refactor(library/algebra/group_bigops.lean,library/data/nat/bigops.lean): add ext principle, clean up file
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Jeremy Avigad
parent
4764f6e8ec
commit
9720d84095
2 changed files with 44 additions and 37 deletions
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@ -99,9 +99,11 @@ section comm_monoid
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theorem Prod_empty (f : A → B) : Prod ∅ f = 1 :=
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Prodl_nil f
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section decidable_eq
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variable [H : decidable_eq A]
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include H
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theorem Prod_mul (s : finset A) (f g : A → B) : Prod s (λx, f x * g x) = Prod s f * Prod s g :=
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quot.induction_on s (take u, !Prodl_mul)
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section deceqA
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include deceqA
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theorem Prod_insert_of_mem (f : A → B) {a : A} {s : finset A} :
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a ∈ s → Prod (insert a s) f = Prod s f :=
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@ -119,10 +121,19 @@ section comm_monoid
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quot.induction_on₂ s₁ s₂
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(λ l₁ l₂ d, Prodl_union f d),
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H1 (disjoint_of_inter_empty disj)
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end decidable_eq
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theorem Prod_mul (s : finset A) (f g : A → B) : Prod s (λx, f x * g x) = Prod s f * Prod s g :=
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quot.induction_on s (take u, !Prodl_mul)
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theorem Prod_ext {s : finset A} {f g : A → B} :
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(∀{x}, x ∈ s → f x = g x) → Prod s f = Prod s g :=
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finset.induction_on s
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(assume H, rfl)
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(take s' x, assume H1 : x ∉ s',
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assume IH : (∀ {x : A}, x ∈ s' → f x = g x) → Prod s' f = Prod s' g,
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assume H2 : ∀{y}, y ∈ insert x s' → f y = g y,
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assert H3 : ∀y, y ∈ s' → f y = g y, from
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take y, assume H', H2 (mem_insert_of_mem _ H'),
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assert H4 : f x = g x, from H2 !mem_insert,
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by rewrite [Prod_insert_of_not_mem f H1, Prod_insert_of_not_mem g H1, IH H3, H4])
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end deceqA
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end comm_monoid
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section add_monoid
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@ -141,16 +152,15 @@ section add_monoid
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theorem Suml_append (l₁ l₂ : list A) (f : A → B) : Suml (l₁++l₂) f = Suml l₁ f + Suml l₂ f :=
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Prodl_append l₁ l₂ f
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section decidable_eq
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variable [H : decidable_eq A]
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include H
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section deceqA
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include deceqA
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theorem Suml_insert_of_mem (f : A → B) {a : A} {l : list A} (H : a ∈ l) :
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Suml (insert a l) f = Suml l f := Prodl_insert_of_mem f H
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theorem Suml_insert_of_not_mem (f : A → B) {a : A} {l : list A} (H : a ∉ l) :
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Suml (insert a l) f = f a + Suml l f := Prodl_insert_of_not_mem f H
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theorem Suml_union {l₁ l₂ : list A} (f : A → B) (d : disjoint l₁ l₂) :
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Suml (union l₁ l₂) f = Suml l₁ f + Suml l₂ f := Prodl_union f d
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end decidable_eq
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end deceqA
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end add_monoid
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section add_comm_monoid
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@ -175,20 +185,20 @@ section add_comm_monoid
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notation `∑` binders `∈` s, r:(scoped f, Sum s f) := r
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theorem Sum_empty (f : A → B) : Sum ∅ f = 0 := Prod_empty f
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theorem Sum_add (s : finset A) (f g : A → B) :
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Sum s (λx, f x + g x) = Sum s f + Sum s g := Prod_mul s f g
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section decidable_eq
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variable [H : decidable_eq A]
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include H
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section deceqA
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include deceqA
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theorem Sum_insert_of_mem (f : A → B) {a : A} {s : finset A} (H : a ∈ s) :
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Sum (insert a s) f = Sum s f := Prod_insert_of_mem f H
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theorem Sum_insert_of_not_mem (f : A → B) {a : A} {s : finset A} (H : a ∉ s) :
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Sum (insert a s) f = f a + Sum s f := Prod_insert_of_not_mem f H
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theorem Sum_union (f : A → B) {s₁ s₂ : finset A} (disj : s₁ ∩ s₂ = ∅) :
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Sum (s₁ ∪ s₂) f = Sum s₁ f + Sum s₂ f := Prod_union f disj
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end decidable_eq
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theorem Sum_add (s : finset A) (f g : A → B) :
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Sum s (λx, f x + g x) = Sum s f + Sum s g := Prod_mul s f g
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theorem Sum_ext {s : finset A} {f g : A → B} (H : ∀x, x ∈ s → f x = g x) :
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Sum s f = Sum s g := Prod_ext H
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end deceqA
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end add_comm_monoid
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end algebra
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@ -23,7 +23,8 @@ theorem Prodl_cons (f : A → nat) (a : A) (l : list A) : Prodl (a::l) f = f a *
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algebra.Prodl_cons f a l
