module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 365
values | kind stringclasses 368
values |
|---|---|---|---|---|---|---|
Mathlib.Algebra.Module.Submodule.Range | {
"line": 275,
"column": 44
} | {
"line": 276,
"column": 59
} | [
{
"pp": "K : Type u_4\nV : Type u_8\nV₂ : Type u_9\ninst✝⁴ : Semifield K\ninst✝³ : AddCommMonoid V\ninst✝² : Module K V\ninst✝¹ : AddCommMonoid V₂\ninst✝ : Module K V₂\nf : V →ₗ[K] V₂\na : K\n⊢ (a • f).range = ⨆ (_ : a ≠ 0), f.range",
"usedConstants": [
"Eq.mpr",
"Submodule",
"RingHomSurje... | by
simpa only [range_eq_map] using Submodule.map_smul' f _ a | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Module.Submodule.Range | {
"line": 434,
"column": 34
} | {
"line": 434,
"column": 59
} | [
{
"pp": "R : Type u_1\nM : Type u_5\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\nM' : Type u_10\ninst✝¹ : AddCommMonoid M'\ninst✝ : Module R M'\nO : Submodule R M\nϕ : ↥O →ₗ[R] M'\nN : Submodule R M\nhNO : N ≤ O\n⊢ Submodule.map ϕ (Submodule.comap O.subtype N) = Submodule.map ϕ (Submodul... | Submodule.range_inclusion | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.ModularLattice | {
"line": 342,
"column": 2
} | {
"line": 344,
"column": 39
} | [
{
"pp": "α : Type u_1\na b c : α\ninst✝² : Lattice α\ninst✝¹ : OrderBot α\ninst✝ : IsModularLattice α\nh : Disjoint b c\nhsup : Disjoint a (b ⊔ c)\n⊢ Disjoint (a ⊔ b) c",
"usedConstants": [
"Eq.mpr",
"Lattice.toSemilatticeSup",
"congrArg",
"Disjoint",
"SemilatticeInf.toPartialO... | rw [disjoint_comm, sup_comm]
apply Disjoint.disjoint_sup_right_of_disjoint_sup_left h.symm
rwa [sup_comm, disjoint_comm] at hsup | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.ModularLattice | {
"line": 342,
"column": 2
} | {
"line": 344,
"column": 39
} | [
{
"pp": "α : Type u_1\na b c : α\ninst✝² : Lattice α\ninst✝¹ : OrderBot α\ninst✝ : IsModularLattice α\nh : Disjoint b c\nhsup : Disjoint a (b ⊔ c)\n⊢ Disjoint (a ⊔ b) c",
"usedConstants": [
"Eq.mpr",
"Lattice.toSemilatticeSup",
"congrArg",
"Disjoint",
"SemilatticeInf.toPartialO... | rw [disjoint_comm, sup_comm]
apply Disjoint.disjoint_sup_right_of_disjoint_sup_left h.symm
rwa [sup_comm, disjoint_comm] at hsup | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Span.Defs | {
"line": 576,
"column": 6
} | {
"line": 576,
"column": 34
} | [
{
"pp": "R : Type u_1\nM : Type u_4\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nι : Sort u_9\np : ι → Submodule R M\nm : M\n⊢ m ∈ ⨆ i, p i ↔ ∀ (N : Submodule R M), (∀ (i : ι), p i ≤ N) → m ∈ N",
"usedConstants": [
"Eq.mpr",
"Submodule",
"congrArg",
"iSup",
... | ← span_singleton_le_iff_mem, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.Atoms | {
"line": 908,
"column": 10
} | {
"line": 910,
"column": 49
} | [
{
"pp": "case refine_2.inl\nι : Sort u_1\nα : Type u_2\nβ : Type u_3\ninst✝² : Lattice α\ninst✝¹ : BoundedOrder α\ninst✝ : IsSimpleOrder α\ns : Set α\nh : ⊥ ∈ upperBounds s\n⊢ (if ⊤ ∈ s then ⊤ else ⊥) ≤ ⊥",
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"Lattice",
"le_refl",
"Lattice.... | rw [if_neg]
intro con
exact bot_ne_top (eq_top_iff.2 (h con)) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.Atoms | {
"line": 908,
"column": 10
} | {
"line": 910,
"column": 49
} | [
{
"pp": "case refine_2.inl\nι : Sort u_1\nα : Type u_2\nβ : Type u_3\ninst✝² : Lattice α\ninst✝¹ : BoundedOrder α\ninst✝ : IsSimpleOrder α\ns : Set α\nh : ⊥ ∈ upperBounds s\n⊢ (if ⊤ ∈ s then ⊤ else ⊥) ≤ ⊥",
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"Lattice",
"le_refl",
"Lattice.... | rw [if_neg]
intro con
exact bot_ne_top (eq_top_iff.2 (h con)) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.Atoms | {
"line": 1298,
"column": 2
} | {
"line": 1298,
"column": 52
} | [
{
"pp": "α : Type u_2\ns : Set α\n⊢ IsCoatom s ↔ ∃ x, s = {x}ᶜ",
"usedConstants": [
"Eq.mpr",
"CompleteBooleanAlgebra.toCompleteDistribLattice",
"congrArg",
"Compl.compl",
"PartialOrder.toPreorder",
"Preorder.toLE",
"Exists",
"BooleanAlgebra.toCompl",
"S... | rw [isCompl_compl.isCoatom_iff_isAtom, isAtom_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Order.SupIndep | {
"line": 451,
"column": 4
} | {
"line": 451,
"column": 19
} | [
{
"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)",
"usedConstants": [
"False",
"Lattice.toSemilatticeSup",
"CompleteLattice.toLattice",
"Iff.of_eq",
"congrAr... | simpa using h 1 | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Order.SupIndep | {
"line": 451,
"column": 4
} | {
"line": 451,
"column": 19
} | [
{
"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)",
"usedConstants": [
"False",
"Lattice.toSemilatticeSup",
"CompleteLattice.toLattice",
"Iff.of_eq",
"congrAr... | simpa using h 1 | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.SupIndep | {
"line": 451,
"column": 4
} | {
"line": 451,
"column": 19
} | [
{
"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)",
"usedConstants": [
"False",
"Lattice.toSemilatticeSup",
"CompleteLattice.toLattice",
"Iff.of_eq",
"congrAr... | simpa using h 1 | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Algebra.Tower | {
"line": 91,
"column": 38
} | {
"line": 91,
"column": 49
} | [
{
"pp": "R : Type u\nA : Type w\nM : Type v₁\ninst✝⁵ : CommSemiring R\ninst✝⁴ : Semiring A\ninst✝³ : Algebra R A\ninst✝² : MulAction A M\ninst✝¹ : SMul R M\ninst✝ : IsScalarTower R A M\nr : R\nx : M\n⊢ (r • 1) • x = r • x",
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
... | smul_assoc, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Algebra.Tower | {
"line": 125,
"column": 65
} | {
"line": 125,
"column": 76
} | [
{
"pp": "R : Type u\nS : Type v\nA : Type w\ninst✝⁶ : CommSemiring R\ninst✝⁵ : CommSemiring S\ninst✝⁴ : Semiring A\ninst✝³ : Algebra R S\ninst✝² : Algebra S A\ninst✝¹ : Algebra R A\ninst✝ : IsScalarTower R S A\nx : R\n⊢ x • 1 = (x • 1) • 1",
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCo... | smul_assoc, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Algebra.Algebra.Tower | {
"line": 220,
"column": 18
} | {
"line": 220,
"column": 67
