module stringlengths 16 90 | startPos dict | endPos dict | nextStartPos dict | goals listlengths 0 96 | goalsAfter listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 375
values | kind stringclasses 379
values |
|---|---|---|---|---|---|---|---|---|
Mathlib.Algebra.Order.BigOperators.Ring.Multiset | {
"line": 25,
"column": 54
} | {
"line": 28,
"column": 43
} | {
"line": 30,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommSemiring R\ninst✝³ : PartialOrder R\ninst✝² : CanonicallyOrderedAdd R\ninst✝¹ : NoZeroDivisors R\ninst✝ : Nontrivial R\nm : Multiset R\n⊢ 0 < m.prod ↔ ∀ (x : R), x ∈ m → 0 < x",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.... | [] | by
rcases m with ⟨l⟩
rw [Multiset.quot_mk_to_coe'', Multiset.prod_coe]
exact CanonicallyOrderedAdd.list_prod_pos | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Finsupp.Basic | {
"line": 549,
"column": 4
} | {
"line": 549,
"column": 25
} | {
"line": 550,
"column": 4
} | [
{
"pp": "case pos\nα : Type u_1\nβ : Type u_2\nM : Type u_5\ninst✝ : Zero M\nf : α ↪ β\ng : β →₀ M\nhg : ↑g.support ⊆ Set.range ⇑f\nb : β\nhb : b ∈ Set.range ⇑f\n⊢ (embDomain f (comapDomain (⇑f) g ⋯)) b = g b",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Finsupp.instFunLike",
... | [
"case pos\nα : Type u_1\nβ : Type u_2\nM : Type u_5\ninst✝ : Zero M\nf : α ↪ β\ng : β →₀ M\nhg : ↑g.support ⊆ Set.range ⇑f\na : α\n⊢ (embDomain f (comapDomain (⇑f) g ⋯)) (f a) = g (f a)"
] | obtain ⟨a, rfl⟩ := hb | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Algebra.Order.BigOperators.Group.Finset | {
"line": 202,
"column": 25
} | {
"line": 202,
"column": 32
} | {
"line": 202,
"column": 32
} | [
{
"pp": "ι : Type u_1\nN : Type u_5\ninst✝² : CommMonoid N\ninst✝¹ : PartialOrder N\nf : ι → N\ns✝ : Finset ι\ninst✝ : MulLeftMono N\na : ι\ns : Finset ι\nha : a ∉ s\nih : (∀ i ∈ s, 1 ≤ f i) → (∏ i ∈ s, f i = 1 ↔ ∀ i ∈ s, f i = 1)\nH : ∀ i ∈ insert a s, 1 ≤ f i\nthis : ∀ i ∈ s, 1 ≤ f i\n⊢ f a = 1 ∧ ∏ i ∈ s, f i... | [
"ι : Type u_1\nN : Type u_5\ninst✝² : CommMonoid N\ninst✝¹ : PartialOrder N\nf : ι → N\ns✝ : Finset ι\ninst✝ : MulLeftMono N\na : ι\ns : Finset ι\nha : a ∉ s\nih : (∀ i ∈ s, 1 ≤ f i) → (∏ i ∈ s, f i = 1 ↔ ∀ i ∈ s, f i = 1)\nH : ∀ i ∈ insert a s, 1 ≤ f i\nthis : ∀ i ∈ s, 1 ≤ f i\n⊢ (f a = 1 ∧ ∀ i ∈ s, f i = 1) ↔ f a... | ih this | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Order.BigOperators.Ring.Finset | {
"line": 69,
"column": 27
} | {
"line": 69,
"column": 50
} | {
"line": 69,
"column": 50
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : PartialOrder R\ninst✝¹ : IsOrderedRing R\nM : Type u_4\ninst✝ : CommMonoid M\nf : M → R\np : M → Prop\nh_nonneg : ∀ (a : M), 0 ≤ f a\nh_one : f 1 ≤ 1\nh_mul : ∀ (a b : M), p a → p b → f (a * b) ≤ f a * f b\nhp_mul : ∀ (a b : M), p a → p b → ... | [] | simp [Multiset.map_map] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Order.BigOperators.Ring.Finset | {
"line": 69,
"column": 27
} | {
"line": 69,
"column": 50
} | {
"line": 69,
"column": 50
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : PartialOrder R\ninst✝¹ : IsOrderedRing R\nM : Type u_4\ninst✝ : CommMonoid M\nf : M → R\np : M → Prop\nh_nonneg : ∀ (a : M), 0 ≤ f a\nh_one : f 1 ≤ 1\nh_mul : ∀ (a b : M), p a → p b → f (a * b) ≤ f a * f b\nhp_mul : ∀ (a b : M), p a → p b → ... | [] | simp [Multiset.map_map] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Order.BigOperators.Ring.Finset | {
"line": 69,
"column": 27
} | {
"line": 69,
"column": 50
} | {
"line": 69,
"column": 50
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : PartialOrder R\ninst✝¹ : IsOrderedRing R\nM : Type u_4\ninst✝ : CommMonoid M\nf : M → R\np : M → Prop\nh_nonneg : ∀ (a : M), 0 ≤ f a\nh_one : f 1 ≤ 1\nh_mul : ∀ (a b : M), p a → p b → f (a * b) ≤ f a * f b\nhp_mul : ∀ (a b : M), p a → p b → ... | [] | simp [Multiset.map_map] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Order.BigOperators.Ring.Finset | {
"line": 79,
"column": 8
} | {
"line": 79,
"column": 31
} | {
"line": 79,
"column": 31
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : PartialOrder R\ninst✝¹ : IsOrderedRing R\nM : Type u_4\ninst✝ : CommMonoid M\nf : M → R\nh_nonneg : ∀ (a : M), 0 ≤ f a\nh_one : f 1 ≤ 1\nh_mul : ∀ (x y : M), f (x * y) ≤ f x * f y\ns : Finset ι\ng : ι → M\n⊢ (Multiset.map f (Multiset.map (fu... | [] | simp [Multiset.map_map] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Order.BigOperators.Ring.Finset | {
"line": 79,
"column": 8
} | {
"line": 79,
"column": 31
} | {
"line": 79,
"column": 31
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : PartialOrder R\ninst✝¹ : IsOrderedRing R\nM : Type u_4\ninst✝ : CommMonoid M\nf : M → R\nh_nonneg : ∀ (a : M), 0 ≤ f a\nh_one : f 1 ≤ 1\nh_mul : ∀ (x y : M), f (x * y) ≤ f x * f y\ns : Finset ι\ng : ι → M\n⊢ (Multiset.map f (Multiset.map (fu... | [] | simp [Multiset.map_map] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Order.BigOperators.Ring.Finset | {
"line": 79,
"column": 8
} | {
"line": 79,
"column": 31
} | {
"line": 79,
"column": 31
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : PartialOrder R\ninst✝¹ : IsOrderedRing R\nM : Type u_4\ninst✝ : CommMonoid M\nf : M → R\nh_nonneg : ∀ (a : M), 0 ≤ f a\nh_one : f 1 ≤ 1\nh_mul : ∀ (x y : M), f (x * y) ≤ f x * f y\ns : Finset ι\ng : ι → M\n⊢ (Multiset.map f (Multiset.map (fu... | [] | simp [Multiset.map_map] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Finsupp.Defs | {
"line": 377,
"column": 4
} | {
"line": 377,
"column": 50
} | {
"line": 379,
"column": 0
} | [
{
"pp": "case inr\nι : Type u_4\nR : Type u_5\nM : Type u_6\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nf : End (End R (ι →₀ M)) (ι →₀ M)\ni : ι\n⊢ ∃ a, (ringHomEndFinsupp ι) a = f",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"RingEquiv.toEquiv",
"Fins... | [] | exact ⟨_, (ringEquivEndFinsupp i).right_inv f⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.LinearAlgebra.Finsupp.Defs | {
"line": 377,
"column": 4
} | {
"line": 377,
"column": 50
} | {
"line": 379,
"column": 0
} | [
{
"pp": "case inr\nι : Type u_4\nR : Type u_5\nM : Type u_6\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nf : End (End R (ι →₀ M)) (ι →₀ M)\ni : ι\n⊢ ∃ a, (ringHomEndFinsupp ι) a = f",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"RingEquiv.toEquiv",
"Fins... | [] | exact ⟨_, (ringEquivEndFinsupp i).right_inv f⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Finsupp.Defs | {
"line": 377,
"column": 4
} | {
"line": 377,
"column": 50
} | {
"line": 379,
"column": 0
} | [
{
"pp": "case inr\nι : Type u_4\nR : Type u_5\nM : Type u_6\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nf : End (End R (ι →₀ M)) (ι →₀ M)\ni : ι\n⊢ ∃ a, (ringHomEndFinsupp ι) a = f",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"RingEquiv.toEquiv",
"Fins... | [] | exact ⟨_, (ringEquivEndFinsupp i).right_inv f⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.BigOperators.Finprod | {
"line": 181,
"column": 2
} | {
"line": 183,
"column": 49
} | {
"line": 185,
"column": 0
} | [
{
"pp": "M : Type u_2\nα : Sort u_4\ninst✝ : CommMonoid M\nf : α → M\nhf : HasFiniteMulSupport (f ∘ PLift.down)\ns : Finset (PLift α)\nhs : Finite.toFinset hf ⊆ s\n⊢ ∏ᶠ (i : α), f i = ∏ i ∈ s, f i.down",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
... | [] | rw [finprod, dif_pos hf]
refine Finset.prod_subset hs fun x _ hxf => ?_
rwa [hf.mem_toFinset, notMem_mulSupport] at hxf | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.BigOperators.Finprod | {
"line": 181,
"column": 2
} | {
"line": 183,
"column": 49
} | {
"line": 185,
"column": 0
} | [
{
"pp": "M : Type u_2\nα : Sort u_4\ninst✝ : CommMonoid M\nf : α → M\nhf : HasFiniteMulSupport (f ∘ PLift.down)\ns : Finset (PLift α)\nhs : Finite.toFinset hf ⊆ s\n⊢ ∏ᶠ (i : α), f i = ∏ i ∈ s, f i.down",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
