module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 366
values | kind stringclasses 370
values |
|---|---|---|---|---|---|---|
Mathlib.Analysis.SpecialFunctions.Pow.NNReal | {
"line": 646,
"column": 4
} | {
"line": 649,
"column": 71
} | [
{
"pp": "case inl\ny z : ℝ\nhyz : 0 < y + z\nhx : 0 ≠ ∞\n⊢ 0 ^ (y + z) = 0 ^ y * 0 ^ z",
"usedConstants": [
"Real.instIsOrderedRing",
"Mathlib.Tactic.Ring.Common.neg_zero",
"Eq.mpr",
"NegZeroClass.toNeg",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Mathlib.Tactic.Ring.Com... | by_cases hy' : 0 < y
· simp [ENNReal.zero_rpow_of_pos hyz, ENNReal.zero_rpow_of_pos hy']
· have hz' : 0 < z := by linarith
simp [ENNReal.zero_rpow_of_pos hyz, ENNReal.zero_rpow_of_pos hz'] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Pow.NNReal | {
"line": 925,
"column": 4
} | {
"line": 926,
"column": 84
} | [
{
"pp": "case coe\nz : ℝ\nhz : 0 < z\nx✝ : ℝ≥0\nhx : 1 ≤ ↑x✝\n⊢ 1 ≤ ↑x✝ ^ z",
"usedConstants": [
"ENNReal.one_le_coe_iff._simp_1",
"Real",
"ENNReal.ofNNReal",
"Real.instZero",
"congrArg",
"ENNReal.instPowReal",
"NNReal.one_le_rpow",
"PartialOrder.toPreorder",
... | simp only [one_le_coe_iff] at hx
simp [← coe_rpow_of_nonneg _ (le_of_lt hz), NNReal.one_le_rpow hx (le_of_lt hz)] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Pow.NNReal | {
"line": 925,
"column": 4
} | {
"line": 926,
"column": 84
} | [
{
"pp": "case coe\nz : ℝ\nhz : 0 < z\nx✝ : ℝ≥0\nhx : 1 ≤ ↑x✝\n⊢ 1 ≤ ↑x✝ ^ z",
"usedConstants": [
"ENNReal.one_le_coe_iff._simp_1",
"Real",
"ENNReal.ofNNReal",
"Real.instZero",
"congrArg",
"ENNReal.instPowReal",
"NNReal.one_le_rpow",
"PartialOrder.toPreorder",
... | simp only [one_le_coe_iff] at hx
simp [← coe_rpow_of_nonneg _ (le_of_lt hz), NNReal.one_le_rpow hx (le_of_lt hz)] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Convex.Segment | {
"line": 413,
"column": 16
} | {
"line": 413,
"column": 34
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁴ : Field 𝕜\ninst✝³ : LinearOrder 𝕜\ninst✝² : IsStrictOrderedRing 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\nx y z : E\nh : SameRay 𝕜 (x - y) (z - x)\na b : 𝕜\nha : 0 ≤ a\nhb : 0 ≤ b\nhab : a + b = 1\nhxy : x - y = a • (z - x + (x - y))\nhzx : z - x = b • (z... | sub_add_sub_cancel | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Convex.Segment | {
"line": 560,
"column": 2
} | {
"line": 560,
"column": 85
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : Field 𝕜\ninst✝¹ : LinearOrder 𝕜\ninst✝ : IsStrictOrderedRing 𝕜\nx y z : 𝕜\nh : x ≤ y\n⊢ z ∈ Icc x y ↔ ∃ a b, 0 ≤ a ∧ 0 ≤ b ∧ a + b = 1 ∧ a * x + b * y = z",
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"instHSMul",
"NonUnitalCommRing.to... | simp only [← segment_eq_Icc h, segment, mem_setOf_eq, smul_eq_mul, exists_and_left] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Convex.Segment | {
"line": 560,
"column": 2
} | {
"line": 560,
"column": 85
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : Field 𝕜\ninst✝¹ : LinearOrder 𝕜\ninst✝ : IsStrictOrderedRing 𝕜\nx y z : 𝕜\nh : x ≤ y\n⊢ z ∈ Icc x y ↔ ∃ a b, 0 ≤ a ∧ 0 ≤ b ∧ a + b = 1 ∧ a * x + b * y = z",
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"instHSMul",
"NonUnitalCommRing.to... | simp only [← segment_eq_Icc h, segment, mem_setOf_eq, smul_eq_mul, exists_and_left] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Convex.Segment | {
"line": 560,
"column": 2
} | {
"line": 560,
"column": 85
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : Field 𝕜\ninst✝¹ : LinearOrder 𝕜\ninst✝ : IsStrictOrderedRing 𝕜\nx y z : 𝕜\nh : x ≤ y\n⊢ z ∈ Icc x y ↔ ∃ a b, 0 ≤ a ∧ 0 ≤ b ∧ a + b = 1 ∧ a * x + b * y = z",
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"instHSMul",
"NonUnitalCommRing.to... | simp only [← segment_eq_Icc h, segment, mem_setOf_eq, smul_eq_mul, exists_and_left] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Convex.Segment | {
"line": 581,
"column": 4
} | {
"line": 582,
"column": 59
} | [
{
"pp": "case refine_2.inl\n𝕜 : Type u_1\ninst✝² : Field 𝕜\ninst✝¹ : LinearOrder 𝕜\ninst✝ : IsStrictOrderedRing 𝕜\nx y : 𝕜\nh : x < y\nb : 𝕜\nhb : 0 < b\nha : 0 ≤ 0\nhab : 0 + b = 1\n⊢ 0 * x + b * y ∈ Ioc x y",
"usedConstants": [
"Eq.mpr",
"Set.Ioc",
"MulOne.toOne",
"Preorder.t... | · rw [zero_add] at hab
rwa [hab, one_mul, zero_mul, zero_add, right_mem_Ioc] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Convex.Star | {
"line": 81,
"column": 2
} | {
"line": 85,
"column": 39
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝³ : Semiring 𝕜\ninst✝² : PartialOrder 𝕜\ninst✝¹ : AddCommMonoid E\ninst✝ : SMul 𝕜 E\nx : E\ns : Set E\n⊢ StarConvex 𝕜 x s ↔ ∀ ⦃y : E⦄, y ∈ s → [x -[𝕜] y] ⊆ s",
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"instHSMul",
"Partia... | constructor
· rintro h y hy z ⟨a, b, ha, hb, hab, rfl⟩
exact h hy ha hb hab
· rintro h y hy a b ha hb hab
exact h hy ⟨a, b, ha, hb, hab, rfl⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Convex.Star | {
"line": 81,
"column": 2
} | {
"line": 85,
"column": 39
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝³ : Semiring 𝕜\ninst✝² : PartialOrder 𝕜\ninst✝¹ : AddCommMonoid E\ninst✝ : SMul 𝕜 E\nx : E\ns : Set E\n⊢ StarConvex 𝕜 x s ↔ ∀ ⦃y : E⦄, y ∈ s → [x -[𝕜] y] ⊆ s",
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"instHSMul",
"Partia... | constructor
· rintro h y hy z ⟨a, b, ha, hb, hab, rfl⟩
exact h hy ha hb hab
· rintro h y hy a b ha hb hab
exact h hy ⟨a, b, ha, hb, hab, rfl⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Defs | {
"line": 103,
"column": 2
} | {
"line": 103,
"column": 10
} | [
{
"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\nh : vectorSpan k s = ⊥\np : P\nhp : p ∈ s\nq : P\nhq : q ∈ s\nhpq : p ≠ q\nhpq' : p -ᵥ q ∈ vectorSpan k s\n⊢ False",
"usedConstants": [
"Submodule",
... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Defs | {
"line": 219,
"column": 43
} | {
"line": 223,
"column": 38
} | [
{
"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\nh : (↑s).Nonempty\n⊢ directionOfNonempty h = s.direction",
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"Submodule",
"vectorSp... | by
refine le_antisymm ?_ (Submodule.span_le.2 Set.Subset.rfl)
