module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 365
values | kind stringclasses 368
values |
|---|---|---|---|---|---|---|
Mathlib.Combinatorics.SimpleGraph.Walks.Operations | {
"line": 538,
"column": 81
} | {
"line": 539,
"column": 59
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nu v : V\np : G.Walk u v\nn : ℕ\n⊢ (p.drop n).darts = List.drop n p.darts",
"usedConstants": [
"SimpleGraph.Walk.darts_copy",
"congrArg",
"SimpleGraph.Adj",
"SimpleGraph.Walk",
"SimpleGraph.Walk.drop._proof_1",
"Prod.mk",
"inst... | by
induction p generalizing n <;> cases n <;> simp [*, drop] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.SimpleGraph.Walks.Operations | {
"line": 541,
"column": 81
} | {
"line": 542,
"column": 59
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nu v : V\np : G.Walk u v\nn : ℕ\n⊢ (p.drop n).edges = List.drop n p.edges",
"usedConstants": [
"SimpleGraph.Walk.edges_copy",
"Sym2.mk",
"congrArg",
"SimpleGraph.Adj",
"SimpleGraph.Walk",
"SimpleGraph.Walk.drop._proof_1",
"inst... | by
induction p generalizing n <;> cases n <;> simp [*, drop] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.SimpleGraph.Walks.Operations | {
"line": 634,
"column": 58
} | {
"line": 635,
"column": 59
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nu v : V\np : G.Walk u v\nn : ℕ\n⊢ (p.drop n).support = List.drop (min n p.length) p.support",
"usedConstants": [
"instDistribLatticeNat",
"congrArg",
"inf_of_le_left",
"Nat.add_min_add_right",
"SimpleGraph.Walk.length",
"SimpleGraph... | by
induction p generalizing n <;> cases n <;> simp [*, drop] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.SimpleGraph.Walks.Subwalks | {
"line": 226,
"column": 6
} | {
"line": 228,
"column": 100
} | [
{
"pp": "case succ.cons\nV : Type u_1\nG : SimpleGraph V\nu v : V\nn k✝ k : ℕ\nh : n ≤ k\nv✝ : V\nh✝ : G.Adj u v✝\np✝ : G.Walk v✝ v\nih : ((cons h✝ p✝).take n).IsSubwalk ((cons h✝ p✝).take k)\n⊢ ((cons h✝ p✝).take k).IsSubwalk ((cons h✝ p✝).take (k + 1))",
"usedConstants": [
"Nat.le_add_right._simp_1"... | cases k
· exact isSubwalk_of_append_left rfl
simp [isSubwalk_iff_support_isInfix, take_support_eq_support_take_succ, List.IsPrefix.isInfix] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SimpleGraph.Walks.Subwalks | {
"line": 226,
"column": 6
} | {
"line": 228,
"column": 100
} | [
{
"pp": "case succ.cons\nV : Type u_1\nG : SimpleGraph V\nu v : V\nn k✝ k : ℕ\nh : n ≤ k\nv✝ : V\nh✝ : G.Adj u v✝\np✝ : G.Walk v✝ v\nih : ((cons h✝ p✝).take n).IsSubwalk ((cons h✝ p✝).take k)\n⊢ ((cons h✝ p✝).take k).IsSubwalk ((cons h✝ p✝).take (k + 1))",
"usedConstants": [
"Nat.le_add_right._simp_1"... | cases k
· exact isSubwalk_of_append_left rfl
simp [isSubwalk_iff_support_isInfix, take_support_eq_support_take_succ, List.IsPrefix.isInfix] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SimpleGraph.Triangle.Basic | {
"line": 103,
"column": 6
} | {
"line": 103,
"column": 54
} | [
{
"pp": "case h.mp\nα : Type u_1\nG : SimpleGraph α\na b : α\nhab✝ : a ≠ b\nc d e : α\nhcd : G.Adj c d\nhce : G.Adj c e\nhde : G.Adj d e\nhab : (a = c ∨ a = d ∨ a = e) ∧ (b = c ∨ b = d ∨ b = e)\n⊢ G.Adj a b ∧ ∃ c_1, G.Adj a c_1 ∧ G.Adj b c_1 ∧ {c, d, e} = {a, b, c_1}",
"usedConstants": []
}
] | obtain ⟨rfl | rfl | rfl, rfl | rfl | rfl⟩ := hab | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Combinatorics.SimpleGraph.Regularity.Chunk | {
"line": 93,
"column": 2
} | {
"line": 118,
"column": 69
} | [
{
"pp": "α : Type u_1\ninst✝² : Fintype α\ninst✝¹ : DecidableEq α\nP : Finpartition univ\nhP : P.IsEquipartition\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\nε : ℝ\nU : Finset α\nhU : U ∈ P.parts\nV : Finset α\nhV : V ∈ P.parts\nhUV : U ≠ V\nh₂ : ¬G.IsUniform ε U V\n⊢ #(G.nonuniformWitness ε U V \\ (star hP ... | have hX : G.nonuniformWitness ε U V ∈ P.nonuniformWitnesses G ε U :=
nonuniformWitness_mem_nonuniformWitnesses h₂ hV hUV
have q : G.nonuniformWitness ε U V \ (star hP G ε hU V).biUnion id ⊆
{B ∈ (atomise U <| P.nonuniformWitnesses G ε U).parts |
B ⊆ G.nonuniformWitness ε U V ∧ B.Nonempty}.biUnion
... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SimpleGraph.Regularity.Chunk | {
"line": 93,
"column": 2
} | {
"line": 118,
"column": 69
} | [
{
"pp": "α : Type u_1\ninst✝² : Fintype α\ninst✝¹ : DecidableEq α\nP : Finpartition univ\nhP : P.IsEquipartition\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\nε : ℝ\nU : Finset α\nhU : U ∈ P.parts\nV : Finset α\nhV : V ∈ P.parts\nhUV : U ≠ V\nh₂ : ¬G.IsUniform ε U V\n⊢ #(G.nonuniformWitness ε U V \\ (star hP ... | have hX : G.nonuniformWitness ε U V ∈ P.nonuniformWitnesses G ε U :=
nonuniformWitness_mem_nonuniformWitnesses h₂ hV hUV
have q : G.nonuniformWitness ε U V \ (star hP G ε hU V).biUnion id ⊆
{B ∈ (atomise U <| P.nonuniformWitnesses G ε U).parts |
B ⊆ G.nonuniformWitness ε U V ∧ B.Nonempty}.biUnion
... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SimpleGraph.Clique | {
"line": 174,
"column": 2
} | {
"line": 176,
"column": 30
} | [
{
"pp": "α : Type u_1\nG : SimpleGraph α\nv w : α\ns : Set α\nhc : (G ⊔ edge v w).IsClique s\n⊢ G.IsClique (s \\ {v})",
"usedConstants": [
"False",
"_private.Mathlib.Combinatorics.SimpleGraph.Clique.0.SimpleGraph.IsClique.sdiff_of_sup_edge._simp_1_2",
"SimpleGraph.edge",
"eq_false",
... | intro _ hx _ hy hxy
have := hc hx.1 hy.1 hxy
simp_all [sup_adj, edge_adj] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SimpleGraph.Clique | {
"line": 174,
"column": 2
} | {
"line": 176,
"column": 30
} | [
{
"pp": "α : Type u_1\nG : SimpleGraph α\nv w : α\ns : Set α\nhc : (G ⊔ edge v w).IsClique s\n⊢ G.IsClique (s \\ {v})",
"usedConstants": [
"False",
"_private.Mathlib.Combinatorics.SimpleGraph.Clique.0.SimpleGraph.IsClique.sdiff_of_sup_edge._simp_1_2",
"SimpleGraph.edge",
"eq_false",
... | intro _ hx _ hy hxy
have := hc hx.1 hy.1 hxy
simp_all [sup_adj, edge_adj] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SimpleGraph.Clique | {
"line": 354,
"column": 2
} | {
"line": 354,
"column": 24
} | [
{
"pp": "case h.Adj.h.h.a\nα : Type u_1\nG : SimpleGraph α\nn : ℕ\nf : ⊤ ↪g G\nv : α\nhv✝ : v ∈ ↑(map f.toEmbedding univ)\nw : α\nhw✝ : w ∈ ↑(map f.toEmbedding univ)\nhv : ∃ x, f.toEmbedding x = v\nhw : ∃ x, f.toEmbedding x = w\n⊢ (induce (↑(map f.toEmbedding univ)) G).Adj ⟨v, hv✝⟩ ⟨w, hw✝⟩ ↔ ⊤.Adj ⟨v, hv✝⟩ ⟨w,... | obtain ⟨v', rfl⟩ := hv | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Combinatorics.SimpleGraph.Regularity.Chunk | {
