module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.LinearAlgebra.Basis.Basic | {
"line": 100,
"column": 4
} | {
"line": 100,
"column": 28
} | {
"line": 100,
"column": 29
} | [
{
"pp": "ι : Type u_1\nR : Type u_3\nR₂ : Type u_4\nM : Type u_5\nM' : Type u_6\ninst✝⁷ : Semiring R\ninst✝⁶ : AddCommMonoid M\ninst✝⁵ : Module R M\ninst✝⁴ : AddCommMonoid M'\ninst✝³ : Module R M'\nb : Basis ι R M\ninst✝² : Semiring R₂\ninst✝¹ : Module R₂ M'\ninst✝ : SMulCommClass R R₂ M'\nv : ι → M'\nhv : Line... | [
"ι : Type u_1\nR : Type u_3\nR₂ : Type u_4\nM : Type u_5\nM' : Type u_6\ninst✝⁷ : Semiring R\ninst✝⁶ : AddCommMonoid M\ninst✝⁵ : Module R M\ninst✝⁴ : AddCommMonoid M'\ninst✝³ : Module R M'\nb : Basis ι R M\ninst✝² : Semiring R₂\ninst✝¹ : Module R₂ M'\ninst✝ : SMulCommClass R R₂ M'\nv : ι → M'\nhv : LinearIndependen... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Basis.Basic | {
"line": 138,
"column": 2
} | {
"line": 138,
"column": 40
} | {
"line": 138,
"column": 41
} | [
{
"pp": "ι : Type u_1\nM : Type u_5\ninst✝² : AddCommMonoid M\nR : Type u_7\ninst✝¹ : CommSemiring R\ninst✝ : Module R M\nb : Basis ι R M\ns : Finset ι\nl₁ l₂ : ι → R\nh : ∑ i ∈ s, l₁ i • b.coord i = ∑ i ∈ s, l₂ i • b.coord i\nj : ι\nhj : j ∈ s\n⊢ l₁ j = l₂ j",
"ppTerm": "?m.41",
"assigned": false,
... | [
"ι : Type u_1\nM : Type u_5\ninst✝² : AddCommMonoid M\nR : Type u_7\ninst✝¹ : CommSemiring R\ninst✝ : Module R M\nb : Basis ι R M\ns : Finset ι\nl₁ l₂ : ι → R\nh : ∑ i ∈ s, l₁ i • b.coord i = ∑ i ∈ s, l₂ i • b.coord i\nj : ι\nhj : j ∈ s\n⊢ l₁ j = l₂ j"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Basis.Basic | {
"line": 159,
"column": 2
} | {
"line": 161,
"column": 48
} | {
"line": 163,
"column": 0
} | [
{
"pp": "ι : Type u_1\nR : Type u_3\nM : Type u_5\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\nv : ι → M\nhli : LinearIndependent R v\nhsp : ⊤ ≤ span R (range v)\ninst✝ : DecidableEq ι\ni j : ι\n⊢ ((Basis.mk hli hsp).coord i) (v j) = if j = i then 1 else 0",
"ppTerm": "?m.46",
"a... | [] | rcases eq_or_ne j i with h | h
· simp only [h, if_true, mk_coord_apply_eq i]
· simp only [h, if_false, mk_coord_apply_ne h] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Basis.Basic | {
"line": 159,
"column": 2
} | {
"line": 161,
"column": 48
} | {
"line": 163,
"column": 0
} | [
{
"pp": "ι : Type u_1\nR : Type u_3\nM : Type u_5\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\nv : ι → M\nhli : LinearIndependent R v\nhsp : ⊤ ≤ span R (range v)\ninst✝ : DecidableEq ι\ni j : ι\n⊢ ((Basis.mk hli hsp).coord i) (v j) = if j = i then 1 else 0",
"ppTerm": "?m.46",
"a... | [] | rcases eq_or_ne j i with h | h
· simp only [h, if_true, mk_coord_apply_eq i]
· simp only [h, if_false, mk_coord_apply_ne h] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Basis.Basic | {
"line": 227,
"column": 22
} | {
"line": 227,
"column": 57
} | {
"line": 227,
"column": 58
} | [
{
"pp": "ι : Type u_1\nR : Type u_3\nM : Type u_5\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ninst✝ : Nontrivial R\nb : Basis ι R M\nw : Set M\nhi : LinearIndependent R Subtype.val\nh : range ⇑b ⊆ w\nx : M\np : x ∈ w\nq : x ∉ range ⇑b\ne : (linearCombination R ⇑b) (b.repr x) = x\ni i' :... | [
"ι : Type u_1\nR : Type u_3\nM : Type u_5\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ninst✝ : Nontrivial R\nb : Basis ι R M\nw : Set M\nhi : LinearIndependent R Subtype.val\nh : range ⇑b ⊆ w\nx : M\np : x ∈ w\nq : x ∉ range ⇑b\ne : (linearCombination R ⇑b) (b.repr x) = x\ni i' : ι\nr : (fun... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Basis.Basic | {
"line": 304,
"column": 4
} | {
"line": 304,
"column": 61
} | {
"line": 304,
"column": 62
} | [
{
"pp": "case mp\nR : Type u_7\nM : Type u_8\ninst✝⁵ : Ring R\ninst✝⁴ : IsDomain R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : IsTorsionFree R M\nι : Type u_9\ninst✝ : Unique ι\nb : Basis ι R M\n⊢ ∀ (y : M), ∃ r, r • b default = y",
"ppTerm": "?mp",
"assigned": false,
"usedConstants": []... | [
"case mp\nR : Type u_7\nM : Type u_8\ninst✝⁵ : Ring R\ninst✝⁴ : IsDomain R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : IsTorsionFree R M\nι : Type u_9\ninst✝ : Unique ι\nb : Basis ι R M\n⊢ ∀ (y : M), ∃ r, r • b default = y"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.ENat.Pow | {
"line": 42,
"column": 2
} | {
"line": 42,
"column": 12
} | {
"line": 42,
"column": 13
} | [
{
"pp": "case coe\nx : ℕ∞\nn : ℕ\n⊢ x ^ ↑n = if ↑n < ⊤ then x ^ (↑n).toNat else if x = 0 then 0 else if x = 1 then 1 else ⊤",
"ppTerm": "?coe",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instAddMonoidWithOneENat",
"LinearOrder.toDecidableEq",
"ENat.instNatCast",
"i... | [] | | coe n => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalCases | null |
Mathlib.LinearAlgebra.LinearIndependent.Defs | {
"line": 144,
"column": 2
} | {
"line": 145,
"column": 9
} | {
"line": 145,
"column": 10
} | [
{
"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\n⊢ Injective v",
"ppTerm": "?m.12",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"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\n⊢ Injective v"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.LinearIndependent.Defs | {
"line": 156,
"column": 50
} | {
"line": 156,
"column": 61
} | {
"line": 156,
"column": 62
} | [
{
"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\ni : ι\nhv : LinearIndependent R v\nh : v i = 0\n⊢ (Finsupp.linearCombination R v) (Finsupp.single i 1) = (Finsupp.linearCombination R v) 0",
"ppTerm": "?m.25... | [
"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\ni : ι\nhv : LinearIndependent R v\nh : v i = 0\n⊢ v i = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.ENat.Pow | {
"line": 93,
"column": 2
} | {
"line": 93,
"column": 13
} | {
"line": 93,
"column": 14
} | [
{
"pp": "x y : ℕ∞\nh : x ≠ 0\n⊢ 1 ≤ x ^ y",
"ppTerm": "?m.12",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"x y : ℕ∞\nh : x ≠ 0\n⊢ 1 ≤ x ^ y"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.LinearIndependent.Defs | {
"line": 224,
"column": 10
} | {
"line": 224,
"column": 70
} | {
"line": 224,
"column": 71
} | [
{
"pp": "ι : Type u'\nR : Type u_2\nM : Type u_4\nv : ι → M\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nhv : ∀ (l₁ l₂ : ι →₀ R), (Finsupp.linearCombination R v) l₁ = (Finsupp.linearCombination R v) l₂ → l₁ = l₂\ns : Finset ι\nf g : ι → R\neq : ∑ i ∈ s, f i • v i = ∑ i ∈ s, g i • v i\ni :... | [
"ι : Type u'\nR : Type u_2\nM : Type u_4\nv : ι → M\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nhv : ∀ (l₁ l₂ : ι →₀ R), (Finsupp.linearCombination R v) l₁ = (Finsupp.linearCombination R v) l₂ → l₁ = l₂\ns : Finset ι\nf g : ι → R\neq : ∑ i ∈ s, f i • v i = ∑ i ∈ s, g i • v i\ni : ι\nhis : i ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Group.ModEq | {
"line": 142,
"column": 2
} | {
"line": 142,
"column": 13
} | {
"line": 142,
"column": 14
} | [
{
"pp": "case h\nM : Type u_1\ninst✝³ : AddCommMonoid M\na b p : M\nN : Type u_2\nF : Type u_3\ninst✝² : AddCommMonoid N\ninst✝¹ : FunLike F M N\ninst✝ : AddMonoidHomClass F M N\nf : F\nm n : ℕ\nh : m • p + a = n • p + b\n⊢ m • f p + f a = n • f p + f b",
"ppTerm": "?h",
"assigned": false,
"usedCons... | [
"case h\nM : Type u_1\ninst✝³ : AddCommMonoid M\na b p : M\nN : Type u_2\nF : Type u_3\ninst✝² : AddCommMonoid N\ninst✝¹ : FunLike F M N\ninst✝ : AddMonoidHomClass F M N\nf : F\nm n : ℕ\nh : m • p + a = n • p + b\n⊢ m • f p + f a = n • f p + f b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.LinearIndependent.Defs | {
