module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.LinearAlgebra.Ray | {
"line": 536,
"column": 25
} | {
"line": 536,
"column": 41
} | {
"line": 536,
"column": 42
} | [
{
"pp": "case pos\nR : Type u_1\ninst✝⁵ : CommRing R\ninst✝⁴ : LinearOrder R\ninst✝³ : IsStrictOrderedRing R\nM : Type u_2\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : IsTorsionFree R M\nx y : M\nhx : ¬x = 0\nhy : y = 0\n⊢ SameRay R x y ∨ SameRay R x (-y) ↔ ¬LinearIndependent R ![x, y]",
"ppTerm":... | [
"case pos\nR : Type u_1\ninst✝⁵ : CommRing R\ninst✝⁴ : LinearOrder R\ninst✝³ : IsStrictOrderedRing R\nM : Type u_2\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : IsTorsionFree R M\nx y : M\nhx : ¬x = 0\nhy : y = 0\n⊢ ¬LinearIndependent R ![x, 0]"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.Segment | {
"line": 317,
"column": 2
} | {
"line": 317,
"column": 71
} | {
"line": 318,
"column": 4
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁴ : CommRing 𝕜\ninst✝³ : PartialOrder 𝕜\ninst✝² : IsStrictOrderedRing 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ny z : E\nθ : 𝕜\nhθ₀ : 0 ≤ θ\nhθ₁ : θ ≤ 1\n⊢ SameRay 𝕜 ((fun θ ↦ y + θ • (z - y)) θ - y) (z - (fun θ ↦ y + θ • (z - y)) θ)",
"ppTerm": "?m.68"... | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝⁴ : CommRing 𝕜\ninst✝³ : PartialOrder 𝕜\ninst✝² : IsStrictOrderedRing 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ny z : E\nθ : 𝕜\nhθ₀ : 0 ≤ θ\nhθ₁ : θ ≤ 1\n⊢ SameRay 𝕜 (θ • (z - y)) (z - y - θ • (z - y))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Ray | {
"line": 555,
"column": 6
} | {
"line": 557,
"column": 45
} | {
"line": 558,
"column": 4
} | [
{
"pp": "case neg.refine_2.inl.inl\nR : Type u_1\ninst✝⁵ : CommRing R\ninst✝⁴ : LinearOrder R\ninst✝³ : IsStrictOrderedRing R\nM : Type u_2\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : IsTorsionFree R M\nx y : M\nhx : ¬x = 0\nhy : ¬y = 0\nm : Fin (Nat.succ 0).succ → R\nhm : m 0 • x = -(m 1 • y)\nhmne ... | [] | refine
Or.inr (Or.inr (Or.inr ⟨-m 0, -m 1, Left.neg_pos_iff.2 hm0, Left.neg_pos_iff.2 hm1, ?_⟩))
linear_combination (norm := module) -hm | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Ray | {
"line": 555,
"column": 6
} | {
"line": 557,
"column": 45
} | {
"line": 558,
"column": 4
} | [
{
"pp": "case neg.refine_2.inl.inl\nR : Type u_1\ninst✝⁵ : CommRing R\ninst✝⁴ : LinearOrder R\ninst✝³ : IsStrictOrderedRing R\nM : Type u_2\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : IsTorsionFree R M\nx y : M\nhx : ¬x = 0\nhy : ¬y = 0\nm : Fin (Nat.succ 0).succ → R\nhm : m 0 • x = -(m 1 • y)\nhmne ... | [] | refine
Or.inr (Or.inr (Or.inr ⟨-m 0, -m 1, Left.neg_pos_iff.2 hm0, Left.neg_pos_iff.2 hm1, ?_⟩))
linear_combination (norm := module) -hm | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.AffineSpace.AffineMap | {
"line": 150,
"column": 2
} | {
"line": 150,
"column": 32
} | {
"line": 151,
"column": 2
} | [
{
"pp": "k : Type u_1\nV1 : Type u_2\nP1 : Type u_3\nV2 : Type u_4\nP2 : Type u_5\ninst✝⁶ : Ring k\ninst✝⁵ : AddCommGroup V1\ninst✝⁴ : Module k V1\ninst✝³ : AffineSpace V1 P1\ninst✝² : AddCommGroup V2\ninst✝¹ : Module k V2\ninst✝ : AffineSpace V2 P2\nf g : P1 →ᵃ[k] P2\nh₁ : f.linear = g.linear\np : P1\nh₂ : f p... | [
"k : Type u_1\nV1 : Type u_2\nP1 : Type u_3\nV2 : Type u_4\nP2 : Type u_5\ninst✝⁶ : Ring k\ninst✝⁵ : AddCommGroup V1\ninst✝⁴ : Module k V1\ninst✝³ : AffineSpace V1 P1\ninst✝² : AddCommGroup V2\ninst✝¹ : Module k V2\ninst✝ : AffineSpace V2 P2\nf g : P1 →ᵃ[k] P2\nh₁ : f.linear = g.linear\np : P1\nh₂ : f p = g p\nq : ... | have := f.map_vadd' q (q -ᵥ p) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Convex.Segment | {
"line": 529,
"column": 4
} | {
"line": 529,
"column": 54
} | {
"line": 530,
"column": 2
} | [
{
"pp": "case inr.refine_1\n𝕜 : Type u_1\ninst✝² : Field 𝕜\ninst✝¹ : LinearOrder 𝕜\ninst✝ : IsStrictOrderedRing 𝕜\nx y z : 𝕜\nhxz : 0 ≤ z - x\nhyz : 0 ≤ y - z\nh : 0 < y - x\n⊢ (y - z) / (y - x) + (z - x) / (y - x) = 1",
"ppTerm": "?inr.refine_1",
"assigned": true,
"usedConstants": [
"Add... | [] | rw [← add_div, sub_add_sub_cancel, div_self h.ne'] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Convex.Segment | {
"line": 529,
"column": 4
} | {
"line": 529,
"column": 54
} | {
"line": 530,
"column": 2
} | [
{
"pp": "case inr.refine_1\n𝕜 : Type u_1\ninst✝² : Field 𝕜\ninst✝¹ : LinearOrder 𝕜\ninst✝ : IsStrictOrderedRing 𝕜\nx y z : 𝕜\nhxz : 0 ≤ z - x\nhyz : 0 ≤ y - z\nh : 0 < y - x\n⊢ (y - z) / (y - x) + (z - x) / (y - x) = 1",
"ppTerm": "?inr.refine_1",
"assigned": true,
"usedConstants": [
"Add... | [] | rw [← add_div, sub_add_sub_cancel, div_self h.ne'] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Convex.Segment | {
"line": 529,
"column": 4
} | {
"line": 529,
"column": 54
} | {
"line": 530,
"column": 2
} | [
{
"pp": "case inr.refine_1\n𝕜 : Type u_1\ninst✝² : Field 𝕜\ninst✝¹ : LinearOrder 𝕜\ninst✝ : IsStrictOrderedRing 𝕜\nx y z : 𝕜\nhxz : 0 ≤ z - x\nhyz : 0 ≤ y - z\nh : 0 < y - x\n⊢ (y - z) / (y - x) + (z - x) / (y - x) = 1",
"ppTerm": "?inr.refine_1",
"assigned": true,
"usedConstants": [
"Add... | [] | rw [← add_div, sub_add_sub_cancel, div_self h.ne'] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Convex.Star | {
"line": 161,
"column": 81
} | {
"line": 170,
"column": 22
} | {
"line": 172,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝³ : Semiring 𝕜\ninst✝² : PartialOrder 𝕜\ninst✝¹ : AddCommMonoid E\ninst✝ : Module 𝕜 E\nx : E\ns : Set E\nhx : x ∈ s\n⊢ StarConvex 𝕜 x s ↔ ∀ ⦃y : E⦄, y ∈ s → ∀ ⦃a b : 𝕜⦄, 0 < a → 0 < b → a + b = 1 → a • x + b • y ∈ s",
"ppTerm": "?m.47",
"assigned": true,
... | [] | by
refine ⟨fun h y hy a b ha hb hab => h hy ha.le hb.le hab, ?_⟩
intro h y hy a b ha hb hab
obtain rfl | ha := ha.eq_or_lt
· rw [zero_add] at hab
rwa [hab, one_smul, zero_smul, zero_add]
obtain rfl | hb := hb.eq_or_lt
· rw [add_zero] at hab
rwa [hab, one_smul, zero_smul, add_zero]
exact h hy ha hb... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.AffineSpace.AffineMap | {
"line": 570,
"column": 96
} | {
"line": 573,
"column": 22
} | {
"line": 575,
"column": 0
} | [
{
"pp": "k : Type u_1\nV1 : Type u_2\nP1 : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V1\ninst✝¹ : Module k V1\ninst✝ : AffineSpace V1 P1\np₀ p₁ : P1\nc : k\n⊢ (lineMap p₀ p₁) (1 - c) = (lineMap p₁ p₀) c",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instHSMul... | [] | by
rw [lineMap_symm p₀, comp_apply]
congr
simp [lineMap_apply] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.Ray | {
"line": 665,
"column": 2
} | {
"line": 665,
"column": 28
} | {
"line": 666,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : Field R\ninst✝³ : LinearOrder R\ninst✝² : IsStrictOrderedRing R\nM : Type u_2\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nx y : M\nhx : x ≠ 0\nhy : y ≠ 0\n⊢ (∃ r, 0 < r ∧ x = r • y) ↔ SameRay R y x",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
"Eq.... | [
"R : Type u_1\ninst✝⁴ : Field R\ninst✝³ : LinearOrder R\ninst✝² : IsStrictOrderedRing R\nM : Type u_2\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nx y : M\nhx : x ≠ 0\nhy : y ≠ 0\n⊢ (∃ r, 0 < r ∧ r • y = x) ↔ SameRay R y x"
] | simp_rw [eq_comm (a := x)] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.LinearAlgebra.Ray | {
"line": 671,
"column": 2
} | {
"line": 671,
"column": 28
} | {
"line": 672,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : Field R\ninst✝³ : LinearOrder R\ninst✝² : IsStrictOrderedRing R\nM : Type u_2\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nx y : M\nhy : y ≠ 0\n⊢ (∃ r, 0 < r ∧ x = r • y) ↔ SameRay R y x ∧ x ≠ 0",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
"Eq.mpr"... | [
"R : Type u_1\ninst✝⁴ : Field R\ninst✝³ : LinearOrder R\ninst✝² : IsStrictOrderedRing R\nM : Type u_2\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nx y : M\nhy : y ≠ 0\n⊢ (∃ r, 0 < r ∧ r • y = x) ↔ SameRay R y x ∧ x ≠ 0"
] | simp_rw [eq_comm (a := x)] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.LinearAlgebra.Ray | {
"line": 677,
"column": 2
} | {
"line": 677,
"column": 28
} | {
"line": 678,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : Field R\ninst✝³ : LinearOrder R\ninst✝² : IsStrictOrderedRing R\nM : Type u_2\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nx y : M\nhy : y ≠ 0\n⊢ (∃ r, 0 ≤ r ∧ x = r • y) ↔ SameRay R y x",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"R : Type u_1\ninst✝⁴ : Field R\ninst✝³ : LinearOrder R\ninst✝² : IsStrictOrderedRing R\nM : Type u_2\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nx y : M\nhy : y ≠ 0\n⊢ (∃ r, 0 ≤ r ∧ r • y = x) ↔ SameRay R y x"
] | simp_rw [eq_comm (a := x)] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Defs | {
"line": 98,
"column": 2
} | {
"line": 103,
"column": 10
} | {
"line": 105,
"column": 0
} | [
{
