module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 366
values | kind stringclasses 370
values |
|---|---|---|---|---|---|---|
Mathlib.Order.Atoms | {
"line": 904,
"column": 8
} | {
"line": 906,
"column": 23
} | [
{
"pp": "case refine_1\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\nx : α\nh : x ∈ s\n⊢ x ≤ if ⊤ ∈ s then ⊤ else ⊥",
"usedConstants": [
"Eq.mpr",
"Lattice",
"le_refl",
"Lattice.toSemilatticeSup",
"co... | rcases eq_bot_or_eq_top x with (rfl | rfl)
· exact bot_le
· rw [if_pos h] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.Atoms | {
"line": 904,
"column": 8
} | {
"line": 906,
"column": 23
} | [
{
"pp": "case refine_1\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\nx : α\nh : x ∈ s\n⊢ x ≤ if ⊤ ∈ s then ⊤ else ⊥",
"usedConstants": [
"Eq.mpr",
"Lattice",
"le_refl",
"Lattice.toSemilatticeSup",
"co... | rcases eq_bot_or_eq_top x with (rfl | rfl)
· exact bot_le
· rw [if_pos h] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.Atoms | {
"line": 1011,
"column": 2
} | {
"line": 1013,
"column": 50
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\ninst✝⁴ : PartialOrder α\ninst✝³ : PartialOrder β\ninst✝² : OrderBot α\ninst✝¹ : IsAtomic α\ninst✝ : OrderBot β\nl : α → β\nu : β → α\ngi : GaloisInsertion l u\nhbot : u ⊥ = ⊥\nh_atom : ∀ (a : α), IsAtom a → u (l a) = a\na : α\nhla : IsAtom (l a)\na' : α\nha' : IsAtom a'\nhab... | have :=
(hla.le_iff.mp <| (gi.l_u_eq (l a) ▸ gi.gc.monotone_l hab' : l a' ≤ l a)).resolve_left fun h =>
ha'.1 (hbot ▸ h_atom a' ha' ▸ congr_arg u h) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Order.Atoms | {
"line": 1200,
"column": 2
} | {
"line": 1200,
"column": 37
} | [
{
"pp": "α : Type u_2\ninst✝³ : Lattice α\ninst✝² : BoundedOrder α\ninst✝¹ : IsModularLattice α\ninst✝ : ComplementedLattice α\nh : IsCoatomic α\n⊢ IsStronglyAtomic α",
"usedConstants": [
"congrArg",
"PartialOrder.toPreorder",
"Preorder.toLE",
"SemilatticeInf.toPartialOrder",
"... | rw [← isAtomic_iff_isCoatomic] at h | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Order.SupIndep | {
"line": 171,
"column": 2
} | {
"line": 183,
"column": 57
} | [
{
"pp": "α : Type u_1\nι : Type u_3\nι' : Type u_4\ninst✝³ : Lattice α\ninst✝² : IsModularLattice α\ninst✝¹ : OrderBot α\ninst✝ : DecidableEq ι\ns : Finset ι'\ng : ι' → Finset ι\nf : ι → α\nhs : s.SupIndep fun i ↦ (g i).sup f\nhg : ∀ i' ∈ s, (g i').SupIndep f\n⊢ (s.biUnion g).SupIndep f",
"usedConstants": [... | classical
intro a ha b hb hab
obtain ⟨i', hi', hb⟩ := mem_biUnion.mp hb
let t := s.erase i'
let u := (g i').erase b
apply Disjoint.mono_right <| calc
a.sup f ≤ (t.biUnion g ∪ u).sup f := by grind
_ ≤ (t.sup fun i => (g i).sup f) ⊔ (u.sup f) := by grind
symm
apply Disjoint.disjoint_sup_left_of_disj... | Lean.Elab.Tactic.evalClassical | Lean.Parser.Tactic.classical |
Mathlib.Order.SupIndep | {
"line": 171,
"column": 2
} | {
"line": 183,
"column": 57
} | [
{
"pp": "α : Type u_1\nι : Type u_3\nι' : Type u_4\ninst✝³ : Lattice α\ninst✝² : IsModularLattice α\ninst✝¹ : OrderBot α\ninst✝ : DecidableEq ι\ns : Finset ι'\ng : ι' → Finset ι\nf : ι → α\nhs : s.SupIndep fun i ↦ (g i).sup f\nhg : ∀ i' ∈ s, (g i').SupIndep f\n⊢ (s.biUnion g).SupIndep f",
"usedConstants": [... | classical
intro a ha b hb hab
obtain ⟨i', hi', hb⟩ := mem_biUnion.mp hb
let t := s.erase i'
let u := (g i').erase b
apply Disjoint.mono_right <| calc
a.sup f ≤ (t.biUnion g ∪ u).sup f := by grind
_ ≤ (t.sup fun i => (g i).sup f) ⊔ (u.sup f) := by grind
symm
apply Disjoint.disjoint_sup_left_of_disj... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.SupIndep | {
"line": 171,
"column": 2
} | {
"line": 183,
"column": 57
} | [
{
"pp": "α : Type u_1\nι : Type u_3\nι' : Type u_4\ninst✝³ : Lattice α\ninst✝² : IsModularLattice α\ninst✝¹ : OrderBot α\ninst✝ : DecidableEq ι\ns : Finset ι'\ng : ι' → Finset ι\nf : ι → α\nhs : s.SupIndep fun i ↦ (g i).sup f\nhg : ∀ i' ∈ s, (g i').SupIndep f\n⊢ (s.biUnion g).SupIndep f",
"usedConstants": [... | classical
intro a ha b hb hab
obtain ⟨i', hi', hb⟩ := mem_biUnion.mp hb
let t := s.erase i'
let u := (g i').erase b
apply Disjoint.mono_right <| calc
a.sup f ≤ (t.biUnion g ∪ u).sup f := by grind
_ ≤ (t.sup fun i => (g i).sup f) ⊔ (u.sup f) := by grind
symm
apply Disjoint.disjoint_sup_left_of_disj... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.SupIndep | {
"line": 485,
"column": 4
} | {
"line": 485,
"column": 43
} | [
{
"pp": "α : Type u_1\nι : Type u_3\ninst✝ : CompleteLattice α\ns : Finset ι\nf : ι → α\n⊢ iSupIndep (f ∘ Subtype.val) ↔ s.SupIndep f",
"usedConstants": [
"Eq.mpr",
"Lattice.toSemilatticeSup",
"iSupIndep",
"congrArg",
"Finset",
"PartialOrder.toPreorder",
"Classical.... | rw [Finset.supIndep_iff_disjoint_erase] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Notation.Indicator | {
"line": 241,
"column": 2
} | {
"line": 242,
"column": 20
} | [
{
"pp": "α : Type u_1\nM : Type u_3\ninst✝ : One M\nt : Set α\ns : Set M\n⊢ t.mulIndicator 1 ⁻¹' s ∈ {univ, ∅}",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Set.preimage_const",
"true_or",
"Set.univ",
"Set.mulIndicator",
"Classical.propDecidable",
"Membership.me... | rw [mulIndicator_one', Pi.one_def, Set.preimage_const]
split_ifs <;> simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Notation.Indicator | {
"line": 241,
"column": 2
} | {
"line": 242,
"column": 20
} | [
{
"pp": "α : Type u_1\nM : Type u_3\ninst✝ : One M\nt : Set α\ns : Set M\n⊢ t.mulIndicator 1 ⁻¹' s ∈ {univ, ∅}",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Set.preimage_const",
"true_or",
"Set.univ",
"Set.mulIndicator",
"Classical.propDecidable",
"Membership.me... | rw [mulIndicator_one', Pi.one_def, Set.preimage_const]
split_ifs <;> simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Algebra.Tower | {
"line": 353,
"column": 95
} | {
"line": 358,
"column": 77
} | [
{
"pp": "R : Type u\nS : Type v\nA : Type w\ninst✝⁶ : Semiring R\ninst✝⁵ : Semiring S\ninst✝⁴ : AddCommMonoid A\ninst✝³ : Module R S\ninst✝² : Module S A\ninst✝¹ : Module R A\ninst✝ : IsScalarTower R S A\ns : Set S\nhs : span R s = ⊤\nt : Set A\nx✝ : A\nhp : x✝ ∈ restrictScalars R (span S t)\ns0 : S\ny : A\nhy ... | by
