internlm/Lean-Github · Datasets at Hugging Face

url stringclasses 147 values	commit stringclasses 147 values	file_path stringlengths 7 101	full_name stringlengths 1 94	start stringlengths 6 10	end stringlengths 6 11	tactic stringlengths 1 11.2k	state_before stringlengths 3 2.09M	state_after stringlengths 6 2.09M
https://github.com/mguaypaq/lean-bruhat.git	1666a1bee2b520d5ba8a662310b4bd257fcf7ac2	Bruhat/Word.lean	stdinversions_inversions	[267, 1]	[278, 65]	simp only [Set.mem_union, Set.mem_preimage, Prod.swap_prod_mk]	case h.mk α : Type u_1 inst✝ : LinearOrder α g : Equiv.Perm α a b : α ⊢ (a, b) ∈ Inversions g ↔ (a, b) ∈ StdInversions g ∪ Prod.swap ⁻¹' StdInversions g	case h.mk α : Type u_1 inst✝ : LinearOrder α g : Equiv.Perm α a b : α ⊢ (a, b) ∈ Inversions g ↔ (a, b) ∈ StdInversions g ∨ (b, a) ∈ StdInversions g
https://github.com/mguaypaq/lean-bruhat.git	1666a1bee2b520d5ba8a662310b4bd257fcf7ac2	Bruhat/Word.lean	stdinversions_inversions	[267, 1]	[278, 65]	by_cases hab : a = b	case h.mk α : Type u_1 inst✝ : LinearOrder α g : Equiv.Perm α a b : α ⊢ (a, b) ∈ Inversions g ↔ (a, b) ∈ StdInversions g ∨ (b, a) ∈ StdInversions g	case pos α : Type u_1 inst✝ : LinearOrder α g : Equiv.Perm α a b : α hab : a = b ⊢ (a, b) ∈ Inversions g ↔ (a, b) ∈ StdInversions g ∨ (b, a) ∈ StdInversions g case neg α : Type u_1 inst✝ : LinearOrder α g : Equiv.Perm α a b : α hab : ¬a = b ⊢ (a, b) ∈ Inversions g ↔ (a, b) ∈ StdInversions g ∨ (b, a) ∈ StdInversions g
https://github.com/mguaypaq/lean-bruhat.git	1666a1bee2b520d5ba8a662310b4bd257fcf7ac2	Bruhat/Word.lean	stdinversions_inversions	[267, 1]	[278, 65]	. subst b simp [not_mem_stdinversions_diag, not_mem_inversions_diag]	case pos α : Type u_1 inst✝ : LinearOrder α g : Equiv.Perm α a b : α hab : a = b ⊢ (a, b) ∈ Inversions g ↔ (a, b) ∈ StdInversions g ∨ (b, a) ∈ StdInversions g case neg α : Type u_1 inst✝ : LinearOrder α g : Equiv.Perm α a b : α hab : ¬a = b ⊢ (a, b) ∈ Inversions g ↔ (a, b) ∈ StdInversions g ∨ (b, a) ∈ StdInversions g	case neg α : Type u_1 inst✝ : LinearOrder α g : Equiv.Perm α a b : α hab : ¬a = b ⊢ (a, b) ∈ Inversions g ↔ (a, b) ∈ StdInversions g ∨ (b, a) ∈ StdInversions g
https://github.com/mguaypaq/lean-bruhat.git	1666a1bee2b520d5ba8a662310b4bd257fcf7ac2	Bruhat/Word.lean	stdinversions_inversions	[267, 1]	[278, 65]	. push_neg at hab obtain (hab \| hba) := hab.lt_or_lt . simp [mem_stdinversions', mem_inversions, hab, hab.not_lt] . rw [mem_inversions_symm] simp [mem_stdinversions', mem_inversions, hba, hba.not_lt]	case neg α : Type u_1 inst✝ : LinearOrder α g : Equiv.Perm α a b : α hab : ¬a = b ⊢ (a, b) ∈ Inversions g ↔ (a, b) ∈ StdInversions g ∨ (b, a) ∈ StdInversions g	no goals
https://github.com/mguaypaq/lean-bruhat.git	1666a1bee2b520d5ba8a662310b4bd257fcf7ac2	Bruhat/Word.lean	stdinversions_inversions	[267, 1]	[278, 65]	subst b	case pos α : Type u_1 inst✝ : LinearOrder α g : Equiv.Perm α a b : α hab : a = b ⊢ (a, b) ∈ Inversions g ↔ (a, b) ∈ StdInversions g ∨ (b, a) ∈ StdInversions g	case pos α : Type u_1 inst✝ : LinearOrder α g : Equiv.Perm α a : α ⊢ (a, a) ∈ Inversions g ↔ (a, a) ∈ StdInversions g ∨ (a, a) ∈ StdInversions g
https://github.com/mguaypaq/lean-bruhat.git	1666a1bee2b520d5ba8a662310b4bd257fcf7ac2	Bruhat/Word.lean	stdinversions_inversions	[267, 1]	[278, 65]	simp [not_mem_stdinversions_diag, not_mem_inversions_diag]	case pos α : Type u_1 inst✝ : LinearOrder α g : Equiv.Perm α a : α ⊢ (a, a) ∈ Inversions g ↔ (a, a) ∈ StdInversions g ∨ (a, a) ∈ StdInversions g	no goals
https://github.com/mguaypaq/lean-bruhat.git	1666a1bee2b520d5ba8a662310b4bd257fcf7ac2	Bruhat/Word.lean	stdinversions_inversions	[267, 1]	[278, 65]	push_neg at hab	case neg α : Type u_1 inst✝ : LinearOrder α g : Equiv.Perm α a b : α hab : ¬a = b ⊢ (a, b) ∈ Inversions g ↔ (a, b) ∈ StdInversions g ∨ (b, a) ∈ StdInversions g	case neg α : Type u_1 inst✝ : LinearOrder α g : Equiv.Perm α a b : α hab : a ≠ b ⊢ (a, b) ∈ Inversions g ↔ (a, b) ∈ StdInversions g ∨ (b, a) ∈ StdInversions g
https://github.com/mguaypaq/lean-bruhat.git	1666a1bee2b520d5ba8a662310b4bd257fcf7ac2	Bruhat/Word.lean	stdinversions_inversions	[267, 1]	[278, 65]	obtain (hab \| hba) := hab.lt_or_lt	case neg α : Type u_1 inst✝ : LinearOrder α g : Equiv.Perm α a b : α hab : a ≠ b ⊢ (a, b) ∈ Inversions g ↔ (a, b) ∈ StdInversions g ∨ (b, a) ∈ StdInversions g	case neg.inl α : Type u_1 inst✝ : LinearOrder α g : Equiv.Perm α a b : α hab✝ : a ≠ b hab : a < b ⊢ (a, b) ∈ Inversions g ↔ (a, b) ∈ StdInversions g ∨ (b, a) ∈ StdInversions g case neg.inr α : Type u_1 inst✝ : LinearOrder α g : Equiv.Perm α a b : α hab : a ≠ b hba : b < a ⊢ (a, b) ∈ Inversions g ↔ (a, b) ∈ StdInversions g ∨ (b, a) ∈ StdInversions g
https://github.com/mguaypaq/lean-bruhat.git	1666a1bee2b520d5ba8a662310b4bd257fcf7ac2	Bruhat/Word.lean	stdinversions_inversions	[267, 1]	[278, 65]	. simp [mem_stdinversions', mem_inversions, hab, hab.not_lt]	case neg.inl α : Type u_1 inst✝ : LinearOrder α g : Equiv.Perm α a b : α hab✝ : a ≠ b hab : a < b ⊢ (a, b) ∈ Inversions g ↔ (a, b) ∈ StdInversions g ∨ (b, a) ∈ StdInversions g case neg.inr α : Type u_1 inst✝ : LinearOrder α g : Equiv.Perm α a b : α hab : a ≠ b hba : b < a ⊢ (a, b) ∈ Inversions g ↔ (a, b) ∈ StdInversions g ∨ (b, a) ∈ StdInversions g	case neg.inr α : Type u_1 inst✝ : LinearOrder α g : Equiv.Perm α a b : α hab : a ≠ b hba : b < a ⊢ (a, b) ∈ Inversions g ↔ (a, b) ∈ StdInversions g ∨ (b, a) ∈ StdInversions g
https://github.com/mguaypaq/lean-bruhat.git	1666a1bee2b520d5ba8a662310b4bd257fcf7ac2	Bruhat/Word.lean	stdinversions_inversions	[267, 1]	[278, 65]	. rw [mem_inversions_symm] simp [mem_stdinversions', mem_inversions, hba, hba.not_lt]	case neg.inr α : Type u_1 inst✝ : LinearOrder α g : Equiv.Perm α a b : α hab : a ≠ b hba : b < a ⊢ (a, b) ∈ Inversions g ↔ (a, b) ∈ StdInversions g ∨ (b, a) ∈ StdInversions g	no goals
https://github.com/mguaypaq/lean-bruhat.git	1666a1bee2b520d5ba8a662310b4bd257fcf7ac2	Bruhat/Word.lean	stdinversions_inversions	[267, 1]	[278, 65]	simp [mem_stdinversions', mem_inversions, hab, hab.not_lt]	case neg.inl α : Type u_1 inst✝ : LinearOrder α g : Equiv.Perm α a b : α hab✝ : a ≠ b hab : a < b ⊢ (a, b) ∈ Inversions g ↔ (a, b) ∈ StdInversions g ∨ (b, a) ∈ StdInversions g	no goals
https://github.com/mguaypaq/lean-bruhat.git	1666a1bee2b520d5ba8a662310b4bd257fcf7ac2	Bruhat/Word.lean	stdinversions_inversions	[267, 1]	[278, 65]	rw [mem_inversions_symm]	case neg.inr α : Type u_1 inst✝ : LinearOrder α g : Equiv.Perm α a b : α hab : a ≠ b hba : b < a ⊢ (a, b) ∈ Inversions g ↔ (a, b) ∈ StdInversions g ∨ (b, a) ∈ StdInversions g	case neg.inr α : Type u_1 inst✝ : LinearOrder α g : Equiv.Perm α a b : α hab : a ≠ b hba : b < a ⊢ (b, a) ∈ Inversions g ↔ (a, b) ∈ StdInversions g ∨ (b, a) ∈ StdInversions g
https://github.com/mguaypaq/lean-bruhat.git	1666a1bee2b520d5ba8a662310b4bd257fcf7ac2	Bruhat/Word.lean	stdinversions_inversions	[267, 1]	[278, 65]	simp [mem_stdinversions', mem_inversions, hba, hba.not_lt]	case neg.inr α : Type u_1 inst✝ : LinearOrder α g : Equiv.Perm α a b : α hab : a ≠ b hba : b < a ⊢ (b, a) ∈ Inversions g ↔ (a, b) ∈ StdInversions g ∨ (b, a) ∈ StdInversions g	no goals
https://github.com/mguaypaq/lean-bruhat.git	1666a1bee2b520d5ba8a662310b4bd257fcf7ac2	Bruhat/Goal.lean	perm_conj	[11, 1]	[13, 37]	rw [Equiv.symm_apply_apply, hab]	α : Type u_1 f g : Equiv.Perm α a b : α hab : f a = b ⊢ g (f (g.symm (g a))) = g b	no goals
https://github.com/mguaypaq/lean-bruhat.git	1666a1bee2b520d5ba8a662310b4bd257fcf7ac2	Bruhat/Goal.lean	swap_conj	[15, 1]	[19, 6]	rw [Equiv.swap_apply_apply]	α : Type u_1 inst✝ : DecidableEq α a b c d : α ⊢ Equiv.swap a b * Equiv.swap c d * Equiv.swap a b = Equiv.swap ((Equiv.swap a b) c) ((Equiv.swap a b) d)	α : Type u_1 inst✝ : DecidableEq α a b c d : α ⊢ Equiv.swap a b * Equiv.swap c d * Equiv.swap a b = Equiv.swap a b * Equiv.swap c d * (Equiv.swap a b)⁻¹
https://github.com/mguaypaq/lean-bruhat.git	1666a1bee2b520d5ba8a662310b4bd257fcf7ac2	Bruhat/Goal.lean	swap_conj	[15, 1]	[19, 6]	rfl	α : Type u_1 inst✝ : DecidableEq α a b c d : α ⊢ Equiv.swap a b * Equiv.swap c d * Equiv.swap a b = Equiv.swap a b * Equiv.swap c d * (Equiv.swap a b)⁻¹	no goals
https://github.com/mguaypaq/lean-bruhat.git	1666a1bee2b520d5ba8a662310b4bd257fcf7ac2	Bruhat/Goal.lean	WordForSwap_toPerm_succ	[42, 1]	[47, 6]	simp [WordForSwap, Word.toPerm]	i k : ℕ ⊢ Word.toPerm (WordForSwap i (k + 1)) = Equiv.swap (i + (k + 1)) (i + (k + 2)) * Word.toPerm (WordForSwap i k) * Equiv.swap (i + (k + 1)) (i + (k + 2))	i k : ℕ ⊢ Equiv.swap (i + k + 1) (i + k + 1 + 1) * (List.prod (List.map (fun i => Equiv.swap i (i + 1)) (WordForSwap i k)) * Equiv.swap (i + k + 1) (i + k + 1 + 1)) = Equiv.swap (i + (k + 1)) (i + (k + 2)) * List.prod (List.map (fun i => Equiv.swap i (i + 1)) (WordForSwap i k)) * Equiv.swap (i + (k + 1)) (i + (k + 2))
https://github.com/mguaypaq/lean-bruhat.git	1666a1bee2b520d5ba8a662310b4bd257fcf7ac2	Bruhat/Goal.lean	WordForSwap_toPerm_succ	[42, 1]	[47, 6]	rfl	i k : ℕ ⊢ Equiv.swap (i + k + 1) (i + k + 1 + 1) * (List.prod (List.map (fun i => Equiv.swap i (i + 1)) (WordForSwap i k)) * Equiv.swap (i + k + 1) (i + k + 1 + 1)) = Equiv.swap (i + (k + 1)) (i + (k + 2)) * List.prod (List.map (fun i => Equiv.swap i (i + 1)) (WordForSwap i k)) * Equiv.swap (i + (k + 1)) (i + (k + 2))	no goals
https://github.com/mguaypaq/lean-bruhat.git	1666a1bee2b520d5ba8a662310b4bd257fcf7ac2	Bruhat/Goal.lean	wordForSwap_eq_swap	[49, 1]	[57, 58]	induction' k with k ih	i k : ℕ ⊢ Word.toPerm (WordForSwap i k) = Equiv.swap i (i + (k + 1))	case zero i : ℕ ⊢ Word.toPerm (WordForSwap i Nat.zero) = Equiv.swap i (i + (Nat.zero + 1)) case succ i k : ℕ ih : Word.toPerm (WordForSwap i k) = Equiv.swap i (i + (k + 1)) ⊢ Word.toPerm (WordForSwap i (Nat.succ k)) = Equiv.swap i (i + (Nat.succ k + 1))
https://github.com/mguaypaq/lean-bruhat.git	1666a1bee2b520d5ba8a662310b4bd257fcf7ac2	Bruhat/Goal.lean	wordForSwap_eq_swap	[49, 1]	[57, 58]	. simp only [Nat.zero_eq, add_zero, Equiv.swap_self] rfl	case zero i : ℕ ⊢ Word.toPerm (WordForSwap i Nat.zero) = Equiv.swap i (i + (Nat.zero + 1)) case succ i k : ℕ ih : Word.toPerm (WordForSwap i k) = Equiv.swap i (i + (k + 1)) ⊢ Word.toPerm (WordForSwap i (Nat.succ k)) = Equiv.swap i (i + (Nat.succ k + 1))	case succ i k : ℕ ih : Word.toPerm (WordForSwap i k) = Equiv.swap i (i + (k + 1)) ⊢ Word.toPerm (WordForSwap i (Nat.succ k)) = Equiv.swap i (i + (Nat.succ k + 1))
https://github.com/mguaypaq/lean-bruhat.git	1666a1bee2b520d5ba8a662310b4bd257fcf7ac2	Bruhat/Goal.lean	wordForSwap_eq_swap	[49, 1]	[57, 58]	. rw [WordForSwap_toPerm_succ, ih] change _ * _ * (Equiv.swap _ _)⁻¹ = _ rw [← Equiv.swap_apply_apply, Equiv.swap_apply_left, Equiv.swap_apply_of_ne_of_ne (by simp) (by simp)]	case succ i k : ℕ ih : Word.toPerm (WordForSwap i k) = Equiv.swap i (i + (k + 1)) ⊢ Word.toPerm (WordForSwap i (Nat.succ k)) = Equiv.swap i (i + (Nat.succ k + 1))	no goals
https://github.com/mguaypaq/lean-bruhat.git	1666a1bee2b520d5ba8a662310b4bd257fcf7ac2	Bruhat/Goal.lean	wordForSwap_eq_swap	[49, 1]	[57, 58]	simp only [Nat.zero_eq, add_zero, Equiv.swap_self]	case zero i : ℕ ⊢ Word.toPerm (WordForSwap i Nat.zero) = Equiv.swap i (i + (Nat.zero + 1))	case zero i : ℕ ⊢ Word.toPerm (WordForSwap i 0) = Equiv.swap i (i + (0 + 1))
https://github.com/mguaypaq/lean-bruhat.git	1666a1bee2b520d5ba8a662310b4bd257fcf7ac2	Bruhat/Goal.lean	wordForSwap_eq_swap	[49, 1]	[57, 58]	rfl	case zero i : ℕ ⊢ Word.toPerm (WordForSwap i 0) = Equiv.swap i (i + (0 + 1))	no goals
https://github.com/mguaypaq/lean-bruhat.git	1666a1bee2b520d5ba8a662310b4bd257fcf7ac2	Bruhat/Goal.lean	wordForSwap_eq_swap	[49, 1]	[57, 58]	rw [WordForSwap_toPerm_succ, ih]	case succ i k : ℕ ih : Word.toPerm (WordForSwap i k) = Equiv.swap i (i + (k + 1)) ⊢ Word.toPerm (WordForSwap i (Nat.succ k)) = Equiv.swap i (i + (Nat.succ k + 1))	case succ i k : ℕ ih : Word.toPerm (WordForSwap i k) = Equiv.swap i (i + (k + 1)) ⊢ Equiv.swap (i + (k + 1)) (i + (k + 2)) * Equiv.swap i (i + (k + 1)) * Equiv.swap (i + (k + 1)) (i + (k + 2)) = Equiv.swap i (i + (Nat.succ k + 1))
https://github.com/mguaypaq/lean-bruhat.git	1666a1bee2b520d5ba8a662310b4bd257fcf7ac2	Bruhat/Goal.lean	wordForSwap_eq_swap	[49, 1]	[57, 58]	change _ * _ * (Equiv.swap _ _)⁻¹ = _	case succ i k : ℕ ih : Word.toPerm (WordForSwap i k) = Equiv.swap i (i + (k + 1)) ⊢ Equiv.swap (i + (k + 1)) (i + (k + 2)) * Equiv.swap i (i + (k + 1)) * Equiv.swap (i + (k + 1)) (i + (k + 2)) = Equiv.swap i (i + (Nat.succ k + 1))	case succ i k : ℕ ih : Word.toPerm (WordForSwap i k) = Equiv.swap i (i + (k + 1)) ⊢ Equiv.swap (i + (k + 1)) (i + (k + 2)) * Equiv.swap i (i + (k + 1)) * (Equiv.swap (i + (k + 1)) (i + (k + 2)))⁻¹ = Equiv.swap i (i + (Nat.succ k + 1))
https://github.com/mguaypaq/lean-bruhat.git	1666a1bee2b520d5ba8a662310b4bd257fcf7ac2	Bruhat/Goal.lean	wordForSwap_eq_swap	[49, 1]	[57, 58]	rw [← Equiv.swap_apply_apply, Equiv.swap_apply_left, Equiv.swap_apply_of_ne_of_ne (by simp) (by simp)]	case succ i k : ℕ ih : Word.toPerm (WordForSwap i k) = Equiv.swap i (i + (k + 1)) ⊢ Equiv.swap (i + (k + 1)) (i + (k + 2)) * Equiv.swap i (i + (k + 1)) * (Equiv.swap (i + (k + 1)) (i + (k + 2)))⁻¹ = Equiv.swap i (i + (Nat.succ k + 1))	no goals
https://github.com/mguaypaq/lean-bruhat.git	1666a1bee2b520d5ba8a662310b4bd257fcf7ac2	Bruhat/Goal.lean	wordForSwap_eq_swap	[49, 1]	[57, 58]	simp	i k : ℕ ih : Word.toPerm (WordForSwap i k) = Equiv.swap i (i + (k + 1)) ⊢ i ≠ i + (k + 1)	no goals
https://github.com/mguaypaq/lean-bruhat.git	1666a1bee2b520d5ba8a662310b4bd257fcf7ac2	Bruhat/Goal.lean	wordForSwap_eq_swap	[49, 1]	[57, 58]	simp	i k : ℕ ih : Word.toPerm (WordForSwap i k) = Equiv.swap i (i + (k + 1)) ⊢ i ≠ i + (k + 2)	no goals