... | [] | rw [finprod, dif_pos hf]
refine Finset.prod_subset hs fun x _ hxf => ?_
rwa [hf.mem_toFinset, notMem_mulSupport] at hxf | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.Interval.Set.Fin | {
"line": 62,
"column": 30
} | {
"line": 62,
"column": 57
} | {
"line": 62,
"column": 57
} | [
{
"pp": "n : ℕ\ni : Fin n\n⊢ val '' val ⁻¹' Iic ↑i = Iic ↑i",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"id",
"Fin.val",
"Set.preimage",
"Nat.instPreorder",
"Set.image_preimage_eq_of_subset",
"Nat",
"Set.image"... | [
"n : ℕ\ni : Fin n\n⊢ Iic ↑i = Iic ↑i",
"n : ℕ\ni : Fin n\n⊢ Iic ↑i ⊆ range val"
] | image_preimage_eq_of_subset | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.Interval.Set.Fin | {
"line": 540,
"column": 36
} | {
"line": 540,
"column": 63
} | {
"line": 540,
"column": 63
} | [
{
"pp": "n m : ℕ\ni : Fin n\n⊢ natAdd m '' natAdd ?m ⁻¹' Ici (natAdd ?m i) = Ici (natAdd m i)",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Fin.natAdd",
"Set.Ici",
"congrArg",
"PartialOrder.toPreorder",
"id",
"instHAdd",
"Fin.ins... | [
"n m : ℕ\ni : Fin n\n⊢ Ici (natAdd m i) = Ici (natAdd m i)",
"n m : ℕ\ni : Fin n\n⊢ Ici (natAdd m i) ⊆ range (natAdd m)"
] | image_preimage_eq_of_subset | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.Interval.Set.Fin | {
"line": 546,
"column": 36
} | {
"line": 546,
"column": 63
} | {
"line": 546,
"column": 63
} | [
{
"pp": "n m : ℕ\ni : Fin n\n⊢ natAdd m '' natAdd ?m ⁻¹' Ioi (natAdd ?m i) = Ioi (natAdd m i)",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Set.Ioi",
"Fin.natAdd",
"congrArg",
"PartialOrder.toPreorder",
"id",
"instHAdd",
"Fin.ins... | [
"n m : ℕ\ni : Fin n\n⊢ Ioi (natAdd m i) = Ioi (natAdd m i)",
"n m : ℕ\ni : Fin n\n⊢ Ioi (natAdd m i) ⊆ range (natAdd m)"
] | image_preimage_eq_of_subset | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.Interval.Set.Fin | {
"line": 552,
"column": 36
} | {
"line": 552,
"column": 63
} | {
"line": 552,
"column": 63
} | [
{
"pp": "n m : ℕ\ni j : Fin n\n⊢ natAdd m '' natAdd ?m ⁻¹' Icc (natAdd ?m i) (natAdd ?m j) = Icc (natAdd m i) (natAdd m j)",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Fin.natAdd",
"congrArg",
"PartialOrder.toPreorder",
"id",
"instHAdd",
... | [
"n m : ℕ\ni j : Fin n\n⊢ Icc (natAdd m i) (natAdd m j) = Icc (natAdd m i) (natAdd m j)",
"n m : ℕ\ni j : Fin n\n⊢ Icc (natAdd m i) (natAdd m j) ⊆ range (natAdd m)"
] | image_preimage_eq_of_subset | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.Interval.Set.Fin | {
"line": 558,
"column": 36
} | {
"line": 558,
"column": 63
} | {
"line": 558,
"column": 63
} | [
{
"pp": "n m : ℕ\ni j : Fin n\n⊢ natAdd m '' natAdd ?m ⁻¹' Ico (natAdd ?m i) (natAdd ?m j) = Ico (natAdd m i) (natAdd m j)",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Fin.natAdd",
"congrArg",
"PartialOrder.toPreorder",
"id",
"Set.Ico",
... | [
"n m : ℕ\ni j : Fin n\n⊢ Ico (natAdd m i) (natAdd m j) = Ico (natAdd m i) (natAdd m j)",
"n m : ℕ\ni j : Fin n\n⊢ Ico (natAdd m i) (natAdd m j) ⊆ range (natAdd m)"
] | image_preimage_eq_of_subset | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.Interval.Set.Fin | {
"line": 564,
"column": 36
} | {
"line": 564,
"column": 63
} | {
"line": 564,
"column": 63
} | [
{
"pp": "n m : ℕ\ni j : Fin n\n⊢ natAdd m '' natAdd ?m ⁻¹' Ioc (natAdd ?m i) (natAdd ?m j) = Ioc (natAdd m i) (natAdd m j)",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Set.Ioc",
"Fin.natAdd",
"congrArg",
"PartialOrder.toPreorder",
"id",
... | [
"n m : ℕ\ni j : Fin n\n⊢ Ioc (natAdd m i) (natAdd m j) = Ioc (natAdd m i) (natAdd m j)",
"n m : ℕ\ni j : Fin n\n⊢ Ioc (natAdd m i) (natAdd m j) ⊆ range (natAdd m)"
] | image_preimage_eq_of_subset | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.Interval.Set.Fin | {
"line": 570,
"column": 36
} | {
"line": 570,
"column": 63
} | {
"line": 570,
"column": 63
} | [
{
"pp": "n m : ℕ\ni j : Fin n\n⊢ natAdd m '' natAdd ?m ⁻¹' Ioo (natAdd ?m i) (natAdd ?m j) = Ioo (natAdd m i) (natAdd m j)",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Fin.natAdd",
"congrArg",
"PartialOrder.toPreorder",
"id",
"instHAdd",
... | [
"n m : ℕ\ni j : Fin n\n⊢ Ioo (natAdd m i) (natAdd m j) = Ioo (natAdd m i) (natAdd m j)",
"n m : ℕ\ni j : Fin n\n⊢ Ioo (natAdd m i) (natAdd m j) ⊆ range (natAdd m)"
] | image_preimage_eq_of_subset | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.Interval.Set.Fin | {
"line": 569,
"column": 59
} | {
"line": 571,
"column": 82
} | {
"line": 573,
"column": 0
} | [
{
"pp": "n m : ℕ\ni j : Fin n\n⊢ natAdd m '' Ioo i j = Ioo (natAdd m i) (natAdd m j)",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Set.Ioi",
"Fin.natAdd",
"congrArg",
"PartialOrder.toPreorder",
"Preorder.toLE",
"SemilatticeInf.toPartia... | [] | by
rw [← preimage_natAdd_Ioo_natAdd, image_preimage_eq_of_subset]
exact Ioo_subset_Ioi_self.trans <| image_natAdd_Ioi m i ▸ image_subset_range _ _ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Order.Interval.Set.Fin | {
"line": 645,
"column": 36
} | {
"line": 645,
"column": 63
} | {
"line": 645,
"column": 63
} | [
{
"pp": "n m : ℕ\ni : Fin n\n⊢ (fun x ↦ x.addNat m) '' (fun x ↦ x.addNat ?m) ⁻¹' Ici (i.addNat ?m) = Ici (i.addNat m)",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Set.Ici",
"congrArg",
"PartialOrder.toPreorder",
"id",
"instHAdd",
"Fin... | [
"n m : ℕ\ni : Fin n\n⊢ Ici (i.addNat m) = Ici (i.addNat m)",
"n m : ℕ\ni : Fin n\n⊢ Ici (i.addNat m) ⊆ range fun x ↦ x.addNat m"
] | image_preimage_eq_of_subset | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.Interval.Set.Fin | {
"line": 653,
"column": 36
} | {
"line": 653,
"column": 63
} | {
"line": 653,
"column": 63
} | [
{
"pp": "n m : ℕ\ni : Fin n\n⊢ (fun x ↦ x.addNat m) '' (fun x ↦ x.addNat ?m) ⁻¹' Ioi (i.addNat ?m) = Ioi (i.addNat m)",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Set.Ioi",
"congrArg",
"PartialOrder.toPreorder",
"id",
"instHAdd",
"Fin... | [
"n m : ℕ\ni : Fin n\n⊢ Ioi (i.addNat m) = Ioi (i.addNat m)",
"n m : ℕ\ni : Fin n\n⊢ Ioi (i.addNat m) ⊆ range fun x ↦ x.addNat m"
] | image_preimage_eq_of_subset | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.Interval.Set.Fin | {
"line": 659,
"column": 36
} | {
"line": 659,
"column": 63
} | {
"line": 659,
"column": 63
} | [
{
"pp": "n m : ℕ\ni j : Fin n\n⊢ (fun x ↦ x.addNat m) '' (fun x ↦ x.addNat ?m) ⁻¹' Icc (i.addNat ?m) (j.addNat ?m) = Icc (i.addNat m) (j.addNat m)",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"PartialOrder.toPreorder",
"id",
"instHAdd"... | [
"n m : ℕ\ni j : Fin n\n⊢ Icc (i.addNat m) (j.addNat m) = Icc (i.addNat m) (j.addNat m)",
"n m : ℕ\ni j : Fin n\n⊢ Icc (i.addNat m) (j.addNat m) ⊆ range fun x ↦ x.addNat m"
] | image_preimage_eq_of_subset | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.BigOperators.Finprod | {
"line": 592,
"column": 4
} | {
"line": 592,
"column": 11
} | {
"line": 593,
"column": 2
} | [
{
"pp": "case hs\nι : Type u_3\nM : Type u_7\ninst✝² : CommMonoid M\ninst✝¹ : PartialOrder M\ninst✝ : IsOrderedCancelMonoid M\nf : ι → M\np : ι → Prop\nh : ∀ (i : ι), p i → 1 ≤ f i\nh' : ∃ i, p i ∧ 1 < f i\nhf : (mulSupport f ∩ {i | p i}).Finite\n⊢ ∃ i ∈ hf.toFinset, 1 < f i",
"ppTerm": "?hs",
"assigned... | [] | · aesop | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Order.Interval.Set.Fin | {
"line": 665,
"column": 36
} | {
"line": 665,
"column": 63
} | {
"line": 665,
"column": 63
} | [
{
"pp": "n m : ℕ\ni j : Fin n\n⊢ (fun x ↦ x.addNat m) '' (fun x ↦ x.addNat ?m) ⁻¹' Ico (i.addNat ?m) (j.addNat ?m) = Ico (i.addNat m) (j.addNat m)",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"PartialOrder.toPreorder",
"id",
"Set.Ico",... | [
"n m : ℕ\ni j : Fin n\n⊢ Ico (i.addNat m) (j.addNat m) = Ico (i.addNat m) (j.addNat m)",
