mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Convex.BetweenList

Goal state before:
pp: R : Type u_1
V : Type u_2
P : Type u_4
inst✝⁵ : Field R
inst✝⁴ : LinearOrder R
inst✝³ : IsStrictOrderedRing R
inst✝² : AddCommGroup V
inst✝¹ : Module R V
inst✝ : AddTorsor V P
l : List P
hl : l ≠ []
⊢ (∃ l', l'.SortedLE ∧ map (⇑(lineMap (l.head hl) (l.getLast hl))) l' = l) ↔ List.Wbtw R l

Tactic:
refine ⟨fun ⟨l', hl's, hl'l⟩ ↦ ?_, fun h ↦ ?_⟩
  · rw [← hl'l]
    exact Wbtw.map hl's.wbtw _
  · suffices ∃ l' : List R, (∀ a ∈ l', 0 ≤ a) ∧ l'.SortedLE ∧
        l'.map (lineMap (l.head hl) (l.getLast hl)) = l by
      rcases this with ⟨l', -, hl'⟩
      exact ⟨l', hl'⟩
    induction l with
    | nil => simp at hl
    | cons head tail ih =>
      by_cases ht : tail = []
      · refine ⟨[0], ?_⟩
        simp [ht, sortedLE_iff_pairwise]
      · rw [wbtw_cons] at h
        replace ih := ih ht h.2
        rcases ih with ⟨l'', hl''0, hl''s, hl''⟩
        simp only [head_cons, getLast_cons ht]
        cases tail with
        | nil => simp at ht
        | cons head2 tail =>
          by_cases ht2 : tail = []
          · exact ⟨[0, 1], by simp [ht2, sortedLE_iff_pairwise]⟩
          · simp only [head_cons, getLast_cons ht2] at hl'' ⊢
            rw [pairwise_cons] at h
            have hw := h.1.1 _ (getLast_mem ht2)
            rcases hw with ⟨r, ⟨hr0, hr1⟩, rfl⟩
            refine ⟨0 :: l''.map fun x ↦ r + (1 - r) * x, ?_, ?_, ?_⟩
            · simp only [mem_cons, mem_map, forall_eq_or_imp, le_refl, forall_exists_index,
                and_imp, forall_apply_eq_imp_iff₂, true_and]
              intro a ha
              have := hl''0 a ha
              nlinarith
            · simp only [sortedLE_iff_pairwise, pairwise_cons, mem_map,
                forall_exists_index, and_imp, forall_apply_eq_imp_iff₂]
              refine ⟨?_, ?_⟩
              · intro a ha
                have := hl''0 a ha
                nlinarith
              · refine hl''s.pairwise.map _ fun a b hab ↦ ?_
                gcongr
            · simp only [map_cons, lineMap_apply_zero, map_map, ← hl'', cons.injEq,
                map_inj_left, Function.comp_apply, lineMap_lineMap_left, lineMap_eq_lineMap_iff,
                true_and]
              ring_nf
              simp

Goal state after:
Module: Mathlib.Analysis.Convex.BetweenList

Goal state before:
pp: case refine_1
R : Type u_1
V : Type u_2
P : Type u_4
inst✝⁶ : Field R
inst✝⁵ : LinearOrder R
inst✝⁴ : IsStrictOrderedRing R
inst✝³ : AddCommGroup V
inst✝² : Module R V
inst✝¹ : AddTorsor V P
inst✝ : Nontrivial P
l : List P
x✝ : ∃ p₁ p₂, p₁ ≠ p₂ ∧ ∃ l', l'.SortedLT ∧ map (⇑(lineMap p₁ p₂)) l' = l
p₁ p₂ : P
hp₁p₂ : p₁ ≠ p₂
l' : List R
hl's : l'.SortedLT
hl'l : map (⇑(lineMap p₁ p₂)) l' = l
⊢ List.Sbtw R l

Tactic:
subst hl'l
    rw [(lineMap_injective _ hp₁p₂).list_sbtw_map_iff]
    exact hl's.sbtw

Goal state after:
Module: Mathlib.Analysis.Convex.BetweenList

Goal state before:
pp: R : Type u_1
V : Type u_2
P : Type u_4
inst✝⁶ : Field R
inst✝⁵ : LinearOrder R
inst✝⁴ : IsStrictOrderedRing R
inst✝³ : AddCommGroup V
inst✝² : Module R V
inst✝¹ : AddTorsor V P
inst✝ : Nontrivial P
l : List P
⊢ (∃ p₁ p₂, p₁ ≠ p₂ ∧ ∃ l', l'.SortedLT ∧ map (⇑(lineMap p₁ p₂)) l' = l) ↔ List.Sbtw R l

