module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Algebra.Order.Ring.WithTop | {
"line": 337,
"column": 29
} | {
"line": 337,
"column": 40
} | {
"line": 337,
"column": 41
} | [
{
"pp": "case neg.coe.bot\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : MulZeroClass α\na✝ : α\nha : ¬↑a✝ = 0\nhb : ¬⊥ = 0\n⊢ unbotD 0 ⊥ = unbotD 0 ↑a✝ * unbotD 0 ⊥",
"ppTerm": "?neg.coe.bot✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"WithBot.some",
"WithBot",
"HMul.h... | [
"case neg.coe.bot\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : MulZeroClass α\na✝ : α\nha : ¬↑a✝ = 0\nhb : ¬⊥ = 0\n⊢ 0 = unbotD 0 ↑a✝ * 0"
] | unbotD_bot, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.SuccPred.Basic | {
"line": 203,
"column": 2
} | {
"line": 204,
"column": 41
} | {
"line": 206,
"column": 0
} | [
{
"pp": "case neg\nα : Type u_1\ninst✝¹ : Preorder α\ninst✝ : SuccOrder α\na b : α\nh : a ≤ b\nhb : ¬IsMax b\n⊢ succ a ≤ succ b",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"Order.succ",
"congrArg",
"Preorder.toLE",
"Order.s... | [] | · rw [succ_le_iff_of_not_isMax fun ha => hb <| ha.mono h]
apply lt_succ_of_le_of_not_isMax h hb | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Order.SuccPred | {
"line": 54,
"column": 2
} | {
"line": 54,
"column": 53
} | {
"line": 56,
"column": 0
} | [
{
"pp": "α : Type u_1\nx y : α\ninst✝³ : Preorder α\ninst✝² : Add α\ninst✝¹ : One α\ninst✝ : SuccAddOrder α\nhx : ¬IsMax x\n⊢ x + 1 ≤ y ↔ x < y",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"Order.succ",
"Order.succ_eq_add_one",
"c... | [] | rw [← succ_eq_add_one, succ_le_iff_of_not_isMax hx] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Order.SuccPred | {
"line": 54,
"column": 2
} | {
"line": 54,
"column": 53
} | {
"line": 56,
"column": 0
} | [
{
"pp": "α : Type u_1\nx y : α\ninst✝³ : Preorder α\ninst✝² : Add α\ninst✝¹ : One α\ninst✝ : SuccAddOrder α\nhx : ¬IsMax x\n⊢ x + 1 ≤ y ↔ x < y",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"Order.succ",
"Order.succ_eq_add_one",
"c... | [] | rw [← succ_eq_add_one, succ_le_iff_of_not_isMax hx] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Order.SuccPred | {
"line": 54,
"column": 2
} | {
"line": 54,
"column": 53
} | {
"line": 56,
"column": 0
} | [
{
"pp": "α : Type u_1\nx y : α\ninst✝³ : Preorder α\ninst✝² : Add α\ninst✝¹ : One α\ninst✝ : SuccAddOrder α\nhx : ¬IsMax x\n⊢ x + 1 ≤ y ↔ x < y",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"Order.succ",
"Order.succ_eq_add_one",
"c... | [] | rw [← succ_eq_add_one, succ_le_iff_of_not_isMax hx] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.BigOperators.Group.Finset.Piecewise | {
"line": 247,
"column": 73
} | {
"line": 249,
"column": 10
} | {
"line": 251,
"column": 0
} | [
{
"pp": "ι : Type u_1\nM : Type u_3\ninst✝¹ : CommMonoid M\ninst✝ : DecidableEq ι\ns : Finset ι\ni : ι\nh : i ∈ s\nf : ι → M\nb : M\n⊢ ∏ x ∈ s, Function.update f i b x = b * ∏ x ∈ s \\ {i}, f x",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Function.update",
"... | [] | by
rw [update_eq_piecewise, prod_piecewise]
simp [h] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Vector.Basic | {
"line": 581,
"column": 6
} | {
"line": 581,
"column": 38
} | {
"line": 583,
"column": 0
} | [
{
"pp": "case x\nα : Type u_1\nn : ℕ\na b : α\ni j : Fin (n + 1)\nh : i ≤ j\nl : List α\nhl : l.length = n\n⊢ ↑j ≤ n",
"ppTerm": "?x✝",
"assigned": true,
"usedConstants": [
"Fin.isLt",
"instOfNatNat",
"Fin.val",
"instHAdd",
"Nat.le_of_succ_le_succ",
"HAdd.hAdd",
... | [] | exact Nat.le_of_succ_le_succ j.2 | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Data.Vector.Basic | {
"line": 750,
"column": 4
} | {
"line": 753,
"column": 35
} | {
"line": 755,
"column": 0
} | [
{
"pp": "case cons\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nn : ℕ\nf : α → β → γ\nn✝ : ℕ\na✝ : α\nb✝ : β\nx✝ : Vector α n✝\ny✝ : Vector β n✝\nih : ∀ (i : Fin n✝), (map₂ f x✝ y✝).get i = f (x✝.get i) (y✝.get i)\ni : Fin n✝.succ\n⊢ (map₂ f (a✝ ::ᵥ x✝) (b✝ ::ᵥ y✝)).get i = f ((a✝ ::ᵥ x✝).get i) ((b✝ ::ᵥ y✝).get ... | [] | rw [map₂_cons]
cases i using Fin.cases
· simp only [get_zero, head_cons]
· simp only [get_cons_succ, ih] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Vector.Basic | {
"line": 750,
"column": 4
} | {
"line": 753,
"column": 35
} | {
"line": 755,
"column": 0
} | [
{
"pp": "case cons\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nn : ℕ\nf : α → β → γ\nn✝ : ℕ\na✝ : α\nb✝ : β\nx✝ : Vector α n✝\ny✝ : Vector β n✝\nih : ∀ (i : Fin n✝), (map₂ f x✝ y✝).get i = f (x✝.get i) (y✝.get i)\ni : Fin n✝.succ\n⊢ (map₂ f (a✝ ::ᵥ x✝) (b✝ ::ᵥ y✝)).get i = f ((a✝ ::ᵥ x✝).get i) ((b✝ ::ᵥ y✝).get ... | [] | rw [map₂_cons]
cases i using Fin.cases
· simp only [get_zero, head_cons]
· simp only [get_cons_succ, ih] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.SetTheory.Cardinal.ENat | {
"line": 296,
"column": 61
} | {
"line": 296,
"column": 93
} | {
"line": 297,
"column": 0
} | [
{
"pp": "c : Cardinal.{u}\nn : ℕ\n⊢ ↑n = toENat c ↔ ↑n = c",
"ppTerm": "?m.4",
"assigned": true,
"usedConstants": [
"ENat.instNatCast",
"Cardinal",
"congrArg",
"CommSemiring.toSemiring",
"Cardinal.commSemiring",
"PartialOrder.toPreorder",
"OrderRingHom.instF... | [] | simp [eq_comm (a := Nat.cast _)] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.SetTheory.Cardinal.ENat | {
"line": 296,
"column": 61
} | {
"line": 296,
"column": 93
} | {
"line": 297,
"column": 0
} | [
{
"pp": "c : Cardinal.{u}\nn : ℕ\n⊢ ↑n = toENat c ↔ ↑n = c",
"ppTerm": "?m.4",
"assigned": true,
"usedConstants": [
"ENat.instNatCast",
"Cardinal",
"congrArg",
"CommSemiring.toSemiring",
"Cardinal.commSemiring",
"PartialOrder.toPreorder",
"OrderRingHom.instF... | [] | simp [eq_comm (a := Nat.cast _)] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.SetTheory.Cardinal.ENat | {
"line": 296,
"column": 61
} | {
"line": 296,
"column": 93
} | {
"line": 297,
"column": 0
} | [
{
"pp": "c : Cardinal.{u}\nn : ℕ\n⊢ ↑n = toENat c ↔ ↑n = c",
"ppTerm": "?m.4",
"assigned": true,
"usedConstants": [
"ENat.instNatCast",
"Cardinal",
"congrArg",
"CommSemiring.toSemiring",
"Cardinal.commSemiring",
"PartialOrder.toPreorder",
"OrderRingHom.instF... | [] | simp [eq_comm (a := Nat.cast _)] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.SetTheory.Cardinal.Basic | {
"line": 285,
"column": 6
} | {
"line": 285,
"column": 26
} | {
"line": 285,
"column": 27
} | [
{
"pp": "n : ℕ\nc : Cardinal.{u_1}\n⊢ c < ↑n + 1 ↔ c ≤ ↑n",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"Cardinal.instOne",
"Order.succ",
"Cardinal",
"congrArg",
"PartialOrder.toPreorder",
"Cardinal.instNoMaxOrder... | [
"n : ℕ\nc : Cardinal.{u_1}\n⊢ c < ↑n + 1 ↔ c < succ ↑n"
] | ← Order.lt_succ_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.GroupWithZero.Action.Hom | {
"line": 25,
"column": 19
} | {
"line": 25,
"column": 53
} | {
"line": 25,
"column": 54
} | [
{
"pp": "M : Type u_1\nN : Type u_2\nA : Type u_3\nB : Type u_4\nC : Type u_5\ninst✝³ : Zero A\ninst✝² : Zero B\ninst✝¹ : Zero C\ninst✝ : SMulZeroClass M B\nr : M\nf : ZeroHom A B\n⊢ r • f 0 = 0",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"ZeroHom.funLike",
"instHSMul",
