module stringlengths 16 90 | startPos dict | endPos dict | nextStartPos dict | goals listlengths 0 96 | goalsAfter listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 375
values | kind stringclasses 379
values |
|---|---|---|---|---|---|---|---|---|
Mathlib.Combinatorics.SetFamily.Compression.UV | {
"line": 136,
"column": 10
} | {
"line": 136,
"column": 12
} | {
"line": 136,
"column": 13
} | [
{
"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\n⊢ a ∈ ↑({a ∈ s | compress u v a ∉ s}) →\n ∀ ⦃x₂ : α⦄, x₂ ∈ ↑({a ∈ s | compress u v a ∉ s}) → compress u v a = compress u v x₂ → a = x₂",
... | [
"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})\n⊢ ∀ ⦃x₂ : α⦄, x₂ ∈ ↑({a ∈ s | compress u v a ∉ s}) → compress u v a = compress u v x₂ → a = x₂"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Combinatorics.SetFamily.AhlswedeZhang | {
"line": 279,
"column": 2
} | {
"line": 279,
"column": 31
} | {
"line": 280,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝³ : DistribLattice α\ninst✝² : DecidableEq α\ns t : Finset α\na : α\ninst✝¹ : DecidableLE α\ninst✝ : BoundedOrder α\nhs : a ∈ lowerClosure ↑s\nht : a ∈ lowerClosure ↑t\n⊢ (({b ∈ s | a ≤ b} ⊼ {b ∈ t | a ≤ b}).sup' ⋯ fun x ↦ id x) =\n ({b ∈ s | a ≤ b} ×ˢ {b ∈ t | a ≤ b}).sup' ⋯ fun ... | [
"α : Type u_1\ninst✝³ : DistribLattice α\ninst✝² : DecidableEq α\ns t : Finset α\na : α\ninst✝¹ : DecidableLE α\ninst✝ : BoundedOrder α\nhs : a ∈ lowerClosure ↑s\nht : a ∈ lowerClosure ↑t\n⊢ ((image (Function.uncurry fun x1 x2 ↦ x1 ⊓ x2) ({b ∈ s | a ≤ b} ×ˢ {b ∈ t | a ≤ b})).sup' ⋯ fun x ↦ id x) =\n ({b ∈ s | a ... | simp_rw [← image_inf_product] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Combinatorics.SetFamily.Compression.UV | {
"line": 150,
"column": 47
} | {
"line": 150,
"column": 57
} | {
"line": 150,
"column": 58
} | [
{
"pp": "α : Type u_1\ninst✝³ : GeneralizedBooleanAlgebra α\ninst✝² : DecidableRel Disjoint\ninst✝¹ : DecidableLE α\ns : Finset α\nu v a : α\ninst✝ : DecidableEq α\n⊢ a ∈ s ∧ compress u v a ∈ s ∨ a ∈ image (compress u v) s ∧ a ∉ s ↔\n a ∈ s ∧ compress u v a ∈ s ∨ a ∉ s ∧ ∃ b ∈ s, compress u v b = a",
"pp... | [
"α : Type u_1\ninst✝³ : GeneralizedBooleanAlgebra α\ninst✝² : DecidableRel Disjoint\ninst✝¹ : DecidableLE α\ns : Finset α\nu v a : α\ninst✝ : DecidableEq α\n⊢ a ∈ s ∧ compress u v a ∈ s ∨ (∃ a_1 ∈ s, compress u v a_1 = a) ∧ a ∉ s ↔\n a ∈ s ∧ compress u v a ∈ s ∨ a ∉ s ∧ ∃ a_1 ∈ s, compress u v a_1 = a"
] | mem_image, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Combinatorics.SetFamily.Shadow | {
"line": 109,
"column": 2
} | {
"line": 110,
"column": 7
} | {
"line": 112,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nt : Finset α\n⊢ t ∈ ∂ 𝒜 ↔ ∃ a ∉ t, insert a t ∈ 𝒜",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"False",
"Finset.shadow",
"Preorder.toLT",
"eq_false",
"and_true",
"... | [] | simp_rw [mem_shadow_iff_exists_sdiff, ← covBy_iff_card_sdiff_eq_one, covBy_iff_exists_insert]
aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SetFamily.Shadow | {
"line": 109,
"column": 2
} | {
"line": 110,
"column": 7
} | {
"line": 112,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nt : Finset α\n⊢ t ∈ ∂ 𝒜 ↔ ∃ a ∉ t, insert a t ∈ 𝒜",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"False",
"Finset.shadow",
"Preorder.toLT",
"eq_false",
"and_true",
"... | [] | simp_rw [mem_shadow_iff_exists_sdiff, ← covBy_iff_card_sdiff_eq_one, covBy_iff_exists_insert]
aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SetFamily.Shadow | {
"line": 205,
"column": 30
} | {
"line": 205,
"column": 40
} | {
"line": 205,
"column": 41
} | [
{
"pp": "α : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\n𝒜 : Finset (Finset α)\nt : Finset α\n⊢ (∃ i ∈ 𝒜, t ∈ image (fun a ↦ insert a i) iᶜ) ↔ ∃ s ∈ 𝒜, ∃ a ∉ s, insert a s = t",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"_private.Mathlib.Combinatorics.... | [
"α : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\n𝒜 : Finset (Finset α)\nt : Finset α\n⊢ (∃ i ∈ 𝒜, ∃ a ∈ iᶜ, insert a i = t) ↔ ∃ s ∈ 𝒜, ∃ a ∉ s, insert a s = t"
] | mem_image, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Combinatorics.SetFamily.AhlswedeZhang | {
"line": 336,
"column": 2
} | {
"line": 337,
"column": 59
} | {
"line": 338,
"column": 2
} | [
{
"pp": "case pos\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\n𝒜 ℬ : Finset (Finset α)\ns : Finset α\nh𝒜 : s ∉ upperClosure ↑𝒜\nhℬ : s ∈ upperClosure ↑ℬ\n⊢ #((𝒜 ∪ ℬ).truncatedInf s) + #((𝒜 ⊻ ℬ).truncatedInf s) = #(𝒜.truncatedInf s) + #(ℬ.truncatedInf s)",
"ppTerm": "?pos✝",
"assigned"... | [
"case neg\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\n𝒜 ℬ : Finset (Finset α)\ns : Finset α\nh𝒜 : s ∉ upperClosure ↑𝒜\nhℬ : s ∉ upperClosure ↑ℬ\n⊢ #((𝒜 ∪ ℬ).truncatedInf s) + #((𝒜 ⊻ ℬ).truncatedInf s) = #(𝒜.truncatedInf s) + #(ℬ.truncatedInf s)"
] | · rw [truncatedInf_union_right h𝒜 hℬ, truncatedInf_of_notMem h𝒜,
truncatedInf_sups_of_notMem fun h ↦ h𝒜 h.1, add_comm] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Combinatorics.SetFamily.Compression.UV | {
"line": 244,
"column": 11
} | {
"line": 244,
"column": 14
} | {
"line": 244,
"column": 15
} | [
{
"pp": "case inr\nα : Type u_1\ninst✝³ : GeneralizedBooleanAlgebra α\ninst✝² : DecidableRel Disjoint\ninst✝¹ : DecidableLE α\ns : Finset α\nu v : α\ninst✝ : DecidableEq α\nb : α\nhb : b ∈ s\nhva : v ≤ compress u v b\nhua : Disjoint u (compress u v b)\nleft✝ : compress u v b ∉ s\nhu : u = ⊥\nhv : v = ⊥\n⊢ (comp... | [
"case inr\nα : Type u_1\ninst✝³ : GeneralizedBooleanAlgebra α\ninst✝² : DecidableRel Disjoint\ninst✝¹ : DecidableLE α\ns : Finset α\nu v : α\ninst✝ : DecidableEq α\nb : α\nhb : b ∈ s\nhva : v ≤ compress u v b\nhua : Disjoint u (compress u v b)\nleft✝ : compress u v b ∉ s\nhu : u = ⊥\nhv : v = ⊥\n⊢ (compress ⊥ ⊥ b ⊔... | hv, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.SetFamily.AhlswedeZhang | {
"line": 420,
"column": 2
} | {
"line": 420,
"column": 28
} | {
"line": 421,
"column": 2
} | [
{
"pp": "case ind.inr.zero\nα : Type u_1\ninst✝² : Fintype α\ninst✝¹ : DecidableEq α\ninst✝ : Nonempty α\n𝒜 : Finset (Finset α)\nh𝒜₁ : 𝒜.Nonempty\nh𝒜₂ : univ ∉ 𝒜\nh𝒜₃ : 𝒜.Nontrivial\nhm : 0 = #𝒜\nih : ∀ (𝒜 : Finset (Finset α)), #𝒜 < 0 → 𝒜.Nonempty → univ ∉ 𝒜 → supSum 𝒜 = ↑(card α) * ∑ k ∈ range (ca... | [
"case ind.inr.succ\nα : Type u_1\ninst✝² : Fintype α\ninst✝¹ : DecidableEq α\ninst✝ : Nonempty α\n𝒜 : Finset (Finset α)\nh𝒜₁ : 𝒜.Nonempty\nh𝒜₂ : univ ∉ 𝒜\nh𝒜₃ : 𝒜.Nontrivial\nn✝ : ℕ\nhm : n✝ + 1 = #𝒜\nih :\n ∀ (𝒜 : Finset (Finset α)), #𝒜 < n✝ + 1 → 𝒜.Nonempty → univ ∉ 𝒜 → supSum 𝒜 = ↑(card α) * ∑ k ∈ ... | · cases h𝒜₁.card_pos.ne hm | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Combinatorics.SetFamily.Shatter | {
"line": 69,
"column": 25
} | {
"line": 69,
"column": 35
} | {
"line": 69,
"column": 36
} | [
{
"pp": "α : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\ns : Finset α\nh : 𝒜.Shatters s\nt : Finset α\n⊢ t ∈ image (fun t ↦ s ∩ t) 𝒜 ↔ t ∈ s.powerset",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Finset.mem_image",
"congrArg",
"Finset",
... | [
"α : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\ns : Finset α\nh : 𝒜.Shatters s\nt : Finset α\n⊢ (∃ a ∈ 𝒜, s ∩ a = t) ↔ t ∈ s.powerset"
] | mem_image, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.SetFamily.KruskalKatona | {
"line": 167,
"column": 43
} | {
"line": 167,
"column": 58
} | {
"line": 167,
"column": 58
} | [
{
"pp": "case inr.refine_2\nα : Type u_1\ninst✝ : LinearOrder α\nU V : Finset α\n𝒜 : Finset (Finset α)\nh₂ : ∀ ⦃U₁ V₁ : Finset α⦄, UsefulCompression U₁ V₁ → #U₁ < #U → IsCompressed U₁ V₁ 𝒜\nUVd : Disjoint U V\nsame_size : #U = #V\nhU : U.Nonempty\nhV : V.Nonempty\nmax_lt : U.max' hU < V.max' hV\nx : α\nHx : x... | [
"case inr.refine_2\nα : Type u_1\ninst✝ : LinearOrder α\nU V : Finset α\n𝒜 : Finset (Finset α)\nh₂ : ∀ ⦃U₁ V₁ : Finset α⦄, UsefulCompression U₁ V₁ → #U₁ < #U → IsCompressed U₁ V₁ 𝒜\nUVd : Disjoint U V\nsame_size : #U = #V\nhU : U.Nonempty\nhV : V.Nonempty\nmax_lt : U.max' hU < V.max' hV\nx : α\nHx : x ∈ U\nhU' : ... | tsub_pos_iff_lt | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.SetFamily.KruskalKatona | {
"line": 168,
"column": 75
} | {
"line": 168,
"column": 90
} | {
"line": 168,
"column": 90
} | [
{
"pp": "case inr.refine_3\nα : Type u_1\ninst✝ : LinearOrder α\nU V : Finset α\n𝒜 : Finset (Finset α)\nh₂ : ∀ ⦃U₁ V₁ : Finset α⦄, UsefulCompression U₁ V₁ → #U₁ < #U → IsCompressed U₁ V₁ 𝒜\nUVd : Disjoint U V\nsame_size : #U = #V\nhU : U.Nonempty\nhV : V.Nonempty\nmax_lt : U.max' hU < V.max' hV\nx : α\nHx : x... | [
