author | wenzelm |
Sat, 07 Apr 2012 16:41:59 +0200 | |
changeset 47389 | e8552cba702d |
parent 42463 | f270e3e18be5 |
child 47994 | d7c0aa802f0d |
permissions | -rw-r--r-- |
42150 | 1 |
(* Title: HOL/MicroJava/BV/JVMType.thy |
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Author: Gerwin Klein |
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Copyright 2000 TUM |
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*) |
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|
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header {* \isaheader{The JVM Type System as Semilattice} *} |
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theory JVMType |
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imports JType |
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begin |
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type_synonym locvars_type = "ty err list" |
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type_synonym opstack_type = "ty list" |
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type_synonym state_type = "opstack_type \<times> locvars_type" |
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type_synonym state = "state_type option err" -- "for Kildall" |
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type_synonym method_type = "state_type option list" -- "for BVSpec" |
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type_synonym class_type = "sig \<Rightarrow> method_type" |
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type_synonym prog_type = "cname \<Rightarrow> class_type" |
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|
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definition stk_esl :: "'c prog \<Rightarrow> nat \<Rightarrow> ty list esl" where |
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"stk_esl S maxs == upto_esl maxs (JType.esl S)" |
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|
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definition reg_sl :: "'c prog \<Rightarrow> nat \<Rightarrow> ty err list sl" where |
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"reg_sl S maxr == Listn.sl maxr (Err.sl (JType.esl S))" |
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|
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definition sl :: "'c prog \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> state sl" where |
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"sl S maxs maxr == |
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Err.sl(Opt.esl(Product.esl (stk_esl S maxs) (Err.esl(reg_sl S maxr))))" |
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|
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definition states :: "'c prog \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> state set" where |
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"states S maxs maxr == fst(sl S maxs maxr)" |
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definition le :: "'c prog \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> state ord" where |
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"le S maxs maxr == fst(snd(sl S maxs maxr))" |
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definition sup :: "'c prog \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> state binop" where |
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"sup S maxs maxr == snd(snd(sl S maxs maxr))" |
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definition sup_ty_opt :: "['code prog,ty err,ty err] \<Rightarrow> bool" |
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("_ |- _ <=o _" [71,71] 70) where |
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"sup_ty_opt G == Err.le (subtype G)" |
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definition sup_loc :: "['code prog,locvars_type,locvars_type] \<Rightarrow> bool" |
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("_ |- _ <=l _" [71,71] 70) where |
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"sup_loc G == Listn.le (sup_ty_opt G)" |
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definition sup_state :: "['code prog,state_type,state_type] \<Rightarrow> bool" |
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("_ |- _ <=s _" [71,71] 70) where |
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"sup_state G == Product.le (Listn.le (subtype G)) (sup_loc G)" |
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definition sup_state_opt :: "['code prog,state_type option,state_type option] \<Rightarrow> bool" |
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("_ |- _ <=' _" [71,71] 70) where |
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"sup_state_opt G == Opt.le (sup_state G)" |
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|
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|
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notation (xsymbols) |
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sup_ty_opt ("_ \<turnstile> _ <=o _" [71,71] 70) and |
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sup_loc ("_ \<turnstile> _ <=l _" [71,71] 70) and |
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sup_state ("_ \<turnstile> _ <=s _" [71,71] 70) and |
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sup_state_opt ("_ \<turnstile> _ <=' _" [71,71] 70) |
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|
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|
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lemma JVM_states_unfold: |
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"states S maxs maxr == err(opt((Union {list n (types S) |n. n <= maxs}) <*> |
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list maxr (err(types S))))" |
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apply (unfold states_def sl_def Opt.esl_def Err.sl_def |
