module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.LinearAlgebra.LinearIndependent.Basic | {
"line": 217,
"column": 26
} | {
"line": 217,
"column": 31
} | {
"line": 217,
"column": 31
} | [
{
"pp": "R : Type u_2\nM : Type u_4\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\ns : Set M\nli : LinearIndependent R Subtype.val\nt : Finset ↑s\nx✝ : ↥(Finset.map (Embedding.subtype fun x ↦ x ∈ s) t)\n⊢ x✝.1 ∈ s",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.LinearAlgebra.LinearIndependent.Basic | {
"line": 217,
"column": 26
} | {
"line": 217,
"column": 31
} | {
"line": 217,
"column": 31
} | [
{
"pp": "R : Type u_2\nM : Type u_4\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\ns : Set M\nli : LinearIndependent R Subtype.val\nt : Finset ↑s\nx✝ : ↥(Finset.map (Embedding.subtype fun x ↦ x ∈ s) t)\n⊢ x✝.1 ∈ s",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.LinearIndependent.Basic | {
"line": 217,
"column": 26
} | {
"line": 217,
"column": 31
} | {
"line": 217,
"column": 31
} | [
{
"pp": "R : Type u_2\nM : Type u_4\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\ns : Set M\nli : LinearIndependent R Subtype.val\nt : Finset ↑s\nx✝ : ↥(Finset.map (Embedding.subtype fun x ↦ x ∈ s) t)\n⊢ x✝.1 ∈ s",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.LinearIndependent.Basic | {
"line": 264,
"column": 2
} | {
"line": 264,
"column": 42
} | {
"line": 265,
"column": 2
} | [
{
"pp": "ι : Type u'\nR : Type u_2\nM : Type u_4\nv : ι → M\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ninst✝ : Nontrivial R\nhv : LinearIndependent R v\nx : ι\nf : ι →₀ R\nh : x ∉ f.support\n⊢ (Finsupp.linearCombination R v) f ≠ v x",
"ppTerm": "?m.24",
"assigned": true,
"u... | [
"ι : Type u'\nR : Type u_2\nM : Type u_4\nv : ι → M\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ninst✝ : Nontrivial R\nhv : LinearIndependent R v\nx : ι\nf : ι →₀ R\nh : x ∉ ↑f.support\n⊢ (Finsupp.linearCombination R v) f ≠ v x"
] | replace h : x ∉ (f.support : Set ι) := h | Lean.Elab.Tactic.evalReplace | Lean.Parser.Tactic.replace |
Mathlib.Data.Nat.ModEq | {
"line": 59,
"column": 2
} | {
"line": 59,
"column": 38
} | {
"line": 61,
"column": 0
} | [
{
"pp": "M : Type u_1\ninst✝ : AddCommMonoidWithOne M\na b n : ℕ\nh : a ≡ b [PMOD n]\n⊢ ↑a ≡ ↑b [PMOD ↑n]",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Nat.castAddMonoidHom",
"AddMonoidHom.instAddMonoidHomClass",
"AddCommMonoidWithOne.toAddCommMonoid",
"AddMonoid... | [] | exact h.map (Nat.castAddMonoidHom M) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.LinearAlgebra.LinearIndependent.Defs | {
"line": 607,
"column": 6
} | {
"line": 608,
"column": 51
} | {
"line": 609,
"column": 4
} | [
{
"pp": "case refine_2.specialize_1\nι : Type u'\nR : Type u_2\nM : Type u_4\nv : ι → M\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\ninst✝³ : LinearOrder R\ninst✝² : CanonicallyOrderedAdd R\ninst✝¹ : AddRightReflectLE R\ninst✝ : IsCancelAdd M\nthis✝ : Sub R := CanonicallyOrderedAdd.toSub... | [] | simp_rw [Finset.disjoint_left, Finset.mem_filter]
exact fun i ⟨_, hi⟩ ⟨_, hi'⟩ => hi.not_gt hi' | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.LinearIndependent.Defs | {
"line": 607,
"column": 6
} | {
"line": 608,
"column": 51
} | {
"line": 609,
"column": 4
} | [
{
"pp": "case refine_2.specialize_1\nι : Type u'\nR : Type u_2\nM : Type u_4\nv : ι → M\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\ninst✝³ : LinearOrder R\ninst✝² : CanonicallyOrderedAdd R\ninst✝¹ : AddRightReflectLE R\ninst✝ : IsCancelAdd M\nthis✝ : Sub R := CanonicallyOrderedAdd.toSub... | [] | simp_rw [Finset.disjoint_left, Finset.mem_filter]
exact fun i ⟨_, hi⟩ ⟨_, hi'⟩ => hi.not_gt hi' | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Nat.ModEq | {
"line": 554,
"column": 29
} | {
"line": 556,
"column": 34
} | {
"line": 558,
"column": 0
} | [
{
"pp": "a b c : ℕ\nh : c ∣ a + b\nha : ¬c ∣ a\nhc : ¬c ≤ a % c + b % c\n⊢ False",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"False",
"Dvd.dvd",
"congrArg",
"Nat.add_mod_of_add_mod_lt",
"False.elim",
"lt_of_not_ge",
"Nat.add_eq_zero_iff._simp_1... | [] | by
have : (a + b) % c = a % c + b % c := add_mod_of_add_mod_lt (lt_of_not_ge hc)
simp_all [dvd_iff_mod_eq_zero] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.LinearIndependent.Basic | {
"line": 479,
"column": 2
} | {
"line": 479,
"column": 7
} | {
"line": 481,
"column": 0
} | [
{
"pp": "case e'_4\nι : Type u'\nR : Type u_2\ns : Set ι\nM : Type u_4\nv : ι → M\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nt : Set ι\nhdj : Disjoint s t\nh : LinearIndepOn R v (s ∪ t)\n⊢ v '' s = (fun x ↦ v ↑x) '' Subtype.val ⁻¹' s",
"ppTerm": "?e'_4",
"assigned": true,
"usedCo... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.LinearAlgebra.LinearIndependent.Basic | {
"line": 479,
"column": 2
} | {
"line": 479,
"column": 7
} | {
"line": 481,
"column": 0
} | [
{
"pp": "case e'_5\nι : Type u'\nR : Type u_2\ns : Set ι\nM : Type u_4\nv : ι → M\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nt : Set ι\nhdj : Disjoint s t\nh : LinearIndepOn R v (s ∪ t)\n⊢ v '' t = (fun x ↦ v ↑x) '' Subtype.val ⁻¹' t",
"ppTerm": "?e'_5",
"assigned": true,
"usedCo... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.SetTheory.Cardinal.Finite | {
"line": 332,
"column": 40
} | {
"line": 332,
"column": 67
} | {
"line": 332,
"column": 68
} | [
{
"pp": "n : ℕ\nc : Cardinal.{u_3}\n⊢ ↑n ≤ toENat c ∧ toENat c ≤ ↑n ↔ ↑n ≤ c ∧ c ≤ ↑n",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"ENat.instNatCast",
"Cardinal",
"instLinearOrderENat",
"congrArg",
"CommSemiring.toSemiring",
"Cardinal.... | [
"n : ℕ\nc : Cardinal.{u_3}\n⊢ ↑n ≤ toENat c ∧ c ≤ ↑n ↔ ↑n ≤ c ∧ c ≤ ↑n"
] | Cardinal.toENat_le_natCast, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.SetTheory.Cardinal.Finite | {
"line": 403,
"column": 4
} | {
"line": 416,
"column": 31
} | {
"line": 418,
"column": 0
} | [
{
"pp": "case inr.inr\nα : Type u_3\nβ : Type u_4\nα_emp : Nonempty α\nh✝ : Infinite α\n⊢ card (α → β) = card β ^ card α",
"ppTerm": "?inr.inr",
"assigned": true,
"usedConstants": [
"Nontrivial",
"Iff.mpr",
"Eq.mpr",
"Eq.ge",
"Preorder.toLT",
"NeZero.one",
"... | [] | rw [card_eq_top_of_infinite (α := α)]
rcases lt_trichotomy (card β) 1 with b_0 | b_1 | b_2
· rw [Order.lt_one_iff, card_eq_zero_iff_empty] at b_0
rw [(card_eq_zero_iff_empty β).2 b_0, zero_epow_top, card_eq_zero_iff_empty]
simp [b_0]
· rw [b_1, one_epow]
apply le_antisymm
· letI := (... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.SetTheory.Cardinal.Finite | {
"line": 403,
"column": 4
} | {
"line": 416,
"column": 31
} | {
"line": 418,
"column": 0
} | [
{
"pp": "case inr.inr\nα : Type u_3\nβ : Type u_4\nα_emp : Nonempty α\nh✝ : Infinite α\n⊢ card (α → β) = card β ^ card α",
"ppTerm": "?inr.inr",
"assigned": true,
"usedConstants": [
