module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Algebra.Group.Submonoid.BigOperators | {
"line": 176,
"column": 16
} | {
"line": 176,
"column": 20
} | {
"line": 177,
"column": 4
} | [
{
"pp": "case mul.e_a\nM : Type u_1\ninst✝ : CommMonoid M\nx : M\ns : Set M\nf : M → ℕ\nt : Finset M\nhts : ↑t ⊆ s\nhf : Function.support f ⊆ ↑t\nhx✝ : ∏ a ∈ t, a ^ f a ∈ closure s\ng : M → ℕ\nu : Finset M\nhus : ↑u ⊆ s\nhg : Function.support g ⊆ ↑u\nhy✝ : ∏ a ∈ u, a ^ g a ∈ closure s\n⊢ ∏ x ∈ t ∪ u, x ^ f x = ... | [
"case mul.e_a\nM : Type u_1\ninst✝ : CommMonoid M\nx : M\ns : Set M\nf : M → ℕ\nt : Finset M\nhts : ↑t ⊆ s\nhf : Function.support f ⊆ ↑t\nhx✝ : ∏ a ∈ t, a ^ f a ∈ closure s\ng : M → ℕ\nu : Finset M\nhus : ↑u ⊆ s\nhg : Function.support g ⊆ ↑u\nhy✝ : ∏ a ∈ u, a ^ g a ∈ closure s\n⊢ ∏ x ∈ t, x ^ f x = ∏ x ∈ t ∪ u, x ^... | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.Algebra.Group.Submonoid.BigOperators | {
"line": 176,
"column": 16
} | {
"line": 176,
"column": 20
} | {
"line": 177,
"column": 4
} | [
{
"pp": "case mul.e_a\nM : Type u_1\ninst✝ : CommMonoid M\nx : M\ns : Set M\nf : M → ℕ\nt : Finset M\nhts : ↑t ⊆ s\nhf : Function.support f ⊆ ↑t\nhx✝ : ∏ a ∈ t, a ^ f a ∈ closure s\ng : M → ℕ\nu : Finset M\nhus : ↑u ⊆ s\nhg : Function.support g ⊆ ↑u\nhy✝ : ∏ a ∈ u, a ^ g a ∈ closure s\n⊢ ∏ x ∈ t ∪ u, x ^ g x = ... | [
"case mul.e_a\nM : Type u_1\ninst✝ : CommMonoid M\nx : M\ns : Set M\nf : M → ℕ\nt : Finset M\nhts : ↑t ⊆ s\nhf : Function.support f ⊆ ↑t\nhx✝ : ∏ a ∈ t, a ^ f a ∈ closure s\ng : M → ℕ\nu : Finset M\nhus : ↑u ⊆ s\nhg : Function.support g ⊆ ↑u\nhy✝ : ∏ a ∈ u, a ^ g a ∈ closure s\n⊢ ∏ x ∈ u, x ^ g x = ∏ x ∈ t ∪ u, x ^... | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.Algebra.Group.Submonoid.BigOperators | {
"line": 182,
"column": 4
} | {
"line": 182,
"column": 30
} | {
"line": 182,
"column": 30
} | [
{
"pp": "M : Type u_1\ninst✝ : CommMonoid M\nx : M\ns : Set M\n⊢ (∃ f t, ↑t ⊆ s ∧ Function.support f ⊆ ↑t ∧ ∏ a ∈ t, a ^ f a = x) → x ∈ closure s",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"Monoid.toMulOneClass",
"Finset",
"AddMonoid.toAddZeroClass",
"Nat.instA... | [
"M : Type u_1\ninst✝ : CommMonoid M\ns : Set M\nn : M → ℕ\nt : Finset M\nhts : ↑t ⊆ s\n⊢ ∏ a ∈ t, a ^ n a ∈ closure s"
] | rintro ⟨n, t, hts, -, rfl⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.Algebra.Group.Subgroup.Pointwise | {
"line": 380,
"column": 32
} | {
"line": 380,
"column": 43
} | {
"line": 380,
"column": 44
} | [
{
"pp": "α : Type u_1\nG : Type u_2\nA : Type u_3\nS : Type u_4\ninst✝² : Group G\ninst✝¹ : AddGroup A\ns : Set G\nι : Sort u_5\nH : ι → Subgroup G\ninst✝ : ∀ (i : ι), (H i).Normal\n⊢ normalizer ↑(⨆ i, H i) = ⊤",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"eq_top_i... | [
"α : Type u_1\nG : Type u_2\nA : Type u_3\nS : Type u_4\ninst✝² : Group G\ninst✝¹ : AddGroup A\ns : Set G\nι : Sort u_5\nH : ι → Subgroup G\ninst✝ : ∀ (i : ι), (H i).Normal\n⊢ ⊤ ≤ normalizer ↑(⨆ i, H i)"
] | eq_top_iff, | Mathlib.Tactic.GRewrite.evalGRewriteSeq | null |
Mathlib.Data.Finset.NoncommProd | {
"line": 411,
"column": 4
} | {
"line": 411,
"column": 42
} | {
"line": 412,
"column": 2
} | [
{
"pp": "ι : Type u_2\nM : ι → Type u_6\ninst✝² : (i : ι) → Monoid (M i)\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\nx : (i : ι) → M i\ni✝ i : ι\na✝² : i ∈ ↑univ\nj : ι\na✝¹ : j ∈ ↑univ\na✝ : i ≠ j\n⊢ (Commute on fun i ↦ (MonoidHom.mulSingle M i) (x i)) i j",
"ppTerm": "?m.64",
"assigned": true,
"us... | [] | exact Pi.mulSingle_apply_commute x i j | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Data.Finset.NoncommProd | {
"line": 418,
"column": 6
} | {
"line": 418,
"column": 38
} | {
"line": 419,
"column": 4
} | [
{
"pp": "case convert_8\nι : Type u_2\nM : ι → Type u_6\ninst✝² : (i : ι) → Monoid (M i)\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\nx : (i : ι) → M i\ni : ι\n⊢ Pi.mulSingle i (x i) i = x i",
"ppTerm": "?convert_8",
"assigned": true,
"usedConstants": [
"MulOne.toOne",
"Pi.mulSingle_eq_sa... | [] | simp only [Pi.mulSingle_eq_same] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Data.Finset.NoncommProd | {
"line": 418,
"column": 6
} | {
"line": 418,
"column": 38
} | {
"line": 419,
"column": 4
} | [
{
"pp": "case convert_8\nι : Type u_2\nM : ι → Type u_6\ninst✝² : (i : ι) → Monoid (M i)\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\nx : (i : ι) → M i\ni : ι\n⊢ Pi.mulSingle i (x i) i = x i",
"ppTerm": "?convert_8",
"assigned": true,
"usedConstants": [
"MulOne.toOne",
"Pi.mulSingle_eq_sa... | [] | simp only [Pi.mulSingle_eq_same] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Finset.NoncommProd | {
"line": 418,
"column": 6
} | {
"line": 418,
"column": 38
} | {
"line": 419,
"column": 4
} | [
{
"pp": "case convert_8\nι : Type u_2\nM : ι → Type u_6\ninst✝² : (i : ι) → Monoid (M i)\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\nx : (i : ι) → M i\ni : ι\n⊢ Pi.mulSingle i (x i) i = x i",
"ppTerm": "?convert_8",
"assigned": true,
"usedConstants": [
"MulOne.toOne",
"Pi.mulSingle_eq_sa... | [] | simp only [Pi.mulSingle_eq_same] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.BigOperators.Group.Finset.Basic | {
"line": 141,
"column": 26
} | {
"line": 141,
"column": 58
} | {
"line": 141,
"column": 58
} | [
{
"pp": "ι : Type u_1\nM : Type u_4\ns₁ s₂ : Finset ι\ninst✝¹ : CommMonoid M\nf : ι → M\ninst✝ : DecidableEq ι\nh : Disjoint s₁ s₂\n⊢ ∏ x ∈ s₁ ∪ s₂, f x = (∏ x ∈ s₁ ∪ s₂, f x) * ∏ x ∈ s₁ ∩ s₂, f x",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"Fin... | [
"ι : Type u_1\nM : Type u_4\ns₁ s₂ : Finset ι\ninst✝¹ : CommMonoid M\nf : ι → M\ninst✝ : DecidableEq ι\nh : Disjoint s₁ s₂\n⊢ ∏ x ∈ s₁ ∪ s₂, f x = (∏ x ∈ s₁ ∪ s₂, f x) * ∏ x ∈ ∅, f x"
] | disjoint_iff_inter_eq_empty.mp h | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.BigOperators.Group.Finset.Basic | {
"line": 240,
"column": 70
} | {
"line": 244,
"column": 80
} | {
"line": 246,
"column": 0
} | [
{
"pp": "ι : Type u_1\nκ : Type u_2\nM : Type u_4\ninst✝¹ : CommMonoid M\ninst✝ : DecidableEq κ\ns : Finset ι\nt : Finset κ\ng : ι → κ\nf : ι → M\n⊢ ∏ j ∈ t, ∏ i ∈ s with g i = j, f i = ∏ i ∈ s with g i ∈ t, f i",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Finset.... | [] | by
rw [← prod_disjiUnion, disjiUnion_filter_eq]
#adaptation_note /-- 2025-09-12 (kmill) copied from private lemma pairwiseDisjoint_fibers -/
intro x' hx y' hy hne
simp_rw [disjoint_left, mem_filter]; rintro i ⟨_, rfl⟩ ⟨_, rfl⟩; exact hne rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.GroupTheory.FreeGroup.Basic | {
"line": 294,
"column": 2
} | {
"line": 294,
"column": 51
} | {
"line": 295,
"column": 2
} | [
{
"pp": "case not\nα : Type u\nL : List (α × Bool)\nx1 : α\nb1 : Bool\nx2 : α\nb2 : Bool\nh : (x1, b1) ≠ (x2, b2)\nL₁✝ L₂✝ : List (α × Bool)\nx✝ : α\nb✝ : Bool\neq : L₁✝ = [] ∧ (x✝ = x1 ∧ b✝ = !b1) ∧ (x✝ = x2 ∧ (!b✝) = b2) ∧ L₂✝ = []\n⊢ False",
"ppTerm": "?not",
"assigned": true,
"usedConstants": [
... | [
"case not\nα : Type u\nL : List (α × Bool)\nb1 : Bool\nx✝ : α\nh : (x✝, b1) ≠ (x✝, !!b1)\n⊢ False"
] | rcases eq with ⟨rfl, ⟨rfl, rfl⟩, ⟨rfl, rfl⟩, rfl⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.GroupTheory.FreeGroup.Basic | {
"line": 318,
"column": 56
} | {
"line": 319,
"column": 15
} | {
"line": 321,
"column": 0
} | [
{
"pp": "α : Type u\nL₁ L₂ : List (α × Bool)\nH : Step L₁ L₂\n⊢ L₂ <+ L₁",
"ppTerm": "?m.3",
"assigned": true,
"usedConstants": [
"FreeGroup.Red.Step.casesOn",
"FreeGroup.Red.Step",
"List.append_sublist_append_left._simp_1",
"Bool.not",
"HEq.refl",
"FreeGroup.Red.... | [] | by
cases H; simp | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.GroupTheory.Finiteness | {