"n m : ℕ\ni j : Fin n\n⊢ Ico (i.addNat m) (j.addNat m) ⊆ range fun x ↦ x.addNat m"
] | image_preimage_eq_of_subset | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.Interval.Set.Fin | {
"line": 671,
"column": 36
} | {
"line": 671,
"column": 63
} | {
"line": 671,
"column": 63
} | [
{
"pp": "n m : ℕ\ni j : Fin n\n⊢ (fun x ↦ x.addNat m) '' (fun x ↦ x.addNat ?m) ⁻¹' Ioc (i.addNat ?m) (j.addNat ?m) = Ioc (i.addNat m) (j.addNat m)",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Set.Ioc",
"congrArg",
"PartialOrder.toPreorder",
"id",... | [
"n m : ℕ\ni j : Fin n\n⊢ Ioc (i.addNat m) (j.addNat m) = Ioc (i.addNat m) (j.addNat m)",
"n m : ℕ\ni j : Fin n\n⊢ Ioc (i.addNat m) (j.addNat m) ⊆ range fun x ↦ x.addNat m"
] | image_preimage_eq_of_subset | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.Interval.Set.Fin | {
"line": 677,
"column": 36
} | {
"line": 677,
"column": 63
} | {
"line": 677,
"column": 63
} | [
{
"pp": "n m : ℕ\ni j : Fin n\n⊢ (fun x ↦ x.addNat m) '' (fun x ↦ x.addNat ?m) ⁻¹' Ioo (i.addNat ?m) (j.addNat ?m) = Ioo (i.addNat m) (j.addNat m)",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"PartialOrder.toPreorder",
"id",
"instHAdd"... | [
"n m : ℕ\ni j : Fin n\n⊢ Ioo (i.addNat m) (j.addNat m) = Ioo (i.addNat m) (j.addNat m)",
"n m : ℕ\ni j : Fin n\n⊢ Ioo (i.addNat m) (j.addNat m) ⊆ range fun x ↦ x.addNat m"
] | image_preimage_eq_of_subset | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Finsupp.LinearCombination | {
"line": 273,
"column": 2
} | {
"line": 273,
"column": 100
} | {
"line": 274,
"column": 2
} | [
{
"pp": "α : Type u_1\nM : Type u_2\nR : Type u_5\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\ns : Finset α\nf : α → R\ng : α → M\nhf : ∀ (a : α), f a ≠ 0 → a ∈ s\n⊢ (linearCombination R g) (onFinset s f hf) = ∑ x ∈ s, f x • g x",
"ppTerm": "?m.33",
"assigned": true,
"usedCons... | [
"α : Type u_1\nM : Type u_2\nR : Type u_5\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\ns : Finset α\nf : α → R\ng : α → M\nhf : ∀ (a : α), f a ≠ 0 → a ∈ s\n⊢ ∑ a ∈ s with f a ≠ 0, f a • g a = ∑ a ∈ s, f a • g a"
] | simp only [linearCombination_apply, Finsupp.sum, Finsupp.onFinset_apply, Finsupp.support_onFinset] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Order.Interval.Set.Fin | {
"line": 806,
"column": 76
} | {
"line": 806,
"column": 93
} | {
"line": 808,
"column": 0
} | [
{
"pp": "n : ℕ\ni x✝ : Fin n\n⊢ x✝ ∈ rev ⁻¹' Iio i ↔ x✝ ∈ Ioi i.rev",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Set.Ioi",
"Preorder.toLT",
"congrArg",
"PartialOrder.toPreorder",
"Membership.mem",
"Set.mem_Ioi._simp_1",
"Set.mem_preimage._simp_... | [] | simp [rev_lt_iff] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.LinearAlgebra.Basis.Defs | {
"line": 471,
"column": 26
} | {
"line": 471,
"column": 40
} | {
"line": 471,
"column": 41
} | [
{
"pp": "ι : Type u_10\nR : Type u_11\nM : Type u_12\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\nb : Basis ι R M\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq M\ni : ι\nh : b i ∈ Finset.image (⇑b) Finset.univ\n⊢ (b.reindexRange.reindex ((Equiv.refl M).subtypeEquiv ⋯)) ⟨b i, h⟩ = b i",
"p... | [
"ι : Type u_10\nR : Type u_11\nM : Type u_12\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\nb : Basis ι R M\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq M\ni : ι\nh : b i ∈ Finset.image (⇑b) Finset.univ\n⊢ b.reindexRange (((Equiv.refl M).subtypeEquiv ⋯).symm ⟨b i, h⟩) = b i"
] | reindex_apply, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.BigOperators.Finprod | {
"line": 664,
"column": 4
} | {
"line": 664,
"column": 15
} | {
"line": 665,
"column": 4
} | [
{
"pp": "α : Type u_1\nM : Type u_5\ninst✝ : CommMonoid M\nf g : α → M\nhf : HasFiniteMulSupport f\nhg : HasFiniteMulSupport g\nx : α\n⊢ f x * g x ≠ 1 → f x ≠ 1 ∨ g x ≠ 1",
"ppTerm": "?m.62",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"Mathlib.Tactic.Contrapose.... | [
"α : Type u_1\nM : Type u_5\ninst✝ : CommMonoid M\nf g : α → M\nhf : HasFiniteMulSupport f\nhg : HasFiniteMulSupport g\nx : α\n⊢ f x = 1 ∧ g x = 1 → f x * g x = 1"
] | contrapose! | Mathlib.Tactic.Contrapose._aux_Mathlib_Tactic_Contrapose___macroRules_Mathlib_Tactic_Contrapose_contrapose!_1 | Mathlib.Tactic.Contrapose.contrapose! |
Mathlib.LinearAlgebra.Finsupp.LinearCombination | {
"line": 363,
"column": 22
} | {
"line": 363,
"column": 87
} | {
"line": 365,
"column": 0
} | [
{
"pp": "α : Type u_1\nM : Type u_2\nR : Type u_3\ninst✝⁶ : Fintype α\ninst✝⁵ : Semiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nS : Type u_4\ninst✝² : Semiring S\ninst✝¹ : Module S M\ninst✝ : SMulCommClass R S M\nv✝ : α → M\nr : S\nv : α → M\n⊢ Fintype.linearCombination R (r • v) = (RingHom.id S) r •... | [] | ext; simp [Fintype.linearCombination, Finset.smul_sum, smul_comm] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Finsupp.LinearCombination | {
"line": 363,
"column": 22
} | {
"line": 363,
"column": 87
} | {
"line": 365,
"column": 0
} | [
{
"pp": "α : Type u_1\nM : Type u_2\nR : Type u_3\ninst✝⁶ : Fintype α\ninst✝⁵ : Semiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nS : Type u_4\ninst✝² : Semiring S\ninst✝¹ : Module S M\ninst✝ : SMulCommClass R S M\nv✝ : α → M\nr : S\nv : α → M\n⊢ Fintype.linearCombination R (r • v) = (RingHom.id S) r •... | [] | ext; simp [Fintype.linearCombination, Finset.smul_sum, smul_comm] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Basis.Defs | {
"line": 626,
"column": 82
} | {
"line": 626,
"column": 98
} | {
"line": 627,
"column": 14
} | [
{
"pp": "ι✝ : Type u_1\nι' : Type u_2\nR✝¹ : Type u_3\nR₂ : Type u_4\nK : Type u_5\nM✝¹ : Type u_6\nM'✝ : Type u_7\nM''✝ : Type u_8\nV : Type u\nV' : Type u_9\ninst✝¹⁶ : Semiring R✝¹\ninst✝¹⁵ : AddCommMonoid M✝¹\ninst✝¹⁴ : Module R✝¹ M✝¹\ninst✝¹³ : AddCommMonoid M'✝\ninst✝¹² : Module R✝¹ M'✝\nι : Type u_10\nR✝ ... | [
"ι✝ : Type u_1\nι' : Type u_2\nR✝¹ : Type u_3\nR₂ : Type u_4\nK : Type u_5\nM✝¹ : Type u_6\nM'✝ : Type u_7\nM''✝ : Type u_8\nV : Type u\nV' : Type u_9\ninst✝¹⁶ : Semiring R✝¹\ninst✝¹⁵ : AddCommMonoid M✝¹\ninst✝¹⁴ : Module R✝¹ M✝¹\ninst✝¹³ : AddCommMonoid M'✝\ninst✝¹² : Module R✝¹ M'✝\nι : Type u_10\nR✝ : Type u_11\... | b'.constr_basis, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Basis.Defs | {
"line": 627,
"column": 35
} | {
"line": 627,
"column": 39
} | {
"line": 627,
"column": 40
} | [
{
"pp": "ι✝ : Type u_1\nι' : Type u_2\nR✝¹ : Type u_3\nR₂ : Type u_4\nK : Type u_5\nM✝¹ : Type u_6\nM'✝ : Type u_7\nM''✝ : Type u_8\nV : Type u\nV' : Type u_9\ninst✝¹⁶ : Semiring R✝¹\ninst✝¹⁵ : AddCommMonoid M✝¹\ninst✝¹⁴ : Module R✝¹ M✝¹\ninst✝¹³ : AddCommMonoid M'✝\ninst✝¹² : Module R✝¹ M'✝\nι : Type u_10\nR✝ ... | [
"ι✝ : Type u_1\nι' : Type u_2\nR✝¹ : Type u_3\nR₂ : Type u_4\nK : Type u_5\nM✝¹ : Type u_6\nM'✝ : Type u_7\nM''✝ : Type u_8\nV : Type u\nV' : Type u_9\ninst✝¹⁶ : Semiring R✝¹\ninst✝¹⁵ : AddCommMonoid M✝¹\ninst✝¹⁴ : Module R✝¹ M✝¹\ninst✝¹³ : AddCommMonoid M'✝\ninst✝¹² : Module R✝¹ M'✝\nι : Type u_10\nR✝ : Type u_11\... | hi', | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Basis.Defs | {
"line": 633,
"column": 38
} | {
"line": 633,
"column": 54
} | {
"line": 633,
"column": 55
} | [
{
"pp": "ι✝ : Type u_1\nι' : Type u_2\nR✝¹ : Type u_3\nR₂ : Type u_4\nK : Type u_5\nM✝¹ : Type u_6\nM'✝ : Type u_7\nM''✝ : Type u_8\nV : Type u\nV' : Type u_9\ninst✝¹⁶ : Semiring R✝¹\ninst✝¹⁵ : AddCommMonoid M✝¹\ninst✝¹⁴ : Module R✝¹ M✝¹\ninst✝¹³ : AddCommMonoid M'✝\ninst✝¹² : Module R✝¹ M'✝\nι : Type u_10\nR✝ ... | [
"ι✝ : Type u_1\nι' : Type u_2\nR✝¹ : Type u_3\nR₂ : Type u_4\nK : Type u_5\nM✝¹ : Type u_6\nM'✝ : Type u_7\nM''✝ : Type u_8\nV : Type u\nV' : Type u_9\ninst✝¹⁶ : Semiring R✝¹\ninst✝¹⁵ : AddCommMonoid M✝¹\ninst✝¹⁴ : Module R✝¹ M✝¹\ninst✝¹³ : AddCommMonoid M'✝\ninst✝¹² : Module R✝¹ M'✝\nι : Type u_10\nR✝ : Type u_11\... | b'.constr_basis, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Basis.Defs | {