"Nontrivial",
"Iff.mpr",
"Eq.mpr",
"Eq.ge",
"Preorder.toLT",
"NeZero.one",
"... | [] | rw [card_eq_top_of_infinite (α := α)]
rcases lt_trichotomy (card β) 1 with b_0 | b_1 | b_2
· rw [Order.lt_one_iff, card_eq_zero_iff_empty] at b_0
rw [(card_eq_zero_iff_empty β).2 b_0, zero_epow_top, card_eq_zero_iff_empty]
simp [b_0]
· rw [b_1, one_epow]
apply le_antisymm
· letI := (... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.BigOperators.Fin | {
"line": 616,
"column": 58
} | {
"line": 616,
"column": 75
} | {
"line": 616,
"column": 75
} | [
{
"pp": "m n : ℕ\ninst✝ : NeZero m\ni : Fin n\nj : Fin m\n⊢ ↑(Pi.single i j i) * m ^ ↑i = ↑j * m ^ ↑i",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"congrArg",
"Nat.instMonoid",
"instDecidableEqFin",
"Zero.ofOfNat0",
"id",
... | [
"m n : ℕ\ninst✝ : NeZero m\ni : Fin n\nj : Fin m\n⊢ ↑j * m ^ ↑i = ↑j * m ^ ↑i",
"m n : ℕ\ninst✝ : NeZero m\ni : Fin n\nj : Fin m\n⊢ ∀ (x : Fin n), x ≠ i → ↑(Pi.single i j x) * m ^ ↑x = 0"
] | Pi.single_eq_same | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.BigOperators.Fin | {
"line": 676,
"column": 52
} | {
"line": 676,
"column": 69
} | {
"line": 676,
"column": 69
} | [
{
"pp": "m : ℕ\nn : Fin m → ℕ\ninst✝ : ∀ (i : Fin m), NeZero (n i)\ni : Fin m\nj : Fin (n i)\n⊢ ↑(Pi.single i j i) * ∏ j, n (Fin.castLE ⋯ j) = ↑j * ∏ j, n (Fin.castLE ⋯ j)",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"Finset.univ",
"congrAr... | [
"m : ℕ\nn : Fin m → ℕ\ninst✝ : ∀ (i : Fin m), NeZero (n i)\ni : Fin m\nj : Fin (n i)\n⊢ ↑j * ∏ j, n (Fin.castLE ⋯ j) = ↑j * ∏ j, n (Fin.castLE ⋯ j)",
"m : ℕ\nn : Fin m → ℕ\ninst✝ : ∀ (i : Fin m), NeZero (n i)\ni : Fin m\nj : Fin (n i)\n⊢ ∀ (x : Fin m), x ≠ i → ↑(Pi.single i j x) * ∏ j, n (Fin.castLE ⋯ j) = 0"
] | Pi.single_eq_same | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Set.Card | {
"line": 165,
"column": 2
} | {
"line": 165,
"column": 26
} | {
"line": 165,
"column": 26
} | [
{
"pp": "α : Type u_1\ns : Set α\nk : ℕ\nh : s.encard ≤ ↑k\n⊢ s.Finite",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Set.encard",
"instTopENat",
"congrArg",
"Set.Finite",
"Set.encard_lt_top_iff",
"id",
"ENat",
"LT.lt",
... | [
"α : Type u_1\ns : Set α\nk : ℕ\nh : s.encard ≤ ↑k\n⊢ s.encard < ⊤"
] | rw [← encard_lt_top_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.LinearIndependent.Defs | {
"line": 866,
"column": 13
} | {
"line": 866,
"column": 43
} | {
"line": 867,
"column": 6
} | [
{
"pp": "ι : Type u'\nR : Type u_2\nM : Type u_4\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nv : ι → M\nH : ∀ (i : ι) (a : R), a • v i ∈ span R (v '' (univ \\ {i})) → a = 0\nl : ι →₀ R\nhl : (Finsupp.linearCombination R v) l = 0\ni : ι\n⊢ l i = 0 i",
"ppTerm": "?m.63",
"assigned": tru... | [
"ι : Type u'\nR : Type u_2\nM : Type u_4\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nv : ι → M\nH : ∀ (i : ι) (a : R), a • v i ∈ span R (v '' (univ \\ {i})) → a = 0\nl : ι →₀ R\nhl : (Finsupp.linearCombination R v) l = 0\ni : ι\n⊢ l i = 0"
] | simp only [Finsupp.zero_apply] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Data.Set.Card | {
"line": 439,
"column": 82
} | {
"line": 439,
"column": 87
} | {
"line": 441,
"column": 0
} | [
{
"pp": "case refine_2\nα : Type u_1\ns : Set α\nx✝ : ∃ x y z, x ≠ y ∧ x ≠ z ∧ y ≠ z ∧ s = {x, y, z}\nx y z : α\nhxy : x ≠ y\nhyz : x ≠ z\nhxz : y ≠ z\nhs : s = {x, y, z}\n⊢ 1 + 1 + 1 = 3",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"instAddMonoidWithOneENat",
"ENat.inst... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Data.Set.Card | {
"line": 439,
"column": 82
} | {
"line": 439,
"column": 87
} | {
"line": 441,
"column": 0
} | [
{
"pp": "case refine_2\nα : Type u_1\ns : Set α\nx✝ : ∃ x y z, x ≠ y ∧ x ≠ z ∧ y ≠ z ∧ s = {x, y, z}\nx y z : α\nhxy : x ≠ y\nhyz : x ≠ z\nhxz : y ≠ z\nhs : s = {x, y, z}\n⊢ y ∉ {z}",
"ppTerm": "?refine_2✝",
"assigned": true,
"usedConstants": [
"False",
"eq_false",
"congrArg",
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Data.Set.Card | {
"line": 439,
"column": 82
} | {
"line": 439,
"column": 87
} | {
"line": 441,
"column": 0
} | [
{
"pp": "case refine_2\nα : Type u_1\ns : Set α\nx✝ : ∃ x y z, x ≠ y ∧ x ≠ z ∧ y ≠ z ∧ s = {x, y, z}\nx y z : α\nhxy : x ≠ y\nhyz : x ≠ z\nhxz : y ≠ z\nhs : s = {x, y, z}\n⊢ x ∉ {y, z}",
"ppTerm": "?refine_2✝",
"assigned": true,
"usedConstants": [
"False",
"eq_false",
"congrArg",
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Data.Set.Card | {
"line": 459,
"column": 2
} | {
"line": 459,
"column": 34
} | {
"line": 460,
"column": 2
} | [
{
"pp": "case e'_2\nk : ℕ\n⊢ {i | i < k}.encard = (↑(Finset.range k)).encard",
"ppTerm": "?e'_2",
"assigned": true,
"usedConstants": [
"Set.Iio_def",
"Eq.mpr",
"Set.encard",
"Preorder.toLT",
"Finset.coe_range",
"congrArg",
"Finset",
"setOf",
"id"... | [
"case e'_3\nk : ℕ\n⊢ ↑k = ↑(Finset.range k).card"
] | · rw [Finset.coe_range, Iio_def] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Data.Set.Card | {
"line": 694,
"column": 42
} | {
"line": 694,
"column": 52
} | {
"line": 696,
"column": 0
} | [
{
"pp": "α : Type u_1\nhs : ∅.ncard ≠ 0\n⊢ False",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"False",
"congrArg",
"False.elim",
"Set.ncard_empty",
"Eq.mp",
"not_true_eq_false",
"Ne",
"instOfNatNat",
"Nat",
"True",
"eq_se... | [] | simp at hs | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Data.Set.Card | {
"line": 700,
"column": 2
} | {
"line": 700,
"column": 72
} | {
"line": 701,
"column": 2
} | [
{
"pp": "α : Type u_1\na : α\ns : Set α\n⊢ ({a} ∩ s).ncard ≤ 1",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Set.encard",
"Set.fintypeSingleton",
"instCharZeroENat",
"instAddMonoidWithOneENat",
"ENat.instNatCast",
"instLinearOrderENat"... | [
"α : Type u_1\na : α\ns : Set α\n⊢ ({a} ∩ s).encard ≤ 1"
] | rw [← Nat.cast_le (α := ℕ∞), (toFinite _).cast_ncard_eq, Nat.cast_one] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Data.Set.Card | {
"line": 1266,
"column": 6
} | {
"line": 1266,
"column": 53
} | {
"line": 1266,
"column": 54
} | [
{
"pp": "case inl\nα : Type u_1\ns t : Set α\nhs : s.Finite\nht : t.Finite\n⊢ (∃ a ∉ s, insert a s = t) ↔ s ⊆ t ∧ hs.toFinset.card + 1 = ht.toFinset.card",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Finset",
"PartialOrder.toPreorder",
... | [
"case inl\nα : Type u_1\ns t : Set α\nhs : s.Finite\nht : t.Finite\n⊢ (∃ a ∉ s, insert a s = t) ↔ hs.toFinset ⊆ ht.toFinset ∧ hs.toFinset.card + 1 = ht.toFinset.card"
] | ← @Finite.toFinset_subset_toFinset _ _ _ hs ht, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Set.Card | {
"line": 1331,
"column": 2
} | {
"line": 1331,
"column": 31
} | {
"line": 1332,
"column": 2
} | [
{
"pp": "α : Type u_1\ns : Set α\nhs : s.Finite\nhn : s.Nonempty\nhe : Even s.ncard\n⊢ 1 < s.ncard",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"congrArg",
"Eq.mp",
