕ a0 : ONote n : ℕ+ a' : ONote e₁ : split o₁ = (oadd a0 n a', m) N₁ : NF (oadd a0 n a') r₁ : repr o₁ = repr (oadd a0 n a') + ↑m this✝ : NF a0 this : NF a' a00 : repr a0 ≠ 0 ad : ω ∣ repr a' al : repr a' + ↑m < ω ^ repr a0 aa : repr (a' + ↑m) = repr a' + ↑m b' : ONote left✝ : NF b' k : ℕ e₂ : split' o₂ = (b', Nat.succ k) r₂ : repr o₂ = ω * repr b' + ↑(Nat.succ k) ⊢ ((ω ^ repr a0) ^ ω) ^ repr b' * ((ω ^ repr a0) ^ ↑k * (ω ^ repr a0 * ↑↑n + repr a') + repr (opowAux 0 a0 (oadd a0 n a' * ↑m) k m)) = ((ω ^ repr a0) ^ ω) ^ repr b' * ((ω ^ repr a0 * ↑↑n + repr a' + ↑m) ^ ↑k * (ω ^ repr a0 * ↑↑n + repr a' + ↑m)) ** congr 1 ** case mk.intro.oadd.intro.mk.intro.succ.e_a o₁ o₂ : ONote inst✝¹ : NF o₁ inst✝ : NF o₂ m : ℕ a0 : ONote n : ℕ+ a' : ONote e₁ : split o₁ = (oadd a0 n a', m) N₁ : NF (oadd a0 n a') r₁ : repr o₁ = repr (oadd a0 n a') + ↑m this✝ : NF a0 this : NF a' a00 : repr a0 ≠ 0 ad : ω ∣ repr a' al : repr a' + ↑m < ω ^ repr a0 aa : repr (a' + ↑m) = repr a' + ↑m b' : ONote left✝ : NF b' k : ℕ e₂ : split' o₂ = (b', Nat.succ k) r₂ : repr o₂ = ω * repr b' + ↑(Nat.succ k) ⊢ (ω ^ repr a0) ^ ↑k * (ω ^ repr a0 * ↑↑n + repr a') + repr (opowAux 0 a0 (oadd a0 n a' * ↑m) k m) = (ω ^ repr a0 * ↑↑n + repr a' + ↑m) ^ ↑k * (ω ^ repr a0 * ↑↑n + repr a' + ↑m) ** rw [← opow_succ] ** case mk.intro.oadd.intro.mk.intro.succ.e_a o₁ o₂ : ONote inst✝¹ : NF o₁ inst✝ : NF o₂ m : ℕ a0 : ONote n : ℕ+ a' : ONote e₁ : split o₁ = (oadd a0 n a', m) N₁ : NF (oadd a0 n a') r₁ : repr o₁ = repr (oadd a0 n a') + ↑m this✝ : NF a0 this : NF a' a00 : repr a0 ≠ 0 ad : ω ∣ repr a' al : repr a' + ↑m < ω ^ repr a0 aa : repr (a' + ↑m) = repr a' + ↑m b' : ONote left✝ : NF b' k : ℕ e₂ : split' o₂ = (b', Nat.succ k) r₂ : repr o₂ = ω * repr b' + ↑(Nat.succ k) ⊢ (ω ^ repr a0) ^ ↑k * (ω ^ repr a0 * ↑↑n + repr a') + repr (opowAux 0 a0 (oadd a0 n a' * ↑m) k m) = (ω ^ repr a0 * ↑↑n + repr a' + ↑m) ^ succ ↑k ** exact (repr_opow_aux₂ _ ad a00 al _ _).2 ** Qed | |
ONote.exists_lt_add ** α : Sort u_1 hα : Nonempty α o : Ordinal.{u_2} f : α → Ordinal.{u_2} H : ∀ ⦃a : Ordinal.{u_2}⦄, a < o → ∃ i, a < f i b a : Ordinal.{u_2} h : a < b + o ⊢ ∃ i, a < b + f i ** cases' lt_or_le a b with h h' ** case inl α : Sort u_1 hα : Nonempty α o : Ordinal.{u_2} f : α → Ordinal.{u_2} H : ∀ ⦃a : Ordinal.{u_2}⦄, a < o → ∃ i, a < f i b a : Ordinal.{u_2} h✝ : a < b + o h : a < b ⊢ ∃ i, a < b + f i ** obtain ⟨i⟩ := id hα ** case inl.intro α : Sort u_1 hα : Nonempty α o : Ordinal.{u_2} f : α → Ordinal.{u_2} H : ∀ ⦃a : Ordinal.{u_2}⦄, a < o → ∃ i, a < f i b a : Ordinal.{u_2} h✝ : a < b + o h : a < b i : α ⊢ ∃ i, a < b + f i ** exact ⟨i, h.trans_le (le_add_right _ _)⟩ ** case inr α : Sort u_1 hα : Nonempty α o : Ordinal.{u_2} f : α → Ordinal.{u_2} H : ∀ ⦃a : Ordinal.{u_2}⦄, a < o → ∃ i, a < f i b a : Ordinal.{u_2} h : a < b + o h' : b ≤ a ⊢ ∃ i, a < b + f i ** rw [← Ordinal.add_sub_cancel_of_le h', add_lt_add_iff_left] at h ** case inr α : Sort u_1 hα : Nonempty α o : Ordinal.{u_2} f : α → Ordinal.{u_2} H : ∀ ⦃a : Ordinal.{u_2}⦄, a < o → ∃ i, a < f i b a : Ordinal.{u_2} h : a - b < o h' : b ≤ a ⊢ ∃ i, a < b + f i ** refine' (H h).imp fun i H => _ ** case inr α : Sort u_1 hα : Nonempty α o : Ordinal.{u_2} f : α → Ordinal.{u_2} H✝ : ∀ ⦃a : Ordinal.{u_2}⦄, a < o → ∃ i, a < f i b a : Ordinal.{u_2} h : a - b < o h' : b ≤ a i : α H : a - b < f i ⊢ a < b + f i ** rwa [← Ordinal.add_sub_cancel_of_le h', add_lt_add_iff_left] ** Qed | |
ONote.exists_lt_mul_omega' ** o a : Ordinal.{u_1} h : a < o * ω ⊢ ∃ i, a < o * ↑i + o ** obtain ⟨i, hi, h'⟩ := (lt_mul_of_limit omega_isLimit).1 h ** case intro.intro o a : Ordinal.{u_1} h : a < o * ω i : Ordinal.{u_1} hi : i < ω h' : a < o * i ⊢ ∃ i, a < o * ↑i + o ** obtain ⟨i, rfl⟩ := lt_omega.1 hi ** case intro.intro.intro o a : Ordinal.{u_1} h : a < o * ω i : ℕ hi : ↑i < ω h' : a < o * ↑i ⊢ ∃ i, a < o * ↑i + o ** exact ⟨i, h'.trans_le (le_add_right _ _)⟩ ** Qed | |
ONote.exists_lt_omega_opow' ** α : Sort u_1 o b : Ordinal.{u_2} hb : 1 < b ho : IsLimit o f : α → Ordinal.{u_2} H : ∀ ⦃a : Ordinal.{u_2}⦄, a < o → ∃ i, a < f i a : Ordinal.{u_2} h : a < b ^ o ⊢ ∃ i, a < b ^ f i ** obtain ⟨d, hd, h'⟩ := (lt_opow_of_limit (zero_lt_one.trans hb).ne' ho).1 h ** case intro.intro α : Sort u_1 o b : Ordinal.{u_2} hb : 1 < b ho : IsLimit o f : α → Ordinal.{u_2} H : ∀ ⦃a : Ordinal.{u_2}⦄, a < o → ∃ i, a < f i a : Ordinal.{u_2} h : a < b ^ o d : Ordinal.{u_2} hd : d < o h' : a < b ^ d ⊢ ∃ i, a < b ^ f i ** exact (H hd).imp fun i hi => h'.trans <| (opow_lt_opow_iff_right hb).2 hi ** Qed | |
ONote.fundamentalSequence_has_prop ** o : ONote ⊢ FundamentalSequenceProp o (fundamentalSequence o) ** induction' o with a m b iha ihb ** case oadd a : ONote m : ℕ+ b : ONote iha : FundamentalSequenceProp a (fundamentalSequence a) ihb : FundamentalSequenceProp b (fundamentalSequence b) ⊢ FundamentalSequenceProp (oadd a m b) (fundamentalSequence (oadd a m b)) ** rw [fundamentalSequence] ** case oadd a : ONote m : ℕ+ b : ONote iha : FundamentalSequenceProp a (fundamentalSequence a) ihb : FundamentalSequenceProp b (fundamentalSequence b) ⊢ FundamentalSequenceProp (oadd a m b) (match fundamentalSequence b with | Sum.inr f => Sum.inr fun i => oadd a m (f i) | Sum.inl (some b') => Sum.inl (some (oadd a m b')) | Sum.inl none => match fundamentalSequence a, PNat.natPred m with | Sum.inl none, 0 => Sum.inl (some zero) | Sum.inl none, Nat.succ m => Sum.inl (some (oadd zero (Nat.succPNat m) zero)) | Sum.inl (some a'), 0 => Sum.inr fun i => oadd a' (Nat.succPNat i) zero | Sum.inl (some a'), Nat.succ m => Sum.inr fun i => oadd a (Nat.succPNat m) (oadd a' (Nat.succPNat i) zero) | Sum.inr f, 0 => Sum.inr fun i => oadd (f i) 1 zero | Sum.inr f, Nat.succ m => Sum.inr fun i => oadd a (Nat.succPNat m) (oadd (f i) 1 zero)) ** rcases e : b.fundamentalSequence with (⟨_ | b'⟩ | f) <;>
simp only [FundamentalSequenceProp] <;>
rw [e, FundamentalSequenceProp] at ihb ** case zero ⊢ FundamentalSequenceProp zero (fundamentalSequence zero) ** exact rfl ** case oadd.inl.none a : ONote m : ℕ+ b : ONote iha : FundamentalSequenceProp a (fundamentalSequence a) ihb : b = 0 e : fundamentalSequence b = Sum.inl none ⊢ FundamentalSequenceProp (oadd a m b) (match fundamentalSequence a, PNat.natPred m with | Sum.inl none, 0 => Sum.inl (some zero) | Sum.inl none, Nat.succ m => Sum.inl (some (oadd zero (Nat.succPNat m) zero)) | Sum.inl (some a'), 0 => Sum.inr fun i => oadd a' (Nat.succPNat i) zero | Sum.inl (some a'), Nat.succ m => Sum.inr fun i => oadd a (Nat.succPNat m) (oadd a' (Nat.succPNat i) zero) | Sum.inr f, 0 => Sum.inr fun i => oadd (f i) 1 zero | Sum.inr f, Nat.succ m => Sum.inr fun i => oadd a (Nat.succPNat m) (oadd (f i) 1 zero)) ** rcases e : a.fundamentalSequence with (⟨_ | a'⟩ | f) <;> cases' e' : m.natPred with m' <;>
simp only [FundamentalSequenceProp] <;>
rw [e, FundamentalSequenceProp] at iha <;>
(try rw [show m = 1 by
have := PNat.natPred_add_one m; rw [e'] at this; exact PNat.coe_inj.1 this.symm]) <;>
(try rw [show m = m'.succ.succPNat by
rw [← e', ← PNat.coe_inj, Nat.succPNat_coe, ← Nat.add_one, PNat.natPred_add_one]]) <;>
simp only [repr, iha, ihb, opow_lt_opow_iff_right one_lt_omega, add_lt_add_iff_left, add_zero,
eq_self_iff_true, lt_add_iff_pos_right, lt_def, mul_one, Nat.cast_zero, Nat.cast_succ,
Nat.succPNat_coe, opow_succ, opow_zero, mul_add_one, PNat.one_coe, succ_zero,
true_and_iff, _root_.zero_add, zero_def] ** case oadd.inl.none.inr.zero a : ONote m : ℕ+ b : ONote ihb : b = 0 e✝ : fundamentalSequence b = Sum.inl none f : ℕ → ONote iha : IsLimit (repr a) ∧ (∀ (i : ℕ), f i < f (i + 1) ∧ f i < a ∧ (NF a → NF (f i))) ∧ ∀ (a_1 : Ordinal.{0}), a_1 < repr a → ∃ i, a_1 < repr (f i) e : fundamentalSequence a = Sum.inr f e' : PNat.natPred m = Nat.zero ⊢ IsLimit (repr (oadd a m b)) ∧ (∀ (i : ℕ), oadd (f i) 1 zero < oadd (f (i + 1)) 1 zero ∧ oadd (f i) 1 zero < oadd a m b ∧ (NF (oadd a m b) → NF (oadd (f i) 1 zero))) ∧ ∀ (a_1 : Ordinal.{0}), a_1 < repr (oadd a m b) → ∃ i, a_1 < repr (oadd (f i) 1 zero) ** try rw [show m = 1 by
have := PNat.natPred_add_one m; rw [e'] at this; exact PNat.coe_inj.1 this.symm] ** case oadd.inl.none.inr.succ a : ONote m : ℕ+ b : ONote ihb : b = 0 e✝ : fundamentalSequence b = Sum.inl none f : ℕ → ONote iha : IsLimit (repr a) ∧ (∀ (i : ℕ), f i < f (i + 1) ∧ f i < a ∧ (NF a → NF (f i))) ∧ ∀ (a_1 : Ordinal.{0}), a_1 < repr a → ∃ i, a_1 < repr (f i) e : fundamentalSequence a = Sum.inr f m' : ℕ e' : PNat.natPred m = Nat.succ m' ⊢ IsLimit (repr (oadd a m b)) ∧ (∀ (i : ℕ), oadd a (Nat.succPNat m') (oadd (f i) 1 zero) < oadd a (Nat.succPNat m') (oadd (f (i + 1)) 1 zero) ∧ oadd a (Nat.succPNat m') (oadd (f i) 1 zero) < oadd a m b ∧ (NF (oadd a m b) → NF (oadd a (Nat.succPNat m') (oadd (f i) 1 zero)))) ∧ ∀ (a_1 : Ordinal.{0}), a_1 < repr (oadd a m b) → ∃ i, a_1 < repr (oadd a (Nat.succPNat m') (oadd (f i) 1 zero)) ** rw [show m = 1 by
have := PNat.natPred_add_one m; rw [e'] at this; exact PNat.coe_inj.1 this.symm] ** a : ONote m : ℕ+ b : ONote ihb : b = 0 e✝ : fundamentalSequence b = Sum.inl none f : ℕ → ONote iha : IsLimit (repr a) ∧ (∀ (i : ℕ), f i < f (i + 1) ∧ f i < a ∧ (NF a → NF (f i))) ∧ ∀ (a_1 : Ordinal.{0}), a_1 < repr a → ∃ i, a_1 < repr (f i) e : fundamentalSequence a = Sum.inr f m' : ℕ e' : PNat.natPred m = Nat.succ m' ⊢ m = 1 ** have := PNat.natPred_add_one m ** a : ONote m : ℕ+ b : ONote ihb : b = 0 e✝ : fundamentalSequence b = Sum.inl none f : ℕ → ONote iha : IsLimit (repr a) ∧ (∀ (i : ℕ), f i < f (i + 1) ∧ f i < a ∧ (NF a → NF (f i))) ∧ ∀ (a_1 : Ordinal.{0}), a_1 < repr a → ∃ i, a_1 < repr (f i) e : fundamentalSequence a = Sum.inr f m' : ℕ e' : PNat.natPred m = Nat.succ m' this : PNat.natPred m + 1 = ↑m ⊢ m = 1 ** rw [e'] at this ** a : ONote m : ℕ+ b : ONote ihb : b = 0 e✝ : fundamentalSequence b = Sum.inl none f : ℕ → ONote iha : IsLimit (repr a) ∧ (∀ (i : ℕ), f i < f (i + 1) ∧ f i < a ∧ (NF a → NF (f i))) ∧ ∀ (a_1 : Ordinal.{0}), a_1 < repr a → ∃ i, a_1 < repr (f i) e : fundamentalSequence a = Sum.inr f e' : PNat.natPred m = Nat.zero this : Nat.zero + 1 = ↑m ⊢ m = 1 ** exact PNat.coe_inj.1 this.symm ** case oadd.inl.none.inr.succ a : ONote m : ℕ+ b : ONote ihb : b = 0 e✝ : fundamentalSequence b = Sum.inl none f : ℕ → ONote iha : IsLimit (repr a) ∧ (∀ (i : ℕ), f i < f (i + 1) ∧ f i < a ∧ (NF a → NF (f i))) ∧ ∀ (a_1 : Ordinal.{0}), a_1 < repr a → ∃ i, a_1 < repr (f i) e : fundamentalSequence a = Sum.inr f m' : ℕ e' : PNat.natPred m = Nat.succ m' ⊢ IsLimit (repr (oadd a m b)) ∧ (∀ (i : ℕ), oadd a (Nat.succPNat m') (oadd (f i) 1 zero) < oadd a (Nat.succPNat m') (oadd (f (i + 1)) 1 zero) ∧ oadd a (Nat.succPNat m') (oadd (f i) 1 zero) < oadd a m b ∧ (NF (oadd a m b) → NF (oadd a (Nat.succPNat m') (oadd (f i) 1 zero)))) ∧ ∀ (a_1 : Ordinal.{0}), a_1 < repr (oadd a m b) → ∃ i, a_1 < repr (oadd a (Nat.succPNat m') (oadd (f i) 1 zero)) ** try rw [show m = m'.succ.succPNat by
rw [← e', ← PNat.coe_inj, Nat.succPNat_coe, ← Nat.add_one, PNat.natPred_add_one]] ** case oadd.inl.none.inr.succ a : ONote m : ℕ+ b : ONote ihb : b = 0 e✝ : fundamentalSequence b = Sum.inl none f : ℕ → ONote iha : IsLimit (repr a) ∧ (∀ (i : ℕ), f i < f (i + 1) ∧ f i < a ∧ (NF a → NF (f i))) ∧ ∀ (a_1 : Ordinal.{0}), a_1 < repr a → ∃ i, a_1 < repr (f i) e : fundamentalSequence a = Sum.inr f m' : ℕ e' : PNat.natPred m = Nat.succ m' ⊢ IsLimit (repr (oadd a m b)) ∧ (∀ (i : ℕ), oadd a (Nat.succPNat m') (oadd (f i) 1 zero) < oadd a (Nat.succPNat m') (oadd (f (i + 1)) 1 zero) ∧ oadd a (Nat.succPNat m') (oadd (f i) 1 zero) < oadd a m b ∧ (NF (oadd a m b) → NF (oadd a (Nat.succPNat m') (oadd (f i) 1 zero)))) ∧ ∀ (a_1 : Ordinal.{0}), a_1 < repr (oadd a m b) → ∃ i, a_1 < repr (oadd a (Nat.succPNat m') (oadd (f i) 1 zero)) ** rw [show m = m'.succ.succPNat by
rw [← e', ← PNat.coe_inj, Nat.succPNat_coe, ← Nat.add_one, PNat.natPred_add_one]] ** a : ONote m : ℕ+ b : ONote ihb : b = 0 e✝ : fundamentalSequence b = Sum.inl none f : ℕ → ONote iha : IsLimit (repr a) ∧ (∀ (i : ℕ), f i < f (i + 1) ∧ f i < a ∧ (NF a → NF (f i))) ∧ ∀ (a_1 : Ordinal.{0}), a_1 < repr a → ∃ i, a_1 < repr (f i) e : fundamentalSequence a = Sum.inr f m' : ℕ e' : PNat.natPred m = Nat.succ m' ⊢ m = Nat.succPNat (Nat.succ m') ** rw [← e', ← PNat.coe_inj, Nat.succPNat_coe, ← Nat.add_one, PNat.natPred_add_one] ** case oadd.inl.none.inl.none.succ a : ONote m : ℕ+ b : ONote iha : a = 0 ihb : b = 0 e✝ : fundamentalSequence b = Sum.inl none e : fundamentalSequence a = Sum.inl none m' : ℕ e' : PNat.natPred m = Nat.succ m' ⊢ 1 * ↑m' + 1 + 1 = succ (1 * ↑m' + 1) ∧ (NF (oadd 0 (Nat.succPNat (Nat.succ m')) 0) → NF (oadd 0 (Nat.succPNat m') 0)) ** exact ⟨rfl, inferInstance⟩ ** case oadd.inl.none.inl.some.zero a : ONote m : ℕ+ b : ONote ihb : b = 0 e✝ : fundamentalSequence b = Sum.inl none a' : ONote iha : repr a = succ (repr a') ∧ (NF a → NF a') e : fundamentalSequence a = Sum.inl (some a') e' : PNat.natPred m = Nat.zero ⊢ IsLimit (ω ^ repr a' * ω) ∧ (∀ (i : ℕ), 0 < ω ^ repr a' ∧ ω ^ repr a' * ↑i + ω ^ repr a' < ω ^ repr a' * ω ∧ (NF (oadd a 1 0) → NF (oadd a' (Nat.succPNat i) 0))) ∧ ∀ (a : Ordinal.{0}), a < ω ^ repr a' * ω → ∃ i, a < ω ^ repr a' * ↑i + ω ^ repr a' ** have := opow_pos (repr a') omega_pos ** case oadd.inl.none.inl.some.zero a : ONote m : ℕ+ b : ONote ihb : b = 0 e✝ : fundamentalSequence b = Sum.inl none a' : ONote iha : repr a = succ (repr a') ∧ (NF a → NF a') e : fundamentalSequence a = Sum.inl (some a') e' : PNat.natPred m = Nat.zero this : 0 < ω ^ repr a' ⊢ IsLimit (ω ^ repr a' * ω) ∧ (∀ (i : ℕ), 0 < ω ^ repr a' ∧ ω ^ repr a' * ↑i + ω ^ repr a' < ω ^ repr a' * ω ∧ (NF (oadd a 1 0) → NF (oadd a' (Nat.succPNat i) 0))) ∧ ∀ (a : Ordinal.{0}), a < ω ^ repr a' * ω → ∃ i, a < ω ^ repr a' * ↑i + ω ^ repr a' ** refine'
⟨mul_isLimit this omega_isLimit, fun i =>
⟨this, _, fun H => @NF.oadd_zero _ _ (iha.2 H.fst)⟩, exists_lt_mul_omega'⟩ ** case oadd.inl.none.inl.some.zero a : ONote m : ℕ+ b : ONote ihb : b = 0 e✝ : fundamentalSequence b = Sum.inl none a' : ONote iha : repr a = succ (repr a') ∧ (NF a → NF a') e : fundamentalSequence a = Sum.inl (some a') e' : PNat.natPred m = Nat.zero this : 0 < ω ^ repr a' i : ℕ ⊢ ω ^ repr a' * ↑i + ω ^ repr a' < ω ^ repr a' * ω ** rw [← mul_succ, ← nat_cast_succ, Ordinal.mul_lt_mul_iff_left this] ** case oadd.inl.none.inl.some.zero a : ONote m : ℕ+ b : ONote ihb : b = 0 e✝ : fundamentalSequence b = Sum.inl none a' : ONote iha : repr a = succ (repr a') ∧ (NF a → NF a') e : fundamentalSequence a = Sum.inl (some a') e' : PNat.natPred m = Nat.zero this : 0 < ω ^ repr a' i : ℕ ⊢ ↑(Nat.succ i) < ω ** apply nat_lt_omega ** case oadd.inl.none.inl.some.succ a : ONote m : ℕ+ b : ONote ihb : b = 0 e✝ : fundamentalSequence b = Sum.inl none a' : ONote iha : repr a = succ (repr a') ∧ (NF a → NF a') e : fundamentalSequence a = Sum.inl (some a') m' : ℕ e' : PNat.natPred m = Nat.succ m' ⊢ IsLimit (ω ^ repr a' * ω * ↑m' + ω ^ repr a' * ω + ω ^ repr a' * ω) ∧ (∀ (i : ℕ), 0 < ω ^ repr a' ∧ ω ^ repr a' * ↑i + ω ^ repr a' < ω ^ repr a' * ω ∧ (NF (oadd a (Nat.succPNat (Nat.succ m')) 0) → NF (oadd a (Nat.succPNat m') (oadd a' (Nat.succPNat i) 0)))) ∧ ∀ (a : Ordinal.{0}), a < ω ^ repr a' * ω * ↑m' + ω ^ repr a' * ω + ω ^ repr a' * ω → ∃ i, a < ω ^ repr a' * ω * ↑m' + ω ^ repr a' * ω + (ω ^ repr a' * ↑i + ω ^ repr a') ** have := opow_pos (repr a') omega_pos ** case oadd.inl.none.inl.some.succ a : ONote m : ℕ+ b : ONote ihb : b = 0 e✝ : fundamentalSequence b = Sum.inl none a' : ONote iha : repr a = succ (repr a') ∧ (NF a → NF a') e : fundamentalSequence a = Sum.inl (some a') m' : ℕ e' : PNat.natPred m = Nat.succ m' this : 0 < ω ^ repr a' ⊢ IsLimit (ω ^ repr a' * ω * ↑m' + ω ^ repr a' * ω + ω ^ repr a' * ω) ∧ (∀ (i : ℕ), 0 < ω ^ repr a' ∧ ω ^ repr a' * ↑i + ω ^ repr a' < ω ^ repr a' * ω ∧ (NF (oadd a (Nat.succPNat (Nat.succ m')) 0) → NF (oadd a (Nat.succPNat m') (oadd a' (Nat.succPNat i) 0)))) ∧ ∀ (a : Ordinal.{0}), a < ω ^ repr a' * ω * ↑m' + ω ^ repr a' * ω + ω ^ repr a' * ω → ∃ i, a < ω ^ repr a' * ω * ↑m' + ω ^ repr a' * ω + (ω ^ repr a' * ↑i + ω ^ repr a') ** refine'
⟨add_isLimit _ (mul_isLimit this omega_isLimit), fun i => ⟨this, _, _⟩,
exists_lt_add exists_lt_mul_omega'⟩ ** case oadd.inl.none.inl.some.succ.refine'_1 a : ONote m : ℕ+ b : ONote ihb : b = 0 e✝ : fundamentalSequence b = Sum.inl none a' : ONote iha : repr a = succ (repr a') ∧ (NF a → NF a') e : fundamentalSequence a = Sum.inl (some a') m' : ℕ e' : PNat.natPred m = Nat.succ m' this : 0 < ω ^ repr a' i : ℕ ⊢ ω ^ repr a' * ↑i + ω ^ repr a' < ω ^ repr a' * ω ** rw [← mul_succ, ← nat_cast_succ, Ordinal.mul_lt_mul_iff_left this] ** case oadd.inl.none.inl.some.succ.refine'_1 a : ONote m : ℕ+ b : ONote ihb : b = 0 e✝ : fundamentalSequence b = Sum.inl none a' : ONote iha : repr a = succ (repr a') ∧ (NF a → NF a') e : fundamentalSequence a = Sum.inl (some a') m' : ℕ e' : PNat.natPred m = Nat.succ m' this : 0 < ω ^ repr a' i : ℕ ⊢ ↑(Nat.succ i) < ω ** apply nat_lt_omega ** case oadd.inl.none.inl.some.succ.refine'_2 a : ONote m : ℕ+ b : ONote ihb : b = 0 e✝ : fundamentalSequence b = Sum.inl none a' : ONote iha : repr a = succ (repr a') ∧ (NF a → NF a') e : fundamentalSequence a = Sum.inl (some a') m' : ℕ e' : PNat.natPred m = Nat.succ m' this : 0 < ω ^ repr a' i : ℕ ⊢ NF (oadd a (Nat.succPNat (Nat.succ m')) 0) → NF (oadd a (Nat.succPNat m') (oadd a' (Nat.succPNat i) 0)) ** refine' fun H => H.fst.oadd _ (NF.below_of_lt' _ (@NF.oadd_zero _ _ (iha.2 H.fst))) ** case oadd.inl.none.inl.some.succ.refine'_2 a : ONote m : ℕ+ b : ONote ihb : b = 0 e✝ : fundamentalSequence b = Sum.inl none a' : ONote iha : repr a = succ (repr a') ∧ (NF a → NF a') e : fundamentalSequence a = Sum.inl (some a') m' : ℕ e' : PNat.natPred m = Nat.succ m' this : 0 < ω ^ repr a' i : ℕ H : NF (oadd a (Nat.succPNat (Nat.succ m')) 0) ⊢ repr (oadd a' (Nat.succPNat i) 0) < ω ^ repr a ** rw [repr, ← zero_def, repr, add_zero, iha.1, opow_succ, Ordinal.mul_lt_mul_iff_left this] ** case oadd.inl.none.inl.some.succ.refine'_2 a : ONote m : ℕ+ b : ONote ihb : b = 0 e✝ : fundamentalSequence b = Sum.inl none a' : ONote iha : repr a = succ (repr a') ∧ (NF a → NF a') e : fundamentalSequence a = Sum.inl (some a') m' : ℕ e' : PNat.natPred m = Nat.succ m' this : 0 < ω ^ repr a' i : ℕ H : NF (oadd a (Nat.succPNat (Nat.succ m')) 0) ⊢ ↑↑(Nat.succPNat i) < ω ** apply nat_lt_omega ** case oadd.inl.none.inr.zero a : ONote m : ℕ+ b : ONote ihb : b = 0 e✝ : fundamentalSequence b = Sum.inl none f : ℕ → ONote iha : IsLimit (repr a) ∧ (∀ (i : ℕ), f i < f (i + 1) ∧ f i < a ∧ (NF a → NF (f i))) ∧ ∀ (a_1 : Ordinal.{0}), a_1 < repr a → ∃ i, a_1 < repr (f i) e : fundamentalSequence a = Sum.inr f e' : PNat.natPred m = Nat.zero ⊢ IsLimit (ω ^ repr a) ∧ (∀ (i : ℕ), repr (f i) < repr (f (i + 1)) ∧ repr (f i) < repr a ∧ (NF (oadd a 1 0) → NF (oadd (f i) 1 0))) ∧ ∀ (a_1 : Ordinal.{0}), a_1 < ω ^ repr a → ∃ i, a_1 < ω ^ repr (f i) ** rcases iha with ⟨h1, h2, h3⟩ ** case oadd.inl.none.inr.zero.intro.intro a : ONote m : ℕ+ b : ONote ihb : b = 0 e✝ : fundamentalSequence b = Sum.inl none f : ℕ → ONote e : fundamentalSequence a = Sum.inr f e' : PNat.natPred m = Nat.zero h1 : IsLimit (repr a) h2 : ∀ (i : ℕ), f i < f (i + 1) ∧ f i < a ∧ (NF a → NF (f i)) h3 : ∀ (a_1 : Ordinal.{0}), a_1 < repr a → ∃ i, a_1 < repr (f i) ⊢ IsLimit (ω ^ repr a) ∧ (∀ (i : ℕ), repr (f i) < repr (f (i + 1)) ∧ repr (f i) < repr a ∧ (NF (oadd a 1 0) → NF (oadd (f i) 1 0))) ∧ ∀ (a_1 : Ordinal.{0}), a_1 < ω ^ repr a → ∃ i, a_1 < ω ^ repr (f i) ** refine' ⟨opow_isLimit one_lt_omega h1, fun i => _, exists_lt_omega_opow' one_lt_omega h1 h3⟩ ** case oadd.inl.none.inr.zero.intro.intro a : ONote m : ℕ+ b : ONote ihb : b = 0 e✝ : fundamentalSequence b = Sum.inl none f : ℕ → ONote e : fundamentalSequence a = Sum.inr f e' : PNat.natPred m = Nat.zero h1 : IsLimit (repr a) h2 : ∀ (i : ℕ), f i < f (i + 1) ∧ f i < a ∧ (NF a → NF (f i)) h3 : ∀ (a_1 : Ordinal.{0}), a_1 < repr a → ∃ i, a_1 < repr (f i) i : ℕ ⊢ repr (f i) < repr (f (i + 1)) ∧ repr (f i) < repr a ∧ (NF (oadd a 1 0) → NF (oadd (f i) 1 0)) ** obtain ⟨h4, h5, h6⟩ := h2 i ** case oadd.inl.none.inr.zero.intro.intro.intro.intro a : ONote m : ℕ+ b : ONote ihb : b = 0 e✝ : fundamentalSequence b = Sum.inl none f : ℕ → ONote e : fundamentalSequence a = Sum.inr f e' : PNat.natPred m = Nat.zero h1 : IsLimit (repr a) h2 : ∀ (i : ℕ), f i < f (i + 1) ∧ f i < a ∧ (NF a → NF (f i)) h3 : ∀ (a_1 : Ordinal.{0}), a_1 < repr a → ∃ i, a_1 < repr (f i) i : ℕ h4 : f i < f (i + 1) h5 : f i < a h6 : NF a → NF (f i) ⊢ repr (f i) < repr (f (i + 1)) ∧ repr (f i) < repr a ∧ (NF (oadd a 1 0) → NF (oadd (f i) 1 0)) ** exact ⟨h4, h5, fun H => @NF.oadd_zero _ _ (h6 H.fst)⟩ ** case oadd.inl.none.inr.succ a : ONote m : ℕ+ b : ONote ihb : b = 0 e✝ : fundamentalSequence b = Sum.inl none f : ℕ → ONote iha : IsLimit (repr a) ∧ (∀ (i : ℕ), f i < f (i + 1) ∧ f i < a ∧ (NF a → NF (f i))) ∧ ∀ (a_1 : Ordinal.{0}), a_1 < repr a → ∃ i, a_1 < repr (f i) e : fundamentalSequence a = Sum.inr f m' : ℕ e' : PNat.natPred m = Nat.succ m' ⊢ IsLimit (ω ^ repr a * ↑m' + ω ^ repr a + ω ^ repr a) ∧ (∀ (i : ℕ), repr (f i) < repr (f (i + 1)) ∧ repr (f i) < repr a ∧ (NF (oadd a (Nat.succPNat (Nat.succ m')) 0) → NF (oadd a (Nat.succPNat m') (oadd (f i) 1 0)))) ∧ ∀ (a_1 : Ordinal.{0}), a_1 < ω ^ repr a * ↑m' + ω ^ repr a + ω ^ repr a → ∃ i, a_1 < ω ^ repr a * ↑m' + ω ^ repr a + ω ^ repr (f i) ** rcases iha with ⟨h1, h2, h3⟩ ** case oadd.inl.none.inr.succ.intro.intro a : ONote m : ℕ+ b : ONote ihb : b = 0 e✝ : fundamentalSequence b = Sum.inl none f : ℕ → ONote e : fundamentalSequence a = Sum.inr f m' : ℕ e' : PNat.natPred m = Nat.succ m' h1 : IsLimit (repr a) h2 : ∀ (i : ℕ), f i < f (i + 1) ∧ f i < a ∧ (NF a → NF (f i)) h3 : ∀ (a_1 : Ordinal.{0}), a_1 < repr a → ∃ i, a_1 < repr (f i) ⊢ IsLimit (ω ^ repr a * ↑m' + ω ^ repr a + ω ^ repr a) ∧ (∀ (i : ℕ), repr (f i) < repr (f (i + 1)) ∧ repr (f i) < repr a ∧ (NF (oadd a (Nat.succPNat (Nat.succ m')) 0) → NF (oadd a (Nat.succPNat m') (oadd (f i) 1 0)))) ∧ ∀ (a_1 : Ordinal.{0}), a_1 < ω ^ repr a * ↑m' + ω ^ repr a + ω ^ repr a → ∃ i, a_1 < ω ^ repr a * ↑m' + ω ^ repr a + ω ^ repr (f i) ** refine'
⟨add_isLimit _ (opow_isLimit one_lt_omega h1), fun i => _,
