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 | input stringlengths 73 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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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) ∈ Invers... |
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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) ∈ StdInversio... | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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) ∈ StdInversio... | 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 | Please generate a tactic in lean4 to solve the state.
STATE:
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 ≠... |
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
f g : Equiv.Perm α
a b : α
hab : f a = b
⊢ g (f (g.symm (g a))) = g b
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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)
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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 +... | Please generate a tactic in lean4 to solve the state.
STATE:
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))
TACTIC:
|
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 +... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
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 => Equ... |
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)) | Please generate a tactic in lean4 to solve the state.
STATE:
i k : ℕ
⊢ Word.toPerm (WordForSwap i k) = Equiv.swap i (i + (k + 1))
TACTIC:
|
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)) | Please generate a tactic in lean4 to solve the state.
STATE:
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))
TACT... |
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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))
TACTIC:
|
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)) | Please generate a tactic in lean4 to solve the state.
STATE:
case zero
i : ℕ
⊢ Word.toPerm (WordForSwap i Nat.zero) = Equiv.swap i (i + (Nat.zero + 1))
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
case zero
i : ℕ
⊢ Word.toPerm (WordForSwap i 0) = Equiv.swap i (i + (0 + 1))
TACTIC:
|
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)) | Please generate a tactic in lean4 to solve the state.
STATE:
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))
TACTIC:
|
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)) | Please generate a tactic in lean4 to solve the state.
STATE:
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))
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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))
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
i k : ℕ
ih : Word.toPerm (WordForSwap i k) = Equiv.swap i (i + (k + 1))
⊢ i ≠ i + (k + 1)
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
i k : ℕ
ih : Word.toPerm (WordForSwap i k) = Equiv.swap i (i + (k + 1))
⊢ i ≠ i + (k + 2)
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝ : DecidableEq α
f : Equiv.Perm α
a : α
⊢ a ∉ support (Equiv.swap a (f a) * f)
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
f g : Equiv.Perm ℕ
⊢ Inversions (f * g) = Prod.map ⇑g.symm ⇑g.symm '' Inversions f ∩ Inversions g
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℕ → ℕ
n x : ℕ
hx : n ≤ x
⊢ ¬x < n
TACTIC:
|
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 ! | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
⊢ Fintype.card { g // ⇑g ≤ ⇑(truncate id n) } = n !
TACTIC:
|
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 ! | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
⊢ Fintype.card ((i : Fin n) → Fin (id ↑i + 1)) = n !
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
⊢ (Finset.prod Finset.univ fun x => ↑x + 1) = n !
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
⊢ (Finset.prod Finset.univ fun x => ↑x + 1) = Finset.prod (Finset.range n) fun x => x + 1
TACTIC:
|
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' | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℝ
f' x a b : ℝ
h : IsLocalMax f a
hf : HasDerivAt f f' a
⊢ f' = 0
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℝ
f' x a b : ℝ
h : IsLocalMax f a
hf : HasDerivAt f f' a
⊢ 0 ≤ f'
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℝ
f' x a b : ℝ
h : IsLocalMax f a
hf : HasDerivAt f f' a
⊢ f' ≤ 0
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC... |
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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 ... |
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
h... | Please generate a tactic in lean4 to solve the state.
STATE:
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) /... |
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') | Please generate a tactic in lean4 to solve the state.
STATE:
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')
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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')
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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 ... |
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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] | sorry | f : ℝ → ℝ
f' x a b : ℝ
h : IsLocalMax f a
hf : HasDerivAt f f' a
⊢ 0 ≤ f' | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℝ
f' x a b : ℝ
h : IsLocalMax f a
hf : HasDerivAt f f' a
⊢ 0 ≤ f'
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℝ
f' x a b : ℝ
h : IsLocalMin f a
hf : HasDerivAt f f' a
⊢ f' = 0
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℝ
f' x a b : ℝ
h : IsLocalExtr f a
hf : HasDerivAt f f' a
⊢ f' = 0
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TAC... |
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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] | 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
h... | Please generate a tactic in lean4 to solve the state.
STATE:
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)... |
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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... | Please generate a tactic in lean4 to solve the state.
STATE:
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... |
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... | Please generate a tactic in lean4 to solve the state.
STATE:
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 ... |
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... | 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... | Please generate a tactic in lean4 to solve the state.
STATE:
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 ... |
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... | 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' ∈ Io... | Please generate a tactic in lean4 to solve the state.
STATE:
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 ... |
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' ∈ Io... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
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... |
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
... | Please generate a tactic in lean4 to solve the state.
STATE:
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... |
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
... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
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 :... |
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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 ... |
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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 ... |
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
ha... | Please generate a tactic in lean4 to solve the state.
STATE:
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 ... |
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
ha... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
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... |
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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... |
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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... |
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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 - ... | Please generate a tactic in lean4 to solve the state.
STATE:
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 -... |
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✝ : 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 ... | Please generate a tactic in lean4 to solve the state.
STATE:
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... |
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 ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
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... |
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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)
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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)
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u
inst✝ : Category C
a b c d e : C
X : Type
x : X
⊢ 𝟙 X x = x
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u
inst✝ : Category C
a : C
h : Initial a
b : C
f g : Hom a b
⊢ f = h.fromInitial b
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u
inst✝ : Category C
a : C
h : Initial a
b : C
f g : Hom a b
⊢ h.fromInitial b = g
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
J : Type u₁
inst✝¹ : Category J
C : Type u₂
inst✝ : Category C
F : Functor J C
i : Shape
⊢ ShapeHom.id i = 𝟙 i
TACTIC:
|
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' | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℝ
f' x a b : ℝ
h : IsLocalMax f a
hf : HasDerivAt f f' a
⊢ f' = 0
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℝ
f' x a b : ℝ
h : IsLocalMax f a
hf : HasDerivAt f f' a
⊢ f' ≤ 0
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC... |
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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 ... |
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
h... | Please generate a tactic in lean4 to solve the state.
STATE:
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) /... |
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') | Please generate a tactic in lean4 to solve the state.
STATE:
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')
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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')
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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 ... |
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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] | 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' | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℝ
f' x a b : ℝ
h : IsLocalMax f a
hf : HasDerivAt f f' a
⊢ 0 ≤ f'
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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'
TACTIC:
|
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