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.Equiv.Basic | {
"line": 295,
"column": 65
} | {
"line": 297,
"column": 5
} | [
{
"pp": "M : Type u_5\nM₂ : Type u_7\ninst✝¹ : AddCommGroup M\ninst✝ : AddCommGroup M₂\nmodM : Module ℤ M\nmodM₂ : Module ℤ M₂\ne : M ≃+ M₂\n⊢ ↑e.toIntLinearEquiv = e",
"usedConstants": [
"AddEquiv.toIntLinearEquiv",
"AddCommGroup.toAddCommMonoid",
"AddEquiv.ext",
"Int",
"AddEq... | by
ext
rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Ring.Subsemiring.Basic | {
"line": 430,
"column": 4
} | {
"line": 430,
"column": 53
} | [
{
"pp": "case refine_2\nR : Type u\ninst✝ : NonAssocSemiring R\nM : Submonoid R\nx✝ : R\nhx : x✝ ∈ AddSubmonoid.closure ↑M\n⊢ x✝ ∈ NonUnitalSubsemiring.closure ↑M",
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"AddMonoid.toAddZeroClass",
"Membership.mem",
"AddSubmono... | induction hx using AddSubmonoid.closure_induction | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.Algebra.Algebra.Hom | {
"line": 257,
"column": 33
} | {
"line": 257,
"column": 71
} | [
{
"pp": "R : Type u\nA : Type v\nB : Type w\nC : Type u₁\nD : Type v₁\ninst✝⁸ : CommSemiring R\ninst✝⁷ : Semiring A\ninst✝⁶ : Semiring B\ninst✝⁵ : Semiring C\ninst✝⁴ : Semiring D\ninst✝³ : Algebra R A\ninst✝² : Algebra R B\ninst✝¹ : Algebra R C\ninst✝ : Algebra R D\nφ : A →ₐ[R] B\nφ₁ : B →ₐ[R] C\nφ₂ : A →ₐ[R] B... | rw [← φ₁.commutes, ← φ₂.commutes]; rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Algebra.Hom | {
"line": 257,
"column": 33
} | {
"line": 257,
"column": 71
} | [
{
"pp": "R : Type u\nA : Type v\nB : Type w\nC : Type u₁\nD : Type v₁\ninst✝⁸ : CommSemiring R\ninst✝⁷ : Semiring A\ninst✝⁶ : Semiring B\ninst✝⁵ : Semiring C\ninst✝⁴ : Semiring D\ninst✝³ : Algebra R A\ninst✝² : Algebra R B\ninst✝¹ : Algebra R C\ninst✝ : Algebra R D\nφ : A →ₐ[R] B\nφ₁ : B →ₐ[R] C\nφ₂ : A →ₐ[R] B... | rw [← φ₁.commutes, ← φ₂.commutes]; rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.GroupWithZero.Associated | {
"line": 316,
"column": 12
} | {
"line": 316,
"column": 17
} | [
{
"pp": "M : Type u_1\ninst✝¹ : CommMonoidWithZero M\ninst✝ : IsCancelMulZero M\na b c d : M\nh : a * b ~ᵤ c * d\nh₁ : a ~ᵤ c\nha : a ≠ 0\nu : Mˣ\nhu : a * b * ↑u = c * d\nv : Mˣ\nhv : c * ↑v = a\n⊢ a * (b * ↑(u * v)) = a * d",
"usedConstants": [
"Units.val",
"Eq.mpr",
"HMul.hMul",
"... | ← hv, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Ring.Subring.Basic | {
"line": 972,
"column": 2
} | {
"line": 974,
"column": 23
} | [
{
"pp": "R : Type u_1\ninst✝ : Ring R\ns : Set R\nC : R → Prop\nh1 : C 1\nhneg1 : C (-1)\nhs : ∀ z ∈ s, ∀ (n : R), C n → C (z * n)\nha : ∀ {x y : R}, C x → C y → C (x + y)\nh0 : C 0\nL : List (List R)\nHL : ∀ t ∈ L, ∀ y ∈ t, y ∈ s ∨ y = -1\n⊢ C (List.map List.prod L).sum",
"usedConstants": [
"NegZeroC... | induction L with
| nil => exact h0
| cons hd tl ih => ?_ | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.Algebra.GroupWithZero.Associated | {
"line": 672,
"column": 2
} | {
"line": 672,
"column": 18
} | [
{
"pp": "case right\nM : Type u_1\ninst✝ : CommMonoidWithZero M\na : Associates M\nx : Associates M\nndvd : ¬a * x ∣ a\n⊢ ¬IsUnit x",
"usedConstants": [
"CommMonoidWithZero.toCommMonoid",
"Semigroup.toMul",
"Associates.instCommMonoid",
"Dvd.dvd",
"HMul.hMul",
"semigroupDv... | contrapose! ndvd | Mathlib.Tactic.Contrapose._aux_Mathlib_Tactic_Contrapose___macroRules_Mathlib_Tactic_Contrapose_contrapose!_1 | Mathlib.Tactic.Contrapose.contrapose! |
Mathlib.Algebra.GroupWithZero.Associated | {
"line": 696,
"column": 6
} | {
"line": 696,
"column": 70
} | [
{
"pp": "case mk.mk.mk\nM : Type u_1\ninst✝¹ : CommMonoidWithZero M\ninst✝ : IsCancelMulZero M\na✝ : Associates M\na : M\nha : Quot.mk (⇑(Associated.setoid M)) a ≠ 0\na₁✝ : Associates M\nb : M\na₂✝ : Associates M\nc : M\nh :\n (fun x ↦ Quot.mk (⇑(Associated.setoid M)) a * x) (Quot.mk (⇑(Associated.setoid M)) b... | exact Quotient.sound' ⟨u, mul_left_cancel₀ (mk_ne_zero.1 ha) hu⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.Module.Submodule.Equiv | {
"line": 85,
"column": 97
} | {
"line": 87,
"column": 5
} | [
{
"pp": "R : Type u_1\nR₂ : Type u_3\nM : Type u_5\nM₂ : Type u_7\ninst✝⁵ : Semiring R\ninst✝⁴ : Semiring R₂\ninst✝³ : AddCommMonoid M\ninst✝² : AddCommMonoid M₂\nσ₁₂ : R →+* R₂\nσ₂₁ : R₂ →+* R\nre₁₂ : RingHomInvPair σ₁₂ σ₂₁\nre₂₁ : RingHomInvPair σ₂₁ σ₁₂\ninst✝¹ : Module R M\ninst✝ : Module R₂ M₂\nf : M ≃ₛₗ[σ₁... | by
ext
rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Order.Closure | {
"line": 76,
"column": 23
} | {
"line": 76,
"column": 92
} | [
{
"pp": "α : Type u_1\nι : Sort u_2\nκ : ι → Sort u_3\ninst✝ : Preorder α\n⊢ Function.Injective fun c ↦ ⇑c.toOrderHom",
"usedConstants": [
"ClosureOperator.mk",
"congrArg",
"ClosureOperator.casesOn",
"OrderHom.toFun",
"Preorder.toLE",
"Eq.rec",
"LE.le",
"Closu... | rintro ⟨⟩ ⟨⟩ h; obtain rfl := DFunLike.ext' h; congr with x; simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.Closure | {
"line": 76,
"column": 23
} | {
"line": 76,
"column": 92
} | [
{
"pp": "α : Type u_1\nι : Sort u_2\nκ : ι → Sort u_3\ninst✝ : Preorder α\n⊢ Function.Injective fun c ↦ ⇑c.toOrderHom",
"usedConstants": [
"ClosureOperator.mk",
"congrArg",
"ClosureOperator.casesOn",
"OrderHom.toFun",
"Preorder.toLE",
"Eq.rec",
"LE.le",
"Closu... | rintro ⟨⟩ ⟨⟩ h; obtain rfl := DFunLike.ext' h; congr with x; simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.SupClosed | {
"line": 145,
"column": 21
} | {
"line": 145,
"column": 51
} | [
{
"pp": "F : Type u_2\nα : Type u_3\nβ : Type u_4\ninst✝³ : SemilatticeInf α\ninst✝² : SemilatticeInf β\ns : Set α\ninst✝¹ : FunLike F β α\ninst✝ : InfHomClass F β α\nhs : InfClosed s\nf : F\na : β\nha : a ∈ ⇑f ⁻¹' s\nb : β\nhb : b ∈ ⇑f ⁻¹' s\n⊢ a ⊓ b ∈ ⇑f ⁻¹' s",
"usedConstants": [
"Eq.mpr",
"c... | simpa [map_inf] using hs ha hb | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Order.SupClosed | {
"line": 145,
"column": 21
