Theory Residuals

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theory Residuals
imports Substitution
begin

(*  Title:      Residuals.thy
    ID:         $Id: Residuals.thy,v 1.12 2005/06/17 14:15:11 haftmann Exp $
    Author:     Ole Rasmussen
    Copyright   1995  University of Cambridge
    Logic Image: ZF

*)

theory Residuals imports Substitution begin

consts
  Sres          :: "i"
  residuals     :: "[i,i,i]=>i"
  "|>"          :: "[i,i]=>i"     (infixl 70)

translations
  "residuals(u,v,w)"  == "<u,v,w> ∈ Sres"

inductive
  domains       "Sres" <= "redexes*redexes*redexes"
  intros
    Res_Var:    "n ∈ nat ==> residuals(Var(n),Var(n),Var(n))"
    Res_Fun:    "[|residuals(u,v,w)|]==>   
                     residuals(Fun(u),Fun(v),Fun(w))"
    Res_App:    "[|residuals(u1,v1,w1);   
                   residuals(u2,v2,w2); b ∈ bool|]==>   
                 residuals(App(b,u1,u2),App(0,v1,v2),App(b,w1,w2))"
    Res_redex:  "[|residuals(u1,v1,w1);   
                   residuals(u2,v2,w2); b ∈ bool|]==>   
                 residuals(App(b,Fun(u1),u2),App(1,Fun(v1),v2),w2/w1)"
  type_intros    subst_type nat_typechecks redexes.intros bool_typechecks

defs
  res_func_def:  "u |> v == THE w. residuals(u,v,w)"


subsection{*Setting up rule lists*}

declare Sres.intros [intro]
declare Sreg.intros [intro]
declare subst_type [intro]

inductive_cases [elim!]:
  "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)"


inductive_cases [elim!]:
  "Var(n) <== u"
  "Fun(n) <== u"
  "u <== Fun(n)"
  "App(1,Fun(t),a) <== u"
  "App(0,t,a) <== u"

inductive_cases [elim!]:
  "Fun(t) ∈ redexes"

declare Sres.intros [simp]

subsection{*residuals is a  partial function*}

lemma residuals_function [rule_format]:
     "residuals(u,v,w) ==> ∀w1. residuals(u,v,w1) --> w1 = w"
by (erule Sres.induct, force+)

lemma residuals_intro [rule_format]:
     "u~v ==> regular(v) --> (∃w. residuals(u,v,w))"
by (erule Scomp.induct, force+)

lemma comp_resfuncD:
     "[| u~v;  regular(v) |] ==> residuals(u, v, THE w. residuals(u, v, w))"
apply (frule residuals_intro, assumption, clarify)
apply (subst the_equality)
apply (blast intro: residuals_function)+
done

subsection{*Residual function*}

lemma res_Var [simp]: "n ∈ nat ==> Var(n) |> Var(n) = Var(n)"
by (unfold res_func_def, blast)

lemma res_Fun [simp]: 
    "[|s~t; regular(t)|]==> Fun(s) |> Fun(t) = Fun(s |> t)"
apply (unfold res_func_def)
apply (blast intro: comp_resfuncD residuals_function) 
done

lemma res_App [simp]: 
    "[|s~u; regular(u); t~v; regular(v); b ∈ bool|]
     ==> App(b,s,t) |> App(0,u,v) = App(b, s |> u, t |> v)"
apply (unfold res_func_def) 
apply (blast dest!: comp_resfuncD intro: residuals_function)
done

lemma res_redex [simp]: 
    "[|s~u; regular(u); t~v; regular(v); b ∈ bool|]
     ==> App(b,Fun(s),t) |> App(1,Fun(u),v) = (t |> v)/ (s |> u)"
apply (unfold res_func_def)
apply (blast elim!: redexes.free_elims dest!: comp_resfuncD 
             intro: residuals_function)
done

lemma resfunc_type [simp]:
     "[|s~t; regular(t)|]==> regular(t) --> s |> t ∈ redexes"
  by (erule Scomp.induct, auto)

subsection{*Commutation theorem*}

lemma sub_comp [simp]: "u<==v ==> u~v"
by (erule Ssub.induct, simp_all)

lemma sub_preserve_reg [rule_format, simp]:
     "u<==v  ==> regular(v) --> regular(u)"
by (erule Ssub.induct, auto)

lemma residuals_lift_rec: "[|u~v; k ∈ nat|]==> regular(v)--> (∀n ∈ nat.   
         lift_rec(u,n) |> lift_rec(v,n) = lift_rec(u |> v,n))"
apply (erule Scomp.induct, safe)
apply (simp_all add: lift_rec_Var subst_Var lift_subst)
done

lemma residuals_subst_rec:
     "u1~u2 ==>  ∀v1 v2. v1~v2 --> regular(v2) --> regular(u2) --> 
                  (∀n ∈ nat. subst_rec(v1,u1,n) |> subst_rec(v2,u2,n) =  
                    subst_rec(v1 |> v2, u1 |> u2,n))"
apply (erule Scomp.induct, safe)
apply (simp_all add: lift_rec_Var subst_Var residuals_lift_rec)
apply (drule_tac psi = "∀x.?P (x) " in asm_rl)
apply (simp add: substitution)
done