Tactic:
refine ⟨fun ⟨p₁, p₂, hp₁p₂, l', hl's, hl'l⟩ ↦ ?_, fun h ↦ ?_⟩
  · subst hl'l
    rw [(lineMap_injective _ hp₁p₂).list_sbtw_map_iff]
    exact hl's.sbtw
  · by_cases hl : l = []
    · rcases exists_pair_ne P with ⟨p₁, p₂, hp₁p₂⟩
      exact ⟨p₁, p₂, hp₁p₂, by simp [hl, sortedLT_iff_pairwise]⟩
    · by_cases hlen : l.length = 1
      · rw [length_eq_one_iff] at hlen
        rcases hlen with ⟨p₁, rfl⟩
        rcases exists_ne p₁ with ⟨p₂, hp₂p₁⟩
        exact ⟨p₁, p₂, hp₂p₁.symm, [0], by simp [sortedLT_iff_pairwise]⟩
      · refine ⟨l.head hl, l.getLast hl, ?_⟩
        rw [← exists_map_eq_of_sorted_nonempty_iff_sbtw hl] at h
        simp only [hlen, false_or] at h
        rcases h with ⟨l', hl's, hl'l, hl⟩
        exact ⟨hl, l', hl's, hl'l⟩

Goal state after:
Module: Mathlib.Analysis.Convex.Between

Goal state before:
pp: R : Type u_1
V : Type u_2
P : Type u_4
inst✝⁷ : Ring R
inst✝⁶ : PartialOrder R
inst✝⁵ : AddCommGroup V
inst✝⁴ : Module R V
inst✝³ : AddTorsor V P
inst✝² : IsOrderedRing R
inst✝¹ : IsDomain R
inst✝ : IsTorsionFree R V
x y : P
r : R
⊢ Wbtw R x ((lineMap x y) r) y ↔ x = y ∨ r ∈ Set.Icc 0 1

Tactic:
by
  by_cases hxy : x = y
  · rw [hxy, lineMap_same_apply]
    simp
  rw [or_iff_right hxy, Wbtw, affineSegment, (lineMap_injective R hxy).mem_set_image]

Goal state after:
Module: Mathlib.Data.Matrix.PEquiv

Goal state before:
pp: l : Type u_2
m : Type u_3
n : Type u_4
α : Type u_5
inst✝² : Fintype m
inst✝¹ : NonAssocSemiring α
inst✝ : DecidableEq n
M : Matrix l m α
f : m ≃. n
i : l
j : n
⊢ (M * f.toMatrix) i j = Option.casesOn (f.symm j) 0 (M i)

Tactic:
by
  dsimp [Matrix.mul_apply, toMatrix_apply]
  rcases h : f.symm j with - | fj
  · simp [h, ← f.eq_some_iff]
  · rw [Finset.sum_eq_single fj]
    · simp [h, ← f.eq_some_iff]
    · rintro b - n
      simp [h, ← f.eq_some_iff, n.symm]
    · simp

Goal state after:
Module: Mathlib.Analysis.Convex.DoublyStochasticMatrix

Goal state before:
pp: R : Type u_1
n : Type u_2
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : Semiring R
inst✝¹ : PartialOrder R
inst✝ : IsOrderedRing R
M : Matrix n n R
⊢ M ∈ doublyStochastic R n ↔ M ∈ rowStochastic R n ∧ M ∈ colStochastic R n

Tactic:
rw [doublyStochastic_eq_rowStochastic_inf_colStochastic, Submonoid.mem_inf]

Goal state after:
Module: Mathlib.Analysis.Convex.Birkhoff

Goal state before:
pp: R : Type u_1
n : Type u_2
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : Semifield R
inst✝¹ : LinearOrder R
inst✝ : IsStrictOrderedRing R
M : Matrix n n R
s : R
hs : 0 < s
hM : (∀ (i j : n), 0 ≤ M i j) ∧ (∀ (i : n), ∑ j, M i j = s) ∧ ∀ (j : n), ∑ i, M i j = s
f : n → Finset n := fun i ↦ {j | M i j ≠ 0}
hf : ∀ (A : Finset n), #A ≤ #(A.biUnion f)
g : n → n
hg : Bijective g
hg' : ∀ (x : n), g x ∈ f x
i : n
hij : M i (g i) = 0
⊢ False