"line": 237,
"column": 6
} | {
"line": 237,
"column": 29
} | {
"line": 237,
"column": 29
} | [
{
"pp": "case h\nM : Type u_1\ninst✝² : Monoid M\nM' : Type u_3\ninst✝¹ : Monoid M'\ninst✝ : FG M\nf : M →* M'\nhf : Function.Surjective ⇑f\ns : Finset M\nhs : Submonoid.closure ↑s = ⊤\n⊢ MonoidHom.mrange f = ⊤",
"ppTerm": "?h",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MonoidHom.i... | [
"case h\nM : Type u_1\ninst✝² : Monoid M\nM' : Type u_3\ninst✝¹ : Monoid M'\ninst✝ : FG M\nf : M →* M'\nhf : Function.Surjective ⇑f\ns : Finset M\nhs : Submonoid.closure ↑s = ⊤\n⊢ Function.Surjective ⇑f"
] | MonoidHom.mrange_eq_top | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.FreeGroup.Basic | {
"line": 416,
"column": 4
} | {
"line": 416,
"column": 8
} | {
"line": 417,
"column": 4
} | [
{
"pp": "α : Type u\na₁ : α\nb₁ b₂ : Bool\nL₁ : List (α × Bool)\nhL₁ : ∀ (L₂ : List (α × Bool)), ¬Red.Step ((a₁, b₁) :: ((a₁, b₁).1, b₂) :: L₁) L₂\n⊢ (a₁, b₁).2 = ((a₁, b₁).1, b₂).2",
"ppTerm": "?m.103",
"assigned": true,
"usedConstants": [
"Prod.mk",
"Prod.fst",
"Bool",
"Eq.... | [
"α : Type u\na₁ : α\nb₁ b₂ : Bool\nL₁ : List (α × Bool)\nhL₁ : ∀ (L₂ : List (α × Bool)), ¬Red.Step ((a₁, b₁) :: ((a₁, b₁).1, b₂) :: L₁) L₂\n⊢ ((a₁, b₁).1, b₂).2 = (a₁, b₁).2"
] | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.GroupTheory.Finiteness | {
"line": 550,
"column": 4
} | {
"line": 551,
"column": 39
} | {
"line": 553,
"column": 0
} | [
{
"pp": "M✝ : Type u_1\nN : Type u_2\ninst✝⁵ : Monoid M✝\nG : Type u_3\nH : Type u_4\ninst✝⁴ : Group G\ninst✝³ : AddGroup H\nι : Type u_5\ninst✝² : Finite ι\nM : ι → Type u_6\ninst✝¹ : (i : ι) → Monoid (M i)\ninst✝ : ∀ (i : ι), Monoid.FG (M i)\n⊢ ⊤.FG",
"ppTerm": "?m.8",
"assigned": true,
"usedConst... | [] | rw [← Submonoid.pi_top Set.univ]
exact .pi fun i => Monoid.FG.fg_top | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.Finiteness | {
"line": 550,
"column": 4
} | {
"line": 551,
"column": 39
} | {
"line": 553,
"column": 0
} | [
{
"pp": "M✝ : Type u_1\nN : Type u_2\ninst✝⁵ : Monoid M✝\nG : Type u_3\nH : Type u_4\ninst✝⁴ : Group G\ninst✝³ : AddGroup H\nι : Type u_5\ninst✝² : Finite ι\nM : ι → Type u_6\ninst✝¹ : (i : ι) → Monoid (M i)\ninst✝ : ∀ (i : ι), Monoid.FG (M i)\n⊢ ⊤.FG",
"ppTerm": "?m.8",
"assigned": true,
"usedConst... | [] | rw [← Submonoid.pi_top Set.univ]
exact .pi fun i => Monoid.FG.fg_top | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.QuotientGroup.Defs | {
"line": 147,
"column": 2
} | {
"line": 148,
"column": 38
} | {
"line": 151,
"column": 0
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nN : Subgroup G\nnN : N.Normal\nx y : G\n⊢ ↑x = ↑y ↔ x / y ∈ N",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"DivInvMonoid.toInv",
"instHDiv",
"Subgroup.Normal.mem_comm_iff",
"HMul.hMul",
"DivInvOneMonoi... | [] | refine eq_comm.trans (QuotientGroup.eq.trans ?_)
rw [nN.mem_comm_iff, div_eq_mul_inv] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.QuotientGroup.Defs | {
"line": 147,
"column": 2
} | {
"line": 148,
"column": 38
} | {
"line": 151,
"column": 0
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nN : Subgroup G\nnN : N.Normal\nx y : G\n⊢ ↑x = ↑y ↔ x / y ∈ N",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"DivInvMonoid.toInv",
"instHDiv",
"Subgroup.Normal.mem_comm_iff",
"HMul.hMul",
"DivInvOneMonoi... | [] | refine eq_comm.trans (QuotientGroup.eq.trans ?_)
rw [nN.mem_comm_iff, div_eq_mul_inv] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.FreeGroup.Basic | {
"line": 716,
"column": 2
} | {
"line": 716,
"column": 59
} | {
"line": 717,
"column": 2
} | [
{
"pp": "α : Type u\nβ : Type v\ninst✝ : Group β\nf : α → β\n⊢ (lift f).range = Subgroup.closure (Set.range f)",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"MonoidHom.range",
"Equiv.instEquivLike",
"Subgroup.closure",
"MonoidHom",
"Monoid.toMulOneClass",
... | [
"α : Type u\nβ : Type v\ninst✝ : Group β\nf : α → β\n⊢ Subgroup.closure (Set.range f) ≤ (lift f).range"
] | apply le_antisymm (range_lift_le Subgroup.subset_closure) | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.GroupTheory.Commutator.Basic | {
"line": 434,
"column": 35
} | {
"line": 434,
"column": 54
} | {
"line": 434,
"column": 55
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\n⊢ ⁅centralizer ↑(commutator G), centralizer ↑(commutator G)⁆ ≤ centralizer Set.univ",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Set.univ",
"Subgroup.centralizer",
"PartialOrder.toPreorder",
... | [
"G : Type u_1\ninst✝ : Group G\n⊢ ⁅centralizer ↑(commutator G), centralizer ↑(commutator G)⁆ ≤ centralizer ↑⊤"
] | ← Subgroup.coe_top, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.OreLocalization.Basic | {
"line": 233,
"column": 2
} | {
"line": 233,
"column": 6
} | {
"line": 233,
"column": 6
} | [
{
"pp": "R : Type u_1\ninst✝² : Monoid R\nS : Submonoid R\ninst✝¹ : OreSet S\nX : Type u_2\ninst✝ : MulAction R X\nr₁ : R\nr₂ : X\ns₁ s₂ u : ↥S\nv : R\nhuv : ↑u * r₁ = v * ↑s₂\nv₀ : R := oreNum r₁ s₂\nu₀ : ↥S := oreDenom r₁ s₂\nh₀ : ↑u₀ * r₁ = v₀ * ↑s₂\nr₃ : R\ns₃ : ↥S\nh₃ : ↑s₃ * ↑u₀ = r₃ * ↑u\nthis : r₃ * v *... | [
"R : Type u_1\ninst✝² : Monoid R\nS : Submonoid R\ninst✝¹ : OreSet S\nX : Type u_2\ninst✝ : MulAction R X\nr₁ : R\nr₂ : X\ns₁ s₂ u : ↥S\nv : R\nhuv : ↑u * r₁ = v * ↑s₂\nv₀ : R := ⋯\nu₀ : ↥S := ⋯\nh₀ : ↑u₀ * r₁ = v₀ * ↑s₂\nr₃ : R\ns₃ : ↥S\nh₃ : ↑s₃ * ↑u₀ = r₃ * ↑u\nthis : r₃ * v * ↑s₂ = ↑s₃ * v₀ * ↑s₂\ns₄ : ↥S\nhs₄ ... | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.GroupTheory.MonoidLocalization.Basic | {
"line": 332,
"column": 2
} | {
"line": 332,
"column": 6
} | {
"line": 333,
"column": 2
} | [
{
"pp": "M : Type u_1\ninst✝ : CommMonoid M\nS : Submonoid M\na : ↥S\n⊢ mk (↑a) a = 1",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"MulOne.toOne",
"Localization.mk",
"Monoid.toMulOneClass",
"Membership.mem",
"Localization",
"MulOneClass.toMulOne",
... | [
"M : Type u_1\ninst✝ : CommMonoid M\nS : Submonoid M\na : ↥S\n⊢ 1 = mk (↑a) a"
] | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.GroupTheory.OreLocalization.Basic | {
"line": 393,
"column": 2
} | {
"line": 393,
"column": 6
} | {
"line": 393,
"column": 6
} | [
{
"pp": "case c.c.c\nR : Type u_1\ninst✝² : Monoid R\nS : Submonoid R\ninst✝¹ : OreSet S\nX : Type u_2\ninst✝ : MulAction R X\nr₁ : R\ns₁ : ↥S\nr₂ : R\ns₂ : ↥S\nr₃ : X\ns₃ : ↥S\nra : R\nsa : ↥S\nha : ↑sa * r₁ = ra * ↑s₂\nrb : R\nsb : ↥S\nhb : ↑sb * r₂ = rb * ↑s₃\nrc : R\nsc : ↥S\nhc : ↑sc * ra = rc * ↑sb\n⊢ rc ... | [
"case c.c.c\nR : Type u_1\ninst✝² : Monoid R\nS : Submonoid R\ninst✝¹ : OreSet S\nX : Type u_2\ninst✝ : MulAction R X\nr₁ : R\ns₁ : ↥S\nr₂ : R\ns₂ : ↥S\nr₃ : X\ns₃ : ↥S\nra : R\nsa : ↥S\nha : ↑sa * r₁ = ra * ↑s₂\nrb : R\nsb : ↥S\nhb : ↑sb * r₂ = rb * ↑s₃\nrc : R\nsc : ↥S\nhc : ↑sc * ra = rc * ↑sb\n⊢ (r₁ /ₒ s₁) • (r... | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.Order.SuccPred.Basic | {
"line": 801,
"column": 6
} | {
"line": 801,
"column": 34
} | {
"line": 802,
"column": 4
} | [
{
"pp": "case top\nα : Type u_1\nβ : Type u_2\ninst✝² : Preorder α\ninst✝¹ : OrderTop α\ninst✝ : PredOrder α\nb : WithTop α\nh : ⊤ < b\n⊢ ⊤ ≤\n match b with\n | none => ↑⊤\n | Option.some a => ↑(pred a)",
"ppTerm": "?top",
"assigned": true,
"usedConstants": [
"WithTop.instPreorder",
... | [] | exact (le_top.not_gt h).elim | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Order.SuccPred.Basic | {
"line": 801,
"column": 6
} | {
"line": 801,
"column": 34
} | {
"line": 802,
"column": 4
} | [
{
"pp": "case top\nα : Type u_1\nβ : Type u_2\ninst✝² : Preorder α\ninst✝¹ : OrderTop α\ninst✝ : PredOrder α\nb : WithTop α\nh : ⊤ < b\n⊢ ⊤ ≤\n match b with\n | none => ↑⊤\n | Option.some a => ↑(pred a)",
