module stringlengths 16 90 | startPos dict | endPos dict | nextStartPos dict | goals listlengths 0 96 | goalsAfter listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 375
values | kind stringclasses 379
values |
|---|---|---|---|---|---|---|---|---|
Mathlib.Analysis.Convex.BetweenList | {
"line": 137,
"column": 33
} | {
"line": 137,
"column": 44
} | {
"line": 137,
"column": 45
} | [
{
"pp": "case cons.refine_2.refine_1.cons.inl.cons\nR : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝⁵ : Ring R\ninst✝⁴ : PartialOrder R\ninst✝³ : AddCommGroup V\ninst✝² : Module R V\ninst✝¹ : AddTorsor V P\ninst✝ : IsOrderedRing R\nhead head3 : P\ntail : List P\nx✝ :\n (Pairwise (Sbtw R head) (head :: head3 :: ... | [
"case cons.refine_2.refine_1.cons.inl.cons\nR : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝⁵ : Ring R\ninst✝⁴ : PartialOrder R\ninst✝³ : AddCommGroup V\ninst✝² : Module R V\ninst✝¹ : AddTorsor V P\ninst✝ : IsOrderedRing R\nhead head3 : P\ntail : List P\nx✝ :\n (Pairwise (Sbtw R head) (head :: head3 :: tail) ∧ Trip... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.Approximation | {
"line": 73,
"column": 2
} | {
"line": 73,
"column": 51
} | {
"line": 74,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ns : Set E\nφ : E → ℝ\ninst✝⁸ : RCLike 𝕜\ninst✝⁷ : TopologicalSpace E\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : Module ℝ E\ninst✝⁴ : Module 𝕜 E\ninst✝³ : IsScalarTower ℝ 𝕜 E\ninst✝² : IsTopologicalAddGroup E\ninst✝¹ : ContinuousSMul 𝕜 E\ninst✝ : LocallyConvexSpace ℝ E\nx : E\na... | [
"𝕜 : Type u_1\nE : Type u_2\ns : Set E\nφ : E → ℝ\ninst✝⁸ : RCLike 𝕜\ninst✝⁷ : TopologicalSpace E\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : Module ℝ E\ninst✝⁴ : Module 𝕜 E\ninst✝³ : IsScalarTower ℝ 𝕜 E\ninst✝² : IsTopologicalAddGroup E\ninst✝¹ : ContinuousSMul 𝕜 E\ninst✝ : LocallyConvexSpace ℝ E\nx : E\na : ℝ\nhx : x... | let A := { p : E × 𝕜 | p.1 ∈ s ∧ φ p.1 ≤ re p.2 } | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.Analysis.Convex.BetweenList | {
"line": 138,
"column": 12
} | {
"line": 138,
"column": 23
} | {
"line": 138,
"column": 24
} | [
{
"pp": "case cons.refine_2.refine_1.cons.inr\nR : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝⁵ : Ring R\ninst✝⁴ : PartialOrder R\ninst✝³ : AddCommGroup V\ninst✝² : Module R V\ninst✝¹ : AddTorsor V P\ninst✝ : IsOrderedRing R\nhead head2 : P\ntail : List P\nx✝ :\n (Pairwise (Sbtw R head) (head2 :: tail) ∧ Tripl... | [
"case cons.refine_2.refine_1.cons.inr\nR : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝⁵ : Ring R\ninst✝⁴ : PartialOrder R\ninst✝³ : AddCommGroup V\ninst✝² : Module R V\ninst✝¹ : AddTorsor V P\ninst✝ : IsOrderedRing R\nhead head2 : P\ntail : List P\nx✝ :\n (Pairwise (Sbtw R head) (head2 :: tail) ∧ Triplewise (Sbtw ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Hall.Finite | {
"line": 114,
"column": 6
} | {
"line": 114,
"column": 27
} | {
"line": 114,
"column": 28
} | [
{
"pp": "ι : Type u\nα : Type v\ninst✝¹ : DecidableEq α\nt : ι → Finset α\ninst✝ : Fintype ι\nn : ℕ\nhn : Fintype.card ι = n + 1\nht : ∀ (s : Finset ι), #s ≤ #(s.biUnion t)\nih :\n ∀ {ι' : Type u} [inst : Fintype ι'] (t' : ι' → Finset α),\n Fintype.card ι' ≤ n →\n (∀ (s' : Finset ι'), #s' ≤ #(s'.biUnio... | [
"ι : Type u\nα : Type v\ninst✝¹ : DecidableEq α\nt : ι → Finset α\ninst✝ : Fintype ι\nn : ℕ\nhn : Fintype.card ι = n + 1\nht : ∀ (s : Finset ι), #s ≤ #(s.biUnion t)\nih :\n ∀ {ι' : Type u} [inst : Fintype ι'] (t' : ι' → Finset α),\n Fintype.card ι' ≤ n →\n (∀ (s' : Finset ι'), #s' ≤ #(s'.biUnion t')) → ∃ f... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.Approximation | {
"line": 84,
"column": 6
} | {
"line": 85,
"column": 29
} | {
"line": 85,
"column": 30
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ns : Set E\nφ : E → ℝ\ninst✝⁸ : RCLike 𝕜\ninst✝⁷ : TopologicalSpace E\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : Module ℝ E\ninst✝⁴ : Module 𝕜 E\ninst✝³ : IsScalarTower ℝ 𝕜 E\ninst✝² : IsTopologicalAddGroup E\ninst✝¹ : ContinuousSMul 𝕜 E\ninst✝ : LocallyConvexSpace ℝ E\nx : E\na... | [
"𝕜 : Type u_1\nE : Type u_2\ns : Set E\nφ : E → ℝ\ninst✝⁸ : RCLike 𝕜\ninst✝⁷ : TopologicalSpace E\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : Module ℝ E\ninst✝⁴ : Module 𝕜 E\ninst✝³ : IsScalarTower ℝ 𝕜 E\ninst✝² : IsTopologicalAddGroup E\ninst✝¹ : ContinuousSMul 𝕜 E\ninst✝ : LocallyConvexSpace ℝ E\nx : E\na : ℝ\nhx : x... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.Approximation | {
"line": 87,
"column": 4
} | {
"line": 88,
"column": 11
} | {
"line": 88,
"column": 12
} | [
{
"pp": "case refine_1\n𝕜 : Type u_1\nE : Type u_2\ns : Set E\nφ : E → ℝ\ninst✝⁸ : RCLike 𝕜\ninst✝⁷ : TopologicalSpace E\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : Module ℝ E\ninst✝⁴ : Module 𝕜 E\ninst✝³ : IsScalarTower ℝ 𝕜 E\ninst✝² : IsTopologicalAddGroup E\ninst✝¹ : ContinuousSMul 𝕜 E\ninst✝ : LocallyConvexSpac... | [
"case refine_1\n𝕜 : Type u_1\nE : Type u_2\ns : Set E\nφ : E → ℝ\ninst✝⁸ : RCLike 𝕜\ninst✝⁷ : TopologicalSpace E\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : Module ℝ E\ninst✝⁴ : Module 𝕜 E\ninst✝³ : IsScalarTower ℝ 𝕜 E\ninst✝² : IsTopologicalAddGroup E\ninst✝¹ : ContinuousSMul 𝕜 E\ninst✝ : LocallyConvexSpace ℝ E\nx : E... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.Approximation | {
"line": 96,
"column": 32
} | {
"line": 96,
"column": 48
} | {
"line": 96,
"column": 49
} | [
{
"pp": "s : Set ℝ\nf : ℝ → ℝ\nx a : ℝ\nhx : x ∈ s\nhax : a < f x\nhsc : IsClosed s\nhfc : LowerSemicontinuousOn f s\nhf : ConvexOn ℝ s f\nl : ℝ →L[ℝ] ℝ\nc' : ℝ\nhlc'_le : s.domRestrict (⇑re ∘ ⇑l) + const (↑s) c' ≤ s.domRestrict f\nhlc'_eq : re (l x) + c' = a\nh1 : ∀ (y : ℝ), l 1 * y = l y\ny : ℝ\nhy : y ∈ s\n⊢... | [
"s : Set ℝ\nf : ℝ → ℝ\nx a : ℝ\nhx : x ∈ s\nhax : a < f x\nhsc : IsClosed s\nhfc : LowerSemicontinuousOn f s\nhf : ConvexOn ℝ s f\nl : ℝ →L[ℝ] ℝ\nc' : ℝ\nhlc'_le : s.domRestrict (⇑re ∘ ⇑l) + const (↑s) c' ≤ s.domRestrict f\nhlc'_eq : re (l x) + c' = a\nh1 : ∀ (y : ℝ), l 1 * y = l y\ny : ℝ\nhy : y ∈ s\n⊢ l y + c' ≤ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.Approximation | {
"line": 96,
"column": 69
} | {
"line": 96,
"column": 85
} | {
"line": 96,
"column": 86
} | [
{
"pp": "s : Set ℝ\nf : ℝ → ℝ\nx a : ℝ\nhx : x ∈ s\nhax : a < f x\nhsc : IsClosed s\nhfc : LowerSemicontinuousOn f s\nhf : ConvexOn ℝ s f\nl : ℝ →L[ℝ] ℝ\nc' : ℝ\nhlc'_le : s.domRestrict (⇑re ∘ ⇑l) + const (↑s) c' ≤ s.domRestrict f\nhlc'_eq : re (l x) + c' = a\nh1 : ∀ (y : ℝ), l 1 * y = l y\n⊢ l 1 * x + c' = a",... | [
"s : Set ℝ\nf : ℝ → ℝ\nx a : ℝ\nhx : x ∈ s\nhax : a < f x\nhsc : IsClosed s\nhfc : LowerSemicontinuousOn f s\nhf : ConvexOn ℝ s f\nl : ℝ →L[ℝ] ℝ\nc' : ℝ\nhlc'_le : s.domRestrict (⇑re ∘ ⇑l) + const (↑s) c' ≤ s.domRestrict f\nhlc'_eq : re (l x) + c' = a\nh1 : ∀ (y : ℝ), l 1 * y = l y\n⊢ l x + c' = a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.Approximation | {
"line": 114,
"column": 2
} | {
"line": 114,
"column": 51
} | {
"line": 115,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ns : Set E\nφ : E → ℝ\ninst✝⁸ : RCLike 𝕜\ninst✝⁷ : TopologicalSpace E\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : Module ℝ E\ninst✝⁴ : Module 𝕜 E\ninst✝³ : IsScalarTower ℝ 𝕜 E\ninst✝² : IsTopologicalAddGroup E\ninst✝¹ : ContinuousSMul 𝕜 E\ninst✝ : LocallyConvexSpace ℝ E\nhsc : Is... | [
"𝕜 : Type u_1\nE : Type u_2\ns : Set E\nφ : E → ℝ\ninst✝⁸ : RCLike 𝕜\ninst✝⁷ : TopologicalSpace E\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : Module ℝ E\ninst✝⁴ : Module 𝕜 E\ninst✝³ : IsScalarTower ℝ 𝕜 E\ninst✝² : IsTopologicalAddGroup E\ninst✝¹ : ContinuousSMul 𝕜 E\ninst✝ : LocallyConvexSpace ℝ E\nhsc : IsClosed s\nhφ... | let A := { p : E × 𝕜 | p.1 ∈ s ∧ φ p.1 ≤ re p.2 } | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.Analysis.Convex.Between | {
"line": 387,
"column": 4
} | {
"line": 387,
"column": 37
} | {
"line": 387,
"column": 38
} | [
{
"pp": "case refine_1\nR : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝⁵ : Ring R\ninst✝⁴ : PartialOrder R\ninst✝³ : AddCommGroup V\ninst✝² : Module R V\ninst✝¹ : AddTorsor V P\ninst✝ : IsOrderedRing R\nx y : P\nh : Wbtw R x y x\n⊢ y = x",
"ppTerm": "?refine_1",
"assigned": false,
"usedConstants": [... | [
"case refine_1\nR : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝⁵ : Ring R\ninst✝⁴ : PartialOrder R\ninst✝³ : AddCommGroup V\ninst✝² : Module R V\ninst✝¹ : AddTorsor V P\ninst✝ : IsOrderedRing R\nx y : P\nh : Wbtw R x y x\n⊢ y = x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Hall.Basic | {
"line": 69,
"column": 2
} | {
"line": 69,
"column": 37
} | {
"line": 69,
"column": 38
} | [
{
"pp": "ι : Type u\nα : Type v\nt : ι → Finset α\nι' ι'' : Finset ι\nh : ι' ⊆ ι''\nf : ↑(hallMatchingsOn t ι'')\nhinj : Injective ↑f\nhc : ∀ (x : ↥ι''), ↑f x ∈ t ↑x\ni : ι\nhi : i ∈ ι'\nj : ι\nhj : j ∈ ι'\nhh : (fun i ↦ ↑f ⟨↑i, ⋯⟩) ⟨i, hi⟩ = (fun i ↦ ↑f ⟨↑i, ⋯⟩) ⟨j, hj⟩\n⊢ ⟨i, hi⟩ = ⟨j, hj⟩",
"ppTerm": "?m... | [
