(* Authors: Christian Urban and Mathilde Arnaud *)
(* *)
(* A formalisation of the Church-Rosser proof by Masako Takahashi.*)
(* This formalisation follows with some very slight exceptions *)
(* the version of this proof given by Randy Pollack in the paper: *)
(* *)
(* Polishing Up the Tait-Martin Löf Proof of the *)
(* Church-Rosser Theorem (1995). *)
theory CR_Takahashi
imports "HOL-Nominal.Nominal"
begin
atom_decl name
nominal_datatype lam =
Var "name"
| App "lam" "lam"
| Lam "\name\lam" ("Lam [_]._" [100,100] 100)
nominal_primrec
subst :: "lam \ name \ lam \ lam" ("_[_::=_]" [100,100,100] 100)
where
"(Var x)[y::=s] = (if x=y then s else (Var x))"
| "(App t\<^sub>1 t\<^sub>2)[y::=s] = App (t\<^sub>1[y::=s]) (t\<^sub>2[y::=s])"
| "x\(y,s) \ (Lam [x].t)[y::=s] = Lam [x].(t[y::=s])"
apply(finite_guess)+
apply(rule TrueI)+
apply(simp add: abs_fresh)
apply(fresh_guess)+
done
section \<open>Lemmas about Capture-Avoiding Substitution\<close>
lemma subst_eqvt[eqvt]:
fixes pi::"name prm"
shows "pi\(t1[x::=t2]) = (pi\t1)[(pi\x)::=(pi\t2)]"
by (nominal_induct t1 avoiding: x t2 rule: lam.strong_induct)
(auto simp add: perm_bij fresh_atm fresh_bij)
lemma forget:
shows "x\t \ t[x::=s] = t"
by (nominal_induct t avoiding: x s rule: lam.strong_induct)
(auto simp add: abs_fresh fresh_atm)
lemma fresh_fact:
fixes z::"name"
shows "\z\s; (z=y \ z\t)\ \ z\t[y::=s]"
by (nominal_induct t avoiding: z y s rule: lam.strong_induct)
(auto simp add: abs_fresh fresh_prod fresh_atm)
lemma substitution_lemma:
assumes a: "x\y" "x\u"
shows "t[x::=s][y::=u] = t[y::=u][x::=s[y::=u]]"
using a
by (nominal_induct t avoiding: x y s u rule: lam.strong_induct)
(auto simp add: fresh_fact forget)
lemma subst_rename:
assumes a: "y\t"
shows "t[x::=s] = ([(y,x)]\t)[y::=s]"
using a
by (nominal_induct t avoiding: x y s rule: lam.strong_induct)
(auto simp add: swap_simps fresh_atm abs_fresh)
section \<open>Beta-Reduction\<close>
inductive
"Beta" :: "lam\lam\bool" (" _ \\<^sub>\ _" [80,80] 80)
where
b1[intro]: "t1 \\<^sub>\ t2 \ App t1 s \\<^sub>\ App t2 s"
| b2[intro]: "s1 \\<^sub>\ s2 \ App t s1 \\<^sub>\ App t s2"
| b3[intro]: "t1 \\<^sub>\ t2 \ Lam [x].t1 \\<^sub>\ Lam [x].t2"
| b4[intro]: "App (Lam [x].t) s \\<^sub>\ t[x::=s]"
section \<open>Transitive Closure of Beta\<close>
inductive
"Beta_star" :: "lam\lam\bool" (" _ \\<^sub>\\<^sup>* _" [80,80] 80)
where
bs1[intro]: "t \\<^sub>\\<^sup>* t"
| bs2[intro]: "t \\<^sub>\ s \ t \\<^sub>\\<^sup>* s"
| bs3[intro,trans]: "\t1\\<^sub>\\<^sup>* t2; t2 \\<^sub>\\<^sup>* t3\ \ t1 \\<^sub>\\<^sup>* t3"
section \<open>One-Reduction\<close>
inductive
One :: "lam\lam\bool" (" _ \\<^sub>1 _" [80,80] 80)
where
o1[intro]: "Var x\\<^sub>1 Var x"