"ppTerm": "?top",
"assigned": true,
"usedConstants": [
"WithTop.instPreorder",
... | [] | exact (le_top.not_gt h).elim | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.SuccPred.Basic | {
"line": 801,
"column": 6
} | {
"line": 801,
"column": 34
} | {
"line": 802,
"column": 4
} | [
{
"pp": "case top\nα : Type u_1\nβ : Type u_2\ninst✝² : Preorder α\ninst✝¹ : OrderTop α\ninst✝ : PredOrder α\nb : WithTop α\nh : ⊤ < b\n⊢ ⊤ ≤\n match b with\n | none => ↑⊤\n | Option.some a => ↑(pred a)",
"ppTerm": "?top",
"assigned": true,
"usedConstants": [
"WithTop.instPreorder",
... | [] | exact (le_top.not_gt h).elim | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.SuccPred.Archimedean | {
"line": 268,
"column": 6
} | {
"line": 268,
"column": 21
} | {
"line": 268,
"column": 21
} | [
{
"pp": "case neg.refine_2\nX : Type u_3\ninst✝² : LinearOrder X\ninst✝¹ : SuccOrder X\ninst✝ : IsSuccArchimedean X\nS : Set X\nm✝ n : X\nhn' : n ≤ m✝\nm : X\nhmn✝ : n ≤ m\nIH : m ∈ upperBounds S → n ∈ S → m ∉ S → ∃ x, IsGreatest S x\nhm : succ m ∈ upperBounds S\nhn : n ∈ S\nhm' : succ m ∉ S\n⊢ ∃ x, IsGreatest ... | [
"case neg.refine_2\nX : Type u_3\ninst✝² : LinearOrder X\ninst✝¹ : SuccOrder X\ninst✝ : IsSuccArchimedean X\nS : Set X\nm✝ n : X\nhn' : n ≤ m✝\nm : X\nhmn✝ : n ≤ m\nIH : (∀ x ∈ S, x ≤ m) → n ∈ S → m ∉ S → ∃ x, IsGreatest S x\nhm : ∀ x ∈ S, x ≤ succ m\nhn : n ∈ S\nhm' : succ m ∉ S\n⊢ ∃ x, IsGreatest S x"
] | mem_upperBounds | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.ENat.Basic | {
"line": 237,
"column": 21
} | {
"line": 237,
"column": 27
} | {
"line": 237,
"column": 27
} | [
{
"pp": "n : ℕ∞\n⊢ ↑⊤.toNat = ⊤ ↔ ⊤ ≠ ⊤",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"instDecidableNot",
"of_decide_eq_true",
"LinearOrder.toDecidableEq",
"ENat.instNatCast",
"instTopENat",
"instLinearOrderENat",
"id",
"Ne",
"Nat.cast... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.Data.ENat.Basic | {
"line": 237,
"column": 21
} | {
"line": 237,
"column": 27
} | {
"line": 237,
"column": 27
} | [
{
"pp": "n : ℕ∞\n⊢ ↑⊤.toNat = ⊤ ↔ ⊤ ≠ ⊤",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"instDecidableNot",
"of_decide_eq_true",
"LinearOrder.toDecidableEq",
"ENat.instNatCast",
"instTopENat",
"instLinearOrderENat",
"id",
"Ne",
"Nat.cast... | [] | decide | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.ENat.Basic | {
"line": 237,
"column": 21
} | {
"line": 237,
"column": 27
} | {
"line": 237,
"column": 27
} | [
{
"pp": "n : ℕ∞\n⊢ ↑⊤.toNat = ⊤ ↔ ⊤ ≠ ⊤",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"instDecidableNot",
"of_decide_eq_true",
"LinearOrder.toDecidableEq",
"ENat.instNatCast",
"instTopENat",
"instLinearOrderENat",
"id",
"Ne",
"Nat.cast... | [] | decide | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.SuccPred.Archimedean | {
"line": 331,
"column": 6
} | {
"line": 334,
"column": 37
} | {
"line": 335,
"column": 6
} | [
{
"pp": "case pos\nα : Type u_1\nβ : Type u_2\ninst✝³ : PartialOrder α\ninst✝² : PredOrder α\ninst✝¹ : IsPredArchimedean α\ns : Set α\ninst✝ : s.OrdConnected\nx✝¹ x✝ : ↑s\nb : α\nhb : b ∈ s\nn : ℕ\nhi : ∀ (c : α) (hc : c ∈ s), b ≤ c → pred^[n] c = b → pred^[n] ⟨c, hc⟩ = ⟨b, hb⟩\nc : α\nhc : c ∈ s\nhbc : b ≤ c\n... | [
"case neg\nα : Type u_1\nβ : Type u_2\ninst✝³ : PartialOrder α\ninst✝² : PredOrder α\ninst✝¹ : IsPredArchimedean α\ns : Set α\ninst✝ : s.OrdConnected\nx✝¹ x✝ : ↑s\nb : α\nhb : b ∈ s\nn : ℕ\nhi : ∀ (c : α) (hc : c ∈ s), b ≤ c → pred^[n] c = b → pred^[n] ⟨c, hc⟩ = ⟨b, hb⟩\nc : α\nhc : c ∈ s\nhbc : b ≤ c\nhn : pred^[n... | · dsimp only at h ⊢
apply hi _ _ _ hn
· rw [← hn]
apply Order.pred_iterate_le | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Data.Fintype.BigOperators | {
"line": 216,
"column": 10
} | {
"line": 216,
"column": 45
} | {
"line": 217,
"column": 10
} | [
{
"pp": "α : Type u_1\ninst✝ : Fintype α\nT : Finset (List α)\ns : ℕ\na : List α\nha : a ∈ {x ∈ T | x.length = s}\n⊢ a ∈ image List.ofFn univ",
"ppTerm": "?m.49",
"assigned": true,
"usedConstants": [
"Finset",
"Membership.mem",
"List",
"And",
"Finset.instSetLike",
... | [
"α : Type u_1\ninst✝ : Fintype α\nT : Finset (List α)\ns : ℕ\na : List α\nha : a ∈ {x ∈ T | x.length = s}\nhlen : a ∈ T ∧ a.length = s := mem_filter.mp ha\n⊢ a ∈ image List.ofFn univ"
] | let hlen := Finset.mem_filter.mp ha | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.Order.InitialSeg | {
"line": 304,
"column": 73
} | {
"line": 306,
"column": 38
} | {
"line": 308,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nr : α → α → Prop\ns : β → β → Prop\nf : r ≺i s\n⊢ Set.SurjOn (⇑f.toRelEmbedding) Set.univ {b | s b f.top}",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Set.image_univ",
"congrArg",
"PrincipalSeg.mem_range_of_rel_top"... | [] | by
intro b h
simpa using mem_range_of_rel_top _ h | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Sum.Order | {
"line": 380,
"column": 6
} | {
"line": 383,
"column": 27
} | {
"line": 383,
"column": 28
} | [
{
"pp": "case refine_2\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝¹ : Preorder α\ninst✝ : Preorder β\na b : α ⊕ₗ β\n⊢ a ≤ b ∧ ¬b ≤ a → a < b",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"LE.le.lt_of_not_ge",
"Sum.Lex.LE",
"Preorder.toLT",
"HEq.refl",
... | [] | · rintro ⟨⟨hab⟩ | ⟨hab⟩ | ⟨a, b⟩, hba⟩
· exact Lex.inl (hab.lt_of_not_ge fun h => hba <| Lex.inl h)
· exact Lex.inr (hab.lt_of_not_ge fun h => hba <| Lex.inr h)
· exact Lex.sep _ _ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Order.UpperLower.Basic | {
"line": 113,
"column": 17
} | {
"line": 113,
"column": 32
} | {
"line": 115,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : LE α\ns : Set α\na : α\nhs : IsUpperSet s\nhas : ∀ b ∈ s, b ≤ a → b = a\n⊢ ∀ b ∈ s, ∀ c ∈ {a}, b ≤ c → b ∈ {a}",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Membership.mem",
"Set.instSingletonSet",
"id",
"LE.le",
... | [] | simpa using has | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Order.UpperLower.Basic | {
"line": 113,
"column": 17
} | {
"line": 113,
"column": 32
} | {
"line": 115,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : LE α\ns : Set α\na : α\nhs : IsUpperSet s\nhas : ∀ b ∈ s, b ≤ a → b = a\n⊢ ∀ b ∈ s, ∀ c ∈ {a}, b ≤ c → b ∈ {a}",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Membership.mem",
"Set.instSingletonSet",
"id",
"LE.le",
... | [] | simpa using has | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.UpperLower.Basic | {
"line": 113,
"column": 17
} | {
"line": 113,
"column": 32
} | {
"line": 115,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : LE α\ns : Set α\na : α\nhs : IsUpperSet s\nhas : ∀ b ∈ s, b ≤ a → b = a\n⊢ ∀ b ∈ s, ∀ c ∈ {a}, b ≤ c → b ∈ {a}",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Membership.mem",
"Set.instSingletonSet",
"id",
"LE.le",
... | [] | simpa using has | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.Hom.Order | {
"line": 136,
"column": 4
} | {
"line": 142,
"column": 45
} | {
"line": 144,
"column": 0
} | [
{
"pp": "case mpr\nα : Type u_3\ninst✝ : SemilatticeSup α\nf : α →o α\nh : ∀ (a₁ a₂ : α), f (a₁ ⊔ a₂) ≤ f a₁ ⊔ a₂\nn₁ n₂ : ℕ\na₁ a₂ : α\nh' : ∀ (n : ℕ) (a₁ a₂ : α), (⇑f)^[n] (a₁ ⊔ a₂) ≤ (⇑f)^[n] a₁ ⊔ a₂\n⊢ (⇑f)^[n₁ + n₂] (a₁ ⊔ a₂) ≤ (⇑f)^[n₁] a₁ ⊔ (⇑f)^[n₂] a₂",
"ppTerm": "?mpr",
"assigned": true,
"... | [] | calc
f^[n₁ + n₂] (a₁ ⊔ a₂) = f^[n₁] (f^[n₂] (a₁ ⊔ a₂)) :=
Function.iterate_add_apply f n₁ n₂ _
_ = f^[n₁] (f^[n₂] (a₂ ⊔ a₁)) := by rw [sup_comm]
_ ≤ f^[n₁] (f^[n₂] a₂ ⊔ a₁) := f.mono.iterate n₁ (h' n₂ _ _)
_ = f^[n₁] (a₁ ⊔ f^[n₂] a₂) := by rw [sup_comm]
_ ≤ f^[n₁] a₁ ⊔ f^[n₂] a₂ :=... | Lean.Elab.Tactic.evalCalc | Lean.calcTactic |
Mathlib.Data.Part | {
"line": 371,
"column": 2
} | {
"line": 371,
"column": 29