rw [← SetLike.coe_subset_coe, directionOfNonempty, direction, Submodule.coe_set_mk,
AddSubmonoid.coe_set_mk]
exact vsub_set_subset_vectorSpan k _ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Defs | {
"line": 394,
"column": 2
} | {
"line": 394,
"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 : Set P\ns₁ : AffineSubspace k P\nh : s ⊆ ↑s₁\np p₁ : P\nhp₁ : p₁ ∈ s\nv : V\nhv : v ∈ vectorSpan k s\nhp : p = v +ᵥ p₁\nhp₁s₁ : p₁ ∈ ↑s₁\nhs : vectorSpan k s ≤ s₁.directi... | rw [SetLike.le_def] at hs | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Defs | {
"line": 727,
"column": 6
} | {
"line": 727,
"column": 26
} | [
{
"pp": "case inr\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\nS : AffineSpace V P\nι : Sort u_4\np₁ p₂ : P\ninst✝ : Subsingleton P\ns : AffineSubspace k P\nh : (↑s).Nonempty\n⊢ ↑s = ∅ ∨ ↑s = univ",
"usedConstants": [
"Set.univ",
"Affi... | exact .inr h.eq_univ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Defs | {
"line": 727,
"column": 6
} | {
"line": 727,
"column": 26
} | [
{
"pp": "case inr\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\nS : AffineSpace V P\nι : Sort u_4\np₁ p₂ : P\ninst✝ : Subsingleton P\ns : AffineSubspace k P\nh : (↑s).Nonempty\n⊢ ↑s = ∅ ∨ ↑s = univ",
"usedConstants": [
"Set.univ",
"Affi... | exact .inr h.eq_univ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Defs | {
"line": 727,
"column": 6
} | {
"line": 727,
"column": 26
} | [
{
"pp": "case inr\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\nS : AffineSpace V P\nι : Sort u_4\np₁ p₂ : P\ninst✝ : Subsingleton P\ns : AffineSubspace k P\nh : (↑s).Nonempty\n⊢ ↑s = ∅ ∨ ↑s = univ",
"usedConstants": [
"Set.univ",
"Affi... | exact .inr h.eq_univ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Defs | {
"line": 832,
"column": 57
} | {
"line": 837,
"column": 99
} | [
{
"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\ns₁ s₂ : AffineSubspace k P\n⊢ s₁.direction ⊔ s₂.direction ≤ (s₁ ⊔ s₂).direction",
"usedConstants": [
"Submodule",
"Lattice.toSemilatticeSup",
"Set.vsub",
... | by
simp only [direction_eq_vectorSpan, vectorSpan_def]
exact
sup_le
(sInf_le_sInf fun p hp => Set.Subset.trans (vsub_self_mono (le_sup_left : s₁ ≤ s₁ ⊔ s₂)) hp)
(sInf_le_sInf fun p hp => Set.Subset.trans (vsub_self_mono (le_sup_right : s₂ ≤ s₁ ⊔ s₂)) hp) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.AffineSpace.AffineEquiv | {
"line": 79,
"column": 2
} | {
"line": 79,
"column": 10
} | [
{
"pp": "k : Type u_1\nP₁ : Type u_2\nP₂ : Type u_3\nV₁ : Type u_6\nV₂ : Type u_7\ninst✝⁶ : Ring k\ninst✝⁵ : AddCommGroup V₁\ninst✝⁴ : AddCommGroup V₂\ninst✝³ : Module k V₁\ninst✝² : Module k V₂\ninst✝¹ : AffineSpace V₁ P₁\ninst✝ : AffineSpace V₂ P₂\ne : P₁ ≃ P₂\nel : V₁ ≃ₗ[k] V₂\nh : ∀ (p : P₁) (v : V₁), e (v ... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.LinearAlgebra.AffineSpace.AffineEquiv | {
"line": 604,
"column": 2
} | {
"line": 605,
"column": 28
} | [
{
"pp": "k : Type u_10\nV : Type u_11\nP : Type u_12\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\np : P\n⊢ ofLinearEquiv (LinearEquiv.refl k V) p p = refl k P",
"usedConstants": [
"AffineEquiv.ofLinearEquiv",
"AffineEquiv.refl",
"congrArg",
... | ext x
simp [ofLinearEquiv_apply] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.AffineSpace.AffineEquiv | {
"line": 604,
"column": 2
} | {
"line": 605,
"column": 28
} | [
{
"pp": "k : Type u_10\nV : Type u_11\nP : Type u_12\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\np : P\n⊢ ofLinearEquiv (LinearEquiv.refl k V) p p = refl k P",
"usedConstants": [
"AffineEquiv.ofLinearEquiv",
"AffineEquiv.refl",
"congrArg",
... | ext x
simp [ofLinearEquiv_apply] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.AffineSpace.AffineMap | {
"line": 432,
"column": 60
} | {
"line": 437,
"column": 54
} | [
{
"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 : P1 →ᵃ[k] P2\n⊢ Function.Surjective ⇑f.linear ↔ Function.S... | by
obtain ⟨p⟩ := (inferInstance : Nonempty P1)
have h : ⇑f.linear = (Equiv.vaddConst (f p)).symm ∘ f ∘ Equiv.vaddConst p := by
ext v
simp [f.map_vadd]
rw [h, Equiv.comp_surjective, Equiv.surjective_comp] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.AffineSpace.AffineMap | {
"line": 762,
"column": 2
} | {
"line": 762,
"column": 53
} | [
{
"pp": "k : Type u_2\nV1 : Type u_3\nP1 : Type u_4\ninst✝⁵ : Ring k\ninst✝⁴ : AddCommGroup V1\ninst✝³ : AffineSpace V1 P1\ninst✝² : Module k V1\nι : Type u_9\nφv : ι → Type u_10\ninst✝¹ : (i : ι) → AddCommGroup (φv i)\ninst✝ : (i : ι) → Module k (φv i)\nfv : (i : ι) → P1 →ᵃ[k] φv i\n⊢ pi fv = 0 ↔ ∀ (i : ι), fv... | simp only [AffineMap.ext_iff, funext_iff, pi_apply] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.LinearAlgebra.AffineSpace.AffineMap | {
"line": 788,
"column": 17
} | {
"line": 788,
"column": 20
} | [
{
"pp": "case h₁.h\nk : Type u_2\nV2 : Type u_5\nP2 : Type u_6\ninst✝⁷ : Ring k\ninst✝⁶ : AddCommGroup V2\ninst✝⁵ : AffineSpace V2 P2\ninst✝⁴ : Module k V2\nι : Type u_9\nφv : ι → Type u_10\ninst✝³ : (i : ι) → AddCommGroup (φv i)\ninst✝² : (i : ι) → Module k (φv i)\ninst✝¹ : Finite ι\ninst✝ : DecidableEq ι\nf g... | h₂, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.LocallyConvex.Basic | {
"line": 199,
"column": 4
} | {
"line": 199,
"column": 12
} | [
{
"pp": "case inl\n𝕜 : Type u_1\n𝕝 : Type u_2\nE : Type u_3\ninst✝⁸ : NormedDivisionRing 𝕜\ninst✝⁷ : AddCommGroup E\ninst✝⁶ : Module 𝕜 E\ns : Set E\ninst✝⁵ : NormedRing 𝕝\ninst✝⁴ : Module 𝕜 𝕝\ninst✝³ : NormSMulClass 𝕜 𝕝\ninst✝² : SMulWithZero 𝕝 E\ninst✝¹ : IsScalarTower 𝕜 𝕝 E\nx : E\ninst✝ : SMulCom... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.LocallyConvex.Basic | {
"line": 199,
"column": 4
} | {
"line": 199,
"column": 12
} | [
{
"pp": "case inl\n𝕜 : Type u_1\n𝕝 : Type u_2\nE : Type u_3\ninst✝⁸ : NormedDivisionRing 𝕜\ninst✝⁷ : AddCommGroup E\ninst✝⁶ : Module 𝕜 E\ns : Set E\ninst✝⁵ : NormedRing 𝕝\ninst✝⁴ : Module 𝕜 𝕝\ninst✝³ : NormSMulClass 𝕜 𝕝\ninst✝² : SMulWithZero 𝕝 E\ninst✝¹ : IsScalarTower 𝕜 𝕝 E\nx : E\ninst✝ : SMulCom... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.LocallyConvex.Basic | {
"line": 199,
"column": 4
} | {
"line": 199,
"column": 12