"instOfNatNat",
"Set.Nonempty",
"Nat",
"LT.lt",
"Even",
"propext",
"... | [
"α : Type u_1\ns : Set α\nhs : s.Finite\nhn : 0 < s.ncard\nhe : Even s.ncard\n⊢ 1 < s.ncard"
] | rw [← Set.ncard_pos hs] at hn | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.Basis.Submodule | {
"line": 183,
"column": 2
} | {
"line": 186,
"column": 63
} | {
"line": 188,
"column": 0
} | [
{
"pp": "ι : Type u_1\nR : Type u_3\nM : Type u_5\nS : Type u_7\ninst✝⁹ : CommRing R\ninst✝⁸ : IsDomain R\ninst✝⁷ : Ring S\ninst✝⁶ : Nontrivial S\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Algebra R S\ninst✝³ : Module S M\ninst✝² : Module R M\ninst✝¹ : IsScalarTower R S M\ninst✝ : IsTorsionFree R S\nb : Basis ι S M\nm ... | [] | simp only [LinearMap.coe_comp, LinearEquiv.coe_toLinearMap, Function.comp_apply, map_one,
Basis.repr_self, Finsupp.mapRange.linearMap_apply, Finsupp.mapRange_single,
Algebra.linearMap_apply, LinearMap.domRestrict_apply,
Basis.restrictScalars_apply, LinearMap.coe_restrictScalars] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Group.Graph | {
"line": 60,
"column": 83
} | {
"line": 60,
"column": 88
} | {
"line": 62,
"column": 0
} | [
{
"pp": "G : Type u_1\nH : Type u_2\ninst✝¹ : Monoid G\ninst✝ : Monoid H\nf : G →* H\n⊢ f.mgraph = mrange ((id G).prod f)",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MonoidHom.mem_mgraph._simp_2",
"MonoidHom.instMonoidHomClass",
"MonoidHom.instFunLike... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Group.Graph | {
"line": 60,
"column": 83
} | {
"line": 60,
"column": 88
} | {
"line": 62,
"column": 0
} | [
{
"pp": "G : Type u_1\nH : Type u_2\ninst✝¹ : Monoid G\ninst✝ : Monoid H\nf : G →* H\n⊢ f.mgraph = mrange ((id G).prod f)",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MonoidHom.mem_mgraph._simp_2",
"MonoidHom.instMonoidHomClass",
"MonoidHom.instFunLike... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Group.Graph | {
"line": 60,
"column": 83
} | {
"line": 60,
"column": 88
} | {
"line": 62,
"column": 0
} | [
{
"pp": "G : Type u_1\nH : Type u_2\ninst✝¹ : Monoid G\ninst✝ : Monoid H\nf : G →* H\n⊢ f.mgraph = mrange ((id G).prod f)",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MonoidHom.mem_mgraph._simp_2",
"MonoidHom.instMonoidHomClass",
"MonoidHom.instFunLike... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Group.Graph | {
"line": 171,
"column": 79
} | {
"line": 171,
"column": 84
} | {
"line": 173,
"column": 0
} | [
{
"pp": "G : Type u_1\nH : Type u_2\ninst✝¹ : Group G\ninst✝ : Group H\nf : G →* H\n⊢ f.graph = ((id G).prod f).range",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MonoidHom.range",
"MonoidHom.instFunLike",
"MonoidHom.mem_range._simp_2",
"MonoidHo... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Group.Graph | {
"line": 171,
"column": 79
} | {
"line": 171,
"column": 84
} | {
"line": 173,
"column": 0
} | [
{
"pp": "G : Type u_1\nH : Type u_2\ninst✝¹ : Group G\ninst✝ : Group H\nf : G →* H\n⊢ f.graph = ((id G).prod f).range",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MonoidHom.range",
"MonoidHom.instFunLike",
"MonoidHom.mem_range._simp_2",
"MonoidHo... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Group.Graph | {
"line": 171,
"column": 79
} | {
"line": 171,
"column": 84
} | {
"line": 173,
"column": 0
} | [
{
"pp": "G : Type u_1\nH : Type u_2\ninst✝¹ : Group G\ninst✝ : Group H\nf : G →* H\n⊢ f.graph = ((id G).prod f).range",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MonoidHom.range",
"MonoidHom.instFunLike",
"MonoidHom.mem_range._simp_2",
"MonoidHo... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Group.Graph | {
"line": 217,
"column": 33
} | {
"line": 218,
"column": 63
} | {
"line": 220,
"column": 0
} | [
{
"pp": "H : Type u_2\nI : Type u_3\ninst✝¹ : Group H\ninst✝ : Group I\nG : Subgroup (H × I)\nhG₁ : Bijective (Prod.fst ∘ ⇑G.subtype)\n⊢ ∃ f, G = f.graph",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MonoidHom.mem_mgraph._simp_2",
"MonoidHom.instFunLike",
... | [] | by
simpa [SetLike.ext_iff] using! Submonoid.exists_eq_mgraph hG₁ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.GroupTheory.Congruence.Basic | {
"line": 94,
"column": 2
} | {
"line": 94,
"column": 31
} | {
"line": 95,
"column": 2
} | [
{
"pp": "M : Type u_4\nN : Type u_5\ninst✝¹ : Mul M\ninst✝ : Mul N\nf : M ≃* N\nrel : N → N → Prop\na b : M\nn1 : N\nfa : f a = n1\nh : (conGen rel) n1 (f b)\n⊢ (conGen fun x y ↦ rel (f x) (f y)) a b",
"ppTerm": "?m.59",
"assigned": true,
"usedConstants": [
"MulEquiv.instEquivLike",
"con... | [
"M : Type u_4\nN : Type u_5\ninst✝¹ : Mul M\ninst✝ : Mul N\nf : M ≃* N\nrel : N → N → Prop\na b : M\nn1 : N\nfa : f a = n1\nn2 : N\nfb : f b = n2\nh : (conGen rel) n1 n2\n⊢ (conGen fun x y ↦ rel (f x) (f y)) a b"
] | generalize fb : f b = n2 at h | Lean.Elab.Tactic.evalGeneralize | Lean.Parser.Tactic.generalize |
Mathlib.LinearAlgebra.Pi | {
"line": 205,
"column": 14
} | {
"line": 205,
"column": 31
} | {
"line": 205,
"column": 31
} | [
{
"pp": "R : Type u\nι : Type x\ninst✝³ : Semiring R\nφ : ι → Type i\ninst✝² : (i : ι) → AddCommMonoid (φ i)\ninst✝¹ : (i : ι) → Module R (φ i)\ninst✝ : DecidableEq ι\nI : Finset ι\nJ : Set ι\nhu : Set.univ ⊆ ↑I ∪ J\nb : (i : ι) → φ i\nhb : ∀ i ∈ J, b i = 0\ni : ι\nhiI : i ∉ I\n⊢ Pi.single i (b i) i = 0",
"... | [
"R : Type u\nι : Type x\ninst✝³ : Semiring R\nφ : ι → Type i\ninst✝² : (i : ι) → AddCommMonoid (φ i)\ninst✝¹ : (i : ι) → Module R (φ i)\ninst✝ : DecidableEq ι\nI : Finset ι\nJ : Set ι\nhu : Set.univ ⊆ ↑I ∪ J\nb : (i : ι) → φ i\nhb : ∀ i ∈ J, b i = 0\ni : ι\nhiI : i ∉ I\n⊢ b i = 0"
] | Pi.single_eq_same | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.Interval.Set.SuccPred | {
"line": 70,
"column": 62
} | {
"line": 70,
"column": 67
} | {
"line": 72,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝¹ : LinearOrder α\ninst✝ : SuccOrder α\na b : α\nh : a ≤ b\nx : α\n⊢ a ≤ x → a = x → x ≤ b",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"PartialOrder.toPreorder",
"Preorder.toLE",
"SemilatticeInf.toPartialOrder",
"DistribLattice.toLatt... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.LinearAlgebra.Quotient.Basic | {
"line": 225,
"column": 5
} | {
"line": 225,
"column": 63
} | {
"line": 225,
"column": 63
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\np : Submodule R M\nR₂ : Type u_3\nM₂ : Type u_4\ninst✝³ : Ring R₂\ninst✝² : AddCommGroup M₂\ninst✝¹ : Module R₂ M₂\nτ₁₂ : R →+* R₂\ninst✝ : RingHomSurjective τ₁₂\nf : M →ₛₗ[τ₁₂] M₂\nh : p ≤ f.ker\nq : Submodule R... | [] | by rintro _ ⟨x, hxq, rfl⟩; exact ⟨Quotient.mk x, hxq, rfl⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.Prod | {
"line": 419,
"column": 4
} | {
"line": 419,
"column": 19
} | {
"line": 420,
"column": 4
} | [
{