"line": 634,
"column": 35
} | {
"line": 634,
"column": 39
} | {
"line": 634,
"column": 40
} | [
{
"pp": "ι✝ : Type u_1\nι' : Type u_2\nR✝¹ : Type u_3\nR₂ : Type u_4\nK : Type u_5\nM✝¹ : Type u_6\nM'✝ : Type u_7\nM''✝ : Type u_8\nV : Type u\nV' : Type u_9\ninst✝¹⁶ : Semiring R✝¹\ninst✝¹⁵ : AddCommMonoid M✝¹\ninst✝¹⁴ : Module R✝¹ M✝¹\ninst✝¹³ : AddCommMonoid M'✝\ninst✝¹² : Module R✝¹ M'✝\nι : Type u_10\nR✝ ... | [
"ι✝ : Type u_1\nι' : Type u_2\nR✝¹ : Type u_3\nR₂ : Type u_4\nK : Type u_5\nM✝¹ : Type u_6\nM'✝ : Type u_7\nM''✝ : Type u_8\nV : Type u\nV' : Type u_9\ninst✝¹⁶ : Semiring R✝¹\ninst✝¹⁵ : AddCommMonoid M✝¹\ninst✝¹⁴ : Module R✝¹ M✝¹\ninst✝¹³ : AddCommMonoid M'✝\ninst✝¹² : Module R✝¹ M'✝\nι : Type u_10\nR✝ : Type u_11\... | hi', | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Fin.VecNotation | {
"line": 254,
"column": 53
} | {
"line": 255,
"column": 56
} | {
"line": 257,
"column": 0
} | [
{
"pp": "α : Type u\nx y : α\nu : Fin 0 → α\n⊢ Set.range (vecCons x (vecCons y u)) = {x, y}",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Matrix.range_cons",
"Eq.mpr",
"congrArg",
"Set.singleton_union",
"Set.instUnion",
"Set.instSingletonSet",
"... | [] | by
rw [range_cons, range_cons_empty, Set.singleton_union] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.BigOperators.Finprod | {
"line": 979,
"column": 2
} | {
"line": 979,
"column": 49
} | {
"line": 980,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nM : Type u_5\ninst✝ : CommMonoid M\nf : α → M\ng : β → M\ne : α → β\nhe₀ : Bijective e\nhe₁ : ∀ (x : α), f x = g (e x)\n⊢ ∏ᶠ (i : α), f i = ∏ᶠ (j : β), g j",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Set.univ... | [
"α : Type u_1\nβ : Type u_2\nM : Type u_5\ninst✝ : CommMonoid M\nf : α → M\ng : β → M\ne : α → β\nhe₀ : Bijective e\nhe₁ : ∀ (x : α), f x = g (e x)\n⊢ ∏ᶠ (i : α) (_ : i ∈ univ), f i = ∏ᶠ (i : β) (_ : i ∈ univ), g i"
] | rw [← finprod_mem_univ f, ← finprod_mem_univ g] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Data.ENat.Pow | {
"line": 53,
"column": 71
} | {
"line": 53,
"column": 97
} | {
"line": 53,
"column": 97
} | [
{
"pp": "case coe\ny✝ : ℕ∞\ny : ℕ\nh : ↑y ≠ 0\n⊢ y ≠ 0",
"ppTerm": "?coe",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instCharZeroENat",
"instAddMonoidWithOneENat",
"congrArg",
"AddMonoid.toAddZeroClass",
"AddZeroClass.toAddZero",
"id",
"AddMonoid... | [
"case coe\ny✝ : ℕ∞\ny : ℕ\nh : ↑y ≠ 0\n⊢ ↑y ≠ 0"
] | ← y.cast_ne_zero (R := ℕ∞) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.ENat.Pow | {
"line": 67,
"column": 69
} | {
"line": 67,
"column": 95
} | {
"line": 67,
"column": 95
} | [
{
"pp": "case coe\ny✝ : ℕ∞\ny : ℕ\nh : ↑y ≠ 0\n⊢ y ≠ 0",
"ppTerm": "?coe",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instCharZeroENat",
"instAddMonoidWithOneENat",
"congrArg",
"AddMonoid.toAddZeroClass",
"AddZeroClass.toAddZero",
"id",
"AddMonoid... | [
"case coe\ny✝ : ℕ∞\ny : ℕ\nh : ↑y ≠ 0\n⊢ ↑y ≠ 0"
] | ← y.cast_ne_zero (R := ℕ∞) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.ENat.Pow | {
"line": 88,
"column": 6
} | {
"line": 88,
"column": 35
} | {
"line": 89,
"column": 4
} | [
{
"pp": "case coe.top.inr.inl\na✝ : ℕ\ny_z : ↑a✝ ≤ ⊤\nh : 1 ≠ 0\n⊢ (fun y ↦ 1 ^ y) ↑a✝ ≤ (fun y ↦ 1 ^ y) ⊤",
"ppTerm": "?coe.top.inr.inl",
"assigned": true,
"usedConstants": [
"_private.Mathlib.Data.ENat.Pow.0.ENat.epow_right_mono._simp_1_1",
"instAddMonoidWithOneENat",
"ENat.instN... | [] | simp only [one_epow, le_refl] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Data.ENat.Pow | {
"line": 88,
"column": 6
} | {
"line": 88,
"column": 35
} | {
"line": 89,
"column": 4
} | [
{
"pp": "case coe.top.inr.inl\na✝ : ℕ\ny_z : ↑a✝ ≤ ⊤\nh : 1 ≠ 0\n⊢ (fun y ↦ 1 ^ y) ↑a✝ ≤ (fun y ↦ 1 ^ y) ⊤",
"ppTerm": "?coe.top.inr.inl",
"assigned": true,
"usedConstants": [
"_private.Mathlib.Data.ENat.Pow.0.ENat.epow_right_mono._simp_1_1",
"instAddMonoidWithOneENat",
"ENat.instN... | [] | simp only [one_epow, le_refl] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.ENat.Pow | {
"line": 88,
"column": 6
} | {
"line": 88,
"column": 35
} | {
"line": 89,
"column": 4
} | [
{
"pp": "case coe.top.inr.inl\na✝ : ℕ\ny_z : ↑a✝ ≤ ⊤\nh : 1 ≠ 0\n⊢ (fun y ↦ 1 ^ y) ↑a✝ ≤ (fun y ↦ 1 ^ y) ⊤",
"ppTerm": "?coe.top.inr.inl",
"assigned": true,
"usedConstants": [
"_private.Mathlib.Data.ENat.Pow.0.ENat.epow_right_mono._simp_1_1",
"instAddMonoidWithOneENat",
"ENat.instN... | [] | simp only [one_epow, le_refl] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.LinearIndependent.Defs | {
"line": 314,
"column": 2
} | {
"line": 314,
"column": 90
} | {
"line": 315,
"column": 2
} | [
{
"pp": "ι : Type u'\nR : Type u_2\nM : Type u_4\nM' : Type u_5\nv : ι → M\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : AddCommMonoid M'\ninst✝¹ : Module R M\ninst✝ : Module R M'\nf : M →ₗ[R] M'\nhfv : LinearIndependent R (⇑f ∘ v)\n⊢ LinearIndependent R v",
"ppTerm": "?m.28",
"assigned": tru... | [
"ι : Type u'\nR : Type u_2\nM : Type u_4\nM' : Type u_5\nv : ι → M\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : AddCommMonoid M'\ninst✝¹ : Module R M\ninst✝ : Module R M'\nf : M →ₗ[R] M'\nhfv : Injective (⇑f ∘ ⇑(Finsupp.linearCombination R v))\n⊢ LinearIndependent R v"
] | rw [LinearIndependent, Finsupp.linearCombination_linear_comp, LinearMap.coe_comp] at hfv | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.BigOperators.Fin | {
"line": 300,
"column": 83
} | {
"line": 301,
"column": 29
} | {
"line": 303,
"column": 0
} | [
{
"pp": "M : Type u_2\ninst✝ : CommMonoid M\nn m : ℕ\nf : Fin (n + m) → M\na b : Fin n\n⊢ ∏ i ∈ Icc (castAdd m a) (castAdd m b), f i = ∏ i ∈ Icc a b, f (castAdd m i)",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Fin.castAddEmb",
"_private.Mathlib.Algebra.BigOperators.Fin.0.F... | [] | by
simp [← map_castAddEmb_Icc] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.LinearIndependent.Basic | {
"line": 129,
"column": 9
} | {
"line": 129,
"column": 13
} | {
"line": 129,
"column": 14
} | [
{
"pp": "case refine_1\nι : Type u'\nR : Type u_2\nM : Type u_4\nv : ι → M\ninst✝⁵ : Semiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nR' : Type u_6\nM' : Type u_7\ninst✝² : Semiring R'\ninst✝¹ : AddCommMonoid M'\ninst✝ : Module R' M'\nhv : LinearIndependent R v\ni : R → R'\nj : M →+ M'\nhi : Surjectiv... | [
"case refine_1\nι : Type u'\nR : Type u_2\nM : Type u_4\nv : ι → M\ninst✝⁵ : Semiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nR' : Type u_6\nM' : Type u_7\ninst✝² : Semiring R'\ninst✝¹ : AddCommMonoid M'\ninst✝ : Module R' M'\nhv : LinearIndependent R v\ni : R → R'\nj : M →+ M'\nhi : Surjective i\nhj : In... | hi', | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.LinearIndependent.Basic | {
"line": 129,
"column": 4
} | {
"line": 129,
"column": 23
} | {
"line": 130,
"column": 2
} | [
{
"pp": "case refine_1\nι : Type u'\nR : Type u_2\nM : Type u_4\nv : ι → M\ninst✝⁵ : Semiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nR' : Type u_6\nM' : Type u_7\ninst✝² : Semiring R'\ninst✝¹ : AddCommMonoid M'\ninst✝ : Module R' M'\nhv : LinearIndependent R v\ni : R → R'\nj : M →+ M'\nhi : Surjectiv... | [] | rwa [hi', hi'] at h | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.LinearAlgebra.LinearIndependent.Basic | {
"line": 130,
"column": 6
} | {
"line": 130,
"column": 18
} | {
"line": 130,
"column": 19
} | [
{
"pp": "case refine_2\nι : Type u'\nR : Type u_2\nM : Type u_4\nv : ι → M\ninst✝⁵ : Semiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nR' : Type u_6\nM' : Type u_7\ninst✝² : Semiring R'\ninst✝¹ : AddCommMonoid M'\ninst✝ : Module R' M'\nhv : LinearIndependent R v\ni : R → R'\nj : M →+ M'\nhi : Surjectiv... | [