refine span_induction (fun x hx ↦ subset_span <| by exact ⟨x, hx, y, hy, rfl⟩) ?_ ?_ ?_
(hs ▸ mem_top : s0 ∈ span R s)
· rw [zero_smul]; apply zero_mem
· intro _ _ _ _; rw [add_smul]; apply add_mem
· intro r s0 _ hy; rw [IsScalarTower.smul_assoc]; exact smul_mem _ r hy | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Order.CompactlyGenerated.Basic | {
"line": 563,
"column": 18
} | {
"line": 563,
"column": 19
} | [
{
"pp": "case eq_top_or_exists_le_coatom.inr.h.refine_1\nα : Type u_2\ninst✝ : CompleteLattice α\nk : α\nh : IsCompactElement k\nb : α\nhbk : b ∈ Iic k\nH : b ≠ k\nS : Set α\nSC : S ⊆ Iio k\ncC : IsChain (fun x1 x2 ↦ x1 ≤ x2) S\n⊢ ∀ y ∈ S, ∃ ub ∈ Iio k, ∀ z ∈ S, z ≤ ub",
"usedConstants": []
}
] | I | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Data.Multiset.Antidiagonal | {
"line": 75,
"column": 30
} | {
"line": 75,
"column": 41
} | [
{
"pp": "α : Type u_1\na : α\ns : Multiset α\nl : List α\n⊢ (powersetAux' l ++ List.map (cons a) (powersetAux' l)).zip\n ((List.map (cons a) (powersetAux' l)).reverse ++ (powersetAux' l).reverse) ~\n (List.map id (powersetAux' l)).zip (List.map (cons a) (powersetAux' l).reverse) ++\n (List.map (con... | zip_append, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Multiset.Antidiagonal | {
"line": 82,
"column": 13
} | {
"line": 82,
"column": 31
} | [
{
"pp": "case cons\nα : Type u_1\ninst✝ : DecidableEq α\na : α\ns : Multiset α\nhs : s.antidiagonal = map (fun t ↦ (s - t, t)) s.powerset\n⊢ (a ::ₘ s).antidiagonal = map (fun t ↦ (a ::ₘ s - t, t)) (a ::ₘ s).powerset",
"usedConstants": [
"Eq.mpr",
"Multiset.map",
"congrArg",
"HSub.hSu... | antidiagonal_cons, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.LinearAlgebra.Span.Basic | {
"line": 560,
"column": 63
} | {
"line": 560,
"column": 72
} | [
{
"pp": "R : Type u_1\nR₂ : Type u_2\nM : Type u_4\nM₂ : Type u_5\ninst✝⁶ : Semiring R\ninst✝⁵ : Semiring R₂\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : AddCommGroup M₂\ninst✝¹ : Module R₂ M₂\nτ₁₂ : R →+* R₂\ninst✝ : RingHomSurjective τ₁₂\nf : M →ₛₗ[τ₁₂] M₂\np : Submodule R₂ M₂\nhp : IsCoatom p\nh :... | comap_top | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Span.Basic | {
"line": 621,
"column": 59
} | {
"line": 621,
"column": 68
} | [
{
"pp": "R : Type u_1\nR₂ : Type u_2\nM : Type u_4\nM₂ : Type u_5\ninst✝⁶ : Semiring R\ninst✝⁵ : Semiring R₂\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : AddCommGroup M₂\ninst✝¹ : Module R₂ M₂\nτ₁₂ : R →+* R₂\ninst✝ : RingHomSurjective τ₁₂\nι : Type u_8\ns : Set ι\np : ι → Submodule R₂ M₂\nhp : ⨆ x, ... | comap_top | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Finsupp.Single | {
"line": 445,
"column": 4
} | {
"line": 447,
"column": 11
} | [
{
"pp": "case h.left\nα : Type u_1\nβ : Type u_2\nM : Type u_5\ninst✝ : Zero M\nl : α →₀ M\nf : α ↪ β\na : β\nb : M\nhb : b ≠ 0\nh : embDomain f l = single a b\nh_map_support : map f l.support = {a}\nha : a ∈ map f l.support\nc : α\n_hc₁ : c ∈ l.support\nhc₂ : f c = a\n⊢ l = single c b",
"usedConstants": [
... | · ext d
rw [← embDomain_apply_self f l, h]
grind | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Order.BigOperators.Group.Finset | {
"line": 388,
"column": 2
} | {
"line": 388,
"column": 48
} | [
{
"pp": "ι : Type u_1\nα : Type u_2\ninst✝ : DecidableEq α\ns : Finset ι\nf : ι → Finset α\nhs : (↑s).PairwiseDisjoint f\nhf : ∀ i ∈ s, (f i).Nonempty\n⊢ ∑ x ∈ s, 1 ≤ ∑ u ∈ s, #(f u)",
"usedConstants": [
"Nat.instIsOrderedAddMonoid",
"Finset",
"Membership.mem",
"Finset.Nonempty.card_... | exact sum_le_sum fun i hi ↦ (hf i hi).card_pos | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.Order.BigOperators.Group.Finset | {
"line": 438,
"column": 28
} | {
"line": 438,
"column": 59
} | [
{
"pp": "ι : Type u_1\nM : Type u_4\ninst✝² : CommMonoid M\ninst✝¹ : Preorder M\ninst✝ : CanonicallyOrderedMul M\nf : ι → M\ns t : Finset ι\nh : ∀ x ∈ s, f x ≠ 1 → x ∈ t\nthis : IsOrderedMonoid M\n⊢ ∀ i ∈ {x ∈ s | f x = 1}, f i ≤ 1",
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"Monoid.toM... | simp only [mem_filter, and_imp] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Data.Finsupp.Basic | {
"line": 557,
"column": 2
} | {
"line": 560,
"column": 42
} | [
{
"pp": "case inr\nα : Type u_1\nβ : Type u_2\nM : Type u_5\ninst✝ : Zero M\nf : α → β\na : α\nm : M\nhif : Set.InjOn f (f ⁻¹' ↑(single (f a) m).support)\nhm : m ≠ 0\n⊢ comapDomain f (single (f a) m) hif = single a m",
"usedConstants": [
"Finsupp.instFunLike",
"Eq.mpr",
"Finsupp.support_si... | · rw [eq_single_iff, comapDomain_apply, comapDomain_support, ← Finset.coe_subset, coe_preimage,
support_single _ hm, coe_singleton, coe_singleton, single_eq_same]
rw [support_single _ hm, coe_singleton] at hif
exact ⟨fun x hx => hif hx rfl hx, rfl⟩ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Order.BigOperators.Group.Finset | {
"line": 682,
"column": 2
} | {
"line": 683,
"column": 99
} | [
{
"pp": "case h.e'_2\nα : Type u_2\ninst✝ : DecidableEq α\nx : Multiset α\n⊢ ((Finset.range (x.card + 1)).sup fun k ↦ powersetCard k x) = ∑ x_1 ∈ Finset.range (x.card + 1), powersetCard x_1 x",
"usedConstants": [
"Iff.mpr",
"Lattice.toSemilatticeSup",
"Finset",
"Multiset.instAddCance... | exact
Eq.symm (finsetSum_eq_sup_iff_disjoint.mpr fun _ _ _ _ h => pairwise_disjoint_powersetCard x h) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Data.Finsupp.Basic | {
"line": 1409,
"column": 55
} | {
"line": 1409,
"column": 88
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nM : Type u_5\ninst✝² : DecidableEq β\ninst✝¹ : AddCommMonoid M\nf : α → β\ninst✝ : Subsingleton (AddUnits M)\nx : α →₀ M\nt : β\nx✝ : ∃ a ∈ x.support, f a = t\ni : α\ni_in : i ∈ x.support\nhi : f i = t\n⊢ ¬x i = 0 ∧ f i = f i ∧ ¬x i = 0",
"usedConstants": [
"Finsup... | by simp [mem_support_iff.mp i_in] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Finsupp.SMul | {
"line": 197,
"column": 35
} | {
"line": 197,
"column": 78
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nM : Type u_3\nN : Type u_4\nG : Type u_5\nR : Type u_6\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\nι : Type u_7\ninst✝ : Module.IsTorsionFree R M\nr : R\nhr : IsRegular r\nf g : ι →₀ M\nhfg : (fun x ↦ r • x) f = (fun x ↦ r • x) g\n⊢ f = g",