lemma commutation [simp]:
     "[|u1~u2; v1~v2; regular(u2); regular(v2)|]
      ==> (v1/u1) |> (v2/u2) = (v1 |> v2)/(u1 |> u2)"
by (simp add: residuals_subst_rec)


subsection{*Residuals are comp and regular*}

lemma residuals_preserve_comp [rule_format, simp]:
     "u~v ==> ∀w. u~w --> v~w --> regular(w) --> (u|>w) ~ (v|>w)"
by (erule Scomp.induct, force+)

lemma residuals_preserve_reg [rule_format, simp]:
     "u~v ==> regular(u) --> regular(v) --> regular(u|>v)"
apply (erule Scomp.induct, auto)
done

subsection{*Preservation lemma*}

lemma union_preserve_comp: "u~v ==> v ~ (u un v)"
by (erule Scomp.induct, simp_all)

lemma preservation [rule_format]:
     "u ~ v ==> regular(v) --> u|>v = (u un v)|>v"
apply (erule Scomp.induct, safe)
apply (drule_tac [3] psi = "Fun (?u) |> ?v = ?w" in asm_rl)
apply (auto simp add: union_preserve_comp comp_sym_iff)
done


declare sub_comp [THEN comp_sym, simp]

subsection{*Prism theorem*}

(* Having more assumptions than needed -- removed below  *)
lemma prism_l [rule_format]:
     "v<==u ==>  
       regular(u) --> (∀w. w~v --> w~u -->   
                            w |> u = (w|>v) |> (u|>v))"
by (erule Ssub.induct, force+)

lemma prism: "[|v <== u; regular(u); w~v|] ==> w |> u = (w|>v) |> (u|>v)"
apply (rule prism_l)
apply (rule_tac [4] comp_trans, auto)
done


subsection{*Levy's Cube Lemma*}

lemma cube: "[|u~v; regular(v); regular(u); w~u|]==>   
           (w|>u) |> (v|>u) = (w|>v) |> (u|>v)"
apply (subst preservation [of u], assumption, assumption)
apply (subst preservation [of v], erule comp_sym, assumption)
apply (subst prism [symmetric, of v])
apply (simp add: union_r comp_sym_iff)
apply (simp add: union_preserve_regular comp_sym_iff)
apply (erule comp_trans, assumption)
apply (simp add: prism [symmetric] union_l union_preserve_regular 
                 comp_sym_iff union_sym)
done


subsection{*paving theorem*}

lemma paving: "[|w~u; w~v; regular(u); regular(v)|]==>  
           ∃uv vu. (w|>u) |> vu = (w|>v) |> uv & (w|>u)~vu & 
             regular(vu) & (w|>v)~uv & regular(uv) "
apply (subgoal_tac "u~v")
apply (safe intro!: exI)
apply (rule cube)
apply (simp_all add: comp_sym_iff)
apply (blast intro: residuals_preserve_comp comp_trans comp_sym)+
done


end


Setting up rule lists

lemmas

  [| residuals(Var(n), Var(n), v); [| n ∈ nat; v = Var(n) |] ==> Q |] ==> Q
  [| residuals(Fun(t), Fun(u), v);
     !!w. [| residuals(t, u, w); v = Fun(w) |] ==> Q |]
  ==> Q
  [| residuals(App(b, u1.0, u2.0), App(0, v1.0, v2.0), v);
     !!w1 w2.
        [| residuals(u1.0, v1.0, w1); residuals(u2.0, v2.0, w2); b ∈ bool;
           v = App(b, w1, w2) |]
        ==> Q |]
  ==> Q
  [| residuals(App(b, u1.0, u2.0), App(1, Fun(v1.0), v2.0), v);
     !!u1 w1 w2.
        [| residuals(u1, v1.0, w1); residuals(u2.0, v2.0, w2); b ∈ bool;
           v = w2 / w1; u1.0 = Fun(u1) |]
        ==> Q |]
  ==> Q
  [| residuals(Var(n), u, v); [| n ∈ nat; u = Var(n); v = Var(n) |] ==> Q |] ==> Q
  [| residuals(Fun(t), u, v);
     !!v w. [| residuals(t, v, w); u = Fun(v); v = Fun(w) |] ==> Q |]
  ==> Q
  [| residuals(App(b, u1.0, u2.0), w, v);
     !!v1 v2 w1 w2.
        [| residuals(u1.0, v1, w1); residuals(u2.0, v2, w2); b ∈ bool;
           w = App(0, v1, v2); v = App(b, w1, w2) |]
        ==> Q;
     !!u1 v1 v2 w1 w2.
        [| residuals(u1, v1, w1); residuals(u2.0, v2, w2); b ∈ bool;
           w = App(1, Fun(v1), v2); v = w2 / w1; u1.0 = Fun(u1) |]
        ==> Q |]
  ==> Q
  [| residuals(u, Var(n), v); [| n ∈ nat; u = Var(n); v = Var(n) |] ==> Q |] ==> Q
  [| residuals(u, Fun(t), v);
     !!u w. [| residuals(u, t, w); u = Fun(u); v = Fun(w) |] ==> Q |]
  ==> Q
  [| residuals(w, App(b, u1.0, u2.0), v);
     !!b u1 u2 w1 w2.
        [| residuals(u1, u1.0, w1); residuals(u2, u2.0, w2); b ∈ bool;
           w = App(b, u1, u2); v = App(b, w1, w2); b = 0 |]
        ==> Q;
     !!b u1 u2 v1 w1 w2.
        [| residuals(u1, v1, w1); residuals(u2, u2.0, w2); b ∈ bool;
           w = App(b, Fun(u1), u2); v = w2 / w1; b = 1; u1.0 = Fun(v1) |]
        ==> Q |]
  ==> Q

