The Coq Proof Assistant

# Library Coq.micromega.Env

Require Import BinInt List.
Set Implicit Arguments.
Local Open Scope positive_scope.
Section S.
Variable D :Type.
Definition Env := positive -> D.
Definition jump (j:positive) (e:Env) := fun x => e (x+j).
Definition nth (n:positive) (e:Env) := e n.
Definition hd (e:Env) := nth 1 e.
Definition tail (e:Env) := jump 1 e.
Lemma jump_add i j l x : jump (i + j) l x = jump i (jump j l) x.
Lemma jump_simpl p l x :
jump p l x =
match p with
| xH => tail l x
| xO p => jump p (jump p l) x
| xI p => jump p (jump p (tail l)) x
end.
Lemma jump_tl j l x : tail (jump j l) x = jump j (tail l) x.
Lemma jump_succ j l x : jump (Pos.succ j) l x = jump 1 (jump j l) x.
Lemma jump_pred_double i l x :
jump (Pos.pred_double i) (tail l) x = jump i (jump i l) x.
Lemma nth_spec p l :
nth p l =
match p with
| xH => hd l
| xO p => nth p (jump p l)
| xI p => nth p (jump p (tail l))
end.
Lemma nth_jump p l : nth p (tail l) = hd (jump p l).
Lemma nth_pred_double p l :
nth (Pos.pred_double p) (tail l) = nth p (jump p l).
End S.
Ltac jump_simpl :=
repeat
match goal with
| |- context [jump xH] => rewrite (jump_simpl xH)
| |- context [jump (xO ?p)] => rewrite (jump_simpl (xO p))
| |- context [jump (xI ?p)] => rewrite (jump_simpl (xI p))
end.