... | [] | by simp only [map_zero, smul_zero] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.GroupWithZero.Action.Hom | {
"line": 69,
"column": 19
} | {
"line": 69,
"column": 53
} | {
"line": 70,
"column": 6
} | [
{
"pp": "M : Type u_1\nN : Type u_2\nA : Type u_3\nB : Type u_4\nC : Type u_5\ninst✝³ : AddZeroClass A\ninst✝² : AddZeroClass B\ninst✝¹ : AddZeroClass C\ninst✝ : DistribSMul M B\nr : M\nf : A →+ B\n⊢ r • f 0 = 0",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants": [
"instHSMul",
"AddM... | [] | by simp only [map_zero, smul_zero] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.GroupTheory.GroupAction.Hom | {
"line": 572,
"column": 26
} | {
"line": 572,
"column": 59
} | {
"line": 572,
"column": 59
} | [
{
"pp": "R : Type u_1\nM : Type u_2\nN : Type u_3\nX : Type u_4\nY : Type u_5\nσ : M → N\ninst✝ : Monoid M\nf : M →ₑ[id] M\nm : M\n⊢ ((fun m ↦ { toFun := fun x ↦ m * x, map_smul' := ⋯ }) ((fun f ↦ f 1) f)) m = f m",
"ppTerm": "?m.73",
"assigned": true,
"usedConstants": [
"MulOne.toOne",
... | [
"R : Type u_1\nM : Type u_2\nN : Type u_3\nX : Type u_4\nY : Type u_5\nσ : M → N\ninst✝ : Monoid M\nf : M →ₑ[id] M\nm : M\n⊢ MulOpposite.op m • f 1 = f m"
] | change MulOpposite.op m • f 1 = _ | Lean.Elab.Tactic.evalChange | Lean.Parser.Tactic.change |
Mathlib.Algebra.Module.Equiv.Defs | {
"line": 632,
"column": 49
} | {
"line": 632,
"column": 90
} | {
"line": 634,
"column": 0
} | [
{
"pp": "S : Type u_14\nR : Type u_15\nV : Type u_16\nW : Type u_17\nG : Type u_18\ninst✝¹⁶ : Semiring R\ninst✝¹⁵ : Semiring S\ninst✝¹⁴ : AddCommMonoid V\ninst✝¹³ : Module R V\ninst✝¹² : Module S V\ninst✝¹¹ : AddCommMonoid W\ninst✝¹⁰ : Module R W\ninst✝⁹ : Module S W\ninst✝⁸ : AddCommMonoid G\ninst✝⁷ : Module R... | [] | simp [LinearMapClass.map_smul_of_tower f] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Module.Submodule.Defs | {
"line": 81,
"column": 4
} | {
"line": 82,
"column": 62
} | {
"line": 83,
"column": 2
} | [
{
"pp": "G : Type u''\nS : Type u'\nR : Type u\nM : Type v\nι : Type w\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nC : Set M\nnonempty : C.Nonempty\nlinearComb : ∀ x ∈ C, ∀ y ∈ C, ∀ (a b : R), a • x + b • y ∈ C\n⊢ 0 ∈ C",
"ppTerm": "?m.61",
"assigned": true,
"usedConstants": ... | [] | obtain ⟨x, hx⟩ := nonempty
simpa [zero_smul, add_zero] using linearComb x hx x hx 0 0 | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Module.Submodule.Defs | {
"line": 81,
"column": 4
} | {
"line": 82,
"column": 62
} | {
"line": 83,
"column": 2
} | [
{
"pp": "G : Type u''\nS : Type u'\nR : Type u\nM : Type v\nι : Type w\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nC : Set M\nnonempty : C.Nonempty\nlinearComb : ∀ x ∈ C, ∀ y ∈ C, ∀ (a b : R), a • x + b • y ∈ C\n⊢ 0 ∈ C",
"ppTerm": "?m.61",
"assigned": true,
"usedConstants": ... | [] | obtain ⟨x, hx⟩ := nonempty
simpa [zero_smul, add_zero] using linearComb x hx x hx 0 0 | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Module.Submodule.Ker | {
"line": 94,
"column": 2
} | {
"line": 94,
"column": 49
} | {
"line": 95,
"column": 2
} | [
{
"pp": "R : Type u_1\nM : Type u_5\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nf g : M →ₗ[R] M\nh : Commute f g\n⊢ f.ker ⊔ g.ker ≤ (f ∘ₗ g).ker",
"ppTerm": "?m.98",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Submodule",
"Lattice.toSemilatticeSup",
... | [
"R : Type u_1\nM : Type u_5\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nf g : M →ₗ[R] M\nh : Commute f g\n⊢ f.ker ≤ (f ∘ₗ g).ker"
] | refine sup_le_iff.mpr ⟨?_, ker_le_ker_comp g f⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Algebra.Module.Submodule.Lattice | {
"line": 318,
"column": 6
} | {
"line": 318,
"column": 25
} | {
"line": 318,
"column": 26
} | [
{
"pp": "R : Type u_1\nM : Type u_3\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\n⊢ ⊥.toAddSubmonoid = ⊤.toAddSubmonoid ↔ ⊥ = ⊤",
"ppTerm": "?m.42",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule",
"AddSubmonoid.instCompleteLattice",
"Submodul... | [
"R : Type u_1\nM : Type u_3\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\n⊢ ⊥ = ⊤.toAddSubmonoid ↔ ⊥ = ⊤"
] | bot_toAddSubmonoid, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Ring.Subsemiring.Defs | {
"line": 244,
"column": 20
} | {
"line": 244,
"column": 58
} | {
"line": 246,
"column": 0
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝¹ : NonAssocSemiring R\ninst✝ : NonAssocSemiring S\ns : Set R\nsm : Submonoid R\nhm : ↑sm = s\nsa : AddSubmonoid R\nha : ↑sa = s\nx y : R\n⊢ x ∈ s → y ∈ s → x * y ∈ s",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
... | [] | by simpa only [← hm] using! sm.mul_mem | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Module.Equiv.Basic | {
"line": 836,
"column": 28
} | {
"line": 836,
"column": 37
} | {
"line": 836,
"column": 38
} | [
{
"pp": "R : Type u_1\nR₂ : Type u_2\nK : Type u_3\nS : Type u_4\nM : Type u_5\nM₁ : Type u_6\nM₂ : Type u_7\nM₃ : Type u_8\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nm : Type u_9\nn : Type u_10\np : Type u_11\ne : m ≃ n\nx : m → M\ni : m\n⊢ (funLeft R M ⇑e ∘ₗ funLeft R M ⇑e.symm) x i =... | [
"R : Type u_1\nR₂ : Type u_2\nK : Type u_3\nS : Type u_4\nM : Type u_5\nM₁ : Type u_6\nM₂ : Type u_7\nM₃ : Type u_8\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nm : Type u_9\nn : Type u_10\np : Type u_11\ne : m ≃ n\nx : m → M\ni : m\n⊢ (funLeft R M ⇑e ∘ₗ funLeft R M ⇑e.symm) x i = x i"
] | id_apply, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Module.Equiv.Basic | {
"line": 838,
"column": 28
} | {
"line": 838,
"column": 37
} | {
"line": 838,
"column": 38
} | [
{
"pp": "R : Type u_1\nR₂ : Type u_2\nK : Type u_3\nS : Type u_4\nM : Type u_5\nM₁ : Type u_6\nM₂ : Type u_7\nM₃ : Type u_8\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nm : Type u_9\nn : Type u_10\np : Type u_11\ne : m ≃ n\nx : n → M\ni : n\n⊢ (funLeft R M ⇑e.symm ∘ₗ funLeft R M ⇑e) x i =... | [
"R : Type u_1\nR₂ : Type u_2\nK : Type u_3\nS : Type u_4\nM : Type u_5\nM₁ : Type u_6\nM₂ : Type u_7\nM₃ : Type u_8\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nm : Type u_9\nn : Type u_10\np : Type u_11\ne : m ≃ n\nx : n → M\ni : n\n⊢ (funLeft R M ⇑e.symm ∘ₗ funLeft R M ⇑e) x i = x i"
] | id_apply, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Ring.Center | {
"line": 55,
"column": 8
} | {
"line": 57,
"column": 84
} | {
"line": 59,
"column": 0
} | [
{
"pp": "M : Type u_1\ninst✝ : NonAssocRing M\nn✝ : ℤ\nx✝¹ x✝ : M\nn : ℕ\n⊢ x✝¹ * x✝ * ↑(Int.negSucc n) = x✝¹ * (x✝ * ↑(Int.negSucc n))",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants": [
"neg_add_rev",
"AddGroup.toSubtractionMonoid",
"Distrib.leftDistribClass",
"Int.ca... | [] | simp only [Int.cast_negSucc, Nat.cast_add, Nat.cast_one, neg_add_rev]
rw [mul_add, mul_add, mul_add, mul_neg, mul_one, mul_neg, mul_neg, mul_one, mul_neg,
add_right_inj, (natCast_mem_center _ n).right_assoc _ _, mul_neg, mul_neg] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Ring.Center | {
"line": 55,
"column": 8
} | {
"line": 57,
"column": 84
} | {
"line": 59,
"column": 0
} | [
{
"pp": "M : Type u_1\ninst✝ : NonAssocRing M\nn✝ : ℤ\nx✝¹ x✝ : M\nn : ℕ\n⊢ x✝¹ * x✝ * ↑(Int.negSucc n) = x✝¹ * (x✝ * ↑(Int.negSucc n))",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants": [