} | {
"line": 372,
"column": 2
} | [
{
"pp": "case inr\nα : Type u_1\nx y z : Part α\nhx : x ≤ z\nhy : y ≤ z\nb : α\nh₀ : x = some b\nb' : α\nh₁ : b' ∈ y\n⊢ b' ∈ x",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Part",
"congrArg",
"Part.eq_some_iff",
"Part.some",
"Membership.mem",
"Eq.mp",
... | [
"case inr\nα : Type u_1\nx y z : Part α\nhx : x ≤ z\nhy : y ≤ z\nb : α\nh₀ : b ∈ x\nb' : α\nh₁ : b' ∈ y\n⊢ b' ∈ x"
] | rw [Part.eq_some_iff] at h₀ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Order.BourbakiWitt | {
"line": 146,
"column": 15
} | {
"line": 146,
"column": 44
} | {
"line": 147,
"column": 4
} | [
{
"pp": "case image_self_subset_self.inr\nα : Type u_1\ninst✝ : ChainCompletePartialOrder α\nx : α\nf : α → α\ny : α\nle_map : ∀ (x : α), x ≤ f x\nhy : IsExtremePt x f y\nz : α\nhz : z ∈ bot x f\nhyz : f y ≤ z\n⊢ f y ≤ f z",
"ppTerm": "?image_self_subset_self.inr",
"assigned": true,
"usedConstants":... | [] | exact le_trans hyz (le_map z) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Order.BourbakiWitt | {
"line": 167,
"column": 15
} | {
"line": 167,
"column": 18
} | {
"line": 168,
"column": 4
} | [
{
"pp": "case image_self_subset_self\nα : Type u_1\ninst✝ : ChainCompletePartialOrder α\nx : α\nf : α → α\nle_map : ∀ (x : α), x ≤ f x\ny : α\nhy : y ∈ {y | IsExtremePt x f y}\nz : α\nhz : z ∈ bot x f\n⊢ z < f y → f z ≤ f y",
"ppTerm": "?image_self_subset_self",
"assigned": true,
"usedConstants": [
... | [
"case image_self_subset_self\nα : Type u_1\ninst✝ : ChainCompletePartialOrder α\nx : α\nf : α → α\nle_map : ∀ (x : α), x ≤ f x\ny : α\nhy : y ∈ {y | IsExtremePt x f y}\nz : α\nhz : z ∈ bot x f\nhzy : z < f y\n⊢ f z ≤ f y"
] | hzy | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.SetTheory.Cardinal.Order | {
"line": 402,
"column": 2
} | {
"line": 402,
"column": 35
} | {
"line": 403,
"column": 2
} | [
{
"pp": "case mk\nα β : Type u_1\nh : #α < #β\nf : α ↪ β\nhf : ¬Surjective ⇑f\n⊢ #α + 1 ≤ #β",
"ppTerm": "?mk",
"assigned": true,
"usedConstants": [
"Function.Surjective.eq_1",
"congrArg",
"Exists",
"Eq.mp",
"Function.Embedding",
"propext",
"Function.instFun... | [
"case mk\nα β : Type u_1\nh : #α < #β\nf : α ↪ β\nhf : ∃ x, ¬∃ a, f a = x\n⊢ #α + 1 ≤ #β"
] | rw [Surjective, not_forall] at hf | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.SetTheory.Cardinal.SchroederBernstein | {
"line": 125,
"column": 10
} | {
"line": 125,
"column": 36
} | {
"line": 126,
"column": 8
} | [
{
"pp": "ι : Type u\nβ : ι → Type v\nI : Nonempty ι\ns : Set ((i : ι) → β i)\nhs : Maximal (fun x ↦ x ∈ sets β) s\nh : ¬∃ i, Surjective fun x ↦ ↑x i\n⊢ ∀ (i : ι), ∃ y, ∀ x ∈ s, x i ≠ y",
"ppTerm": "?m.39",
"assigned": true,
"usedConstants": [
"not_exists._simp_1",
"Iff.of_eq",
"con... | [] | simpa [Surjective] using h | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.SetTheory.Cardinal.SchroederBernstein | {
"line": 125,
"column": 10
} | {
"line": 125,
"column": 36
} | {
"line": 126,
"column": 8
} | [
{
"pp": "ι : Type u\nβ : ι → Type v\nI : Nonempty ι\ns : Set ((i : ι) → β i)\nhs : Maximal (fun x ↦ x ∈ sets β) s\nh : ¬∃ i, Surjective fun x ↦ ↑x i\n⊢ ∀ (i : ι), ∃ y, ∀ x ∈ s, x i ≠ y",
"ppTerm": "?m.39",
"assigned": true,
"usedConstants": [
"not_exists._simp_1",
"Iff.of_eq",
"con... | [] | simpa [Surjective] using h | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.SetTheory.Cardinal.SchroederBernstein | {
"line": 125,
"column": 10
} | {
"line": 125,
"column": 36
} | {
"line": 126,
"column": 8
} | [
{
"pp": "ι : Type u\nβ : ι → Type v\nI : Nonempty ι\ns : Set ((i : ι) → β i)\nhs : Maximal (fun x ↦ x ∈ sets β) s\nh : ¬∃ i, Surjective fun x ↦ ↑x i\n⊢ ∀ (i : ι), ∃ y, ∀ x ∈ s, x i ≠ y",
"ppTerm": "?m.39",
"assigned": true,
"usedConstants": [
"not_exists._simp_1",
"Iff.of_eq",
"con... | [] | simpa [Surjective] using h | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.SetTheory.Cardinal.SchroederBernstein | {
"line": 129,
"column": 39
} | {
"line": 129,
"column": 61
} | {
"line": 129,
"column": 61
} | [
{
"pp": "case inl\nι : Type u\nβ : ι → Type v\nI : Nonempty ι\ns : Set ((i : ι) → β i)\nhs : Maximal (fun x ↦ x ∈ sets β) s\nh✝ : ¬∃ i, Surjective fun x ↦ ↑x i\nh : ∀ (i : ι), ∃ y, ∀ x ∈ s, x i ≠ y\nf : (x : ι) → β x\nhf : ∀ (x : ι), ∀ x_1 ∈ s, x_1 x ≠ f x\ni : ι\nx y : (i : ι) → β i\nhy : y ∈ insert f s\nhx : ... | [
"case inl.inl\nι : Type u\nβ : ι → Type v\nI : Nonempty ι\ns : Set ((i : ι) → β i)\nhs : Maximal (fun x ↦ x ∈ sets β) s\nh✝ : ¬∃ i, Surjective fun x ↦ ↑x i\nh : ∀ (i : ι), ∃ y, ∀ x ∈ s, x i ≠ y\nf : (x : ι) → β x\nhf : ∀ (x : ι), ∀ x_1 ∈ s, x_1 x ≠ f x\ni : ι\nx y : (i : ι) → β i\nhx : x = f\nhy : y = f\n⊢ (fun x ↦... | rcases hy with hy | hy | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.SetTheory.Cardinal.SchroederBernstein | {
"line": 129,
"column": 39
} | {
"line": 129,
"column": 61
} | {
"line": 129,
"column": 61
} | [
{
"pp": "case inr\nι : Type u\nβ : ι → Type v\nI : Nonempty ι\ns : Set ((i : ι) → β i)\nhs : Maximal (fun x ↦ x ∈ sets β) s\nh✝ : ¬∃ i, Surjective fun x ↦ ↑x i\nh : ∀ (i : ι), ∃ y, ∀ x ∈ s, x i ≠ y\nf : (x : ι) → β x\nhf : ∀ (x : ι), ∀ x_1 ∈ s, x_1 x ≠ f x\ni : ι\nx y : (i : ι) → β i\nhy : y ∈ insert f s\nhx : ... | [
"case inr.inl\nι : Type u\nβ : ι → Type v\nI : Nonempty ι\ns : Set ((i : ι) → β i)\nhs : Maximal (fun x ↦ x ∈ sets β) s\nh✝ : ¬∃ i, Surjective fun x ↦ ↑x i\nh : ∀ (i : ι), ∃ y, ∀ x ∈ s, x i ≠ y\nf : (x : ι) → β x\nhf : ∀ (x : ι), ∀ x_1 ∈ s, x_1 x ≠ f x\ni : ι\nx y : (i : ι) → β i\nhx : x ∈ s\nhy : y = f\n⊢ (fun x ↦... | rcases hy with hy | hy | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.SetTheory.Cardinal.ToNat | {
"line": 123,
"column": 22
} | {
"line": 123,
"column": 43
} | {
"line": 123,
"column": 44
} | [
{
"pp": "c : Cardinal.{u}\nn : ℕ\nhn : n ≠ 0\n⊢ (toENat c).toNat = n ↔ c = ↑n",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"ENat.instNatCast",
"Cardinal",
"congrArg",
"CommSemiring.toSemiring",
"Cardinal.commSemiring",
"PartialOrder.toP... | [
"c : Cardinal.{u}\nn : ℕ\nhn : n ≠ 0\n⊢ toENat c = ↑n ↔ c = ↑n"
] | ENat.toNat_eq_iff hn, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.SetTheory.Cardinal.Basic | {
"line": 405,
"column": 2
} | {
"line": 405,
"column": 34
} | {
"line": 407,
"column": 0
} | [
{
"pp": "n : ℕ\nhx : ↑n < ℵ₀\n⊢ 2 ^ ↑n < ℵ₀",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"Preorder.toLT",
"Cardinal.instPowCardinal",
"Cardinal",
"congrArg",
"CommSemiring.toSemiring",
"Nat.instMonoid",... | [] | exact mod_cast natCast_lt_aleph0 | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.SetTheory.Cardinal.Basic | {
"line": 519,
"column": 2
} | {
"line": 519,
"column": 31
} | {
"line": 521,
"column": 0
} | [
{
"pp": "a b : Cardinal.{u}\nthis : ∀ {a : Cardinal.{u}}, ℵ₀ ≤ a → a ≠ 0\n⊢ b ≠ 0 ∧ ℵ₀ ≤ a ∨ a ≠ 0 ∧ ℵ₀ ≤ b ↔ a ≠ 0 ∧ ℵ₀ ≤ b ∨ ℵ₀ ≤ a ∧ b ≠ 0",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"Cardinal",
"congrArg",
"_private.Mathlib.SetTheory.Cardinal.Basic.0.Cardinal.alep... | [] | simp only [and_comm, or_comm] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.SetTheory.Cardinal.Basic | {
"line": 885,
"column": 82
} | {
"line": 887,
"column": 32
} | {
"line": 889,
"column": 0
} | [
{
"pp": "α : Type u\nS T : Set α\nh : T ⊆ S\n⊢ #↑(S \\ T) + #↑T = #↑S",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Cardinal",
"congrArg",
"BooleanAlgebra.toGeneralizedBooleanAlgebra",
"Set.sdiff_union_of_subset",
"Cardinal.mk",
"Set.i... | [] | by
refine (mk_union_of_disjoint <| ?_).symm.trans <| by rw [sdiff_union_of_subset h]
exact disjoint_sdiff_self_left | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Module.LinearMap.Defs | {
"line": 546,
"column": 2
} | {
"line": 549,
"column": 38
} | {
"line": 551,
