module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Combinatorics.SimpleGraph.Walk.Decomp | {
"line": 138,
"column": 4
} | {
"line": 138,
"column": 87
} | {
"line": 140,
"column": 0
} | [
{
"pp": "case mpr\nV : Type u\nG : SimpleGraph V\nu v w : V\nq : G.Walk u w\nr : G.Walk w v\n⊢ w ∈ (q.append r).support",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"_private.Mathlib.Combinatorics.SimpleGraph.Walk.Decomp.0.SimpleGraph.Walk.mem_support_iff_exists_append._simp_1_1",
... | [] | simp only [mem_support_append_iff, end_mem_support, start_mem_support, or_self_iff] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Combinatorics.SimpleGraph.Clique | {
"line": 177,
"column": 2
} | {
"line": 184,
"column": 29
} | {
"line": 186,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nG : SimpleGraph α\nf : α ↪ β\nt : Finset β\nht : t.Nontrivial\n⊢ (SimpleGraph.map (⇑f) G).IsClique ↑t ↔ ∃ s, G.IsClique ↑s ∧ map f s = t",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Function.Injective.injOn",
... | [] | constructor
· rw [isClique_map_iff_of_nontrivial (by simpa)]
rintro ⟨s, hs, hst⟩
obtain ⟨s, rfl⟩ := Set.Finite.exists_finset_coe <|
(show s.Finite from Set.Finite.of_finite_image (by simp [hst]) f.injective.injOn)
exact ⟨s,hs, Finset.coe_inj.1 (by simpa)⟩
rintro ⟨s, hs, rfl⟩
simpa using hs.map (... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SimpleGraph.Clique | {
"line": 177,
"column": 2
} | {
"line": 184,
"column": 29
} | {
"line": 186,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nG : SimpleGraph α\nf : α ↪ β\nt : Finset β\nht : t.Nontrivial\n⊢ (SimpleGraph.map (⇑f) G).IsClique ↑t ↔ ∃ s, G.IsClique ↑s ∧ map f s = t",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Function.Injective.injOn",
... | [] | constructor
· rw [isClique_map_iff_of_nontrivial (by simpa)]
rintro ⟨s, hs, hst⟩
obtain ⟨s, rfl⟩ := Set.Finite.exists_finset_coe <|
(show s.Finite from Set.Finite.of_finite_image (by simp [hst]) f.injective.injOn)
exact ⟨s,hs, Finset.coe_inj.1 (by simpa)⟩
rintro ⟨s, hs, rfl⟩
simpa using hs.map (... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SimpleGraph.Clique | {
"line": 230,
"column": 2
} | {
"line": 230,
"column": 29
} | {
"line": 231,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nG : SimpleGraph α\nf : ⊤.Copy G\n⊢ G.IsClique (Set.range ⇑f)",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"SimpleGraph.Adj",
"Membership.mem",
"SimpleGraph.Copy",
"Ne",
"_private.Mathlib.Combinatorics.SimpleGraph.Clique.0... | [
"α : Type u_1\nβ : Type u_2\nG : SimpleGraph α\nf : ⊤.Copy G\nx✝ : α\nw✝¹ : β\nh : f w✝¹ = x✝\ny✝ : α\nw✝ : β\nh' : f w✝ = y✝\nnh : x✝ ≠ y✝\n⊢ G.Adj x✝ y✝"
] | intro _ ⟨_, h⟩ _ ⟨_, h'⟩ nh | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Combinatorics.SimpleGraph.Paths | {
"line": 456,
"column": 2
} | {
"line": 474,
"column": 7
} | {
"line": 476,
"column": 0
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nu v : V\np : G.Walk u v\n⊢ Set.InjOn p.getVert {i | i ≤ p.length} → p.IsPath",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"False",
"SimpleGraph.Walk.getVert_cons",
"congrArg",
"SimpleGraph.Walk.length",
... | [] | induction p with
| nil => simp
| cons h q ih =>
intro hinj
rw [cons_isPath_iff]
refine ⟨ih (by
intro n hn m hm hnm
simp only [Set.mem_setOf_eq] at hn hm
have := hinj
(by rw [length_cons]; lia : n + 1 ≤ (q.cons h).length)
(by rw [length_cons]; lia : m + 1 ≤ (q.cons h).le... | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.Combinatorics.SimpleGraph.Clique | {
"line": 518,
"column": 15
} | {
"line": 518,
"column": 18
} | {
"line": 518,
"column": 19
} | [
{
"pp": "case pos\nα : Type u_1\nG : SimpleGraph α\nn : ℕ\ninst✝ : DecidableEq α\ns t : α\nh : ¬(G.replaceVertex s t).CliqueFree n\nφ : Fin n ↪ α\nhφ : ∀ {a b : Fin n}, (G.replaceVertex s t).Adj (φ a) (φ b) ↔ (completeGraph (Fin n)).Adj a b\nx : Fin n\nhx : φ x = t\ny : Fin n\nhy : φ y = s\ne : (G.replaceVertex... | [
"case pos\nα : Type u_1\nG : SimpleGraph α\nn : ℕ\ninst✝ : DecidableEq α\ns t : α\nh : ¬(G.replaceVertex s t).CliqueFree n\nφ : Fin n ↪ α\nhφ : ∀ {a b : Fin n}, (G.replaceVertex s t).Adj (φ a) (φ b) ↔ (completeGraph (Fin n)).Adj a b\nx : Fin n\nhx : φ x = t\ny : Fin n\nhy : φ y = s\ne : (G.replaceVertex s t).Adj t ... | hx, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Combinatorics.SimpleGraph.Clique | {
"line": 567,
"column": 2
} | {
"line": 567,
"column": 45
} | {
"line": 568,
"column": 2
} | [
{
"pp": "case right\nα : Type u_1\nG : SimpleGraph α\nn : ℕ\ninst✝ : DecidableEq α\nh : Maximal (fun H ↦ H.CliqueFree (n + 1)) G\nx y : α\nhne : x ≠ y\nhn : ¬G.Adj x y\nt : Finset α\nhc : (G ⊔ edge x y).IsNClique (n + 1) t\n⊢ G.IsNClique n (insert x ((t.erase x).erase y)) ∧ G.IsNClique n (insert y ((t.erase x).... | [
"case right\nα : Type u_1\nG : SimpleGraph α\nn : ℕ\ninst✝ : DecidableEq α\nh : Maximal (fun H ↦ H.CliqueFree (n + 1)) G\nx y : α\nhne : x ≠ y\nhn : ¬G.Adj x y\nt : Finset α\nhc : (G ⊔ edge x y).IsNClique (n + 1) t\nh1 : x ∈ t\n⊢ G.IsNClique n (insert x ((t.erase x).erase y)) ∧ G.IsNClique n (insert y ((t.erase x).... | have h1 := h.1.mem_of_sup_edge_isNClique hc | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Combinatorics.SimpleGraph.Clique | {
"line": 660,
"column": 2
} | {
"line": 660,
"column": 73
} | {
"line": 662,
"column": 0
} | [
{
"pp": "α : Type u_1\nG : SimpleGraph α\nn : ℕ\n⊢ G.cliqueSet n = ∅ ↔ G.CliqueFree n",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"SimpleGraph.IsNClique",
"congrArg",
"Finset",
"_private.Mathlib.Combinatorics.SimpleGraph.Clique.0.SimpleGraph.cliqu... | [] | simp_rw [CliqueFree, Set.eq_empty_iff_forall_notMem, mem_cliqueSet_iff] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Combinatorics.SimpleGraph.Clique | {
"line": 660,
"column": 2
} | {
"line": 660,
"column": 73
} | {
"line": 662,
"column": 0
} | [
{
"pp": "α : Type u_1\nG : SimpleGraph α\nn : ℕ\n⊢ G.cliqueSet n = ∅ ↔ G.CliqueFree n",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"SimpleGraph.IsNClique",
"congrArg",
"Finset",
"_private.Mathlib.Combinatorics.SimpleGraph.Clique.0.SimpleGraph.cliqu... | [] | simp_rw [CliqueFree, Set.eq_empty_iff_forall_notMem, mem_cliqueSet_iff] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SimpleGraph.Clique | {
"line": 660,
"column": 2
} | {
"line": 660,
"column": 73
} | {
"line": 662,
"column": 0
} | [
{
"pp": "α : Type u_1\nG : SimpleGraph α\nn : ℕ\n⊢ G.cliqueSet n = ∅ ↔ G.CliqueFree n",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"SimpleGraph.IsNClique",
"congrArg",
"Finset",
"_private.Mathlib.Combinatorics.SimpleGraph.Clique.0.SimpleGraph.cliqu... | [] | simp_rw [CliqueFree, Set.eq_empty_iff_forall_notMem, mem_cliqueSet_iff] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SimpleGraph.Triangle.Basic | {
"line": 248,
"column": 2
} | {
"line": 249,
"column": 83
} | {
"line": 251,
"column": 0
} | [
{
"pp": "α : Type u_1\n𝕜 : Type u_3\ninst✝⁶ : Field 𝕜\ninst✝⁵ : LinearOrder 𝕜\ninst✝⁴ : IsStrictOrderedRing 𝕜\nG : SimpleGraph α\nε : 𝕜\ninst✝³ : Fintype α\ninst✝² : DecidableRel G.Adj\ninst✝¹ : DecidableEq α\ntris : Finset (Finset α)\nhtris : tris ⊆ G.cliqueFinset 3\npd : (↑tris).Pairwise fun x y ↦ (↑x ∩ ... | [] | exact tris_big.trans
(Nat.cast_le.2 <| farFromTriangleFree_of_disjoint_triangles_aux htris pd hG hH) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Data.Finset.CastCard | {
"line": 44,
"column": 25
} | {
"line": 44,
"column": 36
} | {
"line": 44,
"column": 37
} | [
{
"pp": "α : Type u_1\nR : Type u_2\ns t : Finset α\ninst✝¹ : DecidableEq α\ninst✝ : AddGroupWithOne R\n⊢ ↑(#(s ∩ t)) + ↑(#(s ∪ t)) = ↑(#s) + ↑(#t)",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"AddMonoid.toAddSemigroup",
"Finset.instUnion",
"AddGroupWit... | [