"line": 185,
"column": 60
} | {
"line": 185,
"column": 68
} | [
{
"pp": "α : Type u_1\ninst✝² : Fintype α\ninst✝¹ : DecidableEq α\nP : Finpartition univ\nhP : P.IsEquipartition\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\nε : ℝ\nU : Finset α\nhU : U ∈ P.parts\ns : Finset α\nhs : s ∈ (chunk hP G ε hU).parts\ni : #s = m + 1\n⊢ m ≤ #s",
"usedConstants": [
"Nat.ins... | simp [i] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Combinatorics.SimpleGraph.Regularity.Chunk | {
"line": 185,
"column": 60
} | {
"line": 185,
"column": 68
} | [
{
"pp": "α : Type u_1\ninst✝² : Fintype α\ninst✝¹ : DecidableEq α\nP : Finpartition univ\nhP : P.IsEquipartition\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\nε : ℝ\nU : Finset α\nhU : U ∈ P.parts\ns : Finset α\nhs : s ∈ (chunk hP G ε hU).parts\ni : #s = m + 1\n⊢ m ≤ #s",
"usedConstants": [
"Nat.ins... | simp [i] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SimpleGraph.Regularity.Chunk | {
"line": 185,
"column": 60
} | {
"line": 185,
"column": 68
} | [
{
"pp": "α : Type u_1\ninst✝² : Fintype α\ninst✝¹ : DecidableEq α\nP : Finpartition univ\nhP : P.IsEquipartition\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\nε : ℝ\nU : Finset α\nhU : U ∈ P.parts\ns : Finset α\nhs : s ∈ (chunk hP G ε hU).parts\ni : #s = m + 1\n⊢ m ≤ #s",
"usedConstants": [
"Nat.ins... | simp [i] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SimpleGraph.Regularity.Chunk | {
"line": 188,
"column": 52
} | {
"line": 188,
"column": 60
} | [
{
"pp": "α : Type u_1\ninst✝² : Fintype α\ninst✝¹ : DecidableEq α\nP : Finpartition univ\nhP : P.IsEquipartition\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\nε : ℝ\nU : Finset α\nhU : U ∈ P.parts\ns : Finset α\nhs : s ∈ (chunk hP G ε hU).parts\ni : #s = m\n⊢ #s ≤ m + 1",
"usedConstants": [
"Nat.ins... | simp [i] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Combinatorics.SimpleGraph.Regularity.Chunk | {
"line": 188,
"column": 52
} | {
"line": 188,
"column": 60
} | [
{
"pp": "α : Type u_1\ninst✝² : Fintype α\ninst✝¹ : DecidableEq α\nP : Finpartition univ\nhP : P.IsEquipartition\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\nε : ℝ\nU : Finset α\nhU : U ∈ P.parts\ns : Finset α\nhs : s ∈ (chunk hP G ε hU).parts\ni : #s = m\n⊢ #s ≤ m + 1",
"usedConstants": [
"Nat.ins... | simp [i] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SimpleGraph.Regularity.Chunk | {
"line": 188,
"column": 52
} | {
"line": 188,
"column": 60
} | [
{
"pp": "α : Type u_1\ninst✝² : Fintype α\ninst✝¹ : DecidableEq α\nP : Finpartition univ\nhP : P.IsEquipartition\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\nε : ℝ\nU : Finset α\nhU : U ∈ P.parts\ns : Finset α\nhs : s ∈ (chunk hP G ε hU).parts\ni : #s = m\n⊢ #s ≤ m + 1",
"usedConstants": [
"Nat.ins... | simp [i] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SimpleGraph.Clique | {
"line": 501,
"column": 4
} | {
"line": 501,
"column": 31
} | [
{
"pp": "case mpr\nα : Type u_1\n⊢ ⊥.CliqueFree 2",
"usedConstants": [
"le_rfl",
"SimpleGraph.cliqueFree_bot",
"instOfNatNat",
"Nat.instPreorder",
"Nat",
"OfNat.ofNat"
]
}
] | exact cliqueFree_bot le_rfl | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Combinatorics.SimpleGraph.Clique | {
"line": 678,
"column": 2
} | {
"line": 678,
"column": 18
} | [
{
"pp": "case h\nα : Type u_3\nG : SimpleGraph α\ninst✝ : Finite α\nthis : Fintype α\ny : ℕ\ns : Finset α\nsyc : G.IsClique ↑s ∧ #s = y\n⊢ y ≤ Fintype.card α",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Finset",
"Fintype.card",
"id",
"LE.le",
"instLENat",
"SetL... | rw [← syc.right] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Combinatorics.SimpleGraph.Regularity.Chunk | {
"line": 306,
"column": 8
} | {
"line": 306,
"column": 81
} | [
{
"pp": "case h.refine_1\nα : 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\nhε₁ : ε... | rw [sq, mul_mul_mul_comm, mul_comm (_ / (m : ℝ)), mul_comm (_ / (m : ℝ))] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Combinatorics.SimpleGraph.Triangle.Removal | {
"line": 123,
"column": 27
} | {
"line": 123,
"column": 63
} | [
{
"pp": "α : Type u_1\ninst✝³ : DecidableEq α\ninst✝² : Fintype α\nG : SimpleGraph α\ninst✝¹ : DecidableRel G.Adj\nP : Finpartition univ\nε : ℝ\ninst✝ : Nonempty α\nhε : 0 < ε\nhP : P.IsEquipartition\nhPε : P.IsUniform G (ε / 8)\nhP' : 4 / ε ≤ ↑(#P.parts)\nA : Finset (α × α) :=\n (P.nonUniforms G (ε / 8)).biUn... | gcongr; exact unreduced_edges_subset | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SimpleGraph.Triangle.Removal | {
"line": 123,
"column": 27
} | {
"line": 123,
"column": 63
} | [
{
"pp": "α : Type u_1\ninst✝³ : DecidableEq α\ninst✝² : Fintype α\nG : SimpleGraph α\ninst✝¹ : DecidableRel G.Adj\nP : Finpartition univ\nε : ℝ\ninst✝ : Nonempty α\nhε : 0 < ε\nhP : P.IsEquipartition\nhPε : P.IsUniform G (ε / 8)\nhP' : 4 / ε ≤ ↑(#P.parts)\nA : Finset (α × α) :=\n (P.nonUniforms G (ε / 8)).biUn... | gcongr; exact unreduced_edges_subset | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SimpleGraph.Triangle.Removal | {
"line": 127,
"column": 6
} | {
"line": 127,
"column": 75
} | [
{
"pp": "α : Type u_1\ninst✝³ : DecidableEq α\ninst✝² : Fintype α\nG : SimpleGraph α\ninst✝¹ : DecidableRel G.Adj\nP : Finpartition univ\nε : ℝ\ninst✝ : Nonempty α\nhε : 0 < ε\nhP : P.IsEquipartition\nhPε : P.IsUniform G (ε / 8)\nhP' : 4 / ε ≤ ↑(#P.parts)\nA : Finset (α × α) :=\n (P.nonUniforms G (ε / 8)).biUn... | gcongr; exact hP.sum_nonUniforms_lt univ_nonempty (by positivity) hPε | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SimpleGraph.Triangle.Removal | {
"line": 127,
"column": 6
} | {
"line": 127,
"column": 75
} | [
{
"pp": "α : Type u_1\ninst✝³ : DecidableEq α\ninst✝² : Fintype α\nG : SimpleGraph α\ninst✝¹ : DecidableRel G.Adj\nP : Finpartition univ\nε : ℝ\ninst✝ : Nonempty α\nhε : 0 < ε\nhP : P.IsEquipartition\nhPε : P.IsUniform G (ε / 8)\nhP' : 4 / ε ≤ ↑(#P.parts)\nA : Finset (α × α) :=\n (P.nonUniforms G (ε / 8)).biUn... | gcongr; exact hP.sum_nonUniforms_lt univ_nonempty (by positivity) hPε | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.Additive.Dissociation | {