"case inr.refine_3\nα : Type u_1\ninst✝ : LinearOrder α\nU V : Finset α\n𝒜 : Finset (Finset α)\nh₂ : ∀ ⦃U₁ V₁ : Finset α⦄, UsefulCompression U₁ V₁ → #U₁ < #U → IsCompressed U₁ V₁ 𝒜\nUVd : Disjoint U V\nsame_size : #U = #V\nhU : U.Nonempty\nhV : V.Nonempty\nmax_lt : U.max' hU < V.max' hV\nx : α\nHx : x ∈ U\nhU' : ... | tsub_pos_iff_lt | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.SetFamily.Shatter | {
"line": 188,
"column": 64
} | {
"line": 188,
"column": 87
} | {
"line": 190,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : DecidableEq α\n𝒜 ℬ : Finset (Finset α)\nh𝒜ℬ : 𝒜 ⊆ ℬ\n⊢ 𝒜.vcDim ≤ ℬ.vcDim",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Lattice.toSemilatticeSup",
"instDistribLatticeNat",
"Finset.shatterer_mono",
"Finset",
"DistribLattice.t... | [] | by unfold vcDim; gcongr | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.SimpleGraph.Connectivity.Connected | {
"line": 618,
"column": 2
} | {
"line": 618,
"column": 42
} | {
"line": 619,
"column": 2
} | [
{
"pp": "V : Type u\nG G' : SimpleGraph V\nh : G ≤ G'\nc' : G'.ConnectedComponent\nv : V\n⊢ (∃ i, ∃ (_ : i.supp ⊆ c'.supp), v ∈ i.supp) ↔ v ∈ c'.supp",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Membership.mem",
"Exists",
"LE.le",
"SimpleGraph.ConnectedComponent... | [
"V : Type u\nG G' : SimpleGraph V\nh : G ≤ G'\nc' : G'.ConnectedComponent\nv : V\n⊢ v ∈ c'.supp → ∃ i, ∃ (_ : i.supp ⊆ c'.supp), v ∈ i.supp"
] | refine ⟨fun ⟨_, ⟨hi, hi'⟩⟩ ↦ hi hi', ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Combinatorics.SimpleGraph.Connectivity.Connected | {
"line": 811,
"column": 47
} | {
"line": 811,
"column": 67
} | {
"line": 819,
"column": 2
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\ninst✝ : DecidableEq V\nu v w : V\nhb : ∀ (p : G.Walk v w), s(v, w) ∈ p.edges\nc : G.Walk u u\nhc : c.IsTrail\nhe : s(v, w) ∈ c.edges\nhw : w ∈ (c.takeUntil v ⋯).support\nhv : v ∈ c.support\npuw : G.Walk u w := (c.takeUntil v hv).takeUntil w hw\npwv : G.Walk w v := (c.take... | [] | simp [puw, pwv, pvu] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Combinatorics.SimpleGraph.Coloring.Vertex | {
"line": 498,
"column": 2
} | {
"line": 500,
"column": 52
} | {
"line": 502,
"column": 0
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nu v : V\nhadj : G.Adj u v\n⊢ 2 ≤ G.chromaticNumber",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"False",
"Preorder.toLT",
"instAddMonoidWithOneENat",
"ENat.instNatCast",
"RelHom.instFunLike",
"instLinearOrderENat... | [] | refine le_of_not_gt fun h ↦ ?_
obtain ⟨c⟩ := chromaticNumber_le_iff_colorable.mp (Order.le_of_lt_add_one h)
exact c.valid hadj (Subsingleton.elim (c u) (c v)) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SimpleGraph.Coloring.Vertex | {
"line": 498,
"column": 2
} | {
"line": 500,
"column": 52
} | {
"line": 502,
"column": 0
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nu v : V\nhadj : G.Adj u v\n⊢ 2 ≤ G.chromaticNumber",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"False",
"Preorder.toLT",
"instAddMonoidWithOneENat",
"ENat.instNatCast",
"RelHom.instFunLike",
"instLinearOrderENat... | [] | refine le_of_not_gt fun h ↦ ?_
obtain ⟨c⟩ := chromaticNumber_le_iff_colorable.mp (Order.le_of_lt_add_one h)
exact c.valid hadj (Subsingleton.elim (c u) (c v)) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SimpleGraph.Coloring.Vertex | {
"line": 514,
"column": 84
} | {
"line": 514,
"column": 100
} | {
"line": 515,
"column": 4
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\n⊢ G.chromaticNumber ≤ ↑1 ∧ ¬G.chromaticNumber ≤ 0 ↔ G = ⊥ ∧ ¬IsEmpty V",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instCompleteLinearOrderENat",
"instAddMonoidWithOneENat",
"ChainCompletePartialOrder.instOfCompl... | [
"V : Type u\nG : SimpleGraph V\n⊢ G.chromaticNumber ≤ ↑1 ∧ ¬G.chromaticNumber ≤ ↑0 ↔ G = ⊥ ∧ ¬IsEmpty V"
] | ← Nat.cast_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.SimpleGraph.Bipartite | {
"line": 250,
"column": 4
} | {
"line": 250,
"column": 52
} | {
"line": 250,
"column": 52
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\ns t : Finset V\ninst✝ : G.LocallyFinite\nh : G.IsBipartiteWith ↑s ↑t\n⊢ ∑ x ∈ s, #(bipartiteAbove G.Adj t x) = ∑ w ∈ t.attach, #(bipartiteBelow G.Adj s ↑w)",
"ppTerm": "?m.73",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Fin... | [
"V : Type u_1\nG : SimpleGraph V\ns t : Finset V\ninst✝ : G.LocallyFinite\nh : G.IsBipartiteWith ↑s ↑t\n⊢ ∑ x ∈ s, #(bipartiteAbove G.Adj t x) = ∑ x ∈ t, #(bipartiteBelow G.Adj s x)"
] | sum_attach t fun v ↦ #(bipartiteBelow G.Adj s v) | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Combinatorics.SimpleGraph.Bipartite | {
"line": 258,
"column": 6
} | {
"line": 258,
"column": 18
} | {
"line": 258,
"column": 19
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\ns t : Finset V\ninst✝² : Fintype V\ninst✝¹ : DecidableRel G.Adj\ninst✝ : DecidableEq V\nh : G.IsBipartiteWith ↑s ↑t\nhsub : G.support ⊆ ↑s ∪ ↑t\n⊢ ∑ v ∈ s ∪ t, G.degree v = 2 * #G.edgeFinset",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"Fi... | [
"V : Type u_1\nG : SimpleGraph V\ns t : Finset V\ninst✝² : Fintype V\ninst✝¹ : DecidableRel G.Adj\ninst✝ : DecidableEq V\nh : G.IsBipartiteWith ↑s ↑t\nhsub : G.support ⊆ ↑(s ∪ t)\n⊢ ∑ v ∈ s ∪ t, G.degree v = 2 * #G.edgeFinset"
] | ← coe_union, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.SimpleGraph.Connectivity.EdgeConnectivity | {
"line": 152,
"column": 91
} | {
"line": 153,
"column": 32
} | {
"line": 155,
"column": 0
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\n⊢ G.IsEdgeConnected 2 ↔ ∀ (e : Sym2 V), (G.deleteEdges {e}).Preconnected",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"SimpleGraph.deleteEdges",
"SimpleGraph.isEdgeConnected_one._simp_1",
"False",
"Nat.instMulZeroClass",
... | [] | by
simp [isEdgeConnected_add_one] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.SimpleGraph.Connectivity.Subgraph | {
"line": 212,
"column": 49
} | {
"line": 212,
"column": 60
} | {
"line": 212,
"column": 61
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nu v : V\np : G.Walk u v\ne : Sym2 V\n⊢ e ∈ p.toSubgraph.edgeSet ↔ e ∈ p.edges",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"SimpleGraph.Adj",
"SimpleGraph.Walk",
"Membership.mem",
"SimpleGraph.Walk.toSubgraph",
"List"... | [
"case nil\nV : Type u\nG : SimpleGraph V\nu v : V\ne : Sym2 V\nu✝ : V\n⊢ e ∈ nil.toSubgraph.edgeSet ↔ e ∈ nil.edges",
"case cons\nV : Type u\nG : SimpleGraph V\nu v : V\ne : Sym2 V\nu✝ v✝ w✝ : V\nh✝ : G.Adj u✝ v✝\np✝ : G.Walk v✝ w✝\np_ih✝ : e ∈ p✝.toSubgraph.edgeSet ↔ e ∈ p✝.edges\n⊢ e ∈ (cons h✝ p✝).toSubgraph.e... | induction p | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.Combinatorics.SimpleGraph.Connectivity.Subgraph | {
"line": 224,
"column": 64
} | {
"line": 224,
"column": 75
} | {
"line": 224,
"column": 76
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nu v w : V\np : G.Walk u v\nq : G.Walk v w\n⊢ (p.append q).toSubgraph = p.toSubgraph ⊔ q.toSubgraph",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"SimpleGraph.Subgraph",
"SimpleGraph.Adj",
"SimpleGraph.Walk",
"SimpleGraph.Wal... | [
"case nil\nV : Type u\nG : SimpleGraph V\nu v w u✝ : V\nq : G.Walk u✝ w\n⊢ (nil.append q).toSubgraph = nil.toSubgraph ⊔ q.toSubgraph",
"case cons\nV : Type u\nG : SimpleGraph V\nu v w u✝ v✝ w✝ : V\nh✝ : G.Adj u✝ v✝\np✝ : G.Walk v✝ w✝\np_ih✝ : ∀ (q : G.Walk w✝ w), (p✝.append q).toSubgraph = p✝.toSubgraph ⊔ q.toSub... | induction p | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.Combinatorics.SimpleGraph.Connectivity.Subgraph | {
"line": 243,
"column": 52
} | {
"line": 243,
"column": 63
} | {
"line": 243,
"column": 64
} | [
{
"pp": "V : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\nu v : V\nf : G →g G'\np : G.Walk u v\n⊢ (Walk.map f p).toSubgraph = Subgraph.map f p.toSubgraph",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"SimpleGraph.Walk.map",
"RelHom.instFunLike",
"SimpleG... | [
"case nil\nV : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\nu v : V\nf : G →g G'\nu✝ : V\n⊢ (Walk.map f nil).toSubgraph = Subgraph.map f nil.toSubgraph",
"case cons\nV : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\nu v : V\nf : G →g G'\nu✝ v✝ w✝ : V\nh✝ : G.Adj u✝ v✝\np✝ : G.Walk v... | induction p | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.Combinatorics.SimpleGraph.Connectivity.Subgraph | {
"line": 447,
"column": 6