https://github.com/mguaypaq/lean-bruhat.git	1666a1bee2b520d5ba8a662310b4bd257fcf7ac2	Bruhat/Goal.lean	unswap_support	[73, 1]	[75, 22]	simp [support] at *	α : Type u_1 inst✝ : DecidableEq α f : Equiv.Perm α a : α ⊢ a ∉ support (Equiv.swap a (f a) * f)	no goals
https://github.com/mguaypaq/lean-bruhat.git	1666a1bee2b520d5ba8a662310b4bd257fcf7ac2	Bruhat/Goal.lean	inversions_mul	[155, 1]	[157, 8]	sorry	f g : Equiv.Perm ℕ ⊢ Inversions (f * g) = Prod.map ⇑g.symm ⇑g.symm '' Inversions f ∩ Inversions g	no goals
https://github.com/mguaypaq/lean-bruhat.git	1666a1bee2b520d5ba8a662310b4bd257fcf7ac2	Bruhat/Lehmer.lean	truncate_apply_of_le	[18, 1]	[19, 20]	simpa	f : ℕ → ℕ n x : ℕ hx : n ≤ x ⊢ ¬x < n	no goals
https://github.com/mguaypaq/lean-bruhat.git	1666a1bee2b520d5ba8a662310b4bd257fcf7ac2	Bruhat/Lehmer.lean	card_lehmer_eq_factorial	[73, 1]	[78, 47]	rw [Fintype.card_of_bijective (le_truncate_equiv_prod_fin _ _).bijective]	n : ℕ ⊢ Fintype.card { g // ⇑g ≤ ⇑(truncate id n) } = n !	n : ℕ ⊢ Fintype.card ((i : Fin n) → Fin (id ↑i + 1)) = n !
https://github.com/mguaypaq/lean-bruhat.git	1666a1bee2b520d5ba8a662310b4bd257fcf7ac2	Bruhat/Lehmer.lean	card_lehmer_eq_factorial	[73, 1]	[78, 47]	simp only [id_eq, Fintype.card_pi, Fintype.card_fin]	n : ℕ ⊢ Fintype.card ((i : Fin n) → Fin (id ↑i + 1)) = n !	n : ℕ ⊢ (Finset.prod Finset.univ fun x => ↑x + 1) = n !
https://github.com/mguaypaq/lean-bruhat.git	1666a1bee2b520d5ba8a662310b4bd257fcf7ac2	Bruhat/Lehmer.lean	card_lehmer_eq_factorial	[73, 1]	[78, 47]	rw [← Finset.prod_range_add_one_eq_factorial n]	n : ℕ ⊢ (Finset.prod Finset.univ fun x => ↑x + 1) = n !	n : ℕ ⊢ (Finset.prod Finset.univ fun x => ↑x + 1) = Finset.prod (Finset.range n) fun x => x + 1
https://github.com/mguaypaq/lean-bruhat.git	1666a1bee2b520d5ba8a662310b4bd257fcf7ac2	Bruhat/Lehmer.lean	card_lehmer_eq_factorial	[73, 1]	[78, 47]	exact Fin.prod_univ_eq_prod_range Nat.succ n	n : ℕ ⊢ (Finset.prod Finset.univ fun x => ↑x + 1) = Finset.prod (Finset.range n) fun x => x + 1	no goals
https://github.com/yuma-mizuno/lean-math-workshop.git	4a69b0130b276b45212e2b12b90032b146b56d67	Tutorial/Advanced/Analysis/Lecture2.lean	Tutorial.IsLocalMax.hasDerivAt_eq_zero	[45, 1]	[69, 10]	apply le_antisymm ?right ?left	f : ℝ → ℝ f' x a b : ℝ h : IsLocalMax f a hf : HasDerivAt f f' a ⊢ f' = 0	case right f : ℝ → ℝ f' x a b : ℝ h : IsLocalMax f a hf : HasDerivAt f f' a ⊢ f' ≤ 0 case left f : ℝ → ℝ f' x a b : ℝ h : IsLocalMax f a hf : HasDerivAt f f' a ⊢ 0 ≤ f'
https://github.com/yuma-mizuno/lean-math-workshop.git	4a69b0130b276b45212e2b12b90032b146b56d67	Tutorial/Advanced/Analysis/Lecture2.lean	Tutorial.IsLocalMax.hasDerivAt_eq_zero	[45, 1]	[69, 10]	case left => sorry	f : ℝ → ℝ f' x a b : ℝ h : IsLocalMax f a hf : HasDerivAt f f' a ⊢ 0 ≤ f'	no goals
https://github.com/yuma-mizuno/lean-math-workshop.git	4a69b0130b276b45212e2b12b90032b146b56d67	Tutorial/Advanced/Analysis/Lecture2.lean	Tutorial.IsLocalMax.hasDerivAt_eq_zero	[45, 1]	[69, 10]	have hf : Tendsto (fun x ↦ (f x - f a) / (x - a)) (𝓝[>] a) (𝓝 f') := by rw [hasDerivAt_iff_tendsto_slope] at hf apply hf.mono_left (nhds_right'_le_nhds_ne a)	f : ℝ → ℝ f' x a b : ℝ h : IsLocalMax f a hf : HasDerivAt f f' a ⊢ f' ≤ 0	f : ℝ → ℝ f' x a b : ℝ h : IsLocalMax f a hf✝ : HasDerivAt f f' a hf : Tendsto (fun x => (f x - f a) / (x - a)) (𝓝[>] a) (𝓝 f') ⊢ f' ≤ 0
https://github.com/yuma-mizuno/lean-math-workshop.git	4a69b0130b276b45212e2b12b90032b146b56d67	Tutorial/Advanced/Analysis/Lecture2.lean	Tutorial.IsLocalMax.hasDerivAt_eq_zero	[45, 1]	[69, 10]	suffices ∀ᶠ x in 𝓝[>] a, (f x - f a) / (x - a) ≤ 0 from le_of_tendsto hf this	f : ℝ → ℝ f' x a b : ℝ h : IsLocalMax f a hf✝ : HasDerivAt f f' a hf : Tendsto (fun x => (f x - f a) / (x - a)) (𝓝[>] a) (𝓝 f') ⊢ f' ≤ 0	f : ℝ → ℝ f' x a b : ℝ h : IsLocalMax f a hf✝ : HasDerivAt f f' a hf : Tendsto (fun x => (f x - f a) / (x - a)) (𝓝[>] a) (𝓝 f') ⊢ ∀ᶠ (x : ℝ) in 𝓝[>] a, (f x - f a) / (x - a) ≤ 0
https://github.com/yuma-mizuno/lean-math-workshop.git	4a69b0130b276b45212e2b12b90032b146b56d67	Tutorial/Advanced/Analysis/Lecture2.lean	Tutorial.IsLocalMax.hasDerivAt_eq_zero	[45, 1]	[69, 10]	have ha : ∀ᶠ x in 𝓝[>] a, a < x := eventually_nhdsWithin_of_forall fun x hx ↦ hx	f : ℝ → ℝ f' x a b : ℝ h : IsLocalMax f a hf✝ : HasDerivAt f f' a hf : Tendsto (fun x => (f x - f a) / (x - a)) (𝓝[>] a) (𝓝 f') ⊢ ∀ᶠ (x : ℝ) in 𝓝[>] a, (f x - f a) / (x - a) ≤ 0	f : ℝ → ℝ f' x a b : ℝ h : IsLocalMax f a hf✝ : HasDerivAt f f' a hf : Tendsto (fun x => (f x - f a) / (x - a)) (𝓝[>] a) (𝓝 f') ha : ∀ᶠ (x : ℝ) in 𝓝[>] a, a < x ⊢ ∀ᶠ (x : ℝ) in 𝓝[>] a, (f x - f a) / (x - a) ≤ 0
https://github.com/yuma-mizuno/lean-math-workshop.git	4a69b0130b276b45212e2b12b90032b146b56d67	Tutorial/Advanced/Analysis/Lecture2.lean	Tutorial.IsLocalMax.hasDerivAt_eq_zero	[45, 1]	[69, 10]	have h : ∀ᶠ x in 𝓝[>] a, f x ≤ f a := h.filter_mono nhdsWithin_le_nhds	f : ℝ → ℝ f' x a b : ℝ h : IsLocalMax f a hf✝ : HasDerivAt f f' a hf : Tendsto (fun x => (f x - f a) / (x - a)) (𝓝[>] a) (𝓝 f') ha : ∀ᶠ (x : ℝ) in 𝓝[>] a, a < x ⊢ ∀ᶠ (x : ℝ) in 𝓝[>] a, (f x - f a) / (x - a) ≤ 0	f : ℝ → ℝ f' x a b : ℝ h✝ : IsLocalMax f a hf✝ : HasDerivAt f f' a hf : Tendsto (fun x => (f x - f a) / (x - a)) (𝓝[>] a) (𝓝 f') ha : ∀ᶠ (x : ℝ) in 𝓝[>] a, a < x h : ∀ᶠ (x : ℝ) in 𝓝[>] a, f x ≤ f a ⊢ ∀ᶠ (x : ℝ) in 𝓝[>] a, (f x - f a) / (x - a) ≤ 0
https://github.com/yuma-mizuno/lean-math-workshop.git	4a69b0130b276b45212e2b12b90032b146b56d67	Tutorial/Advanced/Analysis/Lecture2.lean	Tutorial.IsLocalMax.hasDerivAt_eq_zero	[45, 1]	[69, 10]	filter_upwards [ha, h]	f : ℝ → ℝ f' x a b : ℝ h✝ : IsLocalMax f a hf✝ : HasDerivAt f f' a hf : Tendsto (fun x => (f x - f a) / (x - a)) (𝓝[>] a) (𝓝 f') ha : ∀ᶠ (x : ℝ) in 𝓝[>] a, a < x h : ∀ᶠ (x : ℝ) in 𝓝[>] a, f x ≤ f a ⊢ ∀ᶠ (x : ℝ) in 𝓝[>] a, (f x - f a) / (x - a) ≤ 0	case h f : ℝ → ℝ f' x a b : ℝ h✝ : IsLocalMax f a hf✝ : HasDerivAt f f' a hf : Tendsto (fun x => (f x - f a) / (x - a)) (𝓝[>] a) (𝓝 f') ha : ∀ᶠ (x : ℝ) in 𝓝[>] a, a < x h : ∀ᶠ (x : ℝ) in 𝓝[>] a, f x ≤ f a ⊢ ∀ (a_1 : ℝ), a < a_1 → f a_1 ≤ f a → (f a_1 - f a) / (a_1 - a) ≤ 0
https://github.com/yuma-mizuno/lean-math-workshop.git	4a69b0130b276b45212e2b12b90032b146b56d67	Tutorial/Advanced/Analysis/Lecture2.lean	Tutorial.IsLocalMax.hasDerivAt_eq_zero	[45, 1]	[69, 10]	intro x ha h	case h f : ℝ → ℝ f' x a b : ℝ h✝ : IsLocalMax f a hf✝ : HasDerivAt f f' a hf : Tendsto (fun x => (f x - f a) / (x - a)) (𝓝[>] a) (𝓝 f') ha : ∀ᶠ (x : ℝ) in 𝓝[>] a, a < x h : ∀ᶠ (x : ℝ) in 𝓝[>] a, f x ≤ f a ⊢ ∀ (a_1 : ℝ), a < a_1 → f a_1 ≤ f a → (f a_1 - f a) / (a_1 - a) ≤ 0	case h f : ℝ → ℝ f' x✝ a b : ℝ h✝¹ : IsLocalMax f a hf✝ : HasDerivAt f f' a hf : Tendsto (fun x => (f x - f a) / (x - a)) (𝓝[>] a) (𝓝 f') ha✝ : ∀ᶠ (x : ℝ) in 𝓝[>] a, a < x h✝ : ∀ᶠ (x : ℝ) in 𝓝[>] a, f x ≤ f a x : ℝ ha : a < x h : f x ≤ f a ⊢ (f x - f a) / (x - a) ≤ 0
https://github.com/yuma-mizuno/lean-math-workshop.git	4a69b0130b276b45212e2b12b90032b146b56d67	Tutorial/Advanced/Analysis/Lecture2.lean	Tutorial.IsLocalMax.hasDerivAt_eq_zero	[45, 1]	[69, 10]	apply div_nonpos_of_nonpos_of_nonneg	case h f : ℝ → ℝ f' x✝ a b : ℝ h✝¹ : IsLocalMax f a hf✝ : HasDerivAt f f' a hf : Tendsto (fun x => (f x - f a) / (x - a)) (𝓝[>] a) (𝓝 f') ha✝ : ∀ᶠ (x : ℝ) in 𝓝[>] a, a < x h✝ : ∀ᶠ (x : ℝ) in 𝓝[>] a, f x ≤ f a x : ℝ ha : a < x h : f x ≤ f a ⊢ (f x - f a) / (x - a) ≤ 0	case h.ha f : ℝ → ℝ f' x✝ a b : ℝ h✝¹ : IsLocalMax f a hf✝ : HasDerivAt f f' a hf : Tendsto (fun x => (f x - f a) / (x - a)) (𝓝[>] a) (𝓝 f') ha✝ : ∀ᶠ (x : ℝ) in 𝓝[>] a, a < x h✝ : ∀ᶠ (x : ℝ) in 𝓝[>] a, f x ≤ f a x : ℝ ha : a < x h : f x ≤ f a ⊢ f x - f a ≤ 0 case h.hb f : ℝ → ℝ f' x✝ a b : ℝ h✝¹ : IsLocalMax f a hf✝ : HasDerivAt f f' a hf : Tendsto (fun x => (f x - f a) / (x - a)) (𝓝[>] a) (𝓝 f') ha✝ : ∀ᶠ (x : ℝ) in 𝓝[>] a, a < x h✝ : ∀ᶠ (x : ℝ) in 𝓝[>] a, f x ≤ f a x : ℝ ha : a < x h : f x ≤ f a ⊢ 0 ≤ x - a
https://github.com/yuma-mizuno/lean-math-workshop.git	4a69b0130b276b45212e2b12b90032b146b56d67	Tutorial/Advanced/Analysis/Lecture2.lean	Tutorial.IsLocalMax.hasDerivAt_eq_zero	[45, 1]	[69, 10]	rw [hasDerivAt_iff_tendsto_slope] at hf	f : ℝ → ℝ f' x a b : ℝ h : IsLocalMax f a hf : HasDerivAt f f' a ⊢ Tendsto (fun x => (f x - f a) / (x - a)) (𝓝[>] a) (𝓝 f')	f : ℝ → ℝ f' x a b : ℝ h : IsLocalMax f a hf : Tendsto (fun x => (f x - f a) / (x - a)) (𝓝[≠] a) (𝓝 f') ⊢ Tendsto (fun x => (f x - f a) / (x - a)) (𝓝[>] a) (𝓝 f')
https://github.com/yuma-mizuno/lean-math-workshop.git	4a69b0130b276b45212e2b12b90032b146b56d67	Tutorial/Advanced/Analysis/Lecture2.lean	Tutorial.IsLocalMax.hasDerivAt_eq_zero	[45, 1]	[69, 10]	apply hf.mono_left (nhds_right'_le_nhds_ne a)	f : ℝ → ℝ f' x a b : ℝ h : IsLocalMax f a hf : Tendsto (fun x => (f x - f a) / (x - a)) (𝓝[≠] a) (𝓝 f') ⊢ Tendsto (fun x => (f x - f a) / (x - a)) (𝓝[>] a) (𝓝 f')	no goals
https://github.com/yuma-mizuno/lean-math-workshop.git	4a69b0130b276b45212e2b12b90032b146b56d67	Tutorial/Advanced/Analysis/Lecture2.lean	Tutorial.IsLocalMax.hasDerivAt_eq_zero	[45, 1]	[69, 10]	linarith only [h]	case h.ha f : ℝ → ℝ f' x✝ a b : ℝ h✝¹ : IsLocalMax f a hf✝ : HasDerivAt f f' a hf : Tendsto (fun x => (f x - f a) / (x - a)) (𝓝[>] a) (𝓝 f') ha✝ : ∀ᶠ (x : ℝ) in 𝓝[>] a, a < x h✝ : ∀ᶠ (x : ℝ) in 𝓝[>] a, f x ≤ f a x : ℝ ha : a < x h : f x ≤ f a ⊢ f x - f a ≤ 0	no goals
https://github.com/yuma-mizuno/lean-math-workshop.git	4a69b0130b276b45212e2b12b90032b146b56d67	Tutorial/Advanced/Analysis/Lecture2.lean	Tutorial.IsLocalMax.hasDerivAt_eq_zero	[45, 1]	[69, 10]	linarith only [ha]	case h.hb f : ℝ → ℝ f' x✝ a b : ℝ h✝¹ : IsLocalMax f a hf✝ : HasDerivAt f f' a hf : Tendsto (fun x => (f x - f a) / (x - a)) (𝓝[>] a) (𝓝 f') ha✝ : ∀ᶠ (x : ℝ) in 𝓝[>] a, a < x h✝ : ∀ᶠ (x : ℝ) in 𝓝[>] a, f x ≤ f a x : ℝ ha : a < x h : f x ≤ f a ⊢ 0 ≤ x - a	no goals
https://github.com/yuma-mizuno/lean-math-workshop.git	4a69b0130b276b45212e2b12b90032b146b56d67	Tutorial/Advanced/Analysis/Lecture2.lean	Tutorial.IsLocalMax.hasDerivAt_eq_zero	[45, 1]	[69, 10]	sorry	f : ℝ → ℝ f' x a b : ℝ h : IsLocalMax f a hf : HasDerivAt f f' a ⊢ 0 ≤ f'	no goals
https://github.com/yuma-mizuno/lean-math-workshop.git	4a69b0130b276b45212e2b12b90032b146b56d67	Tutorial/Advanced/Analysis/Lecture2.lean	Tutorial.IsLocalMin.hasDerivAt_eq_zero	[72, 1]	[74, 8]	sorry	f : ℝ → ℝ f' x a b : ℝ h : IsLocalMin f a hf : HasDerivAt f f' a ⊢ f' = 0	no goals
https://github.com/yuma-mizuno/lean-math-workshop.git	4a69b0130b276b45212e2b12b90032b146b56d67	Tutorial/Advanced/Analysis/Lecture2.lean	Tutorial.IsLocalExtr.hasDerivAt_eq_zero	[80, 1]	[81, 8]	sorry	f : ℝ → ℝ f' x a b : ℝ h : IsLocalExtr f a hf : HasDerivAt f f' a ⊢ f' = 0	no goals
https://github.com/yuma-mizuno/lean-math-workshop.git	4a69b0130b276b45212e2b12b90032b146b56d67	Tutorial/Advanced/Analysis/Lecture2.lean	Tutorial.exists_local_extr_Ioo	[93, 1]	[115, 55]	suffices ∃ c ∈ Ioo a b, IsExtrOn f (Icc a b) c by rcases this with ⟨c, cmem, hc⟩ exists c, cmem apply hc.isLocalExtr <\| Icc_mem_nhds cmem.1 cmem.2	f : ℝ → ℝ f' x a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hfI : f a = f b ⊢ ∃ c ∈ Ioo a b, IsLocalExtr f c	f : ℝ → ℝ f' x a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hfI : f a = f b ⊢ ∃ c ∈ Ioo a b, IsExtrOn f (Icc a b) c
https://github.com/yuma-mizuno/lean-math-workshop.git	4a69b0130b276b45212e2b12b90032b146b56d67	Tutorial/Advanced/Analysis/Lecture2.lean	Tutorial.exists_local_extr_Ioo	[93, 1]	[115, 55]	have ne : (Icc a b).Nonempty := nonempty_Icc.2 (le_of_lt hab)	f : ℝ → ℝ f' x a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hfI : f a = f b ⊢ ∃ c ∈ Ioo a b, IsExtrOn f (Icc a b) c	f : ℝ → ℝ f' x a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hfI : f a = f b ne : Set.Nonempty (Icc a b) ⊢ ∃ c ∈ Ioo a b, IsExtrOn f (Icc a b) c
https://github.com/yuma-mizuno/lean-math-workshop.git	4a69b0130b276b45212e2b12b90032b146b56d67	Tutorial/Advanced/Analysis/Lecture2.lean	Tutorial.exists_local_extr_Ioo	[93, 1]	[115, 55]	have ⟨C, Cmem, Cge⟩ : ∃ C ∈ Icc a b, IsMaxOn f (Icc a b) C := by sorry	f : ℝ → ℝ f' x a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hfI : f a = f b ne : Set.Nonempty (Icc a b) ⊢ ∃ c ∈ Ioo a b, IsExtrOn f (Icc a b) c	f : ℝ → ℝ f' x a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hfI : f a = f b ne : Set.Nonempty (Icc a b) C : ℝ Cmem : C ∈ Icc a b Cge : IsMaxOn f (Icc a b) C ⊢ ∃ c ∈ Ioo a b, IsExtrOn f (Icc a b) c
https://github.com/yuma-mizuno/lean-math-workshop.git	4a69b0130b276b45212e2b12b90032b146b56d67	Tutorial/Advanced/Analysis/Lecture2.lean	Tutorial.exists_local_extr_Ioo	[93, 1]	[115, 55]	have ⟨c, cmem, cle⟩ : ∃ c ∈ Icc a b, IsMinOn f (Icc a b) c := by sorry	f : ℝ → ℝ f' x a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hfI : f a = f b ne : Set.Nonempty (Icc a b) C : ℝ Cmem : C ∈ Icc a b Cge : IsMaxOn f (Icc a b) C ⊢ ∃ c ∈ Ioo a b, IsExtrOn f (Icc a b) c	f : ℝ → ℝ f' x a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hfI : f a = f b ne : Set.Nonempty (Icc a b) C : ℝ Cmem : C ∈ Icc a b Cge : IsMaxOn f (Icc a b) C c : ℝ cmem : c ∈ Icc a b cle : IsMinOn f (Icc a b) c ⊢ ∃ c ∈ Ioo a b, IsExtrOn f (Icc a b) c
https://github.com/yuma-mizuno/lean-math-workshop.git	4a69b0130b276b45212e2b12b90032b146b56d67	Tutorial/Advanced/Analysis/Lecture2.lean	Tutorial.exists_local_extr_Ioo	[93, 1]	[115, 55]	change ∀ x ∈ Icc a b, f x ≤ f C at Cge	f : ℝ → ℝ f' x a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hfI : f a = f b ne : Set.Nonempty (Icc a b) C : ℝ Cmem : C ∈ Icc a b Cge : IsMaxOn f (Icc a b) C c : ℝ cmem : c ∈ Icc a b cle : IsMinOn f (Icc a b) c ⊢ ∃ c ∈ Ioo a b, IsExtrOn f (Icc a b) c	f : ℝ → ℝ f' x a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hfI : f a = f b ne : Set.Nonempty (Icc a b) C : ℝ Cmem : C ∈ Icc a b c : ℝ cmem : c ∈ Icc a b cle : IsMinOn f (Icc a b) c Cge : ∀ x ∈ Icc a b, f x ≤ f C ⊢ ∃ c ∈ Ioo a b, IsExtrOn f (Icc a b) c
