module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Data.Set.Basic | {
"line": 889,
"column": 37
} | {
"line": 889,
"column": 57
} | {
"line": 889,
"column": 57
} | [
{
"pp": "α : Type u\ns : Set α\np : α → Prop\n⊢ (∀ (x : α), x ∈ s ∧ p x ↔ x ∈ s) ↔ ∀ (x : α), x ∈ s → p x",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"congrArg",
"Membership.mem",
"iff_self",
"_private.Mathlib.Data.Set.Basic.0.Set.sep_eq_self_iff_mem_true._simp_... | [] | and_iff_left_iff_imp | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Data.Set.Insert | {
"line": 297,
"column": 4
} | {
"line": 297,
"column": 24
} | {
"line": 297,
"column": 24
} | [
{
"pp": "α : Type u_1\ns : Set α\nx : α\n⊢ s = ∅ ∧ s ≠ {x} ↔ s = ∅",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Set.instSingletonSet",
"id",
"Ne",
"And",
"Iff",
"propext",
"Set.instEmptyCollection",
"Empt... | [
"α : Type u_1\ns : Set α\nx : α\n⊢ s = ∅ → s ≠ {x}"
] | and_iff_left_iff_imp | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Set.Insert | {
"line": 364,
"column": 49
} | {
"line": 364,
"column": 54
} | {
"line": 366,
"column": 0
} | [
{
"pp": "α : Type u_1\nx y z w : α\n⊢ ((x = z ∨ x = w) ∧ (y = z ∨ y = w)) ∧ (z = x ∨ z = y) ∧ (w = x ∨ w = y) ↔ x = z ∧ y = w ∨ x = w ∧ y = z",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"congrArg",
"and_self",
"true_or",
"Or.casesOn",
"And.casesOn",
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Order.Lattice | {
"line": 523,
"column": 21
} | {
"line": 523,
"column": 43
} | {
"line": 524,
"column": 4
} | [
{
"pp": "α : Type u\ninst✝ : DistribLattice α\nx y z : α\nh₁ : x ⊓ z ≤ y ⊓ z\nh₂ : x ⊔ z ≤ y ⊔ z\n⊢ (y ⊔ x) ⊓ (y ⊔ z) = y ⊔ x ⊓ z",
"ppTerm": "?m.59",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Lattice.toSemilatticeSup",
"congrArg",
"SemilatticeSup.toMax",
"Distrib... | [] | by rw [← sup_inf_left] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Set.Subsingleton | {
"line": 327,
"column": 29
} | {
"line": 327,
"column": 34
} | {
"line": 328,
"column": 2
} | [
{
"pp": "α : Type u\nh : univ = {∅}\nthis : univ ∈ univ\n⊢ IsEmpty α",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"congrArg",
"Set.univ",
"Membership.mem",
"Eq.mp",
"Set.instSingletonSet",
"id",
"Set.univ_eq_empty_iff._simp_1",
"IsEmpty",
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Data.Set.Subsingleton | {
"line": 327,
"column": 29
} | {
"line": 327,
"column": 34
} | {
"line": 328,
"column": 2
} | [
{
"pp": "α : Type u\nh : univ = {∅}\nthis : univ ∈ univ\n⊢ IsEmpty α",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"congrArg",
"Set.univ",
"Membership.mem",
"Eq.mp",
"Set.instSingletonSet",
"id",
"Set.univ_eq_empty_iff._simp_1",
"IsEmpty",
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Set.Subsingleton | {
"line": 327,
"column": 29
} | {
"line": 327,
"column": 34
} | {
"line": 328,
"column": 2
} | [
{
"pp": "α : Type u\nh : univ = {∅}\nthis : univ ∈ univ\n⊢ IsEmpty α",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"congrArg",
"Set.univ",
"Membership.mem",
"Eq.mp",
"Set.instSingletonSet",
"id",
"Set.univ_eq_empty_iff._simp_1",
"IsEmpty",
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.BooleanAlgebra.Set | {
"line": 591,
"column": 34
} | {
"line": 591,
"column": 48
} | {
"line": 591,
"column": 49
} | [
{
"pp": "α : Type u_1\ns t : Set α\na : α\n⊢ insert a (s \\ (t ∪ {a})) = insert a (s \\ t)",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Set.instUnion",
"Set.instSingletonSet",
"id",
"Insert.insert",
"SDiff.sdiff",
"S... | [
"α : Type u_1\ns t : Set α\na : α\n⊢ insert a ((s \\ t) \\ {a}) = insert a (s \\ t)"
] | ← sdiff_sdiff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.Heyting.Basic | {
"line": 972,
"column": 24
} | {
"line": 972,
"column": 37
} | {
"line": 972,
"column": 37
} | [
{
"pp": "case neg\nι : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝¹ : LinearOrder α\ninst✝ : BoundedOrder α\na b c : α\nh : ¬b ≤ c\n⊢ a ≤ c ↔ a ≤ c ∨ b ≤ c",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Lattice.toSemilatticeSup",
"congrArg",
"PartialOrde... | [
"case neg\nι : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝¹ : LinearOrder α\ninst✝ : BoundedOrder α\na b c : α\nh : ¬b ≤ c\n⊢ a ≤ c ↔ a ≤ c"
] | or_iff_left h | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.BooleanAlgebra.Set | {
"line": 700,
"column": 26
} | {
"line": 700,
"column": 70
} | {
"line": 702,
"column": 0
} | [
{
"pp": "case pos\nα : Type u_1\nt s₁ s₂ : Set α\nh : s₂ ⊆ s₁\nx : α\nhx : x ∈ t\n⊢ x ∈ t.ite s₁ s₂ ↔ x ∈ s₁ ∩ t ∪ s₂",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"False",
"Set.ite",
"and_true",
"congrArg",
"or_iff_left_of_imp",
"Membership.mem",
... | [] | simp [*, Set.ite, or_iff_left_of_imp (@h x)] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Order.BooleanAlgebra.Set | {
"line": 700,
"column": 26
} | {
"line": 700,
"column": 70
} | {
"line": 702,
"column": 0
} | [
{
"pp": "case neg\nα : Type u_1\nt s₁ s₂ : Set α\nh : s₂ ⊆ s₁\nx : α\nhx : ¬x ∈ t\n⊢ x ∈ t.ite s₁ s₂ ↔ x ∈ s₁ ∩ t ∪ s₂",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"False",
"Set.ite",
"eq_false",
"and_true",
"congrArg",
"Membership.mem",
"Set.in... | [] | simp [*, Set.ite, or_iff_left_of_imp (@h x)] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Order.BooleanAlgebra.Basic | {
"line": 584,
"column": 41
} | {
"line": 584,
"column": 46
} | {
"line": 586,
"column": 0
} | [
{
"pp": "α : Type u\nx y : α\ninst✝ : BooleanAlgebra α\n⊢ x ⇨ y = ⊤ ∧ x = ⊤ ↔ x = ⊤ ∧ y = ⊤",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"himp_eq_top_iff._simp_1",
"instReflLe",
"congrArg",
"and_self",
"PartialOrder.toPreorder",
"Preor... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Logic.Equiv.Basic | {
"line": 664,
"column": 2
} | {
"line": 664,
"column": 38
} | {
"line": 666,
"column": 0
} | [
{
"pp": "α : Sort u_1\ninst✝ : DecidableEq α\na b x : α\nh : x ≠ a ∧ x ≠ b\n⊢ (swap a b) x = x",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Ne",