} | {
"line": 145,
"column": 51
} | [
{
"pp": "F : Type u_2\nα : Type u_3\nβ : Type u_4\ninst✝³ : SemilatticeInf α\ninst✝² : SemilatticeInf β\ns : Set α\ninst✝¹ : FunLike F β α\ninst✝ : InfHomClass F β α\nhs : InfClosed s\nf : F\na : β\nha : a ∈ ⇑f ⁻¹' s\nb : β\nhb : b ∈ ⇑f ⁻¹' s\n⊢ a ⊓ b ∈ ⇑f ⁻¹' s",
"usedConstants": [
"Eq.mpr",
"c... | simpa [map_inf] using hs ha hb | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.SupClosed | {
"line": 145,
"column": 21
} | {
"line": 145,
"column": 51
} | [
{
"pp": "F : Type u_2\nα : Type u_3\nβ : Type u_4\ninst✝³ : SemilatticeInf α\ninst✝² : SemilatticeInf β\ns : Set α\ninst✝¹ : FunLike F β α\ninst✝ : InfHomClass F β α\nhs : InfClosed s\nf : F\na : β\nha : a ∈ ⇑f ⁻¹' s\nb : β\nhb : b ∈ ⇑f ⁻¹' s\n⊢ a ⊓ b ∈ ⇑f ⁻¹' s",
"usedConstants": [
"Eq.mpr",
"c... | simpa [map_inf] using hs ha hb | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.SupIndep | {
"line": 128,
"column": 97
} | {
"line": 134,
"column": 18
} | [
{
"pp": "α : Type u_1\nι : Type u_3\nι' : Type u_4\ninst✝¹ : Lattice α\ninst✝ : OrderBot α\nf : ι → α\ns : Finset ι'\ng : ι' ↪ ι\n⊢ (map g s).SupIndep f ↔ s.SupIndep (f ∘ ⇑g)",
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"Lattice.toSemilatticeSup",
"congrArg",
"Finset",
"Clas... | by
refine ⟨fun hs t ht i hi hit => ?_, fun hs => ?_⟩
· rw [← sup_map]
exact hs (map_subset_map.2 ht) ((mem_map' _).2 hi) (by rwa [mem_map'])
· classical
rw [map_eq_image]
exact hs.image | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Order.SupIndep | {
"line": 140,
"column": 4
} | {
"line": 140,
"column": 44
} | [
{
"pp": "α : Type u_1\nι : Type u_3\ninst✝² : Lattice α\ninst✝¹ : OrderBot α\nf : ι → α\ninst✝ : DecidableEq ι\ni j : ι\nhij : i ≠ j\nthis : Disjoint (f i) (f j) → Disjoint (f j) (({i, j}.erase j).sup f)\n⊢ {i, j}.SupIndep f ↔ Disjoint (f i) (f j)",
"usedConstants": [
"Eq.mpr",
"False",
"L... | simpa [supIndep_iff_disjoint_erase, hij] | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Order.SupIndep | {
"line": 140,
"column": 4
} | {
"line": 140,
"column": 44
} | [
{
"pp": "α : Type u_1\nι : Type u_3\ninst✝² : Lattice α\ninst✝¹ : OrderBot α\nf : ι → α\ninst✝ : DecidableEq ι\ni j : ι\nhij : i ≠ j\nthis : Disjoint (f i) (f j) → Disjoint (f j) (({i, j}.erase j).sup f)\n⊢ {i, j}.SupIndep f ↔ Disjoint (f i) (f j)",
"usedConstants": [
"Eq.mpr",
"False",
"L... | simpa [supIndep_iff_disjoint_erase, hij] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.SupIndep | {
"line": 140,
"column": 4
} | {
"line": 140,
"column": 44
} | [
{
"pp": "α : Type u_1\nι : Type u_3\ninst✝² : Lattice α\ninst✝¹ : OrderBot α\nf : ι → α\ninst✝ : DecidableEq ι\ni j : ι\nhij : i ≠ j\nthis : Disjoint (f i) (f j) → Disjoint (f j) (({i, j}.erase j).sup f)\n⊢ {i, j}.SupIndep f ↔ Disjoint (f i) (f j)",
"usedConstants": [
"Eq.mpr",
"False",
"L... | simpa [supIndep_iff_disjoint_erase, hij] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.SupIndep | {
"line": 247,
"column": 6
} | {
"line": 247,
"column": 57
} | [
{
"pp": "α : Type u_1\nι : Type u_3\ninst✝³ : Lattice α\ninst✝² : IsModularLattice α\ninst✝¹ : OrderBot α\ninst✝ : DecidableEq ι\ns t : Finset ι\nf : ι → α\nhs : s.SupIndep f\nht : t.SupIndep f\nh : Disjoint (s.sup f) (t.sup f)\n⊢ (s ∪ t).SupIndep f",
"usedConstants": [
"Eq.mpr",
"Finset.instUni... | show s ∪ t = ({s, t} : Finset _).biUnion id by simp | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.CompactlyGenerated.Basic | {
"line": 225,
"column": 19
} | {
"line": 225,
"column": 46
} | [
{
"pp": "α : Type u_2\ninst✝ : CompleteLattice α\nh : IsSupFiniteCompact α\ns : Set α\nhne : s.Nonempty\nhsc : SupClosed s\n⊢ sSup s ∈ s",
"usedConstants": []
}
] | obtain ⟨t, ht₁, ht₂⟩ := h s | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Order.CompactlyGenerated.Basic | {
"line": 405,
"column": 40
} | {
"line": 405,
"column": 51
} | [
{
"pp": "α : Type u_2\ninst✝¹ : CompleteLattice α\ninst✝ : IsCompactlyGenerated α\na : α\ns : Set α\nh : DirectedOn (fun x1 x2 ↦ x1 ≤ x2) s\n⊢ ⨆ b ∈ s, a ⊓ b = ⊥ ↔ ∀ ⦃b : α⦄, b ∈ s → a ⊓ b = ⊥",
"usedConstants": [
"Lattice.toSemilatticeSup",
"CompleteLattice.toLattice",
"congrArg",
"... | iSup_eq_bot | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Order.CompactlyGenerated.Basic | {
"line": 409,
"column": 40
} | {
"line": 409,
"column": 51
} | [
{
"pp": "α : Type u_2\ninst✝¹ : CompleteLattice α\ninst✝ : IsCompactlyGenerated α\na : α\ns : Set α\nh : DirectedOn (fun x1 x2 ↦ x1 ≤ x2) s\n⊢ ⨆ b ∈ s, b ⊓ a = ⊥ ↔ ∀ ⦃b : α⦄, b ∈ s → b ⊓ a = ⊥",
"usedConstants": [
"Lattice.toSemilatticeSup",
"CompleteLattice.toLattice",
"congrArg",
"... | iSup_eq_bot | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Order.CompactlyGenerated.Basic | {
"line": 413,
"column": 40
} | {
"line": 413,
"column": 51
} | [
{
"pp": "ι : Sort u_1\nα : Type u_2\ninst✝¹ : CompleteLattice α\nf : ι → α\ninst✝ : IsCompactlyGenerated α\na : α\nh : Directed (fun x1 x2 ↦ x1 ≤ x2) f\n⊢ ⨆ i, a ⊓ f i = ⊥ ↔ ∀ (i : ι), a ⊓ f i = ⊥",
"usedConstants": [
"Lattice.toSemilatticeSup",
"CompleteLattice.toLattice",
"congrArg",
... | iSup_eq_bot | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Order.CompactlyGenerated.Basic | {
"line": 417,
"column": 40
} | {
"line": 417,
"column": 51
} | [
{
"pp": "ι : Sort u_1\nα : Type u_2\ninst✝¹ : CompleteLattice α\nf : ι → α\ninst✝ : IsCompactlyGenerated α\na : α\nh : Directed (fun x1 x2 ↦ x1 ≤ x2) f\n⊢ ⨆ i, f i ⊓ a = ⊥ ↔ ∀ (i : ι), f i ⊓ a = ⊥",
"usedConstants": [
"Lattice.toSemilatticeSup",
"CompleteLattice.toLattice",
"congrArg",
... | iSup_eq_bot | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Order.CompactlyGenerated.Basic | {
"line": 438,
"column": 55
} | {
"line": 438,
"column": 66
} | [
{
"pp": "α : 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\n⊢ ⨆ t, ⨆ (_ : ↑t ⊆ s \\ {a}), a ⊓ t.sup id = ⊥",
"usedConstants": [
"Eq.mpr",
"Lattice.toSemilatticeSup",
