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Dataset card Data Studio Files
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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 : βˆ€αΆ  x in 𝓝[<] a, x < a := eventually_nhdsWithin_of_forall fun x hx ↦ hx
f : ℝ β†’ 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') ⊒ βˆ€αΆ  (x : ℝ) in 𝓝[<] a, (f x - f a) / (x - a) β‰₯ 0
f : ℝ β†’ 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 : βˆ€αΆ  (x : ℝ) in 𝓝[<] a, x < a ⊒ βˆ€αΆ  (x : ℝ) in 𝓝[<] 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 : βˆ€αΆ  x in 𝓝[<] a, f x ≀ f a := h.filter_mono nhdsWithin_le_nhds
f : ℝ β†’ 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 : βˆ€αΆ  (x : ℝ) in 𝓝[<] a, x < a ⊒ βˆ€αΆ  (x : ℝ) in 𝓝[<] a, (f x - f a) / (x - a) β‰₯ 0
f : ℝ β†’ 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 : βˆ€αΆ  (x : ℝ) in 𝓝[<] a, x < a h : βˆ€αΆ  (x : ℝ) in 𝓝[<] a, f x ≀ f a ⊒ βˆ€αΆ  (x : ℝ) in 𝓝[<] 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 : ℝ β†’ 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 : βˆ€αΆ  (x : ℝ) in 𝓝[<] a, x < a h : βˆ€αΆ  (x : ℝ) in 𝓝[<] a, f x ≀ f a ⊒ βˆ€αΆ  (x : ℝ) in 𝓝[<] a, (f x - f a) / (x - a) β‰₯ 0
case h f : ℝ β†’ 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 : βˆ€αΆ  (x : ℝ) in 𝓝[<] a, x < a h : βˆ€αΆ  (x : ℝ) in 𝓝[<] a, f x ≀ f a ⊒ βˆ€ a_1 < a, f a_1 ≀ 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 : ℝ β†’ 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 : βˆ€αΆ  (x : ℝ) in 𝓝[<] a, x < a h : βˆ€αΆ  (x : ℝ) in 𝓝[<] a, f x ≀ f a ⊒ βˆ€ a_1 < a, f a_1 ≀ f a β†’ (f a_1 - f a) / (a_1 - a) β‰₯ 0
case h f : ℝ β†’ 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✝ : βˆ€αΆ  (x : ℝ) in 𝓝[<] a, x < a h✝ : βˆ€αΆ  (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) 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_nonneg_of_nonpos
case h f : ℝ β†’ 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✝ : βˆ€αΆ  (x : ℝ) in 𝓝[<] a, x < a h✝ : βˆ€αΆ  (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) β‰₯ 0
case h.ha f : ℝ β†’ 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✝ : βˆ€αΆ  (x : ℝ) in 𝓝[<] a, x < a h✝ : βˆ€αΆ  (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 β†’ 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✝ : βˆ€αΆ  (x : ℝ) in 𝓝[<] a, x < a h✝ : βˆ€αΆ  (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 β†’ 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 β†’ ℝ f' x a b : ℝ h : IsLocalMax f a hf : Tendsto (fun x => (f x - f a) / (x - a)) (𝓝[β‰ ] 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 β†’ ℝ f' x a b : ℝ h : IsLocalMax f a hf : Tendsto (fun x => (f x - f a) / (x - a)) (𝓝[β‰ ] 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 β†’ 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✝ : βˆ€αΆ  (x : ℝ) in 𝓝[<] a, x < a h✝ : βˆ€αΆ  (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 β†’ 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✝ : βˆ€αΆ  (x : ℝ) in 𝓝[<] a, x < a h✝ : βˆ€αΆ  (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 β†’ ℝ f' x a b : ℝ h : IsLocalMin f a hf : HasDerivAt f f' a ⊒ f' = 0
f : ℝ β†’ 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 β†’ ℝ f' x a b : ℝ h : IsLocalMin f a hf : HasDerivAt f f' a ⊒ -f' = 0