"ι : Type u\nα : Type v\nt : ι → Finset α\nι' ι'' : Finset ι\nh : ι' ⊆ ι''\nf : ↑(hallMatchingsOn t ι'')\nhinj : Injective ↑f\nhc : ∀ (x : ↥ι''), ↑f x ∈ t ↑x\ni : ι\nhi : i ∈ ι'\nj : ι\nhj : j ∈ ι'\nhh : (fun i ↦ ↑f ⟨↑i, ⋯⟩) ⟨i, hi⟩ = (fun i ↦ ↑f ⟨↑i, ⋯⟩) ⟨j, hj⟩\n⊢ i = j"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Hall.Basic | {
"line": 104,
"column": 4
} | {
"line": 104,
"column": 19
} | {
"line": 104,
"column": 20
} | [
{
"pp": "ι : Type u\nα : Type v\nt : ι → Finset α\nι' : Finset ι\ng : ↑(hallMatchingsOn t ι') → ↥ι' → ↥(ι'.biUnion t) := fun f i ↦ ⟨↑f i, ⋯⟩\nf f' : ↑(hallMatchingsOn t ι')\nh : ∀ (x : ↥ι'), g f x = g f' x\na : ↥ι'\n⊢ ↑f a = ↑f' a",
"ppTerm": "?m.79",
"assigned": false,
"usedConstants": [],
"use... | [
"ι : Type u\nα : Type v\nt : ι → Finset α\nι' : Finset ι\ng : ↑(hallMatchingsOn t ι') → ↥ι' → ↥(ι'.biUnion t) := fun f i ↦ ⟨↑f i, ⋯⟩\nf f' : ↑(hallMatchingsOn t ι')\nh : ∀ (x : ↥ι'), g f x = g f' x\na : ↥ι'\n⊢ ↑f a = ↑f' a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.Approximation | {
"line": 228,
"column": 2
} | {
"line": 228,
"column": 13
} | {
"line": 228,
"column": 14
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nφ : E → ℝ\ninst✝⁹ : RCLike 𝕜\ninst✝⁸ : TopologicalSpace E\ninst✝⁷ : AddCommGroup E\ninst✝⁶ : Module ℝ E\ninst✝⁵ : Module 𝕜 E\ninst✝⁴ : IsScalarTower ℝ 𝕜 E\ninst✝³ : IsTopologicalAddGroup E\ninst✝² : ContinuousSMul 𝕜 E\ninst✝¹ : LocallyConvexSpace ℝ E\ninst✝ : Hereditari... | [
"𝕜 : Type u_1\nE : Type u_2\nφ : E → ℝ\ninst✝⁹ : RCLike 𝕜\ninst✝⁸ : TopologicalSpace E\ninst✝⁷ : AddCommGroup E\ninst✝⁶ : Module ℝ E\ninst✝⁵ : Module 𝕜 E\ninst✝⁴ : IsScalarTower ℝ 𝕜 E\ninst✝³ : IsTopologicalAddGroup E\ninst✝² : ContinuousSMul 𝕜 E\ninst✝¹ : LocallyConvexSpace ℝ E\ninst✝ : HereditarilyLindelofSp... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.CofilteredSystem | {
"line": 282,
"column": 2
} | {
"line": 282,
"column": 13
} | {
"line": 282,
"column": 14
} | [
{
"pp": "J : Type u\ninst✝¹ : Category.{v_1, u} J\nF : J ⥤ Type v\ninst✝ : IsCofilteredOrEmpty J\nFsur : ∀ ⦃i j : J⦄ (f : i ⟶ j), Function.Surjective ⇑(ConcreteCategory.hom (F.map f))\ni j : J\nf g : i ⟶ j\nk : J\nφ : k ⟶ i\nhφ : φ ≫ f = φ ≫ g\nthis :\n (fun x ↦ (ConcreteCategory.hom (F.map f)) ((ConcreteCateg... | [
"J : Type u\ninst✝¹ : Category.{v_1, u} J\nF : J ⥤ Type v\ninst✝ : IsCofilteredOrEmpty J\nFsur : ∀ ⦃i j : J⦄ (f : i ⟶ j), Function.Surjective ⇑(ConcreteCategory.hom (F.map f))\ni j : J\nf g : i ⟶ j\nk : J\nφ : k ⟶ i\nhφ : φ ≫ f = φ ≫ g\nthis :\n (fun x ↦ (ConcreteCategory.hom (F.map f)) ((ConcreteCategory.hom (F.m... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.Permutation | {
"line": 117,
"column": 4
} | {
"line": 117,
"column": 95
} | {
"line": 118,
"column": 6
} | [
{
"pp": "n : Type u_1\ninst✝³ : DecidableEq n\nσ : Perm n\ninst✝² : Fintype n\n𝕜 : Type u_3\ninst✝¹ : RCLike 𝕜\ninst✝ : Nonempty n\ninhabited_h : Inhabited n\n⊢ 1 ≤ ‖Perm.permMatrix 𝕜 σ‖",
"ppTerm": "?m.21",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"n : Type u_1\ninst✝³ : DecidableEq n\nσ : Perm n\ninst✝² : Fintype n\n𝕜 : Type u_3\ninst✝¹ : RCLike 𝕜\ninst✝ : Nonempty n\ninhabited_h : Inhabited n\n⊢ 1 ≤ ‖Perm.permMatrix 𝕜 σ‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.Between | {
"line": 742,
"column": 2
} | {
"line": 742,
"column": 13
} | {
"line": 742,
"column": 14
} | [
{
"pp": "R : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝⁵ : CommRing R\ninst✝⁴ : PartialOrder R\ninst✝³ : IsStrictOrderedRing R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\ninst✝ : AddTorsor V P\nx y z : P\nh : Wbtw R x y z\n⊢ SameRay R (y -ᵥ x) (z -ᵥ y)",
"ppTerm": "?m.25",
"assigned": false,
"us... | [
"R : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝⁵ : CommRing R\ninst✝⁴ : PartialOrder R\ninst✝³ : IsStrictOrderedRing R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\ninst✝ : AddTorsor V P\nx y z : P\nh : Wbtw R x y z\n⊢ SameRay R (y -ᵥ x) (z -ᵥ y)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.Stochastic | {
"line": 51,
"column": 4
} | {
"line": 51,
"column": 36
} | {
"line": 52,
"column": 2
} | [
{
"pp": "R✝ : Type u_1\nn✝ : Type u_2\ninst✝⁹ : Fintype n✝\ninst✝⁸ : DecidableEq n✝\ninst✝⁷ : Semiring R✝\ninst✝⁶ : PartialOrder R✝\ninst✝⁵ : IsOrderedRing R✝\nM✝ : Matrix n✝ n✝ R✝\nx : n✝ → R✝\nR : Type u_3\nn : Type u_4\ninst✝⁴ : Fintype n\ninst✝³ : DecidableEq n\ninst✝² : Semiring R\ninst✝¹ : PartialOrder R\... | [] | rw [← mulVec_mulVec, hN.2, hM.2] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.Matrix.Stochastic | {
"line": 213,
"column": 6
} | {
"line": 213,
"column": 31
} | {
"line": 213,
"column": 31
} | [
{
"pp": "R : Type u_1\nn : Type u_2\ninst✝⁴ : Fintype n\ninst✝³ : DecidableEq n\ninst✝² : Semiring R\ninst✝¹ : PartialOrder R\ninst✝ : IsOrderedRing R\nσ : Equiv.Perm n\n⊢ Equiv.Perm.permMatrix R σ ∈ colStochastic R n",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Matrix.colStochas... | [
"R : Type u_1\nn : Type u_2\ninst✝⁴ : Fintype n\ninst✝³ : DecidableEq n\ninst✝² : Semiring R\ninst✝¹ : PartialOrder R\ninst✝ : IsOrderedRing R\nσ : Equiv.Perm n\n⊢ (∀ (i j : n), 0 ≤ Equiv.Perm.permMatrix R σ i j) ∧ ∀ (j : n), ∑ i, Equiv.Perm.permMatrix R σ i j = 1"
] | mem_colStochastic_iff_sum | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Matrix.Stochastic | {
"line": 238,
"column": 16
} | {
"line": 238,
"column": 27
} | {
"line": 238,
"column": 28
} | [
{
"pp": "R : Type u_1\nn : Type u_2\ninst✝⁶ : Fintype n\ninst✝⁵ : DecidableEq n\ninst✝⁴ : Semiring R\ninst✝³ : PartialOrder R\ninst✝² : IsOrderedRing R\nm : Type u_3\ninst✝¹ : Fintype m\ninst✝ : DecidableEq m\nM : Matrix n n R\ne₁ e₂ : n ≃ m\nhM : M ∈ rowStochastic R n\nx✝¹ x✝ : m\n⊢ 0 ≤ (reindex e₁ e₂) M x✝¹ x... | [
"R : Type u_1\nn : Type u_2\ninst✝⁶ : Fintype n\ninst✝⁵ : DecidableEq n\ninst✝⁴ : Semiring R\ninst✝³ : PartialOrder R\ninst✝² : IsOrderedRing R\nm : Type u_3\ninst✝¹ : Fintype m\ninst✝ : DecidableEq m\nM : Matrix n n R\ne₁ e₂ : n ≃ m\nhM : M ∈ rowStochastic R n\nx✝¹ x✝ : m\n⊢ 0 ≤ M (e₁.symm x✝¹) (e₂.symm x✝)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.DoublyStochasticMatrix | {
"line": 102,
"column": 88
} | {
"line": 105,
"column": 78
} | {
"line": 107,
"column": 0
} | [
{
"pp": "R : Type u_1\nn : Type u_2\ninst✝⁴ : Fintype n\ninst✝³ : DecidableEq n\ninst✝² : Semiring R\ninst✝¹ : PartialOrder R\ninst✝ : IsOrderedRing R\n⊢ Convex R ↑(doublyStochastic R n)",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"_private.Mathlib.Analysis.Convex... | [] | by
intro x hx y hy a b ha hb h
simp only [SetLike.mem_coe, mem_doublyStochastic_iff_sum] at hx hy ⊢
simp [add_nonneg, ha, hb, mul_nonneg, hx, hy, sum_add_distrib, ← mul_sum, h] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Convex.Caratheodory | {
"line": 70,
"column": 4
} | {
"line": 70,
"column": 15
} | {
"line": 71,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u\ninst✝⁵ : Field 𝕜\ninst✝⁴ : LinearOrder 𝕜\ninst✝³ : IsStrictOrderedRing 𝕜\ninst✝² : AddCommGroup E\ninst✝¹ : Module 𝕜 E\ninst✝ : DecidableEq E\nt : Finset E\nf : E → 𝕜\nfpos : ∀ y ∈ t, 0 ≤ f y\nfsum : ∑ y ∈ t, f y = 1\ng : E → 𝕜\ngcombo : ∑ e ∈ t, g e • e = 0\ngsum : ∑ e... | [] | exact mem.2 | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Convex.Birkhoff | {
"line": 74,
"column": 2
} | {
"line": 74,
"column": 22
} | {
"line": 74,
"column": 23
} | [
{
"pp": "R : Type u_1\nn : Type u_2\ninst✝⁴ : Fintype n\ninst✝³ : DecidableEq n\ninst✝² : Semifield R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\nM : Matrix n n R\ns : R\nhs : 0 < s\nhM : (∀ (i j : n), 0 ≤ M i j) ∧ (∀ (i : n), ∑ j, M i j = s) ∧ ∀ (j : n), ∑ i, M i j = s\nf : n → Finset n := fun i ↦ ... | [
"R : Type u_1\nn : Type u_2\ninst✝⁴ : Fintype n\ninst✝³ : DecidableEq n\ninst✝² : Semifield R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\nM : Matrix n n R\ns : R\nhs : 0 < s\nhM : (∀ (i j : n), 0 ≤ M i j) ∧ (∀ (i : n), ∑ j, M i j = s) ∧ ∀ (j : n), ∑ i, M i j = s\nf : n → Finset n := fun i ↦ {j | M i j ≠... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.Birkhoff | {
"line": 110,
"column": 4
} | {
"line": 110,
"column": 38
} | {
"line": 110,
"column": 39
} | [
{
"pp": "R : Type u_1\nn : Type u_2\ninst✝⁴ : Fintype n\ninst✝³ : DecidableEq n\ninst✝² : Field R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\nh✝ : Nonempty n\nd : ℕ\nih :\n ∀ m < d,\n ∀ (M : Matrix n n R) (s : R),\n 0 ≤ s →\n (∃ M' ∈ doublyStochastic R n, M = s • M') →\n #{i... | [
"R : Type u_1\nn : Type u_2\ninst✝⁴ : Fintype n\ninst✝³ : DecidableEq n\ninst✝² : Field R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\nh✝ : Nonempty n\nd : ℕ\nih :\n ∀ m < d,\n ∀ (M : Matrix n n R) (s : R),\n 0 ≤ s →\n (∃ M' ∈ doublyStochastic R n, M = s • M') →\n #{i | M i.1 i.2... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.Between | {
"line": 820,
"column": 2
} | {
"line": 820,
"column": 20
} | {
"line": 821,
"column": 2
} | [
{
"pp": "R : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝⁶ : Ring R\ninst✝⁵ : LinearOrder R\ninst✝⁴ : IsStrictOrderedRing R\ninst✝³ : AddCommGroup V\ninst✝² : Module R V\ninst✝¹ : AddTorsor V P\ninst✝ : IsTorsionFree R V\nt : Affine.Triangle R P\ni₁ i₂ i₃ : Fin 3\nh₁₂ : i₁ ≠ i₂\nh₁₃ : i₁ ≠ i₃\nh₂₃ : i₂ ≠ i₃\nh3 ... | [