Tactic:
simpa [f, hij] using hg' i

Goal state after:
Module: Mathlib.Analysis.Convex.Caratheodory

Goal state before:
pp: case refine_1
𝕜 : Type u_1
E : Type u
inst✝⁵ : Field 𝕜
inst✝⁴ : LinearOrder 𝕜
inst✝³ : IsStrictOrderedRing 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
inst✝ : DecidableEq E
t : Finset E
f : E → 𝕜
fpos : ∀ y ∈ t, 0 ≤ f y
fsum : ∑ y ∈ t, f y = 1
g : E → 𝕜
gcombo : ∑ e ∈ t, g e • e = 0
gsum : ∑ e ∈ t, g e = 0
gpos : ∃ i ∈ t, 0 < g i
s : Finset E := {z ∈ t | 0 < g z}
i₀ : E
mem : i₀ ∈ s
w : ∀ i ∈ s, f i₀ / g i₀ ≤ f i / g i
hg : 0 < g i₀
hi₀ : i₀ ∈ t
k : E → 𝕜 := fun z ↦ f z - f i₀ / g i₀ * g z
hk : k i₀ = 0
ksum : ∑ e ∈ t.erase i₀, k e = 1
e : E
a✝ : ¬e = i₀
⊢ e ∈ t → f i₀ / g i₀ * g e ≤ f e

Tactic:
het

Goal state after:
case refine_1
𝕜 : Type u_1
E : Type u
inst✝⁵ : Field 𝕜
inst✝⁴ : LinearOrder 𝕜
inst✝³ : IsStrictOrderedRing 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
inst✝ : DecidableEq E
t : Finset E
f : E → 𝕜
fpos : ∀ y ∈ t, 0 ≤ f y
fsum : ∑ y ∈ t, f y = 1
g : E → 𝕜
gcombo : ∑ e ∈ t, g e • e = 0
gsum : ∑ e ∈ t, g e = 0
gpos : ∃ i ∈ t, 0 < g i
s : Finset E := {z ∈ t | 0 < g z}
i₀ : E
mem : i₀ ∈ s
w : ∀ i ∈ s, f i₀ / g i₀ ≤ f i / g i
hg : 0 < g i₀
hi₀ : i₀ ∈ t
k : E → 𝕜 := fun z ↦ f z - f i₀ / g i₀ * g z
hk : k i₀ = 0
ksum : ∑ e ∈ t.erase i₀, k e = 1
e : E
a✝ : ¬e = i₀
het : e ∈ t
⊢ f i₀ / g i₀ * g e ≤ f e
Module: Mathlib.Analysis.Convex.Between

Goal state before:
pp: R : Type u_1
V : Type u_2
P : Type u_4
inst✝⁶ : Ring R
inst✝⁵ : LinearOrder R
inst✝⁴ : IsStrictOrderedRing R
inst✝³ : AddCommGroup V
inst✝² : Module R V
inst✝¹ : AddTorsor V P
inst✝ : IsTorsionFree R V
t : Affine.Triangle R P
i₁ i₂ i₃ : Fin 3
h₁₂ : i₁ ≠ i₂
h₁₃ : i₁ ≠ i₃
h₂₃ : i₂ ≠ i₃
h3 : ∀ (i : Fin 3), i = i₁ ∨ i = i₂ ∨ i = i₃
hu : Finset.univ = {i₁, i₂, i₃}
r₁ : R
hr₁0 : 0 < r₁
hr₁1 : r₁ < 1
h₁ : Sbtw R (t.points i₂) ((lineMap (t.points i₂) (t.points i₃)) r₁) (t.points i₃)
r₂ : R
hr₂0 : 0 < r₂
hr₂1 : r₂ < 1
h₂ : Sbtw R (t.points i₁) ((lineMap (t.points i₁) (t.points i₃)) r₂) (t.points i₃)
w : Fin (2 + 1) → R
hw : w i₁ + (w i₂ + w i₃) = 1
hp : (Finset.affineCombination R Finset.univ t.points) w ∈ affineSpan R (Set.range t.points)
h₁' :
  (Finset.affineCombination R Finset.univ t.points) w ∈
    line[R, (Finset.affineCombination R Finset.univ t.points) (Pi.single i₁ 1),
      (Finset.affineCombination R Finset.univ t.points) (Finset.affineCombinationLineMapWeights i₂ i₃ r₁)]
h₂' :
  (Finset.affineCombination R Finset.univ t.points) w ∈ line[R, t.points i₂, (lineMap (t.points i₁) (t.points i₃)) r₂]
h₁s : SignType.sign (w i₂) = SignType.sign (w i₃)
h₂s : SignType.sign (w i₁) = SignType.sign (w i₃)
hss : SignType.sign (w i₃) = 1
hs' : SignType.sign (∑ i, w i) = SignType.sign (w i₃)
hs : ∀ (i : Fin 3), 0 < w i
hle : ¬w i₁ < 1
⊢ False