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stk_esl_def reg_sl_def Product.esl_def |
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Listn.sl_def upto_esl_def JType.esl_def Err.esl_def) |
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by simp |
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|
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|
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lemma JVM_le_unfold: |
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"le S m n == |
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Err.le(Opt.le(Product.le(Listn.le(subtype S))(Listn.le(Err.le(subtype S)))))" |
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apply (unfold le_def sl_def Opt.esl_def Err.sl_def |
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stk_esl_def reg_sl_def Product.esl_def |
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Listn.sl_def upto_esl_def JType.esl_def Err.esl_def) |
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by simp |
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|
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lemma JVM_le_convert: |
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"le G m n (OK t1) (OK t2) = G \<turnstile> t1 <=' t2" |
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by (simp add: JVM_le_unfold Err.le_def lesub_def sup_state_opt_def |
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sup_state_def sup_loc_def sup_ty_opt_def) |
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|
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lemma JVM_le_Err_conv: |
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"le G m n = Err.le (sup_state_opt G)" |
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by (unfold sup_state_opt_def sup_state_def sup_loc_def |
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sup_ty_opt_def JVM_le_unfold) simp |
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|
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lemma zip_map [rule_format]: |
13006 | 92 |
"\<forall>a. length a = length b \<longrightarrow> |
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zip (map f a) (map g b) = map (\<lambda>(x,y). (f x, g y)) (zip a b)" |
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apply (induct b) |
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apply simp |
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apply clarsimp |
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apply (case_tac aa) |
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apply simp+ |
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done |
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|
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lemma [simp]: "Err.le r (OK a) (OK b) = r a b" |
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by (simp add: Err.le_def lesub_def) |
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103 |
|
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lemma stk_convert: |
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"Listn.le (subtype G) a b = G \<turnstile> map OK a <=l map OK b" |
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106 |
proof |
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assume "Listn.le (subtype G) a b" |
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108 |
|
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hence le: "list_all2 (subtype G) a b" |
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110 |
by (unfold Listn.le_def lesub_def) |
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111 |
|
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112 |
{ fix x' y' |
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113 |
assume "length a = length b" |
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"(x',y') \<in> set (zip (map OK a) (map OK b))" |
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then |
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obtain x y where OK: |
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"x' = OK x" "y' = OK y" "(x,y) \<in> set (zip a b)" |
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by (auto simp add: zip_map) |
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119 |
with le |
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120 |
have "subtype G x y" |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
121 |
by (simp add: list_all2_def Ball_def) |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
122 |
with OK |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
123 |
have "G \<turnstile> x' <=o y'" |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
124 |
by (simp add: sup_ty_opt_def) |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
125 |
} |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
126 |
|
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
127 |
with le |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
128 |
show "G \<turnstile> map OK a <=l map OK b" |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
129 |
by (unfold sup_loc_def Listn.le_def lesub_def list_all2_def) auto |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
130 |
next |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
131 |
assume "G \<turnstile> map OK a <=l map OK b" |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
132 |
|
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
133 |
thus "Listn.le (subtype G) a b" |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
134 |
apply (unfold sup_loc_def list_all2_def Listn.le_def lesub_def) |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
135 |
apply (clarsimp simp add: zip_map) |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
136 |
