module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Order.SupIndep | {
"line": 450,
"column": 4
} | {
"line": 450,
"column": 15
} | {
"line": 450,
"column": 16
} | [
{
"pp": "case refine_1\nα : Type u_5\ninst✝ : CompleteLattice α\nf : Fin 3 → α\nh : ∀ (i : Fin 3), Disjoint (f i) (⨆ j, ⨆ (_ : j ≠ i), f j)\n⊢ Disjoint (f 0) (f 1 ⊔ f 2)",
"ppTerm": "?refine_1",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"case refine_1\nα : Type u_5\ninst✝ : CompleteLattice α\nf : Fin 3 → α\nh : ∀ (i : Fin 3), Disjoint (f i) (⨆ j, ⨆ (_ : j ≠ i), f j)\n⊢ Disjoint (f 0) (f 1 ⊔ f 2)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.SupIndep | {
"line": 451,
"column": 4
} | {
"line": 451,
"column": 15
} | {
"line": 451,
"column": 16
} | [
{
"pp": "case refine_2\nα : Type u_5\ninst✝ : CompleteLattice α\nf : Fin 3 → α\nh : ∀ (i : Fin 3), Disjoint (f i) (⨆ j, ⨆ (_ : j ≠ i), f j)\n⊢ Disjoint (f 1) (f 0 ⊔ f 2)",
"ppTerm": "?refine_2",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"case refine_2\nα : Type u_5\ninst✝ : CompleteLattice α\nf : Fin 3 → α\nh : ∀ (i : Fin 3), Disjoint (f i) (⨆ j, ⨆ (_ : j ≠ i), f j)\n⊢ Disjoint (f 1) (f 0 ⊔ f 2)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.SupIndep | {
"line": 452,
"column": 4
} | {
"line": 452,
"column": 15
} | {
"line": 452,
"column": 16
} | [
{
"pp": "case refine_3\nα : Type u_5\ninst✝ : CompleteLattice α\nf : Fin 3 → α\nh : ∀ (i : Fin 3), Disjoint (f i) (⨆ j, ⨆ (_ : j ≠ i), f j)\n⊢ Disjoint (f 2) (f 0 ⊔ f 1)",
"ppTerm": "?refine_3",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"case refine_3\nα : Type u_5\ninst✝ : CompleteLattice α\nf : Fin 3 → α\nh : ∀ (i : Fin 3), Disjoint (f i) (⨆ j, ⨆ (_ : j ≠ i), f j)\n⊢ Disjoint (f 2) (f 0 ⊔ f 1)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Algebra.Tower | {
"line": 138,
"column": 4
} | {
"line": 138,
"column": 90
} | {
"line": 138,
"column": 91
} | [
{
"pp": "S : Type u\nA : Type v\ninst✝¹ : CommSemiring S\ninst✝ : Semiring A\nh1 h2 : Algebra S A\nh :\n ∀ (r : S) (x : A),\n (have I := h1;\n r • x) =\n r • x\nr : S\n⊢ (algebraMap S A) r = (algebraMap S A) r",
"ppTerm": "?m.46",
"assigned": false,
"usedConstants": [],
"usedFVars"... | [
"S : Type u\nA : Type v\ninst✝¹ : CommSemiring S\ninst✝ : Semiring A\nh1 h2 : Algebra S A\nh :\n ∀ (r : S) (x : A),\n (have I := h1;\n r • x) =\n r • x\nr : S\n⊢ (algebraMap S A) r = (algebraMap S A) r"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Algebra.Tower | {
"line": 350,
"column": 2
} | {
"line": 350,
"column": 31
} | {
"line": 350,
"column": 32
} | [
{
"pp": "R : Type u\nA : Type w\nM : Type v₁\ninst✝⁶ : CommSemiring R\ninst✝⁵ : Semiring A\ninst✝⁴ : Algebra R A\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : Module A M\ninst✝ : IsScalarTower R A M\nhsur : Function.Surjective ⇑(algebraMap R A)\nX : Set M\nm✝ : M\nhm✝ : m✝ ∈ restrictScalars R (span A... | [
"R : Type u\nA : Type w\nM : Type v₁\ninst✝⁶ : CommSemiring R\ninst✝⁵ : Semiring A\ninst✝⁴ : Algebra R A\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : Module A M\ninst✝ : IsScalarTower R A M\nhsur : Function.Surjective ⇑(algebraMap R A)\nX : Set M\nm✝ : M\nhm✝ : m✝ ∈ restrictScalars R (span A X)\nm : M\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Algebra.Tower | {
"line": 367,
"column": 36
} | {
"line": 367,
"column": 58
} | {
"line": 367,
"column": 59
} | [
{
"pp": "R✝ : Type u\nS✝ : Type v\nA : Type w\nB : Type u₁\nM✝ : Type v₁\ninst✝¹³ : CommSemiring R✝\ninst✝¹² : Semiring A\ninst✝¹¹ : Algebra R✝ A\ninst✝¹⁰ : AddCommMonoid M✝\ninst✝⁹ : Module R✝ M✝\ninst✝⁸ : Module A M✝\ninst✝⁷ : IsScalarTower R✝ A M✝\nR : Type u_1\nS : Type u_2\nM : Type u_3\ninst✝⁶ : CommRing ... | [
"R✝ : Type u\nS✝ : Type v\nA : Type w\nB : Type u₁\nM✝ : Type v₁\ninst✝¹³ : CommSemiring R✝\ninst✝¹² : Semiring A\ninst✝¹¹ : Algebra R✝ A\ninst✝¹⁰ : AddCommMonoid M✝\ninst✝⁹ : Module R✝ M✝\ninst✝⁸ : Module A M✝\ninst✝⁷ : IsScalarTower R✝ A M✝\nR : Type u_1\nS : Type u_2\nM : Type u_3\ninst✝⁶ : CommRing R\ninst✝⁵ : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Notation.Indicator | {
"line": 247,
"column": 66
} | {
"line": 249,
"column": 44
} | {
"line": 251,
"column": 0
} | [
{
"pp": "α : Type u_1\nM : Type u_3\ninst✝² : One M\nU : Set α\ns : Set M\na : M\ninst✝¹ : Decidable (a ∈ s)\ninst✝ : Decidable (1 ∈ s)\n⊢ (U.mulIndicator fun x ↦ a) ⁻¹' s = (if a ∈ s then U else ∅) ∪ if 1 ∈ s then Uᶜ else ∅",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [] | by
rw [mulIndicator_preimage, Pi.one_def, Set.preimage_const, preimage_const]
split_ifs <;> simp [← compl_eq_univ_sdiff] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.FunLike.IsApply | {
"line": 215,
"column": 4
} | {
"line": 216,
"column": 24
} | {
"line": 218,
"column": 0
} | [
{
"pp": "case mpr\nF : Type u_3\nα : Type u_5\nβ : Type u_6\ninst✝³ : FunLike F α β\ninst✝² : One F\ninst✝¹ : One β\ninst✝ : IsOneApply F α β\nf : F\n⊢ f = 1 → ⇑f = 1",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"congrArg",
"_private.Mathlib.Data.FunLike.IsApply.0.FunLike.coe... | [] | intro h
simp [funext_iff, h] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.FunLike.IsApply | {
"line": 215,
"column": 4
} | {
"line": 216,
"column": 24
} | {
"line": 218,
"column": 0
} | [
{
"pp": "case mpr\nF : Type u_3\nα : Type u_5\nβ : Type u_6\ninst✝³ : FunLike F α β\ninst✝² : One F\ninst✝¹ : One β\ninst✝ : IsOneApply F α β\nf : F\n⊢ f = 1 → ⇑f = 1",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"congrArg",
"_private.Mathlib.Data.FunLike.IsApply.0.FunLike.coe... | [] | intro h
simp [funext_iff, h] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.FunLike.IsApply | {
"line": 251,
"column": 4
} | {
"line": 252,
"column": 24
} | {
"line": 254,
"column": 0
} | [
{
"pp": "case mpr\nF' : Type u_4\nα : Type u_5\ninst✝² : FunLike F' α α\ninst✝¹ : One F'\ninst✝ : IsOneApplyEqSelf F' α\nf : F'\n⊢ f = 1 → ⇑f = id",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"congrArg",
"id",
"_private.Mathlib.Data.FunLike.IsApply.0.FunLike.coe_one_eq_... | [] | intro h
simp [funext_iff, h] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.FunLike.IsApply | {
"line": 251,
"column": 4
} | {
"line": 252,
"column": 24
} | {
"line": 254,
"column": 0
} | [
{
"pp": "case mpr\nF' : Type u_4\nα : Type u_5\ninst✝² : FunLike F' α α\ninst✝¹ : One F'\ninst✝ : IsOneApplyEqSelf F' α\nf : F'\n⊢ f = 1 → ⇑f = id",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"congrArg",
"id",
"_private.Mathlib.Data.FunLike.IsApply.0.FunLike.coe_one_eq_... | [] | intro h
simp [funext_iff, h] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.CompactlyGenerated.Basic | {
"line": 519,
"column": 57
} | {
"line": 519,
"column": 68
} | {
"line": 519,
"column": 69
} | [
{
"pp": "α : Type u_2\ninst✝¹ : CompleteLattice α\ninst✝ : IsCompactlyGenerated α\ns : Set (Set α)\nhs : DirectedOn (fun x1 x2 ↦ x1 ⊆ x2) s\nh : ∀ a ∈ s, sSupIndep a\n⊢ ∀ (i : ↑s), sSupIndep ↑i",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"sSupIndep",
"Eq.mpr",
"Subtyp... | [
"α : Type u_2\ninst✝¹ : CompleteLattice α\ninst✝ : IsCompactlyGenerated α\ns : Set (Set α)\nhs : DirectedOn (fun x1 x2 ↦ x1 ⊆ x2) s\nh : ∀ a ∈ s, sSupIndep a\n⊢ ∀ a ∈ s, sSupIndep a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Span.Basic | {
"line": 119,
"column": 4
} | {
"line": 119,
"column": 30
} | {
"line": 120,
"column": 8
} | [
{
"pp": "case mpr.add\nR : Type u_1\nM : Type u_4\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\nN : Type u_8\ninst✝¹ : AddCommMonoid N\ninst✝ : Module R N\nS : Set M\nf g : ↥(span R S) →ₗ[R] N\nh : ∀ (s : ↑S), f ⟨↑s, ⋯⟩ = g ⟨↑s, ⋯⟩\nx✝ x y : M\nhx : x ∈ span R S\nhy : y ∈ span R S\nhx' : ... | [] | | add x y hx hy hx' hy' => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.LinearAlgebra.Span.Basic | {
"line": 183,
"column": 2
} | {
"line": 183,
"column": 13
} | {
"line": 183,
"column": 14
} | [
{
"pp": "R : Type u_1\nM : Type u_4\nS : Type u_7\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\ninst✝³ : Semiring S\ninst✝² : SMul R S\ninst✝¹ : Module S M\ninst✝ : IsScalarTower R S M\np : Submodule R M\nx y : ↥p\nhxy : ↑((inclusionSpan S p) x) = ↑((inclusionSpan S p) y)\n⊢ x = y",
"... | [