exists_lt_add (exists_lt_omega_opow' one_lt_omega h1 h3)⟩ ** case oadd.inl.none.inr.succ.intro.intro a : ONote m : ℕ+ b : ONote ihb : b = 0 e✝ : fundamentalSequence b = Sum.inl none f : ℕ → ONote e : fundamentalSequence a = Sum.inr f m' : ℕ e' : PNat.natPred m = Nat.succ m' h1 : IsLimit (repr a) h2 : ∀ (i : ℕ), f i < f (i + 1) ∧ f i < a ∧ (NF a → NF (f i)) h3 : ∀ (a_1 : Ordinal.{0}), a_1 < repr a → ∃ i, a_1 < repr (f i) i : ℕ ⊢ repr (f i) < repr (f (i + 1)) ∧ repr (f i) < repr a ∧ (NF (oadd a (Nat.succPNat (Nat.succ m')) 0) → NF (oadd a (Nat.succPNat m') (oadd (f i) 1 0))) ** obtain ⟨h4, h5, h6⟩ := h2 i ** case oadd.inl.none.inr.succ.intro.intro.intro.intro a : ONote m : ℕ+ b : ONote ihb : b = 0 e✝ : fundamentalSequence b = Sum.inl none f : ℕ → ONote e : fundamentalSequence a = Sum.inr f m' : ℕ e' : PNat.natPred m = Nat.succ m' h1 : IsLimit (repr a) h2 : ∀ (i : ℕ), f i < f (i + 1) ∧ f i < a ∧ (NF a → NF (f i)) h3 : ∀ (a_1 : Ordinal.{0}), a_1 < repr a → ∃ i, a_1 < repr (f i) i : ℕ h4 : f i < f (i + 1) h5 : f i < a h6 : NF a → NF (f i) ⊢ repr (f i) < repr (f (i + 1)) ∧ repr (f i) < repr a ∧ (NF (oadd a (Nat.succPNat (Nat.succ m')) 0) → NF (oadd a (Nat.succPNat m') (oadd (f i) 1 0))) ** refine' ⟨h4, h5, fun H => H.fst.oadd _ (NF.below_of_lt' _ (@NF.oadd_zero _ _ (h6 H.fst)))⟩ ** case oadd.inl.none.inr.succ.intro.intro.intro.intro a : ONote m : ℕ+ b : ONote ihb : b = 0 e✝ : fundamentalSequence b = Sum.inl none f : ℕ → ONote e : fundamentalSequence a = Sum.inr f m' : ℕ e' : PNat.natPred m = Nat.succ m' h1 : IsLimit (repr a) h2 : ∀ (i : ℕ), f i < f (i + 1) ∧ f i < a ∧ (NF a → NF (f i)) h3 : ∀ (a_1 : Ordinal.{0}), a_1 < repr a → ∃ i, a_1 < repr (f i) i : ℕ h4 : f i < f (i + 1) h5 : f i < a h6 : NF a → NF (f i) H : NF (oadd a (Nat.succPNat (Nat.succ m')) 0) ⊢ repr (oadd (f i) 1 0) < ω ^ repr a ** rwa [repr, ← zero_def, repr, add_zero, PNat.one_coe, Nat.cast_one, mul_one,
opow_lt_opow_iff_right one_lt_omega] ** case oadd.inl.some a : ONote m : ℕ+ b : ONote iha : FundamentalSequenceProp a (fundamentalSequence a) b' : ONote ihb : repr b = succ (repr b') ∧ (NF b → NF b') e : fundamentalSequence b = Sum.inl (some b') ⊢ repr (oadd a m b) = succ (repr (oadd a m b')) ∧ (NF (oadd a m b) → NF (oadd a m b')) ** refine'
⟨by rw [repr, ihb.1, add_succ, repr], fun H => H.fst.oadd _ (NF.below_of_lt' _ (ihb.2 H.snd))⟩ ** case oadd.inl.some a : ONote m : ℕ+ b : ONote iha : FundamentalSequenceProp a (fundamentalSequence a) b' : ONote ihb : repr b = succ (repr b') ∧ (NF b → NF b') e : fundamentalSequence b = Sum.inl (some b') H : NF (oadd a m b) ⊢ repr b' < ω ^ repr a ** have := H.snd'.repr_lt ** case oadd.inl.some a : ONote m : ℕ+ b : ONote iha : FundamentalSequenceProp a (fundamentalSequence a) b' : ONote ihb : repr b = succ (repr b') ∧ (NF b → NF b') e : fundamentalSequence b = Sum.inl (some b') H : NF (oadd a m b) this : repr b < ω ^ repr a ⊢ repr b' < ω ^ repr a ** rw [ihb.1] at this ** case oadd.inl.some a : ONote m : ℕ+ b : ONote iha : FundamentalSequenceProp a (fundamentalSequence a) b' : ONote ihb : repr b = succ (repr b') ∧ (NF b → NF b') e : fundamentalSequence b = Sum.inl (some b') H : NF (oadd a m b) this : succ (repr b') < ω ^ repr a ⊢ repr b' < ω ^ repr a ** exact (lt_succ _).trans this ** a : ONote m : ℕ+ b : ONote iha : FundamentalSequenceProp a (fundamentalSequence a) b' : ONote ihb : repr b = succ (repr b') ∧ (NF b → NF b') e : fundamentalSequence b = Sum.inl (some b') ⊢ repr (oadd a m b) = succ (repr (oadd a m b')) ** rw [repr, ihb.1, add_succ, repr] ** case oadd.inr a : ONote m : ℕ+ b : ONote iha : FundamentalSequenceProp a (fundamentalSequence a) f : ℕ → ONote ihb : IsLimit (repr b) ∧ (∀ (i : ℕ), f i < f (i + 1) ∧ f i < b ∧ (NF b → NF (f i))) ∧ ∀ (a : Ordinal.{0}), a < repr b → ∃ i, a < repr (f i) e : fundamentalSequence b = Sum.inr f ⊢ IsLimit (repr (oadd a m b)) ∧ (∀ (i : ℕ), oadd a m (f i) < oadd a m (f (i + 1)) ∧ oadd a m (f i) < oadd a m b ∧ (NF (oadd a m b) → NF (oadd a m (f i)))) ∧ ∀ (a_1 : Ordinal.{0}), a_1 < repr (oadd a m b) → ∃ i, a_1 < repr (oadd a m (f i)) ** rcases ihb with ⟨h1, h2, h3⟩ ** case oadd.inr.intro.intro a : ONote m : ℕ+ b : ONote iha : FundamentalSequenceProp a (fundamentalSequence a) f : ℕ → ONote e : fundamentalSequence b = Sum.inr f h1 : IsLimit (repr b) h2 : ∀ (i : ℕ), f i < f (i + 1) ∧ f i < b ∧ (NF b → NF (f i)) h3 : ∀ (a : Ordinal.{0}), a < repr b → ∃ i, a < repr (f i) ⊢ IsLimit (repr (oadd a m b)) ∧ (∀ (i : ℕ), oadd a m (f i) < oadd a m (f (i + 1)) ∧ oadd a m (f i) < oadd a m b ∧ (NF (oadd a m b) → NF (oadd a m (f i)))) ∧ ∀ (a_1 : Ordinal.{0}), a_1 < repr (oadd a m b) → ∃ i, a_1 < repr (oadd a m (f i)) ** simp only [repr] ** case oadd.inr.intro.intro a : ONote m : ℕ+ b : ONote iha : FundamentalSequenceProp a (fundamentalSequence a) f : ℕ → ONote e : fundamentalSequence b = Sum.inr f h1 : IsLimit (repr b) h2 : ∀ (i : ℕ), f i < f (i + 1) ∧ f i < b ∧ (NF b → NF (f i)) h3 : ∀ (a : Ordinal.{0}), a < repr b → ∃ i, a < repr (f i) ⊢ IsLimit (ω ^ repr a * ↑↑m + repr b) ∧ (∀ (i : ℕ), oadd a m (f i) < oadd a m (f (i + 1)) ∧ oadd a m (f i) < oadd a m b ∧ (NF (oadd a m b) → NF (oadd a m (f i)))) ∧ ∀ (a_1 : Ordinal.{0}), a_1 < ω ^ repr a * ↑↑m + repr b → ∃ i, a_1 < ω ^ repr a * ↑↑m + repr (f i) ** exact
⟨Ordinal.add_isLimit _ h1, fun i =>
⟨oadd_lt_oadd_3 (h2 i).1, oadd_lt_oadd_3 (h2 i).2.1, fun H =>
H.fst.oadd _ (NF.below_of_lt' (lt_trans (h2 i).2.1 H.snd'.repr_lt) ((h2 i).2.2 H.snd))⟩,
exists_lt_add h3⟩ ** Qed | |
ONote.fastGrowing_def ** o : ONote x : Option ONote ⊕ (ℕ → ONote) e : fundamentalSequence o = x ⊢ fastGrowing o = match (motive := (x : Option ONote ⊕ (ℕ → ONote)) → FundamentalSequenceProp o x → ℕ → ℕ) x, (_ : FundamentalSequenceProp o x) with | Sum.inl none, x => Nat.succ | Sum.inl (some a), x => fun i => (fastGrowing a)^[i] i | Sum.inr f, x => fun i => fastGrowing (f i) i ** subst x ** o : ONote ⊢ fastGrowing o = match (motive := (x : Option ONote ⊕ (ℕ → ONote)) → FundamentalSequenceProp o x → ℕ → ℕ) fundamentalSequence o, (_ : FundamentalSequenceProp o (fundamentalSequence o)) with | Sum.inl none, x => Nat.succ | Sum.inl (some a), x => fun i => (fastGrowing a)^[i] i | Sum.inr f, x => fun i => fastGrowing (f i) i ** rw [fastGrowing] ** Qed | |
ONote.fastGrowing_zero' ** o : ONote h : fundamentalSequence o = Sum.inl none ⊢ fastGrowing o = Nat.succ ** rw [fastGrowing_def h] ** Qed | |
ONote.fastGrowing_succ ** o a : ONote h : fundamentalSequence o = Sum.inl (some a) ⊢ fastGrowing o = fun i => (fastGrowing a)^[i] i ** rw [fastGrowing_def h] ** Qed | |
ONote.fastGrowing_limit ** o : ONote f : ℕ → ONote h : fundamentalSequence o = Sum.inr f ⊢ fastGrowing o = fun i => fastGrowing (f i) i ** rw [fastGrowing_def h] ** Qed | |
ONote.fastGrowing_one ** ⊢ fastGrowing 1 = fun n => 2 * n ** rw [@fastGrowing_succ 1 0 rfl] ** ⊢ (fun i => (fastGrowing 0)^[i] i) = fun n => 2 * n ** funext i ** case h i : ℕ ⊢ (fastGrowing 0)^[i] i = 2 * i ** rw [two_mul, fastGrowing_zero] ** case h i : ℕ ⊢ Nat.succ^[i] i = i + i ** suffices : ∀ a b, Nat.succ^[a] b = b + a ** case h i : ℕ this : ∀ (a b : ℕ), Nat.succ^[a] b = b + a ⊢ Nat.succ^[i] i = i + i case this i : ℕ ⊢ ∀ (a b : ℕ), Nat.succ^[a] b = b + a ** exact this _ _ ** case this i : ℕ ⊢ ∀ (a b : ℕ), Nat.succ^[a] b = b + a ** intro a b ** case this i a b : ℕ ⊢ Nat.succ^[a] b = b + a ** induction a <;> simp [*, Function.iterate_succ', Nat.add_succ, -Function.iterate_succ] ** Qed | |
ONote.fastGrowing_two ** ⊢ fastGrowing 2 = fun n => 2 ^ n * n ** rw [@fastGrowing_succ 2 1 rfl] ** ⊢ (fun i => (fastGrowing 1)^[i] i) = fun n => 2 ^ n * n ** funext i ** case h i : ℕ ⊢ (fastGrowing 1)^[i] i = 2 ^ i * i ** rw [fastGrowing_one] ** case h i : ℕ ⊢ (fun n => 2 * n)^[i] i = 2 ^ i * i ** suffices : ∀ a b, (fun n : ℕ => 2 * n)^[a] b = (2 ^ a) * b ** case h i : ℕ this : ∀ (a b : ℕ), (fun n => 2 * n)^[a] b = 2 ^ a * b ⊢ (fun n => 2 * n)^[i] i = 2 ^ i * i case this i : ℕ ⊢ ∀ (a b : ℕ), (fun n => 2 * n)^[a] b = 2 ^ a * b ** exact this _ _ ** case this i : ℕ ⊢ ∀ (a b : ℕ), (fun n => 2 * n)^[a] b = 2 ^ a * b ** intro a b ** case this i a b : ℕ ⊢ (fun n => 2 * n)^[a] b = 2 ^ a * b ** induction a <;>
simp [*, Function.iterate_succ', pow_succ, mul_assoc, -Function.iterate_succ] ** Qed | |
ONote.fastGrowingε₀_zero ** ⊢ fastGrowingε₀ 0 = 1 ** simp [fastGrowingε₀] ** Qed | |
ONote.fastGrowingε₀_one ** ⊢ fastGrowingε₀ 1 = 2 ** simp [fastGrowingε₀, show oadd 0 1 0 = 1 from rfl] ** Qed | |
ONote.fastGrowingε₀_two ** ⊢ fastGrowingε₀ 2 = 2048 ** norm_num [fastGrowingε₀, show oadd 0 1 0 = 1 from rfl, @fastGrowing_limit (oadd 1 1 0) _ rfl,
show oadd 0 (2 : Nat).succPNat 0 = 3 from rfl, @fastGrowing_succ 3 2 rfl] ** Qed | |
NONote.cmp_compares ** a : ONote ha : ONote.NF a b : ONote hb : ONote.NF b ⊢ Ordering.Compares (cmp { val := a, property := ha } { val := b, property := hb }) { val := a, property := ha } { val := b, property := hb } ** dsimp [cmp] ** a : ONote ha : ONote.NF a b : ONote hb : ONote.NF b ⊢ Ordering.Compares (ONote.cmp a b) { val := a, property := ha } { val := b, property := hb } ** have := ONote.cmp_compares a b ** a : ONote ha : ONote.NF a b : ONote hb : ONote.NF b this : Ordering.Compares (ONote.cmp a b) a b ⊢ Ordering.Compares (ONote.cmp a b) { val := a, property := ha } { val := b, property := hb } ** cases h: ONote.cmp a b <;> simp only [h] at this <;> try exact this ** case eq a : ONote ha : ONote.NF a b : ONote hb : ONote.NF b h : ONote.cmp a b = Ordering.eq this : Ordering.Compares Ordering.eq a b ⊢ Ordering.Compares Ordering.eq { val := a, property := ha } { val := b, property := hb } ** exact Subtype.mk_eq_mk.2 this ** case gt a : ONote ha : ONote.NF a b : ONote hb : ONote.NF b h : ONote.cmp a b = Ordering.gt this : Ordering.Compares Ordering.gt a b ⊢ Ordering.Compares Ordering.gt { val := a, property := ha } { val := b, property := hb } ** exact this ** Qed | |
TopCat.Presheaf.restrict_restrict ** C✝ : Type u inst✝² : Category.{v, u} C✝ X : TopCat C : Type u_1 inst✝¹ : Category.{u_3, u_1} C inst✝ : ConcreteCategory C F : Presheaf C X U V W : Opens ↑X e₁ : U ≤ V e₂ : V ≤ W x : (forget C).obj (F.obj (op W)) ⊢ x |_ V |_ U = x |_ U ** delta restrictOpen restrict ** C✝ : Type u inst✝² : Category.{v, u} C✝ X : TopCat C : Type u_1 inst✝¹ : Category.{u_3, u_1} C inst✝ : ConcreteCategory C F : Presheaf C X U V W : Opens ↑X e₁ : U ≤ V e₂ : V ≤ W x : (forget C).obj (F.obj (op W)) ⊢ ↑(F.map (homOfLE (_ : ∀ ⦃a : ↑X⦄, a ∈ ↑U → a ∈ ↑V)).op) (↑(F.map (homOfLE (_ : ∀ ⦃a : ↑X⦄, a ∈ ↑V → a ∈ ↑W)).op) x) = ↑(F.map (homOfLE (_ : ∀ ⦃a : ↑X⦄, a ∈ ↑U → a ∈ ↑W)).op) x ** rw [← comp_apply, ← Functor.map_comp] ** C✝ : Type u inst✝² : Category.{v, u} C✝ X : TopCat C : Type u_1 inst✝¹ : Category.{u_3, u_1} C inst✝ : ConcreteCategory C F : Presheaf C X U V W : Opens ↑X e₁ : U ≤ V e₂ : V ≤ W x : (forget C).obj (F.obj (op W)) ⊢ ↑(F.map ((homOfLE (_ : ∀ ⦃a : ↑X⦄, a ∈ ↑V → a ∈ ↑W)).op ≫ (homOfLE (_ : ∀ ⦃a : ↑X⦄, a ∈ ↑U → a ∈ ↑V)).op)) x = ↑(F.map (homOfLE (_ : ∀ ⦃a : ↑X⦄, a ∈ ↑U → a ∈ ↑W)).op) x ** rfl ** Qed | |
TopCat.Presheaf.map_restrict ** C✝ : Type u inst✝² : Category.{v, u} C✝ X : TopCat C : Type u_1 inst✝¹ : Category.{u_3, u_1} C inst✝ : ConcreteCategory C F G : Presheaf C X e : F ⟶ G U V : Opens ↑X h : U ≤ V x : (forget C).obj (F.obj (op V)) ⊢ ↑(e.app (op U)) (x |_ U) = ↑(e.app (op V)) x |_ U ** delta restrictOpen restrict ** C✝ : Type u inst✝² : Category.{v, u} C✝ X : TopCat C : Type u_1 inst✝¹ : Category.{u_3, u_1} C inst✝ : ConcreteCategory C F G : Presheaf C X e : F ⟶ G U V : Opens ↑X h : U ≤ V x : (forget C).obj (F.obj (op V)) ⊢ ↑(e.app (op U)) (↑(F.map (homOfLE (_ : ∀ ⦃a : ↑X⦄, a ∈ ↑U → a ∈ ↑V)).op) x) = ↑(G.map (homOfLE (_ : ∀ ⦃a : ↑X⦄, a ∈ ↑U → a ∈ ↑V)).op) (↑(e.app (op V)) x) ** rw [← comp_apply, NatTrans.naturality, comp_apply] ** Qed | |
TopCat.Presheaf.pushforward_eq' ** C : Type u inst✝ : Category.{v, u} C X Y : TopCat f g : X ⟶ Y h : f = g ℱ : Presheaf C X ⊢ f _* ℱ = g _* ℱ ** rw [h] ** Qed | |
TopCat.Presheaf.pushforwardEq_hom_app ** C : Type u inst✝ : Category.{v, u} C X Y : TopCat f g : X ⟶ Y h : f = g ℱ : Presheaf C X U : (Opens ↑Y)ᵒᵖ ⊢ (Opens.map f).op.obj U ⟶ (Opens.map g).op.obj U ** dsimp [Functor.op] ** C : Type u inst✝ : Category.{v, u} C X Y : TopCat f g : X ⟶ Y h : f = g ℱ : Presheaf C X U : (Opens ↑Y)ᵒᵖ ⊢ op ((Opens.map f).obj U.unop) ⟶ op ((Opens.map g).obj U.unop) ** apply Quiver.Hom.op ** case f C : Type u inst✝ : Category.{v, u} C X Y : TopCat f g : X ⟶ Y h : f = g ℱ : Presheaf C X U : (Opens ↑Y)ᵒᵖ ⊢ (Opens.map g).obj U.unop ⟶ (Opens.map f).obj U.unop ** apply eqToHom ** case f.p C : Type u inst✝ : Category.{v, u} C X Y : TopCat f g : X ⟶ Y h : f = g ℱ : Presheaf C X U : (Opens ↑Y)ᵒᵖ ⊢ (Opens.map g).obj U.unop = (Opens.map f).obj U.unop ** rw [h] ** C : Type u inst✝ : Category.{v, u} C X Y : TopCat f g : X ⟶ Y h : f = g ℱ : Presheaf C X U : (Opens ↑Y)ᵒᵖ ⊢ (pushforwardEq h ℱ).hom.app U = ℱ.map (id (eqToHom (_ : (Opens.map g).obj U.unop = (Opens.map f).obj U.unop)).op) ** simp [pushforwardEq] ** Qed | |
TopCat.Presheaf.pushforward_eq'_hom_app ** C : Type u inst✝ : Category.{v, u} C X Y : TopCat f g : X ⟶ Y h : f = g ℱ : Presheaf C X U : (Opens ↑Y)ᵒᵖ ⊢ (Opens.map f).op.obj U = (Opens.map g).op.obj U ** rw [h] ** C : Type u inst✝ : Category.{v, u} C X Y : TopCat f g : X ⟶ Y h : f = g ℱ : Presheaf C X U : (Opens ↑Y)ᵒᵖ ⊢ (eqToHom (_ : f _* ℱ = g _* ℱ)).app U = ℱ.map (eqToHom (_ : (Opens.map f).op.obj U = (Opens.map g).op.obj U)) ** rw [eqToHom_app, eqToHom_map] ** Qed | |
TopCat.Presheaf.pushforwardEq_rfl ** C : Type u inst✝ : Category.{v, u} C X Y : TopCat f : X ⟶ Y ℱ : Presheaf C X U : Opens ↑Y ⊢ (pushforwardEq (_ : f = f) ℱ).hom.app (op U) = 𝟙 ((f _* ℱ).obj (op U)) ** dsimp [pushforwardEq] ** C : Type u inst✝ : Category.{v, u} C X Y : TopCat f : X ⟶ Y ℱ : Presheaf C X U : Opens ↑Y ⊢ ℱ.map (𝟙 (op ((Opens.map f).obj U))) = 𝟙 (ℱ.obj (op ((Opens.map f).obj U))) ** simp ** Qed | |
TopCat.Presheaf.Pushforward.id_eq ** C : Type u inst✝ : Category.{v, u} C X : TopCat ℱ : Presheaf C X ⊢ 𝟙 X _* ℱ = ℱ ** unfold pushforwardObj ** C : Type u inst✝ : Category.{v, u} C X : TopCat ℱ : Presheaf C X ⊢ (Opens.map (𝟙 X)).op ⋙ ℱ = ℱ ** rw [Opens.map_id_eq] ** C : Type u inst✝ : Category.{v, u} C X : TopCat ℱ : Presheaf C X ⊢ (𝟭 (Opens ↑X)).op ⋙ ℱ = ℱ ** erw [Functor.id_comp] ** Qed | |
TopCat.Presheaf.Pushforward.id_hom_app' ** C : Type u inst✝ : Category.{v, u} C X : TopCat ℱ : Presheaf C X U : Set ↑X p : IsOpen U ⊢ (id ℱ).hom.app (op { carrier := U, is_open' := p }) = ℱ.map (𝟙 (op { carrier := U, is_open' := p })) ** dsimp [id] ** C : Type u inst✝ : Category.{v, u} C X : TopCat ℱ : Presheaf C X U : Set ↑X p : IsOpen U ⊢ (whiskerRight (NatTrans.op (Opens.mapId X).inv) ℱ ≫ (Functor.leftUnitor ℱ).hom).app (op { carrier := U, is_open' := p }) = ℱ.map (𝟙 (op { carrier := U, is_open' := p })) ** simp [CategoryStruct.comp] ** Qed | |
TopCat.Presheaf.Pushforward.id_hom_app ** C : Type u inst✝ : Category.{v, u} C X : TopCat ℱ : Presheaf C X U : (Opens ↑X)ᵒᵖ ⊢ (id ℱ).hom.app U = ℱ.map (eqToHom (_ : (Opens.map (𝟙 X)).op.obj U = U)) ** induction U ** case h C : Type u inst✝ : Category.{v, u} C X : TopCat ℱ : Presheaf C X X✝ : Opens ↑X ⊢ (id ℱ).hom.app (op X✝) = ℱ.map (eqToHom (_ : (Opens.map (𝟙 X)).op.obj (op X✝) = op X✝)) ** apply id_hom_app' ** Qed | |
TopCat.Presheaf.Pushforward.id_inv_app' ** C : Type u inst✝ : Category.{v, u} C X : TopCat ℱ : Presheaf C X U : Set ↑X p : IsOpen U ⊢ (id ℱ).inv.app (op { carrier := U, is_open' := p }) = ℱ.map (𝟙 (op { carrier := U, is_open' := p })) ** dsimp [id] ** C : Type u inst✝ : Category.{v, u} C X : TopCat ℱ : Presheaf C X U : Set ↑X p : IsOpen U ⊢ ((Functor.leftUnitor ℱ).inv ≫ whiskerRight (NatTrans.op (Opens.mapId X).hom) ℱ).app (op { carrier := U, is_open' := p }) = ℱ.map (𝟙 (op { carrier := U, is_open' := p })) ** simp [CategoryStruct.comp] ** Qed | |
TopCat.Presheaf.Pushforward.comp_hom_app ** C : Type u inst✝ : Category.{v, u} C X : TopCat ℱ : Presheaf C X Y Z : TopCat f : X ⟶ Y g : Y ⟶ Z U : (Opens ↑Z)ᵒᵖ ⊢ (comp ℱ f g).hom.app U = 𝟙 (((f ≫ g) _* ℱ).obj U) ** simp [comp] ** Qed | |
TopCat.Presheaf.Pushforward.comp_inv_app ** C : Type u inst✝ : Category.{v, u} C X : TopCat ℱ : Presheaf C X Y Z : TopCat f : X ⟶ Y g : Y ⟶ Z U : (Opens ↑Z)ᵒᵖ ⊢ (comp ℱ f g).inv.app U = 𝟙 ((g _* (f _* ℱ)).obj U) ** simp [comp] ** Qed | |
TopCat.Presheaf.Pullback.id_inv_app ** C : Type u inst✝¹ : Category.{v, u} C inst✝ : HasColimits C X Y : TopCat ℱ : Presheaf C Y U : Opens ↑Y ⊢ (Opens.map (𝟙 Y)).op.obj (op U) = op U ** simp ** C : Type u inst✝¹ : Category.{v, u} C inst✝ : HasColimits C X Y : TopCat ℱ : Presheaf C Y U : Opens ↑Y ⊢ (id ℱ).inv.app (op U) = colimit.ι (Lan.diagram (Opens.map (𝟙 Y)).op ℱ (op U)) (CostructuredArrow.mk (eqToHom (_ : (Opens.map (𝟙 Y)).op.obj (op U) = op U))) ** rw [← Category.id_comp ((id ℱ).inv.app (op U)), ← NatIso.app_inv, Iso.comp_inv_eq] ** C : Type u inst✝¹ : Category.{v, u} C inst✝ : HasColimits C X Y : TopCat ℱ : Presheaf C Y U : Opens ↑Y ⊢ 𝟙 (ℱ.obj (op U)) = colimit.ι (Lan.diagram (Opens.map (𝟙 Y)).op ℱ (op U)) (CostructuredArrow.mk (eqToHom (_ : (Opens.map (𝟙 Y)).op.obj (op U) = op U))) ≫ ((id ℱ).app (op U)).hom ** dsimp [id] ** C : Type u inst✝¹ : Category.{v, u} C inst✝ : HasColimits C X Y : TopCat ℱ : Presheaf C Y U : Opens ↑Y ⊢ 𝟙 (ℱ.obj (op U)) = colimit.ι (Lan.diagram (Opens.map (𝟙 Y)).op ℱ (op U)) (CostructuredArrow.mk (𝟙 (op ((Opens.map (𝟙 Y)).obj U)))) ≫ colimit.desc (Lan.diagram (Opens.map (𝟙 Y)).op ℱ (op U)) (coconeOfDiagramTerminal (IsLimit.mk fun s => CostructuredArrow.homMk (homOfLE (_ : ↑(𝟙 Y) '' ↑((Functor.fromPUnit (op U)).obj s.pt.right).unop ⊆ ↑s.pt.1.unop)).op) (Lan.diagram (Opens.map (𝟙 Y)).op ℱ (op U))) ≫ ℱ.map (eqToHom (_ : op { carrier := ↑(𝟙 Y) '' ↑(op U).unop, is_open' := (_ : IsOpen (↑(𝟙 Y) '' ↑(op U).unop)) } = op U)) ** erw [colimit.ι_desc_assoc] ** C : Type u inst✝¹ : Category.{v, u} C inst✝ : HasColimits C X Y : TopCat ℱ : Presheaf C Y U : Opens ↑Y ⊢ 𝟙 (ℱ.obj (op U)) = (coconeOfDiagramTerminal (IsLimit.mk fun s => CostructuredArrow.homMk (homOfLE (_ : ↑(𝟙 Y) '' ↑((Functor.fromPUnit (op U)).obj s.pt.right).unop ⊆ ↑s.pt.1.unop)).op) (Lan.diagram (Opens.map (𝟙 Y)).op ℱ (op U))).ι.app (CostructuredArrow.mk (𝟙 (op ((Opens.map (𝟙 Y)).obj U)))) ≫ ℱ.map (eqToHom (_ : op { carrier := ↑(𝟙 Y) '' ↑(op U).unop, is_open' := (_ : IsOpen (↑(𝟙 Y) '' ↑(op U).unop)) } = op U)) ** dsimp ** C : Type u inst✝¹ : Category.{v, u} C inst✝ : HasColimits C X Y : TopCat ℱ : Presheaf C Y U : Opens ↑Y ⊢ 𝟙 (ℱ.obj (op U)) = ℱ.map (IsTerminal.from (IsLimit.mk fun s => CostructuredArrow.homMk (homOfLE (_ : ↑(𝟙 Y) '' ↑((Functor.fromPUnit (op U)).obj s.pt.right).unop ⊆ ↑s.pt.1.unop)).op) (CostructuredArrow.mk (𝟙 (op ((Opens.map (𝟙 Y)).obj U))))).left ≫ ℱ.map (eqToHom (_ : op { carrier := ↑(𝟙 Y) '' ↑(op U).unop, is_open' := (_ : IsOpen (↑(𝟙 Y) '' ↑(op U).unop)) } = op U)) ** rw [← ℱ.map_comp, ← ℱ.map_id] ** C : Type u inst✝¹ : Category.{v, u} C inst✝ : HasColimits C X Y : TopCat ℱ : Presheaf C Y U : Opens ↑Y ⊢ ℱ.map (𝟙 (op U)) = ℱ.map ((IsTerminal.from (IsLimit.mk fun s => CostructuredArrow.homMk (homOfLE (_ : ↑(𝟙 Y) '' ↑((Functor.fromPUnit (op U)).obj s.pt.right).unop ⊆ ↑s.pt.1.unop)).op) (CostructuredArrow.mk (𝟙 (op ((Opens.map (𝟙 Y)).obj U))))).left ≫ eqToHom (_ : op { carrier := ↑(𝟙 Y) '' ↑(op U).unop, is_open' := (_ : IsOpen (↑(𝟙 Y) '' ↑(op U).unop)) } = op U)) ** rfl ** Qed | |
TopCat.Presheaf.id_pushforward ** C : Type u inst✝ : Category.{v, u} C X : TopCat ⊢ pushforward C (𝟙 X) = 𝟭 (Presheaf C X) ** apply CategoryTheory.Functor.ext ** case h_map C : Type u inst✝ : Category.{v, u} C X : TopCat ⊢ autoParam (∀ (X_1 Y : Presheaf C X) (f : X_1 ⟶ Y), (pushforward C (𝟙 X)).map f = eqToHom (_ : ?F.obj X_1 = ?G.obj X_1) ≫ (𝟭 (Presheaf C X)).map f ≫ eqToHom (_ : (𝟭 (Presheaf C X)).obj Y = (pushforward C (𝟙 X)).obj Y)) _auto✝ ** intros a b f ** case h_map C : Type u inst✝ : Category.{v, u} C X : TopCat a b : Presheaf C X f : a ⟶ b ⊢ (pushforward C (𝟙 X)).map f = eqToHom (_ : ?F.obj a = ?G.obj a) ≫ (𝟭 (Presheaf C X)).map f ≫ eqToHom (_ : (𝟭 (Presheaf C X)).obj b = (pushforward C (𝟙 X)).obj b) ** ext U ** case h_map.w C : Type u inst✝ : Category.{v, u} C X : TopCat a b : Presheaf C X f : a ⟶ b U : Opens ↑X ⊢ ((pushforward C (𝟙 X)).map f).app (op U) = (eqToHom (_ : ?F.obj a = ?G.obj a) ≫ (𝟭 (Presheaf C X)).map f ≫ eqToHom (_ : (𝟭 (Presheaf C X)).obj b = (pushforward C (𝟙 X)).obj b)).app (op U) ** erw [NatTrans.congr f (Opens.op_map_id_obj (op U))] ** case h_map.w C : Type u inst✝ : Category.{v, u} C X : TopCat a b : Presheaf C X f : a ⟶ b U : Opens ↑X ⊢ a.map (eqToHom (_ : (Opens.map (𝟙 X)).op.obj (op U) = op U)) ≫ f.app (op U) ≫ b.map (eqToHom (_ : op U = (Opens.map (𝟙 X)).op.obj (op U))) = (eqToHom (_ : ?F.obj a = ?G.obj a) ≫ (𝟭 (Presheaf C X)).map f ≫ eqToHom (_ : (𝟭 (Presheaf C X)).obj b = (pushforward C (𝟙 X)).obj b)).app (op U) case h_obj C : Type u inst✝ : Category.{v, u} C X : TopCat ⊢ ∀ (X_1 : Presheaf C X), (pushforward C (𝟙 X)).obj X_1 = (𝟭 (Presheaf C X)).obj X_1 ** simp only [Functor.op_obj, eqToHom_refl, CategoryTheory.Functor.map_id,
Category.comp_id, Category.id_comp, Functor.id_obj, Functor.id_map] ** case h_obj C : Type u inst✝ : Category.{v, u} C X : TopCat ⊢ ∀ (X_1 : Presheaf C X), (pushforward C (𝟙 X)).obj X_1 = (𝟭 (Presheaf C X)).obj X_1 ** apply Pushforward.id_eq ** Qed | |
TopCat.Presheaf.toPushforwardOfIso_app ** C : Type u inst✝ : Category.{v, u} C X Y : TopCat H₁ : X ≅ Y ℱ : Presheaf C X 𝒢 : Presheaf C Y H₂ : H₁.hom _* ℱ ⟶ 𝒢 U : (Opens ↑X)ᵒᵖ ⊢ U = (Opens.map H₁.hom).op.obj (op ((Opens.map H₁.inv).obj U.unop)) ** simp [Opens.map, Set.preimage_preimage] ** C : Type u inst✝ : Category.{v, u} C X Y : TopCat H₁ : X ≅ Y ℱ : Presheaf C X 𝒢 : Presheaf C Y H₂ : H₁.hom _* ℱ ⟶ 𝒢 U : (Opens ↑X)ᵒᵖ ⊢ (toPushforwardOfIso H₁ H₂).app U = ℱ.map (eqToHom (_ : U = op { carrier := ↑H₁.hom ⁻¹' ↑(op { carrier := ↑H₁.inv ⁻¹' ↑U.unop, is_open' := (_ : IsOpen (↑H₁.inv ⁻¹' ↑U.unop)) }).unop, is_open' := (_ : IsOpen (↑H₁.hom ⁻¹' ↑(op { carrier := ↑H₁.inv ⁻¹' ↑U.unop, is_open' := (_ : IsOpen (↑H₁.inv ⁻¹' ↑U.unop)) }).unop)) })) ≫ H₂.app (op ((Opens.map H₁.inv).obj U.unop)) ** delta toPushforwardOfIso ** C : Type u inst✝ : Category.{v, u} C X Y : TopCat H₁ : X ≅ Y ℱ : Presheaf C X 𝒢 : Presheaf C Y H₂ : H₁.hom _* ℱ ⟶ 𝒢 U : (Opens ↑X)ᵒᵖ ⊢ (↑(Adjunction.homEquiv (presheafEquivOfIso C H₁).toAdjunction ℱ 𝒢) H₂).app U = ℱ.map (eqToHom (_ : U = op { carrier := ↑H₁.hom ⁻¹' ↑(op { carrier := ↑H₁.inv ⁻¹' ↑U.unop, is_open' := (_ : IsOpen (↑H₁.inv ⁻¹' ↑U.unop)) }).unop, is_open' := (_ : IsOpen (↑H₁.hom ⁻¹' ↑(op { carrier := ↑H₁.inv ⁻¹' ↑U.unop, is_open' := (_ : IsOpen (↑H₁.inv ⁻¹' ↑U.unop)) }).unop)) })) ≫ H₂.app (op ((Opens.map H₁.inv).obj U.unop)) ** simp only [pushforwardObj_obj, Functor.op_obj, Equivalence.toAdjunction, Adjunction.homEquiv_unit,