"used... | ext i; exact hr.isSMulRegular congr($hfg i) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Finsupp.SMul | {
"line": 197,
"column": 35
} | {
"line": 197,
"column": 78
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nM : Type u_3\nN : Type u_4\nG : Type u_5\nR : Type u_6\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\nι : Type u_7\ninst✝ : Module.IsTorsionFree R M\nr : R\nhr : IsRegular r\nf g : ι →₀ M\nhfg : (fun x ↦ r • x) f = (fun x ↦ r • x) g\n⊢ f = g",
"used... | ext i; exact hr.isSMulRegular congr($hfg i) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Finsupp.Supported | {
"line": 167,
"column": 2
} | {
"line": 167,
"column": 63
} | [
{
"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\n⊢ supported M R (s ∩ t) = supported M R s ⊓ supported M R t",
"usedConstants": [
"cond",
"Eq.mpr",
"Submodule",
"iInf",
"Finsupp.module",
"Co... | rw [Set.inter_eq_iInter, supported_iInter, iInf_bool_eq]; rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Finsupp.Supported | {
"line": 167,
"column": 2
} | {
"line": 167,
"column": 63
} | [
{
"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\n⊢ supported M R (s ∩ t) = supported M R s ⊓ supported M R t",
"usedConstants": [
"cond",
"Eq.mpr",
"Submodule",
"iInf",
"Finsupp.module",
"Co... | rw [Set.inter_eq_iInter, supported_iInter, iInf_bool_eq]; rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Finsupp.Defs | {
"line": 142,
"column": 51
} | {
"line": 142,
"column": 69
} | [
{
"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\na a' : α\nh : a ≠ a'\n⊢ lapply a ∘ₗ lsingle a' = 0",
"usedConstants": [
"False",
"Finsupp.module",
"eq_false",
"LinearMap.ext",
"congrArg",
"AddMonoid.toA... | ext; simp [h.symm] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Finsupp.Defs | {
"line": 142,
"column": 51
} | {
"line": 142,
"column": 69
} | [
{
"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\na a' : α\nh : a ≠ a'\n⊢ lapply a ∘ₗ lsingle a' = 0",
"usedConstants": [
"False",
"Finsupp.module",
"eq_false",
"LinearMap.ext",
"congrArg",
"AddMonoid.toA... | ext; simp [h.symm] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Finsupp.Defs | {
"line": 341,
"column": 8
} | {
"line": 341,
"column": 69
} | [
{
"pp": "case h\nR✝ : Type u_1\nM✝ : Type u_2\nN : Type u_3\ninst✝⁷ : Semiring R✝\ninst✝⁶ : AddCommMonoid M✝\ninst✝⁵ : Module R✝ M✝\ninst✝⁴ : AddCommMonoid N\ninst✝³ : Module R✝ N\nι : Type u_4\nR : Type u_5\nM : Type u_6\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nf : End (End R M) M\ng... | change f (Finsupp.lapply j ∘ₗ g ∘ₗ Finsupp.lsingle i • m) = _ | Lean.Elab.Tactic.evalChange | Lean.Parser.Tactic.change |
Mathlib.Algebra.BigOperators.Finprod | {
"line": 322,
"column": 2
} | {
"line": 324,
"column": 82
} | [
{
"pp": "ι : Sort u_6\nR : Type u_7\nM : Type u_8\ninst✝⁴ : Semiring R\ninst✝³ : IsDomain R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : Module.IsTorsionFree R M\nc : R\nf : ι → M\n⊢ c • ∑ᶠ (i : ι), f i = ∑ᶠ (i : ι), c • f i",
"usedConstants": [
"AddMonoidHom.map_finsum_of_injective",
... | rcases eq_or_ne c 0 with (rfl | hc)
· simp
· exact (smulAddHom R M c).map_finsum_of_injective (smul_right_injective M hc) _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.BigOperators.Finprod | {
"line": 322,
"column": 2
} | {
"line": 324,
"column": 82
} | [
{
"pp": "ι : Sort u_6\nR : Type u_7\nM : Type u_8\ninst✝⁴ : Semiring R\ninst✝³ : IsDomain R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : Module.IsTorsionFree R M\nc : R\nf : ι → M\n⊢ c • ∑ᶠ (i : ι), f i = ∑ᶠ (i : ι), c • f i",
"usedConstants": [
"AddMonoidHom.map_finsum_of_injective",
... | rcases eq_or_ne c 0 with (rfl | hc)
· simp
· exact (smulAddHom R M c).map_finsum_of_injective (smul_right_injective M hc) _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.Interval.Set.Fin | {
"line": 96,
"column": 86
} | {
"line": 96,
"column": 97
} | [
{
"pp": "n : ℕ\ni j : Fin n\n⊢ val '' uIoo i j = uIoo ↑i ↑j",
"usedConstants": [
"Lattice.toSemilatticeSup",
"congrArg",
"PartialOrder.toPreorder",
"SemilatticeInf.toPartialOrder",
"SemilatticeSup.toMax",
"DistribLattice.toLattice",
"SemilatticeInf.toMin",
"Fi... | simp [uIoo] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Order.Interval.Set.Fin | {
"line": 96,
"column": 86
} | {
"line": 96,
"column": 97
} | [
{
"pp": "n : ℕ\ni j : Fin n\n⊢ val '' uIoo i j = uIoo ↑i ↑j",
"usedConstants": [
"Lattice.toSemilatticeSup",
"congrArg",
"PartialOrder.toPreorder",
"SemilatticeInf.toPartialOrder",
"SemilatticeSup.toMax",
"DistribLattice.toLattice",
"SemilatticeInf.toMin",
"Fi... | simp [uIoo] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.Interval.Set.Fin | {
"line": 96,
"column": 86
} | {
"line": 96,
"column": 97
} | [
{
"pp": "n : ℕ\ni j : Fin n\n⊢ val '' uIoo i j = uIoo ↑i ↑j",
"usedConstants": [
"Lattice.toSemilatticeSup",
"congrArg",
"PartialOrder.toPreorder",
"SemilatticeInf.toPartialOrder",
"SemilatticeSup.toMax",
"DistribLattice.toLattice",
"SemilatticeInf.toMin",
"Fi... | simp [uIoo] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.BigOperators.Finprod | {
"line": 484,
"column": 4
} | {
"line": 484,
"column": 15
} | [
{
"pp": "α : Type u_1\nM : Type u_5\ninst✝ : CommMonoid M\nf : α → M\ns : Set α\nt : Finset α\nh : s ∩ mulSupport f = ↑t ∩ mulSupport f\n⊢ ∀ {x : α}, f x ≠ 1 → (x ∈ s ↔ x ∈ t)",
"usedConstants": []
}
] | intro x hxf | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.LinearAlgebra.Finsupp.LinearCombination | {
"line": 219,
"column": 12
} | {
"line": 219,
"column": 38
} | [
{
"pp": "case zero\nM : Type u_2\nR : Type u_5\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nα : Type u_9\nβ : Type u_10\nA : α → M\nB : β → α →₀ R\n⊢ ((sum 0 fun i a ↦ a • B i).sum fun i a ↦ a • A i) = sum 0 fun i a ↦ a • (B i).sum fun i a ↦ a • A i",
"usedConstants": [
"AddMono... | simp only [sum_zero_index] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.LinearAlgebra.Finsupp.LinearCombination | {
"line": 219,
"column": 12
} | {
"line": 219,
"column": 38
} | [
{
