module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Algebra.BigOperators.GroupWithZero.Finset | {
"line": 35,
"column": 2
} | {
"line": 39,
"column": 40
} | {
"line": 41,
"column": 0
} | [
{
"pp": "ι : Type u_1\nM₀ : Type u_4\ninst✝¹ : CommMonoidWithZero M₀\np : ι → Prop\ninst✝ : DecidablePred p\nf : ι → M₀\ns : Finset ι\n⊢ (∏ i ∈ s, if p i then f i else 0) = if ∀ i ∈ s, p i then ∏ i ∈ s, f i else 0",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Push.n... | [] | split_ifs with h
· exact prod_congr rfl fun i hi => by simp [h i hi]
· push Not at h
rcases h with ⟨i, hi, hq⟩
exact prod_eq_zero hi (by simp [hq]) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.BigOperators.GroupWithZero.Finset | {
"line": 35,
"column": 2
} | {
"line": 39,
"column": 40
} | {
"line": 41,
"column": 0
} | [
{
"pp": "ι : Type u_1\nM₀ : Type u_4\ninst✝¹ : CommMonoidWithZero M₀\np : ι → Prop\ninst✝ : DecidablePred p\nf : ι → M₀\ns : Finset ι\n⊢ (∏ i ∈ s, if p i then f i else 0) = if ∀ i ∈ s, p i then ∏ i ∈ s, f i else 0",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Push.n... | [] | split_ifs with h
· exact prod_congr rfl fun i hi => by simp [h i hi]
· push Not at h
rcases h with ⟨i, hi, hq⟩
exact prod_eq_zero hi (by simp [hq]) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.BigOperators.Group.Finset.Pi | {
"line": 39,
"column": 72
} | {
"line": 40,
"column": 75
} | {
"line": 42,
"column": 0
} | [
{
"pp": "ι : Type u_1\nβ : Type u_2\ninst✝² : CommMonoid β\ninst✝¹ : DecidableEq ι\ninst✝ : Fintype ι\nκ : ι → Type u_3\nt : (i : ι) → Finset (κ i)\nf : ((i : ι) → i ∈ univ → κ i) → β\n⊢ ∏ x ∈ univ.pi t, f x = ∏ x ∈ Fintype.piFinset t, f fun a x_1 ↦ x a",
"ppTerm": "?m.30",
"assigned": true,
"usedCo... | [] | by
apply prod_nbij' (fun x i ↦ x i <| mem_univ _) (fun x i _ ↦ x i) <;> simp | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Order.CompactlyGenerated.Basic | {
"line": 626,
"column": 6
} | {
"line": 626,
"column": 10
} | {
"line": 627,
"column": 6
} | [
{
"pp": "ι : Sort u_1\nα : Type u_2\ninst✝³ : CompleteLattice α\nf : ι → α\ninst✝² : IsModularLattice α\ninst✝¹ : IsCompactlyGenerated α\ninst✝ : ComplementedLattice α\nb : α\n⊢ b = sSup {a | IsAtom a ∧ a ≤ b}",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"PartialOrder.toPreorder",... | [
"ι : Sort u_1\nα : Type u_2\ninst✝³ : CompleteLattice α\nf : ι → α\ninst✝² : IsModularLattice α\ninst✝¹ : IsCompactlyGenerated α\ninst✝ : ComplementedLattice α\nb : α\n⊢ sSup {a | IsAtom a ∧ a ≤ b} = b"
] | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.Data.Multiset.Antidiagonal | {
"line": 50,
"column": 4
} | {
"line": 50,
"column": 31
} | {
"line": 51,
"column": 4
} | [
{
"pp": "α : Type u_1\ns : Multiset α\nx : Multiset α × Multiset α\nl : List α\nh : x.1 + x.2 = ↑l\n⊢ x ∈ ↑(powersetAux l).revzip",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Classical.decEq",
"DecidableEq"
],
"usedFVars": [
"α"
],
"usedGoals": [
... | [
"α : Type u_1\ns : Multiset α\nx : Multiset α × Multiset α\nl : List α\nh : x.1 + x.2 = ↑l\nx✝ : DecidableEq α\n⊢ x ∈ ↑(powersetAux l).revzip"
] | have _ := Classical.decEq α | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Algebra.BigOperators.Pi | {
"line": 207,
"column": 2
} | {
"line": 209,
"column": 60
} | {
"line": 211,
"column": 0
} | [
{
"pp": "ι : Type u_1\ninst✝² : Finite ι\ninst✝¹ : DecidableEq ι\nM : ι → Type u_7\ninst✝ : (i : ι) → AddCommMonoid (M i)\np : ((i : ι) → M i) → Prop\nf : (i : ι) → M i\nzero : p 0\nadd : ∀ (f g : (i : ι) → M i), p f → p g → p (f + g)\nsingle : ∀ (i : ι) (m : M i), p (Pi.single i m)\n⊢ p f",
"ppTerm": "?m.1... | [] | cases nonempty_fintype ι
rw [← Finset.univ_sum_single f]
exact Finset.sum_induction _ _ add zero (by simp [single]) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.BigOperators.Pi | {
"line": 207,
"column": 2
} | {
"line": 209,
"column": 60
} | {
"line": 211,
"column": 0
} | [
{
"pp": "ι : Type u_1\ninst✝² : Finite ι\ninst✝¹ : DecidableEq ι\nM : ι → Type u_7\ninst✝ : (i : ι) → AddCommMonoid (M i)\np : ((i : ι) → M i) → Prop\nf : (i : ι) → M i\nzero : p 0\nadd : ∀ (f g : (i : ι) → M i), p f → p g → p (f + g)\nsingle : ∀ (i : ι) (m : M i), p (Pi.single i m)\n⊢ p f",
"ppTerm": "?m.1... | [] | cases nonempty_fintype ι
rw [← Finset.univ_sum_single f]
exact Finset.sum_induction _ _ add zero (by simp [single]) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.BigOperators.Ring.Finset | {
"line": 172,
"column": 82
} | {
"line": 192,
"column": 16
} | {
"line": 194,
"column": 0
} | [
{
"pp": "ι : Type u_1\nR : Type u_4\ninst✝¹ : CommSemiring R\ninst✝ : DecidableEq ι\nf g : ι → R\ns : Finset ι\n⊢ ∏ i ∈ s, (f i + g i) = ∑ t ∈ s.powerset, (∏ i ∈ t, f i) * ∏ i ∈ s \\ t, g i",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instDecidableNot",
"Fal... | [] | by
classical
calc
∏ i ∈ s, (f i + g i) =
∏ i ∈ s, ∑ p ∈ ({True, False} : Finset Prop), if p then f i else g i := by simp
_ = ∑ p ∈ (s.pi fun _ => {True, False} : Finset (∀ a ∈ s, Prop)),
∏ a ∈ s.attach, if p a.1 a.2 then f a.1 else g a.1 := prod_sum _ _ _
_ = ∑ t ∈ s.powerset, (∏ a ∈ t... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Finsupp.Defs | {
"line": 371,
"column": 2
} | {
"line": 372,
"column": 6
} | {
"line": 374,
"column": 0
} | [
{
"pp": "α : Type u_1\nM : Type u_4\nN : Type u_5\ninst✝¹ : Zero M\ninst✝ : Zero N\ne : M → N\nhe₀ : e 0 = 0\nhe : Surjective e\n⊢ Surjective (mapRange e he₀)",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Finsupp.range_mapRange",
"Set.range_eq_univ",
"Finsupp.instFunLi... | [] | rw [← Set.range_eq_univ, range_mapRange, he.range_eq]
simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Finsupp.Defs | {
"line": 371,
"column": 2
} | {
"line": 372,
"column": 6
} | {
"line": 374,
"column": 0
} | [
{
"pp": "α : Type u_1\nM : Type u_4\nN : Type u_5\ninst✝¹ : Zero M\ninst✝ : Zero N\ne : M → N\nhe₀ : e 0 = 0\nhe : Surjective e\n⊢ Surjective (mapRange e he₀)",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Finsupp.range_mapRange",
"Set.range_eq_univ",
"Finsupp.instFunLi... | [] | rw [← Set.range_eq_univ, range_mapRange, he.range_eq]
simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Finsupp.Defs | {
"line": 458,
"column": 78
} | {
"line": 459,
"column": 29
} | {
"line": 461,
"column": 0
} | [
{
"pp": "α : Type u_1\nM : Type u_4\ninst✝ : Zero M\n⊢ embDomain (Embedding.refl α) = id",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"dite_cond_eq_true",
"Finsupp.embDomain_apply",
"Finsupp.instFunLike",
"Finsupp.ext",
"congrArg",
"Exists.choose.congr... | [] | by
ext; simp [embDomain_apply] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Finsupp.Single | {
"line": 58,
"column": 2
} | {
"line": 59,
"column": 60
} | {
"line": 61,
"column": 0
} | [
{
"pp": "α : Type u_1\nM : Type u_5\ninst✝¹ : Zero M\na a' : α\nb : M\ninst✝ : Decidable (a = a')\n⊢ (single a b) a' = if a = a' then b else 0",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Finsupp.instFunLike",
"Eq.mpr",
"congrArg",
"Finset",
"Pi.single_app... | [] | classical
simp_rw [@eq_comm _ a a', single, coe_mk, Pi.single_apply] | Lean.Elab.Tactic.evalClassical | Lean.Parser.Tactic.classical |
Mathlib.Data.Finsupp.Single | {
"line": 58,
"column": 2
} | {
"line": 59,
"column": 60
} | {
"line": 61,
"column": 0
} | [
{
"pp": "α : Type u_1\nM : Type u_5\ninst✝¹ : Zero M\na a' : α\nb : M\ninst✝ : Decidable (a = a')\n⊢ (single a b) a' = if a = a' then b else 0",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Finsupp.instFunLike",
