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/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, x < a := 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, x < a
⊢ ∀ᶠ (x : ℝ) in 𝓝[<] a, (f x - f a) / (x - a) ≥ 0
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture2.lean
|
Tutorial.IsLocalMax.hasDerivAt_eq_zero
|
[45, 1]
|
[80, 15]
|
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, x < a
⊢ ∀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, x < a
h : ∀ᶠ (x : ℝ) in 𝓝[<] a, f x ≤ f a
⊢ ∀ᶠ (x : ℝ) in 𝓝[<] a, (f x - f a) / (x - a) ≥ 0
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture2.lean
|
Tutorial.IsLocalMax.hasDerivAt_eq_zero
|
[45, 1]
|
[80, 15]
|
filter_upwards [ha, h]
|
f : ℝ → ℝ
f' x a b : ℝ
h✝ : IsLocalMax f a
hf✝ : HasDerivAt f f' a
hf : Tendsto (fun x => (f x - f a) / (x - a)) (𝓝[<] a) (𝓝 f')
ha : ∀ᶠ (x : ℝ) in 𝓝[<] a, x < a
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, x < a
h : ∀ᶠ (x : ℝ) in 𝓝[<] a, f x ≤ f a
⊢ ∀ a_1 < a, f a_1 ≤ f a → (f a_1 - f a) / (a_1 - a) ≥ 0
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture2.lean
|
Tutorial.IsLocalMax.hasDerivAt_eq_zero
|
[45, 1]
|
[80, 15]
|
intro x ha h
|
case h
f : ℝ → ℝ
f' x a b : ℝ
h✝ : IsLocalMax f a
hf✝ : HasDerivAt f f' a
hf : Tendsto (fun x => (f x - f a) / (x - a)) (𝓝[<] a) (𝓝 f')
ha : ∀ᶠ (x : ℝ) in 𝓝[<] a, x < a
h : ∀ᶠ (x : ℝ) in 𝓝[<] a, f x ≤ f a
⊢ ∀ a_1 < a, 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, x < a
h✝ : ∀ᶠ (x : ℝ) in 𝓝[<] a, f x ≤ f a
x : ℝ
ha : x < a
h : f x ≤ f a
⊢ (f x - f a) / (x - a) ≥ 0
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture2.lean
|
Tutorial.IsLocalMax.hasDerivAt_eq_zero
|
[45, 1]
|
[80, 15]
|
apply div_nonneg_of_nonpos
|
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, x < a
h✝ : ∀ᶠ (x : ℝ) in 𝓝[<] a, f x ≤ f a
x : ℝ
ha : x < a
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, x < a
h✝ : ∀ᶠ (x : ℝ) in 𝓝[<] a, f x ≤ f a
x : ℝ
ha : x < a
h : f x ≤ f a
⊢ f x - f a ≤ 0
case h.hb
f : ℝ → ℝ
f' x✝ a b : ℝ
h✝¹ : IsLocalMax f a
hf✝ : HasDerivAt f f' a
hf : Tendsto (fun x => (f x - f a) / (x - a)) (𝓝[<] a) (𝓝 f')
ha✝ : ∀ᶠ (x : ℝ) in 𝓝[<] a, x < a
h✝ : ∀ᶠ (x : ℝ) in 𝓝[<] a, f x ≤ f a
x : ℝ
ha : x < a
h : f x ≤ f a
⊢ x - a ≤ 0
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture2.lean
|
Tutorial.IsLocalMax.hasDerivAt_eq_zero
|
[45, 1]
|
[80, 15]
|
rw [hasDerivAt_iff_tendsto_slope] at hf
|
f : ℝ → ℝ
f' x a b : ℝ
h : IsLocalMax f a
hf : HasDerivAt f f' a
⊢ Tendsto (fun x => (f x - f a) / (x - a)) (𝓝[<] a) (𝓝 f')
|
f : ℝ → ℝ
f' x a b : ℝ
h : IsLocalMax f a
hf : Tendsto (fun x => (f x - f a) / (x - a)) (𝓝[≠] a) (𝓝 f')
⊢ Tendsto (fun x => (f x - f a) / (x - a)) (𝓝[<] a) (𝓝 f')
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture2.lean
|
Tutorial.IsLocalMax.hasDerivAt_eq_zero
|
[45, 1]
|
[80, 15]
|
apply hf.mono_left (nhds_left'_le_nhds_ne a)
|
f : ℝ → ℝ
f' x a b : ℝ
h : IsLocalMax f a
hf : Tendsto (fun x => (f x - f a) / (x - a)) (𝓝[≠] a) (𝓝 f')
⊢ Tendsto (fun x => (f x - f a) / (x - a)) (𝓝[<] a) (𝓝 f')
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture2.lean
|
Tutorial.IsLocalMax.hasDerivAt_eq_zero
|
[45, 1]
|
[80, 15]
|
linarith
|
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, x < a
h✝ : ∀ᶠ (x : ℝ) in 𝓝[<] a, f x ≤ f a
x : ℝ
ha : x < a
h : f x ≤ f a
⊢ f x - f a ≤ 0
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture2.lean
|
Tutorial.IsLocalMax.hasDerivAt_eq_zero
|
[45, 1]
|
[80, 15]
|
linarith
|
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, x < a
h✝ : ∀ᶠ (x : ℝ) in 𝓝[<] a, f x ≤ f a
x : ℝ
ha : x < a
h : f x ≤ f a
⊢ x - a ≤ 0
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture2.lean
|
Tutorial.IsLocalMin.hasDerivAt_eq_zero
|
[84, 1]
|
[90, 15]
|
suffices -f' = 0 from neg_eq_zero.mp this
|
f : ℝ → ℝ
f' x a b : ℝ
h : IsLocalMin f a
hf : HasDerivAt f f' a
⊢ f' = 0
|
f : ℝ → ℝ
f' x a b : ℝ
h : IsLocalMin f a
hf : HasDerivAt f f' a
⊢ -f' = 0
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture2.lean
|
Tutorial.IsLocalMin.hasDerivAt_eq_zero
|
[84, 1]
|
[90, 15]
|
apply IsLocalMax.hasDerivAt_eq_zero
|
f : ℝ → ℝ
f' x a b : ℝ
h : IsLocalMin f a
hf : HasDerivAt f f' a
⊢ -f' = 0
|
case h
f : ℝ → ℝ
f' x a b : ℝ
h : IsLocalMin f a
hf : HasDerivAt f f' a
⊢ IsLocalMax ?f ?a
case hf
f : ℝ → ℝ
f' x a b : ℝ
h : IsLocalMin f a
hf : HasDerivAt f f' a
⊢ HasDerivAt ?f (-f') ?a
case f
f : ℝ → ℝ
f' x a b : ℝ
h : IsLocalMin f a
hf : HasDerivAt f f' a
⊢ ℝ → ℝ
case a
f : ℝ → ℝ
f' x a b : ℝ
h : IsLocalMin f a
hf : HasDerivAt f f' a
⊢ ℝ
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture2.lean
|
Tutorial.IsLocalMin.hasDerivAt_eq_zero
|
[84, 1]
|
[90, 15]
|
apply IsLocalMin.neg h
|
case h
f : ℝ → ℝ
f' x a b : ℝ
h : IsLocalMin f a
hf : HasDerivAt f f' a
⊢ IsLocalMax ?f ?a
case hf
f : ℝ → ℝ
f' x a b : ℝ
h : IsLocalMin f a
hf : HasDerivAt f f' a
⊢ HasDerivAt ?f (-f') ?a
case f
f : ℝ → ℝ
f' x a b : ℝ
h : IsLocalMin f a
hf : HasDerivAt f f' a
⊢ ℝ → ℝ
case a
f : ℝ → ℝ
f' x a b : ℝ
h : IsLocalMin f a
hf : HasDerivAt f f' a
⊢ ℝ
|
case hf
f : ℝ → ℝ
f' x a b : ℝ
h : IsLocalMin f a
hf : HasDerivAt f f' a
⊢ HasDerivAt (fun x => -f x) (-f') a
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture2.lean
|
Tutorial.IsLocalMin.hasDerivAt_eq_zero
|
[84, 1]
|
[90, 15]
|
apply hf.neg
|
case hf
f : ℝ → ℝ
f' x a b : ℝ
h : IsLocalMin f a
hf : HasDerivAt f f' a
⊢ HasDerivAt (fun x => -f x) (-f') a
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture2.lean
|
Tutorial.IsLocalExtr.hasDerivAt_eq_zero
|
[97, 1]
|
[103, 45]
|
apply IsLocalExtr.elim h
|
f : ℝ → ℝ
f' x a b : ℝ
h : IsLocalExtr f a
hf : HasDerivAt f f' a
