module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics | {
"line": 94,
"column": 48
} | {
"line": 94,
"column": 53
} | {
"line": 96,
"column": 0
} | [
{
"pp": "b : ℝ\nhb : 1 < b\n⊢ 1 < b",
"ppTerm": "?m.45",
"assigned": true,
"usedConstants": [],
"usedFVars": [
"hb"
],
"usedGoals": []
}
] | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Pow.NNReal | {
"line": 235,
"column": 2
} | {
"line": 239,
"column": 21
} | {
"line": 241,
"column": 0
} | [
{
"pp": "l : List ℝ\nhl : ∀ x ∈ l, 0 ≤ x\nr : ℝ\n⊢ (List.map (fun x ↦ x ^ r) l).prod = l.prod ^ r",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real.instPow",
"Real.instLE",
"Real",
"NNReal.canLift",
"Real.instZero",
"congrArg",
... | [] | lift l to List ℝ≥0 using hl
have := congr_arg ((↑) : ℝ≥0 → ℝ) (NNReal.list_prod_map_rpow l r)
push_cast at this
rw [List.map_map] at this ⊢
exact mod_cast this | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Pow.NNReal | {
"line": 235,
"column": 2
} | {
"line": 239,
"column": 21
} | {
"line": 241,
"column": 0
} | [
{
"pp": "l : List ℝ\nhl : ∀ x ∈ l, 0 ≤ x\nr : ℝ\n⊢ (List.map (fun x ↦ x ^ r) l).prod = l.prod ^ r",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real.instPow",
"Real.instLE",
"Real",
"NNReal.canLift",
"Real.instZero",
"congrArg",
... | [] | lift l to List ℝ≥0 using hl
have := congr_arg ((↑) : ℝ≥0 → ℝ) (NNReal.list_prod_map_rpow l r)
push_cast at this
rw [List.map_map] at this ⊢
exact mod_cast this | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Pow.Real | {
"line": 1022,
"column": 4
} | {
"line": 1022,
"column": 31
} | {
"line": 1024,
"column": 0
} | [
{
"pp": "case hr\nx : ℂ\nhx : x.im < 0\n⊢ x.arg ≤ 0",
"ppTerm": "?hr",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Real",
"Real.instZero",
"Complex.im",
"Real.instLT",
"Complex.arg",
"LT.lt.le",
"Complex.arg_neg_iff",
"LT.lt",
"Zero.to... | [] | exact (arg_neg_iff.2 hx).le | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.SpecialFunctions.Pow.Real | {
"line": 1022,
"column": 4
} | {
"line": 1022,
"column": 31
} | {
"line": 1024,
"column": 0
} | [
{
"pp": "case hr\nx : ℂ\nhx : x.im < 0\n⊢ x.arg ≤ 0",
"ppTerm": "?hr",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Real",
"Real.instZero",
"Complex.im",
"Real.instLT",
"Complex.arg",
"LT.lt.le",
"Complex.arg_neg_iff",
"LT.lt",
"Zero.to... | [] | exact (arg_neg_iff.2 hx).le | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Pow.Real | {
"line": 1022,
"column": 4
} | {
"line": 1022,
"column": 31
} | {
"line": 1024,
"column": 0
} | [
{
"pp": "case hr\nx : ℂ\nhx : x.im < 0\n⊢ x.arg ≤ 0",
"ppTerm": "?hr",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Real",
"Real.instZero",
"Complex.im",
"Real.instLT",
"Complex.arg",
"LT.lt.le",
"Complex.arg_neg_iff",
"LT.lt",
"Zero.to... | [] | exact (arg_neg_iff.2 hx).le | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Pow.Real | {
"line": 1052,
"column": 47
} | {
"line": 1052,
"column": 56
} | {
"line": 1052,
"column": 57
} | [
{
"pp": "b : ℝ\nn : ℕ\nh : IsInt b (Int.negOfNat n)\na : ℝ\n⊢ a ^ (-Int.ofNat n) = (a ^ n)⁻¹",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"DivisionCommMonoid.toDivisionMonoid",
"DivInvOneMonoid.toInvOneClass",
"congrArg",
"Real.instI... | [
"b : ℝ\nn : ℕ\nh : IsInt b (Int.negOfNat n)\na : ℝ\n⊢ (a ^ Int.ofNat n)⁻¹ = (a ^ n)⁻¹"
] | zpow_neg, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Pow.NNReal | {
"line": 616,
"column": 13
} | {
"line": 616,
"column": 53
} | {
"line": 618,
"column": 0
} | [
{
"pp": "y : ℝ\nx : ℝ≥0∞\nhx : 0 < x\nhx' : x ≠ ∞\nh : x ^ y = ∞\n⊢ False",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"False",
"Real",
"eq_false",
"Real.instZero",
"congrArg",
"ENNReal.instPowReal",
"False.elim",
"PartialOrder.toPreorder"... | [] | simp [rpow_eq_top_iff, hx.ne', hx'] at h | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.SpecialFunctions.Pow.NNReal | {
"line": 616,
"column": 13
} | {
"line": 616,
"column": 53
} | {
"line": 618,
"column": 0
} | [
{
"pp": "y : ℝ\nx : ℝ≥0∞\nhx : 0 < x\nhx' : x ≠ ∞\nh : x ^ y = ∞\n⊢ False",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"False",
"Real",
"eq_false",
"Real.instZero",
"congrArg",
"ENNReal.instPowReal",
"False.elim",
"PartialOrder.toPreorder"... | [] | simp [rpow_eq_top_iff, hx.ne', hx'] at h | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Pow.NNReal | {
"line": 616,
"column": 13
} | {
"line": 616,
"column": 53
} | {
"line": 618,
"column": 0
} | [
{
"pp": "y : ℝ\nx : ℝ≥0∞\nhx : 0 < x\nhx' : x ≠ ∞\nh : x ^ y = ∞\n⊢ False",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"False",
"Real",
"eq_false",
"Real.instZero",
"congrArg",
"ENNReal.instPowReal",
"False.elim",
"PartialOrder.toPreorder"... | [] | simp [rpow_eq_top_iff, hx.ne', hx'] at h | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.AffineSpace.Midpoint | {
"line": 196,
"column": 75
} | {
"line": 196,
"column": 96
} | {
"line": 197,
"column": 4
} | [
{
"pp": "R : Type u_1\nV : Type u_2\ninst✝³ : Ring R\ninst✝² : Invertible 2\ninst✝¹ : AddCommGroup V\ninst✝ : Module R V\nx y : V\n⊢ midpoint R x y +ᵥ midpoint R x y = midpoint R x y +ᵥ midpoint R y x",
"ppTerm": "?m.72",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instVAddOfAdd",
... | [] | by rw [midpoint_comm] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.SpecialFunctions.Pow.NNReal | {
"line": 823,
"column": 87
} | {
"line": 826,
"column": 57
} | {
"line": 828,
"column": 0
} | [
{
"pp": "x y : ℝ≥0∞\nz : ℝ\nhz : 0 < z\n⊢ x < y ^ z⁻¹ ↔ x ^ z < y",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"ENNReal.rpow_mul",
"MulOne.toOne",
"ENNReal.rpow_lt_rpow_iff",
"Real.partialOrder",
"Real... | [] | by
nth_rw 1 [← rpow_one x]
nth_rw 1 [← @mul_inv_cancel₀ _ _ z (ne_of_lt hz).symm]
rw [rpow_mul, @rpow_lt_rpow_iff _ _ z⁻¹ (by simp [hz])] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.SpecialFunctions.Pow.NNReal | {
"line": 1196,
"column": 49
} | {
"line": 1196,
"column": 58
} | {
"line": 1196,
"column": 59
} | [
{
"pp": "b : ℝ\nn : ℕ\nh : IsInt b (Int.negOfNat n)\na : ℝ≥0\n⊢ a ^ (-Int.ofNat n) = (a ^ n)⁻¹",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"DivisionCommMonoid.toDivisionMonoid",
"DivInvOneMonoid.toInvOneClass",
"NNReal.zpow",
"congrArg",
"D... | [
"b : ℝ\nn : ℕ\nh : IsInt b (Int.negOfNat n)\na : ℝ≥0\n⊢ (a ^ Int.ofNat n)⁻¹ = (a ^ n)⁻¹"
] | zpow_neg, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Ray | {
"line": 511,
"column": 2
} | {
"line": 511,
"column": 68
} | {
"line": 513,
"column": 0
} | [
{
"pp": "R : 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\nv : M\nr : R\nhv : v ≠ 0\nhr : r ≠ 0\n⊢ SameRay R (-v) (r • v) ↔ r < 0",
"ppTerm": "?m.30",
"assigned": true,
"us... | [] | simp only [sameRay_neg_smul_right_iff, hv, or_false, hr.le_iff_lt] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.LinearAlgebra.Ray | {