} | [
{
"pp": "case inl\n𝕜 : Type u_1\n𝕝 : Type u_2\nE : Type u_3\ninst✝⁸ : NormedDivisionRing 𝕜\ninst✝⁷ : AddCommGroup E\ninst✝⁶ : Module 𝕜 E\ns : Set E\ninst✝⁵ : NormedRing 𝕝\ninst✝⁴ : Module 𝕜 𝕝\ninst✝³ : NormSMulClass 𝕜 𝕝\ninst✝² : SMulWithZero 𝕝 E\ninst✝¹ : IsScalarTower 𝕜 𝕝 E\nx : E\ninst✝ : SMulCom... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.LocallyConvex.Bounded | {
"line": 190,
"column": 52
} | {
"line": 190,
"column": 65
} | [
{
"pp": "𝕜 : Type u_6\nι : Type u_7\nE : ι → Type u_8\ninst✝³ : NormedDivisionRing 𝕜\ninst✝² : (i : ι) → AddCommGroup (E i)\ninst✝¹ : (i : ι) → Module 𝕜 (E i)\ninst✝ : (i : ι) → TopologicalSpace (E i)\nS : Set ((i : ι) → E i)\n⊢ (∀ (i : ι), Tendsto (fun x ↦ (fun a ↦ x • a i) '' S) (𝓝 0) (𝓝 (0 i)).smallSets... | Pi.zero_apply | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Analysis.Convex.Function | {
"line": 404,
"column": 22
} | {
"line": 410,
"column": 28
} | [
{
"pp": "𝕜 : 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 : AddCommMonoid β\ninst✝³ : PartialOrder β\ninst✝² : Module 𝕜 E\ninst✝¹ : Module 𝕜 β\ninst✝ : LinearOrder E\ns : Set E\nf : E → β\nhs : Convex 𝕜 s\nhf :\n ∀ ⦃x : E⦄,\n x ∈ ... | by
refine convexOn_iff_pairwise_pos.2 ⟨hs, fun x hx y hy hxy a b ha hb hab => ?_⟩
wlog h : x < y
· rw [add_comm (a • x), add_comm (a • f x)]
rw [add_comm] at hab
exact this hs hf y hy x hx hxy.symm b a hb ha hab (hxy.lt_or_gt.resolve_left h)
exact hf hx hy h ha hb hab | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Convex.Function | {
"line": 544,
"column": 2
} | {
"line": 549,
"column": 82
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nβ : Type u_5\ninst✝⁸ : Semiring 𝕜\ninst✝⁷ : PartialOrder 𝕜\ninst✝⁶ : AddCommMonoid E\ninst✝⁵ : AddCommMonoid β\ninst✝⁴ : PartialOrder β\ninst✝³ : IsOrderedCancelAddMonoid β\ninst✝² : Module 𝕜 E\ninst✝¹ : Module 𝕜 β\ninst✝ : PosSMulStrictMono 𝕜 β\ns : Set E\nf : E → β\n... | rintro _ ⟨a, b, ha, hb, hab, rfl⟩
refine ⟨hf.1 hp.1 hq.1 ha.le hb.le hab, ?_⟩
calc
f (a • p.1 + b • q.1) ≤ a • f p.1 + b • f q.1 := hf.2 hp.1 hq.1 ha.le hb.le hab
_ < a • p.2 + b • q.2 := add_lt_add_of_lt_of_le
(smul_lt_smul_of_pos_left hp.2 ha) (smul_le_smul_of_nonneg_left hq.2 hb.le) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Convex.Function | {
"line": 544,
"column": 2
} | {
"line": 549,
"column": 82
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nβ : Type u_5\ninst✝⁸ : Semiring 𝕜\ninst✝⁷ : PartialOrder 𝕜\ninst✝⁶ : AddCommMonoid E\ninst✝⁵ : AddCommMonoid β\ninst✝⁴ : PartialOrder β\ninst✝³ : IsOrderedCancelAddMonoid β\ninst✝² : Module 𝕜 E\ninst✝¹ : Module 𝕜 β\ninst✝ : PosSMulStrictMono 𝕜 β\ns : Set E\nf : E → β\n... | rintro _ ⟨a, b, ha, hb, hab, rfl⟩
refine ⟨hf.1 hp.1 hq.1 ha.le hb.le hab, ?_⟩
calc
f (a • p.1 + b • q.1) ≤ a • f p.1 + b • f q.1 := hf.2 hp.1 hq.1 ha.le hb.le hab
_ < a • p.2 + b • q.2 := add_lt_add_of_lt_of_le
(smul_lt_smul_of_pos_left hp.2 ha) (smul_le_smul_of_nonneg_left hq.2 hb.le) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Convex.Function | {
"line": 583,
"column": 47
} | {
"line": 583,
"column": 79
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nβ : Type u_5\ninst✝⁸ : Semiring 𝕜\ninst✝⁷ : PartialOrder 𝕜\ninst✝⁶ : AddCommMonoid E\ninst✝⁵ : AddCommMonoid β\ninst✝⁴ : LinearOrder β\ninst✝³ : IsOrderedAddMonoid β\ninst✝² : SMul 𝕜 E\ninst✝¹ : Module 𝕜 β\ninst✝ : PosSMulStrictMono 𝕜 β\ns : Set E\nf g : E → β\nhf : Co... | by gcongr <;> apply le_sup_right | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Convex.Function | {
"line": 576,
"column": 91
} | {
"line": 583,
"column": 79
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nβ : Type u_5\ninst✝⁸ : Semiring 𝕜\ninst✝⁷ : PartialOrder 𝕜\ninst✝⁶ : AddCommMonoid E\ninst✝⁵ : AddCommMonoid β\ninst✝⁴ : LinearOrder β\ninst✝³ : IsOrderedAddMonoid β\ninst✝² : SMul 𝕜 E\ninst✝¹ : Module 𝕜 β\ninst✝ : PosSMulStrictMono 𝕜 β\ns : Set E\nf g : E → β\nhf : Co... | by
refine ⟨hf.left, fun x hx y hy a b ha hb hab => sup_le ?_ ?_⟩
· calc
f (a • x + b • y) ≤ a • f x + b • f y := hf.right hx hy ha hb hab
_ ≤ a • (f x ⊔ g x) + b • (f y ⊔ g y) := by gcongr <;> apply le_sup_left
· calc
g (a • x + b • y) ≤ a • g x + b • g y := hg.right hx hy ha hb hab
_ ≤ a ... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Convex.Function | {
"line": 599,
"column": 49
} | {
"line": 599,
"column": 81
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nβ : Type u_5\ninst✝⁸ : Semiring 𝕜\ninst✝⁷ : PartialOrder 𝕜\ninst✝⁶ : AddCommMonoid E\ninst✝⁵ : AddCommMonoid β\ninst✝⁴ : LinearOrder β\ninst✝³ : IsOrderedAddMonoid β\ninst✝² : SMul 𝕜 E\ninst✝¹ : Module 𝕜 β\ninst✝ : PosSMulStrictMono 𝕜 β\ns : Set E\nf g : E → β\nhf : St... | by gcongr <;> apply le_sup_right | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.AffineSpace.Centroid | {
"line": 62,
"column": 2
} | {
"line": 62,
"column": 10
} | [
{
"pp": "k : Type u_1\ninst✝¹ : DivisionRing k\nι : Type u_4\ns : Finset ι\ninst✝ : CharZero k\nh : #s ≠ 0\n⊢ ∑ i ∈ s, centroidWeights k s i = 1",
"usedConstants": [
"GroupWithZero.toMonoidWithZero",
"NonAssocSemiring.toAddCommMonoidWithOne",
"MulOne.toOne",
"False",
"instHSMul... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.LinearAlgebra.AffineSpace.Centroid | {
"line": 62,
"column": 2
} | {
"line": 62,
"column": 10
} | [
{
"pp": "k : Type u_1\ninst✝¹ : DivisionRing k\nι : Type u_4\ns : Finset ι\ninst✝ : CharZero k\nh : #s ≠ 0\n⊢ ∑ i ∈ s, centroidWeights k s i = 1",
"usedConstants": [
"GroupWithZero.toMonoidWithZero",
"NonAssocSemiring.toAddCommMonoidWithOne",
"MulOne.toOne",
"False",
"instHSMul... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.AffineSpace.Centroid | {
"line": 62,
"column": 2
} | {
"line": 62,
"column": 10
} | [
{
"pp": "k : Type u_1\ninst✝¹ : DivisionRing k\nι : Type u_4\ns : Finset ι\ninst✝ : CharZero k\nh : #s ≠ 0\n⊢ ∑ i ∈ s, centroidWeights k s i = 1",
"usedConstants": [
"GroupWithZero.toMonoidWithZero",
"NonAssocSemiring.toAddCommMonoidWithOne",
"MulOne.toOne",
"False",
"instHSMul... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.AffineSpace.Centroid | {
"line": 61,
"column": 42
} | {
"line": 62,
"column": 10
} | [
{
