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(* 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) == \ Sres"
inductive
domains "Sres" \<subseteq> "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 \<in> 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 \<in> 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 \<open>|>\<close> 70) where
"u |> v == THE w. residuals(u,v,w)"
subsection\<open>Setting up rule lists\<close>
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\<open>residuals is a partial function\<close>
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\<open>Residual function\<close>
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\<open>Commutation theorem\<close>
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) \
(\<forall>n \<in> 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)" for P 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\<open>Residuals are comp and regular\<close>
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\<open>Preservation lemma\<close>
lemma union_preserve_comp: "u \ v ==> v \ (u \ v)"
by (erule Scomp.induct, simp_all)
lemma preservation [rule_format]:
"u \ v ==> regular(v) \ u|>v = (u \ v)|>v"
apply (erule Scomp.induct, safe)
apply (drule_tac [3] psi = "Fun (u) |> v = w" for 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\<open>Prism theorem\<close>
(* Having more assumptions than needed -- removed below *)
lemma prism_l [rule_format]:
"v \ u \
regular(u) \<longrightarrow> (\<forall>w. w \<sim> v \<longrightarrow> w \<sim> u \<longrightarrow>
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\<open>Levy's Cube Lemma\<close>
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\<open>paving theorem\<close>
lemma paving: "[|w \ u; w \ v; regular(u); regular(v)|]==>
\<exists>uv vu. (w|>u) |> vu = (w|>v) |> uv & (w|>u) \<sim> vu \<and>
regular(vu) & (w|>v) \<sim> uv \<and> 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
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