"Eq.mpr",
"congrArg",
"Finset",
"Pi.single_app... | [] | classical
simp_rw [@eq_comm _ a a', single, coe_mk, Pi.single_apply] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Finsupp.Single | {
"line": 58,
"column": 2
} | {
"line": 59,
"column": 60
} | {
"line": 61,
"column": 0
} | [
{
"pp": "α : Type u_1\nM : Type u_5\ninst✝¹ : Zero M\na a' : α\nb : M\ninst✝ : Decidable (a = a')\n⊢ (single a b) a' = if a = a' then b else 0",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Finsupp.instFunLike",
"Eq.mpr",
"congrArg",
"Finset",
"Pi.single_app... | [] | classical
simp_rw [@eq_comm _ a a', single, coe_mk, Pi.single_apply] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Order.Group.Indicator | {
"line": 194,
"column": 8
} | {
"line": 194,
"column": 31
} | {
"line": 194,
"column": 32
} | [
{
"pp": "case pos\nα : Type u_2\nM : Type u_3\ninst✝³ : CompleteLattice M\ninst✝² : One M\nι : Type u_4\ninst✝¹ : Preorder ι\ninst✝ : IsDirectedOrder ι\nf : ι → α → M\ns : ι → Set α\nh1 : ⊥ = 1\nhf : Monotone f\nhs : Monotone s\na : α\nha : a ∈ ⋃ i, s i\ni : ι\nhi : a ∈ s i\n⊢ (⋃ i, s i).mulIndicator (⨆ i, f i)... | [
"case pos\nα : Type u_2\nM : Type u_3\ninst✝³ : CompleteLattice M\ninst✝² : One M\nι : Type u_4\ninst✝¹ : Preorder ι\ninst✝ : IsDirectedOrder ι\nf : ι → α → M\ns : ι → Set α\nh1 : ⊥ = 1\nhf : Monotone f\nhs : Monotone s\na : α\nha : a ∈ ⋃ i, s i\ni : ι\nhi : a ∈ s i\n⊢ (⨆ i, f i) a ≤ (⨆ i, (s i).mulIndicator (f i))... | mulIndicator_of_mem ha, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.BigOperators.Finsupp.Basic | {
"line": 175,
"column": 6
} | {
"line": 175,
"column": 80
} | {
"line": 177,
"column": 0
} | [
{
"pp": "case neg\nα : Type u_1\nM : Type u_8\nN : Type u_10\ninst✝¹ : Zero M\ninst✝ : CommMonoid N\nf : α →₀ M\ny : α\ng : α → M → N\nhg : ∀ (i : α), g i 0 = 1\nhyf : y ∉ f.support\n⊢ g y (f y) * (erase y f).prod g = f.prod g",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"Finsupp.... | [] | rw [notMem_support_iff.mp hyf, hg y, erase_of_notMem_support hyf, one_mul] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.BigOperators.Finsupp.Basic | {
"line": 175,
"column": 6
} | {
"line": 175,
"column": 80
} | {
"line": 177,
"column": 0
} | [
{
"pp": "case neg\nα : Type u_1\nM : Type u_8\nN : Type u_10\ninst✝¹ : Zero M\ninst✝ : CommMonoid N\nf : α →₀ M\ny : α\ng : α → M → N\nhg : ∀ (i : α), g i 0 = 1\nhyf : y ∉ f.support\n⊢ g y (f y) * (erase y f).prod g = f.prod g",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"Finsupp.... | [] | rw [notMem_support_iff.mp hyf, hg y, erase_of_notMem_support hyf, one_mul] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.BigOperators.Finsupp.Basic | {
"line": 175,
"column": 6
} | {
"line": 175,
"column": 80
} | {
"line": 177,
"column": 0
} | [
{
"pp": "case neg\nα : Type u_1\nM : Type u_8\nN : Type u_10\ninst✝¹ : Zero M\ninst✝ : CommMonoid N\nf : α →₀ M\ny : α\ng : α → M → N\nhg : ∀ (i : α), g i 0 = 1\nhyf : y ∉ f.support\n⊢ g y (f y) * (erase y f).prod g = f.prod g",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"Finsupp.... | [] | rw [notMem_support_iff.mp hyf, hg y, erase_of_notMem_support hyf, one_mul] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Order.BigOperators.GroupWithZero.List | {
"line": 93,
"column": 4
} | {
"line": 94,
"column": 11
} | {
"line": 95,
"column": 4
} | [
{
"pp": "case c0\nR : Type u_1\ninst✝⁴ : CommMonoidWithZero R\ninst✝³ : PartialOrder R\ninst✝² : ZeroLEOneClass R\ninst✝¹ : PosMulStrictMono R\ninst✝ : NeZero 1\nι : Type u_2\ns✝ : List ι\nf g : ι → R\na : ι\ns : List ι\nhs : a :: s ≠ []\nh0 : ∀ (i : ι), i ∈ a :: s → 0 < f i\nh : ∀ (i : ι), i ∈ a :: s → f i < g... | [
"case b0\nR : Type u_1\ninst✝⁴ : CommMonoidWithZero R\ninst✝³ : PartialOrder R\ninst✝² : ZeroLEOneClass R\ninst✝¹ : PosMulStrictMono R\ninst✝ : NeZero 1\nι : Type u_2\ns✝ : List ι\nf g : ι → R\na : ι\ns : List ι\nhs : a :: s ≠ []\nh0 : ∀ (i : ι), i ∈ a :: s → 0 < f i\nh : ∀ (i : ι), i ∈ a :: s → f i < g i\nthis : M... | · apply prod_pos
grind | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Data.Finsupp.Basic | {
"line": 378,
"column": 4
} | {
"line": 381,
"column": 88
} | {
"line": 382,
"column": 4
} | [
{
"pp": "case pos\nα : Type u_1\nβ : Type u_2\nM : Type u_5\ninst✝ : AddCommMonoid M\nS : Set α\nf : α → β\nx : α →₀ M\nhS : ↑x.support ⊆ S\nhf : Set.InjOn f S\na : α\nha : a ∈ S\nhax : a ∈ x.support\n⊢ (∑ x_1 ∈ x.support, if f x_1 = f a then x x_1 else 0) = x a",
"ppTerm": "?pos✝",
"assigned": true,
... | [
"case neg\nα : Type u_1\nβ : Type u_2\nM : Type u_5\ninst✝ : AddCommMonoid M\nS : Set α\nf : α → β\nx : α →₀ M\nhS : ↑x.support ⊆ S\nhf : Set.InjOn f S\na : α\nha : a ∈ S\nhax : a ∉ x.support\n⊢ (∑ x_1 ∈ x.support, if f x_1 = f a then x x_1 else 0) = x a"
] | · rw [← Finset.add_sum_erase _ _ hax, if_pos rfl]
convert! add_zero (x a)
refine Finset.sum_eq_zero fun i hi => if_neg ?_
exact (hf.mono hS).ne (Finset.mem_of_mem_erase hi) hax (Finset.ne_of_mem_erase hi) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Data.Finsupp.Basic | {
"line": 688,
"column": 2
} | {
"line": 688,
"column": 86
} | {
"line": 690,
"column": 0
} | [
{
"pp": "α : Type u_1\nM : Type u_5\ninst✝¹ : Zero M\np : α → Prop\ninst✝ : DecidablePred p\n⊢ filter p 0 = 0",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Finsupp.support_zero",
"congrArg",
"Finset",
"Finset.filter_empty",
"Finsupp.support"... | [] | classical rw [← support_eq_empty, support_filter, support_zero, Finset.filter_empty] | Lean.Elab.Tactic.evalClassical | Lean.Parser.Tactic.classical |
Mathlib.Data.Finsupp.Basic | {
"line": 688,
"column": 2
} | {
"line": 688,
"column": 86
} | {
"line": 690,
"column": 0
} | [
{
"pp": "α : Type u_1\nM : Type u_5\ninst✝¹ : Zero M\np : α → Prop\ninst✝ : DecidablePred p\n⊢ filter p 0 = 0",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Finsupp.support_zero",
"congrArg",
"Finset",
"Finset.filter_empty",
"Finsupp.support"... | [] | classical rw [← support_eq_empty, support_filter, support_zero, Finset.filter_empty] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Finsupp.Basic | {
"line": 688,
"column": 2
} | {
"line": 688,
"column": 86
} | {
"line": 690,
"column": 0
} | [
{
"pp": "α : Type u_1\nM : Type u_5\ninst✝¹ : Zero M\np : α → Prop\ninst✝ : DecidablePred p\n⊢ filter p 0 = 0",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Finsupp.support_zero",
"congrArg",
"Finset",
"Finset.filter_empty",
"Finsupp.support"... | [] | classical rw [← support_eq_empty, support_filter, support_zero, Finset.filter_empty] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Finsupp.Supported | {
"line": 183,
"column": 22
} | {
"line": 183,
"column": 33
} | {
"line": 183,
"column": 34
} | [
{
"pp": "α : Type u_1\nM : Type u_2\nR : Type u_5\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\ns t : Set α\nh : Codisjoint s t\n⊢ supported M R s ⊔ supported M R t = ⊤",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule",
"Lattice.t... | [
"α : Type u_1\nM : Type u_2\nR : Type u_5\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\ns t : Set α\nh : Codisjoint s t\n⊢ ⊤ ≤ supported M R s ⊔ supported M R t"
] | eq_top_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Finsupp.LinearCombination | {
"line": 186,
"column": 2
} | {
"line": 186,
"column": 21
} | {
"line": 187,
"column": 2
} | [
{