Tactic:
rw [not_lt] at hle

Goal state after:
R : Type u_1
V : Type u_2
P : Type u_4
inst✝⁶ : Ring R
inst✝⁵ : LinearOrder R
inst✝⁴ : IsStrictOrderedRing R
inst✝³ : AddCommGroup V
inst✝² : Module R V
inst✝¹ : AddTorsor V P
inst✝ : IsTorsionFree R V
t : Affine.Triangle R P
i₁ i₂ i₃ : Fin 3
h₁₂ : i₁ ≠ i₂
h₁₃ : i₁ ≠ i₃
h₂₃ : i₂ ≠ i₃
h3 : ∀ (i : Fin 3), i = i₁ ∨ i = i₂ ∨ i = i₃
hu : Finset.univ = {i₁, i₂, i₃}
r₁ : R
hr₁0 : 0 < r₁
hr₁1 : r₁ < 1
h₁ : Sbtw R (t.points i₂) ((lineMap (t.points i₂) (t.points i₃)) r₁) (t.points i₃)
r₂ : R
hr₂0 : 0 < r₂
hr₂1 : r₂ < 1
h₂ : Sbtw R (t.points i₁) ((lineMap (t.points i₁) (t.points i₃)) r₂) (t.points i₃)
w : Fin (2 + 1) → R
hw : w i₁ + (w i₂ + w i₃) = 1
hp : (Finset.affineCombination R Finset.univ t.points) w ∈ affineSpan R (Set.range t.points)
h₁' :
  (Finset.affineCombination R Finset.univ t.points) w ∈
    line[R, (Finset.affineCombination R Finset.univ t.points) (Pi.single i₁ 1),
      (Finset.affineCombination R Finset.univ t.points) (Finset.affineCombinationLineMapWeights i₂ i₃ r₁)]
h₂' :
  (Finset.affineCombination R Finset.univ t.points) w ∈ line[R, t.points i₂, (lineMap (t.points i₁) (t.points i₃)) r₂]
h₁s : SignType.sign (w i₂) = SignType.sign (w i₃)
h₂s : SignType.sign (w i₁) = SignType.sign (w i₃)
hss : SignType.sign (w i₃) = 1
hs' : SignType.sign (∑ i, w i) = SignType.sign (w i₃)
hs : ∀ (i : Fin 3), 0 < w i
hle : 1 ≤ w i₁
⊢ False
Module: Mathlib.Analysis.Convex.Caratheodory

Goal state before:
pp: 𝕜 : Type u_1
E : Type u
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s : Set E
x : E
hx : x ∈ (convexHull 𝕜) s
k : ℕ := ⋯
hk : #(minCardFinsetOfMemConvexHull hx) = k + 1
h : ¬AffineIndependent 𝕜 Subtype.val
p : ↑↑(minCardFinsetOfMemConvexHull hx)
hp : x ∈ (convexHull 𝕜) ↑((minCardFinsetOfMemConvexHull hx).erase ↑p)
contra : ¬#((minCardFinsetOfMemConvexHull hx).erase ↑p) < #(minCardFinsetOfMemConvexHull hx)
⊢ #((minCardFinsetOfMemConvexHull hx).erase ↑p) < #(minCardFinsetOfMemConvexHull hx)

Tactic:
rw [card_erase_of_mem p.2, hk]

Goal state after:
𝕜 : Type u_1
E : Type u
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s : Set E
x : E
hx : x ∈ (convexHull 𝕜) s
k : ℕ := #(minCardFinsetOfMemConvexHull hx) - 1
hk : #(minCardFinsetOfMemConvexHull hx) = k + 1
h : ¬AffineIndependent 𝕜 Subtype.val
p : ↑↑(minCardFinsetOfMemConvexHull hx)
hp : x ∈ (convexHull 𝕜) ↑((minCardFinsetOfMemConvexHull hx).erase ↑p)
contra : ¬#((minCardFinsetOfMemConvexHull hx).erase ↑p) < #(minCardFinsetOfMemConvexHull hx)
⊢ k + 1 - 1 < k + 1
Module: Mathlib.Analysis.Convex.Cone.InnerDual

Goal state before:
pp: E : Type u_2
F : Type u_3
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace ℝ E
inst✝³ : CompleteSpace E
inst✝² : NormedAddCommGroup F
inst✝¹ : InnerProductSpace ℝ F
inst✝ : CompleteSpace F
C : ProperCone ℝ E
f : E →L[ℝ] F
b : F
⊢ ∀ (x : ℕ → F) (x_1 : ℕ → E),
    (∀ (x : ℕ), x_1 x ∈ C) →
      (∀ (x_2 : ℕ), f (x_1 x_2) = x x_2) →
        Filter.Tendsto x Filter.atTop (nhds b) → ∀ (y : F), (∀ ⦃x : E⦄, x ∈ C → 0 ≤ ⟪f x, y⟫_ℝ) → 0 ≤ ⟪b, y⟫_ℝ