"line": 77,
"column": 2
} | {
"line": 78,
"column": 90
} | [
{
"pp": "α : Type u_1\ninst✝ : CommGroup α\ns : Set α\nt : Finset α\nht : ↑t ⊆ s\nu : Finset α\nhu : ↑u ⊆ s\nhtu : t ≠ u\nh : ∏ x ∈ t, x = ∏ x ∈ u, x\n⊢ ∃ t u, ↑t ⊆ s ∧ ↑u ⊆ s ∧ Disjoint t u ∧ t ≠ u ∧ ∏ a ∈ t, a = ∏ a ∈ u, a",
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"CancelCommMonoid.toCom... | refine ⟨t \ u, u \ t, ?_, ?_, disjoint_sdiff_sdiff, sdiff_ne_sdiff_iff.2 htu,
Finset.prod_sdiff_eq_prod_sdiff_iff.2 h⟩ <;> push_cast <;> exact diff_subset.trans ‹_› | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Combinatorics.SimpleGraph.Regularity.Chunk | {
"line": 326,
"column": 4
} | {
"line": 327,
"column": 67
} | [
{
"pp": "case left\nα : 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\nhε₁ : ε ≤ 1\n... | rw [sub_le_iff_le_add']
exact sum_density_div_card_le_density_add_eps hPα hPε hε₁ hA hB | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SimpleGraph.Regularity.Chunk | {
"line": 326,
"column": 4
} | {
"line": 327,
"column": 67
} | [
{
"pp": "case left\nα : 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\nhε₁ : ε ≤ 1\n... | rw [sub_le_iff_le_add']
exact sum_density_div_card_le_density_add_eps hPα hPε hε₁ hA hB | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.Additive.DoublingConst | {
"line": 159,
"column": 73
} | {
"line": 160,
"column": 17
} | [
{
"pp": "G' : Type u_2\ninst✝³ : AddGroup G'\ninst✝² : DecidableEq G'\n𝕜 : Type u_3\ninst✝¹ : Semifield 𝕜\ninst✝ : CharZero 𝕜\nA B : Finset G'\n⊢ ↑σ[A, B] = ↑(#(A + B)) / ↑(#A)",
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"instHDiv",
"GroupWithZero.toDivInvMonoid",
... | by
simp [addConst] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.Additive.SubsetSum | {
"line": 39,
"column": 74
} | {
"line": 39,
"column": 90
} | [
{
"pp": "M : Type u_1\ninst✝¹ : DecidableEq M\ninst✝ : AddCommMonoid M\nA : Finset M\na : M\n⊢ a ∈ A.subsetSum ↔ ∃ B ⊆ A, ∑ b ∈ B, b = a",
"usedConstants": [
"Finset.subsetSum",
"congrArg",
"Finset",
"Finset.mem_image._simp_1",
"Membership.mem",
"Exists",
"id",
... | simp [subsetSum] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Combinatorics.Additive.SubsetSum | {
"line": 39,
"column": 74
} | {
"line": 39,
"column": 90
} | [
{
"pp": "M : Type u_1\ninst✝¹ : DecidableEq M\ninst✝ : AddCommMonoid M\nA : Finset M\na : M\n⊢ a ∈ A.subsetSum ↔ ∃ B ⊆ A, ∑ b ∈ B, b = a",
"usedConstants": [
"Finset.subsetSum",
"congrArg",
"Finset",
"Finset.mem_image._simp_1",
"Membership.mem",
"Exists",
"id",
... | simp [subsetSum] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.Additive.SubsetSum | {
"line": 39,
"column": 74
} | {
"line": 39,
"column": 90
} | [
{
"pp": "M : Type u_1\ninst✝¹ : DecidableEq M\ninst✝ : AddCommMonoid M\nA : Finset M\na : M\n⊢ a ∈ A.subsetSum ↔ ∃ B ⊆ A, ∑ b ∈ B, b = a",
"usedConstants": [
"Finset.subsetSum",
"congrArg",
"Finset",
"Finset.mem_image._simp_1",
"Membership.mem",
"Exists",
"id",
... | simp [subsetSum] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.Additive.ErdosGinzburgZiv | {
"line": 82,
"column": 4
} | {
"line": 83,
"column": 72
} | [
{
"pp": "case refine_2.refine_1\nι : Type u_1\np : ℕ\ninst✝ : Fact (Nat.Prime p)\ns : Finset ι\na : ι → ZMod p\nhs : #s = 2 * p - 1\nthis : NeZero p\nN : ℕ := Fintype.card { x // (eval x) (f₁ s a) = 0 ∧ (eval x) (f₂ s a) = 0 }\nzero_sol : { x // (eval x) (f₁ s a) = 0 ∧ (eval x) (f₂ s a) = 0 } := ⟨0, ⋯⟩\nhN₀ : 0... | · rw [← Subtype.coe_ne_coe, Function.ne_iff] at hx
exact hx.imp (fun a ha ↦ mem_filter.2 ⟨Finset.mem_attach _ _, ha⟩) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Combinatorics.Derangements.Finite | {
"line": 100,
"column": 62
} | {
"line": 100,
"column": 70
} | [
{
"pp": "case ind.succ.succ\nn : ℕ\nhyp : ∀ m < n + 1 + 1, card ↑(derangements (Fin m)) = numDerangements m\n⊢ (n + 1) * card ↑(derangements (Fin n)) + (n + 1) * card ↑(derangements (Fin (n + 1))) =\n (n + 1) * (numDerangements n + numDerangements (n + 1))",
"usedConstants": [
"Distrib.leftDistribC... | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.Configuration | {
"line": 503,
"column": 19
} | {
"line": 503,
"column": 30
} | [
{
"pp": "case h\nK : Type u_3\ninst✝ : Field K\nb c d : ℙ K (Fin 3 → K)\nhbc : b.orthogonal c\nhbd : b.orthogonal d\na : Fin 3 → K\nha : a ≠ 0\nhac : (Projectivization.mk K a ha).orthogonal c\nhad : (Projectivization.mk K a ha).orthogonal d\n⊢ Projectivization.mk K a ha = b ∨ c = d",
"usedConstants": [
... | | h a ha => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.Combinatorics.Enumerative.Bell | {
"line": 142,
"column": 2
} | {
"line": 142,
"column": 23
} | [
{
"pp": "n : ℕ\n⊢ uniformBell 0 n = 1",
"usedConstants": [
"MulOne.toOne",
"Monoid.toMulOneClass",
"congrArg",
"instOfNatNat",
"Nat.uniformBell_eq",
"MulOneClass.toMulOne",
"CommMonoid.toMonoid",
"Nat",
"True",
"eq_self",
"Nat.instCommMonoid"... | simp [uniformBell_eq] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Combinatorics.Enumerative.Bell | {
"line": 142,
"column": 2
} | {
"line": 142,
"column": 23
} | [
{
"pp": "n : ℕ\n⊢ uniformBell 0 n = 1",
"usedConstants": [
"MulOne.toOne",
"Monoid.toMulOneClass",
"congrArg",
"instOfNatNat",
"Nat.uniformBell_eq",
"MulOneClass.toMulOne",
"CommMonoid.toMonoid",
"Nat",
"True",
"eq_self",
"Nat.instCommMonoid"... | simp [uniformBell_eq] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.Enumerative.Bell | {
"line": 142,
"column": 2
} | {
"line": 142,
"column": 23
} | [
{
"pp": "n : ℕ\n⊢ uniformBell 0 n = 1",
"usedConstants": [
"MulOne.toOne",
"Monoid.toMulOneClass",
"congrArg",
"instOfNatNat",
"Nat.uniformBell_eq",
"MulOneClass.toMulOne",
"CommMonoid.toMonoid",
"Nat",
"True",
"eq_self",
"Nat.instCommMonoid"... | simp [uniformBell_eq] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.Enumerative.Bell | {
"line": 145,
"column": 2
} | {
"line": 145,
"column": 23
} | [
{
"pp": "m : ℕ\n⊢ m.uniformBell 0 = 1",
"usedConstants": [
"Nat.instCanonicallyOrderedAdd",
"MulOne.toOne",
"Nat.instMulZeroClass",
"Nat.instOrderedSub",
"Nat.choose",
"HMul.hMul",
"Monoid.toMulOneClass",
"congrArg",
"Finset",
"AddMonoid.toAddZeroC... | simp [uniformBell_eq] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Combinatorics.Enumerative.Bell | {