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theorem Prodl_append (l₁ l₂ : list A) (f : A → nat) : Prodl (l₁++l₂) f = Prodl l₁ f * Prodl l₂ f :=
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algebra.Prodl_append l₁ l₂ f
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theorem Prodl_mul (l : list A) (f g : A → nat) :
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Prodl l (λx, f x * g x) = Prodl l f * Prodl l g := algebra.Prodl_mul l f g
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section deceqA
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include deceqA
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theorem Prodl_insert_of_mem (f : A → nat) {a : A} {l : list A} (H : a ∈ l) :
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@ -34,16 +35,14 @@ section deceqA
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Prodl (union l₁ l₂) f = Prodl l₁ f * Prodl l₂ f := algebra.Prodl_union f d
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end deceqA
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theorem Prodl_mul (l : list A) (f g : A → nat) :
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Prodl l (λx, f x * g x) = Prodl l f * Prodl l g := algebra.Prodl_mul l f g
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/- Prod -/
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definition Prod (s : finset A) (f : A → nat) : nat := algebra.Prod s f
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notation `∏` binders `∈` s, r:(scoped f, Prod s f) := r
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theorem Prod_empty (f : A → nat) : Prod ∅ f = 1 := algebra.Prod_empty f
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theorem Prod_mul (s : finset A) (f g : A → nat) : Prod s (λx, f x * g x) = Prod s f * Prod s g :=
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algebra.Prod_mul s f g
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section deceqA
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include deceqA
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theorem Prod_insert_of_mem (f : A → nat) {a : A} {s : finset A} (H : a ∈ s) :
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@ -52,11 +51,10 @@ section deceqA
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Prod (insert a s) f = f a * Prod s f := algebra.Prod_insert_of_not_mem f H
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theorem Prod_union (f : A → nat) {s₁ s₂ : finset A} (disj : s₁ ∩ s₂ = ∅) :
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Prod (s₁ ∪ s₂) f = Prod s₁ f * Prod s₂ f := algebra.Prod_union f disj
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theorem Prod_ext {s : finset A} {f g : A → nat} (H : ∀x, x ∈ s → f x = g x) :
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Prod s f = Prod s g := algebra.Prod_ext H
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end deceqA
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theorem Prod_mul (s : finset A) (f g : A → nat) : Prod s (λx, f x * g x) = Prod s f * Prod s g :=
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algebra.Prod_mul s f g
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/- Suml -/
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definition Suml (l : list A) (f : A → nat) : nat := algebra.Suml l f
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@ -67,7 +65,8 @@ theorem Suml_cons (f : A → nat) (a : A) (l : list A) : Suml (a::l) f = f a + S
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algebra.Suml_cons f a l
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theorem Suml_append (l₁ l₂ : list A) (f : A → nat) : Suml (l₁++l₂) f = Suml l₁ f + Suml l₂ f :=
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algebra.Suml_append l₁ l₂ f
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theorem Suml_add (l : list A) (f g : A → nat) : Suml l (λx, f x + g x) = Suml l f + Suml l g :=
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algebra.Suml_add l f g
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section deceqA
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include deceqA
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theorem Suml_insert_of_mem (f : A → nat) {a : A} {l : list A} (H : a ∈ l) :
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@ -78,26 +77,24 @@ section deceqA
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Suml (union l₁ l₂) f = Suml l₁ f + Suml l₂ f := algebra.Suml_union f d
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end deceqA
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theorem Suml_add (l : list A) (f g : A → nat) : Suml l (λx, f x + g x) = Suml l f + Suml l g :=
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algebra.Suml_add l f g
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/- Sum -/
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definition Sum (s : finset A) (f : A → nat) : nat := algebra.Sum s f
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notation `∑` binders `∈` s, r:(scoped f, Sum s f) := r
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theorem Sum_empty (f : A → nat) : Sum ∅ f = 0 := algebra.Sum_empty f
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theorem Sum_add (s : finset A) (f g : A → nat) : Sum s (λx, f x + g x) = Sum s f + Sum s g :=
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algebra.Sum_add s f g
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section deceqA
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include deceqA
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theorem Sum_insert_of_mem (f : A → nat) {a : A} {s : finset A} (H : a ∈ s) :
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Sum (insert a s) f = Sum s f := algebra.Sum_insert_of_mem f H
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theorem Sum_insert_of_not_mem (f : A → nat) {a : A} {s : finset A} (H : a ∉ s) :
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Sum (insert a s) f = f a + Sum s f := algebra.Sum_insert_of_not_mem f H
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theorem Sum_union (f : A → nat) {s₁ s₂ : finset A} (disj : s₁ ∩ s₂ = ∅) :
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Sum (s₁ ∪ s₂) f = Sum s₁ f + Sum s₂ f := algebra.Sum_union f disj
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Sum (insert a s) f = Sum s f := algebra.Sum_insert_of_mem f H
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theorem Sum_insert_of_not_mem (f : A → nat) {a : A} {s : finset A} (H : a ∉ s) :
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Sum (insert a s) f = f a + Sum s f := algebra.Sum_insert_of_not_mem f H
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theorem Sum_union (f : A → nat) {s₁ s₂ : finset A} (disj : s₁ ∩ s₂ = ∅) :
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Sum (s₁ ∪ s₂) f = Sum s₁ f + Sum s₂ f := algebra.Sum_union f disj
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theorem Sum_ext {s : finset A} {f g : A → nat} (H : ∀x, x ∈ s → f x = g x) :
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Sum s f = Sum s g := algebra.Sum_ext H
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end deceqA
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theorem Sum_add (s : finset A) (f g : A → nat) : Sum s (λx, f x + g x) = Sum s f + Sum s g :=
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algebra.Sum_add s f g
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end nat
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