module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Order.Lattice | {
"line": 1076,
"column": 2
} | {
"line": 1076,
"column": 63
} | {
"line": 1076,
"column": 64
} | [
{
"pp": "α : Type u\ninst✝¹ : LinearOrder α\np : α → α → Prop\ninst✝ : Std.Symm p\n⊢ Pairwise p ↔ ∀ ⦃a b : α⦄, a < b → p a b",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"_private.Mathlib.Order.Lattice.0.pairwise_iff_lt._simp_1_4",
"cong... | [
"α : Type u\ninst✝¹ : LinearOrder α\np : α → α → Prop\ninst✝ : Std.Symm p\n⊢ (∀ (x x_1 : α), x < x_1 → p x x_1) → ∀ (x x_1 : α), x_1 < x → p x x_1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Lattice | {
"line": 1079,
"column": 2
} | {
"line": 1079,
"column": 63
} | {
"line": 1079,
"column": 64
} | [
{
"pp": "α : Type u\ninst✝¹ : LinearOrder α\np : α → α → Prop\ninst✝ : Std.Symm p\n⊢ Pairwise p ↔ ∀ ⦃a b : α⦄, b < a → p a b",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"congrArg",
"PartialOrder.toPreorder",
"_private.Mathlib.Ord... | [
"α : Type u\ninst✝¹ : LinearOrder α\np : α → α → Prop\ninst✝ : Std.Symm p\n⊢ (∀ (x x_1 : α), x_1 < x → p x x_1) → ∀ (x x_1 : α), x < x_1 → p x x_1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Heyting.Basic | {
"line": 359,
"column": 6
} | {
"line": 359,
"column": 18
} | {
"line": 359,
"column": 19
} | [
{
"pp": "α : Type u_2\ninst✝ : GeneralizedHeytingAlgebra α\na b c : α\n⊢ (a ⇨ b) ⊓ (b ⇨ c) ≤ a ⇨ c",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"PartialOrder.toPreorder",
"Preorder.toLE",
"SemilatticeInf.toPartialOrder",
"General... | [
"α : Type u_2\ninst✝ : GeneralizedHeytingAlgebra α\na b c : α\n⊢ (a ⇨ b) ⊓ (b ⇨ c) ⊓ a ≤ c"
] | le_himp_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.Heyting.Basic | {
"line": 371,
"column": 52
} | {
"line": 371,
"column": 64
} | {
"line": 371,
"column": 65
} | [
{
"pp": "ι : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝ : GeneralizedHeytingAlgebra α\na✝ b✝ c✝ d a b c : α\n⊢ b ≤ a ⇨ b ⊓ a ⊔ c ⊓ a ∧ c ≤ a ⇨ b ⊓ a ⊔ c ⊓ a",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Lattice.toSemilatticeSup",
"congrArg",
"PartialOr... | [
"ι : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝ : GeneralizedHeytingAlgebra α\na✝ b✝ c✝ d a b c : α\n⊢ b ⊓ a ≤ b ⊓ a ⊔ c ⊓ a ∧ c ⊓ a ≤ b ⊓ a ⊔ c ⊓ a"
] | le_himp_iff, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Order.Heyting.Basic | {
"line": 481,
"column": 2
} | {
"line": 481,
"column": 13
} | {
"line": 481,
"column": 14
} | [
{
"pp": "α : Type u_2\ninst✝ : GeneralizedCoheytingAlgebra α\na b c : α\n⊢ a ≤ b ⊔ (a \\ c ⊔ c \\ b)",
"ppTerm": "?m.11",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_2\ninst✝ : GeneralizedCoheytingAlgebra α\na b c : α\n⊢ a ≤ b ⊔ (a \\ c ⊔ c \\ b)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Heyting.Basic | {
"line": 547,
"column": 79
} | {
"line": 548,
"column": 58
} | {
"line": 550,
"column": 0
} | [
{
"pp": "α : Type u_2\ninst✝ : GeneralizedCoheytingAlgebra α\na b : α\nh : Disjoint a b\n⊢ (a ⊔ b) \\ a = b",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Lattice.toSemilatticeSup",
"Disjoint.sdiff_eq_right",
"congrArg",
"OrderBot.toBot",
"Par... | [] | by
rw [sup_sdiff, sdiff_self, bot_sup_eq, h.sdiff_eq_right] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Order.Heyting.Basic | {
"line": 652,
"column": 18
} | {
"line": 652,
"column": 30
} | {
"line": 652,
"column": 31
} | [
{
"pp": "α : Type u_2\ninst✝ : HeytingAlgebra α\na b : α\n⊢ a ≤ b ⇨ ⊥ ↔ Disjoint a b",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Lattice.toSemilatticeSup",
"congrArg",
"OrderBot.toBot",
"PartialOrder.toPreorder",
"Preorder.toLE",
"Di... | [
"α : Type u_2\ninst✝ : HeytingAlgebra α\na b : α\n⊢ a ⊓ b ≤ ⊥ ↔ Disjoint a b"
] | le_himp_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.Heyting.Basic | {
"line": 751,
"column": 51
} | {
"line": 751,
"column": 81
} | {
"line": 752,
"column": 4
} | [
{
"pp": "α : Type u_2\ninst✝ : HeytingAlgebra α\na b : α\n⊢ Disjoint aᶜᶜ (bᶜᶜ ⊓ (a ⊓ b)ᶜ)",
"ppTerm": "?m.56",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Compl.compl",
"disjoint_compl_compl_left_iff",
"Disjoint",
"SemilatticeInf.toPartialOrder",
... | [
"α : Type u_2\ninst✝ : HeytingAlgebra α\na b : α\n⊢ Disjoint a (bᶜᶜ ⊓ (a ⊓ b)ᶜ)"
] | disjoint_compl_compl_left_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.Heyting.Basic | {
"line": 752,
"column": 24
} | {
"line": 752,
"column": 54
} | {
"line": 752,
"column": 55
} | [
{
"pp": "α : Type u_2\ninst✝ : HeytingAlgebra α\na b : α\n⊢ Disjoint bᶜᶜ (a ⊓ (a ⊓ b)ᶜ)",
"ppTerm": "?m.68",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Compl.compl",
"disjoint_compl_compl_left_iff",
"Disjoint",
"SemilatticeInf.toPartialOrder",
... | [
"α : Type u_2\ninst✝ : HeytingAlgebra α\na b : α\n⊢ Disjoint b (a ⊓ (a ⊓ b)ᶜ)"
] | disjoint_compl_compl_left_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.Heyting.Basic | {
"line": 757,
"column": 8
} | {
"line": 757,
"column": 20
} | {
"line": 757,
"column": 21
} | [
{
"pp": "case a\nα : Type u_2\ninst✝ : HeytingAlgebra α\na b : α\n⊢ (a ⇨ b)ᶜᶜ ≤ aᶜᶜ ⇨ bᶜᶜ",
"ppTerm": "?a✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Compl.compl",
"PartialOrder.toPreorder",
"Preorder.toLE",
"SemilatticeInf.toPartialOrder",
... | [
"case a\nα : Type u_2\ninst✝ : HeytingAlgebra α\na b : α\n⊢ (a ⇨ b)ᶜᶜ ⊓ aᶜᶜ ≤ bᶜᶜ"
] | le_himp_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Control.EquivFunctor | {
"line": 101,
"column": 31
} | {
"line": 101,
"column": 61
} | {
"line": 101,
"column": 62
} | [
{
"pp": "f : Type u₀ → Type u₁\ninst✝¹ : Applicative f\ninst✝ : LawfulApplicative f\nα β : Type u₀\nh : ∀ (γ : Type u₀), Injective pure\ne₁ e₂ : α ≃ β\nH : mapEquiv f e₁ = mapEquiv f e₂\nx : α\n⊢ pure (e₁ x) = pure (e₂ x)",
"ppTerm": "?m.21",
"assigned": false,
"usedConstants": [],
"usedFVars": ... | [
"f : Type u₀ → Type u₁\ninst✝¹ : Applicative f\ninst✝ : LawfulApplicative f\nα β : Type u₀\nh : ∀ (γ : Type u₀), Injective pure\ne₁ e₂ : α ≃ β\nH : mapEquiv f e₁ = mapEquiv f e₂\nx : α\n⊢ pure (e₁ x) = pure (e₂ x)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Logic.Equiv.Option | {
"line": 133,
"column": 15
} | {
"line": 133,
"column": 26
} | {
"line": 133,
"column": 27
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ne : Option α ≃ Option β\nh : e.symm none = none\n⊢ e none = none",
"ppTerm": "?m.14",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\nβ : Type u_2\ne : Option α ≃ Option β\nh : e.symm none = none\n⊢ e none = none"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Logic.Equiv.Option | {
"line": 133,
"column": 61
} | {
"line": 133,
"column": 72
} | {
"line": 133,
"column": 73
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ne : Option α ≃ Option β\nh : e none = none\n⊢ e.symm none = none",
"ppTerm": "?m.17",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\nβ : Type u_2\ne : Option α ≃ Option β\nh : e none = none\n⊢ e.symm none = none"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Logic.Equiv.Option | {