"line": 273,
"column": 2
} | {
"line": 273,
"column": 13
} | {
"line": 273,
"column": 14
} | [
{
"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✝ : Fintype ι\n⊢ ¬LinearIndependent R v ↔ ∃ f g, ∑ i, f i • v i = ∑ i, g i • v i ∧ ∃ i, f i ≠ g i",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"inst... | [
"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✝ : Fintype ι\n⊢ ¬LinearIndependent R v ↔ ∃ f g, ∑ i, f i • v i = ∑ i, g i • v i ∧ ∃ i, ¬f i = g i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.LinearIndependent.Defs | {
"line": 279,
"column": 26
} | {
"line": 279,
"column": 56
} | {
"line": 279,
"column": 56
} | [
{
"pp": "ι : Type u'\nR : Type u_2\nM : Type u_4\nv : ι → M\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\ns : Finset ι\n⊢ (LinearIndependent R fun (x : ↑↑s) ↦ v ↑x) ↔\n ∀ (f g : ι → R), ∑ i ∈ s, f i • v i = ∑ i ∈ s, g i • v i → ∀ i ∈ s, f i = g i",
"ppTerm": "?m.36",
"assigned":... | [
"ι : Type u'\nR : Type u_2\nM : Type u_4\nv : ι → M\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\ns : Finset ι\n⊢ (∀ (f g : ↑↑s → R), ∑ i, f i • v ↑i = ∑ i, g i • v ↑i → ∀ (i : ↑↑s), f i = g i) ↔\n ∀ (f g : ι → R), ∑ i ∈ s, f i • v i = ∑ i ∈ s, g i • v i → ∀ i ∈ s, f i = g i"
] | Fintype.linearIndependent_iffₛ | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.LinearAlgebra.LinearIndependent.Defs | {
"line": 285,
"column": 4
} | {
"line": 285,
"column": 15
} | {
"line": 285,
"column": 16
} | [
{
"pp": "case mpr\nι : Type u'\nR : Type u_2\nM : Type u_4\nv : ι → M\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\ns : Finset ι\nhv : ∀ (f g : ι → R), ∑ i ∈ s, f i • v i = ∑ i ∈ s, g i • v i → ∀ i ∈ s, f i = g i\nf g : ↑↑s → R\nhfg : ∑ i, f i • v ↑i = ∑ i, g i • v ↑i\ni : ↑↑s\n⊢ f i = g i... | [
"case mpr\nι : Type u'\nR : Type u_2\nM : Type u_4\nv : ι → M\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\ns : Finset ι\nhv : ∀ (f g : ι → R), ∑ i ∈ s, f i • v i = ∑ i ∈ s, g i • v i → ∀ i ∈ s, f i = g i\nf g : ↑↑s → R\nhfg : ∑ i, f i • v ↑i = ∑ i, g i • v ↑i\ni : ↑↑s\n⊢ f i = g i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Group.ModEq | {
"line": 179,
"column": 2
} | {
"line": 179,
"column": 43
} | {
"line": 179,
"column": 44
} | [
{
"pp": "M : Type u_1\ninst✝ : AddCancelCommMonoid M\na b c d p : M\nh : c ≡ d [PMOD p]\n⊢ a + c ≡ b + d [PMOD p] ↔ a ≡ b [PMOD p]",
"ppTerm": "?m.15",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"M : Type u_1\ninst✝ : AddCancelCommMonoid M\na b c d p : M\nh : c ≡ d [PMOD p]\n⊢ a + c ≡ b + d [PMOD p] ↔ a ≡ b [PMOD p]"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Group.ModEq | {
"line": 195,
"column": 2
} | {
"line": 195,
"column": 13
} | {
"line": 195,
"column": 14
} | [
{
"pp": "M : Type u_1\ninst✝ : AddCancelCommMonoid M\na b p : M\n⊢ a + b ≡ a [PMOD p] ↔ b ≡ 0 [PMOD p]",
"ppTerm": "?m.11",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"M : Type u_1\ninst✝ : AddCancelCommMonoid M\na b p : M\n⊢ a + b ≡ a [PMOD p] ↔ b ≡ 0 [PMOD p]"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.LinearIndependent.Defs | {
"line": 291,
"column": 2
} | {
"line": 291,
"column": 13
} | {
"line": 291,
"column": 14
} | [
{
"pp": "ι : Type u'\nR : Type u_2\nM : Type u_4\nv : ι → M\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\ns : Finset ι\n⊢ ¬LinearIndepOn R v ↑s ↔ ∃ f g, ∑ i ∈ s, f i • v i = ∑ i ∈ s, g i • v i ∧ ∃ i ∈ s, f i ≠ g i",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
... | [
"ι : Type u'\nR : Type u_2\nM : Type u_4\nv : ι → M\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\ns : Finset ι\n⊢ ¬LinearIndepOn R v ↑s ↔ ∃ f g, ∑ i ∈ s, f i • v i = ∑ i ∈ s, g i • v i ∧ ∃ i ∈ s, ¬f i = g i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.LinearIndependent.Defs | {
"line": 412,
"column": 12
} | {
"line": 412,
"column": 23
} | {
"line": 412,
"column": 24
} | [
{
"pp": "case refine_1\nι : Type u'\nR : Type u_2\ns : Set ι\nM : Type u_4\nv : ι → M\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nh :\n ∀ (f : ι →₀ R),\n (∀ x ∈ f.support, x ∈ s) →\n ∀ (g : ι →₀ R), (∀ x ∈ g.support, x ∈ s) → ((f.sum fun i a ↦ a • v i) = g.sum fun i a ↦ a • v i)... | [
"case refine_1\nι : Type u'\nR : Type u_2\ns : Set ι\nM : Type u_4\nv : ι → M\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nh :\n ∀ (f : ι →₀ R),\n (∀ x ∈ f.support, x ∈ s) →\n ∀ (g : ι →₀ R), (∀ x ∈ g.support, x ∈ s) → ((f.sum fun i a ↦ a • v i) = g.sum fun i a ↦ a • v i) → f = g\nl₁... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.LinearIndependent.Defs | {
"line": 412,
"column": 12
} | {
"line": 412,
"column": 23
} | {
"line": 412,
"column": 24
} | [
{
"pp": "case refine_2\nι : Type u'\nR : Type u_2\ns : Set ι\nM : Type u_4\nv : ι → M\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nh :\n ∀ (f : ι →₀ R),\n (∀ x ∈ f.support, x ∈ s) →\n ∀ (g : ι →₀ R), (∀ x ∈ g.support, x ∈ s) → ((f.sum fun i a ↦ a • v i) = g.sum fun i a ↦ a • v i)... | [
"case refine_2\nι : Type u'\nR : Type u_2\ns : Set ι\nM : Type u_4\nv : ι → M\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nh :\n ∀ (f : ι →₀ R),\n (∀ x ∈ f.support, x ∈ s) →\n ∀ (g : ι →₀ R), (∀ x ∈ g.support, x ∈ s) → ((f.sum fun i a ↦ a • v i) = g.sum fun i a ↦ a • v i) → f = g\nl₁... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.LinearIndependent.Basic | {
"line": 234,
"column": 13
} | {
"line": 234,
"column": 58
} | {
"line": 234,
"column": 59
} | [
{
"pp": "ι : Type u'\nR : Type u_2\ninst✝² : Semiring R\nM : Type u_6\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nv : ι → M\nli : LinearIndependent R v\nc d : R\ni j : ι\nhc : c ≠ 0\nh : c • v i = d • v j\n⊢ (Finsupp.linearCombination R v) (Finsupp.single i c) = (Finsupp.linearCombination R v) (Finsupp.singl... | [
"ι : Type u'\nR : Type u_2\ninst✝² : Semiring R\nM : Type u_6\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nv : ι → M\nli : LinearIndependent R v\nc d : R\ni j : ι\nhc : c ≠ 0\nh : c • v i = d • v j\n⊢ c • v i = d • v j"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.LinearIndependent.Defs | {
"line": 532,
"column": 55
} | {
"line": 532,
"column": 92
} | {
"line": 532,
"column": 93
} | [
{
"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\ni : ι\nhi : v i ∈ span R (v '' {i}ᶜ)\n⊢ 1 • v i ∈ span R (v '' (univ \\ {i}))",
"ppTerm": "?m.43",
"assigned": true,
"use... | [
"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\ni : ι\nhi : v i ∈ span R (v '' {i}ᶜ)\n⊢ v i ∈ span R (v '' (univ \\ {i}))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.LinearIndependent.Defs | {
"line": 536,
"column": 56
} | {
"line": 536,
"column": 93
} | {
"line": 536,
"column": 94
} | [
{
"pp": "ι : Type u'\nR : Type u_2\ns : Set ι\nM : Type u_4\nv : ι → M\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ni : ι\ninst✝ : Nontrivial R\nhv : LinearIndepOn R v s\nhi : i ∈ s\nhi' : v i ∈ span R (v '' (s \\ {i}))\n⊢ 1 • v i ∈ span R (v '' (s \\ {i}))",
"ppTerm": "?m.48",
"... | [
"ι : Type u'\nR : Type u_2\ns : Set ι\nM : Type u_4\nv : ι → M\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ni : ι\ninst✝ : Nontrivial R\nhv : LinearIndepOn R v s\nhi : i ∈ s\nhi' : v i ∈ span R (v '' (s \\ {i}))\n⊢ v i ∈ span R (v '' (s \\ {i}))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.LinearIndependent.Defs | {
"line": 539,
"column": 32
} | {
"line": 539,
"column": 48
} | {
"line": 539,
"column": 49
} | [
{
"pp": "ι : Type u'\nR : Type u_2\ns : Set ι\nM : Type u_4\nv : ι → M\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ni : ι\ninst✝ : Nontrivial R\nhv : LinearIndepOn R v (insert i s)\nhi : i ∉ s\n⊢ v i ∉ span R (v '' s)",
