src/ZF/Resid/Residuals.ML

Implicit simpsets and clasets for FOL and ZF

(*  Title:      Residuals.ML
    ID:         $Id$
    Author:     Ole Rasmussen
    Copyright   1995  University of Cambridge
    Logic Image: ZF
*)

open Residuals;

(* ------------------------------------------------------------------------- *)
(*       Setting up rule lists                                               *)
(* ------------------------------------------------------------------------- *)

AddIs ([e-mail]@Sreg.intrs@
       [subst_type]@nat_typechecks); 
AddSEs (redexes.free_SEs @
	(map (Sres.mk_cases redexes.con_defs) 
	     ["residuals(Var(n),Var(n),v)",
	      "residuals(Fun(t),Fun(u),v)",
	      "residuals(App(b, u1, u2), App(0, v1, v2),v)",
	      "residuals(App(b, u1, u2), App(1, Fun(v1), v2),v)",
	      "residuals(Var(n),u,v)",
	      "residuals(Fun(t),u,v)",
	      "residuals(App(b, u1, u2), w,v)",
	      "residuals(u,Var(n),v)",
	      "residuals(u,Fun(t),v)",
	      "residuals(w,App(b, u1, u2),v)"]) @
	(map (Ssub.mk_cases redexes.con_defs) 
	     ["Var(n) <== u",
	      "Fun(n) <== u",
	      "u <== Fun(n)",
	      "App(1,Fun(t),a) <== u",
	      "App(0,t,a) <== u"]) @
	[redexes.mk_cases redexes.con_defs "Fun(t):redexes"]);

Addsimps Sres.intrs;
val resD1 =Sres.dom_subset RS subsetD RS SigmaD1;
val resD2 =Sres.dom_subset RS subsetD RS SigmaD2 RS SigmaD1;
val resD3 =Sres.dom_subset RS subsetD RS SigmaD2 RS SigmaD2;


(* ------------------------------------------------------------------------- *)
(*       residuals is a  partial function                                    *)
(* ------------------------------------------------------------------------- *)

goal Residuals.thy 
    "!!u.residuals(u,v,w) ==> ALL w1.residuals(u,v,w1) --> w1 = w";
by (etac Sres.induct 1);
by (ALLGOALS Fast_tac);
val residuals_function = result();
val res_is_func  = residuals_function  RS spec RS mp;

goal Residuals.thy 
    "!!u.u~v ==> regular(v) --> (EX w.residuals(u,v,w))";
by (etac Scomp.induct 1);
by (ALLGOALS Fast_tac);
val residuals_intro = result();

val prems = goal  Residuals.thy 
    "[|u~v; residuals(u,v,THE w. residuals(u, v, w))==> R; \
\      regular(v)|] ==> R";
by (cut_facts_tac prems 1);
by (resolve_tac prems 1);
by (resolve_tac [residuals_intro RS mp RS exE] 1);
by (resolve_tac [the_equality RS ssubst] 3);
by (ALLGOALS (fast_tac (!claset addIs [res_is_func])));
val comp_resfuncE = result();


(* ------------------------------------------------------------------------- *)
(*               Residual function                                           *)
(* ------------------------------------------------------------------------- *)

val resfunc_cs = (!claset addIs  [the_equality,res_is_func] 
                          addSEs [comp_resfuncE]);

goalw Residuals.thy [res_func_def]
    "!!n.n:nat ==> Var(n) |> Var(n) = Var(n)";
by (fast_tac resfunc_cs 1);
val res_Var = result();

goalw Residuals.thy [res_func_def]
    "!!n.[|s~t; regular(t)|]==> Fun(s) |> Fun(t) = Fun(s |> t)";
by (fast_tac resfunc_cs 1);
val res_Fun = result();

goalw Residuals.thy [res_func_def]
    "!!n.[|s~u; regular(u); t~v; regular(v); b:bool|]==> \
\           App(b,s,t) |> App(0,u,v) = App(b, s |> u, t |> v)";
by (fast_tac resfunc_cs 1);
val res_App = result();

goalw Residuals.thy [res_func_def]
    "!!n.[|s~u; regular(u); t~v; regular(v); b:bool|]==> \
\           App(b,Fun(s),t) |> App(1,Fun(u),v) = (t |> v)/ (s |> u)";
by (fast_tac resfunc_cs 1);
val res_redex = result();

goal Residuals.thy
    "!!n.[|s~t; regular(t)|]==> regular(t) --> s |> t:redexes";
by (etac Scomp.induct 1);
by (ALLGOALS (asm_full_simp_tac 
             (!simpset addsimps [res_Var,res_Fun,res_App,res_redex] 
              setloop (SELECT_GOAL (safe_tac (!claset))))));
by (dres_inst_tac [("psi", "Fun(?u) |> ?v : redexes")] asm_rl 1);
by (asm_full_simp_tac 
             (!simpset addsimps [res_Fun] 
              setloop (SELECT_GOAL (safe_tac (!claset)))) 1);
val resfunc_type = result();