"line": 138,
"column": 4
} | {
"line": 138,
"column": 15
} | {
"line": 138,
"column": 16
} | [
{
"pp": "case none\nα : Type u_1\nβ : Type u_2\ne : Option α ≃ Option β\nx : α\nh : e (some x) = none\n⊢ e none = e none ↔ e.symm none = some x",
"ppTerm": "?none",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Equiv.instEquivLike",
"congrArg",
"true_iff",
"Option.som... | [
"case none\nα : Type u_1\nβ : Type u_2\ne : Option α ≃ Option β\nx : α\nh : e (some x) = none\n⊢ e.symm none = some x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Sigma.Basic | {
"line": 76,
"column": 2
} | {
"line": 76,
"column": 26
} | {
"line": 76,
"column": 27
} | [
{
"pp": "α : Type u_1\nβ : α → Type u_4\nγ : Type u_7\ninst✝ : Nonempty γ\na : α\nf : γ → β a\ni : γ\ng : γ → β a\nh : mk a ∘ f = mk a ∘ g\n⊢ a = a ∧ f ≍ g",
"ppTerm": "?m.49",
"assigned": true,
"usedConstants": [
"_private.Mathlib.Data.Sigma.Basic.0.Function.eq_of_sigmaMk_comp._simp_1_1",
... | [
"α : Type u_1\nβ : α → Type u_4\nγ : Type u_7\ninst✝ : Nonempty γ\na : α\nf : γ → β a\ni : γ\ng : γ → β a\nh : mk a ∘ f = mk a ∘ g\n⊢ ∀ (x : γ), f x = g x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Logic.Equiv.Option | {
"line": 188,
"column": 6
} | {
"line": 188,
"column": 17
} | {
"line": 188,
"column": 18
} | [
{
"pp": "case none\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝ : DecidableEq β\nx : β\ne : { e // e none = x }\n⊢ ↑((fun e ↦\n ⟨{ toFun := fun a ↦ a.casesOn' x (Subtype.val ∘ ⇑e),\n invFun := fun b ↦ if h : b = x then none else some (e.symm ⟨b, h⟩), left_inv := ⋯, right_inv := ⋯ }... | [
"case none\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝ : DecidableEq β\nx : β\ne : { e // e none = x }\n⊢ x = ↑e none"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.BooleanAlgebra.Basic | {
"line": 281,
"column": 61
} | {
"line": 282,
"column": 48
} | {
"line": 284,
"column": 0
} | [
{
"pp": "α : Type u\nx y : α\ninst✝ : GeneralizedBooleanAlgebra α\nh : y ≤ x\n⊢ x \\ (x \\ y) = y",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"inf_of_le_right",
"id",
"SemilatticeInf.toMin",
"GeneralizedBooleanAlgebra.toGeneraliz... | [] | by
rw [sdiff_sdiff_right_self, inf_of_le_right h] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Order.SymmDiff | {
"line": 192,
"column": 61
} | {
"line": 193,
"column": 56
} | {
"line": 195,
"column": 0
} | [
{
"pp": "α : Type u_2\ninst✝ : GeneralizedCoheytingAlgebra α\na b : α\n⊢ a ≤ a ∆ b ⊔ b",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Lattice.toSemilatticeSup",
"congrArg",
"HEq.refl",
"OrderBot.toBot",
"PartialOrder.toPreorder",
"Preord... | [] | by
convert! symmDiff_triangle a b ⊥ <;> rw [symmDiff_bot] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Order.SymmDiff | {
"line": 240,
"column": 31
} | {
"line": 240,
"column": 43
} | {
"line": 240,
"column": 44
} | [
{
"pp": "α : Type u_2\ninst✝ : GeneralizedHeytingAlgebra α\na b c : α\n⊢ a ≤ c ⇨ b ∧ a ≤ b ⇨ c ↔ a ⊓ b ≤ c ∧ a ⊓ c ≤ b",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"_private.Mathlib.Order.SymmDiff.0.le_bihimp_iff._simp_1_3",
"PartialOrder.to... | [
"α : Type u_2\ninst✝ : GeneralizedHeytingAlgebra α\na b c : α\n⊢ a ⊓ c ≤ b ∧ a ⊓ b ≤ c ↔ a ⊓ b ≤ c ∧ a ⊓ c ≤ b"
] | le_himp_iff, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Order.BooleanAlgebra.Basic | {
"line": 521,
"column": 43
} | {
"line": 521,
"column": 54
} | {
"line": 521,
"column": 55
} | [
{
"pp": "α : Type u\nx y : α\ninst✝ : BooleanAlgebra α\nh✝ : yᶜ ≤ xᶜ\nh : xᶜᶜ ≤ yᶜᶜ\n⊢ x ≤ y",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"PartialOrder.toPreorder",
"Preorder.toLE",
"SemilatticeInf.toPartialOrder",
"id",
"BiheytingAlgebra.to... | [
"α : Type u\nx y : α\ninst✝ : BooleanAlgebra α\nh✝ : yᶜ ≤ xᶜ\nh : xᶜᶜ ≤ yᶜᶜ\n⊢ x ≤ y"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.BooleanAlgebra.Basic | {
"line": 527,
"column": 2
} | {
"line": 527,
"column": 32
} | {
"line": 527,
"column": 33
} | [
{
"pp": "α : Type u\nx y : α\ninst✝ : BooleanAlgebra α\nh : yᶜ ≤ x\n⊢ xᶜ ≤ y",
"ppTerm": "?m.9",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u\nx y : α\ninst✝ : BooleanAlgebra α\nh : yᶜ ≤ x\n⊢ xᶜ ≤ y"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.BooleanAlgebra.Basic | {
"line": 532,
"column": 53
} | {
"line": 532,
"column": 64
} | {
"line": 532,
"column": 65
} | [
{
"pp": "α : Type u\nx : α\ninst✝ : BooleanAlgebra α\n⊢ xᶜ ≤ x ↔ x = ⊤",
"ppTerm": "?m.8",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u\nx : α\ninst✝ : BooleanAlgebra α\n⊢ xᶜ ≤ x ↔ x = ⊤"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.BooleanAlgebra.Basic | {
"line": 535,
"column": 2
} | {
"line": 535,
"column": 13
} | {
"line": 535,
"column": 14
} | [
{
"pp": "α : Type u\nx : α\ninst✝¹ : BooleanAlgebra α\ninst✝ : Nontrivial α\n⊢ xᶜ < x ↔ x = ⊤",
"ppTerm": "?m.8",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u\nx : α\ninst✝¹ : BooleanAlgebra α\ninst✝ : Nontrivial α\n⊢ xᶜ < x ↔ x = ⊤"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Logic.Equiv.Basic | {
"line": 417,
"column": 4
} | {
"line": 418,
"column": 7
} | {
"line": 419,
"column": 2
} | [
{
"pp": "α : Sort u_1\nα₁ : Sort u_2\nα₂ : Sort u_3\nβ✝ : Sort u_4\nβ₁ : Sort u_5\nβ₂ : Sort u_6\nγ : Sort u_7\nδ : Sort u_8\nβ : α → Sort v\np : (a : α) → β a → Prop\n⊢ LeftInverse (fun f ↦ ⟨fun a ↦ ↑(f a), ⋯⟩) fun f a ↦ ⟨↑f a, ⋯⟩",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Sub... | [] | rintro ⟨f, h⟩
rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Logic.Equiv.Basic | {
"line": 417,
"column": 4
} | {
"line": 418,
"column": 7
} | {
"line": 419,
"column": 2
} | [
{
"pp": "α : Sort u_1\nα₁ : Sort u_2\nα₂ : Sort u_3\nβ✝ : Sort u_4\nβ₁ : Sort u_5\nβ₂ : Sort u_6\nγ : Sort u_7\nδ : Sort u_8\nβ : α → Sort v\np : (a : α) → β a → Prop\n⊢ LeftInverse (fun f ↦ ⟨fun a ↦ ↑(f a), ⋯⟩) fun f a ↦ ⟨↑f a, ⋯⟩",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Sub... | [] | rintro ⟨f, h⟩
rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Set.Restrict | {
"line": 246,
"column": 2
} | {
"line": 248,
"column": 93
} | {
"line": 250,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ns : Set α\nt : Set β\nf : α → β\n⊢ t.restrictPreimage f '' Subtype.val ⁻¹' s = Subtype.val ⁻¹' f '' s",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Set.restrictPreimage",
"Eq.mpr",
"congrArg",
"Subtype.image_preimage_coe",
... | [] | delta Set.restrictPreimage
rw [← (Subtype.coe_injective).image_injective.eq_iff, ← image_comp, MapsTo.restrict_commutes,
image_comp, Subtype.image_preimage_coe, Subtype.image_preimage_coe, image_preimage_inter] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Set.Restrict | {
"line": 246,
"column": 2
} | {
"line": 248,
"column": 93
} | {
"line": 250,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ns : Set α\nt : Set β\nf : α → β\n⊢ t.restrictPreimage f '' Subtype.val ⁻¹' s = Subtype.val ⁻¹' f '' s",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Set.restrictPreimage",