} | {
"line": 447,
"column": 13
} | {
"line": 447,
"column": 14
} | [
{
"pp": "case neg\nV : Type u\nG : SimpleGraph V\nu v : V\np : G.Walk u u\nhpc : p.IsCycle\ni : ℕ\nhi : p.getVert i = v ∧ i ≤ p.length\nhe : i ≠ 0 ∧ i ≠ p.length\n⊢ (p.toSubgraph.neighborSet v).ncard = 2",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
... | [
"case neg\nV : Type u\nG : SimpleGraph V\nu v : V\np : G.Walk u u\nhpc : p.IsCycle\ni : ℕ\nhi : p.getVert i = v ∧ i ≤ p.length\nhe : i ≠ 0 ∧ i ≠ p.length\n⊢ (p.toSubgraph.neighborSet (p.getVert i)).ncard = 2"
] | ← hi.1, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.SimpleGraph.Connectivity.Subgraph | {
"line": 575,
"column": 35
} | {
"line": 577,
"column": 53
} | {
"line": 579,
"column": 0
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nu v : V\nhuv : G.Adj u v\n⊢ (induce {u, v} G).Connected",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"SimpleGraph.Subgraph.top_induce_pair_connected_of_adj",
"congrArg",
"SimpleGraph.Subgraph",
"Set.Elem",
... | [] | by
rw [connected_induce_iff]
exact Subgraph.top_induce_pair_connected_of_adj huv | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.SimpleGraph.Sum | {
"line": 260,
"column": 2
} | {
"line": 260,
"column": 15
} | {
"line": 260,
"column": 16
} | [
{
"pp": "V : Type u_3\nW : Type u_5\nG : SimpleGraph V\nH : SimpleGraph W\nv : V\n⊢ (G ⊕g H).neighborSet (Sum.inl v) = Sum.inl '' G.neighborSet v",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Set.ext",
"Membership.mem",
"SimpleGraph.neighborSet",
"Sum.casesOn",
... | [
"case inl\nV : Type u_3\nW : Type u_5\nG : SimpleGraph V\nH : SimpleGraph W\nv v' : V\n⊢ Sum.inl v' ∈ (G ⊕g H).neighborSet (Sum.inl v) ↔ Sum.inl v' ∈ Sum.inl '' G.neighborSet v",
"case inr\nV : Type u_3\nW : Type u_5\nG : SimpleGraph V\nH : SimpleGraph W\nv : V\nw' : W\n⊢ Sum.inr w' ∈ (G ⊕g H).neighborSet (Sum.in... | ext (v' | w') | _private.Lean.Elab.Tactic.Ext.0.Lean.Elab.Tactic.Ext.evalExt | Lean.Elab.Tactic.Ext.ext |
Mathlib.Combinatorics.SimpleGraph.Sum | {
"line": 263,
"column": 2
} | {
"line": 263,
"column": 15
} | {
"line": 263,
"column": 16
} | [
{
"pp": "V : Type u_3\nW : Type u_5\nG : SimpleGraph V\nH : SimpleGraph W\nw : W\n⊢ (G ⊕g H).neighborSet (Sum.inr w) = Sum.inr '' H.neighborSet w",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Set.ext",
"Membership.mem",
"SimpleGraph.neighborSet",
"Sum.casesOn",
... | [
"case inl\nV : Type u_3\nW : Type u_5\nG : SimpleGraph V\nH : SimpleGraph W\nw : W\nv' : V\n⊢ Sum.inl v' ∈ (G ⊕g H).neighborSet (Sum.inr w) ↔ Sum.inl v' ∈ Sum.inr '' H.neighborSet w",
"case inr\nV : Type u_3\nW : Type u_5\nG : SimpleGraph V\nH : SimpleGraph W\nw w' : W\n⊢ Sum.inr w' ∈ (G ⊕g H).neighborSet (Sum.in... | ext (v' | w') | _private.Lean.Elab.Tactic.Ext.0.Lean.Elab.Tactic.Ext.evalExt | Lean.Elab.Tactic.Ext.ext |
Mathlib.Combinatorics.SimpleGraph.CycleGraph | {
"line": 162,
"column": 2
} | {
"line": 162,
"column": 54
} | {
"line": 164,
"column": 0
} | [
{
"pp": "n : ℕ\nx✝¹ x✝ : Fin (cycle n).tail.support.length\ni : ℕ\nhi✝ : i < (cycle n).tail.support.length\nhi : i < (cycle n).support.tail.length\nj : ℕ\nhj✝ : j < (cycle n).tail.support.length\nhj : j < (cycle n).support.tail.length\nhij : (cycle n).getVert (i + 1) = (cycle n).getVert (j + 1)\n⊢ ⟨i, hi✝⟩ = ⟨j... | [] | grind [← Nat.mod_eq_of_lt, cycleGraph.getVert_cycle] | Lean.Elab.Tactic.evalGrind | Lean.Parser.Tactic.grind |
Mathlib.Combinatorics.SimpleGraph.Cayley | {
"line": 56,
"column": 19
} | {
"line": 56,
"column": 21
} | {
"line": 57,
"column": 4
} | [
{
"pp": "case mp\nM : Type u_1\ns : Set M\ninst✝ : Mul M\nG : SimpleGraph M\nh : ∀ (v w : M), ¬v = w → ∀ x ∈ s, v * x = w ∨ v = w * x → G.Adj v w\ng : M\nhg : g ∈ s\na : M\n⊢ ¬a * g = a → G.Adj (a * g) a",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"HMul.hMul",
"Eq",
"No... | [
"case mp\nM : Type u_1\ns : Set M\ninst✝ : Mul M\nG : SimpleGraph M\nh : ∀ (v w : M), ¬v = w → ∀ x ∈ s, v * x = w ∨ v = w * x → G.Adj v w\ng : M\nhg : g ∈ s\na : M\nha : ¬a * g = a\n⊢ G.Adj (a * g) a"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Combinatorics.SimpleGraph.Cayley | {
"line": 141,
"column": 2
} | {
"line": 142,
"column": 22
} | {
"line": 144,
"column": 0
} | [
{
"pp": "M : Type u_1\ninst✝ : Group M\n⊢ mulCayley Set.univ = ⊤",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"HMul.hMul",
"DivInvOneMonoid.toInvOneClass",
"and_true",
"Monoid.toMulOneClass",
"congrArg",
"Set.mem_univ._simp_1",
"Set.univ",
... | [] | ext _ _
simp [mulCayley_adj] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SimpleGraph.Cayley | {
"line": 141,
"column": 2
} | {
"line": 142,
"column": 22
} | {
"line": 144,
"column": 0
} | [
{
"pp": "M : Type u_1\ninst✝ : Group M\n⊢ mulCayley Set.univ = ⊤",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"HMul.hMul",
"DivInvOneMonoid.toInvOneClass",
"and_true",
"Monoid.toMulOneClass",
"congrArg",
"Set.mem_univ._simp_1",
"Set.univ",
... | [] | ext _ _
simp [mulCayley_adj] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SimpleGraph.Acyclic | {
"line": 313,
"column": 12
} | {
"line": 313,
"column": 14
} | {
"line": 313,
"column": 15
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\ninst✝¹ : Fintype V\ninst✝ : Fintype ↑G.edgeSet\nhG : G.IsTree\nthis✝ : Nonempty V\ninhabited_h : Inhabited V\nthis : {default}ᶜ.card + 1 = Fintype.card V\nf : (x : V) → G.Walk x default\nhf : ∀ (x : V), (fun p ↦ p.IsPath) (f x)\nhf' : ∀ (x : V) (y : G.Walk x default), (... | [
"V : Type u_1\nG : SimpleGraph V\ninst✝¹ : Fintype V\ninst✝ : Fintype ↑G.edgeSet\nhG : G.IsTree\nthis✝ : Nonempty V\ninhabited_h : Inhabited V\nthis : {default}ᶜ.card + 1 = Fintype.card V\nf : (x : V) → G.Walk x default\nhf : ∀ (x : V), (fun p ↦ p.IsPath) (f x)\nhf' : ∀ (x : V) (y : G.Walk x default), (fun p ↦ p.Is... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Combinatorics.SimpleGraph.AdjMatrix | {
"line": 315,
"column": 80
} | {
"line": 316,
"column": 30
} | {
"line": 318,
"column": 0
} | [
{
"pp": "α : Type u_1\nV : Type u_2\nG : SimpleGraph V\ninst✝⁴ : DecidableRel G.Adj\ninst✝³ : DecidableEq V\ninst✝² : DecidableEq α\ninst✝¹ : AddMonoid α\ninst✝ : One α\n⊢ 1 + adjMatrix α G + (adjMatrix α G).compl = of 1",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [] | by
aesop (add simp [add_assoc]) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.SimpleGraph.Acyclic | {
"line": 381,
"column": 61
} | {
"line": 390,
"column": 30
} | {
"line": 392,
"column": 0
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\nu v : V\n⊢ (G ⊔ edge u v).IsAcyclic ↔ G.IsAcyclic ∧ (G.Reachable u v → u = v ∨ G.Adj u v)",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"SimpleGraph.deleteEdges",
"Eq.mpr",
"False",
"Lattice.toSemilatticeSup",
"Simple... | [] | by
by_cases huv : u = v
· grind [sup_eq_left, edge_le, Sym2.mem_diagSet, Sym2.mk_isDiag_iff]
by_cases hadj : G.Adj u v
· grind [sup_eq_left, edge_le, mem_edgeSet]
refine ⟨?_, fun ⟨hacyc, hreach⟩ ↦ hacyc.sup_edge_of_not_reachable <| by grind⟩
refine fun hacyc ↦ ⟨hacyc.anti le_sup_left, fun hreach ↦ False.eli... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.SimpleGraph.Connectivity.Represents | {
"line": 58,
"column": 10
} | {
"line": 58,
"column": 12
} | {
"line": 58,
"column": 13
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nC : Set G.ConnectedComponent\ns : Set V\nc : G.ConnectedComponent\nhrep : Represents s C\nh : c ∉ C\na : V\n⊢ a ∈ s → a ∉ c.supp",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Membership.mem",
"Set.instMembership",
"Set"
],
... | [
"V : Type u\nG : SimpleGraph V\nC : Set G.ConnectedComponent\ns : Set V\nc : G.ConnectedComponent\nhrep : Represents s C\nh : c ∉ C\na : V\nha : a ∈ s\n⊢ a ∉ c.supp"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Combinatorics.SimpleGraph.Extremal.Turan | {
"line": 364,
"column": 4
} | {
"line": 364,
"column": 63
} | {
"line": 365,
"column": 4
} | [
{
"pp": "case e_a.e_a.e_a\nn r : ℕ\n⊢ (∑ x ∈ range n, ∑ x_1 ∈ range r, if x % r ≠ (n + x_1) % r then 1 else 0) = n * (r - 1)",
"ppTerm": "?e_a.e_a.e_a✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instDecidableNot",
"instHSMul",
"instSMulOfMul",
"HMul.hMul",
... | [
"case e_a.e_a.e_a\nn r : ℕ\n⊢ (∑ x ∈ range n, ∑ x_1 ∈ range r, if x % r ≠ (n + x_1) % r then 1 else 0) = ∑ _x ∈ range n, (r - 1)"
] | conv_rhs => rw [← card_range n, ← smul_eq_mul, ← sum_const] | Mathlib.Tactic.Conv._aux_Mathlib_Tactic_Conv___macroRules_Mathlib_Tactic_Conv_convRHS_1 | Mathlib.Tactic.Conv.convRHS |