"neg_add_rev",
"AddGroup.toSubtractionMonoid",
"Distrib.leftDistribClass",
"Int.ca... | [] | simp only [Int.cast_negSucc, Nat.cast_add, Nat.cast_one, neg_add_rev]
rw [mul_add, mul_add, mul_add, mul_neg, mul_one, mul_neg, mul_neg, mul_one, mul_neg,
add_right_inj, (natCast_mem_center _ n).right_assoc _ _, mul_neg, mul_neg] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Ring.Center | {
"line": 71,
"column": 62
} | {
"line": 71,
"column": 74
} | {
"line": 71,
"column": 74
} | [
{
"pp": "M : Type u_1\ninst✝ : NonUnitalNonAssocRing M\na : M\nha : a ∈ center M\nx✝ : M\n⊢ -a * x✝ = a * -x✝",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"HMul.hMul",
"congrArg",
"NonUnitalNonAssocRing.toAddCommGroup",
... | [
"M : Type u_1\ninst✝ : NonUnitalNonAssocRing M\na : M\nha : a ∈ center M\nx✝ : M\n⊢ a * -x✝ = a * -x✝"
] | neg_mul_comm | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Group.Subsemigroup.Operations | {
"line": 528,
"column": 4
} | {
"line": 529,
"column": 21
} | {
"line": 530,
"column": 2
} | [
{
"pp": "case mp.right\nM : Type u_1\nN : Type u_2\ninst✝¹ : Mul M\ninst✝ : Mul N\ns : Subsemigroup M\nt : Subsemigroup N\nu : Subsemigroup (M × N)\nh : u ≤ s.prod t\n⊢ map (snd M N) u ≤ t",
"ppTerm": "?mp.right",
"assigned": true,
"usedConstants": [
"MulHom",
"Subsemigroup.map",
"... | [] | · rintro x ⟨⟨y1, y2⟩, ⟨hy1, rfl⟩⟩
exact (h hy1).2 | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Algebra.Basic | {
"line": 376,
"column": 4
} | {
"line": 376,
"column": 95
} | {
"line": 377,
"column": 2
} | [
{
"pp": "R : Type u_1\nA : Type u_2\ninst✝⁷ : CommSemiring R\ninst✝⁶ : Semiring A\ninst✝⁵ : Algebra R A\ninst✝⁴ : FaithfulSMul R A\nG : Type u_3\ninst✝³ : Monoid G\ninst✝² : MulSemiringAction G A\ninst✝¹ : SMul G R\ninst✝ : SMulDistribClass G R A\nx✝² : G\nx✝¹ x✝ : R\n⊢ (algebraMap R A) (x✝² • (x✝¹ + x✝)) = (al... | [] | rw [algebraMap.smul', map_add, smul_add, ← algebraMap.smul', ← algebraMap.smul', ← map_add] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.NonUnitalSubring.Basic | {
"line": 539,
"column": 4
} | {
"line": 545,
"column": 34
} | {
"line": 545,
"column": 34
} | [
{
"pp": "R : Type u\ninst✝ : NonUnitalNonAssocRing R\ns : Set R\nx : R\nh : x ∈ AddSubgroup.closure ↑(Subsemigroup.closure s)\n⊢ x ∈ closure s",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"NonUnitalSubring.instSetLike",
"NegZeroClass.toN... | [] | induction h using AddSubgroup.closure_induction with
| mem _ hx => induction hx using Subsemigroup.closure_induction with
| mem _ h => exact subset_closure h
| mul _ _ _ _ h₁ h₂ => exact mul_mem h₁ h₂
| zero => exact zero_mem _
| add _ _ _ _ h₁ h₂ => exact add_mem h₁ h₂
| neg _ _ h => exact ... | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.RingTheory.NonUnitalSubring.Basic | {
"line": 539,
"column": 4
} | {
"line": 545,
"column": 34
} | {
"line": 545,
"column": 34
} | [
{
"pp": "R : Type u\ninst✝ : NonUnitalNonAssocRing R\ns : Set R\nx : R\nh : x ∈ AddSubgroup.closure ↑(Subsemigroup.closure s)\n⊢ x ∈ closure s",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"NonUnitalSubring.instSetLike",
"NegZeroClass.toN... | [] | induction h using AddSubgroup.closure_induction with
| mem _ hx => induction hx using Subsemigroup.closure_induction with
| mem _ h => exact subset_closure h
| mul _ _ _ _ h₁ h₂ => exact mul_mem h₁ h₂
| zero => exact zero_mem _
| add _ _ _ _ h₁ h₂ => exact add_mem h₁ h₂
| neg _ _ h => exact ... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.NonUnitalSubring.Basic | {
"line": 539,
"column": 4
} | {
"line": 545,
"column": 34
} | {
"line": 545,
"column": 34
} | [
{
"pp": "R : Type u\ninst✝ : NonUnitalNonAssocRing R\ns : Set R\nx : R\nh : x ∈ AddSubgroup.closure ↑(Subsemigroup.closure s)\n⊢ x ∈ closure s",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"NonUnitalSubring.instSetLike",
"NegZeroClass.toN... | [] | induction h using AddSubgroup.closure_induction with
| mem _ hx => induction hx using Subsemigroup.closure_induction with
| mem _ h => exact subset_closure h
| mul _ _ _ _ h₁ h₂ => exact mul_mem h₁ h₂
| zero => exact zero_mem _
| add _ _ _ _ h₁ h₂ => exact add_mem h₁ h₂
| neg _ _ h => exact ... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Group.Irreducible.Lemmas | {
"line": 42,
"column": 8
} | {
"line": 42,
"column": 30
} | {
"line": 42,
"column": 30
} | [
{
"pp": "case refine_2\nM : Type u_2\ninst✝ : Monoid M\ny : M\nu : Mˣ\nx✝ : ¬IsUnit y\nh : ∀ ⦃a b : M⦄, y = a * b → IsUnit a ∨ IsUnit b\nA B : M\nHAB : ↑u * y = A * B\n⊢ IsUnit A ∨ IsUnit B",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"Units.val",
"Eq.mpr",
"HMul.h... | [
"case refine_2\nM : Type u_2\ninst✝ : Monoid M\ny : M\nu : Mˣ\nx✝ : ¬IsUnit y\nh : ∀ ⦃a b : M⦄, y = a * b → IsUnit a ∨ IsUnit b\nA B : M\nHAB : ↑u * y = A * B\n⊢ IsUnit (↑u⁻¹ * A) ∨ IsUnit B"
] | ← u⁻¹.isUnit_units_mul | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.GroupWithZero.NonZeroDivisors | {
"line": 309,
"column": 89
} | {
"line": 310,
"column": 35
} | {
"line": 312,
"column": 0
} | [
{
"pp": "M₀ : Type u_1\ninst✝ : CommMonoidWithZero M₀\n⊢ nonZeroDivisorsRight M₀ = nonZeroDivisorsLeft M₀",
"ppTerm": "?m.4",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"nonZeroDivisorsRight",
"congrArg",
"nonZeroDivisorsLeft_eq_right",
"id",
"CommMonoidWithZe... | [] | by
rw [nonZeroDivisorsLeft_eq_right] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Module.Submodule.Range | {
"line": 101,
"column": 12
} | {
"line": 101,
"column": 56
} | {
"line": 103,
"column": 0
} | [
{
"pp": "case hs\nR : Type u_1\nR₂ : Type u_2\nM : Type u_5\nM₂ : Type u_6\ninst✝⁶ : Semiring R\ninst✝⁵ : Semiring R₂\ninst✝⁴ : AddCommMonoid M\ninst✝³ : AddCommMonoid M₂\ninst✝² : Module R M\ninst✝¹ : Module R₂ M₂\nτ₁₂ : R →+* R₂\ninst✝ : RingHomSurjective τ₁₂\nf g : M →ₛₗ[τ₁₂] M₂\nw✝ : M\n⊢ f w✝ ∈ f.range",
... | [] | simp only [mem_range, exists_apply_eq_apply] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Module.Submodule.Range | {
"line": 101,
"column": 12
} | {
"line": 101,
"column": 56
} | {
"line": 103,
"column": 0
} | [
{
"pp": "case hs\nR : Type u_1\nR₂ : Type u_2\nM : Type u_5\nM₂ : Type u_6\ninst✝⁶ : Semiring R\ninst✝⁵ : Semiring R₂\ninst✝⁴ : AddCommMonoid M\ninst✝³ : AddCommMonoid M₂\ninst✝² : Module R M\ninst✝¹ : Module R₂ M₂\nτ₁₂ : R →+* R₂\ninst✝ : RingHomSurjective τ₁₂\nf g : M →ₛₗ[τ₁₂] M₂\nw✝ : M\n⊢ f w✝ ∈ f.range",
... | [] | simp only [mem_range, exists_apply_eq_apply] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Module.Submodule.Range | {
"line": 101,
"column": 12
} | {
"line": 101,
"column": 56
} | {
"line": 103,
"column": 0
} | [
{
"pp": "case hs\nR : Type u_1\nR₂ : Type u_2\nM : Type u_5\nM₂ : Type u_6\ninst✝⁶ : Semiring R\ninst✝⁵ : Semiring R₂\ninst✝⁴ : AddCommMonoid M\ninst✝³ : AddCommMonoid M₂\ninst✝² : Module R M\ninst✝¹ : Module R₂ M₂\nτ₁₂ : R →+* R₂\ninst✝ : RingHomSurjective τ₁₂\nf g : M →ₛₗ[τ₁₂] M₂\nw✝ : M\n⊢ f w✝ ∈ f.range",
... | [] | simp only [mem_range, exists_apply_eq_apply] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Module.Submodule.Range | {
"line": 101,
"column": 12
} | {
"line": 101,
"column": 56
} | {
"line": 103,
"column": 0
} | [
{