⊢ f' = 0
|
case a
f : ℝ → ℝ
f' x a b : ℝ
h : IsLocalExtr f a
hf : HasDerivAt f f' a
⊢ IsLocalMin f a → f' = 0
case a
f : ℝ → ℝ
f' x a b : ℝ
h : IsLocalExtr f a
hf : HasDerivAt f f' a
⊢ IsLocalMax f a → f' = 0
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture2.lean
|
Tutorial.IsLocalExtr.hasDerivAt_eq_zero
|
[97, 1]
|
[103, 45]
|
intro h
|
case a
f : ℝ → ℝ
f' x a b : ℝ
h : IsLocalExtr f a
hf : HasDerivAt f f' a
⊢ IsLocalMin f a → f' = 0
|
case a
f : ℝ → ℝ
f' x a b : ℝ
h✝ : IsLocalExtr f a
hf : HasDerivAt f f' a
h : IsLocalMin f a
⊢ f' = 0
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture2.lean
|
Tutorial.IsLocalExtr.hasDerivAt_eq_zero
|
[97, 1]
|
[103, 45]
|
apply IsLocalMin.hasDerivAt_eq_zero h hf
|
case a
f : ℝ → ℝ
f' x a b : ℝ
h✝ : IsLocalExtr f a
hf : HasDerivAt f f' a
h : IsLocalMin f a
⊢ f' = 0
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture2.lean
|
Tutorial.IsLocalExtr.hasDerivAt_eq_zero
|
[97, 1]
|
[103, 45]
|
intro h
|
case a
f : ℝ → ℝ
f' x a b : ℝ
h : IsLocalExtr f a
hf : HasDerivAt f f' a
⊢ IsLocalMax f a → f' = 0
|
case a
f : ℝ → ℝ
f' x a b : ℝ
h✝ : IsLocalExtr f a
hf : HasDerivAt f f' a
h : IsLocalMax f a
⊢ f' = 0
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture2.lean
|
Tutorial.IsLocalExtr.hasDerivAt_eq_zero
|
[97, 1]
|
[103, 45]
|
apply IsLocalMax.hasDerivAt_eq_zero h hf
|
case a
f : ℝ → ℝ
f' x a b : ℝ
h✝ : IsLocalExtr f a
hf : HasDerivAt f f' a
h : IsLocalMax f a
⊢ f' = 0
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture2.lean
|
Tutorial.exists_local_extr_Ioo
|
[116, 1]
|
[142, 55]
|
suffices ∃ c ∈ Ioo a b, IsExtrOn f (Icc a b) c by
rcases this with ⟨c, cmem, hc⟩
exists c, cmem
apply hc.isLocalExtr <| Icc_mem_nhds cmem.1 cmem.2
|
f : ℝ → ℝ
f' x a b : ℝ
hab : a < b
hfc : ContinuousOn f (Icc a b)
hfI : f a = f b
⊢ ∃ c ∈ Ioo a b, IsLocalExtr f c
|
f : ℝ → ℝ
f' x a b : ℝ
hab : a < b
hfc : ContinuousOn f (Icc a b)
hfI : f a = f b
⊢ ∃ c ∈ Ioo a b, IsExtrOn f (Icc a b) c
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture2.lean
|
Tutorial.exists_local_extr_Ioo
|
[116, 1]
|
[142, 55]
|
have ne : (Icc a b).Nonempty := nonempty_Icc.2 (le_of_lt hab)
|
f : ℝ → ℝ
f' x a b : ℝ
hab : a < b
hfc : ContinuousOn f (Icc a b)
hfI : f a = f b
⊢ ∃ c ∈ Ioo a b, IsExtrOn f (Icc a b) c
|
f : ℝ → ℝ
f' x a b : ℝ
hab : a < b
hfc : ContinuousOn f (Icc a b)
hfI : f a = f b
ne : Set.Nonempty (Icc a b)
⊢ ∃ c ∈ Ioo a b, IsExtrOn f (Icc a b) c
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture2.lean
|
Tutorial.exists_local_extr_Ioo
|
[116, 1]
|
[142, 55]
|
have ⟨C, Cmem, Cge⟩ : ∃ C ∈ Icc a b, IsMaxOn f (Icc a b) C := by
apply isCompact_Icc.exists_isMaxOn ne hfc
|
f : ℝ → ℝ
f' x a b : ℝ
hab : a < b
hfc : ContinuousOn f (Icc a b)
hfI : f a = f b
ne : Set.Nonempty (Icc a b)
⊢ ∃ c ∈ Ioo a b, IsExtrOn f (Icc a b) c
|
f : ℝ → ℝ
f' x a b : ℝ
hab : a < b
hfc : ContinuousOn f (Icc a b)
hfI : f a = f b
ne : Set.Nonempty (Icc a b)
C : ℝ
Cmem : C ∈ Icc a b
Cge : IsMaxOn f (Icc a b) C
⊢ ∃ c ∈ Ioo a b, IsExtrOn f (Icc a b) c
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture2.lean
|
Tutorial.exists_local_extr_Ioo
|
[116, 1]
|
[142, 55]
|
have ⟨c, cmem, cle⟩ : ∃ c ∈ Icc a b, IsMinOn f (Icc a b) c := by
apply isCompact_Icc.exists_isMinOn ne hfc
|
f : ℝ → ℝ
f' x a b : ℝ
hab : a < b
hfc : ContinuousOn f (Icc a b)
hfI : f a = f b
ne : Set.Nonempty (Icc a b)
C : ℝ
Cmem : C ∈ Icc a b
Cge : IsMaxOn f (Icc a b) C
⊢ ∃ c ∈ Ioo a b, IsExtrOn f (Icc a b) c
|
f : ℝ → ℝ
f' x a b : ℝ
hab : a < b
hfc : ContinuousOn f (Icc a b)
hfI : f a = f b
ne : Set.Nonempty (Icc a b)
C : ℝ
Cmem : C ∈ Icc a b
Cge : IsMaxOn f (Icc a b) C
c : ℝ
cmem : c ∈ Icc a b
cle : IsMinOn f (Icc a b) c
⊢ ∃ c ∈ Ioo a b, IsExtrOn f (Icc a b) c
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture2.lean
|
Tutorial.exists_local_extr_Ioo
|
[116, 1]
|
[142, 55]
|
change ∀ x ∈ Icc a b, f x ≤ f C at Cge
|
f : ℝ → ℝ
f' x a b : ℝ
hab : a < b
hfc : ContinuousOn f (Icc a b)
hfI : f a = f b
ne : Set.Nonempty (Icc a b)
C : ℝ
Cmem : C ∈ Icc a b
Cge : IsMaxOn f (Icc a b) C
c : ℝ
cmem : c ∈ Icc a b
cle : IsMinOn f (Icc a b) c
⊢ ∃ c ∈ Ioo a b, IsExtrOn f (Icc a b) c
|
f : ℝ → ℝ
f' x a b : ℝ
hab : a < b
hfc : ContinuousOn f (Icc a b)
hfI : f a = f b
ne : Set.Nonempty (Icc a b)
C : ℝ
Cmem : C ∈ Icc a b
c : ℝ
cmem : c ∈ Icc a b
cle : IsMinOn f (Icc a b) c
Cge : ∀ x ∈ Icc a b, f x ≤ f C
⊢ ∃ c ∈ Ioo a b, IsExtrOn f (Icc a b) c
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture2.lean
|
Tutorial.exists_local_extr_Ioo
|
[116, 1]
|
[142, 55]
|
change ∀ x ∈ Icc a b, f c ≤ f x at cle
|
f : ℝ → ℝ
f' x a b : ℝ
hab : a < b
hfc : ContinuousOn f (Icc a b)
hfI : f a = f b
ne : Set.Nonempty (Icc a b)
C : ℝ
Cmem : C ∈ Icc a b
c : ℝ
cmem : c ∈ Icc a b
cle : IsMinOn f (Icc a b) c
Cge : ∀ x ∈ Icc a b, f x ≤ f C
⊢ ∃ c ∈ Ioo a b, IsExtrOn f (Icc a b) c
|
f : ℝ → ℝ
f' x a b : ℝ
hab : a < b
hfc : ContinuousOn f (Icc a b)
hfI : f a = f b
ne : Set.Nonempty (Icc a b)
C : ℝ
Cmem : C ∈ Icc a b
c : ℝ
cmem : c ∈ Icc a b
Cge : ∀ x ∈ Icc a b, f x ≤ f C
cle : ∀ x ∈ Icc a b, f c ≤ f x
⊢ ∃ c ∈ Ioo a b, IsExtrOn f (Icc a b) c
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture2.lean
|
Tutorial.exists_local_extr_Ioo
|
[116, 1]
|
[142, 55]
|
by_cases hc : f c = f a
|
f : ℝ → ℝ
f' x a b : ℝ
hab : a < b
hfc : ContinuousOn f (Icc a b)
hfI : f a = f b
ne : Set.Nonempty (Icc a b)
C : ℝ
Cmem : C ∈ Icc a b
c : ℝ
cmem : c ∈ Icc a b
Cge : ∀ x ∈ Icc a b, f x ≤ f C
cle : ∀ x ∈ Icc a b, f c ≤ f x
⊢ ∃ c ∈ Ioo a b, IsExtrOn f (Icc a b) c
|
case pos
f : ℝ → ℝ
f' x a b : ℝ
hab : a < b
hfc : ContinuousOn f (Icc a b)
hfI : f a = f b