} | [
{
"pp": "R : Type u\nS : Type v\nA : Type w\nB : Type u₁\nM : Type v₁\ninst✝¹⁰ : CommSemiring R\ninst✝⁹ : CommSemiring S\ninst✝⁸ : Semiring A\ninst✝⁷ : Semiring B\ninst✝⁶ : Algebra R S\ninst✝⁵ : Algebra S A\ninst✝⁴ : Algebra S B\ninst✝³ : Algebra R A\ninst✝² : Algebra R B\ninst✝¹ : IsScalarTower R S A\ninst✝ : ... | simp [h.forall, ← IsScalarTower.algebraMap_apply] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Algebra.Tower | {
"line": 220,
"column": 18
} | {
"line": 220,
"column": 67
} | [
{
"pp": "R : Type u\nS : Type v\nA : Type w\nB : Type u₁\nM : Type v₁\ninst✝¹⁰ : CommSemiring R\ninst✝⁹ : CommSemiring S\ninst✝⁸ : Semiring A\ninst✝⁷ : Semiring B\ninst✝⁶ : Algebra R S\ninst✝⁵ : Algebra S A\ninst✝⁴ : Algebra S B\ninst✝³ : Algebra R A\ninst✝² : Algebra R B\ninst✝¹ : IsScalarTower R S A\ninst✝ : ... | simp [h.forall, ← IsScalarTower.algebraMap_apply] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Algebra.Tower | {
"line": 220,
"column": 18
} | {
"line": 220,
"column": 67
} | [
{
"pp": "R : Type u\nS : Type v\nA : Type w\nB : Type u₁\nM : Type v₁\ninst✝¹⁰ : CommSemiring R\ninst✝⁹ : CommSemiring S\ninst✝⁸ : Semiring A\ninst✝⁷ : Semiring B\ninst✝⁶ : Algebra R S\ninst✝⁵ : Algebra S A\ninst✝⁴ : Algebra S B\ninst✝³ : Algebra R A\ninst✝² : Algebra R B\ninst✝¹ : IsScalarTower R S A\ninst✝ : ... | simp [h.forall, ← IsScalarTower.algebraMap_apply] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Notation.Indicator | {
"line": 241,
"column": 37
} | {
"line": 241,
"column": 55
} | [
{
"pp": "α : Type u_1\nM : Type u_3\ninst✝ : One M\nt : Set α\ns : Set M\n⊢ (fun x ↦ 1) ⁻¹' s ∈ {univ, ∅}",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Set.preimage_const",
"Set.univ",
"Classical.propDecidable",
"Membership.mem",
"Set.instSingletonSet",
"id",
... | Set.preimage_const | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.CompactlyGenerated.Basic | {
"line": 446,
"column": 8
} | {
"line": 446,
"column": 53
} | [
{
"pp": "case refine_1\nα : Type u_2\ninst✝¹ : CompleteLattice α\ninst✝ : IsCompactlyGenerated α\ns : Set α\nh : ∀ (t : Finset α), ↑t ⊆ s → sSupIndep ↑t\na : α\nha : a ∈ s\nt : Finset α\nht : ↑t ⊆ s \\ {a}\n⊢ ↑(insert a t) ⊆ s",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Finset",
"Cla... | rw [Finset.coe_insert, Set.insert_subset_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.BigOperators.GroupWithZero.Finset | {
"line": 61,
"column": 10
} | {
"line": 61,
"column": 26
} | [
{
"pp": "ι : Type u_1\nM₀ : Type u_4\ninst✝² : CommMonoidWithZero M₀\nf : ι → M₀\ns : Finset ι\ninst✝¹ : Nontrivial M₀\ninst✝ : NoZeroDivisors M₀\n⊢ ¬∏ x ∈ s, f x = 0 ↔ ∀ a ∈ s, f a ≠ 0",
"usedConstants": [
"CommMonoidWithZero.toCommMonoid",
"Eq.mpr",
"congrArg",
"Finset",
"Mem... | prod_eq_zero_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.CompactlyGenerated.Basic | {
"line": 695,
"column": 2
} | {
"line": 698,
"column": 19
} | [
{
"pp": "α : 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 : sSupIndep ... | · rw [Set.mem_union, Set.mem_singleton_iff] at hx
obtain hx | rfl := hx
· exact s_atoms x hx
· exact ha.symm | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Order.CompactlyGenerated.Basic | {
"line": 728,
"column": 2
} | {
"line": 730,
"column": 44
} | [
{
"pp": "α : Type u_2\ninst✝² : CompleteLattice α\ninst✝¹ : IsModularLattice α\ninst✝ : IsCompactlyGenerated α\n⊢ ComplementedLattice α ↔ IsAtomistic α",
"usedConstants": [
"PartialOrder.toPreorder",
"Preorder.toLE",
"CompleteLattice.toConditionallyCompleteLattice",
"CompleteLattice.... | constructor <;> intros
· exact isAtomistic_of_complementedLattice
· exact complementedLattice_of_isAtomistic | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.CompactlyGenerated.Basic | {
"line": 728,
"column": 2
} | {
"line": 730,
"column": 44
} | [
{
"pp": "α : Type u_2\ninst✝² : CompleteLattice α\ninst✝¹ : IsModularLattice α\ninst✝ : IsCompactlyGenerated α\n⊢ ComplementedLattice α ↔ IsAtomistic α",
"usedConstants": [
"PartialOrder.toPreorder",
"Preorder.toLE",
"CompleteLattice.toConditionallyCompleteLattice",
"CompleteLattice.... | constructor <;> intros
· exact isAtomistic_of_complementedLattice
· exact complementedLattice_of_isAtomistic | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Span.Basic | {
"line": 291,
"column": 2
} | {
"line": 299,
"column": 40
} | [
{
"pp": "case refine_4\nR : Type u_1\nM : Type u_4\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nι : Sort u_8\np : ι → Submodule R M\nx✝¹ : M\nhx✝ : x✝¹ ∈ span R (⋃ i, ↑(p i))\nr : R\nx : M\nx✝ : x ∈ span R (⋃ i, ↑(p i))\nhx : x ∈ AddSubmonoid.closure (⋃ i, ↑(p i))\n⊢ r • x ∈ AddSubmonoid.... | · refine AddSubmonoid.closure_induction ?_ ?_ ?_ hx
· rintro x ⟨_, ⟨i, rfl⟩, hix : x ∈ p i⟩
apply AddSubmonoid.subset_closure (Set.mem_iUnion.mpr ⟨i, _⟩)
exact smul_mem _ r hix
· rw [smul_zero]
exact AddSubmonoid.zero_mem _
· intro x y _ _ hx hy
rw [smul_add]
exact AddSubmonoid... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.LinearAlgebra.Span.Basic | {
"line": 361,
"column": 46
} | {
"line": 361,
"column": 86
} | [
{
"pp": "R : Type u_1\nR₂ : Type u_2\nK : Type u_3\nM : Type u_4\nM₂ : Type u_5\nV : Type u_6\nS : Type u_7\ninst✝⁷ : Semiring R\ninst✝⁶ : AddCommMonoid M\ninst✝⁵ : Module R M\nx✝ : M\np p' : Submodule R M\ninst✝⁴ : Semiring R₂\nσ₁₂ : R →+* R₂\ninst✝³ : AddCommMonoid M₂\ninst✝² : Module R₂ M₂\ns t : Set M\nM' :... | exact ⟨smul_mem _ a hx, smul_mem _ a hy⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.LinearAlgebra.Span.Basic | {
"line": 500,
"column": 36
} | {
"line": 500,
"column": 68
} | [
{
"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... | simpa using sub_eq_zero.2 e.symm | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.LinearAlgebra.Span.Basic | {
"line": 500,
"column": 36
} | {
"line": 500,
"column": 68
} | [
{
"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... | simpa using sub_eq_zero.2 e.symm | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Span.Basic | {
"line": 500,
"column": 36