Tactic:
rintro x seq hmem hx htends y hinner

Goal state after:
E : Type u_2
F : Type u_3
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace ℝ E
inst✝³ : CompleteSpace E
inst✝² : NormedAddCommGroup F
inst✝¹ : InnerProductSpace ℝ F
inst✝ : CompleteSpace F
C : ProperCone ℝ E
f : E →L[ℝ] F
b : F
x : ℕ → F
seq : ℕ → E
hmem : ∀ (x : ℕ), seq x ∈ C
hx : ∀ (x_1 : ℕ), f (seq x_1) = x x_1
htends : Filter.Tendsto x Filter.atTop (nhds b)
y : F
hinner : ∀ ⦃x : E⦄, x ∈ C → 0 ≤ ⟪f x, y⟫_ℝ
⊢ 0 ≤ ⟪b, y⟫_ℝ
Module: Mathlib.Analysis.Convex.Cone.Dual

Goal state before:
pp: R : Type u_1
M : Type u_2
N : Type u_3
inst✝¹¹ : CommRing R
inst✝¹⁰ : PartialOrder R
inst✝⁹ : IsOrderedRing R
inst✝⁸ : TopologicalSpace R
inst✝⁷ : ClosedIciTopology R
inst✝⁶ : AddCommGroup M
inst✝⁵ : Module R M
inst✝⁴ : TopologicalSpace M
inst✝³ : AddCommGroup N
inst✝² : Module R N
inst✝¹ : TopologicalSpace N
p : M →ₗ[R] N →ₗ[R] R
inst✝ : p.IsContPerfPair
x : M
s : Set M
⊢ dual p ({x} ∪ s) = dual p {x} ⊓ dual p s

Tactic:
dual_union

Goal state after:
R : Type u_1
M : Type u_2
N : Type u_3
inst✝¹¹ : CommRing R
inst✝¹⁰ : PartialOrder R
inst✝⁹ : IsOrderedRing R
inst✝⁸ : TopologicalSpace R
inst✝⁷ : ClosedIciTopology R
inst✝⁶ : AddCommGroup M
inst✝⁵ : Module R M
inst✝⁴ : TopologicalSpace M
inst✝³ : AddCommGroup N
inst✝² : Module R N
inst✝¹ : TopologicalSpace N
p : M →ₗ[R] N →ₗ[R] R
inst✝ : p.IsContPerfPair
x : M
s : Set M
⊢ dual p {x} ⊓ dual p s = dual p {x} ⊓ dual p s
Module: Mathlib.Analysis.Convex.Between

Goal state before:
pp: R : Type u_1
V : Type u_2
P : Type u_4
inst✝⁵ : Field R
inst✝⁴ : LinearOrder R
inst✝³ : IsStrictOrderedRing R
inst✝² : AddCommGroup V
inst✝¹ : Module R V
inst✝ : AddTorsor V P
x y : P
h : x ≠ y
⊢ (pointReflection R x) x ≠ (pointReflection R x) y

Tactic:
exact (pointReflection_involutive R x).injective.ne h

Goal state after:
Module: Mathlib.Geometry.Convex.Cone.Dual

Goal state before:
pp: R : Type u_1
inst✝⁶ : CommSemiring R
inst✝⁵ : PartialOrder R
inst✝⁴ : IsOrderedRing R
M : Type u_2
inst✝³ : AddCommMonoid M
inst✝² : Module R M
N : Type u_3
inst✝¹ : AddCommMonoid N
inst✝ : Module R N
p : M →ₗ[R] N →ₗ[R] R
x : M
s : Set M
⊢ dual p ({x} ∪ s) = dual p {x} ⊓ dual p s

Tactic:
dual_union

Goal state after:
R : Type u_1
inst✝⁶ : CommSemiring R
inst✝⁵ : PartialOrder R
inst✝⁴ : IsOrderedRing R
M : Type u_2
inst✝³ : AddCommMonoid M
inst✝² : Module R M
N : Type u_3
inst✝¹ : AddCommMonoid N
inst✝ : Module R N
p : M →ₗ[R] N →ₗ[R] R
x : M
s : Set M
⊢ dual p {x} ⊓ dual p s = dual p {x} ⊓ dual p s
Module: Mathlib.Analysis.Convex.Extrema

Goal state before:
pp: E : Type u_1
β : Type u_2
inst✝¹⁰ : AddCommGroup E
inst✝⁹ : TopologicalSpace E
inst✝⁸ : Module ℝ E
inst✝⁷ : IsTopologicalAddGroup E
inst✝⁶ : ContinuousSMul ℝ E
inst✝⁵ : AddCommGroup β
inst✝⁴ : PartialOrder β
inst✝³ : IsOrderedAddMonoid β
inst✝² : Module ℝ β
inst✝¹ : IsOrderedModule ℝ β
inst✝ : PosSMulReflectLE ℝ β
s : Set E
f : E → β
a : E
a_in_s : a ∈ s
h_localmin : IsLocalMinOn f s a
h_conv : ConvexOn ℝ s f
x : E
x_in_s : x ∈ s
g : ℝ →ᵃ[ℝ] E := AffineMap.lineMap a x
hg0 : g 0 = a
hg1 : g 1 = x
hgc : Continuous[_, inst✝⁹] ⇑g
h_maps : MapsTo (⇑g) (Icc 0 1) s
fg_local_min_on : IsLocalMinOn (f ∘ ⇑g) (Icc 0 1) 0
⊢ ConvexOn ℝ (Icc 0 1) (f ∘ ⇑g)

