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
Ξ± : Type u_1
instβ : LinearOrder Ξ±
g : Equiv.Perm Ξ±
a b : Ξ±
β’ (a, b) β Inversions g β (a, b) β StdInversions g βͺ Prod.swap β»ΒΉ' StdInversions g
|
case h.mk
Ξ± : Type u_1
instβ : LinearOrder Ξ±
g : Equiv.Perm Ξ±
a b : Ξ±
β’ (a, b) β Inversions g β (a, b) β StdInversions g β¨ (b, a) β 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
Ξ± : Type u_1
instβ : LinearOrder Ξ±
g : Equiv.Perm Ξ±
a b : Ξ±
β’ (a, b) β Inversions g β (a, b) β StdInversions g β¨ (b, a) β StdInversions g
|
case pos
Ξ± : Type u_1
instβ : LinearOrder Ξ±
g : Equiv.Perm Ξ±
a b : Ξ±
hab : a = b
β’ (a, b) β Inversions g β (a, b) β StdInversions g β¨ (b, a) β StdInversions g
case neg
Ξ± : Type u_1
instβ : LinearOrder Ξ±
g : Equiv.Perm Ξ±
a b : Ξ±
hab : Β¬a = b
β’ (a, b) β Inversions g β (a, b) β StdInversions g β¨ (b, a) β 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
Ξ± : Type u_1
instβ : LinearOrder Ξ±
g : Equiv.Perm Ξ±
a b : Ξ±
hab : a = b
β’ (a, b) β Inversions g β (a, b) β StdInversions g β¨ (b, a) β StdInversions g
case neg
Ξ± : Type u_1
instβ : LinearOrder Ξ±
g : Equiv.Perm Ξ±
a b : Ξ±
hab : Β¬a = b
β’ (a, b) β Inversions g β (a, b) β StdInversions g β¨ (b, a) β StdInversions g
|
case neg
Ξ± : Type u_1
instβ : LinearOrder Ξ±
g : Equiv.Perm Ξ±
a b : Ξ±
hab : Β¬a = b
β’ (a, b) β Inversions g β (a, b) β StdInversions g β¨ (b, a) β 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
Ξ± : Type u_1
instβ : LinearOrder Ξ±
g : Equiv.Perm Ξ±
a b : Ξ±
hab : Β¬a = b
β’ (a, b) β Inversions g β (a, b) β StdInversions g β¨ (b, a) β 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
Ξ± : Type u_1
instβ : LinearOrder Ξ±
g : Equiv.Perm Ξ±
a b : Ξ±
hab : a = b
β’ (a, b) β Inversions g β (a, b) β StdInversions g β¨ (b, a) β StdInversions g
|
case pos
Ξ± : Type u_1
instβ : LinearOrder Ξ±
g : Equiv.Perm Ξ±
a : Ξ±
β’ (a, a) β Inversions g β (a, a) β StdInversions g β¨ (a, a) β 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
Ξ± : Type u_1
instβ : LinearOrder Ξ±
g : Equiv.Perm Ξ±
a : Ξ±
β’ (a, a) β Inversions g β (a, a) β StdInversions g β¨ (a, a) β 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
Ξ± : Type u_1
instβ : LinearOrder Ξ±
g : Equiv.Perm Ξ±
a b : Ξ±
hab : Β¬a = b
β’ (a, b) β Inversions g β (a, b) β StdInversions g β¨ (b, a) β StdInversions g
|
case neg
Ξ± : Type u_1
instβ : LinearOrder Ξ±
g : Equiv.Perm Ξ±
a b : Ξ±
hab : a β b
β’ (a, b) β Inversions g β (a, b) β StdInversions g β¨ (b, a) β 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
Ξ± : Type u_1
instβ : LinearOrder Ξ±
g : Equiv.Perm Ξ±
a b : Ξ±
hab : a β b
β’ (a, b) β Inversions g β (a, b) β StdInversions g β¨ (b, a) β StdInversions g
|
case neg.inl
Ξ± : Type u_1
instβ : LinearOrder Ξ±
g : Equiv.Perm Ξ±
a b : Ξ±
habβ : a β b
hab : a < b
β’ (a, b) β Inversions g β (a, b) β StdInversions g β¨ (b, a) β StdInversions g
case neg.inr
Ξ± : Type u_1
instβ : LinearOrder Ξ±
g : Equiv.Perm Ξ±
a b : Ξ±
hab : a β b
hba : b < a
β’ (a, b) β Inversions g β (a, b) β StdInversions g β¨ (b, a) β 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
Ξ± : Type u_1
instβ : LinearOrder Ξ±
g : Equiv.Perm Ξ±
a b : Ξ±
habβ : a β b
hab : a < b
β’ (a, b) β Inversions g β (a, b) β StdInversions g β¨ (b, a) β StdInversions g
case neg.inr
Ξ± : Type u_1
instβ : LinearOrder Ξ±
g : Equiv.Perm Ξ±
a b : Ξ±
hab : a β b
hba : b < a
β’ (a, b) β Inversions g β (a, b) β StdInversions g β¨ (b, a) β StdInversions g
|
case neg.inr
Ξ± : Type u_1
instβ : LinearOrder Ξ±
g : Equiv.Perm Ξ±
a b : Ξ±
hab : a β b
hba : b < a
β’ (a, b) β Inversions g β (a, b) β StdInversions g β¨ (b, a) β 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
Ξ± : Type u_1
instβ : LinearOrder Ξ±
g : Equiv.Perm Ξ±
a b : Ξ±
hab : a β b
hba : b < a
β’ (a, b) β Inversions g β (a, b) β StdInversions g β¨ (b, a) β 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
Ξ± : Type u_1
instβ : LinearOrder Ξ±
g : Equiv.Perm Ξ±
a b : Ξ±
habβ : a β b
hab : a < b
β’ (a, b) β Inversions g β (a, b) β StdInversions g β¨ (b, a) β 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
Ξ± : Type u_1
instβ : LinearOrder Ξ±
g : Equiv.Perm Ξ±
a b : Ξ±
hab : a β b
hba : b < a
β’ (a, b) β Inversions g β (a, b) β StdInversions g β¨ (b, a) β StdInversions g
|
case neg.inr
Ξ± : Type u_1
instβ : LinearOrder Ξ±
g : Equiv.Perm Ξ±
a b : Ξ±
hab : a β b
hba : b < a
β’ (b, a) β Inversions g β (a, b) β StdInversions g β¨ (b, a) β 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
Ξ± : Type u_1
instβ : LinearOrder Ξ±