https://github.com/yuma-mizuno/lean-math-workshop.git	4a69b0130b276b45212e2b12b90032b146b56d67	Tutorial/Advanced/Analysis/Lecture2.lean	Tutorial.exists_local_extr_Ioo	[93, 1]	[115, 55]	change ∀ x ∈ Icc a b, f c ≤ f x at cle	f : ℝ → ℝ f' x a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hfI : f a = f b ne : Set.Nonempty (Icc a b) C : ℝ Cmem : C ∈ Icc a b c : ℝ cmem : c ∈ Icc a b cle : IsMinOn f (Icc a b) c Cge : ∀ x ∈ Icc a b, f x ≤ f C ⊢ ∃ c ∈ Ioo a b, IsExtrOn f (Icc a b) c	f : ℝ → ℝ f' x a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hfI : f a = f b ne : Set.Nonempty (Icc a b) C : ℝ Cmem : C ∈ Icc a b c : ℝ cmem : c ∈ Icc a b Cge : ∀ x ∈ Icc a b, f x ≤ f C cle : ∀ x ∈ Icc a b, f c ≤ f x ⊢ ∃ c ∈ Ioo a b, IsExtrOn f (Icc a b) c
https://github.com/yuma-mizuno/lean-math-workshop.git	4a69b0130b276b45212e2b12b90032b146b56d67	Tutorial/Advanced/Analysis/Lecture2.lean	Tutorial.exists_local_extr_Ioo	[93, 1]	[115, 55]	by_cases hc : f c = f a	f : ℝ → ℝ f' x a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hfI : f a = f b ne : Set.Nonempty (Icc a b) C : ℝ Cmem : C ∈ Icc a b c : ℝ cmem : c ∈ Icc a b Cge : ∀ x ∈ Icc a b, f x ≤ f C cle : ∀ x ∈ Icc a b, f c ≤ f x ⊢ ∃ c ∈ Ioo a b, IsExtrOn f (Icc a b) c	case pos f : ℝ → ℝ f' x a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hfI : f a = f b ne : Set.Nonempty (Icc a b) C : ℝ Cmem : C ∈ Icc a b c : ℝ cmem : c ∈ Icc a b Cge : ∀ x ∈ Icc a b, f x ≤ f C cle : ∀ x ∈ Icc a b, f c ≤ f x hc : f c = f a ⊢ ∃ c ∈ Ioo a b, IsExtrOn f (Icc a b) c case neg f : ℝ → ℝ f' x a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hfI : f a = f b ne : Set.Nonempty (Icc a b) C : ℝ Cmem : C ∈ Icc a b c : ℝ cmem : c ∈ Icc a b Cge : ∀ x ∈ Icc a b, f x ≤ f C cle : ∀ x ∈ Icc a b, f c ≤ f x hc : ¬f c = f a ⊢ ∃ c ∈ Ioo a b, IsExtrOn f (Icc a b) c
https://github.com/yuma-mizuno/lean-math-workshop.git	4a69b0130b276b45212e2b12b90032b146b56d67	Tutorial/Advanced/Analysis/Lecture2.lean	Tutorial.exists_local_extr_Ioo	[93, 1]	[115, 55]	rcases this with ⟨c, cmem, hc⟩	f : ℝ → ℝ f' x a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hfI : f a = f b this : ∃ c ∈ Ioo a b, IsExtrOn f (Icc a b) c ⊢ ∃ c ∈ Ioo a b, IsLocalExtr f c	case intro.intro f : ℝ → ℝ f' x a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hfI : f a = f b c : ℝ cmem : c ∈ Ioo a b hc : IsExtrOn f (Icc a b) c ⊢ ∃ c ∈ Ioo a b, IsLocalExtr f c
https://github.com/yuma-mizuno/lean-math-workshop.git	4a69b0130b276b45212e2b12b90032b146b56d67	Tutorial/Advanced/Analysis/Lecture2.lean	Tutorial.exists_local_extr_Ioo	[93, 1]	[115, 55]	exists c, cmem	case intro.intro f : ℝ → ℝ f' x a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hfI : f a = f b c : ℝ cmem : c ∈ Ioo a b hc : IsExtrOn f (Icc a b) c ⊢ ∃ c ∈ Ioo a b, IsLocalExtr f c	case intro.intro f : ℝ → ℝ f' x a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hfI : f a = f b c : ℝ cmem : c ∈ Ioo a b hc : IsExtrOn f (Icc a b) c ⊢ IsLocalExtr f c
https://github.com/yuma-mizuno/lean-math-workshop.git	4a69b0130b276b45212e2b12b90032b146b56d67	Tutorial/Advanced/Analysis/Lecture2.lean	Tutorial.exists_local_extr_Ioo	[93, 1]	[115, 55]	apply hc.isLocalExtr <\| Icc_mem_nhds cmem.1 cmem.2	case intro.intro f : ℝ → ℝ f' x a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hfI : f a = f b c : ℝ cmem : c ∈ Ioo a b hc : IsExtrOn f (Icc a b) c ⊢ IsLocalExtr f c	no goals
https://github.com/yuma-mizuno/lean-math-workshop.git	4a69b0130b276b45212e2b12b90032b146b56d67	Tutorial/Advanced/Analysis/Lecture2.lean	Tutorial.exists_local_extr_Ioo	[93, 1]	[115, 55]	sorry	f : ℝ → ℝ f' x a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hfI : f a = f b ne : Set.Nonempty (Icc a b) ⊢ ∃ C ∈ Icc a b, IsMaxOn f (Icc a b) C	no goals
https://github.com/yuma-mizuno/lean-math-workshop.git	4a69b0130b276b45212e2b12b90032b146b56d67	Tutorial/Advanced/Analysis/Lecture2.lean	Tutorial.exists_local_extr_Ioo	[93, 1]	[115, 55]	sorry	f : ℝ → ℝ f' x a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hfI : f a = f b ne : Set.Nonempty (Icc a b) C : ℝ Cmem : C ∈ Icc a b Cge : IsMaxOn f (Icc a b) C ⊢ ∃ c ∈ Icc a b, IsMinOn f (Icc a b) c	no goals
https://github.com/yuma-mizuno/lean-math-workshop.git	4a69b0130b276b45212e2b12b90032b146b56d67	Tutorial/Advanced/Analysis/Lecture2.lean	Tutorial.exists_local_extr_Ioo	[93, 1]	[115, 55]	by_cases hC : f C = f a	case pos f : ℝ → ℝ f' x a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hfI : f a = f b ne : Set.Nonempty (Icc a b) C : ℝ Cmem : C ∈ Icc a b c : ℝ cmem : c ∈ Icc a b Cge : ∀ x ∈ Icc a b, f x ≤ f C cle : ∀ x ∈ Icc a b, f c ≤ f x hc : f c = f a ⊢ ∃ c ∈ Ioo a b, IsExtrOn f (Icc a b) c	case pos f : ℝ → ℝ f' x a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hfI : f a = f b ne : Set.Nonempty (Icc a b) C : ℝ Cmem : C ∈ Icc a b c : ℝ cmem : c ∈ Icc a b Cge : ∀ x ∈ Icc a b, f x ≤ f C cle : ∀ x ∈ Icc a b, f c ≤ f x hc : f c = f a hC : f C = f a ⊢ ∃ c ∈ Ioo a b, IsExtrOn f (Icc a b) c case neg f : ℝ → ℝ f' x a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hfI : f a = f b ne : Set.Nonempty (Icc a b) C : ℝ Cmem : C ∈ Icc a b c : ℝ cmem : c ∈ Icc a b Cge : ∀ x ∈ Icc a b, f x ≤ f C cle : ∀ x ∈ Icc a b, f c ≤ f x hc : f c = f a hC : ¬f C = f a ⊢ ∃ c ∈ Ioo a b, IsExtrOn f (Icc a b) c
https://github.com/yuma-mizuno/lean-math-workshop.git	4a69b0130b276b45212e2b12b90032b146b56d67	Tutorial/Advanced/Analysis/Lecture2.lean	Tutorial.exists_local_extr_Ioo	[93, 1]	[115, 55]	have : ∀ x ∈ Icc a b, f x = f a := fun x hx ↦ le_antisymm (hC ▸ Cge x hx) (hc ▸ cle x hx)	case pos f : ℝ → ℝ f' x a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hfI : f a = f b ne : Set.Nonempty (Icc a b) C : ℝ Cmem : C ∈ Icc a b c : ℝ cmem : c ∈ Icc a b Cge : ∀ x ∈ Icc a b, f x ≤ f C cle : ∀ x ∈ Icc a b, f c ≤ f x hc : f c = f a hC : f C = f a ⊢ ∃ c ∈ Ioo a b, IsExtrOn f (Icc a b) c	case pos f : ℝ → ℝ f' x a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hfI : f a = f b ne : Set.Nonempty (Icc a b) C : ℝ Cmem : C ∈ Icc a b c : ℝ cmem : c ∈ Icc a b Cge : ∀ x ∈ Icc a b, f x ≤ f C cle : ∀ x ∈ Icc a b, f c ≤ f x hc : f c = f a hC : f C = f a this : ∀ x ∈ Icc a b, f x = f a ⊢ ∃ c ∈ Ioo a b, IsExtrOn f (Icc a b) c
https://github.com/yuma-mizuno/lean-math-workshop.git	4a69b0130b276b45212e2b12b90032b146b56d67	Tutorial/Advanced/Analysis/Lecture2.lean	Tutorial.exists_local_extr_Ioo	[93, 1]	[115, 55]	rcases nonempty_Ioo.2 hab with ⟨c', hc'⟩	case pos f : ℝ → ℝ f' x a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hfI : f a = f b ne : Set.Nonempty (Icc a b) C : ℝ Cmem : C ∈ Icc a b c : ℝ cmem : c ∈ Icc a b Cge : ∀ x ∈ Icc a b, f x ≤ f C cle : ∀ x ∈ Icc a b, f c ≤ f x hc : f c = f a hC : f C = f a this : ∀ x ∈ Icc a b, f x = f a ⊢ ∃ c ∈ Ioo a b, IsExtrOn f (Icc a b) c	case pos.intro f : ℝ → ℝ f' x a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hfI : f a = f b ne : Set.Nonempty (Icc a b) C : ℝ Cmem : C ∈ Icc a b c : ℝ cmem : c ∈ Icc a b Cge : ∀ x ∈ Icc a b, f x ≤ f C cle : ∀ x ∈ Icc a b, f c ≤ f x hc : f c = f a hC : f C = f a this : ∀ x ∈ Icc a b, f x = f a c' : ℝ hc' : c' ∈ Ioo a b ⊢ ∃ c ∈ Ioo a b, IsExtrOn f (Icc a b) c
https://github.com/yuma-mizuno/lean-math-workshop.git	4a69b0130b276b45212e2b12b90032b146b56d67	Tutorial/Advanced/Analysis/Lecture2.lean	Tutorial.exists_local_extr_Ioo	[93, 1]	[115, 55]	refine ⟨c', hc', Or.inl fun x hx ↦ ?_⟩	case pos.intro f : ℝ → ℝ f' x a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hfI : f a = f b ne : Set.Nonempty (Icc a b) C : ℝ Cmem : C ∈ Icc a b c : ℝ cmem : c ∈ Icc a b Cge : ∀ x ∈ Icc a b, f x ≤ f C cle : ∀ x ∈ Icc a b, f c ≤ f x hc : f c = f a hC : f C = f a this : ∀ x ∈ Icc a b, f x = f a c' : ℝ hc' : c' ∈ Ioo a b ⊢ ∃ c ∈ Ioo a b, IsExtrOn f (Icc a b) c	case pos.intro f : ℝ → ℝ f' x✝ a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hfI : f a = f b ne : Set.Nonempty (Icc a b) C : ℝ Cmem : C ∈ Icc a b c : ℝ cmem : c ∈ Icc a b Cge : ∀ x ∈ Icc a b, f x ≤ f C cle : ∀ x ∈ Icc a b, f c ≤ f x hc : f c = f a hC : f C = f a this : ∀ x ∈ Icc a b, f x = f a c' : ℝ hc' : c' ∈ Ioo a b x : ℝ hx : x ∈ Icc a b ⊢ x ∈ {x \| (fun x => f c' ≤ f x) x}
https://github.com/yuma-mizuno/lean-math-workshop.git	4a69b0130b276b45212e2b12b90032b146b56d67	Tutorial/Advanced/Analysis/Lecture2.lean	Tutorial.exists_local_extr_Ioo	[93, 1]	[115, 55]	simp [this x hx, this c' (Ioo_subset_Icc_self hc')]	case pos.intro f : ℝ → ℝ f' x✝ a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hfI : f a = f b ne : Set.Nonempty (Icc a b) C : ℝ Cmem : C ∈ Icc a b c : ℝ cmem : c ∈ Icc a b Cge : ∀ x ∈ Icc a b, f x ≤ f C cle : ∀ x ∈ Icc a b, f c ≤ f x hc : f c = f a hC : f C = f a this : ∀ x ∈ Icc a b, f x = f a c' : ℝ hc' : c' ∈ Ioo a b x : ℝ hx : x ∈ Icc a b ⊢ x ∈ {x \| (fun x => f c' ≤ f x) x}	no goals
https://github.com/yuma-mizuno/lean-math-workshop.git	4a69b0130b276b45212e2b12b90032b146b56d67	Tutorial/Advanced/Analysis/Lecture2.lean	Tutorial.exists_local_extr_Ioo	[93, 1]	[115, 55]	refine ⟨C, ⟨lt_of_le_of_ne Cmem.1 <\| mt ?_ hC, lt_of_le_of_ne Cmem.2 <\| mt ?_ hC⟩, Or.inr Cge⟩	case neg f : ℝ → ℝ f' x a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hfI : f a = f b ne : Set.Nonempty (Icc a b) C : ℝ Cmem : C ∈ Icc a b c : ℝ cmem : c ∈ Icc a b Cge : ∀ x ∈ Icc a b, f x ≤ f C cle : ∀ x ∈ Icc a b, f c ≤ f x hc : f c = f a hC : ¬f C = f a ⊢ ∃ c ∈ Ioo a b, IsExtrOn f (Icc a b) c	case neg.refine_1 f : ℝ → ℝ f' x a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hfI : f a = f b ne : Set.Nonempty (Icc a b) C : ℝ Cmem : C ∈ Icc a b c : ℝ cmem : c ∈ Icc a b Cge : ∀ x ∈ Icc a b, f x ≤ f C cle : ∀ x ∈ Icc a b, f c ≤ f x hc : f c = f a hC : ¬f C = f a ⊢ a = C → f C = f a case neg.refine_2 f : ℝ → ℝ f' x a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hfI : f a = f b ne : Set.Nonempty (Icc a b) C : ℝ Cmem : C ∈ Icc a b c : ℝ cmem : c ∈ Icc a b Cge : ∀ x ∈ Icc a b, f x ≤ f C cle : ∀ x ∈ Icc a b, f c ≤ f x hc : f c = f a hC : ¬f C = f a ⊢ C = b → f C = f a
https://github.com/yuma-mizuno/lean-math-workshop.git	4a69b0130b276b45212e2b12b90032b146b56d67	Tutorial/Advanced/Analysis/Lecture2.lean	Tutorial.exists_local_extr_Ioo	[93, 1]	[115, 55]	exacts [fun h ↦ by rw [h], fun h ↦ by rw [h, hfI]]	case neg.refine_1 f : ℝ → ℝ f' x a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hfI : f a = f b ne : Set.Nonempty (Icc a b) C : ℝ Cmem : C ∈ Icc a b c : ℝ cmem : c ∈ Icc a b Cge : ∀ x ∈ Icc a b, f x ≤ f C cle : ∀ x ∈ Icc a b, f c ≤ f x hc : f c = f a hC : ¬f C = f a ⊢ a = C → f C = f a case neg.refine_2 f : ℝ → ℝ f' x a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hfI : f a = f b ne : Set.Nonempty (Icc a b) C : ℝ Cmem : C ∈ Icc a b c : ℝ cmem : c ∈ Icc a b Cge : ∀ x ∈ Icc a b, f x ≤ f C cle : ∀ x ∈ Icc a b, f c ≤ f x hc : f c = f a hC : ¬f C = f a ⊢ C = b → f C = f a	no goals
https://github.com/yuma-mizuno/lean-math-workshop.git	4a69b0130b276b45212e2b12b90032b146b56d67	Tutorial/Advanced/Analysis/Lecture2.lean	Tutorial.exists_local_extr_Ioo	[93, 1]	[115, 55]	rw [h]	f : ℝ → ℝ f' x a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hfI : f a = f b ne : Set.Nonempty (Icc a b) C : ℝ Cmem : C ∈ Icc a b c : ℝ cmem : c ∈ Icc a b Cge : ∀ x ∈ Icc a b, f x ≤ f C cle : ∀ x ∈ Icc a b, f c ≤ f x hc : f c = f a hC : ¬f C = f a h : a = C ⊢ f C = f a	no goals
https://github.com/yuma-mizuno/lean-math-workshop.git	4a69b0130b276b45212e2b12b90032b146b56d67	Tutorial/Advanced/Analysis/Lecture2.lean	Tutorial.exists_local_extr_Ioo	[93, 1]	[115, 55]	rw [h, hfI]	f : ℝ → ℝ f' x a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hfI : f a = f b ne : Set.Nonempty (Icc a b) C : ℝ Cmem : C ∈ Icc a b c : ℝ cmem : c ∈ Icc a b Cge : ∀ x ∈ Icc a b, f x ≤ f C cle : ∀ x ∈ Icc a b, f c ≤ f x hc : f c = f a hC : ¬f C = f a h : C = b ⊢ f C = f a	no goals
https://github.com/yuma-mizuno/lean-math-workshop.git	4a69b0130b276b45212e2b12b90032b146b56d67	Tutorial/Advanced/Analysis/Lecture2.lean	Tutorial.exists_local_extr_Ioo	[93, 1]	[115, 55]	refine ⟨c, ⟨lt_of_le_of_ne cmem.1 <\| mt ?_ hc, lt_of_le_of_ne cmem.2 <\| mt ?_ hc⟩, Or.inl cle⟩	case neg f : ℝ → ℝ f' x a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hfI : f a = f b ne : Set.Nonempty (Icc a b) C : ℝ Cmem : C ∈ Icc a b c : ℝ cmem : c ∈ Icc a b Cge : ∀ x ∈ Icc a b, f x ≤ f C cle : ∀ x ∈ Icc a b, f c ≤ f x hc : ¬f c = f a ⊢ ∃ c ∈ Ioo a b, IsExtrOn f (Icc a b) c	case neg.refine_1 f : ℝ → ℝ f' x a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hfI : f a = f b ne : Set.Nonempty (Icc a b) C : ℝ Cmem : C ∈ Icc a b c : ℝ cmem : c ∈ Icc a b Cge : ∀ x ∈ Icc a b, f x ≤ f C cle : ∀ x ∈ Icc a b, f c ≤ f x hc : ¬f c = f a ⊢ a = c → f c = f a case neg.refine_2 f : ℝ → ℝ f' x a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hfI : f a = f b ne : Set.Nonempty (Icc a b) C : ℝ Cmem : C ∈ Icc a b c : ℝ cmem : c ∈ Icc a b Cge : ∀ x ∈ Icc a b, f x ≤ f C cle : ∀ x ∈ Icc a b, f c ≤ f x hc : ¬f c = f a ⊢ c = b → f c = f a
https://github.com/yuma-mizuno/lean-math-workshop.git	4a69b0130b276b45212e2b12b90032b146b56d67	Tutorial/Advanced/Analysis/Lecture2.lean	Tutorial.exists_local_extr_Ioo	[93, 1]	[115, 55]	exacts [fun h ↦ by rw [h], fun h ↦ by rw [h, hfI]]	case neg.refine_1 f : ℝ → ℝ f' x a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hfI : f a = f b ne : Set.Nonempty (Icc a b) C : ℝ Cmem : C ∈ Icc a b c : ℝ cmem : c ∈ Icc a b Cge : ∀ x ∈ Icc a b, f x ≤ f C cle : ∀ x ∈ Icc a b, f c ≤ f x hc : ¬f c = f a ⊢ a = c → f c = f a case neg.refine_2 f : ℝ → ℝ f' x a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hfI : f a = f b ne : Set.Nonempty (Icc a b) C : ℝ Cmem : C ∈ Icc a b c : ℝ cmem : c ∈ Icc a b Cge : ∀ x ∈ Icc a b, f x ≤ f C cle : ∀ x ∈ Icc a b, f c ≤ f x hc : ¬f c = f a ⊢ c = b → f c = f a	no goals
https://github.com/yuma-mizuno/lean-math-workshop.git	4a69b0130b276b45212e2b12b90032b146b56d67	Tutorial/Advanced/Analysis/Lecture2.lean	Tutorial.exists_local_extr_Ioo	[93, 1]	[115, 55]	rw [h]	f : ℝ → ℝ f' x a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hfI : f a = f b ne : Set.Nonempty (Icc a b) C : ℝ Cmem : C ∈ Icc a b c : ℝ cmem : c ∈ Icc a b Cge : ∀ x ∈ Icc a b, f x ≤ f C cle : ∀ x ∈ Icc a b, f c ≤ f x hc : ¬f c = f a h : a = c ⊢ f c = f a	no goals