Tactic:
exact (h_conv.comp_affineMap g).subset h_maps (convex_Icc 0 1)

Goal state after:
Module: Mathlib.Analysis.Convex.Independent

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝³ : Semiring 𝕜
inst✝² : PartialOrder 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p : ι → E
hc : ConvexIndependent 𝕜 p
i j : ι
hij : p i = p j
⊢ p j ∈ (convexHull 𝕜) {p j}

Tactic:
convexHull_singleton

Goal state after:
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝³ : Semiring 𝕜
inst✝² : PartialOrder 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p : ι → E
hc : ConvexIndependent 𝕜 p
i j : ι
hij : p i = p j
⊢ p j ∈ {p j}
Module: Mathlib.Analysis.Convex.Independent

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝³ : Semiring 𝕜
inst✝² : PartialOrder 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s : Set E
⊢ ConvexIndependent 𝕜 Subtype.val ↔ ∀ x ∈ s, x ∉ (convexHull 𝕜) (s \ {x})

Tactic:
convexIndependent_set_iff_inter_convexHull_subset

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝³ : Semiring 𝕜
inst✝² : PartialOrder 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s : Set E
⊢ (∀ t ⊆ s, s ∩ (convexHull 𝕜) t ⊆ t) ↔ ∀ x ∈ s, x ∉ (convexHull 𝕜) (s \ {x})
Module: Mathlib.Analysis.Convex.Independent

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p : ι → E
h : ∀ (s : Finset ι) (x : ι), p x ∈ (convexHull 𝕜) ↑(image p s) → x ∈ s
s : Set ι
x : ι
hx : p x ∈ (convexHull 𝕜) (p '' s)
a b : ι
hab : p a = p b
⊢ p b ∈ (convexHull 𝕜) {p b}

Tactic:
convexHull_singleton

Goal state after:
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p : ι → E
h : ∀ (s : Finset ι) (x : ι), p x ∈ (convexHull 𝕜) ↑(image p s) → x ∈ s
s : Set ι
x : ι
hx : p x ∈ (convexHull 𝕜) (p '' s)
a b : ι
hab : p a = p b
⊢ p b ∈ {p b}
Module: Mathlib.Analysis.Convex.Continuous

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
C : Set E
f : E → ℝ
hf : ConvexOn ℝ C f
x₀ : E
hC : C ∈ 𝓝 x₀
r : ℝ
this : Filter.Tendsto (fun y ↦ 2 • x₀ - y) (𝓝 x₀) (𝓝 x₀)
hr : ∀ᶠ (a : E) in 𝓝 x₀, f a ≤ r
y : E
hx : 2 • x₀ - y ∈ C
hx' : y ∈ C
hfr : f y ≤ r
hfr' : f (2 • x₀ - y) ≤ r
⊢ -(2 * |f x₀|) ≤ f y + |r|

Tactic:
← abs_two,

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
C : Set E
f : E → ℝ
hf : ConvexOn ℝ C f
x₀ : E
hC : C ∈ 𝓝 x₀
r : ℝ
this : Filter.Tendsto (fun y ↦ 2 • x₀ - y) (𝓝 x₀) (𝓝 x₀)
hr : ∀ᶠ (a : E) in 𝓝 x₀, f a ≤ r
y : E
hx : 2 • x₀ - y ∈ C
hx' : y ∈ C
hfr : f y ≤ r
hfr' : f (2 • x₀ - y) ≤ r
⊢ -(|2| * |f x₀|) ≤ f y + |r|
Module: Mathlib.Analysis.Convex.Join

Goal state before:
pp: 𝕜 : Type u_2
E : Type u_3
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s : Set E
x : E
hs : s.Nonempty
⊢ convexJoin 𝕜 ((convexHull 𝕜) {x}) ((convexHull 𝕜) s) = convexJoin 𝕜 {x} ((convexHull 𝕜) s)

Tactic:
convexHull_singleton

Goal state after:
𝕜 : Type u_2
E : Type u_3
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s : Set E
x : E
hs : s.Nonempty
⊢ convexJoin 𝕜 {x} ((convexHull 𝕜) s) = convexJoin 𝕜 {x} ((convexHull 𝕜) s)
Module: Mathlib.Analysis.Convex.Join

Goal state before:
pp: 𝕜 : Type u_2
E : Type u_3
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
a b c d : E
⊢ convexJoin 𝕜 {a} (convexJoin 𝕜 ((convexHull 𝕜) {b}) (convexJoin 𝕜 {c} ((convexHull 𝕜) {d}))) =
    convexJoin 𝕜 {a} (convexJoin 𝕜 {b} (convexJoin 𝕜 {c} ((convexHull 𝕜) {d})))

