--- a/src/HOL/Nominal/Examples/CR.thy Tue Aug 13 14:20:22 2013 +0200
+++ b/src/HOL/Nominal/Examples/CR.thy Tue Aug 13 16:25:47 2013 +0200
@@ -146,12 +146,12 @@
section {* Beta Reduction *}
inductive
- "Beta" :: "lam\<Rightarrow>lam\<Rightarrow>bool" (" _ \<longrightarrow>\<^isub>\<beta> _" [80,80] 80)
+ "Beta" :: "lam\<Rightarrow>lam\<Rightarrow>bool" (" _ \<longrightarrow>\<^sub>\<beta> _" [80,80] 80)
where
- b1[intro]: "s1\<longrightarrow>\<^isub>\<beta>s2 \<Longrightarrow> (App s1 t)\<longrightarrow>\<^isub>\<beta>(App s2 t)"
- | b2[intro]: "s1\<longrightarrow>\<^isub>\<beta>s2 \<Longrightarrow> (App t s1)\<longrightarrow>\<^isub>\<beta>(App t s2)"
- | b3[intro]: "s1\<longrightarrow>\<^isub>\<beta>s2 \<Longrightarrow> (Lam [a].s1)\<longrightarrow>\<^isub>\<beta> (Lam [a].s2)"
- | b4[intro]: "a\<sharp>s2 \<Longrightarrow> (App (Lam [a].s1) s2)\<longrightarrow>\<^isub>\<beta>(s1[a::=s2])"
+ b1[intro]: "s1\<longrightarrow>\<^sub>\<beta>s2 \<Longrightarrow> (App s1 t)\<longrightarrow>\<^sub>\<beta>(App s2 t)"
+ | b2[intro]: "s1\<longrightarrow>\<^sub>\<beta>s2 \<Longrightarrow> (App t s1)\<longrightarrow>\<^sub>\<beta>(App t s2)"
+ | b3[intro]: "s1\<longrightarrow>\<^sub>\<beta>s2 \<Longrightarrow> (Lam [a].s1)\<longrightarrow>\<^sub>\<beta> (Lam [a].s2)"
+ | b4[intro]: "a\<sharp>s2 \<Longrightarrow> (App (Lam [a].s1) s2)\<longrightarrow>\<^sub>\<beta>(s1[a::=s2])"
equivariance Beta
@@ -159,29 +159,29 @@
by (simp_all add: abs_fresh fresh_fact')
inductive
- "Beta_star" :: "lam\<Rightarrow>lam\<Rightarrow>bool" (" _ \<longrightarrow>\<^isub>\<beta>\<^sup>* _" [80,80] 80)
+ "Beta_star" :: "lam\<Rightarrow>lam\<Rightarrow>bool" (" _ \<longrightarrow>\<^sub>\<beta>\<^sup>* _" [80,80] 80)
where
- bs1[intro, simp]: "M \<longrightarrow>\<^isub>\<beta>\<^sup>* M"
- | bs2[intro]: "\<lbrakk>M1\<longrightarrow>\<^isub>\<beta>\<^sup>* M2; M2 \<longrightarrow>\<^isub>\<beta> M3\<rbrakk> \<Longrightarrow> M1 \<longrightarrow>\<^isub>\<beta>\<^sup>* M3"
+ bs1[intro, simp]: "M \<longrightarrow>\<^sub>\<beta>\<^sup>* M"
+ | bs2[intro]: "\<lbrakk>M1\<longrightarrow>\<^sub>\<beta>\<^sup>* M2; M2 \<longrightarrow>\<^sub>\<beta> M3\<rbrakk> \<Longrightarrow> M1 \<longrightarrow>\<^sub>\<beta>\<^sup>* M3"
equivariance Beta_star
lemma beta_star_trans:
- assumes a1: "M1\<longrightarrow>\<^isub>\<beta>\<^sup>* M2"
- and a2: "M2\<longrightarrow>\<^isub>\<beta>\<^sup>* M3"
- shows "M1 \<longrightarrow>\<^isub>\<beta>\<^sup>* M3"
+ assumes a1: "M1\<longrightarrow>\<^sub>\<beta>\<^sup>* M2"
+ and a2: "M2\<longrightarrow>\<^sub>\<beta>\<^sup>* M3"
+ shows "M1 \<longrightarrow>\<^sub>\<beta>\<^sup>* M3"
using a2 a1
by (induct) (auto)
section {* One-Reduction *}
inductive
- One :: "lam\<Rightarrow>lam\<Rightarrow>bool" (" _ \<longrightarrow>\<^isub>1 _" [80,80] 80)
+ One :: "lam\<Rightarrow>lam\<Rightarrow>bool" (" _ \<longrightarrow>\<^sub>1 _" [80,80] 80)
where
- o1[intro!]: "M\<longrightarrow>\<^isub>1M"
- | o2[simp,intro!]: "\<lbrakk>t1\<longrightarrow>\<^isub>1t2;s1\<longrightarrow>\<^isub>1s2\<rbrakk> \<Longrightarrow> (App t1 s1)\<longrightarrow>\<^isub>1(App t2 s2)"
- | o3[simp,intro!]: "s1\<longrightarrow>\<^isub>1s2 \<Longrightarrow> (Lam [a].s1)\<longrightarrow>\<^isub>1(Lam [a].s2)"
- | o4[simp,intro!]: "\<lbrakk>a\<sharp>(s1,s2); s1\<longrightarrow>\<^isub>1s2;t1\<longrightarrow>\<^isub>1t2\<rbrakk> \<Longrightarrow> (App (Lam [a].t1) s1)\<longrightarrow>\<^isub>1(t2[a::=s2])"
+ o1[intro!]: "M\<longrightarrow>\<^sub>1M"
+ | o2[simp,intro!]: "\<lbrakk>t1\<longrightarrow>\<^sub>1t2;s1\<longrightarrow>\<^sub>1s2\<rbrakk> \<Longrightarrow> (App t1 s1)\<longrightarrow>\<^sub>1(App t2 s2)"
+ | o3[simp,intro!]: "s1\<longrightarrow>\<^sub>1s2 \<Longrightarrow> (Lam [a].s1)\<longrightarrow>\<^sub>1(Lam [a].s2)"
+ | o4[simp,intro!]: "\<lbrakk>a\<sharp>(s1,s2); s1\<longrightarrow>\<^sub>1s2;t1\<longrightarrow>\<^sub>1t2\<rbrakk> \<Longrightarrow> (App (Lam [a].t1) s1)\<longrightarrow>\<^sub>1(t2[a::=s2])"
equivariance One
@@ -189,23 +189,23 @@
by (simp_all add: abs_fresh fresh_fact')
inductive
- "One_star" :: "lam\<Rightarrow>lam\<Rightarrow>bool" (" _ \<longrightarrow>\<^isub>1\<^sup>* _" [80,80] 80)
+ "One_star" :: "lam\<Rightarrow>lam\<Rightarrow>bool" (" _ \<longrightarrow>\<^sub>1\<^sup>* _" [80,80] 80)
where
- os1[intro, simp]: "M \<longrightarrow>\<^isub>1\<^sup>* M"
- | os2[intro]: "\<lbrakk>M1\<longrightarrow>\<^isub>1\<^sup>* M2; M2 \<longrightarrow>\<^isub>1 M3\<rbrakk> \<Longrightarrow> M1 \<longrightarrow>\<^isub>1\<^sup>* M3"
+ os1[intro, simp]: "M \<longrightarrow>\<^sub>1\<^sup>* M"
+ | os2[intro]: "\<lbrakk>M1\<longrightarrow>\<^sub>1\<^sup>* M2; M2 \<longrightarrow>\<^sub>1 M3\<rbrakk> \<Longrightarrow> M1 \<longrightarrow>\<^sub>1\<^sup>* M3"
equivariance One_star
lemma one_star_trans:
- assumes a1: "M1\<longrightarrow>\<^isub>1\<^sup>* M2"
- and a2: "M2\<longrightarrow>\<^isub>1\<^sup>* M3"
- shows "M1\<longrightarrow>\<^isub>1\<^sup>* M3"
+ assumes a1: "M1\<longrightarrow>\<^sub>1\<^sup>* M2"
+ and a2: "M2\<longrightarrow>\<^sub>1\<^sup>* M3"