case h f : ℝ β†’ ℝ f' x a b : ℝ h : IsLocalMin f a hf : HasDerivAt f f' a ⊒ IsLocalMax ?f ?a case hf f : ℝ β†’ ℝ f' x a b : ℝ h : IsLocalMin f a hf : HasDerivAt f f' a ⊒ HasDerivAt ?f (-f') ?a case f f : ℝ β†’ ℝ f' x a b : ℝ h : IsLocalMin f a hf : HasDerivAt f f' a ⊒ ℝ β†’ ℝ case a f : ℝ β†’ 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 β†’ ℝ f' x a b : ℝ h : IsLocalMin f a hf : HasDerivAt f f' a ⊒ IsLocalMax ?f ?a case hf f : ℝ β†’ ℝ f' x a b : ℝ h : IsLocalMin f a hf : HasDerivAt f f' a ⊒ HasDerivAt ?f (-f') ?a case f f : ℝ β†’ ℝ f' x a b : ℝ h : IsLocalMin f a hf : HasDerivAt f f' a ⊒ ℝ β†’ ℝ case a f : ℝ β†’ ℝ f' x a b : ℝ h : IsLocalMin f a hf : HasDerivAt f f' a ⊒ ℝ
case hf f : ℝ β†’ 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 β†’ 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 β†’ ℝ f' x a b : ℝ h : IsLocalExtr f a hf : HasDerivAt f f' a ⊒ f' = 0
case a f : ℝ β†’ ℝ f' x a b : ℝ h : IsLocalExtr f a hf : HasDerivAt f f' a ⊒ IsLocalMin f a β†’ f' = 0 case a f : ℝ β†’ ℝ f' x a b : ℝ h : IsLocalExtr f a hf : HasDerivAt f f' a ⊒ IsLocalMax f a β†’ 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 β†’ ℝ f' x a b : ℝ h : IsLocalExtr f a hf : HasDerivAt f f' a ⊒ IsLocalMin f a β†’ f' = 0
case a f : ℝ β†’ 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 β†’ 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 β†’ ℝ f' x a b : ℝ h : IsLocalExtr f a hf : HasDerivAt f f' a ⊒ IsLocalMax f a β†’ f' = 0
case a f : ℝ β†’ 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 β†’ 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 βˆƒ 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 β†’ ℝ f' x a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hfI : f a = f b ⊒ βˆƒ c ∈ Ioo a b, IsLocalExtr f c
f : ℝ β†’ ℝ f' x a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hfI : f a = f b ⊒ βˆƒ 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 β†’ ℝ f' x a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hfI : f a = f b ⊒ βˆƒ c ∈ Ioo a b, IsExtrOn f (Icc a b) c
f : ℝ β†’ 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) ⊒ βˆƒ 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⟩ : βˆƒ C ∈ Icc a b, IsMaxOn f (Icc a b) C := by apply isCompact_Icc.exists_isMaxOn ne hfc
f : ℝ β†’ 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) ⊒ βˆƒ c ∈ Ioo a b, IsExtrOn f (Icc a b) c
f : ℝ β†’ 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 ⊒ βˆƒ 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⟩ : βˆƒ c ∈ Icc a b, IsMinOn f (Icc a b) c := by apply isCompact_Icc.exists_isMinOn ne hfc
f : ℝ β†’ 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 ⊒ βˆƒ c ∈ Ioo a b, IsExtrOn f (Icc a b) c
f : ℝ β†’ 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 ⊒ βˆƒ 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 βˆ€ x ∈ Icc a b, f x ≀ f C at Cge
f : ℝ β†’ 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 ⊒ βˆƒ c ∈ Ioo a b, IsExtrOn f (Icc a b) c
f : ℝ β†’ 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 : βˆ€ x ∈ Icc a b, f x ≀ f C ⊒ βˆƒ 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 βˆ€ x ∈ Icc a b, f c ≀ f x at cle
f : ℝ β†’ 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 : βˆ€ x ∈ Icc a b, f x ≀ f C ⊒ βˆƒ c ∈ Ioo a b, IsExtrOn f (Icc a b) c
f : ℝ β†’ 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 : βˆ€ x ∈ Icc a b, f x ≀ f C cle : βˆ€ x ∈ Icc a b, f c ≀ f x ⊒ βˆƒ 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 β†’ 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 : βˆ€ x ∈ Icc a b, f x ≀ f C cle : βˆ€ x ∈ Icc a b, f c ≀ f x ⊒ βˆƒ c ∈ Ioo a b, IsExtrOn f (Icc a b) c
case pos f : ℝ β†’ 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 : βˆ€ x ∈ Icc a b, f x ≀ f C cle : βˆ€ x ∈ Icc a b, f c ≀ f x hc : f c = f a ⊒ βˆƒ c ∈ Ioo a b, IsExtrOn f (Icc a b) c case neg f : ℝ β†’ 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 : βˆ€ x ∈ Icc a b, f x ≀ f C cle : βˆ€ x ∈ Icc a b, f c ≀ f x hc : Β¬f c = f a ⊒ βˆƒ 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 β†’ ℝ f' x a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hfI : f a = f b this : βˆƒ c ∈ Ioo a b, IsExtrOn f (Icc a b) c ⊒ βˆƒ c ∈ Ioo a b, IsLocalExtr f c
