@@ -62,7 +62,7 @@ instance : has_coe_to_fun (Π₀ i, β i) :=
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instance : has_zero (Π₀ i, β i) := ⟨⟦⟨λ i, 0 , ∅, λ i, or.inr rfl⟩⟧⟩
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instance : inhabited (Π₀ i, β i) := ⟨0 ⟩
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- @[simp] lemma zero_apply { i : ι} : (0 : Π₀ i, β i) i = 0 := rfl
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+ @[simp] lemma zero_apply ( i : ι) : (0 : Π₀ i, β i) i = 0 := rfl
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@[ext]
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lemma ext {f g : Π₀ i, β i} (H : ∀ i, f i = g i) : f = g :=
@@ -76,7 +76,7 @@ quotient.lift_on g (λ x, ⟦(⟨λ i, f i (x.1 i), x.2,
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quotient.sound $ λ i, by simp only [H i]
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@[simp] lemma map_range_apply
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- { f : Π i, β₁ i → β₂ i} { hf : ∀ i, f i 0 = 0 } { g : Π₀ i, β₁ i} { i : ι} :
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+ ( f : Π i, β₁ i → β₂ i) ( hf : ∀ i, f i 0 = 0 ) ( g : Π₀ i, β₁ i) ( i : ι) :
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map_range f hf g i = f i (g i) :=
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quotient.induction_on g $ λ x, rfl
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end
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@[simp] lemma zip_with_apply
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- { f : Π i, β₁ i → β₂ i → β i} { hf : ∀ i, f i 0 0 = 0 } { g₁ : Π₀ i, β₁ i} { g₂ : Π₀ i, β₂ i} { i : ι} :
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+ ( f : Π i, β₁ i → β₂ i → β i) ( hf : ∀ i, f i 0 0 = 0 ) ( g₁ : Π₀ i, β₁ i) ( g₂ : Π₀ i, β₂ i) ( i : ι) :
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zip_with f hf g₁ g₂ i = f i (g₁ i) (g₂ i) :=
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quotient.induction_on₂ g₁ g₂ $ λ _ _, rfl
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@@ -107,9 +107,9 @@ section algebra
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instance [Π i, add_monoid (β i)] : has_add (Π₀ i, β i) :=
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⟨zip_with (λ _, (+)) (λ _, add_zero 0 )⟩
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- @[simp] lemma add_apply [Π i, add_monoid (β i)] { g₁ g₂ : Π₀ i, β i} { i : ι} :
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+ @[simp] lemma add_apply [Π i, add_monoid (β i)] ( g₁ g₂ : Π₀ i, β i) ( i : ι) :
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(g₁ + g₂) i = g₁ i + g₂ i :=
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- zip_with_apply
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+ zip_with_apply _ _ g₁ g₂ i
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instance [Π i, add_monoid (β i)] : add_monoid (Π₀ i, β i) :=
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{ add_monoid .