+ shows "M1\<longrightarrow>\<^sub>1\<^sup>* M3"
using a2 a1
by (induct) (auto)
lemma one_fresh_preserv:
fixes a :: "name"
- assumes a: "t\<longrightarrow>\<^isub>1s"
+ assumes a: "t\<longrightarrow>\<^sub>1s"
and b: "a\<sharp>t"
shows "a\<sharp>s"
using a b
@@ -247,7 +247,7 @@
lemma one_fresh_preserv_automatic:
fixes a :: "name"
- assumes a: "t\<longrightarrow>\<^isub>1s"
+ assumes a: "t\<longrightarrow>\<^sub>1s"
and b: "a\<sharp>t"
shows "a\<sharp>s"
using a b
@@ -263,8 +263,8 @@
(auto simp add: calc_atm fresh_atm abs_fresh)
lemma one_abs:
- assumes a: "Lam [a].t\<longrightarrow>\<^isub>1t'"
- shows "\<exists>t''. t'=Lam [a].t'' \<and> t\<longrightarrow>\<^isub>1t''"
+ assumes a: "Lam [a].t\<longrightarrow>\<^sub>1t'"
+ shows "\<exists>t''. t'=Lam [a].t'' \<and> t\<longrightarrow>\<^sub>1t''"
proof -
have "a\<sharp>Lam [a].t" by (simp add: abs_fresh)
with a have "a\<sharp>t'" by (simp add: one_fresh_preserv)
@@ -274,15 +274,15 @@
qed
lemma one_app:
- assumes a: "App t1 t2 \<longrightarrow>\<^isub>1 t'"
- shows "(\<exists>s1 s2. t' = App s1 s2 \<and> t1 \<longrightarrow>\<^isub>1 s1 \<and> t2 \<longrightarrow>\<^isub>1 s2) \<or>
- (\<exists>a s s1 s2. t1 = Lam [a].s \<and> t' = s1[a::=s2] \<and> s \<longrightarrow>\<^isub>1 s1 \<and> t2 \<longrightarrow>\<^isub>1 s2 \<and> a\<sharp>(t2,s2))"
+ assumes a: "App t1 t2 \<longrightarrow>\<^sub>1 t'"
+ shows "(\<exists>s1 s2. t' = App s1 s2 \<and> t1 \<longrightarrow>\<^sub>1 s1 \<and> t2 \<longrightarrow>\<^sub>1 s2) \<or>
+ (\<exists>a s s1 s2. t1 = Lam [a].s \<and> t' = s1[a::=s2] \<and> s \<longrightarrow>\<^sub>1 s1 \<and> t2 \<longrightarrow>\<^sub>1 s2 \<and> a\<sharp>(t2,s2))"
using a by (erule_tac One.cases) (auto simp add: lam.inject)
lemma one_red:
- assumes a: "App (Lam [a].t1) t2 \<longrightarrow>\<^isub>1 M" "a\<sharp>(t2,M)"
- shows "(\<exists>s1 s2. M = App (Lam [a].s1) s2 \<and> t1 \<longrightarrow>\<^isub>1 s1 \<and> t2 \<longrightarrow>\<^isub>1 s2) \<or>
- (\<exists>s1 s2. M = s1[a::=s2] \<and> t1 \<longrightarrow>\<^isub>1 s1 \<and> t2 \<longrightarrow>\<^isub>1 s2)"
+ assumes a: "App (Lam [a].t1) t2 \<longrightarrow>\<^sub>1 M" "a\<sharp>(t2,M)"
+ shows "(\<exists>s1 s2. M = App (Lam [a].s1) s2 \<and> t1 \<longrightarrow>\<^sub>1 s1 \<and> t2 \<longrightarrow>\<^sub>1 s2) \<or>
+ (\<exists>s1 s2. M = s1[a::=s2] \<and> t1 \<longrightarrow>\<^sub>1 s1 \<and> t2 \<longrightarrow>\<^sub>1 s2)"
using a
by (cases rule: One.strong_cases [where a="a" and aa="a"])
(auto dest: one_abs simp add: lam.inject abs_fresh alpha fresh_prod)
@@ -290,31 +290,31 @@
text {* first case in Lemma 3.2.4*}
lemma one_subst_aux:
- assumes a: "N\<longrightarrow>\<^isub>1N'"
- shows "M[x::=N] \<longrightarrow>\<^isub>1 M[x::=N']"
+ assumes a: "N\<longrightarrow>\<^sub>1N'"
+ shows "M[x::=N] \<longrightarrow>\<^sub>1 M[x::=N']"
using a
proof (nominal_induct M avoiding: x N N' rule: lam.strong_induct)
case (Var y)
- thus "Var y[x::=N] \<longrightarrow>\<^isub>1 Var y[x::=N']" by (cases "x=y") auto
+ thus "Var y[x::=N] \<longrightarrow>\<^sub>1 Var y[x::=N']" by (cases "x=y") auto
next
case (App P Q) (* application case - third line *)
- thus "(App P Q)[x::=N] \<longrightarrow>\<^isub>1 (App P Q)[x::=N']" using o2 by simp
+ thus "(App P Q)[x::=N] \<longrightarrow>\<^sub>1 (App P Q)[x::=N']" using o2 by simp
next
case (Lam y P) (* abstraction case - fourth line *)
- thus "(Lam [y].P)[x::=N] \<longrightarrow>\<^isub>1 (Lam [y].P)[x::=N']" using o3 by simp
+ thus "(Lam [y].P)[x::=N] \<longrightarrow>\<^sub>1 (Lam [y].P)[x::=N']" using o3 by simp
qed
lemma one_subst_aux_automatic:
- assumes a: "N\<longrightarrow>\<^isub>1N'"
- shows "M[x::=N] \<longrightarrow>\<^isub>1 M[x::=N']"
+ assumes a: "N\<longrightarrow>\<^sub>1N'"
+ shows "M[x::=N] \<longrightarrow>\<^sub>1 M[x::=N']"
using a
by (nominal_induct M avoiding: x N N' rule: lam.strong_induct)
(auto simp add: fresh_prod fresh_atm)
lemma one_subst:
- assumes a: "M\<longrightarrow>\<^isub>1M'"
- and b: "N\<longrightarrow>\<^isub>1N'"
- shows "M[x::=N]\<longrightarrow>\<^isub>1M'[x::=N']"
+ assumes a: "M\<longrightarrow>\<^sub>1M'"
+ and b: "N\<longrightarrow>\<^sub>1N'"
+ shows "M[x::=N]\<longrightarrow>\<^sub>1M'[x::=N']"
using a b
proof (nominal_induct M M' avoiding: N N' x rule: One.strong_induct)
case (o1 M)
@@ -328,22 +328,22 @@
next
case (o4 a N1 N2 M1 M2 N N' x)
have vc: "a\<sharp>N" "a\<sharp>N'" "a\<sharp>x" "a\<sharp>N1" "a\<sharp>N2" by fact+
- have asm: "N\<longrightarrow>\<^isub>1N'" by fact
+ have asm: "N\<longrightarrow>\<^sub>1N'" by fact
show ?case
proof -
have "(App (Lam [a].M1) N1)[x::=N] = App (Lam [a].(M1[x::=N])) (N1[x::=N])" using vc by simp
- moreover have "App (Lam [a].(M1[x::=N])) (N1[x::=N]) \<longrightarrow>\<^isub>1 M2[x::=N'][a::=N2[x::=N']]"
+ moreover have "App (Lam [a].(M1[x::=N])) (N1[x::=N]) \<longrightarrow>\<^sub>1 M2[x::=N'][a::=N2[x::=N']]"
using o4 asm by (simp add: fresh_fact)
moreover have "M2[x::=N'][a::=N2[x::=N']] = M2[a::=N2][x::=N']"