"line": 511,
"column": 2
} | {
"line": 511,
"column": 68
} | {
"line": 513,
"column": 0
} | [
{
"pp": "R : 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\nv : M\nr : R\nhv : v ≠ 0\nhr : r ≠ 0\n⊢ SameRay R (-v) (r • v) ↔ r < 0",
"ppTerm": "?m.30",
"assigned": true,
"us... | [] | simp only [sameRay_neg_smul_right_iff, hv, or_false, hr.le_iff_lt] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Ray | {
"line": 511,
"column": 2
} | {
"line": 511,
"column": 68
} | {
"line": 513,
"column": 0
} | [
{
"pp": "R : 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\nv : M\nr : R\nhv : v ≠ 0\nhr : r ≠ 0\n⊢ SameRay R (-v) (r • v) ↔ r < 0",
"ppTerm": "?m.30",
"assigned": true,
"us... | [] | simp only [sameRay_neg_smul_right_iff, hv, or_false, hr.le_iff_lt] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Convex.Segment | {
"line": 330,
"column": 2
} | {
"line": 330,
"column": 7
} | {
"line": 332,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁶ : CommRing 𝕜\ninst✝⁵ : PartialOrder 𝕜\ninst✝⁴ : IsOrderedRing 𝕜\ninst✝³ : IsDomain 𝕜\ninst✝² : AddCommGroup E\ninst✝¹ : Module 𝕜 E\nx y : E\nh : LinearIndependent 𝕜 ![x, y]\ns t : 𝕜\nhs : s ≠ t\nc : E\ninst✝ : Nontrivial 𝕜\n⊢ LinearIndependent 𝕜 ![x, y] ∧ -1... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Analysis.Convex.Star | {
"line": 82,
"column": 4
} | {
"line": 83,
"column": 39
} | {
"line": 85,
"column": 0
} | [
{
"pp": "case mpr\n𝕜 : 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⊢ (∀ ⦃y : E⦄, y ∈ s → [x -[𝕜] y] ⊆ s) → StarConvex 𝕜 x s",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"NonAssocSemiring.t... | [] | 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": 82,
"column": 4
} | {
"line": 83,
"column": 39
} | {
"line": 85,
"column": 0
} | [
{
"pp": "case mpr\n𝕜 : 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⊢ (∀ ⦃y : E⦄, y ∈ s → [x -[𝕜] y] ⊆ s) → StarConvex 𝕜 x s",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"NonAssocSemiring.t... | [] | 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.Analysis.Convex.Segment | {
"line": 528,
"column": 2
} | {
"line": 528,
"column": 97
} | {
"line": 529,
"column": 2
} | [
{
"pp": "case inr\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⊢ z ∈ [x -[𝕜] y]",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"NonAssocS... | [
"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",
"case inr.refine_2\n𝕜 : Type u_1\ninst✝² : Field 𝕜\ninst✝¹ : LinearOrder 𝕜\ninst✝ : IsStrictO... | refine ⟨(y - z) / (y - x), (z - x) / (y - x), div_nonneg hyz h.le, div_nonneg hxz h.le, ?_, ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Defs | {
"line": 223,
"column": 40
} | {
"line": 223,
"column": 45
} | {
"line": 224,
"column": 2
} | [
{
"pp": "k : Type u_1\nV : Type u_2\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ns : AffineSubspace k V\ninst✝ : Nonempty ↥s\nh : ↑s -ᵥ ↑s = ↑s\nx : ↥s\nthis : ↑x - ↑x ∈ ↑s\n⊢ 0 ∈ s",
"ppTerm": "?m.48",
"assigned": true,
"usedConstants": [
"SetLike.mem_coe._simp_1",
"s... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Defs | {
"line": 223,
"column": 40
} | {
"line": 223,
"column": 45
} | {
"line": 224,
"column": 2
} | [
{
"pp": "k : Type u_1\nV : Type u_2\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ns : AffineSubspace k V\ninst✝ : Nonempty ↥s\nh : ↑s -ᵥ ↑s = ↑s\nx : ↥s\nthis : ↑x - ↑x ∈ ↑s\n⊢ 0 ∈ s",
"ppTerm": "?m.48",
"assigned": true,
"usedConstants": [
"SetLike.mem_coe._simp_1",
"s... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Defs | {
"line": 223,
"column": 40
} | {
"line": 223,
"column": 45
} | {
"line": 224,
"column": 2
} | [
{
"pp": "k : Type u_1\nV : Type u_2\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ns : AffineSubspace k V\ninst✝ : Nonempty ↥s\nh : ↑s -ᵥ ↑s = ↑s\nx : ↥s\nthis : ↑x - ↑x ∈ ↑s\n⊢ 0 ∈ s",
"ppTerm": "?m.48",
"assigned": true,
"usedConstants": [
"SetLike.mem_coe._simp_1",
"s... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Defs | {
"line": 225,
"column": 2
} | {
"line": 225,
"column": 7
} | {
"line": 227,
"column": 0
} | [
{
"pp": "k : Type u_1\nV : Type u_2\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ns : AffineSubspace k V\ninst✝ : Nonempty ↥s\nh : ↑s -ᵥ ↑s = ↑s\nx : ↥s\n⊢ ∃ x_1 ∈ ↑s, ∃ y ∈ ↑s, x_1 -ᵥ y = ↑x - ↑x",
"ppTerm": "?m.57",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"SetL... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Defs | {
"line": 287,
"column": 33
} | {
"line": 292,
"column": 15
} | {
"line": 294,
"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 : AffineSubspace k P\nv : V\nhv : v ∈ s.direction\np : P\nhp : p ∈ s\n⊢ v +ᵥ p ∈ s",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [] | by
rw [mem_direction_iff_eq_vsub ⟨p, hp⟩] at hv
rcases hv with ⟨p₁, hp₁, p₂, hp₂, hv⟩
rw [hv]
convert s.smul_vsub_vadd_mem 1 hp₁ hp₂ hp
rw [one_smul] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Defs | {
"line": 322,
"column": 2
} | {
"line": 323,
"column": 31
} | {
"line": 325,
"column": 0
} | [
{
"pp": "case refine_2\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\ns : AffineSubspace k P\np : P\nhp : p ∈ s\n⊢ (fun x ↦ x -ᵥ p) '' ↑s ⊆ ↑s -ᵥ ↑s",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
... | [] | · rintro v ⟨p₂, hp₂, rfl⟩
exact ⟨p₂, hp₂, p, hp, rfl⟩ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Defs | {
"line": 331,
"column": 40
} | {
"line": 331,
"column": 54
} | {
"line": 331,
"column": 55
} | [
{
"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\np : P\nhp : p ∈ s\nv : V\n⊢ -v ∈ (fun x ↦ x -ᵥ p) '' ↑s ↔ v ∈ (fun x ↦ p -ᵥ x) '' ↑s",
"ppTerm": "?m.62",
"assigned": true,
"usedConstant... | [
"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\np : P\nhp : p ∈ s\nv : V\n⊢ (∃ x ∈ ↑s, x -ᵥ p = -v) ↔ v ∈ (fun x ↦ p -ᵥ x) '' ↑s"
] | Set.mem_image, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Defs | {
"line": 555,
"column": 8
} | {
"line": 555,
"column": 33
} | {
"line": 556,
"column": 8
} | [
{
"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 : Sort u_4\ns : Set (AffineSubspace k P)\nc : k\np₁ p₂ p₃ : P\nhp₁ : p₁ ∈ ⋂ s' ∈ s, ↑s'\nhp₂ : p₂ ∈ ⋂ s' ∈ s, ↑s'\nhp₃ : p₃ ∈ ⋂ s' ∈ s, ↑s'\ns₂ : AffineSubspace k P\nhs₂ : s₂ ∈... | [
"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 : Sort u_4\ns : Set (AffineSubspace k P)\nc : k\np₁ p₂ p₃ : P\nhp₁ : ∀ i ∈ s, p₁ ∈ ↑i\nhp₂ : ∀ i ∈ s, p₂ ∈ ↑i\nhp₃ : ∀ i ∈ s, p₃ ∈ ↑i\ns₂ : AffineSubspace k P\nhs₂ : s₂ ∈ s\n⊢ c • (p₁ -ᵥ p... | rw [Set.mem_iInter₂] at * | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Convex.Star | {
"line": 430,
"column": 2
} | {
"line": 445,
"column": 38
} | {
"line": 447,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁶ : Semiring 𝕜\ninst✝⁵ : PartialOrder 𝕜\ninst✝⁴ : AddCommMonoid E\ninst✝³ : PartialOrder E\ninst✝² : IsOrderedAddMonoid E\ninst✝¹ : Module 𝕜 E\ninst✝ : PosSMulMono 𝕜 E\nx : E\ns : Set E\nhs : s.OrdConnected\nhx : x ∈ s\nh : ∀ y ∈ s, x ≤ y ∨ y ≤ x\n⊢ StarConvex 𝕜 x... | [] | intro y hy a b ha hb hab
