feat(library/smt/array): define array theory used in SMT

library/smt/array.lean

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+/-
+Copyright (c) 2016 Microsoft Corporation. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Author: Leonardo de Moura
+-/
+namespace smt
+definition array (A B : Type) := A → B
+
+variables {A B : Type}
+open tactic
+
+definition select (a : array A B) (i : A) : B :=
+a i
+
+theorem arrayext (a₁ a₂ : array A B) : (∀ i, select a₁ i = select a₂ i) → a₁ = a₂ :=
+funext
+
+variable [decidable_eq A]
+
+definition store (a : array A B) (i : A) (v : B) : array A B :=
+λ j, if j = i then v else select a j
+
+theorem select_store (a : array A B) (i : A) (v : B) : select (store a i v) i = v :=
+by unfold [`smt.store, `smt.select] >> dsimp >> rewrite `if_pos >> reflexivity
+
+theorem select_store_ne (a : array A B) (i j : A) (v : B) : j ≠ i → select (store a i v) j = select a j :=
+by intros >> unfold [`smt.store, `smt.select] >> dsimp >> rewrite `if_neg >> assumption
+
+end smt