} | {
"line": 500,
"column": 68
} | [
{
"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... | simpa using sub_eq_zero.2 e.symm | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.BigOperators.Ring.Finset | {
"line": 146,
"column": 25
} | {
"line": 146,
"column": 51
} | [
{
"pp": "ι : 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 (... | by rintro rfl; exact ha hv | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.Span.Basic | {
"line": 580,
"column": 6
} | {
"line": 580,
"column": 19
} | [
{
"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\nι : Type u_8\ns : Set ι\np : ι → Submodule R₂ M₂\nhp : ⨆ i ∈... | iSup_subtype' | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Finsupp.Single | {
"line": 90,
"column": 41
} | {
"line": 90,
"column": 70
} | [
{
"pp": "α : Type u_1\nM : Type u_5\ninst✝¹ : Zero M\ninst✝ : DecidableEq α\na : α\nb : M\n⊢ ({a}.indicator fun x ↦ b) = update 0 a b",
"usedConstants": [
"Eq.mpr",
"Function.update",
"congrArg",
"Set.indicator",
"Set.instSingletonSet",
"id",
"Pi.instZero",
"S... | ← Set.piecewise_eq_indicator, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Span.Basic | {
"line": 709,
"column": 18
} | {
"line": 709,
"column": 42
} | [
{
"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✝",
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"Semiring.toModule",
"congrArg",
"Line... | simpa using congr($eq 1) | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.LinearAlgebra.Span.Basic | {
"line": 709,
"column": 18
} | {
"line": 709,
"column": 42
} | [
{
"pp": "R : Type u_1\nM : Type u_4\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nx✝¹ x✝ : M\neq : toSpanSingleton R M x✝¹ = toSpanSingleton R M x✝\n⊢ x✝¹ = x✝",
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"Semiring.toModule",
"congrArg",
"Line... | simpa using congr($eq 1) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Span.Basic | {
"line": 709,
"column": 18
} | {
"line": 709,
"column": 42
} | [
{
"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✝",
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"Semiring.toModule",
"congrArg",
"Line... | simpa using congr($eq 1) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.BigOperators.Ring.Finset | {
"line": 212,
"column": 41
} | {
"line": 212,
"column": 49
} | [
{
"pp": "ι : Type u_1\nR : Type u_4\ninst✝¹ : CommSemiring R\ninst✝ : LinearOrder ι\nf g : ι → R\na : ι\ns : Finset ι\nha : ∀ x ∈ s, x < a\nihs : ∏ i ∈ s, (f i + g i) = ∏ i ∈ s, f i + ∑ i ∈ s, (g i * ∏ j ∈ s with j < i, (f j + g j)) * ∏ j ∈ s with i < j, f j\nha' : a ∉ s\n⊢ f a * (∏ i ∈ s, f i + ∑ i ∈ s, (g i *... | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.BigOperators.Ring.Finset | {
"line": 212,
"column": 50
} | {
"line": 212,
"column": 58
} | [
{
"pp": "ι : Type u_1\nR : Type u_4\ninst✝¹ : CommSemiring R\ninst✝ : LinearOrder ι\nf g : ι → R\na : ι\ns : Finset ι\nha : ∀ x ∈ s, x < a\nihs : ∏ i ∈ s, (f i + g i) = ∏ i ∈ s, f i + ∑ i ∈ s, (g i * ∏ j ∈ s with j < i, (f j + g j)) * ∏ j ∈ s with i < j, f j\nha' : a ∉ s\n⊢ f a * ∏ i ∈ s, f i + f a * ∑ i ∈ s, (... | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.BigOperators.Ring.Finset | {
"line": 236,
"column": 6
} | {
"line": 236,
"column": 17
} | [
{
"pp": "ι : Type u_1\nR : Type u_4\ninst✝ : CommSemiring R\na b : R\ns t : Finset ι\nht : t ∈ s.powerset\n⊢ a ^ #t * b ^ (#s - #t) = (∏ i ∈ t, a) * ∏ i ∈ s \\ t, b",
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"congrArg",
"CommSemiring.toSemiring",
"Finset",
"HSub.hSub",
... | prod_const, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.BigOperators.Ring.Finset | {
"line": 236,
"column": 18
} | {
"line": 236,
"column": 29
} | [
{
"pp": "ι : Type u_1\nR : Type u_4\ninst✝ : CommSemiring R\na b : R\ns t : Finset ι\nht : t ∈ s.powerset\n⊢ a ^ #t * b ^ (#s - #t) = a ^ #t * ∏ i ∈ s \\ t, b",
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"congrArg",
"CommSemiring.toSemiring",
"Finset",
"HSub.hSub",
"... | prod_const, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.BigOperators.Ring.Finset | {
"line": 273,
"column": 77
} | {
"line": 275,
"column": 6
} | [
{
"pp": "ι : Type u_1\nR : Type u_4\ninst✝¹ : CommRing R\ninst✝ : LinearOrder ι\ns : Finset ι\nf : ι → R\n⊢ ∏ i ∈ s, (1 - f i) = 1 - ∑ i ∈ s, f i * ∏ j ∈ s with j < i, (1 - f j)",
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"Preorder.toLT",
"NonUnitalCommRing.toNonUnitalNonAssocComm... | by
rw [prod_sub_ordered]
simp | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Group.Finsupp | {
"line": 170,
"column": 11
} | {
"line": 176,
"column": 14
} | [
{
"pp": "ι : Type u_1\nM : Type u_3\ninst✝¹ : AddZeroClass M\ninst✝ : DecidableEq ι\nf g : ι →₀ M\nh : ∀ x ∈ f.support ∩ g.support, AddCommute (f x) (g x)\n⊢ AddCommute f g",
"usedConstants": [
"Finsupp.instFunLike",
"Finsupp.ext",
"Classical.not_not._simp_1",
"congrArg",
"Fins... | by
ext x
by_cases hf : x ∈ f.support
· by_cases hg : x ∈ g.support
· exact h _ (mem_inter_of_mem hf hg)
· simp_all
· simp_all | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Group.Indicator | {
"line": 132,
"column": 6
} | {
"line": 132,
"column": 60
} | [
{
"pp": "case pos\nα : Type u_1\nM : Type u_3\ninst✝¹ : MulOneClass M\ns : Set α\ninst✝ : DecidablePred fun x ↦ x ∈ s\nf g : α → M\nx : α\nh : x ∈ s\n⊢ f x * sᶜ.mulIndicator g x = f x",
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"MulOne.toOne",
"HMul.hMul",
"congrArg",
"Comp... | Set.mulIndicator_of_notMem (Set.notMem_compl_iff.2 h), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Finsupp.Option | {
"line": 198,
"column": 75
} | {
"line": 207,
"column": 45
} | [
{
"pp": "α : Type u_1\nM : Type u_2\nN : Type u_3\ninst✝¹ : AddZeroClass M\ninst✝ : CommMonoid N\nf : Option α →₀ M\nb : Option α → M → N\nh_zero : ∀ (o : Option α), b o 0 = 1\nh_add : ∀ (o : Option α) (m₁ m₂ : M), b o (m₁ + m₂) = b o m₁ * b o m₂\n⊢ f.prod b = b none (f none) * f.some.prod fun a ↦ b (Option.som... | by
classical
induction f using induction_linear with