case intro.intro f : ℝ β†’ 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 ⊒ βˆƒ 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 β†’ 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 ⊒ βˆƒ c ∈ Ioo a b, IsLocalExtr f c
case intro.intro f : ℝ β†’ 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 β†’ 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 β†’ 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) ⊒ βˆƒ 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 β†’ 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 ⊒ βˆƒ 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 β†’ 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 : βˆ€ x ∈ Icc a b, f x ≀ f C cle : βˆ€ x ∈ Icc a b, f c ≀ f x hc : f c = f a ⊒ βˆƒ c ∈ Ioo a b, IsExtrOn f (Icc a b) c
case pos f : ℝ β†’ 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 : βˆ€ x ∈ Icc a b, f x ≀ f C cle : βˆ€ x ∈ Icc a b, f c ≀ f x hc : f c = f a hC : f C = f a ⊒ βˆƒ c ∈ Ioo a b, IsExtrOn f (Icc a b) c case neg f : ℝ β†’ 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 : βˆ€ x ∈ Icc a b, f x ≀ f C cle : βˆ€ x ∈ Icc a b, f c ≀ f x hc : f c = f a hC : Β¬f C = f a ⊒ βˆƒ 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 : βˆ€ x ∈ Icc a b, f x = f a := fun x hx ↦ le_antisymm (hC β–Έ Cge x hx) (hc β–Έ cle x hx)
case pos f : ℝ β†’ 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 : βˆ€ x ∈ Icc a b, f x ≀ f C cle : βˆ€ x ∈ Icc a b, f c ≀ f x hc : f c = f a hC : f C = f a ⊒ βˆƒ c ∈ Ioo a b, IsExtrOn f (Icc a b) c
case pos f : ℝ β†’ 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 : βˆ€ x ∈ Icc a b, f x ≀ f C cle : βˆ€ x ∈ Icc a b, f c ≀ f x hc : f c = f a hC : f C = f a this : βˆ€ x ∈ Icc a b, f x = f a ⊒ βˆƒ 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 β†’ 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 : βˆ€ x ∈ Icc a b, f x ≀ f C cle : βˆ€ x ∈ Icc a b, f c ≀ f x hc : f c = f a hC : f C = f a this : βˆ€ x ∈ Icc a b, f x = f a ⊒ βˆƒ c ∈ Ioo a b, IsExtrOn f (Icc a b) c
case pos.intro f : ℝ β†’ 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 : βˆ€ x ∈ Icc a b, f x ≀ f C cle : βˆ€ x ∈ Icc a b, f c ≀ f x hc : f c = f a hC : f C = f a this : βˆ€ x ∈ Icc a b, f x = f a c' : ℝ hc' : c' ∈ Ioo a b ⊒ βˆƒ 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 β†’ 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 : βˆ€ x ∈ Icc a b, f x ≀ f C cle : βˆ€ x ∈ Icc a b, f c ≀ f x hc : f c = f a hC : f C = f a this : βˆ€ x ∈ Icc a b, f x = f a c' : ℝ hc' : c' ∈ Ioo a b ⊒ βˆƒ c ∈ Ioo a b, IsExtrOn f (Icc a b) c
case pos.intro f : ℝ β†’ 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 : βˆ€ x ∈ Icc a b, f x ≀ f C cle : βˆ€ x ∈ Icc a b, f c ≀ f x hc : f c = f a hC : f C = f a this : βˆ€ 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 β†’ 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 : βˆ€ x ∈ Icc a b, f x ≀ f C cle : βˆ€ x ∈ Icc a b, f c ≀ f x hc : f c = f a hC : f C = f a this : βˆ€ 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 β†’ 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 : βˆ€ x ∈ Icc a b, f x ≀ f C cle : βˆ€ x ∈ Icc a b, f c ≀ f x hc : f c = f a hC : Β¬f C = f a ⊒ βˆƒ c ∈ Ioo a b, IsExtrOn f (Icc a b) c