"pp": "case ht\nR : Type u_1\nR₂ : Type u_2\nM : Type u_5\nM₂ : Type u_6\ninst✝⁶ : Semiring R\ninst✝⁵ : Semiring R₂\ninst✝⁴ : AddCommMonoid M\ninst✝³ : AddCommMonoid M₂\ninst✝² : Module R M\ninst✝¹ : Module R₂ M₂\nτ₁₂ : R →+* R₂\ninst✝ : RingHomSurjective τ₁₂\nf g : M →ₛₗ[τ₁₂] M₂\nw✝ : M\n⊢ g w✝ ∈ g.range",
... | [] | simp only [mem_range, exists_apply_eq_apply] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Module.Submodule.Range | {
"line": 101,
"column": 12
} | {
"line": 101,
"column": 56
} | {
"line": 103,
"column": 0
} | [
{
"pp": "case ht\nR : Type u_1\nR₂ : Type u_2\nM : Type u_5\nM₂ : Type u_6\ninst✝⁶ : Semiring R\ninst✝⁵ : Semiring R₂\ninst✝⁴ : AddCommMonoid M\ninst✝³ : AddCommMonoid M₂\ninst✝² : Module R M\ninst✝¹ : Module R₂ M₂\nτ₁₂ : R →+* R₂\ninst✝ : RingHomSurjective τ₁₂\nf g : M →ₛₗ[τ₁₂] M₂\nw✝ : M\n⊢ g w✝ ∈ g.range",
... | [] | simp only [mem_range, exists_apply_eq_apply] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Module.Submodule.Range | {
"line": 101,
"column": 12
} | {
"line": 101,
"column": 56
} | {
"line": 103,
"column": 0
} | [
{
"pp": "case ht\nR : Type u_1\nR₂ : Type u_2\nM : Type u_5\nM₂ : Type u_6\ninst✝⁶ : Semiring R\ninst✝⁵ : Semiring R₂\ninst✝⁴ : AddCommMonoid M\ninst✝³ : AddCommMonoid M₂\ninst✝² : Module R M\ninst✝¹ : Module R₂ M₂\nτ₁₂ : R →+* R₂\ninst✝ : RingHomSurjective τ₁₂\nf g : M →ₛₗ[τ₁₂] M₂\nw✝ : M\n⊢ g w✝ ∈ g.range",
... | [] | simp only [mem_range, exists_apply_eq_apply] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.GroupWithZero.Associated | {
"line": 532,
"column": 13
} | {
"line": 532,
"column": 51
} | {
"line": 532,
"column": 51
} | [
{
"pp": "M : Type u_1\ninst✝ : CommMonoid M\na b c d : Associates M\nh₁ : a ≤ b\nh₂ : c ≤ d\nx : Associates M\nhx : b = a * x\ny : Associates M\nhy : d = c * y\n⊢ b * d = a * c * (x * y)",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"Semigroup.toMul",
"Associates.instCommMono... | [] | simp [hx, hy, mul_comm, mul_left_comm] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.GroupWithZero.Associated | {
"line": 532,
"column": 13
} | {
"line": 532,
"column": 51
} | {
"line": 532,
"column": 51
} | [
{
"pp": "M : Type u_1\ninst✝ : CommMonoid M\na b c d : Associates M\nh₁ : a ≤ b\nh₂ : c ≤ d\nx : Associates M\nhx : b = a * x\ny : Associates M\nhy : d = c * y\n⊢ b * d = a * c * (x * y)",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"Semigroup.toMul",
"Associates.instCommMono... | [] | simp [hx, hy, mul_comm, mul_left_comm] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.GroupWithZero.Associated | {
"line": 532,
"column": 13
} | {
"line": 532,
"column": 51
} | {
"line": 532,
"column": 51
} | [
{
"pp": "M : Type u_1\ninst✝ : CommMonoid M\na b c d : Associates M\nh₁ : a ≤ b\nh₂ : c ≤ d\nx : Associates M\nhx : b = a * x\ny : Associates M\nhy : d = c * y\n⊢ b * d = a * c * (x * y)",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"Semigroup.toMul",
"Associates.instCommMono... | [] | simp [hx, hy, mul_comm, mul_left_comm] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Module.Submodule.Equiv | {
"line": 286,
"column": 20
} | {
"line": 286,
"column": 29
} | {
"line": 286,
"column": 29
} | [
{
"pp": "R : Type u_1\nR₁ : Type u_2\nR₂ : Type u_3\nR₃ : Type u_4\nM : Type u_5\nM₁ : Type u_6\nM₂ : Type u_7\nM₃ : Type u_8\nN : Type u_9\ninst✝⁸ : CommSemiring R\ninst✝⁷ : AddCommMonoid M\ninst✝⁶ : AddCommMonoid M₁\ninst✝⁵ : AddCommMonoid M₂\ninst✝⁴ : AddCommMonoid M₃\ninst✝³ : Module R M\ninst✝² : Module R ... | [
"R : Type u_1\nR₁ : Type u_2\nR₂ : Type u_3\nR₃ : Type u_4\nM : Type u_5\nM₁ : Type u_6\nM₂ : Type u_7\nM₃ : Type u_8\nN : Type u_9\ninst✝⁸ : CommSemiring R\ninst✝⁷ : AddCommMonoid M\ninst✝⁶ : AddCommMonoid M₁\ninst✝⁵ : AddCommMonoid M₂\ninst✝⁴ : AddCommMonoid M₃\ninst✝³ : Module R M\ninst✝² : Module R M₁\ninst✝¹ :... | intro t x | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Order.Interval.Set.OrderIso | {
"line": 37,
"column": 2
} | {
"line": 37,
"column": 24
} | {
"line": 39,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : Preorder α\ninst✝ : Preorder β\ne : α ≃o β\na b : β\n⊢ ⇑e ⁻¹' Icc a b = Icc (e.symm a) (e.symm b)",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Set.Ici",
"congrArg",
"Preorder.toLE",
"OrderIso",
"Set.instInter",... | [] | simp [← Ici_inter_Iic] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Order.Interval.Set.OrderIso | {
"line": 37,
"column": 2
} | {
"line": 37,
"column": 24
} | {
"line": 39,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : Preorder α\ninst✝ : Preorder β\ne : α ≃o β\na b : β\n⊢ ⇑e ⁻¹' Icc a b = Icc (e.symm a) (e.symm b)",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Set.Ici",
"congrArg",
"Preorder.toLE",
"OrderIso",
"Set.instInter",... | [] | simp [← Ici_inter_Iic] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.Interval.Set.OrderIso | {
"line": 37,
"column": 2
} | {
"line": 37,
"column": 24
} | {
"line": 39,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : Preorder α\ninst✝ : Preorder β\ne : α ≃o β\na b : β\n⊢ ⇑e ⁻¹' Icc a b = Icc (e.symm a) (e.symm b)",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Set.Ici",
"congrArg",
"Preorder.toLE",
"OrderIso",
"Set.instInter",... | [] | simp [← Ici_inter_Iic] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.Interval.Set.OrderIso | {
"line": 57,
"column": 32
} | {
"line": 57,
"column": 52
} | {
"line": 57,
"column": 53
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : Preorder α\ninst✝ : Preorder β\ne : α ≃o β\na b : α\n⊢ ⇑e.symm ⁻¹' Ioo a b = Ioo (e a) (e b)",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"OrderIso.preimage_Ioo",
"Preorder.toLE",
"id",
... | [
"α : Type u_1\nβ : Type u_2\ninst✝¹ : Preorder α\ninst✝ : Preorder β\ne : α ≃o β\na b : α\n⊢ Ioo (e.symm.symm a) (e.symm.symm b) = Ioo (e a) (e b)"
] | e.symm.preimage_Ioo, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.Atoms | {
"line": 926,
"column": 8
} | {
"line": 928,
"column": 49
} | {
"line": 928,
"column": 50
} | [
{
"pp": "case refine_2.inr\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 : ⊤ ∈ lowerBounds s\n⊢ ⊤ ≤ if ⊥ ∈ s then ⊥ else ⊤",
"ppTerm": "?refine_2.inr",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr"... | [] | · rw [if_neg]
intro con
exact top_ne_bot (eq_bot_iff.2 (h con)) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Order.CompactlyGenerated.Basic | {
"line": 227,
"column": 8
} | {
"line": 227,
"column": 24
} | {
"line": 227,
"column": 24
} | [
{
"pp": "case inl\nα : Type u_2\ninst✝ : CompleteLattice α\ns : Set α\nhne : s.Nonempty\nhsc : SupClosed s\nht₁ : ↑∅ ⊆ s\nht₂ : sSup s = ∅.sup id\n⊢ sSup s ∈ s",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Lattice.toSemilatticeSup",
"congrArg",
"Finset",
"OrderBot... | [
"case inl\nα : Type u_2\ninst✝ : CompleteLattice α\ns : Set α\nhne : s.Nonempty\nhsc : SupClosed s\nht₁ : ↑∅ ⊆ s\nht₂ : sSup s = ⊥\n⊢ sSup s ∈ s"
] | Finset.sup_empty | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.CompactlyGenerated.Basic | {
"line": 246,
"column": 4
} | {
"line": 246,
"column": 25
} | {
"line": 247,
"column": 4
} | [
{