"ppTerm": "?m.30",
"assigned": false,
"usedConstants": [... | [
"ι : Type u'\nR : Type u_2\ns : Set ι\nM : Type u_4\nv : ι → M\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ni : ι\ninst✝ : Nontrivial R\nhv : LinearIndepOn R v (insert i s)\nhi : i ∉ s\n⊢ v i ∉ span R (v '' s)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.LinearIndependent.Basic | {
"line": 259,
"column": 2
} | {
"line": 259,
"column": 13
} | {
"line": 259,
"column": 14
} | [
{
"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\ns : Set ι\nx : ι\nh : x ∉ s\nh' : v x ∈ span R (v '' {x})\nw : v x ∈ span R (v '' s)\n⊢ Disjoint {x} s",
"ppTerm": "?m.77",
"... | [
"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\ns : Set ι\nx : ι\nh : x ∉ s\nh' : v x ∈ span R (v '' {x})\nw : v x ∈ span R (v '' s)\n⊢ x ∉ s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Nat.GCD.Basic | {
"line": 275,
"column": 2
} | {
"line": 276,
"column": 62
} | {
"line": 276,
"column": 63
} | [
{
"pp": "case inr.inr\nn m k : ℕ\nhkm : m ∣ k\nhkn : n ∣ k\nc : ℕ\nhmc : k ∣ m * c\nhnc : k ∣ n * c\nhm : m > 0\nhn : n > 0\n⊢ k / m.gcd n ∣ c",
"ppTerm": "?inr.inr",
"assigned": true,
"usedConstants": [
"Nat.gcd",
"Nat.gcd_dvd_left",
"Eq.mpr",
"Dvd.dvd",
"instHDiv",
... | [
"case inr.inr\nn m k : ℕ\nhkm : m ∣ k\nhkn : n ∣ k\nc : ℕ\nhmc : k ∣ m * c\nhnc : k ∣ n * c\nhm : m > 0\nhn : n > 0\n⊢ k ∣ m.gcd n * c"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.LinearIndependent.Basic | {
"line": 268,
"column": 4
} | {
"line": 268,
"column": 66
} | {
"line": 268,
"column": 67
} | [
{
"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\nw : (Finsupp.linearCombination R v) f = v x\n⊢ ∀ x ∈ Finsupp.supported R R ↑f.support, (Finsup... | [
"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\nw : (Finsupp.linearCombination R v) f = v x\n⊢ ∀ x ∈ Finsupp.supported R R ↑f.support, ¬(Finsupp.linearCom... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Nat.ModEq | {
"line": 49,
"column": 4
} | {
"line": 49,
"column": 15
} | {
"line": 49,
"column": 16
} | [
{
"pp": "case mp\na b n k l : ℕ\nh : k • n + a = l • n + b\n⊢ a % n = b % n",
"ppTerm": "?mp",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"case mp\na b n k l : ℕ\nh : k • n + a = l • n + b\n⊢ a % n = b % n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Nat.ModEq | {
"line": 63,
"column": 2
} | {
"line": 63,
"column": 13
} | {
"line": 63,
"column": 14
} | [
{
"pp": "M : Type u_1\ninst✝¹ : AddCommMonoidWithOne M\ninst✝ : CharZero M\na b n : ℕ\n⊢ ↑a ≡ ↑b [PMOD ↑n] ↔ a ≡ b [MOD n]",
"ppTerm": "?m.4",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"M : Type u_1\ninst✝¹ : AddCommMonoidWithOne M\ninst✝ : CharZero M\na b n : ℕ\n⊢ ↑a ≡ ↑b [PMOD ↑n] ↔ a ≡ b [MOD n]"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.LinearIndependent.Basic | {
"line": 275,
"column": 33
} | {
"line": 275,
"column": 44
} | {
"line": 275,
"column": 45
} | [
{
"pp": "R : Type u_2\nM : Type u_4\nM' : Type u_5\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : AddCommMonoid M'\ninst✝¹ : Module R M\ninst✝ : Module R M'\ns : Set M\nf : M →ₗ[R] M'\nhs : LinearIndepOn R id s\nhf_inj : InjOn ⇑f ↑(span R s)\n⊢ InjOn ⇑f ↑(span R (id '' s))",
"ppTerm": "?m.55",
... | [
"R : Type u_2\nM : Type u_4\nM' : Type u_5\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : AddCommMonoid M'\ninst✝¹ : Module R M\ninst✝ : Module R M'\ns : Set M\nf : M →ₗ[R] M'\nhs : LinearIndepOn R id s\nhf_inj : InjOn ⇑f ↑(span R s)\n⊢ InjOn ⇑f ↑(span R s)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.LinearIndependent.Defs | {
"line": 576,
"column": 4
} | {
"line": 576,
"column": 15
} | {
"line": 576,
"column": 16
} | [
{
"pp": "case mpr\nι : Type w\nR : Type u\ninst✝³ : Semiring R\ninst✝² : Nontrivial R\nM : Type v\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nv : ι → M\ni : LinearIndependent R v\nw : Set M\ni' : LinearIndependent R Subtype.val\nh : range v ⊆ w\np : Surjective fun i ↦ ⟨v i, ⋯⟩\nq : (fun x ↦ ↑⟨v x, ⋯⟩) '' uni... | [
"case mpr\nι : Type w\nR : Type u\ninst✝³ : Semiring R\ninst✝² : Nontrivial R\nM : Type v\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nv : ι → M\ni : LinearIndependent R v\nw : Set M\ni' : LinearIndependent R Subtype.val\nh : range v ⊆ w\np : Surjective fun i ↦ ⟨v i, ⋯⟩\nq : (fun x ↦ ↑⟨v x, ⋯⟩) '' univ = Subtype.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.LinearIndependent.Defs | {
"line": 603,
"column": 6
} | {
"line": 603,
"column": 56
} | {
"line": 603,
"column": 57
} | [
{
"pp": "case refine_1.refine_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\nth... | [
"case refine_1.refine_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\nthis : Ordered... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.LinearIndependent.Defs | {
"line": 604,
"column": 6
} | {
"line": 604,
"column": 57
} | {
"line": 604,
"column": 58
} | [
{
"pp": "case refine_1.refine_2\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\nth... | [
"case refine_1.refine_2\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 : Ordered... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Nat.ModEq | {
"line": 396,
"column": 2
} | {
"line": 396,
"column": 24
} | {
"line": 396,
"column": 25
} | [
{
"pp": "m a b c : ℕ\nhm : 0 < m\nh : a * c ≡ b * c [MOD m]\n⊢ c * a ≡ c * b [MOD m]",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"CommSemiring.toNonUnitalCommSemiring",
"congrArg",
"id",
"CommMagma.toMul",
"instMulNat",
... | [
"m a b c : ℕ\nhm : 0 < m\nh : a * c ≡ b * c [MOD m]\n⊢ a * c ≡ b * c [MOD m]"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Nat.ModEq | {
"line": 413,
"column": 2
} | {
"line": 413,
"column": 19
} | {
"line": 413,
"column": 20
} | [
{
"pp": "case inr\nm a b c : ℕ\nhmc : m.gcd c = 1\nh : c * a ≡ c * b [MOD m]\nhm : m > 0\n⊢ a ≡ b [MOD m]",
"ppTerm": "?inr",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"case inr\nm a b c : ℕ\nhmc : m.gcd c = 1\nh : c * a ≡ c * b [MOD m]\nhm : m > 0\n⊢ a ≡ b [MOD m]"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Nat.ModEq | {
"line": 417,
"column": 35
} | {
"line": 417,
"column": 57
} | {
"line": 417,
"column": 58
} | [
{
"pp": "m a b c : ℕ\nhmc : m.gcd c = 1\nh : a * c ≡ b * c [MOD m]\n⊢ c * a ≡ c * b [MOD m]",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"CommSemiring.toNonUnitalCommSemiring",
"congrArg",
"id",
"CommMagma.toMul",
"instMul... | [
"m a b c : ℕ\nhmc : m.gcd c = 1\nh : a * c ≡ b * c [MOD m]\n⊢ a * c ≡ b * c [MOD m]"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.BigOperators.Fin | {
"line": 86,
"column": 2
} | {
"line": 86,
"column": 24
} | {
"line": 86,
"column": 25
} | [
{
"pp": "M : Type u_2\ninst✝ : CommMonoid M\nn : ℕ\nf : Fin (n + 1) → M\n⊢ ∏ i, f i = (∏ i, f i.castSucc) * f (last n)",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"CommMonoid.toCommSemigroup",
"Finset.univ",
"Monoid.toMulOneClass",
... | [
"M : Type u_2\ninst✝ : CommMonoid M\nn : ℕ\nf : Fin (n + 1) → M\n⊢ ∏ i, f i = f (last n) * ∏ i, f i.castSucc"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Nat.ModEq | {
"line": 483,
"column": 2
} | {
"line": 483,
"column": 41
} | {
"line": 484,
"column": 4
} | [
{
"pp": "m n : ℕ\nco : n.Coprime m\na b z : ℕ\nhzan : z ≡ a [MOD n]\nhzbm : z ≡ b [MOD m]\n⊢ z ≡ ↑(chineseRemainder co a b) [MOD n * m]",
