diff -r 000000000000 -r 7949f97df77a Subst/Unifier.ML --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/Subst/Unifier.ML Thu Sep 16 12:21:07 1993 +0200 @@ -0,0 +1,299 @@ +(* Title: Substitutions/unifier.ML + Author: Martin Coen, Cambridge University Computer Laboratory + Copyright 1993 University of Cambridge + +For unifier.thy. +Properties of unifiers. +Cases for partial correctness of algorithm and conditions for termination. +*) + +open Unifier; + +val unify_defs = + [Idem_def,Unifier_def,MoreGeneral_def,MGUnifier_def,MGIUnifier_def]; + +(**** Unifiers ****) + +goalw Unifier.thy [Unifier_def] "Unifier(s,t,u) = (t <| s = u <| s)"; +by (rtac refl 1); +val Unifier_iff = result(); + +goal Unifier.thy + "Unifier(s,Comb(t,u),Comb(v,w)) --> Unifier(s,t,v) & Unifier(s,u,w)"; +by (simp_tac (subst_ss addsimps [Unifier_iff]) 1); +val Unifier_Comb = result() RS mp RS conjE; + +goal Unifier.thy + "~v : vars_of(t) --> ~v : vars_of(u) -->Unifier(s,t,u) --> \ +\ Unifier(Cons(,s),t,u)"; +by (simp_tac (subst_ss addsimps [Unifier_iff,repl_invariance]) 1); +val Cons_Unifier = result() RS mp RS mp RS mp; + +(**** Most General Unifiers ****) + +goalw Unifier.thy [MoreGeneral_def] "r >> s = (EX q. s =s= r <> q)"; +by (rtac refl 1); +val MoreGen_iff = result(); + +goal Unifier.thy "Nil >> s"; +by (simp_tac (subst_ss addsimps [MoreGen_iff]) 1); +by (fast_tac (set_cs addIs [refl RS subst_refl]) 1); +val MoreGen_Nil = result(); + +goalw Unifier.thy unify_defs + "MGUnifier(s,t,u) = (ALL r.Unifier(r,t,u) = s >> r)"; +by (REPEAT (ares_tac [iffI,allI] 1 ORELSE + eresolve_tac [conjE,allE,mp,exE,ssubst_subst2] 1)); +by (asm_simp_tac (subst_ss addsimps [subst_comp]) 1); +by (fast_tac (set_cs addIs [comp_Nil RS sym RS subst_refl]) 1); +val MGU_iff = result(); + +val [prem] = goal Unifier.thy + "~ Var(v) <: t ==> MGUnifier(Cons(,Nil),Var(v),t)"; +by (simp_tac (subst_ss addsimps [MGU_iff,MoreGen_iff,Unifier_iff]) 1); +by (REPEAT_SOME (step_tac set_cs)); +by (etac subst 1); +by (etac ssubst_subst2 2); +by (rtac (Cons_trivial RS subst_sym) 1); +by (simp_tac (subst_ss addsimps [prem RS Var_not_occs,Var_subst]) 1); +val MGUnifier_Var = result(); + +(**** Most General Idempotent Unifiers ****) + +goal Unifier.thy "r <> r =s= r --> s =s= r <> q --> r <> s =s= s"; +by (simp_tac (subst_ss addsimps [subst_eq_iff,subst_comp]) 1); +val MGIU_iff_lemma = result() RS mp RS mp; + +goalw Unifier.thy unify_defs + "MGIUnifier(s,t,u) = \ +\ (Idem(s) & Unifier(s,t,u) & (ALL r.Unifier(r,t,u) --> s<>r=s=r))"; +by (fast_tac (set_cs addEs [subst_sym,MGIU_iff_lemma]) 1); +val MGIU_iff = result(); + +(**** Idempotence ****) + +goalw Unifier.thy unify_defs "Idem(s) = (s <> s =s= s)"; +by (rtac refl 1); +val raw_Idem_iff = result(); + +goal Unifier.thy "Idem(s) = (sdom(s) Int srange(s) = {})"; +by (simp_tac (subst_ss addsimps [raw_Idem_iff,subst_eq_iff,subst_comp, + invariance,dom_range_disjoint])1); +val Idem_iff = result(); + +goal