module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 366
values | kind stringclasses 370
values |
|---|---|---|---|---|---|---|
Mathlib.Combinatorics.KatonaCircle | {
"line": 99,
"column": 16
} | {
"line": 104,
"column": 15
} | [
{
"pp": "X : Type u_1\ninst✝¹ : Fintype X\nf✝ : Numbering X\ns✝ t : Finset X\ninst✝ : DecidableEq X\ns : Finset X\nf : ↥(prefixed s)\n⊢ (fun x ↦\n match x with\n | (g, g') =>\n ⟨{ toFun := fun x ↦ if hx : x ∈ s then Fin.castLE ⋯ (g ⟨x, hx⟩) else Fin.cast ⋯ ((g' ⟨x, ⋯⟩).addNat #s),\n ... | by
ext x
by_cases hx : x ∈ s
· simp [hx]
· rw [mem_prefixed.1 f.2, not_lt] at hx
simp [hx] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.Matroid.Minor.Contract | {
"line": 402,
"column": 79
} | {
"line": 402,
"column": 96
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nC X : Set α\n⊢ (M ／ C).closure X ⊆ (M ／ C).closure X \\ X ∪ M.closure (C ∪ X) \\ C ∩ X ∧\n (M ／ C).closure X \\ X ∪ M.closure (C ∪ X) \\ C ∩ X ⊆ (M ／ C).closure X",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Set.union_subset_iff",
"Set.instUnio... | union_subset_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.Matroid.Minor.Contract | {
"line": 411,
"column": 72
} | {
"line": 411,
"column": 89
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nX C : Set α\nhC : C ⊆ M.E\n⊢ M.closure (X ∪ C) \\ C = M.E \\ C ∧ X ⊆ M.E \\ C ↔ (M.closure (X ∪ C) = M.E ∧ X ∪ C ⊆ M.E) ∧ Disjoint X C",
"usedConstants": [
"Eq.mpr",
"ChainCompletePartialOrder.instOfCompleteLattice",
"CompleteBooleanAlgebra.toCompleteD... | union_subset_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.Nullstellensatz | {
"line": 122,
"column": 8
} | {
"line": 122,
"column": 41
} | [
{
"pp": "case h.e'_2.h.e'_6.a\nR : Type u_1\ninst✝³ : CommRing R\nσ✝ : Type u_2\ninst✝² : Finite σ✝\ninst✝¹ : IsDomain R\nσ : Type u_2\ninst✝ : Fintype σ\nh :\n ∀ (P : MvPolynomial σ R) (S : σ → Finset R),\n (∀ (i : σ), degreeOf i P < #(S i)) → (∀ (x : σ → R), (∀ (i : σ), x i ∈ S i) → (eval x) P = 0) → P = ... | rw [MvPolynomial.degreeOf_eq_sup] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Combinatorics.Quiver.Path.Weight | {
"line": 105,
"column": 6
} | {
"line": 105,
"column": 45
} | [
{
"pp": "case cons\nV : Type u_1\ninst✝³ : Quiver V\nR : Type u_2\ninst✝² : Semiring R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\nw : {i j : V} → (i ⟶ j) → R\nhw : ∀ {i j : V} (e : i ⟶ j), 0 < w e\ni j b✝ c✝ : V\np : Path i b✝\ne : b✝ ⟶ c✝\nih : 0 < weight (fun {i j} ↦ w) p\nhe : 0 < w e\n⊢ 0 < wei... | simpa [weight_cons] using mul_pos ih he | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Combinatorics.Quiver.Path.Decomposition | {
"line": 51,
"column": 2
} | {
"line": 56,
"column": 60
} | [
{
"pp": "V : Type u_1\ninst✝ : Quiver V\na b : V\np : Path a b\nS : Set V\nha_in_S : a ∈ S\nhb_not_in_S : ¬b ∈ S\n⊢ ∃ u, u ∈ S ∧ ∃ v, ¬v ∈ S ∧ ∃ e p₁ p₂, p = p₁.comp (e.toPath.comp p₂)",
"usedConstants": [
"Eq.mpr",
"Quiver.Hom",
"Quiver.Path.exists_notMem_mem_hom_path_path_of_notMem_mem",... | have ha_not_in_compl : a ∉ Sᶜ := by simpa
have hb_in_compl : b ∈ Sᶜ := by simpa
obtain ⟨u, hu_not_in_compl, v, hv_in_compl, e, p₁, p₂, hp⟩ :=
exists_notMem_mem_hom_path_path_of_notMem_mem p Sᶜ ha_not_in_compl hb_in_compl
simp only [Set.mem_compl_iff, not_not] at hu_not_in_compl hv_in_compl
refine ⟨u, hu_not... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.Quiver.Path.Decomposition | {
"line": 51,
"column": 2
} | {
"line": 56,
"column": 60
} | [
{
"pp": "V : Type u_1\ninst✝ : Quiver V\na b : V\np : Path a b\nS : Set V\nha_in_S : a ∈ S\nhb_not_in_S : ¬b ∈ S\n⊢ ∃ u, u ∈ S ∧ ∃ v, ¬v ∈ S ∧ ∃ e p₁ p₂, p = p₁.comp (e.toPath.comp p₂)",
"usedConstants": [
"Eq.mpr",
"Quiver.Hom",
"Quiver.Path.exists_notMem_mem_hom_path_path_of_notMem_mem",... | have ha_not_in_compl : a ∉ Sᶜ := by simpa
have hb_in_compl : b ∈ Sᶜ := by simpa
obtain ⟨u, hu_not_in_compl, v, hv_in_compl, e, p₁, p₂, hp⟩ :=
exists_notMem_mem_hom_path_path_of_notMem_mem p Sᶜ ha_not_in_compl hb_in_compl
simp only [Set.mem_compl_iff, not_not] at hu_not_in_compl hv_in_compl
refine ⟨u, hu_not... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.Schnirelmann | {
"line": 247,
"column": 4
} | {
"line": 247,
"column": 39
} | [
{
"pp": "m : ℕ\nhm : m ≠ 1\nhm' : m > 0\nn : ℕ\nhn : 0 < n\ny : ℕ\na✝ : 0 ≤ y\nhy' : y ≤ (n - 1) / m\n⊢ (0 < y * m + 1 ∧ y * m + 1 ≤ n) ∧ (y * m + 1) % m = 1",
"usedConstants": [
"instOfNatNat",
"LE.le",
"instLENat",
"Nat",
"Nat.instPartialOrder",
"OfNat.ofNat",
"Ne... | have hm : 2 ≤ m := hm.lt_of_le' hm' | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Combinatorics.Schnirelmann | {
"line": 282,
"column": 2
} | {
"line": 282,
"column": 29
} | [
{
"pp": "A B : Set ℕ\ninst✝¹ : DecidablePred fun x ↦ x ∈ A\ninst✝ : DecidablePred fun x ↦ x ∈ B\nhA : 0 ∈ A\nhB : 0 ∈ B\nh : 1 ≤ schnirelmannDensity A + schnirelmannDensity B\n⊢ A + B = Set.univ",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Set.univ",
"Set.eq_univ_iff_forall",
"M... | rw [Set.eq_univ_iff_forall] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Data.Set.Sups | {
"line": 351,
"column": 4
} | {
"line": 351,
"column": 71
} | [
{
"pp": "case a.h.mp\nα : Type u_2\ninst✝ : SemilatticeSup α\ns t : Set α\na b : α\nhb : b ∈ s\nc : α\nhc : c ∈ t\nha : b ⊔ c ≤ a\n⊢ (∃ a_1 ∈ s, a_1 ≤ a) ∧ ∃ a_1 ∈ t, a_1 ≤ a",
"usedConstants": [
"le_sup_left",
"PartialOrder.toPreorder",
"Preorder.toLE",
"Membership.mem",
"Exis... | exact ⟨⟨b, hb, le_sup_left.trans ha⟩, c, hc, le_sup_right.trans ha⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Combinatorics.SetFamily.Compression.Down | {
"line": 145,
"column": 2
} | {
"line": 145,
"column": 56
} | [
{
"pp": "case h\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\na : α\ns : Finset α\n⊢ s ∈ image (insert a) (memberSubfamily a 𝒜) ↔ s ∈ {s ∈ 𝒜 | a ∈ s}",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Finset.memberSubfamily",
"Finset",
"Membership.mem",
"Exists... | simp only [mem_memberSubfamily, mem_image, mem_filter] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Combinatorics.SetFamily.AhlswedeZhang | {
"line": 268,
"column": 2
} | {
"line": 268,
"column": 56
} | [
{
"pp": "α : Type u_1\ninst✝¹ : DistribLattice α\ninst✝ : DecidableEq α\ns t : Finset α\na : α\n⊢ a ∈ lowerClosure ↑(s ⊼ t) ↔ a ∈ lowerClosure ↑s ∧ a ∈ lowerClosure ↑t",
"usedConstants": [
"Eq.mpr",
"LowerSet.mem_inf_iff",
"Finset.coe_infs",
"lowerClosure_infs",