ne : Set.Nonempty (Icc a b)
C : ℝ
Cmem : C ∈ Icc a b
c : ℝ
cmem : c ∈ Icc a b
Cge : ∀ x ∈ Icc a b, f x ≤ f C
cle : ∀ x ∈ Icc a b, f c ≤ f x
hc : f c = f a
⊢ ∃ c ∈ Ioo a b, IsExtrOn f (Icc a b) c
case neg
f : ℝ → ℝ
f' x a b : ℝ
hab : a < b
hfc : ContinuousOn f (Icc a b)
hfI : f a = f b
ne : Set.Nonempty (Icc a b)
C : ℝ
Cmem : C ∈ Icc a b
c : ℝ
cmem : c ∈ Icc a b
Cge : ∀ x ∈ Icc a b, f x ≤ f C
cle : ∀ x ∈ Icc a b, f c ≤ f x
hc : ¬f c = f a
⊢ ∃ c ∈ Ioo a b, IsExtrOn f (Icc a b) c
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture2.lean
|
Tutorial.exists_local_extr_Ioo
|
[116, 1]
|
[142, 55]
|
rcases this with ⟨c, cmem, hc⟩
|
f : ℝ → ℝ
f' x a b : ℝ
hab : a < b
hfc : ContinuousOn f (Icc a b)
hfI : f a = f b
this : ∃ c ∈ Ioo a b, IsExtrOn f (Icc a b) c
⊢ ∃ c ∈ Ioo a b, IsLocalExtr f c
|
case intro.intro
f : ℝ → ℝ
f' x a b : ℝ
hab : a < b
hfc : ContinuousOn f (Icc a b)
hfI : f a = f b
c : ℝ
cmem : c ∈ Ioo a b
hc : IsExtrOn f (Icc a b) c
⊢ ∃ c ∈ Ioo a b, IsLocalExtr f c
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture2.lean
|
Tutorial.exists_local_extr_Ioo
|
[116, 1]
|
[142, 55]
|
exists c, cmem
|
case intro.intro
f : ℝ → ℝ
f' x a b : ℝ
hab : a < b
hfc : ContinuousOn f (Icc a b)
hfI : f a = f b
c : ℝ
cmem : c ∈ Ioo a b
hc : IsExtrOn f (Icc a b) c
⊢ ∃ c ∈ Ioo a b, IsLocalExtr f c
|
case intro.intro
f : ℝ → ℝ
f' x a b : ℝ
hab : a < b
hfc : ContinuousOn f (Icc a b)
hfI : f a = f b
c : ℝ
cmem : c ∈ Ioo a b
hc : IsExtrOn f (Icc a b) c
⊢ IsLocalExtr f c
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture2.lean
|
Tutorial.exists_local_extr_Ioo
|
[116, 1]
|
[142, 55]
|
apply hc.isLocalExtr <| Icc_mem_nhds cmem.1 cmem.2
|
case intro.intro
f : ℝ → ℝ
f' x a b : ℝ
hab : a < b
hfc : ContinuousOn f (Icc a b)
hfI : f a = f b
c : ℝ
cmem : c ∈ Ioo a b
hc : IsExtrOn f (Icc a b) c
⊢ IsLocalExtr f c
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture2.lean
|
Tutorial.exists_local_extr_Ioo
|
[116, 1]
|
[142, 55]
|
apply isCompact_Icc.exists_isMaxOn ne hfc
|
f : ℝ → ℝ
f' x a b : ℝ
hab : a < b
hfc : ContinuousOn f (Icc a b)
hfI : f a = f b
ne : Set.Nonempty (Icc a b)
⊢ ∃ C ∈ Icc a b, IsMaxOn f (Icc a b) C
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture2.lean
|
Tutorial.exists_local_extr_Ioo
|
[116, 1]
|
[142, 55]
|
apply isCompact_Icc.exists_isMinOn ne hfc
|
f : ℝ → ℝ
f' x a b : ℝ
hab : a < b
hfc : ContinuousOn f (Icc a b)
hfI : f a = f b
ne : Set.Nonempty (Icc a b)
C : ℝ
Cmem : C ∈ Icc a b
Cge : IsMaxOn f (Icc a b) C
⊢ ∃ c ∈ Icc a b, IsMinOn f (Icc a b) c
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture2.lean
|
Tutorial.exists_local_extr_Ioo
|
[116, 1]
|
[142, 55]
|
by_cases hC : f C = f a
|
case pos
f : ℝ → ℝ
f' x a b : ℝ
hab : a < b
hfc : ContinuousOn f (Icc a b)
hfI : f a = f b
ne : Set.Nonempty (Icc a b)
C : ℝ
Cmem : C ∈ Icc a b
c : ℝ
cmem : c ∈ Icc a b
Cge : ∀ x ∈ Icc a b, f x ≤ f C
cle : ∀ x ∈ Icc a b, f c ≤ f x
hc : f c = f a
⊢ ∃ c ∈ Ioo a b, IsExtrOn f (Icc a b) c
|
case pos
f : ℝ → ℝ
f' x a b : ℝ
hab : a < b
hfc : ContinuousOn f (Icc a b)
hfI : f a = f b
ne : Set.Nonempty (Icc a b)
C : ℝ
Cmem : C ∈ Icc a b
c : ℝ
cmem : c ∈ Icc a b
Cge : ∀ x ∈ Icc a b, f x ≤ f C
cle : ∀ x ∈ Icc a b, f c ≤ f x
hc : f c = f a
hC : f C = f a
⊢ ∃ c ∈ Ioo a b, IsExtrOn f (Icc a b) c
case neg
f : ℝ → ℝ
f' x a b : ℝ
hab : a < b
hfc : ContinuousOn f (Icc a b)
hfI : f a = f b
ne : Set.Nonempty (Icc a b)
C : ℝ
Cmem : C ∈ Icc a b
c : ℝ
cmem : c ∈ Icc a b
Cge : ∀ x ∈ Icc a b, f x ≤ f C
cle : ∀ x ∈ Icc a b, f c ≤ f x
hc : f c = f a
hC : ¬f C = f a
⊢ ∃ c ∈ Ioo a b, IsExtrOn f (Icc a b) c
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture2.lean
|
Tutorial.exists_local_extr_Ioo
|
[116, 1]
|
[142, 55]
|
have : ∀ x ∈ Icc a b, f x = f a := fun x hx ↦ le_antisymm (hC ▸ Cge x hx) (hc ▸ cle x hx)
|
case pos
f : ℝ → ℝ
f' x a b : ℝ
hab : a < b
hfc : ContinuousOn f (Icc a b)
hfI : f a = f b
ne : Set.Nonempty (Icc a b)
C : ℝ
Cmem : C ∈ Icc a b
c : ℝ
cmem : c ∈ Icc a b
Cge : ∀ x ∈ Icc a b, f x ≤ f C
cle : ∀ x ∈ Icc a b, f c ≤ f x
hc : f c = f a
hC : f C = f a
⊢ ∃ c ∈ Ioo a b, IsExtrOn f (Icc a b) c
|
case pos
f : ℝ → ℝ
f' x a b : ℝ
hab : a < b
hfc : ContinuousOn f (Icc a b)
hfI : f a = f b
ne : Set.Nonempty (Icc a b)
C : ℝ
Cmem : C ∈ Icc a b
c : ℝ
cmem : c ∈ Icc a b
Cge : ∀ x ∈ Icc a b, f x ≤ f C
cle : ∀ x ∈ Icc a b, f c ≤ f x
hc : f c = f a
hC : f C = f a
this : ∀ x ∈ Icc a b, f x = f a
⊢ ∃ c ∈ Ioo a b, IsExtrOn f (Icc a b) c
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture2.lean
|
Tutorial.exists_local_extr_Ioo
|
[116, 1]
|
[142, 55]
|
rcases nonempty_Ioo.2 hab with ⟨c', hc'⟩
|
case pos
f : ℝ → ℝ
f' x a b : ℝ
hab : a < b
hfc : ContinuousOn f (Icc a b)
hfI : f a = f b
ne : Set.Nonempty (Icc a b)
C : ℝ
Cmem : C ∈ Icc a b
c : ℝ
cmem : c ∈ Icc a b
Cge : ∀ x ∈ Icc a b, f x ≤ f C
cle : ∀ x ∈ Icc a b, f c ≤ f x
hc : f c = f a
hC : f C = f a
this : ∀ x ∈ Icc a b, f x = f a
⊢ ∃ c ∈ Ioo a b, IsExtrOn f (Icc a b) c
|
case pos.intro
f : ℝ → ℝ
f' x a b : ℝ
hab : a < b
hfc : ContinuousOn f (Icc a b)
hfI : f a = f b
ne : Set.Nonempty (Icc a b)
C : ℝ
Cmem : C ∈ Icc a b
c : ℝ
cmem : c ∈ Icc a b
Cge : ∀ x ∈ Icc a b, f x ≤ f C
cle : ∀ x ∈ Icc a b, f c ≤ f x
hc : f c = f a
hC : f C = f a
this : ∀ x ∈ Icc a b, f x = f a
c' : ℝ
hc' : c' ∈ Ioo a b
⊢ ∃ c ∈ Ioo a b, IsExtrOn f (Icc a b) c
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture2.lean
|
Tutorial.exists_local_extr_Ioo
|
[116, 1]
|
[142, 55]
|
refine ⟨c', hc', Or.inl fun x hx ↦ ?_⟩
|
case pos.intro
f : ℝ → ℝ
f' x a b : ℝ
hab : a < b
hfc : ContinuousOn f (Icc a b)
hfI : f a = f b
ne : Set.Nonempty (Icc a b)
C : ℝ
Cmem : C ∈ Icc a b
c : ℝ