"α : Type u_1\nR : Type u_2\ns t : Finset α\ninst✝¹ : DecidableEq α\ninst✝ : AddGroupWithOne R\n⊢ ↑(#(s ∩ t) + #(s ∪ t)) = ↑(#s) + ↑(#t)"
] | ← cast_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Finset.CastCard | {
"line": 47,
"column": 25
} | {
"line": 47,
"column": 36
} | {
"line": 47,
"column": 37
} | [
{
"pp": "α : Type u_1\nR : Type u_2\ns t : Finset α\ninst✝¹ : DecidableEq α\ninst✝ : AddGroupWithOne R\n⊢ ↑(#(s ∪ t)) + ↑(#(s ∩ t)) = ↑(#s) + ↑(#t)",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"AddMonoid.toAddSemigroup",
"Finset.instUnion",
"AddGroupWit... | [
"α : Type u_1\nR : Type u_2\ns t : Finset α\ninst✝¹ : DecidableEq α\ninst✝ : AddGroupWithOne R\n⊢ ↑(#(s ∪ t) + #(s ∩ t)) = ↑(#s) + ↑(#t)"
] | ← cast_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.SimpleGraph.Regularity.Chunk | {
"line": 514,
"column": 11
} | {
"line": 514,
"column": 14
} | {
"line": 514,
"column": 14
} | [
{
"pp": "α : Type u_1\ninst✝³ : Fintype α\ninst✝² : DecidableEq α\nP : Finpartition univ\nhP : P.IsEquipartition\nG : SimpleGraph α\ninst✝¹ : DecidableRel G.Adj\nε : ℝ\nU V : Finset α\ninst✝ : Nonempty α\nhPα : #P.parts * 16 ^ #P.parts ≤ Fintype.card α\nhPε : 100 ≤ 4 ^ #P.parts * ε ^ 5\nhU : U ∈ P.parts\nhV : V... | [
"α : Type u_1\ninst✝³ : Fintype α\ninst✝² : DecidableEq α\nP : Finpartition univ\nhP : P.IsEquipartition\nG : SimpleGraph α\ninst✝¹ : DecidableRel G.Adj\nε : ℝ\nU V : Finset α\ninst✝ : Nonempty α\nhPα : #P.parts * 16 ^ #P.parts ≤ Fintype.card α\nhPε : 100 ≤ 4 ^ #P.parts * ε ^ 5\nhU : U ∈ P.parts\nhV : V ∈ P.parts\n... | key | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.FieldTheory.ChevalleyWarning | {
"line": 90,
"column": 6
} | {
"line": 90,
"column": 82
} | {
"line": 91,
"column": 4
} | [
{
"pp": "K : Type u_1\nσ : Type u_2\ninst✝³ : Fintype K\ninst✝² : Field K\ninst✝¹ : Fintype σ\ninst✝ : DecidableEq σ\nf : MvPolynomial σ K\nh : f.totalDegree < (q - 1) * Fintype.card σ\nthis✝ : DecidableEq K\nd : σ →₀ ℕ\nhd : d ∈ f.support\ni : σ\nhi : d i < q - 1\nx₀ : { j // j ≠ i } → K\ne : K ≃ { x // x ∘ Su... | [] | rw [← e'.prod_comp, Fintype.prod_sum_type, univ_unique, prod_singleton]; rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.FieldTheory.ChevalleyWarning | {
"line": 90,
"column": 6
} | {
"line": 90,
"column": 82
} | {
"line": 91,
"column": 4
} | [
{
"pp": "K : Type u_1\nσ : Type u_2\ninst✝³ : Fintype K\ninst✝² : Field K\ninst✝¹ : Fintype σ\ninst✝ : DecidableEq σ\nf : MvPolynomial σ K\nh : f.totalDegree < (q - 1) * Fintype.card σ\nthis✝ : DecidableEq K\nd : σ →₀ ℕ\nhd : d ∈ f.support\ni : σ\nhi : d i < q - 1\nx₀ : { j // j ≠ i } → K\ne : K ≃ { x // x ∘ Su... | [] | rw [← e'.prod_comp, Fintype.prod_sum_type, univ_unique, prod_singleton]; rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.Colex | {
"line": 185,
"column": 29
} | {
"line": 185,
"column": 43
} | {
"line": 185,
"column": 44
} | [
{
"pp": "α : Type u_1\ninst✝ : PartialOrder α\ns : Finset α\na : α\n⊢ (a ∉ s → ∃ b ∈ s, ¬b = a ∧ a ≤ b) ↔ ∃ x ∈ s, a ≤ x",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Finset",
"PartialOrder.toPreorder",
"Classical.propDecidable",
"Preorder.toLE",
"Membershi... | [
"case pos\nα : Type u_1\ninst✝ : PartialOrder α\ns : Finset α\na : α\nh✝ : a ∈ s\n⊢ (a ∉ s → ∃ b ∈ s, ¬b = a ∧ a ≤ b) ↔ ∃ x ∈ s, a ≤ x",
"case neg\nα : Type u_1\ninst✝ : PartialOrder α\ns : Finset α\na : α\nh✝ : a ∉ s\n⊢ (a ∉ s → ∃ b ∈ s, ¬b = a ∧ a ≤ b) ↔ ∃ x ∈ s, a ≤ x"
] | by_cases a ∈ s | «_aux_Init_ByCases___macroRules_tacticBy_cases_:__1» | «tacticBy_cases_:_» |
Mathlib.LinearAlgebra.Projectivization.Basic | {
"line": 120,
"column": 4
} | {
"line": 121,
"column": 17
} | {
"line": 122,
"column": 2
} | [
{
"pp": "case mp\nK : Type u_1\nV : Type u_2\ninst✝² : DivisionRing K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nv w : V\nhv : v ≠ 0\nhw : w ≠ 0\n⊢ (∃ a, a • w = v) → ∃ a, a • w = v",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"Units.val",
"instHSMul",
"DistribMulActi... | [] | rintro ⟨a, ha⟩
exact ⟨a, ha⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Projectivization.Basic | {
"line": 120,
"column": 4
} | {
"line": 121,
"column": 17
} | {
"line": 122,
"column": 2
} | [
{
"pp": "case mp\nK : Type u_1\nV : Type u_2\ninst✝² : DivisionRing K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nv w : V\nhv : v ≠ 0\nhw : w ≠ 0\n⊢ (∃ a, a • w = v) → ∃ a, a • w = v",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"Units.val",
"instHSMul",
"DistribMulActi... | [] | rintro ⟨a, ha⟩
exact ⟨a, ha⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.CrossProduct | {
"line": 177,
"column": 2
} | {
"line": 182,
"column": 8
} | {
"line": 184,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nu v w : Fin 3 → R\n⊢ (crossProduct ((crossProduct u) v)) w = (u ⬝ᵥ w) • v - (v ⬝ᵥ w) • u",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Mathlib.Tactic.Ring.Common.neg_zero",
"Eq.mpr",
... | [] | simp_rw [cross_apply, vec3_dotProduct]
ext i
fin_cases i <;>
· simp only [Fin.isValue, Nat.succ_eq_add_one, Nat.reduceAdd, Fin.reduceFinMk, cons_val,
Pi.sub_apply, Pi.smul_apply, smul_eq_mul]
ring | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.CrossProduct | {
"line": 177,
"column": 2
} | {
"line": 182,
"column": 8
} | {
"line": 184,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nu v w : Fin 3 → R\n⊢ (crossProduct ((crossProduct u) v)) w = (u ⬝ᵥ w) • v - (v ⬝ᵥ w) • u",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Mathlib.Tactic.Ring.Common.neg_zero",
"Eq.mpr",
... | [] | simp_rw [cross_apply, vec3_dotProduct]
ext i
fin_cases i <;>
· simp only [Fin.isValue, Nat.succ_eq_add_one, Nat.reduceAdd, Fin.reduceFinMk, cons_val,
Pi.sub_apply, Pi.smul_apply, smul_eq_mul]
ring | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.Additive.VerySmallDoubling | {
"line": 67,
"column": 32
} | {
"line": 67,
"column": 46
} | {
"line": 67,
"column": 47
} | [
{
"pp": "case refine_2\nG : Type u_1\ninst✝¹ : Group G\ninst✝ : DecidableEq G\nA : Finset G\na✝ : G\nhA : #(A * A) ≤ #A\nha✝ : a✝ ∈ A\nsmul_A : ∀ {a : G}, a ∈ A → a •> A = A * A\nA_smul : ∀ {a : G}, a ∈ A → A <• a = A * A\nsmul_A_eq_A_smul : ∀ {a : G}, a ∈ A → a •> A = A <• a\nmul_mem_A_comm : ∀ {x a : G}, a ∈ ... | [
"case refine_2\nG : Type u_1\ninst✝¹ : Group G\ninst✝ : DecidableEq G\nA : Finset G\na✝ : G\nhA : #(A * A) ≤ #A\nha✝ : a✝ ∈ A\nsmul_A : ∀ {a : G}, a ∈ A → a •> A = A * A\nA_smul : ∀ {a : G}, a ∈ A → A <• a = A * A\nsmul_A_eq_A_smul : ∀ {a : G}, a ∈ A → a •> A = A <• a\nmul_mem_A_comm : ∀ {x a : G}, a ∈ A → (x * a ∈... | ← smul_eq_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.Additive.VerySmallDoubling | {
"line": 67,
"column": 74
} | {
"line": 67,
"column": 84
} | {
"line": 67,
"column": 84
} | [
{
"pp": "case refine_2\nG : Type u_1\ninst✝¹ : Group G\ninst✝ : DecidableEq G\nA : Finset G\na✝ : G\nhA : #(A * A) ≤ #A\nha✝ : a✝ ∈ A\nsmul_A : ∀ {a : G}, a ∈ A → a •> A = A * A\nA_smul : ∀ {a : G}, a ∈ A → A <• a = A * A\nsmul_A_eq_A_smul : ∀ {a : G}, a ∈ A → a •> A = A <• a\nmul_mem_A_comm : ∀ {x a : G}, a ∈ ... | [
"case refine_2\nG : Type u_1\ninst✝¹ : Group G\ninst✝ : DecidableEq G\nA : Finset G\na✝ : G\nhA : #(A * A) ≤ #A\nha✝ : a✝ ∈ A\nsmul_A : ∀ {a : G}, a ∈ A → a •> A = A * A\nA_smul : ∀ {a : G}, a ∈ A → A <• a = A * A\nsmul_A_eq_A_smul : ∀ {a : G}, a ∈ A → a •> A = A <• a\nmul_mem_A_comm : ∀ {x a : G}, a ∈ A → (x * a ∈... | inv_mem hx | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.Configuration | {
"line": 261,
"column": 6
} | {
"line": 261,
"column": 88
} | {
"line": 262,
"column": 4
} | [
{
"pp": "P : Type u_1\nL : Type u_2\ninst✝³ : Membership P L\ninst✝² : HasLines P L\ninst✝¹ : Fintype P\ninst✝ : Fintype L\nhPL : Fintype.card P = Fintype.card L\np : P\nl : L\nhpl : p ∉ l\nf : L → P\nhf1 : Function.Bijective f\nhf2 : ∀ (l : L), pointCount P l = lineCount L (f l)\ns : Finset (P × L) := {i | i.1... | [] | simp_rw [Finset.sum_const, Finset.card_univ, hPL, sum_lineCount_eq_sum_pointCount] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Combinatorics.Additive.VerySmallDoubling | {