| zero => simp [some_zero, h_zero]
| add f₁ f₂ h₁ h₂ =>
rw [Finsupp.prod_add_index, h₁, h₂, some_add, Finsupp.prod_add_index]
· simp only [h_add, Pi.add_apply, Finsupp.coe_add]
rw [mul_mul_mul_comm]
all_goals simp [h_zero, h_ad... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.BigOperators.Finsupp.Basic | {
"line": 537,
"column": 2
} | {
"line": 538,
"column": 25
} | [
{
"pp": "α : Type u_1\nM : Type u_8\ninst✝ : AddCommMonoid M\ns : Finset α\nf : (a : α) → a ∈ s → M\n⊢ ∑ x ∈ s.attach, single (↑x) ((indicator s f) ↑x) = ∑ x ∈ s.attach, single (↑x) (f ↑x ⋯)",
"usedConstants": [
"Finsupp.instFunLike",
"Finsupp.indicator",
"Eq.mpr",
"congrArg",
... | · refine Finset.sum_congr rfl (fun _ _ => ?_)
rw [indicator_of_mem] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Order.AbsoluteValue.Basic | {
"line": 192,
"column": 9
} | {
"line": 192,
"column": 23
} | [
{
"pp": "case inr.succ\nR : Type u_5\nS : Type u_6\ninst✝⁴ : Semiring R\ninst✝³ : Semiring S\ninst✝² : PartialOrder S\nabv : AbsoluteValue R S\ninst✝¹ : IsDomain S\ninst✝ : IsOrderedRing S\nh✝ : Nontrivial R\nn : ℕ\nih : abv ↑n ≤ ↑n\n⊢ abv ↑(n + 1) ≤ ↑(n + 1)",
"usedConstants": [
"Eq.mpr",
"NonA... | Nat.cast_succ, | Mathlib.Tactic.evalGRewriteSeq | null |
Mathlib.Algebra.Order.AbsoluteValue.Basic | {
"line": 192,
"column": 24
} | {
"line": 192,
"column": 38
} | [
{
"pp": "case inr.succ\nR : Type u_5\nS : Type u_6\ninst✝⁴ : Semiring R\ninst✝³ : Semiring S\ninst✝² : PartialOrder S\nabv : AbsoluteValue R S\ninst✝¹ : IsDomain S\ninst✝ : IsOrderedRing S\nh✝ : Nontrivial R\nn : ℕ\nih : abv ↑n ≤ ↑n\n⊢ abv (↑n + 1) ≤ ↑(n + 1)",
"usedConstants": [
"Eq.mpr",
"NonA... | Nat.cast_succ, | Mathlib.Tactic.evalGRewriteSeq | null |
Mathlib.Algebra.Order.BigOperators.Group.Multiset | {
"line": 107,
"column": 7
} | {
"line": 107,
"column": 33
} | [
{
"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 ∈ ... | by simpa using hs_nonempty | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Order.BigOperators.Group.Multiset | {
"line": 113,
"column": 66
} | {
"line": 113,
"column": 92
} | [
{
"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 ≠ ∅",
"usedConstants": [
"congrArg",
"List.instEmptyCollection",
... | by simpa using hs_nonempty | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Order.BigOperators.GroupWithZero.Multiset | {
"line": 50,
"column": 26
} | {
"line": 53,
"column": 14
} | [
{
"pp": "R : Type u_1\ninst✝³ : CommMonoidWithZero R\ninst✝² : PartialOrder R\ninst✝¹ : ZeroLEOneClass R\ninst✝ : PosMulMono R\nα : Type u_2\ns : Multiset α\nf : α → R\nh : ∀ (a : α), a ∈ s → 0 ≤ f a\n⊢ 0 ≤ (map f s).prod",
"usedConstants": [
"Multiset.map",
"Multiset.mem_map",
"PartialOrd... | by
refine prod_nonneg fun r hr ↦ ?_
obtain ⟨a, ha, rfl⟩ := mem_map.mp hr
exact h a ha | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Order.BigOperators.Group.Finset | {
"line": 348,
"column": 2
} | {
"line": 348,
"column": 63
} | [
{
"pp": "α : Type u_2\ninst✝ : DecidableEq α\ns : Finset α\nB : Finset (Finset α)\nn : ℕ\nh : ∀ a ∈ s, #({b ∈ B | a ∈ b}) ≤ n\n⊢ ∑ t ∈ B, #(s ∩ t) ≤ ∑ x ∈ s, #({b ∈ B | x ∈ b})",
"usedConstants": [
"Eq.mpr",
"Finset.card_eq_sum_ones",
"congrArg",
"Finset",
"_private.Mathlib.Alg... | simp_rw [← filter_mem_eq_inter, card_eq_sum_ones, sum_filter] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Algebra.Order.BigOperators.Group.Finset | {
"line": 362,
"column": 2
} | {
"line": 362,
"column": 63
} | [
{
"pp": "α : Type u_2\ninst✝ : DecidableEq α\ns : Finset α\nB : Finset (Finset α)\nn : ℕ\nh : ∀ a ∈ s, n ≤ #({b ∈ B | a ∈ b})\n⊢ ∑ x ∈ s, #({b ∈ B | x ∈ b}) ≤ ∑ t ∈ B, #(s ∩ t)",
"usedConstants": [
"Eq.mpr",
"Finset.card_eq_sum_ones",
"congrArg",
"Finset",
"AddMonoid.toAddZeroC... | simp_rw [← filter_mem_eq_inter, card_eq_sum_ones, sum_filter] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Data.Finsupp.Basic | {
"line": 398,
"column": 91
} | {
"line": 403,
"column": 68
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nM : Type u_5\ninst✝ : AddCommMonoid M\nf : α ↪ β\nv : α →₀ M\n⊢ embDomain f v = mapDomain (⇑f) v",
"usedConstants": [
"Finsupp.instFunLike",
"Eq.mpr",
"Finsupp.ext",
"congrArg",
"Finsupp.mapDomain",
"AddMonoid.toAddZeroClass",
"C... | by
ext a
by_cases h : a ∈ Set.range f
· rcases h with ⟨a, rfl⟩
rw [mapDomain_apply f.injective, embDomain_apply_self]
· rw [mapDomain_notin_range, embDomain_notin_range] <;> assumption | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Order.BigOperators.Ring.Finset | {
"line": 60,
"column": 4
} | {
"line": 60,
"column": 49
} | [
{
"pp": "case refine_2.h0\nι : Type u_1\nR : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : PartialOrder R\ninst✝ : IsOrderedRing R\ns : Finset ι\ni : ι\nf g h : ι → R\nhi : i ∈ s\nh2i : g i + h i ≤ f i\nhgf : ∀ j ∈ s, j ≠ i → g j ≤ f j\nhhf : ∀ j ∈ s, j ≠ i → h j ≤ f j\nhg : ∀ i ∈ s, 0 ≤ g i\nhh : ∀ i ∈ s, 0 ≤ h ... | simp only [and_imp, mem_sdiff, mem_singleton] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Data.Finsupp.Basic | {
"line": 914,
"column": 2
} | {
"line": 914,
"column": 73
} | [
{
"pp": "case h\nα : Type u_1\nβ : Type u_2\nM : Type u_5\ninst✝ : Zero M\na : α\nb : β\nm : M\nx : α\ny : β\n⊢ (single a (single b m)).uncurry (x, y) = (single (a, b) m) (x, y)",
"usedConstants": [
"Finsupp.instFunLike",
"False",
"Finsupp.single_eq_same",
"eq_false",
"instDeci... | rcases eq_or_ne a x with rfl | hne <;> classical simp [single_apply, *] | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Data.Finsupp.Basic | {
"line": 1160,
"column": 2
} | {
"line": 1160,
"column": 18
} | [
{
"pp": "case h\nα : Type u_1\nM : Type u_12\ninst✝¹ : Zero M\nP : α → Prop\ninst✝ : DecidablePred P\nf : Subtype P →₀ M\na : α\n⊢ f.extendDomain a = (embDomain (Embedding.subtype P) f) a",
"usedConstants": [
"Finsupp.instFunLike",
"Subtype",
"dite",
"Function.Embedding.subtype",
... | by_cases h : P a | «_aux_Init_ByCases___macroRules_tacticBy_cases_:__2» | «tacticBy_cases_:_» |
Mathlib.Data.Finsupp.Basic | {
"line": 1175,
"column": 44
} | {
"line": 1178,