"R : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝⁶ : Ring R\ninst✝⁵ : LinearOrder R\ninst✝⁴ : IsStrictOrderedRing R\ninst✝³ : AddCommGroup V\ninst✝² : Module R V\ninst✝¹ : AddTorsor V P\ninst✝ : IsTorsionFree R V\nt : Affine.Triangle R P\ni₁ i₂ i₃ : Fin 3\nh₁₂ : i₁ ≠ i₂\nh₁₃ : i₁ ≠ i₃\nh₂₃ : i₂ ≠ i₃\nh3 : ∀ (i : Fin... | refine ⟨hs i₁, ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.Convex.Caratheodory | {
"line": 93,
"column": 12
} | {
"line": 93,
"column": 67
} | {
"line": 93,
"column": 68
} | [
{
"pp": "case hb\n𝕜 : Type u_1\nE : Type u\ninst✝⁵ : Field 𝕜\ninst✝⁴ : LinearOrder 𝕜\ninst✝³ : IsStrictOrderedRing 𝕜\ninst✝² : AddCommGroup E\ninst✝¹ : Module 𝕜 E\ninst✝ : DecidableEq E\nt : Finset E\nf : E → 𝕜\nfpos : ∀ y ∈ t, 0 ≤ f y\nfsum : ∑ y ∈ t, f y = 1\ng : E → 𝕜\ngcombo : ∑ e ∈ t, g e • e = 0\ng... | [
"case hb\n𝕜 : Type u_1\nE : Type u\ninst✝⁵ : Field 𝕜\ninst✝⁴ : LinearOrder 𝕜\ninst✝³ : IsStrictOrderedRing 𝕜\ninst✝² : AddCommGroup E\ninst✝¹ : Module 𝕜 E\ninst✝ : DecidableEq E\nt : Finset E\nf : E → 𝕜\nfpos : ∀ y ∈ t, 0 ≤ f y\nfsum : ∑ y ∈ t, f y = 1\ng : E → 𝕜\ngcombo : ∑ e ∈ t, g e • e = 0\ngsum : ∑ e ∈ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.Cone.Basic | {
"line": 192,
"column": 2
} | {
"line": 192,
"column": 45
} | {
"line": 192,
"column": 46
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_3\ninst✝⁹ : AddCommMonoid E\ninst✝⁸ : TopologicalSpace E\ninst✝⁷ : Semifield 𝕜\ninst✝⁶ : LinearOrder 𝕜\ninst✝⁵ : Module 𝕜 E\ninst✝⁴ : TopologicalSpace 𝕜\ninst✝³ : OrderTopology 𝕜\ninst✝² : DenselyOrdered 𝕜\ninst✝¹ : NoMaxOrder 𝕜\ninst✝ : ContinuousSMul 𝕜 E\nC : ConvexC... | [
"𝕜 : Type u_1\nE : Type u_3\ninst✝⁹ : AddCommMonoid E\ninst✝⁸ : TopologicalSpace E\ninst✝⁷ : Semifield 𝕜\ninst✝⁶ : LinearOrder 𝕜\ninst✝⁵ : Module 𝕜 E\ninst✝⁴ : TopologicalSpace 𝕜\ninst✝³ : OrderTopology 𝕜\ninst✝² : DenselyOrdered 𝕜\ninst✝¹ : NoMaxOrder 𝕜\ninst✝ : ContinuousSMul 𝕜 E\nC : ConvexCone 𝕜 E\nhS... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.Between | {
"line": 826,
"column": 4
} | {
"line": 826,
"column": 15
} | {
"line": 826,
"column": 16
} | [
{
"pp": "R : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝⁶ : Ring R\ninst✝⁵ : LinearOrder R\ninst✝⁴ : IsStrictOrderedRing R\ninst✝³ : AddCommGroup V\ninst✝² : Module R V\ninst✝¹ : AddTorsor V P\ninst✝ : IsTorsionFree R V\nt : Affine.Triangle R P\ni₁ i₂ i₃ : Fin 3\nh₁₂ : i₁ ≠ i₂\nh₁₃ : i₁ ≠ i₃\nh₂₃ : i₂ ≠ i₃\nh3 ... | [
"R : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝⁶ : Ring R\ninst✝⁵ : LinearOrder R\ninst✝⁴ : IsStrictOrderedRing R\ninst✝³ : AddCommGroup V\ninst✝² : Module R V\ninst✝¹ : AddTorsor V P\ninst✝ : IsTorsionFree R V\nt : Affine.Triangle R P\ni₁ i₂ i₃ : Fin 3\nh₁₂ : i₁ ≠ i₂\nh₁₃ : i₁ ≠ i₃\nh₂₃ : i₂ ≠ i₃\nh3 : ∀ (i : Fin... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.Between | {
"line": 827,
"column": 4
} | {
"line": 827,
"column": 24
} | {
"line": 827,
"column": 25
} | [
{
"pp": "R : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝⁶ : Ring R\ninst✝⁵ : LinearOrder R\ninst✝⁴ : IsStrictOrderedRing R\ninst✝³ : AddCommGroup V\ninst✝² : Module R V\ninst✝¹ : AddTorsor V P\ninst✝ : IsTorsionFree R V\nt : Affine.Triangle R P\ni₁ i₂ i₃ : Fin 3\nh₁₂ : i₁ ≠ i₂\nh₁₃ : i₁ ≠ i₃\nh₂₃ : i₂ ≠ i₃\nh3 ... | [
"R : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝⁶ : Ring R\ninst✝⁵ : LinearOrder R\ninst✝⁴ : IsStrictOrderedRing R\ninst✝³ : AddCommGroup V\ninst✝² : Module R V\ninst✝¹ : AddTorsor V P\ninst✝ : IsTorsionFree R V\nt : Affine.Triangle R P\ni₁ i₂ i₃ : Fin 3\nh₁₂ : i₁ ≠ i₂\nh₁₃ : i₁ ≠ i₃\nh₂₃ : i₂ ≠ i₃\nh3 : ∀ (i : Fin... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.Birkhoff | {
"line": 132,
"column": 8
} | {
"line": 132,
"column": 42
} | {
"line": 132,
"column": 43
} | [
{
"pp": "R : Type u_1\nn : Type u_2\ninst✝⁴ : Fintype n\ninst✝³ : DecidableEq n\ninst✝² : Field R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\nh✝ : Nonempty n\nd : ℕ\nih :\n ∀ m < d,\n ∀ (M : Matrix n n R) (s : R),\n 0 ≤ s →\n (∃ M' ∈ doublyStochastic R n, M = s • M') →\n #{i... | [
"R : Type u_1\nn : Type u_2\ninst✝⁴ : Fintype n\ninst✝³ : DecidableEq n\ninst✝² : Field R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\nh✝ : Nonempty n\nd : ℕ\nih :\n ∀ m < d,\n ∀ (M : Matrix n n R) (s : R),\n 0 ≤ s →\n (∃ M' ∈ doublyStochastic R n, M = s • M') →\n #{i | M i.1 i.2... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.Between | {
"line": 943,
"column": 2
} | {
"line": 943,
"column": 18
} | {
"line": 943,
"column": 19
} | [
{
"pp": "R : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝⁴ : Field R\ninst✝³ : LinearOrder R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\ninst✝ : AddTorsor V P\nx z : P\ns : AffineSubspace R P\nhx : x ∈ s\nε : R\nhy : (lineMap x z) ε ∈ s\nhxy : x ≠ (lineMap x z) ε\nhε : ε ≠ 0\n⊢ z ∈ s",
"ppTerm": "?m.58",
... | [
"R : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝⁴ : Field R\ninst✝³ : LinearOrder R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\ninst✝ : AddTorsor V P\nx z : P\ns : AffineSubspace R P\nhx : x ∈ s\nε : R\nhy : (lineMap x z) ε ∈ s\nhxy : x ≠ (lineMap x z) ε\nhε : ε ≠ 0\n⊢ z ∈ s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.Between | {
"line": 954,
"column": 4
} | {
"line": 954,
"column": 69
} | {
"line": 955,
"column": 2
} | [
{
"pp": "case inl\nR : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝⁵ : Field R\ninst✝⁴ : LinearOrder R\ninst✝³ : IsStrictOrderedRing R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\ninst✝ : AddTorsor V P\nx : P\nv : V\nr₁ r₂ : R\nhr₁ : 0 ≤ r₁\nhr₂ : 0 ≤ r₂\nh : r₁ ≤ r₂\n⊢ Wbtw R x (r₁ • v +ᵥ x) (r₂ • v +ᵥ x) ∨ W... | [] | exact Or.inl (wbtw_smul_vadd_smul_vadd_of_nonneg_of_le x v hr₁ h) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Convex.Between | {
"line": 954,
"column": 4
} | {
"line": 954,
"column": 69
} | {
"line": 955,
"column": 2
} | [
{
"pp": "case inl\nR : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝⁵ : Field R\ninst✝⁴ : LinearOrder R\ninst✝³ : IsStrictOrderedRing R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\ninst✝ : AddTorsor V P\nx : P\nv : V\nr₁ r₂ : R\nhr₁ : 0 ≤ r₁\nhr₂ : 0 ≤ r₂\nh : r₁ ≤ r₂\n⊢ Wbtw R x (r₁ • v +ᵥ x) (r₂ • v +ᵥ x) ∨ W... | [] | exact Or.inl (wbtw_smul_vadd_smul_vadd_of_nonneg_of_le x v hr₁ h) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Convex.Between | {
"line": 954,
"column": 4
} | {
"line": 954,
"column": 69
} | {
"line": 955,
"column": 2
} | [
{
"pp": "case inl\nR : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝⁵ : Field R\ninst✝⁴ : LinearOrder R\ninst✝³ : IsStrictOrderedRing R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\ninst✝ : AddTorsor V P\nx : P\nv : V\nr₁ r₂ : R\nhr₁ : 0 ≤ r₁\nhr₂ : 0 ≤ r₂\nh : r₁ ≤ r₂\n⊢ Wbtw R x (r₁ • v +ᵥ x) (r₂ • v +ᵥ x) ∨ W... | [] | exact Or.inl (wbtw_smul_vadd_smul_vadd_of_nonneg_of_le x v hr₁ h) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Convex.Birkhoff | {
"line": 158,
"column": 2
} | {
"line": 158,
"column": 55
} | {
"line": 158,
"column": 56
} | [
{
"pp": "case inr\nR : Type u_1\nn : Type u_2\ninst✝⁴ : Fintype n\ninst✝³ : DecidableEq n\ninst✝² : Field R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\nM : Matrix n n R\nhM : M ∈ doublyStochastic R n\nh✝ : Nonempty n\nw : Equiv.Perm n → R\nhw1 : ∀ (σ : Equiv.Perm n), 0 ≤ w σ\nhw3 : ∑ σ, w σ • Equiv.... | [
"case inr\nR : Type u_1\nn : Type u_2\ninst✝⁴ : Fintype n\ninst✝³ : DecidableEq n\ninst✝² : Field R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\nM : Matrix n n R\nhM : M ∈ doublyStochastic R n\nh✝ : Nonempty n\nw : Equiv.Perm n → R\nhw1 : ∀ (σ : Equiv.Perm n), 0 ≤ w σ\nhw3 : ∑ σ, w σ • Equiv.Perm.permMat... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.Between | {
"line": 1056,
"column": 2
} | {
"line": 1056,
"column": 20
} | {
"line": 1056,
"column": 21
} | [
{
"pp": "R : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝⁴ : Field R\ninst✝³ : LinearOrder R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\ninst✝ : AddTorsor V P\nx y z : P\nh : Wbtw R x y z\nthis : {y, x, z} = {x, y, z}\n⊢ Collinear R {x, y, z}",
"ppTerm": "?m.56",
"assigned": false,
"usedConstants"... | [
"R : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝⁴ : Field R\ninst✝³ : LinearOrder R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\ninst✝ : AddTorsor V P\nx y z : P\nh : Wbtw R x y z\nthis : {y, x, z} = {x, y, z}\n⊢ Collinear R {x, y, z}"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.Cone.InnerDual | {
"line": 67,
"column": 29
} | {
"line": 67,
"column": 40
} | {
"line": 67,
"column": 41
} | [
{
"pp": "E : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace ℝ E\ninst✝ : CompleteSpace E\nx : E\nhx : x ∈ innerDual univ\n⊢ x ∈ ⊥",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Real.instIsOrderedRing",
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
... | [
"E : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace ℝ E\ninst✝ : CompleteSpace E\nx : E\nhx : x ∈ innerDual univ\n⊢ x = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.Cone.InnerDual | {