"LowerSet.instM... | rw [coe_infs, lowerClosure_infs, LowerSet.mem_inf_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Combinatorics.SetFamily.AhlswedeZhang | {
"line": 268,
"column": 2
} | {
"line": 268,
"column": 56
} | [
{
"pp": "α : Type u_1\ninst✝¹ : DistribLattice α\ninst✝ : DecidableEq α\ns t : Finset α\na : α\n⊢ a ∈ lowerClosure ↑(s ⊼ t) ↔ a ∈ lowerClosure ↑s ∧ a ∈ lowerClosure ↑t",
"usedConstants": [
"Eq.mpr",
"LowerSet.mem_inf_iff",
"Finset.coe_infs",
"lowerClosure_infs",
"LowerSet.instM... | rw [coe_infs, lowerClosure_infs, LowerSet.mem_inf_iff] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SetFamily.AhlswedeZhang | {
"line": 268,
"column": 2
} | {
"line": 268,
"column": 56
} | [
{
"pp": "α : Type u_1\ninst✝¹ : DistribLattice α\ninst✝ : DecidableEq α\ns t : Finset α\na : α\n⊢ a ∈ lowerClosure ↑(s ⊼ t) ↔ a ∈ lowerClosure ↑s ∧ a ∈ lowerClosure ↑t",
"usedConstants": [
"Eq.mpr",
"LowerSet.mem_inf_iff",
"Finset.coe_infs",
"lowerClosure_infs",
"LowerSet.instM... | rw [coe_infs, lowerClosure_infs, LowerSet.mem_inf_iff] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SetFamily.AhlswedeZhang | {
"line": 331,
"column": 8
} | {
"line": 331,
"column": 33
} | [
{
"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)",
"usedConstants": [
"Eq.mpr"... | truncatedInf_union h𝒜 hℬ, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Finset.Sups | {
"line": 370,
"column": 2
} | {
"line": 371,
"column": 34
} | [
{
"pp": "case h.refine_2\nα : Type u_2\ninst✝ : DecidableEq α\ns t u : Finset α\n⊢ (∃ a ⊆ s, ∃ b ⊆ t, a ∪ b = u) → u ⊆ s ∪ t",
"usedConstants": [
"Finset.instUnion",
"Finset",
"Finset.union_subset_union",
"Exists",
"HasSubset.Subset",
"And.casesOn",
"And",
"Ex... | · rintro ⟨v, hv, w, hw, rfl⟩
exact union_subset_union hv hw | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Combinatorics.SetFamily.Compression.UV | {
"line": 318,
"column": 68
} | {
"line": 318,
"column": 91
} | [
{
"pp": "case pos\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu v : Finset α\nhuv : ∀ x ∈ u, ∃ y ∈ v, IsCompressed (u.erase x) (v.erase y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\nH : ∀ s ∈ ∂ 𝒜', s ∉ ∂ 𝒜 → u ⊆ s ∧ Disjoint v s ∧ (s ∪ v) \\ u ∈ ∂ 𝒜 ∧ (s ∪ v) \\ u ∉ ∂ 𝒜'\nt : Finset α\nHt :... | union_erase_of_mem hat, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.SetFamily.Kleitman | {
"line": 61,
"column": 4
} | {
"line": 61,
"column": 89
} | [
{
"pp": "ι : Type u_1\nα : Type u_2\ninst✝² : Fintype α\ninst✝¹ : DecidableEq α\ninst✝ : Nonempty α\nthis : DecidableEq ι\ni : ι\ns : Finset ι\nhi : i ∉ s\nih :\n ∀ (f : ι → Finset (Finset α)),\n (∀ i ∈ s, (↑(f i)).Intersecting) →\n #s ≤ Fintype.card α → #(s.biUnion f) ≤ 2 ^ Fintype.card α - 2 ^ (Finty... | refine fun j hj ↦ (hf₁ _ hj).2.2.isUpperSet' ((hf₁ _ hj).2.2.is_max_iff_card_eq.2 ?_) | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Combinatorics.SetFamily.Kleitman | {
"line": 63,
"column": 4
} | {
"line": 63,
"column": 24
} | [
{
"pp": "ι : Type u_1\nα : Type u_2\ninst✝² : Fintype α\ninst✝¹ : DecidableEq α\ninst✝ : Nonempty α\nthis : DecidableEq ι\ni : ι\ns : Finset ι\nhi : i ∉ s\nih :\n ∀ (f : ι → Finset (Finset α)),\n (∀ i ∈ s, (↑(f i)).Intersecting) →\n #s ≤ Fintype.card α → #(s.biUnion f) ≤ 2 ^ Fintype.card α - 2 ^ (Finty... | exact (hf₁ _ hj).2.1 | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Combinatorics.SetFamily.LYM | {
"line": 72,
"column": 31
} | {
"line": 72,
"column": 49
} | [
{
"pp": "case refine_1\nα : Type u_2\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\n𝒜 : Finset (Finset α)\nr : ℕ\nh𝒜 : Set.Sized r ↑𝒜\ni : DecidableRel fun x1 x2 ↦ x1 ⊆ x2 := fun x x_1 ↦ Classical.dec ((fun x1 x2 ↦ x1 ⊆ x2) x x_1)\ns : Finset α\nhs : s ∈ 𝒜\n⊢ ∀ x ∈ s, s.erase x ∈ bipartiteBelow (fun x1 x2 ↦ x1... | mem_bipartiteBelow | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Combinatorics.SimpleGraph.Connectivity.Connected | {
"line": 283,
"column": 67
} | {
"line": 286,
"column": 36
} | [
{
"pp": "V : Type u\ninst✝² : Nontrivial V\ninst✝¹ : Fintype V\nG : SimpleGraph V\ninst✝ : DecidableRel G.Adj\nh : G.Preconnected\n⊢ 0 < G.minDegree",
"usedConstants": [
"SimpleGraph.Preconnected.degree_pos_of_nontrivial",
"Eq.mpr",
"congrArg",
"Membership.mem",
"SimpleGraph.ne... | by
obtain ⟨v, hv⟩ := G.exists_minimal_degree_vertex
rw [hv]
exact h.degree_pos_of_nontrivial v | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.SimpleGraph.Connectivity.Connected | {
"line": 605,
"column": 2
} | {
"line": 605,
"column": 44
} | [
{
"pp": "V : Type u\nG G' : SimpleGraph V\nv : V\nh : G ≤ G'\nc' : G'.ConnectedComponent\nhc' : v ∈ c'.supp\nv' : V\nhv' : G.Reachable v' v\n⊢ G'.connectedComponentMk v' = c'",
"usedConstants": [
"Eq.mpr",
"SimpleGraph.connectedComponentMk",
"congrArg",
"id",
"SimpleGraph.Conne... | rw [ConnectedComponent.sound (hv'.mono h)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Combinatorics.SetFamily.FourFunctions | {
"line": 155,
"column": 71
} | {
"line": 235,
"column": 67
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝⁴ : DecidableEq α\ninst✝³ : CommSemiring β\ninst✝² : LinearOrder β\ninst✝¹ : IsStrictOrderedRing β\na : α\nf₁ f₂ f₃ f₄ : Finset α → β\nu : Finset α\ninst✝ : ExistsAddOfLE β\nhu : a ∉ u\nh₁ : 0 ≤ f₁\nh₂ : 0 ≤ f₂\nh₃ : 0 ≤ f₃\nh₄ : 0 ≤ f₄\nh : ∀ ⦃s : Finset α⦄, s ⊆ insert... | by
rintro s hsu t htu
-- Gather a bunch of facts we'll need a lot
have := hsu.trans <| subset_insert a _
have := htu.trans <| subset_insert a _
have := insert_subset_insert a hsu
have := insert_subset_insert a htu
have has := notMem_mono hsu hu
have hat := notMem_mono htu hu
have : a ∉ s ∩ t := notMem... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.SimpleGraph.Bipartite | {
"line": 462,
"column": 2
} | {
"line": 466,
"column": 67
} | [
{
"pp": "V : Type u_1\nw : V\nG : SimpleGraph V\ninst✝² : DecidableEq V\ninst✝¹ : Fintype V\ns : Finset V\ninst✝ : DecidableRel G.Adj\nhw : w ∈ sᶜ\n⊢ G.degree w ≤ (between (↑s) (↑s)ᶜ G).degree w + #sᶜ",
"usedConstants": [
"Eq.mpr",
"Finset.instUnion",
"CompleteBooleanAlgebra.toCompleteDist... | have h_bipartite : (G.between s sᶜ).IsBipartiteWith s ↑(sᶜ) := by
simpa using between_isBipartiteWith disjoint_compl_right
simp_rw [← card_neighborFinset_eq_degree,
← card_union_of_disjoint (isBipartiteWith_neighborFinset_disjoint' h_bipartite hw)]
exact card_le_card (neighborFinset_subset_between_union_com... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SimpleGraph.Bipartite | {
"line": 462,