"ppTerm": "?m.10",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"m n : ℕ\nco : n.Coprime m\na b z : ℕ\nhzan : z ≡ a [MOD n]\nhzbm : z ≡ b [MOD m]\n⊢ z ≡ ↑(chineseRemainder co a b) [MOD n * m]"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Nat.ModEq | {
"line": 564,
"column": 2
} | {
"line": 564,
"column": 25
} | {
"line": 564,
"column": 26
} | [
{
"pp": "n m : ℕ\n⊢ n % 2 = 1 → m % 2 = 1 → n * m % 2 = 1",
"ppTerm": "?m.32",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"n m : ℕ\n⊢ n % 2 = 1 → m % 2 = 1 → n * m % 2 = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.LinearIndependent.Defs | {
"line": 623,
"column": 6
} | {
"line": 623,
"column": 44
} | {
"line": 623,
"column": 45
} | [
{
"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 := ⋯\nthis : OrderedSub R\ns : Finset ι\nf g : ι... | [
"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 := ⋯\nthis : OrderedSub R\ns : Finset ι\nf g : ι → R\nheq : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Nat.ModEq | {
"line": 576,
"column": 2
} | {
"line": 576,
"column": 21
} | {
"line": 576,
"column": 22
} | [
{
"pp": "n : ℕ\n⊢ n % 4 = 1 → n % 2 = 1",
"ppTerm": "?m.19",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"n : ℕ\n⊢ n % 4 = 1 → n % 2 = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Nat.ModEq | {
"line": 579,
"column": 2
} | {
"line": 579,
"column": 21
} | {
"line": 579,
"column": 22
} | [
{
"pp": "n : ℕ\n⊢ n % 4 = 3 → n % 2 = 1",
"ppTerm": "?m.19",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"n : ℕ\n⊢ n % 4 = 3 → n % 2 = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.BigOperators.Fin | {
"line": 464,
"column": 2
} | {
"line": 464,
"column": 13
} | {
"line": 464,
"column": 14
} | [
{
"pp": "n : ℕ\nR : Type u_3\ninst✝ : CommSemiring R\na b : R\n⊢ ∑ s, a ^ #s * b ^ (n - #s) = (a + b) ^ n",
"ppTerm": "?m.34",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"n : ℕ\nR : Type u_3\ninst✝ : CommSemiring R\na b : R\n⊢ ∑ s, a ^ #s * b ^ (n - #s) = (a + b) ^ n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.LinearIndependent.Defs | {
"line": 625,
"column": 6
} | {
"line": 625,
"column": 44
} | {
"line": 625,
"column": 45
} | [
{
"pp": "case neg\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 := ⋯\nthis : OrderedSub R\ns : Finset ι\nf g : ι... | [
"case neg\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 : OrderedSub R\ns : Fin... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.BigOperators.Fin | {
"line": 525,
"column": 4
} | {
"line": 525,
"column": 61
} | {
"line": 526,
"column": 4
} | [
{
"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 ∘ ... | [
"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.succAbove i).s... | simp only [Function.comp_apply, succ_succAbove_succ] at * | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.SetTheory.Cardinal.Finite | {
"line": 138,
"column": 2
} | {
"line": 138,
"column": 59
} | {
"line": 138,
"column": 60
} | [
{
"pp": "α : Type u_3\nn : ℕ\nf : α ≃ Fin n\n⊢ Nat.card α = n",
"ppTerm": "?m.2",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_3\nn : ℕ\nf : α ≃ Fin n\n⊢ Nat.card α = n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Cardinal.Finite | {
"line": 190,
"column": 4
} | {
"line": 190,
"column": 43
} | {
"line": 190,
"column": 44
} | [
{
"pp": "case inl\nα✝ : Type u_1\nβ : Type u_2\nα : Type u_3\nh : Nat.card α ≠ 0\nval✝ : Fintype α\n⊢ α ≃ Fin (Nat.card α)",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Fintype.card",
"id",
"Equiv",
"Nat.card",
"Nat",
... | [
"case inl\nα✝ : Type u_1\nβ : Type u_2\nα : Type u_3\nh : Nat.card α ≠ 0\nval✝ : Fintype α\n⊢ α ≃ Fin (Fintype.card α)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Cardinal.Finite | {
"line": 384,
"column": 40
} | {
"line": 384,
"column": 63
} | {
"line": 384,
"column": 64
} | [
{
"pp": "α : Type u_1\nh : 3 ≤ card α\nx y : α\n⊢ 3 ≤ #α",
"ppTerm": "?m.10",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\nh : 3 ≤ card α\nx y : α\n⊢ 3 ≤ #α"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.LinearIndependent.Defs | {
"line": 653,
"column": 48
} | {
"line": 653,
"column": 64
} | {
"line": 653,
"column": 65
} | [
{
"pp": "case refine_2.refine_1\nι : Type u'\nR : Type u_2\nM : Type u_4\nv : ι → M\ninst✝⁸ : Semiring R\ninst✝⁷ : AddCommMonoid M\ninst✝⁶ : Module R M\ninst✝⁵ : LinearOrder R\ninst✝⁴ : CanonicallyOrderedAdd R\ninst✝³ : AddRightReflectLE R\ninst✝² : IsCancelAdd M\ninst✝¹ : DecidableEq ι\ninst✝ : Fintype ι\nt₁ t... | [
"case refine_2.refine_1\nι : Type u'\nR : Type u_2\nM : Type u_4\nv : ι → M\ninst✝⁸ : Semiring R\ninst✝⁷ : AddCommMonoid M\ninst✝⁶ : Module R M\ninst✝⁵ : LinearOrder R\ninst✝⁴ : CanonicallyOrderedAdd R\ninst✝³ : AddRightReflectLE R\ninst✝² : IsCancelAdd M\ninst✝¹ : DecidableEq ι\ninst✝ : Fintype ι\nt₁ t₂ : Finset ι... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Cardinal.Finite | {
"line": 402,
"column": 6
} | {
"line": 402,
"column": 37
} | {
"line": 403,
"column": 2
} | [
{
"pp": "case inr.inl.inr\nα : Type u_3\nβ : Type u_4\nα_emp : Nonempty α\nh✝¹ : Finite α\nh✝ : Infinite β\n⊢ card α ≠ 0",
"ppTerm": "?inr.inl.inr✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"CommSemiring.toSemiring",
"id",
"Ne",
"ENat",
"p... | [] | rwa [card_ne_zero_iff_nonempty] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.LinearAlgebra.LinearIndependent.Defs | {
"line": 653,
"column": 48
} | {
"line": 653,
"column": 64
} | {
"line": 653,
"column": 65
} | [
{
"pp": "case refine_2.refine_2\nι : Type u'\nR : Type u_2\nM : Type u_4\nv : ι → M\ninst✝⁸ : Semiring R\ninst✝⁷ : AddCommMonoid M\ninst✝⁶ : Module R M\ninst✝⁵ : LinearOrder R\ninst✝⁴ : CanonicallyOrderedAdd R\ninst✝³ : AddRightReflectLE R\ninst✝² : IsCancelAdd M\ninst✝¹ : DecidableEq ι\ninst✝ : Fintype ι\nt₁ t... | [
"case refine_2.refine_2\nι : Type u'\nR : Type u_2\nM : Type u_4\nv : ι → M\ninst✝⁸ : Semiring R\ninst✝⁷ : AddCommMonoid M\ninst✝⁶ : Module R M\ninst✝⁵ : LinearOrder R\ninst✝⁴ : CanonicallyOrderedAdd R\ninst✝³ : AddRightReflectLE R\ninst✝² : IsCancelAdd M\ninst✝¹ : DecidableEq ι\ninst✝ : Fintype ι\nt₁ t₂ : Finset ι... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.LinearIndependent.Basic | {
"line": 569,
"column": 2
} | {
"line": 570,
"column": 36
} | {
"line": 570,
"column": 37
} | [
{
"pp": "ι : Type u'\nR : Type u_2\nM : Type u_4\ninst✝⁵ : Ring R\ninst✝⁴ : IsDomain R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : Module.IsTorsionFree R M\nv : ι → M\ninst✝ : Unique ι\n⊢ LinearIndependent R v → v default ≠ 0",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
... | [
"ι : Type u'\nR : Type u_2\nM : Type u_4\ninst✝⁵ : Ring R\ninst✝⁴ : IsDomain R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : Module.IsTorsionFree R M\nv : ι → M\ninst✝ : Unique ι\n⊢ (∀ (l : ι →₀ R), v default = 0 → l default = 0) → ¬v default = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.LinearIndependent.Basic | {
"line": 570,
"column": 51
} | {
"line": 570,
"column": 62
} | {
"line": 570,
"column": 63
} | [
{
"pp": "ι : Type u'\nR : Type u_2\nM : Type u_4\ninst✝⁵ : Ring R\ninst✝⁴ : IsDomain R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : Module.IsTorsionFree R M\nv : ι → M\ninst✝ : Unique ι\nh : ∀ (l : ι →₀ R), v default = 0 → l default = 0\nhv : v default = 0\n⊢ False",
"ppTerm": "?m.40",
"assig... | [
"ι : Type u'\nR : Type u_2\nM : Type u_4\ninst✝⁵ : Ring R\ninst✝⁴ : IsDomain R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : Module.IsTorsionFree R M\nv : ι → M\ninst✝ : Unique ι\nh : ∀ (l : ι →₀ R), v default = 0 → l default = 0\nhv : v default = 0\n⊢ False"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Dimension.Basic | {