"Eq.mpr",
"congrArg",
"Subtype.image_preimage_coe",
... | [] | delta Set.restrictPreimage
rw [← (Subtype.coe_injective).image_injective.eq_iff, ← image_comp, MapsTo.restrict_commutes,
image_comp, Subtype.image_preimage_coe, Subtype.image_preimage_coe, image_preimage_inter] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Set.Restrict | {
"line": 312,
"column": 32
} | {
"line": 312,
"column": 61
} | {
"line": 312,
"column": 62
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ns : Set α\nt : Set β\nf : α → β\nh : MapsTo f s t\nh' : Surjective (restrict f s t h)\nb : β\nhb : b ∈ t\na : α\nha : a ∈ s\nha' : restrict f s t h ⟨a, ha⟩ = ⟨b, hb⟩\n⊢ f a = b",
"ppTerm": "?m.60",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
... | [
"α : Type u_1\nβ : Type u_2\ns : Set α\nt : Set β\nf : α → β\nh : MapsTo f s t\nh' : Surjective (restrict f s t h)\nb : β\nhb : b ∈ t\na : α\nha : a ∈ s\nha' : restrict f s t h ⟨a, ha⟩ = ⟨b, hb⟩\n⊢ f a = b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Set.Image | {
"line": 311,
"column": 78
} | {
"line": 314,
"column": 71
} | {
"line": 316,
"column": 0
} | [
{
"pp": "α : Type u_1\nf : α → α\nn : ℕ\n⊢ image f^[n] = (image f)^[n]",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.recAux",
"congrArg",
"Function.iterate_succ'",
"Set.image_id_eq",
"Function.comp",
"id",
"instOfNatNat",
... | [] | by
induction n with
| zero => simp
| succ n ih => rw [iterate_succ', iterate_succ', ← ih, image_comp_eq] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Set.Piecewise | {
"line": 109,
"column": 2
} | {
"line": 111,
"column": 64
} | {
"line": 113,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ns s₁ s₂ : Set α\nt t₁ t₂ : Set β\nf₁ f₂ : α → β\ninst✝ : (i : α) → Decidable (i ∈ s)\nh₁ : MapsTo f₁ (s₁ ∩ s) (t₁ ∩ t)\nh₂ : MapsTo f₂ (s₂ ∩ sᶜ) (t₂ ∩ tᶜ)\n⊢ MapsTo (s.piecewise f₁ f₂) (s.ite s₁ s₂) (t.ite t₁ t₂)",
"ppTerm": "?m.35",
"assigned": true,
"usedConsta... | [] | refine (h₁.congr ?_).union_union (h₂.congr ?_)
exacts [(piecewise_eqOn s f₁ f₂).symm.mono inter_subset_right,
(piecewise_eqOn_compl s f₁ f₂).symm.mono inter_subset_right] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Set.Piecewise | {
"line": 109,
"column": 2
} | {
"line": 111,
"column": 64
} | {
"line": 113,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ns s₁ s₂ : Set α\nt t₁ t₂ : Set β\nf₁ f₂ : α → β\ninst✝ : (i : α) → Decidable (i ∈ s)\nh₁ : MapsTo f₁ (s₁ ∩ s) (t₁ ∩ t)\nh₂ : MapsTo f₂ (s₂ ∩ sᶜ) (t₂ ∩ tᶜ)\n⊢ MapsTo (s.piecewise f₁ f₂) (s.ite s₁ s₂) (t.ite t₁ t₂)",
"ppTerm": "?m.35",
"assigned": true,
"usedConsta... | [] | refine (h₁.congr ?_).union_union (h₂.congr ?_)
exacts [(piecewise_eqOn s f₁ f₂).symm.mono inter_subset_right,
(piecewise_eqOn_compl s f₁ f₂).symm.mono inter_subset_right] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Set.Prod | {
"line": 321,
"column": 2
} | {
"line": 321,
"column": 18
} | {
"line": 323,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ns s₁ : Set α\nt t₁ : Set β\nh : (s ×ˢ t).Nonempty\n⊢ ¬s ×ˢ t = ∅",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
"Set.instSProd",
"SProd.sprod",
"Set.Nonempty.ne_empty",
"Prod",
"Set"
],
"usedFVars": [
"α",
... | [] | exact h.ne_empty | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.Group.Action.Pi | {
"line": 88,
"column": 4
} | {
"line": 88,
"column": 15
} | {
"line": 89,
"column": 6
} | [
{
"pp": "ι : Type u_1\nM : Type u_2\nα : ι → Type u_4\ninst✝² : (i : ι) → SMul M (α i)\ninst✝¹ : ∀ (i : ι), Nonempty (α i)\ni : ι\ninst✝ : FaithfulSMul M (α i)\nm₁✝ m₂✝ : M\nh : ∀ (a : (i : ι) → α i), m₁✝ • a = m₂✝ • a\na : α i\n⊢ m₁✝ • a = m₂✝ • a",
"ppTerm": "?m.16",
"assigned": false,
"usedConsta... | [
"ι : Type u_1\nM : Type u_2\nα : ι → Type u_4\ninst✝² : (i : ι) → SMul M (α i)\ninst✝¹ : ∀ (i : ι), Nonempty (α i)\ni : ι\ninst✝ : FaithfulSMul M (α i)\nm₁✝ m₂✝ : M\nh : ∀ (a : (i : ι) → α i), m₁✝ • a = m₂✝ • a\na : α i\n⊢ m₁✝ • a = m₂✝ • a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Set.Function | {
"line": 161,
"column": 2
} | {
"line": 161,
"column": 13
} | {
"line": 161,
"column": 14
} | [
{
"pp": "α : Type u_1\nf : α → α\ns : Set α\nh : MapsTo f s s\nn : ℕ\nx✝ : ↑s\n⊢ ↑((restrict f s s h)^[n] x✝) = ↑(restrict f^[n] s s ⋯ x✝)",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Membership.mem",
"Set.Elem",
"id",
"Set.MapsTo.iterate",
"Nat.iterate",
... | [
"α : Type u_1\nf : α → α\ns : Set α\nh : MapsTo f s s\nn : ℕ\nx✝ : ↑s\n⊢ ↑((restrict f s s h)^[n] x✝) = f^[n] ↑x✝"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Set.Function | {
"line": 198,
"column": 2
} | {
"line": 198,
"column": 33
} | {
"line": 198,
"column": 34
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ns : Set α\nt : Set β\nf : α → β\nh : MapsTo f s t\nx : α\n⊢ MapsTo f (Insert.insert x s) (Insert.insert (f x) t)",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"true_or",
"Set.mapsTo_singleton._simp_1",
... | [
"α : Type u_1\nβ : Type u_2\ns : Set α\nt : Set β\nf : α → β\nh : MapsTo f s t\nx : α\n⊢ MapsTo f s ({f x} ∪ t)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Set.Image | {
"line": 726,
"column": 4
} | {
"line": 726,
"column": 25
} | {
"line": 727,
"column": 4
} | [
{
"pp": "case mp\nα : Type u_1\nβ : Type u_2\ns t : Set α\nf : β → α\nhs : s ⊆ range f\nht : t ⊆ range f\nh : f ⁻¹' s = f ⁻¹' t\n⊢ s = t",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"Set.Subset.antisymm"
],
"usedFVars": [
"α",
"s",
"t"
],
"usedGoals... | [
"case mp.h₁\nα : Type u_1\nβ : Type u_2\ns t : Set α\nf : β → α\nhs : s ⊆ range f\nht : t ⊆ range f\nh : f ⁻¹' s = f ⁻¹' t\n⊢ s ⊆ t",
"case mp.h₂\nα : Type u_1\nβ : Type u_2\ns t : Set α\nf : β → α\nhs : s ⊆ range f\nht : t ⊆ range f\nh : f ⁻¹' s = f ⁻¹' t\n⊢ t ⊆ s"
] | apply Subset.antisymm | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Data.Set.Function | {
"line": 414,
"column": 30
} | {
"line": 414,
"column": 78
} | {
"line": 414,
"column": 79
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ns : Set α\nf : α → β\nh : InjOn f s\nx✝¹ x✝ : ↑s\nx : α\nhx : x ∈ s\ny : α\nhy : y ∈ s\nh' : imageFactorization f s ⟨x, hx⟩ = imageFactorization f s ⟨y, hy⟩\n⊢ ⟨x, hx⟩ = ⟨y, hy⟩",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Memb... | [
"α : Type u_1\nβ : Type u_2\ns : Set α\nf : α → β\nh : InjOn f s\nx✝¹ x✝ : ↑s\nx : α\nhx : x ∈ s\ny : α\nhy : y ∈ s\nh' : imageFactorization f s ⟨x, hx⟩ = imageFactorization f s ⟨y, hy⟩\n⊢ x = y"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Set.Function | {
"line": 417,
"column": 26
} | {
"line": 417,
"column": 37
} | {
"line": 417,
"column": 38
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ns : Set α\nf : α → β\nh : Injective (imageFactorization f s)\nx : α\nhx : x ∈ s\ny : α\nhy : y ∈ s\nx✝ : f x = f y\n⊢ x = y",
"ppTerm": "?m.15",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\nβ : Type u_2\ns : Set α\nf : α → β\nh : Injective (imageFactorization f s)\nx : α\nhx : x ∈ s\ny : α\nhy : y ∈ s\nx✝ : f x = f y\n⊢ x = y"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Set.Prod | {