obtain hxy | hyx := h _ hy
· refine hs.out hx hy (mem_Icc.2 ⟨?_, ?_⟩)
· calc
x = a • x + b • x := (Convex.combo_self hab _).symm
_ ≤ a • x + b • y := by gcongr
calc
a • x + b • y ≤ a • y + b • y := by gcongr
_ = y := Convex.combo_self hab _
· refine hs.... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Convex.Star | {
"line": 430,
"column": 2
} | {
"line": 445,
"column": 38
} | {
"line": 447,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁶ : Semiring 𝕜\ninst✝⁵ : PartialOrder 𝕜\ninst✝⁴ : AddCommMonoid E\ninst✝³ : PartialOrder E\ninst✝² : IsOrderedAddMonoid E\ninst✝¹ : Module 𝕜 E\ninst✝ : PosSMulMono 𝕜 E\nx : E\ns : Set E\nhs : s.OrdConnected\nhx : x ∈ s\nh : ∀ y ∈ s, x ≤ y ∨ y ≤ x\n⊢ StarConvex 𝕜 x... | [] | intro y hy a b ha hb hab
obtain hxy | hyx := h _ hy
· refine hs.out hx hy (mem_Icc.2 ⟨?_, ?_⟩)
· calc
x = a • x + b • x := (Convex.combo_self hab _).symm
_ ≤ a • x + b • y := by gcongr
calc
a • x + b • y ≤ a • y + b • y := by gcongr
_ = y := Convex.combo_self hab _
· refine hs.... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Defs | {
"line": 1058,
"column": 21
} | {
"line": 1058,
"column": 43
} | {
"line": 1058,
"column": 44
} | [
{
"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\np₁ p₂ : P\ns : AffineSubspace k P\nhp₁ : p₁ ∈ s\nhp₂ : p₂ ∈ s\n⊢ {p₁, p₂} ⊆ ↑s",
"ppTerm": "?m.39",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"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\np₁ p₂ : P\ns : AffineSubspace k P\nhp₁ : p₁ ∈ s\nhp₂ : p₂ ∈ s\n⊢ p₁ ∈ ↑s ∧ {p₂} ⊆ ↑s"
] | Set.insert_subset_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.LocallyConvex.Basic | {
"line": 103,
"column": 62
} | {
"line": 103,
"column": 67
} | {
"line": 103,
"column": 67
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_3\ninst✝¹ : SeminormedRing 𝕜\ninst✝ : SMul 𝕜 E\nS : Set (Set E)\nh : ∀ s ∈ S, Balanced 𝕜 s\nx✝³ : 𝕜\nx✝² : ‖x✝³‖ ≤ 1\nx✝¹ : E\nx✝ : x✝¹ ∈ ⋂ s ∈ S, x✝³ • s\n⊢ x✝¹ ∈ ⋂₀ S",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instHSMul",... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Analysis.LocallyConvex.Basic | {
"line": 103,
"column": 62
} | {
"line": 103,
"column": 67
} | {
"line": 103,
"column": 67
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_3\ninst✝¹ : SeminormedRing 𝕜\ninst✝ : SMul 𝕜 E\nS : Set (Set E)\nh : ∀ s ∈ S, Balanced 𝕜 s\nx✝³ : 𝕜\nx✝² : ‖x✝³‖ ≤ 1\nx✝¹ : E\nx✝ : x✝¹ ∈ ⋂ s ∈ S, x✝³ • s\n⊢ x✝¹ ∈ ⋂₀ S",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instHSMul",... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.LocallyConvex.Basic | {
"line": 103,
"column": 62
} | {
"line": 103,
"column": 67
} | {
"line": 103,
"column": 67
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_3\ninst✝¹ : SeminormedRing 𝕜\ninst✝ : SMul 𝕜 E\nS : Set (Set E)\nh : ∀ s ∈ S, Balanced 𝕜 s\nx✝³ : 𝕜\nx✝² : ‖x✝³‖ ≤ 1\nx✝¹ : E\nx✝ : x✝¹ ∈ ⋂ s ∈ S, x✝³ • s\n⊢ x✝¹ ∈ ⋂₀ S",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instHSMul",... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.LocallyConvex.Basic | {
"line": 115,
"column": 2
} | {
"line": 115,
"column": 39
} | {
"line": 117,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_3\nF : Type u_4\ninst✝² : SeminormedRing 𝕜\ninst✝¹ : SMul 𝕜 E\ninst✝ : SMul 𝕜 F\ns : Set F\nhs : Balanced 𝕜 s\nf : E →ₑ[id] F\na : 𝕜\nha : ‖a‖ ≤ 1\nx : E\nx✝ : x ∈ a • ⇑f ⁻¹' s\ny : E\nhy₁ : y ∈ ⇑f ⁻¹' s\nhy₂ : (fun x ↦ a • x) y = x\n⊢ a • f y ∈ s",
"ppTerm": "?m.62",... | [] | exact hs a ha (smul_mem_smul_set hy₁) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Basic | {
"line": 279,
"column": 90
} | {
"line": 281,
"column": 7
} | {
"line": 283,
"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\nι : Type u_4\np : ι → P\ni0 : ι\n⊢ vectorSpan k (range p) = Submodule.span k (range fun i ↦ p i0 -ᵥ p i)",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": ... | [] | by
rw [vectorSpan_eq_span_vsub_set_left k (Set.mem_range_self i0), ← Set.range_comp]
congr | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Seminorm | {
"line": 486,
"column": 10
} | {
"line": 486,
"column": 28
} | {
"line": 487,
"column": 8
} | [
{
"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\np q : Seminorm 𝕜 E\nx : E\ns : Set (Seminorm 𝕜 E)\nh : BddAbove (... | [] | exact map_zero i.1 | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.LinearAlgebra.AffineSpace.Simplex.Basic | {
"line": 396,
"column": 2
} | {
"line": 396,
"column": 61
} | {
"line": 397,
"column": 2
} | [
{
"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\nm n : ℕ\ns : Simplex k P n\ne : Fin (n + 1) ≃ Fin (m + 1)\np : P\n⊢ p ∈ Simplex.setInterior I (s.reindex e) ↔ p ∈ Simplex.setInterior I s",
"ppTerm": "?m.55",... | [
"case refine_1\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\nI : Set k\nm n : ℕ\ns : Simplex k P n\ne : Fin (n + 1) ≃ Fin (m + 1)\np : P\nx✝ : p ∈ Simplex.setInterior I (s.reindex e)\nw : Fin (m + 1) → k\nhw : ∑ i, w i = 1\nhwI : ∀... | refine ⟨fun ⟨w, hw, hwI, h⟩ ↦ ?_, fun ⟨w, hw, hwI, h⟩ ↦ ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.LinearAlgebra.AffineSpace.Independent | {
"line": 539,
"column": 88
} | {
"line": 545,
"column": 41
} | {
"line": 547,
"column": 0
} | [
{
"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\nha : AffineIndependent k p\n⊢ Injective fun s ↦ affineSpan k (p '' s)",
"ppTerm": "?m.27",
"assigned": true,
"use... | [] | by
by_contra hn
rw [not_injective_iff] at hn
obtain ⟨s₁, s₂, hs₁₂, hne⟩ := hn
apply hne
ext i
simp_rw [← ha.mem_affineSpan_iff, hs₁₂] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.AffineSpace.Independent | {
"line": 570,
"column": 2
} | {
"line": 572,
"column": 63
} | {
"line": 573,
"column": 2
} | [
{
"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\nha : AffineIndependent k p\ns₁ s₂ : Set ι\ni : ι\nhis₁ : i ∈ s₁\nhis₂ : i ∉ s₂\nh₁ : s₁.Nontrivial\nj : ι\nhj : j ∈ s₁\nhne :... | [
"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\nha : AffineIndependent k p\ns₁ s₂ : Set ι\ni : ι\nhis₁ : i ∈ s₁\nhis₂ : i ∉ s₂\nh₁ : s₁.Nontrivial\nj : ι\nhj : j ∈ s₁\nhne : j ≠ i\nhe :... | have hs'' : p i -ᵥ p j = fs'.weightedVSub p w'' := by
rw [hs']
exact weightedVSubOfPoint_indicator_subset _ _ _ (by grind) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Seminorm | {
"line": 1228,
"column": 25
} | {
"line": 1228,
"column": 34
} | {
"line": 1228,
"column": 35
} | [
{
"pp": "case refine_2\n𝕜 : Type u_3\nE : Type u_7\ninst✝² : NormedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\np : Seminorm 𝕜 E\nc : 𝕜\nhc : 1 < ‖c‖\nε : ℝ\nεpos : 0 < ε\nx : E\nhx : p x ≠ 0\nxεpos : 0 < p x / ε\nn : ℤ\nhn : p x / ε ∈ Ico (‖c‖ ^ n) (‖c‖ ^ (n + 1))\ncpos : 0 < ‖c‖\ncnpos : 0 < ‖c ... | [
