--- a/src/HOL/Nominal/Examples/Crary.thy Fri Oct 19 23:21:06 2007 +0200
+++ b/src/HOL/Nominal/Examples/Crary.thy Fri Oct 19 23:21:08 2007 +0200
@@ -168,7 +168,7 @@
using assms
proof (nominal_induct t avoiding: x u \<theta> rule: trm.induct)
case (Lam y t x u)
- have fs: "y\<sharp>\<theta>" "y\<sharp>x" "y\<sharp>u" by fact
+ have fs: "y\<sharp>\<theta>" "y\<sharp>x" "y\<sharp>u" by fact+
moreover have "x\<sharp> Lam [y].t" by fact
ultimately have "x\<sharp>t" by (simp add: abs_fresh fresh_atm)
moreover have ih:"\<And>n T. n\<sharp>t \<Longrightarrow> ((n,T)#\<theta>)<t> = \<theta><t>" by fact
@@ -244,7 +244,7 @@
ultimately show ?case by auto
next
case (Lam n t x u \<theta>)
- have fs:"n\<sharp>x" "n\<sharp>u" "n\<sharp>\<theta>" "x\<sharp>\<theta>" by fact
+ have fs:"n\<sharp>x" "n\<sharp>u" "n\<sharp>\<theta>" "x\<sharp>\<theta>" by fact+
have ih:"\<And> y s \<theta>. y\<sharp>\<theta> \<Longrightarrow> ((\<theta><(t[y::=s])>) = ((\<theta><t>)[y::=(\<theta><s>)]))" by fact
have "\<theta> <(Lam [n].t)[x::=u]> = \<theta><Lam [n]. (t [x::=u])>" using fs by auto
then have "\<theta> <(Lam [n].t)[x::=u]> = Lam [n]. \<theta><t [x::=u]>" using fs by auto
@@ -562,7 +562,7 @@
using assms
proof (induct arbitrary: b)
case (QAN_Reduce x t a b)
- have h:"x \<leadsto> t" "t \<Down> a" by fact
+ have h:"x \<leadsto> t" "t \<Down> a" by fact+
have ih:"\<And>b. t \<Down> b \<Longrightarrow> a = b" by fact
have "x \<Down> b" by fact
then obtain t' where "x \<leadsto> t'" and hl:"t' \<Down> b" using h by auto
@@ -659,14 +659,14 @@
case (QAP_Var \<Gamma> x T u T')
have "\<Gamma> \<turnstile> Var x \<leftrightarrow> u : T'" by fact
then have "u=Var x" and "(x,T') \<in> set \<Gamma>" by auto
- moreover have "valid \<Gamma>" "(x,T) \<in> set \<Gamma>" by fact
+ moreover have "valid \<Gamma>" "(x,T) \<in> set \<Gamma>" by fact+
ultimately show "T=T'" using type_unicity_in_context by auto
next
case (QAP_App \<Gamma> p q T\<^isub>1 T\<^isub>2 s t u T\<^isub>2')
have ih:"\<And>u T. \<Gamma> \<turnstile> p \<leftrightarrow> u : T \<Longrightarrow> T\<^isub>1\<rightarrow>T\<^isub>2 = T" by fact
have "\<Gamma> \<turnstile> App p s \<leftrightarrow> u : T\<^isub>2'" by fact
then obtain r t T\<^isub>1' where "u = App r t" "\<Gamma> \<turnstile> p \<leftrightarrow> r : T\<^isub>1' \<rightarrow> T\<^isub>2'" by auto
- then have "T\<^isub>1\<rightarrow>T\<^isub>2 = T\<^isub>1' \<rightarrow> T\<^isub>2'" by auto
+ with ih have "T\<^isub>1\<rightarrow>T\<^isub>2 = T\<^isub>1' \<rightarrow> T\<^isub>2'" by auto
then show "T\<^isub>2=T\<^isub>2'" using ty.inject by auto
qed (auto)
@@ -699,7 +699,7 @@
case (QAT_Arrow x \<Gamma> s t T\<^isub>1 T\<^isub>2 u)
have ih:"(x,T\<^isub>1)#\<Gamma> \<turnstile> App t (Var x) \<Leftrightarrow> App u (Var x) : T\<^isub>2