"pp": "α : Type u_1\nM : Type u_2\nR : Type u_5\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nv : α → M\ns : Set α\n⊢ span R (v '' s) = map (linearCombination R v) (supported R R s)",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
"RingHomSurjective.ids",
... | [
"case h₁\nα : Type u_1\nM : Type u_2\nR : Type u_5\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nv : α → M\ns : Set α\n⊢ v '' s ⊆ ↑(map (linearCombination R v) (supported R R s))",
"case h₂\nα : Type u_1\nM : Type u_2\nR : Type u_5\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Modu... | apply span_eq_of_le | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Algebra.BigOperators.Finprod | {
"line": 293,
"column": 24
} | {
"line": 293,
"column": 34
} | {
"line": 293,
"column": 35
} | [
{
"pp": "case neg\nM : Type u_2\nN : Type u_3\nα : Sort u_4\ninst✝¹ : CommMonoid M\ninst✝ : CommMonoid N\nf : M →* N\nhf : ∀ (x : M), f x = 1 → x = 1\ng : α → M\nhg : ¬HasFiniteMulSupport (g ∘ PLift.down)\n⊢ f 1 = ∏ᶠ (i : α), f (g i)",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"E... | [
"case neg\nM : Type u_2\nN : Type u_3\nα : Sort u_4\ninst✝¹ : CommMonoid M\ninst✝ : CommMonoid N\nf : M →* N\nhf : ∀ (x : M), f x = 1 → x = 1\ng : α → M\nhg : ¬HasFiniteMulSupport (g ∘ PLift.down)\n⊢ 1 = ∏ᶠ (i : α), f (g i)",
"case neg.hnc\nM : Type u_2\nN : Type u_3\nα : Sort u_4\ninst✝¹ : CommMonoid M\ninst✝ : ... | f.map_one, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Finsupp.LinearCombination | {
"line": 344,
"column": 2
} | {
"line": 345,
"column": 76
} | {
"line": 347,
"column": 0
} | [
{
"pp": "α : Type u_1\nM : Type u_2\nR : Type u_3\ninst✝³ : Fintype α\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nv : α → M\n⊢ (Fintype.linearCombination R v).range = span R (Set.range v)",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Pi.Func... | [] | rw [← Finsupp.linearCombination_eq_fintype_linearCombination, LinearMap.range_comp,
LinearEquiv.range, Submodule.map_top, Finsupp.range_linearCombination] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.Finsupp.LinearCombination | {
"line": 344,
"column": 2
} | {
"line": 345,
"column": 76
} | {
"line": 347,
"column": 0
} | [
{
"pp": "α : Type u_1\nM : Type u_2\nR : Type u_3\ninst✝³ : Fintype α\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nv : α → M\n⊢ (Fintype.linearCombination R v).range = span R (Set.range v)",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Pi.Func... | [] | rw [← Finsupp.linearCombination_eq_fintype_linearCombination, LinearMap.range_comp,
LinearEquiv.range, Submodule.map_top, Finsupp.range_linearCombination] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Finsupp.LinearCombination | {
"line": 344,
"column": 2
} | {
"line": 345,
"column": 76
} | {
"line": 347,
"column": 0
} | [
{
"pp": "α : Type u_1\nM : Type u_2\nR : Type u_3\ninst✝³ : Fintype α\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nv : α → M\n⊢ (Fintype.linearCombination R v).range = span R (Set.range v)",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Pi.Func... | [] | rw [← Finsupp.linearCombination_eq_fintype_linearCombination, LinearMap.range_comp,
LinearEquiv.range, Submodule.map_top, Finsupp.range_linearCombination] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Finsupp.LinearCombination | {
"line": 466,
"column": 4
} | {
"line": 466,
"column": 30
} | {
"line": 466,
"column": 30
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\ns : Set M\nx : M\n⊢ (∃ f t, ↑t ⊆ s ∧ Function.support f ⊆ ↑t ∧ ∑ a ∈ t, f a • a = x) → x ∈ span R s",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"Submodule",
"instHSMul",
... | [
"R : Type u_1\nM : Type u_2\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\ns : Set M\nn : M → R\nt : Finset M\nhts : ↑t ⊆ s\n⊢ ∑ a ∈ t, n a • a ∈ span R s"
] | rintro ⟨n, t, hts, -, rfl⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.LinearAlgebra.Finsupp.LinearCombination | {
"line": 493,
"column": 18
} | {
"line": 493,
"column": 34
} | {
"line": 493,
"column": 34
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nm : M\ns : Set M\n| m ∈ span R s",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"Submodule",
"congrArg",
"Membership.mem",
"id",
"Submodule.setLike",
... | [
"R : Type u_1\nM : Type u_2\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nm : M\ns : Set M\n| m ∈ span R (_root_.id '' s)"
] | ← Set.image_id s | Lean.Elab.Tactic.Conv.evalRewrite | null |
Mathlib.Algebra.Group.Fin.Basic | {
"line": 135,
"column": 19
} | {
"line": 135,
"column": 89
} | {
"line": 137,
"column": 0
} | [
{
"pp": "n : ℕ\nk : Fin (n + 1)\n⊢ ¬k - 1 < k ↔ ¬0 < k",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"instNeZeroNatHAdd_1",
"Fin.instSub",
"congrArg",
"_private.Mathlib.Algebra.Group.Fin.Basic.0.Fin.sub_one_lt_iff._simp_1_3",
"PartialOrder.toPreorder",
... | [] | by simp only [lt_def, not_lt, val_fin_le, le_sub_one_iff, le_zero_iff] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Logic.Equiv.Fin.Rotate | {
"line": 44,
"column": 6
} | {
"line": 44,
"column": 38
} | {
"line": 44,
"column": 38
} | [
{
"pp": "n : ℕ\n⊢ (finCongr ⋯) (finAddFlip ⟨n, ⋯⟩) = 0",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instNeZeroNatHAdd_1",
"le_rfl",
"Equiv.instEquivLike",
"congrArg",
"finCongr",
"HSub.hSub",
"finAddFlip_apply_mk_right",
"... | [
"n : ℕ\n⊢ (finCongr ⋯) ⟨n - n, ⋯⟩ = 0"
] | finAddFlip_apply_mk_right le_rfl | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Logic.Equiv.Fin.Rotate | {
"line": 60,
"column": 45
} | {
"line": 60,
"column": 66
} | {
"line": 60,
"column": 66
} | [
{
"pp": "case neg\nn : ℕ\nα : Type u_1\nv : Fin n → α\na : α\nh : n < n + 1\nh' : ¬n < n\n⊢ (if h_1 : ↑⟨n, h⟩ < n then cast ⋯ (v (⟨n, h⟩.castLT h_1)) else cast ⋯ a) = cases a v ⟨0, ⋯⟩",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"lt_irrefl",
... | [
"case neg\nn : ℕ\nα : Type u_1\nv : Fin n → α\na : α\nh : n < n + 1\nh' : ¬n < n\n⊢ cast ⋯ a = cases a v ⟨0, ⋯⟩"
] | dif_neg (lt_irrefl _) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.BigOperators.Finprod | {
"line": 1293,
"column": 2
} | {
"line": 1293,
"column": 6
} | {
"line": 1294,
"column": 2
} | [
{
"pp": "N : Type u_6\ninst✝ : CommMonoid N\nα : Type u_7\nι : Type u_8\nf : ι → α → N\nhf : HasFiniteMulSupport f\na : α\nhf' : HasFiniteMulSupport fun i ↦ f i a\n⊢ ∏ c ∈ Finite.toFinset ⋯, f c a = ∏ c ∈ Finite.toFinset ⋯, f c a",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"MulOn... | [
"N : Type u_6\ninst✝ : CommMonoid N\nα : Type u_7\nι : Type u_8\nf : ι → α → N\nhf : HasFiniteMulSupport f\na : α\nhf' : HasFiniteMulSupport fun i ↦ f i a\n⊢ ∏ c ∈ Finite.toFinset ⋯, f c a = ∏ c ∈ Finite.toFinset ⋯, f c a"
] | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.Data.ENat.Pow | {
"line": 126,
"column": 4
} | {
"line": 126,
"column": 42
} | {
"line": 127,
"column": 4
} | [
{
"pp": "case inl\nx y z : ℕ∞\nx_0 : x < 1\n⊢ 0 ^ (y + z) = 0 ^ y * 0 ^ z",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Preorder.toLT",
"instAddMonoidWithOneENat",
"HMul.hMul",
"instLinearOrderENat",
"CommSemiring.toSemiring",
"instIsBotZeroClass",
... | [
"case inl.inl\nx z : ℕ∞\nx_0 : x < 1\n⊢ 0 ^ (0 + z) = 0 ^ 0 * 0 ^ z",
"case inl.inr\nx y z : ℕ∞\nx_0 : x < 1\ny_0 : 0 < y\n⊢ 0 ^ (y + z) = 0 ^ y * 0 ^ z"
] | rcases eq_zero_or_pos y with rfl | y_0 | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Data.ENat.Pow | {
"line": 153,
"column": 2
} | {
"line": 153,
"column": 14
} | {
"line": 154,
"column": 2
} | [
{
"pp": "case inl\nx y z : ℕ∞\ny_0 : y = 0\n⊢ x ^ (y * z) = (x ^ y) ^ z",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"instAddMonoidWithOneENat",