"case refine_2\n𝕜 : Type u_3\nE : Type u_7\ninst✝² : NormedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\np : Seminorm 𝕜 E\nc : 𝕜\nhc : 1 < ‖c‖\nε : ℝ\nεpos : 0 < ε\nx : E\nhx : p x ≠ 0\nxεpos : 0 < p x / ε\nn : ℤ\nhn : p x / ε ∈ Ico (‖c‖ ^ n) (‖c‖ ^ (n + 1))\ncpos : 0 < ‖c‖\ncnpos : 0 < ‖c ^ (n + 1)‖\n... | zpow_neg, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Seminorm | {
"line": 1232,
"column": 8
} | {
"line": 1232,
"column": 17
} | {
"line": 1232,
"column": 18
} | [
{
"pp": "case refine_3\n𝕜 : Type u_3\nE : Type u_7\ninst✝² : NormedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\np : Seminorm 𝕜 E\nc : 𝕜\nhc : 1 < ‖c‖\nε : ℝ\nεpos : 0 < ε\nx : E\nhx : p x ≠ 0\nxεpos : 0 < p x / ε\nn : ℤ\nhn : p x / ε ∈ Ico (‖c‖ ^ n) (‖c‖ ^ (n + 1))\ncpos : 0 < ‖c‖\ncnpos : 0 < ‖c ... | [
"case refine_3\n𝕜 : Type u_3\nE : Type u_7\ninst✝² : NormedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\np : Seminorm 𝕜 E\nc : 𝕜\nhc : 1 < ‖c‖\nε : ℝ\nεpos : 0 < ε\nx : E\nhx : p x ≠ 0\nxεpos : 0 < p x / ε\nn : ℤ\nhn : p x / ε ∈ Ico (‖c‖ ^ n) (‖c‖ ^ (n + 1))\ncpos : 0 < ‖c‖\ncnpos : 0 < ‖c ^ (n + 1)‖\n... | zpow_neg, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.AffineSpace.Simplex.Basic | {
"line": 629,
"column": 27
} | {
"line": 629,
"column": 32
} | {
"line": 631,
"column": 0
} | [
{
"pp": "case pos\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\nfs : Finset (Fin (n + 1))\nm : ℕ\nh : #fs = m + 1\nw : Fin (n + 1) → k\nhw1 : ∑ i, w ... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.LinearAlgebra.AffineSpace.Simplex.Basic | {
"line": 629,
"column": 27
} | {
"line": 629,
"column": 32
} | {
"line": 631,
"column": 0
} | [
{
"pp": "case neg\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\nfs : Finset (Fin (n + 1))\nm : ℕ\nh : #fs = m + 1\nw : Fin (n + 1) → k\nhw1 : ∑ i, w ... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Analysis.Seminorm | {
"line": 1238,
"column": 8
} | {
"line": 1238,
"column": 17
} | {
"line": 1238,
"column": 18
} | [
{
"pp": "case refine_4\n𝕜 : Type u_3\nE : Type u_7\ninst✝² : NormedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\np : Seminorm 𝕜 E\nc : 𝕜\nhc : 1 < ‖c‖\nε : ℝ\nεpos : 0 < ε\nx : E\nhx : p x ≠ 0\nxεpos : 0 < p x / ε\nn : ℤ\nhn : p x / ε ∈ Ico (‖c‖ ^ n) (‖c‖ ^ (n + 1))\ncpos : 0 < ‖c‖\ncnpos : 0 < ‖c ... | [
"case refine_4\n𝕜 : Type u_3\nE : Type u_7\ninst✝² : NormedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\np : Seminorm 𝕜 E\nc : 𝕜\nhc : 1 < ‖c‖\nε : ℝ\nεpos : 0 < ε\nx : E\nhx : p x ≠ 0\nxεpos : 0 < p x / ε\nn : ℤ\nhn : p x / ε ∈ Ico (‖c‖ ^ n) (‖c‖ ^ (n + 1))\ncpos : 0 < ‖c‖\ncnpos : 0 < ‖c ^ (n + 1)‖\n... | zpow_neg, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Convex.StdSimplex | {
"line": 150,
"column": 2
} | {
"line": 150,
"column": 7
} | {
"line": 152,
"column": 0
} | [
{
"pp": "R : Type u_1\nι : Type u_2\ninst✝⁴ : Field R\ninst✝³ : LinearOrder R\ninst✝² : IsStrictOrderedRing R\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\nx✝¹ x✝ : ι\n⊢ Pi.single x✝¹ 1 x✝ = if x✝¹ = x✝ then 1 else 0",
"ppTerm": "?m.207",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Fal... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Analysis.Convex.StdSimplex | {
"line": 168,
"column": 2
} | {
"line": 168,
"column": 7
} | {
"line": 170,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : Field R\ninst✝³ : LinearOrder R\ninst✝² : IsStrictOrderedRing R\nE : Type u_3\ninst✝¹ : AddCommGroup E\ninst✝ : Module R E\ns : Set E\nhs : s.Finite\nthis : Fintype ↑s := ⋯\n⊢ s = range (⇑(∑ x, (LinearMap.proj x).smulRight ↑x) ∘ fun i j ↦ if i = j then 1 else 0)",
"ppTerm": "... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Analysis.Convex.StdSimplex | {
"line": 280,
"column": 22
} | {
"line": 280,
"column": 27
} | {
"line": 282,
"column": 0
} | [
{
"pp": "S : Type u_1\ninst✝⁴ : Semiring S\ninst✝³ : PartialOrder S\nX : Type u_2\nY : Type u_3\nZ : Type u_4\ninst✝² : Fintype X\ninst✝¹ : Fintype Y\ninst✝ : Fintype Z\n⊢ Function.Injective fun s ↦ ↑s",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Membership.mem",
"Subtype",... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Analysis.Convex.StdSimplex | {
"line": 280,
"column": 22
} | {
"line": 280,
"column": 27
} | {
"line": 282,
"column": 0
} | [
{
"pp": "S : Type u_1\ninst✝⁴ : Semiring S\ninst✝³ : PartialOrder S\nX : Type u_2\nY : Type u_3\nZ : Type u_4\ninst✝² : Fintype X\ninst✝¹ : Fintype Y\ninst✝ : Fintype Z\n⊢ Function.Injective fun s ↦ ↑s",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Membership.mem",
"Subtype",... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Convex.StdSimplex | {
"line": 280,
"column": 22
} | {
"line": 280,
"column": 27
} | {
"line": 282,
"column": 0
} | [
{
"pp": "S : Type u_1\ninst✝⁴ : Semiring S\ninst✝³ : PartialOrder S\nX : Type u_2\nY : Type u_3\nZ : Type u_4\ninst✝² : Fintype X\ninst✝¹ : Fintype Y\ninst✝ : Fintype Z\n⊢ Function.Injective fun s ↦ ↑s",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Membership.mem",
"Subtype",... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Convex.StdSimplex | {
"line": 312,
"column": 32
} | {
"line": 312,
"column": 37
} | {
"line": 312,
"column": 37
} | [
{
"pp": "S : Type u_1\ninst✝⁴ : Semiring S\ninst✝³ : PartialOrder S\nX : Type u_2\nY : Type u_3\ninst✝² : Fintype X\ninst✝¹ : Fintype Y\ninst✝ : IsOrderedRing S\nf : X → Y\ns : X → S\nhs₀ : ∀ (x : X), 0 ≤ s x\nhs₁ : ∑ x, s x = 1\ny : Y\n⊢ ∀ i ∈ {x | f x = y}, 0 ≤ s i",
"ppTerm": "?m.81",
"assigned": tru... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Analysis.Convex.StdSimplex | {
"line": 312,
"column": 32
} | {
"line": 312,
"column": 37
} | {
"line": 312,
"column": 37
} | [
{
"pp": "S : Type u_1\ninst✝⁴ : Semiring S\ninst✝³ : PartialOrder S\nX : Type u_2\nY : Type u_3\ninst✝² : Fintype X\ninst✝¹ : Fintype Y\ninst✝ : IsOrderedRing S\nf : X → Y\ns : X → S\nhs₀ : ∀ (x : X), 0 ≤ s x\nhs₁ : ∑ x, s x = 1\ny : Y\n⊢ ∀ i ∈ {x | f x = y}, 0 ≤ s i",
"ppTerm": "?m.81",
"assigned": tru... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Convex.StdSimplex | {
"line": 312,
"column": 32
} | {
"line": 312,
"column": 37
} | {
"line": 312,
"column": 37
} | [
{
"pp": "S : Type u_1\ninst✝⁴ : Semiring S\ninst✝³ : PartialOrder S\nX : Type u_2\nY : Type u_3\ninst✝² : Fintype X\ninst✝¹ : Fintype Y\ninst✝ : IsOrderedRing S\nf : X → Y\ns : X → S\nhs₀ : ∀ (x : X), 0 ≤ s x\nhs₁ : ∑ x, s x = 1\ny : Y\n⊢ ∀ i ∈ {x | f x = y}, 0 ≤ s i",
"ppTerm": "?m.81",
"assigned": tru... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Convex.StdSimplex | {
"line": 318,
"column": 56
} | {
"line": 318,
"column": 61
} | {
"line": 318,
"column": 61
} | [
{
"pp": "S : Type u_1\ninst✝⁵ : Semiring S\ninst✝⁴ : PartialOrder S\nX : Type u_2\nY : Type u_3\nZ : Type u_4\ninst✝³ : Fintype X\ninst✝² : Fintype Y\ninst✝¹ : Fintype Z\ninst✝ : IsOrderedRing S\nf : X → Y\ns : ↑(stdSimplex S X)\n⊢ (FunOnFinite.linearMap S S f) ⇑s ∈ ⇑(FunOnFinite.linearMap S S f) '' stdSimplex ... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Analysis.Convex.StdSimplex | {