"pp": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns : Set P\n⊢ vectorSpan k s = ⊥ ↔ s.Subsingleton",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Set.not_subsingleton_iff",
"Submodule",
... | [] | refine ⟨fun h ↦ ?_, vectorSpan_of_subsingleton _⟩
by_contra hns
rw [Set.not_subsingleton_iff] at hns
obtain ⟨p, hp, q, hq, hpq⟩ := hns
have hpq' := vsub_mem_vectorSpan k hp hq
simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Defs | {
"line": 98,
"column": 2
} | {
"line": 103,
"column": 10
} | {
"line": 105,
"column": 0
} | [
{
"pp": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns : Set P\n⊢ vectorSpan k s = ⊥ ↔ s.Subsingleton",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Set.not_subsingleton_iff",
"Submodule",
... | [] | refine ⟨fun h ↦ ?_, vectorSpan_of_subsingleton _⟩
by_contra hns
rw [Set.not_subsingleton_iff] at hns
obtain ⟨p, hp, q, hq, hpq⟩ := hns
have hpq' := vsub_mem_vectorSpan k hp hq
simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Defs | {
"line": 216,
"column": 4
} | {
"line": 216,
"column": 15
} | {
"line": 216,
"column": 16
} | [
{
"pp": "case mp\nk : Type u_1\nV : Type u_2\ninst✝² : Ring k\ninst✝¹ : AddCommGroup V\ninst✝ : Module k V\ns : AffineSubspace k V\nhs : 0 ∈ s\na : V\nha : a ∈ ↑s\nb : V\nhb : b ∈ ↑s\n⊢ (fun x1 x2 ↦ x1 -ᵥ x2) a b ∈ ↑s",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Set... | [
"case mp\nk : Type u_1\nV : Type u_2\ninst✝² : Ring k\ninst✝¹ : AddCommGroup V\ninst✝ : Module k V\ns : AffineSubspace k V\nhs : 0 ∈ s\na : V\nha : a ∈ ↑s\nb : V\nhb : b ∈ ↑s\n⊢ a - b ∈ s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.AffineSpace.AffineMap | {
"line": 879,
"column": 17
} | {
"line": 879,
"column": 48
} | {
"line": 879,
"column": 49
} | [
{
"pp": "k : Type u_2\nV1 : Type u_3\nP1 : Type u_4\ninst✝⁵ : CommRing k\ninst✝⁴ : AddCommGroup V1\ninst✝³ : AffineSpace V1 P1\ninst✝² : Module k V1\ninst✝¹ : IsTorsionFree k V1\ninst✝ : IsCancelMulZero k\nc : P1\nr : k\nhr : r ≠ 0\nx✝¹ x✝ : P1\nh : (homothety c r) x✝¹ = (homothety c r) x✝\n⊢ x✝¹ = x✝",
"pp... | [
"k : Type u_2\nV1 : Type u_3\nP1 : Type u_4\ninst✝⁵ : CommRing k\ninst✝⁴ : AddCommGroup V1\ninst✝³ : AffineSpace V1 P1\ninst✝² : Module k V1\ninst✝¹ : IsTorsionFree k V1\ninst✝ : IsCancelMulZero k\nc : P1\nr : k\nhr : r ≠ 0\nx✝¹ x✝ : P1\nh : (homothety c r) x✝¹ = (homothety c r) x✝\n⊢ x✝¹ = x✝"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Defs | {
"line": 303,
"column": 15
} | {
"line": 303,
"column": 26
} | {
"line": 303,
"column": 27
} | [
{
"pp": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns : AffineSubspace k P\nv : V\np : P\nhp : p ∈ s\nh : v +ᵥ p ∈ s\n⊢ v ∈ s.direction",
"ppTerm": "?m.32",
"assigned": false,
"usedConstants": [],
"usedFVars":... | [
"k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns : AffineSubspace k P\nv : V\np : P\nhp : p ∈ s\nh : v +ᵥ p ∈ s\n⊢ v ∈ s.direction"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.Star | {
"line": 245,
"column": 2
} | {
"line": 245,
"column": 29
} | {
"line": 245,
"column": 30
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝³ : Semiring 𝕜\ninst✝² : PartialOrder 𝕜\ninst✝¹ : AddCommMonoid E\ninst✝ : Module 𝕜 E\nx z : E\ns : Set E\nhs : StarConvex 𝕜 (z + x) s\n⊢ StarConvex 𝕜 x ((fun x ↦ x + z) ⁻¹' s)",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝³ : Semiring 𝕜\ninst✝² : PartialOrder 𝕜\ninst✝¹ : AddCommMonoid E\ninst✝ : Module 𝕜 E\nx z : E\ns : Set E\nhs : StarConvex 𝕜 (z + x) s\n⊢ StarConvex 𝕜 x ((fun x ↦ z + x) ⁻¹' s)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.Star | {
"line": 275,
"column": 2
} | {
"line": 275,
"column": 13
} | {
"line": 275,
"column": 14
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝³ : CommSemiring 𝕜\ninst✝² : PartialOrder 𝕜\ninst✝¹ : AddCommMonoid E\ninst✝ : Module 𝕜 E\ns : Set E\nhs : StarConvex 𝕜 0 s\nc : 𝕜\n⊢ StarConvex 𝕜 0 (c • s)",
"ppTerm": "?m.23",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoa... | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝³ : CommSemiring 𝕜\ninst✝² : PartialOrder 𝕜\ninst✝¹ : AddCommMonoid E\ninst✝ : Module 𝕜 E\ns : Set E\nhs : StarConvex 𝕜 0 s\nc : 𝕜\n⊢ StarConvex 𝕜 0 (c • s)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.Star | {
"line": 302,
"column": 4
} | {
"line": 302,
"column": 94
} | {
"line": 303,
"column": 6
} | [
{
"pp": "case refine_1\n𝕜 : Type u_1\nE : Type u_2\ninst✝⁴ : Ring 𝕜\ninst✝³ : PartialOrder 𝕜\ninst✝² : AddRightMono 𝕜\ninst✝¹ : AddCommMonoid E\ninst✝ : SMulWithZero 𝕜 E\ns : Set E\nx : E\nx✝ : x ∈ s\nh : ∀ ⦃a b : 𝕜⦄, 0 ≤ a → 0 ≤ b → a + b = 1 → a • 0 + b • x ∈ s\na : 𝕜\nha₀ : 0 ≤ a\nha₁ : a ≤ 1\n⊢ a • x... | [
"case refine_1\n𝕜 : Type u_1\nE : Type u_2\ninst✝⁴ : Ring 𝕜\ninst✝³ : PartialOrder 𝕜\ninst✝² : AddRightMono 𝕜\ninst✝¹ : AddCommMonoid E\ninst✝ : SMulWithZero 𝕜 E\ns : Set E\nx : E\nx✝ : x ∈ s\nh : ∀ ⦃a b : 𝕜⦄, 0 ≤ a → 0 ≤ b → a + b = 1 → a • 0 + b • x ∈ s\na : 𝕜\nha₀ : 0 ≤ a\nha₁ : a ≤ 1\n⊢ a • x ∈ s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Defs | {
"line": 391,
"column": 4
} | {
"line": 391,
"column": 33
} | {
"line": 392,
"column": 6
} | [
{
"pp": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\np : P\ndirection : Submodule k V\nc : k\np₁ p₂ p₃ : P\nhp₁ : p₁ ∈ {q | q -ᵥ p ∈ direction}\nhp₂ : p₂ ∈ {q | q -ᵥ p ∈ direction}\nhp₃ : p₃ ∈ {q | q -ᵥ p ∈ direction}\n⊢ c • (... | [
"k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\np : P\ndirection : Submodule k V\nc : k\np₁ p₂ p₃ : P\nhp₁ : p₁ ∈ {q | q -ᵥ p ∈ direction}\nhp₂ : p₂ ∈ {q | q -ᵥ p ∈ direction}\nhp₃ : p₃ ∈ {q | q -ᵥ p ∈ direction}\n⊢ c • (p₁ -ᵥ p₂) + ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Defs | {
"line": 425,
"column": 4
} | {
"line": 425,
"column": 15
} | {
"line": 425,
"column": 16
} | [
{
"pp": "case mp\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\np : P\ndirection : Submodule k V\np₁ : P\nhp₁ : p₁ ∈ mk' p direction\np₂ : P\nhp₂ : p₂ ∈ mk' p direction\n⊢ p₁ -ᵥ p₂ ∈ direction",
"ppTerm": "?mp",
"assigne... | [
"case mp\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\np : P\ndirection : Submodule k V\np₁ : P\nhp₁ : p₁ ∈ mk' p direction\np₂ : P\nhp₂ : p₂ ∈ mk' p direction\n⊢ p₁ -ᵥ p₂ ∈ direction"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.Star | {
"line": 321,
"column": 2
} | {
"line": 321,
"column": 61
} | {
"line": 323,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁴ : Ring 𝕜\ninst✝³ : PartialOrder 𝕜\ninst✝² : AddRightMono 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\nx y : E\ns : Set E\nhs : StarConvex 𝕜 x s\nhy : x + y ∈ s\nt : 𝕜\nht₀ : 0 ≤ t\nht₁ : t ≤ 1\nh : x + t • y = (1 - t) • x + t • (x + y)\n⊢ (1 - t) • x + t • (... | [] | exact hs hy (sub_nonneg_of_le ht₁) ht₀ (sub_add_cancel _ _) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Convex.Star | {
"line": 324,
"column": 36
} | {
"line": 324,
"column": 47
} | {
"line": 324,
"column": 48
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁴ : Ring 𝕜\ninst✝³ : PartialOrder 𝕜\ninst✝² : AddRightMono 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\nx : E\ns : Set E\nhs : StarConvex 𝕜 0 s\nhx : x ∈ s\nt : 𝕜\nht₀ : 0 ≤ t\nht₁ : t ≤ 1\n⊢ t • x ∈ s",
"ppTerm": "?m.32",
"assigned": false,
"usedC... | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝⁴ : Ring 𝕜\ninst✝³ : PartialOrder 𝕜\ninst✝² : AddRightMono 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\nx : E\ns : Set E\nhs : StarConvex 𝕜 0 s\nhx : x ∈ s\nt : 𝕜\nht₀ : 0 ≤ t\nht₁ : t ≤ 1\n⊢ t • x ∈ s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.Star | {
"line": 324,
"column": 68
} | {
"line": 324,
"column": 79
} | {
"line": 324,
"column": 80
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁴ : Ring 𝕜\ninst✝³ : PartialOrder 𝕜\ninst✝² : AddRightMono 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\nx : E\ns : Set E\nhs : StarConvex 𝕜 0 s\nhx : x ∈ s\nt : 𝕜\nht₀ : 0 ≤ t\nht₁ : t ≤ 1\n⊢ 0 + x ∈ s",
"ppTerm": "?m.43",
"assigned": true,
"usedCo... | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝⁴ : Ring 𝕜\ninst✝³ : PartialOrder 𝕜\ninst✝² : AddRightMono 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\nx : E\ns : Set E\nhs : StarConvex 𝕜 0 s\nhx : x ∈ s\nt : 𝕜\nht₀ : 0 ≤ t\nht₁ : t ≤ 1\n⊢ x ∈ s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.Basic | {
"line": 239,
"column": 2
} | {
"line": 239,
"column": 29
} | {
"line": 239,
"column": 30
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝³ : Semiring 𝕜\ninst✝² : PartialOrder 𝕜\ninst✝¹ : AddCommMonoid E\ninst✝ : Module 𝕜 E\ns : Set E\nhs : Convex 𝕜 s\nz : E\n⊢ Convex 𝕜 ((fun x ↦ x + z) ⁻¹' s)",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
... | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝³ : Semiring 𝕜\ninst✝² : PartialOrder 𝕜\ninst✝¹ : AddCommMonoid E\ninst✝ : Module 𝕜 E\ns : Set E\nhs : Convex 𝕜 s\nz : E\n⊢ Convex 𝕜 ((fun x ↦ z + x) ⁻¹' s)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.Basic | {
"line": 321,
"column": 56
} | {
"line": 321,