"case refine_2\nι : Type u'\nR : Type u_2\nM : Type u_4\nv : ι → M\ninst✝⁵ : Semiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nR' : Type u_6\nM' : Type u_7\ninst✝² : Semiring R'\ninst✝¹ : AddCommMonoid M'\ninst✝ : Module R' M'\nhv : LinearIndependent R v\ni : R → R'\nj : M →+ M'\nhi : Surjective i\nhj : In... | hc (i' r) m, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.BigOperators.Fin | {
"line": 510,
"column": 8
} | {
"line": 510,
"column": 25
} | {
"line": 510,
"column": 26
} | [
{
"pp": "n : ℕ\nG : Type u_3\ninst✝ : Group G\nf : Fin (n + 1) → G\nx✝ : Fin (n + 1)\nx : Fin n\nhx : f 0 * partialProd (fun i ↦ (f i.castSucc)⁻¹ * f i.succ) x.castSucc = f x.castSucc\n⊢ f 0 * partialProd (fun i ↦ (f i.castSucc)⁻¹ * f i.succ) x.succ = f x.succ",
"ppTerm": "?m.42",
"assigned": true,
... | [
"n : ℕ\nG : Type u_3\ninst✝ : Group G\nf : Fin (n + 1) → G\nx✝ : Fin (n + 1)\nx : Fin n\nhx : f 0 * partialProd (fun i ↦ (f i.castSucc)⁻¹ * f i.succ) x.castSucc = f x.castSucc\n⊢ f 0 * (partialProd (fun i ↦ (f i.castSucc)⁻¹ * f i.succ) x.castSucc * ((f x.castSucc)⁻¹ * f x.succ)) = f x.succ"
] | partialProd_succ, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.BigOperators.Fin | {
"line": 515,
"column": 6
} | {
"line": 515,
"column": 23
} | {
"line": 515,
"column": 24
} | [
{
"pp": "n : ℕ\nG : Type u_3\ninst✝ : Group G\nf : Fin n → G\ni : Fin n\n⊢ (partialProd f i.castSucc)⁻¹ * partialProd f i.succ = f i",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"DivInvOneMonoid.toInvOneClass",
"Fin.succ",
"Monoid.toM... | [
"n : ℕ\nG : Type u_3\ninst✝ : Group G\nf : Fin n → G\ni : Fin n\n⊢ (partialProd f i.castSucc)⁻¹ * (partialProd f i.castSucc * f i) = f i"
] | partialProd_succ, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.BigOperators.Fin | {
"line": 526,
"column": 8
} | {
"line": 526,
"column": 25
} | {
"line": 526,
"column": 26
} | [
{
"pp": "case refine_2\nG : Type u_3\ninst✝ : Monoid G\nn : ℕ\ng : Fin (n + 1) → G\na i✝ : Fin (n + 1)\ni : Fin n\nhi : partialProd (a.contractNth (fun x1 x2 ↦ x1 * x2) g) i.castSucc = partialProd g (a.succ.succAbove i.castSucc)\n⊢ partialProd (a.contractNth (fun x1 x2 ↦ x1 * x2) g) i.succ = partialProd g (a.su... | [
"case refine_2\nG : Type u_3\ninst✝ : Monoid G\nn : ℕ\ng : Fin (n + 1) → G\na i✝ : Fin (n + 1)\ni : Fin n\nhi : partialProd (a.contractNth (fun x1 x2 ↦ x1 * x2) g) i.castSucc = partialProd g (a.succ.succAbove i.castSucc)\n⊢ partialProd (a.contractNth (fun x1 x2 ↦ x1 * x2) g) i.castSucc * a.contractNth (fun x1 x2 ↦ ... | partialProd_succ, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.BigOperators.Fin | {
"line": 526,
"column": 26
} | {
"line": 526,
"column": 43
} | {
"line": 526,
"column": 44
} | [
{
"pp": "case refine_2\nG : Type u_3\ninst✝ : Monoid G\nn : ℕ\ng : Fin (n + 1) → G\na i✝ : Fin (n + 1)\ni : Fin n\nhi : partialProd (a.contractNth (fun x1 x2 ↦ x1 * x2) g) i.castSucc = partialProd g (a.succ.succAbove i.castSucc)\n⊢ partialProd (a.contractNth (fun x1 x2 ↦ x1 * x2) g) i.castSucc * a.contractNth (... | [
"case refine_2\nG : Type u_3\ninst✝ : Monoid G\nn : ℕ\ng : Fin (n + 1) → G\na i✝ : Fin (n + 1)\ni : Fin n\nhi : partialProd (a.contractNth (fun x1 x2 ↦ x1 * x2) g) i.castSucc = partialProd g (a.succ.succAbove i.castSucc)\n⊢ partialProd (a.contractNth (fun x1 x2 ↦ x1 * x2) g) i.castSucc * a.contractNth (fun x1 x2 ↦ ... | partialProd_succ, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.BigOperators.Fin | {
"line": 533,
"column": 49
} | {
"line": 533,
"column": 66
} | {
"line": 533,
"column": 67
} | [
{
"pp": "case refine_2.inr.inl\nG : Type u_3\ninst✝ : Monoid G\nn : ℕ\ng : Fin (n + 1) → G\na i✝ : Fin (n + 1)\ni : Fin n\nhi : partialProd (a.contractNth (fun x1 x2 ↦ x1 * x2) g) i.castSucc = partialProd g (a.succ.succAbove i.castSucc)\nh : ↑i = ↑a\n⊢ partialProd g i.castSucc.castSucc * (g i.castSucc * g i.suc... | [
"case refine_2.inr.inl\nG : Type u_3\ninst✝ : Monoid G\nn : ℕ\ng : Fin (n + 1) → G\na i✝ : Fin (n + 1)\ni : Fin n\nhi : partialProd (a.contractNth (fun x1 x2 ↦ x1 * x2) g) i.castSucc = partialProd g (a.succ.succAbove i.castSucc)\nh : ↑i = ↑a\n⊢ partialProd g i.castSucc.castSucc * (g i.castSucc * g i.succ) =\n pa... | partialProd_succ, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.BigOperators.Fin | {
"line": 562,
"column": 61
} | {
"line": 562,
"column": 78
} | {
"line": 563,
"column": 4
} | [
{
"pp": "case inr.inl\nn : ℕ\nG : Type u_3\ninst✝ : Group G\ng : Fin (n + 1) → G\nj : Fin (n + 1)\nk : Fin n\nh : ↑k = ↑j\n⊢ (partialProd g k.castSucc.castSucc)⁻¹ * partialProd g k.succ.succ = j.contractNth (fun x1 x2 ↦ x1 * x2) g k",
"ppTerm": "?inr.inl",
"assigned": true,
"usedConstants": [
... | [
"case inr.inl\nn : ℕ\nG : Type u_3\ninst✝ : Group G\ng : Fin (n + 1) → G\nj : Fin (n + 1)\nk : Fin n\nh : ↑k = ↑j\n⊢ (partialProd g k.castSucc.castSucc)⁻¹ * (partialProd g k.succ.castSucc * g k.succ) =\n j.contractNth (fun x1 x2 ↦ x1 * x2) g k",
"case inr.inl.h\nn : ℕ\nG : Type u_3\ninst✝ : Group G\ng : Fin (n... | partialProd_succ, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Nat.ModEq | {
"line": 157,
"column": 77
} | {
"line": 162,
"column": 18
} | {
"line": 164,
"column": 0
} | [
{
"pp": "n a b m : ℕ\nh : a ≡ b [MOD n]\n⊢ a ^ m ≡ b ^ m [MOD n]",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"instPowNat",
"Eq.mpr",
"Nat.recAux",
"HMul.hMul",
"congrArg",
"Nat.instMonoid",
"Nat.pow_succ",
"Nat.ModEq.mul",
"id",
... | [] | by
induction m with
| zero => rfl
| succ d hd =>
rw [Nat.pow_succ, Nat.pow_succ]
exact hd.mul h | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.BigOperators.Fin | {
"line": 568,
"column": 61
} | {
"line": 568,
"column": 78
} | {
"line": 568,
"column": 79
} | [
{
"pp": "case inr.inr\nn : ℕ\nG : Type u_3\ninst✝ : Group G\ng : Fin (n + 1) → G\nj : Fin (n + 1)\nk : Fin n\nh : ↑j < ↑k\n⊢ (partialProd g k.castSucc.succ)⁻¹ * partialProd g k.succ.succ = j.contractNth (fun x1 x2 ↦ x1 * x2) g k",
"ppTerm": "?inr.inr",
"assigned": true,
"usedConstants": [
"Eq.... | [
"case inr.inr\nn : ℕ\nG : Type u_3\ninst✝ : Group G\ng : Fin (n + 1) → G\nj : Fin (n + 1)\nk : Fin n\nh : ↑j < ↑k\n⊢ (partialProd g k.castSucc.castSucc * g k.castSucc)⁻¹ * partialProd g k.succ.succ =\n j.contractNth (fun x1 x2 ↦ x1 * x2) g k",
"case inr.inr.h\nn : ℕ\nG : Type u_3\ninst✝ : Group G\ng : Fin (n +... | partialProd_succ, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.BigOperators.Fin | {
"line": 568,
"column": 79
} | {
"line": 568,
"column": 96
} | {
"line": 569,
"column": 6
} | [
{
"pp": "case inr.inr\nn : ℕ\nG : Type u_3\ninst✝ : Group G\ng : Fin (n + 1) → G\nj : Fin (n + 1)\nk : Fin n\nh : ↑j < ↑k\n⊢ (partialProd g k.castSucc.castSucc * g k.castSucc)⁻¹ * partialProd g k.succ.succ =\n j.contractNth (fun x1 x2 ↦ x1 * x2) g k",
"ppTerm": "?inr.inr",
"assigned": true,
"used... | [
"case inr.inr\nn : ℕ\nG : Type u_3\ninst✝ : Group G\ng : Fin (n + 1) → G\nj : Fin (n + 1)\nk : Fin n\nh : ↑j < ↑k\n⊢ (partialProd g k.castSucc.castSucc * g k.castSucc)⁻¹ * (partialProd g k.succ.castSucc * g k.succ) =\n j.contractNth (fun x1 x2 ↦ x1 * x2) g k",
"case inr.inr.h\nn : ℕ\nG : Type u_3\ninst✝ : Grou... | partialProd_succ, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.BigOperators.Fin | {
"line": 569,
"column": 21
} | {
"line": 569,
"column": 38
} | {
"line": 569,
"column": 39
} | [
{
"pp": "case inr.inr\nn : ℕ\nG : Type u_3\ninst✝ : Group G\ng : Fin (n + 1) → G\nj : Fin (n + 1)\nk : Fin n\nh : ↑j < ↑k\n⊢ (partialProd g k.castSucc.castSucc * g k.castSucc)⁻¹ * (partialProd g k.castSucc.succ * g k.succ) =\n j.contractNth (fun x1 x2 ↦ x1 * x2) g k",
"ppTerm": "?inr.inr",
"assigned"... | [
"case inr.inr\nn : ℕ\nG : Type u_3\ninst✝ : Group G\ng : Fin (n + 1) → G\nj : Fin (n + 1)\nk : Fin n\nh : ↑j < ↑k\n⊢ (partialProd g k.castSucc.castSucc * g k.castSucc)⁻¹ * (partialProd g k.castSucc.castSucc * g k.castSucc * g k.succ) =\n j.contractNth (fun x1 x2 ↦ x1 * x2) g k",