"line": 231,
"column": 28
} | {
"line": 231,
"column": 72
} | {
"line": 231,
"column": 72
} | [
{
"pp": "G : Type u_1\ninst✝¹ : Group G\ninst✝ : DecidableEq G\nA : Finset G\na : G\nha : a ∈ A\n⊢ 1 ∈ a •> A⁻¹",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"MulOne.toOne",
"instHSMul",
"DivInvOneMonoid.toInvOneClass",
"Monoid.toMulOneClass",
"congrArg",
... | [] | by simp [← inv_smul_mem_iff, inv_mem_inv ha] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.Additive.VerySmallDoubling | {
"line": 242,
"column": 27
} | {
"line": 242,
"column": 71
} | {
"line": 242,
"column": 71
} | [
{
"pp": "G : Type u_1\ninst✝¹ : Group G\ninst✝ : DecidableEq G\nA : Finset G\nh : ↑(#(A * A)) < 3 / 2 * ↑(#A)\na : G\nha : a ∈ A\nh₁ : A⁻¹ * A * (A⁻¹ * A) = A⁻¹ * A\nh₂ : a •> ((A⁻¹ * A) <• a) = a •> (A⁻¹ * A) * (A⁻¹ * A) <• a\n⊢ 1 ∈ A⁻¹ <• a",
"ppTerm": "?m.209",
"assigned": true,
"usedConstants": ... | [] | by simp [← inv_smul_mem_iff, inv_mem_inv ha] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.Enumerative.Catalan.Basic | {
"line": 105,
"column": 4
} | {
"line": 105,
"column": 23
} | {
"line": 106,
"column": 2
} | [
{
"pp": "n : ℕ\nthis : ↑(catalan n) = ↑n.centralBinom / (↑n + 1)\nh : n + 1 ∣ n.centralBinom\n⊢ catalan n = n.centralBinom / (n + 1)",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Rat.instOfNat",
"Int.cast",
"Int.cast_natCast",
"catalan",
"Int.instDiv",
... | [] | exact mod_cast this | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Combinatorics.Enumerative.Bell | {
"line": 67,
"column": 74
} | {
"line": 68,
"column": 43
} | {
"line": 69,
"column": 6
} | [
{
"pp": "x : ℕ\nhx : x ≠ 0\nc : ℕ\n⊢ x ! ^ (c + 1) * (c + 1)! * ∏ j ∈ Finset.range (c + 1), (j * x + x - 1).choose (x - 1) =\n x ! * (c + 1) * x ! ^ c * c ! * ∏ j ∈ Finset.range (c + 1), (j * x + x - 1).choose (x - 1)",
"ppTerm": "?m.174",
"assigned": true,
"usedConstants": [
"Mathlib.Tacti... | [] | by
rw [factorial_succ, pow_succ]; ring | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.Enumerative.Catalan.Tree | {
"line": 83,
"column": 4
} | {
"line": 85,
"column": 67
} | {
"line": 86,
"column": 4
} | [
{
"pp": "case hi\nn : ℕ\nih : ∀ m ≤ n, #(treesOfNumNodesEq m) = catalan m\n⊢ ∑ u ∈ antidiagonal n, #(pairwiseNode (treesOfNumNodesEq u.1) (treesOfNumNodesEq u.2)) =\n ∑ ij ∈ antidiagonal n, catalan ij.1 * catalan ij.2",
"ppTerm": "?hi",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"... | [
"case hi\nn : ℕ\nih : ∀ m ≤ n, #(treesOfNumNodesEq m) = catalan m\n⊢ (↑(antidiagonal n)).PairwiseDisjoint fun ij ↦ pairwiseNode (treesOfNumNodesEq ij.1) (treesOfNumNodesEq ij.2)"
] | · apply sum_congr rfl
rintro ⟨i, j⟩ H
rw [card_map, card_product, ih _ (fst_le H), ih _ (snd_le H)] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Combinatorics.Enumerative.Bell | {
"line": 206,
"column": 57
} | {
"line": 207,
"column": 15
} | {
"line": 209,
"column": 0
} | [
{
"pp": "n : ℕ\n⊢ (n + 1).bell = ∑ i ∈ Finset.Iic n, n.choose i * (n - i).bell",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.choose",
"HMul.hMul",
"Nat.bell.eq_2",
"congrArg",
"HSub.hSub",
"Nat.instLocallyFiniteOrder",
"Local... | [] | by
rw [Nat.bell] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.Additive.VerySmallDoubling | {
"line": 464,
"column": 16
} | {
"line": 464,
"column": 66
} | {
"line": 465,
"column": 6
} | [
{
"pp": "case hbc\nG : Type u_1\ninst✝¹ : Group G\ninst✝ : DecidableEq G\nK : ℝ\nA : Finset G\nhK₁ : 1 < K\nhKφ : K < φ\nhA₁ : ↑(#(A⁻¹ * A)) ≤ K * ↑(#A)\nhA₂ : ↑(#(A * A⁻¹)) ≤ K * ↑(#A)\nK_pos hK₀ : 0 < K\nhKφ' : 0 < φ - K\nhKψ' : 0 < K - ψ\nhK₂' : 0 < 2 - K\nconst_pos : 0 < K * (2 - K) / ((φ - K) * (K - ψ))\nA... | [] | exact le_of_mul_le_mul_right ineq <| by positivity | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Combinatorics.Enumerative.DyckWord | {
"line": 353,
"column": 11
} | {
"line": 353,
"column": 22
} | {
"line": 353,
"column": 23
} | [
{
"pp": "p q : DyckWord\nh : p ≠ 0\n⊢ (p + q).insidePart = p.insidePart",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"instAddDyckWord",
"DyckWord",
"id",
"instHAdd",
"HAdd.hAdd",
"DyckWord.insidePart",
"Eq"
],
"usedFVars": [
"p",
... | [
"p q : DyckWord\nh : p ≠ 0\n⊢ (if h : p + q = 0 then 0 else ((p + q).take ((p + q).firstReturn + 1) ⋯).denest ⋯) =\n if h : p = 0 then 0 else (p.take (p.firstReturn + 1) ⋯).denest ⋯"
] | insidePart, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Combinatorics.Enumerative.DyckWord | {
"line": 367,
"column": 11
} | {
"line": 367,
"column": 22
} | {
"line": 367,
"column": 23
} | [
{
"pp": "p : DyckWord\n⊢ p.nest.insidePart = p",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"DyckWord",
"id",
"DyckWord.nest",
"DyckWord.insidePart",
"Eq"
],
"usedFVars": [
"p"
],
"usedGoals": [
{
"new": true,
"ind... | [
"p : DyckWord\n⊢ (if h : p.nest = 0 then 0 else (p.nest.take (p.nest.firstReturn + 1) ⋯).denest ⋯) = p"
] | insidePart, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Combinatorics.Enumerative.DyckWord | {
"line": 380,
"column": 11
} | {
"line": 380,
"column": 22
} | {
"line": 380,
"column": 23
} | [
{
"pp": "p : DyckWord\nh : p ≠ 0\n⊢ p.insidePart.nest + p.outsidePart = p",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"instAddDyckWord",
"DyckWord",
"id",
"DyckWord.outsidePart",
"instHAdd",
"DyckWord.nest",
"HAdd.hAdd",
"DyckWord.insidePa... | [
"p : DyckWord\nh : p ≠ 0\n⊢ (if h : p = 0 then 0 else (p.take (p.firstReturn + 1) ⋯).denest ⋯).nest + p.outsidePart = p"
] | insidePart, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Combinatorics.Enumerative.IncidenceAlgebra | {
"line": 554,
"column": 73
} | {
"line": 555,
"column": 34
} | {
"line": 556,
"column": 4
} | [
{
"pp": "𝕜 : Type u_2\nα : Type u_5\ninst✝⁴ : Ring 𝕜\ninst✝³ : PartialOrder α\ninst✝² : OrderTop α\ninst✝¹ : LocallyFiniteOrder α\ninst✝ : DecidableEq α\nf g : α → 𝕜\nh : ∀ (x : α), g x = ∑ y ∈ Ici x, f y\nx : α\nthis : DecidableLE α := Classical.decRel LE.le\n⊢ ∑ z ∈ Ici x, ∑ y ∈ Icc x z, (mu 𝕜) x y * (zet... | [] | by
simp_rw [mul_apply, sum_mul] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.Graph.Subgraph | {
"line": 105,
"column": 2
} | {
"line": 107,
"column": 25
} | {
"line": 109,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nG H : Graph α β\nhHG : H ≤ G\n⊢ EqOn H.IsLink G.IsLink E(H)",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Graph.IsSubgraph.isLink_iff",
"Membership.mem",
"funext",
"Graph.IsLink",
"Graph.edgeSet",
"propext",
... | [] | rintro e he
ext x y
exact isLink_iff hHG he | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.Graph.Subgraph | {
"line": 105,
"column": 2
} | {
"line": 107,
"column": 25
} | {
"line": 109,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nG H : Graph α β\nhHG : H ≤ G\n⊢ EqOn H.IsLink G.IsLink E(H)",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Graph.IsSubgraph.isLink_iff",
"Membership.mem",
"funext",
"Graph.IsLink",
"Graph.edgeSet",
"propext",
... | [] | rintro e he
ext x y
exact isLink_iff hHG he | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.Graph.Delete | {
"line": 144,
"column": 2
} | {
"line": 147,
"column": 64
} | {
"line": 149,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nG : Graph α β\nF₁ F₂ : Set β\n⊢ (G.deleteEdges F₁).deleteEdges F₂ = G.deleteEdges (F₁ ∪ F₂)",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Set.compl_union",
"congrArg",
"Compl.compl",
"Set.inter_self",
"Se... | [] | simp only [← restrict_edgeSet_sdiff_eq_deleteEdges, sdiff_eq_compl_inter, restrict_inter_edgeSet,
edgeSet_restrict, restrict_restrict, compl_union]