"column": 67
} | {
"line": 321,
"column": 68
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝¹⁰ : Semiring 𝕜\ninst✝⁹ : PartialOrder 𝕜\ninst✝⁸ : AddCommMonoid E\ninst✝⁷ : ZeroLEOneClass 𝕜\ninst✝⁶ : Module 𝕜 E\nR : Type u_5\ninst✝⁵ : Semiring R\ninst✝⁴ : PartialOrder R\ninst✝³ : Module R E\ninst✝² : Module R 𝕜\ninst✝¹ : IsScalarTower R 𝕜 E\ninst✝ : SMulPos... | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝¹⁰ : Semiring 𝕜\ninst✝⁹ : PartialOrder 𝕜\ninst✝⁸ : AddCommMonoid E\ninst✝⁷ : ZeroLEOneClass 𝕜\ninst✝⁶ : Module 𝕜 E\nR : Type u_5\ninst✝⁵ : Semiring R\ninst✝⁴ : PartialOrder R\ninst✝³ : Module R E\ninst✝² : Module R 𝕜\ninst✝¹ : IsScalarTower R 𝕜 E\ninst✝ : SMulPosMono R 𝕜\ns... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.Basic | {
"line": 322,
"column": 28
} | {
"line": 322,
"column": 50
} | {
"line": 322,
"column": 51
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝¹⁰ : Semiring 𝕜\ninst✝⁹ : PartialOrder 𝕜\ninst✝⁸ : AddCommMonoid E\ninst✝⁷ : ZeroLEOneClass 𝕜\ninst✝⁶ : Module 𝕜 E\nR : Type u_5\ninst✝⁵ : Semiring R\ninst✝⁴ : PartialOrder R\ninst✝³ : Module R E\ninst✝² : Module R 𝕜\ninst✝¹ : IsScalarTower R 𝕜 E\ninst✝ : SMulPos... | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝¹⁰ : Semiring 𝕜\ninst✝⁹ : PartialOrder 𝕜\ninst✝⁸ : AddCommMonoid E\ninst✝⁷ : ZeroLEOneClass 𝕜\ninst✝⁶ : Module 𝕜 E\nR : Type u_5\ninst✝⁵ : Semiring R\ninst✝⁴ : PartialOrder R\ninst✝³ : Module R E\ninst✝² : Module R 𝕜\ninst✝¹ : IsScalarTower R 𝕜 E\ninst✝ : SMulPosMono R 𝕜\ns... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Bornology.Absorbs | {
"line": 210,
"column": 2
} | {
"line": 210,
"column": 35
} | {
"line": 210,
"column": 36
} | [
{
"pp": "M : Type u_1\nE : Type u_2\ninst✝³ : Monoid M\ninst✝² : AddGroup E\ninst✝¹ : DistribMulAction M E\ninst✝ : Bornology M\ns₁ s₂ t₁ t₂ : Set E\nh₁ : Absorbs M s₁ t₁\nh₂ : Absorbs M s₂ t₂\n⊢ Absorbs M (s₁ - s₂) (t₁ - t₂)",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"AddGroup.... | [
"M : Type u_1\nE : Type u_2\ninst✝³ : Monoid M\ninst✝² : AddGroup E\ninst✝¹ : DistribMulAction M E\ninst✝ : Bornology M\ns₁ s₂ t₁ t₂ : Set E\nh₁ : Absorbs M s₁ t₁\nh₂ : Absorbs M s₂ t₂\n⊢ Absorbs M (s₁ + -s₂) (t₁ + -t₂)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.Basic | {
"line": 446,
"column": 14
} | {
"line": 446,
"column": 25
} | {
"line": 446,
"column": 26
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝³ : Ring 𝕜\ninst✝² : PartialOrder 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns : Set E\na : E\nh : Convex 𝕜 (a +ᵥ s)\n⊢ Convex 𝕜 s",
"ppTerm": "?m.22",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝³ : Ring 𝕜\ninst✝² : PartialOrder 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns : Set E\na : E\nh : Convex 𝕜 (a +ᵥ s)\n⊢ Convex 𝕜 s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.Basic | {
"line": 478,
"column": 2
} | {
"line": 478,
"column": 13
} | {
"line": 478,
"column": 14
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁴ : Ring 𝕜\ninst✝³ : PartialOrder 𝕜\ninst✝² : AddCommGroup E\ninst✝¹ : Module 𝕜 E\ns : Set E\ninst✝ : AddRightMono 𝕜\nhs : Convex 𝕜 s\nx : E\nzero_mem : 0 ∈ s\nhx : x ∈ s\nt : 𝕜\nht : t ∈ Icc 0 1\n⊢ t • x ∈ s",
"ppTerm": "?m.36",
"assigned": false,
"u... | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝⁴ : Ring 𝕜\ninst✝³ : PartialOrder 𝕜\ninst✝² : AddCommGroup E\ninst✝¹ : Module 𝕜 E\ns : Set E\ninst✝ : AddRightMono 𝕜\nhs : Convex 𝕜 s\nx : E\nzero_mem : 0 ∈ s\nhx : x ∈ s\nt : 𝕜\nht : t ∈ Icc 0 1\n⊢ t • x ∈ s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.Basic | {
"line": 478,
"column": 43
} | {
"line": 478,
"column": 54
} | {
"line": 478,
"column": 55
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁴ : Ring 𝕜\ninst✝³ : PartialOrder 𝕜\ninst✝² : AddCommGroup E\ninst✝¹ : Module 𝕜 E\ns : Set E\ninst✝ : AddRightMono 𝕜\nhs : Convex 𝕜 s\nx : E\nzero_mem : 0 ∈ s\nhx : x ∈ s\nt : 𝕜\nht : t ∈ Icc 0 1\n⊢ 0 + x ∈ s",
"ppTerm": "?m.47",
"assigned": true,
"us... | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝⁴ : Ring 𝕜\ninst✝³ : PartialOrder 𝕜\ninst✝² : AddCommGroup E\ninst✝¹ : Module 𝕜 E\ns : Set E\ninst✝ : AddRightMono 𝕜\nhs : Convex 𝕜 s\nx : E\nzero_mem : 0 ∈ s\nhx : x ∈ s\nt : 𝕜\nht : t ∈ Icc 0 1\n⊢ x ∈ s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.Basic | {
"line": 482,
"column": 2
} | {
"line": 482,
"column": 70
} | {
"line": 482,
"column": 71
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁴ : Ring 𝕜\ninst✝³ : PartialOrder 𝕜\ninst✝² : AddCommGroup E\ninst✝¹ : Module 𝕜 E\ns : Set E\ninst✝ : AddRightMono 𝕜\nh : Convex 𝕜 s\nx y : E\nhx : x ∈ s\nhy : y ∈ s\n⊢ MapsTo (⇑(AffineMap.lineMap x y)) (Icc 0 1) s",
"ppTerm": "?m.34",
"assigned": true,
... | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝⁴ : Ring 𝕜\ninst✝³ : PartialOrder 𝕜\ninst✝² : AddCommGroup E\ninst✝¹ : Module 𝕜 E\ns : Set E\ninst✝ : AddRightMono 𝕜\nh : Convex 𝕜 s\nx y : E\nhx : x ∈ s\nhy : y ∈ s\n⊢ ⇑(AffineMap.lineMap x y) '' Icc 0 1 ⊆ s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Defs | {
"line": 940,
"column": 2
} | {
"line": 940,
"column": 13
} | {
"line": 940,
"column": 14
} | [
{
"pp": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝² : Ring k\ninst✝¹ : AddCommGroup V\ninst✝ : Module k V\nS : AffineSpace V P\np : P\nd₁ d₂ : Submodule k V\nh : ∀ ⦃x : P⦄, x -ᵥ p ∈ d₁ → x -ᵥ p ∈ d₂\nx : V\nhx : x ∈ d₁\n⊢ x ∈ d₂",
"ppTerm": "?m.36",
"assigned": false,
"usedConstants": [],
... | [
"k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝² : Ring k\ninst✝¹ : AddCommGroup V\ninst✝ : Module k V\nS : AffineSpace V P\np : P\nd₁ d₂ : Submodule k V\nh : ∀ ⦃x : P⦄, x -ᵥ p ∈ d₁ → x -ᵥ p ∈ d₂\nx : V\nhx : x ∈ d₁\n⊢ x ∈ d₂"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Defs | {
"line": 940,
"column": 44
} | {
"line": 940,
"column": 55
} | {
"line": 940,
"column": 56
} | [
{
"pp": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝² : Ring k\ninst✝¹ : AddCommGroup V\ninst✝ : Module k V\nS : AffineSpace V P\np : P\nd₁ d₂ : Submodule k V\nh : ∀ ⦃x : P⦄, x -ᵥ p ∈ d₁ → x -ᵥ p ∈ d₂\nx : V\nhx : x ∈ d₁\n⊢ (x +ᵥ p) -ᵥ p ∈ d₁",
"ppTerm": "?m.54",
"assigned": true,
"usedConstants... | [
"k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝² : Ring k\ninst✝¹ : AddCommGroup V\ninst✝ : Module k V\nS : AffineSpace V P\np : P\nd₁ d₂ : Submodule k V\nh : ∀ ⦃x : P⦄, x -ᵥ p ∈ d₁ → x -ᵥ p ∈ d₂\nx : V\nhx : x ∈ d₁\n⊢ x ∈ d₁"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.LocallyConvex.BalancedCoreHull | {
"line": 172,
"column": 2
} | {
"line": 172,
"column": 29
} | {
"line": 172,
"column": 30
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : NormedDivisionRing 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns : Set E\nx : E\nhx : x ∈ balancedCoreAux 𝕜 s\n⊢ x ∈ s",
"ppTerm": "?m.10",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝² : NormedDivisionRing 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns : Set E\nx : E\nhx : x ∈ balancedCoreAux 𝕜 s\n⊢ x ∈ s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.LocallyConvex.Basic | {
"line": 136,
"column": 15
} | {
"line": 136,
"column": 26
} | {
"line": 136,
"column": 27
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_3\ninst✝³ : SeminormedRing 𝕜\ninst✝² : AddCommGroup E\ninst✝¹ : Module 𝕜 E\ns : Set E\ninst✝ : NormOneClass 𝕜\nh : Balanced 𝕜 s\nx : E\nhx : -x ∈ s\n⊢ x ∈ s",
"ppTerm": "?m.19",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
... | [
"𝕜 : Type u_1\nE : Type u_3\ninst✝³ : SeminormedRing 𝕜\ninst✝² : AddCommGroup E\ninst✝¹ : Module 𝕜 E\ns : Set E\ninst✝ : NormOneClass 𝕜\nh : Balanced 𝕜 s\nx : E\nhx : -x ∈ s\n⊢ x ∈ s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.LocallyConvex.Basic | {
"line": 137,
"column": 16
} | {
"line": 137,
"column": 27
} | {
"line": 137,
"column": 28
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_3\ninst✝³ : SeminormedRing 𝕜\ninst✝² : AddCommGroup E\ninst✝¹ : Module 𝕜 E\ns : Set E\ninst✝ : NormOneClass 𝕜\nh : Balanced 𝕜 s\nx : E\nhx : x ∈ s\n⊢ -x ∈ s",
"ppTerm": "?m.22",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
... | [
"𝕜 : Type u_1\nE : Type u_3\ninst✝³ : SeminormedRing 𝕜\ninst✝² : AddCommGroup E\ninst✝¹ : Module 𝕜 E\ns : Set E\ninst✝ : NormOneClass 𝕜\nh : Balanced 𝕜 s\nx : E\nhx : x ∈ s\n⊢ -x ∈ s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.LocallyConvex.Basic | {
"line": 208,
"column": 28
} | {
"line": 208,
"column": 39
} | {
"line": 208,
"column": 40
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_3\ninst✝² : NormedDivisionRing 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns : Set E\na : 𝕜\nhs : Balanced 𝕜 s\nha : ‖1‖ ≤ ‖a‖\n⊢ s ⊆ a • s",
"ppTerm": "?m.35",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"𝕜 : Type u_1\nE : Type u_3\ninst✝² : NormedDivisionRing 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns : Set E\na : 𝕜\nhs : Balanced 𝕜 s\nha : ‖1‖ ≤ ‖a‖\n⊢ s ⊆ a • s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.LocallyConvex.Basic | {