"Equiv.swap_apply_of_ne_of_ne",
"And.right",
"And.left"
],
"usedFVars": [
"α",
"inst✝",
"a"... | [] | exact swap_apply_of_ne_of_ne h.1 h.2 | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Logic.Equiv.Basic | {
"line": 747,
"column": 2
} | {
"line": 747,
"column": 7
} | {
"line": 749,
"column": 0
} | [
{
"pp": "α : Type u_9\nβ : Type u_10\ninst✝¹ : DecidableEq α\ninst✝ : DecidableEq β\ni j : α\n⊢ (swap i j).sumCongr (Equiv.refl β) = swap (Sum.inl i) (Sum.inl j)",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"False",
"Decidable.casesOn",
"Equiv.instEquiv... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Logic.Equiv.Basic | {
"line": 747,
"column": 2
} | {
"line": 747,
"column": 7
} | {
"line": 749,
"column": 0
} | [
{
"pp": "α : Type u_9\nβ : Type u_10\ninst✝¹ : DecidableEq α\ninst✝ : DecidableEq β\ni j : α\n⊢ (swap i j).sumCongr (Equiv.refl β) = swap (Sum.inl i) (Sum.inl j)",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"False",
"Decidable.casesOn",
"Equiv.instEquiv... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Logic.Equiv.Basic | {
"line": 747,
"column": 2
} | {
"line": 747,
"column": 7
} | {
"line": 749,
"column": 0
} | [
{
"pp": "α : Type u_9\nβ : Type u_10\ninst✝¹ : DecidableEq α\ninst✝ : DecidableEq β\ni j : α\n⊢ (swap i j).sumCongr (Equiv.refl β) = swap (Sum.inl i) (Sum.inl j)",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"False",
"Decidable.casesOn",
"Equiv.instEquiv... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Logic.Equiv.Basic | {
"line": 752,
"column": 2
} | {
"line": 752,
"column": 7
} | {
"line": 754,
"column": 0
} | [
{
"pp": "α : Type u_9\nβ : Type u_10\ninst✝¹ : DecidableEq α\ninst✝ : DecidableEq β\ni j : β\n⊢ sumCongr (Equiv.refl α) (swap i j) = swap (Sum.inr i) (Sum.inr j)",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"False",
"Decidable.casesOn",
"Equiv.instEquiv... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Logic.Equiv.Basic | {
"line": 752,
"column": 2
} | {
"line": 752,
"column": 7
} | {
"line": 754,
"column": 0
} | [
{
"pp": "α : Type u_9\nβ : Type u_10\ninst✝¹ : DecidableEq α\ninst✝ : DecidableEq β\ni j : β\n⊢ sumCongr (Equiv.refl α) (swap i j) = swap (Sum.inr i) (Sum.inr j)",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"False",
"Decidable.casesOn",
"Equiv.instEquiv... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Logic.Equiv.Basic | {
"line": 752,
"column": 2
} | {
"line": 752,
"column": 7
} | {
"line": 754,
"column": 0
} | [
{
"pp": "α : Type u_9\nβ : Type u_10\ninst✝¹ : DecidableEq α\ninst✝ : DecidableEq β\ni j : β\n⊢ sumCongr (Equiv.refl α) (swap i j) = swap (Sum.inr i) (Sum.inr j)",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"False",
"Decidable.casesOn",
"Equiv.instEquiv... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.SymmDiff | {
"line": 405,
"column": 30
} | {
"line": 405,
"column": 53
} | {
"line": 405,
"column": 53
} | [
{
"pp": "α : Type u_2\ninst✝ : GeneralizedBooleanAlgebra α\na b c : α\n⊢ a \\ (b ⊔ c) ⊔ b \\ (a ⊔ c) ⊔ c \\ (a ⊔ b) ⊔ a ⊓ b ⊓ c = a ∆ (b ∆ c)",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Lattice.toSemilatticeSup",
"congrArg",
"symmDiff_symmDiff_right",... | [
"α : Type u_2\ninst✝ : GeneralizedBooleanAlgebra α\na b c : α\n⊢ a \\ (b ⊔ c) ⊔ b \\ (a ⊔ c) ⊔ c \\ (a ⊔ b) ⊔ a ⊓ b ⊓ c = a \\ (b ⊔ c) ⊔ b \\ (a ⊔ c) ⊔ c \\ (a ⊔ b) ⊔ a ⊓ b ⊓ c"
] | symmDiff_symmDiff_right | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.SymmDiff | {
"line": 533,
"column": 73
} | {
"line": 534,
"column": 42
} | {
"line": 536,
"column": 0
} | [
{
"pp": "α : Type u_2\ninst✝ : BooleanAlgebra α\na b c d : α\n⊢ a ⇔ b ⇔ (c ⇔ d) = a ⇔ c ⇔ (b ⇔ d)",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"bihimp_left_comm",
"id",
"BiheytingAlgebra.toCoheytingAlgebra",
"SemilatticeInf.toMin... | [] | by
simp_rw [bihimp_assoc, bihimp_left_comm] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Set.Image | {
"line": 174,
"column": 17
} | {
"line": 174,
"column": 23
} | {
"line": 175,
"column": 4
} | [
{
"pp": "case mp\nα : Type u_1\ns u v : Set α\nhsuv : s ⊆ u ∪ v\nH : s ∩ (u ∩ v) = ∅\nx : α\nx_in_s : x ∈ s\nx_in_u : ⟨x, x_in_s⟩ ∈ Subtype.val ⁻¹' u\n⊢ ⟨x, x_in_s⟩ ∈ (Subtype.val ⁻¹' v)ᶜ",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"Membership.mem",
"Subtype",
"Subtype.... | [
"case mp\nα : Type u_1\ns u v : Set α\nhsuv : s ⊆ u ∪ v\nH : s ∩ (u ∩ v) = ∅\nx : α\nx_in_s : x ∈ s\nx_in_u : ⟨x, x_in_s⟩ ∈ Subtype.val ⁻¹' u\nx_in_v : ⟨x, x_in_s⟩ ∈ Subtype.val ⁻¹' v\n⊢ False"
] | x_in_v | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Data.Set.Image | {
"line": 212,
"column": 2
} | {
"line": 212,
"column": 7
} | {
"line": 214,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nf g : α → β\ns : Set α\nh : ∀ (a : α), a ∈ s → f a = g a\n⊢ f '' s = g '' s",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Set.ext",
"Eq.mpr",
"congrArg",
"Set.mem_image._simp_1",
"Membership.mem",
"Exists",
"... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Data.Set.Image | {
"line": 212,
"column": 2
} | {
"line": 212,
"column": 7
} | {
"line": 214,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nf g : α → β\ns : Set α\nh : ∀ (a : α), a ∈ s → f a = g a\n⊢ f '' s = g '' s",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Set.ext",
"Eq.mpr",
"congrArg",
"Set.mem_image._simp_1",
"Membership.mem",
"Exists",
"... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Set.Image | {
"line": 212,
"column": 2
} | {
"line": 212,
"column": 7
} | {
"line": 214,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nf g : α → β\ns : Set α\nh : ∀ (a : α), a ∈ s → f a = g a\n⊢ f '' s = g '' s",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Set.ext",
"Eq.mpr",
"congrArg",
"Set.mem_image._simp_1",
"Membership.mem",
"Exists",
"... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Set.Image | {
"line": 224,
"column": 88
} | {
"line": 224,
"column": 93
} | {