"pp": "α : Type u_2\ninst✝ : CompleteLattice α\nh : IsSupClosedCompact α\na : (fun x1 x2 ↦ x1 > x2) ↪r fun x1 x2 ↦ x2 < x1\n⊢ sSup (range ⇑a) ∈ range ⇑a",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Preorder.toLT",
"PartialOrder.toPreorder",
"CompleteLattice.toCondit... | [
"case x\nα : Type u_2\ninst✝ : CompleteLattice α\nh : IsSupClosedCompact α\na : (fun x1 x2 ↦ x1 > x2) ↪r fun x1 x2 ↦ x2 < x1\n⊢ (range ⇑a).Nonempty",
"case a\nα : Type u_2\ninst✝ : CompleteLattice α\nh : IsSupClosedCompact α\na : (fun x1 x2 ↦ x1 > x2) ↪r fun x1 x2 ↦ x2 < x1\n⊢ SupClosed (range ⇑a)"
] | apply h (Set.range a) | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Order.SupIndep | {
"line": 439,
"column": 52
} | {
"line": 439,
"column": 68
} | {
"line": 440,
"column": 2
} | [
{
"pp": "α : Type u_5\ninst✝ : CompleteLattice α\nf : Fin 3 → α\nthis : ⨆ i ∈ Finset.univ, f i = f 0 ⊔ f 1 ⊔ f 2\n⊢ ⨆ i, f i = f 0 ⊔ f 1 ⊔ f 2",
"ppTerm": "?m.51",
"assigned": true,
"usedConstants": [
"Lattice.toSemilatticeSup",
"Finset.univ",
"Iff.of_eq",
"congrArg",
"... | [] | by simp [← this] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Order.CompactlyGenerated.Basic | {
"line": 309,
"column": 23
} | {
"line": 309,
"column": 63
} | {
"line": 309,
"column": 64
} | [
{
"pp": "α : Type u_2\ninst✝¹ : CompleteLattice α\ninst✝ : WellFoundedGT α\ns : Set α\nhs : sSupIndep s\nt : Finset α\nht₁ : ↑t ⊆ s\nht₂ : sSup s = t.sup id\nx : α\nhx₀ : x ∈ s\nhx₁ : x ≠ ⊥\nhx₂ : x ∉ t\n⊢ ↑t ∪ {x} ⊆ s",
"ppTerm": "?m.147",
"assigned": true,
"usedConstants": [
"congrArg",
... | [] | by simp [ht₁, hx₀, -Set.union_singleton] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Order.CompactlyGenerated.Basic | {
"line": 450,
"column": 52
} | {
"line": 476,
"column": 40
} | {
"line": 478,
"column": 0
} | [
{
"pp": "α : Type u_2\ninst✝¹ : CompleteLattice α\ninst✝ : IsCompactlyGenerated α\nι : Type u_3\nf : ι → α\n⊢ iSupIndep f ↔ ∀ (s : Finset ι), s.SupIndep f",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"SetLike.mem_coe._simp_1",
"False",
"... | [] | by
refine ⟨fun h ↦ h.supIndep', fun h ↦ iSupIndep_def'.mpr fun i ↦ ?_⟩
classical
have hf : Set.InjOn f {i : ι | f i ≠ ⊥} := by
by_contra! hf
simp_all only [Set.InjOn, ne_eq, Set.mem_setOf_eq, not_forall]
obtain ⟨x₁, hx₁, x₂, hx₂, hfeq, hneq⟩ := hf
specialize h ({x₁, x₂} : Finset ι)
rw [Finset.... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.BigOperators.Ring.List | {
"line": 72,
"column": 63
} | {
"line": 72,
"column": 72
} | {
"line": 72,
"column": 73
} | [
{
"pp": "M₀ : Type u_4\ninst✝² : MonoidWithZero M₀\ninst✝¹ : Nontrivial M₀\ninst✝ : NoZeroDivisors M₀\na : M₀\nl : List M₀\n⊢ a = 0 ∨ 0 ∈ l ↔ 0 ∈ a :: l",
"ppTerm": "?m.46",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Membership.mem",
"id",
"List.cons",
... | [
"M₀ : Type u_4\ninst✝² : MonoidWithZero M₀\ninst✝¹ : Nontrivial M₀\ninst✝ : NoZeroDivisors M₀\na : M₀\nl : List M₀\n⊢ a = 0 ∨ 0 ∈ l ↔ 0 = a ∨ 0 ∈ l"
] | mem_cons, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Span.Basic | {
"line": 567,
"column": 37
} | {
"line": 567,
"column": 47
} | {
"line": 567,
"column": 47
} | [
{
"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 q : Submodule R M\nhpq : Disjoint p q\n... | [
"R : Type u_1\nR₂ : Type u_2\nM : Type u_4\nM₂ : Type u_5\ninst✝⁶ : Semiring R\ninst✝⁵ : Semiring R₂\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : AddCommGroup M₂\ninst✝¹ : Module R₂ M₂\nτ₁₂ : R →+* R₂\ninst✝ : RingHomSurjective τ₁₂\nf : M →ₛₗ[τ₁₂] M₂\np q : Submodule R M\nhpq : Disjoint p q\nhker : f.ker... | hpq.eq_bot | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.BigOperators.Ring.Finset | {
"line": 135,
"column": 6
} | {
"line": 136,
"column": 80
} | {
"line": 137,
"column": 6
} | [
{
"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 ⋯)\nx : κ a\na✝² : x ∈ t a\ny : κ a\na✝¹ : y ∈ t ... | [
"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 ⋯)\nx : κ a\na✝² : x ∈ t a\ny : κ a\na✝¹ : y ∈ t a\nh : x ≠ y... | have : Pi.cons s a x p₂ a (mem_insert_self _ _)
= Pi.cons s a y p₃ a (mem_insert_self _ _) := by rw [eq₂, eq₃, eq] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Algebra.BigOperators.Ring.Finset | {
"line": 143,
"column": 8
} | {
"line": 143,
"column": 21
} | {
"line": 143,
"column": 22
} | [
{
"pp": "case insert\nι : Type u_1\nR : Type u_4\ninst✝¹ : CommSemiring R\ninst✝ : DecidableEq ι\nκ : ι → Type u_5\nt : (i : ι) → Finset (κ i)\nf : (i : ι) → κ i → R\na : ι\ns : Finset ι\nha : a ∉ s\nih : ∏ a ∈ s, ∑ b ∈ t a, f a b = ∑ p ∈ s.pi t, ∏ x ∈ s.attach, f (↑x) (p ↑x ⋯)\nh₁ : ∀ x ∈ t a, ∀ y ∈ t a, x ≠ y... | [
"case insert\nι : Type u_1\nR : Type u_4\ninst✝¹ : CommSemiring R\ninst✝ : DecidableEq ι\nκ : ι → Type u_5\nt : (i : ι) → Finset (κ i)\nf : (i : ι) → κ i → R\na : ι\ns : Finset ι\nha : a ∉ s\nih : ∏ a ∈ s, ∑ b ∈ t a, f a b = ∑ p ∈ s.pi t, ∏ x ∈ s.attach, f (↑x) (p ↑x ⋯)\nh₁ : ∀ x ∈ t a, ∀ y ∈ t a, x ≠ y → Disjoint ... | sum_image h₂, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Finsupp.Single | {
"line": 454,
"column": 4
} | {
"line": 455,
"column": 42
} | {
"line": 457,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nM : Type u_5\ninst✝ : Zero M\nf : α ↪ β\na : α\nm : M\n⊢ embDomain f (single a m) = single (f a) m",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Finsupp.instFunLike",
"_private.Mathlib.Data.Finsupp.Single.0.Finsupp.embDomain_single._proof... | [] | ext b
by_cases h : b ∈ Set.range f <;> grind | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Finsupp.Single | {
"line": 454,
"column": 4
} | {
"line": 455,
"column": 42
} | {
"line": 457,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nM : Type u_5\ninst✝ : Zero M\nf : α ↪ β\na : α\nm : M\n⊢ embDomain f (single a m) = single (f a) m",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Finsupp.instFunLike",
"_private.Mathlib.Data.Finsupp.Single.0.Finsupp.embDomain_single._proof... | [] | ext b
by_cases h : b ∈ Set.range f <;> grind | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Finsupp.Option | {
"line": 149,
"column": 2
} | {
"line": 152,
"column": 8
} | {
"line": 154,
"column": 0
} | [
{
"pp": "α : Type u_1\nM : Type u_2\ninst✝ : Zero M\nf : Option α →₀ M\n⊢ optionElim (f none) f.some = f",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Finsupp.instFunLike",
"Eq.mpr",
"Finsupp.ext",
"congrArg",
"Option.casesOn",
"Finsupp.some",
"... | [] | ext a
cases a
· rw [optionElim_apply_none]
· simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Finsupp.Option | {
"line": 149,
"column": 2
} | {
"line": 152,
"column": 8
} | {
"line": 154,
"column": 0
} | [
{
"pp": "α : Type u_1\nM : Type u_2\ninst✝ : Zero M\nf : Option α →₀ M\n⊢ optionElim (f none) f.some = f",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Finsupp.instFunLike",
"Eq.mpr",
"Finsupp.ext",
"congrArg",
"Option.casesOn",
"Finsupp.some",
"... | [] | ext a
cases a
· rw [optionElim_apply_none]
· simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.GroupWithZero.Indicator | {
"line": 69,
"column": 20
} | {
"line": 69,
"column": 90
} | {
"line": 69,
"column": 90
} | [
{
"pp": "ι : Type u_1\nM₀ : Type u_4\ninst✝ : MulZeroOneClass M₀\ns t : Set ι\nx✝ : ι\n⊢ (s ∩ t).indicator 1 x✝ = (s.indicator 1 * t.indicator 1) x✝",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"HMul.hMul",