"line": 318,
"column": 56
} | {
"line": 318,
"column": 61
} | {
"line": 318,
"column": 61
} | [
{
"pp": "S : Type u_1\ninst✝⁵ : Semiring S\ninst✝⁴ : PartialOrder S\nX : Type u_2\nY : Type u_3\nZ : Type u_4\ninst✝³ : Fintype X\ninst✝² : Fintype Y\ninst✝¹ : Fintype Z\ninst✝ : IsOrderedRing S\nf : X → Y\ns : ↑(stdSimplex S X)\n⊢ (FunOnFinite.linearMap S S f) ⇑s ∈ ⇑(FunOnFinite.linearMap S S f) '' stdSimplex ... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Convex.StdSimplex | {
"line": 318,
"column": 56
} | {
"line": 318,
"column": 61
} | {
"line": 318,
"column": 61
} | [
{
"pp": "S : Type u_1\ninst✝⁵ : Semiring S\ninst✝⁴ : PartialOrder S\nX : Type u_2\nY : Type u_3\nZ : Type u_4\ninst✝³ : Fintype X\ninst✝² : Fintype Y\ninst✝¹ : Fintype Z\ninst✝ : IsOrderedRing S\nf : X → Y\ns : ↑(stdSimplex S X)\n⊢ (FunOnFinite.linearMap S S f) ⇑s ∈ ⇑(FunOnFinite.linearMap S S f) '' stdSimplex ... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Convex.StdSimplex | {
"line": 326,
"column": 2
} | {
"line": 326,
"column": 7
} | {
"line": 328,
"column": 0
} | [
{
"pp": "S : Type u_1\ninst✝³ : Semiring S\ninst✝² : PartialOrder S\nX : Type u_2\ninst✝¹ : Fintype X\ninst✝ : IsOrderedRing S\nx : ↑(stdSimplex S X)\n⊢ map id x = x",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"LinearMap.id",
"Pi.Function.module",
"Semiring.toModule",... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Analysis.Convex.StdSimplex | {
"line": 326,
"column": 2
} | {
"line": 326,
"column": 7
} | {
"line": 328,
"column": 0
} | [
{
"pp": "S : Type u_1\ninst✝³ : Semiring S\ninst✝² : PartialOrder S\nX : Type u_2\ninst✝¹ : Fintype X\ninst✝ : IsOrderedRing S\nx : ↑(stdSimplex S X)\n⊢ map id x = x",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"LinearMap.id",
"Pi.Function.module",
"Semiring.toModule",... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Convex.StdSimplex | {
"line": 326,
"column": 2
} | {
"line": 326,
"column": 7
} | {
"line": 328,
"column": 0
} | [
{
"pp": "S : Type u_1\ninst✝³ : Semiring S\ninst✝² : PartialOrder S\nX : Type u_2\ninst✝¹ : Fintype X\ninst✝ : IsOrderedRing S\nx : ↑(stdSimplex S X)\n⊢ map id x = x",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"LinearMap.id",
"Pi.Function.module",
"Semiring.toModule",... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Convex.StdSimplex | {
"line": 344,
"column": 2
} | {
"line": 344,
"column": 7
} | {
"line": 346,
"column": 0
} | [
{
"pp": "S : Type u_1\ninst✝⁶ : Semiring S\ninst✝⁵ : PartialOrder S\nX : Type u_2\nY : Type u_3\ninst✝⁴ : Fintype X\ninst✝³ : Fintype Y\ninst✝² : IsOrderedRing S\ninst✝¹ : DecidableEq X\ninst✝ : DecidableEq Y\nf : X → Y\nx : X\n⊢ map f (vertex x) = vertex (f x)",
"ppTerm": "?m.25",
"assigned": true,
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Analysis.Convex.StdSimplex | {
"line": 344,
"column": 2
} | {
"line": 344,
"column": 7
} | {
"line": 346,
"column": 0
} | [
{
"pp": "S : Type u_1\ninst✝⁶ : Semiring S\ninst✝⁵ : PartialOrder S\nX : Type u_2\nY : Type u_3\ninst✝⁴ : Fintype X\ninst✝³ : Fintype Y\ninst✝² : IsOrderedRing S\ninst✝¹ : DecidableEq X\ninst✝ : DecidableEq Y\nf : X → Y\nx : X\n⊢ map f (vertex x) = vertex (f x)",
"ppTerm": "?m.25",
"assigned": true,
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Convex.StdSimplex | {
"line": 344,
"column": 2
} | {
"line": 344,
"column": 7
} | {
"line": 346,
"column": 0
} | [
{
"pp": "S : Type u_1\ninst✝⁶ : Semiring S\ninst✝⁵ : PartialOrder S\nX : Type u_2\nY : Type u_3\ninst✝⁴ : Fintype X\ninst✝³ : Fintype Y\ninst✝² : IsOrderedRing S\ninst✝¹ : DecidableEq X\ninst✝ : DecidableEq Y\nf : X → Y\nx : X\n⊢ map f (vertex x) = vertex (f x)",
"ppTerm": "?m.25",
"assigned": true,
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.AffineSpace.Independent | {
"line": 777,
"column": 6
} | {
"line": 783,
"column": 10
} | {
"line": 784,
"column": 4
} | [
{
"pp": "case inr.refine_2.hd\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\ns : Set P\np : P\nhp : p ∈ s\nb : Set V\nhb₁ : b ⊆ ⇑(Equiv.vaddConst p).symm '' s\nhb₂ : Submodule.span k b = vectorSpan k s\nhb₃ : AffineIndep... | [] | have : Submodule.span k b = Submodule.span k (insert 0 b) := by simp
simp only [direction_affineSpan, ← hb₂, Equiv.coe_vaddConst, Set.singleton_union,
vectorSpan_eq_span_vsub_set_right k (Set.mem_insert p _), this]
congr
change (Equiv.vaddConst p).symm '' insert p (Equiv.vaddConst p '' b) = _
... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.AffineSpace.Independent | {
"line": 777,
"column": 6
} | {
"line": 783,
"column": 10
} | {
"line": 784,
"column": 4
} | [
{
"pp": "case inr.refine_2.hd\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\ns : Set P\np : P\nhp : p ∈ s\nb : Set V\nhb₁ : b ⊆ ⇑(Equiv.vaddConst p).symm '' s\nhb₂ : Submodule.span k b = vectorSpan k s\nhb₃ : AffineIndep... | [] | have : Submodule.span k b = Submodule.span k (insert 0 b) := by simp
simp only [direction_affineSpan, ← hb₂, Equiv.coe_vaddConst, Set.singleton_union,
vectorSpan_eq_span_vsub_set_right k (Set.mem_insert p _), this]
congr
change (Equiv.vaddConst p).symm '' insert p (Equiv.vaddConst p '' b) = _
... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Convex.StdSimplex | {
"line": 379,
"column": 10
} | {
"line": 379,
"column": 25
} | {
"line": 381,
"column": 0
} | [
{
"pp": "S : Type u_1\ninst✝⁶ : Semiring S\ninst✝⁵ : PartialOrder S\nX : Type u_2\nY : Type u_3\nZ : Type u_4\ninst✝⁴ : Fintype X\ninst✝³ : Fintype Y\ninst✝² : Fintype Z\ninst✝¹ : IsOrderedRing S\ninst✝ : Unique X\n⊢ ∀ (a : ↑(stdSimplex S X)), a = ⟨1, ⋯⟩",
"ppTerm": "?m.18",
"assigned": true,
"usedC... | [] | by subsingleton | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Convex.StdSimplex | {
"line": 384,
"column": 30
} | {
"line": 384,
"column": 45
} | {
"line": 385,
"column": 2
} | [
{
"pp": "S : Type u_1\ninst✝⁴ : Semiring S\ninst✝³ : PartialOrder S\nX : Type u_2\ninst✝² : Fintype X\ninst✝¹ : IsOrderedRing S\ninst✝ : Unique X\ns : ↑(stdSimplex S X)\nx : X\n⊢ s = default",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"stdSimplex.instUniqueElemForall",
"Inh... | [] | by subsingleton | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Convex.Combination | {
"line": 173,
"column": 2
} | {
"line": 174,
"column": 35
} | {
"line": 176,
"column": 0
} | [
{
"pp": "R : Type u_1\nE : Type u_3\nι : Type u_5\ninst✝² : Field R\ninst✝¹ : AddCommGroup E\ninst✝ : Module R E\nt : Finset ι\nw : ι → R\nz : ι → E\nw' : ι → R\nh : ∀ i ∈ t, w i = w' i\n⊢ t.centerMass w z = t.centerMass w' z",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Classical... | [] | classical
exact centerMass_congr (by grind) | Lean.Elab.Tactic.evalClassical | Lean.Parser.Tactic.classical |