Functor.id_obj, Functor.comp_obj, Adjunction.mkOfUnitCounit_unit, unop_op, eqToHom_map] ** C : Type u inst✝ : Category.{v, u} C X Y : TopCat H₁ : X ≅ Y ℱ : Presheaf C X 𝒢 : Presheaf C Y H₂ : H₁.hom _* ℱ ⟶ 𝒢 U : (Opens ↑X)ᵒᵖ ⊢ ((presheafEquivOfIso C H₁).unit.app ℱ ≫ (presheafEquivOfIso C H₁).inverse.map H₂).app U = eqToHom (_ : ℱ.obj U = ℱ.obj (op ((Opens.map H₁.hom).obj ((Opens.map H₁.inv).obj U.unop)))) ≫ H₂.app (op ((Opens.map H₁.inv).obj U.unop)) ** rw [NatTrans.comp_app, presheafEquivOfIso_inverse_map_app, Equivalence.Equivalence_mk'_unit] ** C : Type u inst✝ : Category.{v, u} C X Y : TopCat H₁ : X ≅ Y ℱ : Presheaf C X 𝒢 : Presheaf C Y H₂ : H₁.hom _* ℱ ⟶ 𝒢 U : (Opens ↑X)ᵒᵖ ⊢ ((presheafEquivOfIso C H₁).unitIso.hom.app ℱ).app U ≫ H₂.app (op ((Opens.map H₁.inv).obj U.unop)) = eqToHom (_ : ℱ.obj U = ℱ.obj (op ((Opens.map H₁.hom).obj ((Opens.map H₁.inv).obj U.unop)))) ≫ H₂.app (op ((Opens.map H₁.inv).obj U.unop)) ** congr 1 ** case e_a C : Type u inst✝ : Category.{v, u} C X Y : TopCat H₁ : X ≅ Y ℱ : Presheaf C X 𝒢 : Presheaf C Y H₂ : H₁.hom _* ℱ ⟶ 𝒢 U : (Opens ↑X)ᵒᵖ ⊢ ((presheafEquivOfIso C H₁).unitIso.hom.app ℱ).app U = eqToHom (_ : ℱ.obj U = ℱ.obj (op ((Opens.map H₁.hom).obj ((Opens.map H₁.inv).obj U.unop)))) ** simp only [Equivalence.unit, Equivalence.op, CategoryTheory.Equivalence.symm, Opens.mapMapIso,
Functor.id_obj, Functor.comp_obj, Iso.symm_hom, NatIso.op_inv, Iso.symm_inv, NatTrans.op_app,
NatIso.ofComponents_hom_app, eqToIso.hom, eqToHom_op, Equivalence.Equivalence_mk'_unitInv,
Equivalence.Equivalence_mk'_counitInv, NatIso.op_hom, unop_op, op_unop, eqToIso.inv,
NatIso.ofComponents_inv_app, eqToHom_unop, ←ℱ.map_comp, eqToHom_trans, eqToHom_map,
presheafEquivOfIso_unitIso_hom_app_app] ** Qed | |
TopCat.Presheaf.pushforwardToOfIso_app ** C : Type u inst✝ : Category.{v, u} C X Y : TopCat H₁ : X ≅ Y ℱ : Presheaf C Y 𝒢 : Presheaf C X H₂ : ℱ ⟶ H₁.hom _* 𝒢 U : (Opens ↑X)ᵒᵖ ⊢ (Opens.map H₁.hom).op.obj (op ((Opens.map H₁.inv).obj U.unop)) = U ** simp [Opens.map, Set.preimage_preimage] ** C : Type u inst✝ : Category.{v, u} C X Y : TopCat H₁ : X ≅ Y ℱ : Presheaf C Y 𝒢 : Presheaf C X H₂ : ℱ ⟶ H₁.hom _* 𝒢 U : (Opens ↑X)ᵒᵖ ⊢ (pushforwardToOfIso H₁ H₂).app U = H₂.app (op ((Opens.map H₁.inv).obj U.unop)) ≫ 𝒢.map (eqToHom (_ : op { carrier := ↑H₁.hom ⁻¹' ↑(op { carrier := ↑H₁.inv ⁻¹' ↑U.unop, is_open' := (_ : IsOpen (↑H₁.inv ⁻¹' ↑U.unop)) }).unop, is_open' := (_ : IsOpen (↑H₁.hom ⁻¹' ↑(op { carrier := ↑H₁.inv ⁻¹' ↑U.unop, is_open' := (_ : IsOpen (↑H₁.inv ⁻¹' ↑U.unop)) }).unop)) } = U)) ** simp [pushforwardToOfIso, Equivalence.toAdjunction, CategoryStruct.comp] ** Qed | |
SetTheory.PGame.subsingleton_short_example ** xl xr : Type u_1 xL : xl → PGame xR : xr → PGame a b : Short (mk xl xr xL xR) ⊢ a = b ** cases a ** case mk xl xr : Type u_1 xL : xl → PGame xR : xr → PGame b : Short (mk xl xr xL xR) inst✝¹ : Fintype xl inst✝ : Fintype xr x✝¹ : (i : xl) → Short (xL i) x✝ : (j : xr) → Short (xR j) ⊢ Short.mk x✝¹ x✝ = b ** cases b ** case mk.mk xl xr : Type u_1 xL : xl → PGame xR : xr → PGame inst✝³ : Fintype xl inst✝² : Fintype xr x✝³ : (i : xl) → Short (xL i) x✝² : (j : xr) → Short (xR j) inst✝¹ : Fintype xl inst✝ : Fintype xr x✝¹ : (i : xl) → Short (xL i) x✝ : (j : xr) → Short (xR j) ⊢ Short.mk x✝³ x✝² = Short.mk x✝¹ x✝ ** congr! ** case mk.mk.h.e'_5 xl xr : Type u_1 xL : xl → PGame xR : xr → PGame inst✝³ : Fintype xl inst✝² : Fintype xr x✝³ : (i : xl) → Short (xL i) x✝² : (j : xr) → Short (xR j) inst✝¹ : Fintype xl inst✝ : Fintype xr x✝¹ : (i : xl) → Short (xL i) x✝ : (j : xr) → Short (xR j) ⊢ x✝³ = x✝¹ ** funext x ** case mk.mk.h.e'_5.h xl xr : Type u_1 xL : xl → PGame xR : xr → PGame inst✝³ : Fintype xl inst✝² : Fintype xr x✝³ : (i : xl) → Short (xL i) x✝² : (j : xr) → Short (xR j) inst✝¹ : Fintype xl inst✝ : Fintype xr x✝¹ : (i : xl) → Short (xL i) x✝ : (j : xr) → Short (xR j) x : xl ⊢ x✝³ x = x✝¹ x ** apply @Subsingleton.elim _ (subsingleton_short_example (xL x)) ** case mk.mk.h.e'_6 xl xr : Type u_1 xL : xl → PGame xR : xr → PGame inst✝³ : Fintype xl inst✝² : Fintype xr x✝³ : (i : xl) → Short (xL i) x✝² : (j : xr) → Short (xR j) inst✝¹ : Fintype xl inst✝ : Fintype xr x✝¹ : (i : xl) → Short (xL i) x✝ : (j : xr) → Short (xR j) ⊢ x✝² = x✝ ** funext x ** case mk.mk.h.e'_6.h xl xr : Type u_1 xL : xl → PGame xR : xr → PGame inst✝³ : Fintype xl inst✝² : Fintype xr x✝³ : (i : xl) → Short (xL i) x✝² : (j : xr) → Short (xR j) inst✝¹ : Fintype xl inst✝ : Fintype xr x✝¹ : (i : xl) → Short (xL i) x✝ : (j : xr) → Short (xR j) x : xr ⊢ x✝² x = x✝ x ** apply @Subsingleton.elim _ (subsingleton_short_example (xR x)) ** Qed | |
SetTheory.PGame.short_birthday ** case mk xl xr : Type u xL : xl → PGame xR : xr → PGame ihl : ∀ (a : xl) [inst : Short (xL a)], birthday (xL a) < Ordinal.omega ihr : ∀ (a : xr) [inst : Short (xR a)], birthday (xR a) < Ordinal.omega ⊢ ∀ [inst : Short (mk xl xr xL xR)], birthday (mk xl xr xL xR) < Ordinal.omega ** intro hs ** case mk xl xr : Type u xL : xl → PGame xR : xr → PGame ihl : ∀ (a : xl) [inst : Short (xL a)], birthday (xL a) < Ordinal.omega ihr : ∀ (a : xr) [inst : Short (xR a)], birthday (xR a) < Ordinal.omega hs : Short (mk xl xr xL xR) ⊢ birthday (mk xl xr xL xR) < Ordinal.omega ** rcases hs with ⟨sL, sR⟩ ** case mk.mk xl xr : Type u xL : xl → PGame xR : xr → PGame ihl : ∀ (a : xl) [inst : Short (xL a)], birthday (xL a) < Ordinal.omega ihr : ∀ (a : xr) [inst : Short (xR a)], birthday (xR a) < Ordinal.omega inst✝¹ : Fintype xl inst✝ : Fintype xr sL : (i : xl) → Short (xL i) sR : (j : xr) → Short (xR j) ⊢ birthday (mk xl xr xL xR) < Ordinal.omega ** rw [birthday, max_lt_iff] ** case mk.mk xl xr : Type u xL : xl → PGame xR : xr → PGame ihl : ∀ (a : xl) [inst : Short (xL a)], birthday (xL a) < Ordinal.omega ihr : ∀ (a : xr) [inst : Short (xR a)], birthday (xR a) < Ordinal.omega inst✝¹ : Fintype xl inst✝ : Fintype xr sL : (i : xl) → Short (xL i) sR : (j : xr) → Short (xR j) ⊢ (Ordinal.lsub fun i => birthday (xL i)) < Ordinal.omega ∧ (Ordinal.lsub fun i => birthday (xR i)) < Ordinal.omega ** constructor ** case mk.mk.left xl xr : Type u xL : xl → PGame xR : xr → PGame ihl : ∀ (a : xl) [inst : Short (xL a)], birthday (xL a) < Ordinal.omega ihr : ∀ (a : xr) [inst : Short (xR a)], birthday (xR a) < Ordinal.omega inst✝¹ : Fintype xl inst✝ : Fintype xr sL : (i : xl) → Short (xL i) sR : (j : xr) → Short (xR j) ⊢ (Ordinal.lsub fun i => birthday (xL i)) < Ordinal.omega case mk.mk.right xl xr : Type u xL : xl → PGame xR : xr → PGame ihl : ∀ (a : xl) [inst : Short (xL a)], birthday (xL a) < Ordinal.omega ihr : ∀ (a : xr) [inst : Short (xR a)], birthday (xR a) < Ordinal.omega inst✝¹ : Fintype xl inst✝ : Fintype xr sL : (i : xl) → Short (xL i) sR : (j : xr) → Short (xR j) ⊢ (Ordinal.lsub fun i => birthday (xR i)) < Ordinal.omega ** all_goals
rw [← Cardinal.ord_aleph0]
refine'
Cardinal.lsub_lt_ord_of_isRegular.{u, u} Cardinal.isRegular_aleph0
(Cardinal.lt_aleph0_of_finite _) fun i => _
rw [Cardinal.ord_aleph0] ** case mk.mk.right xl xr : Type u xL : xl → PGame xR : xr → PGame ihl : ∀ (a : xl) [inst : Short (xL a)], birthday (xL a) < Ordinal.omega ihr : ∀ (a : xr) [inst : Short (xR a)], birthday (xR a) < Ordinal.omega inst✝¹ : Fintype xl inst✝ : Fintype xr sL : (i : xl) → Short (xL i) sR : (j : xr) → Short (xR j) ⊢ (Ordinal.lsub fun i => birthday (xR i)) < Ordinal.omega ** rw [← Cardinal.ord_aleph0] ** case mk.mk.right xl xr : Type u xL : xl → PGame xR : xr → PGame ihl : ∀ (a : xl) [inst : Short (xL a)], birthday (xL a) < Ordinal.omega ihr : ∀ (a : xr) [inst : Short (xR a)], birthday (xR a) < Ordinal.omega inst✝¹ : Fintype xl inst✝ : Fintype xr sL : (i : xl) → Short (xL i) sR : (j : xr) → Short (xR j) ⊢ (Ordinal.lsub fun i => birthday (xR i)) < Cardinal.ord Cardinal.aleph0 ** refine'
Cardinal.lsub_lt_ord_of_isRegular.{u, u} Cardinal.isRegular_aleph0
(Cardinal.lt_aleph0_of_finite _) fun i => _ ** case mk.mk.right xl xr : Type u xL : xl → PGame xR : xr → PGame ihl : ∀ (a : xl) [inst : Short (xL a)], birthday (xL a) < Ordinal.omega ihr : ∀ (a : xr) [inst : Short (xR a)], birthday (xR a) < Ordinal.omega inst✝¹ : Fintype xl inst✝ : Fintype xr sL : (i : xl) → Short (xL i) sR : (j : xr) → Short (xR j) i : xr ⊢ birthday (xR i) < Cardinal.ord Cardinal.aleph0 ** rw [Cardinal.ord_aleph0] ** case mk.mk.left xl xr : Type u xL : xl → PGame xR : xr → PGame ihl : ∀ (a : xl) [inst : Short (xL a)], birthday (xL a) < Ordinal.omega ihr : ∀ (a : xr) [inst : Short (xR a)], birthday (xR a) < Ordinal.omega inst✝¹ : Fintype xl inst✝ : Fintype xr sL : (i : xl) → Short (xL i) sR : (j : xr) → Short (xR j) i : xl ⊢ birthday (xL i) < Ordinal.omega ** apply ihl ** case mk.mk.right xl xr : Type u xL : xl → PGame xR : xr → PGame ihl : ∀ (a : xl) [inst : Short (xL a)], birthday (xL a) < Ordinal.omega ihr : ∀ (a : xr) [inst : Short (xR a)], birthday (xR a) < Ordinal.omega inst✝¹ : Fintype xl inst✝ : Fintype xr sL : (i : xl) → Short (xL i) sR : (j : xr) → Short (xR j) i : xr ⊢ birthday (xR i) < Ordinal.omega ** apply ihr ** Qed | |
Order.le_cof ** α : Type u_1 r✝ r : α → α → Prop inst✝ : IsRefl α r c : Cardinal.{u_1} ⊢ c ≤ cof r ↔ ∀ {S : Set α}, (∀ (a : α), ∃ b, b ∈ S ∧ r a b) → c ≤ #↑S ** rw [cof, le_csInf_iff'' (cof_nonempty r)] ** α : Type u_1 r✝ r : α → α → Prop inst✝ : IsRefl α r c : Cardinal.{u_1} ⊢ (∀ (b : Cardinal.{u_1}), b ∈ {c | ∃ S, (∀ (a : α), ∃ b, b ∈ S ∧ r a b) ∧ #↑S = c} → c ≤ b) ↔ ∀ {S : Set α}, (∀ (a : α), ∃ b, b ∈ S ∧ r a b) → c ≤ #↑S ** use fun H S h => H _ ⟨S, h, rfl⟩ ** case mpr α : Type u_1 r✝ r : α → α → Prop inst✝ : IsRefl α r c : Cardinal.{u_1} ⊢ (∀ {S : Set α}, (∀ (a : α), ∃ b, b ∈ S ∧ r a b) → c ≤ #↑S) → ∀ (b : Cardinal.{u_1}), b ∈ {c | ∃ S, (∀ (a : α), ∃ b, b ∈ S ∧ r a b) ∧ #↑S = c} → c ≤ b ** rintro H d ⟨S, h, rfl⟩ ** case mpr.intro.intro α : Type u_1 r✝ r : α → α → Prop inst✝ : IsRefl α r c : Cardinal.{u_1} H : ∀ {S : Set α}, (∀ (a : α), ∃ b, b ∈ S ∧ r a b) → c ≤ #↑S S : Set α h : ∀ (a : α), ∃ b, b ∈ S ∧ r a b ⊢ c ≤ #↑S ** exact H h ** Qed | |
RelIso.cof_le_lift ** α✝ : Type u_1 r✝ : α✝ → α✝ → Prop α : Type u β : Type v r : α → α → Prop s : β → β → Prop inst✝ : IsRefl β s f : r ≃r s ⊢ Cardinal.lift.{max u v, u} (cof r) ≤ Cardinal.lift.{max u v, v} (cof s) ** rw [Order.cof, Order.cof, lift_sInf, lift_sInf,
le_csInf_iff'' (nonempty_image_iff.2 (Order.cof_nonempty s))] ** α✝ : Type u_1 r✝ : α✝ → α✝ → Prop α : Type u β : Type v r : α → α → Prop s : β → β → Prop inst✝ : IsRefl β s f : r ≃r s ⊢ ∀ (b : Cardinal.{max u v}), b ∈ Cardinal.lift.{max u v, v} '' {c | ∃ S, (∀ (a : β), ∃ b, b ∈ S ∧ s a b) ∧ #↑S = c} → sInf (Cardinal.lift.{max u v, u} '' {c | ∃ S, (∀ (a : α), ∃ b, b ∈ S ∧ r a b) ∧ #↑S = c}) ≤ b ** rintro - ⟨-, ⟨u, H, rfl⟩, rfl⟩ ** case intro.intro.intro.intro α✝ : Type u_1 r✝ : α✝ → α✝ → Prop α : Type u β : Type v r : α → α → Prop s : β → β → Prop inst✝ : IsRefl β s f : r ≃r s u : Set β H : ∀ (a : β), ∃ b, b ∈ u ∧ s a b ⊢ sInf (Cardinal.lift.{max u v, u} '' {c | ∃ S, (∀ (a : α), ∃ b, b ∈ S ∧ r a b) ∧ #↑S = c}) ≤ Cardinal.lift.{max u v, v} #↑u ** apply csInf_le' ** case intro.intro.intro.intro.h α✝ : Type u_1 r✝ : α✝ → α✝ → Prop α : Type u β : Type v r : α → α → Prop s : β → β → Prop inst✝ : IsRefl β s f : r ≃r s u : Set β H : ∀ (a : β), ∃ b, b ∈ u ∧ s a b ⊢ Cardinal.lift.{max u v, v} #↑u ∈ Cardinal.lift.{max u v, u} '' {c | ∃ S, (∀ (a : α), ∃ b, b ∈ S ∧ r a b) ∧ #↑S = c} ** refine'
⟨_, ⟨f.symm '' u, fun a => _, rfl⟩,
lift_mk_eq.{u, v, max u v}.2 ⟨(f.symm.toEquiv.image u).symm⟩⟩ ** case intro.intro.intro.intro.h α✝ : Type u_1 r✝ : α✝ → α✝ → Prop α : Type u β : Type v r : α → α → Prop s : β → β → Prop inst✝ : IsRefl β s f : r ≃r s u : Set β H : ∀ (a : β), ∃ b, b ∈ u ∧ s a b a : α ⊢ ∃ b, b ∈ ↑(RelIso.symm f) '' u ∧ r a b ** rcases H (f a) with ⟨b, hb, hb'⟩ ** case intro.intro.intro.intro.h.intro.intro α✝ : Type u_1 r✝ : α✝ → α✝ → Prop α : Type u β : Type v r : α → α → Prop s : β → β → Prop inst✝ : IsRefl β s f : r ≃r s u : Set β H : ∀ (a : β), ∃ b, b ∈ u ∧ s a b a : α b : β hb : b ∈ u hb' : s (↑f a) b ⊢ ∃ b, b ∈ ↑(RelIso.symm f) '' u ∧ r a b ** refine' ⟨f.symm b, mem_image_of_mem _ hb, f.map_rel_iff.1 _⟩ ** case intro.intro.intro.intro.h.intro.intro α✝ : Type u_1 r✝ : α✝ → α✝ → Prop α : Type u β : Type v r : α → α → Prop s : β → β → Prop inst✝ : IsRefl β s f : r ≃r s u : Set β H : ∀ (a : β), ∃ b, b ∈ u ∧ s a b a : α b : β hb : b ∈ u hb' : s (↑f a) b ⊢ s (↑f a) (↑f (↑(RelIso.symm f) b)) ** rwa [RelIso.apply_symm_apply] ** Qed | |
Ordinal.le_cof_type ** α : Type u_1 r : α → α → Prop inst✝ : IsWellOrder α r c : Cardinal.{u_1} ⊢ (∀ (S : Set α), Unbounded r S → c ≤ #↑S) → ∀ (b : Cardinal.{u_1}), b ∈ {c | ∃ S, Unbounded r S ∧ #↑S = c} → c ≤ b ** rintro H d ⟨S, h, rfl⟩ ** case intro.intro α : Type u_1 r : α → α → Prop inst✝ : IsWellOrder α r c : Cardinal.{u_1} H : ∀ (S : Set α), Unbounded r S → c ≤ #↑S S : Set α h : Unbounded r S ⊢ c ≤ #↑S ** exact H _ h ** Qed | |
Ordinal.lt_cof_type ** α : Type u_1 r : α → α → Prop inst✝ : IsWellOrder α r S : Set α ⊢ #↑S < cof (type r) → Bounded r S ** simpa using not_imp_not.2 cof_type_le ** Qed | |
Ordinal.ord_cof_eq ** α : Type u_1 r✝ r : α → α → Prop inst✝ : IsWellOrder α r ⊢ ∃ S, Unbounded r S ∧ type (Subrel r S) = ord (cof (type r)) ** let ⟨S, hS, e⟩ := cof_eq r ** α : Type u_1 r✝ r : α → α → Prop inst✝ : IsWellOrder α r S : Set α hS : Unbounded r S e : #↑S = cof (type r) ⊢ ∃ S, Unbounded r S ∧ type (Subrel r S) = ord (cof (type r)) ** let ⟨s, _, e'⟩ := Cardinal.ord_eq S ** α : Type u_1 r✝ r : α → α → Prop inst✝ : IsWellOrder α r S : Set α hS : Unbounded r S e : #↑S = cof (type r) s : ↑S → ↑S → Prop w✝ : IsWellOrder (↑S) s e' : ord #↑S = type s ⊢ ∃ S, Unbounded r S ∧ type (Subrel r S) = ord (cof (type r)) ** let T : Set α := { a | ∃ aS : a ∈ S, ∀ b : S, s b ⟨_, aS⟩ → r b a } ** α : Type u_1 r✝ r : α → α → Prop inst✝ : IsWellOrder α r S : Set α hS : Unbounded r S e : #↑S = cof (type r) s : ↑S → ↑S → Prop w✝ : IsWellOrder (↑S) s e' : ord #↑S = type s T : Set α := {a | ∃ aS, ∀ (b : ↑S), s b { val := a, property := aS } → r (↑b) a} ⊢ ∃ S, Unbounded r S ∧ type (Subrel r S) = ord (cof (type r)) ** suffices : Unbounded r T ** α : Type u_1 r✝ r : α → α → Prop inst✝ : IsWellOrder α r S : Set α hS : Unbounded r S e : #↑S = cof (type r) s : ↑S → ↑S → Prop w✝ : IsWellOrder (↑S) s e' : ord #↑S = type s T : Set α := {a | ∃ aS, ∀ (b : ↑S), s b { val := a, property := aS } → r (↑b) a} this : Unbounded r T ⊢ ∃ S, Unbounded r S ∧ type (Subrel r S) = ord (cof (type r)) ** refine' ⟨T, this, le_antisymm _ (Cardinal.ord_le.2 <| cof_type_le this)⟩ ** α : Type u_1 r✝ r : α → α → Prop inst✝ : IsWellOrder α r S : Set α hS : Unbounded r S e : #↑S = cof (type r) s : ↑S → ↑S → Prop w✝ : IsWellOrder (↑S) s e' : ord #↑S = type s T : Set α := {a | ∃ aS, ∀ (b : ↑S), s b { val := a, property := aS } → r (↑b) a} this : Unbounded r T ⊢ type (Subrel r T) ≤ ord (cof (type r)) ** rw [← e, e'] ** α : Type u_1 r✝ r : α → α → Prop inst✝ : IsWellOrder α r S : Set α hS : Unbounded r S e : #↑S = cof (type r) s : ↑S → ↑S → Prop w✝ : IsWellOrder (↑S) s e' : ord #↑S = type s T : Set α := {a | ∃ aS, ∀ (b : ↑S), s b { val := a, property := aS } → r (↑b) a} this : Unbounded r T ⊢ type (Subrel r T) ≤ type s ** refine'
(RelEmbedding.ofMonotone
(fun a : T =>
(⟨a,
let ⟨aS, _⟩ := a.2
aS⟩ :
S))
fun a b h => _).ordinal_type_le ** α : Type u_1 r✝ r : α → α → Prop inst✝ : IsWellOrder α r S : Set α hS : Unbounded r S e : #↑S = cof (type r) s : ↑S → ↑S → Prop w✝ : IsWellOrder (↑S) s e' : ord #↑S = type s T : Set α := {a | ∃ aS, ∀ (b : ↑S), s b { val := a, property := aS } → r (↑b) a} this : Unbounded r T a b : ↑T h : Subrel r T a b ⊢ s ((fun a => { val := ↑a, property := (_ : ↑a ∈ S) }) a) ((fun a => { val := ↑a, property := (_ : ↑a ∈ S) }) b) ** rcases a with ⟨a, aS, ha⟩ ** case mk.intro α : Type u_1 r✝ r : α → α → Prop inst✝ : IsWellOrder α r S : Set α hS : Unbounded r S e : #↑S = cof (type r) s : ↑S → ↑S → Prop w✝ : IsWellOrder (↑S) s e' : ord #↑S = type s T : Set α := {a | ∃ aS, ∀ (b : ↑S), s b { val := a, property := aS } → r (↑b) a} this : Unbounded r T b : ↑T a : α aS : a ∈ S ha : ∀ (b : ↑S), s b { val := a, property := aS } → r (↑b) a h : Subrel r T { val := a, property := (_ : ∃ aS, ∀ (b : ↑S), s b { val := a, property := aS } → r (↑b) a) } b ⊢ s ((fun a => { val := ↑a, property := (_ : ↑a ∈ S) }) { val := a, property := (_ : ∃ aS, ∀ (b : ↑S), s b { val := a, property := aS } → r (↑b) a) }) ((fun a => { val := ↑a, property := (_ : ↑a ∈ S) }) b) ** rcases b with ⟨b, bS, hb⟩ ** case mk.intro.mk.intro α : Type u_1 r✝ r : α → α → Prop inst✝ : IsWellOrder α r S : Set α hS : Unbounded r S e : #↑S = cof (type r) s : ↑S → ↑S → Prop w✝ : IsWellOrder (↑S) s e' : ord #↑S = type s T : Set α := {a | ∃ aS, ∀ (b : ↑S), s b { val := a, property := aS } → r (↑b) a} this : Unbounded r T a : α aS : a ∈ S ha : ∀ (b : ↑S), s b { val := a, property := aS } → r (↑b) a b : α bS : b ∈ S hb : ∀ (b_1 : ↑S), s b_1 { val := b, property := bS } → r (↑b_1) b h : Subrel r T { val := a, property := (_ : ∃ aS, ∀ (b : ↑S), s b { val := a, property := aS } → r (↑b) a) } { val := b, property := (_ : ∃ aS, ∀ (b_1 : ↑S), s b_1 { val := b, property := aS } → r (↑b_1) b) } ⊢ s ((fun a => { val := ↑a, property := (_ : ↑a ∈ S) }) { val := a, property := (_ : ∃ aS, ∀ (b : ↑S), s b { val := a, property := aS } → r (↑b) a) }) ((fun a => { val := ↑a, property := (_ : ↑a ∈ S) }) { val := b, property := (_ : ∃ aS, ∀ (b_1 : ↑S), s b_1 { val := b, property := aS } → r (↑b_1) b) }) ** change s ⟨a, _⟩ ⟨b, _⟩ ** case mk.intro.mk.intro α : Type u_1 r✝ r : α → α → Prop inst✝ : IsWellOrder α r S : Set α hS : Unbounded r S e : #↑S = cof (type r) s : ↑S → ↑S → Prop w✝ : IsWellOrder (↑S) s e' : ord #↑S = type s T : Set α := {a | ∃ aS, ∀ (b : ↑S), s b { val := a, property := aS } → r (↑b) a} this : Unbounded r T a : α aS : a ∈ S ha : ∀ (b : ↑S), s b { val := a, property := aS } → r (↑b) a b : α bS : b ∈ S hb : ∀ (b_1 : ↑S), s b_1 { val := b, property := bS } → r (↑b_1) b h : Subrel r T { val := a, property := (_ : ∃ aS, ∀ (b : ↑S), s b { val := a, property := aS } → r (↑b) a) } { val := b, property := (_ : ∃ aS, ∀ (b_1 : ↑S), s b_1 { val := b, property := aS } → r (↑b_1) b) } ⊢ s { val := a, property := (_ : ↑{ val := a, property := (_ : ∃ aS, ∀ (b : ↑S), s b { val := a, property := aS } → r (↑b) a) } ∈ S) } { val := b, property := (_ : ↑{ val := b, property := (_ : ∃ aS, ∀ (b_1 : ↑S), s b_1 { val := b, property := aS } → r (↑b_1) b) } ∈ S) } ** refine' ((trichotomous_of s _ _).resolve_left fun hn => _).resolve_left _ ** case mk.intro.mk.intro.refine'_1 α : Type u_1 r✝ r : α → α → Prop inst✝ : IsWellOrder α r S : Set α hS : Unbounded r S e : #↑S = cof (type r) s : ↑S → ↑S → Prop w✝ : IsWellOrder (↑S) s e' : ord #↑S = type s T : Set α := {a | ∃ aS, ∀ (b : ↑S), s b { val := a, property := aS } → r (↑b) a} this : Unbounded r T a : α aS : a ∈ S ha : ∀ (b : ↑S), s b { val := a, property := aS } → r (↑b) a b : α bS : b ∈ S hb : ∀ (b_1 : ↑S), s b_1 { val := b, property := bS } → r (↑b_1) b h : Subrel r T { val := a, property := (_ : ∃ aS, ∀ (b : ↑S), s b { val := a, property := aS } → r (↑b) a) } { val := b, property := (_ : ∃ aS, ∀ (b_1 : ↑S), s b_1 { val := b, property := aS } → r (↑b_1) b) } hn : s { val := b, property := (_ : ↑{ val := b, property := (_ : ∃ aS, ∀ (b_1 : ↑S), s b_1 { val := b, property := aS } → r (↑b_1) b) } ∈ S) } { val := a, property := (_ : ↑{ val := a, property := (_ : ∃ aS, ∀ (b : ↑S), s b { val := a, property := aS } → r (↑b) a) } ∈ S) } ⊢ False ** exact asymm h (ha _ hn) ** case mk.intro.mk.intro.refine'_2 α : Type u_1 r✝ r : α → α → Prop inst✝ : IsWellOrder α r S : Set α hS : Unbounded r S e : #↑S = cof (type r) s : ↑S → ↑S → Prop w✝ : IsWellOrder (↑S) s e' : ord #↑S = type s T : Set α := {a | ∃ aS, ∀ (b : ↑S), s b { val := a, property := aS } → r (↑b) a} this : Unbounded r T a : α aS : a ∈ S ha : ∀ (b : ↑S), s b { val := a, property := aS } → r (↑b) a b : α bS : b ∈ S hb : ∀ (b_1 : ↑S), s b_1 { val := b, property := bS } → r (↑b_1) b h : Subrel r T { val := a, property := (_ : ∃ aS, ∀ (b : ↑S), s b { val := a, property := aS } → r (↑b) a) } { val := b, property := (_ : ∃ aS, ∀ (b_1 : ↑S), s b_1 { val := b, property := aS } → r (↑b_1) b) } ⊢ ¬{ val := b, property := (_ : ↑{ val := b, property := (_ : ∃ aS, ∀ (b_1 : ↑S), s b_1 { val := b, property := aS } → r (↑b_1) b) } ∈ S) } = { val := a, property := (_ : ↑{ val := a, property := (_ : ∃ aS, ∀ (b : ↑S), s b { val := a, property := aS } → r (↑b) a) } ∈ S) } ** intro e ** case mk.intro.mk.intro.refine'_2 α : Type u_1 r✝ r : α → α → Prop inst✝ : IsWellOrder α r S : Set α hS : Unbounded r S e✝ : #↑S = cof (type r) s : ↑S → ↑S → Prop w✝ : IsWellOrder (↑S) s e' : ord #↑S = type s T : Set α := {a | ∃ aS, ∀ (b : ↑S), s b { val := a, property := aS } → r (↑b) a} this : Unbounded r T a : α aS : a ∈ S ha : ∀ (b : ↑S), s b { val := a, property := aS } → r (↑b) a b : α bS : b ∈ S hb : ∀ (b_1 : ↑S), s b_1 { val := b, property := bS } → r (↑b_1) b h : Subrel r T { val := a, property := (_ : ∃ aS, ∀ (b : ↑S), s b { val := a, property := aS } → r (↑b) a) } { val := b, property := (_ : ∃ aS, ∀ (b_1 : ↑S), s b_1 { val := b, property := aS } → r (↑b_1) b) } e : { val := b, property := (_ : ↑{ val := b, property := (_ : ∃ aS, ∀ (b_1 : ↑S), s b_1 { val := b, property := aS } → r (↑b_1) b) } ∈ S) } = { val := a, property := (_ : ↑{ val := a, property := (_ : ∃ aS, ∀ (b : ↑S), s b { val := a, property := aS } → r (↑b) a) } ∈ S) } ⊢ False ** injection e with e ** case mk.intro.mk.intro.refine'_2 α : Type u_1 r✝ r : α → α → Prop inst✝ : IsWellOrder α r S : Set α hS : Unbounded r S e✝ : #↑S = cof (type r) s : ↑S → ↑S → Prop w✝ : IsWellOrder (↑S) s e' : ord #↑S = type s T : Set α := {a | ∃ aS, ∀ (b : ↑S), s b { val := a, property := aS } → r (↑b) a} this : Unbounded r T a : α aS : a ∈ S ha : ∀ (b : ↑S), s b { val := a, property := aS } → r (↑b) a b : α bS : b ∈ S hb : ∀ (b_1 : ↑S), s b_1 { val := b, property := bS } → r (↑b_1) b h : Subrel r T { val := a, property := (_ : ∃ aS, ∀ (b : ↑S), s b { val := a, property := aS } → r (↑b) a) } { val := b, property := (_ : ∃ aS, ∀ (b_1 : ↑S), s b_1 { val := b, property := aS } → r (↑b_1) b) } e : b = a ⊢ False ** subst b ** case mk.intro.mk.intro.refine'_2 α : Type u_1 r✝ r : α → α → Prop inst✝ : IsWellOrder α r S : Set α hS : Unbounded r S e : #↑S = cof (type r) s : ↑S → ↑S → Prop w✝ : IsWellOrder (↑S) s e' : ord #↑S = type s T : Set α := {a | ∃ aS, ∀ (b : ↑S), s b { val := a, property := aS } → r (↑b) a} this : Unbounded r T a : α aS : a ∈ S ha : ∀ (b : ↑S), s b { val := a, property := aS } → r (↑b) a bS : a ∈ S hb : ∀ (b : ↑S), s b { val := a, property := bS } → r (↑b) a h : Subrel r T { val := a, property := (_ : ∃ aS, ∀ (b : ↑S), s b { val := a, property := aS } → r (↑b) a) } { val := a, property := (_ : ∃ aS, ∀ (b : ↑S), s b { val := a, property := aS } → r (↑b) a) } ⊢ False ** exact irrefl _ h ** case this α : Type u_1 r✝ r : α → α → Prop inst✝ : IsWellOrder α r S : Set α hS : Unbounded r S e : #↑S = cof (type r) s : ↑S → ↑S → Prop w✝ : IsWellOrder (↑S) s e' : ord #↑S = type s T : Set α := {a | ∃ aS, ∀ (b : ↑S), s b { val := a, property := aS } → r (↑b) a} ⊢ Unbounded r T ** intro a ** case this α : Type u_1 r✝ r : α → α → Prop inst✝ : IsWellOrder α r S : Set α hS : Unbounded r S e : #↑S = cof (type r) s : ↑S → ↑S → Prop w✝ : IsWellOrder (↑S) s e' : ord #↑S = type s T : Set α := {a | ∃ aS, ∀ (b : ↑S), s b { val := a, property := aS } → r (↑b) a} a : α ⊢ ∃ b, b ∈ T ∧ ¬r b a ** have : { b : S | ¬r b a }.Nonempty :=