"line": 145,
"column": 2
} | {
"line": 145,
"column": 23
} | [
{
"pp": "m : ℕ\n⊢ m.uniformBell 0 = 1",
"usedConstants": [
"Nat.instCanonicallyOrderedAdd",
"MulOne.toOne",
"Nat.instMulZeroClass",
"Nat.instOrderedSub",
"Nat.choose",
"HMul.hMul",
"Monoid.toMulOneClass",
"congrArg",
"Finset",
"AddMonoid.toAddZeroC... | simp [uniformBell_eq] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.Enumerative.Bell | {
"line": 145,
"column": 2
} | {
"line": 145,
"column": 23
} | [
{
"pp": "m : ℕ\n⊢ m.uniformBell 0 = 1",
"usedConstants": [
"Nat.instCanonicallyOrderedAdd",
"MulOne.toOne",
"Nat.instMulZeroClass",
"Nat.instOrderedSub",
"Nat.choose",
"HMul.hMul",
"Monoid.toMulOneClass",
"congrArg",
"Finset",
"AddMonoid.toAddZeroC... | simp [uniformBell_eq] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Finset.Finsupp | {
"line": 57,
"column": 4
} | {
"line": 59,
"column": 86
} | [
{
"pp": "case refine_2\nι : Type u_1\nα : Type u_2\ninst✝ : Zero α\ns : Finset ι\nf : ι →₀ α\nt : ι → Finset α\n⊢ (f.support ⊆ s ∧ ∀ i ∈ s, f i ∈ t i) → ∃ a ∈ s.pi t, { toFun := indicator s, inj' := ⋯ } a = f",
"usedConstants": [
"Iff.mpr",
"Finsupp.instFunLike",
"Finsupp.indicator",
... | refine fun h => ⟨fun i _ => f i, mem_pi.2 h.2, ?_⟩
ext i
exact ite_eq_left_iff.2 fun hi => (notMem_support_iff.1 fun H => hi <| h.1 H).symm | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Finset.Finsupp | {
"line": 57,
"column": 4
} | {
"line": 59,
"column": 86
} | [
{
"pp": "case refine_2\nι : Type u_1\nα : Type u_2\ninst✝ : Zero α\ns : Finset ι\nf : ι →₀ α\nt : ι → Finset α\n⊢ (f.support ⊆ s ∧ ∀ i ∈ s, f i ∈ t i) → ∃ a ∈ s.pi t, { toFun := indicator s, inj' := ⋯ } a = f",
"usedConstants": [
"Iff.mpr",
"Finsupp.instFunLike",
"Finsupp.indicator",
... | refine fun h => ⟨fun i _ => f i, mem_pi.2 h.2, ?_⟩
ext i
exact ite_eq_left_iff.2 fun hi => (notMem_support_iff.1 fun H => hi <| h.1 H).symm | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Finset.Finsupp | {
"line": 75,
"column": 2
} | {
"line": 75,
"column": 26
} | [
{
"pp": "case refine_2\nι : Type u_1\nα : Type u_2\ninst✝ : Zero α\ns : Finset ι\nf : ι →₀ α\nt : ι →₀ Finset α\nht : t.support ⊆ s\ni : ι\nh : f i ∈ t i\nhi : i ∈ f.support\nH : t i = 0\n⊢ f i = 0",
"usedConstants": [
"Finsupp.instFunLike",
"congrArg",
"Finset",
"Finset.mem_zero",
... | · rwa [H, mem_zero] at h | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Combinatorics.Enumerative.DyckWord | {
"line": 115,
"column": 2
} | {
"line": 115,
"column": 21
} | [
{
"pp": "p : DyckWord\nh : ↑p ≠ []\n⊢ (↑p).head h = U",
"usedConstants": []
}
] | rcases p with - | s | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Combinatorics.Enumerative.DyckWord | {
"line": 132,
"column": 4
} | {
"line": 132,
"column": 39
} | [
{
"pp": "p : DyckWord\nh : p ≠ 0\nh' : ↑p ≠ []\n⊢ take 1 (↑p).dropLast = [(↑p).head h']",
"usedConstants": []
}
] | rcases p with - | ⟨s, ⟨- | ⟨t, r⟩⟩⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Combinatorics.Enumerative.DyckWord | {
"line": 206,
"column": 6
} | {
"line": 206,
"column": 41
} | [
{
"pp": "p q : DyckWord\nhn : p.IsNested\ni : ℕ\nh : ↑p ≠ []\nl1 : List.take 1 ↑p = [(↑p).head h]\n⊢ (↑p).length - 1 = (↑p).length - 1 - 1 + 1",
"usedConstants": []
}
] | rcases p with - | ⟨s, ⟨- | ⟨t, r⟩⟩⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Combinatorics.Enumerative.DyckWord | {
"line": 412,
"column": 2
} | {
"line": 414,
"column": 70
} | [
{
"pp": "case neg\np q : DyckWord\nh : ¬p = 0\n⊢ q ≤ p + q",
"usedConstants": [
"DyckWord.semilength_outsidePart_lt",
"instAddDyckWord",
"DyckWord.outsidePart_add",
"DyckWord",
"DyckWord.outsidePart",
"DyckWord.instPreorder",
"instHAdd",
"DyckWord.semilength",... | · have := semilength_outsidePart_lt h
exact (le_add_self p.outsidePart q).trans
(Relation.ReflTransGen.single (Or.inr (outsidePart_add h).symm)) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Combinatorics.Additive.VerySmallDoubling | {
"line": 195,
"column": 4
} | {
"line": 195,
"column": 90
} | [
{
"pp": "G : Type u_1\ninst✝¹ : Group G\ninst✝ : DecidableEq G\nK : ℝ\nA✝ B S : Finset G\na✝ b✝ c✝ d✝ x y : G\nA : Finset G\nh : ↑(#(A * A)) < 3 / 2 * ↑(#A)\nh₁ : ∀ x ∈ A, ∀ y ∈ A, 1 / 2 * ↑(#A) < ↑(#(x •> A ∩ y •> A))\na : G\nha : a⁻¹ ∈ A\nb : G\nhb : b ∈ A\nc : G\nhc : c⁻¹ ∈ A\nd : G\nhd : d ∈ A\nh₂ : 1 / 2 *... | exact ⟨a * b * t, by simp [ht, mul_assoc], ((c * d)⁻¹ * t)⁻¹, by simp [ht, mul_assoc]⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Combinatorics.Additive.VerySmallDoubling | {
"line": 223,
"column": 36
} | {
"line": 223,
"column": 48
} | [
{
"pp": "G : Type u_1\ninst✝¹ : Group G\ninst✝ : DecidableEq G\nA : Finset G\nh : ↑(#(A * A)) < 3 / 2 * ↑(#A)\nh₀ : A.Nonempty\nh₁ : ∀ a ∈ A⁻¹ * A, 1 / 2 * ↑(#A) < ↑(#({xy ∈ A ×ˢ A | xy.1 * xy.2⁻¹ = a}))\nh₂ :\n ∀ x ∈ A ×ˢ A,\n (fun x ↦\n match x with\n | (x, y) => x * y⁻¹)\n x ∈\n ... | by simp [h₀] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.Enumerative.Schroder | {
"line": 90,
"column": 4
} | {
"line": 90,
"column": 60
} | [
{
"pp": "n : ℕ\nhn : n ≠ 0\n⊢ 2 * (n.largeSchroder / 2) = n.largeSchroder",
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Nat.largeSchroder",
"instHDiv",
"HMul.hMul",
"congrArg",
"Nat.instAtLeastTwoHAddOfNat",
"id",
"HDiv.hDiv"... | Nat.mul_div_cancel_left' (even_largeSchroder hn).two_dvd | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.Enumerative.Partition.GenFun | {
"line": 96,
"column": 15
} | {
"line": 114,
"column": 74
} | [
{
"pp": "R : Type u_1\ninst✝² : CommSemiring R\ninst✝¹ : TopologicalSpace R\ninst✝ : T2Space R\nf : ℕ → ℕ → R\nd : ℕ\ns : Finset ℕ\nhs0 : 0 ∉ s\ng : ℕ →₀ ℕ\nhg : g ∈ s.finsuppAntidiag d\nhprod : ∀ i ∈ s, (coeff (g i)) (1 + ∑' (j : ℕ), f i (j + 1) • X ^ (i * (j + 1))) ≠ 0\nx : ℕ\n⊢ x ∣ g x",
"usedConstants":... | by
by_cases hx : x ∈ s
· specialize hprod x hx
contrapose! hprod
have hx0 : x ≠ 0 := fun h ↦ hs0 (h ▸ hx)
rw [map_add, (summable_genFun_term' f hx0).map_tsum _ (WithPiTopology.continuous_coeff _ _)]
rw [show (0 : R) = 0 + ∑' (i : ℕ), 0 by simp]
congrm (?_ + ∑' (i : ℕ), ?_)
· suffices g x ≠ 0... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.Additive.VerySmallDoubling | {
"line": 714,
"column": 6
} | {
"line": 715,
"column": 84
} | [
{