"R : Type u_1\nM : Type u_4\nS : Type u_7\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\ninst✝³ : Semiring S\ninst✝² : SMul R S\ninst✝¹ : Module S M\ninst✝ : IsScalarTower R S M\np : Submodule R M\nx y : ↥p\nhxy : ↑((inclusionSpan S p) x) = ↑((inclusionSpan S p) y)\n⊢ x = y"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Span.Basic | {
"line": 226,
"column": 4
} | {
"line": 226,
"column": 15
} | {
"line": 226,
"column": 16
} | [
{
"pp": "R : Type u_1\nM : Type u_4\nS : Type u_7\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\ninst✝³ : Semiring S\ninst✝² : SMul R S\ninst✝¹ : Module S M\ninst✝ : IsScalarTower R S M\np : Submodule R M\nq : Submodule S M\nh₁ : p ≤ restrictScalars R q\nh₂ : q ≤ span S ↑p\nthis : ⇑q.subty... | [
"R : Type u_1\nM : Type u_4\nS : Type u_7\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\ninst✝³ : Semiring S\ninst✝² : SMul R S\ninst✝¹ : Module S M\ninst✝ : IsScalarTower R S M\np : Submodule R M\nq : Submodule S M\nh₁ : p ≤ restrictScalars R q\nh₂ : q ≤ span S ↑p\nthis : ⇑q.subtype '' ↑(incl... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Span.Basic | {
"line": 228,
"column": 2
} | {
"line": 228,
"column": 31
} | {
"line": 228,
"column": 32
} | [
{
"pp": "R : Type u_1\nM : Type u_4\nS : Type u_7\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\ninst✝³ : Semiring S\ninst✝² : SMul R S\ninst✝¹ : Module S M\ninst✝ : IsScalarTower R S M\np : Submodule R M\nq : Submodule S M\nh₁ : p ≤ restrictScalars R q\nh₂ : q ≤ span S ↑p\nx : M\n⊢ x ∈ ⇑q... | [
"R : Type u_1\nM : Type u_4\nS : Type u_7\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\ninst✝³ : Semiring S\ninst✝² : SMul R S\ninst✝¹ : Module S M\ninst✝ : IsScalarTower R S M\np : Submodule R M\nq : Submodule S M\nh₁ : p ≤ restrictScalars R q\nh₂ : q ≤ span S ↑p\nx : M\n⊢ x ∈ p → x ∈ q"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.BigOperators.Ring.Multiset | {
"line": 78,
"column": 4
} | {
"line": 80,
"column": 71
} | {
"line": 81,
"column": 4
} | [
{
"pp": "case refine_2\nι : Type u_1\nR : Type u_4\ninst✝ : CommSemiring R\ns✝ : Multiset ι\nf g : ι → R\na : ι\ns : Multiset ι\nih : (map (fun i ↦ f i + g i) s).prod = (map (fun p ↦ (map f p.1).prod * (map g p.2).prod) s.antidiagonal).sum\n⊢ (map (fun i ↦ f i + g i) (a ::ₘ s)).prod =\n (map (fun p ↦ (map f ... | [
"case refine_2\nι : Type u_1\nR : Type u_4\ninst✝ : CommSemiring R\ns✝ : Multiset ι\nf g : ι → R\na : ι\ns : Multiset ι\nih : (map (fun i ↦ f i + g i) s).prod = (map (fun p ↦ (map f p.1).prod * (map g p.2).prod) s.antidiagonal).sum\n⊢ (map (fun i ↦ (map f i.1).prod * (f a * (map g i.2).prod)) s.antidiagonal).sum +\... | simp only [map_cons, prod_cons, ih, sum_map_mul_left.symm, add_mul, mul_left_comm (f a),
mul_left_comm (g a), sum_map_add, antidiagonal_cons, Prod.map_fst, Prod.map_snd,
id_eq, map_add, map_map, Function.comp_apply, mul_assoc, sum_add] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.BigOperators.Pi | {
"line": 89,
"column": 29
} | {
"line": 89,
"column": 40
} | {
"line": 89,
"column": 41
} | [
{
"pp": "ι : Type u_1\nκ : Type u_2\nR : Type u_5\ninst✝ : CommSemiring R\ns : Finset ι\nf : ι → Set κ\ng : ι → κ → R\nj : κ\nhj : j ∉ ⋂ x ∈ s, f x\n⊢ ?m.85",
"ppTerm": "?m.90",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"ι : Type u_1\nκ : Type u_2\nR : Type u_5\ninst✝ : CommSemiring R\ns : Finset ι\nf : ι → Set κ\ng : ι → κ → R\nj : κ\nhj : j ∉ ⋂ x ∈ s, f x\n⊢ ?m.85"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.BigOperators.Pi | {
"line": 94,
"column": 9
} | {
"line": 94,
"column": 20
} | {
"line": 94,
"column": 21
} | [
{
"pp": "ι : Type u_1\nκ : Type u_2\nR : Type u_5\ninst✝ : CommSemiring R\ns : Finset ι\nf : ι → Set κ\ng : ι → κ → R\na : κ\n⊢ (∏ i ∈ s, (f i).indicator (g i)) a = (⋂ x ∈ s, f x).indicator (∏ i ∈ s, g i) a",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
... | [
"ι : Type u_1\nκ : Type u_2\nR : Type u_5\ninst✝ : CommSemiring R\ns : Finset ι\nf : ι → Set κ\ng : ι → κ → R\na : κ\n⊢ ∏ c ∈ s, (f c).indicator (g c) a = (⋂ x ∈ s, f x).indicator (∏ i ∈ s, g i) a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Span.Basic | {
"line": 433,
"column": 45
} | {
"line": 433,
"column": 56
} | {
"line": 433,
"column": 57
} | [
{
"pp": "case add_left\nR : Type u_9\nM : Type u_10\nN : Type u_11\nP : Type u_12\ninst✝⁶ : CommSemiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : AddCommMonoid N\ninst✝³ : AddCommMonoid P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nP' : Submodule R P\ns : Set M\nt : Set N\nB : M →ₗ[R] N →ₗ[R] P\... | [
"case add_left\nR : Type u_9\nM : Type u_10\nN : Type u_11\nP : Type u_12\ninst✝⁶ : CommSemiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : AddCommMonoid N\ninst✝³ : AddCommMonoid P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nP' : Submodule R P\ns : Set M\nt : Set N\nB : M →ₗ[R] N →ₗ[R] P\nhB : ∀ x ∈ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Span.Basic | {
"line": 434,
"column": 46
} | {
"line": 434,
"column": 57
} | {
"line": 434,
"column": 58
} | [
{
"pp": "case add_right\nR : Type u_9\nM : Type u_10\nN : Type u_11\nP : Type u_12\ninst✝⁶ : CommSemiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : AddCommMonoid N\ninst✝³ : AddCommMonoid P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nP' : Submodule R P\ns : Set M\nt : Set N\nB : M →ₗ[R] N →ₗ[R] P... | [
"case add_right\nR : Type u_9\nM : Type u_10\nN : Type u_11\nP : Type u_12\ninst✝⁶ : CommSemiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : AddCommMonoid N\ninst✝³ : AddCommMonoid P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nP' : Submodule R P\ns : Set M\nt : Set N\nB : M →ₗ[R] N →ₗ[R] P\nhB : ∀ x ∈... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Span.Basic | {
"line": 435,
"column": 33
} | {
"line": 435,
"column": 44
} | {
"line": 435,
"column": 45
} | [
{
"pp": "case smul_left\nR : Type u_9\nM : Type u_10\nN : Type u_11\nP : Type u_12\ninst✝⁶ : CommSemiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : AddCommMonoid N\ninst✝³ : AddCommMonoid P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nP' : Submodule R P\ns : Set M\nt✝ : Set N\nB : M →ₗ[R] N →ₗ[R] ... | [
"case smul_left\nR : Type u_9\nM : Type u_10\nN : Type u_11\nP : Type u_12\ninst✝⁶ : CommSemiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : AddCommMonoid N\ninst✝³ : AddCommMonoid P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nP' : Submodule R P\ns : Set M\nt✝ : Set N\nB : M →ₗ[R] N →ₗ[R] P\nhB : ∀ x ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.CompactlyGenerated.Basic | {
"line": 677,
"column": 2
} | {
"line": 677,
"column": 71
} | {
"line": 678,
"column": 2
} | [
{
"pp": "case refine_2.refine_2\nα : Type u_2\ninst✝² : CompleteLattice α\ninst✝¹ : IsModularLattice α\ninst✝ : IsCompactlyGenerated α\nb c : α\nhbc : b ≤ c\nh : sSup {a | a ≤ c ∧ IsAtom a} = c\ns : Set α\ns_max : ∀ ⦃t : Set α⦄, t ∈ {s | sSupIndep s ∧ Disjoint b (sSup s) ∧ ∀ a ∈ s, IsAtom a ∧ a ≤ c} → s ⊆ t → s... | [
"case refine_2.refine_2\nα : Type u_2\ninst✝² : CompleteLattice α\ninst✝¹ : IsModularLattice α\ninst✝ : IsCompactlyGenerated α\nb c : α\nhbc : b ≤ c\nh : sSup {a | a ≤ c ∧ IsAtom a} = c\ns : Set α\ns_max : ∀ ⦃t : Set α⦄, t ∈ {s | sSupIndep s ∧ Disjoint b (sSup s) ∧ ∀ a ∈ s, IsAtom a ∧ a ≤ c} → s ⊆ t → s = t\ns_ind ... | rw [s_max ⟨fun x hx => ?_, ?_, fun x hx => ?_⟩ Set.subset_union_left] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.Span.Basic | {
"line": 436,
"column": 34
} | {
"line": 436,
"column": 45
} | {
"line": 436,
"column": 46
} | [
{
"pp": "case smul_right\nR : Type u_9\nM : Type u_10\nN : Type u_11\nP : Type u_12\ninst✝⁶ : CommSemiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : AddCommMonoid N\ninst✝³ : AddCommMonoid P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nP' : Submodule R P\ns : Set M\nt✝ : Set N\nB : M →ₗ[R] N →ₗ[R]... | [
"case smul_right\nR : Type u_9\nM : Type u_10\nN : Type u_11\nP : Type u_12\ninst✝⁶ : CommSemiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : AddCommMonoid N\ninst✝³ : AddCommMonoid P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nP' : Submodule R P\ns : Set M\nt✝ : Set N\nB : M →ₗ[R] N →ₗ[R] P\nhB : ∀ x... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Span.Basic | {
"line": 493,
"column": 4
} | {
"line": 493,
"column": 22
} | {
"line": 493,
"column": 23
} | [
{