"column": 2
} | {
"line": 466,
"column": 67
} | [
{
"pp": "V : Type u_1\nw : V\nG : SimpleGraph V\ninst✝² : DecidableEq V\ninst✝¹ : Fintype V\ns : Finset V\ninst✝ : DecidableRel G.Adj\nhw : w ∈ sᶜ\n⊢ G.degree w ≤ (between (↑s) (↑s)ᶜ G).degree w + #sᶜ",
"usedConstants": [
"Eq.mpr",
"Finset.instUnion",
"CompleteBooleanAlgebra.toCompleteDist... | have h_bipartite : (G.between s sᶜ).IsBipartiteWith s ↑(sᶜ) := by
simpa using between_isBipartiteWith disjoint_compl_right
simp_rw [← card_neighborFinset_eq_degree,
← card_union_of_disjoint (isBipartiteWith_neighborFinset_disjoint' h_bipartite hw)]
exact card_le_card (neighborFinset_subset_between_union_com... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SimpleGraph.Connectivity.Subgraph | {
"line": 248,
"column": 2
} | {
"line": 256,
"column": 22
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nu v w : V\np : G.Walk u v\n⊢ (p.toSubgraph.neighborSet w).Finite",
"usedConstants": [
"SimpleGraph.neighborSet_subgraphOfAdj_subset",
"Set.Finite.union",
"Eq.mpr",
"Set.fintypeSingleton",
"congrArg",
"SimpleGraph.Subgraph",
"F... | induction p with
| nil =>
rw [Walk.toSubgraph, neighborSet_singletonSubgraph]
apply Set.toFinite
| cons ha _ ih =>
rw [Walk.toSubgraph, Subgraph.neighborSet_sup]
refine Set.Finite.union ?_ ih
refine Set.Finite.subset ?_ (neighborSet_subgraphOfAdj_subset ha)
apply Set.toFinite | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.Combinatorics.SimpleGraph.Connectivity.Subgraph | {
"line": 248,
"column": 2
} | {
"line": 256,
"column": 22
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nu v w : V\np : G.Walk u v\n⊢ (p.toSubgraph.neighborSet w).Finite",
"usedConstants": [
"SimpleGraph.neighborSet_subgraphOfAdj_subset",
"Set.Finite.union",
"Eq.mpr",
"Set.fintypeSingleton",
"congrArg",
"SimpleGraph.Subgraph",
"F... | induction p with
| nil =>
rw [Walk.toSubgraph, neighborSet_singletonSubgraph]
apply Set.toFinite
| cons ha _ ih =>
rw [Walk.toSubgraph, Subgraph.neighborSet_sup]
refine Set.Finite.union ?_ ih
refine Set.Finite.subset ?_ (neighborSet_subgraphOfAdj_subset ha)
apply Set.toFinite | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SimpleGraph.Connectivity.Subgraph | {
"line": 248,
"column": 2
} | {
"line": 256,
"column": 22
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nu v w : V\np : G.Walk u v\n⊢ (p.toSubgraph.neighborSet w).Finite",
"usedConstants": [
"SimpleGraph.neighborSet_subgraphOfAdj_subset",
"Set.Finite.union",
"Eq.mpr",
"Set.fintypeSingleton",
"congrArg",
"SimpleGraph.Subgraph",
"F... | induction p with
| nil =>
rw [Walk.toSubgraph, neighborSet_singletonSubgraph]
apply Set.toFinite
| cons ha _ ih =>
rw [Walk.toSubgraph, Subgraph.neighborSet_sup]
refine Set.Finite.union ?_ ih
refine Set.Finite.subset ?_ (neighborSet_subgraphOfAdj_subset ha)
apply Set.toFinite | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SimpleGraph.Metric | {
"line": 314,
"column": 2
} | {
"line": 320,
"column": 5
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\nu v w : V\nhadj : G.Adj v w\n⊢ G.dist u w = G.dist u v ∨ G.dist u w = G.dist u v + 1 ∨ G.dist u w = G.dist u v - 1",
"usedConstants": [
"Iff.mpr",
"SimpleGraph.Adj.symm",
"instDecidableNot",
"SimpleGraph.dist",
"SimpleGraph.Adj",
... | by_cases! huw : ¬G.Reachable u w
· grind [dist_eq_zero_iff_eq_or_not_reachable, Reachable.trans, Adj.reachable]
have : G.dist v w = 1 := dist_eq_one_iff_adj.mpr hadj
have : G.dist w v = 1 := dist_eq_one_iff_adj.mpr hadj.symm
have : G.dist u w ≤ G.dist u v + G.dist v w := hadj.reachable.dist_triangle_right u
h... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SimpleGraph.Metric | {
"line": 314,
"column": 2
} | {
"line": 320,
"column": 5
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\nu v w : V\nhadj : G.Adj v w\n⊢ G.dist u w = G.dist u v ∨ G.dist u w = G.dist u v + 1 ∨ G.dist u w = G.dist u v - 1",
"usedConstants": [
"Iff.mpr",
"SimpleGraph.Adj.symm",
"instDecidableNot",
"SimpleGraph.dist",
"SimpleGraph.Adj",
... | by_cases! huw : ¬G.Reachable u w
· grind [dist_eq_zero_iff_eq_or_not_reachable, Reachable.trans, Adj.reachable]
have : G.dist v w = 1 := dist_eq_one_iff_adj.mpr hadj
have : G.dist w v = 1 := dist_eq_one_iff_adj.mpr hadj.symm
have : G.dist u w ≤ G.dist u v + G.dist v w := hadj.reachable.dist_triangle_right u
h... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SimpleGraph.Connectivity.Subgraph | {
"line": 455,
"column": 2
} | {
"line": 457,
"column": 9
} | [
{
"pp": "case h.inr\nV : Type u\nG : SimpleGraph V\nu : V\ni : ℕ\np : G.Walk u u\nhpc : p.IsCycle\nv : V\nh : ¬i = 0\nh' : i < p.length\nhadj1 :\n ∃ i_1,\n (p.getVert i_1 = p.getVert i ∧ p.getVert (i_1 + 1) = p.getVert (i - 1) ∨\n p.getVert i_1 = p.getVert (i - 1) ∧ p.getVert (i_1 + 1) = p.getVert i)... | · apply hpc.getVert_injOn (by rw [Set.mem_setOf_eq]; lia)
(by rw [Set.mem_setOf_eq]; lia) at hr2
aesop | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Combinatorics.SimpleGraph.Connectivity.Subgraph | {
"line": 700,
"column": 82
} | {
"line": 704,
"column": 56
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nG' : G.Subgraph\nG'' : G'.coe.Subgraph\nf : G'.coe →g G\nhpreconn : G''.Preconnected\n⊢ (Subgraph.map f G'').Preconnected",
"usedConstants": [
"Eq.mpr",
"Subtype.mk.congr_simp",
"RelHom.instFunLike",
"congrArg",
"SimpleGraph.Adj",
"... | by
rw [Subgraph.preconnected_iff]
intro ⟨u', u, hu, hfu⟩ ⟨v', v, hv, hfv⟩
simp_rw [← hfu, ← hfv]
exact (hpreconn.coe ⟨u, hu⟩ ⟨v, hv⟩).coe_subgraphMap f | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.SimpleGraph.Sum | {
"line": 45,
"column": 26
} | {
"line": 45,
"column": 42
} | [
{
"pp": "U : Type u_1\nU' : Type u_2\nV : Type u_3\nV' : Type u_4\nW : Type u_5\nW' : Type u_6\nγ : Type u_7\nG✝ : SimpleGraph V\nH✝ : SimpleGraph W\nI : SimpleGraph U\nG' : SimpleGraph V'\nH' : SimpleGraph W'\nI' : SimpleGraph U'\nv v' : V\nw w' : W\nG : SimpleGraph V\nH : SimpleGraph W\nu : V ⊕ W\n⊢ ¬match u,... | cases u <;> simp | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Combinatorics.SimpleGraph.Sum | {
"line": 45,
"column": 26
} | {
"line": 45,
"column": 42
} | [
{
"pp": "U : Type u_1\nU' : Type u_2\nV : Type u_3\nV' : Type u_4\nW : Type u_5\nW' : Type u_6\nγ : Type u_7\nG✝ : SimpleGraph V\nH✝ : SimpleGraph W\nI : SimpleGraph U\nG' : SimpleGraph V'\nH' : SimpleGraph W'\nI' : SimpleGraph U'\nv v' : V\nw w' : W\nG : SimpleGraph V\nH : SimpleGraph W\nu : V ⊕ W\n⊢ ¬match u,... | cases u <;> simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SimpleGraph.Sum | {