Tactic:
convexHull_singleton

Goal state after:
Module: Mathlib.Geometry.Convex.ConvexSpace.AffineSpace

Goal state before:
pp: R : Type u_1
V : Type u_2
P : Type u_3
inst✝⁵ : Ring R
inst✝⁴ : PartialOrder R
inst✝³ : IsStrictOrderedRing R
inst✝² : AddCommGroup V
inst✝¹ : Module R V
inst✝ : AffineSpace V P
s t : R
hs : 0 ≤ s
ht : 0 ≤ t
h : s + t = 1
x y : P
⊢ ∀ x_1 ∈ (Finsupp.single x s + Finsupp.single y t).support,
    (Finsupp.single x s + Finsupp.single y t) x_1 = (fun x r ↦ r) x_1 ((Finsupp.single x s + Finsupp.single y t) x_1)

Tactic:
intro i _

Goal state after:
R : Type u_1
V : Type u_2
P : Type u_3
inst✝⁵ : Ring R
inst✝⁴ : PartialOrder R
inst✝³ : IsStrictOrderedRing R
inst✝² : AddCommGroup V
inst✝¹ : Module R V
inst✝ : AffineSpace V P
s t : R
hs : 0 ≤ s
ht : 0 ≤ t
h : s + t = 1
x y i : P
a✝ : i ∈ (Finsupp.single x s + Finsupp.single y t).support
⊢ (Finsupp.single x s + Finsupp.single y t) i = (fun x r ↦ r) i ((Finsupp.single x s + Finsupp.single y t) i)
Module: Mathlib.Geometry.Convex.Set

Goal state before:
pp: case pos
R : Type u_3
X : Type u_5
Y : Type u_6
inst✝⁴ : Semiring R
inst✝³ : PartialOrder R
inst✝² : IsStrictOrderedRing R
inst✝¹ : ConvexSpace R X
inst✝ : ConvexSpace R Y
f : X → Y
s : Set X
hf : IsAffineMap R f
hs : IsConvexSet R s
w : StdSimplex R Y
hw : ↑w.weights.support ⊆ f '' s
u : Finset X
hus : ↑u ⊆ s
hfu : InjOn f ↑u
huw : Finset.image f u = w.weights.support
y : Y
hy : y ∈ w.weights.support
⊢ (mapDomain f
        { weights := onFinset u (fun x ↦ if x ∈ u then w.weights (f x) else 0) ⋯, nonneg := ⋯, total := ⋯ }.weights)
      y =
    w.weights y

Tactic:
← huw,

Goal state after:
case pos
R : Type u_3
X : Type u_5
Y : Type u_6
inst✝⁴ : Semiring R
inst✝³ : PartialOrder R
inst✝² : IsStrictOrderedRing R
inst✝¹ : ConvexSpace R X
inst✝ : ConvexSpace R Y
f : X → Y
s : Set X
hf : IsAffineMap R f
hs : IsConvexSet R s
w : StdSimplex R Y
hw : ↑w.weights.support ⊆ f '' s
u : Finset X
hus : ↑u ⊆ s
hfu : InjOn f ↑u
huw : Finset.image f u = w.weights.support
y : Y
hy : y ∈ Finset.image f u
⊢ (mapDomain f
        { weights := onFinset u (fun x ↦ if x ∈ u then w.weights (f x) else 0) ⋯, nonneg := ⋯, total := ⋯ }.weights)
      y =
    w.weights y
Module: Mathlib.Analysis.Convex.Piecewise

Goal state before:
pp: case inl.inr
𝕜 : Type u_1
E : Type u_2
β : Type u_3
inst✝¹¹ : Semiring 𝕜
inst✝¹⁰ : PartialOrder 𝕜
inst✝⁹ : AddCommMonoid E
inst✝⁸ : LinearOrder E
inst✝⁷ : IsOrderedAddMonoid E
inst✝⁶ : Module 𝕜 E
inst✝⁵ : PosSMulMono 𝕜 E
inst✝⁴ : AddCommGroup β
inst✝³ : PartialOrder β
inst✝² : IsOrderedAddMonoid β
inst✝¹ : Module 𝕜 β
inst✝ : PosSMulMono 𝕜 β
e : E
f g : E → β
hf : ConvexOn 𝕜 (Set.Iic e) f
hg : ConvexOn 𝕜 (Set.Ici e) g
h_anti : AntitoneOn f (Set.Iic e)
h_mono : MonotoneOn g (Set.Ici e)
h_eq : f e = g e
x : E
x✝¹ : x ∈ Set.univ
y : E
x✝ : y ∈ Set.univ
a b : 𝕜
ha : 0 ≤ a
hb : 0 ≤ b
hab : a + b = 1
hx : x ≤ e
hy : e < y
⊢ (Set.Iic e).piecewise f g (a • x + b • y) ≤ a • (Set.Iic e).piecewise f g x + b • (Set.Iic e).piecewise f g y