https://github.com/yuma-mizuno/lean-math-workshop.git	4a69b0130b276b45212e2b12b90032b146b56d67	Tutorial/Advanced/Analysis/Lecture2.lean	Tutorial.exists_local_extr_Ioo	[93, 1]	[115, 55]	rw [h, hfI]	f : ℝ → ℝ f' x a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hfI : f a = f b ne : Set.Nonempty (Icc a b) C : ℝ Cmem : C ∈ Icc a b c : ℝ cmem : c ∈ Icc a b Cge : ∀ x ∈ Icc a b, f x ≤ f C cle : ∀ x ∈ Icc a b, f c ≤ f x hc : ¬f c = f a h : c = b ⊢ f c = f a	no goals
https://github.com/yuma-mizuno/lean-math-workshop.git	4a69b0130b276b45212e2b12b90032b146b56d67	Tutorial/Advanced/Analysis/Lecture2.lean	Tutorial.exists_hasDerivAt_eq_zero	[120, 1]	[122, 8]	sorry	f✝ : ℝ → ℝ f'✝ x a✝ b✝ : ℝ f f' g g' : ℝ → ℝ a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hfI : f a = f b hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x ⊢ ∃ c ∈ Ioo a b, f' c = 0	no goals
https://github.com/yuma-mizuno/lean-math-workshop.git	4a69b0130b276b45212e2b12b90032b146b56d67	Tutorial/Advanced/Analysis/Lecture2.lean	Tutorial.exists_ratio_hasDerivAt_eq_ratio_slope	[125, 1]	[132, 8]	let h x := (g b - g a) * f x - (f b - f a) * g x	f✝ : ℝ → ℝ f'✝ x a✝ b✝ : ℝ f f' g g' : ℝ → ℝ a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x hgc : ContinuousOn g (Icc a b) hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x ⊢ ∃ c ∈ Ioo a b, (g b - g a) * f' c = (f b - f a) * g' c	f✝ : ℝ → ℝ f'✝ x a✝ b✝ : ℝ f f' g g' : ℝ → ℝ a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x hgc : ContinuousOn g (Icc a b) hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x h : ℝ → ℝ := fun x => (g b - g a) * f x - (f b - f a) * g x ⊢ ∃ c ∈ Ioo a b, (g b - g a) * f' c = (f b - f a) * g' c
https://github.com/yuma-mizuno/lean-math-workshop.git	4a69b0130b276b45212e2b12b90032b146b56d67	Tutorial/Advanced/Analysis/Lecture2.lean	Tutorial.exists_ratio_hasDerivAt_eq_ratio_slope	[125, 1]	[132, 8]	have hhc : ContinuousOn h (Icc a b) := (continuousOn_const.mul hfc).sub (continuousOn_const.mul hgc)	f✝ : ℝ → ℝ f'✝ x a✝ b✝ : ℝ f f' g g' : ℝ → ℝ a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x hgc : ContinuousOn g (Icc a b) hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x h : ℝ → ℝ := fun x => (g b - g a) * f x - (f b - f a) * g x ⊢ ∃ c ∈ Ioo a b, (g b - g a) * f' c = (f b - f a) * g' c	f✝ : ℝ → ℝ f'✝ x a✝ b✝ : ℝ f f' g g' : ℝ → ℝ a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x hgc : ContinuousOn g (Icc a b) hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x h : ℝ → ℝ := fun x => (g b - g a) * f x - (f b - f a) * g x hhc : ContinuousOn h (Icc a b) ⊢ ∃ c ∈ Ioo a b, (g b - g a) * f' c = (f b - f a) * g' c
https://github.com/yuma-mizuno/lean-math-workshop.git	4a69b0130b276b45212e2b12b90032b146b56d67	Tutorial/Advanced/Analysis/Lecture2.lean	Tutorial.exists_ratio_hasDerivAt_eq_ratio_slope	[125, 1]	[132, 8]	sorry	f✝ : ℝ → ℝ f'✝ x a✝ b✝ : ℝ f f' g g' : ℝ → ℝ a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x hgc : ContinuousOn g (Icc a b) hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x h : ℝ → ℝ := fun x => (g b - g a) * f x - (f b - f a) * g x hhc : ContinuousOn h (Icc a b) ⊢ ∃ c ∈ Ioo a b, (g b - g a) * f' c = (f b - f a) * g' c	no goals
https://github.com/yuma-mizuno/lean-math-workshop.git	4a69b0130b276b45212e2b12b90032b146b56d67	Tutorial/Advanced/Analysis/Lecture2.lean	Tutorial.exists_hasDerivAt_eq_slope	[138, 1]	[141, 8]	sorry	f✝ : ℝ → ℝ f'✝ x a✝ b✝ : ℝ f f' g g' : ℝ → ℝ a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x ⊢ ∃ c ∈ Ioo a b, f' c = (f b - f a) / (b - a)	no goals
https://github.com/yuma-mizuno/lean-math-workshop.git	4a69b0130b276b45212e2b12b90032b146b56d67	Tutorial/Advanced/Category/Lecture1.lean	Tutorial.comp_app	[109, 1]	[110, 6]	rfl	C : Type u inst✝ : Category C a b c d e : C X Y Z : Type f : Hom X Y g : Hom Y Z x : X ⊢ (f ≫ g) x = g (f x)	no goals
https://github.com/yuma-mizuno/lean-math-workshop.git	4a69b0130b276b45212e2b12b90032b146b56d67	Tutorial/Advanced/Category/Lecture1.lean	Tutorial.id_app	[113, 1]	[114, 6]	rfl	C : Type u inst✝ : Category C a b c d e : C X : Type x : X ⊢ 𝟙 X x = x	no goals
https://github.com/yuma-mizuno/lean-math-workshop.git	4a69b0130b276b45212e2b12b90032b146b56d67	Solution/Advanced/Category/Lecture2.lean	Tutorial.Category.Initial.uniq'	[28, 1]	[30, 45]	rw [h.uniq f]	C : Type u inst✝ : Category C a : C h : Initial a b : C f g : Hom a b ⊢ f = h.fromInitial b	no goals
https://github.com/yuma-mizuno/lean-math-workshop.git	4a69b0130b276b45212e2b12b90032b146b56d67	Solution/Advanced/Category/Lecture2.lean	Tutorial.Category.Initial.uniq'	[28, 1]	[30, 45]	rw [h.uniq g]	C : Type u inst✝ : Category C a : C h : Initial a b : C f g : Hom a b ⊢ h.fromInitial b = g	no goals
https://github.com/yuma-mizuno/lean-math-workshop.git	4a69b0130b276b45212e2b12b90032b146b56d67	Solution/Advanced/Category/Lecture2.lean	Tutorial.Coequalizer.hom_id	[329, 1]	[329, 71]	cases i <;> rfl	J : Type u₁ inst✝¹ : Category J C : Type u₂ inst✝ : Category C F : Functor J C i : Shape ⊢ ShapeHom.id i = 𝟙 i	no goals
https://github.com/yuma-mizuno/lean-math-workshop.git	4a69b0130b276b45212e2b12b90032b146b56d67	Solution/Advanced/Analysis/Lecture2.lean	Tutorial.IsLocalMax.hasDerivAt_eq_zero	[45, 1]	[80, 15]	apply le_antisymm ?right ?left	f : ℝ → ℝ f' x a b : ℝ h : IsLocalMax f a hf : HasDerivAt f f' a ⊢ f' = 0	case right f : ℝ → ℝ f' x a b : ℝ h : IsLocalMax f a hf : HasDerivAt f f' a ⊢ f' ≤ 0 case left f : ℝ → ℝ f' x a b : ℝ h : IsLocalMax f a hf : HasDerivAt f f' a ⊢ 0 ≤ f'
https://github.com/yuma-mizuno/lean-math-workshop.git	4a69b0130b276b45212e2b12b90032b146b56d67	Solution/Advanced/Analysis/Lecture2.lean	Tutorial.IsLocalMax.hasDerivAt_eq_zero	[45, 1]	[80, 15]	have hf : Tendsto (fun x ↦ (f x - f a) / (x - a)) (𝓝[>] a) (𝓝 f') := by rw [hasDerivAt_iff_tendsto_slope] at hf apply hf.mono_left (nhds_right'_le_nhds_ne a)	f : ℝ → ℝ f' x a b : ℝ h : IsLocalMax f a hf : HasDerivAt f f' a ⊢ f' ≤ 0	f : ℝ → ℝ f' x a b : ℝ h : IsLocalMax f a hf✝ : HasDerivAt f f' a hf : Tendsto (fun x => (f x - f a) / (x - a)) (𝓝[>] a) (𝓝 f') ⊢ f' ≤ 0
https://github.com/yuma-mizuno/lean-math-workshop.git	4a69b0130b276b45212e2b12b90032b146b56d67	Solution/Advanced/Analysis/Lecture2.lean	Tutorial.IsLocalMax.hasDerivAt_eq_zero	[45, 1]	[80, 15]	suffices ∀ᶠ x in 𝓝[>] a, (f x - f a) / (x - a) ≤ 0 from le_of_tendsto hf this	f : ℝ → ℝ f' x a b : ℝ h : IsLocalMax f a hf✝ : HasDerivAt f f' a hf : Tendsto (fun x => (f x - f a) / (x - a)) (𝓝[>] a) (𝓝 f') ⊢ f' ≤ 0	f : ℝ → ℝ f' x a b : ℝ h : IsLocalMax f a hf✝ : HasDerivAt f f' a hf : Tendsto (fun x => (f x - f a) / (x - a)) (𝓝[>] a) (𝓝 f') ⊢ ∀ᶠ (x : ℝ) in 𝓝[>] a, (f x - f a) / (x - a) ≤ 0
https://github.com/yuma-mizuno/lean-math-workshop.git	4a69b0130b276b45212e2b12b90032b146b56d67	Solution/Advanced/Analysis/Lecture2.lean	Tutorial.IsLocalMax.hasDerivAt_eq_zero	[45, 1]	[80, 15]	have ha : ∀ᶠ x in 𝓝[>] a, a < x := eventually_nhdsWithin_of_forall fun x hx ↦ hx	f : ℝ → ℝ f' x a b : ℝ h : IsLocalMax f a hf✝ : HasDerivAt f f' a hf : Tendsto (fun x => (f x - f a) / (x - a)) (𝓝[>] a) (𝓝 f') ⊢ ∀ᶠ (x : ℝ) in 𝓝[>] a, (f x - f a) / (x - a) ≤ 0	f : ℝ → ℝ f' x a b : ℝ h : IsLocalMax f a hf✝ : HasDerivAt f f' a hf : Tendsto (fun x => (f x - f a) / (x - a)) (𝓝[>] a) (𝓝 f') ha : ∀ᶠ (x : ℝ) in 𝓝[>] a, a < x ⊢ ∀ᶠ (x : ℝ) in 𝓝[>] a, (f x - f a) / (x - a) ≤ 0
https://github.com/yuma-mizuno/lean-math-workshop.git	4a69b0130b276b45212e2b12b90032b146b56d67	Solution/Advanced/Analysis/Lecture2.lean	Tutorial.IsLocalMax.hasDerivAt_eq_zero	[45, 1]	[80, 15]	have h : ∀ᶠ x in 𝓝[>] a, f x ≤ f a := h.filter_mono nhdsWithin_le_nhds	f : ℝ → ℝ f' x a b : ℝ h : IsLocalMax f a hf✝ : HasDerivAt f f' a hf : Tendsto (fun x => (f x - f a) / (x - a)) (𝓝[>] a) (𝓝 f') ha : ∀ᶠ (x : ℝ) in 𝓝[>] a, a < x ⊢ ∀ᶠ (x : ℝ) in 𝓝[>] a, (f x - f a) / (x - a) ≤ 0	f : ℝ → ℝ f' x a b : ℝ h✝ : IsLocalMax f a hf✝ : HasDerivAt f f' a hf : Tendsto (fun x => (f x - f a) / (x - a)) (𝓝[>] a) (𝓝 f') ha : ∀ᶠ (x : ℝ) in 𝓝[>] a, a < x h : ∀ᶠ (x : ℝ) in 𝓝[>] a, f x ≤ f a ⊢ ∀ᶠ (x : ℝ) in 𝓝[>] a, (f x - f a) / (x - a) ≤ 0
https://github.com/yuma-mizuno/lean-math-workshop.git	4a69b0130b276b45212e2b12b90032b146b56d67	Solution/Advanced/Analysis/Lecture2.lean	Tutorial.IsLocalMax.hasDerivAt_eq_zero	[45, 1]	[80, 15]	filter_upwards [ha, h]	f : ℝ → ℝ f' x a b : ℝ h✝ : IsLocalMax f a hf✝ : HasDerivAt f f' a hf : Tendsto (fun x => (f x - f a) / (x - a)) (𝓝[>] a) (𝓝 f') ha : ∀ᶠ (x : ℝ) in 𝓝[>] a, a < x h : ∀ᶠ (x : ℝ) in 𝓝[>] a, f x ≤ f a ⊢ ∀ᶠ (x : ℝ) in 𝓝[>] a, (f x - f a) / (x - a) ≤ 0	case h f : ℝ → ℝ f' x a b : ℝ h✝ : IsLocalMax f a hf✝ : HasDerivAt f f' a hf : Tendsto (fun x => (f x - f a) / (x - a)) (𝓝[>] a) (𝓝 f') ha : ∀ᶠ (x : ℝ) in 𝓝[>] a, a < x h : ∀ᶠ (x : ℝ) in 𝓝[>] a, f x ≤ f a ⊢ ∀ (a_1 : ℝ), a < a_1 → f a_1 ≤ f a → (f a_1 - f a) / (a_1 - a) ≤ 0
https://github.com/yuma-mizuno/lean-math-workshop.git	4a69b0130b276b45212e2b12b90032b146b56d67	Solution/Advanced/Analysis/Lecture2.lean	Tutorial.IsLocalMax.hasDerivAt_eq_zero	[45, 1]	[80, 15]	intro x ha h	case h f : ℝ → ℝ f' x a b : ℝ h✝ : IsLocalMax f a hf✝ : HasDerivAt f f' a hf : Tendsto (fun x => (f x - f a) / (x - a)) (𝓝[>] a) (𝓝 f') ha : ∀ᶠ (x : ℝ) in 𝓝[>] a, a < x h : ∀ᶠ (x : ℝ) in 𝓝[>] a, f x ≤ f a ⊢ ∀ (a_1 : ℝ), a < a_1 → f a_1 ≤ f a → (f a_1 - f a) / (a_1 - a) ≤ 0	case h f : ℝ → ℝ f' x✝ a b : ℝ h✝¹ : IsLocalMax f a hf✝ : HasDerivAt f f' a hf : Tendsto (fun x => (f x - f a) / (x - a)) (𝓝[>] a) (𝓝 f') ha✝ : ∀ᶠ (x : ℝ) in 𝓝[>] a, a < x h✝ : ∀ᶠ (x : ℝ) in 𝓝[>] a, f x ≤ f a x : ℝ ha : a < x h : f x ≤ f a ⊢ (f x - f a) / (x - a) ≤ 0
https://github.com/yuma-mizuno/lean-math-workshop.git	4a69b0130b276b45212e2b12b90032b146b56d67	Solution/Advanced/Analysis/Lecture2.lean	Tutorial.IsLocalMax.hasDerivAt_eq_zero	[45, 1]	[80, 15]	apply div_nonpos_of_nonpos_of_nonneg	case h f : ℝ → ℝ f' x✝ a b : ℝ h✝¹ : IsLocalMax f a hf✝ : HasDerivAt f f' a hf : Tendsto (fun x => (f x - f a) / (x - a)) (𝓝[>] a) (𝓝 f') ha✝ : ∀ᶠ (x : ℝ) in 𝓝[>] a, a < x h✝ : ∀ᶠ (x : ℝ) in 𝓝[>] a, f x ≤ f a x : ℝ ha : a < x h : f x ≤ f a ⊢ (f x - f a) / (x - a) ≤ 0	case h.ha f : ℝ → ℝ f' x✝ a b : ℝ h✝¹ : IsLocalMax f a hf✝ : HasDerivAt f f' a hf : Tendsto (fun x => (f x - f a) / (x - a)) (𝓝[>] a) (𝓝 f') ha✝ : ∀ᶠ (x : ℝ) in 𝓝[>] a, a < x h✝ : ∀ᶠ (x : ℝ) in 𝓝[>] a, f x ≤ f a x : ℝ ha : a < x h : f x ≤ f a ⊢ f x - f a ≤ 0 case h.hb f : ℝ → ℝ f' x✝ a b : ℝ h✝¹ : IsLocalMax f a hf✝ : HasDerivAt f f' a hf : Tendsto (fun x => (f x - f a) / (x - a)) (𝓝[>] a) (𝓝 f') ha✝ : ∀ᶠ (x : ℝ) in 𝓝[>] a, a < x h✝ : ∀ᶠ (x : ℝ) in 𝓝[>] a, f x ≤ f a x : ℝ ha : a < x h : f x ≤ f a ⊢ 0 ≤ x - a
https://github.com/yuma-mizuno/lean-math-workshop.git	4a69b0130b276b45212e2b12b90032b146b56d67	Solution/Advanced/Analysis/Lecture2.lean	Tutorial.IsLocalMax.hasDerivAt_eq_zero	[45, 1]	[80, 15]	rw [hasDerivAt_iff_tendsto_slope] at hf	f : ℝ → ℝ f' x a b : ℝ h : IsLocalMax f a hf : HasDerivAt f f' a ⊢ Tendsto (fun x => (f x - f a) / (x - a)) (𝓝[>] a) (𝓝 f')	f : ℝ → ℝ f' x a b : ℝ h : IsLocalMax f a hf : Tendsto (fun x => (f x - f a) / (x - a)) (𝓝[≠] a) (𝓝 f') ⊢ Tendsto (fun x => (f x - f a) / (x - a)) (𝓝[>] a) (𝓝 f')
https://github.com/yuma-mizuno/lean-math-workshop.git	4a69b0130b276b45212e2b12b90032b146b56d67	Solution/Advanced/Analysis/Lecture2.lean	Tutorial.IsLocalMax.hasDerivAt_eq_zero	[45, 1]	[80, 15]	apply hf.mono_left (nhds_right'_le_nhds_ne a)	f : ℝ → ℝ f' x a b : ℝ h : IsLocalMax f a hf : Tendsto (fun x => (f x - f a) / (x - a)) (𝓝[≠] a) (𝓝 f') ⊢ Tendsto (fun x => (f x - f a) / (x - a)) (𝓝[>] a) (𝓝 f')	no goals
https://github.com/yuma-mizuno/lean-math-workshop.git	4a69b0130b276b45212e2b12b90032b146b56d67	Solution/Advanced/Analysis/Lecture2.lean	Tutorial.IsLocalMax.hasDerivAt_eq_zero	[45, 1]	[80, 15]	linarith only [h]	case h.ha f : ℝ → ℝ f' x✝ a b : ℝ h✝¹ : IsLocalMax f a hf✝ : HasDerivAt f f' a hf : Tendsto (fun x => (f x - f a) / (x - a)) (𝓝[>] a) (𝓝 f') ha✝ : ∀ᶠ (x : ℝ) in 𝓝[>] a, a < x h✝ : ∀ᶠ (x : ℝ) in 𝓝[>] a, f x ≤ f a x : ℝ ha : a < x h : f x ≤ f a ⊢ f x - f a ≤ 0	no goals
https://github.com/yuma-mizuno/lean-math-workshop.git	4a69b0130b276b45212e2b12b90032b146b56d67	Solution/Advanced/Analysis/Lecture2.lean	Tutorial.IsLocalMax.hasDerivAt_eq_zero	[45, 1]	[80, 15]	linarith only [ha]	case h.hb f : ℝ → ℝ f' x✝ a b : ℝ h✝¹ : IsLocalMax f a hf✝ : HasDerivAt f f' a hf : Tendsto (fun x => (f x - f a) / (x - a)) (𝓝[>] a) (𝓝 f') ha✝ : ∀ᶠ (x : ℝ) in 𝓝[>] a, a < x h✝ : ∀ᶠ (x : ℝ) in 𝓝[>] a, f x ≤ f a x : ℝ ha : a < x h : f x ≤ f a ⊢ 0 ≤ x - a	no goals
https://github.com/yuma-mizuno/lean-math-workshop.git	4a69b0130b276b45212e2b12b90032b146b56d67	Solution/Advanced/Analysis/Lecture2.lean	Tutorial.IsLocalMax.hasDerivAt_eq_zero	[45, 1]	[80, 15]	have hf : Tendsto (fun x ↦ (f x - f a) / (x - a)) (𝓝[<] a) (𝓝 f') := by rw [hasDerivAt_iff_tendsto_slope] at hf apply hf.mono_left (nhds_left'_le_nhds_ne a)	f : ℝ → ℝ f' x a b : ℝ h : IsLocalMax f a hf : HasDerivAt f f' a ⊢ 0 ≤ f'	f : ℝ → ℝ f' x a b : ℝ h : IsLocalMax f a hf✝ : HasDerivAt f f' a hf : Tendsto (fun x => (f x - f a) / (x - a)) (𝓝[<] a) (𝓝 f') ⊢ 0 ≤ f'
https://github.com/yuma-mizuno/lean-math-workshop.git	4a69b0130b276b45212e2b12b90032b146b56d67	Solution/Advanced/Analysis/Lecture2.lean	Tutorial.IsLocalMax.hasDerivAt_eq_zero	[45, 1]	[80, 15]	suffices ∀ᶠ x in 𝓝[<] a, (f x - f a) / (x - a) ≥ 0 from ge_of_tendsto hf this	f : ℝ → ℝ f' x a b : ℝ h : IsLocalMax f a hf✝ : HasDerivAt f f' a hf : Tendsto (fun x => (f x - f a) / (x - a)) (𝓝[<] a) (𝓝 f') ⊢ 0 ≤ f'	f : ℝ → ℝ f' x a b : ℝ h : IsLocalMax f a hf✝ : HasDerivAt f f' a hf : Tendsto (fun x => (f x - f a) / (x - a)) (𝓝[<] a) (𝓝 f') ⊢ ∀ᶠ (x : ℝ) in 𝓝[<] a, (f x - f a) / (x - a) ≥ 0