cmem : c ∈ Icc a b
Cge : ∀ x ∈ Icc a b, f x ≤ f C
cle : ∀ x ∈ Icc a b, f c ≤ f x
hc : f c = f a
hC : f C = f a
this : ∀ x ∈ Icc a b, f x = f a
c' : ℝ
hc' : c' ∈ Ioo a b
⊢ ∃ c ∈ Ioo a b, IsExtrOn f (Icc a b) c
|
case pos.intro
f : ℝ → ℝ
f' x✝ a b : ℝ
hab : a < b
hfc : ContinuousOn f (Icc a b)
hfI : f a = f b
ne : Set.Nonempty (Icc a b)
C : ℝ
Cmem : C ∈ Icc a b
c : ℝ
cmem : c ∈ Icc a b
Cge : ∀ x ∈ Icc a b, f x ≤ f C
cle : ∀ x ∈ Icc a b, f c ≤ f x
hc : f c = f a
hC : f C = f a
this : ∀ x ∈ Icc a b, f x = f a
c' : ℝ
hc' : c' ∈ Ioo a b
x : ℝ
hx : x ∈ Icc a b
⊢ x ∈ {x | (fun x => f c' ≤ f x) x}
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture2.lean
|
Tutorial.exists_local_extr_Ioo
|
[116, 1]
|
[142, 55]
|
simp [this x hx, this c' (Ioo_subset_Icc_self hc')]
|
case pos.intro
f : ℝ → ℝ
f' x✝ a b : ℝ
hab : a < b
hfc : ContinuousOn f (Icc a b)
hfI : f a = f b
ne : Set.Nonempty (Icc a b)
C : ℝ
Cmem : C ∈ Icc a b
c : ℝ
cmem : c ∈ Icc a b
Cge : ∀ x ∈ Icc a b, f x ≤ f C
cle : ∀ x ∈ Icc a b, f c ≤ f x
hc : f c = f a
hC : f C = f a
this : ∀ x ∈ Icc a b, f x = f a
c' : ℝ
hc' : c' ∈ Ioo a b
x : ℝ
hx : x ∈ Icc a b
⊢ x ∈ {x | (fun x => f c' ≤ f x) x}
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture2.lean
|
Tutorial.exists_local_extr_Ioo
|
[116, 1]
|
[142, 55]
|
refine ⟨C, ⟨lt_of_le_of_ne Cmem.1 <| mt ?_ hC, lt_of_le_of_ne Cmem.2 <| mt ?_ hC⟩, Or.inr Cge⟩
|
case neg
f : ℝ → ℝ
f' x a b : ℝ
hab : a < b
hfc : ContinuousOn f (Icc a b)
hfI : f a = f b
ne : Set.Nonempty (Icc a b)
C : ℝ
Cmem : C ∈ Icc a b
c : ℝ
cmem : c ∈ Icc a b
Cge : ∀ x ∈ Icc a b, f x ≤ f C
cle : ∀ x ∈ Icc a b, f c ≤ f x
hc : f c = f a
hC : ¬f C = f a
⊢ ∃ c ∈ Ioo a b, IsExtrOn f (Icc a b) c
|
case neg.refine_1
f : ℝ → ℝ
f' x a b : ℝ
hab : a < b
hfc : ContinuousOn f (Icc a b)
hfI : f a = f b
ne : Set.Nonempty (Icc a b)
C : ℝ
Cmem : C ∈ Icc a b
c : ℝ
cmem : c ∈ Icc a b
Cge : ∀ x ∈ Icc a b, f x ≤ f C
cle : ∀ x ∈ Icc a b, f c ≤ f x
hc : f c = f a
hC : ¬f C = f a
⊢ a = C → f C = f a
case neg.refine_2
f : ℝ → ℝ
f' x a b : ℝ
hab : a < b
hfc : ContinuousOn f (Icc a b)
hfI : f a = f b
ne : Set.Nonempty (Icc a b)
C : ℝ
Cmem : C ∈ Icc a b
c : ℝ
cmem : c ∈ Icc a b
Cge : ∀ x ∈ Icc a b, f x ≤ f C
cle : ∀ x ∈ Icc a b, f c ≤ f x
hc : f c = f a
hC : ¬f C = f a
⊢ C = b → f C = f a
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture2.lean
|
Tutorial.exists_local_extr_Ioo
|
[116, 1]
|
[142, 55]
|
exacts [fun h ↦ by rw [h], fun h ↦ by rw [h, hfI]]
|
case neg.refine_1
f : ℝ → ℝ
f' x a b : ℝ
hab : a < b
hfc : ContinuousOn f (Icc a b)
hfI : f a = f b
ne : Set.Nonempty (Icc a b)
C : ℝ
Cmem : C ∈ Icc a b
c : ℝ
cmem : c ∈ Icc a b
Cge : ∀ x ∈ Icc a b, f x ≤ f C
cle : ∀ x ∈ Icc a b, f c ≤ f x
hc : f c = f a
hC : ¬f C = f a
⊢ a = C → f C = f a
case neg.refine_2
f : ℝ → ℝ
f' x a b : ℝ
hab : a < b
hfc : ContinuousOn f (Icc a b)
hfI : f a = f b
ne : Set.Nonempty (Icc a b)
C : ℝ
Cmem : C ∈ Icc a b
c : ℝ
cmem : c ∈ Icc a b
Cge : ∀ x ∈ Icc a b, f x ≤ f C
cle : ∀ x ∈ Icc a b, f c ≤ f x
hc : f c = f a
hC : ¬f C = f a
⊢ C = b → f C = f a
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture2.lean
|
Tutorial.exists_local_extr_Ioo
|
[116, 1]
|
[142, 55]
|
rw [h]
|
f : ℝ → ℝ
f' x a b : ℝ
hab : a < b
hfc : ContinuousOn f (Icc a b)
hfI : f a = f b
ne : Set.Nonempty (Icc a b)
C : ℝ
Cmem : C ∈ Icc a b
c : ℝ
cmem : c ∈ Icc a b
Cge : ∀ x ∈ Icc a b, f x ≤ f C
cle : ∀ x ∈ Icc a b, f c ≤ f x
hc : f c = f a
hC : ¬f C = f a
h : a = C
⊢ f C = f a
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture2.lean
|
Tutorial.exists_local_extr_Ioo
|
[116, 1]
|
[142, 55]
|
rw [h, hfI]
|
f : ℝ → ℝ
f' x a b : ℝ
hab : a < b
hfc : ContinuousOn f (Icc a b)
hfI : f a = f b
ne : Set.Nonempty (Icc a b)
C : ℝ
Cmem : C ∈ Icc a b
c : ℝ
cmem : c ∈ Icc a b
Cge : ∀ x ∈ Icc a b, f x ≤ f C
cle : ∀ x ∈ Icc a b, f c ≤ f x
hc : f c = f a
hC : ¬f C = f a
h : C = b
⊢ f C = f a
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture2.lean
|
Tutorial.exists_local_extr_Ioo
|
[116, 1]
|
[142, 55]
|
refine ⟨c, ⟨lt_of_le_of_ne cmem.1 <| mt ?_ hc, lt_of_le_of_ne cmem.2 <| mt ?_ hc⟩, Or.inl cle⟩
|
case neg
f : ℝ → ℝ
f' x a b : ℝ
hab : a < b
hfc : ContinuousOn f (Icc a b)
hfI : f a = f b
ne : Set.Nonempty (Icc a b)
C : ℝ
Cmem : C ∈ Icc a b
c : ℝ
cmem : c ∈ Icc a b
Cge : ∀ x ∈ Icc a b, f x ≤ f C
cle : ∀ x ∈ Icc a b, f c ≤ f x
hc : ¬f c = f a
⊢ ∃ c ∈ Ioo a b, IsExtrOn f (Icc a b) c
|
case neg.refine_1
f : ℝ → ℝ
f' x a b : ℝ
hab : a < b
hfc : ContinuousOn f (Icc a b)
hfI : f a = f b
ne : Set.Nonempty (Icc a b)
C : ℝ
Cmem : C ∈ Icc a b
c : ℝ
cmem : c ∈ Icc a b
Cge : ∀ x ∈ Icc a b, f x ≤ f C
cle : ∀ x ∈ Icc a b, f c ≤ f x
hc : ¬f c = f a
⊢ a = c → f c = f a
case neg.refine_2
f : ℝ → ℝ
f' x a b : ℝ
hab : a < b
hfc : ContinuousOn f (Icc a b)
hfI : f a = f b
ne : Set.Nonempty (Icc a b)
C : ℝ
Cmem : C ∈ Icc a b
c : ℝ
cmem : c ∈ Icc a b
Cge : ∀ x ∈ Icc a b, f x ≤ f C
cle : ∀ x ∈ Icc a b, f c ≤ f x
hc : ¬f c = f a
⊢ c = b → f c = f a
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture2.lean
|
Tutorial.exists_local_extr_Ioo
|
[116, 1]
|
[142, 55]
|
exacts [fun h ↦ by rw [h], fun h ↦ by rw [h, hfI]]
|
case neg.refine_1
f : ℝ → ℝ
f' x a b : ℝ
hab : a < b
hfc : ContinuousOn f (Icc a b)
hfI : f a = f b
ne : Set.Nonempty (Icc a b)
C : ℝ
Cmem : C ∈ Icc a b
c : ℝ
cmem : c ∈ Icc a b
Cge : ∀ x ∈ Icc a b, f x ≤ f C
cle : ∀ x ∈ Icc a b, f c ≤ f x
hc : ¬f c = f a
⊢ a = c → f c = f a
case neg.refine_2
f : ℝ → ℝ
f' x a b : ℝ
hab : a < b
hfc : ContinuousOn f (Icc a b)
hfI : f a = f b
ne : Set.Nonempty (Icc a b)
C : ℝ
Cmem : C ∈ Icc a b
c : ℝ
cmem : c ∈ Icc a b
Cge : ∀ x ∈ Icc a b, f x ≤ f C
cle : ∀ x ∈ Icc a b, f c ≤ f x