"CompleteLattice.toLattice... | iSup_eq_bot | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.CompactlyGenerated.Basic | {
"line": 455,
"column": 4
} | {
"line": 455,
"column": 66
} | [
{
"pp": "α : Type u_2\ninst✝¹ : CompleteLattice α\ninst✝ : IsCompactlyGenerated α\nι : Type u_3\nf : ι → α\nh : ∀ (s : Finset ι), s.SupIndep f\ni : ι\nhf : ¬InjOn f {i | f i ≠ ⊥}\n⊢ False",
"usedConstants": [
"Lattice.toSemilatticeSup",
"CompleteLattice.toLattice",
"Iff.of_eq",
"cong... | simp_all only [Set.InjOn, ne_eq, Set.mem_setOf_eq, not_forall] | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Algebra.BigOperators.GroupWithZero.Finset | {
"line": 60,
"column": 64
} | {
"line": 62,
"column": 15
} | [
{
"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": [
"Mathlib.Tactic.Push.not_exists._simp_1",
"CommMonoidWithZero.toCommMonoid",
"Eq.mpr... | by
rw [Ne, prod_eq_zero_iff]
push Not; rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.BigOperators.Pi | {
"line": 88,
"column": 29
} | {
"line": 88,
"column": 43
} | [
{
"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",
"usedConstants": [
"congrArg",
"Set.iInter",
"Finset",
"Membership.mem",
"Exists",
"Eq.mp",
"Set.mem_iInter... | simpa using hj | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Algebra.BigOperators.Pi | {
"line": 88,
"column": 29
} | {
"line": 88,
"column": 43
} | [
{
"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",
"usedConstants": [
"congrArg",
"Set.iInter",
"Finset",
"Membership.mem",
"Exists",
"Eq.mp",
"Set.mem_iInter... | simpa using hj | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.BigOperators.Pi | {
"line": 88,
"column": 29
} | {
"line": 88,
"column": 43
} | [
{
"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",
"usedConstants": [
"congrArg",
"Set.iInter",
"Finset",
"Membership.mem",
"Exists",
"Eq.mp",
"Set.mem_iInter... | simpa using hj | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.CompactlyGenerated.Basic | {
"line": 525,
"column": 4
} | {
"line": 525,
"column": 15
} | [
{
"pp": "α : Type u_2\ninst✝¹ : CompleteLattice α\ninst✝ : IsCompactlyGenerated α\nι : Type u_3\nf : ι → α\ns : Set ι\na : α\nhs : ∀ t ⊆ s, t.Finite → Disjoint (⨆ i ∈ t, f i) a\n⊢ ⨆ t, ⨆ (_ : ↑t ⊆ range fun x ↦ f ↑x), a ⊓ t.sup id = ⊥",
"usedConstants": [
"Eq.mpr",
"Lattice.toSemilatticeSup",
... | iSup_eq_bot | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Order.CompactlyGenerated.Basic | {
"line": 532,
"column": 2
} | {
"line": 532,
"column": 19
} | [
{
"pp": "α : Type u_2\ninst✝¹ : CompleteLattice α\ninst✝ : IsCompactlyGenerated α\nι : Type u_3\nf : ι → α\ns : Set ι\na : α\nhs : ∀ t ⊆ s, t.Finite → Disjoint (⨆ i ∈ t, f i) a\nu : Finset α\nhu : ↑u ⊆ range fun x ↦ f ↑x\nt : Set ι\nht : t ⊆ s\nht' : t.Finite\nhtu : u.sup id = ⨆ i ∈ t, f i\n⊢ Disjoint (⨆ i ∈ t,... | exact hs t ht ht' | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.BigOperators.Pi | {
"line": 188,
"column": 6
} | {
"line": 188,
"column": 21
} | [
{
"pp": "ι✝ : Type u_1\nκ : Type u_2\nM✝ : Type u_3\nN : Type u_4\nR : Type u_5\nα : Type u_6\nι : Type u_7\ninst✝³ : Fintype ι\ninst✝² : DecidableEq ι\nM : ι → Type u_8\ninst✝¹ : (i : ι) → CommMonoid (M i)\nM' : Type u_9\ninst✝ : CommMonoid M'\nφ : (i : ι) → M i →* M'\ni : ι\nm : M i\nφ' : (i : ι) → M i → M' :... | enter [1, 2, j] | Lean.Elab.Tactic.Conv.evalEnter | Lean.Parser.Tactic.Conv.enter |
Mathlib.LinearAlgebra.Span.Basic | {
"line": 154,
"column": 4
} | {
"line": 154,
"column": 22
} | [
{
"pp": "case a\nR : Type u_1\nM : Type u_4\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\ns : Set M\nr : R\nhr : IsUnit r\n⊢ span R (r • s) ≤ span R s",
"usedConstants": [
"Submodule.span_smul_le"
]
}
] | apply span_smul_le | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.LinearAlgebra.Span.Basic | {
"line": 154,
"column": 4
} | {
"line": 154,
"column": 22
} | [
{
"pp": "case a\nR : Type u_1\nM : Type u_4\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\ns : Set M\nr : R\nhr : IsUnit r\n⊢ span R (r • s) ≤ span R s",
"usedConstants": [
"Submodule.span_smul_le"
]
}
] | apply span_smul_le | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Span.Basic | {
"line": 154,
"column": 4
} | {
"line": 154,
"column": 22
} | [
{
"pp": "case a\nR : Type u_1\nM : Type u_4\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\ns : Set M\nr : R\nhr : IsUnit r\n⊢ span R (r • s) ≤ span R s",
"usedConstants": [
"Submodule.span_smul_le"
]
}
] | apply span_smul_le | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Span.Basic | {
"line": 331,
"column": 2
} | {
"line": 331,
"column": 66
} | [
{
"pp": "R : Type u_1\nM : Type u_4\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nx : M\nd : Set (Submodule R M)\nhemp : d.Nonempty\nhdir : DirectedOn (fun x1 x2 ↦ x1 ≤ x2) d\nhsup : R ∙ x ≤ sSup d\nthis : x ∈ sSup d\ny : Submodule R M\nhyd : y ∈ d\nhxy : x ∈ y\n⊢ ∃ x_1 ∈ d, R ∙ x ≤ x_1",
... | exact ⟨y, ⟨hyd, by simpa only [span_le, singleton_subset_iff] ⟩⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Data.Finsupp.Single | {
"line": 352,
"column": 2
} | {
"line": 353,
"column": 24
} | [
{
"pp": "α : Type u_1\nM : Type u_5\ninst✝¹ : Zero M\ninst✝ : DecidableEq α\na a' : α\nf : α →₀ M\n⊢ (erase a f) a' = if a' = a then 0 else f a'",
"usedConstants": [
"Finsupp.instFunLike",
"Eq.mpr",
"Finsupp.erase._proof_2",
"Finsupp.erase",
"congrArg",
"Finsupp.support",... | rw [erase, coe_mk]
simp only [ite_eq_ite] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Finsupp.Single | {
"line": 352,
"column": 2
} | {
"line": 353,
"column": 24
} | [
{
"pp": "α : Type u_1\nM : Type u_5\ninst✝¹ : Zero M\ninst✝ : DecidableEq α\na a' : α\nf : α →₀ M\n⊢ (erase a f) a' = if a' = a then 0 else f a'",
"usedConstants": [
"Finsupp.instFunLike",
"Eq.mpr",
"Finsupp.erase._proof_2",
"Finsupp.erase",
"congrArg",
"Finsupp.support",... | rw [erase, coe_mk]
simp only [ite_eq_ite] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Finsupp.Indicator | {