"pp": "G : Type u_1\ninst✝¹ : Group G\ninst✝ : DecidableEq G\nK : ℝ\nS : Finset G\nhK : K < 1\nhS : S.Nonempty\nN : Finset G\nhN : IsAtom K S N\nn : G\nhn : n ∈ N\none_mem_carrier : 1 ∈ n⁻¹ •> N\nself_mem_smul_carrier : ∀ (x : G), x ∈ x •> n⁻¹ •> N\na : G\nha✝ : a ∈ n⁻¹ •> N\nha : 1 ∈ a⁻¹ •> n⁻¹ •> N\nthis : ... | simpa only [← (hN.smul_finset n⁻¹).eq_of_inter_nonempty hK.le hS
((hN.smul_finset n⁻¹).smul_finset a⁻¹) this] using self_mem_smul_carrier a⁻¹ | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Combinatorics.HalesJewett | {
"line": 125,
"column": 6
} | {
"line": 126,
"column": 60
} | [
{
"pp": "case idxFun.h.inr.inl\nη : Type u_5\nα : Type u_6\nι : Type u_7\ninst✝ : Nontrivial α\nl m : Subspace η α ι\ni : ι\nhlm : ∀ (x : η → α) (x_1 : ι), ↑l x x_1 = ↑m x x_1\ne : η\nhl : l.idxFun i = Sum.inr e\na : α\nhm : m.idxFun i = Sum.inl a\n⊢ Sum.inr e = Sum.inl a",
"usedConstants": [
"Eq.mpr"... | obtain ⟨b, hba⟩ := exists_ne a
simpa [hl, hm, hba, coe_apply] using hlm (const _ b) i | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.HalesJewett | {
"line": 125,
"column": 6
} | {
"line": 126,
"column": 60
} | [
{
"pp": "case idxFun.h.inr.inl\nη : Type u_5\nα : Type u_6\nι : Type u_7\ninst✝ : Nontrivial α\nl m : Subspace η α ι\ni : ι\nhlm : ∀ (x : η → α) (x_1 : ι), ↑l x x_1 = ↑m x x_1\ne : η\nhl : l.idxFun i = Sum.inr e\na : α\nhm : m.idxFun i = Sum.inl a\n⊢ Sum.inr e = Sum.inl a",
"usedConstants": [
"Eq.mpr"... | obtain ⟨b, hba⟩ := exists_ne a
simpa [hl, hm, hba, coe_apply] using hlm (const _ b) i | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.Enumerative.Partition.Glaisher | {
"line": 94,
"column": 21
} | {
"line": 94,
"column": 37
} | [
{
"pp": "case h.e'_6\nR : Type u_1\ninst✝² : TopologicalSpace R\ninst✝¹ : T2Space R\ninst✝ : CommSemiring R\nm : ℕ\nhm : 0 < m\na✝ : Nontrivial R\n⊢ (PowerSeries.mk fun n ↦ ↑(#(countRestricted n m))) =\n PowerSeries.mk fun n ↦ ∑ p, (Multiset.toFinsupp p.parts).prod fun i c ↦ if c < m then 1 else 0",
"use... | countRestricted, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Topology.Compactification.StoneCech | {
"line": 118,
"column": 2
} | {
"line": 118,
"column": 52
} | [
{
"pp": "α : Type u\nb : Ultrafilter α\n⊢ Tendsto pure (↑b) (𝓝 b)",
"usedConstants": [
"Pure.pure",
"Eq.mpr",
"Ultrafilter.coe_map",
"congrArg",
"Filter.map",
"PartialOrder.toPreorder",
"Preorder.toLE",
"nhds",
"id",
"LE.le",
"Filter.Tendsto... | rw [Tendsto, ← coe_map, ultrafilter_converges_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.Compactification.StoneCech | {
"line": 183,
"column": 2
} | {
"line": 183,
"column": 78
} | [
{
"pp": "α : Type u\nγ : Type u_1\ninst✝¹ : TopologicalSpace γ\ninst✝ : T2Space γ\nf : α → γ\nthis✝ : TopologicalSpace α := ⊥\nthis : DiscreteTopology α\n⊢ Ultrafilter.extend f ∘ pure = f",
"usedConstants": [
"Pure.pure",
"Lattice.toSemilatticeSup",
"CompleteLattice.toLattice",
"Orde... | exact funext (isDenseInducing_pure.extend_eq continuous_of_discreteTopology) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Combinatorics.Matroid.Minor.Delete | {
"line": 97,
"column": 6
} | {
"line": 97,
"column": 20
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nD₁ D₂ : Set α\n⊢ M ＼ D₁ ＼ D₂ = M ＼ D₂ ＼ D₁",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Set.instUnion",
"id",
"Union.union",
"Matroid.delete_delete",
"Eq",
"Matroid",
"Matroid.delete",
"Set"
]
}
] | delete_delete, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.Matroid.Minor.Delete | {
"line": 122,
"column": 7
} | {
"line": 122,
"column": 21
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nX D : Set α\nhD : D ⊆ M.E\nh : M ＼ D ≤r M\nhX : Disjoint X (M ＼ D).E\n⊢ M ＼ D = M ＼ X ＼ (D \\ X)",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Set.instUnion",
"id",
"SDiff.sdiff",
"Union.union",
"Matroid.delete_delete",
"Eq"... | delete_delete, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.Hindman | {
"line": 149,
"column": 4
} | {
"line": 149,
"column": 34
} | [
{
"pp": "case refine_5\nM : Type u_1\ninst✝ : Semigroup M\na : Stream' M\nS : Set (Ultrafilter M) := ⋂ n, {U | ∀ᶠ (m : M) in ↑U, m ∈ FP (Stream'.drop n a)}\nU : Ultrafilter M\nhU : U ∈ S\nU_idem : U * U = U\n⊢ ∀ᶠ (m : M) in ↑U, m ∈ FP a",
"usedConstants": [
"Eq.mpr",
"Hindman.FP",
"Set.iIn... | convert Set.mem_iInter.mp hU 0 | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___elabRules_Mathlib_Tactic_convert_1 | Mathlib.Tactic.convert |
Mathlib.Combinatorics.Hindman | {
"line": 188,
"column": 2
} | {
"line": 188,
"column": 48
} | [
{
"pp": "M : Type u_1\ninst✝ : Semigroup M\nU : Ultrafilter M\nU_idem : U * U = U\ns₀ : Set M\nsU : s₀ ∈ U\nexists_elem : ∀ {s : Set M}, s ∈ U → (s ∩ {m | ∀ᶠ (m' : M) in ↑U, m * m' ∈ s}).Nonempty\nelem : { s // s ∈ U } → M := fun p ↦ ⋯.some\nsucc : { s // s ∈ U } → { s // s ∈ U } := fun p ↦ ⟨↑p ∩ {m | elem p * ... | use Stream'.corec elem succ (Subtype.mk s₀ sU) | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.Combinatorics.Matroid.Minor.Order | {
"line": 59,
"column": 2
} | {
"line": 62,
"column": 66
} | [
{
"pp": "α : Type u_1\nM N : Matroid α\nh : N ≤m M\n⊢ ∃ C D, C ⊆ M.E ∧ D ⊆ M.E ∧ Disjoint C D ∧ N = M ／ C ＼ D",
"usedConstants": [
"Set.diff_subset",
"CompleteLattice.instOmegaCompletePartialOrder",
"CompleteBooleanAlgebra.toCompleteDistribLattice",
"congrArg",
"Matroid.E",
... | obtain ⟨C, D, rfl⟩ := h
exact ⟨C ∩ M.E, (D ∩ M.E) \ C, inter_subset_right, diff_subset.trans inter_subset_right,
disjoint_sdiff_right.mono_left inter_subset_left,
by simp [delete_eq_delete_iff, inter_assoc, inter_diff_assoc]⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.Matroid.Minor.Order | {
"line": 59,
"column": 2
} | {
"line": 62,
"column": 66
} | [
{
"pp": "α : Type u_1\nM N : Matroid α\nh : N ≤m M\n⊢ ∃ C D, C ⊆ M.E ∧ D ⊆ M.E ∧ Disjoint C D ∧ N = M ／ C ＼ D",
"usedConstants": [
"Set.diff_subset",
"CompleteLattice.instOmegaCompletePartialOrder",
"CompleteBooleanAlgebra.toCompleteDistribLattice",
"congrArg",
"Matroid.E",
... | obtain ⟨C, D, rfl⟩ := h
exact ⟨C ∩ M.E, (D ∩ M.E) \ C, inter_subset_right, diff_subset.trans inter_subset_right,
disjoint_sdiff_right.mono_left inter_subset_left,
by simp [delete_eq_delete_iff, inter_assoc, inter_diff_assoc]⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.Matroid.Minor.Order | {