"case inr.inr.h\nn : ℕ\nG : Typ... | partialProd_succ, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Nat.ModEq | {
"line": 346,
"column": 56
} | {
"line": 346,
"column": 89
} | {
"line": 348,
"column": 0
} | [
{
"pp": "a b : ℕ\n⊢ a ≡ b [MOD 0] ↔ a = b",
"ppTerm": "?m.4",
"assigned": true,
"usedConstants": [
"Nat.ModEq.eq_1",
"Eq.mpr",
"Nat.mod_zero",
"congrArg",
"Iff.rfl",
"id",
"Nat.instMod",
"instHMod",
"instOfNatNat",
"HMod.hMod",
"Iff",... | [] | by rw [ModEq, mod_zero, mod_zero] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.LinearIndependent.Basic | {
"line": 335,
"column": 6
} | {
"line": 335,
"column": 20
} | {
"line": 335,
"column": 21
} | [
{
"pp": "ι : Type u'\nR : Type u_2\nM : Type u_4\ninst✝³ : Ring R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : Fintype ι\nv : ι → M\nhv : LinearIndependent R v\nf g : ι → R\nheq : ∑ i, f i • v i = ∑ i, g i • v i\ni : ι\n⊢ f i = g i",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
... | [
"ι : Type u'\nR : Type u_2\nM : Type u_4\ninst✝³ : Ring R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : Fintype ι\nv : ι → M\nhv : LinearIndependent R v\nf g : ι → R\nheq : ∑ i, f i • v i - ∑ i, g i • v i = 0\ni : ι\n⊢ f i = g i"
] | ← sub_eq_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.LinearIndependent.Basic | {
"line": 350,
"column": 15
} | {
"line": 350,
"column": 51
} | {
"line": 352,
"column": 0
} | [
{
"pp": "ι : Type u'\nR : Type u_2\nM : Type u_4\nM' : Type u_5\nv : ι → M\ninst✝⁴ : Ring R\ninst✝³ : AddCommGroup M\ninst✝² : AddCommGroup M'\ninst✝¹ : Module R M\ninst✝ : Module R M'\nhv : LinearIndependent R v\nf : M →ₗ[R] M'\nhf_inj : f.ker = ⊥\n⊢ Disjoint (span R (range v)) f.ker",
"ppTerm": "?m.53",
... | [] | simp_rw [hf_inj, disjoint_bot_right] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.LinearAlgebra.LinearIndependent.Basic | {
"line": 350,
"column": 15
} | {
"line": 350,
"column": 51
} | {
"line": 352,
"column": 0
} | [
{
"pp": "ι : Type u'\nR : Type u_2\nM : Type u_4\nM' : Type u_5\nv : ι → M\ninst✝⁴ : Ring R\ninst✝³ : AddCommGroup M\ninst✝² : AddCommGroup M'\ninst✝¹ : Module R M\ninst✝ : Module R M'\nhv : LinearIndependent R v\nf : M →ₗ[R] M'\nhf_inj : f.ker = ⊥\n⊢ Disjoint (span R (range v)) f.ker",
"ppTerm": "?m.53",
... | [] | simp_rw [hf_inj, disjoint_bot_right] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.LinearIndependent.Basic | {
"line": 350,
"column": 15
} | {
"line": 350,
"column": 51
} | {
"line": 352,
"column": 0
} | [
{
"pp": "ι : Type u'\nR : Type u_2\nM : Type u_4\nM' : Type u_5\nv : ι → M\ninst✝⁴ : Ring R\ninst✝³ : AddCommGroup M\ninst✝² : AddCommGroup M'\ninst✝¹ : Module R M\ninst✝ : Module R M'\nhv : LinearIndependent R v\nf : M →ₗ[R] M'\nhf_inj : f.ker = ⊥\n⊢ Disjoint (span R (range v)) f.ker",
"ppTerm": "?m.53",
... | [] | simp_rw [hf_inj, disjoint_bot_right] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.LinearIndependent.Basic | {
"line": 381,
"column": 44
} | {
"line": 381,
"column": 80
} | {
"line": 383,
"column": 0
} | [
{
"pp": "ι : Type u'\nR : Type u_2\nM : Type u_4\nM' : Type u_5\nv : ι → M\ninst✝⁴ : Ring R\ninst✝³ : AddCommGroup M\ninst✝² : AddCommGroup M'\ninst✝¹ : Module R M\ninst✝ : Module R M'\nf : M →ₗ[R] M'\nhf_inj : f.ker = ⊥\n⊢ Disjoint (span R (Set.range v)) f.ker",
"ppTerm": "?m.52",
"assigned": true,
... | [] | simp_rw [hf_inj, disjoint_bot_right] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.LinearAlgebra.LinearIndependent.Basic | {
"line": 381,
"column": 44
} | {
"line": 381,
"column": 80
} | {
"line": 383,
"column": 0
} | [
{
"pp": "ι : Type u'\nR : Type u_2\nM : Type u_4\nM' : Type u_5\nv : ι → M\ninst✝⁴ : Ring R\ninst✝³ : AddCommGroup M\ninst✝² : AddCommGroup M'\ninst✝¹ : Module R M\ninst✝ : Module R M'\nf : M →ₗ[R] M'\nhf_inj : f.ker = ⊥\n⊢ Disjoint (span R (Set.range v)) f.ker",
"ppTerm": "?m.52",
"assigned": true,
... | [] | simp_rw [hf_inj, disjoint_bot_right] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.LinearIndependent.Basic | {
"line": 381,
"column": 44
} | {
"line": 381,
"column": 80
} | {
"line": 383,
"column": 0
} | [
{
"pp": "ι : Type u'\nR : Type u_2\nM : Type u_4\nM' : Type u_5\nv : ι → M\ninst✝⁴ : Ring R\ninst✝³ : AddCommGroup M\ninst✝² : AddCommGroup M'\ninst✝¹ : Module R M\ninst✝ : Module R M'\nf : M →ₗ[R] M'\nhf_inj : f.ker = ⊥\n⊢ Disjoint (span R (Set.range v)) f.ker",
"ppTerm": "?m.52",
"assigned": true,
... | [] | simp_rw [hf_inj, disjoint_bot_right] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Nat.ModEq | {
"line": 506,
"column": 2
} | {
"line": 511,
"column": 29
} | {
"line": 513,
"column": 0
} | [
{
"pp": "b a n : ℕ\nh : a * b ≡ 1 [MOD n]\n⊢ a.Coprime n",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Nat.gcd",
"Iff.mpr",
"Nat.gcd_dvd_left",
"Eq.mpr",
"Nat.Coprime",
"Nat.ModEq.mul_right",
"Nat.instMulZeroClass",
"Trans.trans",
"Dv... | [] | obtain ⟨g, hh⟩ := Nat.gcd_dvd_right a n
rw [Nat.coprime_iff_gcd_eq_one, ← Nat.dvd_one, ← Nat.modEq_zero_iff_dvd]
calc
1 ≡ a * b [MOD a.gcd n] := (hh ▸ h).symm.of_mul_right g
_ ≡ 0 * b [MOD a.gcd n] := (Nat.modEq_zero_iff_dvd.mpr (Nat.gcd_dvd_left _ _)).mul_right b
_ = 0 := by rw [zero_mul] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Nat.ModEq | {
"line": 506,
"column": 2
} | {
"line": 511,
"column": 29
} | {
"line": 513,
"column": 0
} | [
{
"pp": "b a n : ℕ\nh : a * b ≡ 1 [MOD n]\n⊢ a.Coprime n",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Nat.gcd",
"Iff.mpr",
"Nat.gcd_dvd_left",
"Eq.mpr",
"Nat.Coprime",
"Nat.ModEq.mul_right",
"Nat.instMulZeroClass",
"Trans.trans",
"Dv... | [] | obtain ⟨g, hh⟩ := Nat.gcd_dvd_right a n
rw [Nat.coprime_iff_gcd_eq_one, ← Nat.dvd_one, ← Nat.modEq_zero_iff_dvd]
calc
1 ≡ a * b [MOD a.gcd n] := (hh ▸ h).symm.of_mul_right g
_ ≡ 0 * b [MOD a.gcd n] := (Nat.modEq_zero_iff_dvd.mpr (Nat.gcd_dvd_left _ _)).mul_right b
_ = 0 := by rw [zero_mul] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.LinearIndependent.Defs | {
"line": 623,
"column": 6
} | {
"line": 624,
"column": 60
} | {
"line": 625,
"column": 4
} | [
{
"pp": "case pos\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\nthis : OrderedSu... | [] | apply hi'.antisymm'
simpa [hi', tsub_eq_zero_iff_le] using h.1 i ⟨hi, hi'⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.LinearIndependent.Defs | {
"line": 623,
"column": 6
} | {
"line": 624,
"column": 60
} | {
"line": 625,
"column": 4
} | [
{
"pp": "case pos\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\nthis : OrderedSu... | [] | apply hi'.antisymm'
simpa [hi', tsub_eq_zero_iff_le] using h.1 i ⟨hi, hi'⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.SetTheory.Cardinal.Finite | {
"line": 71,
"column": 16
} | {
"line": 71,
"column": 48
} | {
"line": 71,
"column": 48
} | [
{
"pp": "α : Type u_1\ninst✝ : Finite α\n⊢ ↑(toNat #α) = #α",
"ppTerm": "?m.4",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.instMulZeroOneClass",
"Cardinal",
"congrArg",
"CommSemiring.toSemiring",
"Cardinal.commSemiring",
"Cardinal.toNat",
"Mon... | [
"α : Type u_1\ninst✝ : Finite α\n⊢ #α = #α",
"α : Type u_1\ninst✝ : Finite α\n⊢ #α < ℵ₀"
] | Cardinal.cast_toNat_of_lt_aleph0 | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.LinearIndependent.Basic | {
"line": 479,
"column": 2
} | {
"line": 480,
"column": 7
} | {
"line": 482,
"column": 0
} | [
{
"pp": "ι : 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⊢ Disjoint (span R (v '' s)) (span R (v '' t))",
"ppTerm": "?m.72",
"assigned": true,
"usedConstants": [... | [] | convert! h.disjoint_span_image (s := (↑) ⁻¹' s) (t := (↑) ⁻¹' t) (hdj.preimage _) <;>
aesop | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.LinearAlgebra.LinearIndependent.Basic | {
"line": 546,
"column": 8
} | {
"line": 546,
"column": 22
} | {
"line": 546,
