Section EXTEN.
Variable valu1: valuation.
Variable upto: valnum.
Variable valu2: valuation.
Hypothesis AGREE: valu_agree valu1 valu2 upto.
Variable ge: genv.
Variable sp: val.
Variable rs: regset.
Variable m: mem.
End EXTEN.
Ltac splitall := repeat (match goal with |- _ /\ _ => split end).
Remark in_remove:
forall (A: Type) (eq: forall (x y: A), {x=y}+{x<>y}) x y l,
In y (List.remove eq x l) <-> x <> y /\ In y l.

Lemma kill_equations_hold:
forall valu ge sp rs m n pred m',
numbering_holds valu ge sp rs m n ->
(forall r v,
pred r = false ->
rhs_eval_to valu ge sp m r v ->
rhs_eval_to valu ge sp m' r v) ->
numbering_holds valu ge sp rs m' (kill_equations pred n).