https://github.com/mguaypaq/lean-bruhat.git

1666a1bee2b520d5ba8a662310b4bd257fcf7ac2

Bruhat/Word.lean

stdinversions_inversions

[267, 1]

[278, 65]

simp only [Set.mem_union, Set.mem_preimage, Prod.swap_prod_mk]

case h.mk α : Type u_1 inst✝ : LinearOrder α g : Equiv.Perm α a b : α ⊢ (a, b) ∈ Inversions g ↔ (a, b) ∈ StdInversions g ∪ Prod.swap ⁻¹' StdInversions g

case h.mk α : Type u_1 inst✝ : LinearOrder α g : Equiv.Perm α a b : α ⊢ (a, b) ∈ Inversions g ↔ (a, b) ∈ StdInversions g ∨ (b, a) ∈ StdInversions g

https://github.com/mguaypaq/lean-bruhat.git

1666a1bee2b520d5ba8a662310b4bd257fcf7ac2

Bruhat/Word.lean

stdinversions_inversions

[267, 1]

[278, 65]

by_cases hab : a = b

case h.mk α : Type u_1 inst✝ : LinearOrder α g : Equiv.Perm α a b : α ⊢ (a, b) ∈ Inversions g ↔ (a, b) ∈ StdInversions g ∨ (b, a) ∈ StdInversions g

case pos α : Type u_1 inst✝ : LinearOrder α g : Equiv.Perm α a b : α hab : a = b ⊢ (a, b) ∈ Inversions g ↔ (a, b) ∈ StdInversions g ∨ (b, a) ∈ StdInversions g case neg α : Type u_1 inst✝ : LinearOrder α g : Equiv.Perm α a b : α hab : ¬a = b ⊢ (a, b) ∈ Inversions g ↔ (a, b) ∈ StdInversions g ∨ (b, a) ∈ StdInversions g

https://github.com/mguaypaq/lean-bruhat.git

1666a1bee2b520d5ba8a662310b4bd257fcf7ac2

Bruhat/Word.lean

stdinversions_inversions

[267, 1]

[278, 65]

. subst b simp [not_mem_stdinversions_diag, not_mem_inversions_diag]

case pos α : Type u_1 inst✝ : LinearOrder α g : Equiv.Perm α a b : α hab : a = b ⊢ (a, b) ∈ Inversions g ↔ (a, b) ∈ StdInversions g ∨ (b, a) ∈ StdInversions g case neg α : Type u_1 inst✝ : LinearOrder α g : Equiv.Perm α a b : α hab : ¬a = b ⊢ (a, b) ∈ Inversions g ↔ (a, b) ∈ StdInversions g ∨ (b, a) ∈ StdInversions g

case neg α : Type u_1 inst✝ : LinearOrder α g : Equiv.Perm α a b : α hab : ¬a = b ⊢ (a, b) ∈ Inversions g ↔ (a, b) ∈ StdInversions g ∨ (b, a) ∈ StdInversions g

https://github.com/mguaypaq/lean-bruhat.git

1666a1bee2b520d5ba8a662310b4bd257fcf7ac2

Bruhat/Word.lean

stdinversions_inversions

[267, 1]

[278, 65]

. push_neg at hab obtain (hab | hba) := hab.lt_or_lt . simp [mem_stdinversions', mem_inversions, hab, hab.not_lt] . rw [mem_inversions_symm] simp [mem_stdinversions', mem_inversions, hba, hba.not_lt]

case neg α : Type u_1 inst✝ : LinearOrder α g : Equiv.Perm α a b : α hab : ¬a = b ⊢ (a, b) ∈ Inversions g ↔ (a, b) ∈ StdInversions g ∨ (b, a) ∈ StdInversions g

no goals

https://github.com/mguaypaq/lean-bruhat.git

1666a1bee2b520d5ba8a662310b4bd257fcf7ac2

Bruhat/Word.lean

stdinversions_inversions

[267, 1]

[278, 65]

subst b

case pos α : Type u_1 inst✝ : LinearOrder α g : Equiv.Perm α a b : α hab : a = b ⊢ (a, b) ∈ Inversions g ↔ (a, b) ∈ StdInversions g ∨ (b, a) ∈ StdInversions g

case pos α : Type u_1 inst✝ : LinearOrder α g : Equiv.Perm α a : α ⊢ (a, a) ∈ Inversions g ↔ (a, a) ∈ StdInversions g ∨ (a, a) ∈ StdInversions g

https://github.com/mguaypaq/lean-bruhat.git

1666a1bee2b520d5ba8a662310b4bd257fcf7ac2

Bruhat/Word.lean

stdinversions_inversions

[267, 1]

[278, 65]

simp [not_mem_stdinversions_diag, not_mem_inversions_diag]

case pos α : Type u_1 inst✝ : LinearOrder α g : Equiv.Perm α a : α ⊢ (a, a) ∈ Inversions g ↔ (a, a) ∈ StdInversions g ∨ (a, a) ∈ StdInversions g

no goals

https://github.com/mguaypaq/lean-bruhat.git

1666a1bee2b520d5ba8a662310b4bd257fcf7ac2

Bruhat/Word.lean

stdinversions_inversions

[267, 1]

[278, 65]

push_neg at hab

case neg α : Type u_1 inst✝ : LinearOrder α g : Equiv.Perm α a b : α hab : ¬a = b ⊢ (a, b) ∈ Inversions g ↔ (a, b) ∈ StdInversions g ∨ (b, a) ∈ StdInversions g

case neg α : Type u_1 inst✝ : LinearOrder α g : Equiv.Perm α a b : α hab : a ≠ b ⊢ (a, b) ∈ Inversions g ↔ (a, b) ∈ StdInversions g ∨ (b, a) ∈ StdInversions g

https://github.com/mguaypaq/lean-bruhat.git

1666a1bee2b520d5ba8a662310b4bd257fcf7ac2

Bruhat/Word.lean

stdinversions_inversions

[267, 1]

[278, 65]

obtain (hab | hba) := hab.lt_or_lt

case neg α : Type u_1 inst✝ : LinearOrder α g : Equiv.Perm α a b : α hab : a ≠ b ⊢ (a, b) ∈ Inversions g ↔ (a, b) ∈ StdInversions g ∨ (b, a) ∈ StdInversions g

case neg.inl α : Type u_1 inst✝ : LinearOrder α g : Equiv.Perm α a b : α hab✝ : a ≠ b hab : a < b ⊢ (a, b) ∈ Inversions g ↔ (a, b) ∈ StdInversions g ∨ (b, a) ∈ StdInversions g case neg.inr α : Type u_1 inst✝ : LinearOrder α g : Equiv.Perm α a b : α hab : a ≠ b hba : b < a ⊢ (a, b) ∈ Inversions g ↔ (a, b) ∈ StdInversions g ∨ (b, a) ∈ StdInversions g

https://github.com/mguaypaq/lean-bruhat.git

1666a1bee2b520d5ba8a662310b4bd257fcf7ac2

Bruhat/Word.lean

stdinversions_inversions

[267, 1]

[278, 65]

. simp [mem_stdinversions', mem_inversions, hab, hab.not_lt]

case neg.inl α : Type u_1 inst✝ : LinearOrder α g : Equiv.Perm α a b : α hab✝ : a ≠ b hab : a < b ⊢ (a, b) ∈ Inversions g ↔ (a, b) ∈ StdInversions g ∨ (b, a) ∈ StdInversions g case neg.inr α : Type u_1 inst✝ : LinearOrder α g : Equiv.Perm α a b : α hab : a ≠ b hba : b < a ⊢ (a, b) ∈ Inversions g ↔ (a, b) ∈ StdInversions g ∨ (b, a) ∈ StdInversions g

case neg.inr α : Type u_1 inst✝ : LinearOrder α g : Equiv.Perm α a b : α hab : a ≠ b hba : b < a ⊢ (a, b) ∈ Inversions g ↔ (a, b) ∈ StdInversions g ∨ (b, a) ∈ StdInversions g

https://github.com/mguaypaq/lean-bruhat.git

1666a1bee2b520d5ba8a662310b4bd257fcf7ac2

Bruhat/Word.lean

stdinversions_inversions

[267, 1]

[278, 65]

. rw [mem_inversions_symm] simp [mem_stdinversions', mem_inversions, hba, hba.not_lt]

case neg.inr α : Type u_1 inst✝ : LinearOrder α g : Equiv.Perm α a b : α hab : a ≠ b hba : b < a ⊢ (a, b) ∈ Inversions g ↔ (a, b) ∈ StdInversions g ∨ (b, a) ∈ StdInversions g

no goals

https://github.com/mguaypaq/lean-bruhat.git

1666a1bee2b520d5ba8a662310b4bd257fcf7ac2

Bruhat/Word.lean

stdinversions_inversions

[267, 1]

[278, 65]

simp [mem_stdinversions', mem_inversions, hab, hab.not_lt]

case neg.inl α : Type u_1 inst✝ : LinearOrder α g : Equiv.Perm α a b : α hab✝ : a ≠ b hab : a < b ⊢ (a, b) ∈ Inversions g ↔ (a, b) ∈ StdInversions g ∨ (b, a) ∈ StdInversions g

no goals

https://github.com/mguaypaq/lean-bruhat.git

1666a1bee2b520d5ba8a662310b4bd257fcf7ac2

Bruhat/Word.lean

stdinversions_inversions

[267, 1]

[278, 65]

rw [mem_inversions_symm]

case neg.inr α : Type u_1 inst✝ : LinearOrder α g : Equiv.Perm α a b : α hab : a ≠ b hba : b < a ⊢ (a, b) ∈ Inversions g ↔ (a, b) ∈ StdInversions g ∨ (b, a) ∈ StdInversions g

case neg.inr α : Type u_1 inst✝ : LinearOrder α g : Equiv.Perm α a b : α hab : a ≠ b hba : b < a ⊢ (b, a) ∈ Inversions g ↔ (a, b) ∈ StdInversions g ∨ (b, a) ∈ StdInversions g

https://github.com/mguaypaq/lean-bruhat.git

1666a1bee2b520d5ba8a662310b4bd257fcf7ac2

Bruhat/Word.lean

stdinversions_inversions

[267, 1]

[278, 65]

simp [mem_stdinversions', mem_inversions, hba, hba.not_lt]

case neg.inr α : Type u_1 inst✝ : LinearOrder α g : Equiv.Perm α a b : α hab : a ≠ b hba : b < a ⊢ (b, a) ∈ Inversions g ↔ (a, b) ∈ StdInversions g ∨ (b, a) ∈ StdInversions g

no goals

https://github.com/mguaypaq/lean-bruhat.git

1666a1bee2b520d5ba8a662310b4bd257fcf7ac2

Bruhat/Goal.lean

perm_conj

[11, 1]

[13, 37]

rw [Equiv.symm_apply_apply, hab]

α : Type u_1 f g : Equiv.Perm α a b : α hab : f a = b ⊢ g (f (g.symm (g a))) = g b

no goals

https://github.com/mguaypaq/lean-bruhat.git

1666a1bee2b520d5ba8a662310b4bd257fcf7ac2

Bruhat/Goal.lean

swap_conj

[15, 1]

[19, 6]

rw [Equiv.swap_apply_apply]

α : Type u_1 inst✝ : DecidableEq α a b c d : α ⊢ Equiv.swap a b * Equiv.swap c d * Equiv.swap a b = Equiv.swap ((Equiv.swap a b) c) ((Equiv.swap a b) d)

α : Type u_1 inst✝ : DecidableEq α a b c d : α ⊢ Equiv.swap a b * Equiv.swap c d * Equiv.swap a b = Equiv.swap a b * Equiv.swap c d * (Equiv.swap a b)⁻¹

https://github.com/mguaypaq/lean-bruhat.git

1666a1bee2b520d5ba8a662310b4bd257fcf7ac2

Bruhat/Goal.lean

swap_conj

[15, 1]

[19, 6]

rfl

α : Type u_1 inst✝ : DecidableEq α a b c d : α ⊢ Equiv.swap a b * Equiv.swap c d * Equiv.swap a b = Equiv.swap a b * Equiv.swap c d * (Equiv.swap a b)⁻¹

no goals

https://github.com/mguaypaq/lean-bruhat.git

1666a1bee2b520d5ba8a662310b4bd257fcf7ac2

Bruhat/Goal.lean

WordForSwap_toPerm_succ

[42, 1]

[47, 6]

simp [WordForSwap, Word.toPerm]

i k : ℕ ⊢ Word.toPerm (WordForSwap i (k + 1)) = Equiv.swap (i + (k + 1)) (i + (k + 2)) * Word.toPerm (WordForSwap i k) * Equiv.swap (i + (k + 1)) (i + (k + 2))

i k : ℕ ⊢ Equiv.swap (i + k + 1) (i + k + 1 + 1) * (List.prod (List.map (fun i => Equiv.swap i (i + 1)) (WordForSwap i k)) * Equiv.swap (i + k + 1) (i + k + 1 + 1)) = Equiv.swap (i + (k + 1)) (i + (k + 2)) * List.prod (List.map (fun i => Equiv.swap i (i + 1)) (WordForSwap i k)) * Equiv.swap (i + (k + 1)) (i + (k + 2))

https://github.com/mguaypaq/lean-bruhat.git

1666a1bee2b520d5ba8a662310b4bd257fcf7ac2

Bruhat/Goal.lean

WordForSwap_toPerm_succ

[42, 1]

[47, 6]

rfl

i k : ℕ ⊢ Equiv.swap (i + k + 1) (i + k + 1 + 1) * (List.prod (List.map (fun i => Equiv.swap i (i + 1)) (WordForSwap i k)) * Equiv.swap (i + k + 1) (i + k + 1 + 1)) = Equiv.swap (i + (k + 1)) (i + (k + 2)) * List.prod (List.map (fun i => Equiv.swap i (i + 1)) (WordForSwap i k)) * Equiv.swap (i + (k + 1)) (i + (k + 2))

no goals

https://github.com/mguaypaq/lean-bruhat.git

1666a1bee2b520d5ba8a662310b4bd257fcf7ac2

Bruhat/Goal.lean

wordForSwap_eq_swap

[49, 1]

[57, 58]

induction' k with k ih

i k : ℕ ⊢ Word.toPerm (WordForSwap i k) = Equiv.swap i (i + (k + 1))

case zero i : ℕ ⊢ Word.toPerm (WordForSwap i Nat.zero) = Equiv.swap i (i + (Nat.zero + 1)) case succ i k : ℕ ih : Word.toPerm (WordForSwap i k) = Equiv.swap i (i + (k + 1)) ⊢ Word.toPerm (WordForSwap i (Nat.succ k)) = Equiv.swap i (i + (Nat.succ k + 1))

https://github.com/mguaypaq/lean-bruhat.git

1666a1bee2b520d5ba8a662310b4bd257fcf7ac2

Bruhat/Goal.lean

wordForSwap_eq_swap

[49, 1]

[57, 58]

. simp only [Nat.zero_eq, add_zero, Equiv.swap_self] rfl

case zero i : ℕ ⊢ Word.toPerm (WordForSwap i Nat.zero) = Equiv.swap i (i + (Nat.zero + 1)) case succ i k : ℕ ih : Word.toPerm (WordForSwap i k) = Equiv.swap i (i + (k + 1)) ⊢ Word.toPerm (WordForSwap i (Nat.succ k)) = Equiv.swap i (i + (Nat.succ k + 1))

case succ i k : ℕ ih : Word.toPerm (WordForSwap i k) = Equiv.swap i (i + (k + 1)) ⊢ Word.toPerm (WordForSwap i (Nat.succ k)) = Equiv.swap i (i + (Nat.succ k + 1))

https://github.com/mguaypaq/lean-bruhat.git

1666a1bee2b520d5ba8a662310b4bd257fcf7ac2

Bruhat/Goal.lean

wordForSwap_eq_swap

[49, 1]

[57, 58]

. rw [WordForSwap_toPerm_succ, ih] change _ * _ * (Equiv.swap _ _)⁻¹ = _ rw [← Equiv.swap_apply_apply, Equiv.swap_apply_left, Equiv.swap_apply_of_ne_of_ne (by simp) (by simp)]

case succ i k : ℕ ih : Word.toPerm (WordForSwap i k) = Equiv.swap i (i + (k + 1)) ⊢ Word.toPerm (WordForSwap i (Nat.succ k)) = Equiv.swap i (i + (Nat.succ k + 1))

no goals

https://github.com/mguaypaq/lean-bruhat.git

1666a1bee2b520d5ba8a662310b4bd257fcf7ac2

Bruhat/Goal.lean

wordForSwap_eq_swap

[49, 1]

[57, 58]

simp only [Nat.zero_eq, add_zero, Equiv.swap_self]

case zero i : ℕ ⊢ Word.toPerm (WordForSwap i Nat.zero) = Equiv.swap i (i + (Nat.zero + 1))

case zero i : ℕ ⊢ Word.toPerm (WordForSwap i 0) = Equiv.swap i (i + (0 + 1))

https://github.com/mguaypaq/lean-bruhat.git

1666a1bee2b520d5ba8a662310b4bd257fcf7ac2

Bruhat/Goal.lean

wordForSwap_eq_swap

[49, 1]

[57, 58]

rfl

case zero i : ℕ ⊢ Word.toPerm (WordForSwap i 0) = Equiv.swap i (i + (0 + 1))

no goals

https://github.com/mguaypaq/lean-bruhat.git

1666a1bee2b520d5ba8a662310b4bd257fcf7ac2

Bruhat/Goal.lean

wordForSwap_eq_swap

[49, 1]

[57, 58]

rw [WordForSwap_toPerm_succ, ih]

case succ i k : ℕ ih : Word.toPerm (WordForSwap i k) = Equiv.swap i (i + (k + 1)) ⊢ Word.toPerm (WordForSwap i (Nat.succ k)) = Equiv.swap i (i + (Nat.succ k + 1))

case succ i k : ℕ ih : Word.toPerm (WordForSwap i k) = Equiv.swap i (i + (k + 1)) ⊢ Equiv.swap (i + (k + 1)) (i + (k + 2)) * Equiv.swap i (i + (k + 1)) * Equiv.swap (i + (k + 1)) (i + (k + 2)) = Equiv.swap i (i + (Nat.succ k + 1))

https://github.com/mguaypaq/lean-bruhat.git

1666a1bee2b520d5ba8a662310b4bd257fcf7ac2

Bruhat/Goal.lean

wordForSwap_eq_swap

[49, 1]

[57, 58]

change _ * _ * (Equiv.swap _ _)⁻¹ = _

case succ i k : ℕ ih : Word.toPerm (WordForSwap i k) = Equiv.swap i (i + (k + 1)) ⊢ Equiv.swap (i + (k + 1)) (i + (k + 2)) * Equiv.swap i (i + (k + 1)) * Equiv.swap (i + (k + 1)) (i + (k + 2)) = Equiv.swap i (i + (Nat.succ k + 1))

case succ i k : ℕ ih : Word.toPerm (WordForSwap i k) = Equiv.swap i (i + (k + 1)) ⊢ Equiv.swap (i + (k + 1)) (i + (k + 2)) * Equiv.swap i (i + (k + 1)) * (Equiv.swap (i + (k + 1)) (i + (k + 2)))⁻¹ = Equiv.swap i (i + (Nat.succ k + 1))

https://github.com/mguaypaq/lean-bruhat.git

1666a1bee2b520d5ba8a662310b4bd257fcf7ac2

Bruhat/Goal.lean

wordForSwap_eq_swap

[49, 1]

[57, 58]

rw [← Equiv.swap_apply_apply, Equiv.swap_apply_left, Equiv.swap_apply_of_ne_of_ne (by simp) (by simp)]

case succ i k : ℕ ih : Word.toPerm (WordForSwap i k) = Equiv.swap i (i + (k + 1)) ⊢ Equiv.swap (i + (k + 1)) (i + (k + 2)) * Equiv.swap i (i + (k + 1)) * (Equiv.swap (i + (k + 1)) (i + (k + 2)))⁻¹ = Equiv.swap i (i + (Nat.succ k + 1))

no goals

https://github.com/mguaypaq/lean-bruhat.git

1666a1bee2b520d5ba8a662310b4bd257fcf7ac2

Bruhat/Goal.lean

wordForSwap_eq_swap

[49, 1]

[57, 58]

simp

i k : ℕ ih : Word.toPerm (WordForSwap i k) = Equiv.swap i (i + (k + 1)) ⊢ i ≠ i + (k + 1)

no goals

https://github.com/mguaypaq/lean-bruhat.git

1666a1bee2b520d5ba8a662310b4bd257fcf7ac2

Bruhat/Goal.lean

wordForSwap_eq_swap

[49, 1]

[57, 58]

simp

i k : ℕ ih : Word.toPerm (WordForSwap i k) = Equiv.swap i (i + (k + 1)) ⊢ i ≠ i + (k + 2)

no goals

https://github.com/mguaypaq/lean-bruhat.git

1666a1bee2b520d5ba8a662310b4bd257fcf7ac2

Bruhat/Goal.lean

unswap_support

[73, 1]

[75, 22]

simp [support] at *

α : Type u_1 inst✝ : DecidableEq α f : Equiv.Perm α a : α ⊢ a ∉ support (Equiv.swap a (f a) * f)

no goals

https://github.com/mguaypaq/lean-bruhat.git

1666a1bee2b520d5ba8a662310b4bd257fcf7ac2

Bruhat/Goal.lean

inversions_mul

[155, 1]

[157, 8]

sorry

f g : Equiv.Perm ℕ ⊢ Inversions (f * g) = Prod.map ⇑g.symm ⇑g.symm '' Inversions f ∩ Inversions g

no goals

https://github.com/mguaypaq/lean-bruhat.git

1666a1bee2b520d5ba8a662310b4bd257fcf7ac2

Bruhat/Lehmer.lean

truncate_apply_of_le

[18, 1]