"line": 45,
"column": 26
} | {
"line": 45,
"column": 42
} | [
{
"pp": "U : Type u_1\nU' : Type u_2\nV : Type u_3\nV' : Type u_4\nW : Type u_5\nW' : Type u_6\nγ : Type u_7\nG✝ : SimpleGraph V\nH✝ : SimpleGraph W\nI : SimpleGraph U\nG' : SimpleGraph V'\nH' : SimpleGraph W'\nI' : SimpleGraph U'\nv v' : V\nw w' : W\nG : SimpleGraph V\nH : SimpleGraph W\nu : V ⊕ W\n⊢ ¬match u,... | cases u <;> simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SimpleGraph.Sum | {
"line": 144,
"column": 52
} | {
"line": 145,
"column": 60
} | [
{
"pp": "V : Type u_3\nW : Type u_5\nG : SimpleGraph V\nH : SimpleGraph W\nv : V\nw : W\nhG : G.Connected\nhH : H.Connected\n⊢ ((G ⊕g H) ⊔ edge (Sum.inl v) (Sum.inr w)).Connected",
"usedConstants": [
"SimpleGraph.Connected.preconnected",
"SimpleGraph.edge",
"SimpleGraph.Connected.mk",
... | by
obtain ⟨hG⟩ := hG; exact ⟨hG.sum_sup_edge hH.preconnected⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.SimpleGraph.Bipartite | {
"line": 607,
"column": 14
} | {
"line": 607,
"column": 50
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\n⊢ Disjoint {v | v.isLeft = true} {w | w.isRight = true}",
"usedConstants": [
"Sum.isRight",
"False",
"Sum.ctorIdx",
"congrArg",
"and_self",
"Sum.inr.injEq",
"False.elim",
"PartialOrder.toPreorder",
"setOf",
... | by simp [Set.disjoint_iff_forall_ne] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.SimpleGraph.Acyclic | {
"line": 384,
"column": 2
} | {
"line": 385,
"column": 82
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\nhtree : G.IsTree\nG' : SimpleGraph V\nh' : G'.Connected\nhle : G' ≤ G\nu v : V\nhadj : G.Adj u v\np : G'.Walk u v\nhp : p.IsPath\n⊢ G'.Adj u v",
"usedConstants": [
"congrArg",
"SimpleGraph.IsAcyclic.path_unique",
"SimpleGraph.Walk",
"Subtype"... | have := congrArg Walk.edges <| congrArg Subtype.val <|
htree.isAcyclic.path_unique ⟨p.mapLe hle, hp.mapLe hle⟩ <| Path.singleton hadj | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Combinatorics.SimpleGraph.Acyclic | {
"line": 547,
"column": 2
} | {
"line": 547,
"column": 70
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\nhconn : G.Connected\nv : V\ninst✝ : Fintype ↑(G.neighborSet v)\nhdeg : G.degree v = 1\n⊢ (induce {v}ᶜ G).Connected",
"usedConstants": [
"SimpleGraph.Adj",
"instOfNatNat",
"ExistsUnique",
"Nat",
"Iff.mp",
"SimpleGraph.degree",
... | obtain ⟨u, adj_vu, hu⟩ := degree_eq_one_iff_existsUnique_adj.mp hdeg | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Combinatorics.SimpleGraph.Ends.Defs | {
"line": 68,
"column": 71
} | {
"line": 71,
"column": 9
} | [
{
"pp": "V : Type u\nK : Set V\nG : SimpleGraph V\nv w : V\nvK : v ∉ K\nwK : w ∉ K\na : G.Adj v w\n⊢ G.componentComplMk vK = G.componentComplMk wK",
"usedConstants": [
"Eq.mpr",
"SimpleGraph.connectedComponentMk",
"congrArg",
"Compl.compl",
"SimpleGraph.Adj.reachable",
"M... | by
rw [ConnectedComponent.eq]
apply Adj.reachable
exact a | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.SimpleGraph.Ends.Defs | {
"line": 145,
"column": 2
} | {
"line": 145,
"column": 29
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nK : Set V\nGc : G.Preconnected\nhK : K.Nonempty\nv : V\nvnK : v ∉ K\nC : G.ComponentCompl K := ⋯\ndis : K ∩ ↑C ⊆ ∅ := ⋯\nh : ∀ (ck : V × V), ck.1 ∈ G.componentComplMk vnK → ck.2 ∈ K → ¬G.Adj ck.1 ck.2\n⊢ ↑C = Set.univ",
"usedConstants": [
"SimpleGraph.ComponentC... | rw [Set.eq_univ_iff_forall] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Combinatorics.SimpleGraph.Diam | {
"line": 454,
"column": 2
} | {
"line": 454,
"column": 29
} | [
{
"pp": "α : Type u_1\nG : SimpleGraph α\ninst✝ : Subsingleton α\n⊢ G.center = Set.univ",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Set.univ",
"Set.eq_univ_iff_forall",
"Membership.mem",
"id",
"SimpleGraph.center",
"propext",
"Eq",
"Set.instMembers... | rw [Set.eq_univ_iff_forall] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Combinatorics.SimpleGraph.Diam | {
"line": 462,
"column": 4
} | {
"line": 462,
"column": 31
} | [
{
"pp": "case inr\nα : Type u_1\nh✝ : Nontrivial α\n⊢ ⊥.center = Set.univ",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Set.univ",
"Set.eq_univ_iff_forall",
"Membership.mem",
"id",
"Bot.bot",
"SimpleGraph.center",
"SimpleGraph",
"propext",
"Sim... | rw [Set.eq_univ_iff_forall] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Combinatorics.SimpleGraph.Diam | {
"line": 469,
"column": 4
} | {
"line": 469,
"column": 31
} | [
{
"pp": "case inr\nα : Type u_1\nh✝ : Nontrivial α\n⊢ ⊤.center = Set.univ",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Set.univ",
"Set.eq_univ_iff_forall",
"Membership.mem",
"id",
"SimpleGraph.center",
"SimpleGraph",
"BooleanAlgebra.toTop",
"propext... | rw [Set.eq_univ_iff_forall] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Data.Set.Card.Arithmetic | {
"line": 50,
"column": 44
} | {
"line": 50,
"column": 79
} | [
{
"pp": "α : Type u_1\ns : Set α\nhs : s.Infinite\nh : ¬1 = 0\n⊢ (s ∩ Function.support fun i ↦ 1).Infinite",
"usedConstants": [
"Eq.mpr",
"Nat.instMulZeroClass",
"congrArg",
"Set.univ",
"AddMonoid.toAddZeroClass",
"Set.inter_univ",
"AddZeroClass.toAddZero",
"i... | by simpa [Function.support_const h] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Set.Card.Arithmetic | {
"line": 74,
"column": 39
} | {
"line": 75,
"column": 49
} | [
{
"pp": "α : Type u_1\ns : Set α\nh : s.Infinite\nthis : Infinite ↑s\n⊢ Nonempty (↑s ≃ ↑s ⊕ ↑s)",
"usedConstants": [
"Eq.mpr",
"Cardinal",
"congrArg",
"Cardinal.mk",
"Set.Elem",
"Sum",
"id",
"Equiv",
"Cardinal.instAdd",
"instHAdd",
"HAdd.hAdd... | by
rw [← Cardinal.eq, ← add_def, add_mk_eq_self] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Set.Card.Arithmetic | {
"line": 102,
"column": 8
} | {
"line": 102,
"column": 20
} | [
{
"pp": "case mpr.inl\nα : Type u_1\nt u : Set α\nhdtu : Disjoint t u\nhctu : #↑t = #↑u\nhfin : (t ∪ u).Finite\n⊢ Even (t ∪ u).ncard",
"usedConstants": [
"congrArg",
"Set.finite_union",
"Set.Finite",
"Set.instUnion",
"Eq.mp",
"And",
"propext",
"Union.union",
... | finite_union | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.SimpleGraph.Hamiltonian | {
"line": 191,
"column": 73
} | {
"line": 191,
"column": 91
} | [
{
"pp": "α : Type u_1\ninst✝¹ : DecidableEq α\nG : SimpleGraph α\na : α\np : G.Walk a a\ninst✝ : Fintype α\nhp : p.IsHamiltonianCycle\n⊢ Fintype.card α - 1 + 1 = Fintype.card α",