"pp": "case codisjoint\nR : Type u\nM : Type v\nM₂ : Type w\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : AddCommMonoid M₂\ninst✝¹ : Module R M\ninst✝ : Module R M₂\n⊢ ⊤ ≤ (inl R M M₂).range ⊔ (inr R M M₂).range",
"ppTerm": "?codisjoint",
"assigned": true,
"usedConstants": [
"Subm... | [
"case codisjoint\nR : Type u\nM : Type v\nM₂ : Type w\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : AddCommMonoid M₂\ninst✝¹ : Module R M\ninst✝ : Module R M₂\nx : M\ny : M₂\n⊢ (x, y) ∈ (inl R M M₂).range ⊔ (inr R M M₂).range"
] | rintro ⟨x, y⟩ - | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.Data.Fintype.Order | {
"line": 297,
"column": 2
} | {
"line": 297,
"column": 61
} | {
"line": 299,
"column": 0
} | [
{
"pp": "case refine_2\nα : Type u_1\nι : Type u_2\ninst✝² : Finite ι\ninst✝¹ : ConditionallyCompleteLattice α\ninst✝ : Nonempty ι\nf : ι → α\na : α\n⊢ a ≤ ⨆ i, f i ⊔ a",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"Lattice.toSemilatticeSup",
"SemilatticeSup.toMax",
... | [] | · exact le_ciSup_of_le (Classical.arbitrary ι) le_sup_right | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.LinearAlgebra.Prod | {
"line": 561,
"column": 26
} | {
"line": 561,
"column": 31
} | {
"line": 561,
"column": 31
} | [
{
"pp": "R : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM₂ : Type w\nV₂ : Type w'\nM₃ : Type y\nV₃ : Type y'\nM₄ : Type z\nι : Type x\nM₅ : Type u_1\nM₆ : Type u_2\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : AddCommMonoid M₂\ninst✝¹ : Module R M\ninst✝ : Module R M₂\np : Submodule R M\nq : Submo... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.LinearAlgebra.Prod | {
"line": 561,
"column": 26
} | {
"line": 561,
"column": 31
} | {
"line": 561,
"column": 31
} | [
{
"pp": "R : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM₂ : Type w\nV₂ : Type w'\nM₃ : Type y\nV₃ : Type y'\nM₄ : Type z\nι : Type x\nM₅ : Type u_1\nM₆ : Type u_2\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : AddCommMonoid M₂\ninst✝¹ : Module R M\ninst✝ : Module R M₂\np : Submodule R M\nq : Submo... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Prod | {
"line": 561,
"column": 26
} | {
"line": 561,
"column": 31
} | {
"line": 561,
"column": 31
} | [
{
"pp": "R : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM₂ : Type w\nV₂ : Type w'\nM₃ : Type y\nV₃ : Type y'\nM₄ : Type z\nι : Type x\nM₅ : Type u_1\nM₆ : Type u_2\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : AddCommMonoid M₂\ninst✝¹ : Module R M\ninst✝ : Module R M₂\np : Submodule R M\nq : Submo... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Prod | {
"line": 568,
"column": 2
} | {
"line": 568,
"column": 7
} | {
"line": 570,
"column": 0
} | [
{
"pp": "R : Type u\nM : Type v\nM₂ : Type w\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : AddCommMonoid M₂\ninst✝¹ : Module R M\ninst✝ : Module R M₂\n⊢ map (LinearMap.fst R M M₂) (fst R M M₂) = ⊤",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"LinearMap.fst",
"Eq.m... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.LinearAlgebra.Prod | {
"line": 568,
"column": 2
} | {
"line": 568,
"column": 7
} | {
"line": 570,
"column": 0
} | [
{
"pp": "R : Type u\nM : Type v\nM₂ : Type w\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : AddCommMonoid M₂\ninst✝¹ : Module R M\ninst✝ : Module R M₂\n⊢ map (LinearMap.fst R M M₂) (fst R M M₂) = ⊤",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"LinearMap.fst",
"Eq.m... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Prod | {
"line": 568,
"column": 2
} | {
"line": 568,
"column": 7
} | {
"line": 570,
"column": 0
} | [
{
"pp": "R : Type u\nM : Type v\nM₂ : Type w\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : AddCommMonoid M₂\ninst✝¹ : Module R M\ninst✝ : Module R M₂\n⊢ map (LinearMap.fst R M M₂) (fst R M M₂) = ⊤",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"LinearMap.fst",
"Eq.m... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Prod | {
"line": 581,
"column": 26
} | {
"line": 581,
"column": 31
} | {
"line": 581,
"column": 31
} | [
{
"pp": "R : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM₂ : Type w\nV₂ : Type w'\nM₃ : Type y\nV₃ : Type y'\nM₄ : Type z\nι : Type x\nM₅ : Type u_1\nM₆ : Type u_2\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : AddCommMonoid M₂\ninst✝¹ : Module R M\ninst✝ : Module R M₂\np : Submodule R M\nq : Submo... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.LinearAlgebra.Prod | {
"line": 581,
"column": 26
} | {
"line": 581,
"column": 31
} | {
"line": 581,
"column": 31
} | [
{
"pp": "R : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM₂ : Type w\nV₂ : Type w'\nM₃ : Type y\nV₃ : Type y'\nM₄ : Type z\nι : Type x\nM₅ : Type u_1\nM₆ : Type u_2\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : AddCommMonoid M₂\ninst✝¹ : Module R M\ninst✝ : Module R M₂\np : Submodule R M\nq : Submo... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Prod | {
"line": 581,
"column": 26
} | {
"line": 581,
"column": 31
} | {
"line": 581,
"column": 31
} | [
{
"pp": "R : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM₂ : Type w\nV₂ : Type w'\nM₃ : Type y\nV₃ : Type y'\nM₄ : Type z\nι : Type x\nM₅ : Type u_1\nM₆ : Type u_2\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : AddCommMonoid M₂\ninst✝¹ : Module R M\ninst✝ : Module R M₂\np : Submodule R M\nq : Submo... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Prod | {
"line": 590,
"column": 2
} | {
"line": 590,
"column": 7
} | {
"line": 592,
"column": 0
} | [
{
"pp": "R : Type u\nM : Type v\nM₂ : Type w\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : AddCommMonoid M₂\ninst✝¹ : Module R M\ninst✝ : Module R M₂\n⊢ map (LinearMap.snd R M M₂) (snd R M M₂) = ⊤",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"LinearMap.fst",
"Eq.m... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.LinearAlgebra.Prod | {
"line": 590,
"column": 2
} | {
"line": 590,
"column": 7
} | {
"line": 592,
"column": 0
} | [
{
"pp": "R : Type u\nM : Type v\nM₂ : Type w\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : AddCommMonoid M₂\ninst✝¹ : Module R M\ninst✝ : Module R M₂\n⊢ map (LinearMap.snd R M M₂) (snd R M M₂) = ⊤",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"LinearMap.fst",
"Eq.m... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Prod | {
"line": 590,
"column": 2
} | {
"line": 590,
"column": 7
} | {
"line": 592,
"column": 0
} | [
{
"pp": "R : Type u\nM : Type v\nM₂ : Type w\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : AddCommMonoid M₂\ninst✝¹ : Module R M\ninst✝ : Module R M₂\n⊢ map (LinearMap.snd R M M₂) (snd R M M₂) = ⊤",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"LinearMap.fst",
"Eq.m... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Prod | {
"line": 601,
"column": 2
} | {
"line": 601,
"column": 7
} | {
"line": 603,
"column": 0
} | [
{