Mathlib.Combinatorics.SimpleGraph.Diam | {
"line": 356,
"column": 6
} | {
"line": 356,
"column": 9
} | {
"line": 356,
"column": 10
} | [
{
"pp": "case h\nα : Type u_1\nG : SimpleGraph α\ninst✝¹ : Nonempty α\ninst✝ : Finite α\nw : α\nhw : G.eccent w = G.radius\nv : α\nhv : G.edist w v = G.eccent w\n⊢ G.edist w v = G.radius",
"ppTerm": "?h",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"id",
"Simpl... | [
"case h\nα : Type u_1\nG : SimpleGraph α\ninst✝¹ : Nonempty α\ninst✝ : Finite α\nw : α\nhw : G.eccent w = G.radius\nv : α\nhv : G.edist w v = G.eccent w\n⊢ G.eccent w = G.radius"
] | hv, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.SimpleGraph.Girth | {
"line": 71,
"column": 66
} | {
"line": 71,
"column": 78
} | {
"line": 72,
"column": 6
} | [
{
"pp": "case refine_2\nα : Type u_1\nG : SimpleGraph α\nh : ⨅ a, ⨅ x, ↑(↑x).length ≠ ⊤\n⊢ ∃ a w, w.IsCycle ∧ ⨅ a, ⨅ x, ↑(↑x).length = ↑w.length",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"iInf",
"instCompleteLinearOrderENat",
"ENat.instNatCast",
... | [
"case refine_2\nα : Type u_1\nG : SimpleGraph α\nh : ⨅ x, ↑(↑x.snd).length ≠ ⊤\n⊢ ∃ a w, w.IsCycle ∧ ⨅ x, ↑(↑x.snd).length = ↑w.length"
] | iInf_sigma', | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Combinatorics.SimpleGraph.Extremal.Zarankiewicz | {
"line": 151,
"column": 4
} | {
"line": 152,
"column": 65
} | {
"line": 153,
"column": 2
} | [
{
"pp": "case hb\nn s t : ℕ\nα : Type u_3\nβ : Type u_4\ninst✝⁴ : Fintype α\ninst✝³ : Fintype β\ninst✝² : Nonempty α\ninst✝¹ : Nonempty β\nhs : Fintype.card α = s\nht : Fintype.card β = t\nG : SimpleGraph (Fin n)\ninst✝ : DecidableRel G.Adj\nh : completeBipartiteGraph (Fin s) (Fin t) ⊑ G.bipartiteDoubleCover\n⊢... | [] | exact completeBipartiteGraphCongr
(Fintype.equivFinOfCardEq hs) (Fintype.equivFinOfCardEq ht) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Data.Set.Card.Arithmetic | {
"line": 127,
"column": 6
} | {
"line": 127,
"column": 75
} | {
"line": 127,
"column": 75
} | [
{
"pp": "case neg\nα : Type u_1\nι : Type u_2\nt : Set ι\nht : t.Finite\ns : ι → Set α\nhs : t.PairwiseDisjoint s\ni : ι\nhi : i ∈ t\nhn : (s i).Infinite\n⊢ (⋃ i_1 ∈ insert i (t \\ {i}), s i_1).encard = ∑ᶠ (i_1 : ι) (_ : i_1 ∈ insert i (t \\ {i})), (s i_1).encard",
"ppTerm": "?neg✝",
"assigned": true,
... | [
"case neg\nα : Type u_1\nι : Type u_2\nt : Set ι\nht : t.Finite\ns : ι → Set α\nhs : t.PairwiseDisjoint s\ni : ι\nhi : i ∈ t\nhn : (s i).Infinite\n⊢ (⋃ i_1 ∈ insert i (t \\ {i}), s i_1).encard = (s i).encard + ∑ᶠ (i_1 : ι) (_ : i_1 ∈ t \\ {i}), (s i_1).encard"
] | finsum_mem_insert _ (notMem_sdiff_of_mem <| mem_singleton i) ht.sdiff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.SimpleGraph.Extremal.ErdosStoneSimonovits | {
"line": 172,
"column": 8
} | {
"line": 172,
"column": 27
} | {
"line": 172,
"column": 27
} | [
{
"pp": "n : ℕ\nG : SimpleGraph (Fin n)\ninst✝ : DecidableRel G.Adj\nε : ℝ\nr t t' : ℕ\nK : G.CompleteEquipartiteSubgraph r t'\nhr_pos : 0 < r\nht'_pos : 0 < t'\nht_lt_t' : t < t'\nhδ : ↑G.minDegree ≥ (1 - 1 / ↑r + ε) * ↑n\nhN : (↑(t'.choose t) ^ r * ↑t + ↑r * ↑t') * (↑t' - ↑t) ≤ ↑n * (↑r * ↑t' * ε - ↑t)\np : F... | [
"n : ℕ\nG : SimpleGraph (Fin n)\ninst✝ : DecidableRel G.Adj\nε : ℝ\nr t t' : ℕ\nK : G.CompleteEquipartiteSubgraph r t'\nhr_pos : 0 < r\nht'_pos : 0 < t'\nht_lt_t' : t < t'\nhδ : ↑G.minDegree ≥ (1 - 1 / ↑r + ε) * ↑n\nhN : (↑(t'.choose t) ^ r * ↑t + ↑r * ↑t') * (↑t' - ↑t) ≤ ↑n * (↑r * ↑t' * ε - ↑t)\np : Finset (Fin n... | K.card_mem_parts hp | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.SimpleGraph.Matching | {
"line": 247,
"column": 98
} | {
"line": 252,
"column": 28
} | {
"line": 254,
"column": 0
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\nM : G.Subgraph\ninst✝ : Fintype ↑M.verts\nh : M.IsMatching\n⊢ Even M.verts.toFinset.card",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"SimpleGraph.sum_degrees_eq_twice_card_e... | [] | by
classical
rw [isMatching_iff_forall_degree] at h
use M.coe.edgeFinset.card
rw [← two_mul, ← M.coe.sum_degrees_eq_twice_card_edges]
simp [h, Finset.card_univ] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.SimpleGraph.LapMatrix | {
"line": 109,
"column": 56
} | {
"line": 109,
"column": 71
} | {
"line": 109,
"column": 72
} | [
{
"pp": "V : Type u_1\nR : Type u_2\ninst✝⁴ : Fintype V\nG : SimpleGraph V\ninst✝³ : DecidableRel G.Adj\ninst✝² : DecidableEq V\ninst✝¹ : Field R\ninst✝ : CharZero R\nx : V → R\n⊢ x ⬝ᵥ (degMatrix R G *ᵥ x - adjMatrix R G *ᵥ x) = (∑ i, ∑ j, if G.Adj i j then (x i - x j) ^ 2 else 0) / 2",
"ppTerm": "?m.88",
... | [
"V : Type u_1\nR : Type u_2\ninst✝⁴ : Fintype V\nG : SimpleGraph V\ninst✝³ : DecidableRel G.Adj\ninst✝² : DecidableEq V\ninst✝¹ : Field R\ninst✝ : CharZero R\nx : V → R\n⊢ x ⬝ᵥ degMatrix R G *ᵥ x - x ⬝ᵥ adjMatrix R G *ᵥ x = (∑ i, ∑ j, if G.Adj i j then (x i - x j) ^ 2 else 0) / 2"
] | dotProduct_sub, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Combinatorics.SimpleGraph.Matching | {
"line": 336,
"column": 2
} | {
"line": 336,
"column": 37
} | {
"line": 337,
"column": 2
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\nM : G.Subgraph\ninst✝ : Finite V\nu : Set V\nhM : M.IsPerfectMatching\nc : ↑(⊤.deleteVerts u).coe.oddComponents\nh : ∀ w ∈ u, ∀ (v : ↑(⊤.deleteVerts u).verts), M.Adj (↑v) w → v ∉ (↑c).supp\nhMmatch : (M.induce (Subtype.val '' (↑c).supp)).IsMatching\n⊢ False",
"ppTer... | [
"V : Type u_1\nG : SimpleGraph V\nM : G.Subgraph\ninst✝ : Finite V\nu : Set V\nhM : M.IsPerfectMatching\nc : ↑(⊤.deleteVerts u).coe.oddComponents\nh : ∀ w ∈ u, ∀ (v : ↑(⊤.deleteVerts u).verts), M.Adj (↑v) w → v ∉ (↑c).supp\nhMmatch : (M.induce (Subtype.val '' (↑c).supp)).IsMatching\n⊢ Even (↑c).supp.ncard"
] | apply Nat.not_even_iff_odd.2 c.prop | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Combinatorics.SimpleGraph.Matching | {
"line": 509,
"column": 2
} | {
"line": 509,
"column": 34
} | {
"line": 510,
"column": 2
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\ninst✝ : Finite V\nw : V\nhcyc : G.IsCycles\n_x : (v : V) ×' (p : G.Walk v w) ×' p.IsPath\na✝² :\n ∀ (y : (v : V) ×' (p : G.Walk v w) ×' p.IsPath),\n InvImage (fun x1 x2 ↦ x1 < x2)\n (fun x ↦ PSigma.casesOn x fun v p ↦ PSigma.casesOn p fun p hp ↦ Nat.card V + ... | [
"V : Type u_1\nG : SimpleGraph V\ninst✝ : Finite V\nw : V\nhcyc : G.IsCycles\n_x : (v : V) ×' (p : G.Walk v w) ×' p.IsPath\na✝² :\n ∀ (y : (v : V) ×' (p : G.Walk v w) ×' p.IsPath),\n InvImage (fun x1 x2 ↦ x1 < x2)\n (fun x ↦ PSigma.casesOn x fun v p ↦ PSigma.casesOn p fun p hp ↦ Nat.card V + 1 - p.length... | have := Walk.IsPath.length_lt hp | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Combinatorics.SimpleGraph.StronglyRegular | {
"line": 184,
"column": 4
} | {
"line": 184,
"column": 14
} | {
"line": 185,
"column": 4
} | [
{
"pp": "case convert_3\nn k ℓ μ : ℕ\nV : Type u\ninst✝¹ : Fintype V\nG : SimpleGraph V\ninst✝ : DecidableRel G.Adj\nh : G.IsSRGWith n k ℓ μ\nthis : DecidableEq V := Classical.decEq V\nv : V\n⊢ ∀ a ∈ G.neighborFinset v, #(bipartiteAbove G.Adj (Gᶜ.neighborFinset v) a) = k - ℓ - 1",
"ppTerm": "?convert_3",
... | [
"case convert_3\nn k ℓ μ : ℕ\nV : Type u\ninst✝¹ : Fintype V\nG : SimpleGraph V\ninst✝ : DecidableRel G.Adj\nh : G.IsSRGWith n k ℓ μ\nthis : DecidableEq V := Classical.decEq V\nv w : V\nhw : w ∈ G.neighborFinset v\n⊢ #(bipartiteAbove G.Adj (Gᶜ.neighborFinset v) w) = k - ℓ - 1"
] | intro w hw | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Combinatorics.SimpleGraph.StronglyRegular | {
"line": 198,
"column": 4
} | {
"line": 198,
"column": 14
} | {
"line": 199,
"column": 4
} | [
{
"pp": "case convert_4\nn k ℓ μ : ℕ\nV : Type u\ninst✝¹ : Fintype V\nG : SimpleGraph V\ninst✝ : DecidableRel G.Adj\nh : G.IsSRGWith n k ℓ μ\nthis : DecidableEq V := Classical.decEq V\nv : V\n⊢ ∀ b ∈ Gᶜ.neighborFinset v, #(bipartiteBelow G.Adj (G.neighborFinset v) b) = μ",
"ppTerm": "?convert_4",
"assig... | [
"case convert_4\nn k ℓ μ : ℕ\nV : Type u\ninst✝¹ : Fintype V\nG : SimpleGraph V\ninst✝ : DecidableRel G.Adj\nh : G.IsSRGWith n k ℓ μ\nthis : DecidableEq V := Classical.decEq V\nv w : V\nhw : w ∈ Gᶜ.neighborFinset v\n⊢ #(bipartiteBelow G.Adj (G.neighborFinset v) w) = μ"
] | intro w hw | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Combinatorics.SimpleGraph.Matching | {
"line": 618,
"column": 4
} | {
"line": 624,
"column": 81
} | {
"line": 626,
"column": 0
} | [
{