"line": 129,
"column": 42
} | {
"line": 129,
"column": 71
} | {
"line": 129,
"column": 72
} | [
{
"pp": "E : Type u_2\nF : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace ℝ E\ninst✝³ : CompleteSpace E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : InnerProductSpace ℝ F\ninst✝ : CompleteSpace F\nC : ProperCone ℝ E\nf : E →L[ℝ] F\nb : F\nseq : ℕ → E\nhmem : ∀ (x : ℕ), seq x ∈ C\ny : F\nhinner ... | [
"E : Type u_2\nF : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace ℝ E\ninst✝³ : CompleteSpace E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : InnerProductSpace ℝ F\ninst✝ : CompleteSpace F\nC : ProperCone ℝ E\nf : E →L[ℝ] F\nb : F\nseq : ℕ → E\nhmem : ∀ (x : ℕ), seq x ∈ C\ny : F\nhinner : ∀ ⦃x : E⦄,... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.Cone.InnerDual | {
"line": 138,
"column": 4
} | {
"line": 139,
"column": 11
} | {
"line": 139,
"column": 12
} | [
{
"pp": "E : Type u_2\nF : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace ℝ E\ninst✝³ : CompleteSpace E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : InnerProductSpace ℝ F\ninst✝ : CompleteSpace F\nC : ProperCone ℝ E\nf : E →L[ℝ] F\nb : F\nh : b ∉ map f C\ny : F\nhxy : ∀ x ∈ map f C, 0 ≤ ⟪x, y⟫_... | [
"E : Type u_2\nF : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace ℝ E\ninst✝³ : CompleteSpace E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : InnerProductSpace ℝ F\ninst✝ : CompleteSpace F\nC : ProperCone ℝ E\nf : E →L[ℝ] F\nb : F\nh : b ∉ map f C\ny : F\nhxy : ∀ x ∈ map f C, 0 ≤ ⟪x, y⟫_ℝ\nhyb : ⟪b,... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.Cone.Dual | {
"line": 83,
"column": 43
} | {
"line": 83,
"column": 54
} | {
"line": 83,
"column": 55
} | [
{
"pp": "R : Type u_1\nM : Type u_2\nN : Type u_3\ninst✝¹³ : CommRing R\ninst✝¹² : PartialOrder R\ninst✝¹¹ : IsOrderedRing R\ninst✝¹⁰ : TopologicalSpace R\ninst✝⁹ : ClosedIciTopology R\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : Module R M\ninst✝⁶ : TopologicalSpace M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\ninst✝... | [
"R : Type u_1\nM : Type u_2\nN : Type u_3\ninst✝¹³ : CommRing R\ninst✝¹² : PartialOrder R\ninst✝¹¹ : IsOrderedRing R\ninst✝¹⁰ : TopologicalSpace R\ninst✝⁹ : ClosedIciTopology R\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : Module R M\ninst✝⁶ : TopologicalSpace M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\ninst✝³ : Topologi... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.Cone.Dual | {
"line": 124,
"column": 25
} | {
"line": 124,
"column": 36
} | {
"line": 124,
"column": 37
} | [
{
"pp": "E : Type u_1\ninst✝⁵ : TopologicalSpace E\ninst✝⁴ : AddCommGroup E\ninst✝³ : IsTopologicalAddGroup E\ninst✝² : Module ℝ E\ninst✝¹ : ContinuousSMul ℝ E\ninst✝ : LocallyConvexSpace ℝ E\nK : Set E\nC : ProperCone ℝ E\nhKconv : Convex ℝ K\nhKcomp : IsCompact K\nhKC : Disjoint K ↑C\nx₀ : E\nhx₀ : x₀ ∈ K\nf ... | [
"E : Type u_1\ninst✝⁵ : TopologicalSpace E\ninst✝⁴ : AddCommGroup E\ninst✝³ : IsTopologicalAddGroup E\ninst✝² : Module ℝ E\ninst✝¹ : ContinuousSMul ℝ E\ninst✝ : LocallyConvexSpace ℝ E\nK : Set E\nC : ProperCone ℝ E\nhKconv : Convex ℝ K\nhKcomp : IsCompact K\nhKC : Disjoint K ↑C\nx₀ : E\nhx₀ : x₀ ∈ K\nf : StrongDual... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.Cone.Dual | {
"line": 127,
"column": 2
} | {
"line": 127,
"column": 22
} | {
"line": 127,
"column": 23
} | [
{
"pp": "case inr\nE : Type u_1\ninst✝⁵ : TopologicalSpace E\ninst✝⁴ : AddCommGroup E\ninst✝³ : IsTopologicalAddGroup E\ninst✝² : Module ℝ E\ninst✝¹ : ContinuousSMul ℝ E\ninst✝ : LocallyConvexSpace ℝ E\nK : Set E\nC : ProperCone ℝ E\nhKconv : Convex ℝ K\nhKcomp : IsCompact K\nhKC : Disjoint K ↑C\nx₀ : E\nhx₀✝ :... | [
"case inr\nE : Type u_1\ninst✝⁵ : TopologicalSpace E\ninst✝⁴ : AddCommGroup E\ninst✝³ : IsTopologicalAddGroup E\ninst✝² : Module ℝ E\ninst✝¹ : ContinuousSMul ℝ E\ninst✝ : LocallyConvexSpace ℝ E\nK : Set E\nC : ProperCone ℝ E\nhKconv : Convex ℝ K\nhKcomp : IsCompact K\nhKC : Disjoint K ↑C\nx₀ : E\nhx₀✝ : x₀ ∈ K\nf :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.Cone.Dual | {
"line": 134,
"column": 2
} | {
"line": 134,
"column": 17
} | {
"line": 134,
"column": 18
} | [
{
"pp": "E : Type u_1\ninst✝⁵ : TopologicalSpace E\ninst✝⁴ : AddCommGroup E\ninst✝³ : IsTopologicalAddGroup E\ninst✝² : Module ℝ E\ninst✝¹ : ContinuousSMul ℝ E\ninst✝ : LocallyConvexSpace ℝ E\nx₀ : E\nC : ProperCone ℝ E\nhx₀ : x₀ ∉ C\n⊢ ∃ f, (∀ x ∈ C, 0 ≤ f x) ∧ f x₀ < 0",
"ppTerm": "?m.44",
"assigned":... | [
"E : Type u_1\ninst✝⁵ : TopologicalSpace E\ninst✝⁴ : AddCommGroup E\ninst✝³ : IsTopologicalAddGroup E\ninst✝² : Module ℝ E\ninst✝¹ : ContinuousSMul ℝ E\ninst✝ : LocallyConvexSpace ℝ E\nx₀ : E\nC : ProperCone ℝ E\nhx₀ : x₀ ∉ C\n⊢ ∃ f, (∀ x ∈ C, 0 ≤ f x) ∧ f x₀ < 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.Cone.Dual | {
"line": 142,
"column": 2
} | {
"line": 142,
"column": 55
} | {
"line": 142,
"column": 56
} | [
{
"pp": "E : Type u_1\nF : Type u_2\ninst✝⁹ : TopologicalSpace E\ninst✝⁸ : AddCommGroup E\ninst✝⁷ : IsTopologicalAddGroup E\ninst✝⁶ : TopologicalSpace F\ninst✝⁵ : AddCommGroup F\ninst✝⁴ : Module ℝ E\ninst✝³ : ContinuousSMul ℝ E\ninst✝² : LocallyConvexSpace ℝ E\ninst✝¹ : Module ℝ F\np : E →ₗ[ℝ] F →ₗ[ℝ] ℝ\ninst✝ ... | [
"E : Type u_1\nF : Type u_2\ninst✝⁹ : TopologicalSpace E\ninst✝⁸ : AddCommGroup E\ninst✝⁷ : IsTopologicalAddGroup E\ninst✝⁶ : TopologicalSpace F\ninst✝⁵ : AddCommGroup F\ninst✝⁴ : Module ℝ E\ninst✝³ : ContinuousSMul ℝ E\ninst✝² : LocallyConvexSpace ℝ E\ninst✝¹ : Module ℝ F\np : E →ₗ[ℝ] F →ₗ[ℝ] ℝ\ninst✝ : p.IsContPe... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Convex.Cone.TensorProduct | {
"line": 125,
"column": 4
} | {
"line": 125,
"column": 40
} | {
"line": 125,
"column": 41
} | [
{
"pp": "case refine_1\nR : Type u_1\ninst✝⁶ : CommRing R\ninst✝⁵ : LinearOrder R\ninst✝⁴ : IsStrictOrderedRing R\nG : Type u_2\ninst✝³ : AddCommGroup G\ninst✝² : Module R G\nH : Type u_3\ninst✝¹ : AddCommGroup H\ninst✝ : Module R H\nC₁ : PointedCone R G\nC₂ : PointedCone R H\nw : G ⊗[R] H\nhw : ∀ φ ∈ dual (Dua... | [
"case refine_1\nR : Type u_1\ninst✝⁶ : CommRing R\ninst✝⁵ : LinearOrder R\ninst✝⁴ : IsStrictOrderedRing R\nG : Type u_2\ninst✝³ : AddCommGroup G\ninst✝² : Module R G\nH : Type u_3\ninst✝¹ : AddCommGroup H\ninst✝ : Module R H\nC₁ : PointedCone R G\nC₂ : PointedCone R H\nw : G ⊗[R] H\nhw : ∀ φ ∈ dual (Dual.eval R G) ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Convex.Cone.TensorProduct | {
"line": 127,
"column": 4
} | {
"line": 127,
"column": 40
} | {
"line": 127,
"column": 41
} | [
{
"pp": "case refine_2\nR : Type u_1\ninst✝⁶ : CommRing R\ninst✝⁵ : LinearOrder R\ninst✝⁴ : IsStrictOrderedRing R\nG : Type u_2\ninst✝³ : AddCommGroup G\ninst✝² : Module R G\nH : Type u_3\ninst✝¹ : AddCommGroup H\ninst✝ : Module R H\nC₁ : PointedCone R G\nC₂ : PointedCone R H\nz : H ⊗[R] G\nhz : ∀ φ ∈ dual (Dua... | [
"case refine_2\nR : Type u_1\ninst✝⁶ : CommRing R\ninst✝⁵ : LinearOrder R\ninst✝⁴ : IsStrictOrderedRing R\nG : Type u_2\ninst✝³ : AddCommGroup G\ninst✝² : Module R G\nH : Type u_3\ninst✝¹ : AddCommGroup H\ninst✝ : Module R H\nC₁ : PointedCone R G\nC₂ : PointedCone R H\nz : H ⊗[R] G\nhz : ∀ φ ∈ dual (Dual.eval R H) ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Convex.Cone.Dual | {
"line": 148,
"column": 43
} | {
"line": 148,
"column": 54
} | {
"line": 148,
"column": 55
} | [
{
"pp": "R : Type u_1\ninst✝⁶ : CommRing R\ninst✝⁵ : PartialOrder R\ninst✝⁴ : IsOrderedRing R\nM : Type u_2\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nN : Type u_3\ninst✝¹ : AddCommMonoid N\ninst✝ : Module R N\np : M →ₗ[R] N →ₗ[R] R\nhp : Injective ⇑p.flip\ny : N\nhy : y ∈ dual p univ\nx : M\n⊢ (p x) y ≤ 0"... | [
"R : Type u_1\ninst✝⁶ : CommRing R\ninst✝⁵ : PartialOrder R\ninst✝⁴ : IsOrderedRing R\nM : Type u_2\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nN : Type u_3\ninst✝¹ : AddCommMonoid N\ninst✝ : Module R N\np : M →ₗ[R] N →ₗ[R] R\nhp : Injective ⇑p.flip\ny : N\nhy : y ∈ dual p univ\nx : M\n⊢ (p x) y ≤ 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Affine.Convex | {
"line": 55,
"column": 46
} | {
"line": 55,
"column": 57
} | {
"line": 55,
"column": 58
} | [
{
"pp": "E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : FiniteDimensional ℝ E\ns : Set E\nx : E\nhs : s ∈ 𝓝 x\nthis :\n ∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] [FiniteDimensional ℝ E] {s : Set E} {x : E},\n s ∈ 𝓝 x → x = 0 → ∃ b, x ∈ interi... | [
"E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : FiniteDimensional ℝ E\ns : Set E\nx : E\nhs : s ∈ 𝓝 x\nthis :\n ∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] [FiniteDimensional ℝ E] {s : Set E} {x : E},\n s ∈ 𝓝 x → x = 0 → ∃ b, x ∈ interior ((convexH... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Affine.Convex | {
"line": 57,
"column": 4
} | {
"line": 58,
"column": 83
} | {
"line": 58,
"column": 84
} | [
{