lemmas

  [| Var(n) <== u; [| n ∈ nat; u = Var(n) |] ==> Q |] ==> Q
  [| Fun(n) <== u; !!v. [| n <== v; u = Fun(v) |] ==> Q |] ==> Q
  [| u <== Fun(n); !!u. [| u <== n; u = Fun(u) |] ==> Q |] ==> Q
  [| App(1, Fun(t), a) <== u;
     !!v1 v2. [| Fun(t) <== v1; a <== v2; u = App(1, v1, v2) |] ==> Q |]
  ==> Q
  [| App(0, t, a) <== u;
     !!b v1 v2. [| t <== v1; a <== v2; b ∈ bool; u = App(b, v1, v2) |] ==> Q |]
  ==> Q

lemmas

  [| Fun(t) ∈ redexes; t ∈ redexes ==> Q |] ==> Q

residuals is a partial function

lemma residuals_function:

  [| residuals(u, v, w); residuals(u, v, w1.0) |] ==> w1.0 = w

lemma residuals_intro:

  [| u ~ v; regular(v) |] ==> ∃w. residuals(u, v, w)

lemma comp_resfuncD:

  [| u ~ v; regular(v) |] ==> residuals(u, v, THE w. residuals(u, v, w))

Residual function

lemma res_Var:

  n ∈ nat ==> Var(n) |> Var(n) = Var(n)

lemma res_Fun:

  [| s ~ t; regular(t) |] ==> Fun(s) |> Fun(t) = Fun(s |> t)

lemma res_App:

  [| s ~ u; regular(u); t ~ v; regular(v); b ∈ bool |]
  ==> App(b, s, t) |> App(0, u, v) = App(b, s |> u, t |> v)

lemma res_redex:

  [| s ~ u; regular(u); t ~ v; regular(v); b ∈ bool |]
  ==> App(b, Fun(s), t) |> App(1, Fun(u), v) = t |> v / (s |> u)

lemma resfunc_type:

  [| s ~ t; regular(t) |] ==> regular(t) --> s |> t ∈ redexes

Commutation theorem

lemma sub_comp:

  u <== v ==> u ~ v

lemma sub_preserve_reg:

  [| u <== v; regular(v) |] ==> regular(u)

lemma residuals_lift_rec:

  [| u ~ v; k ∈ nat |]
  ==> regular(v) -->
      (∀n∈nat. lift_rec(u, n) |> lift_rec(v, n) = lift_rec(u |> v, n))

lemma residuals_subst_rec:

  u1.0 ~ u2.0
  ==> ∀v1 v2.
         v1 ~ v2 -->
         regular(v2) -->
         regular(u2.0) -->
         (∀n∈nat.
             subst_rec(v1, u1.0, n) |> subst_rec(v2, u2.0, n) =
             subst_rec(v1 |> v2, u1.0 |> u2.0, n))

lemma commutation:

  [| u1.0 ~ u2.0; v1.0 ~ v2.0; regular(u2.0); regular(v2.0) |]
  ==> v1.0 / u1.0 |> (v2.0 / u2.0) = v1.0 |> v2.0 / (u1.0 |> u2.0)

Residuals are comp and regular

lemma residuals_preserve_comp:

  [| u ~ v; u ~ w; v ~ w; regular(w) |] ==> u |> w ~ (v |> w)

lemma residuals_preserve_reg:

  [| u ~ v; regular(u); regular(v) |] ==> regular(u |> v)

Preservation lemma

lemma union_preserve_comp:

  u ~ v ==> v ~ (u un v)

lemma preservation:

  [| u ~ v; regular(v) |] ==> u |> v = u un v |> v

Prism theorem

lemma prism_l:

  [| v <== u; regular(u); w ~ v; w ~ u |] ==> w |> u = w |> v |> (u |> v)

lemma prism:

  [| v <== u; regular(u); w ~ v |] ==> w |> u = w |> v |> (u |> v)

Levy's Cube Lemma

lemma cube:

  [| u ~ v; regular(v); regular(u); w ~ u |]
  ==> w |> u |> (v |> u) = w |> v |> (u |> v)

paving theorem

lemma paving:

  [| w ~ u; w ~ v; regular(u); regular(v) |]
  ==> ∃uv vu.
         w |> u |> vu = w |> v |> uvw |> u ~ vu ∧ regular(vu) ∧ w |> v ~ uv ∧ regular(uv)