let ⟨b, bS, ba⟩ := hS a
⟨⟨b, bS⟩, ba⟩ ** case this α : Type u_1 r✝ r : α → α → Prop inst✝ : IsWellOrder α r S : Set α hS : Unbounded r S e : #↑S = cof (type r) s : ↑S → ↑S → Prop w✝ : IsWellOrder (↑S) s e' : ord #↑S = type s T : Set α := {a | ∃ aS, ∀ (b : ↑S), s b { val := a, property := aS } → r (↑b) a} a : α this : Set.Nonempty {b | ¬r (↑b) a} ⊢ ∃ b, b ∈ T ∧ ¬r b a ** let b := (IsWellFounded.wf : WellFounded s).min _ this ** case this α : Type u_1 r✝ r : α → α → Prop inst✝ : IsWellOrder α r S : Set α hS : Unbounded r S e : #↑S = cof (type r) s : ↑S → ↑S → Prop w✝ : IsWellOrder (↑S) s e' : ord #↑S = type s T : Set α := {a | ∃ aS, ∀ (b : ↑S), s b { val := a, property := aS } → r (↑b) a} a : α this : Set.Nonempty {b | ¬r (↑b) a} b : ↑S := WellFounded.min (_ : WellFounded s) {b | ¬r (↑b) a} this ⊢ ∃ b, b ∈ T ∧ ¬r b a ** have ba : ¬r b a := IsWellFounded.wf.min_mem _ this ** case this α : Type u_1 r✝ r : α → α → Prop inst✝ : IsWellOrder α r S : Set α hS : Unbounded r S e : #↑S = cof (type r) s : ↑S → ↑S → Prop w✝ : IsWellOrder (↑S) s e' : ord #↑S = type s T : Set α := {a | ∃ aS, ∀ (b : ↑S), s b { val := a, property := aS } → r (↑b) a} a : α this : Set.Nonempty {b | ¬r (↑b) a} b : ↑S := WellFounded.min (_ : WellFounded s) {b | ¬r (↑b) a} this ba : ¬r (↑b) a ⊢ ∃ b, b ∈ T ∧ ¬r b a ** refine' ⟨b, ⟨b.2, fun c => not_imp_not.1 fun h => _⟩, ba⟩ ** case this α : Type u_1 r✝ r : α → α → Prop inst✝ : IsWellOrder α r S : Set α hS : Unbounded r S e : #↑S = cof (type r) s : ↑S → ↑S → Prop w✝ : IsWellOrder (↑S) s e' : ord #↑S = type s T : Set α := {a | ∃ aS, ∀ (b : ↑S), s b { val := a, property := aS } → r (↑b) a} a : α this : Set.Nonempty {b | ¬r (↑b) a} b : ↑S := WellFounded.min (_ : WellFounded s) {b | ¬r (↑b) a} this ba : ¬r (↑b) a c : ↑S h : ¬r ↑c ↑b ⊢ ¬s c { val := ↑b, property := (_ : ↑b ∈ S) } ** rw [show ∀ b : S, (⟨b, b.2⟩ : S) = b by intro b; cases b; rfl] ** case this α : Type u_1 r✝ r : α → α → Prop inst✝ : IsWellOrder α r S : Set α hS : Unbounded r S e : #↑S = cof (type r) s : ↑S → ↑S → Prop w✝ : IsWellOrder (↑S) s e' : ord #↑S = type s T : Set α := {a | ∃ aS, ∀ (b : ↑S), s b { val := a, property := aS } → r (↑b) a} a : α this : Set.Nonempty {b | ¬r (↑b) a} b : ↑S := WellFounded.min (_ : WellFounded s) {b | ¬r (↑b) a} this ba : ¬r (↑b) a c : ↑S h : ¬r ↑c ↑b ⊢ ¬s c b ** exact IsWellFounded.wf.not_lt_min _ this (IsOrderConnected.neg_trans h ba) ** α : Type u_1 r✝ r : α → α → Prop inst✝ : IsWellOrder α r S : Set α hS : Unbounded r S e : #↑S = cof (type r) s : ↑S → ↑S → Prop w✝ : IsWellOrder (↑S) s e' : ord #↑S = type s T : Set α := {a | ∃ aS, ∀ (b : ↑S), s b { val := a, property := aS } → r (↑b) a} a : α this : Set.Nonempty {b | ¬r (↑b) a} b : ↑S := WellFounded.min (_ : WellFounded s) {b | ¬r (↑b) a} this ba : ¬r (↑b) a c : ↑S h : ¬r ↑c ↑b ⊢ ∀ (b : ↑S), { val := ↑b, property := (_ : ↑b ∈ S) } = b ** intro b ** α : Type u_1 r✝ r : α → α → Prop inst✝ : IsWellOrder α r S : Set α hS : Unbounded r S e : #↑S = cof (type r) s : ↑S → ↑S → Prop w✝ : IsWellOrder (↑S) s e' : ord #↑S = type s T : Set α := {a | ∃ aS, ∀ (b : ↑S), s b { val := a, property := aS } → r (↑b) a} a : α this : Set.Nonempty {b | ¬r (↑b) a} b✝ : ↑S := WellFounded.min (_ : WellFounded s) {b | ¬r (↑b) a} this ba : ¬r (↑b✝) a c : ↑S h : ¬r ↑c ↑b✝ b : ↑S ⊢ { val := ↑b, property := (_ : ↑b ∈ S) } = b ** cases b ** case mk α : Type u_1 r✝ r : α → α → Prop inst✝ : IsWellOrder α r S : Set α hS : Unbounded r S e : #↑S = cof (type r) s : ↑S → ↑S → Prop w✝ : IsWellOrder (↑S) s e' : ord #↑S = type s T : Set α := {a | ∃ aS, ∀ (b : ↑S), s b { val := a, property := aS } → r (↑b) a} a : α this : Set.Nonempty {b | ¬r (↑b) a} b : ↑S := WellFounded.min (_ : WellFounded s) {b | ¬r (↑b) a} this ba : ¬r (↑b) a c : ↑S h : ¬r ↑c ↑b val✝ : α property✝ : val✝ ∈ S ⊢ { val := ↑{ val := val✝, property := property✝ }, property := (_ : ↑{ val := val✝, property := property✝ } ∈ S) } = { val := val✝, property := property✝ } ** rfl ** Qed | |
Ordinal.cof_eq_sInf_lsub ** α : Type u_1 r : α → α → Prop o : Ordinal.{u} ⊢ cof o = sInf {a | ∃ ι f, lsub f = o ∧ #ι = a} ** refine' le_antisymm (le_csInf (cof_lsub_def_nonempty o) _) (csInf_le' _) ** case refine'_1 α : Type u_1 r : α → α → Prop o : Ordinal.{u} ⊢ ∀ (b : Cardinal.{u}), b ∈ {a | ∃ ι f, lsub f = o ∧ #ι = a} → cof o ≤ b ** rintro a ⟨ι, f, hf, rfl⟩ ** case refine'_1.intro.intro.intro α : Type u_1 r : α → α → Prop o : Ordinal.{u} ι : Type u f : ι → Ordinal.{u} hf : lsub f = o ⊢ cof o ≤ #ι ** rw [← type_lt o] ** case refine'_1.intro.intro.intro α : Type u_1 r : α → α → Prop o : Ordinal.{u} ι : Type u f : ι → Ordinal.{u} hf : lsub f = o a : (Quotient.out o).α ⊢ ∃ b, b ∈ (typein fun x x_1 => x < x_1) ⁻¹' range f ∧ ¬b < a ** have := typein_lt_self a ** case refine'_1.intro.intro.intro α : Type u_1 r : α → α → Prop o : Ordinal.{u} ι : Type u f : ι → Ordinal.{u} hf : lsub f = o a : (Quotient.out o).α this : typein (fun x x_1 => x < x_1) a < o ⊢ ∃ b, b ∈ (typein fun x x_1 => x < x_1) ⁻¹' range f ∧ ¬b < a ** simp_rw [← hf, lt_lsub_iff] at this ** case refine'_1.intro.intro.intro α : Type u_1 r : α → α → Prop o : Ordinal.{u} ι : Type u f : ι → Ordinal.{u} hf : lsub f = o a : (Quotient.out o).α this : ∃ i, typein (fun x x_1 => x < x_1) a ≤ f i ⊢ ∃ b, b ∈ (typein fun x x_1 => x < x_1) ⁻¹' range f ∧ ¬b < a ** cases' this with i hi ** α : Type u_1 r : α → α → Prop o : Ordinal.{u} ι : Type u f : ι → Ordinal.{u} hf : lsub f = o s t : ↑((typein fun x x_1 => x < x_1) ⁻¹' range f) hst : (fun s => choose (_ : ↑s ∈ (typein fun x x_1 => x < x_1) ⁻¹' range f)) s = (fun s => choose (_ : ↑s ∈ (typein fun x x_1 => x < x_1) ⁻¹' range f)) t ⊢ s = t ** let H := congr_arg f hst ** α : Type u_1 r : α → α → Prop o : Ordinal.{u} ι : Type u f : ι → Ordinal.{u} hf : lsub f = o s t : ↑((typein fun x x_1 => x < x_1) ⁻¹' range f) hst : (fun s => choose (_ : ↑s ∈ (typein fun x x_1 => x < x_1) ⁻¹' range f)) s = (fun s => choose (_ : ↑s ∈ (typein fun x x_1 => x < x_1) ⁻¹' range f)) t H : f ((fun s => choose (_ : ↑s ∈ (typein fun x x_1 => x < x_1) ⁻¹' range f)) s) = f ((fun s => choose (_ : ↑s ∈ (typein fun x x_1 => x < x_1) ⁻¹' range f)) t) := congr_arg f hst ⊢ s = t ** rwa [Classical.choose_spec s.prop, Classical.choose_spec t.prop, typein_inj,
Subtype.coe_inj] at H ** case refine'_1.intro.intro.intro.intro.refine'_1 α : Type u_1 r : α → α → Prop o : Ordinal.{u} ι : Type u f : ι → Ordinal.{u} hf : lsub f = o a : (Quotient.out o).α i : ι hi : typein (fun x x_1 => x < x_1) a ≤ f i ⊢ f i < type fun x x_1 => x < x_1 ** rw [type_lt, ← hf] ** case refine'_1.intro.intro.intro.intro.refine'_1 α : Type u_1 r : α → α → Prop o : Ordinal.{u} ι : Type u f : ι → Ordinal.{u} hf : lsub f = o a : (Quotient.out o).α i : ι hi : typein (fun x x_1 => x < x_1) a ≤ f i ⊢ f i < lsub f ** apply lt_lsub ** case refine'_1.intro.intro.intro.intro.refine'_2 α : Type u_1 r : α → α → Prop o : Ordinal.{u} ι : Type u f : ι → Ordinal.{u} hf : lsub f = o a : (Quotient.out o).α i : ι hi : typein (fun x x_1 => x < x_1) a ≤ f i ⊢ enum (fun x x_1 => x < x_1) (f i) (_ : f i < type fun x x_1 => x < x_1) ∈ (typein fun x x_1 => x < x_1) ⁻¹' range f ** rw [mem_preimage, typein_enum] ** case refine'_1.intro.intro.intro.intro.refine'_2 α : Type u_1 r : α → α → Prop o : Ordinal.{u} ι : Type u f : ι → Ordinal.{u} hf : lsub f = o a : (Quotient.out o).α i : ι hi : typein (fun x x_1 => x < x_1) a ≤ f i ⊢ f i ∈ range f ** exact mem_range_self i ** case refine'_1.intro.intro.intro.intro.refine'_3 α : Type u_1 r : α → α → Prop o : Ordinal.{u} ι : Type u f : ι → Ordinal.{u} hf : lsub f = o a : (Quotient.out o).α i : ι hi : typein (fun x x_1 => x < x_1) a ≤ f i ⊢ ¬enum (fun x x_1 => x < x_1) (f i) (_ : f i < type fun x x_1 => x < x_1) < a ** rwa [← typein_le_typein, typein_enum] ** case refine'_2.intro.intro α : Type u_1 r : α → α → Prop o : Ordinal.{u} S : Set (Quotient.out o).α hS : Unbounded (fun x x_1 => x < x_1) S hS' : #↑S = cof (type fun x x_1 => x < x_1) ⊢ cof o ∈ {a | ∃ ι f, lsub f = o ∧ #ι = a} ** let f : S → Ordinal := fun s => typein LT.lt s.val ** case refine'_2.intro.intro α : Type u_1 r : α → α → Prop o : Ordinal.{u} S : Set (Quotient.out o).α hS : Unbounded (fun x x_1 => x < x_1) S hS' : #↑S = cof (type fun x x_1 => x < x_1) f : ↑S → Ordinal.{u} := fun s => typein LT.lt ↑s ⊢ cof o ∈ {a | ∃ ι f, lsub f = o ∧ #ι = a} ** refine'
⟨S, f, le_antisymm (lsub_le fun i => typein_lt_self i) (le_of_forall_lt fun a ha => _), by
rwa [type_lt o] at hS'⟩ ** case refine'_2.intro.intro α : Type u_1 r : α → α → Prop o : Ordinal.{u} S : Set (Quotient.out o).α hS : Unbounded (fun x x_1 => x < x_1) S hS' : #↑S = cof (type fun x x_1 => x < x_1) f : ↑S → Ordinal.{u} := fun s => typein LT.lt ↑s a : Ordinal.{u} ha : a < o ⊢ a < lsub f ** rw [← type_lt o] at ha ** case refine'_2.intro.intro.intro.intro α : Type u_1 r : α → α → Prop o : Ordinal.{u} S : Set (Quotient.out o).α hS : Unbounded (fun x x_1 => x < x_1) S hS' : #↑S = cof (type fun x x_1 => x < x_1) f : ↑S → Ordinal.{u} := fun s => typein LT.lt ↑s a : Ordinal.{u} ha : a < type fun x x_1 => x < x_1 b : (Quotient.out o).α hb : b ∈ S hb' : ¬(fun x x_1 => x < x_1) b (enum (fun x x_1 => x < x_1) a ha) ⊢ a < lsub f ** rw [← typein_le_typein, typein_enum] at hb' ** case refine'_2.intro.intro.intro.intro α : Type u_1 r : α → α → Prop o : Ordinal.{u} S : Set (Quotient.out o).α hS : Unbounded (fun x x_1 => x < x_1) S hS' : #↑S = cof (type fun x x_1 => x < x_1) f : ↑S → Ordinal.{u} := fun s => typein LT.lt ↑s a : Ordinal.{u} ha : a < type fun x x_1 => x < x_1 b : (Quotient.out o).α hb : b ∈ S hb' : a ≤ typein LT.lt b ⊢ a < lsub f ** exact hb'.trans_lt (lt_lsub.{u, u} f ⟨b, hb⟩) ** α : Type u_1 r : α → α → Prop o : Ordinal.{u} S : Set (Quotient.out o).α hS : Unbounded (fun x x_1 => x < x_1) S hS' : #↑S = cof (type fun x x_1 => x < x_1) f : ↑S → Ordinal.{u} := fun s => typein LT.lt ↑s ⊢ #↑S = cof o ** rwa [type_lt o] at hS' ** Qed | |
Ordinal.lift_cof ** α : Type u_1 r : α → α → Prop o : Ordinal.{v} ⊢ Cardinal.lift.{u, v} (cof o) = cof (lift.{u, v} o) ** refine' inductionOn o _ ** α : Type u_1 r : α → α → Prop o : Ordinal.{v} ⊢ ∀ (α : Type v) (r : α → α → Prop) [inst : IsWellOrder α r], Cardinal.lift.{u, v} (cof (type r)) = cof (lift.{u, v} (type r)) ** intro α r _ ** α✝ : Type u_1 r✝ : α✝ → α✝ → Prop o : Ordinal.{v} α : Type v r : α → α → Prop inst✝ : IsWellOrder α r ⊢ Cardinal.lift.{u, v} (cof (type r)) = cof (lift.{u, v} (type r)) ** apply le_antisymm ** case a α✝ : Type u_1 r✝ : α✝ → α✝ → Prop o : Ordinal.{v} α : Type v r : α → α → Prop inst✝ : IsWellOrder α r ⊢ Cardinal.lift.{u, v} (cof (type r)) ≤ cof (lift.{u, v} (type r)) ** refine' le_cof_type.2 fun S H => _ ** case a α✝ : Type u_1 r✝ : α✝ → α✝ → Prop o : Ordinal.{v} α : Type v r : α → α → Prop inst✝ : IsWellOrder α r S : Set (ULift.{u, v} { α := α, r := r, wo := inst✝ }.α) H : Unbounded (ULift.down ⁻¹'o { α := α, r := r, wo := inst✝ }.r) S ⊢ Cardinal.lift.{u, v} (cof (type r)) ≤ #↑S ** have : Cardinal.lift.{u, v} #(ULift.up ⁻¹' S) ≤ #(S : Type (max u v)) := by
rw [← Cardinal.lift_umax.{v, u}, ← Cardinal.lift_id'.{v, u} #S]
refine mk_preimage_of_injective_lift.{v, max u v} ULift.up S (ULift.up_injective.{u, v}) ** case a α✝ : Type u_1 r✝ : α✝ → α✝ → Prop o : Ordinal.{v} α : Type v r : α → α → Prop inst✝ : IsWellOrder α r S : Set (ULift.{u, v} { α := α, r := r, wo := inst✝ }.α) H : Unbounded (ULift.down ⁻¹'o { α := α, r := r, wo := inst✝ }.r) S this : Cardinal.lift.{u, v} #↑(ULift.up ⁻¹' S) ≤ #↑S ⊢ Cardinal.lift.{u, v} (cof (type r)) ≤ #↑S ** refine' (Cardinal.lift_le.2 <| cof_type_le _).trans this ** case a α✝ : Type u_1 r✝ : α✝ → α✝ → Prop o : Ordinal.{v} α : Type v r : α → α → Prop inst✝ : IsWellOrder α r S : Set (ULift.{u, v} { α := α, r := r, wo := inst✝ }.α) H : Unbounded (ULift.down ⁻¹'o { α := α, r := r, wo := inst✝ }.r) S this : Cardinal.lift.{u, v} #↑(ULift.up ⁻¹' S) ≤ #↑S ⊢ Unbounded r (ULift.up ⁻¹' S) ** exact fun a =>
let ⟨⟨b⟩, bs, br⟩ := H ⟨a⟩
⟨b, bs, br⟩ ** α✝ : Type u_1 r✝ : α✝ → α✝ → Prop o : Ordinal.{v} α : Type v r : α → α → Prop inst✝ : IsWellOrder α r S : Set (ULift.{u, v} { α := α, r := r, wo := inst✝ }.α) H : Unbounded (ULift.down ⁻¹'o { α := α, r := r, wo := inst✝ }.r) S ⊢ Cardinal.lift.{u, v} #↑(ULift.up ⁻¹' S) ≤ #↑S ** rw [← Cardinal.lift_umax.{v, u}, ← Cardinal.lift_id'.{v, u} #S] ** α✝ : Type u_1 r✝ : α✝ → α✝ → Prop o : Ordinal.{v} α : Type v r : α → α → Prop inst✝ : IsWellOrder α r S : Set (ULift.{u, v} { α := α, r := r, wo := inst✝ }.α) H : Unbounded (ULift.down ⁻¹'o { α := α, r := r, wo := inst✝ }.r) S ⊢ Cardinal.lift.{max v u, v} #↑(ULift.up ⁻¹' S) ≤ Cardinal.lift.{v, max v u} #↑S ** refine mk_preimage_of_injective_lift.{v, max u v} ULift.up S (ULift.up_injective.{u, v}) ** case a α✝ : Type u_1 r✝ : α✝ → α✝ → Prop o : Ordinal.{v} α : Type v r : α → α → Prop inst✝ : IsWellOrder α r ⊢ cof (lift.{u, v} (type r)) ≤ Cardinal.lift.{u, v} (cof (type r)) ** rcases cof_eq r with ⟨S, H, e'⟩ ** case a.intro.intro α✝ : Type u_1 r✝ : α✝ → α✝ → Prop o : Ordinal.{v} α : Type v r : α → α → Prop inst✝ : IsWellOrder α r S : Set α H : Unbounded r S e' : #↑S = cof (type r) ⊢ cof (lift.{u, v} (type r)) ≤ Cardinal.lift.{u, v} (cof (type r)) ** have : #(ULift.down.{u, v} ⁻¹' S) ≤ Cardinal.lift.{u, v} #S :=
⟨⟨fun ⟨⟨x⟩, h⟩ => ⟨⟨x, h⟩⟩, fun ⟨⟨x⟩, h₁⟩ ⟨⟨y⟩, h₂⟩ e => by
simp at e; congr⟩⟩ ** case a.intro.intro α✝ : Type u_1 r✝ : α✝ → α✝ → Prop o : Ordinal.{v} α : Type v r : α → α → Prop inst✝ : IsWellOrder α r S : Set α H : Unbounded r S e' : #↑S = cof (type r) this : #↑(ULift.down ⁻¹' S) ≤ Cardinal.lift.{u, v} #↑S ⊢ cof (lift.{u, v} (type r)) ≤ Cardinal.lift.{u, v} (cof (type r)) ** rw [e'] at this ** case a.intro.intro α✝ : Type u_1 r✝ : α✝ → α✝ → Prop o : Ordinal.{v} α : Type v r : α → α → Prop inst✝ : IsWellOrder α r S : Set α H : Unbounded r S e' : #↑S = cof (type r) this : #↑(ULift.down ⁻¹' S) ≤ Cardinal.lift.{u, v} (cof (type r)) ⊢ cof (lift.{u, v} (type r)) ≤ Cardinal.lift.{u, v} (cof (type r)) ** refine' (cof_type_le _).trans this ** case a.intro.intro α✝ : Type u_1 r✝ : α✝ → α✝ → Prop o : Ordinal.{v} α : Type v r : α → α → Prop inst✝ : IsWellOrder α r S : Set α H : Unbounded r S e' : #↑S = cof (type r) this : #↑(ULift.down ⁻¹' S) ≤ Cardinal.lift.{u, v} (cof (type r)) ⊢ Unbounded (ULift.down ⁻¹'o { α := α, r := r, wo := inst✝ }.r) (ULift.down ⁻¹' S) ** exact fun ⟨a⟩ =>
let ⟨b, bs, br⟩ := H a
⟨⟨b⟩, bs, br⟩ ** α✝ : Type u_1 r✝ : α✝ → α✝ → Prop o : Ordinal.{v} α : Type v r : α → α → Prop inst✝ : IsWellOrder α r S : Set α H : Unbounded r S e' : #↑S = cof (type r) x✝¹ x✝ : ↑(ULift.down ⁻¹' S) x : α h₁ : { down := x } ∈ ULift.down ⁻¹' S y : α h₂ : { down := y } ∈ ULift.down ⁻¹' S e : (fun x => match x with | { val := { down := x }, property := h } => { down := { val := x, property := h } }) { val := { down := x }, property := h₁ } = (fun x => match x with | { val := { down := x }, property := h } => { down := { val := x, property := h } }) { val := { down := y }, property := h₂ } ⊢ { val := { down := x }, property := h₁ } = { val := { down := y }, property := h₂ } ** simp at e ** α✝ : Type u_1 r✝ : α✝ → α✝ → Prop o : Ordinal.{v} α : Type v r : α → α → Prop inst✝ : IsWellOrder α r S : Set α H : Unbounded r S e' : #↑S = cof (type r) x✝¹ x✝ : ↑(ULift.down ⁻¹' S) x : α h₁ : { down := x } ∈ ULift.down ⁻¹' S y : α h₂ : { down := y } ∈ ULift.down ⁻¹' S e : x = y ⊢ { val := { down := x }, property := h₁ } = { val := { down := y }, property := h₂ } ** congr ** Qed | |
Ordinal.cof_le_card ** α : Type u_1 r : α → α → Prop o : Ordinal.{u_2} ⊢ cof o ≤ card o ** rw [cof_eq_sInf_lsub] ** α : Type u_1 r : α → α → Prop o : Ordinal.{u_2} ⊢ sInf {a | ∃ ι f, lsub f = o ∧ #ι = a} ≤ card o ** exact csInf_le' card_mem_cof ** Qed | |
Ordinal.cof_ord_le ** α : Type u_1 r : α → α → Prop c : Cardinal.{u_2} ⊢ cof (ord c) ≤ c ** simpa using cof_le_card c.ord ** Qed | |
Ordinal.exists_lsub_cof ** α : Type u_1 r : α → α → Prop o : Ordinal.{u} ⊢ ∃ ι f, lsub f = o ∧ #ι = cof o ** rw [cof_eq_sInf_lsub] ** α : Type u_1 r : α → α → Prop o : Ordinal.{u} ⊢ ∃ ι f, lsub f = o ∧ #ι = sInf {a | ∃ ι f, lsub f = o ∧ #ι = a} ** exact csInf_mem (cof_lsub_def_nonempty o) ** Qed | |
Ordinal.cof_lsub_le ** α : Type u_1 r : α → α → Prop ι : Type u f : ι → Ordinal.{u} ⊢ cof (lsub f) ≤ #ι ** rw [cof_eq_sInf_lsub] ** α : Type u_1 r : α → α → Prop ι : Type u f : ι → Ordinal.{u} ⊢ sInf {a | ∃ ι_1 f_1, lsub f_1 = lsub f ∧ #ι_1 = a} ≤ #ι ** exact csInf_le' ⟨ι, f, rfl, rfl⟩ ** Qed | |
Ordinal.cof_lsub_le_lift ** α : Type u_1 r : α → α → Prop ι : Type u f : ι → Ordinal.{max u v} ⊢ cof (lsub f) ≤ Cardinal.lift.{v, u} #ι ** rw [← mk_uLift.{u, v}] ** α : Type u_1 r : α → α → Prop ι : Type u f : ι → Ordinal.{max u v} ⊢ cof (lsub f) ≤ #(ULift.{v, u} ι) ** convert cof_lsub_le.{max u v} fun i : ULift.{v, u} ι => f i.down ** case h.e'_3.h.e'_1 α : Type u_1 r : α → α → Prop ι : Type u f : ι → Ordinal.{max u v} ⊢ lsub f = lsub fun i => f i.down ** exact
lsub_eq_of_range_eq.{u, max u v, max u v}
(Set.ext fun x => ⟨fun ⟨i, hi⟩ => ⟨ULift.up.{v, u} i, hi⟩, fun ⟨i, hi⟩ => ⟨_, hi⟩⟩) ** Qed | |
Ordinal.le_cof_iff_lsub ** α : Type u_1 r : α → α → Prop o : Ordinal.{u} a : Cardinal.{u} ⊢ a ≤ cof o ↔ ∀ {ι : Type u} (f : ι → Ordinal.{u}), lsub f = o → a ≤ #ι ** rw [cof_eq_sInf_lsub] ** α : Type u_1 r : α → α → Prop o : Ordinal.{u} a : Cardinal.{u} ⊢ a ≤ sInf {a | ∃ ι f, lsub f = o ∧ #ι = a} ↔ ∀ {ι : Type u} (f : ι → Ordinal.{u}), lsub f = o → a ≤ #ι ** exact
(le_csInf_iff'' (cof_lsub_def_nonempty o)).trans
⟨fun H ι f hf => H _ ⟨ι, f, hf, rfl⟩, fun H b ⟨ι, f, hf, hb⟩ => by
rw [← hb]
exact H _ hf⟩ ** α : Type u_1 r : α → α → Prop o : Ordinal.{u} a : Cardinal.{u} H : ∀ {ι : Type u} (f : ι → Ordinal.{u}), lsub f = o → a ≤ #ι b : Cardinal.{u} x✝ : b ∈ {a | ∃ ι f, lsub f = o ∧ #ι = a} ι : Type u f : ι → Ordinal.{u} hf : lsub f = o hb : #ι = b ⊢ a ≤ b ** rw [← hb] ** α : Type u_1 r : α → α → Prop o : Ordinal.{u} a : Cardinal.{u} H : ∀ {ι : Type u} (f : ι → Ordinal.{u}), lsub f = o → a ≤ #ι b : Cardinal.{u} x✝ : b ∈ {a | ∃ ι f, lsub f = o ∧ #ι = a} ι : Type u f : ι → Ordinal.{u} hf : lsub f = o hb : #ι = b ⊢ a ≤ #ι ** exact H _ hf ** Qed | |
Ordinal.lsub_lt_ord_lift ** α : Type u_1 r : α → α → Prop ι : Type u f : ι → Ordinal.{max u v} c : Ordinal.{max u v} hι : Cardinal.lift.{v, u} #ι < cof c hf : ∀ (i : ι), f i < c h : lsub f = c ⊢ False ** subst h ** α : Type u_1 r : α → α → Prop ι : Type u f : ι → Ordinal.{max u v} hι : Cardinal.lift.{v, u} #ι < cof (lsub f) hf : ∀ (i : ι), f i < lsub f ⊢ False ** exact (cof_lsub_le_lift.{u, v} f).not_lt hι ** Qed | |
Ordinal.lsub_lt_ord ** α : Type u_1 r : α → α → Prop ι : Type u f : ι → Ordinal.{u} c : Ordinal.{u} hι : #ι < cof c ⊢ Cardinal.lift.{u, u} #ι < cof c ** rwa [(#ι).lift_id] ** Qed | |
Ordinal.cof_sup_le_lift ** α : Type u_1 r : α → α → Prop ι : Type u f : ι → Ordinal.{max u v} H : ∀ (i : ι), f i < sup f ⊢ cof (sup f) ≤ Cardinal.lift.{v, u} #ι ** rw [← sup_eq_lsub_iff_lt_sup.{u, v}] at H ** α : Type u_1 r : α → α → Prop ι : Type u f : ι → Ordinal.{max u v} H : (sup fun i => f i) = lsub fun i => f i ⊢ cof (sup f) ≤ Cardinal.lift.{v, u} #ι ** rw [H] ** α : Type u_1 r : α → α → Prop ι : Type u f : ι → Ordinal.{max u v} H : (sup fun i => f i) = lsub fun i => f i ⊢ cof (lsub fun i => f i) ≤ Cardinal.lift.{v, u} #ι ** exact cof_lsub_le_lift f ** Qed | |
Ordinal.cof_sup_le ** α : Type u_1 r : α → α → Prop ι : Type u f : ι → Ordinal.{u} H : ∀ (i : ι), f i < sup f ⊢ cof (sup f) ≤ #ι ** rw [← (#ι).lift_id] ** α : Type u_1 r : α → α → Prop ι : Type u f : ι → Ordinal.{u} H : ∀ (i : ι), f i < sup f ⊢ cof (sup f) ≤ Cardinal.lift.{u, u} #ι ** exact cof_sup_le_lift H ** Qed | |
Ordinal.sup_lt_ord ** α : Type u_1 r : α → α → Prop ι : Type u f : ι → Ordinal.{u} c : Ordinal.{u} hι : #ι < cof c ⊢ Cardinal.lift.{u, u} #ι < cof c ** rwa [(#ι).lift_id] ** Qed | |
Ordinal.iSup_lt_lift ** α : Type u_1 r : α → α → Prop ι : Type u f : ι → Cardinal.{max u v} c : Cardinal.{max u v} hι : Cardinal.lift.{v, u} #ι < cof (ord c) hf : ∀ (i : ι), f i < c ⊢ iSup f < c ** rw [← ord_lt_ord, iSup_ord (Cardinal.bddAbove_range.{u, v} _)] ** α : Type u_1 r : α → α → Prop ι : Type u f : ι → Cardinal.{max u v} c : Cardinal.{max u v} hι : Cardinal.lift.{v, u} #ι < cof (ord c) hf : ∀ (i : ι), f i < c ⊢ ⨆ i, ord (f i) < ord c ** refine' sup_lt_ord_lift hι fun i => _ ** α : Type u_1 r : α → α → Prop ι : Type u f : ι → Cardinal.{max u v} c : Cardinal.{max u v} hι : Cardinal.lift.{v, u} #ι < cof (ord c) hf : ∀ (i : ι), f i < c i : ι ⊢ ord (f i) < ord c ** rw [ord_lt_ord] ** α : Type u_1 r : α → α → Prop ι : Type u f : ι → Cardinal.{max u v} c : Cardinal.{max u v} hι : Cardinal.lift.{v, u} #ι < cof (ord c) hf : ∀ (i : ι), f i < c i : ι ⊢ f i < c ** apply hf ** Qed | |
Ordinal.iSup_lt ** α : Type u_1 r : α → α → Prop ι : Type u_2 f : ι → Cardinal.{u_2} c : Cardinal.{u_2} hι : #ι < cof (ord c) ⊢ Cardinal.lift.{?u.30349, u_2} #ι < cof (ord c) ** rwa [(#ι).lift_id] ** Qed | |
Ordinal.nfpFamily_lt_ord_lift ** α : Type u_1 r : α → α → Prop ι : Type u f : ι → Ordinal.{max u v} → Ordinal.{max u v} c : Ordinal.{max u v} hc : ℵ₀ < cof c hc' : Cardinal.lift.{v, u} #ι < cof c hf : ∀ (i : ι) (b : Ordinal.{max u v}), b < c → f i b < c a : Ordinal.{max u v} ha : a < c ⊢ nfpFamily f a < c ** refine' sup_lt_ord_lift ((Cardinal.lift_le.2 (mk_list_le_max ι)).trans_lt _) fun l => _ ** case refine'_1 α : Type u_1 r : α → α → Prop ι : Type u f : ι → Ordinal.{max u v} → Ordinal.{max u v} c : Ordinal.{max u v} hc : ℵ₀ < cof c hc' : Cardinal.lift.{v, u} #ι < cof c hf : ∀ (i : ι) (b : Ordinal.{max u v}), b < c → f i b < c a : Ordinal.{max u v} ha : a < c ⊢ Cardinal.lift.{v, u} (max ℵ₀ #ι) < cof c ** rw [lift_max] ** case refine'_1 α : Type u_1 r : α → α → Prop ι : Type u f : ι → Ordinal.{max u v} → Ordinal.{max u v} c : Ordinal.{max u v} hc : ℵ₀ < cof c hc' : Cardinal.lift.{v, u} #ι < cof c hf : ∀ (i : ι) (b : Ordinal.{max u v}), b < c → f i b < c a : Ordinal.{max u v} ha : a < c ⊢ max (Cardinal.lift.{v, u} ℵ₀) (Cardinal.lift.{v, u} #ι) < cof c ** apply max_lt _ hc' ** α : Type u_1 r : α → α → Prop ι : Type u f : ι → Ordinal.{max u v} → Ordinal.{max u v} c : Ordinal.{max u v} hc : ℵ₀ < cof c hc' : Cardinal.lift.{v, u} #ι < cof c hf : ∀ (i : ι) (b : Ordinal.{max u v}), b < c → f i b < c a : Ordinal.{max u v} ha : a < c ⊢ Cardinal.lift.{v, u} ℵ₀ < cof c ** rwa [Cardinal.lift_aleph0] ** case refine'_2 α : Type u_1 r : α → α → Prop ι : Type u f : ι → Ordinal.{max u v} → Ordinal.{max u v} c : Ordinal.{max u v} hc : ℵ₀ < cof c hc' : Cardinal.lift.{v, u} #ι < cof c hf : ∀ (i : ι) (b : Ordinal.{max u v}), b < c → f i b < c a : Ordinal.{max u v} ha : a < c l : List ι ⊢ List.foldr f a l < c ** induction' l with i l H ** case refine'_2.nil α : Type u_1 r : α → α → Prop ι : Type u f : ι → Ordinal.{max u v} → Ordinal.{max u v} c : Ordinal.{max u v} hc : ℵ₀ < cof c hc' : Cardinal.lift.{v, u} #ι < cof c hf : ∀ (i : ι) (b : Ordinal.{max u v}), b < c → f i b < c a : Ordinal.{max u v} ha : a < c ⊢ List.foldr f a [] < c ** exact ha ** case refine'_2.cons α : Type u_1 r : α → α → Prop ι : Type u f : ι → Ordinal.{max u v} → Ordinal.{max u v} c : Ordinal.{max u v} hc : ℵ₀ < cof c hc' : Cardinal.lift.{v, u} #ι < cof c hf : ∀ (i : ι) (b : Ordinal.{max u v}), b < c → f i b < c a : Ordinal.{max u v} ha : a < c i : ι l : List ι H : List.foldr f a l < c ⊢ List.foldr f a (i :: l) < c ** exact hf _ _ H ** Qed | |