[19, 20]

simpa

f : ℕ → ℕ n x : ℕ hx : n ≤ x ⊢ ¬x < n

no goals

https://github.com/mguaypaq/lean-bruhat.git

1666a1bee2b520d5ba8a662310b4bd257fcf7ac2

Bruhat/Lehmer.lean

card_lehmer_eq_factorial

[73, 1]

[78, 47]

rw [Fintype.card_of_bijective (le_truncate_equiv_prod_fin _ _).bijective]

n : ℕ ⊢ Fintype.card { g // ⇑g ≤ ⇑(truncate id n) } = n !

n : ℕ ⊢ Fintype.card ((i : Fin n) → Fin (id ↑i + 1)) = n !

https://github.com/mguaypaq/lean-bruhat.git

1666a1bee2b520d5ba8a662310b4bd257fcf7ac2

Bruhat/Lehmer.lean

card_lehmer_eq_factorial

[73, 1]

[78, 47]

simp only [id_eq, Fintype.card_pi, Fintype.card_fin]

n : ℕ ⊢ Fintype.card ((i : Fin n) → Fin (id ↑i + 1)) = n !

n : ℕ ⊢ (Finset.prod Finset.univ fun x => ↑x + 1) = n !

https://github.com/mguaypaq/lean-bruhat.git

1666a1bee2b520d5ba8a662310b4bd257fcf7ac2

Bruhat/Lehmer.lean

card_lehmer_eq_factorial

[73, 1]

[78, 47]

rw [← Finset.prod_range_add_one_eq_factorial n]

n : ℕ ⊢ (Finset.prod Finset.univ fun x => ↑x + 1) = n !

n : ℕ ⊢ (Finset.prod Finset.univ fun x => ↑x + 1) = Finset.prod (Finset.range n) fun x => x + 1

https://github.com/mguaypaq/lean-bruhat.git

1666a1bee2b520d5ba8a662310b4bd257fcf7ac2

Bruhat/Lehmer.lean

card_lehmer_eq_factorial

[73, 1]

[78, 47]

exact Fin.prod_univ_eq_prod_range Nat.succ n

n : ℕ ⊢ (Finset.prod Finset.univ fun x => ↑x + 1) = Finset.prod (Finset.range n) fun x => x + 1

no goals

https://github.com/yuma-mizuno/lean-math-workshop.git

4a69b0130b276b45212e2b12b90032b146b56d67

Tutorial/Advanced/Analysis/Lecture2.lean

Tutorial.IsLocalMax.hasDerivAt_eq_zero

[45, 1]

[69, 10]

apply le_antisymm ?right ?left

f : ℝ → ℝ f' x a b : ℝ h : IsLocalMax f a hf : HasDerivAt f f' a ⊢ f' = 0

case right f : ℝ → ℝ f' x a b : ℝ h : IsLocalMax f a hf : HasDerivAt f f' a ⊢ f' ≤ 0 case left f : ℝ → ℝ f' x a b : ℝ h : IsLocalMax f a hf : HasDerivAt f f' a ⊢ 0 ≤ f'

https://github.com/yuma-mizuno/lean-math-workshop.git

4a69b0130b276b45212e2b12b90032b146b56d67

Tutorial/Advanced/Analysis/Lecture2.lean

Tutorial.IsLocalMax.hasDerivAt_eq_zero

[45, 1]

[69, 10]

case left => sorry

f : ℝ → ℝ f' x a b : ℝ h : IsLocalMax f a hf : HasDerivAt f f' a ⊢ 0 ≤ f'

no goals

https://github.com/yuma-mizuno/lean-math-workshop.git

4a69b0130b276b45212e2b12b90032b146b56d67

Tutorial/Advanced/Analysis/Lecture2.lean

Tutorial.IsLocalMax.hasDerivAt_eq_zero

[45, 1]

[69, 10]

have hf : Tendsto (fun x ↦ (f x - f a) / (x - a)) (𝓝[>] a) (𝓝 f') := by rw [hasDerivAt_iff_tendsto_slope] at hf apply hf.mono_left (nhds_right'_le_nhds_ne a)

f : ℝ → ℝ f' x a b : ℝ h : IsLocalMax f a hf : HasDerivAt f f' a ⊢ f' ≤ 0

f : ℝ → ℝ f' x a b : ℝ h : IsLocalMax f a hf✝ : HasDerivAt f f' a hf : Tendsto (fun x => (f x - f a) / (x - a)) (𝓝[>] a) (𝓝 f') ⊢ f' ≤ 0

https://github.com/yuma-mizuno/lean-math-workshop.git

4a69b0130b276b45212e2b12b90032b146b56d67

Tutorial/Advanced/Analysis/Lecture2.lean

Tutorial.IsLocalMax.hasDerivAt_eq_zero

[45, 1]

[69, 10]

suffices ∀ᶠ x in 𝓝[>] a, (f x - f a) / (x - a) ≤ 0 from le_of_tendsto hf this

f : ℝ → ℝ f' x a b : ℝ h : IsLocalMax f a hf✝ : HasDerivAt f f' a hf : Tendsto (fun x => (f x - f a) / (x - a)) (𝓝[>] a) (𝓝 f') ⊢ f' ≤ 0

f : ℝ → ℝ f' x a b : ℝ h : IsLocalMax f a hf✝ : HasDerivAt f f' a hf : Tendsto (fun x => (f x - f a) / (x - a)) (𝓝[>] a) (𝓝 f') ⊢ ∀ᶠ (x : ℝ) in 𝓝[>] a, (f x - f a) / (x - a) ≤ 0

https://github.com/yuma-mizuno/lean-math-workshop.git

4a69b0130b276b45212e2b12b90032b146b56d67

Tutorial/Advanced/Analysis/Lecture2.lean

Tutorial.IsLocalMax.hasDerivAt_eq_zero

[45, 1]

[69, 10]

have ha : ∀ᶠ x in 𝓝[>] a, a < x := eventually_nhdsWithin_of_forall fun x hx ↦ hx

f : ℝ → ℝ f' x a b : ℝ h : IsLocalMax f a hf✝ : HasDerivAt f f' a hf : Tendsto (fun x => (f x - f a) / (x - a)) (𝓝[>] a) (𝓝 f') ⊢ ∀ᶠ (x : ℝ) in 𝓝[>] a, (f x - f a) / (x - a) ≤ 0

f : ℝ → ℝ f' x a b : ℝ h : IsLocalMax f a hf✝ : HasDerivAt f f' a hf : Tendsto (fun x => (f x - f a) / (x - a)) (𝓝[>] a) (𝓝 f') ha : ∀ᶠ (x : ℝ) in 𝓝[>] a, a < x ⊢ ∀ᶠ (x : ℝ) in 𝓝[>] a, (f x - f a) / (x - a) ≤ 0

https://github.com/yuma-mizuno/lean-math-workshop.git

4a69b0130b276b45212e2b12b90032b146b56d67

Tutorial/Advanced/Analysis/Lecture2.lean

Tutorial.IsLocalMax.hasDerivAt_eq_zero

[45, 1]

[69, 10]

have h : ∀ᶠ x in 𝓝[>] a, f x ≤ f a := h.filter_mono nhdsWithin_le_nhds

f : ℝ → ℝ f' x a b : ℝ h : IsLocalMax f a hf✝ : HasDerivAt f f' a hf : Tendsto (fun x => (f x - f a) / (x - a)) (𝓝[>] a) (𝓝 f') ha : ∀ᶠ (x : ℝ) in 𝓝[>] a, a < x ⊢ ∀ᶠ (x : ℝ) in 𝓝[>] a, (f x - f a) / (x - a) ≤ 0

f : ℝ → ℝ f' x a b : ℝ h✝ : IsLocalMax f a hf✝ : HasDerivAt f f' a hf : Tendsto (fun x => (f x - f a) / (x - a)) (𝓝[>] a) (𝓝 f') ha : ∀ᶠ (x : ℝ) in 𝓝[>] a, a < x h : ∀ᶠ (x : ℝ) in 𝓝[>] a, f x ≤ f a ⊢ ∀ᶠ (x : ℝ) in 𝓝[>] a, (f x - f a) / (x - a) ≤ 0

https://github.com/yuma-mizuno/lean-math-workshop.git

4a69b0130b276b45212e2b12b90032b146b56d67

Tutorial/Advanced/Analysis/Lecture2.lean

Tutorial.IsLocalMax.hasDerivAt_eq_zero

[45, 1]

[69, 10]

filter_upwards [ha, h]

f : ℝ → ℝ f' x a b : ℝ h✝ : IsLocalMax f a hf✝ : HasDerivAt f f' a hf : Tendsto (fun x => (f x - f a) / (x - a)) (𝓝[>] a) (𝓝 f') ha : ∀ᶠ (x : ℝ) in 𝓝[>] a, a < x h : ∀ᶠ (x : ℝ) in 𝓝[>] a, f x ≤ f a ⊢ ∀ᶠ (x : ℝ) in 𝓝[>] a, (f x - f a) / (x - a) ≤ 0

case h f : ℝ → ℝ f' x a b : ℝ h✝ : IsLocalMax f a hf✝ : HasDerivAt f f' a hf : Tendsto (fun x => (f x - f a) / (x - a)) (𝓝[>] a) (𝓝 f') ha : ∀ᶠ (x : ℝ) in 𝓝[>] a, a < x h : ∀ᶠ (x : ℝ) in 𝓝[>] a, f x ≤ f a ⊢ ∀ (a_1 : ℝ), a < a_1 → f a_1 ≤ f a → (f a_1 - f a) / (a_1 - a) ≤ 0

https://github.com/yuma-mizuno/lean-math-workshop.git

4a69b0130b276b45212e2b12b90032b146b56d67

Tutorial/Advanced/Analysis/Lecture2.lean

Tutorial.IsLocalMax.hasDerivAt_eq_zero

[45, 1]

[69, 10]

intro x ha h

case h f : ℝ → ℝ f' x a b : ℝ h✝ : IsLocalMax f a hf✝ : HasDerivAt f f' a hf : Tendsto (fun x => (f x - f a) / (x - a)) (𝓝[>] a) (𝓝 f') ha : ∀ᶠ (x : ℝ) in 𝓝[>] a, a < x h : ∀ᶠ (x : ℝ) in 𝓝[>] a, f x ≤ f a ⊢ ∀ (a_1 : ℝ), a < a_1 → f a_1 ≤ f a → (f a_1 - f a) / (a_1 - a) ≤ 0

case h f : ℝ → ℝ f' x✝ a b : ℝ h✝¹ : IsLocalMax f a hf✝ : HasDerivAt f f' a hf : Tendsto (fun x => (f x - f a) / (x - a)) (𝓝[>] a) (𝓝 f') ha✝ : ∀ᶠ (x : ℝ) in 𝓝[>] a, a < x h✝ : ∀ᶠ (x : ℝ) in 𝓝[>] a, f x ≤ f a x : ℝ ha : a < x h : f x ≤ f a ⊢ (f x - f a) / (x - a) ≤ 0

https://github.com/yuma-mizuno/lean-math-workshop.git

4a69b0130b276b45212e2b12b90032b146b56d67

Tutorial/Advanced/Analysis/Lecture2.lean

Tutorial.IsLocalMax.hasDerivAt_eq_zero

[45, 1]

[69, 10]

apply div_nonpos_of_nonpos_of_nonneg

case h f : ℝ → ℝ f' x✝ a b : ℝ h✝¹ : IsLocalMax f a hf✝ : HasDerivAt f f' a hf : Tendsto (fun x => (f x - f a) / (x - a)) (𝓝[>] a) (𝓝 f') ha✝ : ∀ᶠ (x : ℝ) in 𝓝[>] a, a < x h✝ : ∀ᶠ (x : ℝ) in 𝓝[>] a, f x ≤ f a x : ℝ ha : a < x h : f x ≤ f a ⊢ (f x - f a) / (x - a) ≤ 0

case h.ha f : ℝ → ℝ f' x✝ a b : ℝ h✝¹ : IsLocalMax f a hf✝ : HasDerivAt f f' a hf : Tendsto (fun x => (f x - f a) / (x - a)) (𝓝[>] a) (𝓝 f') ha✝ : ∀ᶠ (x : ℝ) in 𝓝[>] a, a < x h✝ : ∀ᶠ (x : ℝ) in 𝓝[>] a, f x ≤ f a x : ℝ ha : a < x h : f x ≤ f a ⊢ f x - f a ≤ 0 case h.hb f : ℝ → ℝ f' x✝ a b : ℝ h✝¹ : IsLocalMax f a hf✝ : HasDerivAt f f' a hf : Tendsto (fun x => (f x - f a) / (x - a)) (𝓝[>] a) (𝓝 f') ha✝ : ∀ᶠ (x : ℝ) in 𝓝[>] a, a < x h✝ : ∀ᶠ (x : ℝ) in 𝓝[>] a, f x ≤ f a x : ℝ ha : a < x h : f x ≤ f a ⊢ 0 ≤ x - a

https://github.com/yuma-mizuno/lean-math-workshop.git

4a69b0130b276b45212e2b12b90032b146b56d67

Tutorial/Advanced/Analysis/Lecture2.lean

Tutorial.IsLocalMax.hasDerivAt_eq_zero

[45, 1]

[69, 10]

rw [hasDerivAt_iff_tendsto_slope] at hf

f : ℝ → ℝ f' x a b : ℝ h : IsLocalMax f a hf : HasDerivAt f f' a ⊢ Tendsto (fun x => (f x - f a) / (x - a)) (𝓝[>] a) (𝓝 f')

f : ℝ → ℝ f' x a b : ℝ h : IsLocalMax f a hf : Tendsto (fun x => (f x - f a) / (x - a)) (𝓝[≠] a) (𝓝 f') ⊢ Tendsto (fun x => (f x - f a) / (x - a)) (𝓝[>] a) (𝓝 f')

https://github.com/yuma-mizuno/lean-math-workshop.git

4a69b0130b276b45212e2b12b90032b146b56d67

Tutorial/Advanced/Analysis/Lecture2.lean

Tutorial.IsLocalMax.hasDerivAt_eq_zero

[45, 1]

[69, 10]

apply hf.mono_left (nhds_right'_le_nhds_ne a)

f : ℝ → ℝ f' x a b : ℝ h : IsLocalMax f a hf : Tendsto (fun x => (f x - f a) / (x - a)) (𝓝[≠] a) (𝓝 f') ⊢ Tendsto (fun x => (f x - f a) / (x - a)) (𝓝[>] a) (𝓝 f')

no goals

https://github.com/yuma-mizuno/lean-math-workshop.git

4a69b0130b276b45212e2b12b90032b146b56d67

Tutorial/Advanced/Analysis/Lecture2.lean

Tutorial.IsLocalMax.hasDerivAt_eq_zero

[45, 1]

[69, 10]

linarith only [h]

case h.ha f : ℝ → ℝ f' x✝ a b : ℝ h✝¹ : IsLocalMax f a hf✝ : HasDerivAt f f' a hf : Tendsto (fun x => (f x - f a) / (x - a)) (𝓝[>] a) (𝓝 f') ha✝ : ∀ᶠ (x : ℝ) in 𝓝[>] a, a < x h✝ : ∀ᶠ (x : ℝ) in 𝓝[>] a, f x ≤ f a x : ℝ ha : a < x h : f x ≤ f a ⊢ f x - f a ≤ 0

no goals

https://github.com/yuma-mizuno/lean-math-workshop.git

4a69b0130b276b45212e2b12b90032b146b56d67

Tutorial/Advanced/Analysis/Lecture2.lean

Tutorial.IsLocalMax.hasDerivAt_eq_zero

[45, 1]

[69, 10]

linarith only [ha]

case h.hb f : ℝ → ℝ f' x✝ a b : ℝ h✝¹ : IsLocalMax f a hf✝ : HasDerivAt f f' a hf : Tendsto (fun x => (f x - f a) / (x - a)) (𝓝[>] a) (𝓝 f') ha✝ : ∀ᶠ (x : ℝ) in 𝓝[>] a, a < x h✝ : ∀ᶠ (x : ℝ) in 𝓝[>] a, f x ≤ f a x : ℝ ha : a < x h : f x ≤ f a ⊢ 0 ≤ x - a

no goals

https://github.com/yuma-mizuno/lean-math-workshop.git

4a69b0130b276b45212e2b12b90032b146b56d67

Tutorial/Advanced/Analysis/Lecture2.lean

Tutorial.IsLocalMax.hasDerivAt_eq_zero

[45, 1]

[69, 10]

sorry

f : ℝ → ℝ f' x a b : ℝ h : IsLocalMax f a hf : HasDerivAt f f' a ⊢ 0 ≤ f'

no goals

https://github.com/yuma-mizuno/lean-math-workshop.git

4a69b0130b276b45212e2b12b90032b146b56d67

Tutorial/Advanced/Analysis/Lecture2.lean

Tutorial.IsLocalMin.hasDerivAt_eq_zero

[72, 1]

[74, 8]

sorry

f : ℝ → ℝ f' x a b : ℝ h : IsLocalMin f a hf : HasDerivAt f f' a ⊢ f' = 0

no goals

https://github.com/yuma-mizuno/lean-math-workshop.git

4a69b0130b276b45212e2b12b90032b146b56d67

Tutorial/Advanced/Analysis/Lecture2.lean

Tutorial.IsLocalExtr.hasDerivAt_eq_zero

[80, 1]

[81, 8]

sorry

f : ℝ → ℝ f' x a b : ℝ h : IsLocalExtr f a hf : HasDerivAt f f' a ⊢ f' = 0

no goals

https://github.com/yuma-mizuno/lean-math-workshop.git

4a69b0130b276b45212e2b12b90032b146b56d67

Tutorial/Advanced/Analysis/Lecture2.lean

Tutorial.exists_local_extr_Ioo

[93, 1]

[115, 55]

suffices ∃ c ∈ Ioo a b, IsExtrOn f (Icc a b) c by rcases this with ⟨c, cmem, hc⟩ exists c, cmem apply hc.isLocalExtr <| Icc_mem_nhds cmem.1 cmem.2

f : ℝ → ℝ f' x a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hfI : f a = f b ⊢ ∃ c ∈ Ioo a b, IsLocalExtr f c

f : ℝ → ℝ f' x a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hfI : f a = f b ⊢ ∃ c ∈ Ioo a b, IsExtrOn f (Icc a b) c

https://github.com/yuma-mizuno/lean-math-workshop.git

4a69b0130b276b45212e2b12b90032b146b56d67

Tutorial/Advanced/Analysis/Lecture2.lean

Tutorial.exists_local_extr_Ioo

[93, 1]

[115, 55]

have ne : (Icc a b).Nonempty := nonempty_Icc.2 (le_of_lt hab)

f : ℝ → ℝ f' x a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hfI : f a = f b ⊢ ∃ c ∈ Ioo a b, IsExtrOn f (Icc a b) c

f : ℝ → ℝ f' x a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hfI : f a = f b ne : Set.Nonempty (Icc a b) ⊢ ∃ c ∈ Ioo a b, IsExtrOn f (Icc a b) c

https://github.com/yuma-mizuno/lean-math-workshop.git

4a69b0130b276b45212e2b12b90032b146b56d67

Tutorial/Advanced/Analysis/Lecture2.lean

Tutorial.exists_local_extr_Ioo

[93, 1]

[115, 55]

have ⟨C, Cmem, Cge⟩ : ∃ C ∈ Icc a b, IsMaxOn f (Icc a b) C := by sorry

f : ℝ → ℝ f' x a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hfI : f a = f b ne : Set.Nonempty (Icc a b) ⊢ ∃ c ∈ Ioo a b, IsExtrOn f (Icc a b) c

f : ℝ → ℝ f' x a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hfI : f a = f b ne : Set.Nonempty (Icc a b) C : ℝ Cmem : C ∈ Icc a b Cge : IsMaxOn f (Icc a b) C ⊢ ∃ c ∈ Ioo a b, IsExtrOn f (Icc a b) c

https://github.com/yuma-mizuno/lean-math-workshop.git

4a69b0130b276b45212e2b12b90032b146b56d67

Tutorial/Advanced/Analysis/Lecture2.lean

Tutorial.exists_local_extr_Ioo

[93, 1]

[115, 55]

have ⟨c, cmem, cle⟩ : ∃ c ∈ Icc a b, IsMinOn f (Icc a b) c := by sorry

f : ℝ → ℝ f' x a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hfI : f a = f b ne : Set.Nonempty (Icc a b) C : ℝ Cmem : C ∈ Icc a b Cge : IsMaxOn f (Icc a b) C ⊢ ∃ c ∈ Ioo a b, IsExtrOn f (Icc a b) c

f : ℝ → ℝ f' x a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hfI : f a = f b ne : Set.Nonempty (Icc a b) C : ℝ Cmem : C ∈ Icc a b Cge : IsMaxOn f (Icc a b) C c : ℝ cmem : c ∈ Icc a b cle : IsMinOn f (Icc a b) c ⊢ ∃ c ∈ Ioo a b, IsExtrOn f (Icc a b) c

https://github.com/yuma-mizuno/lean-math-workshop.git

4a69b0130b276b45212e2b12b90032b146b56d67

Tutorial/Advanced/Analysis/Lecture2.lean

Tutorial.exists_local_extr_Ioo

[93, 1]