Mathlib.Analysis.Convex.Combination | {
"line": 173,
"column": 2
} | {
"line": 174,
"column": 35
} | {
"line": 176,
"column": 0
} | [
{
"pp": "R : Type u_1\nE : Type u_3\nι : Type u_5\ninst✝² : Field R\ninst✝¹ : AddCommGroup E\ninst✝ : Module R E\nt : Finset ι\nw : ι → R\nz : ι → E\nw' : ι → R\nh : ∀ i ∈ t, w i = w' i\n⊢ t.centerMass w z = t.centerMass w' z",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Classical... | [] | classical
exact centerMass_congr (by grind) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Convex.Combination | {
"line": 173,
"column": 2
} | {
"line": 174,
"column": 35
} | {
"line": 176,
"column": 0
} | [
{
"pp": "R : Type u_1\nE : Type u_3\nι : Type u_5\ninst✝² : Field R\ninst✝¹ : AddCommGroup E\ninst✝ : Module R E\nt : Finset ι\nw : ι → R\nz : ι → E\nw' : ι → R\nh : ∀ i ∈ t, w i = w' i\n⊢ t.centerMass w z = t.centerMass w' z",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Classical... | [] | classical
exact centerMass_congr (by grind) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Convex.Combination | {
"line": 178,
"column": 2
} | {
"line": 179,
"column": 35
} | {
"line": 181,
"column": 0
} | [
{
"pp": "R : Type u_1\nE : Type u_3\nι : Type u_5\ninst✝² : Field R\ninst✝¹ : AddCommGroup E\ninst✝ : Module R E\nt : Finset ι\nw : ι → R\nz z' : ι → E\nh : ∀ i ∈ t, w i ≠ 0 → z i = z' i\n⊢ t.centerMass w z = t.centerMass w z'",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Classica... | [] | classical
exact centerMass_congr (by grind) | Lean.Elab.Tactic.evalClassical | Lean.Parser.Tactic.classical |
Mathlib.Analysis.Convex.Combination | {
"line": 178,
"column": 2
} | {
"line": 179,
"column": 35
} | {
"line": 181,
"column": 0
} | [
{
"pp": "R : Type u_1\nE : Type u_3\nι : Type u_5\ninst✝² : Field R\ninst✝¹ : AddCommGroup E\ninst✝ : Module R E\nt : Finset ι\nw : ι → R\nz z' : ι → E\nh : ∀ i ∈ t, w i ≠ 0 → z i = z' i\n⊢ t.centerMass w z = t.centerMass w z'",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Classica... | [] | classical
exact centerMass_congr (by grind) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Convex.Combination | {
"line": 178,
"column": 2
} | {
"line": 179,
"column": 35
} | {
"line": 181,
"column": 0
} | [
{
"pp": "R : Type u_1\nE : Type u_3\nι : Type u_5\ninst✝² : Field R\ninst✝¹ : AddCommGroup E\ninst✝ : Module R E\nt : Finset ι\nw : ι → R\nz z' : ι → E\nh : ∀ i ∈ t, w i ≠ 0 → z i = z' i\n⊢ t.centerMass w z = t.centerMass w z'",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Classica... | [] | classical
exact centerMass_congr (by grind) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.AffineSpace.Independent | {
"line": 894,
"column": 81
} | {
"line": 894,
"column": 86
} | {
"line": 895,
"column": 2
} | [
{
"pp": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁴ : DivisionRing k\ninst✝³ : AddCommGroup V\ninst✝² : Module k V\ninst✝¹ : AffineSpace V P\nι : Type u_4\ninst✝ : DecidableEq ι\np : ι → P\nha : AffineIndependent k p\ni : ι\np₀ : P\nhp₀ : p₀ ∉ affineSpan k (p '' {x | x ≠ i})\nf : ι → P := update p i p₀\nh... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.LinearAlgebra.AffineSpace.Independent | {
"line": 897,
"column": 68
} | {
"line": 897,
"column": 73
} | {
"line": 899,
"column": 0
} | [
{
"pp": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁴ : DivisionRing k\ninst✝³ : AddCommGroup V\ninst✝² : Module k V\ninst✝¹ : AffineSpace V P\nι : Type u_4\ninst✝ : DecidableEq ι\np : ι → P\ni : ι\np₀ : P\nhp₀ : p₀ ∉ affineSpan k (p '' {x | x ≠ i})\nf : ι → P := update p i p₀\nhf : f = update p i p₀\nh₁ : ... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.LinearAlgebra.AffineSpace.Independent | {
"line": 897,
"column": 68
} | {
"line": 897,
"column": 73
} | {
"line": 899,
"column": 0
} | [
{
"pp": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁴ : DivisionRing k\ninst✝³ : AddCommGroup V\ninst✝² : Module k V\ninst✝¹ : AffineSpace V P\nι : Type u_4\ninst✝ : DecidableEq ι\np : ι → P\ni : ι\np₀ : P\nhp₀ : p₀ ∉ affineSpan k (p '' {x | x ≠ i})\nf : ι → P := update p i p₀\nhf : f = update p i p₀\nh₁ : ... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.AffineSpace.Independent | {
"line": 897,
"column": 68
} | {
"line": 897,
"column": 73
} | {
"line": 899,
"column": 0
} | [
{
"pp": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁴ : DivisionRing k\ninst✝³ : AddCommGroup V\ninst✝² : Module k V\ninst✝¹ : AffineSpace V P\nι : Type u_4\ninst✝ : DecidableEq ι\np : ι → P\ni : ι\np₀ : P\nhp₀ : p₀ ∉ affineSpan k (p '' {x | x ≠ i})\nf : ι → P := update p i p₀\nhf : f = update p i p₀\nh₁ : ... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Convex.Combination | {
"line": 323,
"column": 6
} | {
"line": 323,
"column": 25
} | {
"line": 323,
"column": 26
} | [
{
"pp": "case refine_2.refine_1\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 ∈... | [
"case pos\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\nhw₁' : s'.s... | by_cases hi : i ∈ s | «_aux_Init_ByCases___macroRules_tacticBy_cases_:__2» | «tacticBy_cases_:_» |
Mathlib.Analysis.Convex.Combination | {
"line": 509,
"column": 4
} | {
"line": 509,
"column": 23
} | {
"line": 510,
"column": 4
} | [
{
"pp": "case refine_1\nR : Type u_1\nE : Type u_3\ninst✝⁴ : Field R\ninst✝³ : AddCommGroup E\ninst✝² : Module R E\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\nι : Type u_8\nb : AffineBasis ι R E\ns : Finset ι\nw : ι → R\nhw₀ : ∀ i ∈ s, 0 ≤ w i\nhw₁ : s.sum w = 1\ni : ι\n⊢ 0 ≤ (b.coord i) ((affineCom... | [
"case pos\nR : Type u_1\nE : Type u_3\ninst✝⁴ : Field R\ninst✝³ : AddCommGroup E\ninst✝² : Module R E\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\nι : Type u_8\nb : AffineBasis ι R E\ns : Finset ι\nw : ι → R\nhw₀ : ∀ i ∈ s, 0 ≤ w i\nhw₁ : s.sum w = 1\ni : ι\nhi : i ∈ s\n⊢ 0 ≤ (b.coord i) ((affineCombinat... | by_cases hi : i ∈ s | «_aux_Init_ByCases___macroRules_tacticBy_cases_:__2» | «tacticBy_cases_:_» |
Mathlib.Topology.Algebra.Module.UniformConvergence | {
"line": 64,
"column": 26
} | {
"line": 87,
"column": 94
} | {
"line": 89,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nα : Type u_2\nE : Type u_3\nH : Type u_4\nhom : Type u_5\ninst✝¹⁰ : NormedField 𝕜\ninst✝⁹ : AddCommGroup H\ninst✝⁸ : Module 𝕜 H\ninst✝⁷ : AddCommGroup E\ninst✝⁶ : Module 𝕜 E\ninst✝⁵ : TopologicalSpace H\ninst✝⁴ : UniformSpace E\ninst✝³ : IsUniformAddGroup E\ninst✝² : ContinuousSMul 𝕜... | [] | by
have : IsTopologicalAddGroup H :=
let ofFun' : (α → E) →+ (α →ᵤ E) := AddMonoidHom.id _
IsInducing.topologicalAddGroup (ofFun'.comp (φ : H →+ (α → E))) hφ
have hb : (𝓝 (0 : H)).HasBasis (· ∈ 𝓝 (0 : E)) fun V ↦ {u | ∀ x, φ u x ∈ V} := by
simp only [hφ.nhds_eq_comap, Function.comp_apply, map_zero]
... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Convex.Combination | {
"line": 601,
"column": 12
} | {