"column": 7
} | [
{
"pp": "α : Type u_1\nM : Type u_12\ninst✝¹ : Zero M\nP : α → Prop\ninst✝ : DecidablePred P\nf : α →₀ M\nhf : ∀ a ∈ f.support, P a\n⊢ (subtypeDomain P f).extendDomain = f",
"usedConstants": [
"Finsupp.instFunLike",
"Eq.mpr",
"dite_congr",
"Finsupp.ext",
"congrArg",
"_pri... | by
ext
simp only [extendDomain_apply, subtypeDomain_apply, dite_eq_ite, ite_eq_left_iff]
grind | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Order.Interval.Set.Fin | {
"line": 551,
"column": 59
} | {
"line": 553,
"column": 82
} | [
{
"pp": "n m : ℕ\ni j : Fin n\n⊢ natAdd m '' Icc i j = Icc (natAdd m i) (natAdd m j)",
"usedConstants": [
"Eq.mpr",
"Fin.natAdd",
"Set.Ici",
"Fin.preimage_natAdd_Icc_natAdd",
"congrArg",
"Fin.image_natAdd_Ici",
"PartialOrder.toPreorder",
"Eq.rec",
"HasSu... | by
rw [← preimage_natAdd_Icc_natAdd, image_preimage_eq_of_subset]
exact Icc_subset_Ici_self.trans <| image_natAdd_Ici m i ▸ image_subset_range _ _ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.Finsupp.LinearCombination | {
"line": 130,
"column": 4
} | {
"line": 130,
"column": 32
} | [
{
"pp": "case h.mp\nα : Type u_1\nM : Type u_2\nR : Type u_5\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nv : α → M\nx : M\nl : α →₀ R\nhl : (linearCombination R v) l = x\n⊢ (linearCombination R v) l ∈ span R (Set.range v)",
"usedConstants": [
"Eq.mpr",
"Submodule",
... | rw [linearCombination_apply] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Order.Interval.Set.Fin | {
"line": 809,
"column": 78
} | {
"line": 810,
"column": 46
} | [
{
"pp": "n : ℕ\ni j : Fin n\n⊢ rev ⁻¹' Icc i j = Icc j.rev i.rev",
"usedConstants": [
"Set.ext",
"congrArg",
"PartialOrder.toPreorder",
"Preorder.toLE",
"Membership.mem",
"LE.le",
"Set.mem_preimage._simp_1",
"instLEFin",
"_private.Mathlib.Order.Interval.... | by
ext; simp [le_rev_iff, rev_le_iff, and_comm] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.Finsupp.LinearCombination | {
"line": 221,
"column": 2
} | {
"line": 221,
"column": 13
} | [
{
"pp": "case single\nM : Type u_2\nR : Type u_5\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nα : Type u_9\nβ : Type u_10\nA : α → M\nB : β → α →₀ R\na✝ : β\nb✝ : R\n⊢ (((single a✝ b✝).sum fun i a ↦ a • B i).sum fun i a ↦ a • A i) =\n (single a✝ b✝).sum fun i a ↦ a • (B i).sum fun i a ... | | single => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.LinearAlgebra.Finsupp.LinearCombination | {
"line": 266,
"column": 2
} | {
"line": 266,
"column": 30
} | [
{
"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\nv : α → M\nf : α → α'\nl : α' →₀ R\nhf : InjOn f (f ⁻¹' ↑l.support)\n⊢ (linearCombination R v) (comapDomain f l hf) = ∑ i ∈ l.support.preimage f hf, l (f i) • v i",
"usedConst... | rw [linearCombination_apply] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.BigOperators.Finprod | {
"line": 394,
"column": 21
} | {
"line": 398,
"column": 17
} | [
{
"pp": "α : Type u_1\nM : Type u_5\ninst✝ : CommMonoid M\nf : α → M\nhf : (mulSupport f).Infinite\n⊢ ∏ᶠ (i : α), f i = 1",
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"Monoid.toMulOneClass",
"congrArg",
"Classical.propDecidable",
"Set.Finite",
"id",
"dif_neg... | by
classical
rw [finprod_def]
simp only [HasFiniteMulSupport]
rw [dif_neg hf] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Logic.Equiv.Fin.Basic | {
"line": 74,
"column": 46
} | {
"line": 75,
"column": 68
} | [
{
"pp": "n : ℕ\ni : Fin (n + 1)\nm : Fin n\nh : m.castSucc < i\n⊢ (finSuccEquiv' i) m.castSucc = some m",
"usedConstants": [
"Fin.succAbove",
"Eq.mpr",
"Equiv.instEquivLike",
"congrArg",
"Option.some",
"id",
"Equiv",
"instOfNatNat",
"instHAdd",
"HA... | by
rw [← Fin.succAbove_of_castSucc_lt _ _ h, finSuccEquiv'_succAbove] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Logic.Equiv.Fin.Basic | {
"line": 381,
"column": 4
} | {
"line": 381,
"column": 40
} | [
{
"pp": "m n✝ n : ℕ\ninst✝ : NeZero n\nx✝ : ℤ × Fin n\nq : ℤ\nr : ℕ\nhrn : r < n\n⊢ (fun a ↦ (a / ↑n, Fin.ofNat n (a.natMod ↑n))) ((fun p ↦ p.1 * ↑n + ↑↑p.2) (q, ⟨r, hrn⟩)) = (q, ⟨r, hrn⟩)",
"usedConstants": [
"Eq.mpr",
"Int.instDiv",
"instHDiv",
"HMul.hMul",
"congrArg",
... | simp only [Prod.mk_inj, Fin.ext_iff] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.BigOperators.Finprod | {
"line": 1287,
"column": 2
} | {
"line": 1287,
"column": 30
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nM : Type u_5\ninst✝² : CommMonoid M\ninst✝¹ : DecidableEq α\ninst✝ : DecidableEq β\ns : Finset (α × β)\nf : α × β → M\nthis : ∀ (a : α), ∏ i ∈ Finset.image Prod.snd ({ab ∈ s | ab.1 = a}), f (a, i) = {x ∈ s | x.1 = a}.prod f\n⊢ ∀ i ∈ s, i.1 ∈ Finset.image Prod.fst s",
"us... | simp only [Finset.mem_image] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.BigOperators.Finprod | {
"line": 1277,
"column": 2
} | {
"line": 1288,
"column": 32
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nM : Type u_5\ninst✝² : CommMonoid M\ninst✝¹ : DecidableEq α\ninst✝ : DecidableEq β\ns : Finset (α × β)\nf : α × β → M\n⊢ ∏ᶠ (ab : α × β) (_ : ab ∈ s), f ab =\n ∏ᶠ (a : α) (b : β) (_ : b ∈ Finset.image Prod.snd ({ab ∈ s | ab.1 = a})), f (a, b)",
"usedConstants": [
... | have (a : _) :
∏ i ∈ (s.filter fun ab => Prod.fst ab = a).image Prod.snd, f (a, i) =
(s.filter (Prod.fst · = a)).prod f := by
refine Finset.prod_nbij' (fun b ↦ (a, b)) Prod.snd ?_ ?_ ?_ ?_ ?_ <;> aesop
rw [finprod_mem_finset_eq_prod]
simp_rw [finprod_mem_finset_eq_prod, this]
rw [finprod_eq_prod... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.BigOperators.Finprod | {
"line": 1277,
"column": 2
} | {
"line": 1288,
"column": 32
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nM : Type u_5\ninst✝² : CommMonoid M\ninst✝¹ : DecidableEq α\ninst✝ : DecidableEq β\ns : Finset (α × β)\nf : α × β → M\n⊢ ∏ᶠ (ab : α × β) (_ : ab ∈ s), f ab =\n ∏ᶠ (a : α) (b : β) (_ : b ∈ Finset.image Prod.snd ({ab ∈ s | ab.1 = a})), f (a, b)",
"usedConstants": [
... | have (a : _) :
∏ i ∈ (s.filter fun ab => Prod.fst ab = a).image Prod.snd, f (a, i) =
(s.filter (Prod.fst · = a)).prod f := by