Ordinal.nfpFamily_lt_ord ** α : Type u_1 r : α → α → Prop ι : Type u f : ι → Ordinal.{u} → Ordinal.{u} c : Ordinal.{u} hc : ℵ₀ < cof c hc' : #ι < cof c hf : ∀ (i : ι) (b : Ordinal.{u}), b < c → f i b < c a : Ordinal.{u} ⊢ Cardinal.lift.{u, u} #ι < cof c ** rwa [(#ι).lift_id] ** Qed | |
Ordinal.nfpBFamily_lt_ord_lift ** α : Type u_1 r : α → α → Prop o : Ordinal.{u} f : (a : Ordinal.{u}) → a < o → Ordinal.{max u v} → Ordinal.{max u v} c : Ordinal.{max u v} hc : ℵ₀ < cof c hc' : Cardinal.lift.{v, u} (card o) < cof c hf : ∀ (i : Ordinal.{u}) (hi : i < o) (b : Ordinal.{max u v}), b < c → f i hi b < c a : Ordinal.{max u v} ⊢ Cardinal.lift.{v, u} #(Quotient.out o).α < cof c ** rwa [mk_ordinal_out] ** Qed | |
Ordinal.nfpBFamily_lt_ord ** α : Type u_1 r : α → α → Prop o : Ordinal.{u} f : (a : Ordinal.{u}) → a < o → Ordinal.{u} → Ordinal.{u} c : Ordinal.{u} hc : ℵ₀ < cof c hc' : card o < cof c hf : ∀ (i : Ordinal.{u}) (hi : i < o) (b : Ordinal.{u}), b < c → f i hi b < c a : Ordinal.{u} ⊢ Cardinal.lift.{u, u} (card o) < cof c ** rwa [o.card.lift_id] ** Qed | |
Ordinal.nfp_lt_ord ** α : Type u_1 r : α → α → Prop f : Ordinal.{u_2} → Ordinal.{u_2} c : Ordinal.{u_2} hc : ℵ₀ < cof c hf : ∀ (i : Ordinal.{u_2}), i < c → f i < c a : Ordinal.{u_2} ⊢ Cardinal.lift.{u_2, 0} #Unit < cof c ** simpa using Cardinal.one_lt_aleph0.trans hc ** Qed | |
Ordinal.exists_blsub_cof ** α : Type u_1 r : α → α → Prop o : Ordinal.{u} ⊢ ∃ f, blsub (ord (cof o)) f = o ** rcases exists_lsub_cof o with ⟨ι, f, hf, hι⟩ ** case intro.intro.intro α : Type u_1 r : α → α → Prop o : Ordinal.{u} ι : Type u f : ι → Ordinal.{u} hf : lsub f = o hι : #ι = cof o ⊢ ∃ f, blsub (ord (cof o)) f = o ** rcases Cardinal.ord_eq ι with ⟨r, hr, hι'⟩ ** case intro.intro.intro.intro.intro α : Type u_1 r✝ : α → α → Prop o : Ordinal.{u} ι : Type u f : ι → Ordinal.{u} hf : lsub f = o hι : #ι = cof o r : ι → ι → Prop hr : IsWellOrder ι r hι' : ord #ι = type r ⊢ ∃ f, blsub (ord (cof o)) f = o ** rw [← @blsub_eq_lsub' ι r hr] at hf ** case intro.intro.intro.intro.intro α : Type u_1 r✝ : α → α → Prop o : Ordinal.{u} ι : Type u f : ι → Ordinal.{u} hι : #ι = cof o r : ι → ι → Prop hr : IsWellOrder ι r hf : blsub (type r) (bfamilyOfFamily' r f) = o hι' : ord #ι = type r ⊢ ∃ f, blsub (ord (cof o)) f = o ** rw [← hι, hι'] ** case intro.intro.intro.intro.intro α : Type u_1 r✝ : α → α → Prop o : Ordinal.{u} ι : Type u f : ι → Ordinal.{u} hι : #ι = cof o r : ι → ι → Prop hr : IsWellOrder ι r hf : blsub (type r) (bfamilyOfFamily' r f) = o hι' : ord #ι = type r ⊢ ∃ f, blsub (type r) f = o ** exact ⟨_, hf⟩ ** Qed | |
Ordinal.le_cof_iff_blsub ** α : Type u_1 r : α → α → Prop b : Ordinal.{u} a : Cardinal.{u} H : ∀ {ι : Type u} (f : ι → Ordinal.{u}), lsub f = b → a ≤ #ι o : Ordinal.{u} f : (a : Ordinal.{u}) → a < o → Ordinal.{u} hf : blsub o f = b ⊢ a ≤ card o ** simpa using H _ hf ** α : Type u_1 r : α → α → Prop b : Ordinal.{u} a : Cardinal.{u} H : ∀ {o : Ordinal.{u}} (f : (a : Ordinal.{u}) → a < o → Ordinal.{u}), blsub o f = b → a ≤ card o ι : Type u f : ι → Ordinal.{u} hf : lsub f = b ⊢ a ≤ #ι ** rcases Cardinal.ord_eq ι with ⟨r, hr, hι'⟩ ** case intro.intro α : Type u_1 r✝ : α → α → Prop b : Ordinal.{u} a : Cardinal.{u} H : ∀ {o : Ordinal.{u}} (f : (a : Ordinal.{u}) → a < o → Ordinal.{u}), blsub o f = b → a ≤ card o ι : Type u f : ι → Ordinal.{u} hf : lsub f = b r : ι → ι → Prop hr : IsWellOrder ι r hι' : ord #ι = type r ⊢ a ≤ #ι ** rw [← @blsub_eq_lsub' ι r hr] at hf ** case intro.intro α : Type u_1 r✝ : α → α → Prop b : Ordinal.{u} a : Cardinal.{u} H : ∀ {o : Ordinal.{u}} (f : (a : Ordinal.{u}) → a < o → Ordinal.{u}), blsub o f = b → a ≤ card o ι : Type u f : ι → Ordinal.{u} r : ι → ι → Prop hr : IsWellOrder ι r hf : blsub (type r) (bfamilyOfFamily' r f) = b hι' : ord #ι = type r ⊢ a ≤ #ι ** simpa using H _ hf ** Qed | |
Ordinal.cof_blsub_le_lift ** α : Type u_1 r : α → α → Prop o : Ordinal.{u} f : (a : Ordinal.{u}) → a < o → Ordinal.{max u v} ⊢ cof (blsub o f) ≤ Cardinal.lift.{v, u} (card o) ** rw [← mk_ordinal_out o] ** α : Type u_1 r : α → α → Prop o : Ordinal.{u} f : (a : Ordinal.{u}) → a < o → Ordinal.{max u v} ⊢ cof (blsub o f) ≤ Cardinal.lift.{v, u} #(Quotient.out o).α ** exact cof_lsub_le_lift _ ** Qed | |
Ordinal.cof_blsub_le ** α : Type u_1 r : α → α → Prop o : Ordinal.{u} f : (a : Ordinal.{u}) → a < o → Ordinal.{u} ⊢ cof (blsub o f) ≤ card o ** rw [← o.card.lift_id] ** α : Type u_1 r : α → α → Prop o : Ordinal.{u} f : (a : Ordinal.{u}) → a < o → Ordinal.{u} ⊢ cof (blsub o f) ≤ Cardinal.lift.{u, u} (card o) ** exact cof_blsub_le_lift f ** Qed | |
Ordinal.blsub_lt_ord_lift ** α : Type u_1 r : α → α → Prop o : Ordinal.{u} f : (a : Ordinal.{u}) → a < o → Ordinal.{max u v} c : Ordinal.{max u v} ho : Cardinal.lift.{v, u} (card o) < cof c hf : ∀ (i : Ordinal.{u}) (hi : i < o), f i hi < c h : blsub o f = c ⊢ cof c ≤ Cardinal.lift.{v, u} (card o) ** simpa [← iSup_ord, hf, h] using cof_blsub_le_lift.{u, v} f ** Qed | |
Ordinal.blsub_lt_ord ** α : Type u_1 r : α → α → Prop o : Ordinal.{u} f : (a : Ordinal.{u}) → a < o → Ordinal.{u} c : Ordinal.{u} ho : card o < cof c hf : ∀ (i : Ordinal.{u}) (hi : i < o), f i hi < c ⊢ Cardinal.lift.{u, u} (card o) < cof c ** rwa [o.card.lift_id] ** Qed | |
Ordinal.cof_bsup_le_lift ** α : Type u_1 r : α → α → Prop o : Ordinal.{u} f : (a : Ordinal.{u}) → a < o → Ordinal.{max u v} H : ∀ (i : Ordinal.{u}) (h : i < o), f i h < bsup o f ⊢ cof (bsup o f) ≤ Cardinal.lift.{v, u} (card o) ** rw [← bsup_eq_blsub_iff_lt_bsup.{u, v}] at H ** α : Type u_1 r : α → α → Prop o : Ordinal.{u} f : (a : Ordinal.{u}) → a < o → Ordinal.{max u v} H : (bsup o fun i h => f i h) = blsub o fun i h => f i h ⊢ cof (bsup o f) ≤ Cardinal.lift.{v, u} (card o) ** rw [H] ** α : Type u_1 r : α → α → Prop o : Ordinal.{u} f : (a : Ordinal.{u}) → a < o → Ordinal.{max u v} H : (bsup o fun i h => f i h) = blsub o fun i h => f i h ⊢ cof (blsub o fun i h => f i h) ≤ Cardinal.lift.{v, u} (card o) ** exact cof_blsub_le_lift.{u, v} f ** Qed | |
Ordinal.cof_bsup_le ** α : Type u_1 r : α → α → Prop o : Ordinal.{u} f : (a : Ordinal.{u}) → a < o → Ordinal.{u} ⊢ (∀ (i : Ordinal.{u}) (h : i < o), f i h < bsup o f) → cof (bsup o f) ≤ card o ** rw [← o.card.lift_id] ** α : Type u_1 r : α → α → Prop o : Ordinal.{u} f : (a : Ordinal.{u}) → a < o → Ordinal.{u} ⊢ (∀ (i : Ordinal.{u}) (h : i < o), f i h < bsup o f) → cof (bsup o f) ≤ Cardinal.lift.{u, u} (card o) ** exact cof_bsup_le_lift ** Qed | |
Ordinal.bsup_lt_ord ** α : Type u_1 r : α → α → Prop o : Ordinal.{u} f : (a : Ordinal.{u}) → a < o → Ordinal.{u} c : Ordinal.{u} ho : card o < cof c ⊢ Cardinal.lift.{u, u} (card o) < cof c ** rwa [o.card.lift_id] ** Qed | |
Ordinal.cof_zero ** α : Type u_1 r : α → α → Prop ⊢ cof 0 = 0 ** refine LE.le.antisymm ?_ (Cardinal.zero_le _) ** α : Type u_1 r : α → α → Prop ⊢ cof 0 ≤ 0 ** rw [← card_zero] ** α : Type u_1 r : α → α → Prop ⊢ cof 0 ≤ card 0 ** exact cof_le_card 0 ** Qed | |
Ordinal.cof_eq_zero ** α : Type u_1 r : α → α → Prop o : Ordinal.{u_2} e : o = 0 ⊢ cof o = 0 ** simp [e] ** Qed | |
Ordinal.cof_succ ** α : Type u_1 r : α → α → Prop o : Ordinal.{u_2} ⊢ cof (succ o) = 1 ** apply le_antisymm ** case a α : Type u_1 r : α → α → Prop o : Ordinal.{u_2} ⊢ cof (succ o) ≤ 1 ** refine' inductionOn o fun α r _ => _ ** case a α✝ : Type u_1 r✝ : α✝ → α✝ → Prop o : Ordinal.{u_2} α : Type u_2 r : α → α → Prop x✝ : IsWellOrder α r ⊢ cof (succ (type r)) ≤ 1 ** change cof (type _) ≤ _ ** case a α✝ : Type u_1 r✝ : α✝ → α✝ → Prop o : Ordinal.{u_2} α : Type u_2 r : α → α → Prop x✝ : IsWellOrder α r ⊢ cof (type (Sum.Lex r EmptyRelation)) ≤ 1 ** rw [← (_ : #_ = 1)] ** case a α✝ : Type u_1 r✝ : α✝ → α✝ → Prop o : Ordinal.{u_2} α : Type u_2 r : α → α → Prop x✝ : IsWellOrder α r ⊢ cof (type (Sum.Lex r EmptyRelation)) ≤ #?m.41981 α✝ : Type u_1 r✝ : α✝ → α✝ → Prop o : Ordinal.{u_2} α : Type u_2 r : α → α → Prop x✝ : IsWellOrder α r ⊢ Type u_2 α✝ : Type u_1 r✝ : α✝ → α✝ → Prop o : Ordinal.{u_2} α : Type u_2 r : α → α → Prop x✝ : IsWellOrder α r ⊢ #?m.41981 = 1 ** apply cof_type_le ** case a.h α✝ : Type u_1 r✝ : α✝ → α✝ → Prop o : Ordinal.{u_2} α : Type u_2 r : α → α → Prop x✝ : IsWellOrder α r ⊢ Unbounded (Sum.Lex r EmptyRelation) ?a.S✝ ** refine' fun a => ⟨Sum.inr PUnit.unit, Set.mem_singleton _, _⟩ ** case a.h α✝ : Type u_1 r✝ : α✝ → α✝ → Prop o : Ordinal.{u_2} α : Type u_2 r : α → α → Prop x✝ : IsWellOrder α r a : α ⊕ PUnit.{u_2 + 1} ⊢ ¬Sum.Lex r EmptyRelation (Sum.inr PUnit.unit) a ** rcases a with (a | ⟨⟨⟨⟩⟩⟩) <;> simp [EmptyRelation] ** α✝ : Type u_1 r✝ : α✝ → α✝ → Prop o : Ordinal.{u_2} α : Type u_2 r : α → α → Prop x✝ : IsWellOrder α r ⊢ #↑{Sum.inr PUnit.unit} = 1 ** rw [Cardinal.mk_fintype, Set.card_singleton] ** α✝ : Type u_1 r✝ : α✝ → α✝ → Prop o : Ordinal.{u_2} α : Type u_2 r : α → α → Prop x✝ : IsWellOrder α r ⊢ ↑1 = 1 ** simp ** case a α : Type u_1 r : α → α → Prop o : Ordinal.{u_2} ⊢ 1 ≤ cof (succ o) ** rw [← Cardinal.succ_zero, succ_le_iff] ** case a α : Type u_1 r : α → α → Prop o : Ordinal.{u_2} ⊢ 0 < cof (succ o) ** simpa [lt_iff_le_and_ne, Cardinal.zero_le] using fun h =>
succ_ne_zero o (cof_eq_zero.1 (Eq.symm h)) ** Qed | |
Ordinal.cof_eq_one_iff_is_succ ** α✝ : Type u_1 r✝ : α✝ → α✝ → Prop o : Ordinal.{u} α : Type u r : α → α → Prop x✝ : IsWellOrder α r z : cof (type r) = 1 ⊢ ∃ a, type r = succ a ** rcases cof_eq r with ⟨S, hl, e⟩ ** case intro.intro α✝ : Type u_1 r✝ : α✝ → α✝ → Prop o : Ordinal.{u} α : Type u r : α → α → Prop x✝ : IsWellOrder α r z : cof (type r) = 1 S : Set α hl : Unbounded r S e : #↑S = cof (type r) ⊢ ∃ a, type r = succ a ** rw [z] at e ** case intro.intro α✝ : Type u_1 r✝ : α✝ → α✝ → Prop o : Ordinal.{u} α : Type u r : α → α → Prop x✝ : IsWellOrder α r z : cof (type r) = 1 S : Set α hl : Unbounded r S e : #↑S = 1 ⊢ ∃ a, type r = succ a ** cases' mk_ne_zero_iff.1 (by rw [e]; exact one_ne_zero) with a ** case intro.intro.intro α✝ : Type u_1 r✝ : α✝ → α✝ → Prop o : Ordinal.{u} α : Type u r : α → α → Prop x✝ : IsWellOrder α r z : cof (type r) = 1 S : Set α hl : Unbounded r S e : #↑S = 1 a : ↑S ⊢ ∃ a, type r = succ a ** refine'
⟨typein r a,
Eq.symm <|
Quotient.sound
⟨RelIso.ofSurjective (RelEmbedding.ofMonotone _ fun x y => _) fun x => _⟩⟩ ** α✝ : Type u_1 r✝ : α✝ → α✝ → Prop o : Ordinal.{u} α : Type u r : α → α → Prop x✝ : IsWellOrder α r z : cof (type r) = 1 S : Set α hl : Unbounded r S e : #↑S = 1 ⊢ #?m.45779 ≠ 0 ** rw [e] ** α✝ : Type u_1 r✝ : α✝ → α✝ → Prop o : Ordinal.{u} α : Type u r : α → α → Prop x✝ : IsWellOrder α r z : cof (type r) = 1 S : Set α hl : Unbounded r S e : #↑S = 1 ⊢ 1 ≠ 0 ** exact one_ne_zero ** case intro.intro.intro.refine'_1 α✝ : Type u_1 r✝ : α✝ → α✝ → Prop o : Ordinal.{u} α : Type u r : α → α → Prop x✝ : IsWellOrder α r z : cof (type r) = 1 S : Set α hl : Unbounded r S e : #↑S = 1 a : ↑S ⊢ ↑{b | r b ↑a} ⊕ PUnit.{u + 1} → α ** apply Sum.rec <;> [exact Subtype.val; exact fun _ => a] ** case intro.intro.intro.refine'_2 α✝ : Type u_1 r✝ : α✝ → α✝ → Prop o : Ordinal.{u} α : Type u r : α → α → Prop x✝ : IsWellOrder α r z : cof (type r) = 1 S : Set α hl : Unbounded r S e : #↑S = 1 a : ↑S x y : ↑{b | r b ↑a} ⊕ PUnit.{u + 1} ⊢ Sum.Lex (Subrel r {b | r b ↑a}) EmptyRelation x y → r (Sum.rec Subtype.val (fun x => ↑a) x) (Sum.rec Subtype.val (fun x => ↑a) y) ** rcases x with (x | ⟨⟨⟨⟩⟩⟩) <;> rcases y with (y | ⟨⟨⟨⟩⟩⟩) <;>
simp [Subrel, Order.Preimage, EmptyRelation] ** case intro.intro.intro.refine'_2.inl.inr.unit α✝ : Type u_1 r✝ : α✝ → α✝ → Prop o : Ordinal.{u} α : Type u r : α → α → Prop x✝ : IsWellOrder α r z : cof (type r) = 1 S : Set α hl : Unbounded r S e : #↑S = 1 a : ↑S x : ↑{b | r b ↑a} ⊢ r ↑x ↑a ** exact x.2 ** case intro.intro.intro.refine'_3 α✝ : Type u_1 r✝ : α✝ → α✝ → Prop o : Ordinal.{u} α : Type u r : α → α → Prop x✝ : IsWellOrder α r z : cof (type r) = 1 S : Set α hl : Unbounded r S e : #↑S = 1 a : ↑S x : α ⊢ ∃ a_1, ↑(RelEmbedding.ofMonotone (Sum.rec Subtype.val fun x => ↑a) (_ : ∀ (x y : ↑{b | r b ↑a} ⊕ PUnit.{u + 1}), Sum.Lex (Subrel r {b | r b ↑a}) EmptyRelation x y → r (Sum.rec Subtype.val (fun x => ↑a) x) (Sum.rec Subtype.val (fun x => ↑a) y))) a_1 = x ** suffices : r x a ∨ ∃ _ : PUnit.{u}, ↑a = x ** case this α✝ : Type u_1 r✝ : α✝ → α✝ → Prop o : Ordinal.{u} α : Type u r : α → α → Prop x✝ : IsWellOrder α r z : cof (type r) = 1 S : Set α hl : Unbounded r S e : #↑S = 1 a : ↑S x : α ⊢ r x ↑a ∨ ∃ x_1, ↑a = x ** rcases trichotomous_of r x a with (h | h | h) ** case intro.intro.intro.refine'_3 α✝ : Type u_1 r✝ : α✝ → α✝ → Prop o : Ordinal.{u} α : Type u r : α → α → Prop x✝ : IsWellOrder α r z : cof (type r) = 1 S : Set α hl : Unbounded r S e : #↑S = 1 a : ↑S x : α this : r x ↑a ∨ ∃ x_1, ↑a = x ⊢ ∃ a_1, ↑(RelEmbedding.ofMonotone (Sum.rec Subtype.val fun x => ↑a) (_ : ∀ (x y : ↑{b | r b ↑a} ⊕ PUnit.{u + 1}), Sum.Lex (Subrel r {b | r b ↑a}) EmptyRelation x y → r (Sum.rec Subtype.val (fun x => ↑a) x) (Sum.rec Subtype.val (fun x => ↑a) y))) a_1 = x ** convert this ** case a α✝ : Type u_1 r✝ : α✝ → α✝ → Prop o : Ordinal.{u} α : Type u r : α → α → Prop x✝ : IsWellOrder α r z : cof (type r) = 1 S : Set α hl : Unbounded r S e : #↑S = 1 a : ↑S x : α this : r x ↑a ∨ ∃ x_1, ↑a = x ⊢ (∃ a_1, ↑(RelEmbedding.ofMonotone (Sum.rec Subtype.val fun x => ↑a) (_ : ∀ (x y : ↑{b | r b ↑a} ⊕ PUnit.{u + 1}), Sum.Lex (Subrel r {b | r b ↑a}) EmptyRelation x y → r (Sum.rec Subtype.val (fun x => ↑a) x) (Sum.rec Subtype.val (fun x => ↑a) y))) a_1 = x) ↔ r x ↑a ∨ ∃ x_1, ↑a = x ** dsimp [RelEmbedding.ofMonotone] ** case a α✝ : Type u_1 r✝ : α✝ → α✝ → Prop o : Ordinal.{u} α : Type u r : α → α → Prop x✝ : IsWellOrder α r z : cof (type r) = 1 S : Set α hl : Unbounded r S e : #↑S = 1 a : ↑S x : α this : r x ↑a ∨ ∃ x_1, ↑a = x ⊢ (∃ a_1, Sum.rec Subtype.val (fun x => ↑a) a_1 = x) ↔ r x ↑a ∨ ∃ x_1, ↑a = x ** simp ** case this.inl α✝ : Type u_1 r✝ : α✝ → α✝ → Prop o : Ordinal.{u} α : Type u r : α → α → Prop x✝ : IsWellOrder α r z : cof (type r) = 1 S : Set α hl : Unbounded r S e : #↑S = 1 a : ↑S x : α h : r x ↑a ⊢ r x ↑a ∨ ∃ x_1, ↑a = x ** exact Or.inl h ** case this.inr.inl α✝ : Type u_1 r✝ : α✝ → α✝ → Prop o : Ordinal.{u} α : Type u r : α → α → Prop x✝ : IsWellOrder α r z : cof (type r) = 1 S : Set α hl : Unbounded r S e : #↑S = 1 a : ↑S x : α h : x = ↑a ⊢ r x ↑a ∨ ∃ x_1, ↑a = x ** exact Or.inr ⟨PUnit.unit, h.symm⟩ ** case this.inr.inr α✝ : Type u_1 r✝ : α✝ → α✝ → Prop o : Ordinal.{u} α : Type u r : α → α → Prop x✝ : IsWellOrder α r z : cof (type r) = 1 S : Set α hl : Unbounded r S e : #↑S = 1 a : ↑S x : α h : r (↑a) x ⊢ r x ↑a ∨ ∃ x_1, ↑a = x ** rcases hl x with ⟨a', aS, hn⟩ ** case this.inr.inr.intro.intro α✝ : Type u_1 r✝ : α✝ → α✝ → Prop o : Ordinal.{u} α : Type u r : α → α → Prop x✝ : IsWellOrder α r z : cof (type r) = 1 S : Set α hl : Unbounded r S e : #↑S = 1 a : ↑S x : α h : r (↑a) x a' : α aS : a' ∈ S hn : ¬r a' x ⊢ r x ↑a ∨ ∃ x_1, ↑a = x ** rw [(_ : ↑a = a')] at h ** α✝ : Type u_1 r✝ : α✝ → α✝ → Prop o : Ordinal.{u} α : Type u r : α → α → Prop x✝ : IsWellOrder α r z : cof (type r) = 1 S : Set α hl : Unbounded r S e : #↑S = 1 a : ↑S x : α h : r (↑a) x a' : α aS : a' ∈ S hn : ¬r a' x ⊢ ↑a = a' ** refine' congr_arg Subtype.val (_ : a = ⟨a', aS⟩) ** α✝ : Type u_1 r✝ : α✝ → α✝ → Prop o : Ordinal.{u} α : Type u r : α → α → Prop x✝ : IsWellOrder α r z : cof (type r) = 1 S : Set α hl : Unbounded r S e : #↑S = 1 a : ↑S x : α h : r (↑a) x a' : α aS : a' ∈ S hn : ¬r a' x ⊢ a = { val := a', property := aS } ** haveI := le_one_iff_subsingleton.1 (le_of_eq e) ** α✝ : Type u_1 r✝ : α✝ → α✝ → Prop o : Ordinal.{u} α : Type u r : α → α → Prop x✝ : IsWellOrder α r z : cof (type r) = 1 S : Set α hl : Unbounded r S e : #↑S = 1 a : ↑S x : α h : r (↑a) x a' : α aS : a' ∈ S hn : ¬r a' x this : Subsingleton ↑S ⊢ a = { val := a', property := aS } ** apply Subsingleton.elim ** case this.inr.inr.intro.intro α✝ : Type u_1 r✝ : α✝ → α✝ → Prop o : Ordinal.{u} α : Type u r : α → α → Prop x✝ : IsWellOrder α r z : cof (type r) = 1 S : Set α hl : Unbounded r S e : #↑S = 1 a : ↑S x a' : α h : r a' x aS : a' ∈ S hn : ¬r a' x ⊢ r x ↑a ∨ ∃ x_1, ↑a = x ** exact absurd h hn ** α : Type u_1 r : α → α → Prop o : Ordinal.{u} x✝ : ∃ a, o = succ a a : Ordinal.{u} e : o = succ a ⊢ cof o = 1 ** simp [e] ** Qed | |
Ordinal.IsFundamentalSequence.cof_eq ** α : Type u_1 r : α → α → Prop a o : Ordinal.{u} f : (b : Ordinal.{u}) → b < o → Ordinal.{u} hf : IsFundamentalSequence a o f ⊢ ord (cof a) ≤ o ** rw [← hf.2.2] ** α : Type u_1 r : α → α → Prop a o : Ordinal.{u} f : (b : Ordinal.{u}) → b < o → Ordinal.{u} hf : IsFundamentalSequence a o f ⊢ ord (cof (blsub o f)) ≤ o ** exact (ord_le_ord.2 (cof_blsub_le f)).trans (ord_card_le o) ** Qed | |
Ordinal.IsFundamentalSequence.ord_cof ** α : Type u_1 r : α → α → Prop a o : Ordinal.{u} f : (b : Ordinal.{u}) → b < o → Ordinal.{u} hf : IsFundamentalSequence a o f i : Ordinal.{u} hi : i < ord (cof a) ⊢ ord (cof a) ≤ o ** rw [hf.cof_eq] ** α : Type u_1 r : α → α → Prop a o : Ordinal.{u} f : (b : Ordinal.{u}) → b < o → Ordinal.{u} hf : IsFundamentalSequence a o f ⊢ IsFundamentalSequence a (ord (cof a)) fun i hi => f i (_ : i < o) ** have H := hf.cof_eq ** α : Type u_1 r : α → α → Prop a o : Ordinal.{u} f : (b : Ordinal.{u}) → b < o → Ordinal.{u} hf : IsFundamentalSequence a o f H : ord (cof a) = o ⊢ IsFundamentalSequence a (ord (cof a)) fun i hi => f i (_ : i < o) ** subst H ** α : Type u_1 r : α → α → Prop a : Ordinal.{u} f : (b : Ordinal.{u}) → b < ord (cof a) → Ordinal.{u} hf : IsFundamentalSequence a (ord (cof a)) f ⊢ IsFundamentalSequence a (ord (cof a)) fun i hi => f i (_ : i < ord (cof a)) ** exact hf ** Qed | |
Ordinal.IsFundamentalSequence.zero ** α : Type u_1 r : α → α → Prop a o : Ordinal.{u} f✝ : (b : Ordinal.{u}) → b < o → Ordinal.{u} f : (b : Ordinal.{u_2}) → b < 0 → Ordinal.{u_2} ⊢ 0 ≤ ord (cof 0) ** rw [cof_zero, ord_zero] ** Qed | |
Ordinal.IsFundamentalSequence.succ ** α : Type u_1 r : α → α → Prop a o : Ordinal.{u} f : (b : Ordinal.{u}) → b < o → Ordinal.{u} ⊢ IsFundamentalSequence (succ o) 1 fun x x => o ** refine' ⟨_, @fun i j hi hj h => _, blsub_const Ordinal.one_ne_zero o⟩ ** case refine'_1 α : Type u_1 r : α → α → Prop a o : Ordinal.{u} f : (b : Ordinal.{u}) → b < o → Ordinal.{u} ⊢ 1 ≤ ord (cof (succ o)) ** rw [cof_succ, ord_one] ** case refine'_2 α : Type u_1 r : α → α → Prop a o : Ordinal.{u} f : (b : Ordinal.{u}) → b < o → Ordinal.{u} i j : Ordinal.{u} hi : i < 1 hj : j < 1 h : i < j ⊢ (fun x x => o) i hi < (fun x x => o) j hj ** rw [lt_one_iff_zero] at hi hj ** case refine'_2 α : Type u_1 r : α → α → Prop a o : Ordinal.{u} f : (b : Ordinal.{u}) → b < o → Ordinal.{u} i j : Ordinal.{u} hi✝ : i < 1 hi : i = 0 hj✝ : j < 1 hj : j = 0 h : i < j ⊢ (fun x x => o) i hi✝ < (fun x x => o) j hj✝ ** rw [hi, hj] at h ** case refine'_2 α : Type u_1 r : α → α → Prop a o : Ordinal.{u} f : (b : Ordinal.{u}) → b < o → Ordinal.{u} i j : Ordinal.{u} hi✝ : i < 1 hi : i = 0 hj✝ : j < 1 hj : j = 0 h : 0 < 0 ⊢ (fun x x => o) i hi✝ < (fun x x => o) j hj✝ ** exact h.false.elim ** Qed | |
Ordinal.IsFundamentalSequence.monotone ** α : Type u_1 r : α → α → Prop a o : Ordinal.{u} f : (b : Ordinal.{u}) → b < o → Ordinal.{u} hf : IsFundamentalSequence a o f i j : Ordinal.{u} hi : i < o hj : j < o hij : i ≤ j ⊢ f i hi ≤ f j hj ** rcases lt_or_eq_of_le hij with (hij | rfl) ** case inl α : Type u_1 r : α → α → Prop a o : Ordinal.{u} f : (b : Ordinal.{u}) → b < o → Ordinal.{u} hf : IsFundamentalSequence a o f i j : Ordinal.{u} hi : i < o hj : j < o hij✝ : i ≤ j hij : i < j ⊢ f i hi ≤ f j hj ** exact (hf.2.1 hi hj hij).le ** case inr α : Type u_1 r : α → α → Prop a o : Ordinal.{u} f : (b : Ordinal.{u}) → b < o → Ordinal.{u} hf : IsFundamentalSequence a o f i : Ordinal.{u} hi hj : i < o hij : i ≤ i ⊢ f i hi ≤ f i hj ** rfl ** Qed | |
Ordinal.IsFundamentalSequence.trans ** α : Type u_1 r : α → α → Prop a✝ o✝ : Ordinal.{u} f✝ : (b : Ordinal.{u}) → b < o✝ → Ordinal.{u} a o o' : Ordinal.{u} f : (b : Ordinal.{u}) → b < o → Ordinal.{u} hf : IsFundamentalSequence a o f g : (b : Ordinal.{u}) → b < o' → Ordinal.{u} hg : IsFundamentalSequence o o' g i : Ordinal.{u} hi : i < o' ⊢ g i hi < o ** rw [← hg.2.2] ** α : Type u_1 r : α → α → Prop a✝ o✝ : Ordinal.{u} f✝ : (b : Ordinal.{u}) → b < o✝ → Ordinal.{u} a o o' : Ordinal.{u} f : (b : Ordinal.{u}) → b < o → Ordinal.{u} hf : IsFundamentalSequence a o f g : (b : Ordinal.{u}) → b < o' → Ordinal.{u} hg : IsFundamentalSequence o o' g i : Ordinal.{u} hi : i < o' ⊢ g i hi < blsub o' g ** apply lt_blsub ** α : Type u_1 r : α → α → Prop a✝ o✝ : Ordinal.{u} f✝ : (b : Ordinal.{u}) → b < o✝ → Ordinal.{u} a o o' : Ordinal.{u} f : (b : Ordinal.{u}) → b < o → Ordinal.{u} hf : IsFundamentalSequence a o f g : (b : Ordinal.{u}) → b < o' → Ordinal.{u} hg : IsFundamentalSequence o o' g ⊢ IsFundamentalSequence a o' fun i hi => f (g i hi) (_ : g i hi < o) ** refine' ⟨_, @fun i j _ _ h => hf.2.1 _ _ (hg.2.1 _ _ h), _⟩ ** case refine'_1 α : Type u_1 r : α → α → Prop a✝ o✝ : Ordinal.{u} f✝ : (b : Ordinal.{u}) → b < o✝ → Ordinal.{u} a o o' : Ordinal.{u} f : (b : Ordinal.{u}) → b < o → Ordinal.{u} hf : IsFundamentalSequence a o f g : (b : Ordinal.{u}) → b < o' → Ordinal.{u} hg : IsFundamentalSequence o o' g ⊢ o' ≤ ord (cof a) ** rw [hf.cof_eq] ** case refine'_1 α : Type u_1 r : α → α → Prop a✝ o✝ : Ordinal.{u} f✝ : (b : Ordinal.{u}) → b < o✝ → Ordinal.{u} a o o' : Ordinal.{u} f : (b : Ordinal.{u}) → b < o → Ordinal.{u} hf : IsFundamentalSequence a o f g : (b : Ordinal.{u}) → b < o' → Ordinal.{u} hg : IsFundamentalSequence o o' g ⊢ o' ≤ o ** exact hg.1.trans (ord_cof_le o) ** case refine'_2 α : Type u_1 r : α → α → Prop a✝ o✝ : Ordinal.{u} f✝ : (b : Ordinal.{u}) → b < o✝ → Ordinal.{u} a o o' : Ordinal.{u} f : (b : Ordinal.{u}) → b < o → Ordinal.{u} hf : IsFundamentalSequence a o f g : (b : Ordinal.{u}) → b < o' → Ordinal.{u} hg : IsFundamentalSequence o o' g ⊢ (blsub o' fun i hi => f (g i hi) (_ : g i hi < o)) = a ** rw [@blsub_comp.{u, u, u} o _ f (@IsFundamentalSequence.monotone _ _ f hf)] ** case refine'_2 α : Type u_1 r : α → α → Prop a✝ o✝ : Ordinal.{u} f✝ : (b : Ordinal.{u}) → b < o✝ → Ordinal.{u} a o o' : Ordinal.{u} f : (b : Ordinal.{u}) → b < o → Ordinal.{u} hf : IsFundamentalSequence a o f g : (b : Ordinal.{u}) → b < o' → Ordinal.{u} hg : IsFundamentalSequence o o' g ⊢ blsub o f = a case refine'_2.hg α : Type u_1 r : α → α → Prop a✝ o✝ : Ordinal.{u} f✝ : (b : Ordinal.{u}) → b < o✝ → Ordinal.{u} a o o' : Ordinal.{u} f : (b : Ordinal.{u}) → b < o → Ordinal.{u} hf : IsFundamentalSequence a o f g : (b : Ordinal.{u}) → b < o' → Ordinal.{u} hg : IsFundamentalSequence o o' g ⊢ (blsub o' fun i hi => g i hi) = o ** exact hf.2.2 ** case refine'_2.hg α : Type u_1 r : α → α → Prop a✝ o✝ : Ordinal.{u} f✝ : (b : Ordinal.{u}) → b < o✝ → Ordinal.{u} a o o' : Ordinal.{u} f : (b : Ordinal.{u}) → b < o → Ordinal.{u} hf : IsFundamentalSequence a o f g : (b : Ordinal.{u}) → b < o' → Ordinal.{u} hg : IsFundamentalSequence o o' g ⊢ (blsub o' fun i hi => g i hi) = o ** exact hg.2.2 ** Qed | |