g : Equiv.Perm Ξ±
a b : Ξ±
hab : a β b
hba : b < a
β’ (b, a) β Inversions g β (a, b) β StdInversions g β¨ (b, a) β 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]
|
Ξ± : Type u_1
f g : Equiv.Perm Ξ±
a b : Ξ±
hab : f a = b
β’ 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]
|
Ξ± : Type u_1
instβ : DecidableEq Ξ±
a b c d : Ξ±
β’ Equiv.swap a b * Equiv.swap c d * Equiv.swap a b = Equiv.swap ((Equiv.swap a b) c) ((Equiv.swap a b) d)
|
Ξ± : Type u_1
instβ : DecidableEq Ξ±
a b c d : Ξ±
β’ Equiv.swap a b * Equiv.swap c d * Equiv.swap a b = Equiv.swap a b * Equiv.swap c d * (Equiv.swap a b)β»ΒΉ
|
https://github.com/mguaypaq/lean-bruhat.git
|
1666a1bee2b520d5ba8a662310b4bd257fcf7ac2
|
Bruhat/Goal.lean
|
swap_conj
|
[15, 1]
|
[19, 6]
|
rfl
|
Ξ± : Type u_1
instβ : DecidableEq Ξ±
a b c d : Ξ±
β’ Equiv.swap a b * Equiv.swap c d * Equiv.swap a b = Equiv.swap a b * Equiv.swap c d * (Equiv.swap a b)β»ΒΉ
|
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 : β
β’ 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 : β
β’ 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 : β
β’ 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 : β
β’ Word.toPerm (WordForSwap i k) = Equiv.swap i (i + (k + 1))
|
case zero
i : β
β’ Word.toPerm (WordForSwap i Nat.zero) = Equiv.swap i (i + (Nat.zero + 1))
case succ
i k : β
ih : Word.toPerm (WordForSwap i k) = Equiv.swap i (i + (k + 1))
β’ 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 : β
β’ Word.toPerm (WordForSwap i Nat.zero) = Equiv.swap i (i + (Nat.zero + 1))
case succ
i k : β
ih : Word.toPerm (WordForSwap i k) = Equiv.swap i (i + (k + 1))
β’ Word.toPerm (WordForSwap i (Nat.succ k)) = Equiv.swap i (i + (Nat.succ k + 1))
|
case succ
i k : β
ih : Word.toPerm (WordForSwap i k) = Equiv.swap i (i + (k + 1))
β’ 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 _ _)β»ΒΉ = _
rw [β Equiv.swap_apply_apply, Equiv.swap_apply_left,
Equiv.swap_apply_of_ne_of_ne (by simp) (by simp)]
|
case succ
i k : β
ih : Word.toPerm (WordForSwap i k) = Equiv.swap i (i + (k + 1))
β’ 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 : β
β’ Word.toPerm (WordForSwap i Nat.zero) = Equiv.swap i (i + (Nat.zero + 1))
|
case zero
i : β
β’ 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 : β
β’ 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 : β
ih : Word.toPerm (WordForSwap i k) = Equiv.swap i (i + (k + 1))
β’ Word.toPerm (WordForSwap i (Nat.succ k)) = Equiv.swap i (i + (Nat.succ k + 1))
|
case succ
i k : β
ih : Word.toPerm (WordForSwap i k) = Equiv.swap i (i + (k + 1))
β’ 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 _ _)β»ΒΉ = _
|
case succ
i k : β
ih : Word.toPerm (WordForSwap i k) = Equiv.swap i (i + (k + 1))
β’ 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 : β
ih : Word.toPerm (WordForSwap i k) = Equiv.swap i (i + (k + 1))
β’ Equiv.swap (i + (k + 1)) (i + (k + 2)) * Equiv.swap i (i + (k + 1)) * (Equiv.swap (i + (k + 1)) (i + (k + 2)))β»ΒΉ =
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 [β Equiv.swap_apply_apply, Equiv.swap_apply_left,
Equiv.swap_apply_of_ne_of_ne (by simp) (by simp)]
|
case succ
i k : β
ih : Word.toPerm (WordForSwap i k) = Equiv.swap i (i + (k + 1))
β’ Equiv.swap (i + (k + 1)) (i + (k + 2)) * Equiv.swap i (i + (k + 1)) * (Equiv.swap (i + (k + 1)) (i + (k + 2)))β»ΒΉ =
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 : β
ih : Word.toPerm (WordForSwap i k) = Equiv.swap i (i + (k + 1))
β’ i β 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 : β
ih : Word.toPerm (WordForSwap i k) = Equiv.swap i (i + (k + 1))
β’ i β 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 *
|
Ξ± : Type u_1
instβ : DecidableEq Ξ±
f : Equiv.Perm Ξ±
a : Ξ±
β’ a β 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 β
β’ Inversions (f * g) = Prod.map βg.symm βg.symm '' Inversions f β© 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 : β β β
n x : β
hx : n β€ x
β’ Β¬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 : β
β’ Fintype.card { g // βg β€ β(truncate id n) } = n !
|
n : β
β’ Fintype.card ((i : Fin n) β Fin (id βi + 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 : β
β’ Fintype.card ((i : Fin n) β Fin (id βi + 1)) = n !
|
n : β
β’ (Finset.prod Finset.univ fun x => βx + 1) = n !
|
https://github.com/mguaypaq/lean-bruhat.git
|
1666a1bee2b520d5ba8a662310b4bd257fcf7ac2
|
Bruhat/Lehmer.lean
|
card_lehmer_eq_factorial
|
[73, 1]
|
[78, 47]
|
rw [β Finset.prod_range_add_one_eq_factorial n]
|
n : β
β’ (Finset.prod Finset.univ fun x => βx + 1) = n !