apply (drule bspec, assumption) |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
137 |
apply (auto simp add: sup_ty_opt_def subtype_def) |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
138 |
done |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
139 |
qed |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
140 |
|
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
141 |
|
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
142 |
lemma sup_state_conv: |
12516 | 143 |
"(G \<turnstile> s1 <=s s2) == |
144 |
(G \<turnstile> map OK (fst s1) <=l map OK (fst s2)) \<and> (G \<turnstile> snd s1 <=l snd s2)" |
|
10812
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
145 |
by (auto simp add: sup_state_def stk_convert lesub_def Product.le_def split_beta) |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
146 |
|
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
147 |
|
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
148 |
lemma subtype_refl [simp]: |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
149 |
"subtype G t t" |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
150 |
by (simp add: subtype_def) |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
151 |
|
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
152 |
theorem sup_ty_opt_refl [simp]: |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
153 |
"G \<turnstile> t <=o t" |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
154 |
by (simp add: sup_ty_opt_def Err.le_def lesub_def split: err.split) |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
155 |
|
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
156 |
lemma le_list_refl2 [simp]: |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
157 |
"(\<And>xs. r xs xs) \<Longrightarrow> Listn.le r xs xs" |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
158 |
by (induct xs, auto simp add: Listn.le_def lesub_def) |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
159 |
|
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
160 |
theorem sup_loc_refl [simp]: |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
161 |
"G \<turnstile> t <=l t" |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
162 |
by (simp add: sup_loc_def) |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
163 |
|
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
164 |
theorem sup_state_refl [simp]: |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
165 |
"G \<turnstile> s <=s s" |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
166 |
by (auto simp add: sup_state_def Product.le_def lesub_def) |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
167 |
|
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
168 |
theorem sup_state_opt_refl [simp]: |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
169 |
"G \<turnstile> s <=' s" |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
170 |
by (simp add: sup_state_opt_def Opt.le_def lesub_def split: option.split) |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
171 |
|
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
172 |
|
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
173 |
theorem anyConvErr [simp]: |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
174 |
"(G \<turnstile> Err <=o any) = (any = Err)" |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
175 |
by (simp add: sup_ty_opt_def Err.le_def split: err.split) |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
176 |
|
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
177 |
theorem OKanyConvOK [simp]: |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
178 |
"(G \<turnstile> (OK ty') <=o (OK ty)) = (G \<turnstile> ty' \<preceq> ty)" |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
179 |
by (simp add: sup_ty_opt_def Err.le_def lesub_def subtype_def) |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
180 |
|
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
181 |
theorem sup_ty_opt_OK: |
13006 | 182 |
"G \<turnstile> a <=o (OK b) \<Longrightarrow> \<exists> x. a = OK x" |
10812
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
183 |
by (clarsimp simp add: sup_ty_opt_def Err.le_def split: err.splits) |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
184 |
|
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
185 |
lemma widen_PrimT_conv1 [simp]: |
13006 | 186 |
"\<lbrakk> G \<turnstile> S \<preceq> T; S = PrimT x\<rbrakk> \<Longrightarrow> T = PrimT x" |
22271 | 187 |
by (auto elim: widen.cases) |
10812
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
188 |
|
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
189 |
theorem sup_PTS_eq: |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
190 |
"(G \<turnstile> OK (PrimT p) <=o X) = (X=Err \<or> X = OK (PrimT p))" |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
191 |
by (auto simp add: sup_ty_opt_def Err.le_def lesub_def subtype_def |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
192 |
split: err.splits) |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
193 |
|
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
194 |
theorem sup_loc_Nil [iff]: |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
195 |
"(G \<turnstile> [] <=l XT) = (XT=[])" |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
196 |
by (simp add: sup_loc_def Listn.le_def) |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
197 |
|
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
198 |
theorem sup_loc_Cons [iff]: |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
199 |
"(G \<turnstile> (Y#YT) <=l XT) = (\<exists>X XT'. XT=X#XT' \<and> (G \<turnstile> Y <=o X) \<and> (G \<turnstile> YT <=l XT'))" |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
200 |
by (simp add: sup_loc_def Listn.le_def lesub_def list_all2_Cons1) |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