"line": 245,
"column": 4
} | {
"line": 245,
"column": 78
} | {
"line": 247,
"column": 0
} | [
{
"pp": "case inr\n𝕜 : Type u_1\nE : Type u_3\ninst✝⁴ : NormedField 𝕜\ninst✝³ : AddCommGroup E\ninst✝² : Module 𝕜 E\nA : Set E\ninst✝¹ : TopologicalSpace E\ninst✝ : ContinuousSMul 𝕜 E\nhA : Balanced 𝕜 A\na : 𝕜\nha : ‖a‖ ≤ 1\nh : a ≠ 0\n⊢ (fun x ↦ a • x) '' interior A ⊆ interior A",
"ppTerm": "?inr",
... | [] | exact ((isOpenMap_smul₀ h).mapsTo_interior <| hA.smul_mem ha).image_subset | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.MetricSpace.Equicontinuity | {
"line": 108,
"column": 20
} | {
"line": 108,
"column": 72
} | {
"line": 108,
"column": 73
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : PseudoMetricSpace α\nι : Type u_4\ninst✝ : PseudoMetricSpace β\nb : ℝ → ℝ\nb_lim : Tendsto b (𝓝 0) (𝓝 0)\nF : ι → β → α\nH : ∀ (x y : β) (i : ι), dist (F i x) (F i y) ≤ b (dist x y)\nε : ℝ\nε0 : ε > 0\nδ : ℝ\nδ0 : δ > 0\nhδ : ∀ ⦃x : ℝ⦄, dist x 0 < δ → dist (b x) 0... | [
"α : Type u_1\nβ : Type u_2\ninst✝¹ : PseudoMetricSpace α\nι : Type u_4\ninst✝ : PseudoMetricSpace β\nb : ℝ → ℝ\nb_lim : Tendsto b (𝓝 0) (𝓝 0)\nF : ι → β → α\nH : ∀ (x y : β) (i : ι), dist (F i x) (F i y) ≤ b (dist x y)\nε : ℝ\nε0 : ε > 0\nδ : ℝ\nδ0 : δ > 0\nhδ : ∀ ⦃x : ℝ⦄, dist x 0 < δ → dist (b x) 0 < ε\nx y : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.LocallyConvex.Bounded | {
"line": 204,
"column": 2
} | {
"line": 204,
"column": 47
} | {
"line": 204,
"column": 48
} | [
{
"pp": "E : Type u_3\nF : Type u_4\n𝕜₁ : Type u_6\n𝕜₂ : Type u_7\ninst✝⁹ : NormedDivisionRing 𝕜₁\ninst✝⁸ : NormedDivisionRing 𝕜₂\ninst✝⁷ : AddCommGroup E\ninst✝⁶ : Module 𝕜₁ E\ninst✝⁵ : AddCommGroup F\ninst✝⁴ : Module 𝕜₂ F\ninst✝³ : TopologicalSpace E\ninst✝² : TopologicalSpace F\nσ : 𝕜₁ →+* 𝕜₂\ninst✝¹... | [
"E : Type u_3\nF : Type u_4\n𝕜₁ : Type u_6\n𝕜₂ : Type u_7\ninst✝⁹ : NormedDivisionRing 𝕜₁\ninst✝⁸ : NormedDivisionRing 𝕜₂\ninst✝⁷ : AddCommGroup E\ninst✝⁶ : Module 𝕜₁ E\ninst✝⁵ : AddCommGroup F\ninst✝⁴ : Module 𝕜₂ F\ninst✝³ : TopologicalSpace E\ninst✝² : TopologicalSpace F\nσ : 𝕜₁ →+* 𝕜₂\ninst✝¹ : RingHomSu... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.LocallyConvex.Bounded | {
"line": 309,
"column": 14
} | {
"line": 309,
"column": 25
} | {
"line": 309,
"column": 26
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_3\ninst✝⁵ : NormedField 𝕜\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module 𝕜 E\ninst✝² : TopologicalSpace E\ninst✝¹ : ContinuousSMul 𝕜 E\ninst✝ : ContinuousAdd E\ns : Set E\nx : E\nh : IsVonNBounded 𝕜 (x +ᵥ s)\n⊢ IsVonNBounded 𝕜 s",
"ppTerm": "?m.25",
"assigned": false,
... | [
"𝕜 : Type u_1\nE : Type u_3\ninst✝⁵ : NormedField 𝕜\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module 𝕜 E\ninst✝² : TopologicalSpace E\ninst✝¹ : ContinuousSMul 𝕜 E\ninst✝ : ContinuousAdd E\ns : Set E\nx : E\nh : IsVonNBounded 𝕜 (x +ᵥ s)\n⊢ IsVonNBounded 𝕜 s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.Function | {
"line": 101,
"column": 4
} | {
"line": 101,
"column": 71
} | {
"line": 101,
"column": 72
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nβ : Type u_5\ninst✝⁶ : Semiring 𝕜\ninst✝⁵ : PartialOrder 𝕜\ninst✝⁴ : AddCommMonoid E\ninst✝³ : AddCommMonoid β\ninst✝² : PartialOrder β\ninst✝¹ : SMul 𝕜 E\ninst✝ : SMul 𝕜 β\ns : Set E\nf g : E → β\nhf : ConvexOn 𝕜 s f\nhfg : EqOn f g s\nx : E\nhx : x ∈ s\ny : E\nhy : y... | [
"𝕜 : Type u_1\nE : Type u_2\nβ : Type u_5\ninst✝⁶ : Semiring 𝕜\ninst✝⁵ : PartialOrder 𝕜\ninst✝⁴ : AddCommMonoid E\ninst✝³ : AddCommMonoid β\ninst✝² : PartialOrder β\ninst✝¹ : SMul 𝕜 E\ninst✝ : SMul 𝕜 β\ns : Set E\nf g : E → β\nhf : ConvexOn 𝕜 s f\nhfg : EqOn f g s\nx : E\nhx : x ∈ s\ny : E\nhy : y ∈ s\na b : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.Function | {
"line": 105,
"column": 4
} | {
"line": 105,
"column": 71
} | {
"line": 105,
"column": 72
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nβ : Type u_5\ninst✝⁶ : Semiring 𝕜\ninst✝⁵ : PartialOrder 𝕜\ninst✝⁴ : AddCommMonoid E\ninst✝³ : AddCommMonoid β\ninst✝² : PartialOrder β\ninst✝¹ : SMul 𝕜 E\ninst✝ : SMul 𝕜 β\ns : Set E\nf g : E → β\nhf : ConcaveOn 𝕜 s f\nhfg : EqOn f g s\nx : E\nhx : x ∈ s\ny : E\nhy : ... | [
"𝕜 : Type u_1\nE : Type u_2\nβ : Type u_5\ninst✝⁶ : Semiring 𝕜\ninst✝⁵ : PartialOrder 𝕜\ninst✝⁴ : AddCommMonoid E\ninst✝³ : AddCommMonoid β\ninst✝² : PartialOrder β\ninst✝¹ : SMul 𝕜 E\ninst✝ : SMul 𝕜 β\ns : Set E\nf g : E → β\nhf : ConcaveOn 𝕜 s f\nhfg : EqOn f g s\nx : E\nhx : x ∈ s\ny : E\nhy : y ∈ s\na b :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.Function | {
"line": 110,
"column": 4
} | {
"line": 110,
"column": 77
} | {
"line": 111,
"column": 6
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nβ : Type u_5\ninst✝⁶ : Semiring 𝕜\ninst✝⁵ : PartialOrder 𝕜\ninst✝⁴ : AddCommMonoid E\ninst✝³ : AddCommMonoid β\ninst✝² : PartialOrder β\ninst✝¹ : SMul 𝕜 E\ninst✝ : SMul 𝕜 β\ns : Set E\nf g : E → β\nhf : StrictConvexOn 𝕜 s f\nhfg : EqOn f g s\nx : E\nhx : x ∈ s\ny : E\n... | [
"𝕜 : Type u_1\nE : Type u_2\nβ : Type u_5\ninst✝⁶ : Semiring 𝕜\ninst✝⁵ : PartialOrder 𝕜\ninst✝⁴ : AddCommMonoid E\ninst✝³ : AddCommMonoid β\ninst✝² : PartialOrder β\ninst✝¹ : SMul 𝕜 E\ninst✝ : SMul 𝕜 β\ns : Set E\nf g : E → β\nhf : StrictConvexOn 𝕜 s f\nhfg : EqOn f g s\nx : E\nhx : x ∈ s\ny : E\nhy : y ∈ s\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.Function | {
"line": 116,
"column": 4
} | {
"line": 116,
"column": 77
} | {
"line": 117,
"column": 6
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nβ : Type u_5\ninst✝⁶ : Semiring 𝕜\ninst✝⁵ : PartialOrder 𝕜\ninst✝⁴ : AddCommMonoid E\ninst✝³ : AddCommMonoid β\ninst✝² : PartialOrder β\ninst✝¹ : SMul 𝕜 E\ninst✝ : SMul 𝕜 β\ns : Set E\nf g : E → β\nhf : StrictConcaveOn 𝕜 s f\nhfg : EqOn f g s\nx : E\nhx : x ∈ s\ny : E\... | [
"𝕜 : Type u_1\nE : Type u_2\nβ : Type u_5\ninst✝⁶ : Semiring 𝕜\ninst✝⁵ : PartialOrder 𝕜\ninst✝⁴ : AddCommMonoid E\ninst✝³ : AddCommMonoid β\ninst✝² : PartialOrder β\ninst✝¹ : SMul 𝕜 E\ninst✝ : SMul 𝕜 β\ns : Set E\nf g : E → β\nhf : StrictConcaveOn 𝕜 s f\nhfg : EqOn f g s\nx : E\nhx : x ∈ s\ny : E\nhy : y ∈ s\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.Function | {
"line": 298,
"column": 2
} | {
"line": 298,
"column": 31
} | {
"line": 298,
"column": 32
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nβ : Type u_5\ninst✝⁶ : Semiring 𝕜\ninst✝⁵ : PartialOrder 𝕜\ninst✝⁴ : AddCommMonoid E\ninst✝³ : AddCommMonoid β\ninst✝² : PartialOrder β\ninst✝¹ : Module 𝕜 E\ninst✝ : SMul 𝕜 β\ns : Set E\nf : E → β\nhf : ConvexOn 𝕜 s f\nc : E\n⊢ ConvexOn 𝕜 ((fun z ↦ c + z) ⁻¹' s) (f ∘ ... | [
"𝕜 : Type u_1\nE : Type u_2\nβ : Type u_5\ninst✝⁶ : Semiring 𝕜\ninst✝⁵ : PartialOrder 𝕜\ninst✝⁴ : AddCommMonoid E\ninst✝³ : AddCommMonoid β\ninst✝² : PartialOrder β\ninst✝¹ : Module 𝕜 E\ninst✝ : SMul 𝕜 β\ns : Set E\nf : E → β\nhf : ConvexOn 𝕜 s f\nc : E\n⊢ ConvexOn 𝕜 ((fun z ↦ z + c) ⁻¹' s) (f ∘ fun z ↦ z + ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.Strict | {
"line": 205,
"column": 2
} | {
"line": 205,
"column": 29
} | {
"line": 205,
"column": 30
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_3\ninst✝⁵ : Semiring 𝕜\ninst✝⁴ : PartialOrder 𝕜\ninst✝³ : TopologicalSpace E\ninst✝² : AddCancelCommMonoid E\ninst✝¹ : ContinuousAdd E\ninst✝ : Module 𝕜 E\ns : Set E\nhs : StrictConvex 𝕜 s\nz : E\n⊢ StrictConvex 𝕜 ((fun x ↦ x + z) ⁻¹' s)",
"ppTerm": "?m.24",
"assi... | [
"𝕜 : Type u_1\nE : Type u_3\ninst✝⁵ : Semiring 𝕜\ninst✝⁴ : PartialOrder 𝕜\ninst✝³ : TopologicalSpace E\ninst✝² : AddCancelCommMonoid E\ninst✝¹ : ContinuousAdd E\ninst✝ : Module 𝕜 E\ns : Set E\nhs : StrictConvex 𝕜 s\nz : E\n⊢ StrictConvex 𝕜 ((fun x ↦ z + x) ⁻¹' s)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.Strict | {
"line": 233,
"column": 2
} | {
"line": 233,
"column": 34
} | {
"line": 233,
"column": 35
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_3\ninst✝⁵ : Semiring 𝕜\ninst✝⁴ : PartialOrder 𝕜\ninst✝³ : TopologicalSpace E\ninst✝² : AddCommGroup E\ninst✝¹ : Module 𝕜 E\ninst✝ : ContinuousAdd E\ns : Set E\nhs : StrictConvex 𝕜 s\nz : E\n⊢ StrictConvex 𝕜 ((fun x ↦ z + x) '' s)",
"ppTerm": "?m.26",
"assigned": f... | [
"𝕜 : Type u_1\nE : Type u_3\ninst✝⁵ : Semiring 𝕜\ninst✝⁴ : PartialOrder 𝕜\ninst✝³ : TopologicalSpace E\ninst✝² : AddCommGroup E\ninst✝¹ : Module 𝕜 E\ninst✝ : ContinuousAdd E\ns : Set E\nhs : StrictConvex 𝕜 s\nz : E\n⊢ StrictConvex 𝕜 ((fun x ↦ z + x) '' s)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.Strict | {
"line": 236,
"column": 49
} | {
"line": 236,
"column": 76
} | {
"line": 236,
"column": 77
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_3\ninst✝⁵ : Semiring 𝕜\ninst✝⁴ : PartialOrder 𝕜\ninst✝³ : TopologicalSpace E\ninst✝² : AddCommGroup E\ninst✝¹ : Module 𝕜 E\ninst✝ : ContinuousAdd E\ns : Set E\nhs : StrictConvex 𝕜 s\nz : E\n⊢ StrictConvex 𝕜 ((fun x ↦ x + z) '' s)",