"line": 58,
"column": 73
} | {
"line": 62,
"column": 66
} | [
{
"pp": "α : Type u_1\nM N : Matroid α\nh : N ≤m M\n⊢ ∃ C D, C ⊆ M.E ∧ D ⊆ M.E ∧ Disjoint C D ∧ N = M ／ C ＼ D",
"usedConstants": [
"Set.diff_subset",
"CompleteLattice.instOmegaCompletePartialOrder",
"CompleteBooleanAlgebra.toCompleteDistribLattice",
"congrArg",
"Matroid.E",
... | by
obtain ⟨C, D, rfl⟩ := h
exact ⟨C ∩ M.E, (D ∩ M.E) \ C, inter_subset_right, diff_subset.trans inter_subset_right,
disjoint_sdiff_right.mono_left inter_subset_left,
by simp [delete_eq_delete_iff, inter_assoc, inter_diff_assoc]⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.Hindman | {
"line": 265,
"column": 2
} | {
"line": 272,
"column": 27
} | [
{
"pp": "case H.inr\nM : Type u_1\ninst✝ : CommMonoid M\na : Stream' M\ns : Finset ℕ\nih :\n ∀ s_1 ∈ s, ∀ (hs : (s.erase s_1).Nonempty), ∏ i ∈ s.erase s_1, a.get i ∈ FP (Stream'.drop ((s.erase s_1).min' hs) a)\nhs : s.Nonempty\nh : (s.erase (s.min' hs)).Nonempty\n⊢ (Stream'.drop (s.min' hs) a).head * ∏ x ∈ s.e... | · apply FP.cons
rw [Stream'.tail_eq_drop, Stream'.drop_drop, add_comm]
refine Set.mem_of_subset_of_mem ?_ (ih _ (s.min'_mem hs) h)
have : s.min' hs + 1 ≤ (s.erase (s.min' hs)).min' h :=
Nat.succ_le_of_lt (Finset.min'_lt_of_mem_erase_min' _ _ <| Finset.min'_mem _ _)
obtain ⟨d, hd⟩ := Nat.exists_eq_... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Combinatorics.Matroid.Minor.Contract | {
"line": 131,
"column": 58
} | {
"line": 138,
"column": 69
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nI B : Set α\nhI : M.Indep I\n⊢ (M ／ I).IsBase B ↔ M.IsBase (B ∪ I) ∧ Disjoint B I",
"usedConstants": [
"Eq.mpr",
"Set.diff_subset_iff._simp_1",
"Matroid.dual_isBase_iff'",
"CompleteLattice.instOmegaCompletePartialOrder",
"CompleteBooleanAlg... | by
rw [← dual_delete_dual, dual_isBase_iff', delete_ground, dual_ground, delete_isBase_iff,
subset_diff, ← and_assoc, and_congr_left_iff, ← dual_dual M, dual_isBase_iff', dual_dual,
dual_dual, union_comm, dual_ground, union_subset_iff, and_iff_right hI.subset_ground,
and_congr_left_iff, ← isBase_restrict_... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.Matroid.Minor.Contract | {
"line": 277,
"column": 42
} | {
"line": 277,
"column": 51
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nI J X C : Set α\nh : M.IsBasis I X\nhJC : M.IsBasis' J C\nh_ind : M.Indep (I \\ C ∪ J)\nhIX : I ⊆ X\nhJCss : J ⊆ C\n⊢ X \\ C ∪ J ⊆ X ∪ C",
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"_private.Mathlib.Combinatorics.Matroid.Minor.Contract.0.Matroid.IsBasis.... | tauto_set | Mathlib.Tactic.TautoSet._aux_Mathlib_Tactic_TautoSet___macroRules_Mathlib_Tactic_TautoSet_tacticTauto_set_1 | Mathlib.Tactic.TautoSet.tacticTauto_set |
Mathlib.Combinatorics.Matroid.Minor.Contract | {
"line": 277,
"column": 42
} | {
"line": 277,
"column": 51
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nI J X C : Set α\nh : M.IsBasis I X\nhJC : M.IsBasis' J C\nh_ind : M.Indep (I \\ C ∪ J)\nhIX : I ⊆ X\nhJCss : J ⊆ C\n⊢ X \\ C ∪ J ⊆ X ∪ C",
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"_private.Mathlib.Combinatorics.Matroid.Minor.Contract.0.Matroid.IsBasis.... | tauto_set | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.Matroid.Minor.Contract | {
"line": 277,
"column": 42
} | {
"line": 277,
"column": 51
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nI J X C : Set α\nh : M.IsBasis I X\nhJC : M.IsBasis' J C\nh_ind : M.Indep (I \\ C ∪ J)\nhIX : I ⊆ X\nhJCss : J ⊆ C\n⊢ X \\ C ∪ J ⊆ X ∪ C",
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"_private.Mathlib.Combinatorics.Matroid.Minor.Contract.0.Matroid.IsBasis.... | tauto_set | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.Matroid.Minor.Contract | {
"line": 279,
"column": 12
} | {
"line": 279,
"column": 21
} | [
{
"pp": "case refine_3\nα : Type u_1\nM : Matroid α\nI J X C : Set α\nh : M.IsBasis I X\nhJC : M.IsBasis' J C\nh_ind : M.Indep (I \\ C ∪ J)\nhIX : I ⊆ X\nhJCss : J ⊆ C\n⊢ Disjoint (I \\ C) J",
"usedConstants": [
"Eq.mpr",
"_private.Mathlib.Combinatorics.Matroid.Minor.Contract.0.Matroid.IsBasis.c... | tauto_set | Mathlib.Tactic.TautoSet._aux_Mathlib_Tactic_TautoSet___macroRules_Mathlib_Tactic_TautoSet_tacticTauto_set_1 | Mathlib.Tactic.TautoSet.tacticTauto_set |
Mathlib.Combinatorics.Matroid.Minor.Contract | {
"line": 279,
"column": 12
} | {
"line": 279,
"column": 21
} | [
{
"pp": "case refine_3\nα : Type u_1\nM : Matroid α\nI J X C : Set α\nh : M.IsBasis I X\nhJC : M.IsBasis' J C\nh_ind : M.Indep (I \\ C ∪ J)\nhIX : I ⊆ X\nhJCss : J ⊆ C\n⊢ Disjoint (I \\ C) J",
"usedConstants": [
"Eq.mpr",
"_private.Mathlib.Combinatorics.Matroid.Minor.Contract.0.Matroid.IsBasis.c... | tauto_set | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.Matroid.Minor.Contract | {
"line": 279,
"column": 12
} | {
"line": 279,
"column": 21
} | [
{
"pp": "case refine_3\nα : Type u_1\nM : Matroid α\nI J X C : Set α\nh : M.IsBasis I X\nhJC : M.IsBasis' J C\nh_ind : M.Indep (I \\ C ∪ J)\nhIX : I ⊆ X\nhJCss : J ⊆ C\n⊢ Disjoint (I \\ C) J",
"usedConstants": [
"Eq.mpr",
"_private.Mathlib.Combinatorics.Matroid.Minor.Contract.0.Matroid.IsBasis.c... | tauto_set | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.Matroid.Minor.Contract | {
"line": 279,
"column": 12
} | {
"line": 279,
"column": 21
} | [
{
"pp": "case refine_4\nα : Type u_1\nM : Matroid α\nI J X C : Set α\nh : M.IsBasis I X\nhJC : M.IsBasis' J C\nh_ind : M.Indep (I \\ C ∪ J)\nhIX : I ⊆ X\nhJCss : J ⊆ C\n⊢ Disjoint (X \\ C) J",
"usedConstants": [
"Eq.mpr",
"_private.Mathlib.Combinatorics.Matroid.Minor.Contract.0.Matroid.IsBasis.c... | tauto_set | Mathlib.Tactic.TautoSet._aux_Mathlib_Tactic_TautoSet___macroRules_Mathlib_Tactic_TautoSet_tacticTauto_set_1 | Mathlib.Tactic.TautoSet.tacticTauto_set |
Mathlib.Combinatorics.Matroid.Minor.Contract | {
"line": 279,
"column": 12
} | {
"line": 279,
"column": 21
} | [
{
"pp": "case refine_4\nα : Type u_1\nM : Matroid α\nI J X C : Set α\nh : M.IsBasis I X\nhJC : M.IsBasis' J C\nh_ind : M.Indep (I \\ C ∪ J)\nhIX : I ⊆ X\nhJCss : J ⊆ C\n⊢ Disjoint (X \\ C) J",
"usedConstants": [
"Eq.mpr",
"_private.Mathlib.Combinatorics.Matroid.Minor.Contract.0.Matroid.IsBasis.c... | tauto_set | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.Matroid.Minor.Contract | {