"HMul.hMul",
"congrArg",
"CommSemiring.toSemiring",
"MulZeroClass.zero_mul",
"instDistribOfSemiring",
... | [
"case inr\nx y z : ℕ∞\ny_0 : y ≠ 0\n⊢ x ^ (y * z) = (x ^ y) ^ z"
] | · simp [y_0] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.LinearAlgebra.LinearIndependent.Defs | {
"line": 240,
"column": 15
} | {
"line": 240,
"column": 46
} | {
"line": 240,
"column": 47
} | [
{
"pp": "case refine_1\nι : Type u'\nR : Type u_2\nM : Type u_4\nv : ι → M\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nhv : ∀ (s : Finset ι) (f g : ι → R), ∑ i ∈ s, f i • v i = ∑ i ∈ s, g i • v i → ∀ i ∈ s, f i = g i\nf g : ι →₀ R\nhl : (Finsupp.linearCombination R v) f = (Finsupp.linear... | [
"case refine_1\nι : Type u'\nR : Type u_2\nM : Type u_4\nv : ι → M\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nhv : ∀ (s : Finset ι) (f g : ι → R), ∑ i ∈ s, f i • v i = ∑ i ∈ s, g i • v i → ∀ i ∈ s, f i = g i\nf g : ι →₀ R\nhl : (Finsupp.linearCombination R v) f = (Finsupp.linearCombination ... | ← sum_subset subset_union_left, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Nat.ModEq | {
"line": 152,
"column": 22
} | {
"line": 152,
"column": 34
} | {
"line": 152,
"column": 34
} | [
{
"pp": "case succ\nn a b : ℕ\nh : a ≡ b [MOD n]\nd : ℕ\nhd : a ^ d ≡ b ^ d [MOD n]\n⊢ a ^ d * a ≡ b ^ (d + 1) [MOD n]",
"ppTerm": "?succ",
"assigned": true,
"usedConstants": [
"instPowNat",
"Eq.mpr",
"HMul.hMul",
"congrArg",
"Nat.instMonoid",
"Nat.pow_succ",
... | [
"case succ\nn a b : ℕ\nh : a ≡ b [MOD n]\nd : ℕ\nhd : a ^ d ≡ b ^ d [MOD n]\n⊢ a ^ d * a ≡ b ^ d * b [MOD n]"
] | Nat.pow_succ | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Nat.ModEq | {
"line": 583,
"column": 63
} | {
"line": 583,
"column": 69
} | {
"line": 584,
"column": 2
} | [
{
"pp": "n : ℕ\n⊢ ∀ (m : ℕ), m < 4 → m % 2 = 1 → m = 1 ∨ m = 3",
"ppTerm": "?m.47",
"assigned": true,
"usedConstants": [
"of_decide_eq_true",
"id",
"Nat.instMod",
"instHMod",
"forall_prop_decidable",
"instOfNatNat",
"Bool.true",
"HMod.hMod",
"Nat... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.Data.Nat.ModEq | {
"line": 583,
"column": 63
} | {
"line": 583,
"column": 69
} | {
"line": 584,
"column": 2
} | [
{
"pp": "n : ℕ\n⊢ ∀ (m : ℕ), m < 4 → m % 2 = 1 → m = 1 ∨ m = 3",
"ppTerm": "?m.47",
"assigned": true,
"usedConstants": [
"of_decide_eq_true",
"id",
"Nat.instMod",
"instHMod",
"forall_prop_decidable",
"instOfNatNat",
"Bool.true",
"HMod.hMod",
"Nat... | [] | decide | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Nat.ModEq | {
"line": 583,
"column": 63
} | {
"line": 583,
"column": 69
} | {
"line": 584,
"column": 2
} | [
{
"pp": "n : ℕ\n⊢ ∀ (m : ℕ), m < 4 → m % 2 = 1 → m = 1 ∨ m = 3",
"ppTerm": "?m.47",
"assigned": true,
"usedConstants": [
"of_decide_eq_true",
"id",
"Nat.instMod",
"instHMod",
"forall_prop_decidable",
"instOfNatNat",
"Bool.true",
"HMod.hMod",
"Nat... | [] | decide | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Nat.ModEq | {
"line": 585,
"column": 62
} | {
"line": 585,
"column": 68
} | {
"line": 585,
"column": 69
} | [
{
"pp": "n : ℕ\nhelp : ∀ (m : ℕ), m < 4 → m % 2 = 1 → m = 1 ∨ m = 3\nhn : n % 2 = 1\n⊢ 2 ∣ 4",
"ppTerm": "?m.68",
"assigned": true,
"usedConstants": [
"Dvd.dvd",
"of_decide_eq_true",
"Nat.decidable_dvd",
"id",
"instOfNatNat",
"Bool.true",
"Nat.instDvd",
... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.Data.Nat.ModEq | {
"line": 585,
"column": 62
} | {
"line": 585,
"column": 68
} | {
"line": 585,
"column": 69
} | [
{
"pp": "n : ℕ\nhelp : ∀ (m : ℕ), m < 4 → m % 2 = 1 → m = 1 ∨ m = 3\nhn : n % 2 = 1\n⊢ 2 ∣ 4",
"ppTerm": "?m.68",
"assigned": true,
"usedConstants": [
"Dvd.dvd",
"of_decide_eq_true",
"Nat.decidable_dvd",
"id",
"instOfNatNat",
"Bool.true",
"Nat.instDvd",
... | [] | decide | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Nat.ModEq | {
"line": 585,
"column": 62
} | {
"line": 585,
"column": 68
} | {
"line": 585,
"column": 69
} | [
{
"pp": "n : ℕ\nhelp : ∀ (m : ℕ), m < 4 → m % 2 = 1 → m = 1 ∨ m = 3\nhn : n % 2 = 1\n⊢ 2 ∣ 4",
"ppTerm": "?m.68",
"assigned": true,
"usedConstants": [
"Dvd.dvd",
"of_decide_eq_true",
"Nat.decidable_dvd",
"id",
"instOfNatNat",
"Bool.true",
"Nat.instDvd",
... | [] | decide | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.LinearIndependent.Basic | {
"line": 538,
"column": 4
} | {
"line": 538,
"column": 18
} | {
"line": 539,
"column": 4
} | [
{
"pp": "G : Type u_6\ninst✝² : MulOneClass G\nL : Type u_7\ninst✝¹ : CommRing L\ninst✝ : IsDomain L\nthis✝ : DecidableEq (G →* L) := Classical.decEq (G →* L)\nthis : MulAction L L := Semiring.toModule.toMulAction\na : G →* L\ns : Finset (G →* L)\nhas : a ∉ s\nih : ∀ (g : (G →* L) → L), ∑ i ∈ s, g i • ⇑i = 0 → ... | [
"G : Type u_6\ninst✝² : MulOneClass G\nL : Type u_7\ninst✝¹ : CommRing L\ninst✝ : IsDomain L\nthis✝ : DecidableEq (G →* L) := Classical.decEq (G →* L)\nthis : MulAction L L := Semiring.toModule.toMulAction\na : G →* L\ns : Finset (G →* L)\nhas : a ∉ s\nih : ∀ (g : (G →* L) → L), ∑ i ∈ s, g i • ⇑i = 0 → ∀ i ∈ s, g i... | by_contra! hia | Mathlib.Tactic.ByContra._aux_Mathlib_Tactic_ByContra___macroRules_Mathlib_Tactic_ByContra_byContra!_1 | Mathlib.Tactic.ByContra.byContra! |
Mathlib.Algebra.BigOperators.Fin | {
"line": 616,
"column": 33
} | {
"line": 616,
"column": 57
} | {
"line": 616,
"column": 58
} | [
{
"pp": "m n : ℕ\ninst✝ : NeZero m\ni : Fin n\nj : Fin m\n⊢ ∑ i_1, ↑(Pi.single i j i_1) * m ^ ↑i_1 = ↑j * m ^ ↑i",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"Finset.univ",
"congrArg",
"Nat.instMonoid",
"instDecidableEqFin",
... | [
"m n : ℕ\ninst✝ : NeZero m\ni : Fin n\nj : Fin m\n⊢ ↑(Pi.single i j i) * m ^ ↑i = ↑j * m ^ ↑i",
"m n : ℕ\ninst✝ : NeZero m\ni : Fin n\nj : Fin m\n⊢ ∀ (x : Fin n), x ≠ i → ↑(Pi.single i j x) * m ^ ↑x = 0"
] | Fintype.sum_eq_single i, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.BigOperators.Fin | {
"line": 666,
"column": 8
} | {
"line": 666,
"column": 36
} | {
"line": 667,
"column": 8
} | [
{
"pp": "case refine_2\nι : Type u_1\nM : Type u_2\nm : ℕ\nn✝ : Fin m → ℕ\nn x : ℕ\nxs : Fin n → ℕ\na : Fin (∏ i, Fin.cons x xs i)\nih : ∀ i < ∏ i, xs i, ∑ x, (i / ∏ j, xs (Fin.castLE ⋯ j)) % xs x * ∏ j, xs (Fin.castLE ⋯ j) = i\n⊢ ↑a % x +\n x *\n ∑ i,\n (↑a / x / ∏ i_1, Fin.cons x xs (Fin.... | [
"case e'_2.e'_6\nι : Type u_1\nM : Type u_2\nm : ℕ\nn✝ : Fin m → ℕ\nn x : ℕ\nxs : Fin n → ℕ\na : Fin (∏ i, Fin.cons x xs i)\nih : ∀ i < ∏ i, xs i, ∑ x, (i / ∏ j, xs (Fin.castLE ⋯ j)) % xs x * ∏ j, xs (Fin.castLE ⋯ j) = i\n⊢ ∑ i, (↑a / x / ∏ i_1, Fin.cons x xs (Fin.castLE ⋯ i_1.succ)) % xs i * ∏ i_1, Fin.cons x xs (... | convert! Nat.mod_add_div _ _ | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___macroRules_Mathlib_Tactic_convert!_1 | Mathlib.Tactic.convert! |
Mathlib.Algebra.BigOperators.Fin | {
"line": 676,
"column": 27
} | {
"line": 676,
"column": 51
} | {
"line": 676,
"column": 52
} | [
{
"pp": "m : ℕ\nn : Fin m → ℕ\ninst✝ : ∀ (i : Fin m), NeZero (n i)\ni : Fin m\nj : Fin (n i)\n⊢ ∑ i_1, ↑(Pi.single i j i_1) * ∏ j, n (Fin.castLE ⋯ j) = ↑j * ∏ j, n (Fin.castLE ⋯ j)",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"Finset.univ",
... | [
"m : ℕ\nn : Fin m → ℕ\ninst✝ : ∀ (i : Fin m), NeZero (n i)\ni : Fin m\nj : Fin (n i)\n⊢ ↑(Pi.single i j i) * ∏ j, n (Fin.castLE ⋯ j) = ↑j * ∏ j, n (Fin.castLE ⋯ j)",