Addsimps ([res_Var, res_Fun, res_App, res_redex, lift_rec_preserve_comp,
	  lift_rec_preserve_reg, subst_rec_preserve_comp, resfunc_type,
	  subst_rec_preserve_reg] @
	  redexes.free_iffs);

val res1L_ss = !simpset setloop (SELECT_GOAL (safe_tac (!claset)));

(* ------------------------------------------------------------------------- *)
(*     Commutation theorem                                                   *)
(* ------------------------------------------------------------------------- *)

goal Residuals.thy 
    "!!u.u<==v ==> u~v";
by (etac Ssub.induct 1);
by (ALLGOALS Asm_simp_tac);
val sub_comp = result();

goal Residuals.thy 
    "!!u.u<==v  ==> regular(v) --> regular(u)";
by (etac Ssub.induct 1);
by (ALLGOALS (asm_simp_tac res1L_ss));
val sub_preserve_reg = result();

goal Residuals.thy 
    "!!u.[|u~v; k:nat|]==> regular(v)--> (ALL n:nat.  \
\        lift_rec(u,n) |> lift_rec(v,n) = lift_rec(u |> v,n))";
by (etac Scomp.induct 1);
by (Step_tac 1);
by (ALLGOALS (asm_full_simp_tac ((addsplit (!simpset)) addsimps [lift_subst])));
by (dres_inst_tac [("psi", "ALL x:nat.Fun(?u(x)) |> ?v(x) = ?w(x)")] asm_rl 1);
by (Asm_full_simp_tac 1);
val residuals_lift_rec = result();

goal Residuals.thy 
    "!!u.u1~u2 ==>  ALL v1 v2.v1~v2 --> regular(v2) --> regular(u2) -->\
\   (ALL n:nat.subst_rec(v1,u1,n) |> subst_rec(v2,u2,n) = \
\              subst_rec(v1 |> v2, u1 |> u2,n))";
by (etac Scomp.induct 1);
by (Step_tac 1);
by (ALLGOALS
    (asm_full_simp_tac ((addsplit (!simpset)) addsimps ([residuals_lift_rec]))));
by (dres_inst_tac [("psi", "ALL x.?P(x)")] asm_rl 1);
by (asm_full_simp_tac (!simpset addsimps ([substitution])) 1);
val residuals_subst_rec = result();


goal Residuals.thy 
    "!!u.[|u1~u2; v1~v2; regular(u2); regular(v2)|]==>\
\       (v1/u1) |> (v2/u2) = (v1 |> v2)/(u1 |> u2)";
by (asm_simp_tac (!simpset addsimps ([residuals_subst_rec])) 1);
val commutation = result();

(* ------------------------------------------------------------------------- *)
(*     Residuals are comp and regular                                        *)
(* ------------------------------------------------------------------------- *)

goal Residuals.thy 
    "!!u.u~v ==> ALL w. u~w --> v~w --> regular(w) --> (u|>w) ~ (v|>w)";
by (etac Scomp.induct 1);
by (ALLGOALS (asm_simp_tac res1L_ss));
by (dresolve_tac [spec RS mp RS mp RS mp] 1 
    THEN resolve_tac Scomp.intrs 1 THEN resolve_tac Scomp.intrs 2 
    THEN resolve_tac Sreg.intrs 3);
by (REPEAT(assume_tac 1));
by (asm_full_simp_tac res1L_ss 1);
val residuals_preserve_comp = result();


goal Residuals.thy 
    "!!u.u~v ==> regular(u) --> regular(v) --> regular(u|>v)";
by (etac Scomp.induct 1);
by (safe_tac (!claset));
by (TRYALL (dres_inst_tac [("psi", "regular(Fun(?u) |> ?v)")] asm_rl));
by (ALLGOALS (asm_full_simp_tac res1L_ss));
val residuals_preserve_reg = result();

(* ------------------------------------------------------------------------- *)
(*     Preservation lemma                                                    *)
(* ------------------------------------------------------------------------- *)

goal Residuals.thy 
    "!!u.u~v ==> v ~ (u un v)";
by (etac Scomp.induct 1);
by (ALLGOALS Asm_full_simp_tac);
val union_preserve_comp = result();

goal Residuals.thy 
    "!!u.u ~ v ==> regular(v) --> u|>v = (u un v)|>v";
by (etac Scomp.induct 1);
by (safe_tac (!claset));
by (dres_inst_tac [("psi", "Fun(?u) |> ?v = ?w")] asm_rl 3);
by (ALLGOALS (asm_full_simp_tac (res1L_ss addsimps 
                                 [union_preserve_comp,comp_sym_iff])));
by (asm_full_simp_tac (!simpset addsimps 
                       [union_preserve_comp RS comp_sym,
                        comp_sym RS union_preserve_comp RS comp_sym]) 1);
val preservation1 = result();

val preservation = preservation1 RS mp;