rw [← inter_comm, inter_comm F₁ᶜ, inter_assoc, inter_assoc, inter_self, inter_comm,
inter_assoc, inter_comm, restrict_inter_edgeSet, inter_comm] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.Graph.Delete | {
"line": 144,
"column": 2
} | {
"line": 147,
"column": 64
} | {
"line": 149,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nG : Graph α β\nF₁ F₂ : Set β\n⊢ (G.deleteEdges F₁).deleteEdges F₂ = G.deleteEdges (F₁ ∪ F₂)",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Set.compl_union",
"congrArg",
"Compl.compl",
"Set.inter_self",
"Se... | [] | simp only [← restrict_edgeSet_sdiff_eq_deleteEdges, sdiff_eq_compl_inter, restrict_inter_edgeSet,
edgeSet_restrict, restrict_restrict, compl_union]
rw [← inter_comm, inter_comm F₁ᶜ, inter_assoc, inter_assoc, inter_self, inter_comm,
inter_assoc, inter_comm, restrict_inter_edgeSet, inter_comm] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.Graph.Delete | {
"line": 170,
"column": 2
} | {
"line": 173,
"column": 42
} | {
"line": 175,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nG : Graph α β\n⊢ G.induce V(G) = G",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Graph.IsLink.edge_mem",
"Set.ext",
"Graph.exists_isLink_of_mem_edgeSet",
"instReflLe",
"PartialOrder.toPreorder",
"Graph.IsLink.right_... | [] | refine (Compatible.of_le_le (G := G) (by simp) (by simp)).ext rfl <| Set.ext fun e ↦
⟨fun ⟨_, _, h⟩ ↦ h.1.edge_mem, fun h ↦ ?_⟩
obtain ⟨x, y, h⟩ := exists_isLink_of_mem_edgeSet h
exact ⟨x, y, h, h.left_mem, h.right_mem⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.Graph.Delete | {
"line": 170,
"column": 2
} | {
"line": 173,
"column": 42
} | {
"line": 175,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nG : Graph α β\n⊢ G.induce V(G) = G",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Graph.IsLink.edge_mem",
"Set.ext",
"Graph.exists_isLink_of_mem_edgeSet",
"instReflLe",
"PartialOrder.toPreorder",
"Graph.IsLink.right_... | [] | refine (Compatible.of_le_le (G := G) (by simp) (by simp)).ext rfl <| Set.ext fun e ↦
⟨fun ⟨_, _, h⟩ ↦ h.1.edge_mem, fun h ↦ ?_⟩
obtain ⟨x, y, h⟩ := exists_isLink_of_mem_edgeSet h
exact ⟨x, y, h, h.left_mem, h.right_mem⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.Graph.Maps | {
"line": 102,
"column": 2
} | {
"line": 102,
"column": 98
} | {
"line": 103,
"column": 2
} | [
{
"pp": "α : Type u_1\nα' : Type u_2\nβ : Type u_4\nG : Graph α β\nf g : α → α'\nh : EqOn f g V(G)\n⊢ map f G = map g G",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"_private.Mathlib.Combinatorics.Graph.Maps.0.Graph.map_eq_of_eqOn._proof_1_3",
"Graph.IsLink",
"Iff.intr... | [
"case refine_1\nα : Type u_1\nα' : Type u_2\nβ : Type u_4\nG : Graph α β\nf g : α → α'\nh : EqOn f g V(G)\nx✝³ : β\nx✝² x✝¹ : α'\nx✝ : (map f G).IsLink x✝³ x✝² x✝¹\nw✝¹ w✝ : α\nhvw : G.IsLink x✝³ w✝¹ w✝\nleft✝ : f w✝¹ = x✝²\nright✝ : f w✝ = x✝¹\n⊢ (map g G).IsLink x✝³ x✝² x✝¹",
"case refine_2\nα : Type u_1\nα' : ... | refine Graph.ext (by grind) fun _ _ _ ↦ ⟨fun ⟨_, _, hvw, _, _⟩ ↦ ?_, fun ⟨_, _, hvw, _, _⟩ ↦ ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Combinatorics.Graph.Basic | {
"line": 169,
"column": 2
} | {
"line": 171,
"column": 31
} | {
"line": 173,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nx y : α\ne : β\nG : Graph α β\nx' y' : α\nh : G.IsLink e x y\nh' : G.IsLink e x' y'\n⊢ x = x' ∧ y = y' ∨ x = y' ∧ y = x'",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"congrArg",
"and_self",
"true_or",
"Or.casesOn",
"Graph... | [] | obtain rfl | rfl := h.left_eq_or_eq h'
· simp [h.right_unique h']
simp [h'.symm.right_unique h] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.Graph.Basic | {
"line": 169,
"column": 2
} | {
"line": 171,
"column": 31
} | {
"line": 173,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nx y : α\ne : β\nG : Graph α β\nx' y' : α\nh : G.IsLink e x y\nh' : G.IsLink e x' y'\n⊢ x = x' ∧ y = y' ∨ x = y' ∧ y = x'",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"congrArg",
"and_self",
"true_or",
"Or.casesOn",
"Graph... | [] | obtain rfl | rfl := h.left_eq_or_eq h'
· simp [h.right_unique h']
simp [h'.symm.right_unique h] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.Hypergraph.Basic | {
"line": 226,
"column": 2
} | {
"line": 226,
"column": 16
} | {
"line": 226,
"column": 17
} | [
{
"pp": "case inl\nα : Type u_1\nH : Hypergraph α\nempty : V(H) = ∅ ∧ E(H) = ∅\n⊢ H = ⊥ ∨ H.IsNonempty",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Hypergraph.edgeSet",
"Hypergraph.vertexSet",
"Hypergraph.IsNonempty",
"Hypergraph.ext",
"Bot.bot",
"And... | [] | | inl empty => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalCases | null |
Mathlib.Combinatorics.Matroid.Minor.Delete | {
"line": 190,
"column": 2
} | {
"line": 190,
"column": 27
} | {
"line": 192,
"column": 0
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nD : Set α\ninst✝ : M.RankPos\nhD : M.Coindep D\n⊢ ¬(M.IsBase ∅ ∧ Disjoint ∅ D)",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"False",
"ChainCompletePartialOrder.instOfCompleteLattice",
"eq_false",
"and_true",
"CompleteBo... | [] | simp [M.empty_not_isBase] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Combinatorics.KatonaCircle | {
"line": 88,
"column": 10
} | {
"line": 88,
"column": 25
} | {
"line": 89,
"column": 8
} | [
{
"pp": "case pos\nX : Type u_1\ninst✝¹ : Fintype X\nf : Numbering X\ns✝ t : Finset X\ninst✝ : DecidableEq X\ns : Finset X\nx✝ : Numbering ↥s × Numbering ↥sᶜ\ng : Numbering ↥s\ng' : Numbering ↥sᶜ\nx : X\nhx : x ∈ s\nthis : ↑(g ⟨x, hx⟩) < #s\n⊢ (fun n ↦ if hn : ↑n < #s then ↑((Equiv.symm g) ⟨↑n, ⋯⟩) else ↑((Equi... | [] | simp [hx, this] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Combinatorics.Matroid.Minor.Contract | {
"line": 198,
"column": 4
} | {
"line": 201,
"column": 58
} | {
"line": 202,
"column": 2
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nI J X : Set α\nhIX : M.IsBasis' I X\nhJI : J ⊆ I\nthis : ∀ ⦃K : Set α⦄, Disjoint K J → M.Indep (K ∪ J) → K ⊆ X → I ⊆ K ∪ J → K ⊆ I\n⊢ (M ／ J).IsBasis' (I \\ J) (X \\ J)",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"ChainComplet... | [] | simpa +contextual [IsBasis', (hIX.indep.subset hJI).contract_indep_iff,
subset_sdiff, maximal_subset_iff, disjoint_sdiff_left,
union_eq_self_of_subset_right hJI, hIX.indep, sdiff_subset.trans hIX.subset,
sdiff_subset_iff, subset_antisymm_iff, union_comm J] | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Combinatorics.Matroid.Minor.Contract | {
"line": 198,
"column": 4
} | {
"line": 201,
"column": 58
} | {
"line": 202,
"column": 2
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nI J X : Set α\nhIX : M.IsBasis' I X\nhJI : J ⊆ I\nthis : ∀ ⦃K : Set α⦄, Disjoint K J → M.Indep (K ∪ J) → K ⊆ X → I ⊆ K ∪ J → K ⊆ I\n⊢ (M ／ J).IsBasis' (I \\ J) (X \\ J)",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"ChainComplet... | [] | simpa +contextual [IsBasis', (hIX.indep.subset hJI).contract_indep_iff,
subset_sdiff, maximal_subset_iff, disjoint_sdiff_left,
union_eq_self_of_subset_right hJI, hIX.indep, sdiff_subset.trans hIX.subset,
sdiff_subset_iff, subset_antisymm_iff, union_comm J] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.Matroid.Minor.Contract | {
"line": 198,
"column": 4
} | {
"line": 201,
"column": 58
} | {
"line": 202,
"column": 2
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nI J X : Set α\nhIX : M.IsBasis' I X\nhJI : J ⊆ I\nthis : ∀ ⦃K : Set α⦄, Disjoint K J → M.Indep (K ∪ J) → K ⊆ X → I ⊆ K ∪ J → K ⊆ I\n⊢ (M ／ J).IsBasis' (I \\ J) (X \\ J)",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"ChainComplet... | [] | simpa +contextual [IsBasis', (hIX.indep.subset hJI).contract_indep_iff,
subset_sdiff, maximal_subset_iff, disjoint_sdiff_left,
union_eq_self_of_subset_right hJI, hIX.indep, sdiff_subset.trans hIX.subset,