"m : ℕ\nn : Fin m → ℕ\ninst✝ : ∀ (i : Fin m), NeZero (n i)\ni : Fin m\nj : Fin (n i)\n⊢ ∀ (x : Fin m), x ≠ i → ↑(Pi.single i j x) * ∏ j, n (Fin.cast... | Fintype.sum_eq_single i, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Set.Card | {
"line": 108,
"column": 34
} | {
"line": 108,
"column": 50
} | {
"line": 108,
"column": 50
} | [
{
"pp": "α : Type u_1\ns : Set α\nh : s.encard ≠ 0\n⊢ ¬s = ∅",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Set.encard_eq_zero",
"Eq.mpr",
"Set.encard",
"congrArg",
"CommSemiring.toSemiring",
"id",
"ENat",
"propext",
"instCommSemiring... | [
"α : Type u_1\ns : Set α\nh : s.encard ≠ 0\n⊢ ¬s.encard = 0"
] | ← encard_eq_zero | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.BigOperators.Fin | {
"line": 769,
"column": 10
} | {
"line": 769,
"column": 24
} | {
"line": 769,
"column": 24
} | [
{
"pp": "G : Type u_3\ninst✝ : DivisionCommMonoid G\ng h : G\nL : List G\n⊢ g * (h⁻¹ * ∏ i, L[i] ^ (-1) ^ ↑i) =\n (g :: h :: L)[0] ^ (-1) ^ ↑0 *\n ((g :: h :: L)[Fin.succ 0] ^ (-1) ^ ↑(Fin.succ 0) * ∏ i, (g :: h :: L)[i.succ.succ] ^ (-1) ^ ↑i.succ.succ)",
"ppTerm": "?m.299",
"assigned": true,
... | [] | simp [pow_add] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Data.Set.Card | {
"line": 769,
"column": 45
} | {
"line": 769,
"column": 61
} | {
"line": 771,
"column": 0
} | [
{
"pp": "α : Type u_1\ns : Set α\na : α\nh : a ∈ s\nhs : s.Finite\n⊢ (s \\ {a}).ncard < (s \\ {a}).ncard + 1",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Nat.instIsOrderedAddMonoid",
"Nat.instOne",
"instIsLeftCancelAddOfAddLeftReflectLE",
"lt_add_one",
"Ad... | [] | apply lt_add_one | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.SetTheory.Cardinal.NatCard | {
"line": 51,
"column": 2
} | {
"line": 51,
"column": 28
} | {
"line": 52,
"column": 2
} | [
{
"pp": "α : Type u_4\n⊢ Nat.card α = if x : Finite α then Fintype.card α else 0",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Fintype.ofFinite",
"Finite",
"Classical.propDecidable",
"finite_or_infinite",
"Fintype.card",
"Nat.card",
"instOfNatNa... | [
"case inl\nα : Type u_4\nh✝ : Finite α\n⊢ Nat.card α = if x : Finite α then Fintype.card α else 0",
"case inr\nα : Type u_4\nh✝ : Infinite α\n⊢ Nat.card α = if x : Finite α then Fintype.card α else 0"
] | cases finite_or_infinite α | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalCases | Lean.Parser.Tactic.cases |
Mathlib.SetTheory.Cardinal.NatCard | {
"line": 53,
"column": 4
} | {
"line": 53,
"column": 58
} | {
"line": 54,
"column": 2
} | [
{
"pp": "case inl\nα : Type u_4\nh✝ : Finite α\nthis : Fintype α := Fintype.ofFinite α\n⊢ Nat.card α = if x : Finite α then Fintype.card α else 0",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"dite_congr",
"Fintype.ofFinite",
"instDecidableTrue",
"congrArg",
... | [] | simp only [this, *, Nat.card_eq_fintype_card, dif_pos] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.SetTheory.Cardinal.NatCard | {
"line": 75,
"column": 2
} | {
"line": 76,
"column": 74
} | {
"line": 78,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : Finite α\n⊢ 1 < Nat.card α ↔ Nontrivial α",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"Nontrivial",
"Fintype.ofFinite",
"congrArg",
"Fintype.card",
"Nat.card",
"instOfNatNat",
"iff_self",
"Iff",
"Nat",
... | [] | haveI := Fintype.ofFinite α
simp only [Nat.card_eq_fintype_card, Fintype.one_lt_card_iff_nontrivial] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.SetTheory.Cardinal.NatCard | {
"line": 75,
"column": 2
} | {
"line": 76,
"column": 74
} | {
"line": 78,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : Finite α\n⊢ 1 < Nat.card α ↔ Nontrivial α",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"Nontrivial",
"Fintype.ofFinite",
"congrArg",
"Fintype.card",
"Nat.card",
"instOfNatNat",
"iff_self",
"Iff",
"Nat",
... | [] | haveI := Fintype.ofFinite α
simp only [Nat.card_eq_fintype_card, Fintype.one_lt_card_iff_nontrivial] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Group.Submonoid.Finite | {
"line": 48,
"column": 6
} | {
"line": 50,
"column": 46
} | {
"line": 52,
"column": 0
} | [
{
"pp": "η : Type u_1\nf : η → Type u_2\ninst✝³ : (i : η) → MulOneClass (f i)\nS : Type u_3\ninst✝² : SetLike S ((i : η) → f i)\ninst✝¹ : SubmonoidClass S ((i : η) → f i)\ninst✝ : DecidableEq η\nH : S\ni : η\nI : Finset η\nhnotMem : i ∉ I\nx : (i : η) → f i\nh1 : ∀ i_1 ∉ insert i I, x i_1 = 1\nh2 : ∀ i_1 ∈ inse... | [] | have : j ≠ i := fun h => h ▸ hnotMem <| hj
simp only [ne_eq, this, not_false_eq_true, Function.update_of_ne]
exact h2 _ (Finset.mem_insert_of_mem hj) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Group.Submonoid.Finite | {
"line": 48,
"column": 6
} | {
"line": 50,
"column": 46
} | {
"line": 52,
"column": 0
} | [
{
"pp": "η : Type u_1\nf : η → Type u_2\ninst✝³ : (i : η) → MulOneClass (f i)\nS : Type u_3\ninst✝² : SetLike S ((i : η) → f i)\ninst✝¹ : SubmonoidClass S ((i : η) → f i)\ninst✝ : DecidableEq η\nH : S\ni : η\nI : Finset η\nhnotMem : i ∉ I\nx : (i : η) → f i\nh1 : ∀ i_1 ∉ insert i I, x i_1 = 1\nh2 : ∀ i_1 ∈ inse... | [] | have : j ≠ i := fun h => h ▸ hnotMem <| hj
simp only [ne_eq, this, not_false_eq_true, Function.update_of_ne]
exact h2 _ (Finset.mem_insert_of_mem hj) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.SetTheory.Cardinal.NatCard | {
"line": 237,
"column": 2
} | {
"line": 237,
"column": 28
} | {
"line": 238,
"column": 2
} | [
{
"pp": "α : Type u_1\nl : List α\nh : l.Nodup\n⊢ ↑l.length ≤ ENat.card α",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"ENat.instNatCast",
"Finite",
"finite_or_infinite",
"LE.le",
"Nat.cast",
"instLEENat",
"Or.casesOn",
"ENat",
"Eq.re... | [
"case inl\nα : Type u_1\nl : List α\nh : l.Nodup\nh✝ : Finite α\n⊢ ↑l.length ≤ ENat.card α",
"case inr\nα : Type u_1\nl : List α\nh : l.Nodup\nh✝ : Infinite α\n⊢ ↑l.length ≤ ENat.card α"
] | cases finite_or_infinite α | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalCases | Lean.Parser.Tactic.cases |
Mathlib.Data.Set.Card | {
"line": 881,
"column": 2
} | {
"line": 882,
"column": 47
} | {
"line": 884,
"column": 0
} | [
{
"pp": "α : Type u_1\ns t : Set α\nh : s ⊂ t\nht : t.Finite\n⊢ s.ncard < t.ncard",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Set.encard",
"Preorder.toLT",
"instCharZeroENat",
"instAddMonoidWithOneENat",
"ChainCompletePartialOrder.instOfCom... | [] | rw [← Nat.cast_lt (α := ℕ∞), ht.cast_ncard_eq, (ht.subset h.subset).cast_ncard_eq]
exact (ht.subset h.subset).encard_lt_encard h | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Set.Card | {
"line": 881,
"column": 2
} | {
"line": 882,
"column": 47
} | {
"line": 884,
"column": 0
} | [
{
"pp": "α : Type u_1\ns t : Set α\nh : s ⊂ t\nht : t.Finite\n⊢ s.ncard < t.ncard",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Set.encard",
"Preorder.toLT",
"instCharZeroENat",
"instAddMonoidWithOneENat",
"ChainCompletePartialOrder.instOfCom... | [] | rw [← Nat.cast_lt (α := ℕ∞), ht.cast_ncard_eq, (ht.subset h.subset).cast_ncard_eq]
exact (ht.subset h.subset).encard_lt_encard h | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Set.Card | {
"line": 1028,
"column": 4
} | {
"line": 1031,
"column": 25
} | {
"line": 1032,
"column": 2
} | [
{
"pp": "case inl\nα : Type u_1\ns t : Set α\nh : (s ∪ t).Finite\n⊢ (s ∪ t).ncard ≤ s.ncard + t.ncard",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Set.encard",
"instCharZeroENat",
"instAddMonoidWithOneENat",
"AddMonoid.toAddSemigroup",
"ENat... | [] | to_encard_tac
rw [h.cast_ncard_eq, (h.subset subset_union_left).cast_ncard_eq,
(h.subset subset_union_right).cast_ncard_eq]