"pp": "k : Type u_1\ninst✝¹ : DivisionRing k\nι : Type u_4\ns : Finset ι\ninst✝ : CharZero k\nh : #s ≠ 0\n⊢ ∑ i ∈ s, centroidWeights k s i = 1",
"usedConstants": [
"GroupWithZero.toMonoidWithZero",
"NonAssocSemiring.toAddCommMonoidWithOne",
"MulOne.toOne",
"False",
"instHSMul... | by
simp_all | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.AffineSpace.Independent | {
"line": 202,
"column": 8
} | {
"line": 202,
"column": 16
} | [
{
"pp": "case neg\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\nha : AffineIndependent k p\ns1 s2 : Finset ι\nw1 w2 : ι → k\nhw1 : ∑ i ∈ s1, w1 i = 1\nhw2 : ∑ i ∈ s2, w2 i = 1\nheq : (affineCombination ... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.LinearAlgebra.AffineSpace.Independent | {
"line": 202,
"column": 8
} | {
"line": 202,
"column": 16
} | [
{
"pp": "case neg\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\nha : AffineIndependent k p\ns1 s2 : Finset ι\nw1 w2 : ι → k\nhw1 : ∑ i ∈ s1, w1 i = 1\nhw2 : ∑ i ∈ s2, w2 i = 1\nheq : (affineCombination ... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.AffineSpace.Independent | {
"line": 202,
"column": 8
} | {
"line": 202,
"column": 16
} | [
{
"pp": "case neg\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\nha : AffineIndependent k p\ns1 s2 : Finset ι\nw1 w2 : ι → k\nhw1 : ∑ i ∈ s1, w1 i = 1\nhw2 : ∑ i ∈ s2, w2 i = 1\nheq : (affineCombination ... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Seminorm | {
"line": 89,
"column": 4
} | {
"line": 89,
"column": 39
} | [
{
"pp": "R : Type u_1\nR' : Type u_2\n𝕜 : Type u_3\n𝕜₂ : Type u_4\n𝕜₃ : Type u_5\n𝕝 : Type u_6\nE : Type u_7\nE₂ : Type u_8\nE₃ : Type u_9\nF : Type u_10\nι : Type u_11\ninst✝² : NormedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\nf : E → ℝ\nmap_zero : f 0 = 0\nadd_le : ∀ (x y : E), f (x + y) ≤ f ... | refine le_antisymm (smul_le r x) ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.LinearAlgebra.AffineSpace.Independent | {
"line": 216,
"column": 6
} | {
"line": 216,
"column": 59
} | [
{
"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\nha :\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 p) w1... | replace ha := ha s s w2 w1 hw2 hw1 (hw1s.symm ▸ hw2s) | Lean.Elab.Tactic.evalReplace | Lean.Parser.Tactic.replace |
Mathlib.Analysis.Seminorm | {
"line": 348,
"column": 73
} | {
"line": 348,
"column": 86
} | [
{
"pp": "case empty\n𝕜 : Type u_3\nE : Type u_7\nι : Type u_11\ninst✝² : SeminormedRing 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\np : ι → Seminorm 𝕜 E\nx : E\n⊢ 0 x = ↑0",
"usedConstants": [
"Eq.mpr",
"Real",
"Real.instZero",
"congrArg",
"Pi.zero_apply",
"id",
... | Pi.zero_apply | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.AffineSpace.Independent | {
"line": 231,
"column": 2
} | {
"line": 238,
"column": 30
} | [
{
"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\n⊢ (∀ (w1 w2 : ι → k),\n ∑ i, w1 i = 1 → ∑ i, w2 i = 1 → (affineCombination k univ p) w1 = (affineCombination k un... | · intro h s1 s2 w1 w2 hw1 hw2 hweq
have hw1' : (∑ i, (s1 : Set ι).indicator w1 i) = 1 := by
rwa [Finset.sum_indicator_subset _ (Finset.subset_univ s1)]
have hw2' : (∑ i, (s2 : Set ι).indicator w2 i) = 1 := by
rwa [Finset.sum_indicator_subset _ (Finset.subset_univ s2)]
rw [Finset.affineCombinatio... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Seminorm | {
"line": 469,
"column": 46
} | {
"line": 469,
"column": 76
} | [
{
"pp": "case h\nR : Type u_1\n𝕜 : Type u_3\nE : Type u_7\ninst✝⁵ : NormedField 𝕜\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module 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✝ : E\n⊢ r • ⨅ u, p u + q (x✝ - u) = ⨅ u, r • p u + r • q (x✝ - u)",
"usedCo... | ← smul_one_smul ℝ≥0 r (_ : ℝ), | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.LinearAlgebra.AffineSpace.Independent | {
"line": 302,
"column": 19
} | {
"line": 302,
"column": 29
} | [
{
"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\nι2 : Type u_5\nf : ι2 ↪ ι\np : ι → P\nha : AffineIndependent k p\nfs : Finset ι2\nw : ι2 → k\nhw : ∑ i ∈ fs, w i = 0\nhs✝ : (fs.weightedVSub (p ∘ ⇑f)) w = 0\ni... | dif_pos h, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Analysis.Seminorm | {
"line": 687,
"column": 2
} | {
"line": 690,
"column": 90
} | [
{
"pp": "𝕜 : Type u_3\nE : Type u_7\nι : Type u_11\ninst✝² : SeminormedRing 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : SMul 𝕜 E\np : ι → Seminorm 𝕜 E\ns : Finset ι\nH : s.Nonempty\ne : E\nr : ℝ\n⊢ (s.sup' H p).closedBall e r = s.inf' H fun i ↦ (p i).closedBall e r",
"usedConstants": [
"Real",
"Fins... | induction H using Finset.Nonempty.cons_induction with
| singleton => simp
| cons _ _ _ hs ih =>
simp only [Finset.sup'_cons hs, Finset.inf'_cons hs, closedBall_sup, inf_eq_inter, ih] | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.Analysis.Seminorm | {
"line": 687,
"column": 2
} | {
"line": 690,
"column": 90
} | [
{
"pp": "𝕜 : Type u_3\nE : Type u_7\nι : Type u_11\ninst✝² : SeminormedRing 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : SMul 𝕜 E\np : ι → Seminorm 𝕜 E\ns : Finset ι\nH : s.Nonempty\ne : E\nr : ℝ\n⊢ (s.sup' H p).closedBall e r = s.inf' H fun i ↦ (p i).closedBall e r",
"usedConstants": [
"Real",
"Fins... | induction H using Finset.Nonempty.cons_induction with
| singleton => simp
| cons _ _ _ hs ih =>
simp only [Finset.sup'_cons hs, Finset.inf'_cons hs, closedBall_sup, inf_eq_inter, ih] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Seminorm | {
"line": 687,
"column": 2
} | {
"line": 690,
"column": 90
} | [
{
"pp": "𝕜 : Type u_3\nE : Type u_7\nι : Type u_11\ninst✝² : SeminormedRing 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : SMul 𝕜 E\np : ι → Seminorm 𝕜 E\ns : Finset ι\nH : s.Nonempty\ne : E\nr : ℝ\n⊢ (s.sup' H p).closedBall e r = s.inf' H fun i ↦ (p i).closedBall e r",
"usedConstants": [
"Real",
"Fins... | induction H using Finset.Nonempty.cons_induction with
| singleton => simp
| cons _ _ _ hs ih =>
simp only [Finset.sup'_cons hs, Finset.inf'_cons hs, closedBall_sup, inf_eq_inter, ih] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.AffineSpace.Independent | {
"line": 453,
"column": 2
} | {