"usedConstants": [
"Eq.mpr",
"congrArg",
"HSub.hSub",
"Fintype.card",
"id",
"instSubNat",
... | Nat.sub_add_cancel | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.SimpleGraph.Matching | {
"line": 101,
"column": 2
} | {
"line": 101,
"column": 24
} | [
{
"pp": "V : Type u_1\nG G' : SimpleGraph V\nM : G.Subgraph\nh : M.IsMatching\nhGG' : G ≤ G'\nv✝ : V\nhv✝ : v✝ ∈ (Subgraph.map (Hom.ofLE hGG') M).verts\nw✝ : V\nhv : w✝ ∈ M.verts\nhv' : (Hom.ofLE hGG') w✝ = v✝\n⊢ ∃! w, (Subgraph.map (Hom.ofLE hGG') M).Adj v✝ w",
"usedConstants": []
}
] | obtain ⟨w, hw⟩ := h hv | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Combinatorics.SimpleGraph.LapMatrix | {
"line": 113,
"column": 14
} | {
"line": 113,
"column": 78
} | [
{
"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_1, ∑ x_2, if G.Adj x_1 x_2 then x x_1 * x x_1 - x x_1 * x x_2 else 0) +\n ∑ x_1, ∑ x_2, if G.Adj x_1 x_2 then x x_1 * x x... | enter [1, 2, 2, i, 2, j]; rw [if_congr (adj_comm G i j) rfl rfl] | Lean.Elab.Tactic.Conv.evalConvSeq1Indented | Lean.Parser.Tactic.Conv.convSeq1Indented |
Mathlib.Combinatorics.SimpleGraph.LapMatrix | {
"line": 113,
"column": 14
} | {
"line": 113,
"column": 78
} | [
{
"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_1, ∑ x_2, if G.Adj x_1 x_2 then x x_1 * x x_1 - x x_1 * x x_2 else 0) +\n ∑ x_1, ∑ x_2, if G.Adj x_1 x_2 then x x_1 * x x... | enter [1, 2, 2, i, 2, j]; rw [if_congr (adj_comm G i j) rfl rfl] | Lean.Elab.Tactic.Conv.evalConvSeq | Lean.Parser.Tactic.Conv.convSeq |
Mathlib.Combinatorics.SimpleGraph.Matching | {
"line": 475,
"column": 4
} | {
"line": 476,
"column": 12
} | [
{
"pp": "case pos\nV : Type u_1\nG : SimpleGraph V\ninst✝ : Finite V\nv w : V\nhcyc : G.IsCycles\np : G.Walk v w\nhp : p.IsPath\nhvw : v = w\n⊢ (G \\ p.toSubgraph.spanningCoe).Reachable w v",
"usedConstants": [
"SimpleGraph.Walk",
"SimpleGraph.Walk.toSubgraph",
"Nonempty.intro",
"Sim... | subst hvw
use .nil | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SimpleGraph.Matching | {
"line": 475,
"column": 4
} | {
"line": 476,
"column": 12
} | [
{
"pp": "case pos\nV : Type u_1\nG : SimpleGraph V\ninst✝ : Finite V\nv w : V\nhcyc : G.IsCycles\np : G.Walk v w\nhp : p.IsPath\nhvw : v = w\n⊢ (G \\ p.toSubgraph.spanningCoe).Reachable w v",
"usedConstants": [
"SimpleGraph.Walk",
"SimpleGraph.Walk.toSubgraph",
"Nonempty.intro",
"Sim... | subst hvw
use .nil | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SimpleGraph.Tutte | {
"line": 51,
"column": 2
} | {
"line": 51,
"column": 74
} | [
{
"pp": "case refine_1\nV : Type u_1\nG G' : SimpleGraph V\nx b a c : V\nM : (G ⊔ edge a c).Subgraph\np : G'.Walk a x\nhp : p.IsPath\nhcalt : G'.IsAlternating M.spanningCoe\nhM2nadj : ¬M.Adj x a\nhpac : p.toSubgraph.Adj a c\nhnpxb : ¬p.toSubgraph.Adj x b\nhM2ac : M.Adj a c\nhgadj : G.Adj x a\nhnxc : x ≠ c\nhnab... | · simpa [← hp.snd_of_toSubgraph_adj hadj, hp.snd_of_toSubgraph_adj hpac] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Computability.Primrec.Basic | {
"line": 162,
"column": 12
} | {
"line": 162,
"column": 30
} | [
{
"pp": "case succ\nα : Type u_1\nh : Primcodable α\nn : ℕ\n⊢ (Nat.casesOn (n + 1) 1 fun n ↦ Nat.casesOn (encode (decode n)) 0 fun n ↦ n.succ.succ) = encode (decode (n + 1))",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Option.some",
"Option.encodable",
"id",
"instOfNatNat"... | decode_option_succ | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Computability.Primrec.Basic | {
"line": 352,
"column": 6
} | {
"line": 352,
"column": 25
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nσ : Type u_3\ninst✝² : Primcodable α\ninst✝¹ : Primcodable β\ninst✝ : Primcodable σ\nf g : α → β → σ\nhg : Primrec₂ f\nH : ∀ (a : α) (b : β), f a b = g a b\n⊢ f = g",
"usedConstants": [
"funext"
]
}
] | funext a b; apply H | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Computability.Primrec.Basic | {
"line": 352,
"column": 6
} | {
"line": 352,
"column": 25
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nσ : Type u_3\ninst✝² : Primcodable α\ninst✝¹ : Primcodable β\ninst✝ : Primcodable σ\nf g : α → β → σ\nhg : Primrec₂ f\nH : ∀ (a : α) (b : β), f a b = g a b\n⊢ f = g",
"usedConstants": [
"funext"
]
}
] | funext a b; apply H | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SimpleGraph.Tutte | {
"line": 240,
"column": 2
} | {
"line": 246,
"column": 9
} | [
{
"pp": "case neg\nV : 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.Is... | have hle : p.toSubgraph.spanningCoe ≤ G ⊔ edge a c := by
rw [← sdiff_edge _ (by simpa : ¬p.toSubgraph.spanningCoe.Adj x b), sdiff_le_iff']
intro v w hvw
apply hsupG ▸ sup_le_sup hM1sub hM2sub
have := p.toSubgraph.spanningCoe_le hvw
simp only [cycles, symmDiff_def] at this
aesop | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Computability.Partrec | {
"line": 212,
"column": 17
} | {
"line": 212,
"column": 34
} | [
{
"pp": "f g h : ℕ →. ℕ\nhf : Nat.Partrec f\nhg : Nat.Partrec g\nhh : Nat.Partrec h\na s : ℕ\n⊢ (s ∈\n Nat.pair <$> Part.some a <*> f a >>=\n unpaired fun a n ↦\n Nat.rec (g a)\n (fun y IH ↦ do\n let i ← IH\n h (Nat.pair a (Nat.pair y i)))\n n... | by simp [Seq.seq] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.SimpleGraph.Tutte | {
"line": 248,
"column": 2
} | {
"line": 256,
"column": 16
} | [
{
"pp": "case neg\nV : 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.Is... | have aux {x' : V} (hx' : x' ∈ ({x, b} : Set V)) (c' : V) (hc : c' ≠ a)
(hadj : p.toSubgraph.Adj c' x') : M2.Adj c' x' := by
refine (hadj.adj_sub.resolve_left fun hl ↦ hnpxb ?_).1
obtain ⟨w, -, hw⟩ := hM1.1 (hM1.2 x')
obtain rfl | rfl := hx'
· rw [hw _ hM1xb, ← hw _ hl.1.symm]
exact hadj.symm... | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Computability.Ackermann | {
"line": 384,
"column": 2
} | {
"line": 389,
"column": 8
} | [
{
"pp": "⊢ Computable₂ ack",
"usedConstants": [
"Part",
"ack",
"congrArg",
"Nat.Partrec.Code.primrec_pappAck",
"Primcodable.ofDenumerable",
"Part.some",
"Primrec.to_comp",
"Membership.mem",
"Nat.Partrec.Code",
"Partrec₂.comp₂",
"id",
"P... | apply _root_.Partrec.of_eq_tot
(f := fun p : ℕ × ℕ => (pappAck p.1).eval p.2) (g := fun p : ℕ × ℕ => ack p.1 p.2)
· change Partrec₂ (fun m n => (pappAck m).eval n)
apply_rules only
[Code.eval_part.comp₂, Computable.fst, Computable.snd, primrec_pappAck.to_comp.comp]
· simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Computability.Ackermann | {
"line": 384,
"column": 2