|
n : β
β’ (Finset.prod Finset.univ fun x => βx + 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 : β
β’ (Finset.prod Finset.univ fun x => βx + 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 : β β β
f' x a b : β
h : IsLocalMax f a
hf : HasDerivAt f f' a
β’ f' = 0
|
case right
f : β β β
f' x a b : β
h : IsLocalMax f a
hf : HasDerivAt f f' a
β’ f' β€ 0
case left
f : β β β
f' x a b : β
h : IsLocalMax f a
hf : HasDerivAt f f' a
β’ 0 β€ 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 : β β β
f' x a b : β
h : IsLocalMax f a
hf : HasDerivAt f f' a
β’ 0 β€ 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 β¦ (f x - f a) / (x - a)) (π[>] a) (π f') := by
rw [hasDerivAt_iff_tendsto_slope] at hf
apply hf.mono_left (nhds_right'_le_nhds_ne a)
|
f : β β β
f' x a b : β
h : IsLocalMax f a
hf : HasDerivAt f f' a
β’ f' β€ 0
|
f : β β β
f' x a b : β
h : IsLocalMax f a
hfβ : HasDerivAt f f' a
hf : Tendsto (fun x => (f x - f a) / (x - a)) (π[>] a) (π f')
β’ f' β€ 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 βαΆ x in π[>] a, (f x - f a) / (x - a) β€ 0 from le_of_tendsto hf this
|
f : β β β
f' x a b : β
h : IsLocalMax f a
hfβ : HasDerivAt f f' a
hf : Tendsto (fun x => (f x - f a) / (x - a)) (π[>] a) (π f')
β’ f' β€ 0
|
f : β β β
f' x a b : β
h : IsLocalMax f a
hfβ : HasDerivAt f f' a
hf : Tendsto (fun x => (f x - f a) / (x - a)) (π[>] a) (π f')
β’ βαΆ (x : β) in π[>] a, (f x - f a) / (x - a) β€ 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 : βαΆ x in π[>] a, a < x := eventually_nhdsWithin_of_forall fun x hx β¦ hx
|
f : β β β
f' x a b : β
h : IsLocalMax f a
hfβ : HasDerivAt f f' a
hf : Tendsto (fun x => (f x - f a) / (x - a)) (π[>] a) (π f')
β’ βαΆ (x : β) in π[>] a, (f x - f a) / (x - a) β€ 0
|
f : β β β
f' x a b : β
h : IsLocalMax f a
hfβ : HasDerivAt f f' a
hf : Tendsto (fun x => (f x - f a) / (x - a)) (π[>] a) (π f')
ha : βαΆ (x : β) in π[>] a, a < x
β’ βαΆ (x : β) in π[>] a, (f x - f a) / (x - a) β€ 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 : βαΆ x in π[>] a, f x β€ f a := h.filter_mono nhdsWithin_le_nhds
|
f : β β β
f' x a b : β
h : IsLocalMax f a
hfβ : HasDerivAt f f' a
hf : Tendsto (fun x => (f x - f a) / (x - a)) (π[>] a) (π f')
ha : βαΆ (x : β) in π[>] a, a < x
β’ βαΆ (x : β) in π[>] a, (f x - f a) / (x - a) β€ 0
|
f : β β β
f' x a b : β
hβ : IsLocalMax f a
hfβ : HasDerivAt f f' a
hf : Tendsto (fun x => (f x - f a) / (x - a)) (π[>] a) (π f')
ha : βαΆ (x : β) in π[>] a, a < x
h : βαΆ (x : β) in π[>] a, f x β€ f a
β’ βαΆ (x : β) in π[>] a, (f x - f a) / (x - a) β€ 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 : β β β
f' x a b : β
hβ : IsLocalMax f a
hfβ : HasDerivAt f f' a
hf : Tendsto (fun x => (f x - f a) / (x - a)) (π[>] a) (π f')
ha : βαΆ (x : β) in π[>] a, a < x
h : βαΆ (x : β) in π[>] a, f x β€ f a
β’ βαΆ (x : β) in π[>] a, (f x - f a) / (x - a) β€ 0
|
case h
f : β β β
f' x a b : β
hβ : IsLocalMax f a
hfβ : HasDerivAt f f' a
hf : Tendsto (fun x => (f x - f a) / (x - a)) (π[>] a) (π f')
ha : βαΆ (x : β) in π[>] a, a < x
h : βαΆ (x : β) in π[>] a, f x β€ f a
β’ β (a_1 : β), a < a_1 β f a_1 β€ f a β (f a_1 - f a) / (a_1 - a) β€ 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 : β β β
f' x a b : β
hβ : IsLocalMax f a
hfβ : HasDerivAt f f' a
hf : Tendsto (fun x => (f x - f a) / (x - a)) (π[>] a) (π f')
ha : βαΆ (x : β) in π[>] a, a < x
h : βαΆ (x : β) in π[>] a, f x β€ f a
β’ β (a_1 : β), a < a_1 β f a_1 β€ f a β (f a_1 - f a) / (a_1 - a) β€ 0
|
case h
f : β β β
f' xβ a b : β
hβΒΉ : IsLocalMax f a
hfβ : HasDerivAt f f' a
hf : Tendsto (fun x => (f x - f a) / (x - a)) (π[>] a) (π f')
haβ : βαΆ (x : β) in π[>] a, a < x
hβ : βαΆ (x : β) in π[>] a, f x β€ f a
x : β
ha : a < x
h : f x β€ f a
β’ (f x - f a) / (x - a) β€ 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 : β β β
f' xβ a b : β
hβΒΉ : IsLocalMax f a
hfβ : HasDerivAt f f' a
hf : Tendsto (fun x => (f x - f a) / (x - a)) (π[>] a) (π f')
haβ : βαΆ (x : β) in π[>] a, a < x
hβ : βαΆ (x : β) in π[>] a, f x β€ f a
x : β
ha : a < x
h : f x β€ f a
β’ (f x - f a) / (x - a) β€ 0
|
case h.ha
f : β β β
f' xβ a b : β
hβΒΉ : IsLocalMax f a
hfβ : HasDerivAt f f' a
hf : Tendsto (fun x => (f x - f a) / (x - a)) (π[>] a) (π f')
haβ : βαΆ (x : β) in π[>] a, a < x
hβ : βαΆ (x : β) in π[>] a, f x β€ f a
x : β
ha : a < x
h : f x β€ f a
β’ f x - f a β€ 0
case h.hb
f : β β β
f' xβ a b : β
hβΒΉ : IsLocalMax f a
hfβ : HasDerivAt f f' a
hf : Tendsto (fun x => (f x - f a) / (x - a)) (π[>] a) (π f')
haβ : βαΆ (x : β) in π[>] a, a < x
hβ : βαΆ (x : β) in π[>] a, f x β€ f a
x : β
ha : a < x
h : f x β€ f a
β’ 0 β€ 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 : β β β
f' x a b : β
h : IsLocalMax f a
hf : HasDerivAt f f' a
β’ Tendsto (fun x => (f x - f a) / (x - a)) (π[>] a) (π f')
|
f : β β β