"pp": "case mp\nR : Type u_8\nM : Type u_9\ninst✝² : Semiring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\ns t p : Submodule R M\nhsp : s ≤ p\nhnsp : -s ≤ p\nx : M\nhx : x ∈ p\ny : M\nhy : y ∈ s\nz : M\nhz : z ∈ t\nhyzx✝ : y + z = x\nhyzx : z = -y + x\n⊢ z ∈ p",
"ppTerm": "?mp",
"assigned": true,
... | [
"case mp\nR : Type u_8\nM : Type u_9\ninst✝² : Semiring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\ns t p : Submodule R M\nhsp : s ≤ p\nhnsp : -s ≤ p\nx : M\nhx : x ∈ p\ny : M\nhy : y ∈ s\nz : M\nhz : z ∈ t\nhyzx✝ : y + z = x\nhyzx : z = -y + x\n⊢ -y + x ∈ p"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Span.Basic | {
"line": 496,
"column": 4
} | {
"line": 496,
"column": 24
} | {
"line": 496,
"column": 25
} | [
{
"pp": "case mpr\nR : Type u_8\nM : Type u_9\ninst✝² : Semiring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\ns t p : Submodule R M\nhsp : s ≤ p\nhnsp : -s ≤ p\nx y : M\nhy : y ∈ s\nz : M\nhyzx : y + z = x\nhz : z ∈ t\nhz' : z ∈ p\n⊢ x ∈ p",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
... | [
"case mpr\nR : Type u_8\nM : Type u_9\ninst✝² : Semiring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\ns t p : Submodule R M\nhsp : s ≤ p\nhnsp : -s ≤ p\nx y : M\nhy : y ∈ s\nz : M\nhyzx : y + z = x\nhz : z ∈ t\nhz' : z ∈ p\n⊢ y + z ∈ p"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Span.Basic | {
"line": 526,
"column": 2
} | {
"line": 526,
"column": 53
} | {
"line": 526,
"column": 54
} | [
{
"pp": "R : Type u_10\nM : Type u_11\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\np q r : Submodule R M\nh₁ : IsCompl q r\nh₂ : q ≤ p\n⊢ IsCompl (comap p.subtype q) (comap p.subtype r)",
"ppTerm": "?m.54",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"_private.Mathli... | [
"R : Type u_10\nM : Type u_11\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\np q r : Submodule R M\nh₁ : IsCompl q r\nh₂ : q ≤ p\n⊢ Disjoint (p ⊓ q) (p ⊓ r) ∧ p ⊓ q ⊔ p ⊓ r = p"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Span.Basic | {
"line": 542,
"column": 36
} | {
"line": 542,
"column": 47
} | {
"line": 542,
"column": 48
} | [
{
"pp": "R : Type u_1\nR₂ : Type u_2\nM : Type u_4\nM₂ : Type u_5\ninst✝⁶ : Semiring R\ninst✝⁵ : Semiring R₂\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : AddCommGroup M₂\ninst✝¹ : Module R₂ M₂\nτ₁₂ : R →+* R₂\ninst✝ : RingHomSurjective τ₁₂\nf : M →ₛₗ[τ₁₂] M₂\np : Submodule R M\nx y : M\nhy : y ∈ ↑p\n... | [
"R : Type u_1\nR₂ : Type u_2\nM : Type u_4\nM₂ : Type u_5\ninst✝⁶ : Semiring R\ninst✝⁵ : Semiring R₂\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : AddCommGroup M₂\ninst✝¹ : Module R₂ M₂\nτ₁₂ : R →+* R₂\ninst✝ : RingHomSurjective τ₁₂\nf : M →ₛₗ[τ₁₂] M₂\np : Submodule R M\nx y : M\nhy : y ∈ ↑p\ne : f y = f ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.CompactlyGenerated.Basic | {
"line": 704,
"column": 2
} | {
"line": 705,
"column": 9
} | {
"line": 705,
"column": 10
} | [
{
"pp": "α : Type u_2\ninst✝² : CompleteLattice α\ninst✝¹ : IsModularLattice α\ninst✝ : IsCompactlyGenerated α\nh : sSup {a | IsAtom a} = ⊤\nb : α\n⊢ ∃ s, sSupIndep s ∧ IsCompl b (sSup s) ∧ ∀ ⦃a : α⦄, a ∈ s → IsAtom a",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"sSupIndep",
... | [
"α : Type u_2\ninst✝² : CompleteLattice α\ninst✝¹ : IsModularLattice α\ninst✝ : IsCompactlyGenerated α\nh : sSup {a | IsAtom a} = ⊤\nb : α\n⊢ ∃ s, sSupIndep s ∧ Disjoint b (sSup s) ∧ b ⊔ sSup s = ⊤ ∧ ∀ ⦃a : α⦄, a ∈ s → IsAtom a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.CompactlyGenerated.Basic | {
"line": 705,
"column": 64
} | {
"line": 705,
"column": 75
} | {
"line": 705,
"column": 76
} | [
{
"pp": "α : Type u_2\ninst✝² : CompleteLattice α\ninst✝¹ : IsModularLattice α\ninst✝ : IsCompactlyGenerated α\nh : sSup {a | IsAtom a} = ⊤\nb : α\n⊢ sSup {a | a ≤ ⊤ ∧ IsAtom a} = ⊤",
"ppTerm": "?m.39",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Lattice.toSemilatticeSup",
"Com... | [
"α : Type u_2\ninst✝² : CompleteLattice α\ninst✝¹ : IsModularLattice α\ninst✝ : IsCompactlyGenerated α\nh : sSup {a | IsAtom a} = ⊤\nb : α\n⊢ sSup {a | IsAtom a} = ⊤"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.CompactlyGenerated.Basic | {
"line": 710,
"column": 16
} | {
"line": 710,
"column": 27
} | {
"line": 710,
"column": 28
} | [
{
"pp": "α : Type u_2\ninst✝² : CompleteLattice α\ninst✝¹ : IsModularLattice α\ninst✝ : IsCompactlyGenerated α\nb : α\nh : sSup {a | a ≤ b ∧ IsAtom a} = b\ns : Set α\ns_ind : sSupIndep s\nleft✝ : Disjoint ⊥ (sSup s)\ns_atoms : ⊥ ⊔ sSup s = b ∧ ∀ ⦃a : α⦄, a ∈ s → IsAtom a\n⊢ sSup s = b ∧ ∀ ⦃a : α⦄, a ∈ s → IsAto... | [
"α : Type u_2\ninst✝² : CompleteLattice α\ninst✝¹ : IsModularLattice α\ninst✝ : IsCompactlyGenerated α\nb : α\nh : sSup {a | a ≤ b ∧ IsAtom a} = b\ns : Set α\ns_ind : sSupIndep s\nleft✝ : Disjoint ⊥ (sSup s)\ns_atoms : ⊥ ⊔ sSup s = b ∧ ∀ ⦃a : α⦄, a ∈ s → IsAtom a\n⊢ sSup s = b ∧ ∀ ⦃a : α⦄, a ∈ s → IsAtom a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Multiset.Find | {
"line": 35,
"column": 4
} | {
"line": 35,
"column": 15
} | {
"line": 35,
"column": 16
} | [
{
"pp": "case refine_2\nα : Type u_1\np : α → Prop\ninst✝ : DecidablePred p\ns : Multiset α\nl₁ l₂ : List α\nh : l₁ ≈ l₂\nhp₁ : {x | x ∈ ⟦l₁⟧ ∧ p x}.Subsingleton\nhp₂ : {x | x ∈ ⟦l₂⟧ ∧ p x}.Subsingleton\nx✝ : hp₁ ≍ hp₂\n⊢ {x | x ∈ l₁ ∧ decide (p x) = true}.Subsingleton",
"ppTerm": "?refine_2",
"assigned... | [
"case refine_2\nα : Type u_1\np : α → Prop\ninst✝ : DecidablePred p\ns : Multiset α\nl₁ l₂ : List α\nh : l₁ ≈ l₂\nhp₁ : {x | x ∈ ⟦l₁⟧ ∧ p x}.Subsingleton\nhp₂ : {x | x ∈ ⟦l₂⟧ ∧ p x}.Subsingleton\nx✝ : hp₁ ≍ hp₂\n⊢ {x | x ∈ l₁ ∧ p x}.Subsingleton"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Span.Basic | {
"line": 562,
"column": 2
} | {
"line": 562,
"column": 13
} | {
"line": 562,
"column": 14
} | [
{
"pp": "R : Type u_1\nR₂ : Type u_2\nM : Type u_4\nM₂ : Type u_5\ninst✝⁶ : Semiring R\ninst✝⁵ : Semiring R₂\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : AddCommGroup M₂\ninst✝¹ : Module R₂ M₂\nτ₁₂ : R →+* R₂\ninst✝ : RingHomSurjective τ₁₂\nf : M →ₛₗ[τ₁₂] M₂\np : Submodule R M\nq : Submodule R₂ M₂\nl... | [
"R : Type u_1\nR₂ : Type u_2\nM : Type u_4\nM₂ : Type u_5\ninst✝⁶ : Semiring R\ninst✝⁵ : Semiring R₂\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : AddCommGroup M₂\ninst✝¹ : Module R₂ M₂\nτ₁₂ : R →+* R₂\ninst✝ : RingHomSurjective τ₁₂\nf : M →ₛₗ[τ₁₂] M₂\np : Submodule R M\nq : Submodule R₂ M₂\nle : comap f ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.BigOperators.Ring.Finset | {
"line": 35,
"column": 2
} | {
"line": 35,
"column": 13
} | {
"line": 35,
"column": 14
} | [
{
"pp": "ι : Type u_1\nM : Type u_3\ns : Finset ι\ninst✝¹ : CommMonoid M\ninst✝ : HasDistribNeg M\nf : ι → M\n⊢ ∏ x ∈ s, -f x = (-1) ^ #s * ∏ x ∈ s, f x",
"ppTerm": "?m.28",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"ι : Type u_1\nM : Type u_3\ns : Finset ι\ninst✝¹ : CommMonoid M\ninst✝ : HasDistribNeg M\nf : ι → M\n⊢ ∏ x ∈ s, -f x = (-1) ^ #s * ∏ x ∈ s, f x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Multiset.Find | {
"line": 45,
"column": 2
} | {
"line": 45,
"column": 13
} | {
"line": 45,
"column": 14
} | [
{
"pp": "case h\nα : Type u_1\np : α → Prop\ninst✝ : DecidablePred p\na : α\nl : List α\nhp : {x | x ∈ ⟦l⟧ ∧ p x}.Subsingleton\n⊢ List.find? (fun a ↦ decide (p a)) l = some a → p a",
"ppTerm": "?h",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"case h\nα : Type u_1\np : α → Prop\ninst✝ : DecidablePred p\na : α\nl : List α\nhp : {x | x ∈ ⟦l⟧ ∧ p x}.Subsingleton\n⊢ List.find? (fun a ↦ decide (p a)) l = some a → p a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Span.Basic | {
"line": 578,
"column": 4
} | {
"line": 578,
"column": 56
} | {
"line": 578,
"column": 57
} | [
{
"pp": "R : Type u_1\nR₂ : Type u_2\nM : Type u_4\nM₂ : Type u_5\ninst✝⁶ : Semiring R\ninst✝⁵ : Semiring R₂\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : AddCommGroup M₂\ninst✝¹ : Module R₂ M₂\nτ₁₂ : R →+* R₂\ninst✝ : RingHomSurjective τ₁₂\nf : M →ₛₗ[τ₁₂] M₂\np : Submodule R₂ M₂\nhp : IsCoatom p\nh :... | [
"R : Type u_1\nR₂ : Type u_2\nM : Type u_4\nM₂ : Type u_5\ninst✝⁶ : Semiring R\ninst✝⁵ : Semiring R₂\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : AddCommGroup M₂\ninst✝¹ : Module R₂ M₂\nτ₁₂ : R →+* R₂\ninst✝ : RingHomSurjective τ₁₂\nf : M →ₛₗ[τ₁₂] M₂\np : Submodule R₂ M₂\nhp : IsCoatom p\nh : ¬comap f p ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Group.Indicator | {