using vc by (simp add: substitution_lemma fresh_atm)
- ultimately show "(App (Lam [a].M1) N1)[x::=N] \<longrightarrow>\<^isub>1 M2[a::=N2][x::=N']" by simp
+ ultimately show "(App (Lam [a].M1) N1)[x::=N] \<longrightarrow>\<^sub>1 M2[a::=N2][x::=N']" by simp
qed
qed
lemma one_subst_automatic:
- assumes a: "M\<longrightarrow>\<^isub>1M'"
- and b: "N\<longrightarrow>\<^isub>1N'"
- shows "M[x::=N]\<longrightarrow>\<^isub>1M'[x::=N']"
+ assumes a: "M\<longrightarrow>\<^sub>1M'"
+ and b: "N\<longrightarrow>\<^sub>1N'"
+ shows "M[x::=N]\<longrightarrow>\<^sub>1M'[x::=N']"
using a b
by (nominal_induct M M' avoiding: N N' x rule: One.strong_induct)
(auto simp add: one_subst_aux substitution_lemma fresh_atm fresh_fact)
@@ -351,122 +351,122 @@
lemma diamond[rule_format]:
fixes M :: "lam"
and M1:: "lam"
- assumes a: "M\<longrightarrow>\<^isub>1M1"
- and b: "M\<longrightarrow>\<^isub>1M2"
- shows "\<exists>M3. M1\<longrightarrow>\<^isub>1M3 \<and> M2\<longrightarrow>\<^isub>1M3"
+ assumes a: "M\<longrightarrow>\<^sub>1M1"
+ and b: "M\<longrightarrow>\<^sub>1M2"
+ shows "\<exists>M3. M1\<longrightarrow>\<^sub>1M3 \<and> M2\<longrightarrow>\<^sub>1M3"
using a b
proof (nominal_induct avoiding: M1 M2 rule: One.strong_induct)
case (o1 M) (* case 1 --- M1 = M *)
- thus "\<exists>M3. M\<longrightarrow>\<^isub>1M3 \<and> M2\<longrightarrow>\<^isub>1M3" by blast
+ thus "\<exists>M3. M\<longrightarrow>\<^sub>1M3 \<and> M2\<longrightarrow>\<^sub>1M3" by blast
next
case (o4 x Q Q' P P') (* case 2 --- a beta-reduction occurs*)
have vc: "x\<sharp>Q" "x\<sharp>Q'" "x\<sharp>M2" by fact+
- have i1: "\<And>M2. Q \<longrightarrow>\<^isub>1M2 \<Longrightarrow> (\<exists>M3. Q'\<longrightarrow>\<^isub>1M3 \<and> M2\<longrightarrow>\<^isub>1M3)" by fact
- have i2: "\<And>M2. P \<longrightarrow>\<^isub>1M2 \<Longrightarrow> (\<exists>M3. P'\<longrightarrow>\<^isub>1M3 \<and> M2\<longrightarrow>\<^isub>1M3)" by fact
- have "App (Lam [x].P) Q \<longrightarrow>\<^isub>1 M2" by fact
- hence "(\<exists>P' Q'. M2 = App (Lam [x].P') Q' \<and> P\<longrightarrow>\<^isub>1P' \<and> Q\<longrightarrow>\<^isub>1Q') \<or>
- (\<exists>P' Q'. M2 = P'[x::=Q'] \<and> P\<longrightarrow>\<^isub>1P' \<and> Q\<longrightarrow>\<^isub>1Q')" using vc by (simp add: one_red)
+ have i1: "\<And>M2. Q \<longrightarrow>\<^sub>1M2 \<Longrightarrow> (\<exists>M3. Q'\<longrightarrow>\<^sub>1M3 \<and> M2\<longrightarrow>\<^sub>1M3)" by fact
+ have i2: "\<And>M2. P \<longrightarrow>\<^sub>1M2 \<Longrightarrow> (\<exists>M3. P'\<longrightarrow>\<^sub>1M3 \<and> M2\<longrightarrow>\<^sub>1M3)" by fact
+ have "App (Lam [x].P) Q \<longrightarrow>\<^sub>1 M2" by fact
+ hence "(\<exists>P' Q'. M2 = App (Lam [x].P') Q' \<and> P\<longrightarrow>\<^sub>1P' \<and> Q\<longrightarrow>\<^sub>1Q') \<or>
+ (\<exists>P' Q'. M2 = P'[x::=Q'] \<and> P\<longrightarrow>\<^sub>1P' \<and> Q\<longrightarrow>\<^sub>1Q')" using vc by (simp add: one_red)
moreover (* subcase 2.1 *)
- { assume "\<exists>P' Q'. M2 = App (Lam [x].P') Q' \<and> P\<longrightarrow>\<^isub>1P' \<and> Q\<longrightarrow>\<^isub>1Q'"
+ { assume "\<exists>P' Q'. M2 = App (Lam [x].P') Q' \<and> P\<longrightarrow>\<^sub>1P' \<and> Q\<longrightarrow>\<^sub>1Q'"
then obtain P'' and Q'' where
- b1: "M2=App (Lam [x].P'') Q''" and b2: "P\<longrightarrow>\<^isub>1P''" and b3: "Q\<longrightarrow>\<^isub>1Q''" by blast
- from b2 i2 have "(\<exists>M3. P'\<longrightarrow>\<^isub>1M3 \<and> P''\<longrightarrow>\<^isub>1M3)" by simp
+ b1: "M2=App (Lam [x].P'') Q''" and b2: "P\<longrightarrow>\<^sub>1P''" and b3: "Q\<longrightarrow>\<^sub>1Q''" by blast
+ from b2 i2 have "(\<exists>M3. P'\<longrightarrow>\<^sub>1M3 \<and> P''\<longrightarrow>\<^sub>1M3)" by simp
then obtain P''' where
- c1: "P'\<longrightarrow>\<^isub>1P'''" and c2: "P''\<longrightarrow>\<^isub>1P'''" by force
- from b3 i1 have "(\<exists>M3. Q'\<longrightarrow>\<^isub>1M3 \<and> Q''\<longrightarrow>\<^isub>1M3)" by simp
+ c1: "P'\<longrightarrow>\<^sub>1P'''" and c2: "P''\<longrightarrow>\<^sub>1P'''" by force
+ from b3 i1 have "(\<exists>M3. Q'\<longrightarrow>\<^sub>1M3 \<and> Q''\<longrightarrow>\<^sub>1M3)" by simp
then obtain Q''' where
- d1: "Q'\<longrightarrow>\<^isub>1Q'''" and d2: "Q''\<longrightarrow>\<^isub>1Q'''" by force
+ d1: "Q'\<longrightarrow>\<^sub>1Q'''" and d2: "Q''\<longrightarrow>\<^sub>1Q'''" by force
from c1 c2 d1 d2
- have "P'[x::=Q']\<longrightarrow>\<^isub>1P'''[x::=Q'''] \<and> App (Lam [x].P'') Q'' \<longrightarrow>\<^isub>1 P'''[x::=Q''']"
+ have "P'[x::=Q']\<longrightarrow>\<^sub>1P'''[x::=Q'''] \<and> App (Lam [x].P'') Q'' \<longrightarrow>\<^sub>1 P'''[x::=Q''']"
using vc b3 by (auto simp add: one_subst one_fresh_preserv)
- hence "\<exists>M3. P'[x::=Q']\<longrightarrow>\<^isub>1M3 \<and> M2\<longrightarrow>\<^isub>1M3" using b1 by blast
+ hence "\<exists>M3. P'[x::=Q']\<longrightarrow>\<^sub>1M3 \<and> M2\<longrightarrow>\<^sub>1M3" using b1 by blast
}
moreover (* subcase 2.2 *)
- { assume "\<exists>P' Q'. M2 = P'[x::=Q'] \<and> P\<longrightarrow>\<^isub>1P' \<and> Q\<longrightarrow>\<^isub>1Q'"
