Library Theory

Correctness of the up-to techniques corresponding to the notions of (weak) monotonic functions, or controlled simulations





Require Export WeakMonotonic.




Section Global.




  Variables A X Y: Type.

  Variable TX: reduction_t A X.

  Variable TY: reduction_t A Y.

Correctness with a monotonic function


  Section MonotonicCorrect.




    Variable F: function X Y.

    Hypothesis HF: monotonic TX TY F.




    Variable R: relation2 X Y.

    Hypothesis HR: evolve TX TY R (F R).

predicate for the induction


    Let phi: forall n, evolve TX TY (UExp F R n) (UExp F R (S n)).

Proposition 1.6


    Theorem monotonic_correct: simulation TX TY (UIter F R).




  End MonotonicCorrect.

Correctness with a weakly monotonic function


  Section WeakMonotonicCorrect.




    Variable F: function X Y.

    Hypothesis HF: wmonotonic TX TY F.




    Variable R: relation2 X Y.

    Hypothesis HRt: simulation_t TX TY R.

    Hypothesis HRa: evolve_a TX TY R (F R).

first induction predicate


    Let silent: forall n, simulation_t TX TY (UExp F R n).

correction w.r.t. silent steps


    Lemma wmonotonic_correct_t: simulation_t TX TY (UIter F R).

second induction predicate


    Let visible: forall n, evolve_a TX TY (UExp F R n) (UExp F R (S n)).

Proposition 2.2


    Theorem wmonotonic_correct: simulation TX TY (UIter F R).




  End WeakMonotonicCorrect.

Correctness with a monotonic function, and a weakly monotonic function


  Section UnifiedCorrect.




    Variables F G: function X Y.

    Hypothesis HF : monotonic TX TY F.

    Hypothesis HG : wmonotonic TX TY G.




    Variable R: relation2 X Y.




    Hypothesis HFG: contains F G.




    Hypothesis HRt: evolve_t TX TY R (F R).

    Hypothesis HRa: evolve_a TX TY R (G R).

first induction predicate


    Let pre_silent: forall n, evolve_t TX TY (UExp F R n) (UExp F R (S n)).

    Let silent: simulation_t TX TY (UIter F R).




    Let HFGn: forall n, incl (UExp F R n) (UExp G R n).

second induction predicate


    Let pre_visible: forall n, evolve TX TY (UExp F R n) (UExp G R (S n)).

    Let visible: evolve_a TX TY (UIter F R) (UIter G R).




    Let HGFGn: forall n, incl (UExp (UIter G) (UIter G R) n) (UExp (UIter G) R (S n)).

    Let HGFG: eeq (UIter (UIter G) (UIter F R)) (UIter (UIter G) R).

Proposition 2.4


    Theorem unified_correct: simulation TX TY (UIter (UIter G) R).




  End UnifiedCorrect.

Correctness with a monotonic function, a weakly monotonic function, and a controlled simulation


  Section ControlledCorrect.




    Variable B: relation X.

    Hypothesis HB: controlled TX TY B.




    Variables F G: function X Y.

    Hypothesis HF : monotonic TX TY F.

    Hypothesis HG : wmonotonic TX TY G.




    Hypothesis HBF: transparent B F.

    Hypothesis HFG: contains F G.

    Hypothesis HBG: contains (chaining_l (star B)) G.



    Variable R: relation2 X Y.

    Hypothesis HRt: evolve_t TX TY R (comp (star B) (F R)).

    Hypothesis HRa: evolve_a TX TY R (G R).

first induction predicate


    Let pre_silent: forall n, evolve_t TX TY (UExp F R n) (comp (star B) (UExp F R (S n))).

    Let silent: simulation_t TX TY (comp (star B) (UIter F R)).




    Let HFGn: forall n, incl (UExp F R n) (UExp G R n).

    Let HBGGn: forall R n, incl (comp (star B) (UExp G R n)) (UExp G R (S n)).

second induction predicate


    Let pre_visible: forall n, evolve TX TY (UExp F R n) (UExp G R (2+n)).

    Let visible: evolve_a TX TY (comp (star B) (UIter F R)) (UIter G (comp (star B) (UIter F R))).




    Let HGGBF: eeq (UIter (UIter G) (comp (star B) (UIter F R))) (UIter (UIter G) R).

Proposition 3.4


    Theorem controlled_correct: simulation TX TY (UIter (UIter G) R).




  End ControlledCorrect.




End Global.