"line": 213,
"column": 2
} | {
"line": 213,
"column": 13
} | {
"line": 213,
"column": 14
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝ : One β\ns : γ → Set α\nf : α → β\ni : α\nj : γ\nhj✝ : i ∈ s j\nj' : γ\nh : j' ≠ j\nhj : {i} ⊆ s j\nhj' : {i} ⊆ s j'\nhi : {i} ⊆ mulSupport f\nhs : ∀ ⦃i j : γ⦄, i ≠ j → ∀ ⦃x : Set α⦄, x ⊆ s i ∧ x ⊆ mulSupport f → x ⊆ s j ∧ x ⊆ mulSupport f → x ⊆ ⊥\n⊢ Fals... | [
"α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝ : One β\ns : γ → Set α\nf : α → β\ni : α\nj : γ\nhj✝ : i ∈ s j\nj' : γ\nh : j' ≠ j\nhj : {i} ⊆ s j\nhj' : {i} ⊆ s j'\nhi : {i} ⊆ mulSupport f\nhs : ∀ ⦃i j : γ⦄, i ≠ j → ∀ ⦃x : Set α⦄, x ⊆ s i ∧ x ⊆ mulSupport f → x ⊆ s j ∧ x ⊆ mulSupport f → x ⊆ ⊥\n⊢ False"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Span.Basic | {
"line": 649,
"column": 4
} | {
"line": 649,
"column": 26
} | {
"line": 649,
"column": 27
} | [
{
"pp": "case inl\nR : Type u_1\nR₂ : Type u_2\nM : Type u_4\nM₂ : Type u_5\ninst✝⁶ : Ring R\ninst✝⁵ : Semiring R₂\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : AddCommGroup M₂\ninst✝¹ : Module R₂ M₂\nτ₁₂ : R →+* R₂\ninst✝ : RingHomSurjective τ₁₂\np p' : Submodule R M\nf : M →ₛₗ[τ₁₂] M₂\nhab : p < p'\... | [
"case inl\nR : Type u_1\nR₂ : Type u_2\nM : Type u_4\nM₂ : Type u_5\ninst✝⁶ : Ring R\ninst✝⁵ : Semiring R₂\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : AddCommGroup M₂\ninst✝¹ : Module R₂ M₂\nτ₁₂ : R →+* R₂\ninst✝ : RingHomSurjective τ₁₂\np p' : Submodule R M\nf : M →ₛₗ[τ₁₂] M₂\nhab : p < p'\nh : p ⊓ f.k... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.BigOperators.Ring.Finset | {
"line": 151,
"column": 6
} | {
"line": 152,
"column": 59
} | {
"line": 152,
"column": 60
} | [
{
"pp": "case insert\nι : Type u_1\nR : Type u_4\ninst✝¹ : CommSemiring R\ninst✝ : DecidableEq ι\nκ : ι → Type u_5\nt : (i : ι) → Finset (κ i)\nf : (i : ι) → κ i → R\na : ι\ns : Finset ι\nha : a ∉ s\nih : ∏ a ∈ s, ∑ b ∈ t a, f a b = ∑ p ∈ s.pi t, ∏ x ∈ s.attach, f (↑x) (p ↑x ⋯)\nh₁ : ∀ x ∈ t a, ∀ y ∈ t a, x ≠ y... | [
"case insert\nι : Type u_1\nR : Type u_4\ninst✝¹ : CommSemiring R\ninst✝ : DecidableEq ι\nκ : ι → Type u_5\nt : (i : ι) → Finset (κ i)\nf : (i : ι) → κ i → R\na : ι\ns : Finset ι\nha : a ∉ s\nih : ∏ a ∈ s, ∑ b ∈ t a, f a b = ∑ p ∈ s.pi t, ∏ x ∈ s.attach, f (↑x) (p ↑x ⋯)\nh₁ : ∀ x ∈ t a, ∀ y ∈ t a, x ≠ y → Disjoint ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Span.Basic | {
"line": 677,
"column": 4
} | {
"line": 677,
"column": 45
} | {
"line": 677,
"column": 46
} | [
{
"pp": "K : Type u_3\nV : Type u_6\ninst✝² : DivisionRing K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nx : V\ns q : Submodule K V\nhpq : s < q\nhqp : q < K ∙ x ⊔ s\ny : V\nhyq : y ∈ q\nhyp : y ∉ s\n⊢ ∃ c, ∃ z ∈ s, c • x + z = y",
"ppTerm": "?m.76",
"assigned": false,
"usedConstants": [],
"us... | [
"K : Type u_3\nV : Type u_6\ninst✝² : DivisionRing K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nx : V\ns q : Submodule K V\nhpq : s < q\nhqp : q < K ∙ x ⊔ s\ny : V\nhyq : y ∈ q\nhyp : y ∉ s\n⊢ ∃ c, ∃ z ∈ s, c • x + z = y"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Finsupp.Defs | {
"line": 366,
"column": 2
} | {
"line": 366,
"column": 46
} | {
"line": 366,
"column": 47
} | [
{
"pp": "α : Type u_1\nM : Type u_4\nN : Type u_5\ninst✝¹ : Zero M\ninst✝ : Zero N\ne : M → N\nhe₀ : e 0 = 0\nhe : Injective e\na b : α →₀ M\nh : ∀ (a_1 : α), (mapRange e he₀ a) a_1 = (mapRange e he₀ b) a_1\n⊢ ∀ (a_1 : α), a a_1 = b a_1",
"ppTerm": "?m.35",
"assigned": false,
"usedConstants": [],
... | [
"α : Type u_1\nM : Type u_4\nN : Type u_5\ninst✝¹ : Zero M\ninst✝ : Zero N\ne : M → N\nhe₀ : e 0 = 0\nhe : Injective e\na b : α →₀ M\nh : ∀ (a_1 : α), (mapRange e he₀ a) a_1 = (mapRange e he₀ b) a_1\n⊢ ∀ (a_1 : α), a a_1 = b a_1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Span.Basic | {
"line": 681,
"column": 11
} | {
"line": 681,
"column": 42
} | {
"line": 681,
"column": 43
} | [
{
"pp": "K : Type u_3\nV : Type u_6\ninst✝² : DivisionRing K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nx : V\ns q : Submodule K V\nhpq : s < q\nhqp : q < K ∙ x ⊔ s\nc : K\nz : V\nhz : z ∈ s\nhyq : c • x + z ∈ q\nhyp : c • x + z ∉ s\nhc : c ≠ 0\n⊢ x ∈ q",
"ppTerm": "?m.151",
"assigned": true,
"us... | [
"K : Type u_3\nV : Type u_6\ninst✝² : DivisionRing K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nx : V\ns q : Submodule K V\nhpq : s < q\nhqp : q < K ∙ x ⊔ s\nc : K\nz : V\nhz : z ∈ s\nhyq : c • x ∈ q\nhyp : c • x + z ∉ s\nhc : c ≠ 0\n⊢ x ∈ q"
] | q.add_mem_iff_left (hpq.le hz), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Span.Basic | {
"line": 689,
"column": 2
} | {
"line": 690,
"column": 25
} | {
"line": 690,
"column": 26
} | [
{
"pp": "K : Type u_3\nV : Type u_6\ninst✝² : DivisionRing K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\ns : Submodule K V\nx : V\n⊢ Disjoint s (K ∙ x) ↔ x ∈ s → x = 0",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule",
"False",
"_private.Mathli... | [
"K : Type u_3\nV : Type u_6\ninst✝² : DivisionRing K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\ns : Submodule K V\nx : V\n⊢ (∀ (r : K), ¬r = 0 → x ∈ s → x = 0) ↔ x ∈ s → x = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Span.Basic | {
"line": 702,
"column": 2
} | {
"line": 702,
"column": 61
} | {
"line": 702,
"column": 62
} | [
{
"pp": "K : Type u_3\nV : Type u_6\ninst✝² : DivisionRing K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\ns : Submodule K V\nx : V\nhs : s ⋖ ⊤\nhx : x ∉ s\n⊢ Codisjoint s (K ∙ x)",
"ppTerm": "?m.46",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule",
"Codisjoint",
"... | [
"K : Type u_3\nV : Type u_6\ninst✝² : DivisionRing K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\ns : Submodule K V\nx : V\nhs : s ⋖ ⊤\nhx : x ∉ s\n⊢ s ⊔ K ∙ x = ⊤"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Span.Basic | {
"line": 754,
"column": 18
} | {
"line": 754,
"column": 29
} | {
"line": 754,
"column": 30
} | [
{
"pp": "R : Type u_1\nM : Type u_4\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nx✝¹ x✝ : M\neq : toSpanSingleton R M x✝¹ = toSpanSingleton R M x✝\n⊢ x✝¹ = x✝",
"ppTerm": "?m.11",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u_1\nM : Type u_4\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nx✝¹ x✝ : M\neq : toSpanSingleton R M x✝¹ = toSpanSingleton R M x✝\n⊢ x✝¹ = x✝"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Finsupp.Defs | {
"line": 444,
"column": 33
} | {
"line": 444,
"column": 72
} | {
"line": 444,
"column": 73
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nM : Type u_4\ninst✝ : Zero M\nf : α ↪ β\nl₁ l₂ : α →₀ M\nh : embDomain f l₁ = embDomain f l₂\na : α\n⊢ l₁ a = l₂ a",
"ppTerm": "?m.18",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\nβ : Type u_2\nM : Type u_4\ninst✝ : Zero M\nf : α ↪ β\nl₁ l₂ : α →₀ M\nh : embDomain f l₁ = embDomain f l₂\na : α\n⊢ l₁ a = l₂ a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Finsupp.Single | {
"line": 290,
"column": 4
} | {
"line": 290,
"column": 65
} | {
"line": 291,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst✝ : Zero M\nf✝ : α →₀ M\na✝ : α\nb✝ : M\ni : α\nf : α →₀ M\na : α\nb : M\nthis✝ : DecidableEq α\nthis : DecidableEq M\n⊢ Finset... | [] | exact if b = 0 then f.support.erase a else insert a f.support | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.BigOperators.Ring.Finset | {
"line": 305,
"column": 2
} | {
"line": 305,
"column": 35
} | {
"line": 305,
"column": 36
} | [
{
"pp": "ι : Type u_5\nR : Type u_7\ninst✝² : Fintype ι\ninst✝¹ : CommSemiring R\ninst✝ : DecidableEq ι\nf g : ι → R\n⊢ ∏ a, (f a + g a) = ∑ t, (∏ a ∈ t, f a) * ∏ a ∈ tᶜ, g a",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"Finset.fintype",
"Eq.mpr",
"HMul.hMul",
"F... | [
"ι : Type u_5\nR : Type u_7\ninst✝² : Fintype ι\ninst✝¹ : CommSemiring R\ninst✝ : DecidableEq ι\nf g : ι → R\n⊢ ∏ a, (f a + g a) = ∑ x, (∏ a ∈ x, f a) * ∏ a ∈ univ \\ x, g a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.BigOperators.Group.Finset.Preimage | {
"line": 33,
"column": 34
} | {
"line": 33,
"column": 71
} | {
"line": 33,
"column": 72
} | [
{
"pp": "ι : Type u_1\nκ : Type u_2\nβ : Type u_3\ninst✝¹ : CommMonoid β\nf : ι → κ\ninst✝ : DecidablePred fun x ↦ x ∈ Set.range f\ns : Finset κ\nhf : Set.InjOn f (f ⁻¹' ↑s)\ng : κ → β\n⊢ Set.InjOn f ↑(s.preimage f hf)",