case neg.refine_1 f : ℝ β†’ 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 : βˆ€ x ∈ Icc a b, f x ≀ f C cle : βˆ€ x ∈ Icc a b, f c ≀ f x hc : f c = f a hC : Β¬f C = f a ⊒ a = C β†’ f C = f a case neg.refine_2 f : ℝ β†’ 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 : βˆ€ x ∈ Icc a b, f x ≀ f C cle : βˆ€ x ∈ Icc a b, f c ≀ f x hc : f c = f a hC : Β¬f C = f a ⊒ C = b β†’ 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 β†’ 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 : βˆ€ x ∈ Icc a b, f x ≀ f C cle : βˆ€ x ∈ Icc a b, f c ≀ f x hc : f c = f a hC : Β¬f C = f a ⊒ a = C β†’ f C = f a case neg.refine_2 f : ℝ β†’ 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 : βˆ€ x ∈ Icc a b, f x ≀ f C cle : βˆ€ x ∈ Icc a b, f c ≀ f x hc : f c = f a hC : Β¬f C = f a ⊒ C = b β†’ 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 β†’ 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 : βˆ€ x ∈ Icc a b, f x ≀ f C cle : βˆ€ x ∈ Icc a b, f c ≀ f x hc : f c = f a hC : Β¬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 β†’ 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 : βˆ€ x ∈ Icc a b, f x ≀ f C cle : βˆ€ x ∈ Icc a b, f c ≀ f x hc : f c = f a hC : Β¬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 β†’ 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 : βˆ€ x ∈ Icc a b, f x ≀ f C cle : βˆ€ x ∈ Icc a b, f c ≀ f x hc : Β¬f c = f a ⊒ βˆƒ c ∈ Ioo a b, IsExtrOn f (Icc a b) c
case neg.refine_1 f : ℝ β†’ 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 : βˆ€ x ∈ Icc a b, f x ≀ f C cle : βˆ€ x ∈ Icc a b, f c ≀ f x hc : Β¬f c = f a ⊒ a = c β†’ f c = f a case neg.refine_2 f : ℝ β†’ 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 : βˆ€ x ∈ Icc a b, f x ≀ f C cle : βˆ€ x ∈ Icc a b, f c ≀ f x hc : Β¬f c = f a ⊒ c = b β†’ 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 β†’ 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 : βˆ€ x ∈ Icc a b, f x ≀ f C cle : βˆ€ x ∈ Icc a b, f c ≀ f x hc : Β¬f c = f a ⊒ a = c β†’ f c = f a case neg.refine_2 f : ℝ β†’ 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 : βˆ€ x ∈ Icc a b, f x ≀ f C cle : βˆ€ x ∈ Icc a b, f c ≀ f x hc : Β¬f c = f a ⊒ c = b β†’ 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 β†’ 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 : βˆ€ x ∈ Icc a b, f x ≀ f C cle : βˆ€ x ∈ Icc a b, f c ≀ f x hc : Β¬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 β†’ 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 : βˆ€ x ∈ Icc a b, f x ≀ f C cle : βˆ€ x ∈ Icc a b, f c ≀ f x hc : Β¬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 β†’ ℝ f'✝ x a✝ b✝ : ℝ f f' g g' : ℝ β†’ ℝ a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hfI : f a = f b hff' : βˆ€ x ∈ Ioo a b, HasDerivAt f (f' x) x ⊒ βˆƒ c ∈ Ioo a b, f' c = 0
f✝ : ℝ β†’ ℝ f'✝ x a✝ b✝ : ℝ f f' g g' : ℝ β†’ ℝ a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hfI : f a = f b hff' : βˆ€ x ∈ Ioo a b, HasDerivAt f (f' x) x c : ℝ cmem : c ∈ Ioo a b hc : IsLocalExtr f c ⊒ βˆƒ 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 β†’ ℝ f'✝ x a✝ b✝ : ℝ f f' g g' : ℝ β†’ ℝ a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hfI : f a = f b hff' : βˆ€ x ∈ Ioo a b, HasDerivAt f (f' x) x c : ℝ cmem : c ∈ Ioo a b hc : IsLocalExtr f c ⊒ βˆƒ 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 β†’ ℝ f'✝ x a✝ b✝ : ℝ f f' g g' : ℝ β†’ ℝ a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hff' : βˆ€ x ∈ Ioo a b, HasDerivAt f (f' x) x hgc : ContinuousOn g (Icc a b) hgg' : βˆ€ x ∈ Ioo a b, HasDerivAt g (g' x) x ⊒ βˆƒ c ∈ Ioo a b, (g b - g a) * f' c = (f b - f a) * g' c