apply encard_union_le | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Set.Card | {
"line": 1028,
"column": 4
} | {
"line": 1031,
"column": 25
} | {
"line": 1032,
"column": 2
} | [
{
"pp": "case inl\nα : Type u_1\ns t : Set α\nh : (s ∪ t).Finite\n⊢ (s ∪ t).ncard ≤ s.ncard + t.ncard",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Set.encard",
"instCharZeroENat",
"instAddMonoidWithOneENat",
"AddMonoid.toAddSemigroup",
"ENat... | [] | to_encard_tac
rw [h.cast_ncard_eq, (h.subset subset_union_left).cast_ncard_eq,
(h.subset subset_union_right).cast_ncard_eq]
apply encard_union_le | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.Coset.Basic | {
"line": 230,
"column": 2
} | {
"line": 232,
"column": 33
} | {
"line": 234,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : Group α\ns : Subgroup α\n⊢ ⇑(leftRel s) = LeftCosetEquivalence ↑s",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instHSMul",
"instSMulOfMul",
"LeftCosetEquivalence",
"HMul.hMul",
"DivInvOneMonoid.toInvOneClass",
... | [] | ext
rw [leftRel_eq]
exact (leftCoset_eq_iff s).symm | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.Coset.Basic | {
"line": 230,
"column": 2
} | {
"line": 232,
"column": 33
} | {
"line": 234,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : Group α\ns : Subgroup α\n⊢ ⇑(leftRel s) = LeftCosetEquivalence ↑s",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instHSMul",
"instSMulOfMul",
"LeftCosetEquivalence",
"HMul.hMul",
"DivInvOneMonoid.toInvOneClass",
... | [] | ext
rw [leftRel_eq]
exact (leftCoset_eq_iff s).symm | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.Coset.Basic | {
"line": 297,
"column": 27
} | {
"line": 297,
"column": 44
} | {
"line": 297,
"column": 45
} | [
{
"pp": "α : Type u_1\ninst✝ : Group α\ns : Subgroup α\ng z : α\n⊢ z ∈ {x | ↑x = ↑g} ↔ g⁻¹ * z ∈ ↑s",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"DivInvOneMonoid.toInvOneClass",
"Monoid.toMulOneClass",
"congrArg",
"QuotientGroup... | [
"α : Type u_1\ninst✝ : Group α\ns : Subgroup α\ng z : α\n⊢ ↑z = ↑g ↔ g⁻¹ * z ∈ ↑s"
] | Set.mem_setOf_eq, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.Congruence.Basic | {
"line": 97,
"column": 4
} | {
"line": 98,
"column": 16
} | {
"line": 99,
"column": 2
} | [
{
"pp": "case of\nM : Type u_4\nN : Type u_5\ninst✝¹ : Mul M\ninst✝ : Mul N\nf : M ≃* N\nrel : N → N → Prop\nn1 n2 x y : N\nh : rel x y\na b : M\nfa : f a = x\nfb : f b = y\n⊢ (conGen fun x y ↦ rel (f x) (f y)) a b",
"ppTerm": "?of",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulEqu... | [] | apply ConGen.Rel.of
rwa [fa, fb] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.Congruence.Basic | {
"line": 97,
"column": 4
} | {
"line": 98,
"column": 16
} | {
"line": 99,
"column": 2
} | [
{
"pp": "case of\nM : Type u_4\nN : Type u_5\ninst✝¹ : Mul M\ninst✝ : Mul N\nf : M ≃* N\nrel : N → N → Prop\nn1 n2 x y : N\nh : rel x y\na b : M\nfa : f a = x\nfb : f b = y\n⊢ (conGen fun x y ↦ rel (f x) (f y)) a b",
"ppTerm": "?of",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulEqu... | [] | apply ConGen.Rel.of
rwa [fa, fb] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Pi | {
"line": 288,
"column": 2
} | {
"line": 289,
"column": 38
} | {
"line": 291,
"column": 0
} | [
{
"pp": "R : Type u\nM : Type v\nι : Type x\ninst✝⁶ : Semiring R\nφ : ι → Type i\ninst✝⁵ : (i : ι) → AddCommMonoid (φ i)\ninst✝⁴ : (i : ι) → Module R (φ i)\ninst✝³ : DecidableEq ι\ninst✝² : Finite ι\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nf g : ((i : ι) → φ i) →ₗ[R] M\nh : ∀ (i : ι), f ∘ₗ single R φ i = ... | [] | refine pi_ext fun i x => ?_
convert! LinearMap.congr_fun (h i) x | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Pi | {
"line": 288,
"column": 2
} | {
"line": 289,
"column": 38
} | {
"line": 291,
"column": 0
} | [
{
"pp": "R : Type u\nM : Type v\nι : Type x\ninst✝⁶ : Semiring R\nφ : ι → Type i\ninst✝⁵ : (i : ι) → AddCommMonoid (φ i)\ninst✝⁴ : (i : ι) → Module R (φ i)\ninst✝³ : DecidableEq ι\ninst✝² : Finite ι\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nf g : ((i : ι) → φ i) →ₗ[R] M\nh : ∀ (i : ι), f ∘ₗ single R φ i = ... | [] | refine pi_ext fun i x => ?_
convert! LinearMap.congr_fun (h i) x | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Fintype.Order | {
"line": 111,
"column": 8
} | {
"line": 111,
"column": 26
} | {
"line": 111,
"column": 26
} | [
{
"pp": "case e'_4\nι : Type u_1\nα : Type u_2\ninst✝³ : Fintype ι\ninst✝² : Fintype α\ninst✝¹ : DistribLattice α\ninst✝ : BoundedOrder α\na : α\ns : Set α\n⊢ ⨆ b ∈ s, a ⊓ b = s.toFinset.sup fun i ↦ a ⊓ id i",
"ppTerm": "?e'_4",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Lattice.toS... | [
"case e'_4\nι : Type u_1\nα : Type u_2\ninst✝³ : Fintype ι\ninst✝² : Fintype α\ninst✝¹ : DistribLattice α\ninst✝ : BoundedOrder α\na : α\ns : Set α\n⊢ ⨆ b ∈ s, a ⊓ b = ⨆ a_1 ∈ s.toFinset, a ⊓ id a_1"
] | Finset.sup_eq_iSup | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Fintype.Order | {
"line": 222,
"column": 92
} | {
"line": 226,
"column": 12
} | {
"line": 228,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝³ : Finite β\ninst✝² : Nonempty α\ninst✝¹ : Preorder α\ninst✝ : IsCodirectedOrder α\nf : β → α\n⊢ BddBelow (Set.range f)",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"lowerBounds",
"Preorder.toLE",
"Membership.mem",
"Exist... | [] | by
obtain ⟨M, hM⟩ := Finite.exists_ge f
refine ⟨M, fun a ha => ?_⟩
obtain ⟨b, rfl⟩ := ha
exact hM b | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Fintype.Order | {
"line": 296,
"column": 4
} | {
"line": 296,
"column": 55
} | {
"line": 297,
"column": 2
} | [
{
"pp": "case refine_1\nα : Type u_1\nι : Type u_2\ninst✝² : Finite ι\ninst✝¹ : ConditionallyCompleteLattice α\ninst✝ : Nonempty ι\nf : ι → α\na : α\n⊢ ⨆ i, f i ≤ ⨆ i, f i ⊔ a",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"Lattice.toSemilatticeSup",
"le_sup_left",
"... | [] | exact ciSup_le fun i ↦ le_ciSup_of_le i le_sup_left | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Data.Fintype.Order | {
"line": 296,
"column": 4
} | {
"line": 296,
"column": 55
} | {
"line": 297,
"column": 2
} | [
{
"pp": "case refine_1\nα : Type u_1\nι : Type u_2\ninst✝² : Finite ι\ninst✝¹ : ConditionallyCompleteLattice α\ninst✝ : Nonempty ι\nf : ι → α\na : α\n⊢ ⨆ i, f i ≤ ⨆ i, f i ⊔ a",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"Lattice.toSemilatticeSup",
"le_sup_left",
"... | [] | exact ciSup_le fun i ↦ le_ciSup_of_le i le_sup_left | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Fintype.Order | {
"line": 296,
"column": 4
} | {
"line": 296,
"column": 55
} | {
"line": 297,
"column": 2
} | [
{
"pp": "case refine_1\nα : Type u_1\nι : Type u_2\ninst✝² : Finite ι\ninst✝¹ : ConditionallyCompleteLattice α\ninst✝ : Nonempty ι\nf : ι → α\na : α\n⊢ ⨆ i, f i ≤ ⨆ i, f i ⊔ a",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"Lattice.toSemilatticeSup",
"le_sup_left",
"... | [] | exact ciSup_le fun i ↦ le_ciSup_of_le i le_sup_left | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Order.Interval.Finset.SuccPred | {
"line": 139,
"column": 97
} | {
"line": 140,
"column": 70
} | {
"line": 142,
"column": 0
} | [
{
"pp": "α : Type u_2\ninst✝⁴ : LinearOrder α\ninst✝³ : One α\ninst✝² : LocallyFiniteOrder α\ninst✝¹ : Sub α\ninst✝ : PredSubOrder α\na : α\nha : ¬IsMin a\nb : α\n⊢ Ioo (a - 1) b = Ico a b",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"PredSubOrder.toPredOrder",
"congrArg",
... | [] | by