"pp": "case neg\nV : Type u_1\nG G' : SimpleGraph V\nM : G.Subgraph\nhM : M.IsPerfectMatching\nhG' : G'.IsAlternating M.spanningCoe\nhG'cyc : G'.IsCycles\nv w : V\nhw : (fun w ↦ M.Adj v w) w ∧ ∀ (y : V), (fun w ↦ M.Adj v w) y → y = w\nh : ¬G'.Adj v w\n⊢ ∃! w, ⊤.Adj v w",
"ppTerm": "?neg✝",
"assigned":... | [] | use w
simp only [Subgraph.top_adj, SimpleGraph.sup_adj, sdiff_adj, Subgraph.spanningCoe_adj, hw.1, h,
not_false_eq_true, and_self, not_true_eq_false, or_false, true_and]
rintro y (hl | hr)
· exact hw.2 _ hl.1
· have ⟨w', hw'⟩ := hG'cyc.other_adj_of_adj hr.1
simp_all [show M.Adj v y ↔ ¬M.Adj ... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SimpleGraph.Matching | {
"line": 618,
"column": 4
} | {
"line": 624,
"column": 81
} | {
"line": 626,
"column": 0
} | [
{
"pp": "case neg\nV : Type u_1\nG G' : SimpleGraph V\nM : G.Subgraph\nhM : M.IsPerfectMatching\nhG' : G'.IsAlternating M.spanningCoe\nhG'cyc : G'.IsCycles\nv w : V\nhw : (fun w ↦ M.Adj v w) w ∧ ∀ (y : V), (fun w ↦ M.Adj v w) y → y = w\nh : ¬G'.Adj v w\n⊢ ∃! w, ⊤.Adj v w",
"ppTerm": "?neg✝",
"assigned":... | [] | use w
simp only [Subgraph.top_adj, SimpleGraph.sup_adj, sdiff_adj, Subgraph.spanningCoe_adj, hw.1, h,
not_false_eq_true, and_self, not_true_eq_false, or_false, true_and]
rintro y (hl | hr)
· exact hw.2 _ hl.1
· have ⟨w', hw'⟩ := hG'cyc.other_adj_of_adj hr.1
simp_all [show M.Adj v y ↔ ¬M.Adj ... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SimpleGraph.Tutte | {
"line": 100,
"column": 48
} | {
"line": 100,
"column": 67
} | {
"line": 100,
"column": 67
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\ninst✝ : Finite V\nh : ¬G.IsTutteViolator G.universalVerts\nh' : ∀ (K : G.deleteUniversalVerts.coe.ConnectedComponent), G.deleteUniversalVerts.coe.IsClique K.supp\nhrep :\n ConnectedComponent.Represents (Quot.out '' G.deleteUniversalVerts.coe.oddComponents)\n G.delet... | [
"V : Type u_1\nG : SimpleGraph V\ninst✝ : Finite V\nh : ¬G.IsTutteViolator G.universalVerts\nh' : ∀ (K : G.deleteUniversalVerts.coe.ConnectedComponent), G.deleteUniversalVerts.coe.IsClique K.supp\nhrep :\n ConnectedComponent.Represents (Quot.out '' G.deleteUniversalVerts.coe.oddComponents)\n G.deleteUniversalVe... | hcomplMatch_compl j | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.SimpleGraph.Tutte | {
"line": 110,
"column": 4
} | {
"line": 110,
"column": 55
} | {
"line": 111,
"column": 4
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\ninst✝ : Finite V\nh : ¬G.IsTutteViolator G.universalVerts\nh' : ∀ (K : G.deleteUniversalVerts.coe.ConnectedComponent), G.deleteUniversalVerts.coe.IsClique K.supp\nhrep :\n ConnectedComponent.Represents (Quot.out '' G.deleteUniversalVerts.coe.oddComponents)\n G.delet... | [
"V : Type u_1\nG : SimpleGraph V\ninst✝ : Finite V\nh : ¬G.IsTutteViolator G.universalVerts\nh' : ∀ (K : G.deleteUniversalVerts.coe.ConnectedComponent), G.deleteUniversalVerts.coe.IsClique K.supp\nhrep :\n ConnectedComponent.Represents (Quot.out '' G.deleteUniversalVerts.coe.oddComponents)\n G.deleteUniversalVe... | rw [Set.compl_subset_comm, Set.compl_eq_univ_sdiff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Combinatorics.SimpleGraph.Tutte | {
"line": 195,
"column": 4
} | {
"line": 195,
"column": 90
} | {
"line": 196,
"column": 4
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\ninst✝ : Finite V\nx a b c : V\nM1 : (G ⊔ edge x b).Subgraph\nM2 : (G ⊔ edge a c).Subgraph\nhxa : G.Adj x a\nhab : G.Adj a b\nhnGxb : ¬G.Adj x b\nhnGac : ¬G.Adj a c\nhnxb : x ≠ b\nhnxc : x ≠ c\nhnac : a ≠ c\nhnbc : b ≠ c\nhM1 : M1.IsPerfectMatching\nhM2 : M2.IsPerfectMat... | [
"V : Type u_1\nG : SimpleGraph V\ninst✝ : Finite V\nx a b c : V\nM1 : (G ⊔ edge x b).Subgraph\nM2 : (G ⊔ edge a c).Subgraph\nhxa : G.Adj x a\nhab : G.Adj a b\nhnGxb : ¬G.Adj x b\nhnGac : ¬G.Adj a c\nhnxb : x ≠ b\nhnxc : x ≠ c\nhnac : a ≠ c\nhnbc : b ≠ c\nhM1 : M1.IsPerfectMatching\nhM2 : M2.IsPerfectMatching\nhM1xb... | refine le_trans (spanningCoe_induce_le cycles (cycles.connectedComponentMk c).supp) ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Computability.Primrec.Basic | {
"line": 116,
"column": 35
} | {
"line": 116,
"column": 57
} | {
"line": 118,
"column": 0
} | [
{
"pp": "case zero\np : ℕ\n⊢ Nat.rec 1 (fun y IH ↦ IH * (unpair p).1) 0 = (unpair p).1 ^ 0",
"ppTerm": "?zero",
"assigned": true,
"usedConstants": [
"MulOne.toOne",
"Monoid.toMulOneClass",
"congrArg",
"Nat.unpair",
"Nat.instMonoid",
"instOfNatNat",
"Prod.fst... | [] | simp [*, Nat.pow_succ] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Computability.Primrec.Basic | {
"line": 116,
"column": 35
} | {
"line": 116,
"column": 57
} | {
"line": 118,
"column": 0
} | [
{
"pp": "case succ\np n✝ : ℕ\na✝ : Nat.rec 1 (fun y IH ↦ IH * (unpair p).1) n✝ = (unpair p).1 ^ n✝\n⊢ Nat.rec 1 (fun y IH ↦ IH * (unpair p).1) (n✝ + 1) = (unpair p).1 ^ (n✝ + 1)",
"ppTerm": "?succ",
"assigned": true,
"usedConstants": [
"instPowNat",
"HMul.hMul",
"congrArg",
"... | [] | simp [*, Nat.pow_succ] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Computability.Primrec.Basic | {
"line": 203,
"column": 88
} | {
"line": 204,
"column": 32
} | {
"line": 206,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : Primcodable α\nn : ℕ\n⊢ (Nat.casesOn (encode (decode n)) 0 fun n ↦ n.succ.succ) = encode (Option.map some (decode n))",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Option.casesOn",
"Option.some",
"Option.encodable",
"instOfNatNat",
... | [] | by
cases @decode α _ n <;> simp | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Computability.Primrec.Basic | {
"line": 721,
"column": 20
} | {
"line": 721,
"column": 37
} | {
"line": 722,
"column": 2
} | [
{
"pp": "this : PrimrecRel fun a b ↦ a.2 = 0 ∧ b = 0 ∨ 0 < a.2 ∧ b * a.2 ≤ a.1 ∧ a.1 < (b + 1) * a.2\na k q : ℕ\nH : k = 0\n⊢ (a, k).2 = 0 ∧ q = 0 ∨ 0 < (a, k).2 ∧ q * (a, k).2 ≤ (a, k).1 ∧ (a, k).1 < (q + 1) * (a, k).2 ↔\n (fun x1 x2 ↦ x1 / x2) (a, k).1 (a, k).2 = q",
"ppTerm": "?m.322",
"assigned":... | [] | simp [H, eq_comm] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Computability.Primrec.Basic | {
"line": 721,
"column": 20
} | {
"line": 721,
"column": 37
} | {
"line": 722,
"column": 2
} | [
{
"pp": "this : PrimrecRel fun a b ↦ a.2 = 0 ∧ b = 0 ∨ 0 < a.2 ∧ b * a.2 ≤ a.1 ∧ a.1 < (b + 1) * a.2\na k q : ℕ\nH : k = 0\n⊢ (a, k).2 = 0 ∧ q = 0 ∨ 0 < (a, k).2 ∧ q * (a, k).2 ≤ (a, k).1 ∧ (a, k).1 < (q + 1) * (a, k).2 ↔\n (fun x1 x2 ↦ x1 / x2) (a, k).1 (a, k).2 = q",
"ppTerm": "?m.322",
"assigned":... | [] | simp [H, eq_comm] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Computability.Primrec.Basic | {
"line": 721,
"column": 20
} | {
"line": 721,
"column": 37
} | {
"line": 722,
"column": 2
} | [
{
"pp": "this : PrimrecRel fun a b ↦ a.2 = 0 ∧ b = 0 ∨ 0 < a.2 ∧ b * a.2 ≤ a.1 ∧ a.1 < (b + 1) * a.2\na k q : ℕ\nH : k = 0\n⊢ (a, k).2 = 0 ∧ q = 0 ∨ 0 < (a, k).2 ∧ q * (a, k).2 ≤ (a, k).1 ∧ (a, k).1 < (q + 1) * (a, k).2 ↔\n (fun x1 x2 ↦ x1 / x2) (a, k).1 (a, k).2 = q",
"ppTerm": "?m.322",
"assigned":... | [] | simp [H, eq_comm] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Computability.Ackermann | {
"line": 117,
"column": 4
} | {
"line": 118,
"column": 30
} | {
"line": 120,
"column": 0
} | [
{
"pp": "m n : ℕ\n⊢ 1 < ack (m + 1 + 1) (n + 1)",
"ppTerm": "?m.45",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"ack",
"congrArg",
"id",
"ack_succ_succ",
"instOfNatNat",
"instHAdd",
"HAdd.hAdd",
"Nat",
"LT.lt",
"instAddNat",
... | [] | rw [ack_succ_succ]
apply one_lt_ack_succ_left | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Computability.Ackermann | {
"line": 117,
"column": 4
} | {
"line": 118,
"column": 30
} | {
"line": 120,
"column": 0
} | [
{
"pp": "m n : ℕ\n⊢ 1 < ack (m + 1 + 1) (n + 1)",
"ppTerm": "?m.45",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"ack",
"congrArg",
"id",
"ack_succ_succ",
"instOfNatNat",
"instHAdd",
"HAdd.hAdd",
"Nat",
"LT.lt",
"instAddNat",
... | [] | rw [ack_succ_succ]
apply one_lt_ack_succ_left | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Computability.AkraBazzi.GrowsPolynomially | {
"line": 91,
"column": 6
} | {
"line": 91,
"column": 22
} | {
"line": 91,
"column": 22
} | [
{
"pp": "f : ℝ → ℝ\nhf✝ : GrowsPolynomially f\nhf' : ∃ᶠ (x : ℝ) in atTop, f x = 0\nc₁ : ℝ\nhc₁_mem : c₁ > 0\nc₂ : ℝ\nhc₂_mem : c₂ > 0\nhf : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (1 / 2 * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)\n⊢ ∀ᶠ (x : ℝ) in atTop, f x = 0",
"ppTerm": "?m.54",
"assigned": true,
"us... | [