Unifier.thy "Idem(Nil)"; +by (simp_tac (subst_ss addsimps [raw_Idem_iff,refl RS subst_refl]) 1); +val Idem_Nil = result(); + +goal Unifier.thy "~ (Var(v) <: t) --> Idem(Cons(,Nil))"; +by (simp_tac (subst_ss addsimps [Var_subst,vars_iff_occseq,Idem_iff,srange_iff] + setloop (split_tac [expand_if])) 1); +by (fast_tac set_cs 1); +val Var_Idem = result() RS mp; + +val [prem] = goalw Unifier.thy [Idem_def] + "Idem(r) ==> Unifier(s,t <| r,u <| r) --> Unifier(r <> s,t <| r,u <| r)"; +by (simp_tac (subst_ss addsimps + [Unifier_iff,subst_comp,prem RS comp_subst_subst]) 1); +val Unifier_Idem_subst = result() RS mp; + +val [prem] = goal Unifier.thy + "r <> s =s= s ==> Unifier(s,t,u) --> Unifier(s,t <| r,u <| r)"; +by (simp_tac (subst_ss addsimps + [Unifier_iff,subst_comp,prem RS comp_subst_subst]) 1); +val Unifier_comp_subst = result() RS mp; + +(*** The domain of a MGIU is a subset of the variables in the terms ***) +(*** NB this and one for range are only needed for termination ***) + +val [prem] = goal Unifier.thy + "~ vars_of(Var(x) <| r) = vars_of(Var(x) <| s) ==> ~r =s= s"; +by (rtac (prem RS contrapos) 1); +by (fast_tac (set_cs addEs [subst_subst2]) 1); +val lemma_lemma = result(); + +val prems = goal Unifier.thy + "x : sdom(s) --> ~x : srange(s) --> \ +\ ~vars_of(Var(x) <| s<>Cons(,s)) = \ +\ vars_of(Var(x) <| Cons(,s))"; +by (simp_tac (subst_ss addsimps [not_equal_iff]) 1); +by (REPEAT (resolve_tac [impI,disjI2] 1)); +by(res_inst_tac [("x","x")] exI 1); +br conjI 1; +by (asm_simp_tac (subst_ss addsimps [Var_elim,subst_comp,repl_invariance]) 1); +by (asm_simp_tac (subst_ss addsimps [Var_subst]) 1); +val MGIU_sdom_lemma = result() RS mp RS mp RS lemma_lemma RS notE;; + +goal Unifier.thy "MGIUnifier(s,t,u) --> sdom(s) <= vars_of(t) Un vars_of(u)"; +by (subgoal_tac "! P Q.(P | Q) = (~( ~P & ~Q))" 1); +by (asm_simp_tac (subst_ss addsimps [MGIU_iff,Idem_iff,subset_iff]) 1); +by (safe_tac set_cs); +by (eresolve_tac ([spec] RL [impE]) 1); +by (rtac Cons_Unifier 1); +by (ALLGOALS (fast_tac (set_cs addIs [Cons_Unifier,MGIU_sdom_lemma]))); +val MGIU_sdom = result() RS mp; + +(*** The range of a MGIU is a subset of the variables in the terms ***) + +val prems = goal HOL.thy "P = Q ==> (~P) = (~Q)"; +by (simp_tac (set_ss addsimps prems) 1); +val not_cong = result(); + +val prems = goal Unifier.thy + "~w=x --> x : vars_of(Var(w) <| s) --> w : sdom(s) --> ~w : srange(s) --> \ +\ ~vars_of(Var(w) <| s<>Cons(,s)) = \ +\ vars_of(Var(w) <| Cons(,s))"; +by (simp_tac (subst_ss addsimps [not_equal_iff]) 1); +by (REPEAT (resolve_tac [impI,disjI1] 1)); +by(res_inst_tac [("x","w")] exI 1); +by (ALLGOALS (asm_simp_tac (subst_ss addsimps [Var_elim,subst_comp, + vars_var_iff RS not_cong RS iffD2 RS repl_invariance]) )); +by (fast_tac (set_cs addIs [Var_in_subst]) 1); +val MGIU_srange_lemma =result() RS mp RS mp RS mp RS mp RS lemma_lemma RS notE; + +goal Unifier.thy "MGIUnifier(s,t,u) --> srange(s) <= vars_of(t) Un vars_of(u)"; +by (subgoal_tac "! P Q.