"pp": "case zero\nM : Type u_2\nR : Type u_5\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nα : Type u_9\nβ : Type u_10\nA : α → M\nB : β → α →₀ R\n⊢ ((sum 0 fun i a ↦ a • B i).sum fun i a ↦ a • A i) = sum 0 fun i a ↦ a • (B i).sum fun i a ↦ a • A i",
"usedConstants": [
"AddMono... | simp only [sum_zero_index] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Finsupp.LinearCombination | {
"line": 219,
"column": 12
} | {
"line": 219,
"column": 38
} | [
{
"pp": "case zero\nM : Type u_2\nR : Type u_5\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nα : Type u_9\nβ : Type u_10\nA : α → M\nB : β → α →₀ R\n⊢ ((sum 0 fun i a ↦ a • B i).sum fun i a ↦ a • A i) = sum 0 fun i a ↦ a • (B i).sum fun i a ↦ a • A i",
"usedConstants": [
"AddMono... | simp only [sum_zero_index] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Basis.Defs | {
"line": 418,
"column": 4
} | {
"line": 418,
"column": 34
} | [
{
"pp": "case inl\nι : Type u_10\nR : Type u_11\nM : Type u_12\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nb : Basis ι R M\ni : ι\nh : b i ∈ range ⇑b\nh✝ : Subsingleton R\n⊢ b.reindexRange ⟨b i, h⟩ = b i",
"usedConstants": [
"AddMonoid.toAddZeroClass",
"Module.toMulAction... | let := Module.subsingleton R M | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.LinearAlgebra.Basis.Defs | {
"line": 420,
"column": 2
} | {
"line": 420,
"column": 41
} | [
{
"pp": "case inr\nι : Type u_10\nR : Type u_11\nM : Type u_12\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nb : Basis ι R M\ni : ι\nh : b i ∈ range ⇑b\nh✝ : Nontrivial R\n⊢ b.reindexRange ⟨b i, h⟩ = b i",
"usedConstants": [
"dite_cond_eq_true",
"Nontrivial",
"Module.... | · simp [*, reindexRange, reindex_apply] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.LinearAlgebra.Finsupp.LinearCombination | {
"line": 384,
"column": 2
} | {
"line": 384,
"column": 87
} | [
{
"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\nx : M\n⊢ x ∈ span R (Set.range v) ↔ ∃ x_1, ∑ i, Finsupp.equivFunOnFinite x_1 i • v i = x",
"usedConstants": [
"Finsupp.instFunLike",
"Eq.mpr",
... | simp only [Finsupp.mem_span_range_iff_exists_finsupp, Finsupp.equivFunOnFinite_apply] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Data.Fintype.Fin | {
"line": 37,
"column": 22
} | {
"line": 37,
"column": 49
} | [
{
"pp": "n : ℕ\n⊢ ↑(Ioi 0) = ↑(map (succEmb n) univ)",
"usedConstants": [
"Set.ext",
"Set.image_univ",
"instNeZeroNatHAdd_1",
"Set.Ioi",
"Preorder.toLT",
"Fin.range_succ",
"Finset.univ",
"Finset.coe_univ",
"Finset.Ioi",
"Fin.succ",
"congrArg"... | ext; simp [pos_iff_ne_zero] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Fintype.Fin | {
"line": 37,
"column": 22
} | {
"line": 37,
"column": 49
} | [
{
"pp": "n : ℕ\n⊢ ↑(Ioi 0) = ↑(map (succEmb n) univ)",
"usedConstants": [
"Set.ext",
"Set.image_univ",
"instNeZeroNatHAdd_1",
"Set.Ioi",
"Preorder.toLT",
"Fin.range_succ",
"Finset.univ",
"Finset.coe_univ",
"Finset.Ioi",
"Fin.succ",
"congrArg"... | ext; simp [pos_iff_ne_zero] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Group.Fin.Basic | {
"line": 152,
"column": 2
} | {
"line": 152,
"column": 78
} | [
{
"pp": "n : ℕ\ninst✝ : NeZero n\na : Fin n\nha : ↑a + 1 < n\n⊢ a < a + 1",
"usedConstants": [
"Fin.val_add",
"Eq.mpr",
"congrArg",
"id",
"Fin.instOfNat",
"Nat.instMod",
"instHMod",
"instOfNatNat",
"Nat.add_mod_mod",
"Nat.mod_eq_of_lt",
"Fin.... | rw [lt_def, val_add, coe_ofNat_eq_mod, Nat.add_mod_mod, Nat.mod_eq_of_lt ha] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Data.ENat.Pow | {
"line": 161,
"column": 6
} | {
"line": 161,
"column": 29
} | [
{
"pp": "case inr.inr.inr.inr.coe.top\nx y z : ℕ∞\nx_2 : 1 < x\na✝ : ℕ\ny_0 : ↑a✝ ≠ 0\nz_0 : ⊤ ≠ 0\n⊢ ⊤ = (x ^ ↑a✝) ^ ⊤",
"usedConstants": [
"ENat.instNatCast",
"instTopENat",
"Nat.cast",
"HPow.hPow",
"ENat",
"instHPow",
"Top.top",
"Eq.symm",
"ENat.epow_... | apply (epow_top _).symm | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Algebra.Group.ModEq | {
"line": 73,
"column": 70
} | {
"line": 76,
"column": 22
} | [
{
"pp": "M : Type u_1\ninst✝ : AddCommMonoid M\na b p : M\nh : a ≡ b [PMOD p]\n⊢ b ≡ a [PMOD p]",
"usedConstants": [
"Eq.mpr",
"AddCommGroup.ModEq",
"instHSMul",
"congrArg",
"AddMonoid.toNSMul",
"Exists",
"Eq.mp",
"id",
"instHAdd",
"Exists.casesOn"... | by
rw [modEq_iff_nsmul] at *
rcases h with ⟨m, n, h⟩
exact ⟨n, m, h.symm⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.LinearIndependent.Basic | {
"line": 176,
"column": 16
} | {
"line": 176,
"column": 19
} | [
{
"pp": "ι : Type u'\nR : Type u_2\nM : Type u_4\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\nG : Type u_6\nhG : Group G\ninst✝³ : MulAction G R\ninst✝² : SMul G M\ninst✝¹ : IsScalarTower G R M\ninst✝ : SMulCommClass G R M\nv : ι → M\nhv :\n ∀ (s : Finset ι) (f g : ι → R), (∀ i ∉ s, f i... | hgs | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.LinearAlgebra.LinearIndependent.Basic | {
"line": 179,
"column": 13
} | {
"line": 179,
"column": 21
} | [
{
"pp": "case refine_1\nι : Type u'\nR : Type u_2\nM : Type u_4\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\nG : Type u_6\nhG : Group G\ninst✝³ : MulAction G R\ninst✝² : SMul G M\ninst✝¹ : IsScalarTower G R M\ninst✝ : SMulCommClass G R M\nv : ι → M\nhv :\n ∀ (s : Finset ι) (f g : ι → R)... | hgs i hi | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Algebra.Group.ModEq | {
"line": 324,
"column": 6
} | {
"line": 324,
"column": 17
} | [
{
"pp": "G : Type u_1\ninst✝ : AddCommGroup G\np a : G\n⊢ a ≡ 0 [PMOD p] ↔ ∃ z, a = z • p",
"usedConstants": [
"Eq.mpr",
"AddCommGroup.ModEq",
"instHSMul",
"congrArg",
"AddCommGroup.toAddCommMonoid",
"AddMonoid.toAddZeroClass",
"AddCommGroup.toAddGroup",
"Exis... | modEq_comm, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.LinearIndependent.Basic | {
"line": 195,
"column": 16
} | {
"line": 195,
"column": 19
} | [
{
"pp": "ι : Type u'\nR : Type u_2\nM : Type u_4\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nv : ι → M\nhv :\n ∀ (s : Finset ι) (f g : ι → R), (∀ i ∉ s, f i = g i) → ∑ i ∈ s, f i • v i = ∑ i ∈ s, g i • v i → ∀ (i : ι), f i = g i\nw : ι → Rˣ\ns : Finset ι\ng₁ g₂ : ι → R\n⊢ (∀ i ∉ s, g₁ i... | hgs | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.LinearAlgebra.LinearIndependent.Basic | {