"pp": "R : Type u\nM : Type v\nM₂ : Type w\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : AddCommMonoid M₂\ninst✝¹ : Module R M\ninst✝ : Module R M₂\n⊢ fst R M M₂ ⊓ snd R M M₂ = ⊥",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"LinearMap.fst",
"Eq.mpr",
"Subm... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.LinearAlgebra.Prod | {
"line": 601,
"column": 2
} | {
"line": 601,
"column": 7
} | {
"line": 603,
"column": 0
} | [
{
"pp": "R : Type u\nM : Type v\nM₂ : Type w\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : AddCommMonoid M₂\ninst✝¹ : Module R M\ninst✝ : Module R M₂\n⊢ fst R M M₂ ⊓ snd R M M₂ = ⊥",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"LinearMap.fst",
"Eq.mpr",
"Subm... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Prod | {
"line": 601,
"column": 2
} | {
"line": 601,
"column": 7
} | {
"line": 603,
"column": 0
} | [
{
"pp": "R : Type u\nM : Type v\nM₂ : Type w\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : AddCommMonoid M₂\ninst✝¹ : Module R M\ninst✝ : Module R M₂\n⊢ fst R M M₂ ⊓ snd R M M₂ = ⊥",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"LinearMap.fst",
"Eq.mpr",
"Subm... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Prod | {
"line": 865,
"column": 2
} | {
"line": 865,
"column": 59
} | {
"line": 866,
"column": 2
} | [
{
"pp": "R : Type u\nM : Type v\nM₂ : Type w\nM₃ : Type y\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M₂\ninst✝³ : AddCommGroup M₃\ninst✝² : Module R M\ninst✝¹ : Module R M₂\ninst✝ : Module R M₃\nf : M →ₗ[R] M₂\ng : M →ₗ[R] M₃\nh : f.ker ⊔ g.ker = ⊤\nx y : M\n⊢ ∃ y_1, y_1 - x ∈ f.ker ∧ g y_... | [
"R : Type u\nM : Type v\nM₂ : Type w\nM₃ : Type y\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M₂\ninst✝³ : AddCommGroup M₃\ninst✝² : Module R M\ninst✝¹ : Module R M₂\ninst✝ : Module R M₃\nf : M →ₗ[R] M₂\ng : M →ₗ[R] M₃\nh : f.ker ⊔ g.ker = ⊤\nx y : M\nthis : y - x ∈ f.ker ⊔ g.ker\n⊢ ∃ y_1, y_1 ... | have : y - x ∈ ker f ⊔ ker g := by simp only [h, mem_top] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.LinearAlgebra.Prod | {
"line": 937,
"column": 10
} | {
"line": 937,
"column": 25
} | {
"line": 937,
"column": 26
} | [
{
"pp": "R : Type u_3\nS : Type u_4\nG : Type u_5\nH : Type u_6\nI : Type u_7\ninst✝⁸ : Semiring R\ninst✝⁷ : Semiring S\nσ : R →+* S\ninst✝⁶ : RingHomSurjective σ\ninst✝⁵ : AddCommMonoid G\ninst✝⁴ : Module R G\ninst✝³ : AddCommMonoid H\ninst✝² : Module S H\ninst✝¹ : AddCommMonoid I\ninst✝ : Module S I\nf : G →ₛ... | [
"R : Type u_3\nS : Type u_4\nG : Type u_5\nH : Type u_6\nI : Type u_7\ninst✝⁸ : Semiring R\ninst✝⁷ : Semiring S\nσ : R →+* S\ninst✝⁶ : RingHomSurjective σ\ninst✝⁵ : AddCommMonoid G\ninst✝⁴ : Module R G\ninst✝³ : AddCommMonoid H\ninst✝² : Module S H\ninst✝¹ : AddCommMonoid I\ninst✝ : Module S I\nf : G →ₛₗ[σ] H × I\n... | ← Prod.smul_mk, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.SuccPred.LinearLocallyFinite | {
"line": 255,
"column": 2
} | {
"line": 255,
"column": 45
} | {
"line": 256,
"column": 2
} | [
{
"pp": "ι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : SuccOrder ι\ninst✝¹ : IsSuccArchimedean ι\ninst✝ : PredOrder ι\ni0 : ι\nn : ℕ\n⊢ -↑n ≤ toZ i0 (pred^[n] i0)",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Preorder.toLT",
"PartialOrder.toPreorder",
"Preorder.toLE",... | [
"case inl\nι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : SuccOrder ι\ninst✝¹ : IsSuccArchimedean ι\ninst✝ : PredOrder ι\ni0 : ι\nn : ℕ\nh : i0 ≤ pred^[n] i0\n⊢ -↑n ≤ toZ i0 (pred^[n] i0)",
"case inr\nι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : SuccOrder ι\ninst✝¹ : IsSuccArchimedean ι\ninst✝ : PredOrder ι\ni0... | rcases le_or_gt i0 (pred^[n] i0) with h | h | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Algebra.Order.Interval.Finset.Basic | {
"line": 77,
"column": 6
} | {
"line": 77,
"column": 25
} | {
"line": 77,
"column": 26
} | [
{
"pp": "α : Type u_2\ninst✝⁵ : AddCommMonoid α\ninst✝⁴ : PartialOrder α\ninst✝³ : IsOrderedCancelAddMonoid α\ninst✝² : ExistsAddOfLE α\ninst✝¹ : LocallyFiniteOrder α\ninst✝ : DecidableEq α\na b c : α\n⊢ image (fun x ↦ c + x) (Ioc a b) = Ioc (c + a) (c + b)",
"ppTerm": "?m.29",
"assigned": true,
"us... | [
"α : Type u_2\ninst✝⁵ : AddCommMonoid α\ninst✝⁴ : PartialOrder α\ninst✝³ : IsOrderedCancelAddMonoid α\ninst✝² : ExistsAddOfLE α\ninst✝¹ : LocallyFiniteOrder α\ninst✝ : DecidableEq α\na b c : α\n⊢ image (fun x ↦ c + x) (Ioc a b) = map (addLeftEmbedding c) (Ioc a b)"
] | ← map_add_left_Ioc, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Order.Antidiag.Prod | {
"line": 220,
"column": 76
} | {
"line": 222,
"column": 61
} | {
"line": 224,
"column": 0
} | [
{
"pp": "A : Type u_1\ninst✝⁸ : AddCommMonoid A\ninst✝⁷ : PartialOrder A\ninst✝⁶ : CanonicallyOrderedAdd A\ninst✝⁵ : Sub A\ninst✝⁴ : OrderedSub A\ninst✝³ : AddLeftReflectLE A\ninst✝² : HasAntidiagonal A\nn m : A\ninst✝¹ : DecidablePred fun x ↦ x = m\ninst✝ : Decidable (m ≤ n)\n⊢ {x ∈ antidiagonal n | x.2 = m} =... | [] | by
rw [← map_swap_antidiagonal, filter_map]
simp [filter_fst_eq_antidiagonal, apply_ite (Finset.map _)] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Order.Antidiag.Prod | {
"line": 275,
"column": 82
} | {
"line": 275,
"column": 87
} | {
"line": 275,
"column": 87
} | [
{
"pp": "A✝ : Type u_1\nA : Type u_2\ninst✝¹ : AddMonoid A\ninst✝ : HasAntidiagonal A\na : Multiplicative A\nx✝¹ x✝ : A × A\nh : (fun p ↦ (ofAdd p.1, ofAdd p.2)) x✝¹ = (fun p ↦ (ofAdd p.1, ofAdd p.2)) x✝\n⊢ x✝¹ = x✝",
"ppTerm": "?m.44",
"assigned": true,
"usedConstants": [
"Equiv.instEquivLike... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Order.Antidiag.Prod | {
"line": 275,
"column": 82
} | {
"line": 275,
"column": 87
} | {
"line": 275,
"column": 87
} | [
{
"pp": "A✝ : Type u_1\nA : Type u_2\ninst✝¹ : AddMonoid A\ninst✝ : HasAntidiagonal A\na : Multiplicative A\nx✝¹ x✝ : A × A\nh : (fun p ↦ (ofAdd p.1, ofAdd p.2)) x✝¹ = (fun p ↦ (ofAdd p.1, ofAdd p.2)) x✝\n⊢ x✝¹ = x✝",
"ppTerm": "?m.44",
"assigned": true,
"usedConstants": [
"Equiv.instEquivLike... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Order.Antidiag.Prod | {
"line": 275,
"column": 82
} | {
"line": 275,
"column": 87
} | {
"line": 275,
"column": 87
} | [
{
"pp": "A✝ : Type u_1\nA : Type u_2\ninst✝¹ : AddMonoid A\ninst✝ : HasAntidiagonal A\na : Multiplicative A\nx✝¹ x✝ : A × A\nh : (fun p ↦ (ofAdd p.1, ofAdd p.2)) x✝¹ = (fun p ↦ (ofAdd p.1, ofAdd p.2)) x✝\n⊢ x✝¹ = x✝",
"ppTerm": "?m.44",
"assigned": true,
"usedConstants": [
"Equiv.instEquivLike... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Order.Antidiag.Prod | {
"line": 276,
"column": 34
} | {
"line": 276,
"column": 39
} | {
"line": 278,
"column": 0
} | [
{
"pp": "A✝ : Type u_1\nA : Type u_2\ninst✝¹ : AddMonoid A\ninst✝ : HasAntidiagonal A\na : Multiplicative A\np : Multiplicative A × Multiplicative A\n⊢ p ∈ map { toFun := fun p ↦ (ofAdd p.1, ofAdd p.2), inj' := ⋯ } (antidiagonal (toAdd a)) ↔ p.1 * p.2 = a",