"line": 279,
"column": 12
} | {
"line": 279,
"column": 21
} | [
{
"pp": "case refine_4\nα : Type u_1\nM : Matroid α\nI J X C : Set α\nh : M.IsBasis I X\nhJC : M.IsBasis' J C\nh_ind : M.Indep (I \\ C ∪ J)\nhIX : I ⊆ X\nhJCss : J ⊆ C\n⊢ Disjoint (X \\ C) J",
"usedConstants": [
"Eq.mpr",
"_private.Mathlib.Combinatorics.Matroid.Minor.Contract.0.Matroid.IsBasis.c... | tauto_set | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.Matroid.Minor.Contract | {
"line": 279,
"column": 12
} | {
"line": 279,
"column": 21
} | [
{
"pp": "case refine_5\nα : Type u_1\nM : Matroid α\nI J X C : Set α\nh : M.IsBasis I X\nhJC : M.IsBasis' J C\nh_ind : M.Indep (I \\ C ∪ J)\nhIX : I ⊆ X\nhJCss : J ⊆ C\n⊢ Disjoint (X \\ C) (C \\ J)",
"usedConstants": [
"Eq.mpr",
"_private.Mathlib.Combinatorics.Matroid.Minor.Contract.0.Matroid.Is... | tauto_set | Mathlib.Tactic.TautoSet._aux_Mathlib_Tactic_TautoSet___macroRules_Mathlib_Tactic_TautoSet_tacticTauto_set_1 | Mathlib.Tactic.TautoSet.tacticTauto_set |
Mathlib.Combinatorics.Matroid.Minor.Contract | {
"line": 279,
"column": 12
} | {
"line": 279,
"column": 21
} | [
{
"pp": "case refine_5\nα : Type u_1\nM : Matroid α\nI J X C : Set α\nh : M.IsBasis I X\nhJC : M.IsBasis' J C\nh_ind : M.Indep (I \\ C ∪ J)\nhIX : I ⊆ X\nhJCss : J ⊆ C\n⊢ Disjoint (X \\ C) (C \\ J)",
"usedConstants": [
"Eq.mpr",
"_private.Mathlib.Combinatorics.Matroid.Minor.Contract.0.Matroid.Is... | tauto_set | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.Matroid.Minor.Contract | {
"line": 279,
"column": 12
} | {
"line": 279,
"column": 21
} | [
{
"pp": "case refine_5\nα : Type u_1\nM : Matroid α\nI J X C : Set α\nh : M.IsBasis I X\nhJC : M.IsBasis' J C\nh_ind : M.Indep (I \\ C ∪ J)\nhIX : I ⊆ X\nhJCss : J ⊆ C\n⊢ Disjoint (X \\ C) (C \\ J)",
"usedConstants": [
"Eq.mpr",
"_private.Mathlib.Combinatorics.Matroid.Minor.Contract.0.Matroid.Is... | tauto_set | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.Matroid.Minor.Contract | {
"line": 279,
"column": 12
} | {
"line": 279,
"column": 21
} | [
{
"pp": "case refine_1\nα : Type u_1\nM : Matroid α\nI J X C : Set α\nh : M.IsBasis I X\nhJC : M.IsBasis' J C\nh_ind : M.Indep (I \\ C ∪ J)\nhIX : I ⊆ X\nhJCss : J ⊆ C\n⊢ I \\ C ∪ J ⊆ X \\ C ∪ J",
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"instDecidableNot",
"_private.Mathlib.Combinato... | tauto_set | Mathlib.Tactic.TautoSet._aux_Mathlib_Tactic_TautoSet___macroRules_Mathlib_Tactic_TautoSet_tacticTauto_set_1 | Mathlib.Tactic.TautoSet.tacticTauto_set |
Mathlib.Combinatorics.Matroid.Minor.Contract | {
"line": 279,
"column": 12
} | {
"line": 279,
"column": 21
} | [
{
"pp": "case refine_1\nα : Type u_1\nM : Matroid α\nI J X C : Set α\nh : M.IsBasis I X\nhJC : M.IsBasis' J C\nh_ind : M.Indep (I \\ C ∪ J)\nhIX : I ⊆ X\nhJCss : J ⊆ C\n⊢ I \\ C ∪ J ⊆ X \\ C ∪ J",
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"instDecidableNot",
"_private.Mathlib.Combinato... | tauto_set | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.Matroid.Minor.Contract | {
"line": 279,
"column": 12
} | {
"line": 279,
"column": 21
} | [
{
"pp": "case refine_1\nα : Type u_1\nM : Matroid α\nI J X C : Set α\nh : M.IsBasis I X\nhJC : M.IsBasis' J C\nh_ind : M.Indep (I \\ C ∪ J)\nhIX : I ⊆ X\nhJCss : J ⊆ C\n⊢ I \\ C ∪ J ⊆ X \\ C ∪ J",
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"instDecidableNot",
"_private.Mathlib.Combinato... | tauto_set | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.Matroid.Minor.Contract | {
"line": 337,
"column": 63
} | {
"line": 337,
"column": 72
} | [
{
"pp": "α : Type u_1\nM M' N : Matroid α\ne f : α\nI✝ J✝ R D B X Y Z K✝ C C₁ C₂ : Set α\ninst✝ : M.Finitary\nJ : Set α\nhJ : M.IsBasis' J C\nI : Set α\nhI : ∀ J_1 ⊆ I, J_1.Finite → (M ／ J).Indep J_1\nK : Set α\nhK : K ⊆ I ∪ J\nhKfin : K.Finite\n⊢ K ⊆ K ∩ I ∪ J",
"usedConstants": [
"Iff.mpr",
"E... | tauto_set | Mathlib.Tactic.TautoSet._aux_Mathlib_Tactic_TautoSet___macroRules_Mathlib_Tactic_TautoSet_tacticTauto_set_1 | Mathlib.Tactic.TautoSet.tacticTauto_set |
Mathlib.Combinatorics.Matroid.Minor.Contract | {
"line": 337,
"column": 63
} | {
"line": 337,
"column": 72
} | [
{
"pp": "α : Type u_1\nM M' N : Matroid α\ne f : α\nI✝ J✝ R D B X Y Z K✝ C C₁ C₂ : Set α\ninst✝ : M.Finitary\nJ : Set α\nhJ : M.IsBasis' J C\nI : Set α\nhI : ∀ J_1 ⊆ I, J_1.Finite → (M ／ J).Indep J_1\nK : Set α\nhK : K ⊆ I ∪ J\nhKfin : K.Finite\n⊢ K ⊆ K ∩ I ∪ J",
"usedConstants": [
"Iff.mpr",
"E... | tauto_set | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.Matroid.Minor.Contract | {
"line": 337,
"column": 63
} | {
"line": 337,
"column": 72
} | [
{
"pp": "α : Type u_1\nM M' N : Matroid α\ne f : α\nI✝ J✝ R D B X Y Z K✝ C C₁ C₂ : Set α\ninst✝ : M.Finitary\nJ : Set α\nhJ : M.IsBasis' J C\nI : Set α\nhI : ∀ J_1 ⊆ I, J_1.Finite → (M ／ J).Indep J_1\nK : Set α\nhK : K ⊆ I ∪ J\nhKfin : K.Finite\n⊢ K ⊆ K ∩ I ∪ J",
"usedConstants": [
"Iff.mpr",
"E... | tauto_set | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.Matroid.Minor.Contract | {
"line": 396,
"column": 4
} | {
"line": 396,
"column": 18
} | [
{
"pp": "α✝ α : Type u_1\nM : Matroid α\nC : Set α\nhCE : C ⊆ M.E\nI : Set α\nhI : M.IsBasis I C\n⊢ M ／ I ＼ (M.closure C \\ I) = M ／ I ＼ (C \\ I) ＼ (M.closure C \\ C)",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Set.instUnion",
"id",
"SDiff.sdiff",
"Matroid.closure",
... | delete_delete, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.Matroid.Minor.Contract | {
"line": 428,
"column": 24
} | {
"line": 428,
"column": 33
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nX : Set α\nhX : M.Spanning X\nC : Set α\nhXE : X ⊆ M.E\n⊢ X ⊆ X \\ C ∪ C ∩ M.E",
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"instDecidableNot",
"instDecidableTrue",
"congrArg",
"Matroid.E",
"Compl.compl",
"False.elim",
... | tauto_set | Mathlib.Tactic.TautoSet._aux_Mathlib_Tactic_TautoSet___macroRules_Mathlib_Tactic_TautoSet_tacticTauto_set_1 | Mathlib.Tactic.TautoSet.tacticTauto_set |
Mathlib.Combinatorics.Matroid.Minor.Contract | {
"line": 428,
"column": 24
} | {
"line": 428,
"column": 33
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nX : Set α\nhX : M.Spanning X\nC : Set α\nhXE : X ⊆ M.E\n⊢ X ⊆ X \\ C ∪ C ∩ M.E",
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"instDecidableNot",