"ppTerm": "?m.61",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"... | [
"ι : Type u_1\nκ : Type u_2\nβ : Type u_3\ninst✝¹ : CommMonoid β\nf : ι → κ\ninst✝ : DecidablePred fun x ↦ x ∈ Set.range f\ns : Finset κ\nhf : Set.InjOn f (f ⁻¹' ↑s)\ng : κ → β\n⊢ ∀ ⦃x₁ : ι⦄, f x₁ ∈ s → ∀ ⦃x₂ : ι⦄, f x₂ ∈ s → f x₁ = f x₂ → x₁ = x₂"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Group.Indicator | {
"line": 71,
"column": 4
} | {
"line": 71,
"column": 20
} | {
"line": 71,
"column": 21
} | [
{
"pp": "case pos\nα : Type u_2\nM : Type u_3\ninst✝¹ : LE M\ninst✝ : One M\ns : Set α\nf : α → M\na : α\ny : M\nhfg : a ∈ s → f a ≤ y\nhg : a ∉ s → 1 ≤ y\nha : a ∈ s\n⊢ s.mulIndicator f a ≤ y",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Set.mulI... | [
"case pos\nα : Type u_2\nM : Type u_3\ninst✝¹ : LE M\ninst✝ : One M\ns : Set α\nf : α → M\na : α\ny : M\nhfg : a ∈ s → f a ≤ y\nhg : a ∉ s → 1 ≤ y\nha : a ∈ s\n⊢ f a ≤ y"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Group.Indicator | {
"line": 72,
"column": 4
} | {
"line": 72,
"column": 20
} | {
"line": 72,
"column": 21
} | [
{
"pp": "case neg\nα : Type u_2\nM : Type u_3\ninst✝¹ : LE M\ninst✝ : One M\ns : Set α\nf : α → M\na : α\ny : M\nhfg : a ∈ s → f a ≤ y\nhg : a ∉ s → 1 ≤ y\nha : a ∉ s\n⊢ s.mulIndicator f a ≤ y",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"False",
"eq_false",
... | [
"case neg\nα : Type u_2\nM : Type u_3\ninst✝¹ : LE M\ninst✝ : One M\ns : Set α\nf : α → M\na : α\ny : M\nhfg : a ∈ s → f a ≤ y\nhg : a ∉ s → 1 ≤ y\nha : a ∉ s\n⊢ 1 ≤ y"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Group.Indicator | {
"line": 182,
"column": 4
} | {
"line": 182,
"column": 43
} | {
"line": 182,
"column": 44
} | [
{
"pp": "case neg\nι : Sort u_1\nα : Type u_2\nM : Type u_3\ninst✝² : CompleteLattice M\ninst✝¹ : One M\ninst✝ : Nonempty ι\nh1 : ⊥ = 1\ns : ι → Set α\nf : α → M\nx : α\nj : ι\nhj : x ∉ s j\n⊢ ⨅ i, (s i).mulIndicator f x ≤ 1",
"ppTerm": "?neg✝",
"assigned": false,
"usedConstants": [],
"usedFVars... | [
"case neg\nι : Sort u_1\nα : Type u_2\nM : Type u_3\ninst✝² : CompleteLattice M\ninst✝¹ : One M\ninst✝ : Nonempty ι\nh1 : ⊥ = 1\ns : ι → Set α\nf : α → M\nx : α\nj : ι\nhj : x ∉ s j\n⊢ ⨅ i, (s i).mulIndicator f x ≤ 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Group.Indicator | {
"line": 193,
"column": 40
} | {
"line": 193,
"column": 51
} | {
"line": 193,
"column": 52
} | [
{
"pp": "α : Type u_2\nM : Type u_3\ninst✝³ : CompleteLattice M\ninst✝² : One M\nι : Type u_4\ninst✝¹ : Preorder ι\ninst✝ : IsDirectedOrder ι\nf : ι → α → M\ns : ι → Set α\nh1 : ⊥ = 1\nhf : Monotone f\nhs : Monotone s\na : α\nha : a ∈ ⋃ i, s i\n⊢ ∃ i, a ∈ s i",
"ppTerm": "?m.132",
"assigned": false,
... | [
"α : Type u_2\nM : Type u_3\ninst✝³ : CompleteLattice M\ninst✝² : One M\nι : Type u_4\ninst✝¹ : Preorder ι\ninst✝ : IsDirectedOrder ι\nf : ι → α → M\ns : ι → Set α\nh1 : ⊥ = 1\nhf : Monotone f\nhs : Monotone s\na : α\nha : a ∈ ⋃ i, s i\n⊢ ∃ i, a ∈ s i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.BigOperators.Finsupp.Basic | {
"line": 326,
"column": 6
} | {
"line": 326,
"column": 43
} | {
"line": 326,
"column": 44
} | [
{
"pp": "case neg\nα : Type u_1\nd : α →₀ ℕ\nh1 : (d.sum fun x n ↦ n) = 1\nhd0 : d ≠ 0\na : α\nha : d a ≠ 0 a\nhda : d a = 1\nhda' : ∀ (i : α), i ≠ a → d i = 0\nb : α\nhb : ¬b = a\n⊢ d b = (single a 1) b",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"Finsupp.instFunLike",
"Eq... | [
"case neg\nα : Type u_1\nd : α →₀ ℕ\nh1 : (d.sum fun x n ↦ n) = 1\nhd0 : d ≠ 0\na : α\nha : d a ≠ 0 a\nhda : d a = 1\nhda' : ∀ (i : α), i ≠ a → d i = 0\nb : α\nhb : ¬b = a\n⊢ d b = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.BigOperators.Group.Multiset | {
"line": 33,
"column": 40
} | {
"line": 33,
"column": 51
} | {
"line": 33,
"column": 52
} | [
{
"pp": "α : Type u_2\ninst✝² : CommMonoid α\ninst✝¹ : Preorder α\ns : Multiset α\ninst✝ : MulLeftMono α\nl : List α\nhl : ∀ (x : α), x ∈ ⟦l⟧ → 1 ≤ x\n⊢ 1 ≤ prod ⟦l⟧",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"MulOne.toOne",
"Monoid.toMulOneClass",
"Multiset.prod",
... | [
"α : Type u_2\ninst✝² : CommMonoid α\ninst✝¹ : Preorder α\ns : Multiset α\ninst✝ : MulLeftMono α\nl : List α\nhl : ∀ (x : α), x ∈ ⟦l⟧ → 1 ≤ x\n⊢ 1 ≤ l.prod"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.BigOperators.Group.Multiset | {
"line": 37,
"column": 45
} | {
"line": 37,
"column": 56
} | {
"line": 37,
"column": 57
} | [
{
"pp": "α : Type u_2\ninst✝² : CommMonoid α\ninst✝¹ : Preorder α\ns : Multiset α\ninst✝ : IsOrderedMonoid α\nl : List α\nhl : ∀ (x : α), x ∈ ⟦l⟧ → 1 ≤ x\nx : α\nhx : x ∈ ⟦l⟧\n⊢ x ≤ prod ⟦l⟧",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Multiset.prod",
"Preorder.toLE",
... | [
"α : Type u_2\ninst✝² : CommMonoid α\ninst✝¹ : Preorder α\ns : Multiset α\ninst✝ : IsOrderedMonoid α\nl : List α\nhl : ∀ (x : α), x ∈ ⟦l⟧ → 1 ≤ x\nx : α\nhx : x ∈ ⟦l⟧\n⊢ x ≤ l.prod"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.BigOperators.Group.Multiset | {
"line": 43,
"column": 2
} | {
"line": 43,
"column": 13
} | {
"line": 43,
"column": 14
} | [
{
"pp": "case h\nα : Type u_2\ninst✝² : CommMonoid α\ninst✝¹ : Preorder α\ninst✝ : MulLeftMono α\nn : α\na✝ : List α\nh : ∀ (x : α), x ∈ ⟦a✝⟧ → x ≤ n\n⊢ prod ⟦a✝⟧ ≤ n ^ card ⟦a✝⟧",
"ppTerm": "?h",
"assigned": true,
"usedConstants": [
"Multiset.prod",
"Preorder.toLE",
"id",
"L... | [
"case h\nα : Type u_2\ninst✝² : CommMonoid α\ninst✝¹ : Preorder α\ninst✝ : MulLeftMono α\nn : α\na✝ : List α\nh : ∀ (x : α), x ∈ ⟦a✝⟧ → x ≤ n\n⊢ a✝.prod ≤ n ^ a✝.length"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.BigOperators.Group.Multiset | {
"line": 107,
"column": 10
} | {
"line": 107,
"column": 21
} | {
"line": 107,
"column": 22
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\ninst✝³ : CommMonoid α\ninst✝² : CommMonoid β\ninst✝¹ : Preorder β\ninst✝ : IsOrderedMonoid β\nf : α → β\np : α → Prop\nh_mul : ∀ (a b : α), p a → p b → f (a * b) ≤ f a * f b\nhp_mul : ∀ (a b : α), p a → p b → p (a * b)\nl : List α\nhs_nonempty : ⟦l⟧ ≠ ∅\nhs : ∀ (a : α), a ∈ ... | [
"α : Type u_2\nβ : Type u_3\ninst✝³ : CommMonoid α\ninst✝² : CommMonoid β\ninst✝¹ : Preorder β\ninst✝ : IsOrderedMonoid β\nf : α → β\np : α → Prop\nh_mul : ∀ (a b : α), p a → p b → f (a * b) ≤ f a * f b\nhp_mul : ∀ (a b : α), p a → p b → p (a * b)\nl : List α\nhs_nonempty : ⟦l⟧ ≠ ∅\nhs : ∀ (a : α), a ∈ ⟦l⟧ → p a\n⊢... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.BigOperators.Group.Multiset | {
"line": 113,
"column": 69
} | {
"line": 113,
"column": 80
} | {
"line": 113,
"column": 81
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\ninst✝³ : CommMonoid α\ninst✝² : CommMonoid β\ninst✝¹ : Preorder β\ninst✝ : IsOrderedMonoid β\nf : α → β\nh_mul : ∀ (a b : α), f (a * b) ≤ f a * f b\nl : List α\nhs_nonempty : ⟦l⟧ ≠ ∅\n⊢ l ≠ ∅",
"ppTerm": "?m.47",
"assigned": true,
"usedConstants": [
"List.i... | [
"α : Type u_2\nβ : Type u_3\ninst✝³ : CommMonoid α\ninst✝² : CommMonoid β\ninst✝¹ : Preorder β\ninst✝ : IsOrderedMonoid β\nf : α → β\nh_mul : ∀ (a b : α), f (a * b) ≤ f a * f b\nl : List α\nhs_nonempty : ⟦l⟧ ≠ ∅\n⊢ ¬l = []"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.BigOperators.Group.Multiset | {
"line": 141,
"column": 36
} | {
"line": 141,
"column": 47
} | {
"line": 141,
"column": 48
} | [
{
"pp": "α : Type u_2\ninst✝³ : CommMonoid α\nm : Multiset α\ninst✝² : PartialOrder α\ninst✝¹ : CanonicallyOrderedMul α\ninst✝ : IsOrderedMonoid α\nl : List α\n⊢ prod ⟦l⟧ = 1 ↔ ∀ (x : α), x ∈ ⟦l⟧ → x = 1",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
... | [
"α : Type u_2\ninst✝³ : CommMonoid α\nm : Multiset α\ninst✝² : PartialOrder α\ninst✝¹ : CanonicallyOrderedMul α\ninst✝ : IsOrderedMonoid α\nl : List α\n⊢ l.prod = 1 ↔ ∀ (x : α), x ∈ l → x = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.AbsoluteValue.Basic | {
"line": 138,
"column": 2
} | {
"line": 138,
"column": 41
} | {
"line": 138,
"column": 42
} | [
{
"pp": "R : Type u_5\nS : Type u_6\ninst✝² : Ring R\ninst✝¹ : Semiring S\ninst✝ : PartialOrder S\nabv : AbsoluteValue R S\na b c : R\n⊢ abv (a - c) ≤ abv (a - b) + abv (b - c)",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"AddGroupWithOne.toAddGroup",
"congrA... | [
"R : Type u_5\nS : Type u_6\ninst✝² : Ring R\ninst✝¹ : Semiring S\ninst✝ : PartialOrder S\nabv : AbsoluteValue R S\na b c : R\n⊢ abv (a + -c) ≤ abv (a + -b) + abv (b + -c)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.BigOperators.Group.Multiset | {