hc : ¬f c = f a
⊢ c = b → f c = f a
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture2.lean
|
Tutorial.exists_local_extr_Ioo
|
[116, 1]
|
[142, 55]
|
rw [h]
|
f : ℝ → ℝ
f' x a b : ℝ
hab : a < b
hfc : ContinuousOn f (Icc a b)
hfI : f a = f b
ne : Set.Nonempty (Icc a b)
C : ℝ
Cmem : C ∈ Icc a b
c : ℝ
cmem : c ∈ Icc a b
Cge : ∀ x ∈ Icc a b, f x ≤ f C
cle : ∀ x ∈ Icc a b, f c ≤ f x
hc : ¬f c = f a
h : a = c
⊢ f c = f a
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture2.lean
|
Tutorial.exists_local_extr_Ioo
|
[116, 1]
|
[142, 55]
|
rw [h, hfI]
|
f : ℝ → ℝ
f' x a b : ℝ
hab : a < b
hfc : ContinuousOn f (Icc a b)
hfI : f a = f b
ne : Set.Nonempty (Icc a b)
C : ℝ
Cmem : C ∈ Icc a b
c : ℝ
cmem : c ∈ Icc a b
Cge : ∀ x ∈ Icc a b, f x ≤ f C
cle : ∀ x ∈ Icc a b, f c ≤ f x
hc : ¬f c = f a
h : c = b
⊢ f c = f a
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture2.lean
|
Tutorial.exists_hasDerivAt_eq_zero
|
[147, 1]
|
[151, 68]
|
have ⟨c, cmem, hc⟩ := exists_local_extr_Ioo hab hfc hfI
|
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
|
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 : ℝ
cmem : c ∈ Ioo a b
hc : IsLocalExtr f c
⊢ ∃ c ∈ Ioo a b, f' c = 0
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture2.lean
|
Tutorial.exists_hasDerivAt_eq_zero
|
[147, 1]
|
[151, 68]
|
exact ⟨c, cmem, IsLocalExtr.hasDerivAt_eq_zero hc <| hff' c cmem⟩
|
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 : ℝ
cmem : c ∈ Ioo a b
hc : IsLocalExtr f c
⊢ ∃ c ∈ Ioo a b, f' c = 0
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture2.lean
|
Tutorial.exists_ratio_hasDerivAt_eq_ratio_slope
|
[155, 1]
|
[169, 37]
|
let h x := (g b - g a) * f x - (f b - f a) * g x
|
f✝ : ℝ → ℝ
f'✝ x a✝ b✝ : ℝ
f f' g g' : ℝ → ℝ
a b : ℝ
hab : a < b
hfc : ContinuousOn f (Icc a b)
hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x
hgc : ContinuousOn g (Icc a b)
hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x
⊢ ∃ c ∈ Ioo a b, (g b - g a) * f' c = (f b - f a) * g' c
|
f✝ : ℝ → ℝ
f'✝ x a✝ b✝ : ℝ
f f' g g' : ℝ → ℝ
a b : ℝ
hab : a < b
hfc : ContinuousOn f (Icc a b)
hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x
hgc : ContinuousOn g (Icc a b)
hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x
h : ℝ → ℝ := fun x => (g b - g a) * f x - (f b - f a) * g x
⊢ ∃ c ∈ Ioo a b, (g b - g a) * f' c = (f b - f a) * g' c
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture2.lean
|
Tutorial.exists_ratio_hasDerivAt_eq_ratio_slope
|
[155, 1]
|
[169, 37]
|
have hhc : ContinuousOn h (Icc a b) :=
(continuousOn_const.mul hfc).sub (continuousOn_const.mul hgc)
|
f✝ : ℝ → ℝ
f'✝ x a✝ b✝ : ℝ
f f' g g' : ℝ → ℝ
a b : ℝ
hab : a < b
hfc : ContinuousOn f (Icc a b)
hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x
hgc : ContinuousOn g (Icc a b)
hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x
h : ℝ → ℝ := fun x => (g b - g a) * f x - (f b - f a) * g x
⊢ ∃ c ∈ Ioo a b, (g b - g a) * f' c = (f b - f a) * g' c
|
f✝ : ℝ → ℝ
f'✝ x a✝ b✝ : ℝ
f f' g g' : ℝ → ℝ
a b : ℝ
hab : a < b
hfc : ContinuousOn f (Icc a b)
hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x
hgc : ContinuousOn g (Icc a b)
hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x
h : ℝ → ℝ := fun x => (g b - g a) * f x - (f b - f a) * g x
hhc : ContinuousOn h (Icc a b)
⊢ ∃ c ∈ Ioo a b, (g b - g a) * f' c = (f b - f a) * g' c
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture2.lean
|
Tutorial.exists_ratio_hasDerivAt_eq_ratio_slope
|
[155, 1]
|
[169, 37]
|
have hI : h a = h b := by ring
|
f✝ : ℝ → ℝ
f'✝ x a✝ b✝ : ℝ
f f' g g' : ℝ → ℝ
a b : ℝ
hab : a < b
hfc : ContinuousOn f (Icc a b)
hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x
hgc : ContinuousOn g (Icc a b)
hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x
h : ℝ → ℝ := fun x => (g b - g a) * f x - (f b - f a) * g x
hhc : ContinuousOn h (Icc a b)
⊢ ∃ c ∈ Ioo a b, (g b - g a) * f' c = (f b - f a) * g' c
|
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)
hI : h a = h b
⊢ ∃ c ∈ Ioo a b, (g b - g a) * f' c = (f b - f a) * g' c
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture2.lean
|
Tutorial.exists_ratio_hasDerivAt_eq_ratio_slope
|
[155, 1]
|
[169, 37]
|
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
h : ℝ → ℝ := fun x => (g b - g a) * f x - (f b - f a) * g x
hhc : ContinuousOn h (Icc a b)
hI : h a = h b
⊢ ∃ c ∈ Ioo a b, (g b - g a) * f' c = (f b - f a) * g' c
|
f✝ : ℝ → ℝ
f'✝ x a✝ b✝ : ℝ
f f' g g' : ℝ → ℝ
a b : ℝ
hab : a < b
hfc : ContinuousOn f (Icc a b)
hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x
hgc : ContinuousOn g (Icc a b)
hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x
h : ℝ → ℝ := fun x => (g b - g a) * f x - (f b - f a) * g x
hhc : ContinuousOn h (Icc a b)
hI : h a = h b
h' : ℝ → ℝ := fun x => (g b - g a) * f' x - (f b - f a) * g' x
⊢ ∃ c ∈ Ioo a b, (g b - g a) * f' c = (f b - f a) * g' c
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture2.lean
|
Tutorial.exists_ratio_hasDerivAt_eq_ratio_slope
|
[155, 1]
|
[169, 37]
|
have hhh' : ∀ x ∈ Ioo a b, HasDerivAt h (h' x) x := by
intro x hx
apply ((hff' x hx).const_mul (g b - g a)).sub ((hgg' x hx).const_mul (f b - f a))
|
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)
hI : h a = h b
h' : ℝ → ℝ := fun x => (g b - g a) * f' x - (f b - f a) * g' x
⊢ ∃ c ∈ Ioo a b, (g b - g a) * f' c = (f b - f a) * g' c
|
f✝ : ℝ → ℝ
f'✝ x a✝ b✝ : ℝ
f f' g g' : ℝ → ℝ
a b : ℝ
hab : a < b
hfc : ContinuousOn f (Icc a b)
hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x
hgc : ContinuousOn g (Icc a b)
hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x