(P | Q) = (~( ~P & ~Q))" 1); +by (asm_simp_tac (subst_ss addsimps [MGIU_iff,srange_iff,subset_iff]) 1); +by (simp_tac (subst_ss addsimps [Idem_iff]) 1); +by (safe_tac set_cs); +by (eresolve_tac ([spec] RL [impE]) 1); +by (rtac Cons_Unifier 1); +by (imp_excluded_middle_tac "w=ta" 4); +by (fast_tac (set_cs addEs [MGIU_srange_lemma]) 5); +by (ALLGOALS (fast_tac (set_cs addIs [Var_elim2]))); +val MGIU_srange = result() RS mp; + +(*************** Correctness of a simple unification algorithm ***************) +(* *) +(* fun unify Const(m) Const(n) = if m=n then Nil else Fail *) +(* | unify Const(m) _ = Fail *) +(* | unify Var(v) t = if Var(v)<:t then Fail else Cons(,Nil)*) +(* | unify Comb(t,u) Const(n) = Fail *) +(* | unify Comb(t,u) Var(v) = if Var(v) <: Comb(t,u) then Fail *) +(* else Cons(,Nil) *) +(* | unify Comb(t,u) Comb(v,w) = let s = unify t v *) +(* in if s=Fail then Fail *) +(* else unify (u<|s) (w<|s); *) + +(**** Cases for the partial correctness of the algorithm ****) + +goalw Unifier.thy unify_defs "MGIUnifier(s,t,u) = MGIUnifier(s,u,t)"; +by (safe_tac (HOL_cs addSEs ([sym]@([spec] RL [mp])))); +val Unify_comm = result(); + +goal Unifier.thy "MGIUnifier(Nil,Const(n),Const(n))"; +by (simp_tac (subst_ss addsimps + [MGIU_iff,MGU_iff,Unifier_iff,subst_eq_iff,Idem_Nil]) 1); +val Unify1 = result(); + +goal Unifier.thy "~m=n --> (ALL l.~Unifier(l,Const(m),Const(n)))"; +by (simp_tac (subst_ss addsimps[Unifier_iff]) 1); +val Unify2 = result() RS mp; + +val [prem] = goalw Unifier.thy [MGIUnifier_def] + "~Var(v) <: t ==> MGIUnifier(Cons(,Nil),Var(v),t)"; +by (fast_tac (HOL_cs addSIs [prem RS MGUnifier_Var,prem RS Var_Idem]) 1); +val Unify3 = result(); + +val [prem] = goal Unifier.thy "Var(v) <: t ==> (ALL l.~Unifier(l,Var(v),t))"; +by (simp_tac (subst_ss addsimps + [Unifier_iff,prem RS subst_mono RS occs_irrefl2]) 1); +val Unify4 = result(); + +goal Unifier.thy "ALL l.~Unifier(l,Const(m),Comb(t,u))"; +by (simp_tac (subst_ss addsimps [Unifier_iff]) 1); +val Unify5 = result(); + +goal Unifier.thy + "(ALL l.~Unifier(l,t,v)) --> (ALL l.~Unifier(l,Comb(t,u),Comb(v,w)))"; +by (simp_tac (subst_ss addsimps [Unifier_iff]) 1); +val Unify6 = result() RS mp; + +goal Unifier.thy "MGIUnifier(s,t,v) --> (ALL l.~Unifier(l,u <| s,w <| s)) --> \ +\ (ALL l.~Unifier(l,Comb(t,u),Comb(v,w)))"; +by (simp_tac (subst_ss addsimps [MGIU_iff]) 1); +by (fast_tac (set_cs addIs [Unifier_comp_subst] addSEs [Unifier_Comb]) 1); +val Unify7 = result() RS mp RS mp; + +val [p1,p2,p3] = goal Unifier.thy + "[| Idem(r); Unifier(s,t <| r,u <| r); \ +\ (! q.Unifier(q,t <| r,u <| r) --> s <> q =s= q) |] ==> \ +\ Idem(r <> s)"; +by (cut_facts_tac [p1, + p2 RS (p1 RS Unifier_Idem_subst RS (p3 RS spec RS mp))] 1); +by (REPEAT_SOME (etac rev_mp)); +by (simp_tac (subst_ss addsimps [raw_Idem_iff,subst_eq_iff,subst_comp]) 1); +val Unify8_lemma1 = result(); + +val [p1,p2,p3,p4] = goal