f✝ : ℝ β†’ ℝ f'✝ x a✝ b✝ : ℝ f f' g g' : ℝ β†’ ℝ a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hff' : βˆ€ x ∈ Ioo a b, HasDerivAt f (f' x) x hgc : ContinuousOn g (Icc a b) hgg' : βˆ€ x ∈ Ioo a b, HasDerivAt g (g' x) x h : ℝ β†’ ℝ := fun x => (g b - g a) * f x - (f b - f a) * g x ⊒ βˆƒ 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 β†’ ℝ f'✝ x a✝ b✝ : ℝ f f' g g' : ℝ β†’ ℝ a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hff' : βˆ€ x ∈ Ioo a b, HasDerivAt f (f' x) x hgc : ContinuousOn g (Icc a b) hgg' : βˆ€ x ∈ Ioo a b, HasDerivAt g (g' x) x h : ℝ β†’ ℝ := fun x => (g b - g a) * f x - (f b - f a) * g x ⊒ βˆƒ c ∈ Ioo a b, (g b - g a) * f' c = (f b - f a) * g' c
f✝ : ℝ β†’ ℝ f'✝ x a✝ b✝ : ℝ f f' g g' : ℝ β†’ ℝ a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hff' : βˆ€ x ∈ Ioo a b, HasDerivAt f (f' x) x hgc : ContinuousOn g (Icc a b) hgg' : βˆ€ x ∈ Ioo a b, HasDerivAt g (g' x) x h : ℝ β†’ ℝ := fun x => (g b - g a) * f x - (f b - f a) * g x hhc : ContinuousOn h (Icc a b) ⊒ βˆƒ 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 β†’ ℝ f'✝ x a✝ b✝ : ℝ f f' g g' : ℝ β†’ ℝ a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hff' : βˆ€ x ∈ Ioo a b, HasDerivAt f (f' x) x hgc : ContinuousOn g (Icc a b) hgg' : βˆ€ x ∈ Ioo a b, HasDerivAt g (g' x) x h : ℝ β†’ ℝ := fun x => (g b - g a) * f x - (f b - f a) * g x hhc : ContinuousOn h (Icc a b) ⊒ βˆƒ c ∈ Ioo a b, (g b - g a) * f' c = (f b - f a) * g' c
f✝ : ℝ β†’ ℝ f'✝ x a✝ b✝ : ℝ f f' g g' : ℝ β†’ ℝ a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hff' : βˆ€ x ∈ Ioo a b, HasDerivAt f (f' x) x hgc : ContinuousOn g (Icc a b) hgg' : βˆ€ x ∈ Ioo a b, HasDerivAt g (g' x) x h : ℝ β†’ 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 ⊒ βˆƒ 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 β†’ ℝ f'✝ x a✝ b✝ : ℝ f f' g g' : ℝ β†’ ℝ a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hff' : βˆ€ x ∈ Ioo a b, HasDerivAt f (f' x) x hgc : ContinuousOn g (Icc a b) hgg' : βˆ€ x ∈ Ioo a b, HasDerivAt g (g' x) x h : ℝ β†’ 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 ⊒ βˆƒ c ∈ Ioo a b, (g b - g a) * f' c = (f b - f a) * g' c
f✝ : ℝ β†’ ℝ f'✝ x a✝ b✝ : ℝ f f' g g' : ℝ β†’ ℝ a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hff' : βˆ€ x ∈ Ioo a b, HasDerivAt f (f' x) x hgc : ContinuousOn g (Icc a b) hgg' : βˆ€ x ∈ Ioo a b, HasDerivAt g (g' x) x h : ℝ β†’ 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 β†’ ℝ := fun x => (g b - g a) * f' x - (f b - f a) * g' x ⊒ βˆƒ 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' : βˆ€ 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 β†’ ℝ f'✝ x a✝ b✝ : ℝ f f' g g' : ℝ β†’ ℝ a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hff' : βˆ€ x ∈ Ioo a b, HasDerivAt f (f' x) x hgc : ContinuousOn g (Icc a b) hgg' : βˆ€ x ∈ Ioo a b, HasDerivAt g (g' x) x h : ℝ β†’ 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 β†’ ℝ := fun x => (g b - g a) * f' x - (f b - f a) * g' x ⊒ βˆƒ c ∈ Ioo a b, (g b - g a) * f' c = (f b - f a) * g' c
f✝ : ℝ β†’ ℝ f'✝ x a✝ b✝ : ℝ f f' g g' : ℝ β†’ ℝ a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hff' : βˆ€ x ∈ Ioo a b, HasDerivAt f (f' x) x hgc : ContinuousOn g (Icc a b) hgg' : βˆ€ x ∈ Ioo a b, HasDerivAt g (g' x) x h : ℝ β†’ 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 β†’ ℝ := fun x => (g b - g a) * f' x - (f b - f a) * g' x hhh' : βˆ€ x ∈ Ioo a b, HasDerivAt h (h' x) x ⊒ βˆƒ 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 β†’ ℝ f'✝ x a✝ b✝ : ℝ f f' g g' : ℝ β†’ ℝ a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hff' : βˆ€ x ∈ Ioo a b, HasDerivAt f (f' x) x hgc : ContinuousOn g (Icc a b) hgg' : βˆ€ x ∈ Ioo a b, HasDerivAt g (g' x) x h : ℝ β†’ 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 β†’ ℝ := fun x => (g b - g a) * f' x - (f b - f a) * g' x hhh' : βˆ€ x ∈ Ioo a b, HasDerivAt h (h' x) x ⊒ βˆƒ c ∈ Ioo a b, (g b - g a) * f' c = (f b - f a) * g' c