"line": 91,
"column": 96
} | {
"line": 92,
"column": 19
} | [
{
"pp": "ι : Type u_1\nα : Type u_2\ninst✝ : Zero α\ns : Finset ι\nd : ι →₀ α\n⊢ (d = indicator s fun i x ↦ d i) ↔ d.support ⊆ s",
"usedConstants": [
"_private.Mathlib.Data.Finsupp.Indicator.0.Finsupp.eq_indicator_self_iff._proof_1_3"
]
}
] | by
grind [indicator] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Order.AbsoluteValue.Basic | {
"line": 289,
"column": 39
} | {
"line": 289,
"column": 55
} | [
{
"pp": "R : Type u_5\nS : Type u_6\ninst✝³ : Ring R\ninst✝² : CommRing S\ninst✝¹ : LinearOrder S\ninst✝ : IsStrictOrderedRing S\nabv : AbsoluteValue R S\na b : R\n⊢ abv b - abv a ≤ abv (a - b)",
"usedConstants": [
"Eq.mpr",
"IsDomain.to_noZeroDivisors",
"AddGroupWithOne.toAddGroup",
... | rw [abv.map_sub] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Order.AbsoluteValue.Basic | {
"line": 372,
"column": 4
} | {
"line": 372,
"column": 20
} | [
{
"pp": "case inl\nR : Type u_3\nS : Type u_4\ninst✝⁴ : Field R\ninst✝³ : Semifield S\ninst✝² : LinearOrder S\ninst✝¹ : IsStrictOrderedRing S\ninst✝ : ExistsAddOfLE S\nv : AbsoluteValue R S\nx : R\nhx₀ : x ≠ 0\nhx₁ : v x ≠ 1\nh : v x < 1\n⊢ ∃ x, 1 < v x",
"usedConstants": [
"NonAssocSemiring.toAddComm... | refine ⟨x⁻¹, ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Algebra.Order.BigOperators.Group.Multiset | {
"line": 167,
"column": 80
} | {
"line": 170,
"column": 24
} | [
{
"pp": "ι : Type u_1\nα : Type u_2\ninst✝² : CommMonoid α\ninst✝¹ : LinearOrder α\ninst✝ : IsOrderedMonoid α\ns : Multiset ι\nf g : ι → α\n⊢ (map (fun i ↦ min (f i) (g i)) s).prod ≤ min (map f s).prod (map g s).prod",
"usedConstants": [
"CommMonoid.toCommSemigroup",
"Multiset.map",
"List.... | by
obtain ⟨l⟩ := s
simp_rw [Multiset.quot_mk_to_coe'', Multiset.map_coe, Multiset.prod_coe]
apply List.prod_min_le | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Order.BigOperators.Group.Finset | {
"line": 570,
"column": 4
} | {
"line": 571,
"column": 27
} | [] | ∏ j ∈ s, f j < ∏ j ∈ s, 1 := prod_lt_prod' h₁ ⟨i, m, (h₁ i m).lt_of_ne i_ne⟩
_ = 1 := prod_const_one | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcSteps |
Mathlib.Algebra.Order.BigOperators.Group.Finset | {
"line": 566,
"column": 2
} | {
"line": 571,
"column": 27
} | [
{
"pp": "ι : Type u_1\nM : Type u_4\ninst✝² : CommMonoid M\ninst✝¹ : LinearOrder M\ns : Finset ι\ninst✝ : IsOrderedCancelMonoid M\nf : ι → M\nh₁ : ∏ i ∈ s, f i = 1\nh₂ : ∃ i ∈ s, f i ≠ 1\n⊢ ∃ i ∈ s, 1 < f i",
"usedConstants": [
"Mathlib.Tactic.Push.not_exists._simp_1",
"Eq.mpr",
"Mathlib.T... | contrapose! h₁
obtain ⟨i, m, i_ne⟩ : ∃ i ∈ s, f i ≠ 1 := h₂
apply ne_of_lt
calc
∏ j ∈ s, f j < ∏ j ∈ s, 1 := prod_lt_prod' h₁ ⟨i, m, (h₁ i m).lt_of_ne i_ne⟩
_ = 1 := prod_const_one | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Order.BigOperators.Group.Finset | {
"line": 566,
"column": 2
} | {
"line": 571,
"column": 27
} | [
{
"pp": "ι : Type u_1\nM : Type u_4\ninst✝² : CommMonoid M\ninst✝¹ : LinearOrder M\ns : Finset ι\ninst✝ : IsOrderedCancelMonoid M\nf : ι → M\nh₁ : ∏ i ∈ s, f i = 1\nh₂ : ∃ i ∈ s, f i ≠ 1\n⊢ ∃ i ∈ s, 1 < f i",
"usedConstants": [
"Mathlib.Tactic.Push.not_exists._simp_1",
"Eq.mpr",
"Mathlib.T... | contrapose! h₁
obtain ⟨i, m, i_ne⟩ : ∃ i ∈ s, f i ≠ 1 := h₂
apply ne_of_lt
calc
∏ j ∈ s, f j < ∏ j ∈ s, 1 := prod_lt_prod' h₁ ⟨i, m, (h₁ i m).lt_of_ne i_ne⟩
_ = 1 := prod_const_one | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Finsupp.Basic | {
"line": 539,
"column": 53
} | {
"line": 541,
"column": 5
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nM : Type u_5\ninst✝ : Zero M\nf : α → β\nhif : Set.InjOn f (f ⁻¹' ↑(support 0))\n⊢ comapDomain f 0 hif = 0",
"usedConstants": [
"Finsupp.instFunLike",
"Finsupp.ext",
"Zero.toOfNat0",
"Eq.refl",
"Finsupp.comapDomain",
"Finsupp.instZero"... | by
ext
rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Finsupp.SMul | {
"line": 169,
"column": 60
} | {
"line": 171,
"column": 5
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nM : Type u_3\nR : Type u_6\ninst✝¹ : Zero M\ninst✝ : SMulZeroClass R M\nf : α → β\nr : R\nv : β →₀ M\nhfv : Set.InjOn f (f ⁻¹' ↑v.support)\nhfrv : Set.InjOn f (f ⁻¹' ↑(r • v).support)\n⊢ comapDomain f (r • v) hfrv = r • comapDomain f v hfv",
"usedConstants": [
"Fin... | by
ext
rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Order.BigOperators.Ring.Finset | {
"line": 55,
"column": 2
} | {
"line": 61,
"column": 58
} | [
{
"pp": "ι : 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 i\n⊢ ∏ i ∈ s, g i ... | simp_rw [prod_eq_mul_prod_diff_singleton_of_mem hi]
refine le_trans ?_ (mul_le_mul_of_nonneg_right h2i ?_)
· rw [right_distrib]
gcongr with j hj <;> aesop
· apply prod_nonneg
simp only [and_imp, mem_sdiff, mem_singleton]
exact fun j hj hji ↦ le_trans (hg j hj) (hgf j hj hji) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Order.BigOperators.Ring.Finset | {
"line": 55,
"column": 2
} | {
"line": 61,
"column": 58
} | [
{
"pp": "ι : 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 i\n⊢ ∏ i ∈ s, g i ... | simp_rw [prod_eq_mul_prod_diff_singleton_of_mem hi]
refine le_trans ?_ (mul_le_mul_of_nonneg_right h2i ?_)
· rw [right_distrib]
gcongr with j hj <;> aesop
· apply prod_nonneg
simp only [and_imp, mem_sdiff, mem_singleton]
exact fun j hj hji ↦ le_trans (hg j hj) (hgf j hj hji) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Finsupp.LSum | {
"line": 78,
"column": 31
} | {
"line": 78,
"column": 62
} | [
{
"pp": "α : Type u_1\nM✝ : Type u_2\nN✝ : Type u_3\nP : Type u_4\nR✝ : Type u_5\nR₂ : Type u_6\nR₃ : Type u_7\nS✝ : Type u_8\ninst✝¹⁷ : Semiring R✝\ninst✝¹⁶ : Semiring R₂\ninst✝¹⁵ : Semiring R₃\ninst✝¹⁴ : Semiring S✝\ninst✝¹³ : AddCommMonoid M✝\ninst✝¹² : Module R✝ M✝\ninst✝¹¹ : AddCommMonoid N✝\ninst✝¹⁰ : Mod... | sum_mapRange_index single_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Finsupp.LSum | {
"line": 234,
"column": 52
} | {
"line": 236,