"ppTerm": "?m.45",
"assigned": true,
"used... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Order.Antidiag.Prod | {
"line": 276,
"column": 34
} | {
"line": 276,
"column": 39
} | {
"line": 278,
"column": 0
} | [
{
"pp": "A✝ : Type u_1\nA : Type u_2\ninst✝¹ : AddMonoid A\ninst✝ : HasAntidiagonal A\na : Multiplicative A\np : Multiplicative A × Multiplicative A\n⊢ p ∈ map { toFun := fun p ↦ (ofAdd p.1, ofAdd p.2), inj' := ⋯ } (antidiagonal (toAdd a)) ↔ p.1 * p.2 = a",
"ppTerm": "?m.45",
"assigned": true,
"used... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Order.Antidiag.Prod | {
"line": 276,
"column": 34
} | {
"line": 276,
"column": 39
} | {
"line": 278,
"column": 0
} | [
{
"pp": "A✝ : Type u_1\nA : Type u_2\ninst✝¹ : AddMonoid A\ninst✝ : HasAntidiagonal A\na : Multiplicative A\np : Multiplicative A × Multiplicative A\n⊢ p ∈ map { toFun := fun p ↦ (ofAdd p.1, ofAdd p.2), inj' := ⋯ } (antidiagonal (toAdd a)) ↔ p.1 * p.2 = a",
"ppTerm": "?m.45",
"assigned": true,
"used... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Finset.NatAntidiagonal | {
"line": 113,
"column": 18
} | {
"line": 113,
"column": 41
} | {
"line": 113,
"column": 42
} | [
{
"pp": "case refine_1\nn k : ℕ\nh : k ≤ n\ni j : ℕ\nhi : i + j = n ∧ j ≤ k\n⊢ n - k + (k - j) = i",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.instCanonicallyOrderedAdd",
"Nat.instOrderedSub",
"Nat.instIsOrderedAddMonoid",
"AddLeftCancel... | [
"case refine_1\nn k : ℕ\nh : k ≤ n\ni j : ℕ\nhi : i + j = n ∧ j ≤ k\n⊢ n + (k - j) - k = i"
] | tsub_add_eq_add_tsub h, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Nat.Choose.Sum | {
"line": 50,
"column": 6
} | {
"line": 54,
"column": 68
} | {
"line": 55,
"column": 2
} | [
{
"pp": "case e_a\nR : Type u_1\ninst✝ : Semiring R\nx y : R\nh : Commute x y\nn✝ : ℕ\nt : ℕ → ℕ → R := fun n m ↦ x ^ m * y ^ (n - m) * ↑(n.choose m)\nh_first : ∀ (n : ℕ), t n 0 = y ^ n\nh_last : ∀ (n : ℕ), t n n.succ = 0\nn i : ℕ\nh_mem : i ∈ range n.succ\nh_le : i ≤ n\n⊢ x ^ i.succ * y ^ (n.succ - i.succ) * ↑... | [] | rw [← mul_assoc y, ← mul_assoc y, (h.symm.pow_right i.succ).eq]
by_cases h_eq : i = n
· rw [h_eq, choose_succ_self, cast_zero, mul_zero, mul_zero]
· rw [succ_sub (lt_of_le_of_ne h_le h_eq)]
rw [pow_succ' y, mul_assoc, mul_assoc, mul_assoc, mul_assoc] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Nat.Choose.Sum | {
"line": 50,
"column": 6
} | {
"line": 54,
"column": 68
} | {
"line": 55,
"column": 2
} | [
{
"pp": "case e_a\nR : Type u_1\ninst✝ : Semiring R\nx y : R\nh : Commute x y\nn✝ : ℕ\nt : ℕ → ℕ → R := fun n m ↦ x ^ m * y ^ (n - m) * ↑(n.choose m)\nh_first : ∀ (n : ℕ), t n 0 = y ^ n\nh_last : ∀ (n : ℕ), t n n.succ = 0\nn i : ℕ\nh_mem : i ∈ range n.succ\nh_le : i ≤ n\n⊢ x ^ i.succ * y ^ (n.succ - i.succ) * ↑... | [] | rw [← mul_assoc y, ← mul_assoc y, (h.symm.pow_right i.succ).eq]
by_cases h_eq : i = n
· rw [h_eq, choose_succ_self, cast_zero, mul_zero, mul_zero]
· rw [succ_sub (lt_of_le_of_ne h_le h_eq)]
rw [pow_succ' y, mul_assoc, mul_assoc, mul_assoc, mul_assoc] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Ideal.Prime | {
"line": 114,
"column": 48
} | {
"line": 114,
"column": 53
} | {
"line": 116,
"column": 0
} | [
{
"pp": "α : Type u\ninst✝² : Semiring α\nI : Ideal α\ninst✝¹ : I.IsTwoSided\ninst✝ : I.IsPrime\nx y : α\nhx : y ∉ I\n⊢ x ∈ I ∨ y ∈ I ↔ x ∈ I",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"False",
"Semiring.toModule",
"eq_false",
"congrArg",
"Membership.mem"... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.RingTheory.Ideal.Prime | {
"line": 114,
"column": 48
} | {
"line": 114,
"column": 53
} | {
"line": 116,
"column": 0
} | [
{
"pp": "case hI\nα : Type u\ninst✝² : Semiring α\nI : Ideal α\ninst✝¹ : I.IsTwoSided\ninst✝ : I.IsPrime\nx y : α\nhx : y ∉ I\n⊢ I.IsPrime",
"ppTerm": "?hI",
"assigned": true,
"usedConstants": [],
"usedFVars": [
"inst✝"
],
"usedGoals": []
}
] | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.RingTheory.Ideal.Maximal | {
"line": 62,
"column": 92
} | {
"line": 63,
"column": 93
} | {
"line": 65,
"column": 0
} | [
{
"pp": "α : Type u\ninst✝ : Semiring α\nI : Ideal α\n⊢ I.IsMaximal ↔ 1 ∉ I ∧ ∀ (J : Ideal α) (x : α), I ≤ J → x ∉ I → x ∈ J → 1 ∈ J",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Semiring.toModule",
"congrArg",... | [] | by
simp_rw [isMaximal_def, SetLike.isCoatom_iff, Ideal.ne_top_iff_one, ← Ideal.eq_top_iff_one] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Ideal.Maximal | {
"line": 69,
"column": 6
} | {
"line": 69,
"column": 11
} | {
"line": 69,
"column": 11
} | [
{
"pp": "α : Type u\ninst✝ : Semiring α\nI J : Ideal α\nhI : I.IsMaximal\nhJ : J ≠ ⊤\n⊢ I = J → I ≤ J",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Semiring.toModule",
"instReflLe",
"PartialOrder.toPreorder",
"Preorder.toLE",
"Std.le_refl._simp_1",
"N... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.RingTheory.Ideal.Maximal | {
"line": 69,
"column": 6
} | {
"line": 69,
"column": 11
} | {
"line": 69,
"column": 11
} | [
{
"pp": "α : Type u\ninst✝ : Semiring α\nI J : Ideal α\nhI : I.IsMaximal\nhJ : J ≠ ⊤\n⊢ I = J → I ≤ J",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Semiring.toModule",
"instReflLe",
"PartialOrder.toPreorder",
"Preorder.toLE",
"Std.le_refl._simp_1",
"N... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Ideal.Maximal | {
"line": 69,
"column": 6
} | {
"line": 69,
"column": 11
} | {
"line": 69,
"column": 11
} | [
{
"pp": "α : Type u\ninst✝ : Semiring α\nI J : Ideal α\nhI : I.IsMaximal\nhJ : J ≠ ⊤\n⊢ I = J → I ≤ J",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Semiring.toModule",
"instReflLe",
"PartialOrder.toPreorder",
"Preorder.toLE",
"Std.le_refl._simp_1",
"N... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Ideal.Span | {
"line": 247,
"column": 37
} | {
"line": 247,
"column": 91
} | {
"line": 247,
"column": 91
} | [
{
"pp": "α : Type u\ninst✝ : Ring α\nx y x✝¹ : α\nx✝ : ∃ a b, a * (x + y) + b * y = x✝¹\na b : α\nh : a * (x + y) + b * y = x✝¹\n⊢ a * x + (b + a) * y = x✝¹",
"ppTerm": "?m.49",
"assigned": true,
"usedConstants": [
"add_mul",
"Distrib.leftDistribClass",
"Eq.mpr",
"HMul.hMul",... | [] | by rw [add_comm b, add_mul, ← add_assoc, ← mul_add, h] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Ideal.Maximal | {
"line": 214,
"column": 2
} | {
"line": 214,
"column": 92
} | {
"line": 215,
"column": 2
} | [
{
"pp": "case refine_2\nα : Type u\ninst✝ : CommSemiring α\nI : Ideal α\nS : Submonoid α\ndisjoint : Disjoint ↑I ↑S\np : Ideal α\nhIp : I ≤ p\nhp : Maximal (fun x ↦ x ∈ {p | Disjoint ↑p ↑S}) p\n⊢ ∃ p, p.IsPrime ∧ I ≤ p ∧ Disjoint ↑p ↑S",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
... | [