"line": 226,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nf : β → γ\ng : α → β\na : Set α\n⊢ f ∘ g '' a = f '' g '' a",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Set.ext",
"Eq.mpr",
"congrArg",
"exists_exists_and_eq_and._simp_1",
"Set.mem_image._simp_1",
"... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Data.Set.Image | {
"line": 224,
"column": 88
} | {
"line": 224,
"column": 93
} | {
"line": 226,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nf : β → γ\ng : α → β\na : Set α\n⊢ f ∘ g '' a = f '' g '' a",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Set.ext",
"Eq.mpr",
"congrArg",
"exists_exists_and_eq_and._simp_1",
"Set.mem_image._simp_1",
"... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Set.Image | {
"line": 224,
"column": 88
} | {
"line": 224,
"column": 93
} | {
"line": 226,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nf : β → γ\ng : α → β\na : Set α\n⊢ f ∘ g '' a = f '' g '' a",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Set.ext",
"Eq.mpr",
"congrArg",
"exists_exists_and_eq_and._simp_1",
"Set.mem_image._simp_1",
"... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Set.Image | {
"line": 338,
"column": 25
} | {
"line": 338,
"column": 30
} | {
"line": 338,
"column": 30
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nf : α → β\ns s' : Set β\n⊢ (∃ x, x ∈ range f ∩ s' ∧ ¬x ∈ range f ∩ s) ↔ ∃ x, x ∈ f ⁻¹' s' ∧ ¬x ∈ f ⁻¹' s",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"False",
"eq_false",
"congrArg",
"Membership.mem",
"Ex... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Data.Set.Image | {
"line": 338,
"column": 25
} | {
"line": 338,
"column": 30
} | {
"line": 338,
"column": 30
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nf : α → β\ns s' : Set β\n⊢ (∃ x, x ∈ range f ∩ s' ∧ ¬x ∈ range f ∩ s) ↔ ∃ x, x ∈ f ⁻¹' s' ∧ ¬x ∈ f ⁻¹' s",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"False",
"eq_false",
"congrArg",
"Membership.mem",
"Ex... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Set.Image | {
"line": 338,
"column": 25
} | {
"line": 338,
"column": 30
} | {
"line": 338,
"column": 30
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nf : α → β\ns s' : Set β\n⊢ (∃ x, x ∈ range f ∩ s' ∧ ¬x ∈ range f ∩ s) ↔ ∃ x, x ∈ f ⁻¹' s' ∧ ¬x ∈ f ⁻¹' s",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"False",
"eq_false",
"congrArg",
"Membership.mem",
"Ex... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Set.Image | {
"line": 344,
"column": 4
} | {
"line": 344,
"column": 9
} | {
"line": 346,
"column": 0
} | [
{
"pp": "case mp\nα : Type u_1\nβ : Type u_2\nf : α → β\ns s' : Set β\nx : α\nr : range f ∩ s ⊆ s'\nhx : f x ∈ s\n⊢ f x ∈ s'",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"congrArg",
"and_self",
"Membership.mem",
"Exists",
"exists_apply_eq_apply._simp_1",
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Data.Set.Image | {
"line": 344,
"column": 4
} | {
"line": 344,
"column": 9
} | {
"line": 346,
"column": 0
} | [
{
"pp": "case mpr\nα : Type u_1\nβ : Type u_2\nf : α → β\ns s' : Set β\nr : f ⁻¹' s ⊆ f ⁻¹' s'\nx : β\nhx : (∃ y, f y = x) ∧ x ∈ s\n⊢ x ∈ s'",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"Membership.mem",
"Exists",
"Set.mem_preimage._simp_1",
"And.casesOn",
"... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Data.Set.Prod | {
"line": 125,
"column": 2
} | {
"line": 126,
"column": 54
} | {
"line": 128,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ns₁ s₂ : Set α\nt : Set β\n⊢ (s₁ ∩ s₂) ×ˢ t = s₁ ×ˢ t ∩ s₂ ×ˢ t",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Set.instSProd",
"Set.ext",
"_private.Mathlib.Data.Set.Prod.0.Set.inter_prod._simp_1_3",
"SProd.sprod",
"congrAr... | [] | ext ⟨x, y⟩
simp only [← and_and_right, mem_inter_iff, mem_prod] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Set.Prod | {
"line": 125,
"column": 2
} | {
"line": 126,
"column": 54
} | {
"line": 128,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ns₁ s₂ : Set α\nt : Set β\n⊢ (s₁ ∩ s₂) ×ˢ t = s₁ ×ˢ t ∩ s₂ ×ˢ t",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Set.instSProd",
"Set.ext",
"_private.Mathlib.Data.Set.Prod.0.Set.inter_prod._simp_1_3",
"SProd.sprod",
"congrAr... | [] | ext ⟨x, y⟩
simp only [← and_and_right, mem_inter_iff, mem_prod] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Set.Prod | {
"line": 157,
"column": 2
} | {
"line": 157,
"column": 7
} | {
"line": 159,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nf : α → β\ng : γ → δ\ns : Set α\nt : Set γ\nx✝ : β × δ\n⊢ x✝ ∈ Prod.map f g '' s ×ˢ t ↔ x✝ ∈ (f '' s) ×ˢ (g '' t)",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Set.instSProd",
"Eq.mpr",
"SProd.sprod",
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Data.Set.Image | {
"line": 465,
"column": 50
} | {
"line": 466,
"column": 65
} | {
"line": 468,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nf : α → β\ns : Set α\nt : Set β\n⊢ Disjoint t (f '' s) ↔ Disjoint (f ⁻¹' t) s",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Set.disjoint_image_left",
"Iff.rfl",
"Disjoint",
"SemilatticeInf.toP... | [] | by
rw [disjoint_comm, disjoint_comm (b := s), disjoint_image_left] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Set.Prod | {
"line": 294,
"column": 67
} | {
"line": 294,
"column": 72
} | {
"line": 296,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ns : Set (α × β)\nx✝¹ : α × β\np₁ : α\np₂ : β\nx✝ : (p₁, p₂) ∈ s\n⊢ (p₁, p₂) ∈ (Prod.fst '' s) ×ˢ (Prod.snd '' s)",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Set.instSProd",
"Eq.mpr",
"SProd.sprod",
"congrArg",
"Set.mem... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Data.Set.Prod | {
"line": 294,
"column": 67
} | {
"line": 294,
"column": 72
} | {
"line": 296,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ns : Set (α × β)\nx✝¹ : α × β\np₁ : α\np₂ : β\nx✝ : (p₁, p₂) ∈ s\n⊢ (p₁, p₂) ∈ (Prod.fst '' s) ×ˢ (Prod.snd '' s)",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Set.instSProd",
"Eq.mpr",