"pp": "case h\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : FiniteDimensional ℝ E\ns : Set E\nx : E\nhs : s ∈ 𝓝 x\nthis :\n ∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] [FiniteDimensional ℝ E] {s : Set E} {x : E},\n s ∈ 𝓝 x → x = 0 → ∃ b, x ... | [
"case h\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : FiniteDimensional ℝ E\ns : Set E\nx : E\nhs : s ∈ 𝓝 x\nthis :\n ∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] [FiniteDimensional ℝ E] {s : Set E} {x : E},\n s ∈ 𝓝 x → x = 0 → ∃ b, x ∈ interior (... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Affine.Convex | {
"line": 66,
"column": 4
} | {
"line": 67,
"column": 11
} | {
"line": 67,
"column": 12
} | [
{
"pp": "E✝ : Type u_1\ninst✝⁴ : NormedAddCommGroup E✝\ninst✝³ : NormedSpace ℝ E✝\ns✝ : Set E✝\nx : E✝\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : FiniteDimensional ℝ E\ns : Set E\nhs : s ∈ 𝓝 0\nb : AffineBasis (Fin (finrank ℝ E + 1)) ℝ E\nc : AffineBasis (Fin (finrank ℝ E +... | [
"E✝ : Type u_1\ninst✝⁴ : NormedAddCommGroup E✝\ninst✝³ : NormedSpace ℝ E✝\ns✝ : Set E✝\nx : E✝\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : FiniteDimensional ℝ E\ns : Set E\nhs : s ∈ 𝓝 0\nb : AffineBasis (Fin (finrank ℝ E + 1)) ℝ E\nc : AffineBasis (Fin (finrank ℝ E + 1)) ℝ E := ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Affine.Convex | {
"line": 70,
"column": 4
} | {
"line": 71,
"column": 70
} | {
"line": 71,
"column": 71
} | [
{
"pp": "E✝ : Type u_1\ninst✝⁴ : NormedAddCommGroup E✝\ninst✝³ : NormedSpace ℝ E✝\ns✝ : Set E✝\nx : E✝\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : FiniteDimensional ℝ E\ns : Set E\nhs : s ∈ 𝓝 0\nb : AffineBasis (Fin (finrank ℝ E + 1)) ℝ E\nc : AffineBasis (Fin (finrank ℝ E +... | [
"E✝ : Type u_1\ninst✝⁴ : NormedAddCommGroup E✝\ninst✝³ : NormedSpace ℝ E✝\ns✝ : Set E✝\nx : E✝\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : FiniteDimensional ℝ E\ns : Set E\nhs : s ∈ 𝓝 0\nb : AffineBasis (Fin (finrank ℝ E + 1)) ℝ E\nc : AffineBasis (Fin (finrank ℝ E + 1)) ℝ E := ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.Extrema | {
"line": 61,
"column": 4
} | {
"line": 62,
"column": 11
} | {
"line": 62,
"column": 12
} | [
{
"pp": "E : Type u_1\nβ : Type u_2\ninst✝¹⁰ : AddCommGroup E\ninst✝⁹ : TopologicalSpace E\ninst✝⁸ : Module ℝ E\ninst✝⁷ : IsTopologicalAddGroup E\ninst✝⁶ : ContinuousSMul ℝ E\ninst✝⁵ : AddCommGroup β\ninst✝⁴ : PartialOrder β\ninst✝³ : IsOrderedAddMonoid β\ninst✝² : Module ℝ β\ninst✝¹ : IsOrderedModule ℝ β\ninst... | [
"E : Type u_1\nβ : Type u_2\ninst✝¹⁰ : AddCommGroup E\ninst✝⁹ : TopologicalSpace E\ninst✝⁸ : Module ℝ E\ninst✝⁷ : IsTopologicalAddGroup E\ninst✝⁶ : ContinuousSMul ℝ E\ninst✝⁵ : AddCommGroup β\ninst✝⁴ : PartialOrder β\ninst✝³ : IsOrderedAddMonoid β\ninst✝² : Module ℝ β\ninst✝¹ : IsOrderedModule ℝ β\ninst✝ : PosSMulR... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.Extrema | {
"line": 69,
"column": 2
} | {
"line": 69,
"column": 56
} | {
"line": 69,
"column": 57
} | [
{
"pp": "E : Type u_1\nβ : Type u_2\ninst✝¹⁰ : AddCommGroup E\ninst✝⁹ : TopologicalSpace E\ninst✝⁸ : Module ℝ E\ninst✝⁷ : IsTopologicalAddGroup E\ninst✝⁶ : ContinuousSMul ℝ E\ninst✝⁵ : AddCommGroup β\ninst✝⁴ : PartialOrder β\ninst✝³ : IsOrderedAddMonoid β\ninst✝² : Module ℝ β\ninst✝¹ : IsOrderedModule ℝ β\ninst... | [
"E : Type u_1\nβ : Type u_2\ninst✝¹⁰ : AddCommGroup E\ninst✝⁹ : TopologicalSpace E\ninst✝⁸ : Module ℝ E\ninst✝⁷ : IsTopologicalAddGroup E\ninst✝⁶ : ContinuousSMul ℝ E\ninst✝⁵ : AddCommGroup β\ninst✝⁴ : PartialOrder β\ninst✝³ : IsOrderedAddMonoid β\ninst✝² : Module ℝ β\ninst✝¹ : IsOrderedModule ℝ β\ninst✝ : PosSMulR... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Affine.Convex | {
"line": 83,
"column": 4
} | {
"line": 84,
"column": 11
} | {
"line": 84,
"column": 12
} | [
{
"pp": "E✝ : Type u_1\ninst✝⁴ : NormedAddCommGroup E✝\ninst✝³ : NormedSpace ℝ E✝\ns✝ : Set E✝\nx : E✝\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : FiniteDimensional ℝ E\ns : Set E\nhs : s ∈ 𝓝 0\nb : AffineBasis (Fin (finrank ℝ E + 1)) ℝ E\nc : AffineBasis (Fin (finrank ℝ E +... | [
"E✝ : Type u_1\ninst✝⁴ : NormedAddCommGroup E✝\ninst✝³ : NormedSpace ℝ E✝\ns✝ : Set E✝\nx : E✝\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : FiniteDimensional ℝ E\ns : Set E\nhs : s ∈ 𝓝 0\nb : AffineBasis (Fin (finrank ℝ E + 1)) ℝ E\nc : AffineBasis (Fin (finrank ℝ E + 1)) ℝ E := ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.Exposed | {
"line": 149,
"column": 4
} | {
"line": 150,
"column": 68
} | {
"line": 152,
"column": 0
} | [
{
"pp": "case insert.inr\n𝕜 : Type u_1\nE : Type u_2\ninst✝⁷ : TopologicalSpace 𝕜\ninst✝⁶ : Ring 𝕜\ninst✝⁵ : PartialOrder 𝕜\ninst✝⁴ : AddCommMonoid E\ninst✝³ : TopologicalSpace E\ninst✝² : Module 𝕜 E\nA : Set E\ninst✝¹ : IsOrderedRing 𝕜\ninst✝ : ContinuousAdd 𝕜\nC : Set E\nF : Finset (Set E)\na✝ : C ∉ F\... | [] | · exact (hAF C (Finset.mem_insert_self C F)).inter
(hF' hFnemp fun B hB => hAF B (Finset.mem_insert_of_mem hB)) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Normed.Affine.Convex | {
"line": 105,
"column": 51
} | {
"line": 105,
"column": 62
} | {
"line": 105,
"column": 63
} | [
{
"pp": "E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : FiniteDimensional ℝ E\ns t : Set E\nhs₁ : Convex ℝ s\nhs₂ : IsCompact s\nht : t ∈ 𝓝ˢ s\nU : Set E\nhU₁ : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] U\nhU₂ : s ⊆ U\nhU₃ : U ⊆ t\nV : Set E\nhV₁ : V ∈ 𝓝 0\nhV... | [
"E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : FiniteDimensional ℝ E\ns t : Set E\nhs₁ : Convex ℝ s\nhs₂ : IsCompact s\nht : t ∈ 𝓝ˢ s\nU : Set E\nhU₁ : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] U\nhU₂ : s ⊆ U\nhU₃ : U ⊆ t\nV : Set E\nhV₁ : V ∈ 𝓝 0\nhV₂ : V + s ⊆ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.Continuous | {
"line": 52,
"column": 32
} | {
"line": 52,
"column": 76
} | {
"line": 52,
"column": 76
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : E → ℝ\nx₀ : E\nε r M : ℝ\nhf : ConvexOn ℝ (ball x₀ r) f\nhε : 0 < ε\nhM : ∀ (a : E), dist a x₀ < r → |f a| ≤ M\nK : ℝ := 2 * M / ε\nhK : K = 2 * M / ε\nx y : E\nhx : x ∈ ball x₀ (r - ε)\nhy : y ∈ ball x₀ (r - ε)\nhx₀r : ball x₀ (... | [
"E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : E → ℝ\nx₀ : E\nε r M : ℝ\nhf : ConvexOn ℝ (ball x₀ r) f\nhε : 0 < ε\nhM : ∀ (a : E), dist a x₀ < r → |f a| ≤ M\nK : ℝ := 2 * M / ε\nhK : K = 2 * M / ε\nx y : E\nhx : x ∈ ball x₀ (r - ε)\nhy : y ∈ ball x₀ (r - ε)\nhx₀r : ball x₀ (r - ε) ⊆ bal... | simp [z, a, b, smul_smul, hxy.ne', smul_sub] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Convex.Continuous | {
"line": 72,
"column": 2
} | {
"line": 72,
"column": 13
} | {
"line": 72,
"column": 14
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : E → ℝ\nx₀ : E\nε r M : ℝ\nhf : ConcaveOn ℝ (ball x₀ r) f\nhε : 0 < ε\nhM : ∀ (a : E), dist a x₀ < r → |f a| ≤ M\n⊢ LipschitzOnWith (2 * M / ε).toNNReal f (ball x₀ (r - ε))",
"ppTerm": "?m.48",
"assigned": false,
"used... | [
"E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : E → ℝ\nx₀ : E\nε r M : ℝ\nhf : ConcaveOn ℝ (ball x₀ r) f\nhε : 0 < ε\nhM : ∀ (a : E), dist a x₀ < r → |f a| ≤ M\n⊢ LipschitzOnWith (2 * M / ε).toNNReal f (ball x₀ (r - ε))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.Continuous | {
"line": 72,
"column": 56
} | {
"line": 72,
"column": 67
} | {
"line": 72,
"column": 68
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : E → ℝ\nx₀ : E\nε r M : ℝ\nhf : ConcaveOn ℝ (ball x₀ r) f\nhε : 0 < ε\nhM : ∀ (a : E), dist a x₀ < r → |f a| ≤ M\n⊢ ∀ (a : E), dist a x₀ < r → |(-f) a| ≤ M",
"ppTerm": "?m.70",
"assigned": true,
"usedConstants": [
... | [
"E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : E → ℝ\nx₀ : E\nε r M : ℝ\nhf : ConcaveOn ℝ (ball x₀ r) f\nhε : 0 < ε\nhM : ∀ (a : E), dist a x₀ < r → |f a| ≤ M\n⊢ ∀ (a : E), dist a x₀ < r → |f a| ≤ M"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.GaugeRescale | {
"line": 195,
"column": 2
} | {
"line": 196,
"column": 9
} | {
"line": 196,
"column": 10
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\ns : Set E\nhc : Convex ℝ s\nhne : (interior s).Nonempty\nhb : Bornology.IsBounded s\n⊢ ∃ h,\n ⇑h '' interior s = ball 0 1 ∧\n ⇑h '' closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s = closedBall 0 1 ∧\n ⇑h... | [
"E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\ns : Set E\nhc : Convex ℝ s\nhne : (interior s).Nonempty\nhb : Bornology.IsBounded s\n⊢ ∃ h,\n ⇑h '' interior s = ball 0 1 ∧\n ⇑h '' closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s = closedBall 0 1 ∧\n ⇑h '' frontier... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.Continuous | {
"line": 80,
"column": 6
} | {
"line": 80,
"column": 27
} | {
"line": 80,
"column": 27
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : E → ℝ\nx₀ : E\nr r' : ℝ\nhf : ConvexOn ℝ (ball x₀ r) f\nhr : r' < r\nM : ℝ\nhM : ∀ (a : E), dist a x₀ < r → |f a| < M\n⊢ ∃ K, LipschitzOnWith K f (ball x₀ r')",
"ppTerm": "?m.50",
"assigned": true,
"usedConstants": [
... | [
"E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : E → ℝ\nx₀ : E\nr r' : ℝ\nhf : ConvexOn ℝ (ball x₀ r) f\nhr : r' < r\nM : ℝ\nhM : ∀ (a : E), dist a x₀ < r → |f a| < M\n⊢ ∃ K, LipschitzOnWith K f (ball x₀ (r - (r - r')))"