+ { assume "\<exists>P' Q'. M2 = P'[x::=Q'] \<and> P\<longrightarrow>\<^sub>1P' \<and> Q\<longrightarrow>\<^sub>1Q'"
then obtain P'' Q'' where
- b1: "M2=P''[x::=Q'']" and b2: "P\<longrightarrow>\<^isub>1P''" and b3: "Q\<longrightarrow>\<^isub>1Q''" by blast
- from b2 i2 have "(\<exists>M3. P'\<longrightarrow>\<^isub>1M3 \<and> P''\<longrightarrow>\<^isub>1M3)" by simp
+ b1: "M2=P''[x::=Q'']" and b2: "P\<longrightarrow>\<^sub>1P''" and b3: "Q\<longrightarrow>\<^sub>1Q''" by blast
+ from b2 i2 have "(\<exists>M3. P'\<longrightarrow>\<^sub>1M3 \<and> P''\<longrightarrow>\<^sub>1M3)" by simp
then obtain P''' where
- c1: "P'\<longrightarrow>\<^isub>1P'''" and c2: "P''\<longrightarrow>\<^isub>1P'''" by blast
- from b3 i1 have "(\<exists>M3. Q'\<longrightarrow>\<^isub>1M3 \<and> Q''\<longrightarrow>\<^isub>1M3)" by simp
+ c1: "P'\<longrightarrow>\<^sub>1P'''" and c2: "P''\<longrightarrow>\<^sub>1P'''" by blast
+ from b3 i1 have "(\<exists>M3. Q'\<longrightarrow>\<^sub>1M3 \<and> Q''\<longrightarrow>\<^sub>1M3)" by simp
then obtain Q''' where
- d1: "Q'\<longrightarrow>\<^isub>1Q'''" and d2: "Q''\<longrightarrow>\<^isub>1Q'''" by blast
+ d1: "Q'\<longrightarrow>\<^sub>1Q'''" and d2: "Q''\<longrightarrow>\<^sub>1Q'''" by blast
from c1 c2 d1 d2
- have "P'[x::=Q']\<longrightarrow>\<^isub>1P'''[x::=Q'''] \<and> P''[x::=Q'']\<longrightarrow>\<^isub>1P'''[x::=Q''']"
+ have "P'[x::=Q']\<longrightarrow>\<^sub>1P'''[x::=Q'''] \<and> P''[x::=Q'']\<longrightarrow>\<^sub>1P'''[x::=Q''']"
by (force simp add: one_subst)
- hence "\<exists>M3. P'[x::=Q']\<longrightarrow>\<^isub>1M3 \<and> M2\<longrightarrow>\<^isub>1M3" using b1 by blast
+ hence "\<exists>M3. P'[x::=Q']\<longrightarrow>\<^sub>1M3 \<and> M2\<longrightarrow>\<^sub>1M3" using b1 by blast
}
- ultimately show "\<exists>M3. P'[x::=Q']\<longrightarrow>\<^isub>1M3 \<and> M2\<longrightarrow>\<^isub>1M3" by blast
+ ultimately show "\<exists>M3. P'[x::=Q']\<longrightarrow>\<^sub>1M3 \<and> M2\<longrightarrow>\<^sub>1M3" by blast
next
case (o2 P P' Q Q') (* case 3 *)
- have i0: "P\<longrightarrow>\<^isub>1P'" by fact
- have i0': "Q\<longrightarrow>\<^isub>1Q'" by fact
- have i1: "\<And>M2. Q \<longrightarrow>\<^isub>1M2 \<Longrightarrow> (\<exists>M3. Q'\<longrightarrow>\<^isub>1M3 \<and> M2\<longrightarrow>\<^isub>1M3)" by fact
- have i2: "\<And>M2. P \<longrightarrow>\<^isub>1M2 \<Longrightarrow> (\<exists>M3. P'\<longrightarrow>\<^isub>1M3 \<and> M2\<longrightarrow>\<^isub>1M3)" by fact
- assume "App P Q \<longrightarrow>\<^isub>1 M2"
- hence "(\<exists>P'' Q''. M2 = App P'' Q'' \<and> P\<longrightarrow>\<^isub>1P'' \<and> Q\<longrightarrow>\<^isub>1Q'') \<or>
- (\<exists>x P' P'' Q'. P = Lam [x].P' \<and> M2 = P''[x::=Q'] \<and> P'\<longrightarrow>\<^isub>1 P'' \<and> Q\<longrightarrow>\<^isub>1Q' \<and> x\<sharp>(Q,Q'))"
+ have i0: "P\<longrightarrow>\<^sub>1P'" by fact
+ have i0': "Q\<longrightarrow>\<^sub>1Q'" by fact
+ have i1: "\<And>M2. Q \<longrightarrow>\<^sub>1M2 \<Longrightarrow> (\<exists>M3. Q'\<longrightarrow>\<^sub>1M3 \<and> M2\<longrightarrow>\<^sub>1M3)" by fact
+ have i2: "\<And>M2. P \<longrightarrow>\<^sub>1M2 \<Longrightarrow> (\<exists>M3. P'\<longrightarrow>\<^sub>1M3 \<and> M2\<longrightarrow>\<^sub>1M3)" by fact
+ assume "App P Q \<longrightarrow>\<^sub>1 M2"
+ hence "(\<exists>P'' Q''. M2 = App P'' Q'' \<and> P\<longrightarrow>\<^sub>1P'' \<and> Q\<longrightarrow>\<^sub>1Q'') \<or>
+ (\<exists>x P' P'' Q'. P = Lam [x].P' \<and> M2 = P''[x::=Q'] \<and> P'\<longrightarrow>\<^sub>1 P'' \<and> Q\<longrightarrow>\<^sub>1Q' \<and> x\<sharp>(Q,Q'))"
by (simp add: one_app[simplified])
moreover (* subcase 3.1 *)
- { assume "\<exists>P'' Q''. M2 = App P'' Q'' \<and> P\<longrightarrow>\<^isub>1P'' \<and> Q\<longrightarrow>\<^isub>1Q''"
+ { assume "\<exists>P'' Q''. M2 = App P'' Q'' \<and> P\<longrightarrow>\<^sub>1P'' \<and> Q\<longrightarrow>\<^sub>1Q''"
then obtain P'' and Q'' where
- b1: "M2=App P'' Q''" and b2: "P\<longrightarrow>\<^isub>1P''" and b3: "Q\<longrightarrow>\<^isub>1Q''" by blast
- from b2 i2 have "(\<exists>M3. P'\<longrightarrow>\<^isub>1M3 \<and> P''\<longrightarrow>\<^isub>1M3)" by simp
+ b1: "M2=App P'' Q''" and b2: "P\<longrightarrow>\<^sub>1P''" and b3: "Q\<longrightarrow>\<^sub>1Q''" by blast
+ from b2 i2 have "(\<exists>M3. P'\<longrightarrow>\<^sub>1M3 \<and> P''\<longrightarrow>\<^sub>1M3)" by simp
then obtain P''' where
- c1: "P'\<longrightarrow>\<^isub>1P'''" and c2: "P''\<longrightarrow>\<^isub>1P'''" by blast
- from b3 i1 have "\<exists>M3. Q'\<longrightarrow>\<^isub>1M3 \<and> Q''\<longrightarrow>\<^isub>1M3" by simp
+ c1: "P'\<longrightarrow>\<^sub>1P'''" and c2: "P''\<longrightarrow>\<^sub>1P'''" by blast
+ from b3 i1 have "\<exists>M3. Q'\<longrightarrow>\<^sub>1M3 \<and> Q''\<longrightarrow>\<^sub>1M3" by simp
then obtain Q''' where
- d1: "Q'\<longrightarrow>\<^isub>1Q'''" and d2: "Q''\<longrightarrow>\<^isub>1Q'''" by blast
+ d1: "Q'\<longrightarrow>\<^sub>1Q'''" and d2: "Q''\<longrightarrow>\<^sub>1Q'''" by blast