"line": 97,
"column": 2
} | {
"line": 97,
"column": 13
} | {
"line": 97,
"column": 14
} | [
{
"pp": "R : Type u\nM : Type v\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ninst✝ : Nontrivial R\nι : Type v\nv : ι → M\nhv : LinearIndependent R v\n⊢ #ι ≤ Module.rank R M",
"ppTerm": "?m.13",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
... | [
"R : Type u\nM : Type v\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ninst✝ : Nontrivial R\nι : Type v\nv : ι → M\nhv : LinearIndependent R v\n⊢ #ι ≤ Module.rank R M"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Dimension.Basic | {
"line": 105,
"column": 2
} | {
"line": 105,
"column": 13
} | {
"line": 105,
"column": 14
} | [
{
"pp": "R : Type u\nM : Type v\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ninst✝ : Nontrivial R\nι : Type u_1\nv : ι → M\ns : Set ι\nhs : LinearIndepOn R v s\n⊢ s.encard ≤ toENat (Module.rank R M)",
"ppTerm": "?m.15",
"assigned": false,
"usedConstants": [],
"usedFVars":... | [
"R : Type u\nM : Type v\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ninst✝ : Nontrivial R\nι : Type u_1\nv : ι → M\ns : Set ι\nhs : LinearIndepOn R v s\n⊢ s.encard ≤ toENat (Module.rank R M)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Dimension.Basic | {
"line": 117,
"column": 35
} | {
"line": 117,
"column": 72
} | {
"line": 117,
"column": 73
} | [
{
"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\n⊢ ?m.41 < iSup ?m.43",
"ppTerm": "?m.44",
"assign... | [
"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\n⊢ ?m.41 < iSup ?m.43"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Dimension.Basic | {
"line": 123,
"column": 2
} | {
"line": 123,
"column": 13
} | {
"line": 123,
"column": 14
} | [
{
"pp": "R : Type u\nM : Type v\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nc : Cardinal.{v}\nh : c < Module.rank R M\n⊢ ∃ s, #↑s = c ∧ LinearIndepOn R id s",
"ppTerm": "?m.16",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u\nM : Type v\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nc : Cardinal.{v}\nh : c < Module.rank R M\n⊢ ∃ s, #↑s = c ∧ LinearIndepOn R id s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.LinearIndependent.Defs | {
"line": 684,
"column": 4
} | {
"line": 684,
"column": 15
} | {
"line": 684,
"column": 16
} | [
{
"pp": "case refine_2\nι : Type u'\nR : Type u_2\nM : Type u_4\nv : ι → M\ninst✝⁷ : Semiring R\ninst✝⁶ : AddCommMonoid M\ninst✝⁵ : Module R M\ninst✝⁴ : LinearOrder R\ninst✝³ : CanonicallyOrderedAdd R\ninst✝² : AddRightReflectLE R\ninst✝¹ : IsCancelAdd M\ninst✝ : DecidableEq ι\ns : Finset ι\nt : Finset ↑↑s\nf :... | [
"case refine_2\nι : Type u'\nR : Type u_2\nM : Type u_4\nv : ι → M\ninst✝⁷ : Semiring R\ninst✝⁶ : AddCommMonoid M\ninst✝⁵ : Module R M\ninst✝⁴ : LinearOrder R\ninst✝³ : CanonicallyOrderedAdd R\ninst✝² : AddRightReflectLE R\ninst✝¹ : IsCancelAdd M\ninst✝ : DecidableEq ι\ns : Finset ι\nt : Finset ↑↑s\nf : ↑↑s → R\nhe... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Dimension.Basic | {
"line": 141,
"column": 34
} | {
"line": 141,
"column": 45
} | {
"line": 141,
"column": 46
} | [
{
"pp": "R : Type u\nM : Type v\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ninst✝ : Nontrivial R\nn : ℕ\ns : { s // LinearIndepOn R id s }\nle : ↑(n + 1) ≤ #↑↑s\nt : Finset ↑↑s\nht : n + 1 = t.card\n⊢ (Finset.map (Embedding.subtype fun x ↦ x ∈ ↑s) t).card = n + 1",
"ppTerm": "?m.148... | [
"R : Type u\nM : Type v\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ninst✝ : Nontrivial R\nn : ℕ\ns : { s // LinearIndepOn R id s }\nle : ↑(n + 1) ≤ #↑↑s\nt : Finset ↑↑s\nht : n + 1 = t.card\n⊢ t.card = n + 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Dimension.Basic | {
"line": 142,
"column": 74
} | {
"line": 142,
"column": 85
} | {
"line": 142,
"column": 86
} | [
{
"pp": "R : Type u\nM : Type v\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ninst✝ : Nontrivial R\nn : ℕ\nx✝ : ∃ s, s.card = n ∧ LinearIndepOn R id ↑s\ns : Finset M\ncard_s : s.card = n\nind_s : LinearIndepOn R id ↑s\n⊢ #↑↑s = ↑n",
"ppTerm": "?m.42",
"assigned": true,
"usedCo... | [
"R : Type u\nM : Type v\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ninst✝ : Nontrivial R\nn : ℕ\nx✝ : ∃ s, s.card = n ∧ LinearIndepOn R id ↑s\ns : Finset M\ncard_s : s.card = n\nind_s : LinearIndepOn R id ↑s\n⊢ s.card = n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Dimension.Basic | {
"line": 148,
"column": 4
} | {
"line": 148,
"column": 15
} | {
"line": 148,
"column": 16
} | [
{
"pp": "case refine_2\nR : Type u\nM : Type v\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ninst✝ : Nontrivial R\nn : ℕ\nx✝ : ∃ v, LinearIndependent R v\nv : Fin n → M\nv_ind : LinearIndependent R v\n⊢ LinearIndepOn R id ↑(Finset.map { toFun := v, inj' := ⋯ } Finset.univ)",
"ppTerm":... | [
"case refine_2\nR : Type u\nM : Type v\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ninst✝ : Nontrivial R\nn : ℕ\nx✝ : ∃ v, LinearIndependent R v\nv : Fin n → M\nv_ind : LinearIndependent R v\n⊢ LinearIndepOn R id (range fun a ↦ v a)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.LinearIndependent.Defs | {
"line": 695,
"column": 4
} | {
"line": 695,
"column": 48
} | {
"line": 695,
"column": 49
} | [
{
"pp": "case neg\nι : Type u'\nR : Type u_2\nM : Type u_4\nv : ι → M\ninst✝⁷ : Semiring R\ninst✝⁶ : AddCommMonoid M\ninst✝⁵ : Module R M\ninst✝⁴ : LinearOrder R\ninst✝³ : CanonicallyOrderedAdd R\ninst✝² : AddRightReflectLE R\ninst✝¹ : IsCancelAdd M\ninst✝ : DecidableEq ι\ns : Finset ι\nx✝ : ∃ x, ∃ (_ : x ⊆ s),... | [
"case neg\nι : Type u'\nR : Type u_2\nM : Type u_4\nv : ι → M\ninst✝⁷ : Semiring R\ninst✝⁶ : AddCommMonoid M\ninst✝⁵ : Module R M\ninst✝⁴ : LinearOrder R\ninst✝³ : CanonicallyOrderedAdd R\ninst✝² : AddRightReflectLE R\ninst✝¹ : IsCancelAdd M\ninst✝ : DecidableEq ι\ns : Finset ι\nx✝ : ∃ x, ∃ (_ : x ⊆ s), ∃ x_1, ∃ (_... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.LinearIndependent.Defs | {
"line": 718,
"column": 2
} | {
"line": 718,
"column": 13
} | {
"line": 718,
"column": 14
} | [
{
"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 : LinearIndependent R (-v)\n⊢ LinearIndependent R v",
"ppTerm": "?m.21",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"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 : LinearIndependent R (-v)\n⊢ LinearIndependent R v"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Dimension.Basic | {
"line": 229,
"column": 2
} | {
"line": 229,
"column": 28
} | {
"line": 229,
"column": 29
} | [
{
"pp": "R : Type u\nR' : Type u'\nM M₁ : Type v\ninst✝⁵ : Semiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\ninst✝² : Semiring R'\ninst✝¹ : AddCommMonoid M₁\ninst✝ : Module R' M₁\ni : R' → R\nj : M →+ M₁\nhi : Injective i\nhj : Injective ⇑j\nhc : ∀ (r : R') (m : M), j (i r • m) = r • j m\n⊢ Module.rank... | [
"R : Type u\nR' : Type u'\nM M₁ : Type v\ninst✝⁵ : Semiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\ninst✝² : Semiring R'\ninst✝¹ : AddCommMonoid M₁\ninst✝ : Module R' M₁\ni : R' → R\nj : M →+ M₁\nhi : Injective i\nhj : Injective ⇑j\nhc : ∀ (r : R') (m : M), j (i r • m) = r • j m\n⊢ Module.rank R M ≤ Modul... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Dimension.Basic | {
"line": 236,
"column": 2
} | {
"line": 236,
"column": 28
} | {
"line": 236,
"column": 29
} | [
{
"pp": "R : Type u\nR' : Type u'\nM M₁ : Type v\ninst✝⁵ : Semiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\ninst✝² : Semiring R'\ninst✝¹ : AddCommMonoid M₁\ninst✝ : Module R' M₁\ni : R → R'\nj : M →+ M₁\nhi : Surjective i\nhj : Injective ⇑j\nhc : ∀ (r : R) (m : M), j (r • m) = i r • j m\n⊢ Module.rank... | [