"line": 154,
"column": 2
} | {
"line": 154,
"column": 13
} | {
"line": 154,
"column": 14
} | [
{
"pp": "case h\nα : Type u_4\ninst✝¹ : LinearOrder α\ninst✝ : OrderBot α\nn : α\na✝ : List α\nh : ∀ (x : α), x ∈ ⟦a✝⟧ → x ≤ n\n⊢ fold max ⊥ ⟦a✝⟧ ≤ n",
"ppTerm": "?h",
"assigned": true,
"usedConstants": [
"instCommutativeMax",
"Lattice.toSemilatticeSup",
"instAssociativeMax",
... | [
"case h\nα : Type u_4\ninst✝¹ : LinearOrder α\ninst✝ : OrderBot α\nn : α\na✝ : List α\nh : ∀ (x : α), x ∈ ⟦a✝⟧ → x ≤ n\n⊢ List.foldr max ⊥ a✝ ≤ n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.AbsoluteValue.Basic | {
"line": 209,
"column": 28
} | {
"line": 209,
"column": 39
} | {
"line": 209,
"column": 40
} | [
{
"pp": "R : Type u_5\nS : Type u_6\ninst✝³ : Ring R\ninst✝² : Ring S\ninst✝¹ : PartialOrder S\ninst✝ : IsOrderedRing S\nabv : AbsoluteValue R S\na b : R\n⊢ abv a ≤ abv (a - b) + abv b",
"ppTerm": "?m.28",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u_5\nS : Type u_6\ninst✝³ : Ring R\ninst✝² : Ring S\ninst✝¹ : PartialOrder S\ninst✝ : IsOrderedRing S\nabv : AbsoluteValue R S\na b : R\n⊢ abv a ≤ abv (a - b) + abv b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.AbsoluteValue.Basic | {
"line": 231,
"column": 2
} | {
"line": 231,
"column": 73
} | {
"line": 231,
"column": 74
} | [
{
"pp": "R : Type u_3\nS : Type u_4\ninst✝⁴ : CommRing S\ninst✝³ : PartialOrder S\ninst✝² : IsOrderedRing S\ninst✝¹ : Ring R\nabv : AbsoluteValue R S\ninst✝ : NoZeroDivisors S\na b : R\n⊢ abv a - abv b ≤ abv (a + b)",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Add... | [
"R : Type u_3\nS : Type u_4\ninst✝⁴ : CommRing S\ninst✝³ : PartialOrder S\ninst✝² : IsOrderedRing S\ninst✝¹ : Ring R\nabv : AbsoluteValue R S\ninst✝ : NoZeroDivisors S\na b : R\n⊢ abv a ≤ abv (a + b) + abv b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.AbsoluteValue.Basic | {
"line": 236,
"column": 2
} | {
"line": 236,
"column": 60
} | {
"line": 236,
"column": 61
} | [
{
"pp": "R : Type u_3\nS : Type u_4\ninst✝⁴ : CommRing S\ninst✝³ : PartialOrder S\ninst✝² : IsOrderedRing S\ninst✝¹ : Ring R\nabv : AbsoluteValue R S\ninst✝ : NoZeroDivisors S\na b : R\n⊢ abv (a - b) ≤ abv a + abv b",
"ppTerm": "?m.21",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
... | [
"R : Type u_3\nS : Type u_4\ninst✝⁴ : CommRing S\ninst✝³ : PartialOrder S\ninst✝² : IsOrderedRing S\ninst✝¹ : Ring R\nabv : AbsoluteValue R S\ninst✝ : NoZeroDivisors S\na b : R\n⊢ abv (a - b) ≤ abv a + abv b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.AbsoluteValue.Basic | {
"line": 493,
"column": 2
} | {
"line": 493,
"column": 41
} | {
"line": 493,
"column": 42
} | [
{
"pp": "S : Type u_5\ninst✝³ : Ring S\ninst✝² : PartialOrder S\nR : Type u_6\ninst✝¹ : Ring R\nabv : R → S\ninst✝ : IsAbsoluteValue abv\na b c : R\n⊢ abv (a - c) ≤ abv (a - b) + abv (b - c)",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"AddGroupWithOne.toAddGroup",... | [
"S : Type u_5\ninst✝³ : Ring S\ninst✝² : PartialOrder S\nR : Type u_6\ninst✝¹ : Ring R\nabv : R → S\ninst✝ : IsAbsoluteValue abv\na b c : R\n⊢ abv (a + -c) ≤ abv (a + -b) + abv (b + -c)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.BigOperators.GroupWithZero.List | {
"line": 25,
"column": 2
} | {
"line": 30,
"column": 35
} | {
"line": 33,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝³ : CommMonoidWithZero R\ninst✝² : PartialOrder R\ninst✝¹ : ZeroLEOneClass R\ninst✝ : PosMulMono R\ns : List R\nh : ∀ (a : R), a ∈ s → 0 ≤ a\n⊢ 0 ≤ s.prod",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"mul_nonneg",
"MulOne.toOne",
"MulZeroCla... | [] | induction s with
| nil => simp
| cons head tail hind =>
simp only [prod_cons]
simp only [mem_cons, forall_eq_or_imp] at h
exact mul_nonneg h.1 (hind h.2) | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.Algebra.Order.BigOperators.GroupWithZero.List | {
"line": 25,
"column": 2
} | {
"line": 30,
"column": 35
} | {
"line": 33,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝³ : CommMonoidWithZero R\ninst✝² : PartialOrder R\ninst✝¹ : ZeroLEOneClass R\ninst✝ : PosMulMono R\ns : List R\nh : ∀ (a : R), a ∈ s → 0 ≤ a\n⊢ 0 ≤ s.prod",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"mul_nonneg",
"MulOne.toOne",
"MulZeroCla... | [] | induction s with
| nil => simp
| cons head tail hind =>
simp only [prod_cons]
simp only [mem_cons, forall_eq_or_imp] at h
exact mul_nonneg h.1 (hind h.2) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Order.BigOperators.GroupWithZero.List | {
"line": 25,
"column": 2
} | {
"line": 30,
"column": 35
} | {
"line": 33,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝³ : CommMonoidWithZero R\ninst✝² : PartialOrder R\ninst✝¹ : ZeroLEOneClass R\ninst✝ : PosMulMono R\ns : List R\nh : ∀ (a : R), a ∈ s → 0 ≤ a\n⊢ 0 ≤ s.prod",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"mul_nonneg",
"MulOne.toOne",
"MulZeroCla... | [] | induction s with
| nil => simp
| cons head tail hind =>
simp only [prod_cons]
simp only [mem_cons, forall_eq_or_imp] at h
exact mul_nonneg h.1 (hind h.2) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Order.BigOperators.GroupWithZero.Finset | {
"line": 56,
"column": 2
} | {
"line": 56,
"column": 13
} | {
"line": 56,
"column": 14
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\ninst✝³ : CommMonoidWithZero R\ninst✝² : Preorder R\ninst✝¹ : ZeroLEOneClass R\ninst✝ : PosMulMono R\nf : ι → R\ns : Finset ι\nhf : ∀ i ∈ s, 1 ≤ f i\n⊢ 1 ≤ ∏ i ∈ s, f i",
"ppTerm": "?m.24",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoal... | [
"ι : Type u_1\nR : Type u_2\ninst✝³ : CommMonoidWithZero R\ninst✝² : Preorder R\ninst✝¹ : ZeroLEOneClass R\ninst✝ : PosMulMono R\nf : ι → R\ns : Finset ι\nhf : ∀ i ∈ s, 1 ≤ f i\n⊢ 1 ≤ ∏ i ∈ s, f i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.BigOperators.Ring.List | {
"line": 26,
"column": 67
} | {
"line": 26,
"column": 97
} | {
"line": 27,
"column": 4
} | [
{
"pp": "α : Type u_2\ninst✝⁴ : CommSemiring α\ninst✝³ : PartialOrder α\ninst✝² : CanonicallyOrderedAdd α\ninst✝¹ : NoZeroDivisors α\ninst✝ : Nontrivial α\nx : α\nxs : List α\n⊢ 0 < x * xs.prod ↔ 0 < x ∧ ∀ (x : α), x ∈ xs → 0 < x",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants": [
"Eq.mp... | [
"α : Type u_2\ninst✝⁴ : CommSemiring α\ninst✝³ : PartialOrder α\ninst✝² : CanonicallyOrderedAdd α\ninst✝¹ : NoZeroDivisors α\ninst✝ : Nontrivial α\nx : α\nxs : List α\n⊢ 0 < x ∧ 0 < xs.prod ↔ 0 < x ∧ ∀ (x : α), x ∈ xs → 0 < x"
] | CanonicallyOrderedAdd.mul_pos, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Algebra.Order.BigOperators.Group.Finset | {
"line": 210,
"column": 24
} | {
"line": 212,
"column": 27
} | {
"line": 214,
"column": 0
} | [
{
"pp": "ι : Type u_1\nN : Type u_5\ninst✝² : CommMonoid N\ninst✝¹ : Preorder N\nf : ι → N\ns : Finset ι\ninst✝ : MulLeftMono N\ni j : ι\nhf : ∀ i ∈ s, 1 ≤ f i\nhi : i ∈ s\nhj : j ∈ s\nhne : i ≠ j\n⊢ ∏ k ∈ cons i {j} ⋯, f k ≤ ∏ k ∈ s, f k",
"ppTerm": "?m.54",
"assigned": true,
"usedConstants": [
... | [] | by
refine prod_le_prod_of_subset_of_one_le' ?_ fun k hk _ ↦ hf k hk
simp [cons_subset, *] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Order.BigOperators.Group.Finset | {
"line": 218,
"column": 4
} | {
"line": 218,
"column": 15
} | {
"line": 218,
"column": 16
} | [
{
"pp": "case refine_1\nι : Type u_1\nN : Type u_5\ninst✝² : CommMonoid N\ninst✝¹ : Preorder N\ninst✝ : MulLeftMono N\ns : Finset ι\nf : ι → N\nn : N\nh : ∀ x ∈ s, f x ≤ n\n⊢ ∀ x ∈ Multiset.map f s.val, x ≤ n",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Multis... | [
"case refine_1\nι : Type u_1\nN : Type u_5\ninst✝² : CommMonoid N\ninst✝¹ : Preorder N\ninst✝ : MulLeftMono N\ns : Finset ι\nf : ι → N\nn : N\nh : ∀ x ∈ s, f x ≤ n\n⊢ ∀ a ∈ s, f a ≤ n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Finsupp.Basic | {
"line": 336,
"column": 17
} | {
"line": 336,
"column": 42
} | {
"line": 336,
"column": 43
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nM : Type u_5\ninst✝ : AddCommMonoid M\nf : α ↪ β\nx : α →₀ M\na : α\nh : ¬(mapDomain (⇑f) x) (f a) = 0\n⊢ ¬x a = 0 ∧ f a = f a",
"ppTerm": "?m.60",
"assigned": true,
"usedConstants": [
"Finsupp.instFunLike",
"Eq.mpr",
"and_true",
"congrArg... | [