"column": 5
} | [
{
"pp": "M : Type u_2\nN : Type u_3\nR : Type u_5\nR₂ : Type u_6\ninst✝⁷ : Semiring R\ninst✝⁶ : Semiring R₂\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\ninst✝³ : AddCommMonoid N\ninst✝² : Module R₂ N\nσ : R →+* R₂\nσ_inv : R₂ →+* R\ninst✝¹ : RingHomInvPair σ σ_inv\ninst✝ : RingHomInvPair σ_inv σ\nι : Type u_... | by
ext
rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Order.Interval.Set.Fin | {
"line": 96,
"column": 83
} | {
"line": 96,
"column": 97
} | [
{
"pp": "n : ℕ\ni j : Fin n\n⊢ val '' uIoo i j = uIoo ↑i ↑j",
"usedConstants": [
"Lattice.toSemilatticeSup",
"congrArg",
"PartialOrder.toPreorder",
"SemilatticeInf.toPartialOrder",
"SemilatticeSup.toMax",
"DistribLattice.toLattice",
"SemilatticeInf.toMin",
"Fi... | by simp [uIoo] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Order.Interval.Set.Fin | {
"line": 797,
"column": 71
} | {
"line": 797,
"column": 93
} | [
{
"pp": "n : ℕ\ni : Fin n\n⊢ rev ⁻¹' Ici i = Iic i.rev",
"usedConstants": [
"Set.ext",
"Set.Ici",
"_private.Mathlib.Order.Interval.Set.Fin.0.Fin.preimage_rev_Ici._simp_1_1",
"congrArg",
"PartialOrder.toPreorder",
"Preorder.toLE",
"Set.mem_Iic._simp_2",
"Member... | ext; simp [le_rev_iff] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.Interval.Set.Fin | {
"line": 797,
"column": 71
} | {
"line": 797,
"column": 93
} | [
{
"pp": "n : ℕ\ni : Fin n\n⊢ rev ⁻¹' Ici i = Iic i.rev",
"usedConstants": [
"Set.ext",
"Set.Ici",
"_private.Mathlib.Order.Interval.Set.Fin.0.Fin.preimage_rev_Ici._simp_1_1",
"congrArg",
"PartialOrder.toPreorder",
"Preorder.toLE",
"Set.mem_Iic._simp_2",
"Member... | ext; simp [le_rev_iff] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Finsupp.LinearCombination | {
"line": 315,
"column": 56
} | {
"line": 315,
"column": 66
} | [
{
"pp": "α : Type u_1\nM : Type u_2\nR : Type u_3\ninst✝⁶ : Fintype α\ninst✝⁵ : Semiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nS : Type u_4\ninst✝² : Semiring S\ninst✝¹ : Module S M\ninst✝ : SMulCommClass R S M\nv : α → M\nf g : α → R\n⊢ ∑ i, (f + g) i • v i = ∑ x, (f x • v x + g x • v x)",
"use... | ← add_smul | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.LinearAlgebra.Basis.Defs | {
"line": 439,
"column": 2
} | {
"line": 441,
"column": 54
} | [
{
"pp": "ι : Type u_10\nR : Type u_11\nM : Type u_12\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nb : Basis ι R M\nx : M\ni : ι\na✝ : Nontrivial R\n⊢ ∀ (x y : M),\n (fun x i ↦ (b.reindexRange.repr x) ⟨b i, ⋯⟩) (x + y) =\n (fun x i ↦ (b.reindexRange.repr x) ⟨b i, ⋯⟩) x + (fun x i ↦... | · intro x y
ext i
simp only [Pi.add_apply, map_add, Finsupp.coe_add] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.LinearAlgebra.Finsupp.LinearCombination | {
"line": 503,
"column": 4
} | {
"line": 503,
"column": 45
} | [
{
"pp": "case refine_1\nR : Type u_1\nM : Type u_2\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nm : M\ns : Set M\nh : m ∈ span R s\n⊢ ∃ n f g, ∑ i, f i • ↑(g i) = m",
"usedConstants": [
"Submodule",
"instHSMul",
"DistribMulAction.toDistribSMul",
"Finset",
... | rcases mem_span_set.1 h with ⟨c, cs, rfl⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Algebra.BigOperators.Finprod | {
"line": 843,
"column": 2
} | {
"line": 845,
"column": 46
} | [
{
"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⊢ ∏ᶠ (i : α) (_ : i ∈ insert a s), f i = f a * ∏ᶠ (i : α) (_ : i ∈ s), f i",
"usedConstants": [
"Eq.mpr",
"finprod_mem_union'",
"MulOne.toOne",
"HMul.hMu... | rw [insert_eq, finprod_mem_union' _ _ hs, finprod_mem_singleton]
· rwa [disjoint_singleton_left]
· exact (finite_singleton a).inter_of_left _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.BigOperators.Finprod | {
"line": 843,
"column": 2
} | {
"line": 845,
"column": 46
} | [
{
"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⊢ ∏ᶠ (i : α) (_ : i ∈ insert a s), f i = f a * ∏ᶠ (i : α) (_ : i ∈ s), f i",
"usedConstants": [
"Eq.mpr",
"finprod_mem_union'",
"MulOne.toOne",
"HMul.hMu... | rw [insert_eq, finprod_mem_union' _ _ hs, finprod_mem_singleton]
· rwa [disjoint_singleton_left]
· exact (finite_singleton a).inter_of_left _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Basis.Defs | {
"line": 689,
"column": 2
} | {
"line": 690,
"column": 88
} | [
{
"pp": "ι : Type u_10\nR : Type u_11\nM : Type u_12\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nb : Basis ι R M\ni : ι\n⊢ b.sumCoords (b i) = 1",
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"Semiring.toModule",
"Finsupp.module",
"congrArg",
... | simp only [Basis.sumCoords, LinearMap.id_coe, LinearEquiv.coe_coe, id, Basis.repr_self,
Function.comp_apply, Finsupp.coe_lsum, LinearMap.coe_comp, Finsupp.sum_single_index] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.LinearAlgebra.Basis.Defs | {
"line": 689,
"column": 2
} | {
"line": 690,
"column": 88
} | [
{
"pp": "ι : Type u_10\nR : Type u_11\nM : Type u_12\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nb : Basis ι R M\ni : ι\n⊢ b.sumCoords (b i) = 1",
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"Semiring.toModule",
"Finsupp.module",
"congrArg",
... | simp only [Basis.sumCoords, LinearMap.id_coe, LinearEquiv.coe_coe, id, Basis.repr_self,
Function.comp_apply, Finsupp.coe_lsum, LinearMap.coe_comp, Finsupp.sum_single_index] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Basis.Defs | {
"line": 689,
"column": 2
} | {
"line": 690,
"column": 88
} | [
{
"pp": "ι : Type u_10\nR : Type u_11\nM : Type u_12\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nb : Basis ι R M\ni : ι\n⊢ b.sumCoords (b i) = 1",
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"Semiring.toModule",
"Finsupp.module",
"congrArg",
... | simp only [Basis.sumCoords, LinearMap.id_coe, LinearEquiv.coe_coe, id, Basis.repr_self,
Function.comp_apply, Finsupp.coe_lsum, LinearMap.coe_comp, Finsupp.sum_single_index] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.BigOperators.Finprod | {