sdiff_subset_iff, subset_antisymm_iff, union_comm J] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.Matroid.Sum | {
"line": 300,
"column": 2
} | {
"line": 300,
"column": 79
} | {
"line": 301,
"column": 2
} | [
{
"pp": "α : Type u_1\nM N : Matroid α\nh : Disjoint M.E N.E\nI : Set α\nhI : M.Indep (I ∩ M.E) ∧ N.Indep (I ∩ N.E) ∧ I ⊆ M.E ∪ N.E\n⊢ ∃ IM IN, M.Indep IM ∧ N.Indep IN ∧ Disjoint IM IN ∧ I = IM ∪ IN",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"ChainCompletePartialOrder.instOfComp... | [
"α : Type u_1\nM N : Matroid α\nh : Disjoint M.E N.E\nI : Set α\nhI : M.Indep (I ∩ M.E) ∧ N.Indep (I ∩ N.E) ∧ I ⊆ M.E ∪ N.E\n⊢ I = I ∩ M.E ∪ I ∩ N.E"
] | refine ⟨_, _, hI.1, hI.2.1, h.mono inter_subset_right inter_subset_right, ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Combinatorics.Quiver.Path.Vertices | {
"line": 238,
"column": 4
} | {
"line": 243,
"column": 44
} | {
"line": 245,
"column": 0
} | [
{
"pp": "case cons\nV : Type u_1\ninst✝ : Quiver V\na b : V\np : Path a b\nv b✝ c✝ : V\npPrev : Path a b✝\ne : b✝ ⟶ c✝\nih : v ∈ pPrev.vertices → ∃ p₁ p₂, pPrev = p₁.comp p₂ ∧ ¬v ∈ p₂.vertices.tail\nhv : v ∈ (pPrev.cons e).vertices\nhv' : v ∈ pPrev.vertices ∨ v = (pPrev.cons e).end\nh_case₁ : v = (pPrev.cons e)... | [] | cases hv' with
| inl h_in_prefix =>
by_cases h_eq_end : v = (pPrev.cons e).end
· exact h_case₁ h_eq_end
· exact h_case₂ h_in_prefix h_eq_end
| inr h_eq_end => exact h_case₁ h_eq_end | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalCases | Lean.Parser.Tactic.cases |
Mathlib.Combinatorics.Schnirelmann | {
"line": 77,
"column": 2
} | {
"line": 77,
"column": 71
} | {
"line": 79,
"column": 0
} | [
{
"pp": "case inr\nA : Set ℕ\ninst✝ : DecidablePred fun x ↦ x ∈ A\nn : ℕ\nhn : n ≠ 0\n⊢ schnirelmannDensity A * ↑n ≤ ↑(#({a ∈ Ioc 0 n | a ∈ A}))",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"zero_le",
"GroupWithZero.toMonoidWithZero",
"Nat.instCanonical... | [] | exact (le_div_iff₀ (by positivity)).1 (schnirelmannDensity_le_div hn) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Data.Finset.Sups | {
"line": 600,
"column": 6
} | {
"line": 600,
"column": 15
} | {
"line": 600,
"column": 16
} | [
{
"pp": "α : Type u_2\ninst✝ : BooleanAlgebra α\ns : Finset α\na : α\n⊢ a ∈ sᶜˢ ↔ aᶜ ∈ s",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Compl.compl",
"Finset",
"Finset.compls",
"Membership.mem",
"BooleanAlgebra.toCompl",
... | [
"α : Type u_2\ninst✝ : BooleanAlgebra α\ns : Finset α\na : α\n⊢ aᶜ ∈ s ↔ a ∈ sᶜˢ"
] | Iff.comm, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.SetFamily.Shadow | {
"line": 119,
"column": 2
} | {
"line": 119,
"column": 21
} | {
"line": 121,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nt✝ t : Finset α\nx✝ : t ∈ 𝒜\nhst : t✝ ⊆ t\n⊢ #t✝ ≤ #t",
"ppTerm": "?m.62",
"assigned": true,
"usedConstants": [
"Finset.card_mono"
],
"usedFVars": [
"α",
"t✝",
"t",
"hst"
],
"usedGoal... | [] | exact card_mono hst | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Combinatorics.SetFamily.Shadow | {
"line": 149,
"column": 2
} | {
"line": 149,
"column": 21
} | {
"line": 151,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nt✝ : Finset α\nk : ℕ\nt : Finset α\nx✝ : t ∈ 𝒜\nhst : t✝ ⊆ t\n⊢ #t✝ ≤ #t",
"ppTerm": "?m.62",
"assigned": true,
"usedConstants": [
"Finset.card_mono"
],
"usedFVars": [
"α",
"t✝",
"t",
"hst"
... | [] | exact card_mono hst | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Combinatorics.SetFamily.Shadow | {
"line": 230,
"column": 2
} | {
"line": 230,
"column": 21
} | {
"line": 232,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\n𝒜 : Finset (Finset α)\nt✝ t : Finset α\nx✝ : t ∈ 𝒜\nhst : t ⊆ t✝\n⊢ #t ≤ #t✝",
"ppTerm": "?m.64",
"assigned": true,
"usedConstants": [
"Finset.card_mono"
],
"usedFVars": [
"α",
"t",
"t✝",
"hst"
... | [] | exact card_mono hst | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Combinatorics.Schnirelmann | {
"line": 308,
"column": 2
} | {
"line": 308,
"column": 84
} | {
"line": 309,
"column": 0
} | [
{
"pp": "case neg.inl.inl\nA B : Set ℕ\ninst✝¹ : DecidablePred fun x ↦ x ∈ A\ninst✝ : DecidablePred fun x ↦ x ∈ B\nhA : 0 ∈ A\nhB : 0 ∈ B\nh : 1 ≤ schnirelmannDensity A + schnirelmannDensity B\nm : ℕ\nn : ℕ := m + 1\nhnA : n ∉ A\nhnB : n ∉ B\nf : ℕ ⊕ ℕ → ℕ :=\n fun x ↦\n match x with\n | Sum.inl x => x\n... | [] | first | grind [inr_mem_disjSum] | exact ⟨a, by simp [*], b, by simp [*], by grind⟩ | Lean.Elab.Tactic.evalFirst | Lean.Parser.Tactic.first |
Mathlib.Combinatorics.SetFamily.Compression.UV | {
"line": 137,
"column": 2
} | {
"line": 137,
"column": 35
} | {
"line": 138,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝³ : GeneralizedBooleanAlgebra α\ninst✝² : DecidableRel Disjoint\ninst✝¹ : DecidableLE α\ns : Finset α\nu v : α\ninst✝ : DecidableEq α\na : α\nha : a ∈ ↑({a ∈ s | compress u v a ∉ s})\nb : α\nhb : b ∈ ↑({a ∈ s | compress u v a ∉ s})\nhab : compress u v a = compress u v b\n⊢ a = b",
... | [
"α : Type u_1\ninst✝³ : GeneralizedBooleanAlgebra α\ninst✝² : DecidableRel Disjoint\ninst✝¹ : DecidableLE α\ns : Finset α\nu v : α\ninst✝ : DecidableEq α\na : α\nha : a ∈ s ∧ compress u v a ∉ s\nb : α\nhb : b ∈ s ∧ compress u v b ∉ s\nhab : compress u v a = compress u v b\n⊢ a = b"
] | rw [mem_coe, mem_filter] at ha hb | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Combinatorics.Schnirelmann | {
"line": 308,
"column": 2
} | {
"line": 308,
"column": 84
} | {
"line": 309,
"column": 0
} | [
{
"pp": "case neg.inl.inr\nA B : Set ℕ\ninst✝¹ : DecidablePred fun x ↦ x ∈ A\ninst✝ : DecidablePred fun x ↦ x ∈ B\nhA : 0 ∈ A\nhB : 0 ∈ B\nh : 1 ≤ schnirelmannDensity A + schnirelmannDensity B\nm : ℕ\nn : ℕ := m + 1\nhnA : n ∉ A\nhnB : n ∉ B\nf : ℕ ⊕ ℕ → ℕ :=\n fun x ↦\n match x with\n | Sum.inl x => x\n... | [] | first | grind [inr_mem_disjSum] | exact ⟨a, by simp [*], b, by simp [*], by grind⟩ | Lean.Elab.Tactic.evalFirst | Lean.Parser.Tactic.first |
Mathlib.Combinatorics.SetFamily.Shadow | {
"line": 265,
"column": 2
} | {
"line": 265,
"column": 21
} | {
"line": 267,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\n𝒜 : Finset (Finset α)\nt✝ : Finset α\nk : ℕ\nt : Finset α\nx✝ : t ∈ 𝒜\nhst : t ⊆ t✝\n⊢ #t ≤ #t✝",
"ppTerm": "?m.64",
"assigned": true,
"usedConstants": [
"Finset.card_mono"
],
"usedFVars": [
"α",
"t",
... | [] | exact card_mono hst | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Combinatorics.Schnirelmann | {
"line": 308,
"column": 2
} | {
"line": 308,
"column": 84
} | {
"line": 309,
"column": 0
} | [
{
"pp": "case neg.inr.inl\nA B : Set ℕ\ninst✝¹ : DecidablePred fun x ↦ x ∈ A\ninst✝ : DecidablePred fun x ↦ x ∈ B\nhA : 0 ∈ A\nhB : 0 ∈ B\nh : 1 ≤ schnirelmannDensity A + schnirelmannDensity B\nm : ℕ\nn : ℕ := m + 1\nhnA : n ∉ A\nhnB : n ∉ B\nf : ℕ ⊕ ℕ → ℕ :=\n fun x ↦\n match x with\n | Sum.inl x => x\n... | [] | first | grind [inr_mem_disjSum] | exact ⟨a, by simp [*], b, by simp [*], by grind⟩ | Lean.Elab.Tactic.evalFirst | Lean.Parser.Tactic.first |
Mathlib.Combinatorics.Schnirelmann | {
"line": 308,
"column": 2
} | {
"line": 308,
"column": 84
} | {
"line": 309,
"column": 0
} | [
{
"pp": "case neg.inr.inr\nA B : Set ℕ\ninst✝¹ : DecidablePred fun x ↦ x ∈ A\ninst✝ : DecidablePred fun x ↦ x ∈ B\nhA : 0 ∈ A\nhB : 0 ∈ B\nh : 1 ≤ schnirelmannDensity A + schnirelmannDensity B\nm : ℕ\nn : ℕ := m + 1\nhnA : n ∉ A\nhnB : n ∉ B\nf : ℕ ⊕ ℕ → ℕ :=\n fun x ↦\n match x with\n | Sum.inl x => x\n... | [] | first | grind [inr_mem_disjSum] | exact ⟨a, by simp [*], b, by simp [*], by grind⟩ | Lean.Elab.Tactic.evalFirst | Lean.Parser.Tactic.first |
Mathlib.Order.Irreducible | {
"line": 94,
"column": 53
} | {
"line": 94,
"column": 93
} | {
"line": 96,
"column": 0
} | [
{
"pp": "α : Type u_2\ninst✝¹ : SemilatticeSup α\na : α\ninst✝ : OrderBot α\nha : SupIrred a\n⊢ a ≠ ⊥",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"False",
