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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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