"f : ℝ → ℝ\nhf✝ : GrowsPolynomially f\nhf' : ∀ (a : ℝ), ∃ b, a ≤ b ∧ f b = 0\nc₁ : ℝ\nhc₁_mem : c₁ > 0\nc₂ : ℝ\nhc₂_mem : c₂ > 0\nhf : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (1 / 2 * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)\n⊢ ∀ᶠ (x : ℝ) in atTop, f x = 0"
] | frequently_atTop | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.SimpleGraph.Tutte | {
"line": 292,
"column": 4
} | {
"line": 312,
"column": 34
} | {
"line": 314,
"column": 0
} | [
{
"pp": "case neg\nV : Type u_1\nG : SimpleGraph V\ninst✝ : Finite V\nh : ∀ (M : G.Subgraph), ¬M.IsPerfectMatching\nhvEven : Even (Nat.card V)\nval✝ : Fintype V\nGmax : SimpleGraph V\nhSubgraph : G ≤ Gmax\nhMatchingFree : Gmax.IsMatchingFree\nhMaximal : ∀ G' > Gmax, ∃ M, M.IsPerfectMatching\nhc : ¬Fintype.card ... | [] | obtain ⟨K, hK⟩ := h'
obtain ⟨x, y, hxy⟩ := (not_isClique_iff _).mp hK
obtain ⟨p, hp⟩ := Reachable.exists_path_of_dist (K.connected_toSimpleGraph x y)
obtain ⟨x, a, b, hxa, hxb, hnadjxb, hnxb⟩ := Walk.exists_adj_adj_not_adj_ne hp.2
(p.reachable.one_lt_dist_of_ne_of_not_adj hxy.1 hxy.2)
simp only [C... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SimpleGraph.Tutte | {
"line": 292,
"column": 4
} | {
"line": 312,
"column": 34
} | {
"line": 314,
"column": 0
} | [
{
"pp": "case neg\nV : Type u_1\nG : SimpleGraph V\ninst✝ : Finite V\nh : ∀ (M : G.Subgraph), ¬M.IsPerfectMatching\nhvEven : Even (Nat.card V)\nval✝ : Fintype V\nGmax : SimpleGraph V\nhSubgraph : G ≤ Gmax\nhMatchingFree : Gmax.IsMatchingFree\nhMaximal : ∀ G' > Gmax, ∃ M, M.IsPerfectMatching\nhc : ¬Fintype.card ... | [] | obtain ⟨K, hK⟩ := h'
obtain ⟨x, y, hxy⟩ := (not_isClique_iff _).mp hK
obtain ⟨p, hp⟩ := Reachable.exists_path_of_dist (K.connected_toSimpleGraph x y)
obtain ⟨x, a, b, hxa, hxb, hnadjxb, hnxb⟩ := Walk.exists_adj_adj_not_adj_ne hp.2
(p.reachable.one_lt_dist_of_ne_of_not_adj hxy.1 hxy.2)
simp only [C... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Computability.AkraBazzi.SumTransform | {
"line": 181,
"column": 71
} | {
"line": 191,
"column": 22
} | {
"line": 193,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝¹ : Fintype α\nT : ℕ → ℝ\ng : ℝ → ℝ\na b : α → ℝ\nr : α → ℕ → ℕ\ninst✝ : Nonempty α\nR : AkraBazziRecurrence T g a b r\nC : ℝ\n⊢ ∀ᶠ (n : ℕ) in atTop, ∀ (i : α), C ≤ ↑(r i n)",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Real.instIsOrderedRing",
"E... | [] | by
obtain ⟨c, hc_mem, hc⟩ := R.exists_eventually_const_mul_le_r
filter_upwards [eventually_ge_atTop ⌈C / c⌉₊, hc] with n hn₁ hn₂ i
have h₁ := hc_mem.1
calc C
_ = c * (C / c) := by
rw [← mul_div_assoc]
exact (mul_div_cancel_left₀ _ (by positivity)).symm
_ ≤ c * ⌈C / c⌉₊ := by gcongr; simp [Na... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Computability.AkraBazzi.SumTransform | {
"line": 456,
"column": 6
} | {
"line": 456,
"column": 33
} | {
"line": 457,
"column": 6
} | [
{
"pp": "α : Type u_1\ninst✝¹ : Fintype α\nT : ℕ → ℝ\ng : ℝ → ℝ\na b : α → ℝ\nr : α → ℕ → ℕ\ninst✝ : Nonempty α\nR : AkraBazziRecurrence T g a b r\ni : α\nn : ℕ\nhn : n ≠ 0\n⊢ (log ↑n - log (b i * ↑n)) / (log (b i * ↑n) * log ↑n) =\n (log ↑n - log (b i) - log ↑n) / ((log (b i) + log ↑n) * log ↑n)",
"ppTe... | [
"α : Type u_1\ninst✝¹ : Fintype α\nT : ℕ → ℝ\ng : ℝ → ℝ\na b : α → ℝ\nr : α → ℕ → ℕ\ninst✝ : Nonempty α\nR : AkraBazziRecurrence T g a b r\ni : α\nn : ℕ\nhn : n ≠ 0\nthis : 0 < b i\n⊢ (log ↑n - log (b i * ↑n)) / (log (b i * ↑n) * log ↑n) =\n (log ↑n - log (b i) - log ↑n) / ((log (b i) + log ↑n) * log ↑n)"
] | have : 0 < b i := R.b_pos i | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Computability.AkraBazzi.AkraBazzi | {
"line": 407,
"column": 88
} | {
"line": 425,
"column": 18
} | {
"line": 426,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝¹ : Fintype α\nT : ℕ → ℝ\ng : ℝ → ℝ\na b : α → ℝ\nr : α → ℕ → ℕ\ninst✝ : Nonempty α\nR : AkraBazziRecurrence T g a b r\ni : α\nq : ℝ → ℝ := fun x ↦ x ^ p a b * (1 + ε x)\nh_diff_q : DifferentiableOn ℝ q (Set.Ioi 1)\nh_deriv_q : deriv q =O[atTop] fun x ↦ x ^ (p a b - 1)\nh_main_norm :... | [] | by
filter_upwards [eventually_gt_atTop ⌈(b i)⁻¹⌉₊, eventually_gt_atTop 1] with n hn hn'
refine norm_of_nonneg ?_
have h₁ := R.b_pos i
have h₂ : 0 ≤ ε (b i * n) - ε n := by
refine sub_nonneg_of_le <|
(strictAntiOn_smoothingFn.le_iff_ge ?n_gt_one ?bn_gt_one).mpr ?le
... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Computability.PartrecCode | {
"line": 690,
"column": 10
} | {
"line": 690,
"column": 65
} | {
"line": 691,
"column": 10
} | [
{
"pp": "case neg.succ\nk : ℕ\ncf : Code\nhf : ∀ (n x : ℕ), x ∈ evaln (k + 1) cf n → x ∈ cf.eval n\nn : ℕ\nleft✝ : n ≤ k\nm : ℕ\nh₁ : evaln (k + 1) cf n = some m\nm0 : ¬m = 0\ny : ℕ\nhy₁ : 0 ∈ cf.eval (Nat.pair (unpair n).1 (y + ((unpair n).2 + 1)))\nhy₂ : ∀ {m : ℕ}, m < y → ∃ a ∈ cf.eval (Nat.pair (unpair n).1... | [
"case neg.succ\nk : ℕ\ncf : Code\nhf : ∀ (n x : ℕ), x ∈ evaln (k + 1) cf n → x ∈ cf.eval n\nn : ℕ\nleft✝ : n ≤ k\nm : ℕ\nh₁ : evaln (k + 1) cf n = some m\nm0 : ¬m = 0\ny : ℕ\nhy₁ : 0 ∈ cf.eval (Nat.pair (unpair n).1 (y + ((unpair n).2 + 1)))\nhy₂ : ∀ {m : ℕ}, m < y → ∃ a ∈ cf.eval (Nat.pair (unpair n).1 (m + ((unpa... | rcases hy₂ (Nat.lt_of_succ_lt_succ im) with ⟨z, hz, z0⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Computability.ContextFreeGrammar | {
"line": 213,
"column": 35
} | {
"line": 216,
"column": 68
} | {
"line": 218,
"column": 0
} | [
{
"pp": "T : Type u_1\ng : ContextFreeGrammar T\nv w : List (Symbol T g.NT)\nhvw : g.Derives v w\np : List (Symbol T g.NT)\n⊢ g.Derives (v ++ p) (w ++ p)",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Symbol",
"ContextFreeGrammar.Produces",
"ContextFreeGrammar.Derives.t... | [] | by
induction hvw with
| refl => rfl
| tail _ last ih => exact ih.trans_produces <| last.append_right p | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Nat.Size | {
"line": 61,
"column": 13
} | {
"line": 61,
"column": 33
} | {
"line": 62,
"column": 4
} | [
{
"pp": "case pos.false\nm n : ℕ\nIH : shiftLeft' false m n ≠ 0 → (shiftLeft' false m n).size = m.size + n\nh : ¬bit false 0 = 0\ns0 : shiftLeft' false m n = 0\n⊢ (size 0).succ = (m.size + n).succ",
"ppTerm": "?pos.false✝",
"assigned": true,
"usedConstants": [
"Nat.bit",
"instOfNatNat",
... | [
"case pos.true\nm n : ℕ\nIH : shiftLeft' true m n ≠ 0 → (shiftLeft' true m n).size = m.size + n\nh : ¬bit true 0 = 0\ns0 : shiftLeft' true m n = 0\n⊢ (size 0).succ = (m.size + n).succ"
] | · exact absurd rfl h | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Data.Nat.Bitwise | {
"line": 275,
"column": 8
} | {
"line": 275,
"column": 11
} | {
"line": 275,
"column": 12
} | [
{
"pp": "a b c : ℕ\nv : ℕ := a ^^^ b ^^^ c\nh : v ≠ 0\nhv : v = a ^^^ b ^^^ c\nhab : a ^^^ b = c ^^^ v\nhbc : b ^^^ c = a ^^^ v\n⊢ c ^^^ a = b ^^^ v",
"ppTerm": "?m.103",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.instXorOp",
"congrArg",
"id",
"Nat",
"HXo... | [
"a b c : ℕ\nv : ℕ := a ^^^ b ^^^ c\nh : v ≠ 0\nhv : v = a ^^^ b ^^^ c\nhab : a ^^^ b = c ^^^ v\nhbc : b ^^^ c = a ^^^ v\n⊢ c ^^^ a = b ^^^ (a ^^^ b ^^^ c)"
] | hv, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Computability.PartrecCode | {
"line": 960,
"column": 8
} | {
"line": 960,
"column": 47
} | {
"line": 961,
"column": 8
} | [
{
"pp": "case succ.comp\nx✝ : Unit\np n : ℕ\nthis : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))\nk' : ℕ\nk : ℕ := k' + 1\nnk : n ≤ k'\ncf cg : Code\nhg :\n ∀ {k' : ℕ} {c' : Code} {n : ℕ},\n Nat.pair k' (encode c') < Nat.pair k (encode (cf.comp cg)) →\n lup\n ... | [
"case succ.comp\nx✝ : Unit\np n : ℕ\nthis : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))\nk' : ℕ\nk : ℕ := k' + 1\nnk : n ≤ k'\ncf cg : Code\nhg :\n ∀ {k' : ℕ} {c' : Code} {n : ℕ},\n Nat.pair k' (encode c') < Nat.pair k (encode (cf.comp cg)) →\n lup\n (List.... | obtain ⟨lf, lg⟩ := encode_lt_comp cf cg | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Data.Nat.Bitwise | {
"line": 340,
"column": 4
} | {
"line": 353,
"column": 20
} | {
"line": 355,
"column": 0
} | [
{
"pp": "case succ\nn : ℕ\nih :\n List.foldl (fun x1 x2 ↦ x1 ^^^ x2) 0 (List.range (n + 1)) =\n match Fin.ofNat 4 n with\n | 0 => n\n | 1 => 1\n | 2 => n + 1\n | 3 => 0\n⊢ (match Fin.ofNat 4 n with\n | 0 => n\n | 1 => 1\n | 2 => n + 1\n | 3 => 0) ^^^\n n + 1 =\n match... | [] | match h : Fin.ofNat 4 n with
| 0 =>
rw [Fin.zero_add, ← xor_one_of_even <| even_iff.mpr ?_, xor_xor_cancel_left]
rw [← @mod_mod_of_dvd _ 4 _ <| by simp, ← Fin.val_ofNat 4, h]
rfl
| 1 =>
rw [Nat.xor_comm]
refine xor_one_of_even <| even_iff.mpr ?_