| o2[intro]: "\t1\\<^sub>1t2; s1\\<^sub>1s2\ \ App t1 s1 \\<^sub>1 App t2 s2"
| o3[intro]: "t1\\<^sub>1t2 \ Lam [x].t1 \\<^sub>1 Lam [x].t2"
| o4[intro]: "\x\(s1,s2); t1\\<^sub>1t2; s1\\<^sub>1s2\ \ App (Lam [x].t1) s1 \\<^sub>1 t2[x::=s2]"
equivariance One
nominal_inductive One
by (simp_all add: abs_fresh fresh_fact)
lemma One_refl:
shows "t \\<^sub>1 t"
by (nominal_induct t rule: lam.strong_induct) (auto)
lemma One_subst:
assumes a: "t1 \\<^sub>1 t2" "s1 \\<^sub>1 s2"
shows "t1[x::=s1] \\<^sub>1 t2[x::=s2]"
using a
by (nominal_induct t1 t2 avoiding: s1 s2 x rule: One.strong_induct)
(auto simp add: substitution_lemma fresh_atm fresh_fact)
lemma better_o4_intro:
assumes a: "t1 \\<^sub>1 t2" "s1 \\<^sub>1 s2"
shows "App (Lam [x].t1) s1 \\<^sub>1 t2[x::=s2]"
proof -
obtain y::"name" where fs: "y\(x,t1,s1,t2,s2)" by (rule exists_fresh, rule fin_supp, blast)
have "App (Lam [x].t1) s1 = App (Lam [y].([(y,x)]\t1)) s1" using fs
by (auto simp add: lam.inject alpha' fresh_prod fresh_atm)
also have "\ \\<^sub>1 ([(y,x)]\t2)[y::=s2]" using fs a by (auto simp add: One.eqvt)
also have "\ = t2[x::=s2]" using fs by (simp add: subst_rename[symmetric])
finally show "App (Lam [x].t1) s1 \\<^sub>1 t2[x::=s2]" by simp
qed
lemma One_Var:
assumes a: "Var x \\<^sub>1 M"
shows "M = Var x"
using a by (cases rule: One.cases) (simp_all)
lemma One_Lam:
assumes a: "Lam [x].t \\<^sub>1 s" "x\s"
shows "\t'. s = Lam [x].t' \ t \\<^sub>1 t'"
using a
by (cases rule: One.strong_cases)
(auto simp add: lam.inject abs_fresh alpha)
lemma One_App:
assumes a: "App t s \\<^sub>1 r"
shows "(\t' s'. r = App t' s' \ t \\<^sub>1 t' \ s \\<^sub>1 s') \
(\<exists>x p p' s'. r = p'[x::=s'] \<and> t = Lam [x].p \<and> p \<longrightarrow>\<^sub>1 p' \<and> s \<longrightarrow>\<^sub>1 s' \<and> x\<sharp>(s,s'))"
using a by (cases rule: One.cases) (auto simp add: lam.inject)
lemma One_Redex:
assumes a: "App (Lam [x].t) s \\<^sub>1 r" "x\(s,r)"
shows "(\t' s'. r = App (Lam [x].t') s' \ t \\<^sub>1 t' \ s \\<^sub>1 s') \
(\<exists>t' s'. r = t'[x::=s'] \<and> t \<longrightarrow>\<^sub>1 t' \<and> s \<longrightarrow>\<^sub>1 s')"
using a
by (cases rule: One.strong_cases)
(auto dest: One_Lam simp add: lam.inject abs_fresh alpha fresh_prod)
section \<open>Transitive Closure of One\<close>
inductive
"One_star" :: "lam\lam\bool" (" _ \\<^sub>1\<^sup>* _" [80,80] 80)
where
os1[intro]: "t \\<^sub>1\<^sup>* t"
| os2[intro]: "t \\<^sub>1 s \ t \\<^sub>1\<^sup>* s"
| os3[intro]: "\t1\\<^sub>1\<^sup>* t2; t2 \\<^sub>1\<^sup>* t3\ \ t1 \\<^sub>1\<^sup>* t3"
section \<open>Complete Development Reduction\<close>
inductive
Dev :: "lam \ lam \ bool" (" _ \\<^sub>d _" [80,80]80)
where