} | {
"line": 389,
"column": 8
} | [
{
"pp": "⊢ Computable₂ ack",
"usedConstants": [
"Part",
"ack",
"congrArg",
"Nat.Partrec.Code.primrec_pappAck",
"Primcodable.ofDenumerable",
"Part.some",
"Primrec.to_comp",
"Membership.mem",
"Nat.Partrec.Code",
"Partrec₂.comp₂",
"id",
"P... | apply _root_.Partrec.of_eq_tot
(f := fun p : ℕ × ℕ => (pappAck p.1).eval p.2) (g := fun p : ℕ × ℕ => ack p.1 p.2)
· change Partrec₂ (fun m n => (pappAck m).eval n)
apply_rules only
[Code.eval_part.comp₂, Computable.fst, Computable.snd, primrec_pappAck.to_comp.comp]
· simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Computability.AkraBazzi.GrowsPolynomially | {
"line": 309,
"column": 19
} | {
"line": 309,
"column": 52
} | [
{
"pp": "f g : ℝ → ℝ\nhf✝ : GrowsPolynomially f\nhg✝ : GrowsPolynomially g\nb : ℝ\nhb : b ∈ Set.Ioo 0 1\nc₁ : ℝ\nhc₁_mem : c₁ > 0\nc₂ : ℝ\nhc₂_mem : c₂ > 0\nhf :\n ∀ᶠ (x : ℝ) in atTop,\n ∀ u ∈ Set.Icc (b * x) x, (fun x ↦ |f x|) u ∈ Set.Icc (c₁ * (fun x ↦ |f x|) x) (c₂ * (fun x ↦ |f x|) x)\nc₃ : ℝ\nhc₃_mem :... | by change 0 < c₂ * c₄; positivity | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Computability.AkraBazzi.SumTransform | {
"line": 215,
"column": 33
} | {
"line": 215,
"column": 62
} | [
{
"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 := b (max_bi b) + (1 - b (max_bi b)) / 2\nh_max_bi_pos : 0 < b (max_bi b)\nh_max_bi_lt_one : 0 < 1 - b (max_bi b)\nhc_pos : 0 < c\nh₁ : 0 < (1 - b (max_bi b))... | by gcongr; exact R.b_lt_one _ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Computability.AkraBazzi.AkraBazzi | {
"line": 247,
"column": 39
} | {
"line": 247,
"column": 69
} | [
{
"pp": "case hbc\nα : 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\nq : ℝ → ℝ\nhq_diff : DifferentiableOn ℝ q (Set.Ioi 1)\nhq_poly✝ : GrowsPolynomially fun x ↦ ‖deriv q x‖\ni : α\nb' : ℝ := ⋯\nhb_pos : 0 < b'\nhb_lt_one : ... | exact le_of_lt <| R.b_lt_one i | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Computability.PartrecCode | {
"line": 377,
"column": 37
} | {
"line": 446,
"column": 44
} | [
{
"pp": "α : Type u_1\nσ : Type u_2\ninst✝¹ : Primcodable α\ninst✝ : Primcodable σ\nc : α → Code\nhc : Computable c\nz : α → σ\nhz : Computable z\ns : α → σ\nhs : Computable s\nl : α → σ\nhl : Computable l\nr : α → σ\nhr : Computable r\npr : α → Code × Code × σ × σ → σ\nhpr : Computable₂ pr\nco : α → Code × Cod... | by
-- TODO(Mario): less copy-paste from previous proof
intro _ _ _ _ F
let G₁ : (α × List σ) × ℕ × ℕ → Option σ := fun p =>
letI a := p.1.1; letI IH := p.1.2; letI n := p.2.1; letI m := p.2.2
IH[m]?.bind fun s =>
IH[m.unpair.1]?.bind fun s₁ =>
IH[m.unpair.2]?.map fun s₂ =>
cond n.bodd
(c... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Computability.PartrecCode | {
"line": 536,
"column": 2
} | {
"line": 553,
"column": 57
} | [
{
"pp": "case refine_1\nf : ℕ →. ℕ\nh : Nat.Partrec f\n⊢ ∃ c, c.eval = f",
"usedConstants": [
"Pure.pure",
"Part",
"Unit.unit",
"Nat.rfind",
"Nat.Partrec",
"PFun",
"Nat.Partrec.Code.rfind'",
"congrArg",
"Part.bind",
"Nat.unpair",
"Part.some",... | · induction h with
| zero => exact ⟨zero, rfl⟩
| succ => exact ⟨succ, rfl⟩
| left => exact ⟨left, rfl⟩
| right => exact ⟨right, rfl⟩
| pair pf pg hf hg =>
rcases hf with ⟨cf, rfl⟩; rcases hg with ⟨cg, rfl⟩
exact ⟨pair cf cg, rfl⟩
| comp pf pg hf hg =>
rcases hf with ⟨cf, rfl⟩; ... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Computability.AkraBazzi.GrowsPolynomially | {
"line": 355,
"column": 16
} | {
"line": 355,
"column": 37
} | [
{
"pp": "case h₁.hbc\nf g : ℝ → ℝ\nhf✝¹ : GrowsPolynomially f\nhg✝¹ : GrowsPolynomially g\nhf'✝ : 0 ≤ᶠ[atTop] f\nhg'✝ : 0 ≤ᶠ[atTop] g\nb : ℝ\nhb : b ∈ Set.Ioo 0 1\nc₁ : ℝ\nhc₁_mem : c₁ > 0\nc₂ : ℝ\nhc₂_mem : c₂ > 0\nhf✝ : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)\nc₃ : ℝ\... | exact min_le_left _ _ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Computability.AkraBazzi.GrowsPolynomially | {
"line": 355,
"column": 16
} | {
"line": 355,
"column": 37
} | [
{
"pp": "case h₁.hbc\nf g : ℝ → ℝ\nhf✝¹ : GrowsPolynomially f\nhg✝¹ : GrowsPolynomially g\nhf'✝ : 0 ≤ᶠ[atTop] f\nhg'✝ : 0 ≤ᶠ[atTop] g\nb : ℝ\nhb : b ∈ Set.Ioo 0 1\nc₁ : ℝ\nhc₁_mem : c₁ > 0\nc₂ : ℝ\nhc₂_mem : c₂ > 0\nhf✝ : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)\nc₃ : ℝ\... | exact min_le_left _ _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Computability.AkraBazzi.GrowsPolynomially | {
"line": 355,
"column": 16
} | {
"line": 355,
"column": 37
} | [
{
"pp": "case h₁.hbc\nf g : ℝ → ℝ\nhf✝¹ : GrowsPolynomially f\nhg✝¹ : GrowsPolynomially g\nhf'✝ : 0 ≤ᶠ[atTop] f\nhg'✝ : 0 ≤ᶠ[atTop] g\nb : ℝ\nhb : b ∈ Set.Ioo 0 1\nc₁ : ℝ\nhc₁_mem : c₁ > 0\nc₂ : ℝ\nhc₂_mem : c₂ > 0\nhf✝ : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)\nc₃ : ℝ\... | exact min_le_left _ _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Computability.AkraBazzi.GrowsPolynomially | {
"line": 398,
"column": 12
} | {
"line": 398,
"column": 26
} | [
{
"pp": "case h\nf g : ℝ → ℝ\nhf✝ : GrowsPolynomially f\nb : ℝ\nhb : b ∈ Set.Ioo 0 1\nhb_ub : b < 1\nhf' : ∀ᶠ (x : ℝ) in atTop, 0 ≤ f x\nc₁ : ℝ\nhc₁_mem : 0 < c₁\nc₂ : ℝ\nhc₂_mem : 0 < c₂\nhf : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)\nhfg : ∀ᶠ (x : ℝ) in atTop, ‖g x‖ ≤ ... | ⟨hu_lb, hu_ub⟩ | Lean.Elab.Tactic.evalIntro | Lean.Parser.Term.anonymousCtor |
Mathlib.Computability.AkraBazzi.GrowsPolynomially | {
"line": 402,
"column": 40
} | {
"line": 402,
"column": 88
} | [
{
"pp": "case e_a\nf g : ℝ → ℝ\nhf✝ : GrowsPolynomially f\nb : ℝ\nhb : b ∈ Set.Ioo 0 1\nhb_ub : b < 1\nhf' : ∀ᶠ (x : ℝ) in atTop, 0 ≤ f x\nc₁ : ℝ\nhc₁_mem : 0 < c₁\nc₂ : ℝ\nhc₂_mem : 0 < c₂\nhf : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)\nhfg : ∀ᶠ (x : ℝ) in atTop, ‖g x‖ ... | simp only [norm_eq_abs, abs_eq_self, hfu_nonneg] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Computability.AkraBazzi.AkraBazzi | {
"line": 502,
"column": 6
} | {
"line": 503,
"column": 81
} | [
{
"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\nb' : ℝ := b (min_bi b) / 2\nhb_pos : 0 < b'\nc₁ : ℝ\nhc₁ : c₁ > 0\nh_sumTransform_aux : ∀ᶠ (n : ℕ) in atTop, ∀ (i : α), c₁ * g ↑n ≤ sumTransform (p a b) g (r i n) n... | gcongr (∑ i, C * a i * (?_
* ((1 + (∑ u ∈ range (r i n), g u / u ^ ((p a b) + 1)))))) + g n with i | Mathlib.Tactic.GCongr._aux_Mathlib_Tactic_GCongr_Core___elabRules_Mathlib_Tactic_GCongr_gcongr_1 | Mathlib.Tactic.GCongr.gcongr |
Mathlib.Computability.AkraBazzi.GrowsPolynomially | {