"column": 23
} | [
{
"pp": "G : Type u_6\ninst✝² : MulOneClass G\nL : Type u_7\ninst✝¹ : CommRing L\ninst✝ : IsDomain L\nthis✝ : DecidableEq (G →* L) := Classical.decEq (G →* L)\nthis : MulAction L L := Semiring.toModule.toMulAction\na : G →* L\ns : Finset (G →* L)\nhas : a ∉ s\nih : ∀ (g : (G →* L) → L), ∑ i ∈ s, g i • ⇑i = 0 → ... | [
"G : Type u_6\ninst✝² : MulOneClass G\nL : Type u_7\ninst✝¹ : CommRing L\ninst✝ : IsDomain L\nthis✝ : DecidableEq (G →* L) := Classical.decEq (G →* L)\nthis : MulAction L L := Semiring.toModule.toMulAction\na : G →* L\ns : Finset (G →* L)\nhas : a ∉ s\nih : ∀ (g : (G →* L) → L), ∑ i ∈ s, g i • ⇑i = 0 → ∀ i ∈ s, g i... | ← sub_eq_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Dimension.Basic | {
"line": 118,
"column": 2
} | {
"line": 118,
"column": 67
} | {
"line": 119,
"column": 2
} | [
{
"pp": "R : Type u\nM : Type v\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nc : Cardinal.{w}\nh✝ : lift.{v, w} c < lift.{w, v} (Module.rank R M)\nc' : Cardinal.{v}\nhc' : c' < Module.rank R M\nhcc' : lift.{w, v} c' = lift.{v, w} c\ns : Set M\nhs : LinearIndepOn R id s\nh : c' < #↑↑⟨s, hs... | [
"R : Type u\nM : Type v\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nc : Cardinal.{w}\nh✝ : lift.{v, w} c < lift.{w, v} (Module.rank R M)\nc' : Cardinal.{v}\nhc' : c' < Module.rank R M\nhcc' : lift.{w, v} c' = lift.{v, w} c\ns : Set M\nhs : LinearIndepOn R id s\nh : c' < #↑↑⟨s, hs⟩\nt : Set M... | rcases Cardinal.le_mk_iff_exists_subset.mp h.le with ⟨t, hst, ht⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.LinearAlgebra.Dimension.Basic | {
"line": 211,
"column": 9
} | {
"line": 211,
"column": 13
} | {
"line": 211,
"column": 14
} | [
{
"pp": "case refine_1\nR : Type u\nR' : Type u'\nM : Type v\nM' : Type v'\ninst✝⁵ : Semiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\ninst✝² : Semiring R'\ninst✝¹ : AddCommMonoid M'\ninst✝ : Module R' M'\ni : R → R'\nj : M →+ M'\nhi : Surjective i\nhj : Injective ⇑j\nhc : ∀ (r : R) (m : M), j (r • m) ... | [
"case refine_1\nR : Type u\nR' : Type u'\nM : Type v\nM' : Type v'\ninst✝⁵ : Semiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\ninst✝² : Semiring R'\ninst✝¹ : AddCommMonoid M'\ninst✝ : Module R' M'\ni : R → R'\nj : M →+ M'\nhi : Surjective i\nhj : Injective ⇑j\nhc : ∀ (r : R) (m : M), j (r • m) = i r • j m\... | hi', | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Dimension.Basic | {
"line": 211,
"column": 4
} | {
"line": 211,
"column": 23
} | {
"line": 212,
"column": 2
} | [
{
"pp": "case refine_1\nR : Type u\nR' : Type u'\nM : Type v\nM' : Type v'\ninst✝⁵ : Semiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\ninst✝² : Semiring R'\ninst✝¹ : AddCommMonoid M'\ninst✝ : Module R' M'\ni : R → R'\nj : M →+ M'\nhi : Surjective i\nhj : Injective ⇑j\nhc : ∀ (r : R) (m : M), j (r • m) ... | [] | rwa [hi', hi'] at h | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.LinearAlgebra.Dimension.Basic | {
"line": 212,
"column": 6
} | {
"line": 212,
"column": 18
} | {
"line": 212,
"column": 19
} | [
{
"pp": "case refine_2\nR : Type u\nR' : Type u'\nM : Type v\nM' : Type v'\ninst✝⁵ : Semiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\ninst✝² : Semiring R'\ninst✝¹ : AddCommMonoid M'\ninst✝ : Module R' M'\ni : R → R'\nj : M →+ M'\nhi : Surjective i\nhj : Injective ⇑j\nhc : ∀ (r : R) (m : M), j (r • m) ... | [
"case refine_2\nR : Type u\nR' : Type u'\nM : Type v\nM' : Type v'\ninst✝⁵ : Semiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\ninst✝² : Semiring R'\ninst✝¹ : AddCommMonoid M'\ninst✝ : Module R' M'\ni : R → R'\nj : M →+ M'\nhi : Surjective i\nhj : Injective ⇑j\nhc : ∀ (r : R) (m : M), j (r • m) = i r • j m\... | hc (i' r) m, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Set.Card | {
"line": 156,
"column": 2
} | {
"line": 156,
"column": 13
} | {
"line": 157,
"column": 2
} | [
{
"pp": "α : Type u_1\ns : Set α\n⊢ s.encard = ⊤ ↔ s.Infinite",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Set.encard",
"Mathlib.Tactic.Contrapose.contrapose_iff₁",
"instTopENat",
"congrArg",
"Set.Finite",
"id",
"Ne",
"Iff"... | [
"α : Type u_1\ns : Set α\n⊢ s.encard ≠ ⊤ ↔ s.Finite"
] | contrapose! | Mathlib.Tactic.Contrapose._aux_Mathlib_Tactic_Contrapose___macroRules_Mathlib_Tactic_Contrapose_contrapose!_1 | Mathlib.Tactic.Contrapose.contrapose! |
Mathlib.LinearAlgebra.LinearIndependent.Defs | {
"line": 737,
"column": 8
} | {
"line": 737,
"column": 22
} | {
"line": 737,
"column": 23
} | [
{
"pp": "case refine_2\nι : 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 : ∀ (s : Finset ι) (g : ι → R), ∑ i ∈ s, g i • v i = 0 → ∀ i ∈ s, g i = 0\ns : Finset ι\nf g : ι → R\n⊢ ∑ i ∈ s, f i • v i = ∑ i ∈ s, g i • v i → ∀ i ∈ s, f i = g i",
"... | [
"case refine_2\nι : 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 : ∀ (s : Finset ι) (g : ι → R), ∑ i ∈ s, g i • v i = 0 → ∀ i ∈ s, g i = 0\ns : Finset ι\nf g : ι → R\n⊢ ∑ i ∈ s, f i • v i - ∑ i ∈ s, g i • v i = 0 → ∀ i ∈ s, f i = g i"
] | ← sub_eq_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Set.Card | {
"line": 406,
"column": 2
} | {
"line": 406,
"column": 13
} | {
"line": 406,
"column": 13
} | [
{
"pp": "α : Type u_1\ns : Set α\n⊢ 1 < s.encard ↔ s.Nontrivial",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Set.encard",
"Mathlib.Tactic.Contrapose.contrapose_iff₁",
"instAddMonoidWithOneENat",
"instLinearOrderENat",
"congrArg",
"Part... | [
"α : Type u_1\ns : Set α\n⊢ s.encard ≤ 1 ↔ s.Subsingleton"
] | contrapose! | Mathlib.Tactic.Contrapose._aux_Mathlib_Tactic_Contrapose___macroRules_Mathlib_Tactic_Contrapose_contrapose!_1 | Mathlib.Tactic.Contrapose.contrapose! |
Mathlib.Data.Set.Card | {
"line": 409,
"column": 2
} | {
"line": 409,
"column": 13
} | {
"line": 409,
"column": 13
} | [
{
"pp": "α : Type u_1\ns : Set α\n⊢ 1 < s.encard ↔ ∃ a b, a ∈ s ∧ b ∈ s ∧ a ≠ b",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Push.not_exists._simp_1",
"Eq.mpr",
"Mathlib.Tactic.Push.not_and_eq",
"Set.encard",
"Mathlib.Tactic.Contrapose.contr... | [
"α : Type u_1\ns : Set α\n⊢ s.encard ≤ 1 ↔ ∀ (a b : α), a ∈ s → b ∈ s → a = b"
] | contrapose! | Mathlib.Tactic.Contrapose._aux_Mathlib_Tactic_Contrapose___macroRules_Mathlib_Tactic_Contrapose_contrapose!_1 | Mathlib.Tactic.Contrapose.contrapose! |
Mathlib.Data.Set.Card | {
"line": 430,
"column": 4
} | {
"line": 438,
"column": 58
} | {
"line": 439,
"column": 2
} | [
{
"pp": "case refine_1\nα : Type u_1\ns : Set α\nh : s.encard = 3\n⊢ ∃ x y z, x ≠ y ∧ x ≠ z ∧ y ≠ z ∧ s = {x, y, z}",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"False",
"Set.encard",
"instCharZeroENat",
"instAddMonoidWithOneENat",
"Chai... | [] | obtain ⟨x, hx⟩ := nonempty_of_encard_ne_zero (s := s) (by rw [h]; simp)
rw [← insert_eq_of_mem hx, ← insert_sdiff_singleton,
encard_insert_of_notMem (fun h ↦ h.2 rfl), (by exact rfl : (3 : ℕ∞) = 2 + 1),
(ENat.addLECancellable_of_ne_top ENat.one_ne_top).inj_left, encard_eq_two] at h
obtain ⟨y, z, hne... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Set.Card | {
"line": 430,
"column": 4
} | {
"line": 438,
"column": 58
} | {
"line": 439,
"column": 2
} | [
{
"pp": "case refine_1\nα : Type u_1\ns : Set α\nh : s.encard = 3\n⊢ ∃ x y z, x ≠ y ∧ x ≠ z ∧ y ≠ z ∧ s = {x, y, z}",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"False",
"Set.encard",
"instCharZeroENat",
"instAddMonoidWithOneENat",
"Chai... | [] | obtain ⟨x, hx⟩ := nonempty_of_encard_ne_zero (s := s) (by rw [h]; simp)
rw [← insert_eq_of_mem hx, ← insert_sdiff_singleton,
encard_insert_of_notMem (fun h ↦ h.2 rfl), (by exact rfl : (3 : ℕ∞) = 2 + 1),
(ENat.addLECancellable_of_ne_top ENat.one_ne_top).inj_left, encard_eq_two] at h
obtain ⟨y, z, hne... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Set.Card | {
"line": 479,
"column": 52
} | {
"line": 479,