] | ← sub_sub_cancel r r' | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Convex.Continuous | {
"line": 86,
"column": 2
} | {
"line": 86,
"column": 13
} | {
"line": 86,
"column": 14
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : E → ℝ\nx₀ : E\nr r' : ℝ\nhf : ConcaveOn ℝ (ball x₀ r) f\nhr : r' < r\nhf' : IsBounded ((-f) '' ball x₀ r)\n⊢ ∃ K, LipschitzOnWith K f (ball x₀ r')",
"ppTerm": "?m.106",
"assigned": false,
"usedConstants": [],
"use... | [
"E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : E → ℝ\nx₀ : E\nr r' : ℝ\nhf : ConcaveOn ℝ (ball x₀ r) f\nhr : r' < r\nhf' : IsBounded ((-f) '' ball x₀ r)\n⊢ ∃ K, LipschitzOnWith K f (ball x₀ r')"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.Continuous | {
"line": 111,
"column": 2
} | {
"line": 111,
"column": 38
} | {
"line": 111,
"column": 39
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nC : Set E\nf : E → ℝ\nhf : ConcaveOn ℝ C f\nx₀ : E\nhC : C ∈ 𝓝 x₀\n⊢ Filter.IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) (𝓝 x₀) |f| ↔ Filter.IsBoundedUnder (fun x1 x2 ↦ x1 ≥ x2) (𝓝 x₀) f",
"ppTerm": "?m.40",
"assigned": true,
... | [
"E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nC : Set E\nf : E → ℝ\nhf : ConcaveOn ℝ C f\nx₀ : E\nhC : C ∈ 𝓝 x₀\n⊢ (Filter.IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) (𝓝 x₀) fun i ↦ |f i|) ↔\n Filter.IsBoundedUnder (fun x1 x2 ↦ x2 ≤ x1) (𝓝 x₀) f"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.Integral | {
"line": 80,
"column": 4
} | {
"line": 83,
"column": 55
} | {
"line": 85,
"column": 0
} | [
{
"pp": "case refine_3\nα : Type u_1\nE : Type u_2\nm0 : MeasurableSpace α\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\nμ : Measure α\ns : Set E\nf : α → E\ninst✝ : IsProbabilityMeasure μ\nhs : Convex ℝ s\nhsc : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace]... | [] | simp only [SimpleFunc.mem_range, forall_mem_range]
intro x
apply (range g).inter_subset_right
exact SimpleFunc.approxOn_mem hgm.measurable h₀ _ _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Convex.Integral | {
"line": 80,
"column": 4
} | {
"line": 83,
"column": 55
} | {
"line": 85,
"column": 0
} | [
{
"pp": "case refine_3\nα : Type u_1\nE : Type u_2\nm0 : MeasurableSpace α\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\nμ : Measure α\ns : Set E\nf : α → E\ninst✝ : IsProbabilityMeasure μ\nhs : Convex ℝ s\nhsc : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace]... | [] | simp only [SimpleFunc.mem_range, forall_mem_range]
intro x
apply (range g).inter_subset_right
exact SimpleFunc.approxOn_mem hgm.measurable h₀ _ _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Convex.Integral | {
"line": 122,
"column": 2
} | {
"line": 122,
"column": 77
} | {
"line": 123,
"column": 4
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nm0 : MeasurableSpace α\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : CompleteSpace E\nμ : Measure α\ns : Set E\nf : α → E\ng : E → ℝ\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : NeZero μ\nhg : ConcaveOn ℝ s g\nhgc : ContinuousOn g s\nhsc : IsClosed[PseudoMetricS... | [
"α : Type u_1\nE : Type u_2\nm0 : MeasurableSpace α\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : CompleteSpace E\nμ : Measure α\ns : Set E\nf : α → E\ng : E → ℝ\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : NeZero μ\nhg : ConcaveOn ℝ s g\nhgc : ContinuousOn g s\nhsc : IsClosed[PseudoMetricSpace.toUnifo... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.Integral | {
"line": 169,
"column": 2
} | {
"line": 169,
"column": 77
} | {
"line": 170,
"column": 4
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nm0 : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nμ : Measure α\ns : Set E\nt : Set α\nf : α → E\ng : E → ℝ\nhg : ConcaveOn ℝ s g\nhgc : ContinuousOn g s\nhsc : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpa... | [
"α : Type u_1\nE : Type u_2\nm0 : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nμ : Measure α\ns : Set E\nt : Set α\nf : α → E\ng : E → ℝ\nhg : ConcaveOn ℝ s g\nhgc : ContinuousOn g s\nhsc : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s\nh0 : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.Integral | {
"line": 202,
"column": 2
} | {
"line": 202,
"column": 40
} | {
"line": 202,
"column": 41
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nm0 : MeasurableSpace α\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\nμ : Measure α\ns : Set E\nf : α → E\ng : E → ℝ\ninst✝ : IsProbabilityMeasure μ\nhg : ConvexOn ℝ s g\nhgc : ContinuousOn g s\nhsc : IsClosed[PseudoMetricSpace.toUniformS... | [
"α : Type u_1\nE : Type u_2\nm0 : MeasurableSpace α\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\nμ : Measure α\ns : Set E\nf : α → E\ng : E → ℝ\ninst✝ : IsProbabilityMeasure μ\nhg : ConvexOn ℝ s g\nhgc : ContinuousOn g s\nhsc : IsClosed[PseudoMetricSpace.toUniformSpace.toTopol... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.Integral | {
"line": 211,
"column": 2
} | {
"line": 211,
"column": 40
} | {
"line": 211,
"column": 41
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nm0 : MeasurableSpace α\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\nμ : Measure α\ns : Set E\nf : α → E\ng : E → ℝ\ninst✝ : IsProbabilityMeasure μ\nhg : ConcaveOn ℝ s g\nhgc : ContinuousOn g s\nhsc : IsClosed[PseudoMetricSpace.toUniform... | [
"α : Type u_1\nE : Type u_2\nm0 : MeasurableSpace α\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\nμ : Measure α\ns : Set E\nf : α → E\ng : E → ℝ\ninst✝ : IsProbabilityMeasure μ\nhg : ConcaveOn ℝ s g\nhgc : ContinuousOn g s\nhsc : IsClosed[PseudoMetricSpace.toUniformSpace.toTopo... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.Intrinsic | {
"line": 246,
"column": 9
} | {
"line": 246,
"column": 25
} | {
"line": 246,
"column": 26
} | [
{
"pp": "𝕜 : Type u_1\nV : Type u_2\nW : Type u_3\nQ : Type u_4\nP : Type u_5\ninst✝⁸ : Ring 𝕜\ninst✝⁷ : AddCommGroup V\ninst✝⁶ : Module 𝕜 V\ninst✝⁵ : TopologicalSpace P\ninst✝⁴ : AddTorsor V P\ninst✝³ : AddCommGroup W\ninst✝² : Module 𝕜 W\ninst✝¹ : TopologicalSpace Q\ninst✝ : AddTorsor W Q\nf : P → Q\ns : ... | [
"𝕜 : Type u_1\nV : Type u_2\nW : Type u_3\nQ : Type u_4\nP : Type u_5\ninst✝⁸ : Ring 𝕜\ninst✝⁷ : AddCommGroup V\ninst✝⁶ : Module 𝕜 V\ninst✝⁵ : TopologicalSpace P\ninst✝⁴ : AddTorsor V P\ninst✝³ : AddCommGroup W\ninst✝² : Module 𝕜 W\ninst✝¹ : TopologicalSpace Q\ninst✝ : AddTorsor W Q\nf : P → Q\ns : Set P\ne : ↥... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.Continuous | {
"line": 181,
"column": 2
} | {
"line": 181,
"column": 13
} | {
"line": 181,
"column": 14
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nC : Set E\nf : E → ℝ\nhC : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] C\nhf : ConcaveOn ℝ C f\n⊢ LocallyLipschitzOn C f ↔ ContinuousOn f C",
"ppTerm": "?m.22",
"assigned": false,
"usedConstants": [],
... | [
"E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nC : Set E\nf : E → ℝ\nhC : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] C\nhf : ConcaveOn ℝ C f\n⊢ LocallyLipschitzOn C f ↔ ContinuousOn f C"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.Continuous | {
"line": 196,
"column": 33
} | {
"line": 196,
"column": 44
} | {
"line": 196,
"column": 45
} | [
{
"pp": "E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\nC : Set E\nf : E → ℝ\ninst✝ : FiniteDimensional ℝ E\nhC : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] C\nhf : ConcaveOn ℝ C f\n⊢ LocallyLipschitzOn C f",
"ppTerm": "?m.21",
"assigned": false,
"usedConstan... | [
"E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\nC : Set E\nf : E → ℝ\ninst✝ : FiniteDimensional ℝ E\nhC : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] C\nhf : ConcaveOn ℝ C f\n⊢ LocallyLipschitzOn C f"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.Continuous | {
"line": 219,
"column": 2
} | {
"line": 219,
"column": 13
} | {
"line": 219,
"column": 14
} | [
{
"pp": "E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\nf : E → ℝ\ninst✝ : FiniteDimensional ℝ E\nhf : ConvexOn ℝ univ f\n⊢ LocallyLipschitz f",
"ppTerm": "?m.20",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\nf : E → ℝ\ninst✝ : FiniteDimensional ℝ E\nhf : ConvexOn ℝ univ f\n⊢ LocallyLipschitz f"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.Continuous | {
"line": 222,
"column": 2
} | {
"line": 222,
"column": 13
} | {
"line": 222,
"column": 14
} | [
{
"pp": "E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\nf : E → ℝ\ninst✝ : FiniteDimensional ℝ E\nhf : ConcaveOn ℝ univ f\n⊢ LocallyLipschitz f",
"ppTerm": "?m.20",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\nf : E → ℝ\ninst✝ : FiniteDimensional ℝ E\nhf : ConcaveOn ℝ univ f\n⊢ LocallyLipschitz f"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.Continuous | {
"line": 253,
"column": 2
} | {
"line": 253,
"column": 13
} | {
"line": 253,
"column": 14
} | [
{
"pp": "f : ℝ → ℝ\ny : ℝ\nhf_cnv : ConcaveOn ℝ (Ici y) f\nhf_cont : ContinuousWithinAt f (Ici y) y\n⊢ ContinuousOn f (Ici y)",