201 |
|
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
202 |
theorem sup_loc_Cons2: |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
203 |
"(G \<turnstile> YT <=l (X#XT)) = (\<exists>Y YT'. YT=Y#YT' \<and> (G \<turnstile> Y <=o X) \<and> (G \<turnstile> YT' <=l XT))" |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
204 |
by (simp add: sup_loc_def Listn.le_def lesub_def list_all2_Cons2) |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
205 |
|
13672 | 206 |
lemma sup_state_Cons: |
207 |
"(G \<turnstile> (x#xt, a) <=s (y#yt, b)) = |
|
208 |
((G \<turnstile> x \<preceq> y) \<and> (G \<turnstile> (xt,a) <=s (yt,b)))" |
|
209 |
by (auto simp add: sup_state_def stk_convert lesub_def Product.le_def) |
|
210 |
||
10812
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
211 |
|
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
212 |
theorem sup_loc_length: |
13006 | 213 |
"G \<turnstile> a <=l b \<Longrightarrow> length a = length b" |
10812
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
214 |
proof - |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
215 |
assume G: "G \<turnstile> a <=l b" |
13006 | 216 |
have "\<forall>b. (G \<turnstile> a <=l b) \<longrightarrow> length a = length b" |
10812
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
217 |
by (induct a, auto) |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
218 |
with G |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
219 |
show ?thesis by blast |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
220 |
qed |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
221 |
|
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
222 |
theorem sup_loc_nth: |
13006 | 223 |
"\<lbrakk> G \<turnstile> a <=l b; n < length a \<rbrakk> \<Longrightarrow> G \<turnstile> (a!n) <=o (b!n)" |
10812
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
224 |
proof - |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
225 |
assume a: "G \<turnstile> a <=l b" "n < length a" |
13006 | 226 |
have "\<forall> n b. (G \<turnstile> a <=l b) \<longrightarrow> n < length a \<longrightarrow> (G \<turnstile> (a!n) <=o (b!n))" |
10812
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
227 |
(is "?P a") |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
228 |
proof (induct a) |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
229 |
show "?P []" by simp |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
230 |
|
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
231 |
fix x xs assume IH: "?P xs" |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
232 |
|
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
233 |
show "?P (x#xs)" |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
234 |
proof (intro strip) |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
235 |
fix n b |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
236 |
assume "G \<turnstile> (x # xs) <=l b" "n < length (x # xs)" |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
237 |
with IH |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
238 |
show "G \<turnstile> ((x # xs) ! n) <=o (b ! n)" |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
239 |
by - (cases n, auto) |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
240 |
qed |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
241 |
qed |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
242 |
with a |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
243 |
show ?thesis by blast |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
244 |
qed |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
245 |
|
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
246 |
theorem all_nth_sup_loc: |
13006 | 247 |
"\<forall>b. length a = length b \<longrightarrow> (\<forall> n. n < length a \<longrightarrow> (G \<turnstile> (a!n) <=o (b!n))) |
248 |
\<longrightarrow> (G \<turnstile> a <=l b)" (is "?P a") |
|
10812
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
249 |
proof (induct a) |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
250 |
show "?P []" by simp |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
251 |
|
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
252 |
fix l ls assume IH: "?P ls" |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
253 |
|
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
254 |
show "?P (l#ls)" |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
255 |
proof (intro strip) |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
256 |
fix b |
13006 | 257 |
assume f: "\<forall>n. n < length (l # ls) \<longrightarrow> (G \<turnstile> ((l # ls) ! n) <=o (b ! n))" |
10812
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
258 |
assume l: "length (l#ls) = length b" |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
259 |
|
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
260 |
then obtain b' bs where b: "b = b'#bs" |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
261 |
by - (cases b, simp, simp add: neq_Nil_conv, rule that) |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
262 |
|
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
263 |
with f |
13006 | 264 |
have "\<forall>n. n < length ls \<longrightarrow> (G \<turnstile> (ls!n) <=o (bs!n))" |
10812
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
265 |
by auto |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
266 |
|
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
267 |
with f b l IH |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
268 |
show "G \<turnstile> (l # ls) <=l b" |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
269 |
by auto |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
270 |
qed |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
271 |