[115, 55]

change ∀ x ∈ Icc a b, f x ≤ f C at Cge

f : ℝ → ℝ f' x a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hfI : f a = f b ne : Set.Nonempty (Icc a b) C : ℝ Cmem : C ∈ Icc a b Cge : IsMaxOn f (Icc a b) C c : ℝ cmem : c ∈ Icc a b cle : IsMinOn f (Icc a b) c ⊢ ∃ c ∈ Ioo a b, IsExtrOn f (Icc a b) c

f : ℝ → ℝ f' x a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hfI : f a = f b ne : Set.Nonempty (Icc a b) C : ℝ Cmem : C ∈ Icc a b c : ℝ cmem : c ∈ Icc a b cle : IsMinOn f (Icc a b) c Cge : ∀ x ∈ Icc a b, f x ≤ f C ⊢ ∃ c ∈ Ioo a b, IsExtrOn f (Icc a b) c

https://github.com/yuma-mizuno/lean-math-workshop.git

4a69b0130b276b45212e2b12b90032b146b56d67

Tutorial/Advanced/Analysis/Lecture2.lean

Tutorial.exists_local_extr_Ioo

[93, 1]

[115, 55]

change ∀ x ∈ Icc a b, f c ≤ f x at cle

f : ℝ → ℝ f' x a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hfI : f a = f b ne : Set.Nonempty (Icc a b) C : ℝ Cmem : C ∈ Icc a b c : ℝ cmem : c ∈ Icc a b cle : IsMinOn f (Icc a b) c Cge : ∀ x ∈ Icc a b, f x ≤ f C ⊢ ∃ c ∈ Ioo a b, IsExtrOn f (Icc a b) c

f : ℝ → ℝ f' x a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hfI : f a = f b ne : Set.Nonempty (Icc a b) C : ℝ Cmem : C ∈ Icc a b c : ℝ cmem : c ∈ Icc a b Cge : ∀ x ∈ Icc a b, f x ≤ f C cle : ∀ x ∈ Icc a b, f c ≤ f x ⊢ ∃ c ∈ Ioo a b, IsExtrOn f (Icc a b) c

https://github.com/yuma-mizuno/lean-math-workshop.git

4a69b0130b276b45212e2b12b90032b146b56d67

Tutorial/Advanced/Analysis/Lecture2.lean

Tutorial.exists_local_extr_Ioo

[93, 1]

[115, 55]

by_cases hc : f c = f a

f : ℝ → ℝ f' x a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hfI : f a = f b ne : Set.Nonempty (Icc a b) C : ℝ Cmem : C ∈ Icc a b c : ℝ cmem : c ∈ Icc a b Cge : ∀ x ∈ Icc a b, f x ≤ f C cle : ∀ x ∈ Icc a b, f c ≤ f x ⊢ ∃ c ∈ Ioo a b, IsExtrOn f (Icc a b) c

case pos f : ℝ → ℝ f' x a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hfI : f a = f b ne : Set.Nonempty (Icc a b) C : ℝ Cmem : C ∈ Icc a b c : ℝ cmem : c ∈ Icc a b Cge : ∀ x ∈ Icc a b, f x ≤ f C cle : ∀ x ∈ Icc a b, f c ≤ f x hc : f c = f a ⊢ ∃ c ∈ Ioo a b, IsExtrOn f (Icc a b) c case neg f : ℝ → ℝ f' x a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hfI : f a = f b ne : Set.Nonempty (Icc a b) C : ℝ Cmem : C ∈ Icc a b c : ℝ cmem : c ∈ Icc a b Cge : ∀ x ∈ Icc a b, f x ≤ f C cle : ∀ x ∈ Icc a b, f c ≤ f x hc : ¬f c = f a ⊢ ∃ c ∈ Ioo a b, IsExtrOn f (Icc a b) c

https://github.com/yuma-mizuno/lean-math-workshop.git

4a69b0130b276b45212e2b12b90032b146b56d67

Tutorial/Advanced/Analysis/Lecture2.lean

Tutorial.exists_local_extr_Ioo

[93, 1]

[115, 55]

rcases this with ⟨c, cmem, hc⟩

f : ℝ → ℝ f' x a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hfI : f a = f b this : ∃ c ∈ Ioo a b, IsExtrOn f (Icc a b) c ⊢ ∃ c ∈ Ioo a b, IsLocalExtr f c

case intro.intro f : ℝ → ℝ f' x a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hfI : f a = f b c : ℝ cmem : c ∈ Ioo a b hc : IsExtrOn f (Icc a b) c ⊢ ∃ c ∈ Ioo a b, IsLocalExtr f c

https://github.com/yuma-mizuno/lean-math-workshop.git

4a69b0130b276b45212e2b12b90032b146b56d67

Tutorial/Advanced/Analysis/Lecture2.lean

Tutorial.exists_local_extr_Ioo

[93, 1]

[115, 55]

exists c, cmem

case intro.intro f : ℝ → ℝ f' x a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hfI : f a = f b c : ℝ cmem : c ∈ Ioo a b hc : IsExtrOn f (Icc a b) c ⊢ ∃ c ∈ Ioo a b, IsLocalExtr f c

case intro.intro f : ℝ → ℝ f' x a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hfI : f a = f b c : ℝ cmem : c ∈ Ioo a b hc : IsExtrOn f (Icc a b) c ⊢ IsLocalExtr f c

https://github.com/yuma-mizuno/lean-math-workshop.git

4a69b0130b276b45212e2b12b90032b146b56d67

Tutorial/Advanced/Analysis/Lecture2.lean

Tutorial.exists_local_extr_Ioo

[93, 1]

[115, 55]

apply hc.isLocalExtr <| Icc_mem_nhds cmem.1 cmem.2

case intro.intro f : ℝ → ℝ f' x a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hfI : f a = f b c : ℝ cmem : c ∈ Ioo a b hc : IsExtrOn f (Icc a b) c ⊢ IsLocalExtr f c

no goals

https://github.com/yuma-mizuno/lean-math-workshop.git

4a69b0130b276b45212e2b12b90032b146b56d67

Tutorial/Advanced/Analysis/Lecture2.lean

Tutorial.exists_local_extr_Ioo

[93, 1]

[115, 55]

sorry

f : ℝ → ℝ f' x a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hfI : f a = f b ne : Set.Nonempty (Icc a b) ⊢ ∃ C ∈ Icc a b, IsMaxOn f (Icc a b) C

no goals

https://github.com/yuma-mizuno/lean-math-workshop.git

4a69b0130b276b45212e2b12b90032b146b56d67

Tutorial/Advanced/Analysis/Lecture2.lean

Tutorial.exists_local_extr_Ioo

[93, 1]

[115, 55]

sorry

f : ℝ → ℝ f' x a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hfI : f a = f b ne : Set.Nonempty (Icc a b) C : ℝ Cmem : C ∈ Icc a b Cge : IsMaxOn f (Icc a b) C ⊢ ∃ c ∈ Icc a b, IsMinOn f (Icc a b) c

no goals

https://github.com/yuma-mizuno/lean-math-workshop.git

4a69b0130b276b45212e2b12b90032b146b56d67

Tutorial/Advanced/Analysis/Lecture2.lean

Tutorial.exists_local_extr_Ioo

[93, 1]

[115, 55]

by_cases hC : f C = f a

case pos f : ℝ → ℝ f' x a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hfI : f a = f b ne : Set.Nonempty (Icc a b) C : ℝ Cmem : C ∈ Icc a b c : ℝ cmem : c ∈ Icc a b Cge : ∀ x ∈ Icc a b, f x ≤ f C cle : ∀ x ∈ Icc a b, f c ≤ f x hc : f c = f a ⊢ ∃ c ∈ Ioo a b, IsExtrOn f (Icc a b) c

case pos f : ℝ → ℝ f' x a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hfI : f a = f b ne : Set.Nonempty (Icc a b) C : ℝ Cmem : C ∈ Icc a b c : ℝ cmem : c ∈ Icc a b Cge : ∀ x ∈ Icc a b, f x ≤ f C cle : ∀ x ∈ Icc a b, f c ≤ f x hc : f c = f a hC : f C = f a ⊢ ∃ c ∈ Ioo a b, IsExtrOn f (Icc a b) c case neg f : ℝ → ℝ f' x a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hfI : f a = f b ne : Set.Nonempty (Icc a b) C : ℝ Cmem : C ∈ Icc a b c : ℝ cmem : c ∈ Icc a b Cge : ∀ x ∈ Icc a b, f x ≤ f C cle : ∀ x ∈ Icc a b, f c ≤ f x hc : f c = f a hC : ¬f C = f a ⊢ ∃ c ∈ Ioo a b, IsExtrOn f (Icc a b) c

https://github.com/yuma-mizuno/lean-math-workshop.git

4a69b0130b276b45212e2b12b90032b146b56d67

Tutorial/Advanced/Analysis/Lecture2.lean

Tutorial.exists_local_extr_Ioo

[93, 1]

[115, 55]

have : ∀ x ∈ Icc a b, f x = f a := fun x hx ↦ le_antisymm (hC ▸ Cge x hx) (hc ▸ cle x hx)

case pos f : ℝ → ℝ f' x a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hfI : f a = f b ne : Set.Nonempty (Icc a b) C : ℝ Cmem : C ∈ Icc a b c : ℝ cmem : c ∈ Icc a b Cge : ∀ x ∈ Icc a b, f x ≤ f C cle : ∀ x ∈ Icc a b, f c ≤ f x hc : f c = f a hC : f C = f a ⊢ ∃ c ∈ Ioo a b, IsExtrOn f (Icc a b) c

case pos f : ℝ → ℝ f' x a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hfI : f a = f b ne : Set.Nonempty (Icc a b) C : ℝ Cmem : C ∈ Icc a b c : ℝ cmem : c ∈ Icc a b Cge : ∀ x ∈ Icc a b, f x ≤ f C cle : ∀ x ∈ Icc a b, f c ≤ f x hc : f c = f a hC : f C = f a this : ∀ x ∈ Icc a b, f x = f a ⊢ ∃ c ∈ Ioo a b, IsExtrOn f (Icc a b) c

https://github.com/yuma-mizuno/lean-math-workshop.git

4a69b0130b276b45212e2b12b90032b146b56d67

Tutorial/Advanced/Analysis/Lecture2.lean

Tutorial.exists_local_extr_Ioo

[93, 1]

[115, 55]

rcases nonempty_Ioo.2 hab with ⟨c', hc'⟩

case pos f : ℝ → ℝ f' x a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hfI : f a = f b ne : Set.Nonempty (Icc a b) C : ℝ Cmem : C ∈ Icc a b c : ℝ cmem : c ∈ Icc a b Cge : ∀ x ∈ Icc a b, f x ≤ f C cle : ∀ x ∈ Icc a b, f c ≤ f x hc : f c = f a hC : f C = f a this : ∀ x ∈ Icc a b, f x = f a ⊢ ∃ c ∈ Ioo a b, IsExtrOn f (Icc a b) c

case pos.intro f : ℝ → ℝ f' x a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hfI : f a = f b ne : Set.Nonempty (Icc a b) C : ℝ Cmem : C ∈ Icc a b c : ℝ cmem : c ∈ Icc a b Cge : ∀ x ∈ Icc a b, f x ≤ f C cle : ∀ x ∈ Icc a b, f c ≤ f x hc : f c = f a hC : f C = f a this : ∀ x ∈ Icc a b, f x = f a c' : ℝ hc' : c' ∈ Ioo a b ⊢ ∃ c ∈ Ioo a b, IsExtrOn f (Icc a b) c

https://github.com/yuma-mizuno/lean-math-workshop.git

4a69b0130b276b45212e2b12b90032b146b56d67

Tutorial/Advanced/Analysis/Lecture2.lean

Tutorial.exists_local_extr_Ioo

[93, 1]

[115, 55]

refine ⟨c', hc', Or.inl fun x hx ↦ ?_⟩

case pos.intro f : ℝ → ℝ f' x a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hfI : f a = f b ne : Set.Nonempty (Icc a b) C : ℝ Cmem : C ∈ Icc a b c : ℝ cmem : c ∈ Icc a b Cge : ∀ x ∈ Icc a b, f x ≤ f C cle : ∀ x ∈ Icc a b, f c ≤ f x hc : f c = f a hC : f C = f a this : ∀ x ∈ Icc a b, f x = f a c' : ℝ hc' : c' ∈ Ioo a b ⊢ ∃ c ∈ Ioo a b, IsExtrOn f (Icc a b) c

case pos.intro f : ℝ → ℝ f' x✝ a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hfI : f a = f b ne : Set.Nonempty (Icc a b) C : ℝ Cmem : C ∈ Icc a b c : ℝ cmem : c ∈ Icc a b Cge : ∀ x ∈ Icc a b, f x ≤ f C cle : ∀ x ∈ Icc a b, f c ≤ f x hc : f c = f a hC : f C = f a this : ∀ x ∈ Icc a b, f x = f a c' : ℝ hc' : c' ∈ Ioo a b x : ℝ hx : x ∈ Icc a b ⊢ x ∈ {x | (fun x => f c' ≤ f x) x}

https://github.com/yuma-mizuno/lean-math-workshop.git

4a69b0130b276b45212e2b12b90032b146b56d67

Tutorial/Advanced/Analysis/Lecture2.lean

Tutorial.exists_local_extr_Ioo

[93, 1]

[115, 55]

simp [this x hx, this c' (Ioo_subset_Icc_self hc')]

case pos.intro f : ℝ → ℝ f' x✝ a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hfI : f a = f b ne : Set.Nonempty (Icc a b) C : ℝ Cmem : C ∈ Icc a b c : ℝ cmem : c ∈ Icc a b Cge : ∀ x ∈ Icc a b, f x ≤ f C cle : ∀ x ∈ Icc a b, f c ≤ f x hc : f c = f a hC : f C = f a this : ∀ x ∈ Icc a b, f x = f a c' : ℝ hc' : c' ∈ Ioo a b x : ℝ hx : x ∈ Icc a b ⊢ x ∈ {x | (fun x => f c' ≤ f x) x}

no goals

https://github.com/yuma-mizuno/lean-math-workshop.git

4a69b0130b276b45212e2b12b90032b146b56d67

Tutorial/Advanced/Analysis/Lecture2.lean

Tutorial.exists_local_extr_Ioo

[93, 1]

[115, 55]

refine ⟨C, ⟨lt_of_le_of_ne Cmem.1 <| mt ?_ hC, lt_of_le_of_ne Cmem.2 <| mt ?_ hC⟩, Or.inr Cge⟩

case neg f : ℝ → ℝ f' x a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hfI : f a = f b ne : Set.Nonempty (Icc a b) C : ℝ Cmem : C ∈ Icc a b c : ℝ cmem : c ∈ Icc a b Cge : ∀ x ∈ Icc a b, f x ≤ f C cle : ∀ x ∈ Icc a b, f c ≤ f x hc : f c = f a hC : ¬f C = f a ⊢ ∃ c ∈ Ioo a b, IsExtrOn f (Icc a b) c

case neg.refine_1 f : ℝ → ℝ f' x a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hfI : f a = f b ne : Set.Nonempty (Icc a b) C : ℝ Cmem : C ∈ Icc a b c : ℝ cmem : c ∈ Icc a b Cge : ∀ x ∈ Icc a b, f x ≤ f C cle : ∀ x ∈ Icc a b, f c ≤ f x hc : f c = f a hC : ¬f C = f a ⊢ a = C → f C = f a case neg.refine_2 f : ℝ → ℝ f' x a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hfI : f a = f b ne : Set.Nonempty (Icc a b) C : ℝ Cmem : C ∈ Icc a b c : ℝ cmem : c ∈ Icc a b Cge : ∀ x ∈ Icc a b, f x ≤ f C cle : ∀ x ∈ Icc a b, f c ≤ f x hc : f c = f a hC : ¬f C = f a ⊢ C = b → f C = f a

https://github.com/yuma-mizuno/lean-math-workshop.git

4a69b0130b276b45212e2b12b90032b146b56d67

Tutorial/Advanced/Analysis/Lecture2.lean

Tutorial.exists_local_extr_Ioo

[93, 1]

[115, 55]

exacts [fun h ↦ by rw [h], fun h ↦ by rw [h, hfI]]

case neg.refine_1 f : ℝ → ℝ f' x a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hfI : f a = f b ne : Set.Nonempty (Icc a b) C : ℝ Cmem : C ∈ Icc a b c : ℝ cmem : c ∈ Icc a b Cge : ∀ x ∈ Icc a b, f x ≤ f C cle : ∀ x ∈ Icc a b, f c ≤ f x hc : f c = f a hC : ¬f C = f a ⊢ a = C → f C = f a case neg.refine_2 f : ℝ → ℝ f' x a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hfI : f a = f b ne : Set.Nonempty (Icc a b) C : ℝ Cmem : C ∈ Icc a b c : ℝ cmem : c ∈ Icc a b Cge : ∀ x ∈ Icc a b, f x ≤ f C cle : ∀ x ∈ Icc a b, f c ≤ f x hc : f c = f a hC : ¬f C = f a ⊢ C = b → f C = f a

no goals

https://github.com/yuma-mizuno/lean-math-workshop.git

4a69b0130b276b45212e2b12b90032b146b56d67

Tutorial/Advanced/Analysis/Lecture2.lean

Tutorial.exists_local_extr_Ioo

[93, 1]

[115, 55]

rw [h]

f : ℝ → ℝ f' x a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hfI : f a = f b ne : Set.Nonempty (Icc a b) C : ℝ Cmem : C ∈ Icc a b c : ℝ cmem : c ∈ Icc a b Cge : ∀ x ∈ Icc a b, f x ≤ f C cle : ∀ x ∈ Icc a b, f c ≤ f x hc : f c = f a hC : ¬f C = f a h : a = C ⊢ f C = f a

no goals

https://github.com/yuma-mizuno/lean-math-workshop.git

4a69b0130b276b45212e2b12b90032b146b56d67

Tutorial/Advanced/Analysis/Lecture2.lean

Tutorial.exists_local_extr_Ioo

[93, 1]

[115, 55]

rw [h, hfI]

f : ℝ → ℝ f' x a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hfI : f a = f b ne : Set.Nonempty (Icc a b) C : ℝ Cmem : C ∈ Icc a b c : ℝ cmem : c ∈ Icc a b Cge : ∀ x ∈ Icc a b, f x ≤ f C cle : ∀ x ∈ Icc a b, f c ≤ f x hc : f c = f a hC : ¬f C = f a h : C = b ⊢ f C = f a

no goals

https://github.com/yuma-mizuno/lean-math-workshop.git

4a69b0130b276b45212e2b12b90032b146b56d67

Tutorial/Advanced/Analysis/Lecture2.lean

Tutorial.exists_local_extr_Ioo

[93, 1]

[115, 55]

refine ⟨c, ⟨lt_of_le_of_ne cmem.1 <| mt ?_ hc, lt_of_le_of_ne cmem.2 <| mt ?_ hc⟩, Or.inl cle⟩

case neg f : ℝ → ℝ f' x a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hfI : f a = f b ne : Set.Nonempty (Icc a b) C : ℝ Cmem : C ∈ Icc a b c : ℝ cmem : c ∈ Icc a b Cge : ∀ x ∈ Icc a b, f x ≤ f C cle : ∀ x ∈ Icc a b, f c ≤ f x hc : ¬f c = f a ⊢ ∃ c ∈ Ioo a b, IsExtrOn f (Icc a b) c

case neg.refine_1 f : ℝ → ℝ f' x a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hfI : f a = f b ne : Set.Nonempty (Icc a b) C : ℝ Cmem : C ∈ Icc a b c : ℝ cmem : c ∈ Icc a b Cge : ∀ x ∈ Icc a b, f x ≤ f C cle : ∀ x ∈ Icc a b, f c ≤ f x hc : ¬f c = f a ⊢ a = c → f c = f a case neg.refine_2 f : ℝ → ℝ f' x a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hfI : f a = f b ne : Set.Nonempty (Icc a b) C : ℝ Cmem : C ∈ Icc a b c : ℝ cmem : c ∈ Icc a b Cge : ∀ x ∈ Icc a b, f x ≤ f C cle : ∀ x ∈ Icc a b, f c ≤ f x hc : ¬f c = f a ⊢ c = b → f c = f a

https://github.com/yuma-mizuno/lean-math-workshop.git

4a69b0130b276b45212e2b12b90032b146b56d67

Tutorial/Advanced/Analysis/Lecture2.lean

Tutorial.exists_local_extr_Ioo

[93, 1]

[115, 55]

exacts [fun h ↦ by rw [h], fun h ↦ by rw [h, hfI]]

case neg.refine_1 f : ℝ → ℝ f' x a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hfI : f a = f b ne : Set.Nonempty (Icc a b) C : ℝ Cmem : C ∈ Icc a b c : ℝ cmem : c ∈ Icc a b Cge : ∀ x ∈ Icc a b, f x ≤ f C cle : ∀ x ∈ Icc a b, f c ≤ f x hc : ¬f c = f a ⊢ a = c → f c = f a case neg.refine_2 f : ℝ → ℝ f' x a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hfI : f a = f b ne : Set.Nonempty (Icc a b) C : ℝ Cmem : C ∈ Icc a b c : ℝ cmem : c ∈ Icc a b Cge : ∀ x ∈ Icc a b, f x ≤ f C cle : ∀ x ∈ Icc a b, f c ≤ f x hc : ¬f c = f a ⊢ c = b → f c = f a

no goals

https://github.com/yuma-mizuno/lean-math-workshop.git

4a69b0130b276b45212e2b12b90032b146b56d67

Tutorial/Advanced/Analysis/Lecture2.lean

Tutorial.exists_local_extr_Ioo

[93, 1]

[115, 55]

rw [h]

f : ℝ → ℝ f' x a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hfI : f a = f b ne : Set.Nonempty (Icc a b) C : ℝ Cmem : C ∈ Icc a b c : ℝ cmem : c ∈ Icc a b Cge : ∀ x ∈ Icc a b, f x ≤ f C cle : ∀ x ∈ Icc a b, f c ≤ f x hc : ¬f c = f a h : a = c ⊢ f c = f a

no goals

https://github.com/yuma-mizuno/lean-math-workshop.git

4a69b0130b276b45212e2b12b90032b146b56d67

Tutorial/Advanced/Analysis/Lecture2.lean

Tutorial.exists_local_extr_Ioo

[93, 1]