f✝ : ℝ β†’ ℝ f'✝ x a✝ b✝ : ℝ f f' g g' : ℝ β†’ ℝ a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hff' : βˆ€ x ∈ Ioo a b, HasDerivAt f (f' x) x hgc : ContinuousOn g (Icc a b) hgg' : βˆ€ x ∈ Ioo a b, HasDerivAt g (g' x) x h : ℝ β†’ 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 β†’ ℝ := fun x => (g b - g a) * f' x - (f b - f a) * g' x hhh' : βˆ€ x ∈ Ioo a b, HasDerivAt h (h' x) x c : ℝ cmem : c ∈ Ioo a b hc : h' c = 0 ⊒ βˆƒ 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 β†’ ℝ f'✝ x a✝ b✝ : ℝ f f' g g' : ℝ β†’ ℝ a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hff' : βˆ€ x ∈ Ioo a b, HasDerivAt f (f' x) x hgc : ContinuousOn g (Icc a b) hgg' : βˆ€ x ∈ Ioo a b, HasDerivAt g (g' x) x h : ℝ β†’ 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 β†’ ℝ := fun x => (g b - g a) * f' x - (f b - f a) * g' x hhh' : βˆ€ x ∈ Ioo a b, HasDerivAt h (h' x) x c : ℝ cmem : c ∈ Ioo a b hc : h' c = 0 ⊒ βˆƒ 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 β†’ ℝ f'✝ x a✝ b✝ : ℝ f f' g g' : ℝ β†’ ℝ a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hff' : βˆ€ x ∈ Ioo a b, HasDerivAt f (f' x) x hgc : ContinuousOn g (Icc a b) hgg' : βˆ€ x ∈ Ioo a b, HasDerivAt g (g' x) x h : ℝ β†’ 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 β†’ ℝ f'✝ x a✝ b✝ : ℝ f f' g g' : ℝ β†’ ℝ a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hff' : βˆ€ x ∈ Ioo a b, HasDerivAt f (f' x) x hgc : ContinuousOn g (Icc a b) hgg' : βˆ€ x ∈ Ioo a b, HasDerivAt g (g' x) x h : ℝ β†’ 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 β†’ ℝ := fun x => (g b - g a) * f' x - (f b - f a) * g' x ⊒ βˆ€ x ∈ Ioo a b, HasDerivAt h (h' x) x
f✝ : ℝ β†’ ℝ f'✝ x✝ a✝ b✝ : ℝ f f' g g' : ℝ β†’ ℝ a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hff' : βˆ€ x ∈ Ioo a b, HasDerivAt f (f' x) x hgc : ContinuousOn g (Icc a b) hgg' : βˆ€ x ∈ Ioo a b, HasDerivAt g (g' x) x h : ℝ β†’ 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 β†’ 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 β†’ ℝ f'✝ x✝ a✝ b✝ : ℝ f f' g g' : ℝ β†’ ℝ a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hff' : βˆ€ x ∈ Ioo a b, HasDerivAt f (f' x) x hgc : ContinuousOn g (Icc a b) hgg' : βˆ€ x ∈ Ioo a b, HasDerivAt g (g' x) x h : ℝ β†’ 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 β†’ 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 β†’ ℝ f'✝ x a✝ b✝ : ℝ f f' g g' : ℝ β†’ ℝ a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hff' : βˆ€ x ∈ Ioo a b, HasDerivAt f (f' x) x ⊒ βˆƒ c ∈ Ioo a b, f' c = (f b - f a) / (b - a)
f✝ : ℝ β†’ ℝ f'✝ x a✝ b✝ : ℝ f f' g g' : ℝ β†’ ℝ a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hff' : βˆ€ 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 ⊒ βˆƒ 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 β†’ ℝ f'✝ x a✝ b✝ : ℝ f f' g g' : ℝ β†’ ℝ a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hff' : βˆ€ 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 ⊒ βˆƒ 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 β†’ ℝ f'✝ x a✝ b✝ : ℝ f f' g g' : ℝ β†’ ℝ a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hff' : βˆ€ 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 β†’ ℝ f'✝ x a✝ b✝ : ℝ f f' g g' : ℝ β†’ ℝ a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hff' : βˆ€ 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 β†’ ℝ f'✝ x a✝ b✝ : ℝ f f' g g' : ℝ β†’ ℝ a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hff' : βˆ€ 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 β†’ ℝ f'✝ x a✝ b✝ : ℝ f f' g g' : ℝ β†’ ℝ a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hff' : βˆ€ 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 β‰  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 β€’ a β€’ 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 β€’ a⁻¹ β€’ 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 β€’ x = a β€’ y)