"R : Type u\nR' : Type u'\nM M₁ : Type v\ninst✝⁵ : Semiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\ninst✝² : Semiring R'\ninst✝¹ : AddCommMonoid M₁\ninst✝ : Module R' M₁\ni : R → R'\nj : M →+ M₁\nhi : Surjective i\nhj : Injective ⇑j\nhc : ∀ (r : R) (m : M), j (r • m) = i r • j m\n⊢ Module.rank R M ≤ Modul... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Dimension.Basic | {
"line": 242,
"column": 2
} | {
"line": 242,
"column": 28
} | {
"line": 242,
"column": 29
} | [
{
"pp": "R : Type u\nR' : Type u'\nM M₁ : Type v\ninst✝⁵ : Semiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\ninst✝² : Semiring R'\ninst✝¹ : AddCommMonoid M₁\ninst✝ : Module R' M₁\ni : R → R'\nj : M ≃+ M₁\nhi : Bijective i\nhc : ∀ (r : R) (m : M), j (r • m) = i r • j m\n⊢ Module.rank R M = Module.rank R... | [
"R : Type u\nR' : Type u'\nM M₁ : Type v\ninst✝⁵ : Semiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\ninst✝² : Semiring R'\ninst✝¹ : AddCommMonoid M₁\ninst✝ : Module R' M₁\ni : R → R'\nj : M ≃+ M₁\nhi : Bijective i\nhc : ∀ (r : R) (m : M), j (r • m) = i r • j m\n⊢ Module.rank R M = Module.rank R' M₁"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Dimension.Basic | {
"line": 255,
"column": 4
} | {
"line": 255,
"column": 40
} | {
"line": 255,
"column": 41
} | [
{
"pp": "R : Type u_1\ninst✝ : CommSemiring R\na✝ : Nontrivial R\n⊢ ¬↑2 ≤ Module.rank R R ∧ 1 ≤ Module.rank R R",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Pi.Function.module",
"Semiring.toModule",
"Pi.addCommMonoid",
"Cardinal.instOne",
"... | [
"R : Type u_1\ninst✝ : CommSemiring R\na✝ : Nontrivial R\n⊢ (¬∃ f, Injective ⇑f) ∧ 1 ≤ Module.rank R R"
] | Module.le_rank_iff_exists_linearMap, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Dimension.Basic | {
"line": 259,
"column": 32
} | {
"line": 259,
"column": 43
} | {
"line": 259,
"column": 44
} | [
{
"pp": "R : Type u_1\ninst✝ : CommSemiring R\na✝ : Nontrivial R\nx✝ : ∃ f, Injective ⇑f\nf : (Fin 2 → R) →ₗ[R] R\ninj : Injective ⇑f\nthis : f ![0, 1] • ![1, 0] = f ![1, 0] • ![0, 1]\n⊢ f ![0, 1] = 0",
"ppTerm": "?m.160",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals":... | [
"R : Type u_1\ninst✝ : CommSemiring R\na✝ : Nontrivial R\nx✝ : ∃ f, Injective ⇑f\nf : (Fin 2 → R) →ₗ[R] R\ninj : Injective ⇑f\nthis : f ![0, 1] • ![1, 0] = f ![1, 0] • ![0, 1]\n⊢ f ![0, 1] = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Dimension.Basic | {
"line": 260,
"column": 32
} | {
"line": 260,
"column": 43
} | {
"line": 260,
"column": 44
} | [
{
"pp": "R : Type u_1\ninst✝ : CommSemiring R\na✝ : Nontrivial R\nx✝ : ∃ f, Injective ⇑f\nf : (Fin 2 → R) →ₗ[R] R\ninj : Injective ⇑f\nthis : f ![0, 1] • ![1, 0] = f ![1, 0] • ![0, 1]\nh₁ : f ![0, 1] = 0\n⊢ 0 = f ![1, 0]",
"ppTerm": "?m.191",
"assigned": false,
"usedConstants": [],
"usedFVars": ... | [
"R : Type u_1\ninst✝ : CommSemiring R\na✝ : Nontrivial R\nx✝ : ∃ f, Injective ⇑f\nf : (Fin 2 → R) →ₗ[R] R\ninj : Injective ⇑f\nthis : f ![0, 1] • ![1, 0] = f ![1, 0] • ![0, 1]\nh₁ : f ![0, 1] = 0\n⊢ 0 = f ![1, 0]"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.LinearIndependent.Defs | {
"line": 773,
"column": 19
} | {
"line": 773,
"column": 30
} | {
"line": 773,
"column": 31
} | [
{
"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\nv : ι → M\ninst✝ : Fintype ι\nH : LinearIndependent R v\ng : ι → R\n⊢ ∑ i, g i • v i = 0 → ∀ (i : ι), g i = 0",
"ppTerm": "?m.32",
"assigned": false,
"usedConstants": [],
"usedFVars":... | [
"ι : Type u'\nR : Type u_2\nM : Type u_4\ninst✝³ : Ring R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nv : ι → M\ninst✝ : Fintype ι\nH : LinearIndependent R v\ng : ι → R\n⊢ ∑ i, g i • v i = 0 → ∀ (i : ι), g i = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Dimension.Basic | {
"line": 261,
"column": 33
} | {
"line": 261,
"column": 44
} | {
"line": 261,
"column": 45
} | [
{
"pp": "R : Type u_1\ninst✝ : CommSemiring R\na✝ : Nontrivial R\nx✝ : ∃ f, Injective ⇑f\nf : (Fin 2 → R) →ₗ[R] R\ninj : Injective ⇑f\nthis : f ![0, 1] • ![1, 0] = f ![1, 0] • ![0, 1]\nh₁ : f ![0, 1] = 0\nh₂ : 0 = f ![1, 0]\n⊢ 0 = 1",
"ppTerm": "?m.212",
"assigned": true,
"usedConstants": [
"E... | [
"R : Type u_1\ninst✝ : CommSemiring R\na✝ : Nontrivial R\nx✝ : ∃ f, Injective ⇑f\nf : (Fin 2 → R) →ₗ[R] R\ninj : Injective ⇑f\nthis : f ![0, 1] • ![1, 0] = f ![1, 0] • ![0, 1]\nh₁ : f ![0, 1] = 0\nh₂ : 0 = f ![1, 0]\n⊢ False"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.BigOperators.Fin | {
"line": 683,
"column": 41
} | {
"line": 683,
"column": 52
} | {
"line": 683,
"column": 53
} | [
{
"pp": "ι : Type u_1\nM : Type u_2\nm : ℕ\nn : Fin 0 → ℕ\n⊢ IsEmpty (Fin (∑ i, n i))",
"ppTerm": "?m.82",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Finset.univ",
"congrArg",
"Finset",
"AddMonoid.toAddZeroClass",
"Membership.mem",
"Finset.univ_eq_empty... | [
"ι : Type u_1\nM : Type u_2\nm : ℕ\nn : Fin 0 → ℕ\n⊢ IsEmpty (Fin 0)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Dimension.Basic | {
"line": 286,
"column": 2
} | {
"line": 286,
"column": 28
} | {
"line": 286,
"column": 29
} | [
{
"pp": "R : Type u\nR' : Type u'\nM M₁ : Type v\ninst✝⁵ : Ring R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : Ring R'\ninst✝¹ : AddCommGroup M₁\ninst✝ : Module R' M₁\ni : R' → R\nj : M →+ M₁\nhi : ∀ (r : R'), i r = 0 → r = 0\nhj : Injective ⇑j\nhc : ∀ (r : R') (m : M), j (i r • m) = r • j m\n⊢ Modul... | [
"R : Type u\nR' : Type u'\nM M₁ : Type v\ninst✝⁵ : Ring R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : Ring R'\ninst✝¹ : AddCommGroup M₁\ninst✝ : Module R' M₁\ni : R' → R\nj : M →+ M₁\nhi : ∀ (r : R'), i r = 0 → r = 0\nhj : Injective ⇑j\nhc : ∀ (r : R') (m : M), j (i r • m) = r • j m\n⊢ Module.rank R M ≤... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.LinearIndependent.Defs | {
"line": 781,
"column": 2
} | {
"line": 781,
"column": 13
} | {
"line": 781,
"column": 14
} | [
{
"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\nv : ι → M\ninst✝ : Fintype ι\n⊢ ¬LinearIndependent R v ↔ ∃ g, ∑ i, g i • v i = 0 ∧ ∃ i, g i ≠ 0",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"instHSMul",
"Finset.... | [
"ι : Type u'\nR : Type u_2\nM : Type u_4\ninst✝³ : Ring R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nv : ι → M\ninst✝ : Fintype ι\n⊢ ¬LinearIndependent R v ↔ ∃ g, ∑ i, g i • v i = 0 ∧ ∃ i, ¬g i = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Dimension.Basic | {
"line": 337,
"column": 2
} | {
"line": 337,
"column": 28
} | {
"line": 337,
"column": 29
} | [
{
"pp": "R : Type w\nS : Type v\ninst✝⁵ : CommSemiring R\ninst✝⁴ : Semiring S\ninst✝³ : Algebra R S\nR' : Type w'\ninst✝² : CommSemiring R'\nS' : Type v\ninst✝¹ : Semiring S'\ninst✝ : Algebra R' S'\ni : R' →+* R\nj : S →+* S'\nhi : Injective ⇑i\nhj : Injective ⇑j\nhc : (j.comp (algebraMap R S)).comp i = algebra... | [
"R : Type w\nS : Type v\ninst✝⁵ : CommSemiring R\ninst✝⁴ : Semiring S\ninst✝³ : Algebra R S\nR' : Type w'\ninst✝² : CommSemiring R'\nS' : Type v\ninst✝¹ : Semiring S'\ninst✝ : Algebra R' S'\ni : R' →+* R\nj : S →+* S'\nhi : Injective ⇑i\nhj : Injective ⇑j\nhc : (j.comp (algebraMap R S)).comp i = algebraMap R' S'\n⊢... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Dimension.Basic | {
"line": 344,
"column": 2
} | {
"line": 344,
"column": 28
} | {
"line": 344,
"column": 29
} | [
{