"line": 198,
"column": 13
} | {
"line": 198,
"column": 21
} | [
{
"pp": "case refine_1\nι : Type u'\nR : Type u_2\nM : Type u_4\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nv : ι → M\nhv :\n ∀ (s : Finset ι) (f g : ι → R), (∀ i ∉ s, f i = g i) → ∑ i ∈ s, f i • v i = ∑ i ∈ s, g i • v i → ∀ (i : ι), f i = g i\nw : ι → Rˣ\ns : Finset ι\ng₁ g₂ : ι → R\nh... | hgs i hi | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Algebra.Group.ModEq | {
"line": 327,
"column": 81
} | {
"line": 328,
"column": 41
} | [
{
"pp": "G : Type u_1\ninst✝ : AddCommGroup G\np a b : G\n⊢ ¬a ≡ b [PMOD p] ↔ ∀ (z : ℤ), b ≠ a + z • p",
"usedConstants": [
"Eq.mpr",
"AddCommGroup.ModEq",
"instHSMul",
"congrArg",
"AddCommGroup.toAddCommMonoid",
"Iff.rfl",
"AddCommGroup.toAddGroup",
"Exists",... | by
rw [modEq_iff_eq_add_zsmul, not_exists] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Nat.GCD.Basic | {
"line": 53,
"column": 34
} | {
"line": 53,
"column": 48
} | [
{
"pp": "a b c : ℕ\nha0 : a > 0\nha1 : succ 0 < a\nhb0 : b > 0\nh : b ≤ c\nthis : a ^ (c - b + b) - 1 = a ^ (c - b) * (a ^ b - 1) + (a ^ (c - b) - 1)\n⊢ (a ^ (c - b + b) - 1) % (a ^ b - 1) = a ^ ((c - b + b) % b) - 1",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Nat.instMonoid",
"HSub.... | add_mod_right, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Nat.GCD.Basic | {
"line": 238,
"column": 33
} | {
"line": 238,
"column": 53
} | [
{
"pp": "m n : ℕ\ncop : m.Coprime n\nx : ℕ\nha : x ≠ 0 ∧ n ≠ 0\ny : ℕ\nhb : y ≠ 0 ∧ m ≠ 0\nh : n * x * m + m * y * n = m * n\n⊢ False",
"usedConstants": [
"Nat.instMulZeroClass",
"congrArg",
"Eq.mp",
"Ne",
"instOfNatNat",
"LE.le",
"instLENat",
"And",
"Na... | ← one_le_iff_ne_zero | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Nat.ModEq | {
"line": 52,
"column": 8
} | {
"line": 52,
"column": 31
} | [
{
"pp": "case mpr\na b n : ℕ\nh : a % n = b % n\n⊢ a ≡ b [PMOD n]",
"usedConstants": [
"Eq.mpr",
"AddCommGroup.ModEq",
"instHDiv",
"HMul.hMul",
"congrArg",
"id",
"HDiv.hDiv",
"Nat.instMod",
"instHMod",
"instMulNat",
"Nat.div_add_mod'",
... | ← Nat.div_add_mod' a n, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Nat.ModEq | {
"line": 331,
"column": 2
} | {
"line": 331,
"column": 52
} | [
{
"pp": "n a b : ℕ\nh : n ≤ b\n⊢ a ≡ b - n [MOD n] ↔ a ≡ b [MOD n]",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Iff.rfl",
"HSub.hSub",
"id",
"instSubNat",
"instHAdd",
"Iff",
"Nat.ModEq",
"instHSub",
"Nat.sub_add_cancel",
"HAdd.hAdd",
... | rw [← modEq_add_modulus_iff, Nat.sub_add_cancel h] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Data.Nat.ModEq | {
"line": 331,
"column": 2
} | {
"line": 331,
"column": 52
} | [
{
"pp": "n a b : ℕ\nh : n ≤ b\n⊢ a ≡ b - n [MOD n] ↔ a ≡ b [MOD n]",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Iff.rfl",
"HSub.hSub",
"id",
"instSubNat",
"instHAdd",
"Iff",
"Nat.ModEq",
"instHSub",
"Nat.sub_add_cancel",
"HAdd.hAdd",
... | rw [← modEq_add_modulus_iff, Nat.sub_add_cancel h] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Nat.ModEq | {
"line": 331,
"column": 2
} | {
"line": 331,
"column": 52
} | [
{
"pp": "n a b : ℕ\nh : n ≤ b\n⊢ a ≡ b - n [MOD n] ↔ a ≡ b [MOD n]",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Iff.rfl",
"HSub.hSub",
"id",
"instSubNat",
"instHAdd",
"Iff",
"Nat.ModEq",
"instHSub",
"Nat.sub_add_cancel",
"HAdd.hAdd",
... | rw [← modEq_add_modulus_iff, Nat.sub_add_cancel h] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.BigOperators.Fin | {
"line": 452,
"column": 4
} | {
"line": 452,
"column": 59
} | [
{
"pp": "M : Type u_2\ninst✝ : CommMonoid M\nn : ℕ\nf : Fin (n + 1) → Fin n → M\n⊢ ((∏ i, ∏ j with i.castSucc ≤ j.castSucc, f i.castSucc j) * ∏ j with last n ≤ j.castSucc, f (last n) j) *\n ((∏ j with j.castSucc < 0, f 0 j) * ∏ i, ∏ j with j.castSucc < i.succ, f i.succ j) =\n (∏ i, ∏ j ∈ Ici i, f i.cast... | simp [Finset.filter_le_eq_Ici, Finset.filter_ge_eq_Iic] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.BigOperators.Fin | {
"line": 560,
"column": 6
} | {
"line": 560,
"column": 71
} | [
{
"pp": "case inl.h\nn : ℕ\nG : Type u_3\ninst✝ : Group G\ng : Fin (n + 1) → G\nj : Fin (n + 1)\nk : Fin n\nh : ↑k < ↑j\n⊢ k.castSucc.castSucc < j.succ",
"usedConstants": [
"Eq.mpr",
"Fin.succ",
"congrArg",
"Fin.succ_le_succ_iff",
"id",
"instOfNatNat",
"LE.le",
... | rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.BigOperators.Fin | {
"line": 566,
"column": 6
} | {
"line": 566,
"column": 71
} | [
{
"pp": "case inr.inl.h\nn : ℕ\nG : Type u_3\ninst✝ : Group G\ng : Fin (n + 1) → G\nj : Fin (n + 1)\nk : Fin n\nh : ↑k = ↑j\n⊢ k.castSucc.castSucc < j.succ",
"usedConstants": [
"Eq.mpr",
"Fin.succ",
"congrArg",
"Fin.succ_le_succ_iff",
"id",
"instOfNatNat",
"LE.le",
... | rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.LinearIndependent.Defs | {
"line": 653,
"column": 8
} | {
"line": 653,
"column": 41
} | [
{
"pp": "case refine_2\nι : Type u'\nR : Type u_2\nM : Type u_4\nv : ι → M\ninst✝⁸ : Semiring R\ninst✝⁷ : AddCommMonoid M\ninst✝⁶ : Module R M\ninst✝⁵ : LinearOrder R\ninst✝⁴ : CanonicallyOrderedAdd R\ninst✝³ : AddRightReflectLE R\ninst✝² : IsCancelAdd M\ninst✝¹ : DecidableEq ι\ninst✝ : Fintype ι\nh : ∀ (t : Fi... | simp [Finset.mem_compl.1 hi, hi'] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.LinearAlgebra.LinearIndependent.Defs | {
"line": 732,
"column": 80
} | {
"line": 737,
"column": 81
} | [
{
"pp": "ι : Type u'\nR : Type u_2\nM : Type u_4\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nv : ι → M\n⊢ LinearIndependent R v ↔ ∀ (s : Finset ι) (g : ι → R), ∑ i ∈ s, g i • v i = 0 → ∀ i ∈ s, g i = 0",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"instH... | by
rw [linearIndependent_iff'ₛ]
refine ⟨fun h s f ↦ ?_, fun h s f g ↦ ?_⟩
· convert! h s f 0; simp_rw [Pi.zero_apply, zero_smul, Finset.sum_const_zero]
· rw [← sub_eq_zero, ← Finset.sum_sub_distrib]
convert! h s (f - g) using 3; simp only [Pi.sub_apply, sub_smul, sub_eq_zero] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Set.Card | {
"line": 281,
"column": 91
} | {
"line": 282,
"column": 73
} | [
{
"pp": "α : Type u_1\ns t : Set α\n⊢ s.encard - t.encard ≤ (s \\ t).encard",