G X : Type inst✝¹ : Group G inst✝ : GroupAction G X a : G ⊒ Function.Injective fun x => a β€’ x
G X : Type inst✝¹ : Group G inst✝ : GroupAction G X a : G x y : X h : a β€’ x = a β€’ 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 β€’ x = a β€’ 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 β€’ 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 β†’ 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 β†’* 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 β†’* H ⊒ Function.Injective ⇑(rangeKerLift f)
G H : Type inst✝¹ : Group G inst✝ : Group H f : G β†’* H ⊒ βˆ€ (a : G β§Έ ker f), (rangeKerLift f) a = 1 β†’ 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 β†’* H ⊒ βˆ€ (a : G β§Έ ker f), (rangeKerLift f) a = 1 β†’ a = 1
case mk G H : Type inst✝¹ : Group G inst✝ : Group H f : G β†’* H a✝¹ : G β§Έ ker f a✝ : G ⊒ (rangeKerLift f) (Quot.mk Setoid.r a✝) = 1 β†’ 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 β†’* H a✝¹ : G β§Έ ker f a✝ : G ⊒ (rangeKerLift f) (Quot.mk Setoid.r a✝) = 1 β†’ 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 β†’* H ⊒ Function.Surjective ⇑(rangeKerLift f)
G H : Type inst✝¹ : Group G inst✝ : Group H f : G β†’* H y : H hy : y ∈ range f ⊒ βˆƒ 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 β†’* H y : H hy : y ∈ range f ⊒ βˆƒ a, (rangeKerLift f) a = { val := y, property := hy }
case intro G H : Type inst✝¹ : Group G inst✝ : Group H f : G β†’* H y : H x : G hxy : f x = y ⊒ βˆƒ a, (rangeKerLift f) a = { val := y, property := β‹― }
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 β†’* H y : H x : G hxy : f x = y ⊒ βˆƒ a, (rangeKerLift f) a = { val := y, property := β‹― }
case intro G H : Type inst✝¹ : Group G inst✝ : Group H f : G β†’* H y : H x : G hxy : f x = y ⊒ (rangeKerLift f) (LeftQuotient.mk x) = { val := y, property := β‹― }
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 β†’* H y : H x : G hxy : f x = y ⊒ (rangeKerLift f) (LeftQuotient.mk x) = { val := y, property := β‹― }
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 Gβ‚‚ : Type inst✝¹ : Group G₁ inst✝ : Group Gβ‚‚ f : G₁ β†’* Gβ‚‚ ⊒ f 1 = 1
G₁ Gβ‚‚ : Type inst✝¹ : Group G₁ inst✝ : Group Gβ‚‚ f : G₁ β†’* Gβ‚‚ 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 Gβ‚‚ : Type inst✝¹ : Group G₁ inst✝ : Group Gβ‚‚ f : G₁ β†’* Gβ‚‚ 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 Gβ‚‚ : Type inst✝¹ : Group G₁ inst✝ : Group Gβ‚‚ f : G₁ β†’* Gβ‚‚ ⊒ 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 Gβ‚‚ : Type inst✝¹ : Group G₁ inst✝ : Group Gβ‚‚ f : G₁ β†’* Gβ‚‚ 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 Gβ‚‚ G : Type inst✝² : Group G₁ inst✝¹ : Group Gβ‚‚ inst✝ : Group G f : G₁ β†’* Gβ‚‚ ⊒ Function.Injective ⇑f ↔ βˆ€ (a : G₁), f a = 1 β†’ a = 1