refine Finset.prod_nbij' (fun b ↦ (a, b)) Prod.snd ?_ ?_ ?_ ?_ ?_ <;> aesop
rw [finprod_mem_finset_eq_prod]
simp_rw [finprod_mem_finset_eq_prod, this]
rw [finprod_eq_prod... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Fin.VecNotation | {
"line": 251,
"column": 2
} | {
"line": 251,
"column": 47
} | [
{
"pp": "α : Type u\nx : α\nu : Fin 0 → α\n⊢ Set.range (vecCons x u) = {x}",
"usedConstants": [
"Matrix.range_cons",
"Eq.mpr",
"Set.union_empty",
"congrArg",
"Set.instUnion",
"Set.instSingletonSet",
"id",
"instOfNatNat",
"Matrix.range_empty",
"Nat"... | rw [range_cons, range_empty, Set.union_empty] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Data.Fin.VecNotation | {
"line": 251,
"column": 2
} | {
"line": 251,
"column": 47
} | [
{
"pp": "α : Type u\nx : α\nu : Fin 0 → α\n⊢ Set.range (vecCons x u) = {x}",
"usedConstants": [
"Matrix.range_cons",
"Eq.mpr",
"Set.union_empty",
"congrArg",
"Set.instUnion",
"Set.instSingletonSet",
"id",
"instOfNatNat",
"Matrix.range_empty",
"Nat"... | rw [range_cons, range_empty, Set.union_empty] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Fin.VecNotation | {
"line": 251,
"column": 2
} | {
"line": 251,
"column": 47
} | [
{
"pp": "α : Type u\nx : α\nu : Fin 0 → α\n⊢ Set.range (vecCons x u) = {x}",
"usedConstants": [
"Matrix.range_cons",
"Eq.mpr",
"Set.union_empty",
"congrArg",
"Set.instUnion",
"Set.instSingletonSet",
"id",
"instOfNatNat",
"Matrix.range_empty",
"Nat"... | rw [range_cons, range_empty, Set.union_empty] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Fin.VecNotation | {
"line": 256,
"column": 36
} | {
"line": 256,
"column": 55
} | [
{
"pp": "α : Type u\nx y : α\nu : Fin 0 → α\n⊢ {x} ∪ {y} = {x, y}",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Set.singleton_union",
"Set.instUnion",
"Set.instSingletonSet",
"id",
"Insert.insert",
"Set.instInsert",
"Union.union",
"Singleton.singleto... | Set.singleton_union | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Group.ModEq | {
"line": 343,
"column": 14
} | {
"line": 343,
"column": 35
} | [
{
"pp": "case mp.refine_2\nG : Type u_1\ninst✝ : AddCommGroup G\np a b : G\nn : ℕ\nhn : n ≠ 0\nk : ℤ\nhk : a - b = k • p\n⊢ k • p = (k / ↑n * ↑n + k % ↑n) • p",
"usedConstants": [
"Eq.mpr",
"Int.instDiv",
"instHSMul",
"instHDiv",
"HMul.hMul",
"congrArg",
"AddCommGro... | Int.ediv_mul_add_emod | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Nat.GCD.Basic | {
"line": 54,
"column": 8
} | {
"line": 54,
"column": 17
} | [
{
"pp": "a b c : ℕ\nha0 : a > 0\nha1 : succ 0 < a\nhb0 : b > 0\nh : b ≤ c\nthis : a ^ (c - b + b) - 1 = a ^ (c - b) * (a ^ b - 1) + (a ^ (c - b) - 1)\n⊢ (a ^ (c - b) % (a ^ b - 1) * 0 % (a ^ b - 1) + (a ^ (c - b) - 1) % (a ^ b - 1)) % (a ^ b - 1) = a ^ ((c - b) % b) - 1",
"usedConstants": [
"Eq.mpr",
... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.BigOperators.Fin | {
"line": 273,
"column": 2
} | {
"line": 273,
"column": 28
} | [
{
"pp": "M : Type u_2\ninst✝ : CommMonoid M\nn m : ℕ\nh : n ≤ m\nf : Fin m → M\na b : Fin n\n⊢ ∏ i ∈ Ioo (castLE h a) (castLE h b), f i = ∏ i ∈ Ioo a b, f (castLE h i)",
"usedConstants": [
"congrArg",
"Fin.castLE",
"Finset",
"PartialOrder.toPreorder",
"Fin.instLocallyFiniteOrde... | simp [← map_castLEEmb_Ioo] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.BigOperators.Fin | {
"line": 273,
"column": 2
} | {
"line": 273,
"column": 28
} | [
{
"pp": "M : Type u_2\ninst✝ : CommMonoid M\nn m : ℕ\nh : n ≤ m\nf : Fin m → M\na b : Fin n\n⊢ ∏ i ∈ Ioo (castLE h a) (castLE h b), f i = ∏ i ∈ Ioo a b, f (castLE h i)",
"usedConstants": [
"congrArg",
"Fin.castLE",
"Finset",
"PartialOrder.toPreorder",
"Fin.instLocallyFiniteOrde... | simp [← map_castLEEmb_Ioo] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.BigOperators.Fin | {
"line": 273,
"column": 2
} | {
"line": 273,
"column": 28
} | [
{
"pp": "M : Type u_2\ninst✝ : CommMonoid M\nn m : ℕ\nh : n ≤ m\nf : Fin m → M\na b : Fin n\n⊢ ∏ i ∈ Ioo (castLE h a) (castLE h b), f i = ∏ i ∈ Ioo a b, f (castLE h i)",
"usedConstants": [
"congrArg",
"Fin.castLE",
"Finset",
"PartialOrder.toPreorder",
"Fin.instLocallyFiniteOrde... | simp [← map_castLEEmb_Ioo] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Nat.ModEq | {
"line": 574,
"column": 8
} | {
"line": 574,
"column": 16
} | [
{
"pp": "m n : ℕ\nhm1 : m % 2 = 1\nhn1 : n % 2 = 1\nhn0 : 0 < n\n⊢ 2 * (m * n / 2) = 2 * (m * (n / 2) + m / 2)",
"usedConstants": [
"Distrib.leftDistribClass",
"Eq.mpr",
"instHDiv",
"HMul.hMul",
"congrArg",
"id",
"HDiv.hDiv",
"instMulNat",
"instOfNatNat"... | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.LinearIndependent.Basic | {
"line": 174,
"column": 2
} | {
"line": 179,
"column": 49
} | [
{
"pp": "ι : Type u'\nR : Type u_2\nM : Type u_4\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\nG : Type u_6\nhG : Group G\ninst✝³ : MulAction G R\ninst✝² : SMul G M\ninst✝¹ : IsScalarTower G R M\ninst✝ : SMulCommClass G R M\nv : ι → M\nhv : LinearIndependent R v\nw : ι → G\n⊢ LinearIndepe... | rw [linearIndependent_iff''ₛ] at hv ⊢
intro s g₁ g₂ hgs hsum i
refine (Group.isUnit (w i)).smul_left_cancel.mp ?_
refine hv s (fun i ↦ w i • g₁ i) (fun i ↦ w i • g₂ i) (fun i hi ↦ ?_) ?_ i
· simp_rw [hgs i hi]
· simpa only [smul_assoc, smul_comm] using hsum | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.LinearIndependent.Basic | {
"line": 174,
"column": 2
} | {
"line": 179,
"column": 49
} | [
{
"pp": "ι : Type u'\nR : Type u_2\nM : Type u_4\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\nG : Type u_6\nhG : Group G\ninst✝³ : MulAction G R\ninst✝² : SMul G M\ninst✝¹ : IsScalarTower G R M\ninst✝ : SMulCommClass G R M\nv : ι → M\nhv : LinearIndependent R v\nw : ι → G\n⊢ LinearIndepe... | rw [linearIndependent_iff''ₛ] at hv ⊢
intro s g₁ g₂ hgs hsum i
refine (Group.isUnit (w i)).smul_left_cancel.mp ?_
refine hv s (fun i ↦ w i • g₁ i) (fun i ↦ w i • g₂ i) (fun i hi ↦ ?_) ?_ i
· simp_rw [hgs i hi]
· simpa only [smul_assoc, smul_comm] using hsum | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.LinearIndependent.Defs | {
