41 lines
1.3 KiB
Text
41 lines
1.3 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: Leonardo de Moura
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Big operator for finite sets.
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-/
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import algebra.group data.finset.basic data.list.bigop
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open algebra finset function binary quot subtype
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namespace finset
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variables {A B : Type}
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variable [g : comm_monoid B]
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include g
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definition bigop (s : finset A) (f : A → B) : B :=
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quot.lift_on s
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(λ l, list.bigop (elt_of l) f)
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(λ l₁ l₂ p, list.bigop_of_perm f p)
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theorem bigop_empty (f : A → B) : bigop ∅ f = 1 :=
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list.bigop_nil f
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variable [H : decidable_eq A]
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include H
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theorem bigop_insert_of_mem (f : A → B) {a : A} {s : finset A} : a ∈ s → bigop (insert a s) f = bigop s f :=
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quot.induction_on s
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(λ l ainl, list.bigop_insert_of_mem f ainl)
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theorem bigop_insert_of_not_mem (f : A → B) {a : A} {s : finset A} : a ∉ s → bigop (insert a s) f = f a * bigop s f :=
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quot.induction_on s
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(λ l nainl, list.bigop_insert_of_not_mem f nainl)
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theorem bigop_union (f : A → B) {s₁ s₂ : finset A} (disj : s₁ ∩ s₂ = ∅) :
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bigop (s₁ ∪ s₂) f = bigop s₁ f * bigop s₂ f :=
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have H1 : disjoint s₁ s₂ → bigop (s₁ ∪ s₂) f = bigop s₁ f * bigop s₂ f, from
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quot.induction_on₂ s₁ s₂
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(λ l₁ l₂ d, list.bigop_union f d),
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H1 (disjoint_of_inter_empty disj)
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end finset
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