"MulZeroClass.toMul",
"c... | [
"ι : Type u_1\nM₀ : Type u_4\ninst✝ : MulZeroOneClass M₀\ns t : Set ι\nx✝ : ι\n⊢ (s ∩ t).indicator 1 x✝ = (s ∩ t).indicator (fun j ↦ 1) x✝"
] | simp only [← inter_indicator_mul, Pi.mul_apply, Pi.one_apply, one_mul] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.BigOperators.Finsupp.Basic | {
"line": 553,
"column": 12
} | {
"line": 554,
"column": 31
} | {
"line": 556,
"column": 0
} | [
{
"pp": "α : Type u_1\nM : Type u_8\ninst✝¹ : AddCommMonoid M\nf1 f2 : α →₀ M\nhd : Disjoint f1.support f2.support\nβ : Type u_16\ninst✝ : CommMonoid β\ng : α → M → β\nthis : ∀ {f1 f2 : α →₀ M}, Disjoint f1.support f2.support → ∏ x ∈ f1.support, g x (f1 x + f2 x) = f1.prod g\n⊢ (f1 + f2).prod g = f1.prod g * f2... | [] | simp_rw [← this hd, ← this hd.symm, add_comm (f2 _), Finsupp.prod, support_add_eq hd,
prod_union hd, add_apply] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Algebra.BigOperators.Finsupp.Basic | {
"line": 553,
"column": 12
} | {
"line": 554,
"column": 31
} | {
"line": 556,
"column": 0
} | [
{
"pp": "α : Type u_1\nM : Type u_8\ninst✝¹ : AddCommMonoid M\nf1 f2 : α →₀ M\nhd : Disjoint f1.support f2.support\nβ : Type u_16\ninst✝ : CommMonoid β\ng : α → M → β\nthis : ∀ {f1 f2 : α →₀ M}, Disjoint f1.support f2.support → ∏ x ∈ f1.support, g x (f1 x + f2 x) = f1.prod g\n⊢ (f1 + f2).prod g = f1.prod g * f2... | [] | simp_rw [← this hd, ← this hd.symm, add_comm (f2 _), Finsupp.prod, support_add_eq hd,
prod_union hd, add_apply] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.BigOperators.Finsupp.Basic | {
"line": 553,
"column": 12
} | {
"line": 554,
"column": 31
} | {
"line": 556,
"column": 0
} | [
{
"pp": "α : Type u_1\nM : Type u_8\ninst✝¹ : AddCommMonoid M\nf1 f2 : α →₀ M\nhd : Disjoint f1.support f2.support\nβ : Type u_16\ninst✝ : CommMonoid β\ng : α → M → β\nthis : ∀ {f1 f2 : α →₀ M}, Disjoint f1.support f2.support → ∏ x ∈ f1.support, g x (f1 x + f2 x) = f1.prod g\n⊢ (f1 + f2).prod g = f1.prod g * f2... | [] | simp_rw [← this hd, ← this hd.symm, add_comm (f2 _), Finsupp.prod, support_add_eq hd,
prod_union hd, add_apply] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Order.AbsoluteValue.Basic | {
"line": 352,
"column": 2
} | {
"line": 354,
"column": 5
} | {
"line": 356,
"column": 0
} | [
{
"pp": "R : Type u_5\ninst✝² : Semiring R\nS : Type u_6\ninst✝¹ : Semiring S\ninst✝ : PartialOrder S\nv : AbsoluteValue R S\n⊢ ¬v.IsNontrivial ↔ ∀ (x : R), x ≠ 0 → v x = 1",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Push.not_exists._simp_1",
"Eq.mpr",
... | [] | simp only [IsNontrivial]
push Not
rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Order.AbsoluteValue.Basic | {
"line": 352,
"column": 2
} | {
"line": 354,
"column": 5
} | {
"line": 356,
"column": 0
} | [
{
"pp": "R : Type u_5\ninst✝² : Semiring R\nS : Type u_6\ninst✝¹ : Semiring S\ninst✝ : PartialOrder S\nv : AbsoluteValue R S\n⊢ ¬v.IsNontrivial ↔ ∀ (x : R), x ≠ 0 → v x = 1",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Push.not_exists._simp_1",
"Eq.mpr",
... | [] | simp only [IsNontrivial]
push Not
rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Order.BigOperators.Ring.Multiset | {
"line": 63,
"column": 6
} | {
"line": 63,
"column": 46
} | {
"line": 64,
"column": 6
} | [
{
"pp": "case ha\nα : Type u_1\nβ : Type u_2\ninst✝³ : CommMonoidWithZero β\ninst✝² : PartialOrder β\ninst✝¹ : PosMulMono β\ninst✝ : ZeroLEOneClass β\nf : α → β\nh1 : ∀ (a : α), 1 ≤ f a\na : α\ns' : Multiset α\nha : a ∈ a ::ₘ s'\n⊢ 0 ≤ f a",
"ppTerm": "?ha",
"assigned": true,
"usedConstants": [
... | [
"case hbc\nα : Type u_1\nβ : Type u_2\ninst✝³ : CommMonoidWithZero β\ninst✝² : PartialOrder β\ninst✝¹ : PosMulMono β\ninst✝ : ZeroLEOneClass β\nf : α → β\nh1 : ∀ (a : α), 1 ≤ f a\na : α\ns' : Multiset α\nha : a ∈ a ::ₘ s'\n⊢ 1 ≤ (map f s').prod"
] | · exact le_trans (zero_le_one' β) (h1 a) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Order.BigOperators.Group.Finset | {
"line": 394,
"column": 46
} | {
"line": 394,
"column": 59
} | {
"line": 394,
"column": 59
} | [
{
"pp": "ι : Type u_1\nα : Type u_2\ninst✝ : DecidableEq α\ns : Finset ι\nf : ι → Finset α\nhs : (↑s).PairwiseDisjoint f\n⊢ #({i ∈ s | f i ≠ ∅}.biUnion f) + #({i ∈ s | f i = ∅}) ≤ #(s.biUnion f) + #({i ∈ s | f i = ∅})",
"ppTerm": "?m.102",
"assigned": true,
"usedConstants": [
"instDecidableNot... | [
"ι : Type u_1\nα : Type u_2\ninst✝ : DecidableEq α\ns : Finset ι\nf : ι → Finset α\nhs : (↑s).PairwiseDisjoint f\n⊢ #(s.biUnion f) + #({i ∈ s | f i = ∅}) ≤ #(s.biUnion f) + #({i ∈ s | f i = ∅})"
] | filter_subset | Mathlib.Tactic.GRewrite.evalGRewriteSeq | null |
Mathlib.Data.Finsupp.SMulWithZero | {
"line": 47,
"column": 4
} | {
"line": 48,
"column": 19
} | {
"line": 50,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst✝¹ : Zero M\ninst✝ : SMulZeroClass R M\na : R\n⊢ a • 0 = 0",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": ... | [] | ext
apply smul_zero | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Finsupp.SMulWithZero | {
"line": 47,
"column": 4
} | {
"line": 48,
"column": 19
} | {
"line": 50,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst✝¹ : Zero M\ninst✝ : SMulZeroClass R M\na : R\n⊢ a • 0 = 0",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": ... | [] | ext
apply smul_zero | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Finset.Fin | {
"line": 74,
"column": 47
} | {
"line": 75,
"column": 50
} | {
"line": 77,
"column": 0
} | [
{
"pp": "n : ℕ\ns t : Finset ℕ\nhs : ∀ m ∈ s, m < n\nht : ∀ m ∈ t, m < n\n⊢ s.attachFin hs ⊂ t.attachFin ht ↔ s ⊂ t",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Preorder.toLT",
"congrArg",
"Finset",
"Fin.valEmbedding",
"PartialOrder.toPreorder",
"Fin... | [] | by
simp [← map_ssubset_map (f := Fin.valEmbedding)] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Order.Interval.Set.Fin | {
"line": 545,
"column": 83
} | {
"line": 547,
"column": 82
} | {
"line": 549,
"column": 0
} | [
{
"pp": "n m : ℕ\ni : Fin n\n⊢ natAdd m '' Ioi i = Ioi (natAdd m i)",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Fin.preimage_natAdd_Ioi_natAdd",
"Set.Ioi",
"Fin.natAdd",
"Set.Ici",
"congrArg",
"Fin.image_natAdd_Ici",
"PartialOr... | [] | by
rw [← preimage_natAdd_Ioi_natAdd, image_preimage_eq_of_subset]
exact Ioi_subset_Ici_self.trans <| image_natAdd_Ici m i ▸ image_subset_range _ _ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.Finsupp.LinearCombination | {
"line": 521,
"column": 4
} | {
"line": 522,
"column": 59
} | {
"line": 524,
"column": 0
} | [
{
"pp": "case refine_2\nR : Type u_1\nM : Type u_2\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\ns : Set M\nm : M\nn : ℕ\n⊢ (∃ x, (∀ (i : Fin n), (x i).2 ∈ s) ∧ ∑ i, (x i).1 • (x i).2 = m) → ∃ f g, ∑ i, f i • ↑(g i) = m",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants"... | [] | rintro ⟨f, hf, rfl⟩
exact ⟨fun i ↦ (f i).1, fun i ↦ ⟨(f i).2, (hf i)⟩, rfl⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Finsupp.LinearCombination | {
"line": 521,
"column": 4
} | {
"line": 522,
"column": 59
} | {
"line": 524,
"column": 0
} | [
{
