104 lines
4.9 KiB
Text
104 lines
4.9 KiB
Text
/-
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Copyright (c) 2015 Microsoft Corporation. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Author: Jeremy Avigad
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Finite products and sums on the natural numbers.
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-/
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import data.nat.basic data.nat.order algebra.group_bigops
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open list finset
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namespace nat
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open [classes] algebra
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local attribute nat.comm_semiring [instance]
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variables {A : Type} [deceqA : decidable_eq A]
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/- Prodl -/
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definition Prodl (l : list A) (f : A → nat) : nat := algebra.Prodl l f
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notation `∏` binders `←` l, r:(scoped f, Prodl l f) := r
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theorem Prodl_nil (f : A → nat) : Prodl [] f = 1 := algebra.Prodl_nil f
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theorem Prodl_cons (f : A → nat) (a : A) (l : list A) : Prodl (a::l) f = f a * Prodl l f :=
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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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Prodl (insert a l) f = Prodl l f := algebra.Prodl_insert_of_mem f H
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theorem Prodl_insert_of_not_mem (f : A → nat) {a : A} {l : list A} (H : a ∉ l) :
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Prodl (insert a l) f = f a * Prodl l f := algebra.Prodl_insert_of_not_mem f H
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theorem Prodl_union {l₁ l₂ : list A} (f : A → nat) (d : disjoint l₁ l₂) :
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Prodl (union l₁ l₂) f = Prodl l₁ f * Prodl l₂ f := algebra.Prodl_union f d
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theorem Prodl_one (l : list A) : Prodl l (λ x, nat.succ 0) = 1 := algebra.Prodl_one l
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end deceqA
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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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Prod (insert a s) f = Prod s f := algebra.Prod_insert_of_mem f H
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theorem Prod_insert_of_not_mem (f : A → nat) {a : A} {s : finset A} (H : a ∉ s) :
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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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theorem Prod_one (s : finset A) : Prod s (λ x, nat.succ 0) = 1 := algebra.Prod_one s
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end deceqA
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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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notation `∑` binders `←` l, r:(scoped f, Suml l f) := r
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theorem Suml_nil (f : A → nat) : Suml [] f = 0 := algebra.Suml_nil f
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theorem Suml_cons (f : A → nat) (a : A) (l : list A) : Suml (a::l) f = f a + Suml l f :=
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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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Suml (insert a l) f = Suml l f := algebra.Suml_insert_of_mem f H
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theorem Suml_insert_of_not_mem (f : A → nat) {a : A} {l : list A} (H : a ∉ l) :
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Suml (insert a l) f = f a + Suml l f := algebra.Suml_insert_of_not_mem f H
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theorem Suml_union {l₁ l₂ : list A} (f : A → nat) (d : disjoint l₁ l₂) :
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Suml (union l₁ l₂) f = Suml l₁ f + Suml l₂ f := algebra.Suml_union f d
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theorem Suml_zero (l : list A) : Suml l (λ x, zero) = 0 := algebra.Suml_zero l
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end deceqA
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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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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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theorem Sum_zero (s : finset A) : Sum s (λ x, zero) = 0 := algebra.Sum_zero s
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end deceqA
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end nat
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