"column": 0
} | [
{
"pp": "R₁ : Type u_2\nR₂ : Type u_3\nR₃ : Type u_4\nM₁ : Type u_9\nM₂ : Type u_10\nM₃ : Type u_11\ninst✝⁸ : Semiring R₁\ninst✝⁷ : Semiring R₂\ninst✝⁶ : Semiring R₃\ninst✝⁵ : AddCommMonoid M₁\ninst✝⁴ : AddCommMonoid M₂\ninst✝³ : AddCommMonoid M₃\nmodule_M₁ : Module R₁ M₁\nmodule_M₂ : Module R₂ M₂\nmodule_M₃ : ... | [] | intro h
obtain ⟨f', hf'⟩ := hf
refine ⟨f'.comp h, ?_⟩
simp_rw [← comp_assoc, hf', id_comp] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Module.LinearMap.Defs | {
"line": 546,
"column": 2
} | {
"line": 549,
"column": 38
} | {
"line": 551,
"column": 0
} | [
{
"pp": "R₁ : Type u_2\nR₂ : Type u_3\nR₃ : Type u_4\nM₁ : Type u_9\nM₂ : Type u_10\nM₃ : Type u_11\ninst✝⁸ : Semiring R₁\ninst✝⁷ : Semiring R₂\ninst✝⁶ : Semiring R₃\ninst✝⁵ : AddCommMonoid M₁\ninst✝⁴ : AddCommMonoid M₂\ninst✝³ : AddCommMonoid M₃\nmodule_M₁ : Module R₁ M₁\nmodule_M₂ : Module R₂ M₂\nmodule_M₃ : ... | [] | intro h
obtain ⟨f', hf'⟩ := hf
refine ⟨f'.comp h, ?_⟩
simp_rw [← comp_assoc, hf', id_comp] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Module.Submodule.Lattice | {
"line": 228,
"column": 27
} | {
"line": 228,
"column": 35
} | {
"line": 228,
"column": 36
} | [
{
"pp": "case refine_2\nR : Type u_1\nM : Type u_3\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nι : Type u_4\ns✝ : Finset ι\np : ι → Submodule R M\nthis : DecidableEq ι := Classical.decEq ι\ni : ι\ns : Finset ι\nx✝ : i ∉ s\nih : ↑(s.inf p) = ⋂ i ∈ s, ↑(p i)\n⊢ ↑(p i ⊓ s.inf p) = ⋂ i_1 ∈ i... | [
"case refine_2\nR : Type u_1\nM : Type u_3\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nι : Type u_4\ns✝ : Finset ι\np : ι → Submodule R M\nthis : DecidableEq ι := Classical.decEq ι\ni : ι\ns : Finset ι\nx✝ : i ∉ s\nih : ↑(s.inf p) = ⋂ i ∈ s, ↑(p i)\n⊢ ↑(p i) ∩ ↑(s.inf p) = ⋂ i_1 ∈ insert i s... | coe_inf, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Module.Submodule.Lattice | {
"line": 296,
"column": 33
} | {
"line": 307,
"column": 78
} | {
"line": 307,
"column": 79
} | [
{
"pp": "R : Type u_1\nM : Type u_3\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\ns : Set (Submodule R M)\nt : R\nm : M\nh : m ∈ (sSup (toAddSubmonoid '' s)).carrier\n⊢ t • m ∈ (sSup (toAddSubmonoid '' s)).carrier",
"ppTerm": "?m.55",
"assigned": true,
"usedConstants": [
... | [] | by
simp_rw [AddSubsemigroup.mem_carrier, AddSubmonoid.mem_toSubsemigroup, sSup_eq_iSup'] at h ⊢
induction h using AddSubmonoid.iSup_induction' with
| mem p x hx =>
obtain ⟨-, ⟨p : Submodule R M, hp : p ∈ s, rfl⟩⟩ := p
suffices p.toAddSubmonoid ≤ ⨆ q : toAddSubmonoid '' s, (q ... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Module.Submodule.Ker | {
"line": 144,
"column": 25
} | {
"line": 144,
"column": 36
} | {
"line": 144,
"column": 37
} | [
{
"pp": "R : Type u_1\nR₂ : Type u_2\nM : Type u_5\nM₂ : Type u_7\ninst✝⁵ : Semiring R\ninst✝⁴ : Semiring R₂\ninst✝³ : AddCommMonoid M\ninst✝² : AddCommMonoid M₂\ninst✝¹ : Module R M\ninst✝ : Module R₂ M₂\nτ₁₂ : R →+* R₂\nf : M →ₛₗ[τ₁₂] M₂\ns : Set (Submodule R M)\nhs : sSup s = ⊤\nh : ∀ m ∈ s, f ∘ₛₗ m.subtype ... | [
"R : Type u_1\nR₂ : Type u_2\nM : Type u_5\nM₂ : Type u_7\ninst✝⁵ : Semiring R\ninst✝⁴ : Semiring R₂\ninst✝³ : AddCommMonoid M\ninst✝² : AddCommMonoid M₂\ninst✝¹ : Module R M\ninst✝ : Module R₂ M₂\nτ₁₂ : R →+* R₂\nf : M →ₛₗ[τ₁₂] M₂\ns : Set (Submodule R M)\nhs : sSup s = ⊤\nh : ∀ m ∈ s, f ∘ₛₗ m.subtype = 0\n⊢ ⊤ ≤ f... | eq_top_iff, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Algebra.Ring.CharZero | {
"line": 34,
"column": 51
} | {
"line": 34,
"column": 57
} | {
"line": 36,
"column": 0
} | [
{
"pp": "α : Type u_1\nR : Type u_2\nS : Type u_3\nn : ℕ\ninst✝¹ : AddMonoidWithOne R\ninst✝ : CharZero R\n⊢ 2 ≠ 0",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"instDecidableNot",
"of_decide_eq_true",
"id",
"Ne",
"instOfNatNat",
"Bool.true",
"Na... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.Algebra.Module.Submodule.Map | {
"line": 561,
"column": 4
} | {
"line": 561,
"column": 22
} | {
"line": 562,
"column": 4
} | [
{
"pp": "case mp\nR : Type u_1\nR₂ : Type u_3\nM : Type u_5\nM₂ : Type u_7\ninst✝⁷ : Semiring R\ninst✝⁶ : Semiring R₂\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : AddCommMonoid M₂\ninst✝³ : Module R M\ninst✝² : Module R₂ M₂\nτ₁₂ : R →+* R₂\nτ₂₁ : R₂ →+* R\ninst✝¹ : RingHomInvPair τ₁₂ τ₂₁\ninst✝ : RingHomInvPair τ₂₁ τ₁₂\... | [
"case mp\nR : Type u_1\nR₂ : Type u_3\nM : Type u_5\nM₂ : Type u_7\ninst✝⁷ : Semiring R\ninst✝⁶ : Semiring R₂\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : AddCommMonoid M₂\ninst✝³ : Module R M\ninst✝² : Module R₂ M₂\nτ₁₂ : R →+* R₂\nτ₂₁ : R₂ →+* R\ninst✝¹ : RingHomInvPair τ₁₂ τ₂₁\ninst✝ : RingHomInvPair τ₂₁ τ₁₂\np : Submodu... | rintro ⟨y, hy, hx⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.Algebra.Algebra.Basic | {
"line": 154,
"column": 2
} | {
"line": 154,
"column": 23
} | {
"line": 155,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝² : CommSemiring R\nS : Type u_4\ninst✝¹ : Semiring S\ninst✝ : Algebra R S\n⊢ algebraMapSubmonoid S (IsUnit.submonoid R) ≤ IsUnit.submonoid S",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Algebra.algebraMap",
"CommSemiring.toSemiring",
"Ring... | [
"R : Type u_1\ninst✝² : CommSemiring R\nS : Type u_4\ninst✝¹ : Semiring S\ninst✝ : Algebra R S\ny : R\nhy : y ∈ ↑(IsUnit.submonoid R)\n⊢ (algebraMap R S) y ∈ IsUnit.submonoid S"
] | rintro x ⟨y, hy, rfl⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.Algebra.Algebra.Basic | {
"line": 268,
"column": 2
} | {
"line": 268,
"column": 55
} | {
"line": 270,
"column": 0
} | [
{
"pp": "R : Type u_1\nA : Type u_2\nB : Type u_3\ninst✝⁴ : CommSemiring R\ninst✝³ : Semiring A\ninst✝² : Semiring B\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : A →ₗ[R] B\na : A\nr : R\n⊢ f ((algebraMap R A) r * a) = (algebraMap R B) r * f a",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants"... | [] | rw [← Algebra.smul_def, ← Algebra.smul_def, map_smul] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Algebra.Basic | {
"line": 268,
"column": 2
} | {
"line": 268,
"column": 55
} | {
"line": 270,
"column": 0
} | [
{
"pp": "R : Type u_1\nA : Type u_2\nB : Type u_3\ninst✝⁴ : CommSemiring R\ninst✝³ : Semiring A\ninst✝² : Semiring B\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : A →ₗ[R] B\na : A\nr : R\n⊢ f ((algebraMap R A) r * a) = (algebraMap R B) r * f a",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants"... | [] | rw [← Algebra.smul_def, ← Algebra.smul_def, map_smul] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Algebra.Basic | {
"line": 268,
"column": 2
} | {
"line": 268,
"column": 55
} | {
"line": 270,
"column": 0
} | [
{
"pp": "R : Type u_1\nA : Type u_2\nB : Type u_3\ninst✝⁴ : CommSemiring R\ninst✝³ : Semiring A\ninst✝² : Semiring B\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : A →ₗ[R] B\na : A\nr : R\n⊢ f ((algebraMap R A) r * a) = (algebraMap R B) r * f a",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants"... | [] | rw [← Algebra.smul_def, ← Algebra.smul_def, map_smul] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Algebra.Hom | {
"line": 213,
"column": 26
} | {
"line": 213,
"column": 85
} | {
"line": 213,
"column": 86
} | [
{
"pp": "R : Type u\nA : Type v\nB : Type w\nC : Type u₁\nD : Type v₁\ninst✝⁸ : CommSemiring R\ninst✝⁷ : Semiring A\ninst✝⁶ : Semiring B\ninst✝⁵ : Semiring C\ninst✝⁴ : Semiring D\ninst✝³ : Algebra R A\ninst✝² : Algebra R B\ninst✝¹ : Algebra R C\ninst✝ : Algebra R D\nφ : A →ₐ[R] B\nf : A →+* B\nh : ∀ (c : R) (x ... | [] | by simp only [Algebra.algebraMap_eq_smul_one, h, f.map_one] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.NonUnitalSubring.Basic | {
"line": 522,