rw [add_mod, ← @mod_mod_of_dvd... | Lean.Elab.Tactic.evalMatch | Lean.Parser.Tactic.match |
Mathlib.Data.List.ReduceOption | {
"line": 144,
"column": 46
} | {
"line": 144,
"column": 66
} | {
"line": 144,
"column": 66
} | [
{
"pp": "α : Type u_1\nl : List (Option α)\nx : α\n⊢ some x ∈ l ↔ x ∈ l.reduceOption",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"List.reduceOption_mem_iff",
"Option.some",
"Membership.mem",
"id",
"List",
"Iff",
... | [
"α : Type u_1\nl : List (Option α)\nx : α\n⊢ some x ∈ l ↔ some x ∈ l"
] | reduceOption_mem_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Computability.MyhillNerode | {
"line": 91,
"column": 2
} | {
"line": 91,
"column": 22
} | {
"line": 92,
"column": 2
} | [
{
"pp": "α : Type u\nL : Language α\nx : List α\n⊢ x ∈ L.toDFA.accepts ↔ x ∈ L",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Membership.mem",
"Language.toDFA",
"Set.Elem",
"id",
"Language.leftQuotient",
"List",
... | [
"α : Type u\nL : Language α\nx : List α\n⊢ L.toDFA.eval x ∈ L.toDFA.accept ↔ x ∈ L"
] | rw [DFA.mem_accepts] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Computability.RE | {
"line": 53,
"column": 6
} | {
"line": 53,
"column": 24
} | {
"line": 55,
"column": 0
} | [
{
"pp": "case inr.some\ncf : Code\nhf : Nat.Partrec cf.eval\ncg : Code\nhg : Nat.Partrec cg.eval\nthis✝ : Nat.Partrec fun n ↦ rfindOpt fun k ↦ Code.evaln k cf n <|> Code.evaln k cg n\nn : ℕ\nthis : ∀ x ∈ rfindOpt fun k ↦ Code.evaln k cf n <|> Code.evaln k cg n, x ∈ cf.eval n ∨ x ∈ cg.eval n\nx k : ℕ\ne : Code.e... | [] | exact ⟨y, by simp⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Computability.RE | {
"line": 53,
"column": 6
} | {
"line": 53,
"column": 24
} | {
"line": 55,
"column": 0
} | [
{
"pp": "case inr.some\ncf : Code\nhf : Nat.Partrec cf.eval\ncg : Code\nhg : Nat.Partrec cg.eval\nthis✝ : Nat.Partrec fun n ↦ rfindOpt fun k ↦ Code.evaln k cf n <|> Code.evaln k cg n\nn : ℕ\nthis : ∀ x ∈ rfindOpt fun k ↦ Code.evaln k cf n <|> Code.evaln k cg n, x ∈ cf.eval n ∨ x ∈ cg.eval n\nx k : ℕ\ne : Code.e... | [] | exact ⟨y, by simp⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Computability.RE | {
"line": 53,
"column": 6
} | {
"line": 53,
"column": 24
} | {
"line": 55,
"column": 0
} | [
{
"pp": "case inr.some\ncf : Code\nhf : Nat.Partrec cf.eval\ncg : Code\nhg : Nat.Partrec cg.eval\nthis✝ : Nat.Partrec fun n ↦ rfindOpt fun k ↦ Code.evaln k cf n <|> Code.evaln k cg n\nn : ℕ\nthis : ∀ x ∈ rfindOpt fun k ↦ Code.evaln k cf n <|> Code.evaln k cg n, x ∈ cf.eval n ∨ x ∈ cg.eval n\nx k : ℕ\ne : Code.e... | [] | exact ⟨y, by simp⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Computability.StateTransition | {
"line": 181,
"column": 4
} | {
"line": 193,
"column": 30
} | {
"line": 195,
"column": 0
} | [
{
"pp": "case tail\nσ₁ : Type u_1\nσ₂ : Type u_2\nf₁ : σ₁ → Option σ₁\nf₂ : σ₂ → Option σ₂\ntr : σ₁ → σ₂ → Prop\nH : Respects f₁ f₂ tr\na₁ : σ₁\na₂ : σ₂\naa : tr a₁ a₂\nb₂ b✝ c✝ : σ₂\na✝ : ReflTransGen (fun a b ↦ b ∈ f₂ a) a₂ b✝\ncd : c✝ ∈ f₂ b✝\nIH : ∃ c₁ c₂, Reaches f₂ b✝ c₂ ∧ tr c₁ c₂ ∧ Reaches f₁ a₁ c₁\n⊢ ∃... | [] | rcases IH with ⟨e₁, e₂, ce, ee, ae⟩
rcases ReflTransGen.cases_head ce with (rfl | ⟨d', cd', de⟩)
· have := H ee
revert this
rcases eg : f₁ e₁ with - | g₁ <;> simp only [and_imp, exists_imp]
· intro c0
cases cd.symm.trans c0
· intro g₂ gg cg
rcases TransGen.head'_iff.1 cg ... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Computability.StateTransition | {
"line": 181,
"column": 4
} | {
"line": 193,
"column": 30
} | {
"line": 195,
"column": 0
} | [
{
"pp": "case tail\nσ₁ : Type u_1\nσ₂ : Type u_2\nf₁ : σ₁ → Option σ₁\nf₂ : σ₂ → Option σ₂\ntr : σ₁ → σ₂ → Prop\nH : Respects f₁ f₂ tr\na₁ : σ₁\na₂ : σ₂\naa : tr a₁ a₂\nb₂ b✝ c✝ : σ₂\na✝ : ReflTransGen (fun a b ↦ b ∈ f₂ a) a₂ b✝\ncd : c✝ ∈ f₂ b✝\nIH : ∃ c₁ c₂, Reaches f₂ b✝ c₂ ∧ tr c₁ c₂ ∧ Reaches f₁ a₁ c₁\n⊢ ∃... | [] | rcases IH with ⟨e₁, e₂, ce, ee, ae⟩
rcases ReflTransGen.cases_head ce with (rfl | ⟨d', cd', de⟩)
· have := H ee
revert this
rcases eg : f₁ e₁ with - | g₁ <;> simp only [and_imp, exists_imp]
· intro c0
cases cd.symm.trans c0
· intro g₂ gg cg
rcases TransGen.head'_iff.1 cg ... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Computability.TuringMachine.Tape | {
"line": 300,
"column": 2
} | {
"line": 300,
"column": 53
} | {
"line": 302,
"column": 0
} | [
{
"pp": "Γ : Type u_1\nΓ' : Type u_2\ninst✝¹ : Inhabited Γ\ninst✝ : Inhabited Γ'\nf : PointedMap Γ Γ'\nl : List Γ\n⊢ (List.map f.f l).headI = f.f l.headI",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Inhabited.default",
"List.map",
"List.cons",
"List",
"Lis... | [] | cases l <;> [exact (PointedMap.map_pt f).symm; rfl] | Batteries.Tactic._aux_Batteries_Tactic_SeqFocus___macroRules_Batteries_Tactic_seq_focus_1 | Batteries.Tactic.seq_focus |
Mathlib.Computability.TuringMachine.Tape | {
"line": 300,
"column": 2
} | {
"line": 300,
"column": 53
} | {
"line": 302,
"column": 0
} | [
{
"pp": "Γ : Type u_1\nΓ' : Type u_2\ninst✝¹ : Inhabited Γ\ninst✝ : Inhabited Γ'\nf : PointedMap Γ Γ'\nl : List Γ\n⊢ (List.map f.f l).headI = f.f l.headI",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Inhabited.default",
"List.map",
"List.cons",
"List",
"Lis... | [] | cases l <;> [exact (PointedMap.map_pt f).symm; rfl] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Computability.TuringMachine.Tape | {
"line": 300,
"column": 2
} | {
"line": 300,
"column": 53
} | {
"line": 302,
"column": 0
} | [
{
"pp": "Γ : Type u_1\nΓ' : Type u_2\ninst✝¹ : Inhabited Γ\ninst✝ : Inhabited Γ'\nf : PointedMap Γ Γ'\nl : List Γ\n⊢ (List.map f.f l).headI = f.f l.headI",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Inhabited.default",
"List.map",
"List.cons",
"List",
"Lis... | [] | cases l <;> [exact (PointedMap.map_pt f).symm; rfl] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Computability.TuringMachine.Config | {
"line": 311,
"column": 97
} | {
"line": 314,
"column": 16
} | {
"line": 315,
"column": 6
} | [
{
"pp": "n✝² : ℕ\nf✝¹ : List.Vector ℕ n✝² →. ℕ\nn✝¹ : ℕ\nf✝ : List.Vector ℕ n✝¹ → ℕ\nn✝ : ℕ\nf : List.Vector ℕ n✝ → ℕ\ng : List.Vector ℕ (n✝ + 2) → ℕ\na✝² : Nat.Primrec' f\na✝¹ : Nat.Primrec' g\ncf cg : Code\nv : List.Vector ℕ (n✝ + 1)\nhf : cf.eval (↑v).tail = pure (pure (f v.tail))\nhg : ∀ (a b : ℕ), cg.eval ... | [] | by
have := Part.eq_some_iff.mpr (this _ _ (zero_add _))
simp [prec, Part.bind_assoc, Bind.bind]
simp_all | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Computability.TuringMachine.Config | {
"line": 346,
"column": 34
} | {
"line": 346,
"column": 48
} | {
"line": 346,
"column": 48
} | [
{
"pp": "n✝ : ℕ\nf✝ : List.Vector ℕ n✝ →. ℕ\nn : ℕ\nf : List.Vector ℕ (n + 1) → ℕ\na✝ : Nat.Partrec' ↑f\ncf : Code\nv : List.Vector ℕ n\nhf : ∀ (a : ℕ), cf.eval (a :: ↑v) = Part.some [f (a ::ᵥ v)]\nv' : List ℕ\nh1 :\n v' ∈\n PFun.fix\n (fun v ↦\n (cf.eval v).bind fun y ↦\n Part.some (if... | [] | by rintro _ ⟨⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Category.CompHausLike.Limits | {
"line": 256,
"column": 2
} | {
"line": 256,
"column": 16
} | {
"line": 257,
"column": 2
} | [
{
"pp": "P : TopCat → Prop\nX Y B : CompHausLike P\nf : X ⟶ B\ng : Y ⟶ B\ninst✝ : HasExplicitPullback f g\nZ : CompHausLike P\na b : Z ⟶ pullback f g\nz : ↑Z.toTop\nhfst : (ConcreteCategory.hom (a ≫ fst f g)) z = (ConcreteCategory.hom (b ≫ fst f g)) z\nhsnd : (ConcreteCategory.hom (a ≫ snd f g)) z = (ConcreteCa... | [
"case fst\nP : TopCat → Prop\nX Y B : CompHausLike P\nf : X ⟶ B\ng : Y ⟶ B\ninst✝ : HasExplicitPullback f g\nZ : CompHausLike P\na b : Z ⟶ pullback f g\nz : ↑Z.toTop\nhfst : (ConcreteCategory.hom (a ≫ fst f g)) z = (ConcreteCategory.hom (b ≫ fst f g)) z\nhsnd : (ConcreteCategory.hom (a ≫ snd f g)) z = (ConcreteCate... | apply Prod.ext | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Computability.TuringMachine.Config | {
"line": 532,
"column": 2
} | {
"line": 532,
"column": 19
} | {
"line": 533,
"column": 4
} | [
{
"pp": "case fix\nk' : Cont\na✝¹ : Code\na✝ : Cont\nk_ih : ∀ {v : List ℕ}, (a✝.then k').eval v = a✝.eval v >>= k'.eval\nv : List ℕ\n⊢ ((fix a✝¹ a✝).then k').eval v = (fix a✝¹ a✝).eval v >>= k'.eval",
"ppTerm": "?fix",
"assigned": true,
"usedConstants": [
"Part",
"Eq.mpr",
"PFun",
... | [] | | fix _ _ k_ih => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.Computability.TuringMachine.Config | {
"line": 565,
"column": 2
} | {
"line": 565,
"column": 19
} | {
"line": 566,
"column": 4
} | [
{