"line": 601,
"column": 17
} | {
"line": 601,
"column": 17
} | [
{
"pp": "𝕜 : Type u_1\nι : Type u_2\nE : ι → Type u_3\ninst✝⁵ : Finite ι\ninst✝⁴ : Field 𝕜\ninst✝³ : LinearOrder 𝕜\ninst✝² : IsStrictOrderedRing 𝕜\ninst✝¹ : (i : ι) → AddCommGroup (E i)\ninst✝ : (i : ι) → Module 𝕜 (E i)\ns : Set ι\nt✝ : (i : ι) → Set (E i)\nx : (i : ι) → E i\nval✝ : Fintype ι\nt : (i : ι) ... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Analysis.Convex.Combination | {
"line": 601,
"column": 12
} | {
"line": 601,
"column": 17
} | {
"line": 601,
"column": 17
} | [
{
"pp": "𝕜 : Type u_1\nι : Type u_2\nE : ι → Type u_3\ninst✝⁵ : Finite ι\ninst✝⁴ : Field 𝕜\ninst✝³ : LinearOrder 𝕜\ninst✝² : IsStrictOrderedRing 𝕜\ninst✝¹ : (i : ι) → AddCommGroup (E i)\ninst✝ : (i : ι) → Module 𝕜 (E i)\ns : Set ι\nt✝ : (i : ι) → Set (E i)\nx : (i : ι) → E i\nval✝ : Fintype ι\nt : (i : ι) ... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Convex.Combination | {
"line": 601,
"column": 12
} | {
"line": 601,
"column": 17
} | {
"line": 601,
"column": 17
} | [
{
"pp": "𝕜 : Type u_1\nι : Type u_2\nE : ι → Type u_3\ninst✝⁵ : Finite ι\ninst✝⁴ : Field 𝕜\ninst✝³ : LinearOrder 𝕜\ninst✝² : IsStrictOrderedRing 𝕜\ninst✝¹ : (i : ι) → AddCommGroup (E i)\ninst✝ : (i : ι) → Module 𝕜 (E i)\ns : Set ι\nt✝ : (i : ι) → Set (E i)\nx : (i : ι) → E i\nval✝ : Fintype ι\nt : (i : ι) ... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Convex.Combination | {
"line": 632,
"column": 50
} | {
"line": 632,
"column": 55
} | {
"line": 633,
"column": 2
} | [
{
"pp": "case pos\n𝕜 : Type u_1\nV : Type u_2\ninst✝⁴ : Field 𝕜\ninst✝³ : LinearOrder 𝕜\ninst✝² : IsOrderedRing 𝕜\ninst✝¹ : AddCommGroup V\ninst✝ : Module 𝕜 V\nn : ℕ\ns : Simplex 𝕜 V n\nu : Finset (Fin (n + 1))\nw : Fin (n + 1) → 𝕜\nhw : ∀ i ∈ u, 0 ≤ w i\nhw1 : u.sum w = 1\nhw' : ∀ i ∈ u, w i ≤ 1\nhw1' :... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Analysis.Convex.Combination | {
"line": 632,
"column": 50
} | {
"line": 632,
"column": 55
} | {
"line": 633,
"column": 2
} | [
{
"pp": "case neg\n𝕜 : Type u_1\nV : Type u_2\ninst✝⁴ : Field 𝕜\ninst✝³ : LinearOrder 𝕜\ninst✝² : IsOrderedRing 𝕜\ninst✝¹ : AddCommGroup V\ninst✝ : Module 𝕜 V\nn : ℕ\ns : Simplex 𝕜 V n\nu : Finset (Fin (n + 1))\nw : Fin (n + 1) → 𝕜\nhw : ∀ i ∈ u, 0 ≤ w i\nhw1 : u.sum w = 1\nhw' : ∀ i ∈ u, w i ≤ 1\nhw1' :... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Analysis.LocallyConvex.WithSeminorms | {
"line": 371,
"column": 2
} | {
"line": 371,
"column": 21
} | {
"line": 372,
"column": 2
} | [
{
"pp": "𝕜 : Type u_2\nE : Type u_6\nι : Type u_9\ninst✝⁴ : NormedField 𝕜\ninst✝³ : AddCommGroup E\ninst✝² : Module 𝕜 E\ninst✝¹ : TopologicalSpace E\np : SeminormFamily 𝕜 E ι\ninst✝ : T1Space E\nhp : WithSeminorms p\nx : E\nhx : x ≠ 0\nthis : ∀ ⦃x y : E⦄, x ⤳ y → x = y\nh : ∀ (i : ι), (p i) x = 0\n⊢ False",... | [
"𝕜 : Type u_2\nE : Type u_6\nι : Type u_9\ninst✝⁴ : NormedField 𝕜\ninst✝³ : AddCommGroup E\ninst✝² : Module 𝕜 E\ninst✝¹ : TopologicalSpace E\np : SeminormFamily 𝕜 E ι\ninst✝ : T1Space E\nhp : WithSeminorms p\nx : E\nhx : x ≠ 0\nthis : ∀ ⦃x y : E⦄, x ⤳ y → x = y\nh : ∀ (i : ι), (p i) x = 0\n⊢ x ⤳ 0"
] | refine hx (this ?_) | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.LocallyConvex.WithSeminorms | {
"line": 573,
"column": 2
} | {
"line": 575,
"column": 9
} | {
"line": 576,
"column": 2
} | [
{
"pp": "case mp\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\ni : ι\n⊢ (∃ r > 0, ∀ x ∈ s, (p i) x < r) → ∃ x, ∀ ⦃x_1 : E⦄, x_1 ∈ s → (p i) x_... | [
"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\ni : ι\n⊢ (∃ x, ∀ ⦃x_1 : E⦄, x_1 ∈ s → (p i) x_1 ≤ x) → ∃ r > 0, ∀ x ∈ s, (p i) x < r"
] | · rintro ⟨r, _⟩
use r
grind | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Topology.UniformSpace.CompactConvergence | {
"line": 355,
"column": 50
} | {
"line": 355,
"column": 98
} | {
"line": 355,
"column": 98
} | [
{
"pp": "α : Type u₁\nβ : Type u₂\ninst✝² : TopologicalSpace α\ninst✝¹ : UniformSpace β\ns : Set α\ninst✝ : CompactSpace ↑s\nf : α → β\nhf : ContinuousOn f s\nι : Type u_1\np : Filter ι\nF : ι → α → β\nhF : ∀ (i : ι), ContinuousOn (F i) s\n⊢ TendstoUniformly (fun i a ↦ { toFun := s.restrict (F i), continuous_to... | [
"α : Type u₁\nβ : Type u₂\ninst✝² : TopologicalSpace α\ninst✝¹ : UniformSpace β\ns : Set α\ninst✝ : CompactSpace ↑s\nf : α → β\nhf : ContinuousOn f s\nι : Type u_1\np : Filter ι\nF : ι → α → β\nhF : ∀ (i : ι), ContinuousOn (F i) s\n⊢ TendstoUniformly (fun i a ↦ { toFun := s.restrict (F i), continuous_toFun := ⋯ } a... | tendstoUniformlyOn_iff_tendstoUniformly_comp_coe | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.ConditionalProbability | {
"line": 304,
"column": 2
} | {
"line": 312,
"column": 86
} | {
"line": 314,
"column": 0
} | [
{
"pp": "Ω : Type u_1\nα : Type u_3\nm : MeasurableSpace Ω\ninst✝³ : Fintype α\ninst✝² : MeasurableSpace α\ninst✝¹ : DiscreteMeasurableSpace α\nX : Ω → α\nhX : Measurable X\nμ : Measure Ω\ninst✝ : IsFiniteMeasure μ\nE : Set Ω\nhE : MeasurableSet E\n⊢ (∑ x, μ (X ⁻¹' {x}) • μ[|X ⁻¹' {x}]) E = μ E",
"ppTerm": ... | [] | calc
_ = ∑ x, μ (X ⁻¹' {x} ∩ E) := by
simp only [Measure.coe_finsetSum, Measure.coe_smul, Finset.sum_apply,
Pi.smul_apply, smul_eq_mul]
simp_rw [mul_comm (μ _), cond_mul_eq_inter (hX (.singleton _))]
_ = _ := by
have : ⋃ x ∈ Finset.univ, X ⁻¹' {x} ∩ E = E := by ext; simp
rw [← me... | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcTactic |
Mathlib.Topology.ContinuousMap.Algebra | {
"line": 780,
"column": 4
} | {
"line": 791,
"column": 43
} | {
"line": 791,
"column": 43
} | [
{
"pp": "α : Type u_1\ninst✝⁴ : TopologicalSpace α\nR : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : TopologicalSpace R\ninst✝¹ : IsTopologicalSemiring R\ninst✝ : Subsingleton α\ns₁ s₂ : Subalgebra R C(α, R)\n⊢ s₁ = s₂",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Subalgebra.instSe... | [] | cases isEmpty_or_nonempty α
· have : Subsingleton C(α, R) := DFunLike.coe_injective.subsingleton
subsingleton
· inhabit α
ext f
have h : f = algebraMap R C(α, R) (f default) := by
ext x'
simp only [mul_one, smul_eq_mul, algebraMap_apply]
congr
simp [eq_iff_true_... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.ContinuousMap.Algebra | {
"line": 780,
"column": 4
} | {
"line": 791,
"column": 43
} | {
"line": 791,
"column": 43
} | [
{
"pp": "α : Type u_1\ninst✝⁴ : TopologicalSpace α\nR : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : TopologicalSpace R\ninst✝¹ : IsTopologicalSemiring R\ninst✝ : Subsingleton α\ns₁ s₂ : Subalgebra R C(α, R)\n⊢ s₁ = s₂",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Subalgebra.instSe... | [] | cases isEmpty_or_nonempty α
· have : Subsingleton C(α, R) := DFunLike.coe_injective.subsingleton
subsingleton
· inhabit α
ext f
have h : f = algebraMap R C(α, R) (f default) := by
ext x'
simp only [mul_one, smul_eq_mul, algebraMap_apply]