"line": 446,
"column": 12
} | {
"line": 446,
"column": 26
} | [
{
"pp": "case h\nf g : ℝ → ℝ\nhf✝ : GrowsPolynomially f\nb : ℝ\nhb : b ∈ Set.Ioo 0 1\nhb_ub : b < 1\nhf' : ∀ᶠ (x : ℝ) in atTop, f x ≤ 0\nc₁ : ℝ\nhc₁_mem : 0 < c₁\nc₂ : ℝ\nhc₂_mem : 0 < c₂\nhf : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)\nhfg : ∀ᶠ (x : ℝ) in atTop, ‖g x‖ ≤ ... | ⟨hu_lb, hu_ub⟩ | Lean.Elab.Tactic.evalIntro | Lean.Parser.Term.anonymousCtor |
Mathlib.Computability.PartrecCode | {
"line": 937,
"column": 6
} | {
"line": 937,
"column": 14
} | [
{
"pp": "x✝ : Unit\np n : ℕ\nthis : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))\nk : ℕ\nc : Code\n⊢ n ∈ List.range k →\n Nat.rec Option.none\n (fun n_1 n_ih ↦\n rec (some 0) (some n.succ) (some (unpair n).1) (some (unpair n).2)\n (fun cf cg ... | intro nk | Lean.Elab.Tactic.evalIntro | null |
Mathlib.Computability.PartrecCode | {
"line": 937,
"column": 6
} | {
"line": 937,
"column": 14
} | [
{
"pp": "x✝ : Unit\np n : ℕ\nthis : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))\nk : ℕ\nc : Code\n⊢ n ∈ List.range k →\n Nat.rec Option.none\n (fun n_1 n_ih ↦\n rec (some 0) (some n.succ) (some (unpair n).1) (some (unpair n).2)\n (fun cf cg ... | intro nk | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Computability.AkraBazzi.SumTransform | {
"line": 717,
"column": 16
} | {
"line": 717,
"column": 37
} | [
{
"pp": "case hbc\nα : 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\nc₁ : ℝ\nhc₁_mem : c₁ ∈ Set.Ioo 0 1\nhc₁ : ∀ᶠ (n : ℕ) in atTop, ∀ (i : α), c₁ * ↑n ≤ ↑(r i n)\nc₂ : ℝ\nhc₂_mem : c₂ > 0\nhc₂ : ∀ᶠ (n : ℕ) in atTop, ∀ u ∈ S... | exact min_le_left _ _ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Computability.PartrecCode | {
"line": 992,
"column": 4
} | {
"line": 992,
"column": 59
} | [
{
"pp": "c : Code\nn x : ℕ\n⊢ x ∈ c.eval n ↔ x ∈ rfindOpt fun k ↦ evaln k c n",
"usedConstants": [
"Part",
"Nat.Partrec.Code.evaln",
"Option.instMembership",
"Membership.mem",
"Exists",
"Part.instMembership",
"Nat.rfindOpt_mono",
"Nat",
"Iff.trans",
... | refine evaln_complete.trans (Nat.rfindOpt_mono ?_).symm | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Computability.Language | {
"line": 256,
"column": 6
} | {
"line": 256,
"column": 51
} | [
{
"pp": "case succ.mpr\nα : Type u_1\nl : Language α\nn : ℕ\nihn : ∀ {x : List α}, x ∈ l ^ n ↔ ∃ S, x = S.flatten ∧ S.length = n ∧ ∀ y ∈ S, y ∈ l\nx : List α\n⊢ (∃ S, x = S.flatten ∧ S.length = n + 1 ∧ ∀ y ∈ S, y ∈ l) →\n ∃ a ∈ l, ∃ b, (∃ S, b = S.flatten ∧ S.length = n ∧ ∀ y ∈ S, y ∈ l) ∧ a ++ b = x",
"... | rintro ⟨_ | ⟨a, S⟩, rfl, hn, hS⟩ <;> cases hn | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Computability.Language | {
"line": 271,
"column": 66
} | {
"line": 272,
"column": 76
} | [
{
"pp": "α : Type u_1\nl : Language α\n⊢ l∗ * l = l * l∗",
"usedConstants": [
"instCompleteAtomicBooleanAlgebraLanguage",
"HMul.hMul",
"Monoid.toMulOneClass",
"congrArg",
"iSup",
"KStar.kstar",
"MulOne.toMul",
"instOfNatNat",
"Language.instSemiring",
... | by
simp only [kstar_eq_iSup_pow, mul_iSup, iSup_mul, ← pow_succ, ← pow_succ'] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Computability.Language | {
"line": 295,
"column": 4
} | {
"line": 295,
"column": 36
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nl✝ m✝ : Language α\na b x : List α\nl m : Language α\nh : m * l ≤ m\n⊢ m * l∗ ≤ m",
"usedConstants": [
"Eq.mpr",
"Semigroup.toMul",
"instCompleteAtomicBooleanAlgebraLanguage",
"Lattice.toSemilatticeSup",
"HMul.hMul",
"Com... | rw [kstar_eq_iSup_pow, mul_iSup] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Computability.DFA | {
"line": 312,
"column": 19
} | {
"line": 312,
"column": 60
} | [
{
"pp": "case h.cons\nα : Type u\nσ1 σ2 : Type v\nM1 : DFA α σ1\nM2 : DFA α σ2\na : α\nx : List α\nih :\n ∀ (s1 : σ1) (s2 : σ2),\n (M1.union M2).evalFrom (s1, s2) x ∈ (M1.union M2).accept ↔\n M1.evalFrom s1 x ∈ M1.accept ∨ M2.evalFrom s2 x ∈ M2.accept\ns1 : σ1\ns2 : σ2\n⊢ (M1.union M2).evalFrom (s1, s2... | simp only [evalFrom_cons, union_step, ih] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Computability.DFA | {
"line": 312,
"column": 19
} | {
"line": 312,
"column": 60
} | [
{
"pp": "case h.cons\nα : Type u\nσ1 σ2 : Type v\nM1 : DFA α σ1\nM2 : DFA α σ2\na : α\nx : List α\nih :\n ∀ (s1 : σ1) (s2 : σ2),\n (M1.union M2).evalFrom (s1, s2) x ∈ (M1.union M2).accept ↔\n M1.evalFrom s1 x ∈ M1.accept ∨ M2.evalFrom s2 x ∈ M2.accept\ns1 : σ1\ns2 : σ2\n⊢ (M1.union M2).evalFrom (s1, s2... | simp only [evalFrom_cons, union_step, ih] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Computability.DFA | {
"line": 312,
"column": 19
} | {
"line": 312,
"column": 60
} | [
{
"pp": "case h.cons\nα : Type u\nσ1 σ2 : Type v\nM1 : DFA α σ1\nM2 : DFA α σ2\na : α\nx : List α\nih :\n ∀ (s1 : σ1) (s2 : σ2),\n (M1.union M2).evalFrom (s1, s2) x ∈ (M1.union M2).accept ↔\n M1.evalFrom s1 x ∈ M1.accept ∨ M2.evalFrom s2 x ∈ M2.accept\ns1 : σ1\ns2 : σ2\n⊢ (M1.union M2).evalFrom (s1, s2... | simp only [evalFrom_cons, union_step, ih] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Nat.Bitwise | {
"line": 184,
"column": 16
} | {
"line": 184,
"column": 40
} | [
{
"pp": "b : Bool\nhn : 0 ≠ 0 → ∃ i, testBit 0 i = true ∧ ∀ (j : ℕ), i < j → testBit 0 j = false\nh : bit b 0 ≠ 0\n⊢ (bit true 0).testBit 0 = true",
"usedConstants": [
"Nat.bit",
"Eq.mpr",
"congrArg",
"id",
"instOfNatNat",
"Bool.true",
"Nat.testBit_bit_zero",
... | by rw [testBit_bit_zero] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Computability.AkraBazzi.GrowsPolynomially | {
"line": 658,
"column": 10
} | {
"line": 658,
"column": 51
} | [
{
"pp": "f g : ℝ → ℝ\nhg✝ : GrowsPolynomially g\nhf : f =Θ[atTop] g\nhf' : ∀ᶠ (x : ℝ) in atTop, 0 ≤ f x\nb : ℝ\nhb : b ∈ Set.Ioo 0 1\nhb_pos : 0 < b\nc₁ : ℝ\nhc₁_pos : 0 < c₁\nhf_lb : ∀ᶠ (x : ℝ) in atTop, c₁ * ‖g x‖ ≤ ‖f x‖\nc₂ : ℝ\nhc₂_pos : 0 < c₂\nhf_ub : ∀ᶠ (x : ℝ) in atTop, ‖f x‖ ≤ c₂ * ‖g x‖\nc₃ : ℝ\nhc₃_... | rw [← Real.norm_of_nonneg (hf_pos x hbx)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Data.List.ReduceOption | {
"line": 96,
"column": 4
} | {
"line": 96,
"column": 29
} | [
{
"pp": "case mpr\nα : Type u_1\nl : List (Option α)\nl' : List α\na : α\n⊢ (∃ l₁ l₂, l = l₁ ++ some a :: l₂ ∧ l₁.reduceOption = l' ∧ l₂.reduceOption = []) → l.reduceOption = l' ++ [a]",
"usedConstants": [
"Option.some",
"Exists",
"List.cons",
"instHAppendOfAppend",
"List",
... | intro ⟨_, _, h, hl₁, hl₂⟩ | Lean.Elab.Tactic.evalIntro | null |