from c1 c2 d1 d2
- have "App P' Q'\<longrightarrow>\<^isub>1App P''' Q''' \<and> App P'' Q'' \<longrightarrow>\<^isub>1 App P''' Q'''" by blast
- hence "\<exists>M3. App P' Q'\<longrightarrow>\<^isub>1M3 \<and> M2\<longrightarrow>\<^isub>1M3" using b1 by blast
+ have "App P' Q'\<longrightarrow>\<^sub>1App P''' Q''' \<and> App P'' Q'' \<longrightarrow>\<^sub>1 App P''' Q'''" by blast
+ hence "\<exists>M3. App P' Q'\<longrightarrow>\<^sub>1M3 \<and> M2\<longrightarrow>\<^sub>1M3" using b1 by blast
}
moreover (* subcase 3.2 *)
- { assume "\<exists>x P1 P'' Q''. P = Lam [x].P1 \<and> M2 = P''[x::=Q''] \<and> P1\<longrightarrow>\<^isub>1 P'' \<and> Q\<longrightarrow>\<^isub>1Q'' \<and> x\<sharp>(Q,Q'')"
+ { assume "\<exists>x P1 P'' Q''. P = Lam [x].P1 \<and> M2 = P''[x::=Q''] \<and> P1\<longrightarrow>\<^sub>1 P'' \<and> Q\<longrightarrow>\<^sub>1Q'' \<and> x\<sharp>(Q,Q'')"
then obtain x P1 P1'' Q'' where
b0: "P = Lam [x].P1" and b1: "M2 = P1''[x::=Q'']" and
- b2: "P1\<longrightarrow>\<^isub>1P1''" and b3: "Q\<longrightarrow>\<^isub>1Q''" and vc: "x\<sharp>(Q,Q'')" by blast
- from b0 i0 have "\<exists>P1'. P'=Lam [x].P1' \<and> P1\<longrightarrow>\<^isub>1P1'" by (simp add: one_abs)
- then obtain P1' where g1: "P'=Lam [x].P1'" and g2: "P1\<longrightarrow>\<^isub>1P1'" by blast
- from g1 b0 b2 i2 have "(\<exists>M3. (Lam [x].P1')\<longrightarrow>\<^isub>1M3 \<and> (Lam [x].P1'')\<longrightarrow>\<^isub>1M3)" by simp
+ b2: "P1\<longrightarrow>\<^sub>1P1''" and b3: "Q\<longrightarrow>\<^sub>1Q''" and vc: "x\<sharp>(Q,Q'')" by blast
+ from b0 i0 have "\<exists>P1'. P'=Lam [x].P1' \<and> P1\<longrightarrow>\<^sub>1P1'" by (simp add: one_abs)
+ then obtain P1' where g1: "P'=Lam [x].P1'" and g2: "P1\<longrightarrow>\<^sub>1P1'" by blast
+ from g1 b0 b2 i2 have "(\<exists>M3. (Lam [x].P1')\<longrightarrow>\<^sub>1M3 \<and> (Lam [x].P1'')\<longrightarrow>\<^sub>1M3)" by simp
then obtain P1''' where
- c1: "(Lam [x].P1')\<longrightarrow>\<^isub>1P1'''" and c2: "(Lam [x].P1'')\<longrightarrow>\<^isub>1P1'''" by blast
- from c1 have "\<exists>R1. P1'''=Lam [x].R1 \<and> P1'\<longrightarrow>\<^isub>1R1" by (simp add: one_abs)
- then obtain R1 where r1: "P1'''=Lam [x].R1" and r2: "P1'\<longrightarrow>\<^isub>1R1" by blast
- from c2 have "\<exists>R2. P1'''=Lam [x].R2 \<and> P1''\<longrightarrow>\<^isub>1R2" by (simp add: one_abs)
- then obtain R2 where r3: "P1'''=Lam [x].R2" and r4: "P1''\<longrightarrow>\<^isub>1R2" by blast
+ c1: "(Lam [x].P1')\<longrightarrow>\<^sub>1P1'''" and c2: "(Lam [x].P1'')\<longrightarrow>\<^sub>1P1'''" by blast
+ from c1 have "\<exists>R1. P1'''=Lam [x].R1 \<and> P1'\<longrightarrow>\<^sub>1R1" by (simp add: one_abs)
+ then obtain R1 where r1: "P1'''=Lam [x].R1" and r2: "P1'\<longrightarrow>\<^sub>1R1" by blast
+ from c2 have "\<exists>R2. P1'''=Lam [x].R2 \<and> P1''\<longrightarrow>\<^sub>1R2" by (simp add: one_abs)
+ then obtain R2 where r3: "P1'''=Lam [x].R2" and r4: "P1''\<longrightarrow>\<^sub>1R2" by blast
from r1 r3 have r5: "R1=R2" by (simp add: lam.inject alpha)
- from b3 i1 have "(\<exists>M3. Q'\<longrightarrow>\<^isub>1M3 \<and> Q''\<longrightarrow>\<^isub>1M3)" by simp
+ from b3 i1 have "(\<exists>M3. Q'\<longrightarrow>\<^sub>1M3 \<and> Q''\<longrightarrow>\<^sub>1M3)" by simp
then obtain Q''' where
- d1: "Q'\<longrightarrow>\<^isub>1Q'''" and d2: "Q''\<longrightarrow>\<^isub>1Q'''" by blast
+ d1: "Q'\<longrightarrow>\<^sub>1Q'''" and d2: "Q''\<longrightarrow>\<^sub>1Q'''" by blast
from g1 r2 d1 r4 r5 d2
- have "App P' Q'\<longrightarrow>\<^isub>1R1[x::=Q'''] \<and> P1''[x::=Q'']\<longrightarrow>\<^isub>1R1[x::=Q''']"
+ have "App P' Q'\<longrightarrow>\<^sub>1R1[x::=Q'''] \<and> P1''[x::=Q'']\<longrightarrow>\<^sub>1R1[x::=Q''']"
using vc i0' by (simp add: one_subst one_fresh_preserv)
- hence "\<exists>M3. App P' Q'\<longrightarrow>\<^isub>1M3 \<and> M2\<longrightarrow>\<^isub>1M3" using b1 by blast
+ hence "\<exists>M3. App P' Q'\<longrightarrow>\<^sub>1M3 \<and> M2\<longrightarrow>\<^sub>1M3" using b1 by blast
}
- ultimately show "\<exists>M3. App P' Q'\<longrightarrow>\<^isub>1M3 \<and> M2\<longrightarrow>\<^isub>1M3" by blast
+ ultimately show "\<exists>M3. App P' Q'\<longrightarrow>\<^sub>1M3 \<and> M2\<longrightarrow>\<^sub>1M3" by blast
next
case (o3 P P' x) (* case 4 *)
- have i1: "P\<longrightarrow>\<^isub>1P'" by fact
- have i2: "\<And>M2. P \<longrightarrow>\<^isub>1M2 \<Longrightarrow> (\<exists>M3. P'\<longrightarrow>\<^isub>1M3 \<and> M2\<longrightarrow>\<^isub>1M3)" by fact
- have "(Lam [x].P)\<longrightarrow>\<^isub>1 M2" by fact
- hence "\<exists>P''. M2=Lam [x].P'' \<and> P\<longrightarrow>\<^isub>1P''" by (simp add: one_abs)
- then obtain P'' where b1: "M2=Lam [x].P''" and b2: "P\<longrightarrow>\<^isub>1P''" by blast
- from i2 b1 b2 have "\<exists>M3. (Lam [x].P')\<longrightarrow>\<^isub>1M3 \<and> (Lam [x].P'')\<longrightarrow>\<^isub>1M3" by blast
- then obtain M3 where c1: "(Lam [x].P')\<longrightarrow>\<^isub>1M3" and c2: "(Lam [x].P'')\<longrightarrow>\<^isub>1M3" by blast