"ppTerm": "?m.26",
"assigned": t... | [
"𝕜 : Type u_1\nE : Type u_3\ninst✝⁵ : Semiring 𝕜\ninst✝⁴ : PartialOrder 𝕜\ninst✝³ : TopologicalSpace E\ninst✝² : AddCommGroup E\ninst✝¹ : Module 𝕜 E\ninst✝ : ContinuousAdd E\ns : Set E\nhs : StrictConvex 𝕜 s\nz : E\n⊢ StrictConvex 𝕜 ((fun a ↦ z + a) '' s)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.Function | {
"line": 789,
"column": 4
} | {
"line": 789,
"column": 26
} | {
"line": 789,
"column": 27
} | [
{
"pp": "case mp\n𝕜 : Type u_1\nE : Type u_2\nβ : Type u_5\ninst✝⁷ : Semiring 𝕜\ninst✝⁶ : PartialOrder 𝕜\ninst✝⁵ : AddCommMonoid E\ninst✝⁴ : AddCommGroup β\ninst✝³ : PartialOrder β\ninst✝² : IsOrderedAddMonoid β\ninst✝¹ : SMul 𝕜 E\ninst✝ : Module 𝕜 β\ns : Set E\nf : E → β\nhconv : Convex 𝕜 s\nh :\n ∀ ⦃x ... | [
"case mp\n𝕜 : Type u_1\nE : Type u_2\nβ : Type u_5\ninst✝⁷ : Semiring 𝕜\ninst✝⁶ : PartialOrder 𝕜\ninst✝⁵ : AddCommMonoid E\ninst✝⁴ : AddCommGroup β\ninst✝³ : PartialOrder β\ninst✝² : IsOrderedAddMonoid β\ninst✝¹ : SMul 𝕜 E\ninst✝ : Module 𝕜 β\ns : Set E\nf : E → β\nhconv : Convex 𝕜 s\nh :\n ∀ ⦃x : E⦄,\n x... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.Strict | {
"line": 312,
"column": 2
} | {
"line": 312,
"column": 13
} | {
"line": 312,
"column": 14
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_3\ninst✝⁵ : Ring 𝕜\ninst✝⁴ : PartialOrder 𝕜\ninst✝³ : TopologicalSpace E\ninst✝² : AddCommGroup E\ninst✝¹ : Module 𝕜 E\ns : Set E\nx : E\ninst✝ : AddRightStrictMono 𝕜\nhs : StrictConvex 𝕜 s\nzero_mem : 0 ∈ s\nhx : x ∈ s\nhx₀ : x ≠ 0\nt : 𝕜\nht₀ : 0 < t\nht₁ : t < 1\n⊢ t ... | [
"𝕜 : Type u_1\nE : Type u_3\ninst✝⁵ : Ring 𝕜\ninst✝⁴ : PartialOrder 𝕜\ninst✝³ : TopologicalSpace E\ninst✝² : AddCommGroup E\ninst✝¹ : Module 𝕜 E\ns : Set E\nx : E\ninst✝ : AddRightStrictMono 𝕜\nhs : StrictConvex 𝕜 s\nzero_mem : 0 ∈ s\nhx : x ∈ s\nhx₀ : x ≠ 0\nt : 𝕜\nht₀ : 0 < t\nht₁ : t < 1\n⊢ t • x ∈ interi... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.Strict | {
"line": 312,
"column": 43
} | {
"line": 312,
"column": 54
} | {
"line": 312,
"column": 55
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_3\ninst✝⁵ : Ring 𝕜\ninst✝⁴ : PartialOrder 𝕜\ninst✝³ : TopologicalSpace E\ninst✝² : AddCommGroup E\ninst✝¹ : Module 𝕜 E\ns : Set E\nx : E\ninst✝ : AddRightStrictMono 𝕜\nhs : StrictConvex 𝕜 s\nzero_mem : 0 ∈ s\nhx : x ∈ s\nhx₀ : x ≠ 0\nt : 𝕜\nht₀ : 0 < t\nht₁ : t < 1\n⊢ 0 ... | [
"𝕜 : Type u_1\nE : Type u_3\ninst✝⁵ : Ring 𝕜\ninst✝⁴ : PartialOrder 𝕜\ninst✝³ : TopologicalSpace E\ninst✝² : AddCommGroup E\ninst✝¹ : Module 𝕜 E\ns : Set E\nx : E\ninst✝ : AddRightStrictMono 𝕜\nhs : StrictConvex 𝕜 s\nzero_mem : 0 ∈ s\nhx : x ∈ s\nhx₀ : x ≠ 0\nt : 𝕜\nht₀ : 0 < t\nht₁ : t < 1\n⊢ x ∈ s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.Function | {
"line": 886,
"column": 2
} | {
"line": 886,
"column": 29
} | {
"line": 886,
"column": 30
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nβ : Type u_5\ninst✝⁶ : Semiring 𝕜\ninst✝⁵ : PartialOrder 𝕜\ninst✝⁴ : AddCancelCommMonoid E\ninst✝³ : AddCommMonoid β\ninst✝² : PartialOrder β\ninst✝¹ : Module 𝕜 E\ninst✝ : SMul 𝕜 β\ns : Set E\nf : E → β\nhf : StrictConvexOn 𝕜 s f\nc : E\n⊢ StrictConvexOn 𝕜 ((fun z ↦ c... | [
"𝕜 : Type u_1\nE : Type u_2\nβ : Type u_5\ninst✝⁶ : Semiring 𝕜\ninst✝⁵ : PartialOrder 𝕜\ninst✝⁴ : AddCancelCommMonoid E\ninst✝³ : AddCommMonoid β\ninst✝² : PartialOrder β\ninst✝¹ : Module 𝕜 E\ninst✝ : SMul 𝕜 β\ns : Set E\nf : E → β\nhf : StrictConvexOn 𝕜 s f\nc : E\n⊢ StrictConvexOn 𝕜 ((fun z ↦ c + z) ⁻¹' s)... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.Function | {
"line": 891,
"column": 2
} | {
"line": 891,
"column": 29
} | {
"line": 891,
"column": 30
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nβ : Type u_5\ninst✝⁶ : Semiring 𝕜\ninst✝⁵ : PartialOrder 𝕜\ninst✝⁴ : AddCancelCommMonoid E\ninst✝³ : AddCommMonoid β\ninst✝² : PartialOrder β\ninst✝¹ : Module 𝕜 E\ninst✝ : SMul 𝕜 β\ns : Set E\nf : E → β\nhf : StrictConcaveOn 𝕜 s f\nc : E\n⊢ StrictConcaveOn 𝕜 ((fun z ↦... | [
"𝕜 : Type u_1\nE : Type u_2\nβ : Type u_5\ninst✝⁶ : Semiring 𝕜\ninst✝⁵ : PartialOrder 𝕜\ninst✝⁴ : AddCancelCommMonoid E\ninst✝³ : AddCommMonoid β\ninst✝² : PartialOrder β\ninst✝¹ : Module 𝕜 E\ninst✝ : SMul 𝕜 β\ns : Set E\nf : E → β\nhf : StrictConcaveOn 𝕜 s f\nc : E\n⊢ StrictConcaveOn 𝕜 ((fun z ↦ c + z) ⁻¹' ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.Function | {
"line": 979,
"column": 4
} | {
"line": 979,
"column": 34
} | {
"line": 979,
"column": 35
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nβ : Type u_5\ninst✝⁷ : Field 𝕜\ninst✝⁶ : LinearOrder 𝕜\ninst✝⁵ : IsStrictOrderedRing 𝕜\ninst✝⁴ : AddCommMonoid E\ninst✝³ : AddCommMonoid β\ninst✝² : PartialOrder β\ninst✝¹ : SMul 𝕜 E\ninst✝ : SMul 𝕜 β\ns : Set E\nf : E → β\nh :\n ∀ ⦃x : E⦄,\n x ∈ s →\n ∀ ⦃y : ... | [
"𝕜 : Type u_1\nE : Type u_2\nβ : Type u_5\ninst✝⁷ : Field 𝕜\ninst✝⁶ : LinearOrder 𝕜\ninst✝⁵ : IsStrictOrderedRing 𝕜\ninst✝⁴ : AddCommMonoid E\ninst✝³ : AddCommMonoid β\ninst✝² : PartialOrder β\ninst✝¹ : SMul 𝕜 E\ninst✝ : SMul 𝕜 β\ns : Set E\nf : E → β\nh :\n ∀ ⦃x : E⦄,\n x ∈ s →\n ∀ ⦃y : E⦄,\n ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.Function | {
"line": 997,
"column": 4
} | {
"line": 997,
"column": 34
} | {
"line": 997,
"column": 35
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nβ : Type u_5\ninst✝⁷ : Field 𝕜\ninst✝⁶ : LinearOrder 𝕜\ninst✝⁵ : IsStrictOrderedRing 𝕜\ninst✝⁴ : AddCommMonoid E\ninst✝³ : AddCommMonoid β\ninst✝² : PartialOrder β\ninst✝¹ : SMul 𝕜 E\ninst✝ : SMul 𝕜 β\ns : Set E\nf : E → β\nh :\n ∀ ⦃x : E⦄,\n x ∈ s →\n ∀ ⦃y : ... | [
"𝕜 : Type u_1\nE : Type u_2\nβ : Type u_5\ninst✝⁷ : Field 𝕜\ninst✝⁶ : LinearOrder 𝕜\ninst✝⁵ : IsStrictOrderedRing 𝕜\ninst✝⁴ : AddCommMonoid E\ninst✝³ : AddCommMonoid β\ninst✝² : PartialOrder β\ninst✝¹ : SMul 𝕜 E\ninst✝ : SMul 𝕜 β\ns : Set E\nf : E → β\nh :\n ∀ ⦃x : E⦄,\n x ∈ s →\n ∀ ⦃y : E⦄,\n ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.AffineSpace.Centroid | {
"line": 105,
"column": 6
} | {
"line": 105,
"column": 17
} | {
"line": 105,
"column": 18
} | [
{
"pp": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁵ : DivisionRing k\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module k V\ninst✝² : AffineSpace V P\nι : Type u_4\ninst✝¹ : DecidableEq ι\ninst✝ : Invertible 2\np : ι → P\ni₁ i₂ : ι\nh : ¬i₁ = i₂\n⊢ ↑(1 + 1) ≠ 0",
"ppTerm": "?m.74",
"assigned": true,
"u... | [
"k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁵ : DivisionRing k\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module k V\ninst✝² : AffineSpace V P\nι : Type u_4\ninst✝¹ : DecidableEq ι\ninst✝ : Invertible 2\np : ι → P\ni₁ i₂ : ι\nh : ¬i₁ = i₂\n⊢ ¬2 = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.AffineSpace.Centroid | {
"line": 106,
"column": 4
} | {
"line": 108,
"column": 70
} | {
"line": 109,
"column": 4
} | [
{
"pp": "case neg\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁵ : DivisionRing k\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module k V\ninst✝² : AffineSpace V P\nι : Type u_4\ninst✝¹ : DecidableEq ι\ninst✝ : Invertible 2\np : ι → P\ni₁ i₂ : ι\nh : ¬i₁ = i₂\nhc : ↑(#{i₁, i₂}) ≠ 0\n⊢ centroid k {i₁, i₂} p = 2⁻¹ • (p ... | [
"case neg\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁵ : DivisionRing k\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module k V\ninst✝² : AffineSpace V P\nι : Type u_4\ninst✝¹ : DecidableEq ι\ninst✝ : Invertible 2\np : ι → P\ni₁ i₂ : ι\nh : ¬i₁ = i₂\nhc : ↑(#{i₁, i₂}) ≠ 0\n⊢ ({i₁, i₂}.weightedVSubOfPoint p (p i₁)) (cent... | rw [centroid_def,
affineCombination_eq_weightedVSubOfPoint_vadd_of_sum_eq_one _ _ _
(sum_centroidWeights_eq_one_of_cast_card_ne_zero _ hc) (p i₁)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.AffineSpace.Combination | {
"line": 455,
"column": 2
} | {
"line": 455,
"column": 60
} | {
"line": 457,
"column": 0
} | [
{
"pp": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝² : Ring k\ninst✝¹ : AddCommGroup V\ninst✝ : Module k V\nS : AffineSpace V P\nι : Type u_4\nι₂ : Type u_5\ns₂ : Finset ι₂\ne : ι₂ ↪ ι\nw : ι → k\np : ι → P\n⊢ (affineCombination k (map e s₂) p) w = (affineCombination k s₂ (p ∘ ⇑e)) (w ∘ ⇑e)",
"ppTerm":... | [] | simp_rw [affineCombination_apply, weightedVSubOfPoint_map] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.LinearAlgebra.AffineSpace.Combination | {
"line": 455,
"column": 2
} | {
"line": 455,
"column": 60
} | {
"line": 457,
"column": 0
} | [
{
"pp": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝² : Ring k\ninst✝¹ : AddCommGroup V\ninst✝ : Module k V\nS : AffineSpace V P\nι : Type u_4\nι₂ : Type u_5\ns₂ : Finset ι₂\ne : ι₂ ↪ ι\nw : ι → k\np : ι → P\n⊢ (affineCombination k (map e s₂) p) w = (affineCombination k s₂ (p ∘ ⇑e)) (w ∘ ⇑e)",
"ppTerm":... | [] | simp_rw [affineCombination_apply, weightedVSubOfPoint_map] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.AffineSpace.Combination | {