Mathlib.Data.List.ReduceOption | {
"line": 96,
"column": 4
} | {
"line": 96,
"column": 29
} | [
{
"pp": "case mpr\nα : Type u_1\nl : List (Option α)\nl' : List α\na : α\n⊢ (∃ l₁ l₂, l = l₁ ++ some a :: l₂ ∧ l₁.reduceOption = l' ∧ l₂.reduceOption = []) → l.reduceOption = l' ++ [a]",
"usedConstants": [
"Option.some",
"Exists",
"List.cons",
"instHAppendOfAppend",
"List",
... | intro ⟨_, _, h, hl₁, hl₂⟩ | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Computability.AkraBazzi.GrowsPolynomially | {
"line": 672,
"column": 16
} | {
"line": 672,
"column": 57
} | [
{
"pp": "case hbc\nf g : ℝ → ℝ\nhg✝ : GrowsPolynomially g\nhf : f =Θ[atTop] g\nhf' : ∀ᶠ (x : ℝ) in atTop, 0 ≤ f x\nb : ℝ\nhb : b ∈ Set.Ioo 0 1\nhb_pos : 0 < b\nc₁ : ℝ\nhc₁_pos : 0 < c₁\nhf_lb : ∀ᶠ (x : ℝ) in atTop, c₁ * ‖g x‖ ≤ ‖f x‖\nc₂ : ℝ\nhc₂_pos : 0 < c₂\nhf_ub : ∀ᶠ (x : ℝ) in atTop, ‖f x‖ ≤ c₂ * ‖g x‖\nc₃... | rw [← Real.norm_of_nonneg (hf_pos x hbx)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Computability.NFA | {
"line": 363,
"column": 4
} | {
"line": 363,
"column": 24
} | [
{
"pp": "case h.mp\nα : Type u\nσ : Type v\nM : DFA α σ\nx : List α\n⊢ (∃ S ∈ M.toNFA.accept, S ∈ {M.evalFrom M.start x}) → x ∈ M.accepts",
"usedConstants": [
"Membership.mem",
"Exists",
"Set.instSingletonSet",
"DFA.evalFrom",
"DFA.start",
"DFA.toNFA",
"NFA.accept",... | rintro ⟨S, hS₁, hS₂⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.Computability.EpsilonNFA | {
"line": 202,
"column": 6
} | {
"line": 202,
"column": 44
} | [
{
"pp": "case h\nα : Type u\nσ : Type v\nM : εNFA α σ\ns₁ s₂ s✝ t✝ : σ\na✝¹ : t✝ ∈ M.step s✝ none\na✝ : M.εClosure {s₁} s✝\nn : ℕ\nh✝ : M.IsPath s₁ s✝ (List.replicate n none)\n⊢ M.IsPath s₁ t✝ (List.replicate (n + 1) none)",
"usedConstants": [
"Eq.mpr",
"List.replicate",
"congrArg",
... | rw [List.replicate_add, isPath_append] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Computability.RE | {
"line": 73,
"column": 2
} | {
"line": 75,
"column": 18
} | [
{
"pp": "α : Type u_1\nσ : Type u_4\ninst✝¹ : Primcodable α\ninst✝ : Primcodable σ\nf g : α →. σ\nhf : Partrec f\nhg : Partrec g\nk : ℕ →. ℕ\nhk : Nat.Partrec k\nH :\n ∀ (a : ℕ),\n (∀ x ∈ k a,\n (x ∈ (↑(decode₂ α a)).bind fun a ↦ Part.map encode (f a)) ∨\n x ∈ (↑(decode₂ α a)).bind fun a ↦ P... | refine
⟨k', ((nat_iff.2 hk).comp Computable.encode).bind (Computable.decode.ofOption.comp snd).to₂,
fun a => ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Data.Num.Lemmas | {
"line": 124,
"column": 2
} | {
"line": 125,
"column": 11
} | [
{
"pp": "m : PosNum\n⊢ ∀ (n : PosNum), (m.cmp n).swap = n.cmp m",
"usedConstants": [
"PosNum.cmp",
"PosNum.rec",
"Ordering",
"Ordering.swap",
"PosNum",
"Eq"
]
}
] | induction m with | one => ?_ | bit1 m IH => ?_ | bit0 m IH => ?_ <;>
intro n | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Computability.TuringMachine.Tape | {
"line": 73,
"column": 47
} | {
"line": 73,
"column": 65
} | [
{
"pp": "case h\nΓ : Type u_1\ninst✝ : Inhabited Γ\nl₁ l₂ : List Γ\ni j : ℕ\ne : l₁ ++ List.replicate i default = l₂ ++ List.replicate j default\nh : l₁.length ≤ l₂.length\n⊢ l₁ ++ List.replicate i default = l₁ ++ List.replicate (i - j + j) default",
"usedConstants": [
"Eq.mpr",
"Inhabited.defau... | Nat.sub_add_cancel | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Computability.TuringMachine.PostTuringMachine | {
"line": 235,
"column": 6
} | {
"line": 236,
"column": 9
} | [
{
"pp": "case some.write\nΓ : Type u_1\ninst✝³ : Inhabited Γ\nΓ' : Type u_2\ninst✝² : Inhabited Γ'\nΛ : Type u_3\ninst✝¹ : Inhabited Λ\nΛ' : Type u_4\ninst✝ : Inhabited Λ'\nM : Machine Γ Λ\nf₁ : PointedMap Γ Γ'\nf₂ : PointedMap Γ' Γ\ng₁ : Λ → Λ'\ng₂ : Λ' → Λ\nS : Set Λ\nf₂₁ : Function.RightInverse f₁.f f₂.f\ng₂... | simp only [Option.map_some, Tape.map_write]
rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Computability.TuringMachine.PostTuringMachine | {
"line": 235,
"column": 6
} | {
"line": 236,
"column": 9
} | [
{
"pp": "case some.write\nΓ : Type u_1\ninst✝³ : Inhabited Γ\nΓ' : Type u_2\ninst✝² : Inhabited Γ'\nΛ : Type u_3\ninst✝¹ : Inhabited Λ\nΛ' : Type u_4\ninst✝ : Inhabited Λ'\nM : Machine Γ Λ\nf₁ : PointedMap Γ Γ'\nf₂ : PointedMap Γ' Γ\ng₁ : Λ → Λ'\ng₂ : Λ' → Λ\nS : Set Λ\nf₂₁ : Function.RightInverse f₁.f f₂.f\ng₂... | simp only [Option.map_some, Tape.map_write]
rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Computability.RegularExpressions | {
"line": 293,
"column": 12
} | {
"line": 293,
"column": 28
} | [
{
"pp": "case mpr.cons.cons.cons.refine_1\nα : Type u_1\ninst✝ : DecidableEq α\nP : RegularExpression α\na : α\nx : List α\nIH :\n ∀ (t : List α),\n t.length < (a :: x).length → (P.star.rmatch t = true ↔ ∃ S, t = S.flatten ∧ ∀ t ∈ S, t ≠ [] ∧ P.rmatch t = true)\nU : List (List α)\nb : α\nt : List α\nhsum : ... | convert! helem.2 | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___macroRules_Mathlib_Tactic_convert!_1 | Mathlib.Tactic.convert! |
Mathlib.Computability.RegularExpressions | {
"line": 306,
"column": 33
} | {
"line": 306,
"column": 50
} | [
{
"pp": "case epsilon\nα : Type u_1\ninst✝ : DecidableEq α\nx : List α\n⊢ x = [] ↔ x ∈ matches' 1",
"usedConstants": [
"Eq.mpr",
"Language.instOne",
"congrArg",
"RegularExpression",
"Membership.mem",
"id",
"List",
"Iff",
"RegularExpression.matches'",
... | matches'_epsilon, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Computability.TuringMachine.Config | {
"line": 320,
"column": 26
} | {
"line": 320,
"column": 56
} | [
{
"pp": "case comp\nn : ℕ\nf : List.Vector ℕ n →. ℕ\nm✝ n✝ : ℕ\nf✝ : List.Vector ℕ n✝ →. ℕ\ng : Fin n✝ → List.Vector ℕ m✝ →. ℕ\na✝¹ : Nat.Partrec' f✝\na✝ : ∀ (i : Fin n✝), Nat.Partrec' (g i)\nIHf : ∃ c, ∀ (v : List.Vector ℕ n✝), c.eval ↑v = pure <$> f✝ v\nIHg : ∀ (i : Fin n✝), ∃ c, ∀ (v : List.Vector ℕ m✝), c.e... | exact exists_code.comp IHf IHg | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Computability.TuringMachine.Config | {
"line": 320,
"column": 26
} | {
"line": 320,
"column": 56
} | [
{
"pp": "case comp\nn : ℕ\nf : List.Vector ℕ n →. ℕ\nm✝ n✝ : ℕ\nf✝ : List.Vector ℕ n✝ →. ℕ\ng : Fin n✝ → List.Vector ℕ m✝ →. ℕ\na✝¹ : Nat.Partrec' f✝\na✝ : ∀ (i : Fin n✝), Nat.Partrec' (g i)\nIHf : ∃ c, ∀ (v : List.Vector ℕ n✝), c.eval ↑v = pure <$> f✝ v\nIHg : ∀ (i : Fin n✝), ∃ c, ∀ (v : List.Vector ℕ m✝), c.e... | exact exists_code.comp IHf IHg | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
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