d1[intro]: "Var x \\<^sub>d Var x"
| d2[intro]: "t \\<^sub>d s \ Lam [x].t \\<^sub>d Lam[x].s"
| d3[intro]: "\\(\y t'. t1 = Lam [y].t'); t1 \\<^sub>d t2; s1 \\<^sub>d s2\ \ App t1 s1 \\<^sub>d App t2 s2"
| d4[intro]: "\x\(s1,s2); t1 \\<^sub>d t2; s1 \\<^sub>d s2\ \ App (Lam [x].t1) s1 \\<^sub>d t2[x::=s2]"
equivariance Dev
nominal_inductive Dev
by (simp_all add: abs_fresh fresh_fact)
lemma better_d4_intro:
assumes a: "t1 \\<^sub>d t2" "s1 \\<^sub>d s2"
shows "App (Lam [x].t1) s1 \\<^sub>d t2[x::=s2]"
proof -
obtain y::"name" where fs: "y\(x,t1,s1,t2,s2)" by (rule exists_fresh, rule fin_supp,blast)
have "App (Lam [x].t1) s1 = App (Lam [y].([(y,x)]\t1)) s1" using fs
by (auto simp add: lam.inject alpha' fresh_prod fresh_atm)
also have "\ \\<^sub>d ([(y,x)]\t2)[y::=s2]" using fs a by (auto simp add: Dev.eqvt)
also have "\ = t2[x::=s2]" using fs by (simp add: subst_rename[symmetric])
finally show "App (Lam [x].t1) s1 \\<^sub>d t2[x::=s2]" by simp
qed
lemma Dev_preserves_fresh:
fixes x::"name"
assumes a: "M\\<^sub>d N"
shows "x\M \ x\N"
using a
by (induct) (auto simp add: abs_fresh fresh_fact)
lemma Dev_Lam:
assumes a: "Lam [x].M \\<^sub>d N"
shows "\N'. N = Lam [x].N' \ M \\<^sub>d N'"
proof -
from a have "x\Lam [x].M" by (simp add: abs_fresh)
with a have "x\N" by (simp add: Dev_preserves_fresh)
with a show "\N'. N = Lam [x].N' \ M \\<^sub>d N'"
by (cases rule: Dev.strong_cases)
(auto simp add: lam.inject abs_fresh alpha)
qed
lemma Development_existence:
shows "\M'. M \\<^sub>d M'"
by (nominal_induct M rule: lam.strong_induct)
(auto dest!: Dev_Lam intro: better_d4_intro)
lemma Triangle:
assumes a: "t \\<^sub>d t1" "t \\<^sub>1 t2"
shows "t2 \\<^sub>1 t1"
using a
proof(nominal_induct avoiding: t2 rule: Dev.strong_induct)
case (d4 x s1 s2 t1 t1' t2)
have fc: "x\t2" "x\s1" by fact+
have "App (Lam [x].t1) s1 \\<^sub>1 t2" by fact
then obtain t' s' where reds:
"(t2 = App (Lam [x].t') s' \ t1 \\<^sub>1 t' \ s1 \\<^sub>1 s') \
(t2 = t'[x::=s'] \<and> t1 \<longrightarrow>\<^sub>1 t' \<and> s1 \<longrightarrow>\<^sub>1 s')"
using fc by (auto dest!: One_Redex)
have ih1: "t1 \\<^sub>1 t' \ t' \\<^sub>1 t1'" by fact
have ih2: "s1 \\<^sub>1 s' \ s' \\<^sub>1 s2" by fact
{ assume "t1 \\<^sub>1 t'" "s1 \\<^sub>1 s'"
then have "App (Lam [x].t') s' \\<^sub>1 t1'[x::=s2]"
using ih1 ih2 by (auto intro: better_o4_intro)
}
moreover
{ assume "t1 \\<^sub>1 t'" "s1 \\<^sub>1 s'"
then have "t'[x::=s'] \\<^sub>1 t1'[x::=s2]"
using ih1 ih2 by (auto intro: One_subst)
}
ultimately show "t2 \\<^sub>1 t1'[x::=s2]" using reds by auto
qed (auto dest!: One_Lam One_Var One_App)
lemma Diamond_for_One:
assumes a: "t \\<^sub>1 t1" "t \\<^sub>1 t2"
shows "\t3. t2 \\<^sub>1 t3 \ t1 \\<^sub>1 t3"