"line": 943,
"column": 2
} | {
"line": 943,
"column": 33
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nM : Type u_5\ninst✝ : CommMonoid M\ns : Set α\nt : Set β\nf : α → M\ng : β → M\ne : α → β\nhe₀ : BijOn e s t\nhe₁ : ∀ x ∈ s, f x = g (e x)\n⊢ ∏ᶠ (i : α) (_ : i ∈ s), f i = ∏ᶠ (j : α) (_ : j ∈ s), g (e j)",
"usedConstants": [
"finprod_mem_congr",
"rfl",
... | exact finprod_mem_congr rfl he₁ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Data.Fintype.Fin | {
"line": 63,
"column": 59
} | {
"line": 63,
"column": 78
} | [
{
"pp": "case cons\nα : Type u_1\nn✝ : ℕ\ninst✝ : DecidableEq α\na : α\nn : ℕ\nx : α\nxs : List.Vector α n\nhxs : #{i | xs.get i = a} = List.count a xs.toList\n⊢ (if x = a then 1 else 0) + #{x_1 | (x ::ᵥ xs).get x_1.succ = a} = List.count a (x ::ᵥ xs).toList",
"usedConstants": [
"List.Vector.get",
... | Vector.toList_cons, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Algebra.BigOperators.Finprod | {
"line": 1084,
"column": 6
} | {
"line": 1084,
"column": 22
} | [
{
"pp": "case neg\nα : Type u_1\nM : Type u_5\ninst✝ : CommMonoid M\nf : α → M\na : α\nhf : HasFiniteMulSupport f\nh : ∀ (x : α), f x ≠ 1 → (x ≠ a ↔ x ∈ Finite.toFinset hf \\ {a})\nha : f a = 1\n⊢ f a * ∏ i ∈ (Finite.toFinset hf).erase a, f i = ∏ i ∈ Finite.toFinset hf, f i",
"usedConstants": [
"Eq.mp... | rw [ha, one_mul] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Data.Fin.VecNotation | {
"line": 229,
"column": 2
} | {
"line": 229,
"column": 16
} | [
{
"pp": "case h\nα : Type u\nm : ℕ\nx : α\nu : Fin m → α\nx✝ : Fin m\n⊢ vecTail (vecCons x u) x✝ = u x✝",
"usedConstants": [
"congrArg",
"Matrix.cons_val_succ",
"True",
"eq_self",
"of_eq_true",
"congrFun'",
"Matrix.vecTail",
"Eq",
"Matrix.vecCons",
... | simp [vecTail] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.BigOperators.Finprod | {
"line": 1332,
"column": 42
} | {
"line": 1332,
"column": 66
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nM : Type u_5\ninst✝¹ : CommMonoid M\nf : α → β\nhf : Injective f\ninst✝ : DecidablePred fun x ↦ x ∈ range f\ng : α → M\n⊢ (∏ᶠ (j : α), if h' : f j ∈ range f then g (Classical.choose h') else 1) = ∏ᶠ (a : α), g a",
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
... | finprod_congr fun a => _ | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Basis.Basic | {
"line": 66,
"column": 2
} | {
"line": 66,
"column": 55
} | [
{
"pp": "ι : Type u_1\nR : Type u_3\nM : Type u_5\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\nb : Basis ι R M\ninst✝ : Nontrivial R\ni : ι\ns : Set ι\n⊢ b i ∈ span R (⇑b '' s) ↔ i ∈ s",
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"Submodule",
"Fal... | simp [mem_span_image, Finsupp.support_single_ne_zero] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.LinearAlgebra.Basis.Basic | {
"line": 66,
"column": 2
} | {
"line": 66,
"column": 55
} | [
{
"pp": "ι : Type u_1\nR : Type u_3\nM : Type u_5\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\nb : Basis ι R M\ninst✝ : Nontrivial R\ni : ι\ns : Set ι\n⊢ b i ∈ span R (⇑b '' s) ↔ i ∈ s",
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"Submodule",
"Fal... | simp [mem_span_image, Finsupp.support_single_ne_zero] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Basis.Basic | {
"line": 66,
"column": 2
} | {
"line": 66,
"column": 55
} | [
{
"pp": "ι : Type u_1\nR : Type u_3\nM : Type u_5\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\nb : Basis ι R M\ninst✝ : Nontrivial R\ni : ι\ns : Set ι\n⊢ b i ∈ span R (⇑b '' s) ↔ i ∈ s",
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"Submodule",
"Fal... | simp [mem_span_image, Finsupp.support_single_ne_zero] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.ENat.Pow | {
"line": 124,
"column": 6
} | {
"line": 124,
"column": 46
} | [
{
"pp": "case inl.inl\nx z : ℕ∞\nx_0 : x < 1\n⊢ 0 ^ (0 + z) = 0 ^ 0 * 0 ^ z",
"usedConstants": [
"instAddMonoidWithOneENat",
"HMul.hMul",
"congrArg",
"CommSemiring.toSemiring",
"AddMonoid.toAddZeroClass",
"NonUnitalNonAssocSemiring.toMulZeroClass",
"AddZeroClass.toA... | simp only [zero_add, epow_zero, one_mul] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Data.ENat.Pow | {
"line": 124,
"column": 6
} | {
"line": 124,
"column": 46
} | [
{
"pp": "case inl.inl\nx z : ℕ∞\nx_0 : x < 1\n⊢ 0 ^ (0 + z) = 0 ^ 0 * 0 ^ z",
"usedConstants": [
"instAddMonoidWithOneENat",
"HMul.hMul",
"congrArg",
"CommSemiring.toSemiring",
"AddMonoid.toAddZeroClass",
"NonUnitalNonAssocSemiring.toMulZeroClass",
"AddZeroClass.toA... | simp only [zero_add, epow_zero, one_mul] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.ENat.Pow | {
"line": 124,
"column": 6
} | {
"line": 124,
"column": 46
} | [
{
"pp": "case inl.inl\nx z : ℕ∞\nx_0 : x < 1\n⊢ 0 ^ (0 + z) = 0 ^ 0 * 0 ^ z",
"usedConstants": [
"instAddMonoidWithOneENat",
"HMul.hMul",
"congrArg",
"CommSemiring.toSemiring",
"AddMonoid.toAddZeroClass",
"NonUnitalNonAssocSemiring.toMulZeroClass",
"AddZeroClass.toA... | simp only [zero_add, epow_zero, one_mul] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Nat.GCD.Basic | {
"line": 47,
"column": 2
} | {
"line": 47,
"column": 40
} | [
{
"pp": "case inr.inr\na b c : ℕ\nha0 : a > 0\nha1 : succ 0 < a\n⊢ (a ^ c - 1) % (a ^ b - 1) = a ^ (c % b) - 1",
"usedConstants": [
"Nat.eq_zero_or_pos"
]
}
] | rcases eq_zero_or_pos b with rfl | hb0 | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Algebra.BigOperators.Fin | {
"line": 277,
"column": 82
} | {
"line": 278,
"column": 29
} | [
{
"pp": "M : Type u_2\ninst✝ : CommMonoid M\nn m : ℕ\nh : n ≤ m\nf : Fin m → M\na b : Fin n\n⊢ ∏ i ∈ uIcc (castLE h a) (castLE h b), f i = ∏ i ∈ uIcc a b, f (castLE h i)",
"usedConstants": [
"congrArg",
"Fin.castLE",
"Finset",
"Fin.instLocallyFiniteOrder",
"Fin.castLEEmb",
... | by
simp [← map_castLEEmb_uIcc] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Nat.ModEq | {
"line": 545,
"column": 97
} | {
"line": 546,
"column": 52