"line": 480,
"column": 70
} | {
"line": 481,
"column": 48
} | [
{
"pp": "ι : Type u'\nR : Type u_2\nM : Type u_4\nv : ι → M\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nhv : LinearIndependent R v\n⊢ hv.repr.range = ⊤",
"usedConstants": [
"Eq.mpr",
"Submodule",
"RingHomSurjective.ids",
"LinearEquiv.symm",
"Semiring.toM... | by
rw [LinearIndependent.repr, LinearEquiv.range] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.LinearIndependent.Defs | {
"line": 847,
"column": 2
} | {
"line": 848,
"column": 82
} | [
{
"pp": "case refine_2\nι : Type u'\nR : Type u_2\ns : Set ι\nM : Type u_4\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nv : ι → M\nh : ∀ (t : Finset ι) (g : ι → R), ↑t ⊆ s → ∑ i ∈ t, g i • v i = 0 → ∀ i ∈ t, g i = 0\nt : Finset ↑s\ng : ↑s → R\nh0 : ∑ i ∈ t, g i • v ↑i = 0\ni : ↑s\nhit : i ∈ t\... | replace h : ∀ i (hi : i ∈ s), ⟨i, hi⟩ ∈ t → ∀ (h : i ∈ s), g ⟨i, h⟩ = 0 := by
simpa [h0] using h (t.image (↑)) (fun i ↦ if hi : i ∈ s then g ⟨i, hi⟩ else 0) | Lean.Elab.Tactic.evalReplace | Lean.Parser.Tactic.replace |
Mathlib.LinearAlgebra.LinearIndependent.Defs | {
"line": 855,
"column": 2
} | {
"line": 857,
"column": 79
} | [
{
"pp": "ι : Type u'\nR : Type u_2\ns : Set ι\nM : Type u_4\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nv : ι → M\n⊢ LinearIndepOn R v s ↔\n ∀ (t : Finset ι) (g : ι → R), ↑t ⊆ s → (∀ i ∉ t, g i = 0) → ∑ i ∈ t, g i • v i = 0 → ∀ i ∈ t, g i = 0",
"usedConstants": [
"instHSMul",
... | classical
exact linearIndepOn_iff'.trans ⟨fun h t g hts htg h0 ↦ h _ _ hts h0, fun h t g hts h0 ↦
by simpa +contextual [h0] using h t (fun i ↦ if i ∈ t then g i else 0) hts⟩ | Lean.Elab.Tactic.evalClassical | Lean.Parser.Tactic.classical |
Mathlib.LinearAlgebra.LinearIndependent.Defs | {
"line": 855,
"column": 2
} | {
"line": 857,
"column": 79
} | [
{
"pp": "ι : Type u'\nR : Type u_2\ns : Set ι\nM : Type u_4\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nv : ι → M\n⊢ LinearIndepOn R v s ↔\n ∀ (t : Finset ι) (g : ι → R), ↑t ⊆ s → (∀ i ∉ t, g i = 0) → ∑ i ∈ t, g i • v i = 0 → ∀ i ∈ t, g i = 0",
"usedConstants": [
"instHSMul",
... | classical
exact linearIndepOn_iff'.trans ⟨fun h t g hts htg h0 ↦ h _ _ hts h0, fun h t g hts h0 ↦
by simpa +contextual [h0] using h t (fun i ↦ if i ∈ t then g i else 0) hts⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.LinearIndependent.Defs | {
"line": 855,
"column": 2
} | {
"line": 857,
"column": 79
} | [
{
"pp": "ι : Type u'\nR : Type u_2\ns : Set ι\nM : Type u_4\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nv : ι → M\n⊢ LinearIndepOn R v s ↔\n ∀ (t : Finset ι) (g : ι → R), ↑t ⊆ s → (∀ i ∉ t, g i = 0) → ∑ i ∈ t, g i • v i = 0 → ∀ i ∈ t, g i = 0",
"usedConstants": [
"instHSMul",
... | classical
exact linearIndepOn_iff'.trans ⟨fun h t g hts htg h0 ↦ h _ _ hts h0, fun h t g hts h0 ↦
by simpa +contextual [h0] using h t (fun i ↦ if i ∈ t then g i else 0) hts⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Set.Card | {
"line": 529,
"column": 2
} | {
"line": 529,
"column": 28
} | [
{
"pp": "case inr.inr\nα : Type u_1\nβ : Type u_2\ninst✝ : Nonempty β\ns : Set α\nt : Set β\nhs : s.Finite\nhle : s.encard ≤ t.encard\na : α\nhas : a ∈ s\nb : β\nhbt : b ∈ t\nhle' : (s \\ {a}).encard ≤ (t \\ {b}).encard\nf₀ : α → β\nhinj : InjOn f₀ (s \\ {a})\nhf₀s : ∀ x ∈ s, ¬x = a → f₀ x ∈ t ∧ ¬f₀ x = b\n⊢ ∃ ... | use Function.update f₀ a b | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.Data.Set.Card | {
"line": 1102,
"column": 2
} | {
"line": 1102,
"column": 69
} | [
{
"pp": "α : Type u_1\ns : Set α\nn : ℕ\nhns : n ≤ s.ncard\n⊢ ∃ t ⊆ s, t.ncard = n",
"usedConstants": [
"Nat.instCanonicallyOrderedAdd",
"congrArg",
"Set.exists_subsuperset_card_eq",
"zero_le._simp_1",
"AddMonoid.toAddZeroClass",
"Set.ncard_empty",
"Nat.instAddMonoi... | simpa using exists_subsuperset_card_eq s.empty_subset (by simp) hns | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Data.Set.Card | {
"line": 1102,
"column": 2
} | {
"line": 1102,
"column": 69
} | [
{
"pp": "α : Type u_1\ns : Set α\nn : ℕ\nhns : n ≤ s.ncard\n⊢ ∃ t ⊆ s, t.ncard = n",
"usedConstants": [
"Nat.instCanonicallyOrderedAdd",
"congrArg",
"Set.exists_subsuperset_card_eq",
"zero_le._simp_1",
"AddMonoid.toAddZeroClass",
"Set.ncard_empty",
"Nat.instAddMonoi... | simpa using exists_subsuperset_card_eq s.empty_subset (by simp) hns | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Set.Card | {
"line": 1102,
"column": 2
} | {
"line": 1102,
"column": 69
} | [
{
"pp": "α : Type u_1\ns : Set α\nn : ℕ\nhns : n ≤ s.ncard\n⊢ ∃ t ⊆ s, t.ncard = n",
"usedConstants": [
"Nat.instCanonicallyOrderedAdd",
"congrArg",
"Set.exists_subsuperset_card_eq",
"zero_le._simp_1",
"AddMonoid.toAddZeroClass",
"Set.ncard_empty",
"Nat.instAddMonoi... | simpa using exists_subsuperset_card_eq s.empty_subset (by simp) hns | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.Coset.Basic | {
"line": 443,
"column": 27
} | {
"line": 443,
"column": 62
} | [
{
"pp": "α : Type u_1\ninst✝ : Group α\ns t : Subgroup α\nι : Type u_2\nf : ι → Subgroup α\nH : Subgroup α\n⊢ ∀ (a₁ a₂ : ↥H),\n (∀ (x : ι), quotientSubgroupOfMapOfLE H ⋯ (Quotient.mk'' a₁) = quotientSubgroupOfMapOfLE H ⋯ (Quotient.mk'' a₂)) →\n Quotient.mk'' a₁ = Quotient.mk'' a₂",
"usedConstants": ... | quotientSubgroupOfMapOfLE_apply_mk, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.GroupTheory.Congruence.Basic | {
"line": 223,
"column": 29
} | {
"line": 223,
"column": 54
} | [
{
"pp": "M : Type u_1\nN : Type u_2\nP : Type u_3\ninst✝² : MulOneClass M\ninst✝¹ : MulOneClass N\ninst✝ : MulOneClass P\nc : Con M\nx✝ y : M\nf✝ f : M →* P\ng : P → M\nhf : RightInverse g ⇑f\nx : P\n⊢ (kerLift f) ((toQuotient ∘ g) x) = x",
"usedConstants": [
"Eq.mpr",
"MonoidHom.instMonoidHomCl... | (conv_rhs => rw [← hf x]) | Lean.Elab.Tactic.evalParen | Lean.Parser.Tactic.paren |
Mathlib.Order.Interval.Set.SuccPred | {
"line": 159,
"column": 37
} | {
"line": 159,
"column": 54
} | [
{
"pp": "α : Type u_1\ninst✝¹ : LinearOrder α\ninst✝ : PredOrder α\na b : α\nh : a ≤ b\nha : ¬IsMin a\n⊢ insert a (Ioc a b) = Icc a b",