congr
simp [eq_iff_true_... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Function.AEEqFun | {
"line": 607,
"column": 2
} | {
"line": 607,
"column": 42
} | {
"line": 608,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝³ : MeasurableSpace α\nμ : Measure α\ninst✝² : TopologicalSpace β\ninst✝¹ : SemilatticeInf β\ninst✝ : ContinuousInf β\nf g : α →ₘ[μ] β\n⊢ ↑(f ⊓ g) ≤ᵐ[μ] ↑f",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"MeasureTheory.ae",
"MeasureTheo... | [
"α : Type u_1\nβ : Type u_2\ninst✝³ : MeasurableSpace α\nμ : Measure α\ninst✝² : TopologicalSpace β\ninst✝¹ : SemilatticeInf β\ninst✝ : ContinuousInf β\nf g : α →ₘ[μ] β\na✝ : α\nha : ↑(f ⊓ g) a✝ = ↑f a✝ ⊓ ↑g a✝\n⊢ ↑(f ⊓ g) a✝ ≤ ↑f a✝"
] | filter_upwards [coeFn_inf f g] with _ ha | Mathlib.Tactic._aux_Mathlib_Order_Filter_Defs___elabRules_Mathlib_Tactic_filterUpwards_1 | Mathlib.Tactic.filterUpwards |
Mathlib.MeasureTheory.Function.AEEqFun | {
"line": 613,
"column": 2
} | {
"line": 613,
"column": 42
} | {
"line": 614,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝³ : MeasurableSpace α\nμ : Measure α\ninst✝² : TopologicalSpace β\ninst✝¹ : SemilatticeInf β\ninst✝ : ContinuousInf β\nf g : α →ₘ[μ] β\n⊢ ↑(f ⊓ g) ≤ᵐ[μ] ↑g",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"MeasureTheory.ae",
"MeasureTheo... | [
"α : Type u_1\nβ : Type u_2\ninst✝³ : MeasurableSpace α\nμ : Measure α\ninst✝² : TopologicalSpace β\ninst✝¹ : SemilatticeInf β\ninst✝ : ContinuousInf β\nf g : α →ₘ[μ] β\na✝ : α\nha : ↑(f ⊓ g) a✝ = ↑f a✝ ⊓ ↑g a✝\n⊢ ↑(f ⊓ g) a✝ ≤ ↑g a✝"
] | filter_upwards [coeFn_inf f g] with _ ha | Mathlib.Tactic._aux_Mathlib_Order_Filter_Defs___elabRules_Mathlib_Tactic_filterUpwards_1 | Mathlib.Tactic.filterUpwards |
Mathlib.MeasureTheory.Function.SpecialFunctions.Basic | {
"line": 44,
"column": 20
} | {
"line": 47,
"column": 30
} | {
"line": 49,
"column": 0
} | [
{
"pp": "α : Type u_1\nx✝ : MeasurableSpace α\nf : α → ℝ\nhf : Measurable fun x ↦ rexp (f x)\n⊢ Measurable f",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"Real.log_exp",
"Measurable.comp",
"congrArg",
"Measurable",
"id",
... | [] | by
have : f = fun x ↦ log (exp (f x)) := by ext; rw [log_exp]
rw [this]
exact measurable_log.comp hf | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Constructions.Pi | {
"line": 373,
"column": 2
} | {
"line": 373,
"column": 66
} | {
"line": 374,
"column": 2
} | [
{
"pp": "ι : Type u_1\nα : ι → Type u_3\ninst✝² : Fintype ι\ninst✝¹ : (i : ι) → MeasurableSpace (α i)\nμ : (i : ι) → Measure (α i)\ninst✝ : ∀ (i : ι), SigmaFinite (μ i)\ni : ι\n⊢ QuasiMeasurePreserving (eval i) (Measure.pi μ) (μ i)",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Mea... | [
"ι : Type u_1\nα : ι → Type u_3\ninst✝² : Fintype ι\ninst✝¹ : (i : ι) → MeasurableSpace (α i)\nμ : (i : ι) → Measure (α i)\ninst✝ : ∀ (i : ι), SigmaFinite (μ i)\ni : ι\ns : Set (α i)\nhs : MeasurableSet s\nh2s : (μ i) s = 0\n⊢ (map (eval i) (Measure.pi μ)) s = 0"
] | refine ⟨by fun_prop, AbsolutelyContinuous.mk fun s hs h2s => ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.MeasureTheory.Function.LpSeminorm.Monotonicity | {
"line": 74,
"column": 37
} | {
"line": 74,
"column": 61
} | {
"line": 75,
"column": 6
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nε' : Type u_6\ninst✝³ : TopologicalSpace ε'\ninst✝² : ContinuousENorm ε'\nε : Type u_7\ninst✝¹ : TopologicalSpace ε\ninst✝ : ESeminormedAddMonoid ε\nf : α → ε\nc : ℝ≥0∞\ng : α → ε'\np : ℝ\nhg : AEStronglyMeasurable g μ\nh : ∀ᵐ (x : α) ∂μ, ‖f x‖ₑ ≤ c *... | [] | simpa [hp, hp'] using hx | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.MeasureTheory.Function.LpSeminorm.Monotonicity | {
"line": 74,
"column": 37
} | {
"line": 74,
"column": 61
} | {
"line": 75,
"column": 6
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nε' : Type u_6\ninst✝³ : TopologicalSpace ε'\ninst✝² : ContinuousENorm ε'\nε : Type u_7\ninst✝¹ : TopologicalSpace ε\ninst✝ : ESeminormedAddMonoid ε\nf : α → ε\nc : ℝ≥0∞\ng : α → ε'\np : ℝ\nhg : AEStronglyMeasurable g μ\nh : ∀ᵐ (x : α) ∂μ, ‖f x‖ₑ ≤ c *... | [] | simpa [hp, hp'] using hx | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Function.LpSeminorm.Monotonicity | {
"line": 74,
"column": 37
} | {
"line": 74,
"column": 61
} | {
"line": 75,
"column": 6
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nε' : Type u_6\ninst✝³ : TopologicalSpace ε'\ninst✝² : ContinuousENorm ε'\nε : Type u_7\ninst✝¹ : TopologicalSpace ε\ninst✝ : ESeminormedAddMonoid ε\nf : α → ε\nc : ℝ≥0∞\ng : α → ε'\np : ℝ\nhg : AEStronglyMeasurable g μ\nh : ∀ᵐ (x : α) ∂μ, ‖f x‖ₑ ≤ c *... | [] | simpa [hp, hp'] using hx | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Function.LpSeminorm.Indicator | {
"line": 174,
"column": 4
} | {
"line": 174,
"column": 36
} | {
"line": 175,
"column": 4
} | [
{
"pp": "case neg.inr.left\nα : Type u_1\nm0 : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\nε : Type u_7\ninst✝² : TopologicalSpace ε\ninst✝¹ : ESeminormedAddMonoid ε\ns : Set α\nf : α → ε\ninst✝ : DecidablePred fun x ↦ x ∈ s\ng : α → ε\nhs : MeasurableSet s\nhf : MemLp f p (μ.restrict s)\nhg : MemLp g p (μ.rest... | [
"case neg.inr.left\nα : Type u_1\nm0 : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\nε : Type u_7\ninst✝² : TopologicalSpace ε\ninst✝¹ : ESeminormedAddMonoid ε\ns : Set α\nf : α → ε\ninst✝ : DecidablePred fun x ↦ x ∈ s\ng : α → ε\nhs : MeasurableSet s\nhf : MemLp f p (μ.restrict s)\nhg : MemLp g p (μ.restrict sᶜ)\nhp... | rw [setLIntegral_congr_fun hs h] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Convex.Slope | {
"line": 191,
"column": 2
} | {
"line": 191,
"column": 51
} | {
"line": 193,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : Field 𝕜\ninst✝¹ : LinearOrder 𝕜\ninst✝ : IsStrictOrderedRing 𝕜\ns : Set 𝕜\nf : 𝕜 → 𝕜\nhf : ConvexOn 𝕜 s f\nx y z : 𝕜\nhx : x ∈ s\nhz : z ∈ s\nhxy : x < y\nhyz : y < z\nhxy' : 0 < y - x\nhxz' : 0 < z - x\n⊢ (f y - f x) * (z - x) ≤ (y - x) * (f z - f x)",
"ppTerm": "?m... | [] | linarith only [hf.secant_mono_aux1 hx hz hxy hyz] | Mathlib.Tactic._aux_Mathlib_Tactic_Linarith_Frontend___elabRules_Mathlib_Tactic_linarith_1 | Mathlib.Tactic.linarith |
Mathlib.Analysis.Convex.Slope | {
"line": 198,
"column": 2
} | {
"line": 198,
"column": 51
} | {
"line": 200,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : Field 𝕜\ninst✝¹ : LinearOrder 𝕜\ninst✝ : IsStrictOrderedRing 𝕜\ns : Set 𝕜\nf : 𝕜 → 𝕜\nhf : ConvexOn 𝕜 s f\nx y z : 𝕜\nhx : x ∈ s\nhz : z ∈ s\nhxy : x < y\nhyz : y < z\nhyz' : 0 < z - y\nhxz' : 0 < z - x\n⊢ (f z - f x) * (z - y) ≤ (z - x) * (f z - f y)",
"ppTerm": "?m... | [] | linarith only [hf.secant_mono_aux1 hx hz hxy hyz] | Mathlib.Tactic._aux_Mathlib_Tactic_Linarith_Frontend___elabRules_Mathlib_Tactic_linarith_1 | Mathlib.Tactic.linarith |
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