"pp": "case refine_2\nR : Type u_1\nM : Type u_2\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\ns : Set M\nm : M\nn : ℕ\n⊢ (∃ x, (∀ (i : Fin n), (x i).2 ∈ s) ∧ ∑ i, (x i).1 • (x i).2 = m) → ∃ f g, ∑ i, f i • ↑(g i) = m",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants"... | [] | rintro ⟨f, hf, rfl⟩
exact ⟨fun i ↦ (f i).1, fun i ↦ ⟨(f i).2, (hf i)⟩, rfl⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Logic.Equiv.Fin.Basic | {
"line": 196,
"column": 2
} | {
"line": 196,
"column": 25
} | {
"line": 198,
"column": 0
} | [
{
"pp": "n : ℕ\n⊢ finSuccEquivLast (Fin.last n) = none",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"Equiv.instEquivLike",
"congrArg",
"Equiv",
"finSuccEquiv'_at",
"instOfNatNat",
"Option.none",
"instHAdd",
"HAdd.hAdd",
"Nat",
"... | [] | simp [finSuccEquivLast] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Logic.Equiv.Fin.Basic | {
"line": 196,
"column": 2
} | {
"line": 196,
"column": 25
} | {
"line": 198,
"column": 0
} | [
{
"pp": "n : ℕ\n⊢ finSuccEquivLast (Fin.last n) = none",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"Equiv.instEquivLike",
"congrArg",
"Equiv",
"finSuccEquiv'_at",
"instOfNatNat",
"Option.none",
"instHAdd",
"HAdd.hAdd",
"Nat",
"... | [] | simp [finSuccEquivLast] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Logic.Equiv.Fin.Basic | {
"line": 196,
"column": 2
} | {
"line": 196,
"column": 25
} | {
"line": 198,
"column": 0
} | [
{
"pp": "n : ℕ\n⊢ finSuccEquivLast (Fin.last n) = none",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"Equiv.instEquivLike",
"congrArg",
"Equiv",
"finSuccEquiv'_at",
"instOfNatNat",
"Option.none",
"instHAdd",
"HAdd.hAdd",
"Nat",
"... | [] | simp [finSuccEquivLast] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.BigOperators.Finprod | {
"line": 1119,
"column": 63
} | {
"line": 1119,
"column": 94
} | {
"line": 1119,
"column": 94
} | [
{
"pp": "α : 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})\n⊢ f a * ∏ i ∈ Finite.toFinset hf \\ {a}, f i = ∏ i ∈ Finite.toFinset hf, f i",
"ppTerm": "?m.81",
"assigned": true,
"usedConstant... | [
"α : 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})\n⊢ f a * ∏ i ∈ (Finite.toFinset hf).erase a, f i = ∏ i ∈ Finite.toFinset hf, f i"
] | Finset.sdiff_singleton_eq_erase | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.BigOperators.Finprod | {
"line": 1180,
"column": 6
} | {
"line": 1180,
"column": 48
} | {
"line": 1180,
"column": 49
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nM : Type u_5\ninst✝ : CommMonoid M\ns : Finset β\nf : α → β → M\nh : ∀ b ∈ s, HasFiniteMulSupport fun a ↦ f a b\nhU : (mulSupport fun a ↦ ∏ b ∈ s, f a b) ⊆ ↑⋯.toFinset\n⊢ ∏ᶠ (a : α), ∏ b ∈ s, f a b = ∏ b ∈ s, ∏ᶠ (a : α), f a b",
"ppTerm": "?m.56",
"assigned": true,
... | [
"α : Type u_1\nβ : Type u_2\nM : Type u_5\ninst✝ : CommMonoid M\ns : Finset β\nf : α → β → M\nh : ∀ b ∈ s, HasFiniteMulSupport fun a ↦ f a b\nhU : (mulSupport fun a ↦ ∏ b ∈ s, f a b) ⊆ ↑⋯.toFinset\n⊢ ∏ i ∈ ⋯.toFinset, ∏ b ∈ s, f i b = ∏ b ∈ s, ∏ᶠ (a : α), f a b"
] | finprod_eq_prod_of_mulSupport_subset _ hU, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.ENat.Pow | {
"line": 135,
"column": 33
} | {
"line": 135,
"column": 40
} | {
"line": 135,
"column": 40
} | [
{
"pp": "case inr.inr.coe.top\nx y z : ℕ∞\nx_2 : 1 < x\na✝ : ℕ\n⊢ ⊤ = x ^ ↑a✝ * ⊤",
"ppTerm": "?inr.inr.coe.top",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"ENat.instNatCast",
"instTopENat",
"congrArg",
"CommSemiring.toSemiring",
"id",
... | [
"case inr.inr.coe.top\nx y z : ℕ∞\nx_2 : 1 < x\na✝ : ℕ\n⊢ ⊤ = ⊤",
"case inr.inr.coe.top\nx y z : ℕ∞\nx_2 : 1 < x\na✝ : ℕ\n⊢ x ^ ↑a✝ ≠ 0"
] | mul_top | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Group.ModEq | {
"line": 198,
"column": 65
} | {
"line": 198,
"column": 85
} | {
"line": 200,
"column": 0
} | [
{
"pp": "M : Type u_1\ninst✝ : AddCancelCommMonoid M\na b p : M\n⊢ a + b ≡ b [PMOD p] ↔ a ≡ 0 [PMOD p]",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"AddCommGroup.ModEq",
"congrArg",
"AddMonoid.toAddZeroClass",
"AddCommGroup.add_modEq_left._simp_1",
"AddZero... | [] | by simp [add_comm a] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.LinearIndependent.Basic | {
"line": 150,
"column": 2
} | {
"line": 152,
"column": 77
} | {
"line": 154,
"column": 0
} | [
{
"pp": "ι : Type u'\nι' : Type u_1\nR : Type u_2\ns : Set ι\nM : Type u_4\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nf : ι → ι'\ng : ι' → M\nhs : LinearIndepOn R (g ∘ f) s\n⊢ LinearIndepOn R g (f '' s)",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Nontriv... | [] | nontriviality R
have : InjOn f s := injOn_iff_injective.2 hs.injective.of_comp
exact (linearIndependent_equiv' (Equiv.Set.imageOfInjOn f s this) rfl).1 hs | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.LinearIndependent.Basic | {
"line": 150,
"column": 2
} | {
"line": 152,
"column": 77
} | {
"line": 154,
"column": 0
} | [
{
"pp": "ι : Type u'\nι' : Type u_1\nR : Type u_2\ns : Set ι\nM : Type u_4\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nf : ι → ι'\ng : ι' → M\nhs : LinearIndepOn R (g ∘ f) s\n⊢ LinearIndepOn R g (f '' s)",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Nontriv... | [] | nontriviality R
have : InjOn f s := injOn_iff_injective.2 hs.injective.of_comp
exact (linearIndependent_equiv' (Equiv.Set.imageOfInjOn f s this) rfl).1 hs | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.SetTheory.Cardinal.Finite | {
"line": 108,
"column": 2
} | {
"line": 114,
"column": 89
} | {
"line": 116,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝ : Finite β\nf : α → β\n⊢ Bijective f ↔ Injective f ∧ Nat.card α = Nat.card β",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Fintype.ofFinite",
"congrArg",
"Iff.rfl",
"Fintype.card",
"id",
"Nat.c... | [] | rw [Bijective, and_congr_right_iff]
intro h
have := Fintype.ofFinite β
have := Fintype.ofInjective f h
revert h
rw [← and_congr_right_iff, ← Bijective,
card_eq_fintype_card, card_eq_fintype_card, Fintype.bijective_iff_injective_and_card] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.SetTheory.Cardinal.Finite | {
"line": 108,
"column": 2
} | {
"line": 114,
"column": 89
} | {
"line": 116,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝ : Finite β\nf : α → β\n⊢ Bijective f ↔ Injective f ∧ Nat.card α = Nat.card β",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Fintype.ofFinite",
"congrArg",
"Iff.rfl",
"Fintype.card",
"id",
"Nat.c... | [] | rw [Bijective, and_congr_right_iff]
intro h
have := Fintype.ofFinite β
have := Fintype.ofInjective f h
revert h
rw [← and_congr_right_iff, ← Bijective,
card_eq_fintype_card, card_eq_fintype_card, Fintype.bijective_iff_injective_and_card] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Dimension.Basic | {
"line": 460,
"column": 2
} | {
"line": 460,
"column": 21
} | {
"line": 462,
"column": 0
} | [
{
"pp": "R : Type u\nM M₁ : Type v\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : AddCommMonoid M₁\ninst✝ : Module R M₁\nf : M →ₗ[R] M₁\nh : Surjective ⇑f\n⊢ Module.rank R ↥f.range ≤ Module.rank R M",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants": [
"rank_... | [] | apply rank_range_le | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Data.Set.Card | {
"line": 435,
"column": 4
} | {
"line": 435,
"column": 37
} | {
