Theory Residuals

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theory Residuals
imports Substitution

(*  Title:      ZF/Resid/Residuals.thy
Author: Ole Rasmussen
Copyright 1995 University of Cambridge
*)


theory Residuals imports Substitution begin

consts
Sres :: "i"


abbreviation
"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


definition
res_func :: "[i,i]=>i" (infixl "|>" 70) where
"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