\<Longrightarrow> (x,T\<^isub>1)#\<Gamma> \<turnstile> App s (Var x) \<Leftrightarrow> App u (Var x) : T\<^isub>2" by fact
- have fs: "x\<sharp>\<Gamma>" "x\<sharp>s" "x\<sharp>t" "x\<sharp>u" by fact
+ have fs: "x\<sharp>\<Gamma>" "x\<sharp>s" "x\<sharp>t" "x\<sharp>u" by fact+
have "\<Gamma> \<turnstile> t \<Leftrightarrow> u : T\<^isub>1\<rightarrow>T\<^isub>2" by fact
then have "(x,T\<^isub>1)#\<Gamma> \<turnstile> App t (Var x) \<Leftrightarrow> App u (Var x) : T\<^isub>2" using fs
by (simp add: Q_Arrow_strong_inversion)
@@ -732,7 +732,7 @@
and "\<Gamma> \<turnstile> s \<leftrightarrow> t : T \<Longrightarrow> \<Gamma> \<subseteq> \<Gamma>' \<Longrightarrow> valid \<Gamma>' \<Longrightarrow> \<Gamma>' \<turnstile> s \<leftrightarrow> t : T"
proof (nominal_induct \<Gamma> s t T and \<Gamma> s t T avoiding: \<Gamma>' rule: alg_equiv_alg_path_equiv.strong_inducts)
case (QAT_Arrow x \<Gamma> s t T\<^isub>1 T\<^isub>2 \<Gamma>')
- have fs:"x\<sharp>\<Gamma>" "x\<sharp>s" "x\<sharp>t" "x\<sharp>\<Gamma>'"by fact
+ have fs:"x\<sharp>\<Gamma>" "x\<sharp>s" "x\<sharp>t" "x\<sharp>\<Gamma>'" by fact+
have h2:"\<Gamma> \<subseteq> \<Gamma>'" by fact
have ih:"\<And>\<Gamma>'. \<lbrakk>(x,T\<^isub>1)#\<Gamma> \<subseteq> \<Gamma>'; valid \<Gamma>'\<rbrakk> \<Longrightarrow> \<Gamma>' \<turnstile> App s (Var x) \<Leftrightarrow> App t (Var x) : T\<^isub>2" by fact
have "valid \<Gamma>'" by fact
@@ -782,7 +782,7 @@
case (3 \<Gamma> s t T\<^isub>1 T\<^isub>2 \<Gamma>')
have "\<Gamma> \<turnstile> s is t : T\<^isub>1\<rightarrow>T\<^isub>2"
and "\<Gamma> \<subseteq> \<Gamma>'"
- and "valid \<Gamma>'" by fact
+ and "valid \<Gamma>'" by fact+
then show "\<Gamma>' \<turnstile> s is t : T\<^isub>1\<rightarrow>T\<^isub>2" by simp
qed (auto)
@@ -939,7 +939,7 @@
moreover
{
assume "(y,U) \<in> set [(x,T)]"
- then have "\<Gamma>' \<turnstile> ((x,s)#\<theta>)<Var y> is ((x,t)#\<theta>')<Var y> : U" by auto
+ with h2 have "\<Gamma>' \<turnstile> ((x,s)#\<theta>)<Var y> is ((x,t)#\<theta>')<Var y> : U" by auto
}
moreover
{
@@ -964,7 +964,7 @@
using a1 a2 a3
proof (nominal_induct \<Gamma> t T avoiding: \<theta> \<theta>' arbitrary: \<Gamma>' rule: typing.strong_induct)
case (T_Lam x \<Gamma> T\<^isub>1 t\<^isub>2 T\<^isub>2 \<theta> \<theta>' \<Gamma>')
- have vc: "x\<sharp>\<theta>" "x\<sharp>\<theta>'" "x\<sharp>\<Gamma>" by fact
+ have vc: "x\<sharp>\<theta>" "x\<sharp>\<theta>'" "x\<sharp>\<Gamma>" by fact+
have asm1: "\<Gamma>' \<turnstile> \<theta> is \<theta>' over \<Gamma>" by fact
have ih:"\<And>\<theta> \<theta>' \<Gamma>'. \<lbrakk>\<Gamma>' \<turnstile> \<theta> is \<theta>' over (x,T\<^isub>1)#\<Gamma>; valid \<Gamma>'\<rbrakk> \<Longrightarrow> \<Gamma>' \<turnstile> \<theta><t\<^isub>2> is \<theta>'<t\<^isub>2> : T\<^isub>2" by fact