"line": 454,
"column": 65
} | [
{
"pp": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁶ : Ring k\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module k V\ninst✝³ : AffineSpace V P\nV₂ : Type u_5\nP₂ : Type u_6\ninst✝² : AddCommGroup V₂\ninst✝¹ : Module k V₂\ninst✝ : AffineSpace V₂ P₂\ns : Set P\ne : P ≃ᵃ[k] P₂\n⊢ AffineIndependent k Subtype.val ↔ Affi... | have : e ∘ ((↑) : s → P) = ((↑) : e '' s → P₂) ∘ (e : P ≃ P₂).image s := rfl
simp [← e.affineIndependent_iff, this, affineIndependent_equiv] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.AffineSpace.Independent | {
"line": 453,
"column": 2
} | {
"line": 454,
"column": 65
} | [
{
"pp": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁶ : Ring k\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module k V\ninst✝³ : AffineSpace V P\nV₂ : Type u_5\nP₂ : Type u_6\ninst✝² : AddCommGroup V₂\ninst✝¹ : Module k V₂\ninst✝ : AffineSpace V₂ P₂\ns : Set P\ne : P ≃ᵃ[k] P₂\n⊢ AffineIndependent k Subtype.val ↔ Affi... | have : e ∘ ((↑) : s → P) = ((↑) : e '' s → P₂) ∘ (e : P ≃ P₂).image s := rfl
simp [← e.affineIndependent_iff, this, affineIndependent_equiv] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.AffineSpace.Independent | {
"line": 473,
"column": 6
} | {
"line": 473,
"column": 59
} | [
{
"pp": "case h.mp.refine_1\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✝ : 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\nh... | rw [← hw₁, ← fs₁.sum_inter_add_sum_diff fs₂, eq_comm] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Basic | {
"line": 953,
"column": 4
} | {
"line": 953,
"column": 12
} | [
{
"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 : P\nhp₁₂ : p₁ ≠ p₂\nr₂ : k\nhp₂ : r₂ • (p₄ -ᵥ p₃) = p₂ -ᵥ p₃\nhp₁ : r₂ • (p₄ -ᵥ p₃) = p₁ -ᵥ p₃\n⊢ False",
"usedConstants": [
"False",
"i... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Seminorm | {
"line": 926,
"column": 2
} | {
"line": 927,
"column": 74
} | [
{
"pp": "case h\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\nx✝ : E\n⊢ x✝ ∈ k • p.ball 0 r ↔ x✝ ∈ p.ball 0 (‖k‖ * r)",
"usedConstants": [
"Seminorm.instSeminormClass",
"Iff.mpr",
"Seminorm... | rw [mem_smul_set_iff_inv_smul_mem₀ hk, p.mem_ball_zero, p.mem_ball_zero, map_smul_eq_mul,
norm_inv, ← div_eq_inv_mul, div_lt_iff₀ (norm_pos_iff.2 hk), mul_comm] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.AffineSpace.Basis | {
"line": 205,
"column": 2
} | {
"line": 207,
"column": 37
} | [
{
"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\ns : Finset ι\ni : ι\nhi : i ∈ s\nw : ι → k\nhw : s.sum w = 1\n⊢ (b.coord i) ((Finset.affineCombination k s ⇑b) w) = w i",
"usedConst... | classical simp only [coord_apply, hi, Finset.affineCombination_eq_linear_combination, if_true,
mul_boole, hw, Function.comp_apply, smul_eq_mul, s.sum_ite_eq,
s.map_affineCombination b w hw] | Lean.Elab.Tactic.evalClassical | Lean.Parser.Tactic.classical |
Mathlib.LinearAlgebra.AffineSpace.Basis | {
"line": 205,
"column": 2
} | {
"line": 207,
"column": 37
} | [
{
"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\ns : Finset ι\ni : ι\nhi : i ∈ s\nw : ι → k\nhw : s.sum w = 1\n⊢ (b.coord i) ((Finset.affineCombination k s ⇑b) w) = w i",
"usedConst... | classical simp only [coord_apply, hi, Finset.affineCombination_eq_linear_combination, if_true,
mul_boole, hw, Function.comp_apply, smul_eq_mul, s.sum_ite_eq,
s.map_affineCombination b w hw] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.AffineSpace.Basis | {
"line": 205,
"column": 2
} | {
"line": 207,
"column": 37
} | [
{
"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\ns : Finset ι\ni : ι\nhi : i ∈ s\nw : ι → k\nhw : s.sum w = 1\n⊢ (b.coord i) ((Finset.affineCombination k s ⇑b) w) = w i",
"usedConst... | classical simp only [coord_apply, hi, Finset.affineCombination_eq_linear_combination, if_true,
mul_boole, hw, Function.comp_apply, smul_eq_mul, s.sum_ite_eq,
s.map_affineCombination b w hw] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.AffineSpace.Basis | {
"line": 328,
"column": 43
} | {
"line": 328,
"column": 70
} | [
{
"pp": "ι : Type u_1\nι' : Type u_2\nG : Type u_3\nG' : Type u_4\nk : Type u_5\nV : Type u_6\nP : Type u_7\ninst✝⁹ : AddCommGroup V\ninst✝⁸ : AffineSpace V P\ninst✝⁷ : Ring k\ninst✝⁶ : Module k V\nb✝ : AffineBasis ι k P\ns : Finset ι\ni j : ι\ne : ι ≃ ι'\ninst✝⁵ : Group G\ninst✝⁴ : Group G'\ninst✝³ : DistribMu... | ← AffineSubspace.smul_span, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.AffineSpace.Independent | {
"line": 626,
"column": 4
} | {
"line": 627,
"column": 81
} | [
{
"pp": "case refine_1\nk : Type u_1\nV : Type u_2\ninst✝² : Ring k\ninst✝¹ : AddCommGroup V\ninst✝ : Module k V\nt : Finset V\nw : ↥t → k\nhw : ∑ i, w i = 0\ni : ↥t\nhi : ¬w i = 0\nhwt : ∑ x, w x • ↑x = 0\nf : (x : V) → x ∈ t → k := fun x hx ↦ w ⟨x, hx⟩\n⊢ (∑ e ∈ t, if hx : e ∈ t then f e hx • e else 0) = 0",
... | all_goals
simp only [f, Finset.sum_dite_of_true fun _ h => h, Finset.mk_coe, hwt, hw] | Lean.Elab.Tactic.evalAllGoals | Lean.Parser.Tactic.allGoals |
Mathlib.Analysis.Convex.PathConnected | {
"line": 114,
"column": 4
} | {
"line": 116,
"column": 8
} | [
{
"pp": "E : Type u_1\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module ℝ E\ninst✝² : TopologicalSpace E\ninst✝¹ : ContinuousAdd E\ninst✝ : ContinuousSMul ℝ E\np q : Submodule ℝ E\nhpq : IsCompl p q\nhpc : IsPathConnected {0}ᶜ\n⊢ (↑q)ᶜ = ⇑(p.prodEquivOfIsCompl q hpq) '' {0}ᶜ ×ˢ univ",
"usedConstants": [
"Set.... | rw [prod_univ, LinearEquiv.image_eq_preimage_symm]
ext
simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Convex.PathConnected | {
"line": 114,
"column": 4
} | {
"line": 116,
"column": 8
} | [
{
"pp": "E : Type u_1\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module ℝ E\ninst✝² : TopologicalSpace E\ninst✝¹ : ContinuousAdd E\ninst✝ : ContinuousSMul ℝ E\np q : Submodule ℝ E\nhpq : IsCompl p q\nhpc : IsPathConnected {0}ᶜ\n⊢ (↑q)ᶜ = ⇑(p.prodEquivOfIsCompl q hpq) '' {0}ᶜ ×ˢ univ",
"usedConstants": [
"Set.... | rw [prod_univ, LinearEquiv.image_eq_preimage_symm]
ext
simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.AffineSpace.Simplex.Basic | {
"line": 429,