"line": 721,
"column": 8
} | {
"line": 721,
"column": 23
} | {
"line": 721,
"column": 24
} | [
{
"pp": "case refine_1.left\nι : Type u_1\nα : ι → Type u_2\np : ι → Prop\nh✝ : DecidablePred p\ns : Set ι\nt₁ t₂ : (i : ι) → Set (α i)\nf : (i : ι) → α i\nh : f ∈ s.pi fun i ↦ if p i then t₁ i else t₂ i\ni : ι\nhis : i ∈ s\nhpi : p i\n⊢ f i ∈ t₁ i",
"ppTerm": "?refine_1.left",
"assigned": false,
"u... | [
"case refine_1.left\nι : Type u_1\nα : ι → Type u_2\np : ι → Prop\nh✝ : DecidablePred p\ns : Set ι\nt₁ t₂ : (i : ι) → Set (α i)\nf : (i : ι) → α i\nh : f ∈ s.pi fun i ↦ if p i then t₁ i else t₂ i\ni : ι\nhis : i ∈ s\nhpi : p i\n⊢ f i ∈ t₁ i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Set.Prod | {
"line": 721,
"column": 8
} | {
"line": 721,
"column": 23
} | {
"line": 721,
"column": 24
} | [
{
"pp": "case refine_1.right\nι : Type u_1\nα : ι → Type u_2\np : ι → Prop\nh✝ : DecidablePred p\ns : Set ι\nt₁ t₂ : (i : ι) → Set (α i)\nf : (i : ι) → α i\nh : f ∈ s.pi fun i ↦ if p i then t₁ i else t₂ i\ni : ι\nhis : i ∈ s\nhpi : ¬p i\n⊢ f i ∈ t₂ i",
"ppTerm": "?refine_1.right",
"assigned": false,
... | [
"case refine_1.right\nι : Type u_1\nα : ι → Type u_2\np : ι → Prop\nh✝ : DecidablePred p\ns : Set ι\nt₁ t₂ : (i : ι) → Set (α i)\nf : (i : ι) → α i\nh : f ∈ s.pi fun i ↦ if p i then t₁ i else t₂ i\ni : ι\nhis : i ∈ s\nhpi : ¬p i\n⊢ f i ∈ t₂ i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Set.Image | {
"line": 1204,
"column": 17
} | {
"line": 1204,
"column": 33
} | {
"line": 1206,
"column": 0
} | [
{
"pp": "α : Type u_1\ns : Set α\nt : Set ↑s\nx : α\nx✝ : x ∈ val '' t\ny : { x // x ∈ s }\nleft✝ : y ∈ t\nyvaleq : ↑y = x\n⊢ ↑y ∈ s",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"Membership.mem",
"Set.instMembership",
"Subtype.property",
"Set"
],
"usedFVa... | [] | exact y.property | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Data.Set.Function | {
"line": 624,
"column": 14
} | {
"line": 624,
"column": 25
} | {
"line": 624,
"column": 26
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ns : Set α\nf : α → β\nh : BijOn f s ∅\n⊢ s = ∅",
"ppTerm": "?m.11",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\nβ : Type u_2\ns : Set α\nf : α → β\nh : BijOn f s ∅\n⊢ s = ∅"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Set.Function | {
"line": 624,
"column": 36
} | {
"line": 624,
"column": 70
} | {
"line": 624,
"column": 70
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ns : Set α\nf : α → β\n⊢ s = ∅ → BijOn f s ∅",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Set.bijOn_empty",
"Set.BijOn",
"Set.instEmptyCollection",
"Eq.ndrec",
"EmptyCollection.emptyCollection",
"Eq.symm",
"E... | [] | by rintro rfl; exact bijOn_empty f | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Set.Function | {
"line": 627,
"column": 14
} | {
"line": 627,
"column": 25
} | {
"line": 627,
"column": 26
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nt : Set β\nf : α → β\nh : BijOn f ∅ t\n⊢ t = ∅",
"ppTerm": "?m.11",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\nβ : Type u_2\nt : Set β\nf : α → β\nh : BijOn f ∅ t\n⊢ t = ∅"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Set.Function | {
"line": 627,
"column": 36
} | {
"line": 627,
"column": 70
} | {
"line": 627,
"column": 70
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nt : Set β\nf : α → β\n⊢ t = ∅ → BijOn f ∅ t",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Set.bijOn_empty",
"Set.BijOn",
"Set.instEmptyCollection",
"Eq.ndrec",
"EmptyCollection.emptyCollection",
"Eq.symm",
"E... | [] | by rintro rfl; exact bijOn_empty f | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Group.Hom.Defs | {
"line": 511,
"column": 4
} | {
"line": 511,
"column": 15
} | {
"line": 511,
"column": 16
} | [
{
"pp": "M : Type u_4\nN : Type u_5\ninst✝² : Monoid M\ninst✝¹ : Monoid N\ninst✝ : IsMulTorsionFree N\nf : M →* N\nhf : Injective ⇑f\nn : ℕ\nhn : n ≠ 0\nx y : M\nhxy : (fun a ↦ a ^ n) x = (fun a ↦ a ^ n) y\n⊢ (fun a ↦ a ^ n) (f x) = (fun a ↦ a ^ n) (f y)",
"ppTerm": "?m.23",
"assigned": true,
"usedC... | [
"M : Type u_4\nN : Type u_5\ninst✝² : Monoid M\ninst✝¹ : Monoid N\ninst✝ : IsMulTorsionFree N\nf : M →* N\nhf : Injective ⇑f\nn : ℕ\nhn : n ≠ 0\nx y : M\nhxy : (fun a ↦ a ^ n) x = (fun a ↦ a ^ n) y\n⊢ f x ^ n = f y ^ n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Set.Prod | {
"line": 842,
"column": 2
} | {
"line": 842,
"column": 72
} | {
"line": 843,
"column": 4
} | [
{
"pp": "ι : Type u_1\nα : ι → Type u_2\ns : Set ι\nt₁ t₂ : (i : ι) → Set (α i)\nh : s.pi t₁ ⊆ s.pi t₂\nhne : (s.pi t₁).Nonempty\ni : ι\nhi : i ∈ s\n⊢ t₁ i ⊆ t₂ i",
"ppTerm": "?m.57",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"ι : Type u_1\nα : ι → Type u_2\ns : Set ι\nt₁ t₂ : (i : ι) → Set (α i)\nh : s.pi t₁ ⊆ s.pi t₂\nhne : (s.pi t₁).Nonempty\ni : ι\nhi : i ∈ s\n⊢ t₁ i ⊆ t₂ i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Group.Hom.Defs | {
"line": 712,
"column": 33
} | {
"line": 712,
"column": 62
} | {
"line": 712,
"column": 63
} | [
{
"pp": "M : Type u_4\nN : Type u_5\nF : Type u_9\ninst✝⁴ : MulOne M\ninst✝³ : MulOne N\ninst✝² : FunLike F M N\ninst✝¹ : MonoidHomClass F M N\nf : F\nhf : Injective ⇑f\ninst✝ : IsDedekindFiniteMonoid N\na✝ b✝ : M\neq : a✝ * b✝ = 1\n⊢ f (b✝ * a✝) = f 1",
"ppTerm": "?m.14",
"assigned": true,
"usedCon... | [
"M : Type u_4\nN : Type u_5\nF : Type u_9\ninst✝⁴ : MulOne M\ninst✝³ : MulOne N\ninst✝² : FunLike F M N\ninst✝¹ : MonoidHomClass F M N\nf : F\nhf : Injective ⇑f\ninst✝ : IsDedekindFiniteMonoid N\na✝ b✝ : M\neq : a✝ * b✝ = 1\n⊢ f a✝ * f b✝ = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Group.Hom.Defs | {
"line": 721,
"column": 39
} | {
"line": 721,
"column": 50
} | {
"line": 721,
"column": 51
} | [
{
"pp": "ι : 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\nF : Type u_9\ninst✝⁵ : MulOne M✝\ninst✝⁴ : MulOne N✝\ninst✝³ : FunLike F M✝ N✝\ninst✝² : MonoidHomClass F M✝ N✝\nM : Type u_10\nN : Type u_11\ninst✝¹ : Monoid M\ninst✝ : LeftCancelMonoid N\... | [
"ι : 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\nF : Type u_9\ninst✝⁵ : MulOne M✝\ninst✝⁴ : MulOne N✝\ninst✝³ : FunLike F M✝ N✝\ninst✝² : MonoidHomClass F M✝ N✝\nM : Type u_10\nN : Type u_11\ninst✝¹ : Monoid M\ninst✝ : LeftCancelMonoid N\nf : M →ₙ* N... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Group.Hom.Defs | {
"line": 728,
"column": 39
} | {
"line": 728,
"column": 50
} | {
"line": 728,
"column": 51
} | [
{
"pp": "ι : 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\nF : Type u_9\ninst✝⁵ : MulOne M✝\ninst✝⁴ : MulOne N✝\ninst✝³ : FunLike F M✝ N✝\ninst✝² : MonoidHomClass F M✝ N✝\nM : Type u_10\nN : Type u_11\ninst✝¹ : Monoid M\ninst✝ : RightCancelMonoid N... | [