"SProd.sprod",
"congrArg",
"Set.mem... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Set.Prod | {
"line": 294,
"column": 67
} | {
"line": 294,
"column": 72
} | {
"line": 296,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ns : Set (α × β)\nx✝¹ : α × β\np₁ : α\np₂ : β\nx✝ : (p₁, p₂) ∈ s\n⊢ (p₁, p₂) ∈ (Prod.fst '' s) ×ˢ (Prod.snd '' s)",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Set.instSProd",
"Eq.mpr",
"SProd.sprod",
"congrArg",
"Set.mem... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Set.Image | {
"line": 634,
"column": 80
} | {
"line": 634,
"column": 85
} | {
"line": 636,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nι : Sort u_4\ng : α → β\nf : ι → α\n⊢ range (g ∘ f) = g '' range f",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Set.ext",
"congrArg",
"Set.mem_image._simp_1",
"exists_exists_eq_and._simp_1",
"Function.comp",
"Memb... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Data.Set.Image | {
"line": 634,
"column": 80
} | {
"line": 634,
"column": 85
} | {
"line": 636,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nι : Sort u_4\ng : α → β\nf : ι → α\n⊢ range (g ∘ f) = g '' range f",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Set.ext",
"congrArg",
"Set.mem_image._simp_1",
"exists_exists_eq_and._simp_1",
"Function.comp",
"Memb... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Set.Image | {
"line": 634,
"column": 80
} | {
"line": 634,
"column": 85
} | {
"line": 636,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nι : Sort u_4\ng : α → β\nf : ι → α\n⊢ range (g ∘ f) = g '' range f",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Set.ext",
"congrArg",
"Set.mem_image._simp_1",
"exists_exists_eq_and._simp_1",
"Function.comp",
"Memb... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Group.Action.Pi | {
"line": 137,
"column": 2
} | {
"line": 139,
"column": 19
} | {
"line": 141,
"column": 0
} | [
{
"pp": "ι : Type u_1\nM : Type u_7\nα : Type u_8\nβ : Type u_9\ninst✝ : SMul M β\nr : M\nf : ι → α\ng : ι → β\ne : α → β\nx : α\n⊢ extend f (r • g) (r • e) x = (r • extend f g e) x",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"dite_congr",
"instHSMul",
... | [] | classical
simp only [extend_def, Pi.smul_apply]
split_ifs <;> rfl | Lean.Elab.Tactic.evalClassical | Lean.Parser.Tactic.classical |
Mathlib.Data.Set.Function | {
"line": 255,
"column": 90
} | {
"line": 255,
"column": 95
} | {
"line": 257,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nf : α → β\na b : α\n⊢ (∀ ⦃x₁ : α⦄, x₁ ∈ {a, b} → ∀ ⦃x₂ : α⦄, x₂ ∈ {a, b} → f x₁ = f x₂ → x₁ = x₂) ↔ f a = f b → a = b",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"and_true",
"congrArg",
"forall_eq_or_imp._simp_1",
"Membership... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Data.Set.Function | {
"line": 331,
"column": 29
} | {
"line": 331,
"column": 34
} | {
"line": 331,
"column": 34
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nf : α → β\ng : β → γ\ns : Set β\nh✝ : InjOn g s\nhs : range f ⊆ s\nh : Injective f\nx✝ : α\n⊢ ∀ ⦃a₂ : α⦄, (g ∘ f) x✝ = (g ∘ f) a₂ → x✝ = a₂",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"congrArg",
"Function.comp",
"Mem... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Data.Set.Function | {
"line": 331,
"column": 29
} | {
"line": 331,
"column": 34
} | {
"line": 331,
"column": 34
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nf : α → β\ng : β → γ\ns : Set β\nh✝ : InjOn g s\nhs : range f ⊆ s\nh : Injective f\nx✝ : α\n⊢ ∀ ⦃a₂ : α⦄, (g ∘ f) x✝ = (g ∘ f) a₂ → x✝ = a₂",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"congrArg",
"Function.comp",
"Mem... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Set.Function | {
"line": 331,
"column": 29
} | {
"line": 331,
"column": 34
} | {
"line": 331,
"column": 34
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nf : α → β\ng : β → γ\ns : Set β\nh✝ : InjOn g s\nhs : range f ⊆ s\nh : Injective f\nx✝ : α\n⊢ ∀ ⦃a₂ : α⦄, (g ∘ f) x✝ = (g ∘ f) a₂ → x✝ = a₂",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"congrArg",
"Function.comp",
"Mem... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Set.Image | {
"line": 938,
"column": 2
} | {
"line": 938,
"column": 7
} | {
"line": 940,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nx : α\ns : Set α\nf : ↑(insert x s) → β\n⊢ range f = insert (f ⟨x, ⋯⟩) (range fun y ↦ f ⟨↑y, ⋯⟩)",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Set.ext",
"Eq.mpr",
"Iff.of_eq",
"congrArg",
"true_or",
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Data.Set.Image | {
"line": 938,
"column": 2
} | {
"line": 938,
"column": 7
} | {
"line": 940,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nx : α\ns : Set α\nf : ↑(insert x s) → β\n⊢ range f = insert (f ⟨x, ⋯⟩) (range fun y ↦ f ⟨↑y, ⋯⟩)",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Set.ext",
"Eq.mpr",
"Iff.of_eq",
"congrArg",
"true_or",
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Set.Image | {
"line": 938,
"column": 2
} | {
"line": 938,
"column": 7
} | {
"line": 940,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nx : α\ns : Set α\nf : ↑(insert x s) → β\n⊢ range f = insert (f ⟨x, ⋯⟩) (range fun y ↦ f ⟨↑y, ⋯⟩)",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Set.ext",
"Eq.mpr",
"Iff.of_eq",
"congrArg",
"true_or",
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Set.Prod | {
"line": 747,
"column": 30
} | {
"line": 747,
"column": 86
} | {
"line": 747,
"column": 86
} | [
{
"pp": "ι : Type u_1\nα : ι → Type u_2\nβ : ι → Type u_3\ns : Set ι\ni : ι\ninst✝ : DecidableEq ι\nhi : i ∈ s\nf : (j : ι) → α j\na : α i\nt : (j : ι) → α j → Set (β j)\n⊢ (s.pi fun j ↦ t j (update f i a j)) = ({i} ∪ s).pi fun j ↦ t j (update f i a j)",
"ppTerm": "?m.84",
"assigned": true,
"usedCon... | [
"ι : Type u_1\nα : ι → Type u_2\nβ : ι → Type u_3\ns : Set ι\ni : ι\ninst✝ : DecidableEq ι\nhi : i ∈ s\nf : (j : ι) → α j\na : α i\nt : (j : ι) → α j → Set (β j)\n⊢ (s.pi fun j ↦ t j (update f i a j)) = s.pi fun j ↦ t j (update f i a j)"