f' x a b : β
h : IsLocalMax f a
hf : Tendsto (fun x => (f x - f a) / (x - a)) (π[β ] a) (π f')
β’ Tendsto (fun x => (f x - f a) / (x - a)) (π[>] a) (π 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 : β β β
f' x a b : β
h : IsLocalMax f a
hf : Tendsto (fun x => (f x - f a) / (x - a)) (π[β ] a) (π f')
β’ Tendsto (fun x => (f x - f a) / (x - a)) (π[>] a) (π 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 : β β β
f' xβ a b : β
hβΒΉ : IsLocalMax f a
hfβ : HasDerivAt f f' a
hf : Tendsto (fun x => (f x - f a) / (x - a)) (π[>] a) (π f')
haβ : βαΆ (x : β) in π[>] a, a < x
hβ : βαΆ (x : β) in π[>] a, f x β€ f a
x : β
ha : a < x
h : f x β€ f a
β’ f x - f a β€ 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 : β β β
f' xβ a b : β
hβΒΉ : IsLocalMax f a
hfβ : HasDerivAt f f' a
hf : Tendsto (fun x => (f x - f a) / (x - a)) (π[>] a) (π f')
haβ : βαΆ (x : β) in π[>] a, a < x
hβ : βαΆ (x : β) in π[>] a, f x β€ f a
x : β
ha : a < x
h : f x β€ f a
β’ 0 β€ 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 : β β β
f' x a b : β
h : IsLocalMax f a
hf : HasDerivAt f f' a
β’ 0 β€ 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 : β β β
f' x a b : β
h : IsLocalMin f a
hf : HasDerivAt f f' a
β’ 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 : β β β
f' x a b : β
h : IsLocalExtr f a
hf : HasDerivAt f f' a
β’ 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 β c β Ioo a b, IsExtrOn f (Icc a b) c by
rcases this with β¨c, cmem, hcβ©
exists c, cmem
apply hc.isLocalExtr <| Icc_mem_nhds cmem.1 cmem.2
|
f : β β β
f' x a b : β
hab : a < b
hfc : ContinuousOn f (Icc a b)
hfI : f a = f b
β’ β c β Ioo a b, IsLocalExtr f c
|
f : β β β
f' x a b : β
hab : a < b
hfc : ContinuousOn f (Icc a b)
hfI : f a = f b
β’ β c β 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 : β β β
f' x a b : β
hab : a < b
hfc : ContinuousOn f (Icc a b)
hfI : f a = f b
β’ β c β Ioo a b, IsExtrOn f (Icc a b) c
|
f : β β β
f' x a b : β
hab : a < b
hfc : ContinuousOn f (Icc a b)
hfI : f a = f b
ne : Set.Nonempty (Icc a b)
β’ β c β 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 β¨C, Cmem, Cgeβ© : β C β Icc a b, IsMaxOn f (Icc a b) C := by
sorry
|
f : β β β
f' x a b : β
hab : a < b
hfc : ContinuousOn f (Icc a b)
hfI : f a = f b
ne : Set.Nonempty (Icc a b)
β’ β c β Ioo a b, IsExtrOn f (Icc a b) c
|
f : β β β
f' x a b : β
hab : a < b
hfc : ContinuousOn f (Icc a b)
hfI : f a = f b
ne : Set.Nonempty (Icc a b)
C : β
Cmem : C β Icc a b
Cge : IsMaxOn f (Icc a b) C
β’ β c β 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 β¨c, cmem, cleβ© : β c β Icc a b, IsMinOn f (Icc a b) c := by
sorry
|
f : β β β
f' x a b : β
hab : a < b
hfc : ContinuousOn f (Icc a b)
hfI : f a = f b
ne : Set.Nonempty (Icc a b)
C : β
Cmem : C β Icc a b
Cge : IsMaxOn f (Icc a b) C
β’ β c β Ioo a b, IsExtrOn f (Icc a b) c
|
f : β β β
f' x a b : β
hab : a < b
hfc : ContinuousOn f (Icc a b)
hfI : f a = f b
ne : Set.Nonempty (Icc a b)
C : β
Cmem : C β Icc a b
Cge : IsMaxOn f (Icc a b) C
c : β
cmem : c β Icc a b
cle : IsMinOn f (Icc a b) c
β’ β c β 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 β x β Icc a b, f x β€ f C at Cge
|
f : β β β
f' x a b : β
hab : a < b
hfc : ContinuousOn f (Icc a b)
hfI : f a = f b
ne : Set.Nonempty (Icc a b)
C : β
Cmem : C β Icc a b
Cge : IsMaxOn f (Icc a b) C
c : β
cmem : c β Icc a b
cle : IsMinOn f (Icc a b) c
β’ β c β Ioo a b, IsExtrOn f (Icc a b) c
|
f : β β β
f' x a b : β
hab : a < b
hfc : ContinuousOn f (Icc a b)
hfI : f a = f b
ne : Set.Nonempty (Icc a b)
C : β
Cmem : C β Icc a b
c : β
cmem : c β Icc a b
cle : IsMinOn f (Icc a b) c
Cge : β x β Icc a b, f x β€ f C
β’ β c β 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 β x β Icc a b, f c β€ f x at cle
|
f : β β β
f' x a b : β
hab : a < b
hfc : ContinuousOn f (Icc a b)
hfI : f a = f b
ne : Set.Nonempty (Icc a b)
C : β
Cmem : C β Icc a b
c : β
cmem : c β Icc a b
cle : IsMinOn f (Icc a b) c
Cge : β x β Icc a b, f x β€ f C
β’ β c β Ioo a b, IsExtrOn f (Icc a b) c
|
f : β β β
f' x a b : β
hab : a < b
hfc : ContinuousOn f (Icc a b)
hfI : f a = f b
ne : Set.Nonempty (Icc a b)
C : β
Cmem : C β Icc a b
c : β
cmem : c β Icc a b
Cge : β x β Icc a b, f x β€ f C
cle : β x β Icc a b, f c β€ f x
β’ β c β 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 : β β β
f' x a b : β
hab : a < b
hfc : ContinuousOn f (Icc a b)
hfI : f a = f b
ne : Set.Nonempty (Icc a b)
C : β
Cmem : C β Icc a b
c : β
cmem : c β Icc a b
Cge : β x β Icc a b, f x β€ f C
cle : β x β Icc a b, f c β€ f x
β’ β c β Ioo a b, IsExtrOn f (Icc a b) c
|
case pos
f : β β β
f' x a b : β
hab : a < b
hfc : ContinuousOn f (Icc a b)
hfI : f a = f b
ne : Set.Nonempty (Icc a b)
C : β
Cmem : C β Icc a b
c : β
cmem : c β Icc a b
Cge : β x β Icc a b, f x β€ f C
cle : β x β Icc a b, f c β€ f x
hc : f c = f a