"OrderBot.toBot",
"PartialOrder.toPreorder",
"Preorder.toLE",
"not_supIrred_bot",
"Bot.bot",
... | [] | by rintro rfl; exact not_supIrred_bot ha | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.SetFamily.Compression.UV | {
"line": 184,
"column": 2
} | {
"line": 187,
"column": 56
} | {
"line": 189,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝³ : GeneralizedBooleanAlgebra α\ninst✝² : DecidableRel Disjoint\ninst✝¹ : DecidableLE α\ninst✝ : DecidableEq α\nu v : α\ns : Finset α\n⊢ 𝓒 u v (𝓒 u v s) = 𝓒 u v s",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instDecidableNot",
... | [] | have h : {a ∈ 𝓒 u v s | compress u v a ∉ 𝓒 u v s} = ∅ :=
filter_false_of_mem fun a ha h ↦ h <| compress_mem_compression_of_mem_compression ha
rw [compression, filter_image, h, image_empty, ← h]
exact filter_union_filter_not_eq _ (compression u v s) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SetFamily.Compression.UV | {
"line": 184,
"column": 2
} | {
"line": 187,
"column": 56
} | {
"line": 189,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝³ : GeneralizedBooleanAlgebra α\ninst✝² : DecidableRel Disjoint\ninst✝¹ : DecidableLE α\ninst✝ : DecidableEq α\nu v : α\ns : Finset α\n⊢ 𝓒 u v (𝓒 u v s) = 𝓒 u v s",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instDecidableNot",
... | [] | have h : {a ∈ 𝓒 u v s | compress u v a ∉ 𝓒 u v s} = ∅ :=
filter_false_of_mem fun a ha h ↦ h <| compress_mem_compression_of_mem_compression ha
rw [compression, filter_image, h, image_empty, ← h]
exact filter_union_filter_not_eq _ (compression u v s) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SetFamily.AhlswedeZhang | {
"line": 397,
"column": 6
} | {
"line": 397,
"column": 20
} | {
"line": 397,
"column": 21
} | [
{
"pp": "α : Type u_1\ninst✝² : Fintype α\ninst✝¹ : DecidableEq α\ns : Finset α\ninst✝ : Nonempty α\nhs : s ≠ univ\nthis :\n ∀ (t : Finset α),\n (↑(card α) - ↑(#({s}.truncatedSup t))) / ((↑(card α) - ↑(#t)) * ↑((card α).choose #t)) =\n if t ⊆ s then (↑(card α) - ↑(#s)) / ((↑(card α) - ↑(#t)) * ↑((card ... | [
"α : Type u_1\ninst✝² : Fintype α\ninst✝¹ : DecidableEq α\ns : Finset α\ninst✝ : Nonempty α\nhs : s ≠ univ\nthis :\n ∀ (t : Finset α),\n (↑(card α) - ↑(#({s}.truncatedSup t))) / ((↑(card α) - ↑(#t)) * ↑((card α).choose #t)) =\n if t ⊆ s then (↑(card α) - ↑(#s)) / ((↑(card α) - ↑(#t)) * ↑((card α).choose #t... | mul_div_assoc, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.SetFamily.Compression.UV | {
"line": 297,
"column": 4
} | {
"line": 297,
"column": 30
} | {
"line": 298,
"column": 4
} | [
{
"pp": "α : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu v : Finset α\nhuv : ∀ x ∈ u, ∃ y ∈ v, IsCompressed (u.erase x) (v.erase y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\nH : ∀ s ∈ ∂ 𝒜', s ∉ ∂ 𝒜 → u ⊆ s ∧ Disjoint v s ∧ (s ∪ v) \\ u ∈ ∂ 𝒜 ∧ (s ∪ v) \\ u ∉ ∂ 𝒜'\ns : Finset α\nhs' : s ∈ ∂ 𝒜... | [
"case pos\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu v : Finset α\nhuv : ∀ x ∈ u, ∃ y ∈ v, IsCompressed (u.erase x) (v.erase y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\nH : ∀ s ∈ ∂ 𝒜', s ∉ ∂ 𝒜 → u ⊆ s ∧ Disjoint v s ∧ (s ∪ v) \\ u ∈ ∂ 𝒜 ∧ (s ∪ v) \\ u ∉ ∂ 𝒜'\ns : Finset α\nhs' : s ∈ ∂ 𝒜'\... | by_cases hs : s ∈ 𝒜.shadow | «_aux_Init_ByCases___macroRules_tacticBy_cases_:__2» | «tacticBy_cases_:_» |
Mathlib.Combinatorics.SetFamily.AhlswedeZhang | {
"line": 428,
"column": 4
} | {
"line": 428,
"column": 12
} | {
"line": 429,
"column": 2
} | [
{
"pp": "case ind.inr.succ.h𝒜₁\nα : Type u_1\ninst✝² : Fintype α\ninst✝¹ : DecidableEq α\ninst✝ : Nonempty α\ns : Finset α\n𝒜 : Finset (Finset α)\nhs : s ∉ 𝒜\nh𝒜₁ : (insert s 𝒜).Nonempty\nh𝒜₂ : univ ∉ insert s 𝒜\nh𝒜₃ : (insert s 𝒜).Nontrivial\nih :\n ∀ (𝒜_1 : Finset (Finset α)),\n #𝒜_1 < #𝒜 + 1 ... | [] | exact h𝒜 | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Combinatorics.SetFamily.AhlswedeZhang | {
"line": 428,
"column": 4
} | {
"line": 428,
"column": 12
} | {
"line": 429,
"column": 2
} | [
{
"pp": "case ind.inr.succ.h𝒜₁\nα : Type u_1\ninst✝² : Fintype α\ninst✝¹ : DecidableEq α\ninst✝ : Nonempty α\ns : Finset α\n𝒜 : Finset (Finset α)\nhs : s ∉ 𝒜\nh𝒜₁ : (insert s 𝒜).Nonempty\nh𝒜₂ : univ ∉ insert s 𝒜\nh𝒜₃ : (insert s 𝒜).Nontrivial\nih :\n ∀ (𝒜_1 : Finset (Finset α)),\n #𝒜_1 < #𝒜 + 1 ... | [] | exact h𝒜 | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SetFamily.AhlswedeZhang | {
"line": 428,
"column": 4
} | {
"line": 428,
"column": 12
} | {
"line": 429,
"column": 2
} | [
{
"pp": "case ind.inr.succ.h𝒜₁\nα : Type u_1\ninst✝² : Fintype α\ninst✝¹ : DecidableEq α\ninst✝ : Nonempty α\ns : Finset α\n𝒜 : Finset (Finset α)\nhs : s ∉ 𝒜\nh𝒜₁ : (insert s 𝒜).Nonempty\nh𝒜₂ : univ ∉ insert s 𝒜\nh𝒜₃ : (insert s 𝒜).Nontrivial\nih :\n ∀ (𝒜_1 : Finset (Finset α)),\n #𝒜_1 < #𝒜 + 1 ... | [] | exact h𝒜 | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SetFamily.KruskalKatona | {
"line": 71,
"column": 76
} | {
"line": 71,
"column": 85
} | {
"line": 71,
"column": 85
} | [
{
"pp": "case mpr\nα : Type u_1\ninst✝¹ : LinearOrder α\ns : Finset α\ninst✝ : Fintype α\nhs : s.Nonempty\nt : Finset α\n⊢ #(s.erase (s.min' hs)) = #t ∧\n (toColex t = toColex (s.erase (s.min' hs)) ∨\n ∃ w, (w ∈ s.erase (s.min' hs) ∧ w ∉ t) ∧ ∀ ⦃a : α⦄, w < a → (a ∈ t ↔ a ∈ s.erase (s.min' hs))) →\n... | [
"case mpr\nα : Type u_1\ninst✝¹ : LinearOrder α\ns : Finset α\ninst✝ : Fintype α\nhs : s.Nonempty\nt : Finset α\n⊢ #(s.erase (s.min' hs)) = #t ∧\n (toColex t = toColex (s.erase (s.min' hs)) ∨\n ∃ w ∈ s.erase (s.min' hs), w ∉ t ∧ ∀ ⦃a : α⦄, w < a → (a ∈ t ↔ a ∈ s.erase (s.min' hs))) →\n ∃ a ∉ t,\n ... | and_assoc | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Combinatorics.SetFamily.KruskalKatona | {
"line": 135,
"column": 43
} | {
"line": 135,
"column": 52
} | {
"line": 135,
"column": 52
} | [
{
"pp": "α : Type u_1\ninst✝ : LinearOrder α\ns U V : Finset α\nhU : U.Nonempty\nhV : V.Nonempty\nh : U.max' hU < V.max' hV\nhA : (Disjoint U s ∧ V ⊆ s) ∧ (s ⊔ U) \\ V ≠ s\n⊢ ∃ w, (w ∈ s ∧ w ∉ (s ⊔ U) \\ V) ∧ ∀ ⦃a : α⦄, w < a → (a ∈ (s ⊔ U) \\ V ↔ a ∈ s)",
"ppTerm": "?m.67",
"assigned": true,
"usedC... | [
"α : Type u_1\ninst✝ : LinearOrder α\ns U V : Finset α\nhU : U.Nonempty\nhV : V.Nonempty\nh : U.max' hU < V.max' hV\nhA : (Disjoint U s ∧ V ⊆ s) ∧ (s ⊔ U) \\ V ≠ s\n⊢ ∃ w ∈ s, w ∉ (s ⊔ U) \\ V ∧ ∀ ⦃a : α⦄, w < a → (a ∈ (s ⊔ U) \\ V ↔ a ∈ s)"
] | and_assoc | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Combinatorics.SimpleGraph.Connectivity.Connected | {
"line": 68,
"column": 33
} | {
"line": 68,
"column": 79
} | {
"line": 70,
"column": 0
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\np : Prop\nu v : V\nh : G.Reachable u v\nhp : ∀ (a : G.Path u v), p\n⊢ p",
"ppTerm": "?m.3",
"assigned": true,
"usedConstants": [
"SimpleGraph.Reachable.elim",
"Classical.propDecidable",
"SimpleGraph.Walk",
"SimpleGraph.Walk.toPath",
... | [] | by classical exact h.elim fun q => hp q.toPath | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.SetFamily.KruskalKatona | {
"line": 177,
"column": 80
} | {
"line": 195,
"column": 72
} | {
"line": 197,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : LinearOrder α\nℬ : Finset (Finset α)\nr : ℕ\nh₁ : Set.Sized r ↑ℬ\nh₂ : ∀ (U V : Finset α), UsefulCompression U V → IsCompressed U V ℬ\n⊢ IsInitSeg ℬ r",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"False",
"Finse... | [] | by
refine ⟨h₁, ?_⟩
rintro A B hA ⟨hBA, sizeA⟩
by_contra hB
have hAB : A ≠ B := ne_of_mem_of_not_mem hA hB
have hAB' : #A = #B := (h₁ hA).trans sizeA.symm
have hU : (A \ B).Nonempty := sdiff_nonempty.2 fun h ↦ hAB <| eq_of_subset_of_card_le h hAB'.ge
have hV : (B \ A).Nonempty :=
sdiff_nonempty.2 fun h... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.SimpleGraph.Connectivity.Connected | {
"line": 501,
"column": 2