] | union_eq_self_of_subset_left (singleton_subset_iff.2 hi) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Group.Equiv.Defs | {
"line": 169,
"column": 4
} | {
"line": 169,
"column": 35
} | {
"line": 171,
"column": 0
} | [
{
"pp": "case mk.mk\nF : Type u_1\nα : Type u_2\nβ : Type u_3\nM : Type u_4\nN : Type u_5\nP : Type u_6\nG : Type u_7\nH : Type u_8\ninst✝³ : EquivLike F α β\ninst✝² : Mul M\ninst✝¹ : Mul N\ninst✝ : Mul P\ntoEquiv✝¹ : M ≃ N\nmap_mul'✝¹ : ∀ (x y : M), toEquiv✝¹.toFun (x * y) = toEquiv✝¹.toFun x * toEquiv✝¹.toFun... | [] | apply Equiv.coe_fn_injective h₁ | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Data.Set.Prod | {
"line": 972,
"column": 2
} | {
"line": 972,
"column": 7
} | {
"line": 974,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\ns : Set α\nt : Set β\nf : α → γ\ng : β → δ\n⊢ graphOn f s ×ˢ graphOn g t = ⇑(Equiv.prodProdProdComm α γ β δ) ⁻¹' graphOn (Prod.map f g) (s ×ˢ t)",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Set.instSProd",
"Se... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Data.Set.Prod | {
"line": 972,
"column": 2
} | {
"line": 972,
"column": 7
} | {
"line": 974,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\ns : Set α\nt : Set β\nf : α → γ\ng : β → δ\n⊢ graphOn f s ×ˢ graphOn g t = ⇑(Equiv.prodProdProdComm α γ β δ) ⁻¹' graphOn (Prod.map f g) (s ×ˢ t)",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Set.instSProd",
"Se... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Set.Prod | {
"line": 972,
"column": 2
} | {
"line": 972,
"column": 7
} | {
"line": 974,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\ns : Set α\nt : Set β\nf : α → γ\ng : β → δ\n⊢ graphOn f s ×ˢ graphOn g t = ⇑(Equiv.prodProdProdComm α γ β δ) ⁻¹' graphOn (Prod.map f g) (s ×ˢ t)",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Set.instSProd",
"Se... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Set.Prod | {
"line": 976,
"column": 2
} | {
"line": 976,
"column": 7
} | {
"line": 978,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\ns : Set α\nt : Set β\nf : α → γ\ng : β → δ\n⊢ graphOn (Prod.map f g) (s ×ˢ t) = ⇑(Equiv.prodProdProdComm α β γ δ) ⁻¹' graphOn f s ×ˢ graphOn g t",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Set.instSProd",
"Se... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Data.Set.Prod | {
"line": 976,
"column": 2
} | {
"line": 976,
"column": 7
} | {
"line": 978,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\ns : Set α\nt : Set β\nf : α → γ\ng : β → δ\n⊢ graphOn (Prod.map f g) (s ×ˢ t) = ⇑(Equiv.prodProdProdComm α β γ δ) ⁻¹' graphOn f s ×ˢ graphOn g t",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Set.instSProd",
"Se... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Set.Prod | {
"line": 976,
"column": 2
} | {
"line": 976,
"column": 7
} | {
"line": 978,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\ns : Set α\nt : Set β\nf : α → γ\ng : β → δ\n⊢ graphOn (Prod.map f g) (s ×ˢ t) = ⇑(Equiv.prodProdProdComm α β γ δ) ⁻¹' graphOn f s ×ˢ graphOn g t",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Set.instSProd",
"Se... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Group.Equiv.Defs | {
"line": 480,
"column": 62
} | {
"line": 480,
"column": 94
} | {
"line": 482,
"column": 0
} | [
{
"pp": "M : Type u_4\nN : Type u_5\nP : Type u_6\ninst✝² : MulOneClass M\ninst✝¹ : MulOneClass N\ninst✝ : MulOneClass P\ne : M ≃* N\nf : P →* M\n⊢ (↑e.symm).comp ((↑e).comp f) = f",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"MulEquiv.coe_monoidHom_symm_comp_coe_monoidHom",
... | [] | by simp [← MonoidHom.comp_assoc] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Set.Function | {
"line": 832,
"column": 2
} | {
"line": 833,
"column": 61
} | {
"line": 834,
"column": 2
} | [
{
"pp": "case h₁\nα : Type u_1\nβ : Type u_2\ns s₁ : Set α\nf : α → β\nf' : β → α\nhf : LeftInvOn f' f s\n⊢ f '' (s₁ ∩ s) ⊆ f' ⁻¹' s₁ ∩ f '' s",
"ppTerm": "?h₁",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Membership.mem",
"id",
"Set.instInter",
"I... | [
"case h₂\nα : Type u_1\nβ : Type u_2\ns s₁ : Set α\nf : α → β\nf' : β → α\nhf : LeftInvOn f' f s\n⊢ f' ⁻¹' s₁ ∩ f '' s ⊆ f '' (s₁ ∩ s)"
] | · rintro _ ⟨x, ⟨h₁, h⟩, rfl⟩
exact ⟨by rwa [mem_preimage, hf h], mem_image_of_mem _ h⟩ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Data.Set.Function | {
"line": 1130,
"column": 67
} | {
"line": 1130,
"column": 72
} | {
"line": 1130,
"column": 72
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ns : Set α\nt : Set β\nf : α → β\nx : ⦃a : β⦄ → a ∈ t → α\nhmem : ∀ ⦃a : β⦄ (a_1 : a ∈ t), x a_1 ∈ s\nheq : ∀ ⦃a : β⦄ (a_1 : a ∈ t), f (x a_1) = a\n⊢ (f '' range fun a ↦ x ⋯) = t",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Set... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Data.Set.Function | {
"line": 1130,
"column": 67
} | {
"line": 1130,
"column": 72
} | {
"line": 1130,
"column": 72
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ns : Set α\nt : Set β\nf : α → β\nx : ⦃a : β⦄ → a ∈ t → α\nhmem : ∀ ⦃a : β⦄ (a_1 : a ∈ t), x a_1 ∈ s\nheq : ∀ ⦃a : β⦄ (a_1 : a ∈ t), f (x a_1) = a\n⊢ (f '' range fun a ↦ x ⋯) = t",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Set... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Set.Function | {
"line": 1130,
"column": 67
} | {
"line": 1130,
"column": 72
} | {
"line": 1130,
"column": 72
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ns : Set α\nt : Set β\nf : α → β\nx : ⦃a : β⦄ → a ∈ t → α\nhmem : ∀ ⦃a : β⦄ (a_1 : a ∈ t), x a_1 ∈ s\nheq : ∀ ⦃a : β⦄ (a_1 : a ∈ t), f (x a_1) = a\n⊢ (f '' range fun a ↦ x ⋯) = t",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Set... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Set.Function | {
"line": 1280,
"column": 18
} | {
"line": 1280,
"column": 23
} | {
"line": 1280,
"column": 23
} | [
{