"line": 436,
"column": 4
} | [
{
"pp": "case refine_1\nα : Type u_1\ns : Set α\nx : α\nhx : x ∈ s\ny z : α\nhne : y ≠ z\nhs : s \\ {x} = {y, z}\n⊢ ∃ x y z, x ≠ y ∧ x ≠ z ∧ y ≠ z ∧ s = {x, y, z}",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"Exists",
"Set.instSingletonSet",
"Insert.insert",
... | [
"case refine_1.refine_1\nα : Type u_1\ns : Set α\nx : α\nhx : x ∈ s\ny z : α\nhne : y ≠ z\nhs : s \\ {x} = {y, z}\n⊢ x ≠ y",
"case refine_1.refine_2\nα : Type u_1\ns : Set α\nx : α\nhx : x ∈ s\ny z : α\nhne : y ≠ z\nhs : s \\ {x} = {y, z}\n⊢ x ≠ z",
"case refine_1.refine_3\nα : Type u_1\ns : Set α\nx : α\nhx : ... | refine ⟨x, y, z, ?_, ?_, hne, ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Algebra.EuclideanDomain.Defs | {
"line": 244,
"column": 2
} | {
"line": 245,
"column": 27
} | {
"line": 247,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝¹ : EuclideanDomain R\ninst✝ : DecidableEq R\ns : R\n⊢ gcdB 0 s = 1",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"EuclideanDomain.xgcd.eq_1",
"congrArg",
"CommSemiring.toSemiring",
"AddGroupWithOne.toAddMonoidWithOne",
... | [] | unfold gcdB
rw [xgcd, xgcd_zero_left] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.EuclideanDomain.Defs | {
"line": 244,
"column": 2
} | {
"line": 245,
"column": 27
} | {
"line": 247,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝¹ : EuclideanDomain R\ninst✝ : DecidableEq R\ns : R\n⊢ gcdB 0 s = 1",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"EuclideanDomain.xgcd.eq_1",
"congrArg",
"CommSemiring.toSemiring",
"AddGroupWithOne.toAddMonoidWithOne",
... | [] | unfold gcdB
rw [xgcd, xgcd_zero_left] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Set.Card | {
"line": 480,
"column": 15
} | {
"line": 480,
"column": 25
} | {
"line": 480,
"column": 25
} | [
{
"pp": "case top\nα : Type u_1\ns : Set α\na✝ : ∀ (n : ℕ), ↑n ≤ s.encard → ∃ t ⊆ s, t.encard = ↑n\nhk : ⊤ ≤ s.encard\n⊢ ∃ t ⊆ s, t.encard = ⊤",
"ppTerm": "?top",
"assigned": true,
"usedConstants": [
"Set.encard",
"instLinearOrderENat",
"congrArg",
"PartialOrder.toPreorder",
... | [
"case top\nα : Type u_1\ns : Set α\na✝ : ∀ (n : ℕ), ↑n ≤ s.encard → ∃ t ⊆ s, t.encard = ↑n\nhk : s.encard = ⊤\n⊢ ∃ t ⊆ s, t.encard = ⊤"
] | top_le_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Set.Card | {
"line": 486,
"column": 12
} | {
"line": 486,
"column": 22
} | {
"line": 486,
"column": 22
} | [
{
"pp": "case inl\nα : Type u_1\ns t : Set α\nk : ℕ∞\nhst : s ⊆ t\nhsk : ⊤ ≤ k\nhkt : k ≤ t.encard\nhs : s.encard = ⊤\n⊢ ∃ r, s ⊆ r ∧ r ⊆ t ∧ r.encard = k",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Set.encard",
"instTopENat",
"instLinearOrderENat",
"congrArg",
... | [
"case inl\nα : Type u_1\ns t : Set α\nk : ℕ∞\nhst : s ⊆ t\nhsk : k = ⊤\nhkt : k ≤ t.encard\nhs : s.encard = ⊤\n⊢ ∃ r, s ⊆ r ∧ r ⊆ t ∧ r.encard = k"
] | top_le_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Set.Card | {
"line": 522,
"column": 17
} | {
"line": 527,
"column": 81
} | {
"line": 529,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ns : Set α\nf : α → β\nhs : s.Finite\nh : (f '' s).encard = s.encard\n⊢ InjOn f s",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Set.encard",
"Function.invFunOn",
"congrArg",
"Eq.mp",
"instPreorderENat",
... | [] | by
obtain (h' | hne) := isEmpty_or_nonempty α
· simp
rw [← (f.invFunOn_injOn_image s).encard_image] at h
rw [injOn_iff_invFunOn_image_image_eq_self]
exact hs.eq_of_subset_of_encard_le' (f.invFunOn_image_image_subset s) h.symm.le | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Set.Card | {
"line": 649,
"column": 2
} | {
"line": 651,
"column": 53
} | {
"line": 653,
"column": 0
} | [
{
"pp": "α : Type u_1\ns : Set α\nk : ℕ\n⊢ s.encard ≤ ↑k ↔ s.Finite ∧ s.ncard ≤ k",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Set.encard",
"Set.ncard_def",
"ENat.instNatCast",
"congrArg",
"Set.Finite",
"Exists",
"id",
"and... | [] | rw [encard_le_coe_iff, and_congr_right_iff]
exact fun hfin ↦ ⟨fun ⟨n₀, hn₀, hle⟩ ↦ by rwa [ncard_def, hn₀, ENat.toNat_coe],
fun h ↦ ⟨s.ncard, by rw [hfin.cast_ncard_eq], h⟩⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Set.Card | {
"line": 649,
"column": 2
} | {
"line": 651,
"column": 53
} | {
"line": 653,
"column": 0
} | [
{
"pp": "α : Type u_1\ns : Set α\nk : ℕ\n⊢ s.encard ≤ ↑k ↔ s.Finite ∧ s.ncard ≤ k",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Set.encard",
"Set.ncard_def",
"ENat.instNatCast",
"congrArg",
"Set.Finite",
"Exists",
"id",
"and... | [] | rw [encard_le_coe_iff, and_congr_right_iff]
exact fun hfin ↦ ⟨fun ⟨n₀, hn₀, hle⟩ ↦ by rwa [ncard_def, hn₀, ENat.toNat_coe],
fun h ↦ ⟨s.ncard, by rw [hfin.cast_ncard_eq], h⟩⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Group.Submonoid.Finite | {
"line": 71,
"column": 5
} | {
"line": 78,
"column": 30
} | {
"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\n⊢ (pi univ fun i ↦ closure (s i)) ≤ closure (univ.pi fun i ↦ s i)",
"ppTerm": "?m.52",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mp... | [] | by
classical
exact pi_le_iff.mpr fun i => map_le_of_le_comap _ <| closure_le.2 fun _x hx =>
subset_closure <| mem_univ_pi.mpr fun j => by
by_cases H : j = i
· subst H
simpa
· simpa [H] using hs _ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.GroupTheory.Coset.Basic | {
"line": 196,
"column": 2
} | {
"line": 206,
"column": 33
} | {
"line": 208,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : Group α\ns : Subgroup α\nx y : α\n⊢ x • ↑s = y • ↑s ↔ x⁻¹ * y ∈ s",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"mul_inv_cancel_right",
"Eq.mpr",
"MulOne.toOne",
"Semigroup.toMul",
"DivInvMonoid.toInv",
"... | [] | rw [Set.ext_iff]
simp_rw [mem_leftCoset_iff, SetLike.mem_coe]
constructor
· intro h
apply (h y).mpr
rw [inv_mul_cancel]
exact s.one_mem
· intro h z
rw [← mul_inv_cancel_right x⁻¹ y]
rw [mul_assoc]
exact s.mul_mem_cancel_left h | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.Coset.Basic | {
"line": 196,
"column": 2
} | {
"line": 206,
"column": 33
} | {
"line": 208,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : Group α\ns : Subgroup α\nx y : α\n⊢ x • ↑s = y • ↑s ↔ x⁻¹ * y ∈ s",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"mul_inv_cancel_right",
"Eq.mpr",
"MulOne.toOne",
"Semigroup.toMul",
"DivInvMonoid.toInv",
"... | [] | rw [Set.ext_iff]
simp_rw [mem_leftCoset_iff, SetLike.mem_coe]
constructor
· intro h
apply (h y).mpr
rw [inv_mul_cancel]
exact s.one_mem
· intro h z
rw [← mul_inv_cancel_right x⁻¹ y]
rw [mul_assoc]
exact s.mul_mem_cancel_left h | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Basis.Submodule | {
"line": 216,
"column": 7
} | {
"line": 216,
"column": 90
} | {
"line": 216,
"column": 90
} | [
{
"pp": "ι✝ : Type u_1\nι' : Type u_2\nR✝ : Type u_3\nR₂ : Type u_4\nM✝ : Type u_5\nM' : Type u_6\nM : Type u_7\nR : Type u_8\ninst✝⁴ : Ring R\ninst✝³ : Nontrivial R\ninst✝² : IsAddTorsionFree R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nA : AddSubgroup M\nι : Type u_9\nb : Basis ι R M\nh : A = AddSubgroup.c... | [] | by rw [h, ← Submodule.span_int_eq_addSubgroupClosure, toAddSubgroup_toIntSubmodule] | [anonymous] | Lean.Parser.Term.byTactic |
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