Unifier.thy + "[| Unifier(q,t,v); Unifier(q,u,w); (! q.Unifier(q,t,v) --> r <> q =s= q); \ +\ (! q.Unifier(q,u <| r,w <| r) --> s <> q =s= q) |] ==> \ +\ r <> s <> q =s= q"; +val pp = p1 RS (p3 RS spec RS mp); +by (cut_facts_tac [pp, + p2 RS (pp RS Unifier_comp_subst) RS (p4 RS spec RS mp)] 1); +by (REPEAT_SOME (etac rev_mp)); +by (simp_tac (subst_ss addsimps [subst_eq_iff,subst_comp]) 1); +val Unify8_lemma2 = result(); + +goal Unifier.thy "MGIUnifier(r,t,v) --> MGIUnifier(s,u <| r,w <| r) --> \ +\ MGIUnifier(r <> s,Comb(t,u),Comb(v,w))"; +by (simp_tac (subst_ss addsimps [MGIU_iff,subst_comp,comp_assoc]) 1); +by (safe_tac HOL_cs); +by (REPEAT (etac rev_mp 2)); +by (simp_tac (subst_ss addsimps + [Unifier_iff,MGIU_iff,subst_comp,comp_assoc]) 2); +by (ALLGOALS (fast_tac (set_cs addEs + [Unifier_Comb,Unify8_lemma1,Unify8_lemma2]))); +val Unify8 = result(); + + +(********************** Termination of the algorithm *************************) +(* *) +(*UWFD is a well-founded relation that orders the 2 recursive calls in unify *) +(* NB well-foundedness of UWFD isn't proved *) + + +goalw Unifier.thy [UWFD_def] "UWFD(t,t',Comb(t,u),Comb(t',u'))"; +by (simp_tac subst_ss 1); +by (fast_tac set_cs 1); +val UnifyWFD1 = result(); + +val [prem] = goal Unifier.thy + "MGIUnifier(s,t,t') ==> vars_of(u <| s) Un vars_of(u' <| s) <= \ +\ vars_of(Comb(t,u)) Un vars_of(Comb(t',u'))"; +by (subgoal_tac "vars_of(u <| s) Un vars_of(u' <| s) <= \ +\ srange(s) Un vars_of(u) Un srange(s) Un vars_of(u')" 1); +by (etac subset_trans 1); +by (ALLGOALS (simp_tac (subst_ss addsimps [Var_intro,subset_iff]))); +by (ALLGOALS (fast_tac (set_cs addDs + [Var_intro,prem RS MGIU_srange RS subsetD]))); +val UWFD2_lemma1 = result(); + +val [major,minor] = goal Unifier.thy + "[| MGIUnifier(s,t,t'); ~ u <| s = u |] ==> \ +\ ~ vars_of(u <| s) Un vars_of(u' <| s) = \ +\ (vars_of(t) Un vars_of(u)) Un (vars_of(t') Un vars_of(u'))"; +by (cut_facts_tac + [major RS (MGIU_iff RS iffD1) RS conjunct1 RS (Idem_iff RS iffD1)] 1); +by (rtac (minor RS subst_not_empty RS exE) 1); +by (rtac (make_elim ((major RS MGIU_sdom) RS subsetD)) 1 THEN assume_tac 1); +by (rtac (disjI2 RS (not_equal_iff RS iffD2)) 1); +by (REPEAT (etac rev_mp 1)); +by (asm_simp_tac subst_ss 1); +by (fast_tac (set_cs addIs [Var_elim2]) 1); +val UWFD2_lemma2 = result(); + +val [prem] = goalw Unifier.thy [UWFD_def] + "MGIUnifier(s,t,t') ==> UWFD(u <| s,u' <| s,Comb(t,u),Comb(t',u'))"; +by (cut_facts_tac + [prem RS UWFD2_lemma1 RS (subseteq_iff_subset_eq RS iffD1)] 1); +by (imp_excluded_middle_tac "u <| s = u" 1); +by (simp_tac (set_ss addsimps [occs_Comb2] ) 1); +by (rtac impI 1 THEN etac subst 1 THEN assume_tac 1); +by (rtac impI 1); +by (rtac (conjI RS (ssubset_iff RS iffD2) RS disjI1) 1); +by (asm_simp_tac (set_ss addsimps [subseteq_iff_subset_eq]) 1); +by (asm_simp_tac subst_ss 1); +by (fast_tac (set_cs addDs [prem RS UWFD2_lemma2]) 1); +val UnifyWFD2 = result();