"pp": "β₁ : Type u_8\nβ₂ : Type u_9\nt₁ : Set β₁\nt₂ : Set β₂\nthis : SurjOn Prod.fst (t₁ ×ˢ t₂) t₁ ∧ t₁.Nonempty ∧ t₂ = ∅\n⊢ False",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Set.instSProd",
"False",
"SProd.sprod",
"congrArg",
"False.elim",
"Set.... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Data.Set.Function | {
"line": 1283,
"column": 18
} | {
"line": 1283,
"column": 23
} | {
"line": 1283,
"column": 23
} | [
{
"pp": "β₁ : Type u_8\nβ₂ : Type u_9\nt₁ : Set β₁\nt₂ : Set β₂\nthis : SurjOn Prod.snd (t₁ ×ˢ t₂) t₂ ∧ t₁ = ∅ ∧ t₂.Nonempty\n⊢ False",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Set.instSProd",
"False",
"SProd.sprod",
"congrArg",
"False.elim",
"Set.... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Group.Torsion | {
"line": 80,
"column": 6
} | {
"line": 80,
"column": 34
} | {
"line": 80,
"column": 35
} | [
{
"pp": "G : Type u_2\ninst✝¹ : Group G\ninst✝ : IsMulTorsionFree G\nn : ℤ\na : G\nhn : n ≠ 0\n⊢ a ^ n = 1 ↔ a = 1",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"InvOneClass.toOne",
"DivInvOneMonoid.toInvOneClass",
"congrArg",
"DivInvMonoid.toZPow"... | [
"G : Type u_2\ninst✝¹ : Group G\ninst✝ : IsMulTorsionFree G\nn : ℤ\na : G\nhn : n ≠ 0\n⊢ a ^ n = 1 ↔ a ^ n = 1 ^ n"
] | ← zpow_left_inj (a := a) hn, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Group.Basic | {
"line": 270,
"column": 87
} | {
"line": 270,
"column": 92
} | {
"line": 271,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : CancelCommMonoid α\na b c d : α\nh : a * b = c * d\n⊢ a = c ↔ b = d",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"CancelCommMonoid.toCommMonoid",
"CancelMonoid.toRightCancelMonoid",
"HMul.hMul",
"Monoid.toMulOneClass",
"congrA... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Group.Basic | {
"line": 270,
"column": 87
} | {
"line": 270,
"column": 92
} | {
"line": 271,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : CancelCommMonoid α\na b c d : α\nh : a * b = c * d\n⊢ a = c ↔ b = d",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"CancelCommMonoid.toCommMonoid",
"CancelMonoid.toRightCancelMonoid",
"HMul.hMul",
"Monoid.toMulOneClass",
"congrA... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Group.Basic | {
"line": 270,
"column": 87
} | {
"line": 270,
"column": 92
} | {
"line": 271,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : CancelCommMonoid α\na b c d : α\nh : a * b = c * d\n⊢ a = c ↔ b = d",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"CancelCommMonoid.toCommMonoid",
"CancelMonoid.toRightCancelMonoid",
"HMul.hMul",
"Monoid.toMulOneClass",
"congrA... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Group.Basic | {
"line": 271,
"column": 87
} | {
"line": 271,
"column": 92
} | {
"line": 273,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : CancelCommMonoid α\na b c d : α\nh : a * b = c * d\n⊢ a ≠ c ↔ b ≠ d",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"CancelCommMonoid.toCommMonoid",
"CancelMonoid.toRightCancelMonoid",
"False",
"HMul.hMul",
"eq_false",
"Mon... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Group.Basic | {
"line": 271,
"column": 87
} | {
"line": 271,
"column": 92
} | {
"line": 273,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : CancelCommMonoid α\na b c d : α\nh : a * b = c * d\n⊢ a ≠ c ↔ b ≠ d",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"CancelCommMonoid.toCommMonoid",
"CancelMonoid.toRightCancelMonoid",
"False",
"HMul.hMul",
"eq_false",
"Mon... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Group.Basic | {
"line": 271,
"column": 87
} | {
"line": 271,
"column": 92
} | {
"line": 273,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : CancelCommMonoid α\na b c d : α\nh : a * b = c * d\n⊢ a ≠ c ↔ b ≠ d",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"CancelCommMonoid.toCommMonoid",
"CancelMonoid.toRightCancelMonoid",
"False",
"HMul.hMul",
"eq_false",
"Mon... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Group.Basic | {
"line": 372,
"column": 49
} | {
"line": 372,
"column": 74
} | {
"line": 374,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : DivisionMonoid α\na b : α\nh : a / b = 1\n⊢ a * b⁻¹ = 1",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"DivInvMonoid.toInv",
"instHDiv",
"HMul.hMul",
"Monoid.toMulOneClass",
"congrArg",
... | [] | by rwa [← div_eq_mul_inv] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Group.TypeTags.Basic | {
"line": 306,
"column": 92
} | {
"line": 307,
"column": 40
} | {
"line": 309,
"column": 0
} | [
{
"pp": "α : Type u\ninst✝ : Monoid α\na : α\n⊢ IsAddRegular (ofMul a) ↔ IsRegular a",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"IsAddRightRegular",
"Equiv.instEquivLike",
"isAddLeftRegular_ofMul._simp_1",
"Monoid.toMulOneClass",
"congrArg",
"Additiv... | [] | by
simp [isAddRegular_iff, isRegular_iff] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Group.TypeTags.Basic | {
"line": 309,
"column": 87
} | {
"line": 310,
"column": 40
} | {
"line": 312,
"column": 0
} | [
{
"pp": "α : Type u\ninst✝ : Monoid α\na : Additive α\n⊢ IsRegular (toMul a) ↔ IsAddRegular a",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"IsAddRightRegular",
"Equiv.instEquivLike",
"Monoid.toMulOneClass",
"congrArg",
"Additive",
"isLeftRegular_toMul.... | [] | by
simp [isAddRegular_iff, isRegular_iff] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Group.TypeTags.Basic | {
"line": 332,
"column": 95
} | {
"line": 333,
"column": 40
} | {
"line": 335,
"column": 0
} | [
{
"pp": "α : Type u\ninst✝ : AddMonoid α\na : α\n⊢ IsRegular (ofAdd a) ↔ IsAddRegular a",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"IsAddRightRegular",
"Equiv.instEquivLike",
"AddMonoid.toAddSemigroup",
"_private.Mathlib.Algebra.Group.TypeTags.Basic.0.isRegular_... | [] | by
simp [isAddRegular_iff, isRegular_iff] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Group.TypeTags.Basic | {
"line": 335,
"column": 96
} | {
"line": 336,
"column": 40
} | {
"line": 338,
"column": 0
} | [
{