"pp": "R : Type w\nS : Type v\ninst✝⁵ : CommSemiring R\ninst✝⁴ : Semiring S\ninst✝³ : Algebra R S\nR' : Type w'\ninst✝² : CommSemiring R'\nS' : Type v\ninst✝¹ : Semiring S'\ninst✝ : Algebra R' S'\ni : R →+* R'\nj : S →+* S'\nhi : Surjective ⇑i\nhj : Injective ⇑j\nhc : (algebraMap R' S').comp i = j.comp (algeb... | [
"R : Type w\nS : Type v\ninst✝⁵ : CommSemiring R\ninst✝⁴ : Semiring S\ninst✝³ : Algebra R S\nR' : Type w'\ninst✝² : CommSemiring R'\nS' : Type v\ninst✝¹ : Semiring S'\ninst✝ : Algebra R' S'\ni : R →+* R'\nj : S →+* S'\nhi : Surjective ⇑i\nhj : Injective ⇑j\nhc : (algebraMap R' S').comp i = j.comp (algebraMap R S)\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Dimension.Basic | {
"line": 350,
"column": 2
} | {
"line": 350,
"column": 28
} | {
"line": 350,
"column": 29
} | [
{
"pp": "R : Type w\nS : Type v\ninst✝⁵ : CommSemiring R\ninst✝⁴ : Semiring S\ninst✝³ : Algebra R S\nR' : Type w'\ninst✝² : CommSemiring R'\nS' : Type v\ninst✝¹ : Semiring S'\ninst✝ : Algebra R' S'\ni : R ≃+* R'\nj : S ≃+* S'\nhc : (algebraMap R' S').comp i.toRingHom = j.toRingHom.comp (algebraMap R S)\n⊢ Modul... | [
"R : Type w\nS : Type v\ninst✝⁵ : CommSemiring R\ninst✝⁴ : Semiring S\ninst✝³ : Algebra R S\nR' : Type w'\ninst✝² : CommSemiring R'\nS' : Type v\ninst✝¹ : Semiring S'\ninst✝ : Algebra R' S'\ni : R ≃+* R'\nj : S ≃+* S'\nhc : (algebraMap R' S').comp i.toRingHom = j.toRingHom.comp (algebraMap R S)\n⊢ Module.rank R S =... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.BigOperators.Fin | {
"line": 717,
"column": 43
} | {
"line": 717,
"column": 54
} | {
"line": 717,
"column": 55
} | [
{
"pp": "n : Fin 1 → ℕ\nij : (i : Fin 1) × Fin (n i)\n⊢ IsEmpty (Fin ↑ij.fst)",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"id",
"Sigma.fst",
"instOfNatNat",
"IsEmpty",
"Fin.val",
"Fin.val_eq_zero",
"Nat",
... | [
"n : Fin 1 → ℕ\nij : (i : Fin 1) × Fin (n i)\n⊢ IsEmpty (Fin 0)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.LinearIndependent.Defs | {
"line": 792,
"column": 4
} | {
"line": 792,
"column": 15
} | {
"line": 792,
"column": 16
} | [
{
"pp": "case mpr\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\ns : Finset ι\nhv : ∀ (f : ι → R), ∑ i ∈ s, f i • v i = 0 → ∀ i ∈ s, f i = 0\nf : ↑↑s → R\nhf₀ : ∑ i, f i • v ↑i = 0\ni : ↑↑s\n⊢ f i = 0",
"ppTerm": "?mpr",
"assigned": fal... | [
"case mpr\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\ns : Finset ι\nhv : ∀ (f : ι → R), ∑ i ∈ s, f i • v i = 0 → ∀ i ∈ s, f i = 0\nf : ↑↑s → R\nhf₀ : ∑ i, f i • v ↑i = 0\ni : ↑↑s\n⊢ f i = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Set.Card | {
"line": 292,
"column": 2
} | {
"line": 292,
"column": 62
} | {
"line": 294,
"column": 0
} | [
{
"pp": "α : Type u_1\ns t : Set α\n⊢ (s \\ t).encard + t.encard = (s ∪ t).encard",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Set.encard_union_eq",
"Set.encard",
"congrArg",
"Set.instUnion",
"instAddENat",
"id",
"Set.sdiff_unio... | [] | rw [← encard_union_eq disjoint_sdiff_left, sdiff_union_self] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Data.Set.Card | {
"line": 292,
"column": 2
} | {
"line": 292,
"column": 62
} | {
"line": 294,
"column": 0
} | [
{
"pp": "α : Type u_1\ns t : Set α\n⊢ (s \\ t).encard + t.encard = (s ∪ t).encard",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Set.encard_union_eq",
"Set.encard",
"congrArg",
"Set.instUnion",
"instAddENat",
"id",
"Set.sdiff_unio... | [] | rw [← encard_union_eq disjoint_sdiff_left, sdiff_union_self] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Set.Card | {
"line": 292,
"column": 2
} | {
"line": 292,
"column": 62
} | {
"line": 294,
"column": 0
} | [
{
"pp": "α : Type u_1\ns t : Set α\n⊢ (s \\ t).encard + t.encard = (s ∪ t).encard",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Set.encard_union_eq",
"Set.encard",
"congrArg",
"Set.instUnion",
"instAddENat",
"id",
"Set.sdiff_unio... | [] | rw [← encard_union_eq disjoint_sdiff_left, sdiff_union_self] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Set.Card | {
"line": 335,
"column": 2
} | {
"line": 336,
"column": 76
} | {
"line": 338,
"column": 0
} | [
{
"pp": "α : Type u_1\ns : Set α\na : α\nh : a ∈ s\n⊢ (s \\ {a}).encard = s.encard - 1",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Set.encard",
"Set.encard_sdiff_singleton_add_one",
"instAddMonoidWithOneENat",
"instLinearOrderENat",
"instS... | [] | rw [← encard_sdiff_singleton_add_one h,
(ENat.addLECancellable_of_ne_top ENat.one_ne_top).add_tsub_cancel_right] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.Dimension.Basic | {
"line": 388,
"column": 2
} | {
"line": 388,
"column": 13
} | {
"line": 388,
"column": 14
} | [
{
"pp": "R : Type u\nM M₁ : Type v\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : AddCommMonoid M₁\ninst✝ : Module R M₁\nf : M →ₗ[R] M₁\n⊢ Module.rank R ↥f.range ≤ Module.rank R M",
"ppTerm": "?m.40",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"us... | [
"R : Type u\nM M₁ : Type v\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : AddCommMonoid M₁\ninst✝ : Module R M₁\nf : M →ₗ[R] M₁\n⊢ Module.rank R ↥f.range ≤ Module.rank R M"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Set.Card | {
"line": 335,
"column": 2
} | {
"line": 336,
"column": 76
} | {
"line": 338,
"column": 0
} | [
{
"pp": "α : Type u_1\ns : Set α\na : α\nh : a ∈ s\n⊢ (s \\ {a}).encard = s.encard - 1",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Set.encard",
"Set.encard_sdiff_singleton_add_one",
"instAddMonoidWithOneENat",
"instLinearOrderENat",
"instS... | [] | rw [← encard_sdiff_singleton_add_one h,
(ENat.addLECancellable_of_ne_top ENat.one_ne_top).add_tsub_cancel_right] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Set.Card | {
"line": 335,
"column": 2
} | {
"line": 336,
"column": 76
} | {
"line": 338,
"column": 0
} | [
{
"pp": "α : Type u_1\ns : Set α\na : α\nh : a ∈ s\n⊢ (s \\ {a}).encard = s.encard - 1",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Set.encard",
"Set.encard_sdiff_singleton_add_one",
"instAddMonoidWithOneENat",
"instLinearOrderENat",
"instS... | [] | rw [← encard_sdiff_singleton_add_one h,
(ENat.addLECancellable_of_ne_top ENat.one_ne_top).add_tsub_cancel_right] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.LinearIndependent.Defs | {
"line": 797,
"column": 2
} | {
"line": 797,
"column": 13
} | {
"line": 797,
"column": 14
} | [
{
"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\ns : Finset ι\n⊢ ¬LinearIndepOn R v ↑s ↔ ∃ f, ∑ i ∈ s, f i • v i = 0 ∧ ∃ i ∈ s, f i ≠ 0",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"instHSMul",
"Distri... | [
"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\ns : Finset ι\n⊢ ¬LinearIndepOn R v ↑s ↔ ∃ f, ∑ i ∈ s, f i • v i = 0 ∧ ∃ i ∈ s, ¬f i = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Dimension.Basic | {
"line": 396,
"column": 52
} | {
"line": 396,
"column": 63
} | {
"line": 396,
"column": 64
} | [
{
"pp": "R : Type u\nM M₁ : Type v\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : AddCommMonoid M₁\ninst✝ : Module R M₁\nf : M →ₗ[R] M₁\np : Submodule R M\n⊢ Module.rank R ↥(Submodule.map f p) ≤ Module.rank R ↥p",
"ppTerm": "?m.43",
"assigned": false,
"usedConstants": ... | [
"R : Type u\nM M₁ : Type v\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : AddCommMonoid M₁\ninst✝ : Module R M₁\nf : M →ₗ[R] M₁\np : Submodule R M\n⊢ Module.rank R ↥(Submodule.map f p) ≤ Module.rank R ↥p"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Dimension.Basic | {
"line": 396,
"column": 49
} | {
"line": 396,
"column": 84
} | {
"line": 398,
"column": 0
} | [
{