"ι : 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\nF : Type u_9\ninst✝⁵ : MulOne M✝\ninst✝⁴ : MulOne N✝\ninst✝³ : FunLike F M✝ N✝\ninst✝² : MonoidHomClass F M✝ N✝\nM : Type u_10\nN : Type u_11\ninst✝¹ : Monoid M\ninst✝ : RightCancelMonoid N\nf : M →ₙ* ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Set.Prod | {
"line": 905,
"column": 2
} | {
"line": 905,
"column": 37
} | {
"line": 905,
"column": 38
} | [
{
"pp": "ι : Type u_1\nι' : Type u_2\nα : ι → Type u_3\nf : ι' ≃ ι\nt : (i : ι) → Set (α i)\n⊢ (⇑(piCongrLeft α f).symm ⁻¹' univ.pi fun i' ↦ t (f i')) = univ.pi t",
"ppTerm": "?m.18",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"ι : Type u_1\nι' : Type u_2\nα : ι → Type u_3\nf : ι' ≃ ι\nt : (i : ι) → Set (α i)\n⊢ (⇑(piCongrLeft α f).symm ⁻¹' univ.pi fun i' ↦ t (f i')) = univ.pi t"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Set.Image | {
"line": 1377,
"column": 2
} | {
"line": 1377,
"column": 13
} | {
"line": 1377,
"column": 14
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nf : α → β\ns : Set β\nh : Disjoint s (range f)\n⊢ f ⁻¹' s = ∅",
"ppTerm": "?m.11",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\nβ : Type u_2\nf : α → β\ns : Set β\nh : Disjoint s (range f)\n⊢ f ⁻¹' s = ∅"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Set.Prod | {
"line": 915,
"column": 2
} | {
"line": 915,
"column": 37
} | {
"line": 915,
"column": 38
} | [
{
"pp": "ι : Type u_1\nι' : Type u_2\nα : ι → Type u_3\nf : ι' ≃ ι\nt : (i : ι) → Set (α i)\n⊢ ⇑(piCongrLeft α f) ⁻¹' univ.pi t = univ.pi fun i ↦ t (f i)",
"ppTerm": "?m.16",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"ι : Type u_1\nι' : Type u_2\nα : ι → Type u_3\nf : ι' ≃ ι\nt : (i : ι) → Set (α i)\n⊢ ⇑(piCongrLeft α f) ⁻¹' univ.pi t = univ.pi fun i ↦ t (f i)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Set.Function | {
"line": 831,
"column": 2
} | {
"line": 831,
"column": 23
} | {
"line": 832,
"column": 2
} | [
{
"pp": "α : 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": "?m.14",
"assigned": true,
"usedConstants": [
"Set.Subset.antisymm",
"Set.instInter",
"Inter.inter",
"Set.preimage",
"Set.im... | [
"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",
"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)"
] | apply Subset.antisymm | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Data.Set.Prod | {
"line": 1024,
"column": 2
} | {
"line": 1024,
"column": 13
} | {
"line": 1024,
"column": 14
} | [
{
"pp": "β : Type u_2\nγ : Type u_3\ns : Set (β × γ)\nhs₁ : Bijective (Prod.fst ∘ Subtype.val)\n⊢ ∃ f, s = graphOn f univ",
"ppTerm": "?m.16",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"β : Type u_2\nγ : Type u_3\ns : Set (β × γ)\nhs₁ : Bijective (Prod.fst ∘ Subtype.val)\n⊢ ∃ f, s = graphOn f univ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Int.Init | {
"line": 86,
"column": 19
} | {
"line": 86,
"column": 78
} | {
"line": 86,
"column": 79
} | [
{
"pp": "case negSucc.succ\nmotive : ℤ → Prop\nzero : motive 0\nsucc : ∀ (i : ℕ), motive ↑i → motive (↑i + 1)\npred : ∀ (i : ℕ), motive (-↑i) → motive (-↑i - 1)\ni n : ℕ\nih : motive (-↑n)\n⊢ motive (-↑(n + 1))",
"ppTerm": "?negSucc.succ",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"... | [
"case negSucc.succ\nmotive : ℤ → Prop\nzero : motive 0\nsucc : ∀ (i : ℕ), motive ↑i → motive (↑i + 1)\npred : ∀ (i : ℕ), motive (-↑i) → motive (-↑i - 1)\ni n : ℕ\nih : motive (-↑n)\n⊢ motive (-↑n + -1)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Set.Function | {
"line": 912,
"column": 4
} | {
"line": 912,
"column": 24
} | {
"line": 912,
"column": 25
} | [
{
"pp": "case mpr\nα : Type u_1\nβ : Type u_2\nf : α → β\ng : β → α\ns : Set α\nhgf : LeftInvOn g f s\nginj : InjOn g (g ⁻¹' s)\nx : β\nhx : x ∈ g ⁻¹' s\ny : α\nhy : y ∈ s\n⊢ f y ∈ g ⁻¹' s",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Membership.me... | [
"case mpr\nα : Type u_1\nβ : Type u_2\nf : α → β\ng : β → α\ns : Set α\nhgf : LeftInvOn g f s\nginj : InjOn g (g ⁻¹' s)\nx : β\nhx : x ∈ g ⁻¹' s\ny : α\nhy : y ∈ s\n⊢ y ∈ s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Int.Init | {
"line": 126,
"column": 4
} | {
"line": 126,
"column": 75
} | {
"line": 127,
"column": 4
} | [
{
"pp": "case pos\nmotive : ℤ → Sort u_1\nz b : ℤ\nzero : motive b\nsucc : (k : ℤ) → b ≤ k → motive k → motive (k + 1)\npred : (k : ℤ) → k ≤ b → motive k → motive (k - 1)\nhz : z ≤ b\nh✝ : z = b\n⊢ cast ⋯\n (match z - 1 - b with\n | ofNat n => inductionOn'.pos b zero succ n\n | -[n+1] =>\n ... | [
"case pos\nmotive : ℤ → Sort u_1\nz b : ℤ\nzero : motive b\nsucc : (k : ℤ) → b ≤ k → motive k → motive (k + 1)\npred : (k : ℤ) → k ≤ b → motive k → motive (k - 1)\nhz : z ≤ b\nh✝ : z = b\n⊢ cast ⋯\n (match -[(b - z).toNat+1] with\n | ofNat n => inductionOn'.pos b zero succ n\n | -[n+1] =>\n ma... | rw! [show z - 1 - b = -[(b - z).toNat+1] by lia, show z - b = 0 by lia] | Mathlib.Tactic.DepRewrite.evalDepRwSeq | Mathlib.Tactic.DepRewrite.depRwSeq |
Mathlib.Data.Set.Function | {
"line": 1095,
"column": 2
} | {
"line": 1095,
"column": 13
} | {
"line": 1095,
"column": 14
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ns : Set α\nt : Set β\nf : α → β\nt' : Set β\nh : BijOn f s t\nhtt' : t ⊆ t'\nht' : t' ⊆ range f\n⊢ ∃ s', s ⊆ s' ∧ BijOn f s' t'",
"ppTerm": "?m.15",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\nβ : Type u_2\ns : Set α\nt : Set β\nf : α → β\nt' : Set β\nh : BijOn f s t\nhtt' : t ⊆ t'\nht' : t' ⊆ range f\n⊢ ∃ s', s ⊆ s' ∧ BijOn f s' t'"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Set.Function | {
"line": 1224,
"column": 2
} | {
"line": 1224,
"column": 41
} | {
"line": 1224,
"column": 42
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nf : α → β\nb : β\nh : range f = {b}\na : α\n⊢ f a = b",
"ppTerm": "?m.10",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\nβ : Type u_2\nf : α → β\nb : β\nh : range f = {b}\na : α\n⊢ f a = b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Group.Hom.Basic | {
"line": 225,
"column": 20
} | {
"line": 225,
"column": 53
} | {
"line": 225,
"column": 54
} | [
{
"pp": "α : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\nG : Type u_5\nH✝ : Type u_6\nF : Type u_7\ninst✝¹ : Group G\nH : Type u_8\ninst✝ : Group H\nf : G → H\nhf : ∀ (x y : G), f (x / y) = f x / f y\n⊢ ∀ (a b : G), f (a * b⁻¹) = f a * (f b)⁻¹",
"ppTerm": "?m.19",
"assigned": false,
"usedCon... | [
"α : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\nG : Type u_5\nH✝ : Type u_6\nF : Type u_7\ninst✝¹ : Group G\nH : Type u_8\ninst✝ : Group H\nf : G → H\nhf : ∀ (x y : G), f (x / y) = f x / f y\n⊢ ∀ (a b : G), f (a * b⁻¹) = f a * (f b)⁻¹"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Group.Basic | {
"line": 99,
"column": 31
} | {
"line": 99,
"column": 42
} | {
"line": 99,
"column": 43
} | [
{
"pp": "case mp\nM : Type u_4\ninst✝ : MulOneClass M\nb : M\nh : 1 * b = 1\n⊢ b = 1",
"ppTerm": "?mp",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"case mp\nM : Type u_4\ninst✝ : MulOneClass M\nb : M\nh : 1 * b = 1\n⊢ b = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Group.Basic | {