β’ β c β Ioo a b, IsExtrOn f (Icc a b) c
case neg
f : β β β
f' x a b : β
hab : a < b
hfc : ContinuousOn f (Icc a b)
hfI : f a = f b
ne : Set.Nonempty (Icc a b)
C : β
Cmem : C β Icc a b
c : β
cmem : c β Icc a b
Cge : β x β Icc a b, f x β€ f C
cle : β x β Icc a b, f c β€ f x
hc : Β¬f c = f a
β’ β c β 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 β¨c, cmem, hcβ©
|
f : β β β
f' x a b : β
hab : a < b
hfc : ContinuousOn f (Icc a b)
hfI : f a = f b
this : β c β Ioo a b, IsExtrOn f (Icc a b) c
β’ β c β Ioo a b, IsLocalExtr f c
|
case intro.intro
f : β β β
f' x a b : β
hab : a < b
hfc : ContinuousOn f (Icc a b)
hfI : f a = f b
c : β
cmem : c β Ioo a b
hc : IsExtrOn f (Icc a b) c
β’ β c β 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 : β β β
f' x a b : β
hab : a < b
hfc : ContinuousOn f (Icc a b)
hfI : f a = f b
c : β
cmem : c β Ioo a b
hc : IsExtrOn f (Icc a b) c
β’ β c β Ioo a b, IsLocalExtr f c
|
case intro.intro
f : β β β
f' x a b : β
hab : a < b
hfc : ContinuousOn f (Icc a b)
hfI : f a = f b
c : β
cmem : c β Ioo a b
hc : IsExtrOn f (Icc a b) c
β’ 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 : β β β
f' x a b : β
hab : a < b
hfc : ContinuousOn f (Icc a b)
hfI : f a = f b
c : β
cmem : c β Ioo a b
hc : IsExtrOn f (Icc a b) c
β’ 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 : β β β
f' x a b : β
hab : a < b
hfc : ContinuousOn f (Icc a b)
hfI : f a = f b
ne : Set.Nonempty (Icc a b)
β’ β C β 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 : β β β
f' x a b : β
hab : a < b
hfc : ContinuousOn f (Icc a b)
hfI : f a = f b
ne : Set.Nonempty (Icc a b)
C : β
Cmem : C β Icc a b
Cge : IsMaxOn f (Icc a b) C
β’ β c β 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 : β β β
f' x a b : β
hab : a < b
hfc : ContinuousOn f (Icc a b)
hfI : f a = f b
ne : Set.Nonempty (Icc a b)
C : β
Cmem : C β Icc a b
c : β
cmem : c β Icc a b
Cge : β x β Icc a b, f x β€ f C
cle : β x β Icc a b, f c β€ f x
hc : f c = f a
β’ β c β Ioo a b, IsExtrOn f (Icc a b) c
|
case pos
f : β β β
f' x a b : β
hab : a < b
hfc : ContinuousOn f (Icc a b)
hfI : f a = f b
ne : Set.Nonempty (Icc a b)
C : β
Cmem : C β Icc a b
c : β
cmem : c β Icc a b
Cge : β x β Icc a b, f x β€ f C
cle : β x β Icc a b, f c β€ f x
hc : f c = f a
hC : f C = f a
β’ β c β Ioo a b, IsExtrOn f (Icc a b) c
case neg
f : β β β
f' x a b : β
hab : a < b
hfc : ContinuousOn f (Icc a b)
hfI : f a = f b
ne : Set.Nonempty (Icc a b)
C : β
Cmem : C β Icc a b
c : β
cmem : c β Icc a b
Cge : β x β Icc a b, f x β€ f C
cle : β x β Icc a b, f c β€ f x
hc : f c = f a
hC : Β¬f C = f a
β’ β c β 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 : β x β Icc a b, f x = f a := fun x hx β¦ le_antisymm (hC βΈ Cge x hx) (hc βΈ cle x hx)
|
case pos
f : β β β
f' x a b : β
hab : a < b
hfc : ContinuousOn f (Icc a b)
hfI : f a = f b
ne : Set.Nonempty (Icc a b)
C : β
Cmem : C β Icc a b
c : β
cmem : c β Icc a b
Cge : β x β Icc a b, f x β€ f C
cle : β x β Icc a b, f c β€ f x
hc : f c = f a
hC : f C = f a
β’ β c β Ioo a b, IsExtrOn f (Icc a b) c
|
case pos
f : β β β
f' x a b : β
hab : a < b
hfc : ContinuousOn f (Icc a b)
hfI : f a = f b
ne : Set.Nonempty (Icc a b)
C : β
Cmem : C β Icc a b
c : β
cmem : c β Icc a b
Cge : β x β Icc a b, f x β€ f C
cle : β x β Icc a b, f c β€ f x
hc : f c = f a
hC : f C = f a
this : β x β Icc a b, f x = f a
β’ β c β 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 β¨c', hc'β©
|
case pos
f : β β β
f' x a b : β
hab : a < b
hfc : ContinuousOn f (Icc a b)
hfI : f a = f b
ne : Set.Nonempty (Icc a b)
C : β
Cmem : C β Icc a b
c : β
cmem : c β Icc a b
Cge : β x β Icc a b, f x β€ f C
cle : β x β Icc a b, f c β€ f x
hc : f c = f a
hC : f C = f a
this : β x β Icc a b, f x = f a
β’ β c β Ioo a b, IsExtrOn f (Icc a b) c
|
case pos.intro
f : β β β
f' x a b : β
hab : a < b
hfc : ContinuousOn f (Icc a b)
hfI : f a = f b
ne : Set.Nonempty (Icc a b)
C : β
Cmem : C β Icc a b
c : β
cmem : c β Icc a b
Cge : β x β Icc a b, f x β€ f C
cle : β x β Icc a b, f c β€ f x
hc : f c = f a
hC : f C = f a
this : β x β Icc a b, f x = f a
c' : β
hc' : c' β Ioo a b
β’ β c β 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 β¨c', hc', Or.inl fun x hx β¦ ?_β©
|
case pos.intro
f : β β β
f' x a b : β
hab : a < b
hfc : ContinuousOn f (Icc a b)
hfI : f a = f b
ne : Set.Nonempty (Icc a b)
C : β
Cmem : C β Icc a b
c : β
cmem : c β Icc a b
Cge : β x β Icc a b, f x β€ f C
cle : β x β Icc a b, f c β€ f x
hc : f c = f a
hC : f C = f a
this : β x β Icc a b, f x = f a
c' : β
hc' : c' β Ioo a b
β’ β c β Ioo a b, IsExtrOn f (Icc a b) c
|
case pos.intro
f : β β β
f' xβ a b : β
hab : a < b
hfc : ContinuousOn f (Icc a b)
hfI : f a = f b
ne : Set.Nonempty (Icc a b)
C : β
Cmem : C β Icc a b
c : β
cmem : c β Icc a b
Cge : β x β Icc a b, f x β€ f C
cle : β x β Icc a b, f c β€ f x