"pp": "R : Type u\nM M₁ : Type v\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : AddCommMonoid M₁\ninst✝ : Module R M₁\nf : M →ₗ[R] M₁\np : Submodule R M\n⊢ Module.rank R ↥(Submodule.map f p) ≤ Module.rank R ↥p",
"ppTerm": "?m.43",
"assigned": true,
"usedConstants": [... | [] | by simpa using lift_rank_map_le f p | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Set.Card | {
"line": 369,
"column": 2
} | {
"line": 370,
"column": 94
} | {
"line": 371,
"column": 2
} | [
{
"pp": "α : Type u_1\ns : Set α\n⊢ s = ∅ ∨ s.encard = ⊤ ∨ ∃ a ∈ s, (s \\ {a}).encard < s.encard",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Set.encard",
"Set.finite_or_infinite",
"instTopENat",
"Set.Finite",
"Function.comp",
"Membership.mem",
... | [
"α : Type u_1\ns : Set α\nx✝ : s.Nonempty\na : α\nha : a ∈ s\nhfin : s.Finite\n⊢ (s \\ {a}).encard < s.encard"
] | refine s.eq_empty_or_nonempty.elim Or.inl (Or.inr ∘ fun ⟨a,ha⟩ ↦
(s.finite_or_infinite.elim (fun hfin ↦ Or.inr ⟨a, ha, ?_⟩) (Or.inl ∘ Infinite.encard_eq))) | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.LinearAlgebra.Dimension.Basic | {
"line": 445,
"column": 60
} | {
"line": 446,
"column": 43
} | {
"line": 448,
"column": 0
} | [
{
"pp": "R : Type u\nM : Type v\nM' : Type v'\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : AddCommMonoid M'\ninst✝ : Module R M'\nf : M →ₗ[R] M'\nh : Surjective ⇑f\n⊢ Module.rank R ↥f.range = Module.rank R M'",
"ppTerm": "?m.42",
"assigned": true,
"usedConstants": [
... | [] | by
rw [LinearMap.range_eq_top.2 h, rank_top] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.Dimension.Basic | {
"line": 467,
"column": 19
} | {
"line": 467,
"column": 30
} | {
"line": 467,
"column": 31
} | [
{
"pp": "R : Type u\nM : Type v\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\nS : Type u_1\ninst✝³ : CommSemiring S\ninst✝² : Algebra S R\ninst✝¹ : Module S M\ninst✝ : IsScalarTower S R M\nL L' : Submodule R M\ns : S\nhr : IsSMulRegular M s\nh✝ : ∀ x ∈ L, s • x ∈ L'\nx✝¹ x✝ : ↥L\nh : (Lin... | [
"R : Type u\nM : Type v\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\nS : Type u_1\ninst✝³ : CommSemiring S\ninst✝² : Algebra S R\ninst✝¹ : Module S M\ninst✝ : IsScalarTower S R M\nL L' : Submodule R M\ns : S\nhr : IsSMulRegular M s\nh✝ : ∀ x ∈ L, s • x ∈ L'\nx✝¹ x✝ : ↥L\nh : (LinearMap.restr... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Dimension.Basic | {
"line": 489,
"column": 2
} | {
"line": 489,
"column": 13
} | {
"line": 489,
"column": 14
} | [
{
"pp": "R : Type u\nR' : Type u'\ninst✝⁸ : Semiring R\ninst✝⁷ : Semiring R'\nT : Type u'\ninst✝⁶ : Module R R'\ninst✝⁵ : NonAssocSemiring T\ninst✝⁴ : Module R T\ninst✝³ : Module R' T\ninst✝² : IsScalarTower R' T T\ninst✝¹ : FaithfulSMul R' T\ninst✝ : IsScalarTower R R' T\n⊢ Module.rank R R' ≤ Module.rank R T",... | [
"R : Type u\nR' : Type u'\ninst✝⁸ : Semiring R\ninst✝⁷ : Semiring R'\nT : Type u'\ninst✝⁶ : Module R R'\ninst✝⁵ : NonAssocSemiring T\ninst✝⁴ : Module R T\ninst✝³ : Module R' T\ninst✝² : IsScalarTower R' T T\ninst✝¹ : FaithfulSMul R' T\ninst✝ : IsScalarTower R R' T\n⊢ Module.rank R R' ≤ Module.rank R T"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.LinearIndependent.Defs | {
"line": 843,
"column": 4
} | {
"line": 843,
"column": 86
} | {
"line": 844,
"column": 4
} | [
{
"pp": "case refine_1\nι : Type u'\nR : Type u_2\ns : Set ι\nM : Type u_4\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nv : ι → M\nh : ∀ (s_1 : Finset ↑s) (g : ↑s → R), ∑ i ∈ s_1, g i • v ↑i = 0 → ∀ i ∈ s_1, g i = 0\nt : Finset ι\ng : ι → R\nhts : ↑t ⊆ s\nh0 : ∑ i ∈ t, g i • v i = 0\ni : ι\nhi... | [
"case refine_1\nι : Type u'\nR : Type u_2\ns : Set ι\nM : Type u_4\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nv : ι → M\nh : ∀ (s_1 : Finset ↑s) (g : ↑s → R), ∑ i ∈ s_1, g i • v ↑i = 0 → ∀ i ∈ s_1, g i = 0\nt : Finset ι\ng : ι → R\nhts : ↑t ⊆ s\nh0 : ∑ i ∈ t, g i • v i = 0\ni : ι\nhit : i ∈ t\n⊢... | rwa [t.sum_preimage ((↑) : s → ι) Subtype.val_injective.injOn (fun i ↦ g i • v i)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.LinearAlgebra.Dimension.Finrank | {
"line": 97,
"column": 2
} | {
"line": 97,
"column": 13
} | {
"line": 97,
"column": 14
} | [
{
"pp": "R : Type u\nM : Type v\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nh : 1 < finrank R M\n⊢ 1 < Module.rank R M",
"ppTerm": "?m.19",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u\nM : Type v\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nh : 1 < finrank R M\n⊢ 1 < Module.rank R M"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.LinearIndependent.Defs | {
"line": 847,
"column": 4
} | {
"line": 847,
"column": 20
} | {
"line": 847,
"column": 21
} | [
{
"pp": "ι : Type u'\nR : Type u_2\ns : Set ι\nM : Type u_4\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nv : ι → M\nh : ∀ (t : Finset ι) (g : ι → R), ↑t ⊆ s → ∑ i ∈ t, g i • v i = 0 → ∀ i ∈ t, g i = 0\nt : Finset ↑s\ng : ↑s → R\nh0 : ∑ i ∈ t, g i • v ↑i = 0\ni : ↑s\nhit : i ∈ t\n⊢ ∀ (i : ι) (h... | [
"ι : Type u'\nR : Type u_2\ns : Set ι\nM : Type u_4\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nv : ι → M\nh : ∀ (t : Finset ι) (g : ι → R), ↑t ⊆ s → ∑ i ∈ t, g i • v i = 0 → ∀ i ∈ t, g i = 0\nt : Finset ↑s\ng : ↑s → R\nh0 : ∑ i ∈ t, g i • v ↑i = 0\ni : ↑s\nhit : i ∈ t\n⊢ ∀ (i : ι) (hi : i ∈ s), ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.EuclideanDomain.Defs | {
"line": 152,
"column": 43
} | {
"line": 152,
"column": 80
} | {
"line": 152,
"column": 81
} | [
{
"pp": "R : Type u\ninst✝ : EuclideanDomain R\na : R\n⊢ a % 0 = a",
"ppTerm": "?m.8",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u\ninst✝ : EuclideanDomain R\na : R\n⊢ a % 0 = a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.EuclideanDomain.Defs | {
"line": 156,
"column": 28
} | {
"line": 156,
"column": 54
} | {
"line": 156,
"column": 55
} | [
{
"pp": "R : Type u\ninst✝ : EuclideanDomain R\na : R\nthis : (p : Prop) → Decidable p\nh : ¬a = 0\n⊢ ¬a ≺ 1",
"ppTerm": "?m.14",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u\ninst✝ : EuclideanDomain R\na : R\nthis : (p : Prop) → Decidable p\nh : ¬a = 0\n⊢ ¬a ≺ 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.LinearIndependent.Defs | {
"line": 856,
"column": 7
} | {
"line": 856,
"column": 35
} | {
"line": 856,
"column": 36
} | [
{
"pp": "ι : Type u'\nR : Type u_2\ns : Set ι\nM : Type u_4\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nv : ι → M\nh : ∀ (t : Finset ι) (g : ι → R), ↑t ⊆ s → (∀ i ∉ t, g i = 0) → ∑ i ∈ t, g i • v i = 0 → ∀ i ∈ t, g i = 0\nt : Finset ι\ng : ι → R\nhts : ↑t ⊆ s\nh0 : ∑ i ∈ t, g i • v i = 0\n⊢ ∀... | [
"ι : Type u'\nR : Type u_2\ns : Set ι\nM : Type u_4\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nv : ι → M\nh : ∀ (t : Finset ι) (g : ι → R), ↑t ⊆ s → (∀ i ∉ t, g i = 0) → ∑ i ∈ t, g i • v i = 0 → ∀ i ∈ t, g i = 0\nt : Finset ι\ng : ι → R\nhts : ↑t ⊆ s\nh0 : ∑ i ∈ t, g i • v i = 0\n⊢ ∀ i ∈ t, g i ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.EuclideanDomain.Int | {
"line": 28,
"column": 6
} | {
"line": 29,
"column": 32
} | {
"line": 30,
"column": 4
} | [
{
"pp": "a b : ℤ\nb0 : b ≠ 0\n⊢ ↑(a % b).natAbs < ↑b.natAbs",
"ppTerm": "?m.48",
"assigned": true,
"usedConstants": [
"Int.emod_lt_abs",
"Eq.mpr",
"abs",
"congrArg",
"id",
"instHMod",
"Int",
"Nat.cast",
"Int.natAbs_of_nonneg",
"Int.instLTIn... | [] | rw [Int.natAbs_of_nonneg (Int.emod_nonneg _ b0), ← Int.abs_eq_natAbs]
exact Int.emod_lt_abs _ b0 | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
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