"line": 455,
"column": 2
} | {
"line": 455,
"column": 60
} | {
"line": 457,
"column": 0
} | [
{
"pp": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝² : Ring k\ninst✝¹ : AddCommGroup V\ninst✝ : Module k V\nS : AffineSpace V P\nι : Type u_4\nι₂ : Type u_5\ns₂ : Finset ι₂\ne : ι₂ ↪ ι\nw : ι → k\np : ι → P\n⊢ (affineCombination k (map e s₂) p) w = (affineCombination k s₂ (p ∘ ⇑e)) (w ∘ ⇑e)",
"ppTerm":... | [] | simp_rw [affineCombination_apply, weightedVSubOfPoint_map] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.AffineSpace.Independent | {
"line": 76,
"column": 4
} | {
"line": 76,
"column": 20
} | {
"line": 76,
"column": 21
} | [
{
"pp": "case mpr\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁴ : Ring k\ninst✝³ : AddCommGroup V\ninst✝² : Module k V\ninst✝¹ : AffineSpace V P\nι : Type u_4\ninst✝ : Fintype ι\np : ι → P\ns : Finset ι\nw : ι → k\nhw : ∑ i, (↑s).indicator w i = 0\nhs : (univ.weightedVSub p) ((↑s).indicator w) = 0\ni : ι\nh... | [
"case mpr\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁴ : Ring k\ninst✝³ : AddCommGroup V\ninst✝² : Module k V\ninst✝¹ : AffineSpace V P\nι : Type u_4\ninst✝ : Fintype ι\np : ι → P\ns : Finset ι\nw : ι → k\nhw : ∑ i, (↑s).indicator w i = 0\nhs : (univ.weightedVSub p) ((↑s).indicator w) = 0\ni : ι\nhi : i ∈ s\nh... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Basic | {
"line": 671,
"column": 2
} | {
"line": 671,
"column": 22
} | {
"line": 671,
"column": 23
} | [
{
"pp": "k : Type u_1\nV₁ : Type u_2\nP₁ : Type u_3\ninst✝⁶ : Ring k\ninst✝⁵ : AddCommGroup V₁\ninst✝⁴ : Module k V₁\ninst✝³ : AffineSpace V₁ P₁\nV₂ : Type u_8\nP₂ : Type u_9\ninst✝² : AddCommGroup V₂\ninst✝¹ : Module k V₂\ninst✝ : AffineSpace V₂ P₂\ns : Set P₁\nf g : P₁ →ᵃ[k] P₂\nh_span : affineSpan k s = ⊤\nh... | [
"k : Type u_1\nV₁ : Type u_2\nP₁ : Type u_3\ninst✝⁶ : Ring k\ninst✝⁵ : AddCommGroup V₁\ninst✝⁴ : Module k V₁\ninst✝³ : AffineSpace V₁ P₁\nV₂ : Type u_8\nP₂ : Type u_9\ninst✝² : AddCommGroup V₂\ninst✝¹ : Module k V₂\ninst✝ : AffineSpace V₂ P₂\ns : Set P₁\nf g : P₁ →ᵃ[k] P₂\nh_span : affineSpan k s = ⊤\nh_agree : EqO... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Basic | {
"line": 860,
"column": 2
} | {
"line": 860,
"column": 55
} | {
"line": 860,
"column": 56
} | [
{
"pp": "k : Type u_1\nV : Type u_2\nW : Type u_3\nP : Type u_4\nQ : Type u_5\ninst✝⁶ : Ring k\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module k V\ninst✝³ : AffineSpace V P\ninst✝² : AddCommGroup W\ninst✝¹ : Module k W\ninst✝ : AffineSpace W Q\ns : Set P\nt : Set Q\n⊢ vectorSpan k (s ×ˢ t) ≤ (vectorSpan k s).prod (ve... | [
"k : Type u_1\nV : Type u_2\nW : Type u_3\nP : Type u_4\nQ : Type u_5\ninst✝⁶ : Ring k\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module k V\ninst✝³ : AffineSpace V P\ninst✝² : AddCommGroup W\ninst✝¹ : Module k W\ninst✝ : AffineSpace W Q\ns : Set P\nt : Set Q\n⊢ Submodule.span k ((s -ᵥ s) ×ˢ (t -ᵥ t)) ≤ (Submodule.span k (... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Basic | {
"line": 864,
"column": 2
} | {
"line": 864,
"column": 49
} | {
"line": 864,
"column": 50
} | [
{
"pp": "k : Type u_1\nV : Type u_2\nW : Type u_3\nP : Type u_4\nQ : Type u_5\ninst✝⁶ : Ring k\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module k V\ninst✝³ : AffineSpace V P\ninst✝² : AddCommGroup W\ninst✝¹ : Module k W\ninst✝ : AffineSpace W Q\ns : AffineSubspace k P\nt : AffineSubspace k Q\n⊢ (s.prod t).direction ≤ ... | [
"k : Type u_1\nV : Type u_2\nW : Type u_3\nP : Type u_4\nQ : Type u_5\ninst✝⁶ : Ring k\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module k V\ninst✝³ : AffineSpace V P\ninst✝² : AddCommGroup W\ninst✝¹ : Module k W\ninst✝ : AffineSpace W Q\ns : AffineSubspace k P\nt : AffineSubspace k Q\n⊢ vectorSpan k (↑s ×ˢ ↑t) ≤ (vectorSp... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Seminorm | {
"line": 230,
"column": 4
} | {
"line": 230,
"column": 84
} | {
"line": 231,
"column": 6
} | [
{
"pp": "R : Type u_1\n𝕜 : Type u_3\nE : Type u_7\ninst✝⁵ : SeminormedRing 𝕜\ninst✝⁴ : AddGroup E\ninst✝³ : SMul 𝕜 E\ninst✝² : SMul R ℝ\ninst✝¹ : SMul R ℝ≥0\ninst✝ : IsScalarTower R ℝ≥0 ℝ\nr : R\np q : Seminorm 𝕜 E\nx y : ℝ\n⊢ r • max x y = max (r • x) (r • y)",
"ppTerm": "?m.49",
"assigned": false,... | [
"R : Type u_1\n𝕜 : Type u_3\nE : Type u_7\ninst✝⁵ : SeminormedRing 𝕜\ninst✝⁴ : AddGroup E\ninst✝³ : SMul 𝕜 E\ninst✝² : SMul R ℝ\ninst✝¹ : SMul R ℝ≥0\ninst✝ : IsScalarTower R ℝ≥0 ℝ\nr : R\np q : Seminorm 𝕜 E\nx y : ℝ\n⊢ r • max x y = max (r • x) (r • y)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.AffineSpace.Independent | {
"line": 220,
"column": 6
} | {
"line": 220,
"column": 22
} | {
"line": 220,
"column": 23
} | [
{
"pp": "case mpr\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ns : Finset ι\nw : ι → k\nhw : ∑ i ∈ s, w i = 0\nhs : (s.weightedVSub p) w = 0\ni0 : ι\nhi0 : i0 ∈ s\nw1 : ι → k := update (const ι 0) i0 1... | [
"case mpr\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ns : Finset ι\nw : ι → k\nhw : ∑ i ∈ s, w i = 0\nhs : (s.weightedVSub p) w = 0\ni0 : ι\nhi0 : i0 ∈ s\nw1 : ι → k := update (const ι 0) i0 1\nhw1 : ∑ i ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Basic | {
"line": 966,
"column": 8
} | {
"line": 966,
"column": 19
} | {
"line": 966,
"column": 20
} | [
{
"pp": "case neg.refine_2\nk : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns₁ s₂ : AffineSubspace k P\nhd : s₁.direction = s₂.direction\nhb : s₁ = ⊥ ↔ s₂ = ⊥\nhs₁ : ¬s₁ = ⊥\nhs₂ : s₂ ≠ ⊥\np₁ : P\nhp₁ : p₁ ∈ ↑s₁\np₂ : P\nhp₂ : p₂ ... | [
"case neg.refine_2\nk : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns₁ s₂ : AffineSubspace k P\nhd : s₁.direction = s₂.direction\nhb : s₁ = ⊥ ↔ s₂ = ⊥\nhs₁ : ¬s₁ = ⊥\nhs₂ : s₂ ≠ ⊥\np₁ : P\nhp₁ : p₁ ∈ ↑s₁\np₂ : P\nhp₂ : p₂ ∈ ↑s₂\n⊢ s₂.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.AffineSpace.Pointwise | {
"line": 142,
"column": 9
} | {
"line": 142,
"column": 43
} | {
"line": 142,
"column": 44
} | [
{
"pp": "M : Type u_1\nk : Type u_2\nV : Type u_3\ninst✝⁵ : Ring k\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module k V\ninst✝² : Monoid M\ninst✝¹ : DistribMulAction M V\ninst✝ : SMulCommClass M k V\na : M\nha : IsUnit a\nx : V\n⊢ x ∈ a • ⊤ ↔ x ∈ ⊤",
"ppTerm": "?m.46",
"assigned": true,
"usedConstants": [
... | [
"M : Type u_1\nk : Type u_2\nV : Type u_3\ninst✝⁵ : Ring k\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module k V\ninst✝² : Monoid M\ninst✝¹ : DistribMulAction M V\ninst✝ : SMulCommClass M k V\na : M\nha : IsUnit a\nx : V\n⊢ ∃ y, a • y = x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.AffineSpace.Basis | {
"line": 124,
"column": 4
} | {
"line": 124,
"column": 84
} | {
"line": 124,
"column": 85
} | [
{
"pp": "ι : Type u_1\nk : Type u_5\nV : Type u_6\nP : Type u_7\ninst✝³ : AddCommGroup V\ninst✝² : AffineSpace V P\ninst✝¹ : Ring k\ninst✝ : Module k V\nb : AffineBasis ι k P\nhι : IsEmpty ι\n⊢ False",
"ppTerm": "?m.13",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": [... | [
"ι : Type u_1\nk : Type u_5\nV : Type u_6\nP : Type u_7\ninst✝³ : AddCommGroup V\ninst✝² : AffineSpace V P\ninst✝¹ : Ring k\ninst✝ : Module k V\nb : AffineBasis ι k P\nhι : IsEmpty ι\n⊢ False"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.AffineSpace.Independent | {
"line": 230,
"column": 4
} | {
"line": 230,
"column": 58
} | {
"line": 230,
"column": 59
} | [
{
"pp": "case mp\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁴ : Ring k\ninst✝³ : AddCommGroup V\ninst✝² : Module k V\ninst✝¹ : AffineSpace V P\nι : Type u_4\ninst✝ : Fintype ι\np : ι → P\nh :\n ∀ (s1 s2 : Finset ι) (w1 w2 : ι → k),\n ∑ i ∈ s1, w1 i = 1 →\n ∑ i ∈ s2, w2 i = 1 →\n (affineComb... | [
"case mp\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁴ : Ring k\ninst✝³ : AddCommGroup V\ninst✝² : Module k V\ninst✝¹ : AffineSpace V P\nι : Type u_4\ninst✝ : Fintype ι\np : ι → P\nh :\n ∀ (s1 s2 : Finset ι) (w1 w2 : ι → k),\n ∑ i ∈ s1, w1 i = 1 →\n ∑ i ∈ s2, w2 i = 1 →\n (affineCombination k s1... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Basic | {
"line": 1074,
"column": 6
} | {
"line": 1074,
"column": 28
} | {
"line": 1074,
"column": 29
} | [
{
"pp": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\np₁ p₂ p₃ p₄ p₅ p₆ : P\nh₂ : p₂ ∉ line[k, p₁, p₃]\nr₁ : kˣ\nhr₁ : ↑r₁ • (p₂ -ᵥ p₁) = p₅ -ᵥ p₄\nr₂ : k\nhr₂ : r₂ • (p₃ -ᵥ p₂) = p₆ -ᵥ p₅\nr₃ : k\nhr₃ : r₃ • (p₁ -ᵥ p₃)... | [
"k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\np₁ p₂ p₃ p₄ p₅ p₆ : P\nh₂ : p₂ ∉ line[k, p₁, p₃]\nr₁ : kˣ\nhr₁ : ↑r₁ • (p₂ -ᵥ p₁) = p₅ -ᵥ p₄\nr₂ : k\nhr₂ : r₂ • (p₃ -ᵥ p₂) = p₆ -ᵥ p₅\nr₃ : k\nhr₃ : r₃ • (p₁ -ᵥ p₃) = p₄ -ᵥ p₆\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.AffineSpace.Independent | {
"line": 276,
"column": 58
} | {
"line": 276,
"column": 74
} | {
"line": 276,
"column": 75
} | [
{