"pp": "α : Type u\ninst✝ : AddMonoid α\na : Multiplicative α\n⊢ IsAddRegular (toAdd a) ↔ IsRegular a",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"IsAddRightRegular",
"Equiv.instEquivLike",
"AddMonoid.toAddSemigroup",
"congrArg",
"_private.Mathlib.Algebra.... | [] | by
simp [isAddRegular_iff, isRegular_iff] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Group.TypeTags.Basic | {
"line": 506,
"column": 2
} | {
"line": 506,
"column": 49
} | {
"line": 507,
"column": 2
} | [
{
"pp": "case inl\nι : Type u_1\ninst✝¹ : DecidableEq ι\nM : ι → Type u_2\ninst✝ : (i : ι) → AddMonoid (M i)\nj : ι\na : M j\n⊢ mulSingle j (ofAdd a) j = ofAdd (single j a j)",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Equiv.instEquivLike",
"Pi.mulSingle_eq_same",
"co... | [
"case inr\nι : Type u_1\ninst✝¹ : DecidableEq ι\nM : ι → Type u_2\ninst✝ : (i : ι) → AddMonoid (M i)\ni : ι\na : M i\nj : ι\nh : j ≠ i\n⊢ mulSingle i (ofAdd a) j = ofAdd (single i a j)"
] | · simp only [mulSingle_eq_same, single_eq_same] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Group.TypeTags.Basic | {
"line": 514,
"column": 2
} | {
"line": 514,
"column": 49
} | {
"line": 515,
"column": 2
} | [
{
"pp": "case inl\nι : Type u_1\ninst✝¹ : DecidableEq ι\nM : ι → Type u_2\ninst✝ : (i : ι) → Monoid (M i)\nj : ι\na : M j\n⊢ single j (ofMul a) j = ofMul (mulSingle j a j)",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"MulOne.toOne",
"Equiv.instEquivLike",
"Pi.mulSingle_... | [
"case inr\nι : Type u_1\ninst✝¹ : DecidableEq ι\nM : ι → Type u_2\ninst✝ : (i : ι) → Monoid (M i)\ni : ι\na : M i\nj : ι\nh : j ≠ i\n⊢ single i (ofMul a) j = ofMul (mulSingle i a j)"
] | · simp only [mulSingle_eq_same, single_eq_same] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Group.Basic | {
"line": 473,
"column": 70
} | {
"line": 473,
"column": 79
} | {
"line": 473,
"column": 80
} | [
{
"pp": "α : Type u_1\ninst✝ : DivisionMonoid α\na : α\nm n : ℕ\n⊢ a ^ (-↑(m * n.succ)) = (a ^ (m * (n + 1)))⁻¹",
"ppTerm": "?m.110",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"DivInvMonoid.toInv",
"HMul.hMul",
"DivInvOneMonoid.toInvOneClass",
"congrArg",
"Di... | [
"α : Type u_1\ninst✝ : DivisionMonoid α\na : α\nm n : ℕ\n⊢ (a ^ ↑(m * n.succ))⁻¹ = (a ^ (m * (n + 1)))⁻¹"
] | zpow_neg, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Group.Basic | {
"line": 476,
"column": 81
} | {
"line": 476,
"column": 90
} | {
"line": 476,
"column": 91
} | [
{
"pp": "α : Type u_1\ninst✝ : DivisionMonoid α\na : α\nm n : ℕ\n⊢ a ^ (-↑(m.succ * n)) = a⁻¹ ^ ((m + 1) * n)",
"ppTerm": "?m.149",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"DivInvOneMonoid.toInvOneClass",
"congrArg",
"DivInvMonoid.toZPow",
"Div... | [
"α : Type u_1\ninst✝ : DivisionMonoid α\na : α\nm n : ℕ\n⊢ (a ^ ↑(m.succ * n))⁻¹ = a⁻¹ ^ ((m + 1) * n)"
] | zpow_neg, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Group.Basic | {
"line": 730,
"column": 72
} | {
"line": 731,
"column": 56
} | {
"line": 733,
"column": 0
} | [
{
"pp": "G : Type u_3\ninst✝ : Group G\na b c : G\n⊢ a * b * (b⁻¹ * c) = a * c",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"Semigroup.toMul",
"DivInvMonoid.toInv",
"HMul.hMul",
"DivInvOneMonoid.toInvOneClass",
"Monoid.... | [] | by
rw [mul_assoc, ← mul_assoc b, mul_inv_cancel, one_mul] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Group.Basic | {
"line": 785,
"column": 58
} | {
"line": 785,
"column": 77
} | {
"line": 787,
"column": 0
} | [
{
"pp": "G : Type u_3\ninst✝ : Group G\na : G\nn : ℤ\n⊢ a ^ n.natAbs = 1 ↔ a ^ n = 1",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"zpow_natCast",
"DivInvMonoid.toInv",
"InvOneClass.toOne",
"DivInvOneMonoid.toInvOneClass",
"congrArg",
"zpow_negSucc",
... | [] | by cases n <;> simp | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Group.Basic | {
"line": 808,
"column": 46
} | {
"line": 808,
"column": 64
} | {
"line": 808,
"column": 64
} | [
{
"pp": "G : Type u_3\ninst✝ : Group G\na : G\nn : ℤ\n⊢ a ^ (n - 1 + 1) * a⁻¹ = a ^ n * a⁻¹",
"ppTerm": "?m.79",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"DivInvOneMonoid.toInvOneClass",
"Monoid.toMulOneClass",
"congrArg",
"HSub.hSub",
"Di... | [
"G : Type u_3\ninst✝ : Group G\na : G\nn : ℤ\n⊢ a ^ n * a⁻¹ = a ^ n * a⁻¹"
] | Int.sub_add_cancel | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Group.Units.Hom | {
"line": 76,
"column": 45
} | {
"line": 76,
"column": 64
} | {
"line": 76,
"column": 64
} | [
{
"pp": "M : Type u\nN : Type v\nP : Type w\ninst✝² : Monoid M\ninst✝¹ : Monoid N\ninst✝ : Monoid P\nf : M →* N\nu : Mˣ\n⊢ f ↑u * f u.inv = 1",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"Units.val",
"MonoidHom.instMonoidHomClass",
"MulOne.toOne",
"MonoidHom.inst... | [] | by simp [← map_mul] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Group.Units.Hom | {
"line": 76,
"column": 66
} | {
"line": 76,
"column": 85
} | {
"line": 76,
"column": 85
} | [
{
"pp": "M : Type u\nN : Type v\nP : Type w\ninst✝² : Monoid M\ninst✝¹ : Monoid N\ninst✝ : Monoid P\nf : M →* N\nu : Mˣ\n⊢ f u.inv * f ↑u = 1",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"Units.val",
"MonoidHom.instMonoidHomClass",
"MulOne.toOne",
"MonoidHom.inst... | [] | by simp [← map_mul] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Group.Units.Hom | {
"line": 219,
"column": 37
} | {
"line": 219,
"column": 56
} | {
"line": 221,
"column": 0
} | [
{
"pp": "F : Type u_1\nM : Type u_3\nN : Type u_4\ninst✝³ : FunLike F M N\ninst✝² : Monoid M\ninst✝¹ : Monoid N\ninst✝ : MonoidHomClass F M N\nf : F\nx : M\nh : IsUnit x\n⊢ f ↑h.unit⁻¹ * ↑⋯.unit = 1",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Units.val",
"MulOne.toOne",
... | [] | by simp [← map_mul] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Group.Units.Defs | {
"line": 602,
"column": 83
} | {
"line": 603,
"column": 57
} | {
"line": 605,
"column": 0
} | [
{
"pp": "α : Type u\ninst✝ : DivisionMonoid α\nb : α\nh : IsUnit b\na : α\n⊢ b / (a * b) = a⁻¹",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"DivInvMonoid.toInv",