"column": 12
} | {
"line": 537,
"column": 62
} | {
"line": 537,
"column": 62
} | [
{
"pp": "R : Type u\ninst✝ : NonUnitalNonAssocRing R\ns : Set R\nx : R\nh : x ∈ closure s\n⊢ x ∈ AddSubgroup.closure ↑(Subsemigroup.closure s)",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"add_mul",
"AddGroup.toSubtractionMonoid",
"Distrib.leftDistribClass",
"Non... | [] | by
induction h using closure_induction with
| mem _ hx => exact AddSubgroup.subset_closure (Subsemigroup.subset_closure hx)
| zero => exact zero_mem _
| add _ _ _ _ hx hy => exact add_mem hx hy
| neg x _ hx => exact neg_mem hx
| mul _ _ _hx _hy hx hy =>
clear _hx _hy
induction hx, hy... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Prime.Defs | {
"line": 75,
"column": 2
} | {
"line": 85,
"column": 22
} | {
"line": 87,
"column": 0
} | [
{
"pp": "M : Type u_1\ninst✝ : CommMonoidWithZero M\np : M\nhp : Prime p\na : M\nn : ℕ\nh : p ∣ a ^ n\n⊢ p ∣ a",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"CommMonoidWithZero.toCommMonoid",
"MulOne.toOne",
"Nat.recAux",
"Dvd.dvd",
"HMul.hMul",
"Prime... | [] | induction n with
| zero =>
rw [pow_zero] at h
have := isUnit_of_dvd_one h
have := not_unit hp
contradiction
| succ n ih =>
rw [pow_succ'] at h
rcases dvd_or_dvd hp h with dvd_a | dvd_pow
· assumption
· exact ih dvd_pow | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.Algebra.Prime.Defs | {
"line": 75,
"column": 2
} | {
"line": 85,
"column": 22
} | {
"line": 87,
"column": 0
} | [
{
"pp": "M : Type u_1\ninst✝ : CommMonoidWithZero M\np : M\nhp : Prime p\na : M\nn : ℕ\nh : p ∣ a ^ n\n⊢ p ∣ a",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"CommMonoidWithZero.toCommMonoid",
"MulOne.toOne",
"Nat.recAux",
"Dvd.dvd",
"HMul.hMul",
"Prime... | [] | induction n with
| zero =>
rw [pow_zero] at h
have := isUnit_of_dvd_one h
have := not_unit hp
contradiction
| succ n ih =>
rw [pow_succ'] at h
rcases dvd_or_dvd hp h with dvd_a | dvd_pow
· assumption
· exact ih dvd_pow | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Prime.Defs | {
"line": 75,
"column": 2
} | {
"line": 85,
"column": 22
} | {
"line": 87,
"column": 0
} | [
{
"pp": "M : Type u_1\ninst✝ : CommMonoidWithZero M\np : M\nhp : Prime p\na : M\nn : ℕ\nh : p ∣ a ^ n\n⊢ p ∣ a",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"CommMonoidWithZero.toCommMonoid",
"MulOne.toOne",
"Nat.recAux",
"Dvd.dvd",
"HMul.hMul",
"Prime... | [] | induction n with
| zero =>
rw [pow_zero] at h
have := isUnit_of_dvd_one h
have := not_unit hp
contradiction
| succ n ih =>
rw [pow_succ'] at h
rcases dvd_or_dvd hp h with dvd_a | dvd_pow
· assumption
· exact ih dvd_pow | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Group.Irreducible.Lemmas | {
"line": 44,
"column": 19
} | {
"line": 44,
"column": 25
} | {
"line": 44,
"column": 26
} | [
{
"pp": "case refine_2\nM : Type u_2\ninst✝ : Monoid M\ny : M\nu : Mˣ\nx✝ : ¬IsUnit y\nh : ∀ ⦃a b : M⦄, y = a * b → IsUnit a ∨ IsUnit b\nA B : M\nHAB : ↑u * y = A * B\n⊢ y = ↑u⁻¹ * (A * B)",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"Units.val",
"Eq.mpr",
"Semigro... | [
"case refine_2\nM : Type u_2\ninst✝ : Monoid M\ny : M\nu : Mˣ\nx✝ : ¬IsUnit y\nh : ∀ ⦃a b : M⦄, y = a * b → IsUnit a ∨ IsUnit b\nA B : M\nHAB : ↑u * y = A * B\n⊢ y = ↑u⁻¹ * (↑u * y)"
] | ← HAB, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Group.Irreducible.Lemmas | {
"line": 60,
"column": 21
} | {
"line": 60,
"column": 27
} | {
"line": 60,
"column": 28
} | [
{
"pp": "case refine_2\nM : Type u_2\ninst✝ : Monoid M\ny : M\nu : Mˣ\nx✝ : ¬IsUnit y\nh : ∀ ⦃a b : M⦄, y = a * b → IsUnit a ∨ IsUnit b\nA B : M\nHAB : y * ↑u = A * B\n⊢ y = A * B * ↑u⁻¹",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"Units.val",
"Eq.mpr",
"Semigroup... | [
"case refine_2\nM : Type u_2\ninst✝ : Monoid M\ny : M\nu : Mˣ\nx✝ : ¬IsUnit y\nh : ∀ ⦃a b : M⦄, y = a * b → IsUnit a ∨ IsUnit b\nA B : M\nHAB : y * ↑u = A * B\n⊢ y = y * ↑u * ↑u⁻¹"
] | ← HAB, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Group.Irreducible.Lemmas | {
"line": 106,
"column": 32
} | {
"line": 106,
"column": 38
} | {
"line": 106,
"column": 38
} | [
{
"pp": "M : Type u_2\ninst✝ : Monoid M\ny : M\nha : Irreducible (y ^ 2)\n⊢ 2 ≠ 1",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"instDecidableNot",
"of_decide_eq_true",
"id",
"Ne",
"instOfNatNat",
"Bool.true",
"Nat",
"Bool",
"Eq.refl"... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.Algebra.Group.Irreducible.Lemmas | {
"line": 106,
"column": 32
} | {
"line": 106,
"column": 38
} | {
"line": 106,
"column": 38
} | [
{
"pp": "M : Type u_2\ninst✝ : Monoid M\ny : M\nha : Irreducible (y ^ 2)\n⊢ 2 ≠ 1",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"instDecidableNot",
"of_decide_eq_true",
"id",
"Ne",
"instOfNatNat",
"Bool.true",
"Nat",
"Bool",
"Eq.refl"... | [] | decide | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Group.Irreducible.Lemmas | {
"line": 106,
"column": 32
} | {
"line": 106,
"column": 38
} | {
"line": 106,
"column": 38
} | [
{
"pp": "M : Type u_2\ninst✝ : Monoid M\ny : M\nha : Irreducible (y ^ 2)\n⊢ 2 ≠ 1",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"instDecidableNot",
"of_decide_eq_true",
"id",
"Ne",
"instOfNatNat",
"Bool.true",
"Nat",
"Bool",
"Eq.refl"... | [] | decide | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Ring.Subring.Basic | {
"line": 559,
"column": 4
} | {
"line": 559,
"column": 19
} | {
"line": 559,
"column": 20
} | [
{
"pp": "case neg\nR : Type u\ninst✝ : NonAssocRing R\ns : Set R\nx x✝ : R\nhx✝ : x✝ ∈ closure s\nhx : x✝ ∈ AddSubgroup.closure ↑(Submonoid.closure s)\n⊢ -x✝ ∈ AddSubgroup.closure ↑(Submonoid.closure s)",
"ppTerm": "?neg",
"assigned": true,
"usedConstants": [
"NegZeroClass.toNeg",
"AddGr... | [] | | neg _ _ hx => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.Algebra.GroupWithZero.Associated | {
"line": 336,
"column": 4
} | {
"line": 336,
"column": 32
} | {
"line": 337,
"column": 2
} | [
{
"pp": "M : Type u_1\ninst✝¹ : CommMonoidWithZero M\ninst✝ : IsCancelMulZero M\np₁ p₂ : M\nk₁ k₂ : ℕ\nhp₁ : Prime p₁\nhp₂ : Prime p₂\nhk₁ : 0 < k₁\nh : p₁ ^ k₁ ~ᵤ p₂ ^ k₂\n⊢ p₁ ∣ p₁ ^ k₁",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"instOfNatNat",
"CommMonoidWithZero.toMono... | [] | apply dvd_pow_self _ hk₁.ne' | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Algebra.Module.Submodule.Range | {
"line": 172,
"column": 58
} | {
"line": 173,
"column": 50
} | {
"line": 175,
"column": 0
} | [
{
"pp": "R : Type u_1\nR₂ : Type u_2\nM : Type u_5\nM₂ : Type u_6\ninst✝⁶ : Semiring R\ninst✝⁵ : Semiring R₂\ninst✝⁴ : AddCommMonoid M\ninst✝³ : AddCommMonoid M₂\ninst✝² : Module R M\ninst✝¹ : Module R₂ M₂\nτ₁₂ : R →+* R₂\ninst✝ : RingHomSurjective τ₁₂\nf : M →ₛₗ[τ₁₂] M₂\nq : Submodule R₂ M₂\n⊢ f.range ⊓ q ≤ ma... | [] | by
rintro _ ⟨⟨x, _, rfl⟩, hx⟩; exact ⟨x, hx, rfl⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.GroupWithZero.Associated | {
"line": 652,
"column": 2
} | {
"line": 653,
"column": 54
} | {
"line": 654,
"column": 2
} | [
{
"pp": "M : Type u_1\ninst✝ : CommMonoidWithZero M\na : M\n⊢ Irreducible (Associates.mk a) ↔ Irreducible a",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Associated.comm",
"CommMonoidWithZero.toCommMonoid",
"_private.Mathlib.Algebra.GroupWithZero.Associated.0.Associates... | [
"M : Type u_1\ninst✝ : CommMonoidWithZero M\na : M\n⊢ (¬IsUnit a ∧ ∀ (a_1 a_2 : M), a_1 * a_2 ~ᵤ a → IsUnit a_1 ∨ IsUnit a_2) ↔\n ¬IsUnit a ∧ ∀ ⦃a_1 b : M⦄, a = a_1 * b → IsUnit a_1 ∨ IsUnit b"
] | simp only [irreducible_iff, isUnit_mk, forall_associated, isUnit_mk, mk_mul_mk,
mk_eq_mk_iff_associated, Associated.comm (x := a)] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.GroupWithZero.Associated | {