"line": 99,
"column": 31
} | {
"line": 99,
"column": 42
} | {
"line": 99,
"column": 43
} | [
{
"pp": "case mpr\nM : Type u_4\ninst✝ : MulOneClass M\na : M\nh : a * 1 = 1\n⊢ a = 1",
"ppTerm": "?mpr",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"case mpr\nM : Type u_4\ninst✝ : MulOneClass M\na : M\nh : a * 1 = 1\n⊢ a = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Group.Basic | {
"line": 375,
"column": 59
} | {
"line": 375,
"column": 70
} | {
"line": 375,
"column": 71
} | [
{
"pp": "α : Type u_1\ninst✝ : DivisionMonoid α\na b : α\nh : a⁻¹ * b = 1\n⊢ a = b",
"ppTerm": "?m.11",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\ninst✝ : DivisionMonoid α\na b : α\nh : a⁻¹ * b = 1\n⊢ a = b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Group.Basic | {
"line": 378,
"column": 59
} | {
"line": 378,
"column": 70
} | {
"line": 378,
"column": 71
} | [
{
"pp": "α : Type u_1\ninst✝ : DivisionMonoid α\na b : α\nh : a * b⁻¹ = 1\n⊢ a = b",
"ppTerm": "?m.11",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\ninst✝ : DivisionMonoid α\na b : α\nh : a * b⁻¹ = 1\n⊢ a = b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Group.Basic | {
"line": 408,
"column": 18
} | {
"line": 408,
"column": 53
} | {
"line": 408,
"column": 54
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nG : Type u_3\nM : Type u_4\ninst✝ : DivisionMonoid α\na b c d : α\n⊢ 1⁻¹ = 1",
"ppTerm": "?m.7",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\nβ : Type u_2\nG : Type u_3\nM : Type u_4\ninst✝ : DivisionMonoid α\na b c d : α\n⊢ 1⁻¹ = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Group.Opposite | {
"line": 239,
"column": 2
} | {
"line": 239,
"column": 34
} | {
"line": 239,
"column": 35
} | [
{
"pp": "α : Type u_1\ninst✝ : MulOne α\n⊢ (∀ (a a_1 : α), opEquiv a * opEquiv a_1 = 1 → opEquiv a_1 * opEquiv a = 1) ↔ ∀ {a b : α}, a * b = 1 → b * a = 1",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"Equiv.instEquivLike",
"HMul.hMul",
... | [
"α : Type u_1\ninst✝ : MulOne α\n⊢ (∀ (a a_1 : α), a_1 * a = 1 → a * a_1 = 1) ↔ ∀ {a b : α}, a * b = 1 → b * a = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Group.Basic | {
"line": 832,
"column": 2
} | {
"line": 832,
"column": 30
} | {
"line": 832,
"column": 31
} | [
{
"pp": "G : Type u_3\ninst✝ : Group G\na : G\nm n : ℕ\n⊢ a ^ (↑m - ↑n) = a ^ m / a ^ n",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"DivInvMonoid.toInv",
"instHDiv",
"HMul.hMul",
"Monoid.toMulOneClass",
"congrArg",
"HSub.hSub",
... | [
"G : Type u_3\ninst✝ : Group G\na : G\nm n : ℕ\n⊢ a ^ (↑m - ↑n) = a ^ m * (a ^ n)⁻¹"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Group.Basic | {
"line": 836,
"column": 2
} | {
"line": 836,
"column": 30
} | {
"line": 836,
"column": 31
} | [
{
"pp": "G : Type u_3\ninst✝ : Group G\na : G\nn : ℕ\n⊢ a ^ (↑n - 1) = a ^ n / a",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"DivInvMonoid.toInv",
"instHDiv",
"HMul.hMul",
"Monoid.toMulOneClass",
"congrArg",
"HSub.hSub",
"DivInv... | [
"G : Type u_3\ninst✝ : Group G\na : G\nn : ℕ\n⊢ a ^ (↑n - 1) = a ^ n * a⁻¹"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Group.InjSurj | {
"line": 80,
"column": 57
} | {
"line": 80,
"column": 79
} | {
"line": 80,
"column": 80
} | [
{
"pp": "M₁ : Type u_1\nM₂ : Type u_2\ninst✝² : Mul M₁\ninst✝¹ : Mul M₂\ninst✝ : IsLeftCancelMul M₂\nf : M₁ → M₂\nhf : Injective f\nmul : ∀ (x y : M₁), f (x * y) = f x * f y\nx y z : M₁\nH : (fun x_1 ↦ x * x_1) y = (fun x_1 ↦ x * x_1) z\n⊢ ?m.31 x y z H * f y = ?m.31 x y z H * f z",
"ppTerm": "?m.30",
"... | [
"M₁ : Type u_1\nM₂ : Type u_2\ninst✝² : Mul M₁\ninst✝¹ : Mul M₂\ninst✝ : IsLeftCancelMul M₂\nf : M₁ → M₂\nhf : Injective f\nmul : ∀ (x y : M₁), f (x * y) = f x * f y\nx y z : M₁\nH : (fun x_1 ↦ x * x_1) y = (fun x_1 ↦ x * x_1) z\n⊢ ?m.31 x y z H * f y = ?m.31 x y z H * f z"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Group.Basic | {
"line": 840,
"column": 2
} | {
"line": 840,
"column": 30
} | {
"line": 840,
"column": 31
} | [
{
"pp": "G : Type u_3\ninst✝ : Group G\na : G\nn : ℕ\n⊢ a ^ (1 - ↑n) = a / a ^ n",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"DivInvMonoid.toInv",
"instHDiv",
"HMul.hMul",
"Monoid.toMulOneClass",
"congrArg",
"HSub.hSub",
"DivInv... | [
"G : Type u_3\ninst✝ : Group G\na : G\nn : ℕ\n⊢ a ^ (1 - ↑n) = a * (a ^ n)⁻¹"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Group.InjSurj | {
"line": 88,
"column": 59
} | {
"line": 88,
"column": 81
} | {
"line": 88,
"column": 82
} | [
{
"pp": "M₁ : Type u_1\nM₂ : Type u_2\ninst✝² : Mul M₁\ninst✝¹ : Mul M₂\ninst✝ : IsRightCancelMul M₂\nf : M₁ → M₂\nhf : Injective f\nmul : ∀ (x y : M₁), f (x * y) = f x * f y\nx y z : M₁\nH : (fun x_1 ↦ x_1 * x) y = (fun x_1 ↦ x_1 * x) z\n⊢ f y * ?m.31 x y z H = f z * ?m.31 x y z H",
"ppTerm": "?m.30",
... | [
"M₁ : Type u_1\nM₂ : Type u_2\ninst✝² : Mul M₁\ninst✝¹ : Mul M₂\ninst✝ : IsRightCancelMul M₂\nf : M₁ → M₂\nhf : Injective f\nmul : ∀ (x y : M₁), f (x * y) = f x * f y\nx y z : M₁\nH : (fun x_1 ↦ x_1 * x) y = (fun x_1 ↦ x_1 * x) z\n⊢ f y * ?m.31 x y z H = f z * ?m.31 x y z H"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Group.Units.Defs | {
"line": 116,
"column": 4
} | {
"line": 116,
"column": 49
} | {
"line": 116,
"column": 50
} | [
{
"pp": "case e_inv\nα : Type u\ninst✝ : Monoid α\nv i₁ : α\nvi₁ : v * i₁ = 1\niv₁ : i₁ * v = 1\ni₂ : α\nvi₂ : v * i₂ = 1\niv₂ : i₂ * v = 1\n⊢ i₁ = i₂",
"ppTerm": "?e_inv",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"case e_inv\nα : Type u\ninst✝ : Monoid α\nv i₁ : α\nvi₁ : v * i₁ = 1\niv₁ : i₁ * v = 1\ni₂ : α\nvi₂ : v * i₂ = 1\niv₂ : i₂ * v = 1\n⊢ i₁ = i₂"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Group.Units.Hom | {
"line": 224,
"column": 2
} | {
"line": 224,
"column": 26
} | {
"line": 224,
"column": 27
} | [
{
"pp": "F : Type u_1\nG : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁴ : FunLike F M N\ninst✝³ : FunLike G N M\ninst✝² : Monoid M\ninst✝¹ : Monoid N\ninst✝ : MonoidHomClass G N M\nf : F\nx : M\ng : G\nhfg : LeftInverse ⇑g ⇑f\nh : IsUnit (f x)\n⊢ IsUnit x",
"ppTerm": "?m.13",
"assigned": false,
"use... | [
"F : Type u_1\nG : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁴ : FunLike F M N\ninst✝³ : FunLike G N M\ninst✝² : Monoid M\ninst✝¹ : Monoid N\ninst✝ : MonoidHomClass G N M\nf : F\nx : M\ng : G\nhfg : LeftInverse ⇑g ⇑f\nh : IsUnit (f x)\n⊢ IsUnit x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Group.Units.Basic | {
"line": 64,
"column": 15
} | {
"line": 64,
"column": 53
} | {
"line": 64,
"column": 54
} | [
{
"pp": "α : Type u\ninst✝ : Monoid α\na : αˣ\nb c : α\nh : ↑a * b = ↑a * c\n⊢ b = c",
"ppTerm": "?m.18",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u\ninst✝ : Monoid α\na : αˣ\nb c : α\nh : ↑a * b = ↑a * c\n⊢ b = c"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Group.Units.Basic | {
"line": 69,
"column": 15
} | {
"line": 69,
"column": 54
} | {
"line": 69,
"column": 55
} | [
{