- from c1 have "\<exists>R1. M3=Lam [x].R1 \<and> P'\<longrightarrow>\<^isub>1R1" by (simp add: one_abs)
- then obtain R1 where r1: "M3=Lam [x].R1" and r2: "P'\<longrightarrow>\<^isub>1R1" by blast
- from c2 have "\<exists>R2. M3=Lam [x].R2 \<and> P''\<longrightarrow>\<^isub>1R2" by (simp add: one_abs)
- then obtain R2 where r3: "M3=Lam [x].R2" and r4: "P''\<longrightarrow>\<^isub>1R2" by blast
+ have i1: "P\<longrightarrow>\<^sub>1P'" by fact
+ have i2: "\<And>M2. P \<longrightarrow>\<^sub>1M2 \<Longrightarrow> (\<exists>M3. P'\<longrightarrow>\<^sub>1M3 \<and> M2\<longrightarrow>\<^sub>1M3)" by fact
+ have "(Lam [x].P)\<longrightarrow>\<^sub>1 M2" by fact
+ hence "\<exists>P''. M2=Lam [x].P'' \<and> P\<longrightarrow>\<^sub>1P''" by (simp add: one_abs)
+ then obtain P'' where b1: "M2=Lam [x].P''" and b2: "P\<longrightarrow>\<^sub>1P''" by blast
+ from i2 b1 b2 have "\<exists>M3. (Lam [x].P')\<longrightarrow>\<^sub>1M3 \<and> (Lam [x].P'')\<longrightarrow>\<^sub>1M3" by blast
+ then obtain M3 where c1: "(Lam [x].P')\<longrightarrow>\<^sub>1M3" and c2: "(Lam [x].P'')\<longrightarrow>\<^sub>1M3" by blast
+ from c1 have "\<exists>R1. M3=Lam [x].R1 \<and> P'\<longrightarrow>\<^sub>1R1" by (simp add: one_abs)
+ then obtain R1 where r1: "M3=Lam [x].R1" and r2: "P'\<longrightarrow>\<^sub>1R1" by blast
+ from c2 have "\<exists>R2. M3=Lam [x].R2 \<and> P''\<longrightarrow>\<^sub>1R2" by (simp add: one_abs)
+ then obtain R2 where r3: "M3=Lam [x].R2" and r4: "P''\<longrightarrow>\<^sub>1R2" by blast
from r1 r3 have r5: "R1=R2" by (simp add: lam.inject alpha)
- from r2 r4 have "(Lam [x].P')\<longrightarrow>\<^isub>1(Lam [x].R1) \<and> (Lam [x].P'')\<longrightarrow>\<^isub>1(Lam [x].R2)"
+ from r2 r4 have "(Lam [x].P')\<longrightarrow>\<^sub>1(Lam [x].R1) \<and> (Lam [x].P'')\<longrightarrow>\<^sub>1(Lam [x].R2)"
by (simp add: one_subst)
- thus "\<exists>M3. (Lam [x].P')\<longrightarrow>\<^isub>1M3 \<and> M2\<longrightarrow>\<^isub>1M3" using b1 r5 by blast
+ thus "\<exists>M3. (Lam [x].P')\<longrightarrow>\<^sub>1M3 \<and> M2\<longrightarrow>\<^sub>1M3" using b1 r5 by blast
qed
lemma one_lam_cong:
- assumes a: "t1\<longrightarrow>\<^isub>\<beta>\<^sup>*t2"
- shows "(Lam [a].t1)\<longrightarrow>\<^isub>\<beta>\<^sup>*(Lam [a].t2)"
+ assumes a: "t1\<longrightarrow>\<^sub>\<beta>\<^sup>*t2"
+ shows "(Lam [a].t1)\<longrightarrow>\<^sub>\<beta>\<^sup>*(Lam [a].t2)"
using a
proof induct
case bs1 thus ?case by simp
@@ -476,8 +476,8 @@
qed
lemma one_app_congL:
- assumes a: "t1\<longrightarrow>\<^isub>\<beta>\<^sup>*t2"
- shows "App t1 s\<longrightarrow>\<^isub>\<beta>\<^sup>* App t2 s"
+ assumes a: "t1\<longrightarrow>\<^sub>\<beta>\<^sup>*t2"
+ shows "App t1 s\<longrightarrow>\<^sub>\<beta>\<^sup>* App t2 s"
using a
proof induct
case bs1 thus ?case by simp
@@ -486,8 +486,8 @@
qed
lemma one_app_congR:
- assumes a: "t1\<longrightarrow>\<^isub>\<beta>\<^sup>*t2"
- shows "App s t1 \<longrightarrow>\<^isub>\<beta>\<^sup>* App s t2"
+ assumes a: "t1\<longrightarrow>\<^sub>\<beta>\<^sup>*t2"
+ shows "App s t1 \<longrightarrow>\<^sub>\<beta>\<^sup>* App s t2"
using a
proof induct
case bs1 thus ?case by simp
@@ -496,19 +496,19 @@
qed
lemma one_app_cong:
- assumes a1: "t1\<longrightarrow>\<^isub>\<beta>\<^sup>*t2"
- and a2: "s1\<longrightarrow>\<^isub>\<beta>\<^sup>*s2"
- shows "App t1 s1\<longrightarrow>\<^isub>\<beta>\<^sup>* App t2 s2"
+ assumes a1: "t1\<longrightarrow>\<^sub>\<beta>\<^sup>*t2"
+ and a2: "s1\<longrightarrow>\<^sub>\<beta>\<^sup>*s2"
+ shows "App t1 s1\<longrightarrow>\<^sub>\<beta>\<^sup>* App t2 s2"
proof -
- have "App t1 s1 \<longrightarrow>\<^isub>\<beta>\<^sup>* App t2 s1" using a1 by (rule one_app_congL)
+ have "App t1 s1 \<longrightarrow>\<^sub>\<beta>\<^sup>* App t2 s1" using a1 by (rule one_app_congL)
moreover
- have "App t2 s1 \<longrightarrow>\<^isub>\<beta>\<^sup>* App t2 s2" using a2 by (rule one_app_congR)
+ have "App t2 s1 \<longrightarrow>\<^sub>\<beta>\<^sup>* App t2 s2" using a2 by (rule one_app_congR)
ultimately show ?thesis by (rule beta_star_trans)
qed
lemma one_beta_star:
- assumes a: "(t1\<longrightarrow>\<^isub>1t2)"
- shows "(t1\<longrightarrow>\<^isub>\<beta>\<^sup>*t2)"
+ assumes a: "(t1\<longrightarrow>\<^sub>1t2)"
+ shows "(t1\<longrightarrow>\<^sub>\<beta>\<^sup>*t2)"
using a
proof(nominal_induct rule: One.strong_induct)
case o1 thus ?case by simp
@@ -519,16 +519,16 @@
next
case (o4 a s1 s2 t1 t2)
have vc: "a\<sharp>s1" "a\<sharp>s2" by fact+
- have a1: "t1\<longrightarrow>\<^isub>\<beta>\<^sup>*t2" and a2: "s1\<longrightarrow>\<^isub>\<beta>\<^sup>*s2" by fact+
- have c1: "(App (Lam [a].t2) s2) \<longrightarrow>\<^isub>\<beta> (t2 [a::= s2])" using vc by (simp add: b4)
- from a1 a2 have c2: "App (Lam [a].t1 ) s1 \<longrightarrow>\<^isub>\<beta>\<^sup>* App (Lam [a].t2 ) s2"
+ have a1: "t1\<longrightarrow>\<^sub>\<beta>\<^sup>*t2" and a2: "s1\<longrightarrow>\<^sub>\<beta>\<^sup>*s2" by fact+
+ have c1: "(App (Lam [a].t2) s2) \<longrightarrow>\<^sub>\<beta> (t2 [a::= s2])" using vc by (simp add: b4)