"usedConstants": [
"Eq.mpr",
"Set.Ioc",
"congrArg",
"Set.Ioc_insert_left",
"PartialOrder.toPreorder",
"SemilatticeInf.toPartialOrder",
... | Ioc_insert_left h | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Pi | {
"line": 212,
"column": 2
} | {
"line": 215,
"column": 30
} | [
{
"pp": "R : Type u\nι : Type x\ninst✝³ : Semiring R\nφ : ι → Type i\ninst✝² : (i : ι) → AddCommMonoid (φ i)\ninst✝¹ : (i : ι) → Module R (φ i)\ninst✝ : DecidableEq ι\nI J : Set ι\nhd : Disjoint I J\nhu : Set.univ ⊆ I ∪ J\nhI : I.Finite\n⊢ ⨆ i ∈ I, (single R φ i).range = ⨅ i ∈ J, (proj i).ker",
"usedConstan... | refine le_antisymm (iSup_range_single_le_iInf_ker_proj _ _ _ _ hd) ?_
have : Set.univ ⊆ ↑hI.toFinset ∪ J := by rwa [hI.coe_toFinset]
refine le_trans (iInf_ker_proj_le_iSup_range_single R φ this) (iSup_mono fun i => ?_)
rw [Set.Finite.mem_toFinset] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Pi | {
"line": 212,
"column": 2
} | {
"line": 215,
"column": 30
} | [
{
"pp": "R : Type u\nι : Type x\ninst✝³ : Semiring R\nφ : ι → Type i\ninst✝² : (i : ι) → AddCommMonoid (φ i)\ninst✝¹ : (i : ι) → Module R (φ i)\ninst✝ : DecidableEq ι\nI J : Set ι\nhd : Disjoint I J\nhu : Set.univ ⊆ I ∪ J\nhI : I.Finite\n⊢ ⨆ i ∈ I, (single R φ i).range = ⨅ i ∈ J, (proj i).ker",
"usedConstan... | refine le_antisymm (iSup_range_single_le_iInf_ker_proj _ _ _ _ hd) ?_
have : Set.univ ⊆ ↑hI.toFinset ∪ J := by rwa [hI.coe_toFinset]
refine le_trans (iInf_ker_proj_le_iSup_range_single R φ this) (iSup_mono fun i => ?_)
rw [Set.Finite.mem_toFinset] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Pi | {
"line": 235,
"column": 8
} | {
"line": 235,
"column": 40
} | [
{
"pp": "case pos\nR : Type u\nι : Type x\ninst✝³ : Semiring R\nφ : ι → Type i\ninst✝² : (i : ι) → AddCommMonoid (φ i)\ninst✝¹ : (i : ι) → Module R (φ i)\ninst✝ : DecidableEq ι\nI J : Set ι\nh : Disjoint I J\nb : (i : ι) → φ i\nhI : ∀ i ∈ Iᶜ, b i = 0\nhJ : ∀ i ∈ Jᶜ, b i = 0\ni : ι\nhiI : i ∈ I\nhiJ : i ∈ J\n⊢ b... | exact (h.le_bot ⟨hiI, hiJ⟩).elim | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.LinearAlgebra.Pi | {
"line": 235,
"column": 8
} | {
"line": 235,
"column": 40
} | [
{
"pp": "case pos\nR : Type u\nι : Type x\ninst✝³ : Semiring R\nφ : ι → Type i\ninst✝² : (i : ι) → AddCommMonoid (φ i)\ninst✝¹ : (i : ι) → Module R (φ i)\ninst✝ : DecidableEq ι\nI J : Set ι\nh : Disjoint I J\nb : (i : ι) → φ i\nhI : ∀ i ∈ Iᶜ, b i = 0\nhJ : ∀ i ∈ Jᶜ, b i = 0\ni : ι\nhiI : i ∈ I\nhiJ : i ∈ J\n⊢ b... | exact (h.le_bot ⟨hiI, hiJ⟩).elim | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Pi | {
"line": 235,
"column": 8
} | {
"line": 235,
"column": 40
} | [
{
"pp": "case pos\nR : Type u\nι : Type x\ninst✝³ : Semiring R\nφ : ι → Type i\ninst✝² : (i : ι) → AddCommMonoid (φ i)\ninst✝¹ : (i : ι) → Module R (φ i)\ninst✝ : DecidableEq ι\nI J : Set ι\nh : Disjoint I J\nb : (i : ι) → φ i\nhI : ∀ i ∈ Iᶜ, b i = 0\nhJ : ∀ i ∈ Jᶜ, b i = 0\ni : ι\nhiI : i ∈ I\nhiJ : i ∈ J\n⊢ b... | exact (h.le_bot ⟨hiI, hiJ⟩).elim | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Quotient.Basic | {
"line": 426,
"column": 73
} | {
"line": 426,
"column": 87
} | [
{
"pp": "R : Type u_1\nM : Type u_2\nr : R\nx y : M\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\np p' P Q : Submodule R M\nhf : map (↑(LinearEquiv.refl R M)) P = Q\n⊢ P = Q",
"usedConstants": [
"Submodule",
"RingHomSurjective.ids",
"congrArg",
"AddCommGroup.toAddCom... | simpa using hf | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.LinearAlgebra.Quotient.Basic | {
"line": 426,
"column": 73
} | {
"line": 426,
"column": 87
} | [
{
"pp": "R : Type u_1\nM : Type u_2\nr : R\nx y : M\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\np p' P Q : Submodule R M\nhf : map (↑(LinearEquiv.refl R M)) P = Q\n⊢ P = Q",
"usedConstants": [
"Submodule",
"RingHomSurjective.ids",
"congrArg",
"AddCommGroup.toAddCom... | simpa using hf | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Quotient.Basic | {
"line": 426,
"column": 73
} | {
"line": 426,
"column": 87
} | [
{
"pp": "R : Type u_1\nM : Type u_2\nr : R\nx y : M\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\np p' P Q : Submodule R M\nhf : map (↑(LinearEquiv.refl R M)) P = Q\n⊢ P = Q",
"usedConstants": [
"Submodule",
"RingHomSurjective.ids",
"congrArg",
"AddCommGroup.toAddCom... | simpa using hf | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Prod | {
"line": 213,
"column": 7
} | {
"line": 213,
"column": 74
} | [
{
"pp": "case h\nR : Type u\nM : Type v\nM₂ : Type w\nM₃ : Type y\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : AddCommMonoid M₂\ninst✝³ : AddCommMonoid M₃\ninst✝² : Module R M\ninst✝¹ : Module R M₂\ninst✝ : Module R M₃\nf : M →ₗ[R] M₃\ng : M₂ →ₗ[R] M₃\nx✝ : M₂\n⊢ (f.coprod g ∘ₗ inr R M M₂) x✝ = g x✝... | simp only [map_zero, coprod_apply, inr_apply, zero_add, comp_apply] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Order.Interval.Finset.SuccPred | {
"line": 56,
"column": 2
} | {
"line": 56,
"column": 70
} | [
{
"pp": "α : Type u_2\ninst✝⁴ : LinearOrder α\ninst✝³ : One α\ninst✝² : LocallyFiniteOrder α\ninst✝¹ : Add α\ninst✝ : SuccAddOrder α\na : α\nha : ¬IsMax a\nb : α\n⊢ Icc (a + 1) b = Ioc a b",
"usedConstants": [
"Order.succ",
"Order.succ_eq_add_one",
"congrArg",
"Finset.Icc_succ_left_e... | simpa [succ_eq_add_one] using Icc_succ_left_eq_Ioc_of_not_isMax ha b | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Algebra.Order.Interval.Finset.SuccPred | {
"line": 56,
"column": 2
} | {
"line": 56,
"column": 70
} | [
{
"pp": "α : Type u_2\ninst✝⁴ : LinearOrder α\ninst✝³ : One α\ninst✝² : LocallyFiniteOrder α\ninst✝¹ : Add α\ninst✝ : SuccAddOrder α\na : α\nha : ¬IsMax a\nb : α\n⊢ Icc (a + 1) b = Ioc a b",
"usedConstants": [
"Order.succ",
"Order.succ_eq_add_one",
"congrArg",
"Finset.Icc_succ_left_e... | simpa [succ_eq_add_one] using Icc_succ_left_eq_Ioc_of_not_isMax ha b | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
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