"pp": "α : Type u\ninst✝ : Monoid α\na : αˣ\nb c : α\nh : b * ↑a = c * ↑a\n⊢ b = c",
"ppTerm": "?m.18",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u\ninst✝ : Monoid α\na : αˣ\nb c : α\nh : b * ↑a = c * ↑a\n⊢ b = c"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Group.Units.Basic | {
"line": 334,
"column": 10
} | {
"line": 334,
"column": 33
} | {
"line": 334,
"column": 34
} | [
{
"pp": "M : Type u_1\ninst✝ : Monoid M\na : M\nh : Bijective fun x ↦ a * x\n⊢ (fun x ↦ a * x) (⋯.choose * a) = (fun x ↦ a * x) 1",
"ppTerm": "?m.49",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"HMul.hMul",
"Monoid.toMulOneClass",
"congrArg",
"... | [
"M : Type u_1\ninst✝ : Monoid M\na : M\nh : Bijective fun x ↦ a * x\n⊢ a * (⋯.choose * a) = a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Group.Units.Basic | {
"line": 340,
"column": 28
} | {
"line": 340,
"column": 51
} | {
"line": 340,
"column": 52
} | [
{
"pp": "M : Type u_1\ninst✝ : Monoid M\na : M\nh : Bijective fun x ↦ x * a\n⊢ (fun x ↦ x * a) (a * ⋯.choose) = (fun x ↦ x * a) 1",
"ppTerm": "?m.43",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"Semigroup.toMul",
"HMul.hMul",
"Monoid.toMulOneClass",
... | [
"M : Type u_1\ninst✝ : Monoid M\na : M\nh : Bijective fun x ↦ x * a\n⊢ a * (⋯.choose * a) = a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Group.Units.Basic | {
"line": 454,
"column": 6
} | {
"line": 454,
"column": 18
} | {
"line": 454,
"column": 19
} | [
{
"pp": "α : Type u\ninst✝ : DivisionCommMonoid α\nb d : α\nhb : IsUnit b\nhd : IsUnit d\na c : α\nh : a / b = c / d\n⊢ a * d = c * b",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"HMul.hMul",
"DivisionCommMonoid.toDivisionMonoid",
... | [
"α : Type u\ninst✝ : DivisionCommMonoid α\nb d : α\nhb : IsUnit b\nhd : IsUnit d\na c : α\nh : a / b = c / d\n⊢ a * 1 * d = c * b"
] | ← mul_one a, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Group.Units.Equiv | {
"line": 210,
"column": 4
} | {
"line": 210,
"column": 15
} | {
"line": 210,
"column": 16
} | [
{
"pp": "F : Type u_1\nM : Type u_3\nN : Type u_4\ninst✝³ : Monoid M\ninst✝² : Monoid N\ninst✝¹ : EquivLike F M N\ninst✝ : MulEquivClass F M N\nf : F\nx : M\nhx : IsUnit (f x)\n⊢ IsUnit x",
"ppTerm": "?m.12",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"F : Type u_1\nM : Type u_3\nN : Type u_4\ninst✝³ : Monoid M\ninst✝² : Monoid N\ninst✝¹ : EquivLike F M N\ninst✝ : MulEquivClass F M N\nf : F\nx : M\nhx : IsUnit (f x)\n⊢ IsUnit x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Group.Prod | {
"line": 253,
"column": 15
} | {
"line": 253,
"column": 70
} | {
"line": 255,
"column": 0
} | [
{
"pp": "M : Type u_3\nN : Type u_4\nP : Type u_5\ninst✝² : Mul M\ninst✝¹ : Mul N\ninst✝ : Mul P\nf : M →ₙ* N × P\nx : M\n⊢ (((fst N P).comp f).prod ((snd N P).comp f)) x = f x",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"MulHom",
"Prod.instMul",
"Prod.mk",
"Pro... | [] | by simp only [prod_apply, coe_fst, coe_snd, comp_apply] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.GroupWithZero.Defs | {
"line": 256,
"column": 55
} | {
"line": 256,
"column": 71
} | {
"line": 256,
"column": 72
} | [
{
"pp": "M₀ : Type u_1\ninst✝¹ : MonoidWithZero M₀\ninst✝ : IsRightCancelMulZero M₀\nx : M₀\nhx : x ^ 2 = x\nx✝ : ¬x = 0\n⊢ (fun a ↦ a * x) x = (fun a ↦ a * x) 1",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"HMul.hMul",
"MulZeroClass.toM... | [
"M₀ : Type u_1\ninst✝¹ : MonoidWithZero M₀\ninst✝ : IsRightCancelMulZero M₀\nx : M₀\nhx : x ^ 2 = x\nx✝ : ¬x = 0\n⊢ x * x = x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.GroupWithZero.NeZero | {
"line": 49,
"column": 2
} | {
"line": 49,
"column": 22
} | {
"line": 49,
"column": 23
} | [
{
"pp": "G₀ : Type u_3\ninst✝ : GroupWithZero G₀\na : G₀\nh : a ≠ 0\na_eq_0 : a⁻¹ = 0\n⊢ False",
"ppTerm": "?m.10",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"G₀ : Type u_3\ninst✝ : GroupWithZero G₀\na : G₀\nh : a ≠ 0\na_eq_0 : a⁻¹ = 0\n⊢ False"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.GroupWithZero.Basic | {
"line": 105,
"column": 6
} | {
"line": 105,
"column": 18
} | {
"line": 105,
"column": 19
} | [
{
"pp": "M₀ : Type u_1\ninst✝ : MulZeroOneClass M₀\nh : 0 = 1\na : M₀\n⊢ a = 0",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"HMul.hMul",
"congrArg",
"id",
"MulOne.toMul",
"MulZeroOneClass.toMulOneClass",
"MulOneCla... | [
"M₀ : Type u_1\ninst✝ : MulZeroOneClass M₀\nh : 0 = 1\na : M₀\n⊢ a * 1 = 0"
] | ← mul_one a, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.GroupWithZero.Basic | {
"line": 211,
"column": 2
} | {
"line": 211,
"column": 51
} | {
"line": 211,
"column": 52
} | [
{
"pp": "R : Type u_5\ninst✝¹ : Zero R\ninst✝ : Pow R ℕ\nh : ¬IsReduced R\n⊢ ∃ x, x ≠ 0 ∧ IsNilpotent x",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Exists",
"id",
"Ne",
"funext",
"And",
"_private.Mathlib.Algebra.Gro... | [
"R : Type u_5\ninst✝¹ : Zero R\ninst✝ : Pow R ℕ\nh : ¬IsReduced R\n⊢ ∃ x, IsNilpotent x ∧ ¬x = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.GroupWithZero.Basic | {
"line": 248,
"column": 14
} | {
"line": 248,
"column": 25
} | {
"line": 248,
"column": 26
} | [
{
"pp": "case zero\nM₀ : Type u_1\nG₀ : Type u_2\ninst✝¹ : MonoidWithZero M₀\na✝ : M₀\nn : ℕ\ninst✝ : NoZeroDivisors M₀\na : M₀\nx✝ : IsNilpotent a\nha : a ^ 0 = 0\n⊢ a = 0",
"ppTerm": "?zero",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"case zero\nM₀ : Type u_1\nG₀ : Type u_2\ninst✝¹ : MonoidWithZero M₀\na✝ : M₀\nn : ℕ\ninst✝ : NoZeroDivisors M₀\na : M₀\nx✝ : IsNilpotent a\nha : a ^ 0 = 0\n⊢ a = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.GroupWithZero.Basic | {
"line": 352,
"column": 2
} | {
"line": 352,
"column": 60
} | {
"line": 352,
"column": 61
} | [
{
"pp": "G₀ : Type u_2\ninst✝ : GroupWithZero G₀\nx : G₀\nh : x ≠ 0\ny y' : G₀\nw : (fun y ↦ x * y) y = (fun y ↦ x * y) y'\n⊢ y = y'",
"ppTerm": "?m.14",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"G₀ : Type u_2\ninst✝ : GroupWithZero G₀\nx : G₀\nh : x ≠ 0\ny y' : G₀\nw : (fun y ↦ x * y) y = (fun y ↦ x * y) y'\n⊢ y = y'"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.GroupWithZero.Basic | {
"line": 356,
"column": 2
} | {
"line": 356,
"column": 58
} | {
"line": 356,
"column": 59
} | [
{
"pp": "G₀ : Type u_2\ninst✝ : GroupWithZero G₀\nx : G₀\nh : x ≠ 0\ny y' : G₀\nw : (fun y ↦ y * x) y = (fun y ↦ y * x) y'\n⊢ y = y'",
"ppTerm": "?m.14",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"G₀ : Type u_2\ninst✝ : GroupWithZero G₀\nx : G₀\nh : x ≠ 0\ny y' : G₀\nw : (fun y ↦ y * x) y = (fun y ↦ y * x) y'\n⊢ y = y'"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.GroupWithZero.Basic | {
"line": 456,
"column": 2
} | {
"line": 456,
"column": 28
} | {
"line": 456,
"column": 29
} | [
{
"pp": "G₀ : Type u_2\ninst✝ : GroupWithZero G₀\na : G₀\nh : a ≠ 0\n⊢ 1 / a ≠ 0",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"DivInvMonoid.toInv",
"instHDiv",
"GroupWithZero.toDivisionMonoid",
"InvOneClass.... | [