"ppTerm": "?m.29",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"f : ℝ → ℝ\ny : ℝ\nhf_cnv : ConcaveOn ℝ (Ici y) f\nhf_cont : ContinuousWithinAt f (Ici y) y\n⊢ ContinuousOn f (Ici y)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.Continuous | {
"line": 269,
"column": 2
} | {
"line": 269,
"column": 13
} | {
"line": 269,
"column": 14
} | [
{
"pp": "f : ℝ → ℝ\ny : ℝ\nhf_cnv : ConcaveOn ℝ (Iic y) f\nhf_cont : ContinuousWithinAt f (Iic y) y\n⊢ ContinuousOn f (Iic y)",
"ppTerm": "?m.29",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"f : ℝ → ℝ\ny : ℝ\nhf_cnv : ConcaveOn ℝ (Iic y) f\nhf_cont : ContinuousWithinAt f (Iic y) y\n⊢ ContinuousOn f (Iic y)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.Continuous | {
"line": 268,
"column": 30
} | {
"line": 269,
"column": 53
} | {
"line": 271,
"column": 0
} | [
{
"pp": "f : ℝ → ℝ\ny : ℝ\nhf_cnv : ConcaveOn ℝ (Iic y) f\nhf_cont : ContinuousWithinAt f (Iic y) y\n⊢ ContinuousOn f (Iic y)",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"NegZeroClass.toNeg",
"Real.partialOrder",
"Real",
"Pi.instNeg",
"NonUnitalCommRing.to... | [] | by
simpa using hf_cnv.neg.continuousOn_Iic hf_cont.neg | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Convex.Continuous | {
"line": 286,
"column": 2
} | {
"line": 286,
"column": 13
} | {
"line": 286,
"column": 14
} | [
{
"pp": "f : ℝ → ℝ\ny z : ℝ\nhf_cnv : ConcaveOn ℝ (Ioc y z) f\nhf_cont : ContinuousWithinAt f (Iic z) z\n⊢ ContinuousOn f (Ioc y z)",
"ppTerm": "?m.29",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"f : ℝ → ℝ\ny z : ℝ\nhf_cnv : ConcaveOn ℝ (Ioc y z) f\nhf_cont : ContinuousWithinAt f (Iic z) z\n⊢ ContinuousOn f (Ioc y z)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.Continuous | {
"line": 303,
"column": 2
} | {
"line": 303,
"column": 13
} | {
"line": 303,
"column": 14
} | [
{
"pp": "f : ℝ → ℝ\ny z : ℝ\nhf_cnv : ConcaveOn ℝ (Ico y z) f\nhf_cont : ContinuousWithinAt f (Ici y) y\n⊢ ContinuousOn f (Ico y z)",
"ppTerm": "?m.29",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"f : ℝ → ℝ\ny z : ℝ\nhf_cnv : ConcaveOn ℝ (Ico y z) f\nhf_cont : ContinuousWithinAt f (Ici y) y\n⊢ ContinuousOn f (Ico y z)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.Continuous | {
"line": 326,
"column": 2
} | {
"line": 326,
"column": 13
} | {
"line": 326,
"column": 14
} | [
{
"pp": "f : ℝ → ℝ\ny z : ℝ\nhf_cnv : ConcaveOn ℝ (Icc y z) f\nhyz : y < z\nhfy : ContinuousWithinAt f (Ici y) y\nhfz : ContinuousWithinAt f (Iic z) z\n⊢ ContinuousOn f (Icc y z)",
"ppTerm": "?m.37",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"f : ℝ → ℝ\ny z : ℝ\nhf_cnv : ConcaveOn ℝ (Icc y z) f\nhyz : y < z\nhfy : ContinuousWithinAt f (Ici y) y\nhfz : ContinuousWithinAt f (Iic z) z\n⊢ ContinuousOn f (Icc y z)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Convex.ConvexSpace.Defs | {
"line": 102,
"column": 4
} | {
"line": 102,
"column": 21
} | {
"line": 102,
"column": 22
} | [
{
"pp": "case e_b\nR : Type u\ninst✝² : PartialOrder R\ninst✝¹ : Semiring R\nM : Type u_9\nw : StdSimplex R M\nx : M\ninst✝ : IsStrictOrderedRing R\na : R\nha : a ≠ 0\nhwa : w.weights = Finsupp.single x a\n⊢ a = 1",
"ppTerm": "?e_b",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"... | [
"case e_b\nR : Type u\ninst✝² : PartialOrder R\ninst✝¹ : Semiring R\nM : Type u_9\nw : StdSimplex R M\nx : M\ninst✝ : IsStrictOrderedRing R\na : R\nha : a ≠ 0\nhwa : w.weights = Finsupp.single x a\n⊢ a = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Convex.ConvexSpace.Defs | {
"line": 124,
"column": 2
} | {
"line": 124,
"column": 47
} | {
"line": 126,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝² : PartialOrder R\ninst✝¹ : Semiring R\nM : Type u_9\nN : Type u_10\ninst✝ : IsStrictOrderedRing R\nf : StdSimplex R M\nx : N\n⊢ map (fun x_1 ↦ x) f = single x",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Finsupp.instFunLike",
"NonAssocSemiring.to... | [] | ext a; by_cases x = a <;> simp [*, mapDomain] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Convex.ConvexSpace.Defs | {
"line": 124,
"column": 2
} | {
"line": 124,
"column": 47
} | {
"line": 126,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝² : PartialOrder R\ninst✝¹ : Semiring R\nM : Type u_9\nN : Type u_10\ninst✝ : IsStrictOrderedRing R\nf : StdSimplex R M\nx : N\n⊢ map (fun x_1 ↦ x) f = single x",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Finsupp.instFunLike",
"NonAssocSemiring.to... | [] | ext a; by_cases x = a <;> simp [*, mapDomain] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Convex.Integral | {
"line": 273,
"column": 2
} | {
"line": 292,
"column": 40
} | {
"line": 294,
"column": 0
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nm0 : MeasurableSpace α\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\nμ : Measure α\ns : Set E\nf : α → E\ng : E → ℝ\ninst✝ : IsFiniteMeasure μ\nhg : StrictConvexOn ℝ s g\nhgc : ContinuousOn g s\nhsc : IsClosed[PseudoMetricSpace.toUniform... | [] | have : ∀ {t}, μ t ≠ 0 → (⨍ x in t, f x ∂μ) ∈ s ∧ g (⨍ x in t, f x ∂μ) ≤ ⨍ x in t, g (f x) ∂μ :=
fun ht =>
hg.convexOn.set_average_mem_epigraph hgc hsc ht (by finiteness) (ae_restrict_of_ae hfs)
hfi.integrableOn hgi.integrableOn
refine (ae_eq_const_or_exists_average_ne_compl hfi).imp_right ?_
rintro ⟨t... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Convex.Integral | {
"line": 273,
"column": 2
} | {
"line": 292,
"column": 40
} | {
"line": 294,
"column": 0
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nm0 : MeasurableSpace α\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\nμ : Measure α\ns : Set E\nf : α → E\ng : E → ℝ\ninst✝ : IsFiniteMeasure μ\nhg : StrictConvexOn ℝ s g\nhgc : ContinuousOn g s\nhsc : IsClosed[PseudoMetricSpace.toUniform... | [] | have : ∀ {t}, μ t ≠ 0 → (⨍ x in t, f x ∂μ) ∈ s ∧ g (⨍ x in t, f x ∂μ) ≤ ⨍ x in t, g (f x) ∂μ :=
fun ht =>
hg.convexOn.set_average_mem_epigraph hgc hsc ht (by finiteness) (ae_restrict_of_ae hfs)
hfi.integrableOn hgi.integrableOn
refine (ae_eq_const_or_exists_average_ne_compl hfi).imp_right ?_
rintro ⟨t... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Convex.Integral | {
"line": 301,
"column": 2
} | {
"line": 301,
"column": 62
} | {
"line": 302,
"column": 4
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nm0 : MeasurableSpace α\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\nμ : Measure α\ns : Set E\nf : α → E\ng : E → ℝ\ninst✝ : IsFiniteMeasure μ\nhg : StrictConcaveOn ℝ s g\nhgc : ContinuousOn g s\nhsc : IsClosed[PseudoMetricSpace.toUnifor... | [
"α : Type u_1\nE : Type u_2\nm0 : MeasurableSpace α\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\nμ : Measure α\ns : Set E\nf : α → E\ng : E → ℝ\ninst✝ : IsFiniteMeasure μ\nhg : StrictConcaveOn ℝ s g\nhgc : ContinuousOn g s\nhsc : IsClosed[PseudoMetricSpace.toUniformSpace.toTop... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Convex.ConvexSpace.Defs | {
"line": 217,
"column": 2
} | {
"line": 217,
"column": 13
} | {
"line": 217,
"column": 14
} | [
{
"pp": "X : Type u_2\nK : Type u_8\ninst✝³ : Semifield K\ninst✝² : LinearOrder K\ninst✝¹ : IsStrictOrderedRing K\ninst✝ : IsDomain K\nw : StdSimplex K X\na✝ : X\nhx : ∃ x ∈ {a✝}, w.weights x ≠ 0\n⊢ ¬w.weights a✝ = 0",
"ppTerm": "?m.40",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
... | [
"X : Type u_2\nK : Type u_8\ninst✝³ : Semifield K\ninst✝² : LinearOrder K\ninst✝¹ : IsStrictOrderedRing K\ninst✝ : IsDomain K\nw : StdSimplex K X\na✝ : X\nhx : ∃ x ∈ {a✝}, w.weights x ≠ 0\n⊢ ¬w.weights a✝ = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Convex.ConvexSpace.Defs | {
"line": 382,
"column": 18
} | {
"line": 382,
"column": 29
} | {
"line": 382,
"column": 30
} | [
{
"pp": "R : Type u_1\nM : Type u_3\nI : Type u_6\ninst✝³ : PartialOrder R\ninst✝² : Semiring R\ninst✝¹ : IsStrictOrderedRing R\ninst✝ : ConvexSpace R M\nw : StdSimplex R I\nf g : I → M\nhfg : ∀ (i : I), w.weights i ≠ 0 → f i = g i\ni✝ : M\ni : I\nhi : i ∈ w.weights.support\n⊢ w.weights i ≠ 0",
"ppTerm": "?... | [
"R : Type u_1\nM : Type u_3\nI : Type u_6\ninst✝³ : PartialOrder R\ninst✝² : Semiring R\ninst✝¹ : IsStrictOrderedRing R\ninst✝ : ConvexSpace R M\nw : StdSimplex R I\nf g : I → M\nhfg : ∀ (i : I), w.weights i ≠ 0 → f i = g i\ni✝ : M\ni : I\nhi : i ∈ w.weights.support\n⊢ ¬w.weights i = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.Integral | {
"line": 319,
"column": 2
} | {
"line": 319,
"column": 69
} | {
"line": 320,
"column": 4
} | [
{
"pp": "case pos.inr\nα : Type u_1\nE : Type u_2\nm0 : MeasurableSpace α\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\nμ : Measure α\nf : α → E\nC : ℝ\ninst✝ : StrictConvexSpace ℝ E\nhC0 : 0 < C\nhfi : Integrable f μ\nhμt : μ univ < ∞\nthis : IsFiniteMeasure μ\nh_le : ∀ᵐ (... | [
"case pos.inr\nα : Type u_1\nE : Type u_2\nm0 : MeasurableSpace α\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\nμ : Measure α\nf : α → E\nC : ℝ\ninst✝ : StrictConvexSpace ℝ E\nhC0 : 0 < C\nhfi : Integrable f μ\nhμt : μ univ < ∞\nthis : IsFiniteMeasure μ\nh_le : ∀ᵐ (x : α) ∂μ, f... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Convex.Star | {