hc : f c = f a
hC : f C = f a
this : β x β Icc a b, f x = f a
c' : β
hc' : c' β Ioo a b
x : β
hx : x β Icc a b
β’ x β {x | (fun x => f c' β€ 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 : β β β
f' xβ a b : β
hab : a < b
hfc : ContinuousOn f (Icc a b)
hfI : f a = f b
ne : Set.Nonempty (Icc a b)
C : β
Cmem : C β Icc a b
c : β
cmem : c β Icc a b
Cge : β x β Icc a b, f x β€ f C
cle : β x β Icc a b, f c β€ f x
hc : f c = f a
hC : f C = f a
this : β x β Icc a b, f x = f a
c' : β
hc' : c' β Ioo a b
x : β
hx : x β Icc a b
β’ x β {x | (fun x => f c' β€ 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 β¨C, β¨lt_of_le_of_ne Cmem.1 <| mt ?_ hC, lt_of_le_of_ne Cmem.2 <| mt ?_ hCβ©, Or.inr Cgeβ©
|
case neg
f : β β β
f' x a b : β
hab : a < b
hfc : ContinuousOn f (Icc a b)
hfI : f a = f b
ne : Set.Nonempty (Icc a b)
C : β
Cmem : C β Icc a b
c : β
cmem : c β Icc a b
Cge : β x β Icc a b, f x β€ f C
cle : β x β Icc a b, f c β€ f x
hc : f c = f a
hC : Β¬f C = f a
β’ β c β Ioo a b, IsExtrOn f (Icc a b) c
|
case neg.refine_1
f : β β β
f' x a b : β
hab : a < b
hfc : ContinuousOn f (Icc a b)
hfI : f a = f b
ne : Set.Nonempty (Icc a b)
C : β
Cmem : C β Icc a b
c : β
cmem : c β Icc a b
Cge : β x β Icc a b, f x β€ f C
cle : β x β Icc a b, f c β€ f x
hc : f c = f a
hC : Β¬f C = f a
β’ a = C β f C = f a
case neg.refine_2
f : β β β
f' x a b : β
hab : a < b
hfc : ContinuousOn f (Icc a b)
hfI : f a = f b
ne : Set.Nonempty (Icc a b)
C : β
Cmem : C β Icc a b
c : β
cmem : c β Icc a b
Cge : β x β Icc a b, f x β€ f C
cle : β x β Icc a b, f c β€ f x
hc : f c = f a
hC : Β¬f C = f a
β’ C = b β 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 β¦ by rw [h], fun h β¦ by rw [h, hfI]]
|
case neg.refine_1
f : β β β
f' x a b : β
hab : a < b
hfc : ContinuousOn f (Icc a b)
hfI : f a = f b
ne : Set.Nonempty (Icc a b)
C : β
Cmem : C β Icc a b
c : β
cmem : c β Icc a b
Cge : β x β Icc a b, f x β€ f C
cle : β x β Icc a b, f c β€ f x
hc : f c = f a
hC : Β¬f C = f a
β’ a = C β f C = f a
case neg.refine_2
f : β β β
f' x a b : β
hab : a < b
hfc : ContinuousOn f (Icc a b)
hfI : f a = f b
ne : Set.Nonempty (Icc a b)
C : β
Cmem : C β Icc a b
c : β
cmem : c β Icc a b
Cge : β x β Icc a b, f x β€ f C
cle : β x β Icc a b, f c β€ f x
hc : f c = f a
hC : Β¬f C = f a
β’ C = b β 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 : β β β
f' x a b : β
hab : a < b
hfc : ContinuousOn f (Icc a b)
hfI : f a = f b
ne : Set.Nonempty (Icc a b)
C : β
Cmem : C β Icc a b
c : β
cmem : c β Icc a b
Cge : β x β Icc a b, f x β€ f C
cle : β x β Icc a b, f c β€ f x
hc : f c = f a
hC : Β¬f C = f a
h : a = C
β’ 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 : β β β
f' x a b : β
hab : a < b
hfc : ContinuousOn f (Icc a b)
hfI : f a = f b
ne : Set.Nonempty (Icc a b)
C : β
Cmem : C β Icc a b
c : β
cmem : c β Icc a b
Cge : β x β Icc a b, f x β€ f C
cle : β x β Icc a b, f c β€ f x
hc : f c = f a
hC : Β¬f C = f a
h : C = b
β’ 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 β¨c, β¨lt_of_le_of_ne cmem.1 <| mt ?_ hc, lt_of_le_of_ne cmem.2 <| mt ?_ hcβ©, Or.inl cleβ©
|
case neg
f : β β β
f' x a b : β
hab : a < b
hfc : ContinuousOn f (Icc a b)
hfI : f a = f b
ne : Set.Nonempty (Icc a b)
C : β
Cmem : C β Icc a b
c : β
cmem : c β Icc a b
Cge : β x β Icc a b, f x β€ f C
cle : β x β Icc a b, f c β€ f x
hc : Β¬f c = f a
β’ β c β Ioo a b, IsExtrOn f (Icc a b) c
|
case neg.refine_1
f : β β β
f' x a b : β
hab : a < b
hfc : ContinuousOn f (Icc a b)
hfI : f a = f b
ne : Set.Nonempty (Icc a b)
C : β
Cmem : C β Icc a b
c : β
cmem : c β Icc a b
Cge : β x β Icc a b, f x β€ f C
cle : β x β Icc a b, f c β€ f x
hc : Β¬f c = f a
β’ a = c β f c = f a
case neg.refine_2
f : β β β
f' x a b : β
hab : a < b
hfc : ContinuousOn f (Icc a b)
hfI : f a = f b
ne : Set.Nonempty (Icc a b)
C : β
Cmem : C β Icc a b
c : β
cmem : c β Icc a b
Cge : β x β Icc a b, f x β€ f C
cle : β x β Icc a b, f c β€ f x
hc : Β¬f c = f a
β’ c = b β 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 β¦ by rw [h], fun h β¦ by rw [h, hfI]]
|
case neg.refine_1
f : β β β
f' x a b : β
hab : a < b
hfc : ContinuousOn f (Icc a b)
hfI : f a = f b
ne : Set.Nonempty (Icc a b)
C : β
Cmem : C β Icc a b
c : β
cmem : c β Icc a b
Cge : β x β Icc a b, f x β€ f C
cle : β x β Icc a b, f c β€ f x
hc : Β¬f c = f a
β’ a = c β f c = f a
case neg.refine_2
f : β β β
f' x a b : β
hab : a < b
hfc : ContinuousOn f (Icc a b)
hfI : f a = f b
ne : Set.Nonempty (Icc a b)
C : β
Cmem : C β Icc a b
c : β
cmem : c β Icc a b
Cge : β x β Icc a b, f x β€ f C
cle : β x β Icc a b, f c β€ f x
hc : Β¬f c = f a
β’ c = b β 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 : β β β
f' x a b : β
hab : a < b
hfc : ContinuousOn f (Icc a b)
hfI : f a = f b
ne : Set.Nonempty (Icc a b)
C : β
Cmem : C β Icc a b
c : β
cmem : c β Icc a b