qed |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
272 |
|
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
273 |
|
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
274 |
theorem sup_loc_append: |
13006 | 275 |
"length a = length b \<Longrightarrow> |
10812
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
276 |
(G \<turnstile> (a@x) <=l (b@y)) = ((G \<turnstile> a <=l b) \<and> (G \<turnstile> x <=l y))" |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
277 |
proof - |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
278 |
assume l: "length a = length b" |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
279 |
|
13006 | 280 |
have "\<forall>b. length a = length b \<longrightarrow> (G \<turnstile> (a@x) <=l (b@y)) = ((G \<turnstile> a <=l b) \<and> |
10812
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
281 |
(G \<turnstile> x <=l y))" (is "?P a") |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
282 |
proof (induct a) |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
283 |
show "?P []" by simp |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
284 |
|
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
285 |
fix l ls assume IH: "?P ls" |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
286 |
show "?P (l#ls)" |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
287 |
proof (intro strip) |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
288 |
fix b |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
289 |
assume "length (l#ls) = length (b::ty err list)" |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
290 |
with IH |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
291 |
show "(G \<turnstile> ((l#ls)@x) <=l (b@y)) = ((G \<turnstile> (l#ls) <=l b) \<and> (G \<turnstile> x <=l y))" |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
292 |
by - (cases b, auto) |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
293 |
qed |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
294 |
qed |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
295 |
with l |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
296 |
show ?thesis by blast |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
297 |
qed |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
298 |
|
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
299 |
theorem sup_loc_rev [simp]: |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
300 |
"(G \<turnstile> (rev a) <=l rev b) = (G \<turnstile> a <=l b)" |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
301 |
proof - |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
302 |
have "\<forall>b. (G \<turnstile> (rev a) <=l rev b) = (G \<turnstile> a <=l b)" (is "\<forall>b. ?Q a b" is "?P a") |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
303 |
proof (induct a) |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
304 |
show "?P []" by simp |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
305 |
|
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
306 |
fix l ls assume IH: "?P ls" |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
307 |
|
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
308 |
{ |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
309 |
fix b |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
310 |
have "?Q (l#ls) b" |
25362 | 311 |
proof (cases b) |
10812
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
312 |
case Nil |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
313 |
thus ?thesis by (auto dest: sup_loc_length) |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
314 |
next |
25362 | 315 |
case (Cons a list) |
10812
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
316 |
show ?thesis |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
317 |
proof |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
318 |
assume "G \<turnstile> (l # ls) <=l b" |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
319 |
thus "G \<turnstile> rev (l # ls) <=l rev b" |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
320 |
by (clarsimp simp add: Cons IH sup_loc_length sup_loc_append) |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
321 |
next |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
322 |
assume "G \<turnstile> rev (l # ls) <=l rev b" |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
323 |
hence G: "G \<turnstile> (rev ls @ [l]) <=l (rev list @ [a])" |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
324 |
by (simp add: Cons) |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
325 |
|
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
326 |
hence "length (rev ls) = length (rev list)" |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
327 |
by (auto dest: sup_loc_length) |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
328 |
|
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
329 |
from this G |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
330 |
obtain "G \<turnstile> rev ls <=l rev list" "G \<turnstile> l <=o a" |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
331 |
by (simp add: sup_loc_append) |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
332 |
|
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
333 |
thus "G \<turnstile> (l # ls) <=l b" |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
334 |
by (simp add: Cons IH) |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
335 |
qed |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
336 |
qed |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
337 |
} |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
338 |
thus "?P (l#ls)" by blast |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
339 |
qed |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
340 |
|
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