Ordinal.exists_fundamental_sequence ** α : Type u_1 r : α → α → Prop a : Ordinal.{u} ⊢ ∃ f, IsFundamentalSequence a (ord (cof a)) f ** suffices h : ∃ o f, IsFundamentalSequence a o f ** case h α : Type u_1 r : α → α → Prop a : Ordinal.{u} ⊢ ∃ o f, IsFundamentalSequence a o f ** rcases exists_lsub_cof a with ⟨ι, f, hf, hι⟩ ** case h.intro.intro.intro α : Type u_1 r : α → α → Prop a : Ordinal.{u} ι : Type u f : ι → Ordinal.{u} hf : lsub f = a hι : #ι = cof a ⊢ ∃ o f, IsFundamentalSequence a o f ** rcases ord_eq ι with ⟨r, wo, hr⟩ ** case h.intro.intro.intro.intro.intro α : Type u_1 r✝ : α → α → Prop a : Ordinal.{u} ι : Type u f : ι → Ordinal.{u} hf : lsub f = a hι : #ι = cof a r : ι → ι → Prop wo : IsWellOrder ι r hr : ord #ι = type r ⊢ ∃ o f, IsFundamentalSequence a o f ** haveI := wo ** case h.intro.intro.intro.intro.intro α : Type u_1 r✝ : α → α → Prop a : Ordinal.{u} ι : Type u f : ι → Ordinal.{u} hf : lsub f = a hι : #ι = cof a r : ι → ι → Prop wo : IsWellOrder ι r hr : ord #ι = type r this : IsWellOrder ι r ⊢ ∃ o f, IsFundamentalSequence a o f ** let r' := Subrel r { i | ∀ j, r j i → f j < f i } ** case h.intro.intro.intro.intro.intro α : Type u_1 r✝ : α → α → Prop a : Ordinal.{u} ι : Type u f : ι → Ordinal.{u} hf : lsub f = a hι : #ι = cof a r : ι → ι → Prop wo : IsWellOrder ι r hr : ord #ι = type r this : IsWellOrder ι r r' : ↑{i | ∀ (j : ι), r j i → f j < f i} → ↑{i | ∀ (j : ι), r j i → f j < f i} → Prop := Subrel r {i | ∀ (j : ι), r j i → f j < f i} ⊢ ∃ o f, IsFundamentalSequence a o f ** let hrr' : r' ↪r r := Subrel.relEmbedding _ _ ** case h.intro.intro.intro.intro.intro α : Type u_1 r✝ : α → α → Prop a : Ordinal.{u} ι : Type u f : ι → Ordinal.{u} hf : lsub f = a hι : #ι = cof a r : ι → ι → Prop wo : IsWellOrder ι r hr : ord #ι = type r this : IsWellOrder ι r r' : ↑{i | ∀ (j : ι), r j i → f j < f i} → ↑{i | ∀ (j : ι), r j i → f j < f i} → Prop := Subrel r {i | ∀ (j : ι), r j i → f j < f i} hrr' : r' ↪r r := Subrel.relEmbedding r {i | ∀ (j : ι), r j i → f j < f i} ⊢ ∃ o f, IsFundamentalSequence a o f ** haveI := hrr'.isWellOrder ** case h.intro.intro.intro.intro.intro α : Type u_1 r✝ : α → α → Prop a : Ordinal.{u} ι : Type u f : ι → Ordinal.{u} hf : lsub f = a hι : #ι = cof a r : ι → ι → Prop wo : IsWellOrder ι r hr : ord #ι = type r this✝ : IsWellOrder ι r r' : ↑{i | ∀ (j : ι), r j i → f j < f i} → ↑{i | ∀ (j : ι), r j i → f j < f i} → Prop := Subrel r {i | ∀ (j : ι), r j i → f j < f i} hrr' : r' ↪r r := Subrel.relEmbedding r {i | ∀ (j : ι), r j i → f j < f i} this : IsWellOrder (↑{i | ∀ (j : ι), r j i → f j < f i}) r' ⊢ ∃ o f, IsFundamentalSequence a o f ** refine'
⟨_, _, hrr'.ordinal_type_le.trans _, @fun i j _ h _ => (enum r' j h).prop _ _,
le_antisymm (blsub_le fun i hi => lsub_le_iff.1 hf.le _) _⟩ ** α : Type u_1 r : α → α → Prop a : Ordinal.{u} h : ∃ o f, IsFundamentalSequence a o f ⊢ ∃ f, IsFundamentalSequence a (ord (cof a)) f ** rcases h with ⟨o, f, hf⟩ ** case intro.intro α : Type u_1 r : α → α → Prop a o : Ordinal.{u} f : (b : Ordinal.{u}) → b < o → Ordinal.{u} hf : IsFundamentalSequence a o f ⊢ ∃ f, IsFundamentalSequence a (ord (cof a)) f ** exact ⟨_, hf.ord_cof⟩ ** case h.intro.intro.intro.intro.intro.refine'_1 α : Type u_1 r✝ : α → α → Prop a : Ordinal.{u} ι : Type u f : ι → Ordinal.{u} hf : lsub f = a hι : #ι = cof a r : ι → ι → Prop wo : IsWellOrder ι r hr : ord #ι = type r this✝ : IsWellOrder ι r r' : ↑{i | ∀ (j : ι), r j i → f j < f i} → ↑{i | ∀ (j : ι), r j i → f j < f i} → Prop := Subrel r {i | ∀ (j : ι), r j i → f j < f i} hrr' : r' ↪r r := Subrel.relEmbedding r {i | ∀ (j : ι), r j i → f j < f i} this : IsWellOrder (↑{i | ∀ (j : ι), r j i → f j < f i}) r' ⊢ type r ≤ ord (cof a) ** rw [← hι, hr] ** case h.intro.intro.intro.intro.intro.refine'_2 α : Type u_1 r✝ : α → α → Prop a : Ordinal.{u} ι : Type u f : ι → Ordinal.{u} hf : lsub f = a hι : #ι = cof a r : ι → ι → Prop wo : IsWellOrder ι r hr : ord #ι = type r this✝ : IsWellOrder ι r r' : ↑{i | ∀ (j : ι), r j i → f j < f i} → ↑{i | ∀ (j : ι), r j i → f j < f i} → Prop := Subrel r {i | ∀ (j : ι), r j i → f j < f i} hrr' : r' ↪r r := Subrel.relEmbedding r {i | ∀ (j : ι), r j i → f j < f i} this : IsWellOrder (↑{i | ∀ (j : ι), r j i → f j < f i}) r' i j : Ordinal.{u} x✝¹ : i < type r' h : j < type r' x✝ : i < j ⊢ r ↑(enum r' i x✝¹) ↑(enum r' j h) ** change r (hrr'.1 _) (hrr'.1 _) ** case h.intro.intro.intro.intro.intro.refine'_2 α : Type u_1 r✝ : α → α → Prop a : Ordinal.{u} ι : Type u f : ι → Ordinal.{u} hf : lsub f = a hι : #ι = cof a r : ι → ι → Prop wo : IsWellOrder ι r hr : ord #ι = type r this✝ : IsWellOrder ι r r' : ↑{i | ∀ (j : ι), r j i → f j < f i} → ↑{i | ∀ (j : ι), r j i → f j < f i} → Prop := Subrel r {i | ∀ (j : ι), r j i → f j < f i} hrr' : r' ↪r r := Subrel.relEmbedding r {i | ∀ (j : ι), r j i → f j < f i} this : IsWellOrder (↑{i | ∀ (j : ι), r j i → f j < f i}) r' i j : Ordinal.{u} x✝¹ : i < type r' h : j < type r' x✝ : i < j ⊢ r (↑hrr'.toEmbedding (enum r' i x✝¹)) (↑hrr'.toEmbedding (enum r' j h)) ** rwa [hrr'.2, @enum_lt_enum _ r'] ** case h.intro.intro.intro.intro.intro.refine'_3 α : Type u_1 r✝ : α → α → Prop a : Ordinal.{u} ι : Type u f : ι → Ordinal.{u} hf : lsub f = a hι : #ι = cof a r : ι → ι → Prop wo : IsWellOrder ι r hr : ord #ι = type r this✝ : IsWellOrder ι r r' : ↑{i | ∀ (j : ι), r j i → f j < f i} → ↑{i | ∀ (j : ι), r j i → f j < f i} → Prop := Subrel r {i | ∀ (j : ι), r j i → f j < f i} hrr' : r' ↪r r := Subrel.relEmbedding r {i | ∀ (j : ι), r j i → f j < f i} this : IsWellOrder (↑{i | ∀ (j : ι), r j i → f j < f i}) r' ⊢ a ≤ blsub (type r') fun j h => f ↑(enum r' j h) ** rw [← hf, lsub_le_iff] ** case h.intro.intro.intro.intro.intro.refine'_3 α : Type u_1 r✝ : α → α → Prop a : Ordinal.{u} ι : Type u f : ι → Ordinal.{u} hf : lsub f = a hι : #ι = cof a r : ι → ι → Prop wo : IsWellOrder ι r hr : ord #ι = type r this✝ : IsWellOrder ι r r' : ↑{i | ∀ (j : ι), r j i → f j < f i} → ↑{i | ∀ (j : ι), r j i → f j < f i} → Prop := Subrel r {i | ∀ (j : ι), r j i → f j < f i} hrr' : r' ↪r r := Subrel.relEmbedding r {i | ∀ (j : ι), r j i → f j < f i} this : IsWellOrder (↑{i | ∀ (j : ι), r j i → f j < f i}) r' ⊢ ∀ (i : ι), f i < blsub (type r') fun j h => f ↑(enum r' j h) ** intro i ** case h.intro.intro.intro.intro.intro.refine'_3 α : Type u_1 r✝ : α → α → Prop a : Ordinal.{u} ι : Type u f : ι → Ordinal.{u} hf : lsub f = a hι : #ι = cof a r : ι → ι → Prop wo : IsWellOrder ι r hr : ord #ι = type r this✝ : IsWellOrder ι r r' : ↑{i | ∀ (j : ι), r j i → f j < f i} → ↑{i | ∀ (j : ι), r j i → f j < f i} → Prop := Subrel r {i | ∀ (j : ι), r j i → f j < f i} hrr' : r' ↪r r := Subrel.relEmbedding r {i | ∀ (j : ι), r j i → f j < f i} this : IsWellOrder (↑{i | ∀ (j : ι), r j i → f j < f i}) r' i : ι ⊢ f i < blsub (type r') fun j h => f ↑(enum r' j h) ** suffices h : ∃ i' hi', f i ≤ bfamilyOfFamily' r' (fun i => f i) i' hi' ** case h α : Type u_1 r✝ : α → α → Prop a : Ordinal.{u} ι : Type u f : ι → Ordinal.{u} hf : lsub f = a hι : #ι = cof a r : ι → ι → Prop wo : IsWellOrder ι r hr : ord #ι = type r this✝ : IsWellOrder ι r r' : ↑{i | ∀ (j : ι), r j i → f j < f i} → ↑{i | ∀ (j : ι), r j i → f j < f i} → Prop := Subrel r {i | ∀ (j : ι), r j i → f j < f i} hrr' : r' ↪r r := Subrel.relEmbedding r {i | ∀ (j : ι), r j i → f j < f i} this : IsWellOrder (↑{i | ∀ (j : ι), r j i → f j < f i}) r' i : ι ⊢ ∃ i' hi', f i ≤ bfamilyOfFamily' r' (fun i => f ↑i) i' hi' ** by_cases h : ∀ j, r j i → f j < f i ** case h.intro.intro.intro.intro.intro.refine'_3 α : Type u_1 r✝ : α → α → Prop a : Ordinal.{u} ι : Type u f : ι → Ordinal.{u} hf : lsub f = a hι : #ι = cof a r : ι → ι → Prop wo : IsWellOrder ι r hr : ord #ι = type r this✝ : IsWellOrder ι r r' : ↑{i | ∀ (j : ι), r j i → f j < f i} → ↑{i | ∀ (j : ι), r j i → f j < f i} → Prop := Subrel r {i | ∀ (j : ι), r j i → f j < f i} hrr' : r' ↪r r := Subrel.relEmbedding r {i | ∀ (j : ι), r j i → f j < f i} this : IsWellOrder (↑{i | ∀ (j : ι), r j i → f j < f i}) r' i : ι h : ∃ i' hi', f i ≤ bfamilyOfFamily' r' (fun i => f ↑i) i' hi' ⊢ f i < blsub (type r') fun j h => f ↑(enum r' j h) ** rcases h with ⟨i', hi', hfg⟩ ** case h.intro.intro.intro.intro.intro.refine'_3.intro.intro α : Type u_1 r✝ : α → α → Prop a : Ordinal.{u} ι : Type u f : ι → Ordinal.{u} hf : lsub f = a hι : #ι = cof a r : ι → ι → Prop wo : IsWellOrder ι r hr : ord #ι = type r this✝ : IsWellOrder ι r r' : ↑{i | ∀ (j : ι), r j i → f j < f i} → ↑{i | ∀ (j : ι), r j i → f j < f i} → Prop := Subrel r {i | ∀ (j : ι), r j i → f j < f i} hrr' : r' ↪r r := Subrel.relEmbedding r {i | ∀ (j : ι), r j i → f j < f i} this : IsWellOrder (↑{i | ∀ (j : ι), r j i → f j < f i}) r' i : ι i' : Ordinal.{u} hi' : i' < type r' hfg : f i ≤ bfamilyOfFamily' r' (fun i => f ↑i) i' hi' ⊢ f i < blsub (type r') fun j h => f ↑(enum r' j h) ** exact hfg.trans_lt (lt_blsub _ _ _) ** case pos α : Type u_1 r✝ : α → α → Prop a : Ordinal.{u} ι : Type u f : ι → Ordinal.{u} hf : lsub f = a hι : #ι = cof a r : ι → ι → Prop wo : IsWellOrder ι r hr : ord #ι = type r this✝ : IsWellOrder ι r r' : ↑{i | ∀ (j : ι), r j i → f j < f i} → ↑{i | ∀ (j : ι), r j i → f j < f i} → Prop := Subrel r {i | ∀ (j : ι), r j i → f j < f i} hrr' : r' ↪r r := Subrel.relEmbedding r {i | ∀ (j : ι), r j i → f j < f i} this : IsWellOrder (↑{i | ∀ (j : ι), r j i → f j < f i}) r' i : ι h : ∀ (j : ι), r j i → f j < f i ⊢ ∃ i' hi', f i ≤ bfamilyOfFamily' r' (fun i => f ↑i) i' hi' ** refine' ⟨typein r' ⟨i, h⟩, typein_lt_type _ _, _⟩ ** case pos α : Type u_1 r✝ : α → α → Prop a : Ordinal.{u} ι : Type u f : ι → Ordinal.{u} hf : lsub f = a hι : #ι = cof a r : ι → ι → Prop wo : IsWellOrder ι r hr : ord #ι = type r this✝ : IsWellOrder ι r r' : ↑{i | ∀ (j : ι), r j i → f j < f i} → ↑{i | ∀ (j : ι), r j i → f j < f i} → Prop := Subrel r {i | ∀ (j : ι), r j i → f j < f i} hrr' : r' ↪r r := Subrel.relEmbedding r {i | ∀ (j : ι), r j i → f j < f i} this : IsWellOrder (↑{i | ∀ (j : ι), r j i → f j < f i}) r' i : ι h : ∀ (j : ι), r j i → f j < f i ⊢ f i ≤ bfamilyOfFamily' r' (fun i => f ↑i) (typein r' { val := i, property := h }) (_ : typein r' { val := i, property := h } < type r') ** rw [bfamilyOfFamily'_typein] ** case neg α : Type u_1 r✝ : α → α → Prop a : Ordinal.{u} ι : Type u f : ι → Ordinal.{u} hf : lsub f = a hι : #ι = cof a r : ι → ι → Prop wo : IsWellOrder ι r hr : ord #ι = type r this✝ : IsWellOrder ι r r' : ↑{i | ∀ (j : ι), r j i → f j < f i} → ↑{i | ∀ (j : ι), r j i → f j < f i} → Prop := Subrel r {i | ∀ (j : ι), r j i → f j < f i} hrr' : r' ↪r r := Subrel.relEmbedding r {i | ∀ (j : ι), r j i → f j < f i} this : IsWellOrder (↑{i | ∀ (j : ι), r j i → f j < f i}) r' i : ι h : ¬∀ (j : ι), r j i → f j < f i ⊢ ∃ i' hi', f i ≤ bfamilyOfFamily' r' (fun i => f ↑i) i' hi' ** push_neg at h ** case neg α : Type u_1 r✝ : α → α → Prop a : Ordinal.{u} ι : Type u f : ι → Ordinal.{u} hf : lsub f = a hι : #ι = cof a r : ι → ι → Prop wo : IsWellOrder ι r hr : ord #ι = type r this✝ : IsWellOrder ι r r' : ↑{i | ∀ (j : ι), r j i → f j < f i} → ↑{i | ∀ (j : ι), r j i → f j < f i} → Prop := Subrel r {i | ∀ (j : ι), r j i → f j < f i} hrr' : r' ↪r r := Subrel.relEmbedding r {i | ∀ (j : ι), r j i → f j < f i} this : IsWellOrder (↑{i | ∀ (j : ι), r j i → f j < f i}) r' i : ι h : ∃ j, r j i ∧ f i ≤ f j ⊢ ∃ i' hi', f i ≤ bfamilyOfFamily' r' (fun i => f ↑i) i' hi' ** cases' wo.wf.min_mem _ h with hji hij ** case neg.intro α : Type u_1 r✝ : α → α → Prop a : Ordinal.{u} ι : Type u f : ι → Ordinal.{u} hf : lsub f = a hι : #ι = cof a r : ι → ι → Prop wo : IsWellOrder ι r hr : ord #ι = type r this✝ : IsWellOrder ι r r' : ↑{i | ∀ (j : ι), r j i → f j < f i} → ↑{i | ∀ (j : ι), r j i → f j < f i} → Prop := Subrel r {i | ∀ (j : ι), r j i → f j < f i} hrr' : r' ↪r r := Subrel.relEmbedding r {i | ∀ (j : ι), r j i → f j < f i} this : IsWellOrder (↑{i | ∀ (j : ι), r j i → f j < f i}) r' i : ι h : ∃ j, r j i ∧ f i ≤ f j hji : r (WellFounded.min (_ : WellFounded r) (fun x => r x i ∧ f i ≤ f x) h) i hij : f i ≤ f (WellFounded.min (_ : WellFounded r) (fun x => r x i ∧ f i ≤ f x) h) ⊢ ∃ i' hi', f i ≤ bfamilyOfFamily' r' (fun i => f ↑i) i' hi' ** refine' ⟨typein r' ⟨_, fun k hkj => lt_of_lt_of_le _ hij⟩, typein_lt_type _ _, _⟩ ** case neg.intro.refine'_1 α : Type u_1 r✝ : α → α → Prop a : Ordinal.{u} ι : Type u f : ι → Ordinal.{u} hf : lsub f = a hι : #ι = cof a r : ι → ι → Prop wo : IsWellOrder ι r hr : ord #ι = type r this✝ : IsWellOrder ι r r' : ↑{i | ∀ (j : ι), r j i → f j < f i} → ↑{i | ∀ (j : ι), r j i → f j < f i} → Prop := Subrel r {i | ∀ (j : ι), r j i → f j < f i} hrr' : r' ↪r r := Subrel.relEmbedding r {i | ∀ (j : ι), r j i → f j < f i} this : IsWellOrder (↑{i | ∀ (j : ι), r j i → f j < f i}) r' i : ι h : ∃ j, r j i ∧ f i ≤ f j hji : r (WellFounded.min (_ : WellFounded r) (fun x => r x i ∧ f i ≤ f x) h) i hij : f i ≤ f (WellFounded.min (_ : WellFounded r) (fun x => r x i ∧ f i ≤ f x) h) k : ι hkj : r k (WellFounded.min (_ : WellFounded r) (fun x => r x i ∧ f i ≤ f x) h) ⊢ f k < f i ** by_contra' H ** case neg.intro.refine'_1 α : Type u_1 r✝ : α → α → Prop a : Ordinal.{u} ι : Type u f : ι → Ordinal.{u} hf : lsub f = a hι : #ι = cof a r : ι → ι → Prop wo : IsWellOrder ι r hr : ord #ι = type r this✝ : IsWellOrder ι r r' : ↑{i | ∀ (j : ι), r j i → f j < f i} → ↑{i | ∀ (j : ι), r j i → f j < f i} → Prop := Subrel r {i | ∀ (j : ι), r j i → f j < f i} hrr' : r' ↪r r := Subrel.relEmbedding r {i | ∀ (j : ι), r j i → f j < f i} this : IsWellOrder (↑{i | ∀ (j : ι), r j i → f j < f i}) r' i : ι h : ∃ j, r j i ∧ f i ≤ f j hji : r (WellFounded.min (_ : WellFounded r) (fun x => r x i ∧ f i ≤ f x) h) i hij : f i ≤ f (WellFounded.min (_ : WellFounded r) (fun x => r x i ∧ f i ≤ f x) h) k : ι hkj : r k (WellFounded.min (_ : WellFounded r) (fun x => r x i ∧ f i ≤ f x) h) H : f i ≤ f k ⊢ False ** exact (wo.wf.not_lt_min _ h ⟨IsTrans.trans _ _ _ hkj hji, H⟩) hkj ** case neg.intro.refine'_2 α : Type u_1 r✝ : α → α → Prop a : Ordinal.{u} ι : Type u f : ι → Ordinal.{u} hf : lsub f = a hι : #ι = cof a r : ι → ι → Prop wo : IsWellOrder ι r hr : ord #ι = type r this✝ : IsWellOrder ι r r' : ↑{i | ∀ (j : ι), r j i → f j < f i} → ↑{i | ∀ (j : ι), r j i → f j < f i} → Prop := Subrel r {i | ∀ (j : ι), r j i → f j < f i} hrr' : r' ↪r r := Subrel.relEmbedding r {i | ∀ (j : ι), r j i → f j < f i} this : IsWellOrder (↑{i | ∀ (j : ι), r j i → f j < f i}) r' i : ι h : ∃ j, r j i ∧ f i ≤ f j hji : r (WellFounded.min (_ : WellFounded r) (fun x => r x i ∧ f i ≤ f x) h) i hij : f i ≤ f (WellFounded.min (_ : WellFounded r) (fun x => r x i ∧ f i ≤ f x) h) ⊢ f i ≤ bfamilyOfFamily' r' (fun i => f ↑i) (typein r' { val := WellFounded.min (_ : WellFounded r) (fun x => r x i ∧ f i ≤ f x) h, property := (_ : ∀ (k : ι), r k (WellFounded.min (_ : WellFounded r) (fun x => r x i ∧ f i ≤ f x) h) → f k < f (WellFounded.min (_ : WellFounded r) (fun x => r x i ∧ f i ≤ f x) h)) }) (_ : typein r' { val := WellFounded.min (_ : WellFounded r) (fun x => r x i ∧ f i ≤ f x) h, property := (_ : ∀ (k : ι), r k (WellFounded.min (_ : WellFounded r) (fun x => r x i ∧ f i ≤ f x) h) → f k < f (WellFounded.min (_ : WellFounded r) (fun x => r x i ∧ f i ≤ f x) h)) } < type r') ** rwa [bfamilyOfFamily'_typein] ** Qed | |
Ordinal.cof_cof ** α : Type u_1 r : α → α → Prop a : Ordinal.{u} ⊢ cof (ord (cof a)) = cof a ** cases' exists_fundamental_sequence a with f hf ** case intro α : Type u_1 r : α → α → Prop a : Ordinal.{u} f : (b : Ordinal.{u}) → b < ord (cof a) → Ordinal.{u} hf : IsFundamentalSequence a (ord (cof a)) f ⊢ cof (ord (cof a)) = cof a ** cases' exists_fundamental_sequence a.cof.ord with g hg ** case intro.intro α : Type u_1 r : α → α → Prop a : Ordinal.{u} f : (b : Ordinal.{u}) → b < ord (cof a) → Ordinal.{u} hf : IsFundamentalSequence a (ord (cof a)) f g : (b : Ordinal.{u}) → b < ord (cof (ord (cof a))) → Ordinal.{u} hg : IsFundamentalSequence (ord (cof a)) (ord (cof (ord (cof a)))) g ⊢ cof (ord (cof a)) = cof a ** exact ord_injective (hf.trans hg).cof_eq.symm ** Qed | |
Ordinal.IsNormal.isFundamentalSequence ** α : Type u_1 r : α → α → Prop f : Ordinal.{u} → Ordinal.{u} hf : IsNormal f a o : Ordinal.{u} ha : IsLimit a g : (b : Ordinal.{u}) → b < o → Ordinal.{u} hg : IsFundamentalSequence a o g ⊢ IsFundamentalSequence (f a) o fun b hb => f (g b hb) ** refine' ⟨_, @fun i j _ _ h => hf.strictMono (hg.2.1 _ _ h), _⟩ ** case refine'_1 α : Type u_1 r : α → α → Prop f : Ordinal.{u} → Ordinal.{u} hf : IsNormal f a o : Ordinal.{u} ha : IsLimit a g : (b : Ordinal.{u}) → b < o → Ordinal.{u} hg : IsFundamentalSequence a o g ⊢ o ≤ ord (cof (f a)) ** rcases exists_lsub_cof (f a) with ⟨ι, f', hf', hι⟩ ** case refine'_1.intro.intro.intro α : Type u_1 r : α → α → Prop f : Ordinal.{u} → Ordinal.{u} hf : IsNormal f a o : Ordinal.{u} ha : IsLimit a g : (b : Ordinal.{u}) → b < o → Ordinal.{u} hg : IsFundamentalSequence a o g ι : Type u f' : ι → Ordinal.{u} hf' : lsub f' = f a hι : #ι = cof (f a) ⊢ o ≤ ord (cof (f a)) ** rw [← hg.cof_eq, ord_le_ord, ← hι] ** case refine'_1.intro.intro.intro α : Type u_1 r : α → α → Prop f : Ordinal.{u} → Ordinal.{u} hf : IsNormal f a o : Ordinal.{u} ha : IsLimit a g : (b : Ordinal.{u}) → b < o → Ordinal.{u} hg : IsFundamentalSequence a o g ι : Type u f' : ι → Ordinal.{u} hf' : lsub f' = f a hι : #ι = cof (f a) ⊢ cof a ≤ #ι ** suffices (lsub.{u, u} fun i => sInf { b : Ordinal | f' i ≤ f b }) = a by
rw [← this]
apply cof_lsub_le ** case refine'_1.intro.intro.intro α : Type u_1 r : α → α → Prop f : Ordinal.{u} → Ordinal.{u} hf : IsNormal f a o : Ordinal.{u} ha : IsLimit a g : (b : Ordinal.{u}) → b < o → Ordinal.{u} hg : IsFundamentalSequence a o g ι : Type u f' : ι → Ordinal.{u} hf' : lsub f' = f a hι : #ι = cof (f a) ⊢ (lsub fun i => sInf {b | f' i ≤ f b}) = a ** have H : ∀ i, ∃ b < a, f' i ≤ f b := fun i => by
have := lt_lsub.{u, u} f' i
rw [hf', ← IsNormal.blsub_eq.{u, u} hf ha, lt_blsub_iff] at this
simpa using this ** case refine'_1.intro.intro.intro α : Type u_1 r : α → α → Prop f : Ordinal.{u} → Ordinal.{u} hf : IsNormal f a o : Ordinal.{u} ha : IsLimit a g : (b : Ordinal.{u}) → b < o → Ordinal.{u} hg : IsFundamentalSequence a o g ι : Type u f' : ι → Ordinal.{u} hf' : lsub f' = f a hι : #ι = cof (f a) H : ∀ (i : ι), ∃ b, b < a ∧ f' i ≤ f b ⊢ (lsub fun i => sInf {b | f' i ≤ f b}) = a ** refine' (lsub_le fun i => _).antisymm (le_of_forall_lt fun b hb => _) ** α : Type u_1 r : α → α → Prop f : Ordinal.{u} → Ordinal.{u} hf : IsNormal f a o : Ordinal.{u} ha : IsLimit a g : (b : Ordinal.{u}) → b < o → Ordinal.{u} hg : IsFundamentalSequence a o g ι : Type u f' : ι → Ordinal.{u} hf' : lsub f' = f a hι : #ι = cof (f a) this : (lsub fun i => sInf {b | f' i ≤ f b}) = a ⊢ cof a ≤ #ι ** rw [← this] ** α : Type u_1 r : α → α → Prop f : Ordinal.{u} → Ordinal.{u} hf : IsNormal f a o : Ordinal.{u} ha : IsLimit a g : (b : Ordinal.{u}) → b < o → Ordinal.{u} hg : IsFundamentalSequence a o g ι : Type u f' : ι → Ordinal.{u} hf' : lsub f' = f a hι : #ι = cof (f a) this : (lsub fun i => sInf {b | f' i ≤ f b}) = a ⊢ cof (lsub fun i => sInf {b | f' i ≤ f b}) ≤ #ι ** apply cof_lsub_le ** α : Type u_1 r : α → α → Prop f : Ordinal.{u} → Ordinal.{u} hf : IsNormal f a o : Ordinal.{u} ha : IsLimit a g : (b : Ordinal.{u}) → b < o → Ordinal.{u} hg : IsFundamentalSequence a o g ι : Type u f' : ι → Ordinal.{u} hf' : lsub f' = f a hι : #ι = cof (f a) i : ι ⊢ ∃ b, b < a ∧ f' i ≤ f b ** have := lt_lsub.{u, u} f' i ** α : Type u_1 r : α → α → Prop f : Ordinal.{u} → Ordinal.{u} hf : IsNormal f a o : Ordinal.{u} ha : IsLimit a g : (b : Ordinal.{u}) → b < o → Ordinal.{u} hg : IsFundamentalSequence a o g ι : Type u f' : ι → Ordinal.{u} hf' : lsub f' = f a hι : #ι = cof (f a) i : ι this : f' i < lsub f' ⊢ ∃ b, b < a ∧ f' i ≤ f b ** rw [hf', ← IsNormal.blsub_eq.{u, u} hf ha, lt_blsub_iff] at this ** α : Type u_1 r : α → α → Prop f : Ordinal.{u} → Ordinal.{u} hf : IsNormal f a o : Ordinal.{u} ha : IsLimit a g : (b : Ordinal.{u}) → b < o → Ordinal.{u} hg : IsFundamentalSequence a o g ι : Type u f' : ι → Ordinal.{u} hf' : lsub f' = f a hι : #ι = cof (f a) i : ι this : ∃ i_1 hi, f' i ≤ f i_1 ⊢ ∃ b, b < a ∧ f' i ≤ f b ** simpa using this ** case refine'_1.intro.intro.intro.refine'_1 α : Type u_1 r : α → α → Prop f : Ordinal.{u} → Ordinal.{u} hf : IsNormal f a o : Ordinal.{u} ha : IsLimit a g : (b : Ordinal.{u}) → b < o → Ordinal.{u} hg : IsFundamentalSequence a o g ι : Type u f' : ι → Ordinal.{u} hf' : lsub f' = f a hι : #ι = cof (f a) H : ∀ (i : ι), ∃ b, b < a ∧ f' i ≤ f b i : ι ⊢ sInf {b | f' i ≤ f b} < a ** rcases H i with ⟨b, hb, hb'⟩ ** case refine'_1.intro.intro.intro.refine'_1.intro.intro α : Type u_1 r : α → α → Prop f : Ordinal.{u} → Ordinal.{u} hf : IsNormal f a o : Ordinal.{u} ha : IsLimit a g : (b : Ordinal.{u}) → b < o → Ordinal.{u} hg : IsFundamentalSequence a o g ι : Type u f' : ι → Ordinal.{u} hf' : lsub f' = f a hι : #ι = cof (f a) H : ∀ (i : ι), ∃ b, b < a ∧ f' i ≤ f b i : ι b : Ordinal.{u} hb : b < a hb' : f' i ≤ f b ⊢ sInf {b | f' i ≤ f b} < a ** exact lt_of_le_of_lt (csInf_le' hb') hb ** case refine'_1.intro.intro.intro.refine'_2 α : Type u_1 r : α → α → Prop f : Ordinal.{u} → Ordinal.{u} hf : IsNormal f a o : Ordinal.{u} ha : IsLimit a g : (b : Ordinal.{u}) → b < o → Ordinal.{u} hg : IsFundamentalSequence a o g ι : Type u f' : ι → Ordinal.{u} hf' : lsub f' = f a hι : #ι = cof (f a) H : ∀ (i : ι), ∃ b, b < a ∧ f' i ≤ f b b : Ordinal.{u} hb : b < a ⊢ b < lsub fun i => sInf {b | f' i ≤ f b} ** have := hf.strictMono hb ** case refine'_1.intro.intro.intro.refine'_2 α : Type u_1 r : α → α → Prop f : Ordinal.{u} → Ordinal.{u} hf : IsNormal f a o : Ordinal.{u} ha : IsLimit a g : (b : Ordinal.{u}) → b < o → Ordinal.{u} hg : IsFundamentalSequence a o g ι : Type u f' : ι → Ordinal.{u} hf' : lsub f' = f a hι : #ι = cof (f a) H : ∀ (i : ι), ∃ b, b < a ∧ f' i ≤ f b b : Ordinal.{u} hb : b < a this : f b < f a ⊢ b < lsub fun i => sInf {b | f' i ≤ f b} ** rw [← hf', lt_lsub_iff] at this ** case refine'_1.intro.intro.intro.refine'_2 α : Type u_1 r : α → α → Prop f : Ordinal.{u} → Ordinal.{u} hf : IsNormal f a o : Ordinal.{u} ha : IsLimit a g : (b : Ordinal.{u}) → b < o → Ordinal.{u} hg : IsFundamentalSequence a o g ι : Type u f' : ι → Ordinal.{u} hf' : lsub f' = f a hι : #ι = cof (f a) H : ∀ (i : ι), ∃ b, b < a ∧ f' i ≤ f b b : Ordinal.{u} hb : b < a this : ∃ i, f b ≤ f' i ⊢ b < lsub fun i => sInf {b | f' i ≤ f b} ** cases' this with i hi ** case refine'_1.intro.intro.intro.refine'_2.intro α : Type u_1 r : α → α → Prop f : Ordinal.{u} → Ordinal.{u} hf : IsNormal f a o : Ordinal.{u} ha : IsLimit a g : (b : Ordinal.{u}) → b < o → Ordinal.{u} hg : IsFundamentalSequence a o g ι : Type u f' : ι → Ordinal.{u} hf' : lsub f' = f a hι : #ι = cof (f a) H : ∀ (i : ι), ∃ b, b < a ∧ f' i ≤ f b b : Ordinal.{u} hb : b < a i : ι hi : f b ≤ f' i ⊢ b < lsub fun i => sInf {b | f' i ≤ f b} ** rcases H i with ⟨b, _, hb⟩ ** case refine'_1.intro.intro.intro.refine'_2.intro.intro.intro α : Type u_1 r : α → α → Prop f : Ordinal.{u} → Ordinal.{u} hf : IsNormal f a o : Ordinal.{u} ha : IsLimit a g : (b : Ordinal.{u}) → b < o → Ordinal.{u} hg : IsFundamentalSequence a o g ι : Type u f' : ι → Ordinal.{u} hf' : lsub f' = f a hι : #ι = cof (f a) H : ∀ (i : ι), ∃ b, b < a ∧ f' i ≤ f b b✝ : Ordinal.{u} hb✝ : b✝ < a i : ι hi : f b✝ ≤ f' i b : Ordinal.{u} left✝ : b < a hb : f' i ≤ f b ⊢ b✝ < lsub fun i => sInf {b | f' i ≤ f b} ** exact