"column": 70
} | {
"line": 429,
"column": 89
} | [
{
"pp": "case pos\nk : Type u_1\nV : Type u_2\nV₂ : Type u_3\nP : Type u_4\nP₂ : Type u_5\ninst✝⁶ : Ring k\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module k V\ninst✝³ : AffineSpace V P\ninst✝² : AddCommGroup V₂\ninst✝¹ : Module k V₂\ninst✝ : AffineSpace V₂ P₂\nI : Set k\nn : ℕ\ns : Simplex k P n\nf : P →ᵃ[k] P₂\nhf :... | Simplex.map_points, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.AffineSpace.Simplex.Basic | {
"line": 498,
"column": 38
} | {
"line": 498,
"column": 46
} | [
{
"pp": "case inl\nk : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝⁵ : Ring k\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module k V\ninst✝² : AffineSpace V P\ninst✝¹ : PartialOrder k\ninst✝ : ZeroLEOneClass k\nn : ℕ\ns : Simplex k P n\nj : Fin (n + 1)\n⊢ Pi.single j 1 j ∈ Set.Icc 0 1",
"usedConstants": [
"Ring... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.LinearAlgebra.AffineSpace.Simplex.Basic | {
"line": 498,
"column": 38
} | {
"line": 498,
"column": 46
} | [
{
"pp": "case inr\nk : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝⁵ : Ring k\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module k V\ninst✝² : AffineSpace V P\ninst✝¹ : PartialOrder k\ninst✝ : ZeroLEOneClass k\nn : ℕ\ns : Simplex k P n\ni j : Fin (n + 1)\nhj : j ≠ i\n⊢ Pi.single i 1 j ∈ Set.Icc 0 1",
"usedConstants":... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Convex.Topology | {
"line": 268,
"column": 12
} | {
"line": 268,
"column": 56
} | [
{
"pp": "case a\n𝕜 : Type u_2\nE : Type u_3\ninst✝⁹ : Field 𝕜\ninst✝⁸ : LinearOrder 𝕜\ninst✝⁷ : IsStrictOrderedRing 𝕜\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : Module 𝕜 E\ninst✝⁴ : TopologicalSpace E\ninst✝³ : IsTopologicalAddGroup E\ninst✝² : TopologicalSpace 𝕜\ninst✝¹ : OrderTopology 𝕜\ninst✝ : ContinuousSMul... | ← AffineMap.lineMap_apply_zero (k := 𝕜) x y, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Algebra.Module.LocallyConvex | {
"line": 115,
"column": 2
} | {
"line": 115,
"column": 60
} | [
{
"pp": "case refine_1\nE : Type u_2\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : TopologicalSpace E\ninst✝³ : IsTopologicalAddGroup E\ninst✝² : Module ℝ E\ninst✝¹ : ContinuousSMul ℝ E\ninst✝ : LocallyConvexSpace ℝ E\nS : Set E\nhS : Convex ℝ S\nx : ↑S\ns : Set ↑S\nhs : s ∈ 𝓝 x\nt : Set E\nht : t ∈ 𝓝 ↑x ∧ Subtype.val ⁻... | · exact continuousAt_subtype_val.preimage_mem_nhds ht'.1.1 | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Topology.Algebra.Module.LocallyConvex | {
"line": 242,
"column": 8
} | {
"line": 242,
"column": 16
} | [
{
"pp": "case inr.inl.refine_1\nR : Type u_1\ninst✝⁵ : TopologicalSpace R\ninst✝⁴ : Semiring R\ninst✝³ : LinearOrder R\ninst✝² : IsStrictOrderedRing R\ninst✝¹ : OrderTopology R\nx : R\nhl : ∃ b, b < x\nhu : IsTop x\ninst✝ : OrderTop R\ni✝ : R\na✝ : i✝ < ⊤\n⊢ Ioi i✝ ∈ 𝓝 ⊤ ∧ Convex R (Ioi i✝)",
"usedConstant... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Normed.Module.Convex | {
"line": 71,
"column": 73
} | {
"line": 72,
"column": 76
} | [
{
"pp": "E : Type u_1\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\na : E\nr : ℝ\n⊢ Convex ℝ (closedBall a r)",
"usedConstants": [
"IsOrderedModule.toPosSMulMono",
"Real.partialOrder",
"Real",
"Set.sep_univ",
"Semiring.toModule",
"convexOn_univ_dist",
... | by
simpa only [closedBall, sep_univ] using (convexOn_univ_dist a).convex_le r | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Normed.Module.Convex | {
"line": 105,
"column": 59
} | {
"line": 105,
"column": 67
} | [
{
"pp": "F : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : Nontrivial F\nx✝¹ : F\nr : ℝ\nhr : 0 ≤ r\nx : F\nh : x ∈ closedBall 0 r\nU : Set F\nhU_sub : sphere 0 r ⊆ U\nhU : Convex ℝ U\nx✝ : Invertible 2\nz : F\nhz : z ∈ sphere 0 r\n⊢ -z ∈ sphere 0 r",
"usedConstants": [
"N... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Normed.Module.Convex | {
"line": 105,
"column": 59
} | {
"line": 105,
"column": 67
} | [
{
"pp": "F : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : Nontrivial F\nx✝¹ : F\nr : ℝ\nhr : 0 ≤ r\nx : F\nh : x ∈ closedBall 0 r\nU : Set F\nhU_sub : sphere 0 r ⊆ U\nhU : Convex ℝ U\nx✝ : Invertible 2\nz : F\nhz : z ∈ sphere 0 r\n⊢ -z ∈ sphere 0 r",
"usedConstants": [
"N... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Module.Convex | {
"line": 105,
"column": 59
} | {
"line": 105,
"column": 67
} | [
{
"pp": "F : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : Nontrivial F\nx✝¹ : F\nr : ℝ\nhr : 0 ≤ r\nx : F\nh : x ∈ closedBall 0 r\nU : Set F\nhU_sub : sphere 0 r ⊆ U\nhU : Convex ℝ U\nx✝ : Invertible 2\nz : F\nhz : z ∈ sphere 0 r\n⊢ -z ∈ sphere 0 r",
"usedConstants": [
"N... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Module.Convex | {
"line": 107,
"column": 4
} | {
"line": 107,
"column": 12
} | [
{
"pp": "case pos\nF : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : Nontrivial F\nx✝ : F\nr : ℝ\nhr : 0 ≤ r\nx : F\nh : x ∈ closedBall 0 r\nU : Set F\nhU_sub : sphere 0 r ⊆ U\nhU : Convex ℝ U\nzero_mem : 0 ∈ U\nhr₀ : r = 0\n⊢ x ∈ U",
"usedConstants": [
"Real",
"Real... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Normed.Module.Convex | {
"line": 107,
"column": 4
} | {
"line": 107,
"column": 12
} | [
{
"pp": "case pos\nF : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : Nontrivial F\nx✝ : F\nr : ℝ\nhr : 0 ≤ r\nx : F\nh : x ∈ closedBall 0 r\nU : Set F\nhU_sub : sphere 0 r ⊆ U\nhU : Convex ℝ U\nzero_mem : 0 ∈ U\nhr₀ : r = 0\n⊢ x ∈ U",
"usedConstants": [
"Real",
"Real... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Module.Convex | {
"line": 107,
"column": 4
} | {
"line": 107,
"column": 12
} | [
{
"pp": "case pos\nF : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : Nontrivial F\nx✝ : F\nr : ℝ\nhr : 0 ≤ r\nx : F\nh : x ∈ closedBall 0 r\nU : Set F\nhU_sub : sphere 0 r ⊆ U\nhU : Convex ℝ U\nzero_mem : 0 ∈ U\nhr₀ : r = 0\n⊢ x ∈ U",
"usedConstants": [
"Real",
"Real... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Module.Convex | {
"line": 119,
"column": 54
} | {
"line": 119,
"column": 62
} | [
{