341 |
thus ?thesis by blast |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
342 |
qed |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
343 |
|
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
344 |
|
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
345 |
theorem sup_loc_update [rule_format]: |
13006 | 346 |
"\<forall> n y. (G \<turnstile> a <=o b) \<longrightarrow> n < length y \<longrightarrow> (G \<turnstile> x <=l y) \<longrightarrow> |
10812
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
347 |
(G \<turnstile> x[n := a] <=l y[n := b])" (is "?P x") |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
348 |
proof (induct x) |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
349 |
show "?P []" by simp |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
350 |
|
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
351 |
fix l ls assume IH: "?P ls" |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
352 |
show "?P (l#ls)" |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
353 |
proof (intro strip) |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
354 |
fix n y |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
355 |
assume "G \<turnstile>a <=o b" "G \<turnstile> (l # ls) <=l y" "n < length y" |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
356 |
with IH |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
357 |
show "G \<turnstile> (l # ls)[n := a] <=l y[n := b]" |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
358 |
by - (cases n, auto simp add: sup_loc_Cons2 list_all2_Cons1) |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
359 |
qed |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
360 |
qed |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
361 |
|
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
362 |
|
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
363 |
theorem sup_state_length [simp]: |
13006 | 364 |
"G \<turnstile> s2 <=s s1 \<Longrightarrow> |
10812
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
365 |
length (fst s2) = length (fst s1) \<and> length (snd s2) = length (snd s1)" |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
366 |
by (auto dest: sup_loc_length simp add: sup_state_def stk_convert lesub_def Product.le_def); |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
367 |
|
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
368 |
theorem sup_state_append_snd: |
13006 | 369 |
"length a = length b \<Longrightarrow> |
10812
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
370 |
(G \<turnstile> (i,a@x) <=s (j,b@y)) = ((G \<turnstile> (i,a) <=s (j,b)) \<and> (G \<turnstile> (i,x) <=s (j,y)))" |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
371 |
by (auto simp add: sup_state_def stk_convert lesub_def Product.le_def sup_loc_append) |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
372 |
|
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
373 |
theorem sup_state_append_fst: |
13006 | 374 |
"length a = length b \<Longrightarrow> |
10812
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
375 |
(G \<turnstile> (a@x,i) <=s (b@y,j)) = ((G \<turnstile> (a,i) <=s (b,j)) \<and> (G \<turnstile> (x,i) <=s (y,j)))" |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
376 |
by (auto simp add: sup_state_def stk_convert lesub_def Product.le_def sup_loc_append) |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
377 |
|
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
378 |
theorem sup_state_Cons1: |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
379 |
"(G \<turnstile> (x#xt, a) <=s (yt, b)) = |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
380 |
(\<exists>y yt'. yt=y#yt' \<and> (G \<turnstile> x \<preceq> y) \<and> (G \<turnstile> (xt,a) <=s (yt',b)))" |
14025 | 381 |
by (auto simp add: sup_state_def stk_convert lesub_def Product.le_def) |
10812
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
382 |
|
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
383 |
theorem sup_state_Cons2: |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
384 |
"(G \<turnstile> (xt, a) <=s (y#yt, b)) = |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
385 |
(\<exists>x xt'. xt=x#xt' \<and> (G \<turnstile> x \<preceq> y) \<and> (G \<turnstile> (xt',a) <=s (yt,b)))" |
14025 | 386 |
by (auto simp add: sup_state_def stk_convert lesub_def Product.le_def sup_loc_Cons2) |
10812
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
387 |
|
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
388 |
theorem sup_state_ignore_fst: |
13006 | 389 |
"G \<turnstile> (a, x) <=s (b, y) \<Longrightarrow> G \<turnstile> (c, x) <=s (c, y)" |
10812
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
390 |
by (simp add: sup_state_def lesub_def Product.le_def) |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
391 |
|
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
392 |
theorem sup_state_rev_fst: |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
393 |
"(G \<turnstile> (rev a, x) <=s (rev b, y)) = (G \<turnstile> (a, x) <=s (b, y))" |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
394 |
proof - |
13006 | 395 |
have m: "\<And>f x. map f (rev x) = rev (map f x)" by (simp add: rev_map) |
10812
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
396 |
show ?thesis by (simp add: m sup_state_def stk_convert lesub_def Product.le_def) |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
397 |
qed |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
398 |
|
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
399 |
|
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
400 |
lemma sup_state_opt_None_any [iff]: |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
401 |
"(G \<turnstile> None <=' any) = True" |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