} | [
{
"pp": "a b c : ℕ\nhca : c ∣ b\n⊢ (a + b) / c = a / c + b / c",
"usedConstants": [
"Eq.mpr",
"instHDiv",
"congrArg",
"id",
"HDiv.hDiv",
"add_comm",
"instHAdd",
"HAdd.hAdd",
"Nat",
"Nat.instDiv",
"Nat.add_div_of_dvd_right",
"instAddNat"... | by
rwa [add_comm, Nat.add_div_of_dvd_right, add_comm] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.SetTheory.Cardinal.Finite | {
"line": 179,
"column": 2
} | {
"line": 180,
"column": 22
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nf : α → β\nhf : Injective f\n⊢ Nat.card ↑(range f) = Nat.card α",
"usedConstants": [
"Eq.mpr",
"le_rfl",
"CompleteLattice.instOmegaCompletePartialOrder",
"congrArg",
"Set.univ",
"PartialOrder.toPreorder",
"Set.Elem",
"id",
... | rw [← Nat.card_preimage_of_injective hf le_rfl]
simp [Nat.card_univ] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.SetTheory.Cardinal.Finite | {
"line": 179,
"column": 2
} | {
"line": 180,
"column": 22
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nf : α → β\nhf : Injective f\n⊢ Nat.card ↑(range f) = Nat.card α",
"usedConstants": [
"Eq.mpr",
"le_rfl",
"CompleteLattice.instOmegaCompletePartialOrder",
"congrArg",
"Set.univ",
"PartialOrder.toPreorder",
"Set.Elem",
"id",
... | rw [← Nat.card_preimage_of_injective hf le_rfl]
simp [Nat.card_univ] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.SetTheory.Cardinal.Finite | {
"line": 397,
"column": 6
} | {
"line": 398,
"column": 89
} | [
{
"pp": "case inr.inl.inr\nα : Type u_3\nβ : Type u_4\nα_emp : Nonempty α\nh✝¹ : Finite α\nh✝ : Infinite β\n⊢ card (α → β) = card β ^ card α",
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"instAddMonoidWithOneENat",
"instTopENat",
"congrArg",
"CommSemiring.toSemiring",
"... | simp only [card_eq_top_of_infinite]
exact (top_epow (one_le_iff_ne_zero.1 ((one_le_card_iff_nonempty α).2 α_emp))).symm | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.SetTheory.Cardinal.Finite | {
"line": 397,
"column": 6
} | {
"line": 398,
"column": 89
} | [
{
"pp": "case inr.inl.inr\nα : Type u_3\nβ : Type u_4\nα_emp : Nonempty α\nh✝¹ : Finite α\nh✝ : Infinite β\n⊢ card (α → β) = card β ^ card α",
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"instAddMonoidWithOneENat",
"instTopENat",
"congrArg",
"CommSemiring.toSemiring",
"... | simp only [card_eq_top_of_infinite]
exact (top_epow (one_le_iff_ne_zero.1 ((one_le_card_iff_nonempty α).2 α_emp))).symm | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.LinearIndependent.Basic | {
"line": 467,
"column": 2
} | {
"line": 468,
"column": 7
} | [
{
"pp": "ι : Type u'\nR : Type u_2\ns : Set ι\nM : Type u_4\nv : ι → M\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nt : Set ι\nhdj : Disjoint s t\nh : LinearIndepOn R v (s ∪ t)\n⊢ Disjoint (span R (v '' s)) (span R (v '' t))",
"usedConstants": [
"Set.ext",
"Eq.mpr",
"Subm... | convert h.disjoint_span_image (s := (↑) ⁻¹' s) (t := (↑) ⁻¹' t) (hdj.preimage _) <;>
aesop | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.LinearAlgebra.Dimension.Basic | {
"line": 239,
"column": 2
} | {
"line": 240,
"column": 78
} | [
{
"pp": "R : Type u_1\ninst✝ : CommSemiring R\na✝ : Nontrivial R\n⊢ Module.rank R R = 1",
"usedConstants": [
"Eq.mpr",
"Pi.Function.module",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Preorder.toLT",
"Semiring.toModule",
"Pi.addCommMonoid",
"Cardinal.instOne",
... | rw [le_antisymm_iff, ← not_lt, ← Order.succ_le_iff, ← Nat.cast_one, ← nat_succ,
Module.le_rank_iff_exists_linearMap, Nat.cast_one, Module.one_le_rank_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.Dimension.Finrank | {
"line": 81,
"column": 4
} | {
"line": 81,
"column": 38
} | [
{
"pp": "case hc\nR : Type u\nM : Type v\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nn : ℕ\nh : Module.rank R M ≤ ↑n\n⊢ Module.rank R M < ℵ₀",
"usedConstants": [
"Cardinal",
"PartialOrder.toPreorder",
"Cardinal.aleph0",
"Cardinal.natCast_lt_aleph0",
"Nat... | exact h.trans_lt natCast_lt_aleph0 | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.LinearAlgebra.Dimension.Finrank | {
"line": 81,
"column": 4
} | {
"line": 81,
"column": 38
} | [
{
"pp": "case hc\nR : Type u\nM : Type v\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nn : ℕ\nh : Module.rank R M ≤ ↑n\n⊢ Module.rank R M < ℵ₀",
"usedConstants": [
"Cardinal",
"PartialOrder.toPreorder",
"Cardinal.aleph0",
"Cardinal.natCast_lt_aleph0",
"Nat... | exact h.trans_lt natCast_lt_aleph0 | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Dimension.Finrank | {
"line": 81,
"column": 4
} | {
"line": 81,
"column": 38
} | [
{
"pp": "case hc\nR : Type u\nM : Type v\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nn : ℕ\nh : Module.rank R M ≤ ↑n\n⊢ Module.rank R M < ℵ₀",
"usedConstants": [
"Cardinal",
"PartialOrder.toPreorder",
"Cardinal.aleph0",
"Cardinal.natCast_lt_aleph0",
"Nat... | exact h.trans_lt natCast_lt_aleph0 | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Set.Card | {
"line": 71,
"column": 2
} | {
"line": 71,
"column": 49
} | [
{
"pp": "α : Type u_3\n⊢ univ.encard = ENat.card α",
"usedConstants": [
"ENat.card_congr",
"Eq.mpr",
"Set.encard",
"congrArg",
"Set.univ",
"Set.Elem",
"id",
"ENat",
"Eq.refl",
"Set.encard.eq_1",
"ENat.card",
"Eq",
"Equiv.Set.univ"... | rw [encard, ENat.card_congr (Equiv.Set.univ α)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Data.Set.Card | {
"line": 71,
"column": 2
} | {
"line": 71,
"column": 49
} | [
{
"pp": "α : Type u_3\n⊢ univ.encard = ENat.card α",
"usedConstants": [
"ENat.card_congr",
"Eq.mpr",
"Set.encard",
"congrArg",
"Set.univ",
"Set.Elem",
"id",
"ENat",
"Eq.refl",
"Set.encard.eq_1",
"ENat.card",
"Eq",
"Equiv.Set.univ"... | rw [encard, ENat.card_congr (Equiv.Set.univ α)] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Set.Card | {
"line": 71,
"column": 2
} | {
"line": 71,
"column": 49
} | [
{
"pp": "α : Type u_3\n⊢ univ.encard = ENat.card α",
"usedConstants": [
"ENat.card_congr",