show "\<Gamma>' \<turnstile> \<theta><Lam [x].t\<^isub>2> is \<theta>'<Lam [x].t\<^isub>2> : T\<^isub>1\<rightarrow>T\<^isub>2" using vc
@@ -996,14 +996,14 @@
proof (nominal_induct \<Gamma> s t T avoiding: \<Gamma>' \<theta> \<theta>' rule: def_equiv.strong_induct)
case (Q_Refl \<Gamma> t T \<Gamma>' \<theta> \<theta>')
have "\<Gamma> \<turnstile> t : T"
- and "valid \<Gamma>'" by fact
+ and "valid \<Gamma>'" by fact+
moreover
have "\<Gamma>' \<turnstile> \<theta> is \<theta>' over \<Gamma>" by fact
ultimately show "\<Gamma>' \<turnstile> \<theta><t> is \<theta>'<t> : T" using fundamental_theorem_1 by blast
next
case (Q_Symm \<Gamma> t s T \<Gamma>' \<theta> \<theta>')
have "\<Gamma>' \<turnstile> \<theta> is \<theta>' over \<Gamma>"
- and "valid \<Gamma>'" by fact
+ and "valid \<Gamma>'" by fact+
moreover
have ih: "\<And> \<Gamma>' \<theta> \<theta>'. \<lbrakk>\<Gamma>' \<turnstile> \<theta> is \<theta>' over \<Gamma>; valid \<Gamma>'\<rbrakk> \<Longrightarrow> \<Gamma>' \<turnstile> \<theta><t> is \<theta>'<s> : T" by fact
ultimately show "\<Gamma>' \<turnstile> \<theta><s> is \<theta>'<t> : T" using logical_symmetry by blast
@@ -1012,7 +1012,7 @@
have ih1: "\<And> \<Gamma>' \<theta> \<theta>'. \<lbrakk>\<Gamma>' \<turnstile> \<theta> is \<theta>' over \<Gamma>; valid \<Gamma>'\<rbrakk> \<Longrightarrow> \<Gamma>' \<turnstile> \<theta><s> is \<theta>'<t> : T" by fact
have ih2: "\<And> \<Gamma>' \<theta> \<theta>'. \<lbrakk>\<Gamma>' \<turnstile> \<theta> is \<theta>' over \<Gamma>; valid \<Gamma>'\<rbrakk> \<Longrightarrow> \<Gamma>' \<turnstile> \<theta><t> is \<theta>'<u> : T" by fact
have h: "\<Gamma>' \<turnstile> \<theta> is \<theta>' over \<Gamma>"
- and v: "valid \<Gamma>'" by fact
+ and v: "valid \<Gamma>'" by fact+
then have "\<Gamma>' \<turnstile> \<theta>' is \<theta>' over \<Gamma>" using logical_pseudo_reflexivity by auto
then have "\<Gamma>' \<turnstile> \<theta>'<t> is \<theta>'<u> : T" using ih2 v by auto
moreover have "\<Gamma>' \<turnstile> \<theta><s> is \<theta>'<t> : T" using ih1 h v by auto
@@ -1020,9 +1020,9 @@
next
case (Q_Abs x \<Gamma> T\<^isub>1 s\<^isub>2 t\<^isub>2 T\<^isub>2 \<Gamma>' \<theta> \<theta>')
have fs:"x\<sharp>\<Gamma>" by fact
- have fs2: "x\<sharp>\<theta>" "x\<sharp>\<theta>'" by fact
+ have fs2: "x\<sharp>\<theta>" "x\<sharp>\<theta>'" by fact+
have h2: "\<Gamma>' \<turnstile> \<theta> is \<theta>' over \<Gamma>"
- and h3: "valid \<Gamma>'" by fact
+ and h3: "valid \<Gamma>'" by fact+
have ih:"\<And>\<Gamma>' \<theta> \<theta>'. \<lbrakk>\<Gamma>' \<turnstile> \<theta> is \<theta>' over (x,T\<^isub>1)#\<Gamma>; valid \<Gamma>'\<rbrakk> \<Longrightarrow> \<Gamma>' \<turnstile> \<theta><s\<^isub>2> is \<theta>'<t\<^isub>2> : T\<^isub>2" by fact
{
fix \<Gamma>'' s' t'
@@ -1036,7 +1036,7 @@