"usedConstants": [
"Eq.mpr",
"Set.encard",
"instSubENat",
"congrArg",
"CommSemiring.toSemiring",
"HSub.hSub",
"Preorder.toLE",
"instPreorderENat",
"id",
"LE.le",
"t... | by
rw [tsub_le_iff_left, add_comm]; apply encard_le_encard_diff_add_encard | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Set.Card | {
"line": 464,
"column": 2
} | {
"line": 464,
"column": 82
} | [
{
"pp": "case inr\nα : Type u_1\ns t : Set α\nhst : s ⊆ t\nhs : s.encard ≠ ⊤\nk' : ℕ∞\nr' : Set α\nhr' : r' ⊆ t \\ s\nhsk : s.encard ≤ s.encard + r'.encard\nhkt : s.encard + r'.encard ≤ t.encard\nhk' : t.encard = s.encard + r'.encard + k'\nhk : r'.encard ≤ (t \\ s).encard\n⊢ ∃ r, s ⊆ r ∧ r ⊆ t ∧ r.encard = s.en... | refine ⟨s ∪ r', subset_union_left, union_subset hst (hr'.trans diff_subset), ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Data.Set.Card | {
"line": 490,
"column": 50
} | {
"line": 490,
"column": 81
} | [
{
"pp": "case inr\nα : Type u_1\nβ : Type u_2\nf : α → β\ns : Set α\nh : Nonempty α\n⊢ (Function.invFunOn f s '' f '' s).encard ≤ s.encard",
"usedConstants": [
"le_refl",
"Set.encard",
"Function.invFunOn",
"instPreorderENat",
"le_imp_le_of_le_of_le",
"ENat",
"Set.im... | f.invFunOn_image_image_subset s | Mathlib.Tactic.evalGRewriteSeq | null |
Mathlib.LinearAlgebra.Basis.Submodule | {
"line": 141,
"column": 6
} | {
"line": 148,
"column": 26
} | [
{
"pp": "ι : Type u_1\nR : Type u_3\nA : Type u_7\ninst✝⁵ : Semiring R\ninst✝⁴ : NonUnitalNonAssocSemiring A\ninst✝³ : Module R A\ninst✝² : SMulCommClass R A A\ninst✝¹ : SMulCommClass R R A\ninst✝ : IsScalarTower R A A\nb : Basis ι R A\nz : A\nh : (∀ (i : ι), Commute (b i) z) ∧ ∀ (i j : ι), z * (b i * b j) = z ... | intro c d
rw [← b.linearCombination_repr c, ← b.linearCombination_repr d, linearCombination_apply,
linearCombination_apply, sum, sum, Finset.sum_mul, Finset.mul_sum, Finset.mul_sum,
Finset.mul_sum]
simp_rw [smul_mul_assoc, Finset.mul_sum, Finset.sum_mul, mul_smul_comm, Finset.mul_sum,
... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Basis.Submodule | {
"line": 141,
"column": 6
} | {
"line": 148,
"column": 26
} | [
{
"pp": "ι : Type u_1\nR : Type u_3\nA : Type u_7\ninst✝⁵ : Semiring R\ninst✝⁴ : NonUnitalNonAssocSemiring A\ninst✝³ : Module R A\ninst✝² : SMulCommClass R A A\ninst✝¹ : SMulCommClass R R A\ninst✝ : IsScalarTower R A A\nb : Basis ι R A\nz : A\nh : (∀ (i : ι), Commute (b i) z) ∧ ∀ (i j : ι), z * (b i * b j) = z ... | intro c d
rw [← b.linearCombination_repr c, ← b.linearCombination_repr d, linearCombination_apply,
linearCombination_apply, sum, sum, Finset.sum_mul, Finset.mul_sum, Finset.mul_sum,
Finset.mul_sum]
simp_rw [smul_mul_assoc, Finset.mul_sum, Finset.sum_mul, mul_smul_comm, Finset.mul_sum,
... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Set.Card | {
"line": 914,
"column": 2
} | {
"line": 929,
"column": 60
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ns : Set α\nt : Set β\nf : (a : α) → a ∈ s → β\nhf : ∀ (a : α) (ha : a ∈ s), f a ha ∈ t\nhinj : ∀ (a₁ a₂ : α) (ha₁ : a₁ ∈ s) (ha₂ : a₂ ∈ s), f a₁ ha₁ = f a₂ ha₂ → a₁ = a₂\nhst : t.ncard ≤ s.ncard\nht : t.Finite\n⊢ ∀ b ∈ t, ∃ a, ∃ (ha : a ∈ s), b = f a ha",
"usedConstants"... | intro b hb
set f' : s → t := fun x ↦ ⟨f x.1 x.2, hf _ _⟩
have finj : f'.Injective := by
rintro ⟨x, hx⟩ ⟨y, hy⟩ hxy
simp only [f', Subtype.mk.injEq] at hxy ⊢
apply hinj _ _ hx hy hxy
have hft := ht.fintype
have hft' := Fintype.ofInjective f' finj
set f'' : ∀ a, a ∈ s.toFinset → β := fun a h ↦ f a (... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Set.Card | {
"line": 914,
"column": 2
} | {
"line": 929,
"column": 60
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ns : Set α\nt : Set β\nf : (a : α) → a ∈ s → β\nhf : ∀ (a : α) (ha : a ∈ s), f a ha ∈ t\nhinj : ∀ (a₁ a₂ : α) (ha₁ : a₁ ∈ s) (ha₂ : a₂ ∈ s), f a₁ ha₁ = f a₂ ha₂ → a₁ = a₂\nhst : t.ncard ≤ s.ncard\nht : t.Finite\n⊢ ∀ b ∈ t, ∃ a, ∃ (ha : a ∈ s), b = f a ha",
"usedConstants"... | intro b hb
set f' : s → t := fun x ↦ ⟨f x.1 x.2, hf _ _⟩
have finj : f'.Injective := by
rintro ⟨x, hx⟩ ⟨y, hy⟩ hxy
simp only [f', Subtype.mk.injEq] at hxy ⊢
apply hinj _ _ hx hy hxy
have hft := ht.fintype
have hft' := Fintype.ofInjective f' finj
set f'' : ∀ a, a ∈ s.toFinset → β := fun a h ↦ f a (... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Set.Card | {
"line": 1164,
"column": 54
} | {
"line": 1164,
"column": 83
} | [
{
"pp": "case inl\nα : Type u_1\ns t : Set α\nhs : s.Finite\nht : t.Finite\n⊢ (∃ a ∉ s, insert a s = t) ↔ hs.toFinset ⊆ ht.toFinset ∧ hs.toFinset.card + 1 = ht.toFinset.card",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Finset",
"Classical.propDecidable",
"Membership.mem",
... | ← Finset.exists_eq_insert_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Set.Card | {
"line": 1235,
"column": 2
} | {
"line": 1235,
"column": 84
} | [
{
"pp": "α : Type u_1\ns : Set α\nhs : s.Finite\n⊢ 2 < s.ncard ↔ ∃ a b c, a ∈ s ∧ b ∈ s ∧ c ∈ s ∧ a ≠ b ∧ a ≠ c ∧ b ≠ c",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Finset",
"Membership.mem",
"Exists",
"id",
"Ne",
"instOfNatNat",
"iff_self",
"funext... | simp_rw [ncard_eq_toFinset_card _ hs, Finset.two_lt_card_iff, Finite.mem_toFinset] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Data.Set.Card | {
"line": 1235,
"column": 2
} | {
"line": 1235,
"column": 84
} | [
{
"pp": "α : Type u_1\ns : Set α\nhs : s.Finite\n⊢ 2 < s.ncard ↔ ∃ a b c, a ∈ s ∧ b ∈ s ∧ c ∈ s ∧ a ≠ b ∧ a ≠ c ∧ b ≠ c",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Finset",
"Membership.mem",
"Exists",
"id",
"Ne",
"instOfNatNat",
"iff_self",
"funext... | simp_rw [ncard_eq_toFinset_card _ hs, Finset.two_lt_card_iff, Finite.mem_toFinset] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Set.Card | {
"line": 1235,
"column": 2
} | {
"line": 1235,
"column": 84
} | [
{
"pp": "α : Type u_1\ns : Set α\nhs : s.Finite\n⊢ 2 < s.ncard ↔ ∃ a b c, a ∈ s ∧ b ∈ s ∧ c ∈ s ∧ a ≠ b ∧ a ≠ c ∧ b ≠ c",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Finset",
"Membership.mem",
"Exists",
"id",
"Ne",
"instOfNatNat",
"iff_self",
"funext... | simp_rw [ncard_eq_toFinset_card _ hs, Finset.two_lt_card_iff, Finite.mem_toFinset] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Set.Card | {