+ from a1 a2 have c2: "App (Lam [a].t1 ) s1 \<longrightarrow>\<^sub>\<beta>\<^sup>* App (Lam [a].t2 ) s2"
by (blast intro!: one_app_cong one_lam_cong)
show ?case using c2 c1 by (blast intro: beta_star_trans)
qed
lemma one_star_lam_cong:
- assumes a: "t1\<longrightarrow>\<^isub>1\<^sup>*t2"
- shows "(Lam [a].t1)\<longrightarrow>\<^isub>1\<^sup>* (Lam [a].t2)"
+ assumes a: "t1\<longrightarrow>\<^sub>1\<^sup>*t2"
+ shows "(Lam [a].t1)\<longrightarrow>\<^sub>1\<^sup>* (Lam [a].t2)"
using a
proof induct
case os1 thus ?case by simp
@@ -537,8 +537,8 @@
qed
lemma one_star_app_congL:
- assumes a: "t1\<longrightarrow>\<^isub>1\<^sup>*t2"
- shows "App t1 s\<longrightarrow>\<^isub>1\<^sup>* App t2 s"
+ assumes a: "t1\<longrightarrow>\<^sub>1\<^sup>*t2"
+ shows "App t1 s\<longrightarrow>\<^sub>1\<^sup>* App t2 s"
using a
proof induct
case os1 thus ?case by simp
@@ -547,8 +547,8 @@
qed
lemma one_star_app_congR:
- assumes a: "t1\<longrightarrow>\<^isub>1\<^sup>*t2"
- shows "App s t1 \<longrightarrow>\<^isub>1\<^sup>* App s t2"
+ assumes a: "t1\<longrightarrow>\<^sub>1\<^sup>*t2"
+ shows "App s t1 \<longrightarrow>\<^sub>1\<^sup>* App s t2"
using a
proof induct
case os1 thus ?case by simp
@@ -557,8 +557,8 @@
qed
lemma beta_one_star:
- assumes a: "t1\<longrightarrow>\<^isub>\<beta>t2"
- shows "t1\<longrightarrow>\<^isub>1\<^sup>*t2"
+ assumes a: "t1\<longrightarrow>\<^sub>\<beta>t2"
+ shows "t1\<longrightarrow>\<^sub>1\<^sup>*t2"
using a
proof(induct)
case b1 thus ?case by (blast intro!: one_star_app_congL)
@@ -571,88 +571,88 @@
qed
lemma trans_closure:
- shows "(M1\<longrightarrow>\<^isub>1\<^sup>*M2) = (M1\<longrightarrow>\<^isub>\<beta>\<^sup>*M2)"
+ shows "(M1\<longrightarrow>\<^sub>1\<^sup>*M2) = (M1\<longrightarrow>\<^sub>\<beta>\<^sup>*M2)"
proof
- assume "M1 \<longrightarrow>\<^isub>1\<^sup>* M2"
- then show "M1\<longrightarrow>\<^isub>\<beta>\<^sup>*M2"
+ assume "M1 \<longrightarrow>\<^sub>1\<^sup>* M2"
+ then show "M1\<longrightarrow>\<^sub>\<beta>\<^sup>*M2"
proof induct
- case (os1 M1) thus "M1\<longrightarrow>\<^isub>\<beta>\<^sup>*M1" by simp
+ case (os1 M1) thus "M1\<longrightarrow>\<^sub>\<beta>\<^sup>*M1" by simp
next
case (os2 M1 M2 M3)
- have "M2\<longrightarrow>\<^isub>1M3" by fact
- then have "M2\<longrightarrow>\<^isub>\<beta>\<^sup>*M3" by (rule one_beta_star)
- moreover have "M1\<longrightarrow>\<^isub>\<beta>\<^sup>*M2" by fact
- ultimately show "M1\<longrightarrow>\<^isub>\<beta>\<^sup>*M3" by (auto intro: beta_star_trans)
+ have "M2\<longrightarrow>\<^sub>1M3" by fact
+ then have "M2\<longrightarrow>\<^sub>\<beta>\<^sup>*M3" by (rule one_beta_star)
+ moreover have "M1\<longrightarrow>\<^sub>\<beta>\<^sup>*M2" by fact
+ ultimately show "M1\<longrightarrow>\<^sub>\<beta>\<^sup>*M3" by (auto intro: beta_star_trans)
qed
next
- assume "M1 \<longrightarrow>\<^isub>\<beta>\<^sup>* M2"
- then show "M1\<longrightarrow>\<^isub>1\<^sup>*M2"
+ assume "M1 \<longrightarrow>\<^sub>\<beta>\<^sup>* M2"
+ then show "M1\<longrightarrow>\<^sub>1\<^sup>*M2"
proof induct
- case (bs1 M1) thus "M1\<longrightarrow>\<^isub>1\<^sup>*M1" by simp
+ case (bs1 M1) thus "M1\<longrightarrow>\<^sub>1\<^sup>*M1" by simp
next
case (bs2 M1 M2 M3)
- have "M2\<longrightarrow>\<^isub>\<beta>M3" by fact
- then have "M2\<longrightarrow>\<^isub>1\<^sup>*M3" by (rule beta_one_star)
- moreover have "M1\<longrightarrow>\<^isub>1\<^sup>*M2" by fact
- ultimately show "M1\<longrightarrow>\<^isub>1\<^sup>*M3" by (auto intro: one_star_trans)
+ have "M2\<longrightarrow>\<^sub>\<beta>M3" by fact
+ then have "M2\<longrightarrow>\<^sub>1\<^sup>*M3" by (rule beta_one_star)
+ moreover have "M1\<longrightarrow>\<^sub>1\<^sup>*M2" by fact
+ ultimately show "M1\<longrightarrow>\<^sub>1\<^sup>*M3" by (auto intro: one_star_trans)
qed
qed
lemma cr_one:
- assumes a: "t\<longrightarrow>\<^isub>1\<^sup>*t1"
- and b: "t\<longrightarrow>\<^isub>1t2"
- shows "\<exists>t3. t1\<longrightarrow>\<^isub>1t3 \<and> t2\<longrightarrow>\<^isub>1\<^sup>*t3"
+ assumes a: "t\<longrightarrow>\<^sub>1\<^sup>*t1"
+ and b: "t\<longrightarrow>\<^sub>1t2"
+ shows "\<exists>t3. t1\<longrightarrow>\<^sub>1t3 \<and> t2\<longrightarrow>\<^sub>1\<^sup>*t3"
using a b
proof (induct arbitrary: t2)
case os1 thus ?case by force
next
case (os2 t s1 s2 t2)
- have b: "s1 \<longrightarrow>\<^isub>1 s2" by fact
- have h: "\<And>t2. t \<longrightarrow>\<^isub>1 t2 \<Longrightarrow> (\<exists>t3. s1 \<longrightarrow>\<^isub>1 t3 \<and> t2 \<longrightarrow>\<^isub>1\<^sup>* t3)" by fact
- have c: "t \<longrightarrow>\<^isub>1 t2" by fact
- show "\<exists>t3. s2 \<longrightarrow>\<^isub>1 t3 \<and> t2 \<longrightarrow>\<^isub>1\<^sup>* t3"
+ have b: "s1 \<longrightarrow>\<^sub>1 s2" by fact
+ have h: "\<And>t2. t \<longrightarrow>\<^sub>1 t2 \<Longrightarrow> (\<exists>t3. s1 \<longrightarrow>\<^sub>1 t3 \<and> t2 \<longrightarrow>\<^sub>1\<^sup>* t3)" by fact
+ have c: "t \<longrightarrow>\<^sub>1 t2" by fact