h : ℝ → ℝ := fun x => (g b - g a) * f x - (f b - f a) * g x
hhc : ContinuousOn h (Icc a b)
hI : h a = h b
h' : ℝ → ℝ := fun x => (g b - g a) * f' x - (f b - f a) * g' x
hhh' : ∀ x ∈ Ioo a b, HasDerivAt h (h' x) x
⊢ ∃ c ∈ Ioo a b, (g b - g a) * f' c = (f b - f a) * g' c
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture2.lean
|
Tutorial.exists_ratio_hasDerivAt_eq_ratio_slope
|
[155, 1]
|
[169, 37]
|
have ⟨c, cmem, hc⟩ := exists_hasDerivAt_eq_zero hab hhc hI hhh'
|
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)
hI : h a = h b
h' : ℝ → ℝ := fun x => (g b - g a) * f' x - (f b - f a) * g' x
hhh' : ∀ x ∈ Ioo a b, HasDerivAt h (h' 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
hhc : ContinuousOn h (Icc a b)
hI : h a = h b
h' : ℝ → ℝ := fun x => (g b - g a) * f' x - (f b - f a) * g' x
hhh' : ∀ x ∈ Ioo a b, HasDerivAt h (h' x) x
c : ℝ
cmem : c ∈ Ioo a b
hc : h' c = 0
⊢ ∃ c ∈ Ioo a b, (g b - g a) * f' c = (f b - f a) * g' c
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture2.lean
|
Tutorial.exists_ratio_hasDerivAt_eq_ratio_slope
|
[155, 1]
|
[169, 37]
|
exact ⟨c, cmem, sub_eq_zero.mp hc⟩
|
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)
hI : h a = h b
h' : ℝ → ℝ := fun x => (g b - g a) * f' x - (f b - f a) * g' x
hhh' : ∀ x ∈ Ioo a b, HasDerivAt h (h' x) x
c : ℝ
cmem : c ∈ Ioo a b
hc : h' c = 0
⊢ ∃ c ∈ Ioo a b, (g b - g a) * f' c = (f b - f a) * g' c
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture2.lean
|
Tutorial.exists_ratio_hasDerivAt_eq_ratio_slope
|
[155, 1]
|
[169, 37]
|
ring
|
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)
⊢ h a = h b
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture2.lean
|
Tutorial.exists_ratio_hasDerivAt_eq_ratio_slope
|
[155, 1]
|
[169, 37]
|
intro x hx
|
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)
hI : h a = h b
h' : ℝ → ℝ := fun x => (g b - g a) * f' x - (f b - f a) * g' x
⊢ ∀ x ∈ Ioo a b, HasDerivAt h (h' x) 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
h : ℝ → ℝ := fun x => (g b - g a) * f x - (f b - f a) * g x
hhc : ContinuousOn h (Icc a b)
hI : h a = h b
h' : ℝ → ℝ := fun x => (g b - g a) * f' x - (f b - f a) * g' x
x : ℝ
hx : x ∈ Ioo a b
⊢ HasDerivAt h (h' x) x
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture2.lean
|
Tutorial.exists_ratio_hasDerivAt_eq_ratio_slope
|
[155, 1]
|
[169, 37]
|
apply ((hff' x hx).const_mul (g b - g a)).sub ((hgg' x hx).const_mul (f b - f a))
|
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)
hI : h a = h b
h' : ℝ → ℝ := fun x => (g b - g a) * f' x - (f b - f a) * g' x
x : ℝ
hx : x ∈ Ioo a b
⊢ HasDerivAt h (h' x) x
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture2.lean
|
Tutorial.exists_hasDerivAt_eq_slope
|
[176, 1]
|
[182, 78]
|
have ⟨c, cmem, hc⟩ := exists_ratio_hasDerivAt_eq_ratio_slope hab hfc hff'
continuousOn_id fun x _ ↦ hasDerivAt_id 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
⊢ ∃ c ∈ Ioo a b, f' c = (f b - f a) / (b - a)
|
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 : ℝ
cmem : c ∈ Ioo a b
hc : (id b - id a) * f' c = (f b - f a) * 1
⊢ ∃ c ∈ Ioo a b, f' c = (f b - f a) / (b - a)
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture2.lean
|
Tutorial.exists_hasDerivAt_eq_slope
|
[176, 1]
|
[182, 78]
|
exact ⟨c, cmem, by rw [eq_div_iff (by linarith), mul_comm]; simpa using hc⟩
|
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 : ℝ
cmem : c ∈ Ioo a b
hc : (id b - id a) * f' c = (f b - f a) * 1
⊢ ∃ c ∈ Ioo a b, f' c = (f b - f a) / (b - a)
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture2.lean
|
Tutorial.exists_hasDerivAt_eq_slope
|
[176, 1]
|
[182, 78]
|
rw [eq_div_iff (by linarith), mul_comm]
|
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 : ℝ
cmem : c ∈ Ioo a b
hc : (id b - id a) * f' c = (f b - f a) * 1
⊢ f' c = (f b - f a) / (b - a)
|
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 : ℝ
cmem : c ∈ Ioo a b
hc : (id b - id a) * f' c = (f b - f a) * 1
⊢ (b - a) * f' c = f b - f a
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture2.lean
|
Tutorial.exists_hasDerivAt_eq_slope
|
[176, 1]
|
[182, 78]
|
simpa using hc
|
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 : ℝ
cmem : c ∈ Ioo a b
hc : (id b - id a) * f' c = (f b - f a) * 1
⊢ (b - a) * f' c = f b - f a
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture2.lean
|
Tutorial.exists_hasDerivAt_eq_slope
|
[176, 1]
|
[182, 78]
|
linarith
|
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 : ℝ
cmem : c ∈ Ioo a b
hc : (id b - id a) * f' c = (f b - f a) * 1
⊢ b - a ≠ 0
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Algebra/Lecture3.lean
|
Tutorial.inv_smul_smul
|
[53, 1]
|
[54, 8]
|
sorry
|
G X : Type
inst✝¹ : Group G
inst✝ : GroupAction G X
a : G
x : X
⊢ a⁻¹ • a • x = x
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Algebra/Lecture3.lean
|
Tutorial.smul_inv_smul
|
[57, 1]
|
[58, 8]
|
sorry
|
G X : Type
inst✝¹ : Group G
inst✝ : GroupAction G X
a : G
x : X
⊢ a • a⁻¹ • x = x
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Algebra/Lecture3.lean
|
Tutorial.GroupAction.injective
|
[61, 1]
|
[64, 8]
|
intro x y (h : a • x = a • y)
|
G X : Type
inst✝¹ : Group G
inst✝ : GroupAction G X
a : G
⊢ Function.Injective fun x => a • x
|
G X : Type
inst✝¹ : Group G
inst✝ : GroupAction G X
a : G
x y : X
h : a • x = a • y
⊢ x = y
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Algebra/Lecture3.lean
|
Tutorial.GroupAction.injective
|
[61, 1]
|
[64, 8]
|
sorry
|
G X : Type
inst✝¹ : Group G
inst✝ : GroupAction G X
a : G
x y : X
h : a • x = a • y
⊢ x = y
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Algebra/Lecture3.lean