"pp": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\nha : AffineIndependent k p\nw₁ w₂ : ι → k\ns : Finset ι\nhw₁ : ∑ i ∈ s, w₁ i = 1\nhw₂ : ∑ i ∈ s, w₂ i = 1\nh : (affineCombination k s p) w₁ = (affin... | [
"k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\nha : AffineIndependent k p\nw₁ w₂ : ι → k\ns : Finset ι\nhw₁ : ∑ i ∈ s, w₁ i = 1\nhw₂ : ∑ i ∈ s, w₂ i = 1\nh : (affineCombination k s p) w₁ = (affineCombination... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.AffineSpace.Independent | {
"line": 408,
"column": 2
} | {
"line": 408,
"column": 13
} | {
"line": 408,
"column": 14
} | [
{
"pp": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁵ : Ring k\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module k V\ninst✝² : AffineSpace V P\nι : Type u_4\ninst✝¹ : Fintype ι\nι₂ : Type u_5\ninst✝ : Fintype ι₂\np : ι → P\nha : AffineIndependent k p\nw₁ : ι → k\nw₂ : ι₂ → k\nhw₁ : ∑ i, w₁ i = 1\nhw₂ : ∑ i, w₂ i = ... | [
"k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁵ : Ring k\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module k V\ninst✝² : AffineSpace V P\nι : Type u_4\ninst✝¹ : Fintype ι\nι₂ : Type u_5\ninst✝ : Fintype ι₂\np : ι → P\nha : AffineIndependent k p\nw₁ : ι → k\nw₂ : ι₂ → k\nhw₁ : ∑ i, w₁ i = 1\nhw₂ : ∑ i, w₂ i = 1\ne : ι₂ ↪ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.AffineSpace.Simplex.Basic | {
"line": 203,
"column": 58
} | {
"line": 203,
"column": 69
} | {
"line": 203,
"column": 70
} | [
{
"pp": "k : Type u_1\nV : Type u_2\nP : Type u_5\ninst✝⁴ : Ring k\ninst✝³ : AddCommGroup V\ninst✝² : Module k V\ninst✝¹ : AffineSpace V P\nn : ℕ\ninst✝ : NeZero n\ns : Simplex k P n\nw : Fin (n + 1) → k\nhw : ∑ i, w i = 1\ni : Fin (n + 1)\nh : w i = 0\nhk : Nontrivial k\n⊢ ∀ i_1 ∈ univ, i_1 ∉ {i}ᶜ → w i_1 = 0"... | [
"k : Type u_1\nV : Type u_2\nP : Type u_5\ninst✝⁴ : Ring k\ninst✝³ : AddCommGroup V\ninst✝² : Module k V\ninst✝¹ : AffineSpace V P\nn : ℕ\ninst✝ : NeZero n\ns : Simplex k P n\nw : Fin (n + 1) → k\nhw : ∑ i, w i = 1\ni : Fin (n + 1)\nh : w i = 0\nhk : Nontrivial k\n⊢ w i = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.AffineSpace.Independent | {
"line": 481,
"column": 73
} | {
"line": 481,
"column": 84
} | {
"line": 481,
"column": 85
} | [
{
"pp": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁴ : Ring k\ninst✝³ : AddCommGroup V\ninst✝² : Module k V\ninst✝¹ : AffineSpace V P\nι : Type u_4\ninst✝ : Nontrivial k\np : ι → P\ns₁ s₂ : Set ι\nfs₁ : Finset ι\nhfs₁ : ↑fs₁ ⊆ s₁\nw₁ : ι → k\nhw₁ : ∑ i ∈ fs₁, w₁ i = 1\nfs₂ : Finset ι\nhfs₂ : ↑fs₂ ⊆ s₂\nw₂ ... | [
"k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁴ : Ring k\ninst✝³ : AddCommGroup V\ninst✝² : Module k V\ninst✝¹ : AffineSpace V P\nι : Type u_4\ninst✝ : Nontrivial k\np : ι → P\ns₁ s₂ : Set ι\nfs₁ : Finset ι\nhfs₁ : ↑fs₁ ⊆ s₁\nw₁ : ι → k\nhw₁ : ∑ i ∈ fs₁, w₁ i = 1\nfs₂ : Finset ι\nhfs₂ : ↑fs₂ ⊆ s₂\nw₂ : ι → k\nhw₂... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.AffineSpace.Independent | {
"line": 485,
"column": 49
} | {
"line": 485,
"column": 60
} | {
"line": 485,
"column": 61
} | [
{
"pp": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁴ : Ring k\ninst✝³ : AddCommGroup V\ninst✝² : Module k V\ninst✝¹ : AffineSpace V P\nι : Type u_4\ninst✝ : Nontrivial k\np : ι → P\ns₁ s₂ : Set ι\nfs₁ : Finset ι\nhfs₁ : ↑fs₁ ⊆ s₁\nw₁ : ι → k\nhw₁ : ∑ i ∈ fs₁, w₁ i = 1\nfs₂ : Finset ι\nhfs₂ : ↑fs₂ ⊆ s₂\nw₂ ... | [
"k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁴ : Ring k\ninst✝³ : AddCommGroup V\ninst✝² : Module k V\ninst✝¹ : AffineSpace V P\nι : Type u_4\ninst✝ : Nontrivial k\np : ι → P\ns₁ s₂ : Set ι\nfs₁ : Finset ι\nhfs₁ : ↑fs₁ ⊆ s₁\nw₁ : ι → k\nhw₁ : ∑ i ∈ fs₁, w₁ i = 1\nfs₂ : Finset ι\nhfs₂ : ↑fs₂ ⊆ s₂\nw₂ : ι → k\nhw₂... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Seminorm | {
"line": 891,
"column": 8
} | {
"line": 891,
"column": 23
} | {
"line": 891,
"column": 24
} | [
{
"pp": "case inl\n𝕜 : Type u_3\nE : Type u_7\ninst✝² : NormedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\nι : Sort u_12\np : ι → Seminorm 𝕜 E\nhp : BddAbove (range p)\ne : E\nr : ℝ\nhr : 0 < r\nh✝ : IsEmpty ι\n⊢ (⨆ i, p i).closedBall e r = ⋂ i, (p i).closedBall e r",
"ppTerm": "?inl",
"ass... | [
"case inl\n𝕜 : Type u_3\nE : Type u_7\ninst✝² : NormedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\nι : Sort u_12\np : ι → Seminorm 𝕜 E\nhp : BddAbove (range p)\ne : E\nr : ℝ\nhr : 0 < r\nh✝ : IsEmpty ι\n⊢ (sSup ∅).closedBall e r = ⋂ i, (p i).closedBall e r"
] | iSup_of_empty', | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Seminorm | {
"line": 902,
"column": 2
} | {
"line": 908,
"column": 75
} | {
"line": 910,
"column": 0
} | [
{
"pp": "case inr\n𝕜 : Type u_3\nE : Type u_7\ninst✝² : NormedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\np : Seminorm 𝕜 E\nk : 𝕜\nr : ℝ\nhk : k ≠ 0\n⊢ p.ball 0 (‖k‖ * r) ⊆ k • p.ball 0 r",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Seminorm.instSeminormClass",
... | [] | · intro x
rw [Set.mem_smul_set, Seminorm.mem_ball_zero]
refine fun hx => ⟨k⁻¹ • x, ?_, ?_⟩
· rwa [Seminorm.mem_ball_zero, map_smul_eq_mul, norm_inv, ←
mul_lt_mul_iff_right₀ <| norm_pos_iff.mpr hk, ← mul_assoc, ← div_eq_mul_inv ‖k‖ ‖k‖,
div_self (ne_of_gt <| norm_pos_iff.mpr hk), one_mul]
... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.LinearAlgebra.AffineSpace.Simplex.Basic | {
"line": 521,
"column": 2
} | {
"line": 521,
"column": 17
} | {
"line": 521,
"column": 18
} | [
{
"pp": "k : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝⁴ : Ring k\ninst✝³ : AddCommGroup V\ninst✝² : Module k V\ninst✝¹ : AffineSpace V P\ninst✝ : PartialOrder k\ns : Simplex k P 0\np : P\nw : Fin 1 → k\nh : w 0 = 1\nhi : ∀ (i : Fin 1), 0 < w i ∧ w i < 1\n⊢ ¬(affineCombination k {0} s.points) w = p",
"ppTe... | [
"k : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝⁴ : Ring k\ninst✝³ : AddCommGroup V\ninst✝² : Module k V\ninst✝¹ : AffineSpace V P\ninst✝ : PartialOrder k\ns : Simplex k P 0\np : P\nw : Fin 1 → k\nh : w 0 = 1\nhi : ∀ (i : Fin 1), 0 < w i ∧ w i < 1\n⊢ ¬(affineCombination k {0} s.points) w = p"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Seminorm | {
"line": 1067,
"column": 13
} | {
"line": 1067,
"column": 24
} | {
"line": 1067,
"column": 25
} | [
{
"pp": "case h\n𝕜 : Type u_3\nE : Type u_7\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : AddCommGroup E\ninst✝² : Module 𝕜 E\ninst✝¹ : TopologicalSpace E\ninst✝ : ContinuousConstSMul 𝕜 E\np : Seminorm 𝕜 E\nr : ℝ\nhp : p.closedBall 0 r ∈ 𝓝 0\nε : ℝ\nhε : ε > 0\nhr : r ≤ 0\n⊢ 0 < ‖1‖ ∧ ‖1‖ * r < ε",
"p... | [
"case h\n𝕜 : Type u_3\nE : Type u_7\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : AddCommGroup E\ninst✝² : Module 𝕜 E\ninst✝¹ : TopologicalSpace E\ninst✝ : ContinuousConstSMul 𝕜 E\np : Seminorm 𝕜 E\nr : ℝ\nhp : p.closedBall 0 r ∈ 𝓝 0\nε : ℝ\nhε : ε > 0\nhr : r ≤ 0\n⊢ r < ε"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Seminorm | {
"line": 1068,
"column": 6
} | {
"line": 1068,
"column": 34
} | {
"line": 1068,
"column": 35
} | [
{
"pp": "case inr\n𝕜 : Type u_3\nE : Type u_7\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : AddCommGroup E\ninst✝² : Module 𝕜 E\ninst✝¹ : TopologicalSpace E\ninst✝ : ContinuousConstSMul 𝕜 E\np : Seminorm 𝕜 E\nr : ℝ\nhp : p.closedBall 0 r ∈ 𝓝 0\nε : ℝ\nhε : ε > 0\nhr : 0 < r\n⊢ ∃ k, 0 < ‖k‖ ∧ ‖k‖ * r < ε",... | [
"case inr\n𝕜 : Type u_3\nE : Type u_7\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : AddCommGroup E\ninst✝² : Module 𝕜 E\ninst✝¹ : TopologicalSpace E\ninst✝ : ContinuousConstSMul 𝕜 E\np : Seminorm 𝕜 E\nr : ℝ\nhp : p.closedBall 0 r ∈ 𝓝 0\nε : ℝ\nhε : ε > 0\nhr : 0 < r\n⊢ ∃ k, ¬k = 0 ∧ ‖k‖ * r < ε"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Seminorm | {
"line": 1068,
"column": 4
} | {
"line": 1068,
"column": 67
} | {
"line": 1069,
"column": 2
} | [
{
"pp": "case inr\n𝕜 : Type u_3\nE : Type u_7\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : AddCommGroup E\ninst✝² : Module 𝕜 E\ninst✝¹ : TopologicalSpace E\ninst✝ : ContinuousConstSMul 𝕜 E\np : Seminorm 𝕜 E\nr : ℝ\nhp : p.closedBall 0 r ∈ 𝓝 0\nε : ℝ\nhε : ε > 0\nhr : 0 < r\n⊢ ∃ k, 0 < ‖k‖ ∧ ‖k‖ * r < ε",... | [] | · simpa [lt_div_iff₀ hr] using exists_norm_lt 𝕜 (div_pos hε hr) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.LinearAlgebra.AffineSpace.Simplex.Basic | {
"line": 555,
"column": 30
} | {
"line": 555,
"column": 41
} | {
"line": 555,
"column": 42
} | [
{
"pp": "k : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nI : Set k\nn : ℕ\ns : Simplex k P n\nfs : Finset (Fin (n + 1))\nm : ℕ\nh : #fs = m + 1\nw : Fin (n + 1) → k\nhw : ∑ i, w i = 1\nx✝ : (∀ i ∈ fs, w i ∈ I) ∧ ∀ i ∉ fs, w i = 0\... | [
"k : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nI : Set k\nn : ℕ\ns : Simplex k P n\nfs : Finset (Fin (n + 1))\nm : ℕ\nh : #fs = m + 1\nw : Fin (n + 1) → k\nhw : ∑ i, w i = 1\nx✝ : (∀ i ∈ fs, w i ∈ I) ∧ ∀ i ∉ fs, w i = 0\nhii : ∀ i ∈... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.AffineSpace.Simplex.Basic | {