"pp": "case fix\nk' : Cont\na✝¹ : Code\na✝ : Cont\nk_ih : ∀ {v : List ℕ}, stepRet (a✝.then k') v = (stepRet a✝ v).then k'\nv : List ℕ\n⊢ (if v.headI = 0 then (stepRet a✝ v.tail).then k' else stepNormal a✝¹ (Cont.fix a✝¹ (a✝.then k')) v.tail) =\n (if v.headI = 0 then stepRet a✝ v.tail else stepNormal a✝¹ (C... | [] | | fix _ _ k_ih => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.Topology.ExtremallyDisconnected | {
"line": 113,
"column": 6
} | {
"line": 113,
"column": 60
} | {
"line": 114,
"column": 2
} | [
{
"pp": "case neg\nX : Type u\ninst✝² : TopologicalSpace X\ninst✝¹ : CompactSpace X\ninst✝ : T2Space X\nh : Projective X\nU : Set X\nhU : IsOpen U\nZ₁ : Set (X × Bool) := Uᶜ ×ˢ {true}\nZ₂ : Set (X × Bool) := closure U ×ˢ {false}\nZ : Set (X × Bool) := Z₁ ∪ Z₂\nhZ₁₂ : Disjoint Z₁ Z₂\nhZ₁ : IsClosed Z₁\nhZ₂ : IsC... | [] | exact ⟨⟨(x, true), Or.inl ⟨hx, mem_singleton _⟩⟩, rfl⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.ExtremallyDisconnected | {
"line": 113,
"column": 6
} | {
"line": 113,
"column": 60
} | {
"line": 114,
"column": 2
} | [
{
"pp": "case neg\nX : Type u\ninst✝² : TopologicalSpace X\ninst✝¹ : CompactSpace X\ninst✝ : T2Space X\nh : Projective X\nU : Set X\nhU : IsOpen U\nZ₁ : Set (X × Bool) := Uᶜ ×ˢ {true}\nZ₂ : Set (X × Bool) := closure U ×ˢ {false}\nZ : Set (X × Bool) := Z₁ ∪ Z₂\nhZ₁₂ : Disjoint Z₁ Z₂\nhZ₁ : IsClosed Z₁\nhZ₂ : IsC... | [] | exact ⟨⟨(x, true), Or.inl ⟨hx, mem_singleton _⟩⟩, rfl⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.ExtremallyDisconnected | {
"line": 113,
"column": 6
} | {
"line": 113,
"column": 60
} | {
"line": 114,
"column": 2
} | [
{
"pp": "case neg\nX : Type u\ninst✝² : TopologicalSpace X\ninst✝¹ : CompactSpace X\ninst✝ : T2Space X\nh : Projective X\nU : Set X\nhU : IsOpen U\nZ₁ : Set (X × Bool) := Uᶜ ×ˢ {true}\nZ₂ : Set (X × Bool) := closure U ×ˢ {false}\nZ : Set (X × Bool) := Z₁ ∪ Z₂\nhZ₁₂ : Disjoint Z₁ Z₂\nhZ₁ : IsClosed Z₁\nhZ₂ : IsC... | [] | exact ⟨⟨(x, true), Or.inl ⟨hx, mem_singleton _⟩⟩, rfl⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.ExtremallyDisconnected | {
"line": 188,
"column": 12
} | {
"line": 188,
"column": 14
} | {
"line": 189,
"column": 4
} | [
{
"pp": "case neg\nA E : Type u\ninst✝¹ : TopologicalSpace A\ninst✝ : TopologicalSpace E\nρ : E → A\nρ_cont : Continuous ρ\nρ_surj : Surjective ρ\nzorn_subset : ∀ (E₀ : Set E), E₀ ≠ univ → IsClosed E₀ → ρ '' E₀ ≠ univ\nG : Set E\nhG : IsOpen G\nG_empty : ¬G = ∅\na : A\n⊢ a ∈ ρ '' G → a ∈ closure (ρ '' Gᶜ)ᶜ",
... | [
"case neg\nA E : Type u\ninst✝¹ : TopologicalSpace A\ninst✝ : TopologicalSpace E\nρ : E → A\nρ_cont : Continuous ρ\nρ_surj : Surjective ρ\nzorn_subset : ∀ (E₀ : Set E), E₀ ≠ univ → IsClosed E₀ → ρ '' E₀ ≠ univ\nG : Set E\nhG : IsOpen G\nG_empty : ¬G = ∅\na : A\nha : a ∈ ρ '' G\n⊢ a ∈ closure (ρ '' Gᶜ)ᶜ"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Topology.Category.Profinite.EffectiveEpi | {
"line": 82,
"column": 2
} | {
"line": 91,
"column": 13
} | {
"line": 93,
"column": 0
} | [
{
"pp": "α : Type\ninst✝ : Finite α\nB : Profinite\nX : α → Profinite\nπ : (a : α) → X a ⟶ B\n⊢ [EffectiveEpiFamily X π, Epi (Sigma.desc π), ∀ (b : ↑B.toTop), ∃ a x, (ConcreteCategory.hom (π a)) x = b].TFAE",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"List.IsChai... | [] | tfae_have 2 → 1
| _ => by
simpa [← effectiveEpi_desc_iff_effectiveEpiFamily, (effectiveEpi_tfae (Sigma.desc π)).out 0 1]
tfae_have 1 → 2 := fun _ ↦ inferInstance
tfae_have 3 ↔ 1 := by
erw [((CompHaus.effectiveEpiFamily_tfae
(fun a ↦ profiniteToCompHaus.obj (X a)) (fun a ↦ profiniteToCompHaus.map (π ... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Category.Profinite.EffectiveEpi | {
"line": 82,
"column": 2
} | {
"line": 91,
"column": 13
} | {
"line": 93,
"column": 0
} | [
{
"pp": "α : Type\ninst✝ : Finite α\nB : Profinite\nX : α → Profinite\nπ : (a : α) → X a ⟶ B\n⊢ [EffectiveEpiFamily X π, Epi (Sigma.desc π), ∀ (b : ↑B.toTop), ∃ a x, (ConcreteCategory.hom (π a)) x = b].TFAE",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"List.IsChai... | [] | tfae_have 2 → 1
| _ => by
simpa [← effectiveEpi_desc_iff_effectiveEpiFamily, (effectiveEpi_tfae (Sigma.desc π)).out 0 1]
tfae_have 1 → 2 := fun _ ↦ inferInstance
tfae_have 3 ↔ 1 := by
erw [((CompHaus.effectiveEpiFamily_tfae
(fun a ↦ profiniteToCompHaus.obj (X a)) (fun a ↦ profiniteToCompHaus.map (π ... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Computability.TuringMachine.ToPartrec | {
"line": 730,
"column": 53
} | {
"line": 730,
"column": 67
} | {
"line": 730,
"column": 67
} | [
{
"pp": "q : Λ'\ns : Option Γ'\nL₁ : List ℕ\nL₃ : List Γ'\n⊢ ∀ x ∈ [], natEnd x = false",
"ppTerm": "?m.66",
"assigned": true,
"usedConstants": [
"Turing.PartrecToTM2.Γ'",
"HEq.refl",
"List.Mem.tail",
"False.elim",
"noConfusion_of_Nat",
"Membership.mem",
"Tu... | [] | by rintro _ ⟨⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Computability.TuringMachine.ToPartrec | {
"line": 801,
"column": 4
} | {
"line": 801,
"column": 45
} | {
"line": 802,
"column": 4
} | [
{
"pp": "case cons.zero\nq₁ q₂ : Λ'\ns : Option Γ'\nc d : List Γ'\nv : List ℕ\n⊢ ∃ s',\n Reaches₁ (TM2.step tr) { l := some (q₁.pred q₂), var := s, stk := elim (trList (0 :: v)) [] c d }\n (Nat.rec { l := some q₁, var := s', stk := elim (trList (0 :: v).tail) [] c d }\n (fun n x ↦ { l := some q₂,... | [
"case cons.zero\nq₁ q₂ : Λ'\ns : Option Γ'\nc d : List Γ'\nv : List ℕ\n⊢ Nat.rec { l := some q₁, var := some Γ'.cons, stk := elim (trList (0 :: v).tail) [] c d }\n (fun n x ↦ { l := some q₂, var := some Γ'.cons, stk := elim (trList (n :: (0 :: v).tail)) [] c d })\n (0 :: v).headI ∈\n TM2.step tr { l :=... | refine ⟨some Γ'.cons, TransGen.single ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Topology.ClopenBox | {
"line": 60,
"column": 2
} | {
"line": 69,
"column": 27
} | {
"line": 71,
"column": 0
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\ninst✝¹ : CompactSpace Y\ninst✝ : CompactSpace X\nW : Clopens (X × Y)\n⊢ ∃ I, W = I.sup fun i ↦ i.1 ×ˢ i.2",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Set.instSProd",
"Eq.mpr",
... | [] | choose! U hxU V hxV hUV using fun x ↦ W.exists_prod_subset (a := x)
rcases W.2.1.isCompact.elim_nhds_subcover (fun x ↦ U x ×ˢ V x) (fun x hx ↦
(U x ×ˢ V x).2.isOpen.mem_nhds ⟨hxU x hx, hxV x hx⟩) with ⟨I, hIW, hWI⟩
classical
use I.image fun x ↦ (U x, V x)
rw [Finset.sup_image]
refine le_antisymm (fun x hx... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.ClopenBox | {
"line": 60,
"column": 2
} | {
"line": 69,
"column": 27
} | {
"line": 71,
"column": 0
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\ninst✝¹ : CompactSpace Y\ninst✝ : CompactSpace X\nW : Clopens (X × Y)\n⊢ ∃ I, W = I.sup fun i ↦ i.1 ×ˢ i.2",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Set.instSProd",
"Eq.mpr",
... | [] | choose! U hxU V hxV hUV using fun x ↦ W.exists_prod_subset (a := x)
rcases W.2.1.isCompact.elim_nhds_subcover (fun x ↦ U x ×ˢ V x) (fun x hx ↦
(U x ×ˢ V x).2.isOpen.mem_nhds ⟨hxU x hx, hxV x hx⟩) with ⟨I, hIW, hWI⟩
classical
use I.image fun x ↦ (U x, V x)
rw [Finset.sup_image]
refine le_antisymm (fun x hx... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Computability.TuringMachine.ToPartrec | {
"line": 982,
"column": 4
} | {
"line": 982,
"column": 18
} | {
"line": 983,
"column": 4
} | [
{
"pp": "case read\nq' : Λ'\nq : Option Γ' → Λ'\nq_ih : ∀ (a : Option Γ'), q' ∈ trStmts₁ (q a) → trStmts₁ q' ⊆ trStmts₁ (q a)\n⊢ ∀ (x : Option Γ'), q' ∈ trStmts₁ (q x) → trStmts₁ q' ⊆ insert (Λ'.read q) (Finset.univ.biUnion fun s ↦ trStmts₁ (q s))",
"ppTerm": "?read",
"assigned": true,
"usedConstant... | [
"case read\nq' : Λ'\nq : Option Γ' → Λ'\nq_ih : ∀ (a : Option Γ'), q' ∈ trStmts₁ (q a) → trStmts₁ q' ⊆ trStmts₁ (q a)\ns : Option Γ'\nh : q' ∈ trStmts₁ (q s)\nx : Λ'\nh' : x ∈ trStmts₁ q'\n⊢ x ∈ insert (Λ'.read q) (Finset.univ.biUnion fun s ↦ trStmts₁ (q s))"
] | intro s h x h' | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Computability.TuringMachine.ToPartrec | {
"line": 987,
"column": 6
} | {
"line": 987,
"column": 32
} | {
"line": 988,
"column": 4
} | [
{
"pp": "case succ.left\nq : Λ'\nq_ih : unrev q ∈ trStmts₁ q → trStmts₁ (unrev q) ⊆ trStmts₁ q\n⊢ trStmts₁ (unrev q) ⊆ insert q.succ (insert (unrev q) (trStmts₁ q))",
"ppTerm": "?succ.left",
"assigned": true,
"usedConstants": [
"Turing.PartrecToTM2.trStmts₁",
"Turing.PartrecToTM2.Λ'.inst... | [] | apply Finset.subset_insert | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
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