|
Tutorial.GroupAction.surjective
|
[72, 1]
|
[73, 8]
|
sorry
|
G X : Type
inst✝¹ : Group G
inst✝ : GroupAction G X
a : G
⊢ Function.Surjective fun x => a • x
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Algebra/Lecture3.lean
|
Tutorial.orbit_eq_orbit_iff_mem_orbit
|
[234, 1]
|
[235, 8]
|
sorry
|
G X : Type
inst✝¹ : Group G
inst✝ : GroupAction G X
x y : X
⊢ orbit G x = orbit G y ↔ y ∈ orbit G x
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture5.lean
|
Tutorial.Subgroup.mem_comm
|
[31, 1]
|
[47, 44]
|
intro hab
|
G : Type
inst✝¹ : Group G
N : Subgroup G
inst✝ : Normal N
a b : G
⊢ a * b ∈ N → b * a ∈ N
|
G : Type
inst✝¹ : Group G
N : Subgroup G
inst✝ : Normal N
a b : G
hab : a * b ∈ N
⊢ b * a ∈ N
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture5.lean
|
Tutorial.Subgroup.mem_comm
|
[31, 1]
|
[47, 44]
|
calc
b * a = b * a⁻¹⁻¹ := by simp
_ = a⁻¹ * (a * b) * a⁻¹⁻¹ := by simp
_ ∈ N := by
apply Normal.conj_mem a⁻¹ (a * b) hab
|
G : Type
inst✝¹ : Group G
N : Subgroup G
inst✝ : Normal N
a b : G
hab : a * b ∈ N
⊢ b * a ∈ N
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture5.lean
|
Tutorial.Subgroup.mem_comm
|
[31, 1]
|
[47, 44]
|
simp
|
G : Type
inst✝¹ : Group G
N : Subgroup G
inst✝ : Normal N
a b : G
hab : a * b ∈ N
⊢ b * a = b * a⁻¹⁻¹
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture5.lean
|
Tutorial.Subgroup.mem_comm
|
[31, 1]
|
[47, 44]
|
simp
|
G : Type
inst✝¹ : Group G
N : Subgroup G
inst✝ : Normal N
a b : G
hab : a * b ∈ N
⊢ b * a⁻¹⁻¹ = a⁻¹ * (a * b) * a⁻¹⁻¹
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture5.lean
|
Tutorial.Subgroup.mem_comm
|
[31, 1]
|
[47, 44]
|
apply Normal.conj_mem a⁻¹ (a * b) hab
|
G : Type
inst✝¹ : Group G
N : Subgroup G
inst✝ : Normal N
a b : G
hab : a * b ∈ N
⊢ a⁻¹ * (a * b) * a⁻¹⁻¹ ∈ N
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture5.lean
|
Tutorial.mem_of_eq_one
|
[104, 1]
|
[106, 23]
|
simp [N.inv_mem_iff]
|
G : Type
inst✝¹ : Group G
N : Subgroup G
inst✝ : Subgroup.Normal N
a : G
⊢ LeftQuotient.mk a = 1 ↔ a ∈ N
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture5.lean
|
Tutorial.Subgroup.coe_one
|
[203, 1]
|
[203, 48]
|
simp
|
G H : Type
inst✝¹ : Group G
inst✝ : Group H
f : G →* H
K : Subgroup G
⊢ 1 ∈ K
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture5.lean
|
Tutorial.GroupHom.rangeKerLift_injective
|
[238, 1]
|
[242, 11]
|
rw [injective_iff_map_eq_one]
|
G H : Type
inst✝¹ : Group G
inst✝ : Group H
f : G →* H
⊢ Function.Injective ⇑(rangeKerLift f)
|
G H : Type
inst✝¹ : Group G
inst✝ : Group H
f : G →* H
⊢ ∀ (a : G ⧸ ker f), (rangeKerLift f) a = 1 → a = 1
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture5.lean
|
Tutorial.GroupHom.rangeKerLift_injective
|
[238, 1]
|
[242, 11]
|
rintro ⟨_⟩
|
G H : Type
inst✝¹ : Group G
inst✝ : Group H
f : G →* H
⊢ ∀ (a : G ⧸ ker f), (rangeKerLift f) a = 1 → a = 1
|
case mk
G H : Type
inst✝¹ : Group G
inst✝ : Group H
f : G →* H
a✝¹ : G ⧸ ker f
a✝ : G
⊢ (rangeKerLift f) (Quot.mk Setoid.r a✝) = 1 → Quot.mk Setoid.r a✝ = 1
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture5.lean
|
Tutorial.GroupHom.rangeKerLift_injective
|
[238, 1]
|
[242, 11]
|
simp_all
|
case mk
G H : Type
inst✝¹ : Group G
inst✝ : Group H
f : G →* H
a✝¹ : G ⧸ ker f
a✝ : G
⊢ (rangeKerLift f) (Quot.mk Setoid.r a✝) = 1 → Quot.mk Setoid.r a✝ = 1
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture5.lean
|
Tutorial.GroupHom.rangeKerLift_surjective
|
[246, 1]
|
[252, 8]
|
intro ⟨y, hy⟩
|
G H : Type
inst✝¹ : Group G
inst✝ : Group H
f : G →* H
⊢ Function.Surjective ⇑(rangeKerLift f)
|
G H : Type
inst✝¹ : Group G
inst✝ : Group H
f : G →* H
y : H
hy : y ∈ range f
⊢ ∃ a, (rangeKerLift f) a = { val := y, property := hy }
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture5.lean
|
Tutorial.GroupHom.rangeKerLift_surjective
|
[246, 1]
|
[252, 8]
|
rcases hy with ⟨x, hxy⟩
|
G H : Type
inst✝¹ : Group G
inst✝ : Group H
f : G →* H
y : H
hy : y ∈ range f
⊢ ∃ a, (rangeKerLift f) a = { val := y, property := hy }
|
case intro
G H : Type
inst✝¹ : Group G
inst✝ : Group H
f : G →* H
y : H
x : G
hxy : f x = y
⊢ ∃ a, (rangeKerLift f) a = { val := y, property := ⋯ }
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture5.lean
|
Tutorial.GroupHom.rangeKerLift_surjective
|
[246, 1]
|
[252, 8]
|
exists LeftQuotient.mk x
|
case intro
G H : Type
inst✝¹ : Group G
inst✝ : Group H
f : G →* H
y : H
x : G
hxy : f x = y
⊢ ∃ a, (rangeKerLift f) a = { val := y, property := ⋯ }
|
case intro
G H : Type
inst✝¹ : Group G
inst✝ : Group H
f : G →* H
y : H
x : G
hxy : f x = y
⊢ (rangeKerLift f) (LeftQuotient.mk x) = { val := y, property := ⋯ }
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture5.lean
|
Tutorial.GroupHom.rangeKerLift_surjective
|
[246, 1]
|
[252, 8]
|
simpa
|
case intro
G H : Type
inst✝¹ : Group G
inst✝ : Group H
f : G →* H
y : H
x : G
hxy : f x = y
⊢ (rangeKerLift f) (LeftQuotient.mk x) = { val := y, property := ⋯ }
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Algebra/Lecture2.lean
|
Tutorial.map_one
|
[45, 1]
|
[48, 8]
|
have h : f 1 * f 1 = f 1 * 1 := by
sorry
|
G₁ G₂ : Type
inst✝¹ : Group G₁
inst✝ : Group G₂
f : G₁ →* G₂
⊢ f 1 = 1
|
G₁ G₂ : Type
inst✝¹ : Group G₁
inst✝ : Group G₂
f : G₁ →* G₂
h : f 1 * f 1 = f 1 * 1
⊢ f 1 = 1
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Algebra/Lecture2.lean
|
Tutorial.map_one
|
[45, 1]
|
[48, 8]
|
sorry
|
G₁ G₂ : Type
inst✝¹ : Group G₁
inst✝ : Group G₂
f : G₁ →* G₂
h : f 1 * f 1 = f 1 * 1
⊢ f 1 = 1
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Algebra/Lecture2.lean
|
Tutorial.map_one
|