"pp": "F : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : Nontrivial F\nx✝ : F\nr : ℝ\nhr : 0 ≤ r\nx : F\nh : x ∈ closedBall 0 r\nU : Set F\nhU_sub : sphere 0 r ⊆ U\nhU : Convex ℝ U\nzero_mem : 0 ∈ U\nhr₀ : ¬r = 0\nx_zero : ¬x = 0\nz : F := (r * ‖x‖⁻¹) • x\nhz_def : z = (r * ‖x‖⁻¹)... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Normed.Module.Convex | {
"line": 119,
"column": 54
} | {
"line": 119,
"column": 62
} | [
{
"pp": "F : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : Nontrivial F\nx✝ : F\nr : ℝ\nhr : 0 ≤ r\nx : F\nh : x ∈ closedBall 0 r\nU : Set F\nhU_sub : sphere 0 r ⊆ U\nhU : Convex ℝ U\nzero_mem : 0 ∈ U\nhr₀ : ¬r = 0\nx_zero : ¬x = 0\nz : F := (r * ‖x‖⁻¹) • x\nhz_def : z = (r * ‖x‖⁻¹)... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Module.Convex | {
"line": 119,
"column": 54
} | {
"line": 119,
"column": 62
} | [
{
"pp": "F : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : Nontrivial F\nx✝ : F\nr : ℝ\nhr : 0 ≤ r\nx : F\nh : x ∈ closedBall 0 r\nU : Set F\nhU_sub : sphere 0 r ⊆ U\nhU : Convex ℝ U\nzero_mem : 0 ∈ U\nhr₀ : ¬r = 0\nx_zero : ¬x = 0\nz : F := (r * ‖x‖⁻¹) • x\nhz_def : z = (r * ‖x‖⁻¹)... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Convex.Combination | {
"line": 286,
"column": 2
} | {
"line": 286,
"column": 42
} | [
{
"pp": "R : Type u_1\nE : Type u_3\nι : Type u_5\ninst✝⁴ : Field R\ninst✝³ : AddCommGroup E\ninst✝² : Module R E\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\ns : Finset ι\nv : ι → E\nw : ι → R\nhw₀ : ∀ i ∈ s, 0 ≤ w i\nhw₁ : s.sum w = 1\n⊢ (affineCombination R s v) w ∈ (convexHull R) (Set.range v)",
... | rw [affineCombination_eq_centerMass hw₁] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Convex.Combination | {
"line": 320,
"column": 32
} | {
"line": 320,
"column": 40
} | [
{
"pp": "case h\nR : Type u_1\nE : Type u_3\nι : Type u_5\ninst✝⁴ : Field R\ninst✝³ : AddCommGroup E\ninst✝² : Module R E\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\nv : ι → E\ns : Finset ι\nw : ι → R\nhw₀ : ∀ i ∈ s, 0 ≤ w i\nhw₁ : s.sum w = 1\ns' : Finset ι\nw' : ι → R\nhw₀' : ∀ i ∈ s', 0 ≤ w' i\nh... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Convex.Combination | {
"line": 320,
"column": 32
} | {
"line": 320,
"column": 40
} | [
{
"pp": "case h\nR : Type u_1\nE : Type u_3\nι : Type u_5\ninst✝⁴ : Field R\ninst✝³ : AddCommGroup E\ninst✝² : Module R E\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\nv : ι → E\ns : Finset ι\nw : ι → R\nhw₀ : ∀ i ∈ s, 0 ≤ w i\nhw₁ : s.sum w = 1\ns' : Finset ι\nw' : ι → R\nhw₀' : ∀ i ∈ s', 0 ≤ w' i\nh... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Convex.Combination | {
"line": 320,
"column": 32
} | {
"line": 320,
"column": 40
} | [
{
"pp": "R : Type u_1\nE : Type u_3\nι : Type u_5\ninst✝⁴ : Field R\ninst✝³ : AddCommGroup E\ninst✝² : Module R E\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\nv : ι → E\ns : Finset ι\nw : ι → R\nhw₀ : ∀ i ∈ s, 0 ≤ w i\nhw₁ : s.sum w = 1\ns' : Finset ι\nw' : ι → R\nhw₀' : ∀ i ∈ s', 0 ≤ w' i\nhw₁' : s'... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Convex.Combination | {
"line": 320,
"column": 32
} | {
"line": 320,
"column": 40
} | [
{
"pp": "R : Type u_1\nE : Type u_3\nι : Type u_5\ninst✝⁴ : Field R\ninst✝³ : AddCommGroup E\ninst✝² : Module R E\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\nv : ι → E\ns : Finset ι\nw : ι → R\nhw₀ : ∀ i ∈ s, 0 ≤ w i\nhw₁ : s.sum w = 1\ns' : Finset ι\nw' : ι → R\nhw₀' : ∀ i ∈ s', 0 ≤ w' i\nhw₁' : s'... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Convex.Jensen | {
"line": 222,
"column": 58
} | {
"line": 222,
"column": 66
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nβ : Type u_4\nι : Type u_5\ninst✝⁹ : Field 𝕜\ninst✝⁸ : LinearOrder 𝕜\ninst✝⁷ : IsStrictOrderedRing 𝕜\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : AddCommGroup β\ninst✝⁴ : PartialOrder β\ninst✝³ : IsOrderedAddMonoid β\ninst✝² : Module 𝕜 E\ninst✝¹ : Module 𝕜 β\ninst✝ : IsStrictOrd... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Convex.Jensen | {
"line": 223,
"column": 63
} | {
"line": 223,
"column": 71
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nβ : Type u_4\nι : Type u_5\ninst✝⁹ : Field 𝕜\ninst✝⁸ : LinearOrder 𝕜\ninst✝⁷ : IsStrictOrderedRing 𝕜\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : AddCommGroup β\ninst✝⁴ : PartialOrder β\ninst✝³ : IsOrderedAddMonoid β\ninst✝² : Module 𝕜 E\ninst✝¹ : Module 𝕜 β\ninst✝ : IsStrictOrd... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.LocallyConvex.WithSeminorms | {
"line": 131,
"column": 2
} | {
"line": 131,
"column": 81
} | [
{
"pp": "case h\nR : Type u_1\nE : Type u_6\nι : Type u_9\ninst✝² : SeminormedRing R\ninst✝¹ : AddCommGroup E\ninst✝ : Module R E\np : SeminormFamily R E ι\nU : Set E\nhU✝ : U ∈ p.basisSets\ns : Finset ι\nr : ℝ\nhr : 0 < r\nhU : U = (s.sup p).ball 0 r\n⊢ (s.sup p).ball 0 (r / 2) + (s.sup p).ball 0 (r / 2) ⊆ U",... | refine Set.Subset.trans (ball_add_ball_subset (s.sup p) (r / 2) (r / 2) 0 0) ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.LocallyConvex.WithSeminorms | {
"line": 194,
"column": 6
} | {
"line": 195,
"column": 34
} | [
{
"pp": "case refine_1.refine_1\n𝕜 : Type u_2\nF : Type u_7\nι : Type u_9\ninst✝² : NormedField 𝕜\ninst✝¹ : AddCommGroup F\ninst✝ : Module 𝕜 F\np : SeminormFamily 𝕜 F ι\ni : ι\nε : ℝ\nhε : 0 < ε\n⊢ (p i).ball 0 ε ∈ p.moduleFilterBasis.toFilterBasis",
"usedConstants": [
"Eq.mpr",
"NormedCommR... | rw [← (Finset.sup_singleton : _ = p i)]
exact p.basisSets_mem {i} hε | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.LocallyConvex.WithSeminorms | {
"line": 194,
"column": 6
} | {
"line": 195,
"column": 34
} | [
{
"pp": "case refine_1.refine_1\n𝕜 : Type u_2\nF : Type u_7\nι : Type u_9\ninst✝² : NormedField 𝕜\ninst✝¹ : AddCommGroup F\ninst✝ : Module 𝕜 F\np : SeminormFamily 𝕜 F ι\ni : ι\nε : ℝ\nhε : 0 < ε\n⊢ (p i).ball 0 ε ∈ p.moduleFilterBasis.toFilterBasis",
"usedConstants": [
"Eq.mpr",
"NormedCommR... | rw [← (Finset.sup_singleton : _ = p i)]
exact p.basisSets_mem {i} hε | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.LocallyConvex.WithSeminorms | {
"line": 550,
"column": 2
} | {
"line": 550,
"column": 12
} | [
{