"G₀ : Type u_2\ninst✝ : GroupWithZero G₀\na : G₀\nh : a ≠ 0\n⊢ a⁻¹ ≠ 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.GroupWithZero.Basic | {
"line": 469,
"column": 6
} | {
"line": 469,
"column": 24
} | {
"line": 469,
"column": 24
} | [
{
"pp": "G₀ : Type u_2\ninst✝ : GroupWithZero G₀\na : G₀\n⊢ a / (a / a) = a",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instHDiv",
"GroupWithZero.toDivisionMonoid",
"HMul.hMul",
"GroupWithZero.toDivInvMonoid",
"Monoid.toMulOneClass",
... | [
"G₀ : Type u_2\ninst✝ : GroupWithZero G₀\na : G₀\n⊢ a * a / a = a"
] | div_div_eq_mul_div | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.GroupWithZero.Basic | {
"line": 486,
"column": 50
} | {
"line": 486,
"column": 83
} | {
"line": 486,
"column": 84
} | [
{
"pp": "G₀ : Type u_2\ninst✝ : GroupWithZero G₀\nn : ℕ\nh : ↑n ≠ 0\n⊢ n ≠ 0",
"ppTerm": "?m.48",
"assigned": true,
"usedConstants": [
"id",
"Ne",
"instOfNatNat",
"Nat",
"OfNat.ofNat"
],
"usedFVars": [
"n"
],
"usedGoals": [
{
"new": t... | [
"G₀ : Type u_2\ninst✝ : GroupWithZero G₀\nn : ℕ\nh : ↑n ≠ 0\n⊢ ¬n = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Group.End | {
"line": 218,
"column": 4
} | {
"line": 218,
"column": 15
} | {
"line": 218,
"column": 16
} | [
{
"pp": "case left\nα : Type u_7\nβ : Type u_8\nfst✝¹ : Perm α\nsnd✝¹ : Perm β\nfst✝ : Perm α\nsnd✝ : Perm β\nh : (sumCongrHom α β) (fst✝¹, snd✝¹) = (sumCongrHom α β) (fst✝, snd✝)\ni : α\n⊢ fst✝¹ i = fst✝ i",
"ppTerm": "?left",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGo... | [
"case left\nα : Type u_7\nβ : Type u_8\nfst✝¹ : Perm α\nsnd✝¹ : Perm β\nfst✝ : Perm α\nsnd✝ : Perm β\nh : (sumCongrHom α β) (fst✝¹, snd✝¹) = (sumCongrHom α β) (fst✝, snd✝)\ni : α\n⊢ fst✝¹ i = fst✝ i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Group.End | {
"line": 219,
"column": 4
} | {
"line": 219,
"column": 15
} | {
"line": 219,
"column": 16
} | [
{
"pp": "case right\nα : Type u_7\nβ : Type u_8\nfst✝¹ : Perm α\nsnd✝¹ : Perm β\nfst✝ : Perm α\nsnd✝ : Perm β\nh : (sumCongrHom α β) (fst✝¹, snd✝¹) = (sumCongrHom α β) (fst✝, snd✝)\ni : β\n⊢ snd✝¹ i = snd✝ i",
"ppTerm": "?right",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"used... | [
"case right\nα : Type u_7\nβ : Type u_8\nfst✝¹ : Perm α\nsnd✝¹ : Perm β\nfst✝ : Perm α\nsnd✝ : Perm β\nh : (sumCongrHom α β) (fst✝¹, snd✝¹) = (sumCongrHom α β) (fst✝, snd✝)\ni : β\n⊢ snd✝¹ i = snd✝ i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Group.End | {
"line": 263,
"column": 2
} | {
"line": 263,
"column": 13
} | {
"line": 263,
"column": 14
} | [
{
"pp": "α : Type u_7\nβ : α → Type u_8\nx y : (a : α) → Perm (β a)\nh : (sigmaCongrRightHom β) x = (sigmaCongrRightHom β) y\na : α\nb : β a\n⊢ (x a) b = (y a) b",
"ppTerm": "?m.23",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_7\nβ : α → Type u_8\nx y : (a : α) → Perm (β a)\nh : (sigmaCongrRightHom β) x = (sigmaCongrRightHom β) y\na : α\nb : β a\n⊢ (x a) b = (y a) b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Group.End | {
"line": 277,
"column": 28
} | {
"line": 277,
"column": 39
} | {
"line": 277,
"column": 40
} | [
{
"pp": "case left\nα : Type u_4\np : α → Prop\ninst✝ : DecidablePred p\nfst✝¹ : Perm { a // p a }\nsnd✝¹ : Perm { a // ¬p a }\nfst✝ : Perm { a // p a }\nsnd✝ : Perm { a // ¬p a }\nh : (subtypeCongrHom p) (fst✝¹, snd✝¹) = (subtypeCongrHom p) (fst✝, snd✝)\ni : { a // p a }\n⊢ ↑(fst✝¹ i) = ↑(fst✝ i)",
"ppTerm... | [
"case left\nα : Type u_4\np : α → Prop\ninst✝ : DecidablePred p\nfst✝¹ : Perm { a // p a }\nsnd✝¹ : Perm { a // ¬p a }\nfst✝ : Perm { a // p a }\nsnd✝ : Perm { a // ¬p a }\nh : (subtypeCongrHom p) (fst✝¹, snd✝¹) = (subtypeCongrHom p) (fst✝, snd✝)\ni : { a // p a }\n⊢ ↑(fst✝¹ i) = ↑(fst✝ i)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Group.End | {
"line": 277,
"column": 28
} | {
"line": 277,
"column": 39
} | {
"line": 277,
"column": 40
} | [
{
"pp": "case right\nα : Type u_4\np : α → Prop\ninst✝ : DecidablePred p\nfst✝¹ : Perm { a // p a }\nsnd✝¹ : Perm { a // ¬p a }\nfst✝ : Perm { a // p a }\nsnd✝ : Perm { a // ¬p a }\nh : (subtypeCongrHom p) (fst✝¹, snd✝¹) = (subtypeCongrHom p) (fst✝, snd✝)\ni : { a // ¬p a }\n⊢ ↑(snd✝¹ i) = ↑(snd✝ i)",
"ppTe... | [
"case right\nα : Type u_4\np : α → Prop\ninst✝ : DecidablePred p\nfst✝¹ : Perm { a // p a }\nsnd✝¹ : Perm { a // ¬p a }\nfst✝ : Perm { a // p a }\nsnd✝ : Perm { a // ¬p a }\nh : (subtypeCongrHom p) (fst✝¹, snd✝¹) = (subtypeCongrHom p) (fst✝, snd✝)\ni : { a // ¬p a }\n⊢ ↑(snd✝¹ i) = ↑(snd✝ i)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Group.End | {
"line": 370,
"column": 49
} | {
"line": 370,
"column": 60
} | {
"line": 370,
"column": 61
} | [
{
"pp": "A : Type u_1\nM : Type u_2\nG : Type u_3\nα : Type u_4\nβ : Type u_5\nγ : Type u_6\np : α → Prop\nf✝ f : Perm α\nh : ∀ (x : α), p (f x) ↔ p x\nx : { x // p x }\n⊢ p (f (f⁻¹ ↑x))",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Equiv.apply_symm_apply",
"... | [
"A : Type u_1\nM : Type u_2\nG : Type u_3\nα : Type u_4\nβ : Type u_5\nγ : Type u_6\np : α → Prop\nf✝ f : Perm α\nh : ∀ (x : α), p (f x) ↔ p x\nx : { x // p x }\n⊢ p ↑x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Group.End | {
"line": 469,
"column": 4
} | {
"line": 469,
"column": 80
} | {
"line": 469,
"column": 81
} | [
{
"pp": "α : Type u_4\np : α → Prop\ninst✝ : DecidablePred p\nf : Perm (Subtype p)\nx : α\nh : p x\n⊢ p ((ofSubtype f) x) ↔ p x",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MonoidHom.instFunLike",
"Equiv.instEquivLike",
"MonoidHom",
"Monoid.toMul... | [
"α : Type u_4\np : α → Prop\ninst✝ : DecidablePred p\nf : Perm (Subtype p)\nx : α\nh : p x\n⊢ p ↑(f ⟨x, h⟩)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.GroupWithZero.Action.Defs | {
"line": 157,
"column": 4
} | {
"line": 157,
"column": 17
} | {
"line": 158,
"column": 2
} | [
{
"pp": "M : Type u_1\nM₀ : Type u_2\nM₀' : Type u_3\nG₀ : Type u_4\nG₀' : Type u_5\nN : Type u_6\nA : Type u_7\nA' : Type u_8\nB : Type u_9\nα : Type u_10\nβ : Type u_11\ninst✝⁵ : Zero M₀\ninst✝⁴ : Zero A\ninst✝³ : SMulWithZero M₀ A\na : M₀\nb : A\ninst✝² : Zero M₀'\ninst✝¹ : Zero A'\ninst✝ : SMul M₀ A'\nf : Z... | [] | simp [← smul] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.GroupWithZero.Action.Defs | {
"line": 226,
"column": 51
} | {
"line": 226,
"column": 63
} | {
"line": 226,
"column": 63
} | [
{
"pp": "M₀ : Type u_2\nA : Type u_7\ninst✝³ : MonoidWithZero M₀\ninst✝² : Zero A\ninst✝¹ : MulActionWithZero M₀ A\nι : Type u_12\ninst✝ : DecidableEq ι\nx : A\ni j : ι\n⊢ (if j = i then x else 0) = single i x j",
"ppTerm": "?m.48",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrAr... | [
"M₀ : Type u_2\nA : Type u_7\ninst✝³ : MonoidWithZero M₀\ninst✝² : Zero A\ninst✝¹ : MulActionWithZero M₀ A\nι : Type u_12\ninst✝ : DecidableEq ι\nx : A\ni j : ι\n⊢ (if j = i then x else 0) = if j = i then x else 0"
] | single_apply | Lean.Elab.Tactic.evalRewriteSeq | null |
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