[45, 1]
|
[48, 8]
|
sorry
|
G₁ G₂ : Type
inst✝¹ : Group G₁
inst✝ : Group G₂
f : G₁ →* G₂
⊢ f 1 * f 1 = f 1 * 1
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Algebra/Lecture2.lean
|
Tutorial.map_inv
|
[53, 1]
|
[54, 8]
|
sorry
|
G₁ G₂ : Type
inst✝¹ : Group G₁
inst✝ : Group G₂
f : G₁ →* G₂
a : G₁
⊢ f a⁻¹ = (f a)⁻¹
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Algebra/Lecture2.lean
|
Tutorial.injective_iff_map_eq_one
|
[177, 1]
|
[180, 10]
|
constructor
|
G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
⊢ Function.Injective ⇑f ↔ ∀ (a : G₁), f a = 1 → a = 1
|
case mp
G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
⊢ Function.Injective ⇑f → ∀ (a : G₁), f a = 1 → a = 1
case mpr
G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
⊢ (∀ (a : G₁), f a = 1 → a = 1) → Function.Injective ⇑f
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Algebra/Lecture2.lean
|
Tutorial.injective_iff_map_eq_one
|
[177, 1]
|
[180, 10]
|
sorry
|
case mp
G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
⊢ Function.Injective ⇑f → ∀ (a : G₁), f a = 1 → a = 1
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Algebra/Lecture2.lean
|
Tutorial.injective_iff_map_eq_one
|
[177, 1]
|
[180, 10]
|
sorry
|
case mpr
G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
⊢ (∀ (a : G₁), f a = 1 → a = 1) → Function.Injective ⇑f
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Algebra/Lecture2.lean
|
Tutorial.GroupHom.ker_eq_bot
|
[185, 1]
|
[190, 10]
|
rw [injective_iff_map_eq_one]
|
G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
⊢ ker f = ⊥ ↔ Function.Injective ⇑f
|
G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
⊢ ker f = ⊥ ↔ ∀ (a : G₁), f a = 1 → a = 1
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Algebra/Lecture2.lean
|
Tutorial.GroupHom.ker_eq_bot
|
[185, 1]
|
[190, 10]
|
constructor
|
G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
⊢ ker f = ⊥ ↔ ∀ (a : G₁), f a = 1 → a = 1
|
case mp
G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
⊢ ker f = ⊥ → ∀ (a : G₁), f a = 1 → a = 1
case mpr
G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
⊢ (∀ (a : G₁), f a = 1 → a = 1) → ker f = ⊥
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Algebra/Lecture2.lean
|
Tutorial.GroupHom.ker_eq_bot
|
[185, 1]
|
[190, 10]
|
sorry
|
case mp
G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
⊢ ker f = ⊥ → ∀ (a : G₁), f a = 1 → a = 1
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Algebra/Lecture2.lean
|
Tutorial.GroupHom.ker_eq_bot
|
[185, 1]
|
[190, 10]
|
sorry
|
case mpr
G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
⊢ (∀ (a : G₁), f a = 1 → a = 1) → ker f = ⊥
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Algebra/Lecture2.lean
|
Tutorial.GroupHom.range_eq_top
|
[193, 1]
|
[200, 10]
|
constructor
|
G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
⊢ range f = ⊤ ↔ Function.Surjective ⇑f
|
case mp
G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
⊢ range f = ⊤ → Function.Surjective ⇑f
case mpr
G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
⊢ Function.Surjective ⇑f → range f = ⊤
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Algebra/Lecture2.lean
|
Tutorial.GroupHom.range_eq_top
|
[193, 1]
|
[200, 10]
|
intro hrange y
|
case mp
G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
⊢ range f = ⊤ → Function.Surjective ⇑f
|
case mp
G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
hrange : range f = ⊤
y : G₂
⊢ ∃ a, f a = y
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Algebra/Lecture2.lean
|
Tutorial.GroupHom.range_eq_top
|
[193, 1]
|
[200, 10]
|
have hy : y ∈ (⊤ : Subgroup G₂) := by
sorry
|
case mp
G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
hrange : range f = ⊤
y : G₂
⊢ ∃ a, f a = y
|
case mp
G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
hrange : range f = ⊤
y : G₂
hy : y ∈ ⊤
⊢ ∃ a, f a = y
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Algebra/Lecture2.lean
|
Tutorial.GroupHom.range_eq_top
|
[193, 1]
|
[200, 10]
|
sorry
|
case mp
G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
hrange : range f = ⊤
y : G₂
hy : y ∈ ⊤
⊢ ∃ a, f a = y
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Algebra/Lecture2.lean
|
Tutorial.GroupHom.range_eq_top
|
[193, 1]
|
[200, 10]
|
sorry
|
G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
hrange : range f = ⊤
y : G₂
⊢ y ∈ ⊤
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Algebra/Lecture2.lean
|
Tutorial.GroupHom.range_eq_top
|
[193, 1]
|
[200, 10]
|
intro hsurj
|
case mpr
G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
⊢ Function.Surjective ⇑f → range f = ⊤
|
case mpr
G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
hsurj : Function.Surjective ⇑f
⊢ range f = ⊤
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Algebra/Lecture2.lean
|
Tutorial.GroupHom.range_eq_top
|
[193, 1]
|
[200, 10]
|
sorry
|
case mpr
G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
hsurj : Function.Surjective ⇑f
⊢ range f = ⊤
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Algebra/Lecture2.lean
|
Tutorial.homToPerm_injective
|
[264, 1]
|
[275, 12]
|
rw [injective_iff_map_eq_one]
|
G : Type
inst✝ : Group G
⊢ Function.Injective ⇑(homToPerm G)
|
G : Type
inst✝ : Group G
⊢ ∀ (a : G), (homToPerm G) a = 1 → a = 1
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Algebra/Lecture2.lean
|
Tutorial.homToPerm_injective
|
[264, 1]
|
[275, 12]
|
intro a h
|
G : Type
inst✝ : Group G
⊢ ∀ (a : G), (homToPerm G) a = 1 → a = 1
|
G : Type
inst✝ : Group G
a : G
h : (homToPerm G) a = 1
⊢ a = 1
|
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