"case refine_1\nα : Type u\ninst✝ : CommSemiring α\nI : Ideal α\nS : Submonoid α\ndisjoint : Disjoint ↑I ↑S\nc : Set (Ideal α)\nhc : c ⊆ {p | Disjoint ↑p ↑S}\nhc' : IsChain (fun x1 x2 ↦ x1 ≤ x2) c\nx : Ideal α\nhx : x ∈ c\n⊢ ∃ ub ∈ {p | Disjoint ↑p ↑S}, ∀ z ∈ c, z ≤ ub"
] | · exact ⟨p, isPrime_of_maximally_disjoint _ _ hp.1 (fun _ ↦ hp.not_prop_of_gt), hIp, hp.1⟩ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Order.Filter.Basic | {
"line": 532,
"column": 2
} | {
"line": 532,
"column": 39
} | {
"line": 534,
"column": 0
} | [
{
"pp": "case inr\nα : Type u\nι : Sort x\nf : ι → Filter α\nhn : Nonempty α\nhd : Directed (fun x1 x2 ↦ x1 ≥ x2) f\nhb : ∀ (i : ι), (f i).NeBot\nh✝ : Nonempty ι\n⊢ (iInf f).NeBot",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Filter.iInf_neBot_of_directed'"
],
"usedFVars": ... | [] | · exact iInf_neBot_of_directed' hd hb | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Order.Filter.Bases.Basic | {
"line": 541,
"column": 2
} | {
"line": 541,
"column": 79
} | {
"line": 543,
"column": 0
} | [
{
"pp": "α : Type u_1\ns t : Set α\n⊢ Disjoint (𝓟 s) (𝓟 t) ↔ Disjoint s t",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
"Eq.mpr",
"Filter.disjoint_principal_left",
"CompleteBooleanAlgebra.toCompleteDistribLattice",
"congrArg",
... | [] | rw [← subset_compl_iff_disjoint_left, disjoint_principal_left, mem_principal] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Order.Filter.Bases.Basic | {
"line": 541,
"column": 2
} | {
"line": 541,
"column": 79
} | {
"line": 543,
"column": 0
} | [
{
"pp": "α : Type u_1\ns t : Set α\n⊢ Disjoint (𝓟 s) (𝓟 t) ↔ Disjoint s t",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
"Eq.mpr",
"Filter.disjoint_principal_left",
"CompleteBooleanAlgebra.toCompleteDistribLattice",
"congrArg",
... | [] | rw [← subset_compl_iff_disjoint_left, disjoint_principal_left, mem_principal] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.Filter.Bases.Basic | {
"line": 541,
"column": 2
} | {
"line": 541,
"column": 79
} | {
"line": 543,
"column": 0
} | [
{
"pp": "α : Type u_1\ns t : Set α\n⊢ Disjoint (𝓟 s) (𝓟 t) ↔ Disjoint s t",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
"Eq.mpr",
"Filter.disjoint_principal_left",
"CompleteBooleanAlgebra.toCompleteDistribLattice",
"congrArg",
... | [] | rw [← subset_compl_iff_disjoint_left, disjoint_principal_left, mem_principal] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Finsupp.Pi | {
"line": 271,
"column": 2
} | {
"line": 271,
"column": 7
} | {
"line": 273,
"column": 0
} | [
{
"pp": "case e_f\nM : Type u_5\ninst✝³ : AddCommMonoid M\nX : Type u_6\nY : Type u_7\ninst✝² : Fintype X\ninst✝¹ : Finite Y\ninst✝ : DecidableEq Y\nf : X → Y\ny : Y\ns : X →₀ M\n⊢ (fun c ↦ (Finsupp.single (f c) (s c)) y) = fun a ↦ if f a = y then s a else 0",
"ppTerm": "?e_f",
"assigned": true,
"us... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.LinearAlgebra.Finsupp.Pi | {
"line": 316,
"column": 2
} | {
"line": 316,
"column": 7
} | {
"line": 318,
"column": 0
} | [
{
"pp": "R : Type u_5\nM : Type u_6\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\nX : Type u_7\ninst✝ : Finite X\n⊢ linearMap R M _root_.id = LinearMap.id",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"LinearMap.id",
"Pi.Function.module",
"Pi.addC... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.LinearAlgebra.Finsupp.Pi | {
"line": 316,
"column": 2
} | {
"line": 316,
"column": 7
} | {
"line": 318,
"column": 0
} | [
{
"pp": "R : Type u_5\nM : Type u_6\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\nX : Type u_7\ninst✝ : Finite X\n⊢ linearMap R M _root_.id = LinearMap.id",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"LinearMap.id",
"Pi.Function.module",
"Pi.addC... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Finsupp.Pi | {
"line": 316,
"column": 2
} | {
"line": 316,
"column": 7
} | {
"line": 318,
"column": 0
} | [
{
"pp": "R : Type u_5\nM : Type u_6\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\nX : Type u_7\ninst✝ : Finite X\n⊢ linearMap R M _root_.id = LinearMap.id",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"LinearMap.id",
"Pi.Function.module",
"Pi.addC... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Finsupp.Pi | {
"line": 321,
"column": 2
} | {
"line": 321,
"column": 7
} | {
"line": 323,
"column": 0
} | [
{
"pp": "R : Type u_5\nM : Type u_6\ninst✝⁵ : Semiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nX : Type u_7\nY : Type u_8\nZ : Type u_9\ninst✝² : Finite X\ninst✝¹ : Finite Y\ninst✝ : Finite Z\nf : X → Y\ng : Y → Z\n⊢ linearMap R M (g ∘ f) = linearMap R M g ∘ₗ linearMap R M f",
"ppTerm": "?m.53",
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.LinearAlgebra.Finsupp.Pi | {
"line": 321,
"column": 2
} | {
"line": 321,
"column": 7
} | {
"line": 323,
"column": 0
} | [
{
"pp": "R : Type u_5\nM : Type u_6\ninst✝⁵ : Semiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nX : Type u_7\nY : Type u_8\nZ : Type u_9\ninst✝² : Finite X\ninst✝¹ : Finite Y\ninst✝ : Finite Z\nf : X → Y\ng : Y → Z\n⊢ linearMap R M (g ∘ f) = linearMap R M g ∘ₗ linearMap R M f",
"ppTerm": "?m.53",
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Finsupp.Pi | {
"line": 321,
"column": 2
} | {
"line": 321,
"column": 7
} | {
"line": 323,
"column": 0
} | [
{
"pp": "R : Type u_5\nM : Type u_6\ninst✝⁵ : Semiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nX : Type u_7\nY : Type u_8\nZ : Type u_9\ninst✝² : Finite X\ninst✝¹ : Finite Y\ninst✝ : Finite Z\nf : X → Y\ng : Y → Z\n⊢ linearMap R M (g ∘ f) = linearMap R M g ∘ₗ linearMap R M f",
"ppTerm": "?m.53",
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.Filter.AtTopBot.Defs | {
"line": 79,
"column": 2
} | {
"line": 81,
"column": 38
} | {
"line": 83,
"column": 0
} | [
{
"pp": "α : Type u_3\ninst✝ : Preorder α\np : α → Prop\nh : ∃ᶠ (x : α) in atTop, p x\na : α\n⊢ ∃ b, a ≤ b ∧ p b",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Push.not_exists._simp_1",
"Eq.mpr",
"Mathlib.Tactic.Push.not_and_eq",
"Mathlib.Tactic.Cont... | [] | rw [Filter.Frequently] at h
contrapose! h
exact (eventually_ge_atTop a).mono h | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.Filter.AtTopBot.Defs | {
"line": 79,
"column": 2
} | {
"line": 81,
"column": 38
} | {
"line": 83,
"column": 0
} | [
{
"pp": "α : Type u_3\ninst✝ : Preorder α\np : α → Prop\nh : ∃ᶠ (x : α) in atTop, p x\na : α\n⊢ ∃ b, a ≤ b ∧ p b",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Push.not_exists._simp_1",
"Eq.mpr",
"Mathlib.Tactic.Push.not_and_eq",
"Mathlib.Tactic.Cont... | [] | rw [Filter.Frequently] at h
contrapose! h
exact (eventually_ge_atTop a).mono h | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.Interval.Set.Disjoint | {
"line": 95,
"column": 59
} | {
"line": 96,
"column": 69
} | {
"line": 98,
"column": 0
} | [
{
"pp": "α : Type v\ninst✝ : Preorder α\na : α\n⊢ ⋃ b, Icc a b = Ici a",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Set.iUnion_Iic",