"line": 1243,
"column": 96
} | {
"line": 1244,
"column": 86
} | [
{
"pp": "α : Type u_1\ns : Set α\nhs : s.Finite\n⊢ 3 < s.ncard ↔ ∃ a b c d, a ∈ s ∧ b ∈ s ∧ c ∈ s ∧ d ∈ s ∧ a ≠ b ∧ a ≠ c ∧ a ≠ d ∧ b ≠ c ∧ b ≠ d ∧ c ≠ d",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Finset",
"Membership.mem",
"Exists",
"id",
"_private.Mathlib.Data.Se... | by
simp_rw [ncard_eq_toFinset_card _ hs, Finset.three_lt_card_iff, Finite.mem_toFinset] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.GroupTheory.QuotientGroup.Basic | {
"line": 480,
"column": 61
} | {
"line": 480,
"column": 72
} | [
{
"pp": "ι : Type u_1\nA : ι → Type u_2\ninst✝ : (i : ι) → CommGroup (A i)\nn : ℕ\nφ : ((i : ι) → A i) →* (i : ι) → A i ⧸ (powMonoidHom n).range :=\n { toFun := fun x x_1 ↦ ↑(x x_1), map_one' := ⋯, map_mul' := ⋯ }\ny : (i : ι) → A i ⧸ (powMonoidHom n).range\n⊢ (φ fun i ↦ Quotient.out (y i)) = y",
"usedCons... | by simp [φ] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Field.IsField | {
"line": 105,
"column": 18
} | {
"line": 105,
"column": 35
} | [
{
"pp": "R : Type u\ninst✝ : Ring R\nhf : IsField R\nx : R\nhx : x ≠ 0\ny z : R\nhxy : x * y = 1\nhxz : x * z = 1\n⊢ x * y * z = z",
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"HMul.hMul",
"congrArg",
"AddGroupWithOne.toAddMonoidWithOne",
"id",
"MulOne.toMul",
... | rw [hxy, one_mul] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Field.IsField | {
"line": 105,
"column": 18
} | {
"line": 105,
"column": 35
} | [
{
"pp": "R : Type u\ninst✝ : Ring R\nhf : IsField R\nx : R\nhx : x ≠ 0\ny z : R\nhxy : x * y = 1\nhxz : x * z = 1\n⊢ x * y * z = z",
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"HMul.hMul",
"congrArg",
"AddGroupWithOne.toAddMonoidWithOne",
"id",
"MulOne.toMul",
... | rw [hxy, one_mul] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Field.IsField | {
"line": 105,
"column": 18
} | {
"line": 105,
"column": 35
} | [
{
"pp": "R : Type u\ninst✝ : Ring R\nhf : IsField R\nx : R\nhx : x ≠ 0\ny z : R\nhxy : x * y = 1\nhxz : x * z = 1\n⊢ x * y * z = z",
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"HMul.hMul",
"congrArg",
"AddGroupWithOne.toAddMonoidWithOne",
"id",
"MulOne.toMul",
... | rw [hxy, one_mul] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Order.Interval.Finset.Basic | {
"line": 30,
"column": 2
} | {
"line": 30,
"column": 43
} | [
{
"pp": "α : Type u_2\ninst✝⁴ : AddCommMonoid α\ninst✝³ : PartialOrder α\ninst✝² : IsOrderedCancelAddMonoid α\ninst✝¹ : ExistsAddOfLE α\ninst✝ : LocallyFiniteOrder α\na b c : α\n⊢ map (addLeftEmbedding c) (Icc a b) = Icc (c + a) (c + b)",
"usedConstants": [
"addLeftEmbedding",
"Eq.mpr",
"c... | rw [← coe_inj, coe_map, coe_Icc, coe_Icc] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Order.Interval.Finset.Basic | {
"line": 35,
"column": 2
} | {
"line": 35,
"column": 43
} | [
{
"pp": "α : Type u_2\ninst✝⁴ : AddCommMonoid α\ninst✝³ : PartialOrder α\ninst✝² : IsOrderedCancelAddMonoid α\ninst✝¹ : ExistsAddOfLE α\ninst✝ : LocallyFiniteOrder α\na b c : α\n⊢ map (addRightEmbedding c) (Icc a b) = Icc (a + c) (b + c)",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Finset",... | rw [← coe_inj, coe_map, coe_Icc, coe_Icc] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Order.BigOperators.Group.LocallyFinite | {
"line": 93,
"column": 2
} | {
"line": 93,
"column": 33
} | [
{
"pp": "α : Type u_1\nM : Type u_2\ninst✝² : CommMonoid M\nf : α → M\ninst✝¹ : PartialOrder α\ninst✝ : LocallyFiniteOrderBot α\na : α\n⊢ f a * ∏ x ∈ Iio a, f x = ∏ x ∈ Iic a, f x",
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"Monoid.toMulOneClass",
"Finset.cons",
"congrArg",
... | rw [Iic_eq_cons_Iio, prod_cons] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Order.BigOperators.Group.LocallyFinite | {
"line": 93,
"column": 2
} | {
"line": 93,
"column": 33
} | [
{
"pp": "α : Type u_1\nM : Type u_2\ninst✝² : CommMonoid M\nf : α → M\ninst✝¹ : PartialOrder α\ninst✝ : LocallyFiniteOrderBot α\na : α\n⊢ f a * ∏ x ∈ Iio a, f x = ∏ x ∈ Iic a, f x",
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"Monoid.toMulOneClass",
"Finset.cons",
"congrArg",
... | rw [Iic_eq_cons_Iio, prod_cons] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Order.BigOperators.Group.LocallyFinite | {
"line": 93,
"column": 2
} | {
"line": 93,
"column": 33
} | [
{
"pp": "α : Type u_1\nM : Type u_2\ninst✝² : CommMonoid M\nf : α → M\ninst✝¹ : PartialOrder α\ninst✝ : LocallyFiniteOrderBot α\na : α\n⊢ f a * ∏ x ∈ Iio a, f x = ∏ x ∈ Iic a, f x",
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"Monoid.toMulOneClass",
"Finset.cons",
"congrArg",
... | rw [Iic_eq_cons_Iio, prod_cons] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.Disjointed | {
"line": 122,
"column": 2
} | {
"line": 153,
"column": 28
} | [
{
"pp": "α : Type u_1\nι : Type u_2\ninst✝² : GeneralizedBooleanAlgebra α\ninst✝¹ : PartialOrder ι\ninst✝ : LocallyFiniteOrderBot ι\nf : ι → α\n⊢ partialSups (disjointed f) = partialSups f",
"usedConstants": [
"partialSups_apply",
"partialSups._proof_2",
"Eq.mpr",
"Finset.mem_Iic._si... | suffices ∀ r i (hi : #(Iio i) ≤ r), partialSups (disjointed f) i = partialSups f i from
OrderHom.ext _ _ (funext fun i ↦ this _ i le_rfl)
intro r i hi
induction r generalizing i with
| zero =>
-- Base case: `n` is minimal, so `partialSups f i = partialSups (disjointed f) n = f i`.
simp only [Nat.le_ze... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.Disjointed | {
"line": 122,
"column": 2
} | {
"line": 153,
"column": 28
} | [
{
"pp": "α : Type u_1\nι : Type u_2\ninst✝² : GeneralizedBooleanAlgebra α\ninst✝¹ : PartialOrder ι\ninst✝ : LocallyFiniteOrderBot ι\nf : ι → α\n⊢ partialSups (disjointed f) = partialSups f",
"usedConstants": [
"partialSups_apply",
"partialSups._proof_2",
"Eq.mpr",
"Finset.mem_Iic._si... | suffices ∀ r i (hi : #(Iio i) ≤ r), partialSups (disjointed f) i = partialSups f i from
OrderHom.ext _ _ (funext fun i ↦ this _ i le_rfl)
intro r i hi
induction r generalizing i with
| zero =>
-- Base case: `n` is minimal, so `partialSups f i = partialSups (disjointed f) n = f i`.