case mp G₁ Gβ‚‚ G : Type inst✝² : Group G₁ inst✝¹ : Group Gβ‚‚ inst✝ : Group G f : G₁ β†’* Gβ‚‚ ⊒ Function.Injective ⇑f β†’ βˆ€ (a : G₁), f a = 1 β†’ a = 1 case mpr G₁ Gβ‚‚ G : Type inst✝² : Group G₁ inst✝¹ : Group Gβ‚‚ inst✝ : Group G f : G₁ β†’* Gβ‚‚ ⊒ (βˆ€ (a : G₁), f a = 1 β†’ a = 1) β†’ 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 Gβ‚‚ G : Type inst✝² : Group G₁ inst✝¹ : Group Gβ‚‚ inst✝ : Group G f : G₁ β†’* Gβ‚‚ ⊒ Function.Injective ⇑f β†’ βˆ€ (a : G₁), f a = 1 β†’ 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 Gβ‚‚ G : Type inst✝² : Group G₁ inst✝¹ : Group Gβ‚‚ inst✝ : Group G f : G₁ β†’* Gβ‚‚ ⊒ (βˆ€ (a : G₁), f a = 1 β†’ a = 1) β†’ 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 Gβ‚‚ G : Type inst✝² : Group G₁ inst✝¹ : Group Gβ‚‚ inst✝ : Group G f : G₁ β†’* Gβ‚‚ ⊒ ker f = βŠ₯ ↔ Function.Injective ⇑f
G₁ Gβ‚‚ G : Type inst✝² : Group G₁ inst✝¹ : Group Gβ‚‚ inst✝ : Group G f : G₁ β†’* Gβ‚‚ ⊒ ker f = βŠ₯ ↔ βˆ€ (a : G₁), f a = 1 β†’ 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 Gβ‚‚ G : Type inst✝² : Group G₁ inst✝¹ : Group Gβ‚‚ inst✝ : Group G f : G₁ β†’* Gβ‚‚ ⊒ ker f = βŠ₯ ↔ βˆ€ (a : G₁), f a = 1 β†’ a = 1
case mp G₁ Gβ‚‚ G : Type inst✝² : Group G₁ inst✝¹ : Group Gβ‚‚ inst✝ : Group G f : G₁ β†’* Gβ‚‚ ⊒ ker f = βŠ₯ β†’ βˆ€ (a : G₁), f a = 1 β†’ a = 1 case mpr G₁ Gβ‚‚ G : Type inst✝² : Group G₁ inst✝¹ : Group Gβ‚‚ inst✝ : Group G f : G₁ β†’* Gβ‚‚ ⊒ (βˆ€ (a : G₁), f a = 1 β†’ a = 1) β†’ ker f = βŠ₯
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 Gβ‚‚ G : Type inst✝² : Group G₁ inst✝¹ : Group Gβ‚‚ inst✝ : Group G f : G₁ β†’* Gβ‚‚ ⊒ ker f = βŠ₯ β†’ βˆ€ (a : G₁), f a = 1 β†’ 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 Gβ‚‚ G : Type inst✝² : Group G₁ inst✝¹ : Group Gβ‚‚ inst✝ : Group G f : G₁ β†’* Gβ‚‚ ⊒ (βˆ€ (a : G₁), f a = 1 β†’ a = 1) β†’ ker f = βŠ₯
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 Gβ‚‚ G : Type inst✝² : Group G₁ inst✝¹ : Group Gβ‚‚ inst✝ : Group G f : G₁ β†’* Gβ‚‚ ⊒ range f = ⊀ ↔ Function.Surjective ⇑f
case mp G₁ Gβ‚‚ G : Type inst✝² : Group G₁ inst✝¹ : Group Gβ‚‚ inst✝ : Group G f : G₁ β†’* Gβ‚‚ ⊒ range f = ⊀ β†’ Function.Surjective ⇑f case mpr G₁ Gβ‚‚ G : Type inst✝² : Group G₁ inst✝¹ : Group Gβ‚‚ inst✝ : Group G f : G₁ β†’* Gβ‚‚ ⊒ Function.Surjective ⇑f β†’ 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 Gβ‚‚ G : Type inst✝² : Group G₁ inst✝¹ : Group Gβ‚‚ inst✝ : Group G f : G₁ β†’* Gβ‚‚ ⊒ range f = ⊀ β†’ Function.Surjective ⇑f
case mp G₁ Gβ‚‚ G : Type inst✝² : Group G₁ inst✝¹ : Group Gβ‚‚ inst✝ : Group G f : G₁ β†’* Gβ‚‚ hrange : range f = ⊀ y : Gβ‚‚ ⊒ βˆƒ 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 Gβ‚‚) := by sorry
case mp G₁ Gβ‚‚ G : Type inst✝² : Group G₁ inst✝¹ : Group Gβ‚‚ inst✝ : Group G f : G₁ β†’* Gβ‚‚ hrange : range f = ⊀ y : Gβ‚‚ ⊒ βˆƒ a, f a = y
case mp G₁ Gβ‚‚ G : Type inst✝² : Group G₁ inst✝¹ : Group Gβ‚‚ inst✝ : Group G f : G₁ β†’* Gβ‚‚ hrange : range f = ⊀ y : Gβ‚‚ hy : y ∈ ⊀ ⊒ βˆƒ 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 Gβ‚‚ G : Type inst✝² : Group G₁ inst✝¹ : Group Gβ‚‚ inst✝ : Group G f : G₁ β†’* Gβ‚‚ hrange : range f = ⊀ y : Gβ‚‚ hy : y ∈ ⊀ ⊒ βˆƒ 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 Gβ‚‚ G : Type inst✝² : Group G₁ inst✝¹ : Group Gβ‚‚ inst✝ : Group G f : G₁ β†’* Gβ‚‚ hrange : range f = ⊀ y : Gβ‚‚ ⊒ 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 Gβ‚‚ G : Type inst✝² : Group G₁ inst✝¹ : Group Gβ‚‚ inst✝ : Group G f : G₁ β†’* Gβ‚‚ ⊒ Function.Surjective ⇑f β†’ range f = ⊀
case mpr G₁ Gβ‚‚ G : Type inst✝² : Group G₁ inst✝¹ : Group Gβ‚‚ inst✝ : Group G f : G₁ β†’* Gβ‚‚ 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 Gβ‚‚ G : Type inst✝² : Group G₁ inst✝¹ : Group Gβ‚‚ inst✝ : Group G f : G₁ β†’* Gβ‚‚ 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 ⊒ βˆ€ (a : G), (homToPerm G) a = 1 β†’ 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 ⊒ βˆ€ (a : G), (homToPerm G) a = 1 β†’ a = 1
G : Type inst✝ : Group G a : G h : (homToPerm G) a = 1 ⊒ a = 1