"_private.Mathlib.Order.Interval.Set.Disjoint.0.Set.iUnion_Icc_right._simp_1_2",
"Set.Ici",
"congrArg",
"Set.univ",
"... | [] | by
simp only [← Ici_inter_Iic, ← inter_iUnion, iUnion_Iic, inter_univ] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Order.Filter.Map | {
"line": 871,
"column": 8
} | {
"line": 871,
"column": 23
} | {
"line": 871,
"column": 23
} | [
{
"pp": "case refine_1\nα : Type u_1\nβ : Type u_2\ng : α → β\nf : Filter α\ns : Set α\nhs : s ∈ f\n⊢ g '' s ∈ (pure g).seq f",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"Pure.pure",
"Filter.instMembership",
"Eq.mpr",
"congrArg",
"Filter.seq",
"M... | [
"case refine_1\nα : Type u_1\nβ : Type u_2\ng : α → β\nf : Filter α\ns : Set α\nhs : s ∈ f\n⊢ {g}.seq s ∈ (pure g).seq f"
] | ← singleton_seq | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.Filter.AtTopBot.Basic | {
"line": 71,
"column": 2
} | {
"line": 71,
"column": 88
} | {
"line": 72,
"column": 2
} | [
{
"pp": "α : Type u_6\ninst✝ : Preorder α\n⊢ atTop.NeBot ↔ Nonempty α ∧ IsDirectedOrder α",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Filter.NeBot",
"Filter.atTop_neBot",
"Preorder.toLE",
"LE.le",
"Filter.nonempty_of_neBot",
"Filter.atTop",
"Is... | [
"α : Type u_6\ninst✝ : Preorder α\nh : atTop.NeBot\nx y : α\n⊢ ∃ c, x ≤ c ∧ y ≤ c"
] | refine ⟨fun h ↦ ⟨nonempty_of_neBot atTop, ⟨fun x y ↦ ?_⟩⟩, fun ⟨h₁, h₂⟩ ↦ atTop_neBot⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Order.Filter.AtTopBot.Basic | {
"line": 127,
"column": 2
} | {
"line": 127,
"column": 67
} | {
"line": 128,
"column": 2
} | [
{
"pp": "P : ℕ → ℕ → Prop\nu : ℕ → ℕ → ℕ\nhu : ∀ (n a : ℕ), u n a > a\nhu' : ∀ (n a : ℕ), P n (u n a)\n⊢ ∃ φ, StrictMono φ ∧ ∀ (n : ℕ), P n (φ n)",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"StrictMono",
"instOfNatNat",
"instHAdd",
"And",
"HAdd.hAdd",
... | [
"case h\nP : ℕ → ℕ → Prop\nu : ℕ → ℕ → ℕ\nhu : ∀ (n a : ℕ), u n a > a\nhu' : ∀ (n a : ℕ), P n (u n a)\n⊢ (StrictMono fun n ↦ Nat.recOn n (u 0 0) fun n v ↦ u (n + 1) v) ∧\n ∀ (n : ℕ), P n ((fun n ↦ Nat.recOn n (u 0 0) fun n v ↦ u (n + 1) v) n)"
] | use (fun n => Nat.recOn n (u 0 0) fun n v => u (n + 1) v : ℕ → ℕ) | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.Order.Filter.AtTopBot.Basic | {
"line": 288,
"column": 2
} | {
"line": 290,
"column": 93
} | {
"line": 292,
"column": 0
} | [
{
"pp": "α : Type u_3\ninst✝¹ : Preorder α\ninst✝ : IsDirectedOrder α\na : α\n⊢ atTop = comap Subtype.val atTop",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"Set.Ioi",
"Preorder.toLT",
"Set.Ici",
"congrArg",
"Filter.map",
... | [] | rcases isEmpty_or_nonempty (Ioi a) with h | ⟨⟨b, hb⟩⟩
· subsingleton
· rw [← map_val_atTop_of_Ici_subset (Ici_subset_Ioi.2 hb), comap_map Subtype.coe_injective] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.Filter.AtTopBot.Basic | {
"line": 288,
"column": 2
} | {
"line": 290,
"column": 93
} | {
"line": 292,
"column": 0
} | [
{
"pp": "α : Type u_3\ninst✝¹ : Preorder α\ninst✝ : IsDirectedOrder α\na : α\n⊢ atTop = comap Subtype.val atTop",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"Set.Ioi",
"Preorder.toLT",
"Set.Ici",
"congrArg",
"Filter.map",
... | [] | rcases isEmpty_or_nonempty (Ioi a) with h | ⟨⟨b, hb⟩⟩
· subsingleton
· rw [← map_val_atTop_of_Ici_subset (Ici_subset_Ioi.2 hb), comap_map Subtype.coe_injective] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.Filter.AtTopBot.Basic | {
"line": 366,
"column": 28
} | {
"line": 366,
"column": 33
} | {
"line": 367,
"column": 2
} | [
{
"pp": "α : Type u_3\nβ : Type u_4\ninst✝² : Preorder β\nl : Filter α\ninst✝¹ : l.NeBot\nf : α → β\ninst✝ : NoMaxOrder β\nh : Tendsto f l atTop\nM : β\nhM : M ∈ upperBounds (range f)\n⊢ ∀ (x : α), f x ≤ M",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Membership.mem",
"Exist... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Order.IsNormal | {
"line": 218,
"column": 4
} | {
"line": 218,
"column": 9
} | {
"line": 220,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nf : α → β\ninst✝⁴ : LinearOrder α\ninst✝³ : WellFoundedLT α\ninst✝² : SuccOrder α\ninst✝¹ : LinearOrder β\ninst✝ : OrderBot α\ng : α → β\nhf : IsNormal f\nhg : IsNormal g\nH₁ : f ⊥ = g ⊥\nH₂ : ∀ (a : α), f a = g a → f (succ a) = g (succ a)\na : α\nha : IsSuccLimit a\nIH : ∀ ... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Order.Module.Defs | {
"line": 1046,
"column": 29
} | {
"line": 1046,
"column": 43
} | {
"line": 1046,
"column": 44
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\na : α\nb₁ b₂ : β\ninst✝⁷ : Ring α\ninst✝⁶ : PartialOrder α\ninst✝⁵ : IsOrderedRing α\ninst✝⁴ : AddCommGroup β\ninst✝³ : PartialOrder β\ninst✝² : IsOrderedAddMonoid β\ninst✝¹ : Module α β\ninst✝ : PosSMulMono α β\nh : b₁ ≤ b₂\nha : a ≤ 0\n⊢ -(-a • b₂) ≤ - -a • b₁",
"ppTer... | [
"α : Type u_1\nβ : Type u_2\na : α\nb₁ b₂ : β\ninst✝⁷ : Ring α\ninst✝⁶ : PartialOrder α\ninst✝⁵ : IsOrderedRing α\ninst✝⁴ : AddCommGroup β\ninst✝³ : PartialOrder β\ninst✝² : IsOrderedAddMonoid β\ninst✝¹ : Module α β\ninst✝ : PosSMulMono α β\nh : b₁ ≤ b₂\nha : a ≤ 0\n⊢ -(-a • b₂) ≤ -(-a • b₁)"
] | neg_smul (-a), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Order.Module.Defs | {
"line": 1062,
"column": 29
} | {
"line": 1062,
"column": 43
} | {
"line": 1062,
"column": 44
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\na : α\nb₁ b₂ : β\ninst✝⁷ : Ring α\ninst✝⁶ : PartialOrder α\ninst✝⁵ : IsOrderedRing α\ninst✝⁴ : AddCommGroup β\ninst✝³ : PartialOrder β\ninst✝² : IsOrderedAddMonoid β\ninst✝¹ : Module α β\ninst✝ : PosSMulStrictMono α β\nhb : b₁ < b₂\nha : a < 0\n⊢ -(-a • b₂) < - -a • b₁",
... | [
"α : Type u_1\nβ : Type u_2\na : α\nb₁ b₂ : β\ninst✝⁷ : Ring α\ninst✝⁶ : PartialOrder α\ninst✝⁵ : IsOrderedRing α\ninst✝⁴ : AddCommGroup β\ninst✝³ : PartialOrder β\ninst✝² : IsOrderedAddMonoid β\ninst✝¹ : Module α β\ninst✝ : PosSMulStrictMono α β\nhb : b₁ < b₂\nha : a < 0\n⊢ -(-a • b₂) < -(-a • b₁)"
] | neg_smul (-a), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Order.Module.Defs | {
"line": 1076,
"column": 29
} | {
"line": 1076,
"column": 43
} | {
"line": 1076,
"column": 44
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\na : α\nb₁ b₂ : β\ninst✝⁷ : Ring α\ninst✝⁶ : PartialOrder α\ninst✝⁵ : IsOrderedRing α\ninst✝⁴ : AddCommGroup β\ninst✝³ : PartialOrder β\ninst✝² : IsOrderedAddMonoid β\ninst✝¹ : Module α β\ninst✝ : PosSMulReflectLE α β\nh : -(-a • b₁) ≤ - -a • b₂\nha : a < 0\n⊢ b₂ ≤ b₁",
"... | [
"α : Type u_1\nβ : Type u_2\na : α\nb₁ b₂ : β\ninst✝⁷ : Ring α\ninst✝⁶ : PartialOrder α\ninst✝⁵ : IsOrderedRing α\ninst✝⁴ : AddCommGroup β\ninst✝³ : PartialOrder β\ninst✝² : IsOrderedAddMonoid β\ninst✝¹ : Module α β\ninst✝ : PosSMulReflectLE α β\nh : -(-a • b₁) ≤ -(-a • b₂)\nha : a < 0\n⊢ b₂ ≤ b₁"
] | neg_smul (-a), | Lean.Elab.Tactic.evalRewriteSeq | null |
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