proof -
obtain tc where "t \\<^sub>d tc" using Development_existence by blast
with a have "t2 \\<^sub>1 tc" and "t1 \\<^sub>1 tc" by (simp_all add: Triangle)
then show "\t3. t2 \\<^sub>1 t3 \ t1 \\<^sub>1 t3" by blast
qed
lemma Rectangle_for_One:
assumes a: "t \\<^sub>1\<^sup>* t1" "t \\<^sub>1 t2"
shows "\t3. t1 \\<^sub>1 t3 \ t2 \\<^sub>1\<^sup>* t3"
using a Diamond_for_One by (induct arbitrary: t2) (blast)+
lemma CR_for_One_star:
assumes a: "t \\<^sub>1\<^sup>* t1" "t \\<^sub>1\<^sup>* t2"
shows "\t3. t2 \\<^sub>1\<^sup>* t3 \ t1 \\<^sub>1\<^sup>* t3"
using a Rectangle_for_One by (induct arbitrary: t2) (blast)+
section \<open>Establishing the Equivalence of Beta-star and One-star\<close>
lemma Beta_Lam_cong:
assumes a: "t1 \\<^sub>\\<^sup>* t2"
shows "Lam [x].t1 \\<^sub>\\<^sup>* Lam [x].t2"
using a by (induct) (blast)+
lemma Beta_App_cong_aux:
assumes a: "t1 \\<^sub>\\<^sup>* t2"
shows "App t1 s\\<^sub>\\<^sup>* App t2 s"
and "App s t1 \\<^sub>\\<^sup>* App s t2"
using a by (induct) (blast)+
lemma Beta_App_cong:
assumes a: "t1 \\<^sub>\\<^sup>* t2" "s1 \\<^sub>\\<^sup>* s2"
shows "App t1 s1 \\<^sub>\\<^sup>* App t2 s2"
using a by (blast intro: Beta_App_cong_aux)
lemmas Beta_congs = Beta_Lam_cong Beta_App_cong
lemma One_implies_Beta_star:
assumes a: "t \\<^sub>1 s"
shows "t \\<^sub>\\<^sup>* s"
using a by (induct) (auto intro!: Beta_congs)
lemma One_congs:
assumes a: "t1 \\<^sub>1\<^sup>* t2"
shows "Lam [x].t1 \\<^sub>1\<^sup>* Lam [x].t2"
and "App t1 s \\<^sub>1\<^sup>* App t2 s"
and "App s t1 \\<^sub>1\<^sup>* App s t2"
using a by (induct) (auto intro: One_refl)
lemma Beta_implies_One_star:
assumes a: "t1 \\<^sub>\ t2"
shows "t1 \\<^sub>1\<^sup>* t2"
using a by (induct) (auto intro: One_refl One_congs better_o4_intro)
lemma Beta_star_equals_One_star:
shows "t1 \\<^sub>1\<^sup>* t2 = t1 \\<^sub>\\<^sup>* t2"
proof
assume "t1 \\<^sub>1\<^sup>* t2"
then show "t1 \\<^sub>\\<^sup>* t2" by (induct) (auto intro: One_implies_Beta_star)
next
assume "t1 \\<^sub>\\<^sup>* t2"
then show "t1 \\<^sub>1\<^sup>* t2" by (induct) (auto intro: Beta_implies_One_star)
qed
section \<open>The Church-Rosser Theorem\<close>
theorem CR_for_Beta_star:
assumes a: "t \\<^sub>\\<^sup>* t1" "t\\<^sub>\\<^sup>* t2"
shows "\t3. t1 \\<^sub>\\<^sup>* t3 \ t2 \\<^sub>\\<^sup>* t3"
proof -
from a have "t \\<^sub>1\<^sup>* t1" and "t\\<^sub>1\<^sup>* t2" by (simp_all add: Beta_star_equals_One_star)
then have "\t3. t1 \\<^sub>1\<^sup>* t3 \ t2 \\<^sub>1\<^sup>* t3" by (simp add: CR_for_One_star)
then show "\t3. t1 \\<^sub>\\<^sup>* t3 \ t2 \\<^sub>\\<^sup>* t3" by (simp add: Beta_star_equals_One_star)
qed
end
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