"column": 90
} | {
"line": 479,
"column": 90
} | [
{
"pp": "α : Type u_1\ns : Set α\nn : ℕ\nIH : ↑n ≤ s.encard → ∃ t ⊆ s, t.encard = ↑n\nt₀ : Set α\nht₀s : t₀ ⊆ s\nht₀ : t₀.encard = ↑n\nhk : ↑n + 1 ≤ s.encard\nhne : t₀ ≠ s\nx : α\nhx : x ∈ s ∧ x ∉ t₀\n⊢ (insert x t₀).encard = ↑n + 1",
"ppTerm": "?m.141",
"assigned": true,
"usedConstants": [
"E... | [] | rw [encard_insert_of_notMem hx.2, ht₀] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Data.Set.Card | {
"line": 479,
"column": 52
} | {
"line": 479,
"column": 90
} | {
"line": 479,
"column": 90
} | [
{
"pp": "α : Type u_1\ns : Set α\nn : ℕ\nIH : ↑n ≤ s.encard → ∃ t ⊆ s, t.encard = ↑n\nt₀ : Set α\nht₀s : t₀ ⊆ s\nht₀ : t₀.encard = ↑n\nhk : ↑n + 1 ≤ s.encard\nhne : t₀ ≠ s\nx : α\nhx : x ∈ s ∧ x ∉ t₀\n⊢ (insert x t₀).encard = ↑n + 1",
"ppTerm": "?m.141",
"assigned": true,
"usedConstants": [
"E... | [] | rw [encard_insert_of_notMem hx.2, ht₀] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Set.Card | {
"line": 479,
"column": 52
} | {
"line": 479,
"column": 90
} | {
"line": 479,
"column": 90
} | [
{
"pp": "α : Type u_1\ns : Set α\nn : ℕ\nIH : ↑n ≤ s.encard → ∃ t ⊆ s, t.encard = ↑n\nt₀ : Set α\nht₀s : t₀ ⊆ s\nht₀ : t₀.encard = ↑n\nhk : ↑n + 1 ≤ s.encard\nhne : t₀ ≠ s\nx : α\nhx : x ∈ s ∧ x ∉ t₀\n⊢ (insert x t₀).encard = ↑n + 1",
"ppTerm": "?m.141",
"assigned": true,
"usedConstants": [
"E... | [] | rw [encard_insert_of_notMem hx.2, ht₀] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Set.Card | {
"line": 1023,
"column": 2
} | {
"line": 1023,
"column": 54
} | {
"line": 1025,
"column": 0
} | [
{
"pp": "α : Type u_1\ns t : Set α\nhs : s.Finite\nht : t.Finite\n⊢ (s ∩ t).ncard + (s ∪ t).ncard = s.ncard + t.ncard",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Set.instUnion",
"id",
"Set.instInter",
"add_comm",
"Inter.i... | [] | rw [add_comm, ncard_union_add_ncard_inter _ _ hs ht] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Data.Set.Card | {
"line": 1023,
"column": 2
} | {
"line": 1023,
"column": 54
} | {
"line": 1025,
"column": 0
} | [
{
"pp": "α : Type u_1\ns t : Set α\nhs : s.Finite\nht : t.Finite\n⊢ (s ∩ t).ncard + (s ∪ t).ncard = s.ncard + t.ncard",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Set.instUnion",
"id",
"Set.instInter",
"add_comm",
"Inter.i... | [] | rw [add_comm, ncard_union_add_ncard_inter _ _ hs ht] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Set.Card | {
"line": 1023,
"column": 2
} | {
"line": 1023,
"column": 54
} | {
"line": 1025,
"column": 0
} | [
{
"pp": "α : Type u_1\ns t : Set α\nhs : s.Finite\nht : t.Finite\n⊢ (s ∩ t).ncard + (s ∪ t).ncard = s.ncard + t.ncard",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Set.instUnion",
"id",
"Set.instInter",
"add_comm",
"Inter.i... | [] | rw [add_comm, ncard_union_add_ncard_inter _ _ hs ht] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Set.Card | {
"line": 1361,
"column": 6
} | {
"line": 1361,
"column": 35
} | {
"line": 1361,
"column": 36
} | [
{
"pp": "α : Type u_1\ns : Set α\nn : ℕ\nh : s.ncard = n + 1\nhsf : s.Finite\n⊢ ∃ a t, a ∉ t ∧ insert a t = s ∧ t.ncard = n",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
"congrArg",
"Eq.mp",
"instOfNatNat",
"instHAdd",
"HAdd.hAdd",
"Set.Finite.toFinset... | [
"α : Type u_1\ns : Set α\nn : ℕ\nhsf : s.Finite\nh : hsf.toFinset.card = n + 1\n⊢ ∃ a t, a ∉ t ∧ insert a t = s ∧ t.ncard = n"
] | ncard_eq_toFinset_card _ hsf, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Pi | {
"line": 290,
"column": 2
} | {
"line": 290,
"column": 29
} | {
"line": 291,
"column": 2
} | [
{
"pp": "R : Type u\nM : Type v\nι : Type x\ninst✝⁶ : Semiring R\nφ : ι → Type i\ninst✝⁵ : (i : ι) → AddCommMonoid (φ i)\ninst✝⁴ : (i : ι) → Module R (φ i)\ninst✝³ : DecidableEq ι\ninst✝² : Finite ι\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nf g : ((i : ι) → φ i) →ₗ[R] M\nh : ∀ (i : ι), f ∘ₗ single R φ i = ... | [
"R : Type u\nM : Type v\nι : Type x\ninst✝⁶ : Semiring R\nφ : ι → Type i\ninst✝⁵ : (i : ι) → AddCommMonoid (φ i)\ninst✝⁴ : (i : ι) → Module R (φ i)\ninst✝³ : DecidableEq ι\ninst✝² : Finite ι\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nf g : ((i : ι) → φ i) →ₗ[R] M\nh : ∀ (i : ι), f ∘ₗ single R φ i = g ∘ₗ single ... | refine pi_ext fun i x => ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.LinearAlgebra.Prod | {
"line": 476,
"column": 27
} | {
"line": 476,
"column": 41
} | {
"line": 476,
"column": 42
} | [
{
"pp": "case h\nR : Type u\nM : Type v\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\nM₂ : Type u_3\ninst✝³ : AddCommGroup M₂\ninst✝² : Module R M₂\nM₃ : Type u_4\ninst✝¹ : AddCommGroup M₃\ninst✝ : Module R M₃\nf : M →ₗ[R] M₃\ng : M₂ →ₗ[R] M₃\nhd : Disjoint f.range g.range\ny : M\nz : M₂\... | [
"case h\nR : Type u\nM : Type v\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\nM₂ : Type u_3\ninst✝³ : AddCommGroup M₂\ninst✝² : Module R M₂\nM₃ : Type u_4\ninst✝¹ : AddCommGroup M₃\ninst✝ : Module R M₃\nf : M →ₗ[R] M₃\ng : M₂ →ₗ[R] M₃\nhd : Disjoint f.range g.range\ny : M\nz : M₂\nh : f y + g... | ← sub_eq_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.SuccPred.LinearLocallyFinite | {
"line": 80,
"column": 10
} | {
"line": 80,
"column": 35
} | {
"line": 80,
"column": 36
} | [
{
"pp": "case succ.hin\nι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : SuccOrder ι\ninst✝¹ : PredOrder ι\ninst✝ : IsSuccArchimedean ι\ni j : ι\nhij : i ≤ j\nh_exists : ∃ n, succ^[n] i = j\nn : ℕ\nhn_eq : succ^[n + 1] i = j\nhn_lt_ne : ∀ m < n + 1, succ^[m] i ≠ j\n⊢ ¬IsMax (succ^[n + 1 - 1] i)",
"ppTerm": "?... | [
"case succ.hin\nι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : SuccOrder ι\ninst✝¹ : PredOrder ι\ninst✝ : IsSuccArchimedean ι\ni j : ι\nhij : i ≤ j\nh_exists : ∃ n, succ^[n] i = j\nn : ℕ\nhn_eq : succ^[n + 1] i = j\nhn_lt_ne : ∀ m < n + 1, succ^[m] i ≠ j\n⊢ ¬IsMax (succ^[n - 0] i)"
] | Nat.succ_sub_succ_eq_sub, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Order.Interval.Finset.Basic | {
"line": 74,
"column": 6
} | {
"line": 74,
"column": 25
} | {
"line": 74,
"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) (Ico a b) = Ico (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) (Ico a b) = map (addLeftEmbedding c) (Ico a b)"
] | ← map_add_left_Ico, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Order.Interval.Finset.Basic | {
"line": 83,
"column": 41
} | {
"line": 83,
"column": 59
} | {
"line": 83,
"column": 60
} | [
{
"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 ↦ x + c) (Icc a b) = image (⇑(addRightEmbedding c)) (Icc a b)",
"ppTerm": "?m.45",
"ass... | [
"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 ↦ x + c) (Icc a b) = image (⇑{ toFun := fun h ↦ h + c, inj' := ⋯ }) (Icc a b)"
] | addRightEmbedding, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Order.Interval.Finset.Basic | {
"line": 86,
"column": 41
} | {
"line": 86,
"column": 59
} | {
"line": 86,
"column": 60
} | [
{
"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 ↦ x + c) (Ico a b) = image (⇑(addRightEmbedding c)) (Ico a b)",
"ppTerm": "?m.45",
"ass... | [
"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 ↦ x + c) (Ico a b) = image (⇑{ toFun := fun h ↦ h + c, inj' := ⋯ }) (Ico a b)"
] | addRightEmbedding, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Order.Interval.Finset.Basic | {
"line": 89,
"column": 41
} | {
"line": 89,
"column": 59
} | {
"line": 89,
"column": 60
} | [
{
"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 ↦ x + c) (Ioc a b) = image (⇑(addRightEmbedding c)) (Ioc a b)",
"ppTerm": "?m.45",
"ass... | [
"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 ↦ x + c) (Ioc a b) = image (⇑{ toFun := fun h ↦ h + c, inj' := ⋯ }) (Ioc a b)"
] | addRightEmbedding, | Lean.Elab.Tactic.evalRewriteSeq | null |
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