+ show "\<exists>t3. s2 \<longrightarrow>\<^sub>1 t3 \<and> t2 \<longrightarrow>\<^sub>1\<^sup>* t3"
proof -
- from c h have "\<exists>t3. s1 \<longrightarrow>\<^isub>1 t3 \<and> t2 \<longrightarrow>\<^isub>1\<^sup>* t3" by blast
- then obtain t3 where c1: "s1 \<longrightarrow>\<^isub>1 t3" and c2: "t2 \<longrightarrow>\<^isub>1\<^sup>* t3" by blast
- have "\<exists>t4. s2 \<longrightarrow>\<^isub>1 t4 \<and> t3 \<longrightarrow>\<^isub>1 t4" using b c1 by (blast intro: diamond)
+ from c h have "\<exists>t3. s1 \<longrightarrow>\<^sub>1 t3 \<and> t2 \<longrightarrow>\<^sub>1\<^sup>* t3" by blast
+ then obtain t3 where c1: "s1 \<longrightarrow>\<^sub>1 t3" and c2: "t2 \<longrightarrow>\<^sub>1\<^sup>* t3" by blast
+ have "\<exists>t4. s2 \<longrightarrow>\<^sub>1 t4 \<and> t3 \<longrightarrow>\<^sub>1 t4" using b c1 by (blast intro: diamond)
thus ?thesis using c2 by (blast intro: one_star_trans)
qed
qed
lemma cr_one_star:
- assumes a: "t\<longrightarrow>\<^isub>1\<^sup>*t2"
- and b: "t\<longrightarrow>\<^isub>1\<^sup>*t1"
- shows "\<exists>t3. t1\<longrightarrow>\<^isub>1\<^sup>*t3\<and>t2\<longrightarrow>\<^isub>1\<^sup>*t3"
+ assumes a: "t\<longrightarrow>\<^sub>1\<^sup>*t2"
+ and b: "t\<longrightarrow>\<^sub>1\<^sup>*t1"
+ shows "\<exists>t3. t1\<longrightarrow>\<^sub>1\<^sup>*t3\<and>t2\<longrightarrow>\<^sub>1\<^sup>*t3"
using a b
proof (induct arbitrary: t1)
case (os1 t) then show ?case by force
next
case (os2 t s1 s2 t1)
- have c: "t \<longrightarrow>\<^isub>1\<^sup>* s1" by fact
- have c': "t \<longrightarrow>\<^isub>1\<^sup>* t1" by fact
- have d: "s1 \<longrightarrow>\<^isub>1 s2" by fact
- have "t \<longrightarrow>\<^isub>1\<^sup>* t1 \<Longrightarrow> (\<exists>t3. t1 \<longrightarrow>\<^isub>1\<^sup>* t3 \<and> s1 \<longrightarrow>\<^isub>1\<^sup>* t3)" by fact
- then obtain t3 where f1: "t1 \<longrightarrow>\<^isub>1\<^sup>* t3"
- and f2: "s1 \<longrightarrow>\<^isub>1\<^sup>* t3" using c' by blast
- from cr_one d f2 have "\<exists>t4. t3\<longrightarrow>\<^isub>1t4 \<and> s2\<longrightarrow>\<^isub>1\<^sup>*t4" by blast
- then obtain t4 where g1: "t3\<longrightarrow>\<^isub>1t4"
- and g2: "s2\<longrightarrow>\<^isub>1\<^sup>*t4" by blast
- have "t1\<longrightarrow>\<^isub>1\<^sup>*t4" using f1 g1 by (blast intro: one_star_trans)
+ have c: "t \<longrightarrow>\<^sub>1\<^sup>* s1" by fact
+ have c': "t \<longrightarrow>\<^sub>1\<^sup>* t1" by fact
+ have d: "s1 \<longrightarrow>\<^sub>1 s2" by fact
+ have "t \<longrightarrow>\<^sub>1\<^sup>* t1 \<Longrightarrow> (\<exists>t3. t1 \<longrightarrow>\<^sub>1\<^sup>* t3 \<and> s1 \<longrightarrow>\<^sub>1\<^sup>* t3)" by fact
+ then obtain t3 where f1: "t1 \<longrightarrow>\<^sub>1\<^sup>* t3"
+ and f2: "s1 \<longrightarrow>\<^sub>1\<^sup>* t3" using c' by blast
+ from cr_one d f2 have "\<exists>t4. t3\<longrightarrow>\<^sub>1t4 \<and> s2\<longrightarrow>\<^sub>1\<^sup>*t4" by blast
+ then obtain t4 where g1: "t3\<longrightarrow>\<^sub>1t4"
+ and g2: "s2\<longrightarrow>\<^sub>1\<^sup>*t4" by blast
+ have "t1\<longrightarrow>\<^sub>1\<^sup>*t4" using f1 g1 by (blast intro: one_star_trans)
thus ?case using g2 by blast
qed
lemma cr_beta_star:
- assumes a1: "t\<longrightarrow>\<^isub>\<beta>\<^sup>*t1"
- and a2: "t\<longrightarrow>\<^isub>\<beta>\<^sup>*t2"
- shows "\<exists>t3. t1\<longrightarrow>\<^isub>\<beta>\<^sup>*t3\<and>t2\<longrightarrow>\<^isub>\<beta>\<^sup>*t3"
+ assumes a1: "t\<longrightarrow>\<^sub>\<beta>\<^sup>*t1"
+ and a2: "t\<longrightarrow>\<^sub>\<beta>\<^sup>*t2"
+ shows "\<exists>t3. t1\<longrightarrow>\<^sub>\<beta>\<^sup>*t3\<and>t2\<longrightarrow>\<^sub>\<beta>\<^sup>*t3"
proof -
- from a1 have "t\<longrightarrow>\<^isub>1\<^sup>*t1" by (simp only: trans_closure)
+ from a1 have "t\<longrightarrow>\<^sub>1\<^sup>*t1" by (simp only: trans_closure)
moreover
- from a2 have "t\<longrightarrow>\<^isub>1\<^sup>*t2" by (simp only: trans_closure)
- ultimately have "\<exists>t3. t1\<longrightarrow>\<^isub>1\<^sup>*t3 \<and> t2\<longrightarrow>\<^isub>1\<^sup>*t3" by (blast intro: cr_one_star)
- then obtain t3 where "t1\<longrightarrow>\<^isub>1\<^sup>*t3" and "t2\<longrightarrow>\<^isub>1\<^sup>*t3" by blast
- hence "t1\<longrightarrow>\<^isub>\<beta>\<^sup>*t3" and "t2\<longrightarrow>\<^isub>\<beta>\<^sup>*t3" by (simp_all only: trans_closure)
- then show "\<exists>t3. t1\<longrightarrow>\<^isub>\<beta>\<^sup>*t3\<and>t2\<longrightarrow>\<^isub>\<beta>\<^sup>*t3" by blast
+ from a2 have "t\<longrightarrow>\<^sub>1\<^sup>*t2" by (simp only: trans_closure)
+ ultimately have "\<exists>t3. t1\<longrightarrow>\<^sub>1\<^sup>*t3 \<and> t2\<longrightarrow>\<^sub>1\<^sup>*t3" by (blast intro: cr_one_star)
+ then obtain t3 where "t1\<longrightarrow>\<^sub>1\<^sup>*t3" and "t2\<longrightarrow>\<^sub>1\<^sup>*t3" by blast
+ hence "t1\<longrightarrow>\<^sub>\<beta>\<^sup>*t3" and "t2\<longrightarrow>\<^sub>\<beta>\<^sup>*t3" by (simp_all only: trans_closure)
+ then show "\<exists>t3. t1\<longrightarrow>\<^sub>\<beta>\<^sup>*t3\<and>t2\<longrightarrow>\<^sub>\<beta>\<^sup>*t3" by blast
qed
end