((le_csInf_iff'' ⟨b, by exact hb⟩).2 fun c hc =>
hf.strictMono.le_iff_le.1 (hi.trans hc)).trans_lt (lt_lsub _ i) ** α : Type u_1 r : α → α → Prop f : Ordinal.{u} → Ordinal.{u} hf : IsNormal f a o : Ordinal.{u} ha : IsLimit a g : (b : Ordinal.{u}) → b < o → Ordinal.{u} hg : IsFundamentalSequence a o g ι : Type u f' : ι → Ordinal.{u} hf' : lsub f' = f a hι : #ι = cof (f a) H : ∀ (i : ι), ∃ b, b < a ∧ f' i ≤ f b b✝ : Ordinal.{u} hb✝ : b✝ < a i : ι hi : f b✝ ≤ f' i b : Ordinal.{u} left✝ : b < a hb : f' i ≤ f b ⊢ b ∈ fun c => Quot.lift (fun a₁ => Quotient.lift ((fun x x_1 => match x with | { α := α, r := r, wo := wo } => match x_1 with | { α := α_1, r := s, wo := wo } => Nonempty (r ≼i s)) a₁) (_ : ∀ (a b : WellOrder), a ≈ b → (match a₁ with | { α := α, r := r, wo := wo } => match a with | { α := α_1, r := s, wo := wo } => Nonempty (r ≼i s)) = match a₁ with | { α := α, r := r, wo := wo } => match b with | { α := α_1, r := s, wo := wo } => Nonempty (r ≼i s)) (f c)) (_ : ∀ (a b : WellOrder), a ≈ b → (fun a₁ => Quotient.lift (fun x => match a₁ with | { α := α, r := r, wo := wo } => match x with | { α := α_1, r := s, wo := wo } => Nonempty (r ≼i s)) (_ : ∀ (a b : WellOrder), a ≈ b → (fun x x_1 => match x with | { α := α, r := r, wo := wo } => match x_1 with | { α := α_1, r := s, wo := wo } => Nonempty (r ≼i s)) a₁ a = (fun x x_1 => match x with | { α := α, r := r, wo := wo } => match x_1 with | { α := α_1, r := s, wo := wo } => Nonempty (r ≼i s)) a₁ b) (f c)) a = (fun a₁ => Quotient.lift (fun x => match a₁ with | { α := α, r := r, wo := wo } => match x with | { α := α_1, r := s, wo := wo } => Nonempty (r ≼i s)) (_ : ∀ (a b : WellOrder), a ≈ b → (fun x x_1 => match x with | { α := α, r := r, wo := wo } => match x_1 with | { α := α_1, r := s, wo := wo } => Nonempty (r ≼i s)) a₁ a = (fun x x_1 => match x with | { α := α, r := r, wo := wo } => match x_1 with | { α := α_1, r := s, wo := wo } => Nonempty (r ≼i s)) a₁ b) (f c)) b) (f' i) ** exact hb ** case refine'_2 α : Type u_1 r : α → α → Prop f : Ordinal.{u} → Ordinal.{u} hf : IsNormal f a o : Ordinal.{u} ha : IsLimit a g : (b : Ordinal.{u}) → b < o → Ordinal.{u} hg : IsFundamentalSequence a o g ⊢ (blsub o fun b hb => f (g b hb)) = f a ** rw [@blsub_comp.{u, u, u} a _ (fun b _ => f b) (@fun i j _ _ h => hf.strictMono.monotone h) g
hg.2.2] ** case refine'_2 α : Type u_1 r : α → α → Prop f : Ordinal.{u} → Ordinal.{u} hf : IsNormal f a o : Ordinal.{u} ha : IsLimit a g : (b : Ordinal.{u}) → b < o → Ordinal.{u} hg : IsFundamentalSequence a o g ⊢ (blsub a fun b x => f b) = f a ** exact IsNormal.blsub_eq.{u, u} hf ha ** Qed | |
Ordinal.IsNormal.cof_le ** α : Type u_1 r : α → α → Prop f : Ordinal.{u_2} → Ordinal.{u_2} hf : IsNormal f a : Ordinal.{u_2} ⊢ cof a ≤ cof (f a) ** rcases zero_or_succ_or_limit a with (rfl | ⟨b, rfl⟩ | ha) ** case inl α : Type u_1 r : α → α → Prop f : Ordinal.{u_2} → Ordinal.{u_2} hf : IsNormal f ⊢ cof 0 ≤ cof (f 0) ** rw [cof_zero] ** case inl α : Type u_1 r : α → α → Prop f : Ordinal.{u_2} → Ordinal.{u_2} hf : IsNormal f ⊢ 0 ≤ cof (f 0) ** exact zero_le _ ** case inr.inl.intro α : Type u_1 r : α → α → Prop f : Ordinal.{u_2} → Ordinal.{u_2} hf : IsNormal f b : Ordinal.{u_2} ⊢ cof (succ b) ≤ cof (f (succ b)) ** rw [cof_succ, Cardinal.one_le_iff_ne_zero, cof_ne_zero, ← Ordinal.pos_iff_ne_zero] ** case inr.inl.intro α : Type u_1 r : α → α → Prop f : Ordinal.{u_2} → Ordinal.{u_2} hf : IsNormal f b : Ordinal.{u_2} ⊢ 0 < f (succ b) ** exact (Ordinal.zero_le (f b)).trans_lt (hf.1 b) ** case inr.inr α : Type u_1 r : α → α → Prop f : Ordinal.{u_2} → Ordinal.{u_2} hf : IsNormal f a : Ordinal.{u_2} ha : IsLimit a ⊢ cof a ≤ cof (f a) ** rw [hf.cof_eq ha] ** Qed | |
Ordinal.cof_add ** α : Type u_1 r : α → α → Prop a b : Ordinal.{u_2} h : b ≠ 0 ⊢ cof (a + b) = cof b ** rcases zero_or_succ_or_limit b with (rfl | ⟨c, rfl⟩ | hb) ** case inl α : Type u_1 r : α → α → Prop a : Ordinal.{u_2} h : 0 ≠ 0 ⊢ cof (a + 0) = cof 0 ** contradiction ** case inr.inl.intro α : Type u_1 r : α → α → Prop a c : Ordinal.{u_2} h : succ c ≠ 0 ⊢ cof (a + succ c) = cof (succ c) ** rw [add_succ, cof_succ, cof_succ] ** case inr.inr α : Type u_1 r : α → α → Prop a b : Ordinal.{u_2} h : b ≠ 0 hb : IsLimit b ⊢ cof (a + b) = cof b ** exact (add_isNormal a).cof_eq hb ** Qed | |
Ordinal.aleph0_le_cof ** α : Type u_1 r : α → α → Prop o : Ordinal.{u_2} ⊢ ℵ₀ ≤ cof o ↔ IsLimit o ** rcases zero_or_succ_or_limit o with (rfl | ⟨o, rfl⟩ | l) ** case inl α : Type u_1 r : α → α → Prop ⊢ ℵ₀ ≤ cof 0 ↔ IsLimit 0 ** simp [not_zero_isLimit, Cardinal.aleph0_ne_zero] ** case inr.inl.intro α : Type u_1 r : α → α → Prop o : Ordinal.{u_2} ⊢ ℵ₀ ≤ cof (succ o) ↔ IsLimit (succ o) ** simp [not_succ_isLimit, Cardinal.one_lt_aleph0] ** case inr.inr α : Type u_1 r : α → α → Prop o : Ordinal.{u_2} l : IsLimit o ⊢ ℵ₀ ≤ cof o ↔ IsLimit o ** simp [l] ** case inr.inr α : Type u_1 r : α → α → Prop o : Ordinal.{u_2} l : IsLimit o ⊢ ℵ₀ ≤ cof o ** refine' le_of_not_lt fun h => _ ** case inr.inr α : Type u_1 r : α → α → Prop o : Ordinal.{u_2} l : IsLimit o h : cof o < ℵ₀ ⊢ False ** cases' Cardinal.lt_aleph0.1 h with n e ** case inr.inr.intro α : Type u_1 r : α → α → Prop o : Ordinal.{u_2} l : IsLimit o h : cof o < ℵ₀ n : ℕ e : cof o = ↑n ⊢ False ** have := cof_cof o ** case inr.inr.intro α : Type u_1 r : α → α → Prop o : Ordinal.{u_2} l : IsLimit o h : cof o < ℵ₀ n : ℕ e : cof o = ↑n this : cof (ord (cof o)) = cof o ⊢ False ** rw [e, ord_nat] at this ** case inr.inr.intro α : Type u_1 r : α → α → Prop o : Ordinal.{u_2} l : IsLimit o h : cof o < ℵ₀ n : ℕ e : cof o = ↑n this : cof ↑n = ↑n ⊢ False ** cases n ** case inr.inr.intro.zero α : Type u_1 r : α → α → Prop o : Ordinal.{u_2} l : IsLimit o h : cof o < ℵ₀ e : cof o = ↑Nat.zero this : cof ↑Nat.zero = ↑Nat.zero ⊢ False ** simp at e ** case inr.inr.intro.zero α : Type u_1 r : α → α → Prop o : Ordinal.{u_2} l : IsLimit o h : cof o < ℵ₀ this : cof ↑Nat.zero = ↑Nat.zero e : o = 0 ⊢ False ** simp [e, not_zero_isLimit] at l ** case inr.inr.intro.succ α : Type u_1 r : α → α → Prop o : Ordinal.{u_2} l : IsLimit o h : cof o < ℵ₀ n✝ : ℕ e : cof o = ↑(Nat.succ n✝) this : cof ↑(Nat.succ n✝) = ↑(Nat.succ n✝) ⊢ False ** rw [nat_cast_succ, cof_succ] at this ** case inr.inr.intro.succ α : Type u_1 r : α → α → Prop o : Ordinal.{u_2} l : IsLimit o h : cof o < ℵ₀ n✝ : ℕ e : cof o = ↑(Nat.succ n✝) this : 1 = ↑(Nat.succ n✝) ⊢ False ** rw [← this, cof_eq_one_iff_is_succ] at e ** case inr.inr.intro.succ α : Type u_1 r : α → α → Prop o : Ordinal.{u_2} l : IsLimit o h : cof o < ℵ₀ n✝ : ℕ e : ∃ a, o = succ a this : 1 = ↑(Nat.succ n✝) ⊢ False ** rcases e with ⟨a, rfl⟩ ** case inr.inr.intro.succ.intro α : Type u_1 r : α → α → Prop n✝ : ℕ this : 1 = ↑(Nat.succ n✝) a : Ordinal.{u_2} l : IsLimit (succ a) h : cof (succ a) < ℵ₀ ⊢ False ** exact not_succ_isLimit _ l ** Qed | |
Ordinal.cof_omega ** α : Type u_1 r : α → α → Prop ⊢ cof ω ≤ ℵ₀ ** rw [← card_omega] ** α : Type u_1 r : α → α → Prop ⊢ cof ω ≤ card ω ** apply cof_le_card ** Qed | |
Ordinal.cof_eq' ** α : Type u_1 r✝ r : α → α → Prop inst✝ : IsWellOrder α r h✝ : IsLimit (type r) S : Set α H : Unbounded r S e : #↑S = cof (type r) a : α a' : α := enum r (succ (typein r a)) (_ : succ (typein r a) < type r) b : α h : b ∈ S ab : ¬r b a' ⊢ typein r a < typein r a' ** rw [typein_enum] ** α : Type u_1 r✝ r : α → α → Prop inst✝ : IsWellOrder α r h✝ : IsLimit (type r) S : Set α H : Unbounded r S e : #↑S = cof (type r) a : α a' : α := enum r (succ (typein r a)) (_ : succ (typein r a) < type r) b : α h : b ∈ S ab : ¬r b a' ⊢ typein r a < succ (typein r a) ** exact lt_succ (typein _ _) ** Qed | |
Ordinal.cof_univ ** α : Type u_1 r : α → α → Prop ⊢ Cardinal.univ ≤ cof univ ** refine' le_of_forall_lt fun c h => _ ** α : Type u_1 r : α → α → Prop c : Cardinal.{max (u + 1) v} h : c < Cardinal.univ ⊢ c < cof univ ** rcases lt_univ'.1 h with ⟨c, rfl⟩ ** case intro.intro.intro α : Type u_1 r : α → α → Prop c : Cardinal.{u} h : Cardinal.lift.{max (u + 1) v, u} c < Cardinal.univ S : Set Ordinal.{u} H : Unbounded (fun x x_1 => x < x_1) S Se : #↑S = cof (type fun x x_1 => x < x_1) ⊢ Cardinal.lift.{max (u + 1) v, u} c < cof univ ** rw [univ, ← lift_cof, ← Cardinal.lift_lift.{u+1, v, u}, Cardinal.lift_lt, ← Se] ** case intro.intro.intro α : Type u_1 r : α → α → Prop c : Cardinal.{u} h : Cardinal.lift.{max (u + 1) v, u} c < Cardinal.univ S : Set Ordinal.{u} H : Unbounded (fun x x_1 => x < x_1) S Se : #↑S = cof (type fun x x_1 => x < x_1) ⊢ Cardinal.lift.{u + 1, u} c < #↑S ** refine' lt_of_not_ge fun h => _ ** case intro.intro.intro α : Type u_1 r : α → α → Prop c : Cardinal.{u} h✝ : Cardinal.lift.{max (u + 1) v, u} c < Cardinal.univ S : Set Ordinal.{u} H : Unbounded (fun x x_1 => x < x_1) S Se : #↑S = cof (type fun x x_1 => x < x_1) h : Cardinal.lift.{u + 1, u} c ≥ #↑S ⊢ False ** cases' Cardinal.lift_down h with a e ** case intro.intro.intro.intro α : Type u_1 r : α → α → Prop c : Cardinal.{u} h✝ : Cardinal.lift.{max (u + 1) v, u} c < Cardinal.univ S : Set Ordinal.{u} H : Unbounded (fun x x_1 => x < x_1) S Se : #↑S = cof (type fun x x_1 => x < x_1) h : Cardinal.lift.{u + 1, u} c ≥ #↑S a : Cardinal.{u} e : Cardinal.lift.{u + 1, u} a = #↑S ⊢ False ** refine' Quotient.inductionOn a (fun α e => _) e ** case intro.intro.intro.intro α✝ : Type u_1 r : α✝ → α✝ → Prop c : Cardinal.{u} h✝ : Cardinal.lift.{max (u + 1) v, u} c < Cardinal.univ S : Set Ordinal.{u} H : Unbounded (fun x x_1 => x < x_1) S Se : #↑S = cof (type fun x x_1 => x < x_1) h : Cardinal.lift.{u + 1, u} c ≥ #↑S a : Cardinal.{u} e✝ : Cardinal.lift.{u + 1, u} a = #↑S α : Type u e : Cardinal.lift.{u + 1, u} (Quotient.mk Cardinal.isEquivalent α) = #↑S ⊢ False ** cases' Quotient.exact e with f ** case intro.intro.intro.intro.intro α✝ : Type u_1 r : α✝ → α✝ → Prop c : Cardinal.{u} h✝ : Cardinal.lift.{max (u + 1) v, u} c < Cardinal.univ S : Set Ordinal.{u} H : Unbounded (fun x x_1 => x < x_1) S Se : #↑S = cof (type fun x x_1 => x < x_1) h : Cardinal.lift.{u + 1, u} c ≥ #↑S a : Cardinal.{u} e✝ : Cardinal.lift.{u + 1, u} a = #↑S α : Type u e : Cardinal.lift.{u + 1, u} (Quotient.mk Cardinal.isEquivalent α) = #↑S f : ULift.{u + 1, u} α ≃ ↑S ⊢ False ** have f := Equiv.ulift.symm.trans f ** case intro.intro.intro.intro.intro α✝ : Type u_1 r : α✝ → α✝ → Prop c : Cardinal.{u} h✝ : Cardinal.lift.{max (u + 1) v, u} c < Cardinal.univ S : Set Ordinal.{u} H : Unbounded (fun x x_1 => x < x_1) S Se : #↑S = cof (type fun x x_1 => x < x_1) h : Cardinal.lift.{u + 1, u} c ≥ #↑S a : Cardinal.{u} e✝ : Cardinal.lift.{u + 1, u} a = #↑S α : Type u e : Cardinal.lift.{u + 1, u} (Quotient.mk Cardinal.isEquivalent α) = #↑S f✝ : ULift.{u + 1, u} α ≃ ↑S f : α ≃ ↑S ⊢ False ** let g a := (f a).1 ** case intro.intro.intro.intro.intro α✝ : Type u_1 r : α✝ → α✝ → Prop c : Cardinal.{u} h✝ : Cardinal.lift.{max (u + 1) v, u} c < Cardinal.univ S : Set Ordinal.{u} H : Unbounded (fun x x_1 => x < x_1) S Se : #↑S = cof (type fun x x_1 => x < x_1) h : Cardinal.lift.{u + 1, u} c ≥ #↑S a : Cardinal.{u} e✝ : Cardinal.lift.{u + 1, u} a = #↑S α : Type u e : Cardinal.lift.{u + 1, u} (Quotient.mk Cardinal.isEquivalent α) = #↑S f✝ : ULift.{u + 1, u} α ≃ ↑S f : α ≃ ↑S g : α → Ordinal.{u} := fun a => ↑(↑f a) ⊢ False ** let o := succ (sup.{u, u} g) ** case intro.intro.intro.intro.intro α✝ : Type u_1 r : α✝ → α✝ → Prop c : Cardinal.{u} h✝ : Cardinal.lift.{max (u + 1) v, u} c < Cardinal.univ S : Set Ordinal.{u} H : Unbounded (fun x x_1 => x < x_1) S Se : #↑S = cof (type fun x x_1 => x < x_1) h : Cardinal.lift.{u + 1, u} c ≥ #↑S a : Cardinal.{u} e✝ : Cardinal.lift.{u + 1, u} a = #↑S α : Type u e : Cardinal.lift.{u + 1, u} (Quotient.mk Cardinal.isEquivalent α) = #↑S f✝ : ULift.{u + 1, u} α ≃ ↑S f : α ≃ ↑S g : α → Ordinal.{u} := fun a => ↑(↑f a) o : Ordinal.{u} := succ (sup g) ⊢ False ** rcases H o with ⟨b, h, l⟩ ** case intro.intro.intro.intro.intro.intro.intro α✝ : Type u_1 r : α✝ → α✝ → Prop c : Cardinal.{u} h✝¹ : Cardinal.lift.{max (u + 1) v, u} c < Cardinal.univ S : Set Ordinal.{u} H : Unbounded (fun x x_1 => x < x_1) S Se : #↑S = cof (type fun x x_1 => x < x_1) h✝ : Cardinal.lift.{u + 1, u} c ≥ #↑S a : Cardinal.{u} e✝ : Cardinal.lift.{u + 1, u} a = #↑S α : Type u e : Cardinal.lift.{u + 1, u} (Quotient.mk Cardinal.isEquivalent α) = #↑S f✝ : ULift.{u + 1, u} α ≃ ↑S f : α ≃ ↑S g : α → Ordinal.{u} := fun a => ↑(↑f a) o : Ordinal.{u} := succ (sup g) b : Ordinal.{u} h : b ∈ S l : ¬(fun x x_1 => x < x_1) b o ⊢ False ** refine' l (lt_succ_iff.2 _) ** case intro.intro.intro.intro.intro.intro.intro α✝ : Type u_1 r : α✝ → α✝ → Prop c : Cardinal.{u} h✝¹ : Cardinal.lift.{max (u + 1) v, u} c < Cardinal.univ S : Set Ordinal.{u} H : Unbounded (fun x x_1 => x < x_1) S Se : #↑S = cof (type fun x x_1 => x < x_1) h✝ : Cardinal.lift.{u + 1, u} c ≥ #↑S a : Cardinal.{u} e✝ : Cardinal.lift.{u + 1, u} a = #↑S α : Type u e : Cardinal.lift.{u + 1, u} (Quotient.mk Cardinal.isEquivalent α) = #↑S f✝ : ULift.{u + 1, u} α ≃ ↑S f : α ≃ ↑S g : α → Ordinal.{u} := fun a => ↑(↑f a) o : Ordinal.{u} := succ (sup g) b : Ordinal.{u} h : b ∈ S l : ¬(fun x x_1 => x < x_1) b o ⊢ b ≤ sup g ** rw [← show g (f.symm ⟨b, h⟩) = b by simp] ** case intro.intro.intro.intro.intro.intro.intro α✝ : Type u_1 r : α✝ → α✝ → Prop c : Cardinal.{u} h✝¹ : Cardinal.lift.{max (u + 1) v, u} c < Cardinal.univ S : Set Ordinal.{u} H : Unbounded (fun x x_1 => x < x_1) S Se : #↑S = cof (type fun x x_1 => x < x_1) h✝ : Cardinal.lift.{u + 1, u} c ≥ #↑S a : Cardinal.{u} e✝ : Cardinal.lift.{u + 1, u} a = #↑S α : Type u e : Cardinal.lift.{u + 1, u} (Quotient.mk Cardinal.isEquivalent α) = #↑S f✝ : ULift.{u + 1, u} α ≃ ↑S f : α ≃ ↑S g : α → Ordinal.{u} := fun a => ↑(↑f a) o : Ordinal.{u} := succ (sup g) b : Ordinal.{u} h : b ∈ S l : ¬(fun x x_1 => x < x_1) b o ⊢ g (↑f.symm { val := b, property := h }) ≤ sup g ** apply le_sup ** α✝ : Type u_1 r : α✝ → α✝ → Prop c : Cardinal.{u} h✝¹ : Cardinal.lift.{max (u + 1) v, u} c < Cardinal.univ S : Set Ordinal.{u} H : Unbounded (fun x x_1 => x < x_1) S Se : #↑S = cof (type fun x x_1 => x < x_1) h✝ : Cardinal.lift.{u + 1, u} c ≥ #↑S a : Cardinal.{u} e✝ : Cardinal.lift.{u + 1, u} a = #↑S α : Type u e : Cardinal.lift.{u + 1, u} (Quotient.mk Cardinal.isEquivalent α) = #↑S f✝ : ULift.{u + 1, u} α ≃ ↑S f : α ≃ ↑S g : α → Ordinal.{u} := fun a => ↑(↑f a) o : Ordinal.{u} := succ (sup g) b : Ordinal.{u} h : b ∈ S l : ¬(fun x x_1 => x < x_1) b o ⊢ g (↑f.symm { val := b, property := h }) = b ** simp ** Qed | |
Ordinal.unbounded_of_unbounded_sUnion ** α : Type u_1 r✝ r : α → α → Prop wo : IsWellOrder α r s : Set (Set α) h₁ : Unbounded r (⋃₀ s) h₂ : #↑s < StrictOrder.cof r ⊢ ∃ x, x ∈ s ∧ Unbounded r x ** by_contra' h ** α : Type u_1 r✝ r : α → α → Prop wo : IsWellOrder α r s : Set (Set α) h₁ : Unbounded r (⋃₀ s) h₂ : #↑s < StrictOrder.cof r h : ∀ (x : Set α), x ∈ s → ¬Unbounded r x ⊢ False ** simp_rw [not_unbounded_iff] at h ** α : Type u_1 r✝ r : α → α → Prop wo : IsWellOrder α r s : Set (Set α) h₁ : Unbounded r (⋃₀ s) h₂ : #↑s < StrictOrder.cof r h : ∀ (x : Set α), x ∈ s → Bounded r x ⊢ False ** let f : s → α := fun x : s => wo.wf.sup x (h x.1 x.2) ** α : Type u_1 r✝ r : α → α → Prop wo : IsWellOrder α r s : Set (Set α) h₁ : Unbounded r (⋃₀ s) h₂ : #↑s < StrictOrder.cof r h : ∀ (x : Set α), x ∈ s → Bounded r x f : ↑s → α := fun x => WellFounded.sup (_ : WellFounded r) ↑x (_ : Bounded r ↑x) ⊢ False ** refine' h₂.not_le (le_trans (csInf_le' ⟨range f, fun x => _, rfl⟩) mk_range_le) ** α : Type u_1 r✝ r : α → α → Prop wo : IsWellOrder α r s : Set (Set α) h₁ : Unbounded r (⋃₀ s) h₂ : #↑s < StrictOrder.cof r h : ∀ (x : Set α), x ∈ s → Bounded r x f : ↑s → α := fun x => WellFounded.sup (_ : WellFounded r) ↑x (_ : Bounded r ↑x) x : α ⊢ ∃ b, b ∈ range f ∧ swap rᶜ x b ** rcases h₁ x with ⟨y, ⟨c, hc, hy⟩, hxy⟩ ** case intro.intro.intro.intro α : Type u_1 r✝ r : α → α → Prop wo : IsWellOrder α r s : Set (Set α) h₁ : Unbounded r (⋃₀ s) h₂ : #↑s < StrictOrder.cof r h : ∀ (x : Set α), x ∈ s → Bounded r x f : ↑s → α := fun x => WellFounded.sup (_ : WellFounded r) ↑x (_ : Bounded r ↑x) x y : α hxy : ¬r y x c : Set α hc : c ∈ s hy : y ∈ c ⊢ ∃ b, b ∈ range f ∧ swap rᶜ x b ** exact ⟨f ⟨c, hc⟩, mem_range_self _, fun hxz => hxy (Trans.trans (wo.wf.lt_sup _ hy) hxz)⟩ ** Qed | |
Ordinal.unbounded_of_unbounded_iUnion ** α✝ : Type u_1 r✝ : α✝ → α✝ → Prop α β : Type u r : α → α → Prop wo : IsWellOrder α r s : β → Set α h₁ : Unbounded r (⋃ x, s x) h₂ : #β < StrictOrder.cof r ⊢ ∃ x, Unbounded r (s x) ** rw [← sUnion_range] at h₁ ** α✝ : Type u_1 r✝ : α✝ → α✝ → Prop α β : Type u r : α → α → Prop wo : IsWellOrder α r s : β → Set α h₁ : Unbounded r (⋃₀ range fun x => s x) h₂ : #β < StrictOrder.cof r ⊢ ∃ x, Unbounded r (s x) ** rcases unbounded_of_unbounded_sUnion r h₁ (mk_range_le.trans_lt h₂) with ⟨_, ⟨x, rfl⟩, u⟩ ** case intro.intro.intro α✝ : Type u_1 r✝ : α✝ → α✝ → Prop α β : Type u r : α → α → Prop wo : IsWellOrder α r s : β → Set α h₁ : Unbounded r (⋃₀ range fun x => s x) h₂ : #β < StrictOrder.cof r x : β u : Unbounded r ((fun x => s x) x) ⊢ ∃ x, Unbounded r (s x) ** exact ⟨x, u⟩ ** Qed | |
Ordinal.infinite_pigeonhole ** α✝ : Type u_1 r : α✝ → α✝ → Prop β α : Type u f : β → α h₁ : ℵ₀ ≤ #β h₂ : #α < cof (ord #β) ⊢ ∃ a, #↑(f ⁻¹' {a}) = #β ** have : ∃ a, #β ≤ #(f ⁻¹' {a}) := by
by_contra' h
apply mk_univ.not_lt
rw [← preimage_univ, ← iUnion_of_singleton, preimage_iUnion]
exact
mk_iUnion_le_sum_mk.trans_lt
((sum_le_iSup _).trans_lt <| mul_lt_of_lt h₁ (h₂.trans_le <| cof_ord_le _) (iSup_lt h₂ h)) ** α✝ : Type u_1 r : α✝ → α✝ → Prop β α : Type u f : β → α h₁ : ℵ₀ ≤ #β h₂ : #α < cof (ord #β) this : ∃ a, #β ≤ #↑(f ⁻¹' {a}) ⊢ ∃ a, #↑(f ⁻¹' {a}) = #β ** cases' this with x h ** case intro α✝ : Type u_1 r : α✝ → α✝ → Prop β α : Type u f : β → α h₁ : ℵ₀ ≤ #β h₂ : #α < cof (ord #β) x : α h : #β ≤ #↑(f ⁻¹' {x}) ⊢ ∃ a, #↑(f ⁻¹' {a}) = #β ** refine' ⟨x, h.antisymm' _⟩ ** case intro α✝ : Type u_1 r : α✝ → α✝ → Prop β α : Type u f : β → α h₁ : ℵ₀ ≤ #β h₂ : #α < cof (ord #β) x : α h : #β ≤ #↑(f ⁻¹' {x}) ⊢ #↑(f ⁻¹' {x}) ≤ #β ** rw [le_mk_iff_exists_set] ** case intro α✝ : Type u_1 r : α✝ → α✝ → Prop β α : Type u f : β → α h₁ : ℵ₀ ≤ #β h₂ : #α < cof (ord #β) x : α h : #β ≤ #↑(f ⁻¹' {x}) ⊢ ∃ p, #↑p = #↑(f ⁻¹' {x}) ** exact ⟨_, rfl⟩ ** α✝ : Type u_1 r : α✝ → α✝ → Prop β α : Type u f : β → α h₁ : ℵ₀ ≤ #β h₂ : #α < cof (ord #β) ⊢ ∃ a, #β ≤ #↑(f ⁻¹' {a}) ** by_contra' h ** α✝ : Type u_1 r : α✝ → α✝ → Prop β α : Type u f : β → α h₁ : ℵ₀ ≤ #β h₂ : #α < cof (ord #β) h : ∀ (a : α), #↑(f ⁻¹' {a}) < #β ⊢ False ** apply mk_univ.not_lt ** α✝ : Type u_1 r : α✝ → α✝ → Prop β α : Type u f : β → α h₁ : ℵ₀ ≤ #β h₂ : #α < cof (ord #β) h : ∀ (a : α), #↑(f ⁻¹' {a}) < #β ⊢ #↑Set.univ < #?m.109558 α✝ : Type u_1 r : α✝ → α✝ → Prop β α : Type u f : β → α h₁ : ℵ₀ ≤ #β h₂ : #α < cof (ord #β) h : ∀ (a : α), #↑(f ⁻¹' {a}) < #β ⊢ Type ?u.109557 ** rw [← preimage_univ, ← iUnion_of_singleton, preimage_iUnion] ** α✝ : Type u_1 r : α✝ → α✝ → Prop β α : Type u f : β → α h₁ : ℵ₀ ≤ #β h₂ : #α < cof (ord #β) h : ∀ (a : α), #↑(f ⁻¹' {a}) < #β ⊢ #↑(⋃ i, ?m.109626 ⁻¹' {i}) < #?m.109623 α✝ : Type u_1 r : α✝ → α✝ → Prop β α : Type u f : β → α h₁ : ℵ₀ ≤ #β h₂ : #α < cof (ord #β) h : ∀ (a : α), #↑(f ⁻¹' {a}) < #β ⊢ Type ?u.109620 α✝ : Type u_1 r : α✝ → α✝ → Prop β α : Type u f : β → α h₁ : ℵ₀ ≤ #β h₂ : #α < cof (ord #β) h : ∀ (a : α), #↑(f ⁻¹' {a}) < #β ⊢ Type ?u.109621 α✝ : Type u_1 r : α✝ → α✝ → Prop β α : Type u f : β → α h₁ : ℵ₀ ≤ #β h₂ : #α < cof (ord #β) h : ∀ (a : α), #↑(f ⁻¹' {a}) < #β ⊢ ?m.109623 → ?m.109625 ** exact
mk_iUnion_le_sum_mk.trans_lt
((sum_le_iSup _).trans_lt <| mul_lt_of_lt h₁ (h₂.trans_le <| cof_ord_le _) (iSup_lt h₂ h)) ** Qed | |
Ordinal.infinite_pigeonhole_card ** α✝ : Type u_1 r : α✝ → α✝ → Prop β α : Type u f : β → α θ : Cardinal.{u} hθ : θ ≤ #β h₁ : ℵ₀ ≤ θ h₂ : #α < cof (ord θ) ⊢ ∃ a, θ ≤ #↑(f ⁻¹' {a}) ** rcases le_mk_iff_exists_set.1 hθ with ⟨s, rfl⟩ ** case intro α✝ : Type u_1 r : α✝ → α✝ → Prop β α : Type u f : β → α s : Set β hθ : #↑s ≤ #β h₁ : ℵ₀ ≤ #↑s h₂ : #α < cof (ord #↑s) ⊢ ∃ a, #↑s ≤ #↑(f ⁻¹' {a}) ** cases' infinite_pigeonhole (f ∘ Subtype.val : s → α) h₁ h₂ with a ha ** case intro.intro α✝ : Type u_1 r : α✝ → α✝ → Prop β α : Type u f : β → α s : Set β hθ : #↑s ≤ #β h₁ : ℵ₀ ≤ #↑s h₂ : #α < cof (ord #↑s) a : α ha : #↑(f ∘ Subtype.val ⁻¹' {a}) = #↑s ⊢ ∃ a, #↑s ≤ #↑(f ⁻¹' {a}) ** use a ** case h α✝ : Type u_1 r : α✝ → α✝ → Prop β α : Type u f : β → α s : Set β hθ : #↑s ≤ #β h₁ : ℵ₀ ≤ #↑s h₂ : #α < cof (ord #↑s) a : α ha : #↑(f ∘ Subtype.val ⁻¹' {a}) = #↑s ⊢ #↑s ≤ #↑(f ⁻¹' {a}) ** rw [← ha, @preimage_comp _ _ _ Subtype.val f] ** case h α✝ : Type u_1 r : α✝ → α✝ → Prop β α : Type u f : β → α s : Set β hθ : #↑s ≤ #β h₁ : ℵ₀ ≤ #↑s h₂ : #α < cof (ord #↑s) a : α ha : #↑(f ∘ Subtype.val ⁻¹' {a}) = #↑s ⊢ #↑(Subtype.val ⁻¹' (f ⁻¹' {a})) ≤ #↑(f ⁻¹' {a}) ** exact mk_preimage_of_injective _ _ Subtype.val_injective ** Qed | |