} | {
"line": 501,
"column": 26
} | {
"line": 502,
"column": 2
} | [
{
"pp": "V : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\nφ : G ≃g G'\nv : V\nC : G.ConnectedComponent\n⊢ G'.connectedComponentMk (φ v) = map (RelIso.toRelEmbedding φ).toRelHom C ↔ G.connectedComponentMk v = C",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"SimpleGra... | [
"V : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\nφ : G ≃g G'\nv : V\nC : G.ConnectedComponent\nu : V\n⊢ G'.connectedComponentMk (φ v) = map (RelIso.toRelEmbedding φ).toRelHom (G.connectedComponentMk u) ↔\n G.connectedComponentMk v = G.connectedComponentMk u"
] | refine C.ind fun u => ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Combinatorics.SimpleGraph.Connectivity.Connected | {
"line": 508,
"column": 2
} | {
"line": 508,
"column": 26
} | {
"line": 509,
"column": 2
} | [
{
"pp": "V : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\nφ : G ≃g G'\nv' : V'\nC : G.ConnectedComponent\n⊢ G.connectedComponentMk (φ.symm v') = C ↔ G'.connectedComponentMk v' = map (RelIso.toRelEmbedding φ).toRelHom C",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"... | [
"V : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\nφ : G ≃g G'\nv' : V'\nC : G.ConnectedComponent\nu : V\n⊢ G.connectedComponentMk (φ.symm v') = G.connectedComponentMk u ↔\n G'.connectedComponentMk v' = map (RelIso.toRelEmbedding φ).toRelHom (G.connectedComponentMk u)"
] | refine C.ind fun u => ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Combinatorics.SetFamily.KruskalKatona | {
"line": 282,
"column": 4
} | {
"line": 282,
"column": 53
} | {
"line": 283,
"column": 2
} | [
{
"pp": "case inl\nn r : ℕ\n𝒜 𝒞 : Finset (Finset (Fin n))\nh𝒜r : Set.Sized r ↑𝒜\nh𝒞𝒜 : #𝒞 ≤ #𝒜\nh𝒞 : IsInitSeg 𝒞 r\n𝒜' : Finset (Finset (Fin n))\nh𝒜 : 𝒜' ⊆ 𝒜\nh𝒜𝒞 : #𝒜' = #𝒞\nℬ : Finset (Finset (Fin n))\nhℬ𝒜 : #(∂ ℬ) ≤ #(∂ 𝒜')\nh𝒜ℬ : #𝒜' = #ℬ\nhℬr : Set.Sized r ↑ℬ\nhℬ : ∀ (U V : Finset (Fi... | [] | exact (eq_of_subset_of_card_le h𝒞ℬ hcard.le).symm | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Combinatorics.SetFamily.KruskalKatona | {
"line": 282,
"column": 4
} | {
"line": 282,
"column": 53
} | {
"line": 283,
"column": 2
} | [
{
"pp": "case inl\nn r : ℕ\n𝒜 𝒞 : Finset (Finset (Fin n))\nh𝒜r : Set.Sized r ↑𝒜\nh𝒞𝒜 : #𝒞 ≤ #𝒜\nh𝒞 : IsInitSeg 𝒞 r\n𝒜' : Finset (Finset (Fin n))\nh𝒜 : 𝒜' ⊆ 𝒜\nh𝒜𝒞 : #𝒜' = #𝒞\nℬ : Finset (Finset (Fin n))\nhℬ𝒜 : #(∂ ℬ) ≤ #(∂ 𝒜')\nh𝒜ℬ : #𝒜' = #ℬ\nhℬr : Set.Sized r ↑ℬ\nhℬ : ∀ (U V : Finset (Fi... | [] | exact (eq_of_subset_of_card_le h𝒞ℬ hcard.le).symm | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SetFamily.KruskalKatona | {
"line": 282,
"column": 4
} | {
"line": 282,
"column": 53
} | {
"line": 283,
"column": 2
} | [
{
"pp": "case inl\nn r : ℕ\n𝒜 𝒞 : Finset (Finset (Fin n))\nh𝒜r : Set.Sized r ↑𝒜\nh𝒞𝒜 : #𝒞 ≤ #𝒜\nh𝒞 : IsInitSeg 𝒞 r\n𝒜' : Finset (Finset (Fin n))\nh𝒜 : 𝒜' ⊆ 𝒜\nh𝒜𝒞 : #𝒜' = #𝒞\nℬ : Finset (Finset (Fin n))\nhℬ𝒜 : #(∂ ℬ) ≤ #(∂ 𝒜')\nh𝒜ℬ : #𝒜' = #ℬ\nhℬr : Set.Sized r ↑ℬ\nhℬ : ∀ (U V : Finset (Fi... | [] | exact (eq_of_subset_of_card_le h𝒞ℬ hcard.le).symm | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SimpleGraph.Coloring.Vertex | {
"line": 236,
"column": 91
} | {
"line": 237,
"column": 36
} | {
"line": 239,
"column": 0
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\n⊢ G.chromaticNumber = ⨅ n, ↑↑n",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"iInf",
"instCompleteLinearOrderENat",
"ENat.instNatCast",
"congrArg",
"CompletelyDistribLattice.toCompleteLattice",
"s... | [] | by
rw [chromaticNumber, iInf_subtype] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.SetFamily.KruskalKatona | {
"line": 312,
"column": 6
} | {
"line": 326,
"column": 78
} | {
"line": 327,
"column": 4
} | [
{
"pp": "n r k i : ℕ\n𝒜 : Finset (Finset (Fin n))\nhir : i ≤ r\nhrk : r ≤ k\nhkn : k ≤ n\nh₁ : Set.Sized r ↑𝒜\nh₂ : k.choose r ≤ #𝒜\nrange'k : Finset (Fin n) := (range k).attachFin ⋯\n𝒞 : Finset (Finset (Fin n)) := powersetCard r range'k\nthis : Set.Sized r ↑𝒞\n⊢ #(powersetCard (r - i) range'k) = #(∂^[i] �... | [] | congr!
ext B
rw [mem_powersetCard, mem_shadow_iterate_iff_exists_sdiff]
constructor
· rintro ⟨hBk, hB⟩
have := exists_subsuperset_card_eq hBk (Nat.le_add_left _ i) <| by
rwa [hB, card_attachFin, card_range, ← Nat.add_sub_assoc hir, Nat.add_sub_cancel_left]
obtain ⟨C, Bs... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SetFamily.KruskalKatona | {
"line": 312,
"column": 6
} | {
"line": 326,
"column": 78
} | {
"line": 327,
"column": 4
} | [
{
"pp": "n r k i : ℕ\n𝒜 : Finset (Finset (Fin n))\nhir : i ≤ r\nhrk : r ≤ k\nhkn : k ≤ n\nh₁ : Set.Sized r ↑𝒜\nh₂ : k.choose r ≤ #𝒜\nrange'k : Finset (Fin n) := (range k).attachFin ⋯\n𝒞 : Finset (Finset (Fin n)) := powersetCard r range'k\nthis : Set.Sized r ↑𝒞\n⊢ #(powersetCard (r - i) range'k) = #(∂^[i] �... | [] | congr!
ext B
rw [mem_powersetCard, mem_shadow_iterate_iff_exists_sdiff]
constructor
· rintro ⟨hBk, hB⟩
have := exists_subsuperset_card_eq hBk (Nat.le_add_left _ i) <| by
rwa [hB, card_attachFin, card_range, ← Nat.add_sub_assoc hir, Nat.add_sub_cancel_left]
obtain ⟨C, Bs... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SimpleGraph.Coloring.Vertex | {
"line": 403,
"column": 2
} | {
"line": 403,
"column": 43
} | {
"line": 404,
"column": 2
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nV' : Type u_4\nG' : SimpleGraph V'\nh : ∀ (n : ℕ), G'.Colorable n → G.Colorable n\n⊢ ⨅ n ∈ setOf G.Colorable, ↑n ≤ ⨅ n ∈ setOf G'.Colorable, ↑n",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"iInf",
"instCompleteLinearOrd... | [
"V : Type u\nG : SimpleGraph V\nV' : Type u_4\nG' : SimpleGraph V'\nh : ∀ (n : ℕ), G'.Colorable n → G.Colorable n\n⊢ ∀ (i : ℕ), G'.Colorable i → ⨅ n, ⨅ (_ : G.Colorable n), ↑n ≤ ↑i"
] | simp only [Set.mem_setOf_eq, le_iInf_iff] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Combinatorics.SimpleGraph.Coloring.Vertex | {
"line": 480,
"column": 80
} | {
"line": 480,
"column": 90
} | {
"line": 480,
"column": 90
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\ninst✝ : Fintype V\nh : G.chromaticNumber = ↑(card V)\nhh : G ≠ ⊤\na b : V\nhne : a ≠ b\nright✝ : ¬G.Adj a b\nx : V\nh' : ¬x ≠ b\n⊢ a ∈ Finset.univ.erase b",
"ppTerm": "?m.110",
"assigned": true,
"usedConstants": [
"False",
"Finset.univ",
"eq_... | [] | simp [hne] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Combinatorics.SimpleGraph.Coloring.Vertex | {
"line": 480,
"column": 80
} | {
"line": 480,
"column": 90
} | {
"line": 480,
"column": 90
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\ninst✝ : Fintype V\nh : G.chromaticNumber = ↑(card V)\nhh : G ≠ ⊤\na b : V\nhne : a ≠ b\nright✝ : ¬G.Adj a b\nx : V\nh' : ¬x ≠ b\n⊢ a ∈ Finset.univ.erase b",
"ppTerm": "?m.110",
"assigned": true,
"usedConstants": [
"False",
"Finset.univ",
"eq_... | [] | simp [hne] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SimpleGraph.Coloring.Vertex | {
"line": 480,
"column": 80
} | {
"line": 480,
"column": 90
} | {
"line": 480,
"column": 90
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\ninst✝ : Fintype V\nh : G.chromaticNumber = ↑(card V)\nhh : G ≠ ⊤\na b : V\nhne : a ≠ b\nright✝ : ¬G.Adj a b\nx : V\nh' : ¬x ≠ b\n⊢ a ∈ Finset.univ.erase b",
"ppTerm": "?m.110",
"assigned": true,
"usedConstants": [
"False",
"Finset.univ",
"eq_... | [] | simp [hne] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SetFamily.KruskalKatona | {
"line": 356,
"column": 8
} | {
"line": 356,
"column": 32
} | {
"line": 356,
"column": 33
} | [
{
"pp": "case e'_3\nn : ℕ\n𝒜 : Finset (Finset (Fin n))\nr : ℕ\nh𝒜 : (↑𝒜).Intersecting\nh₂ : Set.Sized r ↑𝒜\nh₃ : r ≤ n / 2\nb : r = 0\nA : Finset (Fin n)\nHA : A ∈ 𝒜\n⊢ Disjoint A A",
"ppTerm": "?e'_3",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Finset.disjoint_self_iff_empty",... | [