"line": 648,
"column": 67
} | {
"line": 648,
"column": 78
} | {
"line": 648,
"column": 79
} | [
{
"pp": "k : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝⁶ : Ring k\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module k V\ninst✝³ : AffineSpace V P\ninst✝² : PartialOrder k\ninst✝¹ : Nontrivial k\ninst✝ : ZeroLEOneClass k\nn : ℕ\ns : Simplex k P n\nfs : Finset (Fin (n + 1))\nhfs : fs ≠ univ\nm : ℕ\nh : #fs = m + 1\na : ... | [
"k : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝⁶ : Ring k\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module k V\ninst✝³ : AffineSpace V P\ninst✝² : PartialOrder k\ninst✝¹ : Nontrivial k\ninst✝ : ZeroLEOneClass k\nn : ℕ\ns : Simplex k P n\nfs : Finset (Fin (n + 1))\nhfs : fs ≠ univ\nm : ℕ\nh : #fs = m + 1\na : Fin (n + 1)\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Seminorm | {
"line": 1259,
"column": 2
} | {
"line": 1260,
"column": 9
} | {
"line": 1260,
"column": 10
} | [
{
"pp": "𝕜 : Type u_3\nE : Type u_7\ninst✝² : NormedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\np q : Seminorm 𝕜 E\nε C : ℝ\nε_pos : 0 < ε\nc : 𝕜\nhc : 1 < ‖c‖\nhf : ∀ (x : E), ε / ‖c‖ ≤ p x → p x < ε → q x ≤ C * p x\nx : E\nhx : p x ≠ 0\nδ : 𝕜\nhδ : δ ≠ 0\nδxle : p (δ • x) < ε\nleδx : ε / ‖c‖ ≤... | [
"𝕜 : Type u_3\nE : Type u_7\ninst✝² : NormedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\np q : Seminorm 𝕜 E\nε C : ℝ\nε_pos : 0 < ε\nc : 𝕜\nhc : 1 < ‖c‖\nhf : ∀ (x : E), ε / ‖c‖ ≤ p x → p x < ε → q x ≤ C * p x\nx : E\nhx : p x ≠ 0\nδ : 𝕜\nhδ : δ ≠ 0\nδxle : p (δ • x) < ε\nleδx : ε / ‖c‖ ≤ p (δ • x)\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.AffineSpace.Simplex.Basic | {
"line": 708,
"column": 30
} | {
"line": 708,
"column": 41
} | {
"line": 708,
"column": 42
} | [
{
"pp": "k : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝⁷ : Ring k\ninst✝⁶ : AddCommGroup V\ninst✝⁵ : Module k V\ninst✝⁴ : AffineSpace V P\ninst✝³ : LinearOrder k\ninst✝² : IsOrderedAddMonoid k\ninst✝¹ : ZeroLEOneClass k\nn : ℕ\ninst✝ : NeZero n\ns : Simplex k P n\nw : Fin (n + 1) → k\nhw1 : ∑ i, w i = 1\nhp : ... | [
"k : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝⁷ : Ring k\ninst✝⁶ : AddCommGroup V\ninst✝⁵ : Module k V\ninst✝⁴ : AffineSpace V P\ninst✝³ : LinearOrder k\ninst✝² : IsOrderedAddMonoid k\ninst✝¹ : ZeroLEOneClass k\nn : ℕ\ninst✝ : NeZero n\ns : Simplex k P n\nw : Fin (n + 1) → k\nhw1 : ∑ i, w i = 1\nhp : ∀ (i : Fin (... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.AffineSpace.Simplex.Basic | {
"line": 715,
"column": 2
} | {
"line": 715,
"column": 48
} | {
"line": 716,
"column": 4
} | [
{
"pp": "k : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝⁷ : Ring k\ninst✝⁶ : AddCommGroup V\ninst✝⁵ : Module k V\ninst✝⁴ : AffineSpace V P\ninst✝³ : LinearOrder k\ninst✝² : IsOrderedAddMonoid k\ninst✝¹ : ZeroLEOneClass k\nn : ℕ\ninst✝ : NeZero n\ns : Simplex k P n\n⊢ s.closedInterior \\ s.interior = ⋃ i, (s.fac... | [
"k : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝⁷ : Ring k\ninst✝⁶ : AddCommGroup V\ninst✝⁵ : Module k V\ninst✝⁴ : AffineSpace V P\ninst✝³ : LinearOrder k\ninst✝² : IsOrderedAddMonoid k\ninst✝¹ : ZeroLEOneClass k\nn : ℕ\ninst✝ : NeZero n\ns : Simplex k P n\n⊢ ∀ (i : Fin (n + 1)), Disjoint (s.faceOpposite i).closedI... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.AffineSpace.Independent | {
"line": 740,
"column": 6
} | {
"line": 740,
"column": 23
} | {
"line": 740,
"column": 24
} | [
{
"pp": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nh : AffineIndependent k fun p ↦ ↑p\np₁ : P\nhsv : Basis (↑(Basis.ofVectorSpaceIndex k V)) k V := Basis.ofVectorSpace k V\nhsvi : LinearIndependent (ι := ↑(Basis.ofVe... | [
"k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nh : AffineIndependent k fun p ↦ ↑p\np₁ : P\nhsv : Basis (↑(Basis.ofVectorSpaceIndex k V)) k V := Basis.ofVectorSpace k V\nhsvi : LinearIndependent (ι := ↑(Basis.ofVectorSpaceInd... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.StdSimplex | {
"line": 77,
"column": 2
} | {
"line": 77,
"column": 13
} | {
"line": 77,
"column": 14
} | [
{
"pp": "𝕜 : Type u_2\nι : Type u_1\ninst✝³ : Semiring 𝕜\ninst✝² : PartialOrder 𝕜\ninst✝¹ : Fintype ι\ninst✝ : IsOrderedAddMonoid 𝕜\nf : ι → 𝕜\nh : f ∈ stdSimplex 𝕜 ι\n⊢ ∀ (i : ι), f ∈ Function.eval i ⁻¹' Icc (0 i) (1 i)",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"Eq.mpr",... | [
"𝕜 : Type u_2\nι : Type u_1\ninst✝³ : Semiring 𝕜\ninst✝² : PartialOrder 𝕜\ninst✝¹ : Fintype ι\ninst✝ : IsOrderedAddMonoid 𝕜\nf : ι → 𝕜\nh : f ∈ stdSimplex 𝕜 ι\n⊢ ∀ (i : ι), 0 ≤ f i ∧ f i ≤ 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.StdSimplex | {
"line": 85,
"column": 2
} | {
"line": 85,
"column": 52
} | {
"line": 85,
"column": 53
} | [
{
"pp": "𝕜 : Type u_2\nι : Type u_1\ninst✝⁴ : Semiring 𝕜\ninst✝³ : PartialOrder 𝕜\ninst✝² : Fintype ι\ninst✝¹ : DecidableEq ι\ninst✝ : ZeroLEOneClass 𝕜\ni : ι\n⊢ (fun x ↦ if i = x then 1 else 0) ∈ stdSimplex 𝕜 ι",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"No... | [
"𝕜 : Type u_2\nι : Type u_1\ninst✝⁴ : Semiring 𝕜\ninst✝³ : PartialOrder 𝕜\ninst✝² : Fintype ι\ninst✝¹ : DecidableEq ι\ninst✝ : ZeroLEOneClass 𝕜\ni : ι\n⊢ (fun x ↦ Pi.single i 1 x) ∈ stdSimplex 𝕜 ι"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.StdSimplex | {
"line": 118,
"column": 39
} | {
"line": 118,
"column": 50
} | {
"line": 118,
"column": 51
} | [
{
"pp": "𝕜 : Type ?u.3\ninst✝² : Ring 𝕜\ninst✝¹ : PartialOrder 𝕜\ninst✝ : IsOrderedRing 𝕜\nf : ↑(stdSimplex 𝕜 (Fin 2))\n⊢ ↑f 0 + ↑f 1 = 1",
"ppTerm": "?m.125",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"𝕜 : Type ?u.3\ninst✝² : Ring 𝕜\ninst✝¹ : PartialOrder 𝕜\ninst✝ : IsOrderedRing 𝕜\nf : ↑(stdSimplex 𝕜 (Fin 2))\n⊢ ↑f 0 + ↑f 1 = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.AffineSpace.Independent | {
"line": 754,
"column": 6
} | {
"line": 754,
"column": 23
} | {
"line": 754,
"column": 24
} | [
{
"pp": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns : Set P\np₁ : P\nh✝ : LinearIndependent k fun (v : ↑((fun p ↦ p -ᵥ p₁) '' (s \\ {p₁}))) ↦ ↑v\nh : LinearIndepOn k id ((fun p ↦ p -ᵥ p₁) '' (s \\ {p₁}))\nhp₁ : p₁ ∈... | [
"k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns : Set P\np₁ : P\nh✝ : LinearIndependent k fun (v : ↑((fun p ↦ p -ᵥ p₁) '' (s \\ {p₁}))) ↦ ↑v\nh : LinearIndepOn k id ((fun p ↦ p -ᵥ p₁) '' (s \\ {p₁}))\nhp₁ : p₁ ∈ s\nbsv : Ba... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.StdSimplex | {
"line": 295,
"column": 2
} | {
"line": 295,
"column": 37
} | {
"line": 295,
"column": 38
} | [
{
"pp": "S : Type u_1\ninst✝¹ : Semiring S\ninst✝ : PartialOrder S\ns : ↑(stdSimplex S (Fin 2))\n⊢ s 0 + s 1 = 1",
"ppTerm": "?m.20",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"S : Type u_1\ninst✝¹ : Semiring S\ninst✝ : PartialOrder S\ns : ↑(stdSimplex S (Fin 2))\n⊢ s 0 + s 1 = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.StdSimplex | {
"line": 304,
"column": 2
} | {
"line": 304,
"column": 18
} | {
"line": 304,
"column": 19
} | [
{
"pp": "S : Type u_1\ninst✝³ : Semiring S\ninst✝² : PartialOrder S\nX : Type u_2\ninst✝¹ : Fintype X\ninst✝ : IsOrderedRing S\ns : ↑(stdSimplex S X)\nx : X\n⊢ s x ≤ ∑ x, s x",
"ppTerm": "?m.18",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"S : Type u_1\ninst✝³ : Semiring S\ninst✝² : PartialOrder S\nX : Type u_2\ninst✝¹ : Fintype X\ninst✝ : IsOrderedRing S\ns : ↑(stdSimplex S X)\nx : X\n⊢ s x ≤ ∑ x, s x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.AffineSpace.Independent | {
"line": 820,
"column": 6
} | {
"line": 820,
"column": 72
} | {
"line": 821,
"column": 6
} | [
{
"pp": "case pos\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni : ι\nha : AffineIndependent k fun x ↦ p ↑x\nhi : p i ∉ affineSpan k (p '' {x | x ≠ i})\ns : Finset ι\nw : ι → k\nhw : ∑ i ∈ s, w... | [
"case pos\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni : ι\nha : AffineIndependent k fun x ↦ p ↑x\nhi : p i ∉ affineSpan k (p '' {x | x ≠ i})\ns : Finset ι\nw : ι → k\nhw : ∑ i ∈ s, w i = 0\nhs :... | have hwm : ∑ i ∈ s, wm i = 0 := by simp [wm, ← Finset.mul_sum, hw] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.LinearAlgebra.AffineSpace.Independent | {
"line": 826,
"column": 8
} | {
"line": 827,
"column": 32
} | {
"line": 827,
"column": 33
} | [
{
"pp": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni : ι\nha : AffineIndependent k fun x ↦ p ↑x\nhi : p i ∉ affineSpan k (p '' {x | x ≠ i})\ns : Finset ι\nw : ι → k\nhw : ∑ i ∈ s, w i = 0\nhs... | [
"k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni : ι\nha : AffineIndependent k fun x ↦ p ↑x\nhi : p i ∉ affineSpan k (p '' {x | x ≠ i})\ns : Finset ι\nw : ι → k\nhw : ∑ i ∈ s, w i = 0\nhs : (s.weight... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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