Ordinal.infinite_pigeonhole_set ** α✝ : Type u_1 r : α✝ → α✝ → Prop β α : Type u s : Set β f : ↑s → α θ : Cardinal.{u} hθ : θ ≤ #↑s h₁ : ℵ₀ ≤ θ h₂ : #α < cof (ord θ) ⊢ ∃ a t h, θ ≤ #↑t ∧ ∀ ⦃x : β⦄ (hx : x ∈ t), f { val := x, property := (_ : x ∈ s) } = a ** cases' infinite_pigeonhole_card f θ hθ h₁ h₂ with a ha ** case intro α✝ : Type u_1 r : α✝ → α✝ → Prop β α : Type u s : Set β f : ↑s → α θ : Cardinal.{u} hθ : θ ≤ #↑s h₁ : ℵ₀ ≤ θ h₂ : #α < cof (ord θ) a : α ha : θ ≤ #↑(f ⁻¹' {a}) ⊢ ∃ a t h, θ ≤ #↑t ∧ ∀ ⦃x : β⦄ (hx : x ∈ t), f { val := x, property := (_ : x ∈ s) } = a ** refine' ⟨a, { x | ∃ h, f ⟨x, h⟩ = a }, _, _, _⟩ ** case intro.refine'_3 α✝ : Type u_1 r : α✝ → α✝ → Prop β α : Type u s : Set β f : ↑s → α θ : Cardinal.{u} hθ : θ ≤ #↑s h₁ : ℵ₀ ≤ θ h₂ : #α < cof (ord θ) a : α ha : θ ≤ #↑(f ⁻¹' {a}) ⊢ ∀ ⦃x : β⦄ (hx : x ∈ {x | ∃ h, f { val := x, property := h } = a}), f { val := x, property := (_ : x ∈ s) } = a ** rintro x ⟨_, hx'⟩ ** case intro.refine'_3.intro α✝ : Type u_1 r : α✝ → α✝ → Prop β α : Type u s : Set β f : ↑s → α θ : Cardinal.{u} hθ : θ ≤ #↑s h₁ : ℵ₀ ≤ θ h₂ : #α < cof (ord θ) a : α ha : θ ≤ #↑(f ⁻¹' {a}) x : β w✝ : x ∈ s hx' : f { val := x, property := w✝ } = a ⊢ f { val := x, property := (_ : x ∈ s) } = a ** exact hx' ** case intro.refine'_1 α✝ : Type u_1 r : α✝ → α✝ → Prop β α : Type u s : Set β f : ↑s → α θ : Cardinal.{u} hθ : θ ≤ #↑s h₁ : ℵ₀ ≤ θ h₂ : #α < cof (ord θ) a : α ha : θ ≤ #↑(f ⁻¹' {a}) ⊢ {x | ∃ h, f { val := x, property := h } = a} ⊆ s ** rintro x ⟨hx, _⟩ ** case intro.refine'_1.intro α✝ : Type u_1 r : α✝ → α✝ → Prop β α : Type u s : Set β f : ↑s → α θ : Cardinal.{u} hθ : θ ≤ #↑s h₁ : ℵ₀ ≤ θ h₂ : #α < cof (ord θ) a : α ha : θ ≤ #↑(f ⁻¹' {a}) x : β hx : x ∈ s h✝ : f { val := x, property := hx } = a ⊢ x ∈ s ** exact hx ** case intro.refine'_2 α✝ : Type u_1 r : α✝ → α✝ → Prop β α : Type u s : Set β f : ↑s → α θ : Cardinal.{u} hθ : θ ≤ #↑s h₁ : ℵ₀ ≤ θ h₂ : #α < cof (ord θ) a : α ha : θ ≤ #↑(f ⁻¹' {a}) ⊢ θ ≤ #↑{x | ∃ h, f { val := x, property := h } = a} ** refine'
ha.trans
(ge_of_eq <|
Quotient.sound ⟨Equiv.trans _ (Equiv.subtypeSubtypeEquivSubtypeExists _ _).symm⟩) ** case intro.refine'_2 α✝ : Type u_1 r : α✝ → α✝ → Prop β α : Type u s : Set β f : ↑s → α θ : Cardinal.{u} hθ : θ ≤ #↑s h₁ : ℵ₀ ≤ θ h₂ : #α < cof (ord θ) a : α ha : θ ≤ #↑(f ⁻¹' {a}) ⊢ ↑{x | ∃ h, f { val := x, property := h } = a} ≃ { a_1 // ∃ h, { val := a_1, property := h } ∈ f ⁻¹' {a} } ** simp only [coe_eq_subtype, mem_singleton_iff, mem_preimage, mem_setOf_eq] ** case intro.refine'_2 α✝ : Type u_1 r : α✝ → α✝ → Prop β α : Type u s : Set β f : ↑s → α θ : Cardinal.{u} hθ : θ ≤ #↑s h₁ : ℵ₀ ≤ θ h₂ : #α < cof (ord θ) a : α ha : θ ≤ #↑(f ⁻¹' {a}) ⊢ { x // ∃ h, f { val := x, property := h } = a } ≃ { a_1 // ∃ h, f { val := a_1, property := (_ : a_1 ∈ s) } = a } ** rfl ** Qed | |
Cardinal.isStrongLimit_aleph0 ** α : Type u_1 r : α → α → Prop x : Cardinal.{u_2} hx : x < ℵ₀ ⊢ 2 ^ x < ℵ₀ ** rcases lt_aleph0.1 hx with ⟨n, rfl⟩ ** case intro α : Type u_1 r : α → α → Prop n : ℕ hx : ↑n < ℵ₀ ⊢ 2 ^ ↑n < ℵ₀ ** exact_mod_cast nat_lt_aleph0 (2 ^ n) ** Qed | |
Cardinal.isStrongLimit_beth ** α : Type u_1 r : α → α → Prop o : Ordinal.{u_2} H : IsSuccLimit o ⊢ IsStrongLimit (beth o) ** rcases eq_or_ne o 0 with (rfl | h) ** case inl α : Type u_1 r : α → α → Prop H : IsSuccLimit 0 ⊢ IsStrongLimit (beth 0) ** rw [beth_zero] ** case inl α : Type u_1 r : α → α → Prop H : IsSuccLimit 0 ⊢ IsStrongLimit ℵ₀ ** exact isStrongLimit_aleph0 ** case inr α : Type u_1 r : α → α → Prop o : Ordinal.{u_2} H : IsSuccLimit o h : o ≠ 0 ⊢ IsStrongLimit (beth o) ** refine' ⟨beth_ne_zero o, fun a ha => _⟩ ** case inr α : Type u_1 r : α → α → Prop o : Ordinal.{u_2} H : IsSuccLimit o h : o ≠ 0 a : Cardinal.{u_2} ha : a < beth o ⊢ 2 ^ a < beth o ** rw [beth_limit ⟨h, isSuccLimit_iff_succ_lt.1 H⟩] at ha ** case inr α : Type u_1 r : α → α → Prop o : Ordinal.{u_2} H : IsSuccLimit o h : o ≠ 0 a : Cardinal.{u_2} ha : a < ⨆ a, beth ↑a ⊢ 2 ^ a < beth o ** rcases exists_lt_of_lt_ciSup' ha with ⟨⟨i, hi⟩, ha⟩ ** case inr.intro.mk α : Type u_1 r : α → α → Prop o : Ordinal.{u_2} H : IsSuccLimit o h : o ≠ 0 a : Cardinal.{u_2} ha✝ : a < ⨆ a, beth ↑a i : Ordinal.{u_2} hi : i ∈ Iio o ha : a < beth ↑{ val := i, property := hi } ⊢ 2 ^ a < beth o ** have := power_le_power_left two_ne_zero ha.le ** case inr.intro.mk α : Type u_1 r : α → α → Prop o : Ordinal.{u_2} H : IsSuccLimit o h : o ≠ 0 a : Cardinal.{u_2} ha✝ : a < ⨆ a, beth ↑a i : Ordinal.{u_2} hi : i ∈ Iio o ha : a < beth ↑{ val := i, property := hi } this : 2 ^ a ≤ 2 ^ beth ↑{ val := i, property := hi } ⊢ 2 ^ a < beth o ** rw [← beth_succ] at this ** case inr.intro.mk α : Type u_1 r : α → α → Prop o : Ordinal.{u_2} H : IsSuccLimit o h : o ≠ 0 a : Cardinal.{u_2} ha✝ : a < ⨆ a, beth ↑a i : Ordinal.{u_2} hi : i ∈ Iio o ha : a < beth ↑{ val := i, property := hi } this : 2 ^ a ≤ beth (succ ↑{ val := i, property := hi }) ⊢ 2 ^ a < beth o ** exact this.trans_lt (beth_lt.2 (H.succ_lt hi)) ** Qed | |
Cardinal.mk_bounded_subset ** α✝ : Type u_1 r✝ : α✝ → α✝ → Prop α : Type u_2 h : ∀ (x : Cardinal.{u_2}), x < #α → 2 ^ x < #α r : α → α → Prop inst✝ : IsWellOrder α r hr : ord #α = type r ⊢ #{ s // Bounded r s } = #α ** rcases eq_or_ne #α 0 with (ha | ha) ** case inr α✝ : Type u_1 r✝ : α✝ → α✝ → Prop α : Type u_2 h : ∀ (x : Cardinal.{u_2}), x < #α → 2 ^ x < #α r : α → α → Prop inst✝ : IsWellOrder α r hr : ord #α = type r ha : #α ≠ 0 ⊢ #{ s // Bounded r s } = #α ** have h' : IsStrongLimit #α := ⟨ha, h⟩ ** case inr α✝ : Type u_1 r✝ : α✝ → α✝ → Prop α : Type u_2 h : ∀ (x : Cardinal.{u_2}), x < #α → 2 ^ x < #α r : α → α → Prop inst✝ : IsWellOrder α r hr : ord #α = type r ha : #α ≠ 0 h' : IsStrongLimit #α ⊢ #{ s // Bounded r s } = #α ** have ha := h'.isLimit.aleph0_le ** case inr α✝ : Type u_1 r✝ : α✝ → α✝ → Prop α : Type u_2 h : ∀ (x : Cardinal.{u_2}), x < #α → 2 ^ x < #α r : α → α → Prop inst✝ : IsWellOrder α r hr : ord #α = type r ha✝ : #α ≠ 0 h' : IsStrongLimit #α ha : ℵ₀ ≤ #α ⊢ #{ s // Bounded r s } = #α ** apply le_antisymm ** case inl α✝ : Type u_1 r✝ : α✝ → α✝ → Prop α : Type u_2 h : ∀ (x : Cardinal.{u_2}), x < #α → 2 ^ x < #α r : α → α → Prop inst✝ : IsWellOrder α r hr : ord #α = type r ha : #α = 0 ⊢ #{ s // Bounded r s } = #α ** rw [ha] ** case inl α✝ : Type u_1 r✝ : α✝ → α✝ → Prop α : Type u_2 h : ∀ (x : Cardinal.{u_2}), x < #α → 2 ^ x < #α r : α → α → Prop inst✝ : IsWellOrder α r hr : ord #α = type r ha : #α = 0 ⊢ #{ s // Bounded r s } = 0 ** haveI := mk_eq_zero_iff.1 ha ** case inl α✝ : Type u_1 r✝ : α✝ → α✝ → Prop α : Type u_2 h : ∀ (x : Cardinal.{u_2}), x < #α → 2 ^ x < #α r : α → α → Prop inst✝ : IsWellOrder α r hr : ord #α = type r ha : #α = 0 this : IsEmpty α ⊢ #{ s // Bounded r s } = 0 ** rw [mk_eq_zero_iff] ** case inl α✝ : Type u_1 r✝ : α✝ → α✝ → Prop α : Type u_2 h : ∀ (x : Cardinal.{u_2}), x < #α → 2 ^ x < #α r : α → α → Prop inst✝ : IsWellOrder α r hr : ord #α = type r ha : #α = 0 this : IsEmpty α ⊢ IsEmpty { s // Bounded r s } ** constructor ** case inl.false α✝ : Type u_1 r✝ : α✝ → α✝ → Prop α : Type u_2 h : ∀ (x : Cardinal.{u_2}), x < #α → 2 ^ x < #α r : α → α → Prop inst✝ : IsWellOrder α r hr : ord #α = type r ha : #α = 0 this : IsEmpty α ⊢ { s // Bounded r s } → False ** rintro ⟨s, hs⟩ ** case inl.false.mk α✝ : Type u_1 r✝ : α✝ → α✝ → Prop α : Type u_2 h : ∀ (x : Cardinal.{u_2}), x < #α → 2 ^ x < #α r : α → α → Prop inst✝ : IsWellOrder α r hr : ord #α = type r ha : #α = 0 this : IsEmpty α s : Set α hs : Bounded r s ⊢ False ** exact (not_unbounded_iff s).2 hs (unbounded_of_isEmpty s) ** case inr.a α✝ : Type u_1 r✝ : α✝ → α✝ → Prop α : Type u_2 h : ∀ (x : Cardinal.{u_2}), x < #α → 2 ^ x < #α r : α → α → Prop inst✝ : IsWellOrder α r hr : ord #α = type r ha✝ : #α ≠ 0 h' : IsStrongLimit #α ha : ℵ₀ ≤ #α ⊢ #{ s // Bounded r s } ≤ #α ** have : { s : Set α | Bounded r s } = ⋃ i, 𝒫{ j | r j i } := setOf_exists _ ** case inr.a α✝ : Type u_1 r✝ : α✝ → α✝ → Prop α : Type u_2 h : ∀ (x : Cardinal.{u_2}), x < #α → 2 ^ x < #α r : α → α → Prop inst✝ : IsWellOrder α r hr : ord #α = type r ha✝ : #α ≠ 0 h' : IsStrongLimit #α ha : ℵ₀ ≤ #α this : {s | Bounded r s} = ⋃ i, 𝒫{j | r j i} ⊢ #{ s // Bounded r s } ≤ #α ** rw [← coe_setOf, this] ** case inr.a α✝ : Type u_1 r✝ : α✝ → α✝ → Prop α : Type u_2 h : ∀ (x : Cardinal.{u_2}), x < #α → 2 ^ x < #α r : α → α → Prop inst✝ : IsWellOrder α r hr : ord #α = type r ha✝ : #α ≠ 0 h' : IsStrongLimit #α ha : ℵ₀ ≤ #α this : {s | Bounded r s} = ⋃ i, 𝒫{j | r j i} ⊢ #↑(⋃ i, 𝒫{j | r j i}) ≤ #α ** refine mk_iUnion_le_sum_mk.trans ((sum_le_iSup (fun i => #(𝒫{ j | r j i }))).trans
((mul_le_max_of_aleph0_le_left ha).trans ?_)) ** case inr.a α✝ : Type u_1 r✝ : α✝ → α✝ → Prop α : Type u_2 h : ∀ (x : Cardinal.{u_2}), x < #α → 2 ^ x < #α r : α → α → Prop inst✝ : IsWellOrder α r hr : ord #α = type r ha✝ : #α ≠ 0 h' : IsStrongLimit #α ha : ℵ₀ ≤ #α this : {s | Bounded r s} = ⋃ i, 𝒫{j | r j i} ⊢ max (#α) (⨆ i, #↑(𝒫{j | r j i})) ≤ #α ** rw [max_eq_left] ** case inr.a α✝ : Type u_1 r✝ : α✝ → α✝ → Prop α : Type u_2 h : ∀ (x : Cardinal.{u_2}), x < #α → 2 ^ x < #α r : α → α → Prop inst✝ : IsWellOrder α r hr : ord #α = type r ha✝ : #α ≠ 0 h' : IsStrongLimit #α ha : ℵ₀ ≤ #α this : {s | Bounded r s} = ⋃ i, 𝒫{j | r j i} ⊢ ⨆ i, #↑(𝒫{j | r j i}) ≤ #α ** apply ciSup_le' _ ** α✝ : Type u_1 r✝ : α✝ → α✝ → Prop α : Type u_2 h : ∀ (x : Cardinal.{u_2}), x < #α → 2 ^ x < #α r : α → α → Prop inst✝ : IsWellOrder α r hr : ord #α = type r ha✝ : #α ≠ 0 h' : IsStrongLimit #α ha : ℵ₀ ≤ #α this : {s | Bounded r s} = ⋃ i, 𝒫{j | r j i} ⊢ ∀ (i : α), #↑(𝒫{j | r j i}) ≤ #α ** intro i ** α✝ : Type u_1 r✝ : α✝ → α✝ → Prop α : Type u_2 h : ∀ (x : Cardinal.{u_2}), x < #α → 2 ^ x < #α r : α → α → Prop inst✝ : IsWellOrder α r hr : ord #α = type r ha✝ : #α ≠ 0 h' : IsStrongLimit #α ha : ℵ₀ ≤ #α this : {s | Bounded r s} = ⋃ i, 𝒫{j | r j i} i : α ⊢ #↑(𝒫{j | r j i}) ≤ #α ** rw [mk_powerset] ** α✝ : Type u_1 r✝ : α✝ → α✝ → Prop α : Type u_2 h : ∀ (x : Cardinal.{u_2}), x < #α → 2 ^ x < #α r : α → α → Prop inst✝ : IsWellOrder α r hr : ord #α = type r ha✝ : #α ≠ 0 h' : IsStrongLimit #α ha : ℵ₀ ≤ #α this : {s | Bounded r s} = ⋃ i, 𝒫{j | r j i} i : α ⊢ 2 ^ #↑{j | r j i} ≤ #α ** apply (h'.two_power_lt _).le ** α✝ : Type u_1 r✝ : α✝ → α✝ → Prop α : Type u_2 h : ∀ (x : Cardinal.{u_2}), x < #α → 2 ^ x < #α r : α → α → Prop inst✝ : IsWellOrder α r hr : ord #α = type r ha✝ : #α ≠ 0 h' : IsStrongLimit #α ha : ℵ₀ ≤ #α this : {s | Bounded r s} = ⋃ i, 𝒫{j | r j i} i : α ⊢ #↑{j | r j i} < #α ** rw [coe_setOf, card_typein, ← lt_ord, hr] ** α✝ : Type u_1 r✝ : α✝ → α✝ → Prop α : Type u_2 h : ∀ (x : Cardinal.{u_2}), x < #α → 2 ^ x < #α r : α → α → Prop inst✝ : IsWellOrder α r hr : ord #α = type r ha✝ : #α ≠ 0 h' : IsStrongLimit #α ha : ℵ₀ ≤ #α this : {s | Bounded r s} = ⋃ i, 𝒫{j | r j i} i : α ⊢ typein (fun x => r x) i < type r ** apply typein_lt_type ** case inr.a α✝ : Type u_1 r✝ : α✝ → α✝ → Prop α : Type u_2 h : ∀ (x : Cardinal.{u_2}), x < #α → 2 ^ x < #α r : α → α → Prop inst✝ : IsWellOrder α r hr : ord #α = type r ha✝ : #α ≠ 0 h' : IsStrongLimit #α ha : ℵ₀ ≤ #α ⊢ #α ≤ #{ s // Bounded r s } ** refine' @mk_le_of_injective α _ (fun x => Subtype.mk {x} _) _ ** case inr.a.refine'_1 α✝ : Type u_1 r✝ : α✝ → α✝ → Prop α : Type u_2 h : ∀ (x : Cardinal.{u_2}), x < #α → 2 ^ x < #α r : α → α → Prop inst✝ : IsWellOrder α r hr : ord #α = type r ha✝ : #α ≠ 0 h' : IsStrongLimit #α ha : ℵ₀ ≤ #α x : α ⊢ Bounded r {x} ** apply bounded_singleton ** case inr.a.refine'_1.hr α✝ : Type u_1 r✝ : α✝ → α✝ → Prop α : Type u_2 h : ∀ (x : Cardinal.{u_2}), x < #α → 2 ^ x < #α r : α → α → Prop inst✝ : IsWellOrder α r hr : ord #α = type r ha✝ : #α ≠ 0 h' : IsStrongLimit #α ha : ℵ₀ ≤ #α x : α ⊢ Ordinal.IsLimit (type r) ** rw [← hr] ** case inr.a.refine'_1.hr α✝ : Type u_1 r✝ : α✝ → α✝ → Prop α : Type u_2 h : ∀ (x : Cardinal.{u_2}), x < #α → 2 ^ x < #α r : α → α → Prop inst✝ : IsWellOrder α r hr : ord #α = type r ha✝ : #α ≠ 0 h' : IsStrongLimit #α ha : ℵ₀ ≤ #α x : α ⊢ Ordinal.IsLimit (ord #α) ** apply ord_isLimit ha ** case inr.a.refine'_2 α✝ : Type u_1 r✝ : α✝ → α✝ → Prop α : Type u_2 h : ∀ (x : Cardinal.{u_2}), x < #α → 2 ^ x < #α r : α → α → Prop inst✝ : IsWellOrder α r hr : ord #α = type r ha✝ : #α ≠ 0 h' : IsStrongLimit #α ha : ℵ₀ ≤ #α ⊢ Injective fun x => { val := {x}, property := (_ : Bounded r {x}) } ** intro a b hab ** case inr.a.refine'_2 α✝ : Type u_1 r✝ : α✝ → α✝ → Prop α : Type u_2 h : ∀ (x : Cardinal.{u_2}), x < #α → 2 ^ x < #α r : α → α → Prop inst✝ : IsWellOrder α r hr : ord #α = type r ha✝ : #α ≠ 0 h' : IsStrongLimit #α ha : ℵ₀ ≤ #α a b : α hab : (fun x => { val := {x}, property := (_ : Bounded r {x}) }) a = (fun x => { val := {x}, property := (_ : Bounded r {x}) }) b ⊢ a = b ** simpa [singleton_eq_singleton_iff] using hab ** Qed | |
Cardinal.mk_subset_mk_lt_cof ** α✝ : Type u_1 r : α✝ → α✝ → Prop α : Type u_2 h : ∀ (x : Cardinal.{u_2}), x < #α → 2 ^ x < #α ⊢ #{ s // #↑s < Ordinal.cof (ord #α) } = #α ** rcases eq_or_ne #α 0 with (ha | ha) ** case inr α✝ : Type u_1 r : α✝ → α✝ → Prop α : Type u_2 h : ∀ (x : Cardinal.{u_2}), x < #α → 2 ^ x < #α ha : #α ≠ 0 ⊢ #{ s // #↑s < Ordinal.cof (ord #α) } = #α ** have h' : IsStrongLimit #α := ⟨ha, h⟩ ** case inr α✝ : Type u_1 r : α✝ → α✝ → Prop α : Type u_2 h : ∀ (x : Cardinal.{u_2}), x < #α → 2 ^ x < #α ha : #α ≠ 0 h' : IsStrongLimit #α ⊢ #{ s // #↑s < Ordinal.cof (ord #α) } = #α ** rcases ord_eq α with ⟨r, wo, hr⟩ ** case inr.intro.intro α✝ : Type u_1 r✝ : α✝ → α✝ → Prop α : Type u_2 h : ∀ (x : Cardinal.{u_2}), x < #α → 2 ^ x < #α ha : #α ≠ 0 h' : IsStrongLimit #α r : α → α → Prop wo : IsWellOrder α r hr : ord #α = type r ⊢ #{ s // #↑s < Ordinal.cof (ord #α) } = #α ** haveI := wo ** case inr.intro.intro α✝ : Type u_1 r✝ : α✝ → α✝ → Prop α : Type u_2 h : ∀ (x : Cardinal.{u_2}), x < #α → 2 ^ x < #α ha : #α ≠ 0 h' : IsStrongLimit #α r : α → α → Prop wo : IsWellOrder α r hr : ord #α = type r this : IsWellOrder α r ⊢ #{ s // #↑s < Ordinal.cof (ord #α) } = #α ** apply le_antisymm ** case inl α✝ : Type u_1 r : α✝ → α✝ → Prop α : Type u_2 h : ∀ (x : Cardinal.{u_2}), x < #α → 2 ^ x < #α ha : #α = 0 ⊢ #{ s // #↑s < Ordinal.cof (ord #α) } = #α ** rw [ha] ** case inl α✝ : Type u_1 r : α✝ → α✝ → Prop α : Type u_2 h : ∀ (x : Cardinal.{u_2}), x < #α → 2 ^ x < #α ha : #α = 0 ⊢ #{ s // #↑s < Ordinal.cof (ord 0) } = 0 ** simp [fun s => (Cardinal.zero_le s).not_lt] ** case inr.intro.intro.a α✝ : Type u_1 r✝ : α✝ → α✝ → Prop α : Type u_2 h : ∀ (x : Cardinal.{u_2}), x < #α → 2 ^ x < #α ha : #α ≠ 0 h' : IsStrongLimit #α r : α → α → Prop wo : IsWellOrder α r hr : ord #α = type r this : IsWellOrder α r ⊢ #{ s // #↑s < Ordinal.cof (ord #α) } ≤ #α ** conv_rhs => rw [← mk_bounded_subset h hr] ** case inr.intro.intro.a α✝ : Type u_1 r✝ : α✝ → α✝ → Prop α : Type u_2 h : ∀ (x : Cardinal.{u_2}), x < #α → 2 ^ x < #α ha : #α ≠ 0 h' : IsStrongLimit #α r : α → α → Prop wo : IsWellOrder α r hr : ord #α = type r this : IsWellOrder α r ⊢ #{ s // #↑s < Ordinal.cof (ord #α) } ≤ #{ s // Bounded r s } ** apply mk_le_mk_of_subset ** case inr.intro.intro.a.h α✝ : Type u_1 r✝ : α✝ → α✝ → Prop α : Type u_2 h : ∀ (x : Cardinal.{u_2}), x < #α → 2 ^ x < #α ha : #α ≠ 0 h' : IsStrongLimit #α r : α → α → Prop wo : IsWellOrder α r hr : ord #α = type r this : IsWellOrder α r ⊢ (fun x => Quotient.liftOn₂ (#↑x) (Ordinal.cof (ord #α)) (fun α β => Nonempty (α ↪ β)) instLECardinal.proof_1 ∧ ¬Quotient.liftOn₂ (Ordinal.cof (ord #α)) (#↑x) (fun α β => Nonempty (α ↪ β)) instLECardinal.proof_1) ⊆ fun x => ∃ a, ∀ (b : α), b ∈ x → r b a ** intro s hs ** case inr.intro.intro.a.h α✝ : Type u_1 r✝ : α✝ → α✝ → Prop α : Type u_2 h : ∀ (x : Cardinal.{u_2}), x < #α → 2 ^ x < #α ha : #α ≠ 0 h' : IsStrongLimit #α r : α → α → Prop wo : IsWellOrder α r hr : ord #α = type r this : IsWellOrder α r s : Set α hs : s ∈ fun x => Quotient.liftOn₂ (#↑x) (Ordinal.cof (ord #α)) (fun α β => Nonempty (α ↪ β)) instLECardinal.proof_1 ∧ ¬Quotient.liftOn₂ (Ordinal.cof (ord #α)) (#↑x) (fun α β => Nonempty (α ↪ β)) instLECardinal.proof_1 ⊢ s ∈ fun x => ∃ a, ∀ (b : α), b ∈ x → r b a ** rw [hr] at hs ** case inr.intro.intro.a.h α✝ : Type u_1 r✝ : α✝ → α✝ → Prop α : Type u_2 h : ∀ (x : Cardinal.{u_2}), x < #α → 2 ^ x < #α ha : #α ≠ 0 h' : IsStrongLimit #α r : α → α → Prop wo : IsWellOrder α r hr : ord #α = type r this : IsWellOrder α r s : Set α hs : s ∈ fun x => Quotient.liftOn₂ (#↑x) (Ordinal.cof (type r)) (fun α β => Nonempty (α ↪ β)) instLECardinal.proof_1 ∧ ¬Quotient.liftOn₂ (Ordinal.cof (type r)) (#↑x) (fun α β => Nonempty (α ↪ β)) instLECardinal.proof_1 ⊢ s ∈ fun x => ∃ a, ∀ (b : α), b ∈ x → r b a ** exact lt_cof_type hs ** case inr.intro.intro.a α✝ : Type u_1 r✝ : α✝ → α✝ → Prop α : Type u_2 h : ∀ (x : Cardinal.{u_2}), x < #α → 2 ^ x < #α ha : #α ≠ 0 h' : IsStrongLimit #α r : α → α → Prop wo : IsWellOrder α r hr : ord #α = type r this : IsWellOrder α r ⊢ #α ≤ #{ s // #↑s < Ordinal.cof (ord #α) } ** refine' @mk_le_of_injective α _ (fun x => Subtype.mk {x} _) _ ** case inr.intro.intro.a.refine'_1 α✝ : Type u_1 r✝ : α✝ → α✝ → Prop α : Type u_2 h : ∀ (x : Cardinal.{u_2}), x < #α → 2 ^ x < #α ha : #α ≠ 0 h' : IsStrongLimit #α r : α → α → Prop wo : IsWellOrder α r hr : ord #α = type r this : IsWellOrder α r x : α ⊢ #↑{x} < Ordinal.cof (ord #α) ** rw [mk_singleton] ** case inr.intro.intro.a.refine'_1 α✝ : Type u_1 r✝ : α✝ → α✝ → Prop α : Type u_2 h : ∀ (x : Cardinal.{u_2}), x < #α → 2 ^ x < #α ha : #α ≠ 0 h' : IsStrongLimit #α r : α → α → Prop wo : IsWellOrder α r hr : ord #α = type r this : IsWellOrder α r x : α ⊢ 1 < Ordinal.cof (ord #α) ** exact one_lt_aleph0.trans_le (aleph0_le_cof.2 (ord_isLimit h'.isLimit.aleph0_le)) ** case inr.intro.intro.a.refine'_2 α✝ : Type u_1 r✝ : α✝ → α✝ → Prop α : Type u_2 h : ∀ (x : Cardinal.{u_2}), x < #α → 2 ^ x < #α ha : #α ≠ 0 h' : IsStrongLimit #α r : α → α → Prop wo : IsWellOrder α r hr : ord #α = type r this : IsWellOrder α r ⊢ Injective fun x => { val := {x}, property := (_ : #↑{x} < Ordinal.cof (ord #α)) } ** intro a b hab ** case inr.intro.intro.a.refine'_2 α✝ : Type u_1 r✝ : α✝ → α✝ → Prop α : Type u_2 h : ∀ (x : Cardinal.{u_2}), x < #α → 2 ^ x < #α ha : #α ≠ 0 h' : IsStrongLimit #α r : α → α → Prop wo : IsWellOrder α r hr : ord #α = type r this : IsWellOrder α r a b : α hab : (fun x => { val := {x}, property := (_ : #↑{x} < Ordinal.cof (ord #α)) }) a = (fun x => { val := {x}, property := (_ : #↑{x} < Ordinal.cof (ord #α)) }) b ⊢ a = b ** simpa [singleton_eq_singleton_iff] using hab ** Qed | |
Cardinal.IsRegular.ord_pos ** α : Type u_1 r : α → α → Prop c : Cardinal.{u_2} H : IsRegular c ⊢ 0 < ord c ** rw [Cardinal.lt_ord, card_zero] ** α : Type u_1 r : α → α → Prop c : Cardinal.{u_2} H : IsRegular c ⊢ 0 < c ** exact H.pos ** Qed | |
Cardinal.isRegular_aleph0 ** α : Type u_1 r : α → α → Prop ⊢ ℵ₀ ≤ Ordinal.cof (ord ℵ₀) ** simp ** Qed | |
Cardinal.isRegular_succ ** α : Type u_1 r : α → α → Prop c : Cardinal.{u} h : ℵ₀ ≤ c ⊢ c < Ordinal.cof (ord (succ c)) ** cases' Quotient.exists_rep (@succ Cardinal _ _ c) with α αe ** case intro α✝ : Type u_1 r : α✝ → α✝ → Prop c : Cardinal.{u} h : ℵ₀ ≤ c α : Type u αe : Quotient.mk isEquivalent α = succ c ⊢ c < Ordinal.cof (ord (succ c)) ** simp at αe ** case intro α✝ : Type u_1 r : α✝ → α✝ → Prop c : Cardinal.{u} h : ℵ₀ ≤ c α : Type u αe : #α = succ c ⊢ c < Ordinal.cof (ord (succ c)) ** rcases ord_eq α with ⟨r, wo, re⟩ ** case intro.intro.intro α✝ : Type u_1 r✝ : α✝ → α✝ → Prop c : Cardinal.{u} h : ℵ₀ ≤ c α : Type u αe : #α = succ c r : α → α → Prop wo : IsWellOrder α r re : ord #α = type r ⊢ c < Ordinal.cof (ord (succ c)) ** have := ord_isLimit (h.trans (le_succ _)) ** case intro.intro.intro α✝ : Type u_1 r✝ : α✝ → α✝ → Prop c : Cardinal.{u} h : ℵ₀ ≤ c α : Type u αe : #α = succ c r : α → α → Prop wo : IsWellOrder α r re : ord #α = type r this : Ordinal.IsLimit (ord (succ c)) ⊢ c < Ordinal.cof (ord (succ c)) ** rw [← αe, re] at this ⊢ ** case intro.intro.intro α✝ : Type u_1 r✝ : α✝ → α✝ → Prop c : Cardinal.{u} h : ℵ₀ ≤ c α : Type u αe : #α = succ c r : α → α → Prop wo : IsWellOrder α r re : ord #α = type r this : Ordinal.IsLimit (type r) ⊢ c < Ordinal.cof (type r) ** rcases cof_eq' r this with ⟨S, H, Se⟩ ** case intro.intro.intro.intro.intro α✝ : Type u_1 r✝ : α✝ → α✝ → Prop c : Cardinal.{u} h : ℵ₀ ≤ c α : Type u αe : #α = succ c r : α → α → Prop wo : IsWellOrder α r re : ord #α = type r this : Ordinal.IsLimit (type r) S : Set α H : ∀ (a : α), ∃ b, b ∈ S ∧ r a b Se : #↑S = Ordinal.cof (type r) ⊢ c < Ordinal.cof (type r) ** rw [← Se] ** case intro.intro.intro.intro.intro α✝ : Type u_1 r✝ : α✝ → α✝ → Prop c : Cardinal.{u} h : ℵ₀ ≤ c α : Type u αe : #α = succ c r : α → α → Prop wo : IsWellOrder α r re : ord #α = type r this : Ordinal.IsLimit (type r) S : Set α H : ∀ (a : α), ∃ b, b ∈ S ∧ r a b Se : #↑S = Ordinal.cof (type r) ⊢ c < #↑S ** apply lt_imp_lt_of_le_imp_le fun h => mul_le_mul_right' h c ** case intro.intro.intro.intro.intro α✝ : Type u_1 r✝ : α✝ → α✝ → Prop c : Cardinal.{u} h : ℵ₀ ≤ c α : Type u αe : #α = succ c r : α → α → Prop wo : IsWellOrder α r re : ord #α = type r this : Ordinal.IsLimit (type r) S : Set α H : ∀ (a : α), ∃ b, b ∈ S ∧ r a b Se : #↑S = Ordinal.cof (type r) ⊢ c * c < #↑S * c ** rw [mul_eq_self h, ← succ_le_iff, ← αe, ← sum_const'] ** case intro.intro.intro.intro.intro α✝ : Type u_1 r✝ : α✝ → α✝ → Prop c : Cardinal.{u} h : ℵ₀ ≤ c α : Type u αe : #α = succ c r : α → α → Prop wo : IsWellOrder α r re : ord #α = type r this : Ordinal.IsLimit (type r) S : Set α H : ∀ (a : α), ∃ b, b ∈ S ∧ r a b Se : #↑S = Ordinal.cof (type r) ⊢ #α ≤ sum fun x => c ** refine' le_trans _ (sum_le_sum (fun (x : S) => card (typein r (x : α))) _ fun i => _) ** case intro.intro.intro.intro.intro.refine'_1 α✝ : Type u_1 r✝ : α✝ → α✝ → Prop c : Cardinal.{u} h : ℵ₀ ≤ c α : Type u αe : #α = succ c r : α → α → Prop wo : IsWellOrder α r re : ord #α = type r this : Ordinal.IsLimit (type r) S : Set α H : ∀ (a : α), ∃ b, b ∈ S ∧ r a b Se : #↑S = Ordinal.cof (type r) ⊢ #α ≤ sum fun x => card (typein r ↑x) ** simp only [← card_typein, ← mk_sigma] ** case intro.intro.intro.intro.intro.refine'_1 α✝ : Type u_1 r✝ : α✝ → α✝ → Prop c : Cardinal.{u} h : ℵ₀ ≤ c α : Type u αe : #α = succ c r : α → α → Prop wo : IsWellOrder α r re : ord #α = type r this : Ordinal.IsLimit (type r) S : Set α H : ∀ (a : α), ∃ b, b ∈ S ∧ r a b Se : #↑S = Ordinal.cof (type r) ⊢ #α ≤ #((i : ↑S) × { y // r y ↑i }) ** exact
⟨Embedding.ofSurjective (fun x => x.2.1) fun a =>
let ⟨b, h, ab⟩ := H a
⟨⟨⟨_, h⟩, _, ab⟩, rfl⟩⟩ ** case intro.intro.intro.intro.intro.refine'_2 α✝ : Type u_1 r✝ : α✝ → α✝ → Prop c : Cardinal.{u} h : ℵ₀ ≤ c α : Type u αe : #α = succ c r : α → α → Prop wo : IsWellOrder α r re : ord #α = type r this : Ordinal.IsLimit (type r) S : Set α H : ∀ (a : α), ∃ b, b ∈ S ∧ r a b Se : #↑S = Ordinal.cof (type r) i : ↑S ⊢ (fun x => card (typein r ↑x)) i ≤ c ** rw [← lt_succ_iff, ← lt_ord, ← αe, re] ** case intro.intro.intro.intro.intro.refine'_2 α✝ : Type u_1 r✝ : α✝ → α✝ → Prop c : Cardinal.{u} h : ℵ₀ ≤ c α : Type u αe : #α = succ c r : α → α → Prop wo : IsWellOrder α r re : ord #α = type r this : Ordinal.IsLimit (type r) S : Set α H : ∀ (a : α), ∃ b, b ∈ S ∧ r a b Se : #↑S = Ordinal.cof (type r) i : ↑S ⊢ typein r ↑i < type r ** apply typein_lt_type ** Qed |
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