[115, 55]

rw [h, hfI]

f : ℝ → ℝ f' x a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hfI : f a = f b ne : Set.Nonempty (Icc a b) C : ℝ Cmem : C ∈ Icc a b c : ℝ cmem : c ∈ Icc a b Cge : ∀ x ∈ Icc a b, f x ≤ f C cle : ∀ x ∈ Icc a b, f c ≤ f x hc : ¬f c = f a h : c = b ⊢ f c = f a

no goals

https://github.com/yuma-mizuno/lean-math-workshop.git

4a69b0130b276b45212e2b12b90032b146b56d67

Tutorial/Advanced/Analysis/Lecture2.lean

Tutorial.exists_hasDerivAt_eq_zero

[120, 1]

[122, 8]

sorry

f✝ : ℝ → ℝ f'✝ x a✝ b✝ : ℝ f f' g g' : ℝ → ℝ a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hfI : f a = f b hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x ⊢ ∃ c ∈ Ioo a b, f' c = 0

no goals

https://github.com/yuma-mizuno/lean-math-workshop.git

4a69b0130b276b45212e2b12b90032b146b56d67

Tutorial/Advanced/Analysis/Lecture2.lean

Tutorial.exists_ratio_hasDerivAt_eq_ratio_slope

[125, 1]

[132, 8]

let h x := (g b - g a) * f x - (f b - f a) * g x

f✝ : ℝ → ℝ f'✝ x a✝ b✝ : ℝ f f' g g' : ℝ → ℝ a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x hgc : ContinuousOn g (Icc a b) hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x ⊢ ∃ c ∈ Ioo a b, (g b - g a) * f' c = (f b - f a) * g' c

f✝ : ℝ → ℝ f'✝ x a✝ b✝ : ℝ f f' g g' : ℝ → ℝ a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x hgc : ContinuousOn g (Icc a b) hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x h : ℝ → ℝ := fun x => (g b - g a) * f x - (f b - f a) * g x ⊢ ∃ c ∈ Ioo a b, (g b - g a) * f' c = (f b - f a) * g' c

https://github.com/yuma-mizuno/lean-math-workshop.git

4a69b0130b276b45212e2b12b90032b146b56d67

Tutorial/Advanced/Analysis/Lecture2.lean

Tutorial.exists_ratio_hasDerivAt_eq_ratio_slope

[125, 1]

[132, 8]

have hhc : ContinuousOn h (Icc a b) := (continuousOn_const.mul hfc).sub (continuousOn_const.mul hgc)

f✝ : ℝ → ℝ f'✝ x a✝ b✝ : ℝ f f' g g' : ℝ → ℝ a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x hgc : ContinuousOn g (Icc a b) hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x h : ℝ → ℝ := fun x => (g b - g a) * f x - (f b - f a) * g x ⊢ ∃ c ∈ Ioo a b, (g b - g a) * f' c = (f b - f a) * g' c

f✝ : ℝ → ℝ f'✝ x a✝ b✝ : ℝ f f' g g' : ℝ → ℝ a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x hgc : ContinuousOn g (Icc a b) hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x h : ℝ → ℝ := fun x => (g b - g a) * f x - (f b - f a) * g x hhc : ContinuousOn h (Icc a b) ⊢ ∃ c ∈ Ioo a b, (g b - g a) * f' c = (f b - f a) * g' c

https://github.com/yuma-mizuno/lean-math-workshop.git

4a69b0130b276b45212e2b12b90032b146b56d67

Tutorial/Advanced/Analysis/Lecture2.lean

Tutorial.exists_ratio_hasDerivAt_eq_ratio_slope

[125, 1]

[132, 8]

sorry

f✝ : ℝ → ℝ f'✝ x a✝ b✝ : ℝ f f' g g' : ℝ → ℝ a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x hgc : ContinuousOn g (Icc a b) hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x h : ℝ → ℝ := fun x => (g b - g a) * f x - (f b - f a) * g x hhc : ContinuousOn h (Icc a b) ⊢ ∃ c ∈ Ioo a b, (g b - g a) * f' c = (f b - f a) * g' c

no goals

https://github.com/yuma-mizuno/lean-math-workshop.git

4a69b0130b276b45212e2b12b90032b146b56d67

Tutorial/Advanced/Analysis/Lecture2.lean

Tutorial.exists_hasDerivAt_eq_slope

[138, 1]

[141, 8]

sorry

f✝ : ℝ → ℝ f'✝ x a✝ b✝ : ℝ f f' g g' : ℝ → ℝ a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x ⊢ ∃ c ∈ Ioo a b, f' c = (f b - f a) / (b - a)

no goals

https://github.com/yuma-mizuno/lean-math-workshop.git

4a69b0130b276b45212e2b12b90032b146b56d67

Tutorial/Advanced/Category/Lecture1.lean

Tutorial.comp_app

[109, 1]

[110, 6]

rfl

C : Type u inst✝ : Category C a b c d e : C X Y Z : Type f : Hom X Y g : Hom Y Z x : X ⊢ (f ≫ g) x = g (f x)

no goals

https://github.com/yuma-mizuno/lean-math-workshop.git

4a69b0130b276b45212e2b12b90032b146b56d67

Tutorial/Advanced/Category/Lecture1.lean

Tutorial.id_app

[113, 1]

[114, 6]

rfl

C : Type u inst✝ : Category C a b c d e : C X : Type x : X ⊢ 𝟙 X x = x

no goals

https://github.com/yuma-mizuno/lean-math-workshop.git

4a69b0130b276b45212e2b12b90032b146b56d67

Solution/Advanced/Category/Lecture2.lean

Tutorial.Category.Initial.uniq'

[28, 1]

[30, 45]

rw [h.uniq f]

C : Type u inst✝ : Category C a : C h : Initial a b : C f g : Hom a b ⊢ f = h.fromInitial b

no goals

https://github.com/yuma-mizuno/lean-math-workshop.git

4a69b0130b276b45212e2b12b90032b146b56d67

Solution/Advanced/Category/Lecture2.lean

Tutorial.Category.Initial.uniq'

[28, 1]

[30, 45]

rw [h.uniq g]

C : Type u inst✝ : Category C a : C h : Initial a b : C f g : Hom a b ⊢ h.fromInitial b = g

no goals

https://github.com/yuma-mizuno/lean-math-workshop.git

4a69b0130b276b45212e2b12b90032b146b56d67

Solution/Advanced/Category/Lecture2.lean

Tutorial.Coequalizer.hom_id

[329, 1]

[329, 71]

cases i <;> rfl

J : Type u₁ inst✝¹ : Category J C : Type u₂ inst✝ : Category C F : Functor J C i : Shape ⊢ ShapeHom.id i = 𝟙 i

no goals

https://github.com/yuma-mizuno/lean-math-workshop.git

4a69b0130b276b45212e2b12b90032b146b56d67

Solution/Advanced/Analysis/Lecture2.lean

Tutorial.IsLocalMax.hasDerivAt_eq_zero

[45, 1]

[80, 15]

apply le_antisymm ?right ?left

f : ℝ → ℝ f' x a b : ℝ h : IsLocalMax f a hf : HasDerivAt f f' a ⊢ f' = 0

case right f : ℝ → ℝ f' x a b : ℝ h : IsLocalMax f a hf : HasDerivAt f f' a ⊢ f' ≤ 0 case left f : ℝ → ℝ f' x a b : ℝ h : IsLocalMax f a hf : HasDerivAt f f' a ⊢ 0 ≤ f'

https://github.com/yuma-mizuno/lean-math-workshop.git

4a69b0130b276b45212e2b12b90032b146b56d67

Solution/Advanced/Analysis/Lecture2.lean

Tutorial.IsLocalMax.hasDerivAt_eq_zero

[45, 1]

[80, 15]

have hf : Tendsto (fun x ↦ (f x - f a) / (x - a)) (𝓝[>] a) (𝓝 f') := by rw [hasDerivAt_iff_tendsto_slope] at hf apply hf.mono_left (nhds_right'_le_nhds_ne a)

f : ℝ → ℝ f' x a b : ℝ h : IsLocalMax f a hf : HasDerivAt f f' a ⊢ f' ≤ 0

f : ℝ → ℝ f' x a b : ℝ h : IsLocalMax f a hf✝ : HasDerivAt f f' a hf : Tendsto (fun x => (f x - f a) / (x - a)) (𝓝[>] a) (𝓝 f') ⊢ f' ≤ 0

https://github.com/yuma-mizuno/lean-math-workshop.git

4a69b0130b276b45212e2b12b90032b146b56d67

Solution/Advanced/Analysis/Lecture2.lean

Tutorial.IsLocalMax.hasDerivAt_eq_zero

[45, 1]

[80, 15]

suffices ∀ᶠ x in 𝓝[>] a, (f x - f a) / (x - a) ≤ 0 from le_of_tendsto hf this

f : ℝ → ℝ f' x a b : ℝ h : IsLocalMax f a hf✝ : HasDerivAt f f' a hf : Tendsto (fun x => (f x - f a) / (x - a)) (𝓝[>] a) (𝓝 f') ⊢ f' ≤ 0

f : ℝ → ℝ f' x a b : ℝ h : IsLocalMax f a hf✝ : HasDerivAt f f' a hf : Tendsto (fun x => (f x - f a) / (x - a)) (𝓝[>] a) (𝓝 f') ⊢ ∀ᶠ (x : ℝ) in 𝓝[>] a, (f x - f a) / (x - a) ≤ 0

https://github.com/yuma-mizuno/lean-math-workshop.git

4a69b0130b276b45212e2b12b90032b146b56d67

Solution/Advanced/Analysis/Lecture2.lean

Tutorial.IsLocalMax.hasDerivAt_eq_zero

[45, 1]

[80, 15]

have ha : ∀ᶠ x in 𝓝[>] a, a < x := eventually_nhdsWithin_of_forall fun x hx ↦ hx

f : ℝ → ℝ f' x a b : ℝ h : IsLocalMax f a hf✝ : HasDerivAt f f' a hf : Tendsto (fun x => (f x - f a) / (x - a)) (𝓝[>] a) (𝓝 f') ⊢ ∀ᶠ (x : ℝ) in 𝓝[>] a, (f x - f a) / (x - a) ≤ 0

f : ℝ → ℝ f' x a b : ℝ h : IsLocalMax f a hf✝ : HasDerivAt f f' a hf : Tendsto (fun x => (f x - f a) / (x - a)) (𝓝[>] a) (𝓝 f') ha : ∀ᶠ (x : ℝ) in 𝓝[>] a, a < x ⊢ ∀ᶠ (x : ℝ) in 𝓝[>] a, (f x - f a) / (x - a) ≤ 0

https://github.com/yuma-mizuno/lean-math-workshop.git

4a69b0130b276b45212e2b12b90032b146b56d67

Solution/Advanced/Analysis/Lecture2.lean

Tutorial.IsLocalMax.hasDerivAt_eq_zero

[45, 1]

[80, 15]

have h : ∀ᶠ x in 𝓝[>] a, f x ≤ f a := h.filter_mono nhdsWithin_le_nhds

f : ℝ → ℝ f' x a b : ℝ h : IsLocalMax f a hf✝ : HasDerivAt f f' a hf : Tendsto (fun x => (f x - f a) / (x - a)) (𝓝[>] a) (𝓝 f') ha : ∀ᶠ (x : ℝ) in 𝓝[>] a, a < x ⊢ ∀ᶠ (x : ℝ) in 𝓝[>] a, (f x - f a) / (x - a) ≤ 0

f : ℝ → ℝ f' x a b : ℝ h✝ : IsLocalMax f a hf✝ : HasDerivAt f f' a hf : Tendsto (fun x => (f x - f a) / (x - a)) (𝓝[>] a) (𝓝 f') ha : ∀ᶠ (x : ℝ) in 𝓝[>] a, a < x h : ∀ᶠ (x : ℝ) in 𝓝[>] a, f x ≤ f a ⊢ ∀ᶠ (x : ℝ) in 𝓝[>] a, (f x - f a) / (x - a) ≤ 0

https://github.com/yuma-mizuno/lean-math-workshop.git

4a69b0130b276b45212e2b12b90032b146b56d67

Solution/Advanced/Analysis/Lecture2.lean

Tutorial.IsLocalMax.hasDerivAt_eq_zero

[45, 1]

[80, 15]

filter_upwards [ha, h]

f : ℝ → ℝ f' x a b : ℝ h✝ : IsLocalMax f a hf✝ : HasDerivAt f f' a hf : Tendsto (fun x => (f x - f a) / (x - a)) (𝓝[>] a) (𝓝 f') ha : ∀ᶠ (x : ℝ) in 𝓝[>] a, a < x h : ∀ᶠ (x : ℝ) in 𝓝[>] a, f x ≤ f a ⊢ ∀ᶠ (x : ℝ) in 𝓝[>] a, (f x - f a) / (x - a) ≤ 0

case h f : ℝ → ℝ f' x a b : ℝ h✝ : IsLocalMax f a hf✝ : HasDerivAt f f' a hf : Tendsto (fun x => (f x - f a) / (x - a)) (𝓝[>] a) (𝓝 f') ha : ∀ᶠ (x : ℝ) in 𝓝[>] a, a < x h : ∀ᶠ (x : ℝ) in 𝓝[>] a, f x ≤ f a ⊢ ∀ (a_1 : ℝ), a < a_1 → f a_1 ≤ f a → (f a_1 - f a) / (a_1 - a) ≤ 0

https://github.com/yuma-mizuno/lean-math-workshop.git

4a69b0130b276b45212e2b12b90032b146b56d67

Solution/Advanced/Analysis/Lecture2.lean

Tutorial.IsLocalMax.hasDerivAt_eq_zero

[45, 1]

[80, 15]

intro x ha h

case h f : ℝ → ℝ f' x a b : ℝ h✝ : IsLocalMax f a hf✝ : HasDerivAt f f' a hf : Tendsto (fun x => (f x - f a) / (x - a)) (𝓝[>] a) (𝓝 f') ha : ∀ᶠ (x : ℝ) in 𝓝[>] a, a < x h : ∀ᶠ (x : ℝ) in 𝓝[>] a, f x ≤ f a ⊢ ∀ (a_1 : ℝ), a < a_1 → f a_1 ≤ f a → (f a_1 - f a) / (a_1 - a) ≤ 0

case h f : ℝ → ℝ f' x✝ a b : ℝ h✝¹ : IsLocalMax f a hf✝ : HasDerivAt f f' a hf : Tendsto (fun x => (f x - f a) / (x - a)) (𝓝[>] a) (𝓝 f') ha✝ : ∀ᶠ (x : ℝ) in 𝓝[>] a, a < x h✝ : ∀ᶠ (x : ℝ) in 𝓝[>] a, f x ≤ f a x : ℝ ha : a < x h : f x ≤ f a ⊢ (f x - f a) / (x - a) ≤ 0

https://github.com/yuma-mizuno/lean-math-workshop.git

4a69b0130b276b45212e2b12b90032b146b56d67

Solution/Advanced/Analysis/Lecture2.lean

Tutorial.IsLocalMax.hasDerivAt_eq_zero

[45, 1]

[80, 15]

apply div_nonpos_of_nonpos_of_nonneg

case h f : ℝ → ℝ f' x✝ a b : ℝ h✝¹ : IsLocalMax f a hf✝ : HasDerivAt f f' a hf : Tendsto (fun x => (f x - f a) / (x - a)) (𝓝[>] a) (𝓝 f') ha✝ : ∀ᶠ (x : ℝ) in 𝓝[>] a, a < x h✝ : ∀ᶠ (x : ℝ) in 𝓝[>] a, f x ≤ f a x : ℝ ha : a < x h : f x ≤ f a ⊢ (f x - f a) / (x - a) ≤ 0

case h.ha f : ℝ → ℝ f' x✝ a b : ℝ h✝¹ : IsLocalMax f a hf✝ : HasDerivAt f f' a hf : Tendsto (fun x => (f x - f a) / (x - a)) (𝓝[>] a) (𝓝 f') ha✝ : ∀ᶠ (x : ℝ) in 𝓝[>] a, a < x h✝ : ∀ᶠ (x : ℝ) in 𝓝[>] a, f x ≤ f a x : ℝ ha : a < x h : f x ≤ f a ⊢ f x - f a ≤ 0 case h.hb f : ℝ → ℝ f' x✝ a b : ℝ h✝¹ : IsLocalMax f a hf✝ : HasDerivAt f f' a hf : Tendsto (fun x => (f x - f a) / (x - a)) (𝓝[>] a) (𝓝 f') ha✝ : ∀ᶠ (x : ℝ) in 𝓝[>] a, a < x h✝ : ∀ᶠ (x : ℝ) in 𝓝[>] a, f x ≤ f a x : ℝ ha : a < x h : f x ≤ f a ⊢ 0 ≤ x - a

https://github.com/yuma-mizuno/lean-math-workshop.git

4a69b0130b276b45212e2b12b90032b146b56d67

Solution/Advanced/Analysis/Lecture2.lean

Tutorial.IsLocalMax.hasDerivAt_eq_zero

[45, 1]

[80, 15]

rw [hasDerivAt_iff_tendsto_slope] at hf

f : ℝ → ℝ f' x a b : ℝ h : IsLocalMax f a hf : HasDerivAt f f' a ⊢ Tendsto (fun x => (f x - f a) / (x - a)) (𝓝[>] a) (𝓝 f')

f : ℝ → ℝ f' x a b : ℝ h : IsLocalMax f a hf : Tendsto (fun x => (f x - f a) / (x - a)) (𝓝[≠] a) (𝓝 f') ⊢ Tendsto (fun x => (f x - f a) / (x - a)) (𝓝[>] a) (𝓝 f')

https://github.com/yuma-mizuno/lean-math-workshop.git

4a69b0130b276b45212e2b12b90032b146b56d67

Solution/Advanced/Analysis/Lecture2.lean

Tutorial.IsLocalMax.hasDerivAt_eq_zero

[45, 1]

[80, 15]

apply hf.mono_left (nhds_right'_le_nhds_ne a)

f : ℝ → ℝ f' x a b : ℝ h : IsLocalMax f a hf : Tendsto (fun x => (f x - f a) / (x - a)) (𝓝[≠] a) (𝓝 f') ⊢ Tendsto (fun x => (f x - f a) / (x - a)) (𝓝[>] a) (𝓝 f')

no goals

https://github.com/yuma-mizuno/lean-math-workshop.git

4a69b0130b276b45212e2b12b90032b146b56d67

Solution/Advanced/Analysis/Lecture2.lean

Tutorial.IsLocalMax.hasDerivAt_eq_zero

[45, 1]

[80, 15]

linarith only [h]

case h.ha f : ℝ → ℝ f' x✝ a b : ℝ h✝¹ : IsLocalMax f a hf✝ : HasDerivAt f f' a hf : Tendsto (fun x => (f x - f a) / (x - a)) (𝓝[>] a) (𝓝 f') ha✝ : ∀ᶠ (x : ℝ) in 𝓝[>] a, a < x h✝ : ∀ᶠ (x : ℝ) in 𝓝[>] a, f x ≤ f a x : ℝ ha : a < x h : f x ≤ f a ⊢ f x - f a ≤ 0

no goals

https://github.com/yuma-mizuno/lean-math-workshop.git

4a69b0130b276b45212e2b12b90032b146b56d67

Solution/Advanced/Analysis/Lecture2.lean

Tutorial.IsLocalMax.hasDerivAt_eq_zero

[45, 1]

[80, 15]

linarith only [ha]

case h.hb f : ℝ → ℝ f' x✝ a b : ℝ h✝¹ : IsLocalMax f a hf✝ : HasDerivAt f f' a hf : Tendsto (fun x => (f x - f a) / (x - a)) (𝓝[>] a) (𝓝 f') ha✝ : ∀ᶠ (x : ℝ) in 𝓝[>] a, a < x h✝ : ∀ᶠ (x : ℝ) in 𝓝[>] a, f x ≤ f a x : ℝ ha : a < x h : f x ≤ f a ⊢ 0 ≤ x - a

no goals

https://github.com/yuma-mizuno/lean-math-workshop.git

4a69b0130b276b45212e2b12b90032b146b56d67

Solution/Advanced/Analysis/Lecture2.lean

Tutorial.IsLocalMax.hasDerivAt_eq_zero

[45, 1]

[80, 15]

have hf : Tendsto (fun x ↦ (f x - f a) / (x - a)) (𝓝[<] a) (𝓝 f') := by rw [hasDerivAt_iff_tendsto_slope] at hf apply hf.mono_left (nhds_left'_le_nhds_ne a)

f : ℝ → ℝ f' x a b : ℝ h : IsLocalMax f a hf : HasDerivAt f f' a ⊢ 0 ≤ f'

f : ℝ → ℝ f' x a b : ℝ h : IsLocalMax f a hf✝ : HasDerivAt f f' a hf : Tendsto (fun x => (f x - f a) / (x - a)) (𝓝[<] a) (𝓝 f') ⊢ 0 ≤ f'

https://github.com/yuma-mizuno/lean-math-workshop.git

4a69b0130b276b45212e2b12b90032b146b56d67

Solution/Advanced/Analysis/Lecture2.lean

Tutorial.IsLocalMax.hasDerivAt_eq_zero

[45, 1]

[80, 15]

suffices ∀ᶠ x in 𝓝[<] a, (f x - f a) / (x - a) ≥ 0 from ge_of_tendsto hf this

f : ℝ → ℝ f' x a b : ℝ h : IsLocalMax f a hf✝ : HasDerivAt f f' a hf : Tendsto (fun x => (f x - f a) / (x - a)) (𝓝[<] a) (𝓝 f') ⊢ 0 ≤ f'

f : ℝ → ℝ f' x a b : ℝ h : IsLocalMax f a hf✝ : HasDerivAt f f' a hf : Tendsto (fun x => (f x - f a) / (x - a)) (𝓝[<] a) (𝓝 f') ⊢ ∀ᶠ (x : ℝ) in 𝓝[<] a, (f x - f a) / (x - a) ≥ 0