"α : Type u_1\nβ : Type u_2\nM : Type u_5\ninst✝ : AddCommMonoid M\nf : α ↪ β\nx : α →₀ M\na : α\nh : ¬(mapDomain (⇑f) x) (f a) = 0\n⊢ ¬x a = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Finsupp.Basic | {
"line": 376,
"column": 13
} | {
"line": 376,
"column": 25
} | {
"line": 376,
"column": 25
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nM : Type u_5\ninst✝ : AddCommMonoid M\nS : Set α\nf : α → β\nx : α →₀ M\nhS : ↑x.support ⊆ S\nhf : Set.InjOn f S\na : α\nha : a ∈ S\n⊢ ∑ a_1 ∈ x.support, (single (f a_1) (x a_1)) (f a) = x a",
"ppTerm": "?m.48",
"assigned": true,
"usedConstants": [
"Finsupp... | [
"α : Type u_1\nβ : Type u_2\nM : Type u_5\ninst✝ : AddCommMonoid M\nS : Set α\nf : α → β\nx : α →₀ M\nhS : ↑x.support ⊆ S\nhf : Set.InjOn f S\na : α\nha : a ∈ S\n⊢ (∑ x_1 ∈ x.support, if f x_1 = f a then x x_1 else 0) = x a"
] | single_apply | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Algebra.Order.BigOperators.Group.Finset | {
"line": 534,
"column": 2
} | {
"line": 534,
"column": 31
} | {
"line": 534,
"column": 32
} | [
{
"pp": "ι : Type u_1\nM : Type u_4\ninst✝³ : CommMonoid M\ninst✝² : Preorder M\ninst✝¹ : IsOrderedCancelMonoid M\nf : ι → M\ns t : Finset ι\ninst✝ : DecidableEq ι\n⊢ Disjoint (t \\ s) (s ∩ t)",
"ppTerm": "?m.65",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"ι : Type u_1\nM : Type u_4\ninst✝³ : CommMonoid M\ninst✝² : Preorder M\ninst✝¹ : IsOrderedCancelMonoid M\nf : ι → M\ns t : Finset ι\ninst✝ : DecidableEq ι\n⊢ Disjoint (t \\ s) (s ∩ t)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.BigOperators.Group.Finset | {
"line": 541,
"column": 2
} | {
"line": 541,
"column": 31
} | {
"line": 541,
"column": 32
} | [
{
"pp": "ι : Type u_9\nM : Type u_10\ninst✝³ : CommMonoid M\ninst✝² : PartialOrder M\ninst✝¹ : IsOrderedCancelMonoid M\ninst✝ : DecidableEq ι\ns t : Finset ι\nf : ι → M\n⊢ Disjoint (t \\ s) (s ∩ t)",
"ppTerm": "?m.67",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
... | [
"ι : Type u_9\nM : Type u_10\ninst✝³ : CommMonoid M\ninst✝² : PartialOrder M\ninst✝¹ : IsOrderedCancelMonoid M\ninst✝ : DecidableEq ι\ns t : Finset ι\nf : ι → M\n⊢ Disjoint (t \\ s) (s ∩ t)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.BigOperators.Group.Finset | {
"line": 582,
"column": 2
} | {
"line": 582,
"column": 53
} | {
"line": 582,
"column": 54
} | [
{
"pp": "ι : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝⁴ : SemilatticeSup α\ninst✝³ : OrderBot α\ninst✝² : AddCommMonoid β\ninst✝¹ : Preorder β\ninst✝ : AddLeftMono β\nf : α → β\nzero : f ⊥ = 0\nih : ∀ {s t : α}, f (s ⊔ t) ≤ f s + f t\ns : ι → α\nt✝ : Finset ι\ni : ι\nt : Finset ι\nit : i ∉ t\nh : f (t.sup s) ... | [
"ι : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝⁴ : SemilatticeSup α\ninst✝³ : OrderBot α\ninst✝² : AddCommMonoid β\ninst✝¹ : Preorder β\ninst✝ : AddLeftMono β\nf : α → β\nzero : f ⊥ = 0\nih : ∀ {s t : α}, f (s ⊔ t) ≤ f s + f t\ns : ι → α\nt✝ : Finset ι\ni : ι\nt : Finset ι\nit : i ∉ t\nh : f (t.sup s) ≤ ∑ i ∈ t, f... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.BigOperators.Group.Finset | {
"line": 640,
"column": 48
} | {
"line": 640,
"column": 59
} | {
"line": 640,
"column": 60
} | [
{
"pp": "ι : Type u_1\nM : Type u_4\ninst✝³ : Fintype ι\ninst✝² : CommMonoid M\ninst✝¹ : PartialOrder M\ninst✝ : IsOrderedCancelMonoid M\nf : ι → M\nhf : 1 < f\n⊢ ∃ i ∈ Finset.univ, 1 < f i",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"Preorde... | [
"ι : Type u_1\nM : Type u_4\ninst✝³ : Fintype ι\ninst✝² : CommMonoid M\ninst✝¹ : PartialOrder M\ninst✝ : IsOrderedCancelMonoid M\nf : ι → M\nhf : 1 < f\n⊢ ∃ i, 1 < f i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.BigOperators.Group.Finset | {
"line": 644,
"column": 48
} | {
"line": 644,
"column": 59
} | {
"line": 644,
"column": 60
} | [
{
"pp": "ι : Type u_1\nM : Type u_4\ninst✝³ : Fintype ι\ninst✝² : CommMonoid M\ninst✝¹ : PartialOrder M\ninst✝ : IsOrderedCancelMonoid M\nf : ι → M\nhf : f < 1\n⊢ ∃ i ∈ Finset.univ, f i < 1",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"Preorde... | [
"ι : Type u_1\nM : Type u_4\ninst✝³ : Fintype ι\ninst✝² : CommMonoid M\ninst✝¹ : PartialOrder M\ninst✝ : IsOrderedCancelMonoid M\nf : ι → M\nhf : f < 1\n⊢ ∃ i, f i < 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Finsupp.Basic | {
"line": 662,
"column": 67
} | {
"line": 662,
"column": 79
} | {
"line": 662,
"column": 79
} | [
{
"pp": "α : Type u_1\nM : Type u_5\ninst✝¹ : Zero M\ninst✝ : DecidableEq α\nf : α →₀ M\na a✝ : α\n⊢ (if a = a✝ then f a✝ else 0) = (single a (f a)) a✝",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"Finsupp.instFunLike",
"Eq.mpr",
"congrArg",
"id",
"Zero.toO... | [
"α : Type u_1\nM : Type u_5\ninst✝¹ : Zero M\ninst✝ : DecidableEq α\nf : α →₀ M\na a✝ : α\n⊢ (if a = a✝ then f a✝ else 0) = if a = a✝ then f a else 0"
] | single_apply | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Finsupp.Basic | {
"line": 702,
"column": 2
} | {
"line": 702,
"column": 45
} | {
"line": 703,
"column": 2
} | [
{
"pp": "α : Type u_1\nM : Type u_5\nN : Type u_6\ninst✝² : Zero M\np : α → Prop\ninst✝¹ : DecidablePred p\nf : α →₀ M\ninst✝ : CommMonoid N\ng : α → M → N\n⊢ (filter p f).prod g = ∏ x ∈ (filter p f).support, g x (f x)",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Finsupp.instFunL... | [
"α : Type u_1\nM : Type u_5\nN : Type u_6\ninst✝² : Zero M\np : α → Prop\ninst✝¹ : DecidablePred p\nf : α →₀ M\ninst✝ : CommMonoid N\ng : α → M → N\nx : α\nhx : x ∈ (filter p f).support\n⊢ g x ((filter p f) x) = g x (f x)"
] | refine Finset.prod_congr rfl fun x hx => ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Data.Finsupp.SMul | {
"line": 119,
"column": 39
} | {
"line": 119,
"column": 50
} | {
"line": 119,
"column": 51
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nM : Type u_3\nN : Type u_4\nG : Type u_5\nR : Type u_6\ninst✝³ : Nonempty α\ninst✝² : Zero M\ninst✝¹ : SMulZeroClass R M\ninst✝ : FaithfulSMul R M\nm₁✝ m₂✝ : R\nh : ∀ (a : α →₀ M), m₁✝ • a = m₂✝ • a\na : α\nm : M\n⊢ m₁✝ • m = m₂✝ • m",
"ppTerm": "?m.19",
"assigned": ... | [
"α : Type u_1\nβ : Type u_2\nM : Type u_3\nN : Type u_4\nG : Type u_5\nR : Type u_6\ninst✝³ : Nonempty α\ninst✝² : Zero M\ninst✝¹ : SMulZeroClass R M\ninst✝ : FaithfulSMul R M\nm₁✝ m₂✝ : R\nh : ∀ (a : α →₀ M), m₁✝ • a = m₂✝ • a\na : α\nm : M\n⊢ m₁✝ • m = m₂✝ • m"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.BigOperators.Ring.Finset | {
"line": 132,
"column": 6
} | {
"line": 132,
"column": 59
} | {
"line": 132,
"column": 60
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : LinearOrder R\ninst✝¹ : IsStrictOrderedRing R\ninst✝ : ExistsAddOfLE R\ns : Finset ι\nr f g : ι → R\nhf : ∀ i ∈ s, 0 ≤ f i\nhg : ∀ i ∈ s, 0 ≤ g i\nht : ∀ i ∈ s, r i ^ 2 ≤ f i * g i\nh : ∑ i ∈ s, g i = 0\ni : ι\nhi : i ∈ s\n⊢ r i = 0",
"p... | [
"ι : Type u_1\nR : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : LinearOrder R\ninst✝¹ : IsStrictOrderedRing R\ninst✝ : ExistsAddOfLE R\ns : Finset ι\nr f g : ι → R\nhf : ∀ i ∈ s, 0 ≤ f i\nhg : ∀ i ∈ s, 0 ≤ g i\nht : ∀ i ∈ s, r i ^ 2 ≤ f i * g i\nh : ∑ i ∈ s, g i = 0\ni : ι\nhi : i ∈ s\n⊢ r i = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Finsupp.Basic | {
"line": 1421,
"column": 4
} | {
"line": 1421,
"column": 46
} | {
"line": 1421,
"column": 47
} | [
{
"pp": "case refine_1\nα : Type u_1\nβ : Type u_2\nM : Type u_5\ninst✝² : DecidableEq β\ninst✝¹ : AddCommMonoid M\nf : α → β\ninst✝ : Subsingleton (AddUnits M)\nx : α →₀ M\nt : β\n⊢ ¬(mapDomain f x) t = 0 → ∃ a ∈ x.support, f a = t",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
... | [
"case refine_1\nα : Type u_1\nβ : Type u_2\nM : Type u_5\ninst✝² : DecidableEq β\ninst✝¹ : AddCommMonoid M\nf : α → β\ninst✝ : Subsingleton (AddUnits M)\nx : α →₀ M\nt : β\n⊢ ∀ (x_1 : α), ¬x x_1 = 0 → f x_1 = t → ¬x x_1 = 0 → ∃ a, ¬x a = 0 ∧ f a = t"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Finsupp.Basic | {
"line": 1422,
"column": 2
} | {
"line": 1422,
"column": 50
} | {
"line": 1422,
"column": 51
} | [
{
"pp": "case refine_2\nα : Type u_1\nβ : Type u_2\nM : Type u_5\ninst✝² : DecidableEq β\ninst✝¹ : AddCommMonoid M\nf : α → β\ninst✝ : Subsingleton (AddUnits M)\nx : α →₀ M\nt : β\nx✝ : ∃ a ∈ x.support, f a = t\ni : α\ni_in : i ∈ x.support\nhi : f i = t\n⊢ ¬(mapDomain f x) t = 0",
"ppTerm": "?refine_2",
... | [