Cge : β x β Icc a b, f x β€ f C
cle : β x β Icc a b, f c β€ f x
hc : Β¬f c = f a
h : a = c
β’ 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 : β β β
f' x a b : β
hab : a < b
hfc : ContinuousOn f (Icc a b)
hfI : f a = f b
ne : Set.Nonempty (Icc a b)
C : β
Cmem : C β Icc a b
c : β
cmem : c β Icc a b
Cge : β x β Icc a b, f x β€ f C
cle : β x β Icc a b, f c β€ f x
hc : Β¬f c = f a
h : c = b
β’ 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
|
fβ : β β β
f'β x aβ bβ : β
f f' g g' : β β β
a b : β
hab : a < b
hfc : ContinuousOn f (Icc a b)
hfI : f a = f b
hff' : β x β Ioo a b, HasDerivAt f (f' x) x
β’ β c β 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
|
fβ : β β β
f'β x aβ bβ : β
f f' g g' : β β β
a b : β
hab : a < b
hfc : ContinuousOn f (Icc a b)
hff' : β x β Ioo a b, HasDerivAt f (f' x) x
hgc : ContinuousOn g (Icc a b)
hgg' : β x β Ioo a b, HasDerivAt g (g' x) x
β’ β c β Ioo a b, (g b - g a) * f' c = (f b - f a) * g' c
|
fβ : β β β
f'β x aβ bβ : β
f f' g g' : β β β
a b : β
hab : a < b
hfc : ContinuousOn f (Icc a b)
hff' : β x β Ioo a b, HasDerivAt f (f' x) x
hgc : ContinuousOn g (Icc a b)
hgg' : β x β Ioo a b, HasDerivAt g (g' x) x
h : β β β := fun x => (g b - g a) * f x - (f b - f a) * g x
β’ β c β 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)
|
fβ : β β β
f'β x aβ bβ : β
f f' g g' : β β β
a b : β
hab : a < b
hfc : ContinuousOn f (Icc a b)
hff' : β x β Ioo a b, HasDerivAt f (f' x) x
hgc : ContinuousOn g (Icc a b)
hgg' : β x β Ioo a b, HasDerivAt g (g' x) x
h : β β β := fun x => (g b - g a) * f x - (f b - f a) * g x
β’ β c β Ioo a b, (g b - g a) * f' c = (f b - f a) * g' c
|
fβ : β β β
f'β x aβ bβ : β
f f' g g' : β β β
a b : β
hab : a < b
hfc : ContinuousOn f (Icc a b)
hff' : β x β Ioo a b, HasDerivAt f (f' x) x
hgc : ContinuousOn g (Icc a b)
hgg' : β x β Ioo a b, HasDerivAt g (g' x) x
h : β β β := fun x => (g b - g a) * f x - (f b - f a) * g x
hhc : ContinuousOn h (Icc a b)
β’ β c β 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
|
fβ : β β β
f'β x aβ bβ : β
f f' g g' : β β β
a b : β
hab : a < b
hfc : ContinuousOn f (Icc a b)
hff' : β x β Ioo a b, HasDerivAt f (f' x) x
hgc : ContinuousOn g (Icc a b)
hgg' : β x β Ioo a b, HasDerivAt g (g' x) x
h : β β β := fun x => (g b - g a) * f x - (f b - f a) * g x
hhc : ContinuousOn h (Icc a b)
β’ β c β 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
|
fβ : β β β
f'β x aβ bβ : β
f f' g g' : β β β
a b : β
hab : a < b
hfc : ContinuousOn f (Icc a b)
hff' : β x β Ioo a b, HasDerivAt f (f' x) x
β’ β c β 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
instβ : Category C
a b c d e : C
X Y Z : Type
f : Hom X Y
g : Hom Y Z
x : X
β’ (f β« 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
instβ : Category C
a b c d e : C
X : Type
x : X
β’ π 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
instβ : Category C
a : C
h : Initial a
b : C
f g : Hom a b
β’ 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
instβ : Category C
a : C
h : Initial a
b : C
f g : Hom a b
β’ 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 uβ
instβΒΉ : Category J
C : Type uβ
instβ : Category C
F : Functor J C
i : Shape
β’ ShapeHom.id i = π 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 : β β β
f' x a b : β
h : IsLocalMax f a
hf : HasDerivAt f f' a
β’ f' = 0
|
case right
f : β β β
f' x a b : β
h : IsLocalMax f a
hf : HasDerivAt f f' a
β’ f' β€ 0
case left
f : β β β
f' x a b : β
h : IsLocalMax f a
hf : HasDerivAt f f' a
β’ 0 β€ 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 β¦ (f x - f a) / (x - a)) (π[>] a) (π f') := by
rw [hasDerivAt_iff_tendsto_slope] at hf
apply hf.mono_left (nhds_right'_le_nhds_ne a)
|
f : β β β
f' x a b : β
h : IsLocalMax f a
hf : HasDerivAt f f' a
β’ f' β€ 0
|
f : β β β
f' x a b : β
h : IsLocalMax f a
hfβ : HasDerivAt f f' a
hf : Tendsto (fun x => (f x - f a) / (x - a)) (π[>] a) (π f')
β’ f' β€ 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 βαΆ x in π[>] a, (f x - f a) / (x - a) β€ 0 from le_of_tendsto hf this
|
f : β β β
f' x a b : β
h : IsLocalMax f a
hfβ : HasDerivAt f f' a
hf : Tendsto (fun x => (f x - f a) / (x - a)) (π[>] a) (π f')
β’ f' β€ 0
|
f : β β β
f' x a b : β
h : IsLocalMax f a
hfβ : HasDerivAt f f' a
hf : Tendsto (fun x => (f x - f a) / (x - a)) (π[>] a) (π f')
β’ βαΆ (x : β) in π[>] a, (f x - f a) / (x - a) β€ 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 : βαΆ x in π[>] a, a < x := eventually_nhdsWithin_of_forall fun x hx β¦ hx
|
f : β β β
f' x a b : β
h : IsLocalMax f a
hfβ : HasDerivAt f f' a
hf : Tendsto (fun x => (f x - f a) / (x - a)) (π[>] a) (π f')
β’ βαΆ (x : β) in π[>] a, (f x - f a) / (x - a) β€ 0
|
f : β β β
f' x a b : β
h : IsLocalMax f a
hfβ : HasDerivAt f f' a
hf : Tendsto (fun x => (f x - f a) / (x - a)) (π[>] a) (π f')