Tactic:
rw [Set.piecewise_eq_of_mem (Set.Iic e) f g hx,
      Set.piecewise_eq_of_notMem (Set.Iic e) f g (Set.notMem_Iic.mpr hy)]

Goal state after:
case inl.inr
𝕜 : Type u_1
E : Type u_2
β : Type u_3
inst✝¹¹ : Semiring 𝕜
inst✝¹⁰ : PartialOrder 𝕜
inst✝⁹ : AddCommMonoid E
inst✝⁸ : LinearOrder E
inst✝⁷ : IsOrderedAddMonoid E
inst✝⁶ : Module 𝕜 E
inst✝⁵ : PosSMulMono 𝕜 E
inst✝⁴ : AddCommGroup β
inst✝³ : PartialOrder β
inst✝² : IsOrderedAddMonoid β
inst✝¹ : Module 𝕜 β
inst✝ : PosSMulMono 𝕜 β
e : E
f g : E → β
hf : ConvexOn 𝕜 (Set.Iic e) f
hg : ConvexOn 𝕜 (Set.Ici e) g
h_anti : AntitoneOn f (Set.Iic e)
h_mono : MonotoneOn g (Set.Ici e)
h_eq : f e = g e
x : E
x✝¹ : x ∈ Set.univ
y : E
x✝ : y ∈ Set.univ
a b : 𝕜
ha : 0 ≤ a
hb : 0 ≤ b
hab : a + b = 1
hx : x ≤ e
hy : e < y
⊢ (Set.Iic e).piecewise f g (a • x + b • y) ≤ a • f x + b • g y
Module: Mathlib.Analysis.Convex.Piecewise

Goal state before:
pp: case inr.inl
𝕜 : Type u_1
E : Type u_2
β : Type u_3
inst✝¹¹ : Semiring 𝕜
inst✝¹⁰ : PartialOrder 𝕜
inst✝⁹ : AddCommMonoid E
inst✝⁸ : LinearOrder E
inst✝⁷ : IsOrderedAddMonoid E
inst✝⁶ : Module 𝕜 E
inst✝⁵ : PosSMulMono 𝕜 E
inst✝⁴ : AddCommGroup β
inst✝³ : PartialOrder β
inst✝² : IsOrderedAddMonoid β
inst✝¹ : Module 𝕜 β
inst✝ : PosSMulMono 𝕜 β
e : E
f g : E → β
hf : ConvexOn 𝕜 (Set.Iic e) f
hg : ConvexOn 𝕜 (Set.Ici e) g
h_anti : AntitoneOn f (Set.Iic e)
h_mono : MonotoneOn g (Set.Ici e)
h_eq : f e = g e
x : E
x✝¹ : x ∈ Set.univ
y : E
x✝ : y ∈ Set.univ
a b : 𝕜
ha : 0 ≤ a
hb : 0 ≤ b
hab : a + b = 1
hx : e < x
hy : y ≤ e
⊢ (Set.Iic e).piecewise f g (a • x + b • y) ≤ a • g x + b • (Set.Iic e).piecewise f g y

Tactic:
Set.piecewise_eq_of_mem (Set.Iic e) f g hy

Goal state after:
case inr.inl
𝕜 : Type u_1
E : Type u_2
β : Type u_3
inst✝¹¹ : Semiring 𝕜
inst✝¹⁰ : PartialOrder 𝕜
inst✝⁹ : AddCommMonoid E
inst✝⁸ : LinearOrder E
inst✝⁷ : IsOrderedAddMonoid E
inst✝⁶ : Module 𝕜 E
inst✝⁵ : PosSMulMono 𝕜 E
inst✝⁴ : AddCommGroup β
inst✝³ : PartialOrder β
inst✝² : IsOrderedAddMonoid β
inst✝¹ : Module 𝕜 β
inst✝ : PosSMulMono 𝕜 β
e : E
f g : E → β
hf : ConvexOn 𝕜 (Set.Iic e) f
hg : ConvexOn 𝕜 (Set.Ici e) g
h_anti : AntitoneOn f (Set.Iic e)
h_mono : MonotoneOn g (Set.Ici e)
h_eq : f e = g e
x : E
x✝¹ : x ∈ Set.univ
y : E
x✝ : y ∈ Set.univ
a b : 𝕜
ha : 0 ≤ a
hb : 0 ≤ b
hab : a + b = 1
hx : e < x
hy : y ≤ e
⊢ (Set.Iic e).piecewise f g (a • x + b • y) ≤ a • g x + b • f y