"case refine_2\nα : Type u_1\nβ : Type u_2\nM : Type u_5\ninst✝² : DecidableEq β\ninst✝¹ : AddCommMonoid M\nf : α → β\ninst✝ : Subsingleton (AddUnits M)\nx : α →₀ M\nt : β\nx✝ : ∃ a ∈ x.support, f a = t\ni : α\ni_in : i ∈ x.support\nhi : f i = t\n⊢ ∃ x_1, ¬x x_1 = 0 ∧ f x_1 = f i ∧ ¬x x_1 = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Finsupp.Supported | {
"line": 87,
"column": 2
} | {
"line": 87,
"column": 13
} | {
"line": 87,
"column": 14
} | [
{
"pp": "α : Type u_1\nR : Type u_5\ninst✝¹ : Semiring R\ninst✝ : Nontrivial R\na : α\ns : Set α\nh : ↑{a} ⊆ s\n⊢ a ∈ s",
"ppTerm": "?m.83",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\nR : Type u_5\ninst✝¹ : Semiring R\ninst✝ : Nontrivial R\na : α\ns : Set α\nh : ↑{a} ⊆ s\n⊢ a ∈ s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Finsupp.Supported | {
"line": 118,
"column": 2
} | {
"line": 118,
"column": 17
} | {
"line": 118,
"column": 18
} | [
{
"pp": "case neg\nα : Type u_1\nM : Type u_2\nR : Type u_5\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ns : Set α\ninst✝ : DecidablePred fun x ↦ x ∈ s\nl : ↥(supported M R s)\na : α\nh : a ∉ s\n⊢ ↑((restrictDom M R s ∘ₗ (supported M R s).subtype) l) a = ↑(LinearMap.id l) a",
"ppTerm... | [
"case neg\nα : Type u_1\nM : Type u_2\nR : Type u_5\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ns : Set α\ninst✝ : DecidablePred fun x ↦ x ∈ s\nl : ↥(supported M R s)\na : α\nh : a ∉ s\n⊢ 0 = ↑l a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Finsupp.Supported | {
"line": 191,
"column": 2
} | {
"line": 191,
"column": 66
} | {
"line": 191,
"column": 67
} | [
{
"pp": "α : Type u_1\nM : Type u_2\nR : Type u_5\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ninst✝ : Nontrivial M\ns t : Set α\nh : ∀ (x : α →₀ M), x ∈ supported M R (s ∪ t)\na : α\nx : M\nhx : x ≠ 0\n⊢ a ∈ ⊤ → a ∈ s ⊔ t",
"ppTerm": "?m.103",
"assigned": true,
"usedConstant... | [
"α : Type u_1\nM : Type u_2\nR : Type u_5\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ninst✝ : Nontrivial M\ns t : Set α\nh : ∀ (x : α →₀ M), x ∈ supported M R (s ∪ t)\na : α\nx : M\nhx : x ≠ 0\n⊢ a ∈ s ∨ a ∈ t"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Finsupp.Supported | {
"line": 239,
"column": 36
} | {
"line": 239,
"column": 47
} | {
"line": 239,
"column": 48
} | [
{
"pp": "α : Type u_1\nM : Type u_2\nR : Type u_5\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nα' : Type u_7\nf : α → α'\ns : Set α\nh✝ : Nonempty α\nl : α' →₀ M\nhl : l ∈ supported M R (f '' s)\nc : α'\nhc : c ∈ l.support\nhx : Function.invFunOn f s c ∈ ↑((lmapDomain M R (Function.invFun... | [
"α : Type u_1\nM : Type u_2\nR : Type u_5\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nα' : Type u_7\nf : α → α'\ns : Set α\nh✝ : Nonempty α\nl : α' →₀ M\nhl : l ∈ supported M R (f '' s)\nc : α'\nhc : c ∈ l.support\nhx : Function.invFunOn f s c ∈ ↑((lmapDomain M R (Function.invFunOn f s)) l).... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Finsupp.Supported | {
"line": 242,
"column": 35
} | {
"line": 242,
"column": 46
} | {
"line": 242,
"column": 47
} | [
{
"pp": "α : Type u_1\nM : Type u_2\nR : Type u_5\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nα' : Type u_7\nf : α → α'\ns : Set α\nh✝ : Nonempty α\nl : α' →₀ M\nhl : l ∈ supported M R (f '' s)\nc : α'\nhc : c ∈ l.support\n⊢ ∃ a ∈ s, f a = c",
"ppTerm": "?m.267",
"assigned": fals... | [
"α : Type u_1\nM : Type u_2\nR : Type u_5\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nα' : Type u_7\nf : α → α'\ns : Set α\nh✝ : Nonempty α\nl : α' →₀ M\nhl : l ∈ supported M R (f '' s)\nc : α'\nhc : c ∈ l.support\n⊢ ∃ a ∈ s, f a = c"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Interval.Set.Fin | {
"line": 95,
"column": 83
} | {
"line": 95,
"column": 97
} | {
"line": 96,
"column": 0
} | [
{
"pp": "n : ℕ\ni j : Fin n\n⊢ val '' uIoc i j = uIoc ↑i ↑j",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Set.Ioc",
"Lattice.toSemilatticeSup",
"congrArg",
"PartialOrder.toPreorder",
"Set.uIoc",
"SemilatticeInf.toPartialOrder",
"SemilatticeSup.to... | [] | by simp [uIoc] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.BigOperators.Finprod | {
"line": 214,
"column": 4
} | {
"line": 214,
"column": 41
} | {
"line": 214,
"column": 42
} | [
{
"pp": "M : Type u_2\nα : Sort u_4\ninst✝ : CommMonoid M\nf : α → M\na : α\nha : ∀ (x : α), x ≠ a → f x = 1\nx : PLift α\n⊢ x ∉ ↑{{ down := a }} → x ∉ mulSupport (f ∘ PLift.down)",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"Function.mem_mulS... | [
"M : Type u_2\nα : Sort u_4\ninst✝ : CommMonoid M\nf : α → M\na : α\nha : ∀ (x : α), x ≠ a → f x = 1\nx : PLift α\n⊢ ¬x.down = a → f x.down = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.BigOperators.Finprod | {
"line": 523,
"column": 53
} | {
"line": 523,
"column": 77
} | {
"line": 525,
"column": 0
} | [
{
"pp": "α : Type u_1\nM : Type u_5\ninst✝¹ : CommMonoid M\nf : α → M\ns : Set α\ninst✝ : Fintype ↑s\n⊢ s ∩ mulSupport f = ↑s.toFinset ∩ mulSupport f",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"MulOne.toOne",
"Monoid.toMulOneClass",
"congrArg",
"Finset",
... | [] | simp_rw [coe_toFinset s] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Algebra.BigOperators.Finprod | {
"line": 523,
"column": 53
} | {
"line": 523,
"column": 77
} | {
"line": 525,
"column": 0
} | [
{
"pp": "α : Type u_1\nM : Type u_5\ninst✝¹ : CommMonoid M\nf : α → M\ns : Set α\ninst✝ : Fintype ↑s\n⊢ s ∩ mulSupport f = ↑s.toFinset ∩ mulSupport f",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"MulOne.toOne",
"Monoid.toMulOneClass",
"congrArg",
"Finset",
... | [] | simp_rw [coe_toFinset s] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.BigOperators.Finprod | {
"line": 523,
"column": 53
} | {
"line": 523,
"column": 77
} | {
"line": 525,
"column": 0
} | [
{
"pp": "α : Type u_1\nM : Type u_5\ninst✝¹ : CommMonoid M\nf : α → M\ns : Set α\ninst✝ : Fintype ↑s\n⊢ s ∩ mulSupport f = ↑s.toFinset ∩ mulSupport f",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"MulOne.toOne",
"Monoid.toMulOneClass",
"congrArg",
"Finset",
... | [] | simp_rw [coe_toFinset s] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Logic.Equiv.Fin.Basic | {
"line": 184,
"column": 2
} | {
"line": 184,
"column": 33
} | {
"line": 184,
"column": 34
} | [
{
"pp": "n : ℕ\np : Fin (n + 1)\nh : p ≠ Fin.last n\nx : { x // x ≠ p }\n⊢ some ((finSuccAboveEquiv p).symm x) = some ((p.castLT ⋯).predAbove ↑x)",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Fin.succAbove",
"Eq.mpr",
"Equiv.instEquivLike",
"congrArg",
"ins... | [
"n : ℕ\np : Fin (n + 1)\nh : p ≠ Fin.last n\nx : { x // x ≠ p }\n⊢ ↑x = p.succAbove ((p.castLT ⋯).predAbove ↑x)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.BigOperators.Finprod | {
"line": 870,
"column": 2
} | {
"line": 870,
"column": 33
} | {
"line": 871,
"column": 2
} | [
{
"pp": "α : Type u_1\nM : Type u_5\ninst✝ : CommMonoid M\na : α\ns : Set α\nf : α → M\nh : a ∉ s\nhs : (s ∩ mulSupport f).Finite\n⊢ Disjoint {a} s",
"ppTerm": "?m.51",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CompleteBooleanAlgebra.toCompleteDistribLattice",
"congrArg",
... | [
"α : Type u_1\nM : Type u_5\ninst✝ : CommMonoid M\na : α\ns : Set α\nf : α → M\nh : a ∉ s\nhs : (s ∩ mulSupport f).Finite\n⊢ ({a} ∩ mulSupport f).Finite"
] | · rwa [disjoint_singleton_left] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Logic.Equiv.Fin.Rotate | {
"line": 98,
"column": 2
} | {
"line": 98,
"column": 13
} | {
"line": 98,
"column": 14
} | [
{
"pp": "n : ℕ\ni : Fin (n + 1)\n⊢ i < (finRotate (n + 1)) i ↔ i ≠ Fin.last n",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Equiv.instEquivLike",
"congrArg",
"Fin.neZero",
"id",
"Fin.instOfNat",
"Ne",
"instOfNatNat",
"finRo... | [
"n : ℕ\ni : Fin (n + 1)\n⊢ i < Fin.last n ↔ ¬i = Fin.last n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Basis.Basic | {
"line": 79,
"column": 2
} | {
"line": 79,
"column": 13
} | {
"line": 79,
"column": 14
} | [
{
"pp": "ι : Type u_1\nR : Type u_3\nM : Type u_5\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nb : Basis ι R M\np : Submodule R M\nh : range ⇑b ⊆ ↑p\n⊢ p = ⊤",
"ppTerm": "?m.53",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"ι : Type u_1\nR : Type u_3\nM : Type u_5\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nb : Basis ι R M\np : Submodule R M\nh : range ⇑b ⊆ ↑p\n⊢ p = ⊤"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.