"instDecidableTrue",
"congrArg",
"Matroid.E",
"Compl.compl",
"False.elim",
... | tauto_set | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.Matroid.Minor.Contract | {
"line": 428,
"column": 24
} | {
"line": 428,
"column": 33
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nX : Set α\nhX : M.Spanning X\nC : Set α\nhXE : X ⊆ M.E\n⊢ X ⊆ X \\ C ∪ C ∩ M.E",
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"instDecidableNot",
"instDecidableTrue",
"congrArg",
"Matroid.E",
"Compl.compl",
"False.elim",
... | tauto_set | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.Matroid.Minor.Contract | {
"line": 428,
"column": 39
} | {
"line": 428,
"column": 48
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nX : Set α\nhX : M.Spanning X\nC : Set α\nhXE : X ⊆ M.E\n⊢ X \\ C ∪ C ∩ M.E ⊆ M.E",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Matroid.E",
"Compl.compl",
"False.elim",
"Classical.propDecidable",
"Membership.mem",
"Set.instUn... | tauto_set | Mathlib.Tactic.TautoSet._aux_Mathlib_Tactic_TautoSet___macroRules_Mathlib_Tactic_TautoSet_tacticTauto_set_1 | Mathlib.Tactic.TautoSet.tacticTauto_set |
Mathlib.Combinatorics.Matroid.Minor.Contract | {
"line": 428,
"column": 39
} | {
"line": 428,
"column": 48
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nX : Set α\nhX : M.Spanning X\nC : Set α\nhXE : X ⊆ M.E\n⊢ X \\ C ∪ C ∩ M.E ⊆ M.E",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Matroid.E",
"Compl.compl",
"False.elim",
"Classical.propDecidable",
"Membership.mem",
"Set.instUn... | tauto_set | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.Matroid.Minor.Contract | {
"line": 428,
"column": 39
} | {
"line": 428,
"column": 48
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nX : Set α\nhX : M.Spanning X\nC : Set α\nhXE : X ⊆ M.E\n⊢ X \\ C ∪ C ∩ M.E ⊆ M.E",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Matroid.E",
"Compl.compl",
"False.elim",
"Classical.propDecidable",
"Membership.mem",
"Set.instUn... | tauto_set | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.Matroid.Minor.Contract | {
"line": 513,
"column": 2
} | {
"line": 513,
"column": 11
} | [
{
"pp": "α : Type u_1\nR C : Set α\nM : Matroid α\nh : Disjoint C R\nI : Set α\nhI : I ⊆ R\nJ : Set α\nhJ : (M ↾ (R ∪ C)).IsBasis' J C\nhJ' : M.IsBasis' J C\nhJC : J ⊆ C\n⊢ (M.Indep (I ∪ J) ∧ Disjoint C I) ∧ I ⊆ R ↔ (M.Indep (I ∪ J) ∧ I ∪ J ⊆ R ∪ C) ∧ Disjoint C I",
"usedConstants": [
"Iff.mpr",
... | tauto_set | Mathlib.Tactic.TautoSet._aux_Mathlib_Tactic_TautoSet___macroRules_Mathlib_Tactic_TautoSet_tacticTauto_set_1 | Mathlib.Tactic.TautoSet.tacticTauto_set |
Mathlib.RingTheory.MvPolynomial.Groebner | {
"line": 123,
"column": 12
} | {
"line": 123,
"column": 23
} | [
{
"pp": "case right\nσ : Type u_1\nm : MonomialOrder σ\nR : Type u_2\ninst✝ : CommRing R\nf b : MvPolynomial σ R\nhb : IsUnit (m.leadingCoeff b)\nhbf : m.degree b ≤ m.degree f\nhf : m.degree f ≠ 0\nH : m.degree f = m.degree ((monomial (m.degree f - m.degree b)) (↑hb.unit⁻¹ * m.leadingCoeff f)) + m.degree b\nH' ... | degree_zero | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.Schnirelmann | {
"line": 204,
"column": 49
} | {
"line": 204,
"column": 74
} | [
{
"pp": "A : Finset ℕ\nε : ℝ\nhε : 0 < ε\nhε₁ : ε ≤ 1\nn : ℕ := ⌊↑(#A) / ε⌋₊ + 1\nhn : 0 < n\n⊢ ↑(#({a ∈ Ioc 0 n | a ∈ ↑A})) / ε ≤ ↑(#A) / ε",
"usedConstants": [
"SetLike.mem_coe._simp_1",
"Real.partialOrder",
"Real",
"Finset.mem_filter._simp_1",
"Preorder.toLT",
"NonUnit... | gcongr; simp [subset_iff] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.Schnirelmann | {
"line": 204,
"column": 49
} | {
"line": 204,
"column": 74
} | [
{
"pp": "A : Finset ℕ\nε : ℝ\nhε : 0 < ε\nhε₁ : ε ≤ 1\nn : ℕ := ⌊↑(#A) / ε⌋₊ + 1\nhn : 0 < n\n⊢ ↑(#({a ∈ Ioc 0 n | a ∈ ↑A})) / ε ≤ ↑(#A) / ε",
"usedConstants": [
"SetLike.mem_coe._simp_1",
"Real.partialOrder",
"Real",
"Finset.mem_filter._simp_1",
"Preorder.toLT",
"NonUnit... | gcongr; simp [subset_iff] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.Schnirelmann | {
"line": 262,
"column": 2
} | {
"line": 262,
"column": 24
} | [
{
"pp": "case inl\n⊢ schnirelmannDensity {n | n ≡ 1 [MOD 1]} = (↑1)⁻¹",
"usedConstants": [
"Set.decidableUniv",
"Real",
"InvOneClass.toOne",
"DivisionCommMonoid.toDivisionMonoid",
"DivInvOneMonoid.toInvOneClass",
"inv_one",
"Nat.instDecidableModEq",
"schnirelm... | · simp [Nat.modEq_one] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Combinatorics.SetFamily.Compression.Down | {
"line": 171,
"column": 2
} | {
"line": 171,
"column": 56
} | [
{
"pp": "α : Type u_1\ninst✝ : DecidableEq α\np : Finset (Finset α) → Prop\n𝒜 : Finset (Finset α)\nempty : p ∅\nsingleton_empty : p {∅}\nsubfamily : ∀ (a : α) ⦃𝒜 : Finset (Finset α)⦄, p (nonMemberSubfamily a 𝒜) → p (memberSubfamily a 𝒜) → p 𝒜\nu : Finset α\nhu : ∀ s ∈ 𝒜, s ⊆ u\n⊢ p 𝒜",
"usedConstants... | induction u using Finset.induction generalizing 𝒜 with | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.Combinatorics.SetFamily.AhlswedeZhang | {
"line": 394,
"column": 2
} | {
"line": 395,
"column": 76
} | [
{
"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 ... | rw [sum_congr rfl fun t _ ↦ this t, sum_ite, sum_const_zero, add_zero, filter_subset_univ,
sum_powerset, ← binomial_sum_eq ((card_lt_iff_ne_univ _).2 hs), eq_comm] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Combinatorics.SetFamily.HarrisKleitman | {
"line": 69,
"column": 67
} | {
"line": 69,
"column": 75
} | [
{
"pp": "case insert\nα : Type u_1\ninst✝ : DecidableEq α\na : α\ns : Finset α\nhs : a ∉ s\nih :\n ∀ {𝒜 ℬ : Finset (Finset α)},\n IsLowerSet ↑𝒜 → IsLowerSet ↑ℬ → (∀ t ∈ 𝒜, t ⊆ s) → (∀ t ∈ ℬ, t ⊆ s) → #𝒜 * #ℬ ≤ 2 ^ #s * #(𝒜 ∩ ℬ)\n𝒜 ℬ : Finset (Finset α)\nh𝒜 : IsLowerSet ↑𝒜\nhℬ : IsLowerSet ↑ℬ\nh𝒜s :... | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.SetFamily.HarrisKleitman | {
"line": 69,
"column": 76
} | {
"line": 69,
"column": 84
} | [
{
"pp": "case insert\nα : Type u_1\ninst✝ : DecidableEq α\na : α\ns : Finset α\nhs : a ∉ s\nih :\n ∀ {𝒜 ℬ : Finset (Finset α)},\n IsLowerSet ↑𝒜 → IsLowerSet ↑ℬ → (∀ t ∈ 𝒜, t ⊆ s) → (∀ t ∈ ℬ, t ⊆ s) → #𝒜 * #ℬ ≤ 2 ^ #s * #(𝒜 ∩ ℬ)\n𝒜 ℬ : Finset (Finset α)\nh𝒜 : IsLowerSet ↑𝒜\nhℬ : IsLowerSet ↑ℬ\nh𝒜s :... | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.