"line": 82,
"column": 2
} | {
"line": 82,
"column": 68
} | {
"line": 84,
"column": 0
} | [
{
"pp": "R : Type u_1\nX : Type u_2\ninst✝³ : Semiring R\ninst✝² : PartialOrder R\ninst✝¹ : IsStrictOrderedRing R\ninst✝ : ConvexSpace R X\nx : X\nS : Set (Set X)\nhS : ∀ s ∈ S, IsStarConvexSet R x s\n⊢ IsStarConvexSet R x (⋃₀ S)",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"NonAs... | [] | rintro y ⟨s, hs, hy⟩ a ha b hb hab; exact ⟨s, hs, hS _ hs hy _ ..⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Convex.Star | {
"line": 82,
"column": 2
} | {
"line": 82,
"column": 68
} | {
"line": 84,
"column": 0
} | [
{
"pp": "R : Type u_1\nX : Type u_2\ninst✝³ : Semiring R\ninst✝² : PartialOrder R\ninst✝¹ : IsStrictOrderedRing R\ninst✝ : ConvexSpace R X\nx : X\nS : Set (Set X)\nhS : ∀ s ∈ S, IsStarConvexSet R x s\n⊢ IsStarConvexSet R x (⋃₀ S)",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"NonAs... | [] | rintro y ⟨s, hs, hy⟩ a ha b hb hab; exact ⟨s, hs, hS _ hs hy _ ..⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Convex.Star | {
"line": 96,
"column": 25
} | {
"line": 96,
"column": 36
} | {
"line": 96,
"column": 37
} | [
{
"pp": "R : Type u_1\nX : Type u_2\ninst✝³ : Semiring R\ninst✝² : PartialOrder R\ninst✝¹ : IsStrictOrderedRing R\ninst✝ : ConvexSpace R X\nx : X\ns : Set X\nhs : IsStarConvexSet R x s\ny : X\nhy : y ∈ s\n⊢ x ∈ s",
"ppTerm": "?m.28",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"... | [
"R : Type u_1\nX : Type u_2\ninst✝³ : Semiring R\ninst✝² : PartialOrder R\ninst✝¹ : IsStrictOrderedRing R\ninst✝ : ConvexSpace R X\nx : X\ns : Set X\nhs : IsStarConvexSet R x s\ny : X\nhy : y ∈ s\n⊢ x ∈ s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Convex.Star | {
"line": 101,
"column": 30
} | {
"line": 101,
"column": 79
} | {
"line": 101,
"column": 80
} | [
{
"pp": "R : Type u_1\nX : Type u_2\nY : Type u_3\ninst✝⁴ : Semiring R\ninst✝³ : PartialOrder R\ninst✝² : IsStrictOrderedRing R\ninst✝¹ : ConvexSpace R X\ninst✝ : ConvexSpace R Y\nf : X → Y\nx : X\ns : Set Y\nhf : IsAffineMap R f\nhs : IsStarConvexSet R (f x) s\ny : X\nhy : y ∈ f ⁻¹' s\na b : R\nha : 0 ≤ a\nhb ... | [
"R : Type u_1\nX : Type u_2\nY : Type u_3\ninst✝⁴ : Semiring R\ninst✝³ : PartialOrder R\ninst✝² : IsStrictOrderedRing R\ninst✝¹ : ConvexSpace R X\ninst✝ : ConvexSpace R Y\nf : X → Y\nx : X\ns : Set Y\nhf : IsAffineMap R f\nhs : IsStarConvexSet R (f x) s\ny : X\nhy : y ∈ f ⁻¹' s\na b : R\nha : 0 ≤ a\nhb : 0 ≤ b\nhab... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Convex.Star | {
"line": 111,
"column": 50
} | {
"line": 111,
"column": 61
} | {
"line": 111,
"column": 62
} | [
{
"pp": "R : Type u_1\nX : Type u_2\nY : Type u_3\ninst✝⁴ : Semiring R\ninst✝³ : PartialOrder R\ninst✝² : IsStrictOrderedRing R\ninst✝¹ : ConvexSpace R X\ninst✝ : ConvexSpace R Y\nx : X\ns : Set X\nt : Set Y\ny : Y\nhs : IsStarConvexSet R x s\nht : IsStarConvexSet R y t\nw : X\nz : Y\nhw : (w, z).1 ∈ s\nhz : (w... | [
"R : Type u_1\nX : Type u_2\nY : Type u_3\ninst✝⁴ : Semiring R\ninst✝³ : PartialOrder R\ninst✝² : IsStrictOrderedRing R\ninst✝¹ : ConvexSpace R X\ninst✝ : ConvexSpace R Y\nx : X\ns : Set X\nt : Set Y\ny : Y\nhs : IsStarConvexSet R x s\nht : IsStarConvexSet R y t\nw : X\nz : Y\nhw : (w, z).1 ∈ s\nhz : (w, z).2 ∈ t\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Convex.ConvexSpace.Defs | {
"line": 524,
"column": 2
} | {
"line": 524,
"column": 13
} | {
"line": 524,
"column": 14
} | [
{
"pp": "R : Type u_1\nM : Type u_3\nJ : Type u_7\ninst✝³ : PartialOrder R\ninst✝² : Semiring R\ninst✝¹ : IsStrictOrderedRing R\ninst✝ : ConvexSpace R M\ns t : R\nhs : 0 ≤ s\nht : 0 ≤ t\nh : s + t = 1\ng : StdSimplex R J\ne : J → M\nm : M\n⊢ convexCombPair s t hs ht h (iConvexComb g e) m =\n sConvexComb (con... | [
"R : Type u_1\nM : Type u_3\nJ : Type u_7\ninst✝³ : PartialOrder R\ninst✝² : Semiring R\ninst✝¹ : IsStrictOrderedRing R\ninst✝ : ConvexSpace R M\ns t : R\nhs : 0 ≤ s\nht : 0 ≤ t\nh : s + t = 1\ng : StdSimplex R J\ne : J → M\nm : M\n⊢ convexCombPair s t hs ht h (iConvexComb g e) m =\n sConvexComb (convexCombPair ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Convex.ConvexSpace.Defs | {
"line": 529,
"column": 2
} | {
"line": 529,
"column": 13
} | {
"line": 529,
"column": 14
} | [
{
"pp": "R : Type u_1\nM : Type u_3\nJ : Type u_7\ninst✝³ : PartialOrder R\ninst✝² : Semiring R\ninst✝¹ : IsStrictOrderedRing R\ninst✝ : ConvexSpace R M\ns t : R\nhs : 0 ≤ s\nht : 0 ≤ t\nh : s + t = 1\nm : M\ng : StdSimplex R J\ne : J → M\n⊢ convexCombPair s t hs ht h m (iConvexComb g e) =\n sConvexComb (con... | [
"R : Type u_1\nM : Type u_3\nJ : Type u_7\ninst✝³ : PartialOrder R\ninst✝² : Semiring R\ninst✝¹ : IsStrictOrderedRing R\ninst✝ : ConvexSpace R M\ns t : R\nhs : 0 ≤ s\nht : 0 ≤ t\nh : s + t = 1\nm : M\ng : StdSimplex R J\ne : J → M\n⊢ convexCombPair s t hs ht h m (iConvexComb g e) =\n sConvexComb (convexCombPair ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Convex.Star | {
"line": 111,
"column": 77
} | {
"line": 111,
"column": 88
} | {
"line": 111,
"column": 89
} | [
{
"pp": "R : Type u_1\nX : Type u_2\nY : Type u_3\ninst✝⁴ : Semiring R\ninst✝³ : PartialOrder R\ninst✝² : IsStrictOrderedRing R\ninst✝¹ : ConvexSpace R X\ninst✝ : ConvexSpace R Y\nx : X\ns : Set X\nt : Set Y\ny : Y\nhs : IsStarConvexSet R x s\nht : IsStarConvexSet R y t\nw : X\nz : Y\nhw : (w, z).1 ∈ s\nhz : (w... | [
"R : Type u_1\nX : Type u_2\nY : Type u_3\ninst✝⁴ : Semiring R\ninst✝³ : PartialOrder R\ninst✝² : IsStrictOrderedRing R\ninst✝¹ : ConvexSpace R X\ninst✝ : ConvexSpace R Y\nx : X\ns : Set X\nt : Set Y\ny : Y\nhs : IsStarConvexSet R x s\nht : IsStarConvexSet R y t\nw : X\nz : Y\nhw : (w, z).1 ∈ s\nhz : (w, z).2 ∈ t\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Convex.Star | {
"line": 117,
"column": 35
} | {
"line": 117,
"column": 46
} | {
"line": 117,
"column": 47
} | [
{
"pp": "R : Type u_1\ninst✝³ : Semiring R\ninst✝² : PartialOrder R\ninst✝¹ : IsStrictOrderedRing R\nι : Type u_6\nX : ι → Type u_7\ninst✝ : (i : ι) → ConvexSpace R (X i)\ns : Set ι\nx : (i : ι) → X i\nt : (i : ι) → Set (X i)\nht : ∀ i ∈ s, IsStarConvexSet R (x i) (t i)\ny : (i : ι) → X i\nhy : y ∈ s.pi t\na b ... | [
"R : Type u_1\ninst✝³ : Semiring R\ninst✝² : PartialOrder R\ninst✝¹ : IsStrictOrderedRing R\nι : Type u_6\nX : ι → Type u_7\ninst✝ : (i : ι) → ConvexSpace R (X i)\ns : Set ι\nx : (i : ι) → X i\nt : (i : ι) → Set (X i)\nht : ∀ i ∈ s, IsStarConvexSet R (x i) (t i)\ny : (i : ι) → X i\nhy : y ∈ s.pi t\na b : R\nha : 0 ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Convex.ConvexSpace.Defs | {
"line": 584,
"column": 2
} | {
"line": 584,
"column": 13
} | {
"line": 584,
"column": 14
} | [
{
"pp": "R : Type u_9\nM : Type u_10\nI : Type u_11\ninst✝³ : PartialOrder R\ninst✝² : CommSemiring R\ninst✝¹ : IsStrictOrderedRing R\ninst✝ : ConvexSpace R M\ns t : R\nhs : 0 ≤ s\nht : 0 ≤ t\nh : s + t = 1\nf : StdSimplex R I\nm : M\ne : I → M\n⊢ (iConvexComb f fun x ↦ convexCombPair s t hs ht h (e x) m) = con... | [
"R : Type u_9\nM : Type u_10\nI : Type u_11\ninst✝³ : PartialOrder R\ninst✝² : CommSemiring R\ninst✝¹ : IsStrictOrderedRing R\ninst✝ : ConvexSpace R M\ns t : R\nhs : 0 ≤ s\nht : 0 ≤ t\nh : s + t = 1\nf : StdSimplex R I\nm : M\ne : I → M\n⊢ (iConvexComb f fun x ↦ convexCombPair s t hs ht h (e x) m) = convexCombPair ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Convex.ConvexSpace.Defs | {
"line": 589,
"column": 2
} | {
"line": 589,
"column": 13
} | {
"line": 589,
"column": 14
} | [
{
"pp": "R : Type u_9\nM : Type u_10\nI : Type u_11\ninst✝³ : PartialOrder R\ninst✝² : CommSemiring R\ninst✝¹ : IsStrictOrderedRing R\ninst✝ : ConvexSpace R M\ns t : R\nhs : 0 ≤ s\nht : 0 ≤ t\nh : s + t = 1\nf : StdSimplex R I\nm : M\ne : I → M\n⊢ (iConvexComb f fun x ↦ convexCombPair s t hs ht h m (e x)) = con... | [
"R : Type u_9\nM : Type u_10\nI : Type u_11\ninst✝³ : PartialOrder R\ninst✝² : CommSemiring R\ninst✝¹ : IsStrictOrderedRing R\ninst✝ : ConvexSpace R M\ns t : R\nhs : 0 ≤ s\nht : 0 ≤ t\nh : s + t = 1\nf : StdSimplex R I\nm : M\ne : I → M\n⊢ (iConvexComb f fun x ↦ convexCombPair s t hs ht h m (e x)) = convexCombPair ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Convex.Set | {
"line": 197,
"column": 35
} | {
"line": 197,
"column": 46
} | {
"line": 197,
"column": 47
} | [
{
"pp": "K : Type u_4\nX : Type u_5\ninst✝³ : Field K\ninst✝² : LinearOrder K\ninst✝¹ : IsStrictOrderedRing K\ninst✝ : ConvexSpace K X\ns : Set X\nhs : ∀ (a b : K) (ha : 0 ≤ a) (hb : 0 ≤ b) (hab : a + b = 1), ∀ x ∈ s, ∀ y ∈ s, convexCombPair a b ha hb hab x y ∈ s\nt✝ : Finset X\nx : X\nt : Finset X\nhx : x ∉ t\... | [
"K : Type u_4\nX : Type u_5\ninst✝³ : Field K\ninst✝² : LinearOrder K\ninst✝¹ : IsStrictOrderedRing K\ninst✝ : ConvexSpace K X\ns : Set X\nhs : ∀ (a b : K) (ha : 0 ≤ a) (hb : 0 ≤ b) (hab : a + b = 1), ∀ x ∈ s, ∀ y ∈ s, convexCombPair a b ha hb hab x y ∈ s\nt✝ : Finset X\nx : X\nt : Finset X\nhx : x ∉ t\nht : t.None... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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