simpa [pred_eq_sub_one] using Ioo_pred_left_eq_Ioc_of_not_isMin ha b | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Order.PartialSups | {
"line": 137,
"column": 2
} | {
"line": 137,
"column": 10
} | {
"line": 137,
"column": 10
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nι : Type u_3\ninst✝⁵ : SemilatticeSup α\ninst✝⁴ : SemilatticeSup β\ninst✝³ : Preorder ι\ninst✝² : LocallyFiniteOrderBot ι\nF : Type u_4\ninst✝¹ : FunLike F α β\ninst✝ : SupHomClass F α β\nf : ι → α\ng : F\n⊢ ⇑(partialSups (⇑g ∘ f)) = ⇑g ∘ ⇑(partialSups f)",
"ppTerm": "?m... | [
"α : Type u_1\nβ : Type u_2\nι : Type u_3\ninst✝⁵ : SemilatticeSup α\ninst✝⁴ : SemilatticeSup β\ninst✝³ : Preorder ι\ninst✝² : LocallyFiniteOrderBot ι\nF : Type u_4\ninst✝¹ : FunLike F α β\ninst✝ : SupHomClass F α β\nf : ι → α\ng : F\nx✝ : ι\n⊢ (partialSups (⇑g ∘ f)) x✝ = (⇑g ∘ ⇑(partialSups f)) x✝"
] | funext _ | _aux_Init_NotationExtra___macroRules_tacticFunext____1 | tacticFunext___ |
Mathlib.LinearAlgebra.Prod | {
"line": 647,
"column": 2
} | {
"line": 647,
"column": 40
} | {
"line": 648,
"column": 2
} | [
{
"pp": "R : Type u\nM : Type v\nM₂ : Type w\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : AddCommMonoid M₂\ninst✝¹ : Module R M\ninst✝ : Module R M₂\ns : Set M\nt : Set M₂\nhs : 0 ∈ s\nht : 0 ∈ t\n⊢ (span R s).prod (span R t) ≤ span R (s ×ˢ t)",
"ppTerm": "?m.59",
"assigned": true,
"used... | [
"R : Type u\nM : Type v\nM₂ : Type w\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : AddCommMonoid M₂\ninst✝¹ : Module R M\ninst✝ : Module R M₂\ns : Set M\nt : Set M₂\nhs : 0 ∈ s\nht : 0 ∈ t\n⊢ span R ((fun a ↦ (a, 0)) '' s) ≤ span R (s ×ˢ t) ∧ span R ((fun a ↦ (0, a)) '' t) ≤ span R (s ×ˢ t)"
] | simp [Submodule.prod_le_iff, map_span] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.LinearAlgebra.Prod | {
"line": 896,
"column": 2
} | {
"line": 898,
"column": 32
} | {
"line": 900,
"column": 0
} | [
{
"pp": "R : Type u\nM₃ : Type y\nM₄ : Type z\ninst✝⁴ : Semiring R\ninst✝³ : AddCommGroup M₃\ninst✝² : AddCommGroup M₄\ninst✝¹ : Module R M₃\ninst✝ : Module R M₄\ng : M₃ →ₗ[R] M₄\n⊢ g.graph = ((-g).coprod id).ker",
"ppTerm": "?m.65",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtraction... | [] | ext x
change _ = _ ↔ -g x.1 + x.2 = _
rw [add_comm, add_neg_eq_zero] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Prod | {
"line": 896,
"column": 2
} | {
"line": 898,
"column": 32
} | {
"line": 900,
"column": 0
} | [
{
"pp": "R : Type u\nM₃ : Type y\nM₄ : Type z\ninst✝⁴ : Semiring R\ninst✝³ : AddCommGroup M₃\ninst✝² : AddCommGroup M₄\ninst✝¹ : Module R M₃\ninst✝ : Module R M₄\ng : M₃ →ₗ[R] M₄\n⊢ g.graph = ((-g).coprod id).ker",
"ppTerm": "?m.65",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtraction... | [] | ext x
change _ = _ ↔ -g x.1 + x.2 = _
rw [add_comm, add_neg_eq_zero] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Order.Interval.Set.Monoid | {
"line": 49,
"column": 2
} | {
"line": 49,
"column": 97
} | {
"line": 51,
"column": 0
} | [
{
"pp": "M : Type u_1\ninst✝³ : AddCommMonoid M\ninst✝² : PartialOrder M\ninst✝¹ : IsOrderedCancelAddMonoid M\ninst✝ : ExistsAddOfLE M\na b d : M\n⊢ BijOn (fun x ↦ x + d) (Ioi a ∩ Iio b) (Ioi (a + d) ∩ Iio (b + d))",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"IsRightCancelAdd.add... | [] | exact (Ioi_add_bij a d).inter_mapsTo (by simp [MapsTo]) fun x hx => lt_of_add_lt_add_right hx.2 | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Order.SuccPred.LinearLocallyFinite | {
"line": 256,
"column": 2
} | {
"line": 258,
"column": 7
} | {
"line": 259,
"column": 2
} | [
{
"pp": "case inl\nι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : SuccOrder ι\ninst✝¹ : IsSuccArchimedean ι\ninst✝ : PredOrder ι\ni0 : ι\nn : ℕ\nh : i0 ≤ pred^[n] i0\n⊢ -↑n ≤ toZ i0 (pred^[n] i0)",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"toZ_of_eq",
"congr... | [
"case inr\nι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : SuccOrder ι\ninst✝¹ : IsSuccArchimedean ι\ninst✝ : PredOrder ι\ni0 : ι\nn : ℕ\nh : pred^[n] i0 < i0\n⊢ -↑n ≤ toZ i0 (pred^[n] i0)"
] | · have h_eq : pred^[n] i0 = i0 := le_antisymm (pred_iterate_le _ _) h
rw [h_eq, toZ_of_eq]
lia | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.Ideal.Defs | {
"line": 82,
"column": 30
} | {
"line": 82,
"column": 36
} | {
"line": 82,
"column": 36
} | [
{
"pp": "α : Type u\ninst✝ : Semiring α\nI : Ideal α\na : α\nha : a ∈ I\nn : ℕ\nhn : 0 < n\n⊢ ¬0 < Nat.zero",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"instDecidableNot",
"of_decide_eq_true",
"id",
"instOfNatNat",
"Bool.true",
"Nat",
"LT.lt",
... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.RingTheory.Ideal.Defs | {
"line": 82,
"column": 30
} | {
"line": 82,
"column": 36
} | {
"line": 82,
"column": 36
} | [
{
"pp": "α : Type u\ninst✝ : Semiring α\nI : Ideal α\na : α\nha : a ∈ I\nn : ℕ\nhn : 0 < n\n⊢ ¬0 < Nat.zero",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"instDecidableNot",
"of_decide_eq_true",
"id",
"instOfNatNat",
"Bool.true",
"Nat",
"LT.lt",
... | [] | decide | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Ideal.Defs | {
"line": 82,
"column": 30
} | {
"line": 82,
"column": 36
} | {
"line": 82,
"column": 36
} | [
{
"pp": "α : Type u\ninst✝ : Semiring α\nI : Ideal α\na : α\nha : a ∈ I\nn : ℕ\nhn : 0 < n\n⊢ ¬0 < Nat.zero",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"instDecidableNot",
"of_decide_eq_true",
"id",
"instOfNatNat",
"Bool.true",
"Nat",
"LT.lt",
... | [] | decide | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Ring.Idempotent | {
"line": 136,
"column": 2
} | {
"line": 136,
"column": 97
} | {
"line": 137,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝¹ : NonUnitalRing R\ninst✝ : IsAddTorsionFree R\np q : R\nhp : IsIdempotentElem p\nhq : IsIdempotentElem q\nhqp : IsIdempotentElem (q - p)\n⊢ p * q = p ∧ q * p = p",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"HMul.hMul",
"AddMonoid.toAddZeroClass... | [
"R : Type u_1\ninst✝¹ : NonUnitalRing R\ninst✝ : IsAddTorsionFree R\np q : R\nhp : IsIdempotentElem p\nhq : IsIdempotentElem q\nhqp : IsIdempotentElem (q - p)\nh : p * (q - p) + (q - p) * p = 0\n⊢ p * q = p ∧ q * p = p"
] | have h : p * (q - p) + (q - p) * p = 0 := hp.add_iff hqp |>.mp ((add_sub_cancel p q).symm ▸ hq) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Data.Nat.Choose.Sum | {
"line": 109,
"column": 12
} | {
"line": 109,
"column": 25
} | {
"line": 109,
"column": 26
} | [
{
"pp": "m : ℕ\nthis : ∑ i ∈ range (m + 1), (2 * m + 1).choose (2 * m + 1 - i) = ∑ i ∈ range (m + 1), (2 * m + 1).choose i\n⊢ ∑ i ∈ range (m + 1), (2 * m + 1).choose i + ∑ i ∈ range (m + 1), (2 * m + 1).choose (2 * m + 1 - i) =\n ∑ i ∈ range (m + 1), (2 * m + 1).choose i + ∑ i ∈ Ico (m + 1) (2 * m + 2), (2 *... | [
"m : ℕ\nthis : ∑ i ∈ range (m + 1), (2 * m + 1).choose (2 * m + 1 - i) = ∑ i ∈ range (m + 1), (2 * m + 1).choose i\n⊢ ∑ i ∈ Ico 0 (m + 1), (2 * m + 1).choose i + ∑ i ∈ Ico 0 (m + 1), (2 * m + 1).choose (2 * m + 1 - i) =\n ∑ i ∈ Ico 0 (m + 1), (2 * m + 1).choose i + ∑ i ∈ Ico (m + 1) (2 * m + 2), (2 * m + 1).choo... | range_eq_Ico, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Ideal.Span | {
"line": 130,
"column": 69
} | {
"line": 132,
"column": 84
} | {
"line": 134,
"column": 0
} | [
{
"pp": "α : Type u\ninst✝ : Semiring α\ns : Set α\n⊢ span s = ⊤ ↔ ∃ s', ↑s' ⊆ s ∧ span ↑s' = ⊤",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"_private.Mathlib.RingTheory.Ideal.Span.0.Ideal.span_eq_top_iff_finite.match_1_2",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWi... | [] | by