402 |
by (simp add: sup_state_opt_def Opt.le_def split: option.split) |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
403 |
|
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
404 |
lemma sup_state_opt_any_None [iff]: |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
405 |
"(G \<turnstile> any <=' None) = (any = None)" |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
406 |
by (simp add: sup_state_opt_def Opt.le_def split: option.split) |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
407 |
|
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
408 |
lemma sup_state_opt_Some_Some [iff]: |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
409 |
"(G \<turnstile> (Some a) <=' (Some b)) = (G \<turnstile> a <=s b)" |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
410 |
by (simp add: sup_state_opt_def Opt.le_def lesub_def del: split_paired_Ex) |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
411 |
|
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
412 |
lemma sup_state_opt_any_Some [iff]: |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
413 |
"(G \<turnstile> (Some a) <=' any) = (\<exists>b. any = Some b \<and> G \<turnstile> a <=s b)" |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
414 |
by (simp add: sup_state_opt_def Opt.le_def lesub_def split: option.split) |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
415 |
|
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
416 |
lemma sup_state_opt_Some_any: |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
417 |
"(G \<turnstile> any <=' (Some b)) = (any = None \<or> (\<exists>a. any = Some a \<and> G \<turnstile> a <=s b))" |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
418 |
by (simp add: sup_state_opt_def Opt.le_def lesub_def split: option.split) |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
419 |
|
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
420 |
|
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
421 |
theorem sup_ty_opt_trans [trans]: |
13006 | 422 |
"\<lbrakk>G \<turnstile> a <=o b; G \<turnstile> b <=o c\<rbrakk> \<Longrightarrow> G \<turnstile> a <=o c" |
10812
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
423 |
by (auto intro: widen_trans |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
424 |
simp add: sup_ty_opt_def Err.le_def lesub_def subtype_def |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
425 |
split: err.splits) |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
426 |
|
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
427 |
theorem sup_loc_trans [trans]: |
13006 | 428 |
"\<lbrakk>G \<turnstile> a <=l b; G \<turnstile> b <=l c\<rbrakk> \<Longrightarrow> G \<turnstile> a <=l c" |
10812
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
429 |
proof - |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
430 |
assume G: "G \<turnstile> a <=l b" "G \<turnstile> b <=l c" |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
431 |
|
13006 | 432 |
hence "\<forall> n. n < length a \<longrightarrow> (G \<turnstile> (a!n) <=o (c!n))" |
10812
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
433 |
proof (intro strip) |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
434 |
fix n |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
435 |
assume n: "n < length a" |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
436 |
with G |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
437 |
have "G \<turnstile> (a!n) <=o (b!n)" |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
438 |
by - (rule sup_loc_nth) |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
439 |
also |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
440 |
from n G |
13006 | 441 |
have "G \<turnstile> \<dots> <=o (c!n)" |
10812
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
442 |
by - (rule sup_loc_nth, auto dest: sup_loc_length) |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
443 |
finally |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
444 |
show "G \<turnstile> (a!n) <=o (c!n)" . |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
445 |
qed |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
446 |
|
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
447 |
with G |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
448 |
show ?thesis |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
449 |
by (auto intro!: all_nth_sup_loc [rule_format] dest!: sup_loc_length) |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
450 |
qed |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
451 |
|
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
452 |
|
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
453 |
theorem sup_state_trans [trans]: |
13006 | 454 |
"\<lbrakk>G \<turnstile> a <=s b; G \<turnstile> b <=s c\<rbrakk> \<Longrightarrow> G \<turnstile> a <=s c" |
10812
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
455 |
by (auto intro: sup_loc_trans simp add: sup_state_def stk_convert Product.le_def lesub_def) |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
456 |
|
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
457 |
theorem sup_state_opt_trans [trans]: |
13006 | 458 |
"\<lbrakk>G \<turnstile> a <=' b; G \<turnstile> b <=' c\<rbrakk> \<Longrightarrow> G \<turnstile> a <=' c" |
10812
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
459 |
by (auto intro: sup_state_trans |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
460 |
simp add: sup_state_opt_def Opt.le_def lesub_def |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
461 |
split: option.splits) |
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
462 |
|
ead84e90bfeb
merged semilattice orders with <=' from Convert.thy (now defined in JVMType.thy)
kleing
parents:
diff
changeset
|
463 |
end |