simp only [Nat.le_ze... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.Disjointed | {
"line": 158,
"column": 2
} | {
"line": 162,
"column": 69
} | [
{
"pp": "α : Type u_1\nι : Type u_2\ninst✝³ : GeneralizedBooleanAlgebra α\ninst✝² : PartialOrder ι\ninst✝¹ : LocallyFiniteOrderBot ι\ninst✝ : Fintype ι\nf : ι → α\n⊢ univ.sup (disjointed f) = univ.sup f",
"usedConstants": [
"partialSups_apply",
"Eq.mpr",
"Lattice.toSemilatticeSup",
"... | have hun : univ.biUnion Iic = (univ : Finset ι) := by
ext r; simpa only [mem_biUnion, mem_univ, mem_Iic, true_and, iff_true] using ⟨r, le_rfl⟩
rw [← hun, sup_biUnion, sup_biUnion, sup_congr rfl (fun i _ ↦ ?_)]
rw [← sup'_eq_sup nonempty_Iic, ← sup'_eq_sup nonempty_Iic,
← partialSups_apply, ← partialSups_app... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.Disjointed | {
"line": 158,
"column": 2
} | {
"line": 162,
"column": 69
} | [
{
"pp": "α : Type u_1\nι : Type u_2\ninst✝³ : GeneralizedBooleanAlgebra α\ninst✝² : PartialOrder ι\ninst✝¹ : LocallyFiniteOrderBot ι\ninst✝ : Fintype ι\nf : ι → α\n⊢ univ.sup (disjointed f) = univ.sup f",
"usedConstants": [
"partialSups_apply",
"Eq.mpr",
"Lattice.toSemilatticeSup",
"... | have hun : univ.biUnion Iic = (univ : Finset ι) := by
ext r; simpa only [mem_biUnion, mem_univ, mem_Iic, true_and, iff_true] using ⟨r, le_rfl⟩
rw [← hun, sup_biUnion, sup_biUnion, sup_congr rfl (fun i _ ↦ ?_)]
rw [← sup'_eq_sup nonempty_Iic, ← sup'_eq_sup nonempty_Iic,
← partialSups_apply, ← partialSups_app... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.List.NatAntidiagonal | {
"line": 71,
"column": 26
} | {
"line": 71,
"column": 43
} | [
{
"pp": "n : ℕ\n⊢ map (Prod.map id Nat.succ) (antidiagonal (n + 1)) ++ [(n + 1 + 1, 0)] =\n (0, n + 2) :: map (Prod.map Nat.succ Nat.succ) (antidiagonal n) ++ [(n + 2, 0)]",
"usedConstants": [
"Eq.mpr",
"congrArg",
"List.map",
"id",
"Prod.mk",
"Prod.map",
"instOf... | antidiagonal_succ | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Finset.NatAntidiagonal | {
"line": 159,
"column": 36
} | {
"line": 159,
"column": 54
} | [
{
"pp": "n : ℕ\nx✝ : Fin (n + 1)\ni : ℕ\nh : i < n + 1\n⊢ (i, n - i).2 + (i, n - i).1 = n",
"usedConstants": [
"Eq.mpr",
"congrArg",
"HSub.hSub",
"id",
"Prod.mk",
"instSubNat",
"Prod.fst",
"instHAdd",
"instHSub",
"Nat.sub_add_cancel",
"HAdd.h... | Nat.sub_add_cancel | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Ideal.Quotient.Basic | {
"line": 112,
"column": 12
} | {
"line": 112,
"column": 44
} | [
{
"pp": "case h.mk\nR : Type u_3\ninst✝¹ : Ring R\nI : Ideal R\ninst✝ : I.IsTwoSided\nhI : I.IsMaximal\na✝ : R ⧸ I\na : R\nh : Quot.mk (⇑(Submodule.quotientRel I)) a ≠ 0\nb c : R\nhc : 1 - b * a ∈ I\nabc : c = 1 - b * a\n⊢ b * a - 1 ∈ I",
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
... | ← neg_mem_iff (G := R) (H := I), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.Filter.Bases.Basic | {
"line": 354,
"column": 2
} | {
"line": 357,
"column": 38
} | [
{
"pp": "α : Type u_1\nι : Sort u_4\nl : Filter α\np : ι → Prop\ns : ι → Set α\nh : l.HasBasis p s\nq : ι → Prop\nhq : ∀ (i : ι), p i → ∃ j, p j ∧ q j ∧ s j ⊆ s i\n⊢ l.HasBasis (fun i ↦ p i ∧ q i) s",
"usedConstants": [
"Filter.instMembership",
"Iff.mpr",
"_private.Mathlib.Order.Filter.Bas... | refine ⟨fun t => ⟨fun ht => ?_, fun ⟨i, hpi, hti⟩ => h.mem_iff.2 ⟨i, hpi.1, hti⟩⟩⟩
rcases h.mem_iff.1 ht with ⟨i, hpi, hti⟩
rcases hq i hpi with ⟨j, hpj, hqj, hji⟩
exact ⟨j, ⟨hpj, hqj⟩, hji.trans hti⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.Filter.Bases.Basic | {
"line": 354,
"column": 2
} | {
"line": 357,
"column": 38
} | [
{
"pp": "α : Type u_1\nι : Sort u_4\nl : Filter α\np : ι → Prop\ns : ι → Set α\nh : l.HasBasis p s\nq : ι → Prop\nhq : ∀ (i : ι), p i → ∃ j, p j ∧ q j ∧ s j ⊆ s i\n⊢ l.HasBasis (fun i ↦ p i ∧ q i) s",
"usedConstants": [
"Filter.instMembership",
"Iff.mpr",
"_private.Mathlib.Order.Filter.Bas... | refine ⟨fun t => ⟨fun ht => ?_, fun ⟨i, hpi, hti⟩ => h.mem_iff.2 ⟨i, hpi.1, hti⟩⟩⟩
rcases h.mem_iff.1 ht with ⟨i, hpi, hti⟩
rcases hq i hpi with ⟨j, hpj, hqj, hji⟩
exact ⟨j, ⟨hpj, hqj⟩, hji.trans hti⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.Filter.Map | {
"line": 435,
"column": 6
} | {
"line": 435,
"column": 15
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nf : α → β\nl : Filter α\n⊢ l ≤ comap f ⊤",
"usedConstants": [
"Eq.mpr",
"congrArg",
"PartialOrder.toPreorder",
"Preorder.toLE",
"id",
"LE.le",
"Filter.comap_top",
"Filter.instTop",
"Top.top",
"Eq",
"Filter... | comap_top | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.Interval.Set.Disjoint | {
"line": 218,
"column": 2
} | {
"line": 219,
"column": 37
} | [
{
"pp": "ι : Sort u\nR : Type u_1\ninst✝ : CompleteLinearOrder R\nf : ι → R\nhas_least_elem : ⨅ i, f i ∈ range f\n⊢ ⋃ i, Ici (f i) = Ici (⨅ i, f i)",
"usedConstants": [
"CompleteLinearOrder.toLinearOrder",
"iInf",
"Set.Ici",
"IsGLB.biUnion_Ici_eq_Ici",
"Iff.of_eq",
"congr... | simp only [← IsGLB.biUnion_Ici_eq_Ici (@isGLB_iInf _ _ _ f) has_least_elem, mem_range,
iUnion_exists, iUnion_iUnion_eq'] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Order.Interval.Set.Disjoint | {
"line": 218,
"column": 2
} | {
"line": 219,
"column": 37
} | [
{
"pp": "ι : Sort u\nR : Type u_1\ninst✝ : CompleteLinearOrder R\nf : ι → R\nhas_least_elem : ⨅ i, f i ∈ range f\n⊢ ⋃ i, Ici (f i) = Ici (⨅ i, f i)",
"usedConstants": [
"CompleteLinearOrder.toLinearOrder",
"iInf",
"Set.Ici",
"IsGLB.biUnion_Ici_eq_Ici",
"Iff.of_eq",
"congr... | simp only [← IsGLB.biUnion_Ici_eq_Ici (@isGLB_iInf _ _ _ f) has_least_elem, mem_range,
iUnion_exists, iUnion_iUnion_eq'] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.Interval.Set.Disjoint | {
"line": 218,
"column": 2
} | {
"line": 219,
"column": 37
} | [
{
"pp": "ι : Sort u\nR : Type u_1\ninst✝ : CompleteLinearOrder R\nf : ι → R\nhas_least_elem : ⨅ i, f i ∈ range f\n⊢ ⋃ i, Ici (f i) = Ici (⨅ i, f i)",
"usedConstants": [
"CompleteLinearOrder.toLinearOrder",
"iInf",
"Set.Ici",
"IsGLB.biUnion_Ici_eq_Ici",
"Iff.of_eq",
"congr... | simp only [← IsGLB.biUnion_Ici_eq_Ici (@isGLB_iInf _ _ _ f) has_least_elem, mem_range,
iUnion_exists, iUnion_iUnion_eq'] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.Filter.Bases.Basic | {
"line": 459,
"column": 61
} | {
"line": 459,
"column": 78
} | [
{
"pp": "α : Type u_1\nι : Sort u_4\nι' : Sort u_5\nl l' : Filter α\np : ι → Prop\ns : ι → Set α\np' : ι' → Prop\ns' : ι' → Set α\nhl : l.HasBasis p s\nhl' : l'.HasBasis p' s'\nt : Set α\n⊢ ((∃ i, p i ∧ s i ⊆ t) ∧ ∃ i, p' i ∧ s' i ⊆ t) ↔ ∃ a b, (p a ∧ p' b) ∧ s a ∪ s' b ⊆ t",
"usedConstants": [
"Eq.mp... | union_subset_iff, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Order.Filter.Bases.Basic | {
"line": 571,
"column": 44
} | {
"line": 571,
"column": 83
} | [
{
"pp": "α : Type u_1\nι : Sort u_4\nl : Filter α\np : ι → Prop\ns : ι → Set α\nh : l.HasBasis p s\nx✝ : Set α\n⊢ x✝ ∈ l ↔ ∃ i, p i ∧ x✝ ∈ 𝓟 (s i)",
"usedConstants": [
"Filter.instMembership",
"Filter.HasBasis.mem_iff",
"congrArg",
"Membership.mem",
"Exists",
"_private.M... | by simp only [h.mem_iff, mem_principal] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Order.Filter.AtTopBot.Tendsto | {
"line": 103,
"column": 2
} | {
"line": 103,
"column": 24
} | [
{
"pp": "α : Type u_3\nβ : Type u_4\nγ : Type u_5\ninst✝² : Preorder β\ninst✝¹ : Preorder γ\nl : Filter α\ninst✝ : l.NeBot\nf : β → γ\nhf : Monotone f\ng : α → β\nhg : Tendsto g l atTop\n⊢ upperBounds (range (f ∘ g)) ⊆ upperBounds (range f)",
"usedConstants": []
}
] | rintro c hc _ ⟨b, rfl⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
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