"line": 658,
"column": 4
} | {
"line": 658,
"column": 25
} | {
"line": 659,
"column": 4
} | [
{
"pp": "case mpr\nM : Type u_1\ninst✝ : CommMonoidWithZero M\na : M\n⊢ (∀ ⦃a_1 b : M⦄, a = a_1 * b → IsUnit a_1 ∨ IsUnit b) → ∀ (a_2 a_3 : M), a_2 * a_3 ~ᵤ a → IsUnit a_2 ∨ IsUnit a_3",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"CommMonoidWithZero.toCommMonoid",
"Units.val"... | [
"case mpr\nM : Type u_1\ninst✝ : CommMonoidWithZero M\nx y : M\nu : Mˣ\nh : ∀ ⦃a b : M⦄, x * y * ↑u = a * b → IsUnit a ∨ IsUnit b\n⊢ IsUnit x ∨ IsUnit y"
] | rintro h x y ⟨u, rfl⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.Algebra.Ring.Subring.Basic | {
"line": 897,
"column": 2
} | {
"line": 898,
"column": 20
} | {
"line": 900,
"column": 0
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝¹ : NonAssocRing R\ninst✝ : NonAssocRing S\nf : R →+* S\ns : Set R\nx : R\nhx : x ∈ closure s\n⊢ f x ∈ closure (⇑f '' s)",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Subring.instSetLike",
"congrArg",
"Subring.map",... | [] | rw [← f.map_closure, Subring.mem_map]
exact ⟨x, hx, rfl⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Ring.Subring.Basic | {
"line": 897,
"column": 2
} | {
"line": 898,
"column": 20
} | {
"line": 900,
"column": 0
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝¹ : NonAssocRing R\ninst✝ : NonAssocRing S\nf : R →+* S\ns : Set R\nx : R\nhx : x ∈ closure s\n⊢ f x ∈ closure (⇑f '' s)",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Subring.instSetLike",
"congrArg",
"Subring.map",... | [] | rw [← f.map_closure, Subring.mem_map]
exact ⟨x, hx, rfl⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Span.Defs | {
"line": 121,
"column": 8
} | {
"line": 121,
"column": 27
} | {
"line": 121,
"column": 28
} | [
{
"pp": "case pos\nR : Type u_1\nM : Type u_4\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\ns : Set M\nh : 0 ∈ s\n⊢ span R (s \\ {0}) = span R s",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule",
"congrArg",
"AddMonoid.toAdd... | [
"case pos\nR : Type u_1\nM : Type u_4\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\ns : Set M\nh : 0 ∈ s\n⊢ span R (insert 0 (s \\ {0})) = span R s"
] | ← span_insert_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.ModularLattice | {
"line": 398,
"column": 2
} | {
"line": 398,
"column": 44
} | {
"line": 400,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝³ : Lattice α\ninst✝² : IsModularLattice α\na b : α\ninst✝¹ : BoundedOrder α\ninst✝ : ComplementedLattice α\nh : a ≤ b\n⊢ ∃ a', a ⊓ a' = ⊥ ∧ a ⊔ a' = b",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"IsModularLattice.exists_inf_eq_and_sup_eq",
"Orde... | [] | apply exists_inf_eq_and_sup_eq (by simp) h | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.LinearAlgebra.Span.Defs | {
"line": 384,
"column": 4
} | {
"line": 384,
"column": 39
} | {
"line": 385,
"column": 4
} | [
{
"pp": "R : Type u_1\nM : Type u_4\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nx : M\np p' : Submodule R M\nh : x ∈ span R (↑p ∪ ↑p')\n⊢ ∃ y ∈ p, ∃ z ∈ p', y + z = x",
"ppTerm": "?m.76",
"assigned": true,
"usedConstants": [
"Submodule",
"Submodule.span_induction"... | [
"case refine_1\nR : Type u_1\nM : Type u_4\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nx : M\np p' : Submodule R M\nh : x ∈ span R (↑p ∪ ↑p')\n⊢ ∀ x ∈ ↑p ∪ ↑p', ∃ y ∈ p, ∃ z ∈ p', y + z = x",
"case refine_2\nR : Type u_1\nM : Type u_4\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ :... | refine span_induction ?_ ?_ ?_ ?_ h | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Order.SupClosed | {
"line": 292,
"column": 6
} | {
"line": 292,
"column": 44
} | {
"line": 293,
"column": 6
} | [
{
"pp": "α : Type u_3\nβ : Type u_4\ninst✝¹ : SemilatticeSup α\ninst✝ : SemilatticeSup β\ns : Set α\nt : Set β\nu : Finset α\nhu : u.Nonempty\nhus : ↑u ⊆ s\nv : Finset β\nhv : v.Nonempty\nhvt : ↑v ⊆ t\n⊢ (u.sup' hu id, v.sup' hv id) ∈ supClosure (s ×ˢ t)",
"ppTerm": "?m.133",
"assigned": true,
"used... | [
"case refine_1\nα : Type u_3\nβ : Type u_4\ninst✝¹ : SemilatticeSup α\ninst✝ : SemilatticeSup β\ns : Set α\nt : Set β\nu : Finset α\nhu : u.Nonempty\nhus : ↑u ⊆ s\nv : Finset β\nhv : v.Nonempty\nhvt : ↑v ⊆ t\n⊢ ↑(u ×ˢ v) ⊆ s ×ˢ t",
"case refine_2\nα : Type u_3\nβ : Type u_4\ninst✝¹ : SemilatticeSup α\ninst✝ : Sem... | refine ⟨u ×ˢ v, hu.product hv, ?_, ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.LinearAlgebra.Span.Defs | {
"line": 449,
"column": 4
} | {
"line": 449,
"column": 39
} | {
"line": 450,
"column": 4
} | [
{
"pp": "R : Type u_1\nM : Type u_4\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nx y : M\nh : x ∈ R ∙ y\n⊢ ∃ a, a • y = x",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": [
"Submodule",
"instHSMul",
"DistribMulAction.toDistribSMul",
"AddMonoid.to... | [
"case refine_1\nR : Type u_1\nM : Type u_4\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nx y : M\nh : x ∈ R ∙ y\n⊢ ∀ x ∈ {y}, ∃ a, a • y = x",
"case refine_2\nR : Type u_1\nM : Type u_4\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nx y : M\nh : x ∈ R ∙ y\n⊢ ∃ a, a • y =... | refine span_induction ?_ ?_ ?_ ?_ h | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Order.SupClosed | {
"line": 350,
"column": 18
} | {
"line": 350,
"column": 56
} | {
"line": 351,
"column": 2
} | [
{
"pp": "case left\nα : Type u_3\nβ : Type u_4\ninst✝¹ : Lattice α\ninst✝ : Lattice β\ns : Set α\nf : α → β\nmap_sup : ∀ (a b : α), f (a ⊔ b) = f a ⊔ f b\nmap_inf : ∀ (a b : α), f (a ⊓ b) = f a ⊓ f b\n⊢ latticeClosure s ⊆ f ⁻¹' latticeClosure (f '' s)",
"ppTerm": "?left",
"assigned": true,
"usedCons... | [
"case left.mem\nα : Type u_3\nβ : Type u_4\ninst✝¹ : Lattice α\ninst✝ : Lattice β\ns : Set α\nf : α → β\nmap_sup : ∀ (a b : α), f (a ⊔ b) = f a ⊔ f b\nmap_inf : ∀ (a b : α), f (a ⊓ b) = f a ⊓ f b\n⊢ ∀ a ∈ s, a ∈ f ⁻¹' latticeClosure (f '' s)",
"case sup\nα : Type u_3\nβ : Type u_4\ninst✝¹ : Lattice α\ninst✝ : Lat... | apply latticeClosure_sup_inf_induction | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Order.SupClosed | {
"line": 350,
"column": 18
} | {
"line": 350,
"column": 56
} | {
"line": 351,
"column": 2
} | [
{
"pp": "case right\nα : Type u_3\nβ : Type u_4\ninst✝¹ : Lattice α\ninst✝ : Lattice β\ns : Set α\nf : α → β\nmap_sup : ∀ (a b : α), f (a ⊔ b) = f a ⊔ f b\nmap_inf : ∀ (a b : α), f (a ⊓ b) = f a ⊓ f b\n⊢ latticeClosure (f '' s) ⊆ f '' latticeClosure s",
"ppTerm": "?right",
"assigned": true,
"usedCon... | [
"case right.mem\nα : Type u_3\nβ : Type u_4\ninst✝¹ : Lattice α\ninst✝ : Lattice β\ns : Set α\nf : α → β\nmap_sup : ∀ (a b : α), f (a ⊔ b) = f a ⊔ f b\nmap_inf : ∀ (a b : α), f (a ⊓ b) = f a ⊓ f b\n⊢ ∀ a ∈ f '' s, a ∈ f '' latticeClosure s",
"case sup\nα : Type u_3\nβ : Type u_4\ninst✝¹ : Lattice α\ninst✝ : Latti... | apply latticeClosure_sup_inf_induction | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.LinearAlgebra.Span.Defs | {
"line": 472,
"column": 6
} | {
"line": 472,
"column": 17
} | {
"line": 472,
"column": 18
} | [
{
"pp": "R : Type u_1\nM : Type u_4\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nx : M\n⊢ R ∙ x = ⊤ ↔ ∀ (v : M), ∃ r, r • x = v",
"ppTerm": "?m.39",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule",
"instHSMul",
"eq_top_iff",
"congrArg",... | [
"R : Type u_1\nM : Type u_4\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nx : M\n⊢ ⊤ ≤ R ∙ x ↔ ∀ (v : M), ∃ r, r • x = v"
] | eq_top_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Span.Defs | {
"line": 563,
"column": 4
} | {
"line": 563,
"column": 14
} | {
"line": 564,
"column": 4
} | [
{
"pp": "case a\nR : Type u_1\nM : Type u_4\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\np : Submodule R M\nS : Set (Submodule R M) := ⋯\n⊢ p ≤ sSup {T | ∃ m ∈ p, m ≠ 0 ∧ T = R ∙ m}",