ultimately have "\<Gamma>'' \<turnstile> App (Lam [x]. \<theta><s\<^isub>2>) s' is App (Lam [x].\<theta>'<t\<^isub>2>) t' : T\<^isub>2"
using logical_weak_head_closure by auto
}
- moreover have "valid \<Gamma>'" using h2 by auto
+ moreover have "valid \<Gamma>'" by fact
ultimately have "\<Gamma>' \<turnstile> Lam [x].\<theta><s\<^isub>2> is Lam [x].\<theta>'<t\<^isub>2> : T\<^isub>1\<rightarrow>T\<^isub>2" by auto
then show "\<Gamma>' \<turnstile> \<theta><Lam [x].s\<^isub>2> is \<theta>'<Lam [x].t\<^isub>2> : T\<^isub>1\<rightarrow>T\<^isub>2" using fs2 by auto
next
@@ -1045,9 +1045,9 @@
next
case (Q_Beta x \<Gamma> s\<^isub>2 t\<^isub>2 T\<^isub>1 s12 t12 T\<^isub>2 \<Gamma>' \<theta> \<theta>')
have h: "\<Gamma>' \<turnstile> \<theta> is \<theta>' over \<Gamma>"
- and h': "valid \<Gamma>'" by fact
+ and h': "valid \<Gamma>'" by fact+
have fs: "x\<sharp>\<Gamma>" by fact
- have fs2: " x\<sharp>\<theta>" "x\<sharp>\<theta>'" by fact
+ have fs2: " x\<sharp>\<theta>" "x\<sharp>\<theta>'" by fact+
have ih1: "\<And>\<Gamma>' \<theta> \<theta>'. \<lbrakk>\<Gamma>' \<turnstile> \<theta> is \<theta>' over \<Gamma>; valid \<Gamma>'\<rbrakk> \<Longrightarrow> \<Gamma>' \<turnstile> \<theta><s\<^isub>2> is \<theta>'<t\<^isub>2> : T\<^isub>1" by fact
have ih2: "\<And>\<Gamma>' \<theta> \<theta>'. \<lbrakk>\<Gamma>' \<turnstile> \<theta> is \<theta>' over (x,T\<^isub>1)#\<Gamma>; valid \<Gamma>'\<rbrakk> \<Longrightarrow> \<Gamma>' \<turnstile> \<theta><s12> is \<theta>'<t12> : T\<^isub>2" by fact
have "\<Gamma>' \<turnstile> \<theta><s\<^isub>2> is \<theta>'<t\<^isub>2> : T\<^isub>1" using ih1 h' h by auto
@@ -1062,8 +1062,8 @@
next
case (Q_Ext x \<Gamma> s t T\<^isub>1 T\<^isub>2 \<Gamma>' \<theta> \<theta>')
have h2: "\<Gamma>' \<turnstile> \<theta> is \<theta>' over \<Gamma>"
- and h2': "valid \<Gamma>'" by fact
- have fs:"x\<sharp>\<Gamma>" "x\<sharp>s" "x\<sharp>t" by fact
+ and h2': "valid \<Gamma>'" by fact+
+ have fs:"x\<sharp>\<Gamma>" "x\<sharp>s" "x\<sharp>t" by fact+
have ih:"\<And>\<Gamma>' \<theta> \<theta>'. \<lbrakk>\<Gamma>' \<turnstile> \<theta> is \<theta>' over (x,T\<^isub>1)#\<Gamma>; valid \<Gamma>'\<rbrakk>
\<Longrightarrow> \<Gamma>' \<turnstile> \<theta><App s (Var x)> is \<theta>'<App t (Var x)> : T\<^isub>2" by fact
{
@@ -1078,7 +1078,7 @@
then have "\<Gamma>'' \<turnstile> App ((x,s')#\<theta>)<s> s' is App ((x,t')#\<theta>')<t> t' : T\<^isub>2" by auto
then have "\<Gamma>'' \<turnstile> App (\<theta><s>) s' is App (\<theta>'<t>) t' : T\<^isub>2" using fs fresh_psubst_simp by auto
}
- moreover have "valid \<Gamma>'" using h2 by auto
+ moreover have "valid \<Gamma>'" by fact
ultimately show "\<Gamma>' \<turnstile> \<theta><s> is \<theta>'<t> : T\<^isub>1\<rightarrow>T\<^isub>2" by auto
next
case (Q_Unit \<Gamma> s t \<Gamma>' \<theta> \<theta>')