"instHDiv",
"HMul.hMul",
"Monoid.toMulOneClass",
"congrArg",
"id",
... | [] | by
rw [div_eq_mul_inv, mul_inv_rev, h.mul_inv_cancel_left] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Group.Units.Basic | {
"line": 276,
"column": 10
} | {
"line": 276,
"column": 25
} | {
"line": 276,
"column": 25
} | [
{
"pp": "M : Type u_1\ninst✝¹ : Monoid M\ninst✝ : Subsingleton M\na : M\n⊢ a * a = 1",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"MulOne.toOne",
"HMul.hMul",
"Monoid.toMulOneClass",
"MulOne.toMul",
"MulOneClass.toMulOne",
"One.toOfNat1",
"Subsin... | [] | by subsingleton | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Group.Units.Basic | {
"line": 276,
"column": 27
} | {
"line": 276,
"column": 42
} | {
"line": 276,
"column": 42
} | [
{
"pp": "M : Type u_1\ninst✝¹ : Monoid M\ninst✝ : Subsingleton M\na : M\n⊢ a * a = 1",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"MulOne.toOne",
"HMul.hMul",
"Monoid.toMulOneClass",
"MulOne.toMul",
"MulOneClass.toMulOne",
"One.toOfNat1",
"Subsin... | [] | by subsingleton | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Group.Units.Basic | {
"line": 285,
"column": 33
} | {
"line": 285,
"column": 48
} | {
"line": 285,
"column": 48
} | [
{
"pp": "α : Type u\nM : Type u_1\ninst✝¹ : Monoid M\ninst✝ : Subsingleton M\nx✝ : Mˣ\n⊢ ↑x✝ = 1",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Units.val",
"MulOne.toOne",
"Monoid.toMulOneClass",
"MulOneClass.toMulOne",
"One.toOfNat1",
"Subsingleton.el... | [] | by subsingleton | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Group.Prod | {
"line": 107,
"column": 23
} | {
"line": 107,
"column": 63
} | {
"line": 108,
"column": 2
} | [
{
"pp": "G : Type u_1\nH : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\ninst✝¹ : DivInvMonoid G\ninst✝ : DivInvMonoid H\nx✝¹ : ℕ\nx✝ : G × H\n⊢ x✝ ^ ↑x✝¹.succ = x✝ ^ ↑x✝¹ * x✝",
"ppTerm": "?m.57",
"assigned": true,
"usedConstants": [
"Semigroup.toMul",
"HMul.hMul",
"ZPow.zpo... | [] | ext <;> exact DivInvMonoid.zpow_succ' .. | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Algebra.Group.Prod | {
"line": 107,
"column": 23
} | {
"line": 107,
"column": 63
} | {
"line": 108,
"column": 2
} | [
{
"pp": "G : Type u_1\nH : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\ninst✝¹ : DivInvMonoid G\ninst✝ : DivInvMonoid H\nx✝¹ : ℕ\nx✝ : G × H\n⊢ x✝ ^ ↑x✝¹.succ = x✝ ^ ↑x✝¹ * x✝",
"ppTerm": "?m.57",
"assigned": true,
"usedConstants": [
"Semigroup.toMul",
"HMul.hMul",
"ZPow.zpo... | [] | ext <;> exact DivInvMonoid.zpow_succ' .. | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Group.Prod | {
"line": 107,
"column": 23
} | {
"line": 107,
"column": 63
} | {
"line": 108,
"column": 2
} | [
{
"pp": "G : Type u_1\nH : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\ninst✝¹ : DivInvMonoid G\ninst✝ : DivInvMonoid H\nx✝¹ : ℕ\nx✝ : G × H\n⊢ x✝ ^ ↑x✝¹.succ = x✝ ^ ↑x✝¹ * x✝",
"ppTerm": "?m.57",
"assigned": true,
"usedConstants": [
"Semigroup.toMul",
"HMul.hMul",
"ZPow.zpo... | [] | ext <;> exact DivInvMonoid.zpow_succ' .. | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.GroupWithZero.Basic | {
"line": 501,
"column": 24
} | {
"line": 501,
"column": 33
} | {
"line": 501,
"column": 34
} | [
{
"pp": "G₀ : Type u_2\ninst✝ : GroupWithZero G₀\na : G₀\nha : a ≠ 0\nn : ℕ\n⊢ a ^ (-(↑(n + 1) + 1) + 1) = a ^ (-(↑(n + 1) + 1)) * a",
"ppTerm": "?m.64",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"GroupWithZero.toDivisionMonoid",
"HMul.h... | [
"G₀ : Type u_2\ninst✝ : GroupWithZero G₀\na : G₀\nha : a ≠ 0\nn : ℕ\n⊢ a ^ (-(↑(n + 1) + 1) + 1) = (a ^ (↑(n + 1) + 1))⁻¹ * a"
] | zpow_neg, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.GroupWithZero.Basic | {
"line": 501,
"column": 73
} | {
"line": 501,
"column": 82
} | {
"line": 502,
"column": 6
} | [
{
"pp": "G₀ : Type u_2\ninst✝ : GroupWithZero G₀\na : G₀\nha : a ≠ 0\nn : ℕ\n⊢ a ^ (-↑(n + 1)) = (a ^ (↑(n + 1) + 1))⁻¹ * a",
"ppTerm": "?m.75",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"GroupWithZero.toDivisionMonoid",
"HMul.hMul",
... | [
"G₀ : Type u_2\ninst✝ : GroupWithZero G₀\na : G₀\nha : a ≠ 0\nn : ℕ\n⊢ (a ^ ↑(n + 1))⁻¹ = (a ^ (↑(n + 1) + 1))⁻¹ * a"
] | zpow_neg, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.GroupWithZero.Basic | {
"line": 508,
"column": 50
} | {
"line": 508,
"column": 68
} | {
"line": 508,
"column": 68
} | [
{
"pp": "G₀ : Type u_2\ninst✝ : GroupWithZero G₀\na : G₀\nha : a ≠ 0\nn : ℤ\n⊢ a ^ (n - 1 + 1) * a⁻¹ = a ^ n * a⁻¹",
"ppTerm": "?m.90",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"GroupWithZero.toDivisionMonoid",
"HMul.hMul",
"Group... | [
"G₀ : Type u_2\ninst✝ : GroupWithZero G₀\na : G₀\nha : a ≠ 0\nn : ℤ\n⊢ a ^ n * a⁻¹ = a ^ n * a⁻¹"
] | Int.sub_add_cancel | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.GroupWithZero.Basic | {
"line": 525,
"column": 23
} | {
"line": 525,
"column": 38
} | {
"line": 525,
"column": 39
} | [
{
"pp": "case pos\nG₀ : Type u_2\ninst✝ : GroupWithZero G₀\nm n : ℤ\nhm : ¬m = 0\nhn : ¬n = 0\nh : ¬m + n = 0\n⊢ 0 = 0 ^ m * 0 ^ n",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"zero_zpow",
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"HMul.hMul",
"GroupWithZ... | [
"case pos\nG₀ : Type u_2\ninst✝ : GroupWithZero G₀\nm n : ℤ\nhm : ¬m = 0\nhn : ¬n = 0\nh : ¬m + n = 0\n⊢ 0 = 0 * 0 ^ n"
] | zero_zpow _ hm, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.GroupWithZero.Action.Defs | {
"line": 424,
"column": 15
} | {
"line": 424,
"column": 53
} | {
"line": 424,
"column": 53
} | [
{
"pp": "α : Type u_10\nβ : Type u_11\ninst✝² : Group α\ninst✝¹ : AddMonoid β\ninst✝ : DistribMulAction α β\na : α\nx : β\nh : a • x = 0\n⊢ x = 0",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"DivInvMonoid.toInv",
"instHSMul",
"congrArg",
"DistribM... | [] | rw [← inv_smul_smul a x, h, smul_zero] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
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