"case e'_3\nn : ℕ\n𝒜 : Finset (Finset (Fin n))\nr : ℕ\nh𝒜 : (↑𝒜).Intersecting\nh₂ : Set.Sized r ↑𝒜\nh₃ : r ≤ n / 2\nb : r = 0\nA : Finset (Fin n)\nHA : A ∈ 𝒜\n⊢ A = ∅"
] | disjoint_self_iff_empty, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.SimpleGraph.Coloring.Vertex | {
"line": 550,
"column": 2
} | {
"line": 550,
"column": 95
} | {
"line": 552,
"column": 0
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nn : ℕ\ns : Finset V\nh : G.IsClique ↑s\nhc : G.Colorable n\n⊢ s.card ≤ n",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Finset",
"SimpleGraph.Adj",
"Subtype.forall._simp_1",
"Membership.mem",... | [] | simpa using! hc.card_le_of_pairwise_adj (Subtype.val : s → V) <| by simpa [Pairwise] using! h | Lean.Elab.Tactic.Simpa.evalSimpaUsingBang | Lean.Parser.Tactic.simpaUsingBang |
Mathlib.Combinatorics.SimpleGraph.Coloring.Vertex | {
"line": 550,
"column": 2
} | {
"line": 550,
"column": 95
} | {
"line": 552,
"column": 0
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nn : ℕ\ns : Finset V\nh : G.IsClique ↑s\nhc : G.Colorable n\n⊢ s.card ≤ n",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Finset",
"SimpleGraph.Adj",
"Subtype.forall._simp_1",
"Membership.mem",... | [] | simpa using! hc.card_le_of_pairwise_adj (Subtype.val : s → V) <| by simpa [Pairwise] using! h | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SimpleGraph.Coloring.Vertex | {
"line": 550,
"column": 2
} | {
"line": 550,
"column": 95
} | {
"line": 552,
"column": 0
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nn : ℕ\ns : Finset V\nh : G.IsClique ↑s\nhc : G.Colorable n\n⊢ s.card ≤ n",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Finset",
"SimpleGraph.Adj",
"Subtype.forall._simp_1",
"Membership.mem",... | [] | simpa using! hc.card_le_of_pairwise_adj (Subtype.val : s → V) <| by simpa [Pairwise] using! h | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SimpleGraph.Connectivity.Subgraph | {
"line": 101,
"column": 2
} | {
"line": 106,
"column": 93
} | {
"line": 108,
"column": 0
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nH K : G.Subgraph\nhH : H.Preconnected\nhK : K.Preconnected\nhn : (H ⊓ K).verts.Nonempty\n⊢ (H ⊔ K).Connected",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Lattice.toSemilatticeSup",
"le_sup_left",
"congrArg",
... | [] | rw [Subgraph.connected_iff', connected_iff_exists_forall_reachable]
obtain ⟨u, hu, hu'⟩ := hn
exists ⟨u, Or.inl hu⟩
rintro ⟨v, (hv | hv)⟩
· exact Reachable.map (Subgraph.inclusion (le_sup_left : H ≤ H ⊔ K)) (hH ⟨u, hu⟩ ⟨v, hv⟩)
· exact Reachable.map (Subgraph.inclusion (le_sup_right : K ≤ H ⊔ K)) (hK ⟨u, hu'⟩... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SimpleGraph.Connectivity.Subgraph | {
"line": 101,
"column": 2
} | {
"line": 106,
"column": 93
} | {
"line": 108,
"column": 0
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nH K : G.Subgraph\nhH : H.Preconnected\nhK : K.Preconnected\nhn : (H ⊓ K).verts.Nonempty\n⊢ (H ⊔ K).Connected",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Lattice.toSemilatticeSup",
"le_sup_left",
"congrArg",
... | [] | rw [Subgraph.connected_iff', connected_iff_exists_forall_reachable]
obtain ⟨u, hu, hu'⟩ := hn
exists ⟨u, Or.inl hu⟩
rintro ⟨v, (hv | hv)⟩
· exact Reachable.map (Subgraph.inclusion (le_sup_left : H ≤ H ⊔ K)) (hH ⟨u, hu⟩ ⟨v, hv⟩)
· exact Reachable.map (Subgraph.inclusion (le_sup_right : K ≤ H ⊔ K)) (hK ⟨u, hu'⟩... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SetFamily.FourFunctions | {
"line": 275,
"column": 4
} | {
"line": 276,
"column": 98
} | {
"line": 277,
"column": 2
} | [
{
"pp": "case empty\nα : Type u_1\nβ : Type u_2\ninst✝⁴ : DecidableEq α\ninst✝³ : CommSemiring β\ninst✝² : LinearOrder β\ninst✝¹ : IsStrictOrderedRing β\ninst✝ : ExistsAddOfLE β\nf₁ f₂ f₃ f₄ : Finset α → β\nh₁ : 0 ≤ f₁\nh₂ : 0 ≤ f₂\nh₃ : 0 ≤ f₃\nh₄ : 0 ≤ f₄\nh : ∀ ⦃s : Finset α⦄, s ⊆ ∅ → ∀ ⦃t : Finset α⦄, t ⊆ ∅... | [] | simp only [Finset.powerset_empty, Finset.subset_singleton_iff] at h𝒜 hℬ
obtain rfl | rfl := h𝒜 <;> obtain rfl | rfl := hℬ <;> simp; exact h (subset_refl ∅) subset_rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SetFamily.FourFunctions | {
"line": 275,
"column": 4
} | {
"line": 276,
"column": 98
} | {
"line": 277,
"column": 2
} | [
{
"pp": "case empty\nα : Type u_1\nβ : Type u_2\ninst✝⁴ : DecidableEq α\ninst✝³ : CommSemiring β\ninst✝² : LinearOrder β\ninst✝¹ : IsStrictOrderedRing β\ninst✝ : ExistsAddOfLE β\nf₁ f₂ f₃ f₄ : Finset α → β\nh₁ : 0 ≤ f₁\nh₂ : 0 ≤ f₂\nh₃ : 0 ≤ f₃\nh₄ : 0 ≤ f₄\nh : ∀ ⦃s : Finset α⦄, s ⊆ ∅ → ∀ ⦃t : Finset α⦄, t ⊆ ∅... | [] | simp only [Finset.powerset_empty, Finset.subset_singleton_iff] at h𝒜 hℬ
obtain rfl | rfl := h𝒜 <;> obtain rfl | rfl := hℬ <;> simp; exact h (subset_refl ∅) subset_rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SimpleGraph.Bipartite | {
"line": 477,
"column": 2
} | {
"line": 477,
"column": 73
} | {
"line": 478,
"column": 2
} | [
{
"pp": "W₁ : Type u_2\nW₂ : Type u_3\n⊢ (completeBipartiteGraph W₁ W₂).edgeSet = Set.range fun x ↦ s(Sum.inl x.1, Sum.inr x.2)",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Set.ext",
"Sym2.mk",
"Membership.mem",
"_private.Mathlib.Combinatorics.SimpleGraph.Bipart... | [
"case refine_1\nW₁ : Type u_2\nW₂ : Type u_3\nu v : W₁ ⊕ W₂\nh : s(u, v) ∈ (completeBipartiteGraph W₁ W₂).edgeSet\n⊢ s(u, v) ∈ Set.range fun x ↦ s(Sum.inl x.1, Sum.inr x.2)",
"case refine_2\nW₁ : Type u_2\nW₂ : Type u_3\nu v : W₁ ⊕ W₂\nx✝ : s(u, v) ∈ Set.range fun x ↦ s(Sum.inl x.1, Sum.inr x.2)\na : W₁\nb : W₂\n... | refine Set.ext <| Sym2.ind fun u v ↦ ⟨fun h ↦ ?_, fun ⟨⟨a, b⟩, z⟩ ↦ ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Combinatorics.SimpleGraph.Metric | {
"line": 196,
"column": 2
} | {
"line": 196,
"column": 51
} | {
"line": 198,
"column": 0
} | [
{
"pp": "case neg\nV : Type u_1\nG : SimpleGraph V\nu v : V\nG' : SimpleGraph V\nh : G ≤ G'\nhr : ¬G.Reachable u v\n⊢ G'.edist u v ≤ G.edist u v",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"instTopENat",
"SimpleGraph.edist_eq_top_of_not_reachable",
"Eq.rec",
"LE... | [] | · exact edist_eq_top_of_not_reachable hr ▸ le_top | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Combinatorics.SimpleGraph.Bipartite | {
"line": 573,
"column": 31
} | {
"line": 573,
"column": 50
} | {
"line": 573,
"column": 51
} | [
{
"pp": "case inl.inl.inr\nV : Type u_1\nG : SimpleGraph V\nα : Type u_2\nβ : Type u_3\ninst✝³ : Finite α\ninst✝² : Finite β\ninst✝¹ : Nonempty α\ninst✝ : Nonempty β\nthis✝ : Fintype α\nthis : Fintype β\nx✝ :\n ∃ left right, #left = Fintype.card α ∧ #right = Fintype.card β ∧ G.bipartiteDoubleCover.IsCompleteBe... | [] | simpa using h hl hr | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Combinatorics.SimpleGraph.Bipartite | {
"line": 573,
"column": 31
} | {
"line": 573,
"column": 50
} | {
"line": 573,
"column": 51
} | [
{
"pp": "case inl.inl.inr\nV : Type u_1\nG : SimpleGraph V\nα : Type u_2\nβ : Type u_3\ninst✝³ : Finite α\ninst✝² : Finite β\ninst✝¹ : Nonempty α\ninst✝ : Nonempty β\nthis✝ : Fintype α\nthis : Fintype β\nx✝ :\n ∃ left right, #left = Fintype.card α ∧ #right = Fintype.card β ∧ G.bipartiteDoubleCover.IsCompleteBe... | [] | simpa using h hl hr | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SimpleGraph.Bipartite | {
"line": 573,
"column": 31
} | {
"line": 573,
"column": 50
} | {
"line": 573,
"column": 51
} | [
{
"pp": "case inl.inl.inr\nV : Type u_1\nG : SimpleGraph V\nα : Type u_2\nβ : Type u_3\ninst✝³ : Finite α\ninst✝² : Finite β\ninst✝¹ : Nonempty α\ninst✝ : Nonempty β\nthis✝ : Fintype α\nthis : Fintype β\nx✝ :\n ∃ left right, #left = Fintype.card α ∧ #right = Fintype.card β ∧ G.bipartiteDoubleCover.IsCompleteBe... | [] | simpa using h hl hr | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
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