ha : βαΆ (x : β) in π[>] a, a < x
β’ βαΆ (x : β) in π[>] a, (f x - f a) / (x - a) β€ 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 : βαΆ x in π[>] a, f x β€ f a := h.filter_mono nhdsWithin_le_nhds
|
f : β β β
f' x a b : β
h : IsLocalMax f a
hfβ : HasDerivAt f f' a
hf : Tendsto (fun x => (f x - f a) / (x - a)) (π[>] a) (π f')
ha : βαΆ (x : β) in π[>] a, a < x
β’ βαΆ (x : β) in π[>] a, (f x - f a) / (x - a) β€ 0
|
f : β β β
f' x a b : β
hβ : IsLocalMax f a
hfβ : HasDerivAt f f' a
hf : Tendsto (fun x => (f x - f a) / (x - a)) (π[>] a) (π f')
ha : βαΆ (x : β) in π[>] a, a < x
h : βαΆ (x : β) in π[>] a, f x β€ f a
β’ βαΆ (x : β) in π[>] a, (f x - f a) / (x - a) β€ 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 : β β β
f' x a b : β
hβ : IsLocalMax f a
hfβ : HasDerivAt f f' a
hf : Tendsto (fun x => (f x - f a) / (x - a)) (π[>] a) (π f')
ha : βαΆ (x : β) in π[>] a, a < x
h : βαΆ (x : β) in π[>] a, f x β€ f a
β’ βαΆ (x : β) in π[>] a, (f x - f a) / (x - a) β€ 0
|
case h
f : β β β
f' x a b : β
hβ : IsLocalMax f a
hfβ : HasDerivAt f f' a
hf : Tendsto (fun x => (f x - f a) / (x - a)) (π[>] a) (π f')
ha : βαΆ (x : β) in π[>] a, a < x
h : βαΆ (x : β) in π[>] a, f x β€ f a
β’ β (a_1 : β), a < a_1 β f a_1 β€ f a β (f a_1 - f a) / (a_1 - a) β€ 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 : β β β
f' x a b : β
hβ : IsLocalMax f a
hfβ : HasDerivAt f f' a
hf : Tendsto (fun x => (f x - f a) / (x - a)) (π[>] a) (π f')
ha : βαΆ (x : β) in π[>] a, a < x
h : βαΆ (x : β) in π[>] a, f x β€ f a
β’ β (a_1 : β), a < a_1 β f a_1 β€ f a β (f a_1 - f a) / (a_1 - a) β€ 0
|
case h
f : β β β
f' xβ a b : β
hβΒΉ : IsLocalMax f a
hfβ : HasDerivAt f f' a
hf : Tendsto (fun x => (f x - f a) / (x - a)) (π[>] a) (π f')
haβ : βαΆ (x : β) in π[>] a, a < x
hβ : βαΆ (x : β) in π[>] a, f x β€ f a
x : β
ha : a < x
h : f x β€ f a
β’ (f x - f a) / (x - a) β€ 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 : β β β
f' xβ a b : β
hβΒΉ : IsLocalMax f a
hfβ : HasDerivAt f f' a
hf : Tendsto (fun x => (f x - f a) / (x - a)) (π[>] a) (π f')
haβ : βαΆ (x : β) in π[>] a, a < x
hβ : βαΆ (x : β) in π[>] a, f x β€ f a
x : β
ha : a < x
h : f x β€ f a
β’ (f x - f a) / (x - a) β€ 0
|
case h.ha
f : β β β
f' xβ a b : β
hβΒΉ : IsLocalMax f a
hfβ : HasDerivAt f f' a
hf : Tendsto (fun x => (f x - f a) / (x - a)) (π[>] a) (π f')
haβ : βαΆ (x : β) in π[>] a, a < x
hβ : βαΆ (x : β) in π[>] a, f x β€ f a
x : β
ha : a < x
h : f x β€ f a
β’ f x - f a β€ 0
case h.hb
f : β β β
f' xβ a b : β
hβΒΉ : IsLocalMax f a
hfβ : HasDerivAt f f' a
hf : Tendsto (fun x => (f x - f a) / (x - a)) (π[>] a) (π f')
haβ : βαΆ (x : β) in π[>] a, a < x
hβ : βαΆ (x : β) in π[>] a, f x β€ f a
x : β
ha : a < x
h : f x β€ f a
β’ 0 β€ 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 : β β β
f' x a b : β
h : IsLocalMax f a
hf : HasDerivAt f f' a
β’ Tendsto (fun x => (f x - f a) / (x - a)) (π[>] a) (π f')
|
f : β β β
f' x a b : β
h : IsLocalMax f a
hf : Tendsto (fun x => (f x - f a) / (x - a)) (π[β ] a) (π f')
β’ Tendsto (fun x => (f x - f a) / (x - a)) (π[>] a) (π 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 : β β β
f' x a b : β
h : IsLocalMax f a
hf : Tendsto (fun x => (f x - f a) / (x - a)) (π[β ] a) (π f')
β’ Tendsto (fun x => (f x - f a) / (x - a)) (π[>] a) (π 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 : β β β
f' xβ a b : β
hβΒΉ : IsLocalMax f a
hfβ : HasDerivAt f f' a
hf : Tendsto (fun x => (f x - f a) / (x - a)) (π[>] a) (π f')
haβ : βαΆ (x : β) in π[>] a, a < x
hβ : βαΆ (x : β) in π[>] a, f x β€ f a
x : β
ha : a < x
h : f x β€ f a
β’ f x - f a β€ 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 : β β β
f' xβ a b : β
hβΒΉ : IsLocalMax f a
hfβ : HasDerivAt f f' a
hf : Tendsto (fun x => (f x - f a) / (x - a)) (π[>] a) (π f')
haβ : βαΆ (x : β) in π[>] a, a < x
hβ : βαΆ (x : β) in π[>] a, f x β€ f a
x : β
ha : a < x
h : f x β€ f a
β’ 0 β€ 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 β¦ (f x - f a) / (x - a)) (π[<] a) (π f') := by
rw [hasDerivAt_iff_tendsto_slope] at hf
apply hf.mono_left (nhds_left'_le_nhds_ne a)
|
f : β β β
f' x a b : β
h : IsLocalMax f a
hf : HasDerivAt f f' a
β’ 0 β€ f'
|
f : β β β
f' x a b : β
h : IsLocalMax f a
hfβ : HasDerivAt f f' a
hf : Tendsto (fun x => (f x - f a) / (x - a)) (π[<] a) (π f')
β’ 0 β€ 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 βαΆ x in π[<] a, (f x - f a) / (x - a) β₯ 0 from ge_of_tendsto hf this
|
f : β β β
f' x a b : β
h : IsLocalMax f a
hfβ : HasDerivAt f f' a
hf : Tendsto (fun x => (f x - f a) / (x - a)) (π[<] a) (π f')
β’ 0 β€ f'
|
f : β β β
f' x a b : β
h : IsLocalMax f a
hfβ : HasDerivAt f f' a
hf : Tendsto (fun x => (f x - f a) / (x - a)) (π[<] a) (π f')
β’ βαΆ (x : β) in π[<] a, (f x - f a) / (x - a) β₯ 0
|
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