"Eq.mpr",
"Set.encard",
"congrArg",
"Set.univ",
"Set.Elem",
"id",
"ENat",
"Eq.refl",
"Set.encard.eq_1",
"ENat.card",
"Eq",
"Equiv.Set.univ"... | rw [encard, ENat.card_congr (Equiv.Set.univ α)] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Set.Card | {
"line": 337,
"column": 6
} | {
"line": 337,
"column": 40
} | [
{
"pp": "α : Type u_1\ns : Set α\nx✝ : s.Nonempty\na : α\nha : a ∈ s\nhfin : s.Finite\n⊢ (s \\ {a}).encard < s.encard",
"usedConstants": [
"Eq.mpr",
"Set.encard",
"instAddMonoidWithOneENat",
"congrArg",
"Set.instSingletonSet",
"instAddENat",
"id",
"AddMonoidWi... | ← encard_diff_singleton_add_one ha | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Group.Submonoid.Finite | {
"line": 75,
"column": 8
} | {
"line": 78,
"column": 30
} | [
{
"pp": "η : Type u_1\nf : η → Type u_2\ninst✝¹ : (i : η) → MulOneClass (f i)\ninst✝ : Finite η\ns : (i : η) → Set (f i)\nhs : ∀ (i : η), 1 ∈ s i\ni : η\n_x : f i\nhx : _x ∈ s i\nj : η\n⊢ (MonoidHom.mulSingle f i) _x j ∈ s j",
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"False",
"Mo... | by_cases H : j = i
· subst H
simpa
· simpa [H] using hs _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Group.Submonoid.Finite | {
"line": 75,
"column": 8
} | {
"line": 78,
"column": 30
} | [
{
"pp": "η : Type u_1\nf : η → Type u_2\ninst✝¹ : (i : η) → MulOneClass (f i)\ninst✝ : Finite η\ns : (i : η) → Set (f i)\nhs : ∀ (i : η), 1 ∈ s i\ni : η\n_x : f i\nhx : _x ∈ s i\nj : η\n⊢ (MonoidHom.mulSingle f i) _x j ∈ s j",
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"False",
"Mo... | by_cases H : j = i
· subst H
simpa
· simpa [H] using hs _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Set.Card | {
"line": 719,
"column": 33
} | {
"line": 720,
"column": 60
} | [
{
"pp": "α : Type u_1\ns : Set α\na : α\nh : a ∈ s\nhs : s.Finite\n⊢ (s \\ {a}).ncard < s.ncard",
"usedConstants": [
"Eq.mpr",
"Nat.instIsOrderedAddMonoid",
"Nat.instOne",
"congrArg",
"instIsLeftCancelAddOfAddLeftReflectLE",
"lt_add_one",
"AddMonoid.toAddZeroClass",... | by
rw [← ncard_diff_singleton_add_one h hs]; apply lt_add_one | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Set.Card | {
"line": 888,
"column": 2
} | {
"line": 888,
"column": 54
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ns : Set α\nt : Set β\nhc : t.ncard < s.ncard\nf : α → β\nhf : ∀ a ∈ s, f a ∈ t\nht : t.Finite\nh' : ∀ x ∈ s, ∀ x_1 ∈ s, f x = f x_1 → x = x_1\n⊢ False",
"usedConstants": [
"LE.le.not_gt",
"Nat.instPreorder",
"Nat",
"Set.ncard",
"Set.ncard_le... | exact (ncard_le_ncard_of_injOn f hf h' ht).not_gt hc | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.LinearAlgebra.Basis.Submodule | {
"line": 46,
"column": 31
} | {
"line": 46,
"column": 83
} | [
{
"pp": "ι : Type u_1\nR : Type u_3\nM : Type u_5\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ninst✝ : Fintype ι\nP : Submodule R M\nb : Basis ι R ↥P\nx : M\nc : ι → R\n⊢ (x = (equivFunOnFinite.symm c).sum fun i x ↦ x • ↑(b i)) ↔ x = ∑ i, c i • ↑(b i)",
"usedConstants": [
"Fins... | simp [Finsupp.sum_fintype, Finsupp.equivFunOnFinite] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.LinearAlgebra.Basis.Submodule | {
"line": 46,
"column": 31
} | {
"line": 46,
"column": 83
} | [
{
"pp": "ι : Type u_1\nR : Type u_3\nM : Type u_5\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ninst✝ : Fintype ι\nP : Submodule R M\nb : Basis ι R ↥P\nx : M\nc : ι → R\n⊢ (x = (equivFunOnFinite.symm c).sum fun i x ↦ x • ↑(b i)) ↔ x = ∑ i, c i • ↑(b i)",
"usedConstants": [
"Fins... | simp [Finsupp.sum_fintype, Finsupp.equivFunOnFinite] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Basis.Submodule | {
"line": 46,
"column": 31
} | {
"line": 46,
"column": 83
} | [
{
"pp": "ι : Type u_1\nR : Type u_3\nM : Type u_5\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ninst✝ : Fintype ι\nP : Submodule R M\nb : Basis ι R ↥P\nx : M\nc : ι → R\n⊢ (x = (equivFunOnFinite.symm c).sum fun i x ↦ x • ↑(b i)) ↔ x = ∑ i, c i • ↑(b i)",
"usedConstants": [
"Fins... | simp [Finsupp.sum_fintype, Finsupp.equivFunOnFinite] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Set.Card | {
"line": 1149,
"column": 4
} | {
"line": 1153,
"column": 38
} | [
{
"pp": "case inl\nα : Type u_1\ns t : Set α\nhs : s.Finite\nht : t.Finite\n⊢ (∃ a ∉ s, insert a s = t) ↔ s ⊆ t ∧ s.ncard + 1 = t.ncard",
"usedConstants": [
"Eq.mpr",
"congrArg",
"HEq.refl",
"Finset",
"Iff.rfl",
"_private.Mathlib.Data.Set.Card.0.Set.exists_eq_insert_iff_n... | rw [ncard_eq_toFinset_card _ hs, ncard_eq_toFinset_card _ ht,
← @Finite.toFinset_subset_toFinset _ _ _ hs ht, ← Finset.exists_eq_insert_iff]
convert Iff.rfl using 2; simp only [Finite.mem_toFinset]
ext x
simp [Finset.ext_iff, Set.ext_iff] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Set.Card | {
"line": 1149,
"column": 4
} | {
"line": 1153,
"column": 38
} | [
{
"pp": "case inl\nα : Type u_1\ns t : Set α\nhs : s.Finite\nht : t.Finite\n⊢ (∃ a ∉ s, insert a s = t) ↔ s ⊆ t ∧ s.ncard + 1 = t.ncard",
"usedConstants": [
"Eq.mpr",
"congrArg",
"HEq.refl",
"Finset",
"Iff.rfl",
"_private.Mathlib.Data.Set.Card.0.Set.exists_eq_insert_iff_n... | rw [ncard_eq_toFinset_card _ hs, ncard_eq_toFinset_card _ ht,
← @Finite.toFinset_subset_toFinset _ _ _ hs ht, ← Finset.exists_eq_insert_iff]
convert Iff.rfl using 2; simp only [Finite.mem_toFinset]
ext x
simp [Finset.ext_iff, Set.ext_iff] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Set.Card | {
"line": 1225,
"column": 2
} | {
"line": 1225,
"column": 50
} | [
{
"pp": "α : Type u_1\ns : Set α\nhs : s.Finite\n⊢ 2 < s.ncard ↔ ∃ a ∈ s, ∃ b ∈ s, ∃ c ∈ s, a ≠ b ∧ a ≠ c ∧ b ≠ c",
"usedConstants": [
"congrArg",
"Membership.mem",
"Exists",
"Ne",
"_private.Mathlib.Data.Set.Card.0.Set.two_lt_ncard._simp_1_1",
"instOfNatNat",
"iff_s... | simp only [two_lt_ncard_iff hs, exists_and_left] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
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