"pp": "case mpr\n𝕜 : Type u_2\nE : Type u_6\nι : Type u_9\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : AddCommGroup E\ninst✝¹ : Module 𝕜 E\np : SeminormFamily 𝕜 E ι\ninst✝ : TopologicalSpace E\ns : Set E\nhp : WithSeminorms p\n⊢ (∀ (i : ι), ∃ r > 0, ∀ x ∈ s, (p i) x < r) → ∀ (I : Finset ι), ∃ r > 0, ∀ x ... | intro hi I | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Topology.Algebra.Module.Spaces.UniformConvergenceCLM | {
"line": 378,
"column": 2
} | {
"line": 378,
"column": 27
} | [
{
"pp": "𝕜₁ : Type u_1\n𝕜₂ : Type u_2\ninst✝¹² : NormedField 𝕜₁\ninst✝¹¹ : NormedField 𝕜₂\nσ : 𝕜₁ →+* 𝕜₂\nE : Type u_3\nF✝ : Type u_4\ninst✝¹⁰ : AddCommGroup E\ninst✝⁹ : Module 𝕜₁ E\ninst✝⁸ : TopologicalSpace E\ninst✝⁷ : AddCommGroup F✝\ninst✝⁶ : Module 𝕜₂ F✝\n𝔖 : Set (Set E)\nh𝔖 : IsCoherentWith 𝔖\n... | apply IsClosed.isComplete | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Analysis.Normed.Operator.Basic | {
"line": 68,
"column": 25
} | {
"line": 68,
"column": 33
} | [
{
"pp": "𝕜 : Type u_1\n𝕜₂ : Type u_2\nE : Type u_4\nF : Type u_5\n𝓕 : Type u_8\ninst✝⁸ : SeminormedAddCommGroup E\ninst✝⁷ : SeminormedAddCommGroup F\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NontriviallyNormedField 𝕜₂\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedSpace 𝕜₂ F\nσ₁₂ : 𝕜 →+* 𝕜₂\ninst✝² : Fu... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Normed.Operator.Basic | {
"line": 68,
"column": 25
} | {
"line": 68,
"column": 33
} | [
{
"pp": "𝕜 : Type u_1\n𝕜₂ : Type u_2\nE : Type u_4\nF : Type u_5\n𝓕 : Type u_8\ninst✝⁸ : SeminormedAddCommGroup E\ninst✝⁷ : SeminormedAddCommGroup F\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NontriviallyNormedField 𝕜₂\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedSpace 𝕜₂ F\nσ₁₂ : 𝕜 →+* 𝕜₂\ninst✝² : Fu... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Operator.Basic | {
"line": 68,
"column": 25
} | {
"line": 68,
"column": 33
} | [
{
"pp": "𝕜 : Type u_1\n𝕜₂ : Type u_2\nE : Type u_4\nF : Type u_5\n𝓕 : Type u_8\ninst✝⁸ : SeminormedAddCommGroup E\ninst✝⁷ : SeminormedAddCommGroup F\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NontriviallyNormedField 𝕜₂\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedSpace 𝕜₂ F\nσ₁₂ : 𝕜 →+* 𝕜₂\ninst✝² : Fu... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Operator.Basic | {
"line": 74,
"column": 2
} | {
"line": 74,
"column": 10
} | [
{
"pp": "case a\n𝕜 : Type u_1\n𝕜₂ : Type u_2\nE : Type u_4\nF : Type u_5\n𝓕 : Type u_8\ninst✝⁸ : SeminormedAddCommGroup E\ninst✝⁷ : SeminormedAddCommGroup F\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NontriviallyNormedField 𝕜₂\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedSpace 𝕜₂ F\nσ₁₂ : 𝕜 →+* 𝕜₂\nins... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Normed.Operator.Basic | {
"line": 79,
"column": 7
} | {
"line": 79,
"column": 15
} | [
{
"pp": "𝕜 : Type u_1\n𝕜₂ : Type u_2\nE : Type u_4\nF : Type u_5\n𝓕 : Type u_8\ninst✝⁸ : SeminormedAddCommGroup E\ninst✝⁷ : SeminormedAddCommGroup F\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NontriviallyNormedField 𝕜₂\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedSpace 𝕜₂ F\nσ₁₂ : 𝕜 →+* 𝕜₂\ninst✝² : Fu... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Normed.Operator.Basic | {
"line": 79,
"column": 7
} | {
"line": 79,
"column": 15
} | [
{
"pp": "𝕜 : Type u_1\n𝕜₂ : Type u_2\nE : Type u_4\nF : Type u_5\n𝓕 : Type u_8\ninst✝⁸ : SeminormedAddCommGroup E\ninst✝⁷ : SeminormedAddCommGroup F\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NontriviallyNormedField 𝕜₂\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedSpace 𝕜₂ F\nσ₁₂ : 𝕜 →+* 𝕜₂\ninst✝² : Fu... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Operator.Basic | {
"line": 79,
"column": 7
} | {
"line": 79,
"column": 15
} | [
{
"pp": "𝕜 : Type u_1\n𝕜₂ : Type u_2\nE : Type u_4\nF : Type u_5\n𝓕 : Type u_8\ninst✝⁸ : SeminormedAddCommGroup E\ninst✝⁷ : SeminormedAddCommGroup F\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NontriviallyNormedField 𝕜₂\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedSpace 𝕜₂ F\nσ₁₂ : 𝕜 →+* 𝕜₂\ninst✝² : Fu... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Operator.Basic | {
"line": 86,
"column": 25
} | {
"line": 86,
"column": 33
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_4\ninst✝⁸ : SeminormedAddCommGroup E\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedSpace 𝕜 E\nF' : Type u_9\n𝓕' : Type u_10\ninst✝⁵ : NormedAddCommGroup F'\ninst✝⁴ : NormedSpace ℝ F'\ninst✝³ : Nontrivial F'\nτ : 𝕜 →+* ℝ\ninst✝² : FunLike 𝓕' E F'\ninst✝¹ : Semilinear... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Normed.Operator.Basic | {
"line": 86,
"column": 25
} | {
"line": 86,
"column": 33
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_4\ninst✝⁸ : SeminormedAddCommGroup E\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedSpace 𝕜 E\nF' : Type u_9\n𝓕' : Type u_10\ninst✝⁵ : NormedAddCommGroup F'\ninst✝⁴ : NormedSpace ℝ F'\ninst✝³ : Nontrivial F'\nτ : 𝕜 →+* ℝ\ninst✝² : FunLike 𝓕' E F'\ninst✝¹ : Semilinear... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Operator.Basic | {
"line": 86,
"column": 25
} | {
"line": 86,
"column": 33
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_4\ninst✝⁸ : SeminormedAddCommGroup E\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedSpace 𝕜 E\nF' : Type u_9\n𝓕' : Type u_10\ninst✝⁵ : NormedAddCommGroup F'\ninst✝⁴ : NormedSpace ℝ F'\ninst✝³ : Nontrivial F'\nτ : 𝕜 →+* ℝ\ninst✝² : FunLike 𝓕' E F'\ninst✝¹ : Semilinear... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Algebra.Module.Spaces.UniformConvergenceCLM | {
"line": 482,
"column": 4
} | {
"line": 482,
"column": 71
} | [
{
"pp": "𝕜₁✝ : Type u_1\n𝕜₂✝ : Type u_2\ninst✝²¹ : NormedField 𝕜₁✝\ninst✝²⁰ : NormedField 𝕜₂✝\nσ✝ : 𝕜₁✝ →+* 𝕜₂✝\nE✝ : Type u_3\nF✝ : Type u_4\nG✝ : Type u_5\ninst✝¹⁹ : AddCommGroup E✝\ninst✝¹⁸ : Module 𝕜₁✝ E✝\ninst✝¹⁷ : TopologicalSpace E✝\ninst✝¹⁶ : AddCommGroup F✝\ninst✝¹⁵ : Module 𝕜₂✝ F✝\n𝕜₁ : Type ... | rw [(UniformConvergenceCLM.isEmbedding_coeFn _ _ _).continuous_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
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