"ppTerm": "?a✝",
"assigned": true,
"usedConstants": [
"Submodule",
"Membership.me... | [
"case a\nR : Type u_1\nM : Type u_4\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\np : Submodule R M\nS : Set (Submodule R M) := {T | ∃ m ∈ p, m ≠ 0 ∧ T = R ∙ m}\nm : M\nhm : m ∈ p\n⊢ m ∈ sSup {T | ∃ m ∈ p, m ≠ 0 ∧ T = R ∙ m}"
] | intro m hm | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Order.Interval.Set.OrderIso | {
"line": 45,
"column": 2
} | {
"line": 45,
"column": 24
} | {
"line": 47,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : Preorder α\ninst✝ : Preorder β\ne : α ≃o β\na b : β\n⊢ ⇑e ⁻¹' Ioo a b = Ioo (e.symm a) (e.symm b)",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Set.Ioi",
"congrArg",
"Preorder.toLE",
"OrderIso",
"OrderIso.preima... | [] | simp [← Ioi_inter_Iio] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Order.Interval.Set.OrderIso | {
"line": 45,
"column": 2
} | {
"line": 45,
"column": 24
} | {
"line": 47,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : Preorder α\ninst✝ : Preorder β\ne : α ≃o β\na b : β\n⊢ ⇑e ⁻¹' Ioo a b = Ioo (e.symm a) (e.symm b)",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Set.Ioi",
"congrArg",
"Preorder.toLE",
"OrderIso",
"OrderIso.preima... | [] | simp [← Ioi_inter_Iio] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.Interval.Set.OrderIso | {
"line": 45,
"column": 2
} | {
"line": 45,
"column": 24
} | {
"line": 47,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : Preorder α\ninst✝ : Preorder β\ne : α ≃o β\na b : β\n⊢ ⇑e ⁻¹' Ioo a b = Ioo (e.symm a) (e.symm b)",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Set.Ioi",
"congrArg",
"Preorder.toLE",
"OrderIso",
"OrderIso.preima... | [] | simp [← Ioi_inter_Iio] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.SupIndep | {
"line": 122,
"column": 18
} | {
"line": 122,
"column": 21
} | {
"line": 123,
"column": 2
} | [
{
"pp": "α : Type u_1\nι : Type u_3\nι' : Type u_4\ninst✝² : Lattice α\ninst✝¹ : OrderBot α\nf : ι → α\ninst✝ : DecidableEq ι\ns : Finset ι'\ng : ι' → ι\nhs : s.SupIndep (f ∘ g)\nt : Finset ι\nht : t ⊆ image g s\ni : ι\nhi : i ∈ image g s\n⊢ i ∉ t → Disjoint (f i) (t.sup f)",
"ppTerm": "?m.23",
"assigne... | [
"α : Type u_1\nι : Type u_3\nι' : Type u_4\ninst✝² : Lattice α\ninst✝¹ : OrderBot α\nf : ι → α\ninst✝ : DecidableEq ι\ns : Finset ι'\ng : ι' → ι\nhs : s.SupIndep (f ∘ g)\nt : Finset ι\nht : t ⊆ image g s\ni : ι\nhi : i ∈ image g s\nhit : i ∉ t\n⊢ Disjoint (f i) (t.sup f)"
] | hit | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Order.SupIndep | {
"line": 179,
"column": 2
} | {
"line": 179,
"column": 6
} | {
"line": 180,
"column": 2
} | [
{
"pp": "α : Type u_1\nι : Type u_3\nι' : Type u_4\ninst✝³ : Lattice α\ninst✝² : IsModularLattice α\ninst✝¹ : OrderBot α\ninst✝ : DecidableEq ι\ns : Finset ι'\ng : ι' → Finset ι\nf : ι → α\nhs : s.SupIndep fun i ↦ (g i).sup f\nhg : ∀ i' ∈ s, (g i').SupIndep f\na : Finset ι\nha : a ⊆ s.biUnion g\nb : ι\nhb✝ : b ... | [
"α : Type u_1\nι : Type u_3\nι' : Type u_4\ninst✝³ : Lattice α\ninst✝² : IsModularLattice α\ninst✝¹ : OrderBot α\ninst✝ : DecidableEq ι\ns : Finset ι'\ng : ι' → Finset ι\nf : ι → α\nhs : s.SupIndep fun i ↦ (g i).sup f\nhg : ∀ i' ∈ s, (g i').SupIndep f\na : Finset ι\nha : a ⊆ s.biUnion g\nb : ι\nhb✝ : b ∈ s.biUnion ... | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.Order.CompactlyGenerated.Basic | {
"line": 154,
"column": 10
} | {
"line": 154,
"column": 47
} | {
"line": 154,
"column": 47
} | [
{
"pp": "case mp\nα : Type u\ninst✝ : CompleteLattice α\nk : α\nH : ∀ (s : Set α), k ≤ sSup s → ∃ t, ↑t ⊆ s ∧ k ≤ t.sup id\nι : Type u\ns : ι → α\nhs : k ≤ iSup s\nt : Finset α\nht : ↑t ⊆ range s\nht' : k ≤ t.sup id\nf : ↥t → ι\nhf : ∀ (x : ↥t), s (f x) = ↑x\nb : α\nhb : b ∈ t\n⊢ id b ≤ (Finset.image f Finset.u... | [
"case mp\nα : Type u\ninst✝ : CompleteLattice α\nk : α\nH : ∀ (s : Set α), k ≤ sSup s → ∃ t, ↑t ⊆ s ∧ k ≤ t.sup id\nι : Type u\ns : ι → α\nhs : k ≤ iSup s\nt : Finset α\nht : ↑t ⊆ range s\nht' : k ≤ t.sup id\nf : ↥t → ι\nhf : ∀ (x : ↥t), s (f x) = ↑x\nb : α\nhb : b ∈ t\n⊢ s (f ⟨b, hb⟩) ≤ (Finset.image f Finset.univ... | ← show s (f ⟨b, hb⟩) = id b from hf _ | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.CompactlyGenerated.Basic | {
"line": 220,
"column": 4
} | {
"line": 221,
"column": 26
} | {
"line": 223,
"column": 0
} | [
{
"pp": "case refine_2\nα : Type u_2\ninst✝¹ : CompleteLattice α\ninst✝ : WellFoundedGT α\ns : Set α\nS : Set α := {x | ∃ t, ↑t ⊆ s ∧ t.sup id = x}\nt : Finset α\nht₁ : ↑t ⊆ s\nhm : ∀ x ∈ S, ¬t.sup id < x\n⊢ t.sup id ≤ sSup s",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"Eq.mp... | [] | rw [Finset.sup_id_eq_sSup]
exact sSup_le_sSup ht₁ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.CompactlyGenerated.Basic | {
"line": 220,
"column": 4
} | {
"line": 221,
"column": 26
} | {
"line": 223,
"column": 0
} | [
{
"pp": "case refine_2\nα : Type u_2\ninst✝¹ : CompleteLattice α\ninst✝ : WellFoundedGT α\ns : Set α\nS : Set α := {x | ∃ t, ↑t ⊆ s ∧ t.sup id = x}\nt : Finset α\nht₁ : ↑t ⊆ s\nhm : ∀ x ∈ S, ¬t.sup id < x\n⊢ t.sup id ≤ sSup s",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"Eq.mp... | [] | rw [Finset.sup_id_eq_sSup]
exact sSup_le_sSup ht₁ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.SupIndep | {
"line": 487,
"column": 8
} | {
"line": 487,
"column": 26
} | {
"line": 487,
"column": 26
} | [
{
"pp": "α : Type u_1\nι : Type u_3\ninst✝ : CompleteLattice α\ns : Finset ι\nf : ι → α\na : ι\nb : a ∈ s\n⊢ Disjoint ((f ∘ Subtype.val) ⟨a, b⟩) (⨆ j, ⨆ (_ : j ≠ ⟨a, b⟩), (f ∘ Subtype.val) j) ↔\n Disjoint (f a) ((s.erase a).sup f)",
"ppTerm": "?m.39",
"assigned": true,
"usedConstants": [
"E... | [
"α : Type u_1\nι : Type u_3\ninst✝ : CompleteLattice α\ns : Finset ι\nf : ι → α\na : ι\nb : a ∈ s\n⊢ Disjoint ((f ∘ Subtype.val) ⟨a, b⟩) (⨆ j, ⨆ (_ : j ≠ ⟨a, b⟩), (f ∘ Subtype.val) j) ↔\n Disjoint (f a) (⨆ a_1 ∈ s.erase a, f a_1)"
] | Finset.sup_eq_iSup | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Algebra.Tower | {
"line": 399,
"column": 58
} | {
"line": 399,
"column": 83
} | {
"line": 399,
"column": 83
} | [
{
"pp": "R : Type u\nS : Type v\nA : Type w\ninst✝⁶ : Semiring R\ninst✝⁵ : Semiring S\ninst✝⁴ : AddCommMonoid A\ninst✝³ : Module R S\ninst✝² : Module S A\ninst✝¹ : Module R A\ninst✝ : IsScalarTower R S A\ns : Set S\nhs : span R s = ⊤\nt : Set A\nx✝ : A\nhp : x✝ ∈ restrictScalars R (span S t)\ns0 : S\ny : A\nhy ... | [] | exact ⟨x, hx, y, hy, rfl⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.Algebra.Tower | {
"line": 399,
"column": 58
} | {
"line": 399,
"column": 83
} | {
"line": 399,
"column": 83
} | [
{
"pp": "R : Type u\nS : Type v\nA : Type w\ninst✝⁶ : Semiring R\ninst✝⁵ : Semiring S\ninst✝⁴ : AddCommMonoid A\ninst✝³ : Module R S\ninst✝² : Module S A\ninst✝¹ : Module R A\ninst✝ : IsScalarTower R S A\ns : Set S\nhs : span R s = ⊤\nt : Set A\nx✝ : A\nhp : x✝ ∈ restrictScalars R (span S t)\ns0 : S\ny : A\nhy ... | [] | exact ⟨x, hx, y, hy, rfl⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Algebra.Tower | {
"line": 399,
"column": 58
} | {
"line": 399,
"column": 83
} | {
"line": 399,
"column": 83
} | [
{
"pp": "R : Type u\nS : Type v\nA : Type w\ninst✝⁶ : Semiring R\ninst✝⁵ : Semiring S\ninst✝⁴ : AddCommMonoid A\ninst✝³ : Module R S\ninst✝² : Module S A\ninst✝¹ : Module R A\ninst✝ : IsScalarTower R S A\ns : Set S\nhs : span R s = ⊤\nt : Set A\nx✝ : A\nhp : x✝ ∈ restrictScalars R (span S t)\ns0 : S\ny : A\nhy ... | [] | exact ⟨x, hx, y, hy, rfl⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
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