simp_rw [eq_top_iff_one]
exact ⟨Submodule.mem_span_finite_of_mem_span, fun ⟨s', h₁, h₂⟩ => span_mono h₁ h₂⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Ideal.Span | {
"line": 164,
"column": 21
} | {
"line": 164,
"column": 46
} | {
"line": 164,
"column": 46
} | [
{
"pp": "α : Type u\ninst✝ : Semiring α\nι : Type u_1\nx : ι → α\n⊢ span (range x) = span (⋃ i, {x i})",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Set.iUnion_singleton_eq_range",
"Set.instSingletonSet",
"id",
"Ideal",
"Id... | [
"α : Type u\ninst✝ : Semiring α\nι : Type u_1\nx : ι → α\n⊢ span (range x) = span (range x)"
] | iUnion_singleton_eq_range | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Ideal.Maximal | {
"line": 176,
"column": 10
} | {
"line": 176,
"column": 21
} | {
"line": 176,
"column": 22
} | [
{
"pp": "α : Type u\ninst✝ : CommSemiring α\ns : Set α\nhs : span s = ⊤\nI : Ideal α\nhI : I ≠ ⊤\nM : Ideal α\nhM : M.IsMaximal\nle : I ≤ M\nthis : ¬M = ⊤\n⊢ ∃ r ∈ s, Disjoint ↑I ↑(Submonoid.powers r)",
"ppTerm": "?m.64",
"assigned": true,
"usedConstants": [
"Semiring.toModule",
"eq_top_... | [
"α : Type u\ninst✝ : CommSemiring α\ns : Set α\nhs : span s = ⊤\nI : Ideal α\nhI : I ≠ ⊤\nM : Ideal α\nhM : M.IsMaximal\nle : I ≤ M\nthis : ¬⊤ ≤ M\n⊢ ∃ r ∈ s, Disjoint ↑I ↑(Submonoid.powers r)"
] | eq_top_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Ideal.Quotient.Defs | {
"line": 138,
"column": 62
} | {
"line": 138,
"column": 70
} | {
"line": 138,
"column": 70
} | [
{
"pp": "R : Type u\ninst✝¹ : Ring R\nI : Ideal R\ninst✝ : I.IsTwoSided\ns : Set R\nx : R\nx✝ : ∃ i, ∃ x_1 ∈ s, ↑i + x_1 = x\ni : R\nhi : i ∈ I\na : R\nha : a ∈ s\nEq : ↑⟨i, hi⟩ + a = x\n⊢ 0 - ↑⟨i, hi⟩ ∈ I",
"ppTerm": "?m.146",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Semiring.toM... | [
"R : Type u\ninst✝¹ : Ring R\nI : Ideal R\ninst✝ : I.IsTwoSided\ns : Set R\nx : R\nx✝ : ∃ i, ∃ x_1 ∈ s, ↑i + x_1 = x\ni : R\nhi : i ∈ I\na : R\nha : a ∈ s\nEq : ↑⟨i, hi⟩ + a = x\n⊢ -↑⟨i, hi⟩ ∈ I"
] | zero_sub | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.Filter.Map | {
"line": 566,
"column": 2
} | {
"line": 566,
"column": 73
} | {
"line": 568,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ng₁ g₂ : Filter β\nm : α → β\nh : Surjective m\n⊢ Disjoint (comap m g₁) (comap m g₂) ↔ Disjoint g₁ g₂",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CompleteLattice.toLattice",
"congrArg",
"Filter.instCompleteLatticeFi... | [] | rw [disjoint_iff, disjoint_iff, ← comap_inf, comap_surjective_eq_bot h] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Order.Filter.Map | {
"line": 566,
"column": 2
} | {
"line": 566,
"column": 73
} | {
"line": 568,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ng₁ g₂ : Filter β\nm : α → β\nh : Surjective m\n⊢ Disjoint (comap m g₁) (comap m g₂) ↔ Disjoint g₁ g₂",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CompleteLattice.toLattice",
"congrArg",
"Filter.instCompleteLatticeFi... | [] | rw [disjoint_iff, disjoint_iff, ← comap_inf, comap_surjective_eq_bot h] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.Filter.Map | {
"line": 566,
"column": 2
} | {
"line": 566,
"column": 73
} | {
"line": 568,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ng₁ g₂ : Filter β\nm : α → β\nh : Surjective m\n⊢ Disjoint (comap m g₁) (comap m g₂) ↔ Disjoint g₁ g₂",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CompleteLattice.toLattice",
"congrArg",
"Filter.instCompleteLatticeFi... | [] | rw [disjoint_iff, disjoint_iff, ← comap_inf, comap_surjective_eq_bot h] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Finsupp.Pi | {
"line": 223,
"column": 4
} | {
"line": 223,
"column": 45
} | {
"line": 224,
"column": 4
} | [
{
"pp": "case refine_1\nR : Type u_1\nM : Type u_2\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nα : Type u_5\np : α → Submodule R M\nx : α →₀ M\nhx : x ∈ submodule p\n⊢ x.sum single ∈ ⨆ i, map (lsingle i) (p i)",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
... | [
"case refine_1\nR : Type u_1\nM : Type u_2\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nα : Type u_5\np : α → Submodule R M\nx : α →₀ M\nhx : x ∈ submodule p\ni : α\nx✝ : i ∈ x.support\n⊢ single i (x i) ∈ ⨆ i, map (lsingle i) (p i)"
] | refine Submodule.sum_mem _ (fun i _ ↦ ?_) | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Algebra.Order.Module.Defs | {
"line": 684,
"column": 4
} | {
"line": 684,
"column": 52
} | {
"line": 686,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝⁸ : Zero α\ninst✝⁷ : Zero β\ninst✝⁶ : SMulWithZero α β\ninst✝⁵ : Preorder α\ninst✝⁴ : Preorder β\ninst✝³ : MulOneClass β\ninst✝² : PosMulMono β\ninst✝¹ : MulPosMono β\ninst✝ : IsScalarTower α β β\nh : Monotone fun x ↦ x • 1\nx✝² : β\nha : 0 ≤ x✝²\nx✝¹ x✝ : α\nhb : x✝¹ ≤... | [] | simpa using mul_le_mul_of_nonneg_right (h hb) ha | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Algebra.Order.Module.Defs | {
"line": 684,
"column": 4
} | {
"line": 684,
"column": 52
} | {
"line": 686,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝⁸ : Zero α\ninst✝⁷ : Zero β\ninst✝⁶ : SMulWithZero α β\ninst✝⁵ : Preorder α\ninst✝⁴ : Preorder β\ninst✝³ : MulOneClass β\ninst✝² : PosMulMono β\ninst✝¹ : MulPosMono β\ninst✝ : IsScalarTower α β β\nh : Monotone fun x ↦ x • 1\nx✝² : β\nha : 0 ≤ x✝²\nx✝¹ x✝ : α\nhb : x✝¹ ≤... | [] | simpa using mul_le_mul_of_nonneg_right (h hb) ha | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Order.Module.Defs | {
"line": 684,
"column": 4
} | {
"line": 684,
"column": 52
} | {
"line": 686,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝⁸ : Zero α\ninst✝⁷ : Zero β\ninst✝⁶ : SMulWithZero α β\ninst✝⁵ : Preorder α\ninst✝⁴ : Preorder β\ninst✝³ : MulOneClass β\ninst✝² : PosMulMono β\ninst✝¹ : MulPosMono β\ninst✝ : IsScalarTower α β β\nh : Monotone fun x ↦ x • 1\nx✝² : β\nha : 0 ≤ x✝²\nx✝¹ x✝ : α\nhb : x✝¹ ≤... | [] | simpa using mul_le_mul_of_nonneg_right (h hb) ha | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.IsNormal | {
"line": 138,
"column": 24
} | {
"line": 138,
"column": 50
} | {
"line": 139,
"column": 2
} | [
{
"pp": "case a\nα : Type u_1\ninst✝¹ : ConditionallyCompleteLinearOrder α\ninst✝ : WellFoundedLT α\nf : α → α\na : α\nhf : IsNormal f\nhf' : BddAbove (range fun n ↦ f^[n] a)\n⊢ ⨆ i, f (f^[i] a) ≤ ⨆ n, f^[n] a",
"ppTerm": "?a✝",
"assigned": true,
"usedConstants": [
"iSup",
"instInhabited... | [
"case a\nα : Type u_1\ninst✝¹ : ConditionallyCompleteLinearOrder α\ninst✝ : WellFoundedLT α\nf : α → α\na : α\nhf : IsNormal f\nhf' : BddAbove (range fun n ↦ f^[n] a)\nn : ℕ\n⊢ f (f^[n] a) ≤ ⨆ n, f^[n] a"
] | refine ciSup_le fun n ↦ ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Order.IsNormal | {
"line": 138,
"column": 24
} | {
"line": 138,
"column": 50
} | {
"line": 139,
"column": 2
} | [
{
"pp": "case a\nα : Type u_1\ninst✝¹ : ConditionallyCompleteLinearOrder α\ninst✝ : WellFoundedLT α\nf : α → α\na : α\nhf : IsNormal f\nhf' : BddAbove (range fun n ↦ f^[n] a)\n⊢ ⨆ n, f^[n] a ≤ ⨆ i, f (f^[i] a)",
"ppTerm": "?a✝",
"assigned": true,
"usedConstants": [
"iSup",
"instInhabited... | [
"case a\nα : Type u_1\ninst✝¹ : ConditionallyCompleteLinearOrder α\ninst✝ : WellFoundedLT α\nf : α → α\na : α\nhf : IsNormal f\nhf' : BddAbove (range fun n ↦ f^[n] a)\nn : ℕ\n⊢ f^[n] a ≤ ⨆ i, f (f^[i] a)"
] | refine ciSup_le fun n ↦ ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
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