@@ -120,7 +120,7 @@ instance [Π i, add_monoid (β i)] : add_monoid (Π₀ i, β i) :=
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add_zero := λ f, ext $ λ i, by simp only [add_apply, zero_apply, add_zero] }
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instance [Π i, add_monoid (β i)] {i : ι} : is_add_monoid_hom (λ g : Π₀ i : ι, β i, g i) :=
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- { map_add := λ _ _ , add_apply, map_zero := zero_apply }
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+ { map_add := λ f g , add_apply f g i , map_zero := zero_apply i }
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instance [Π i, add_group (β i)] : has_neg (Π₀ i, β i) :=
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⟨λ f, f.map_range (λ _, has_neg.neg) (λ _, neg_zero)⟩
@@ -129,15 +129,15 @@ instance [Π i, add_comm_monoid (β i)] : add_comm_monoid (Π₀ i, β i) :=
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{ add_comm := λ f g, ext $ λ i, by simp only [add_apply, add_comm],
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.. dfinsupp.add_monoid }
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- @[simp] lemma neg_apply [Π i, add_group (β i)] { g : Π₀ i, β i} { i : ι} : (- g) i = - g i :=
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- map_range_apply
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+ @[simp] lemma neg_apply [Π i, add_group (β i)] ( g : Π₀ i, β i) ( i : ι) : (- g) i = - g i :=
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+ map_range_apply _ _ g i
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instance [Π i, add_group (β i)] : add_group (Π₀ i, β i) :=
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{ add_left_neg := λ f, ext $ λ i, by simp only [add_apply, neg_apply, zero_apply, add_left_neg],
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.. dfinsupp.add_monoid,
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.. (infer_instance : has_neg (Π₀ i, β i)) }
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- @[simp] lemma sub_apply [Π i, add_group (β i)] { g₁ g₂ : Π₀ i, β i} { i : ι} :
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+ @[simp] lemma sub_apply [Π i, add_group (β i)] ( g₁ g₂ : Π₀ i, β i) ( i : ι) :
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(g₁ - g₂) i = g₁ i - g₂ i :=
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by rw [sub_eq_add_neg]; simp [sub_eq_add_neg]
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@@ -153,9 +153,9 @@ def to_has_scalar {γ : Type w} [semiring γ] [Π i, add_comm_group (β i)] [Π
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local attribute [instance] to_has_scalar
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@[simp] lemma smul_apply {γ : Type w} [semiring γ] [Π i, add_comm_group (β i)]
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- [Π i, semimodule γ (β i)] {i : ι} { b : γ} { v : Π₀ i, β i} :
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+ [Π i, semimodule γ (β i)] ( b : γ) ( v : Π₀ i, β i) (i : ι) :
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(b • v) i = b • (v i) :=
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- map_range_apply
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+ map_range_apply _ _ v i
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/-- Dependent functions with finite support inherit a semimodule structure from such a structure on
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each coordinate. -/
@@ -179,22 +179,22 @@ quotient.lift_on f (λ x, ⟦(⟨λ i, if p i then x.1 i else 0, x.2,
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quotient.sound $ λ i, by simp only [H i]
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@[simp] lemma filter_apply [Π i, has_zero (β i)]
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- { p : ι → Prop } [decidable_pred p] { i : ι} { f : Π₀ i, β i} :
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+ ( p : ι → Prop ) [decidable_pred p] ( i : ι) ( f : Π₀ i, β i) :
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f.filter p i = if p i then f i else 0 :=
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quotient.induction_on f $ λ x, rfl
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lemma filter_apply_pos [Π i, has_zero (β i)]
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- {p : ι → Prop } [decidable_pred p] { f : Π₀ i, β i} {i : ι} (h : p i) :
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+ {p : ι → Prop } [decidable_pred p] ( f : Π₀ i, β i) {i : ι} (h : p i) :
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f.filter p i = f i :=
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by simp only [filter_apply, if_pos h]
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lemma filter_apply_neg [Π i, has_zero (β i)]
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- {p : ι → Prop } [decidable_pred p] { f : Π₀ i, β i} {i : ι} (h : ¬ p i) :
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+ {p : ι → Prop } [decidable_pred p] ( f : Π₀ i, β i) {i : ι} (h : ¬ p i) :
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f.filter p i = 0 :=
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by simp only [filter_apply, if_neg h]
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- lemma filter_pos_add_filter_neg [Π i, add_monoid (β i)] { f : Π₀ i, β i}
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- { p : ι → Prop } [decidable_pred p] :
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+ lemma filter_pos_add_filter_neg [Π i, add_monoid (β i)] ( f : Π₀ i, β i)
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+ ( p : ι → Prop ) [decidable_pred p] :
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f.filter p + f.filter (λi, ¬ p i) = f :=
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ext $ λ i, by simp only [add_apply, filter_apply]; split_ifs; simp only [add_zero, zero_add]
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