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34.2k
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| defs
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FeatherweightJava/FJSound
|
FJSound.A_1_2(1)
|
lemma A_1_2:
assumes "CT OK"
and "\<Gamma> = \<Gamma>1 ++ \<Gamma>2"
and "\<Gamma>2 = [xs [\<mapsto>] Bs]"
and "length xs = length ds"
and "length Bs = length ds"
and "\<exists>As. CT;\<Gamma>1 \<turnstile>+ ds : As \<and> CT \<turnstile>+ As <: Bs"
shows "CT;\<Gamma> \<turnstile>+ es:Ds \<Longrightarrow> \<exists>Cs. (CT;\<Gamma>1 \<turnstile>+ ([ds/xs]es):Cs \<and> CT \<turnstile>+ Cs <: Ds)" (is "?TYPINGS \<Longrightarrow> ?P1")
and "CT;\<Gamma> \<turnstile> e:D \<Longrightarrow> \<exists>C. (CT;\<Gamma>1 \<turnstile> ((ds/xs)e):C \<and> CT \<turnstile> C <: D)" (is "?TYPING \<Longrightarrow> ?P2")
|
?CT OK \<Longrightarrow> ?\<Gamma> = ?\<Gamma>1.0 ++ ?\<Gamma>2.0 \<Longrightarrow> ?\<Gamma>2.0 = [ ?xs [\<mapsto>] ?Bs] \<Longrightarrow> length ?xs = length ?ds \<Longrightarrow> length ?Bs = length ?ds \<Longrightarrow> \<exists>As. ?CT; ?\<Gamma>1.0 \<turnstile>+ ?ds : As \<and> ?CT \<turnstile>+ As <: ?Bs \<Longrightarrow> ?CT; ?\<Gamma> \<turnstile>+ ?es : ?Ds \<Longrightarrow> \<exists>Cs. ?CT; ?\<Gamma>1.0 \<turnstile>+ map (substs [ ?xs [\<mapsto>] ?ds]) ?es : Cs \<and> ?CT \<turnstile>+ Cs <: ?Ds
|
\<lbrakk> ?H1 x_1; x_2 = ?H2 x_3 x_4; x_4 = ?H3 (\<lambda>y_0. ?H4) x_5 x_6; ?H5 x_5 = ?H6 x_7; ?H5 x_6 = ?H6 x_7; \<exists>y_1. ?H7 x_1 x_3 x_7 y_1 \<and> ?H8 x_1 y_1 x_6; ?H7 x_1 x_2 x_8 x_9\<rbrakk> \<Longrightarrow> \<exists>y_2. ?H7 x_1 x_3 (?H9 (?H10 (?H11 (\<lambda>y_3. ?H12) x_5 x_7)) x_8) y_2 \<and> ?H8 x_1 y_2 x_9
|
[
"FJDefs.substs",
"List.list.map",
"FJDefs.subtypings",
"FJDefs.typings",
"List.length",
"Option.option.None",
"Map.map_upds",
"Map.map_add",
"FJDefs.ct_typing"
] |
[
"(nat \\<Rightarrow> exp option) \\<Rightarrow> exp \\<Rightarrow> exp",
"('a \\<Rightarrow> 'b) \\<Rightarrow> 'a list \\<Rightarrow> 'b list",
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> nat list \\<Rightarrow> nat list \\<Rightarrow> bool",
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> (nat \\<Rightarrow> nat option) \\<Rightarrow> exp list \\<Rightarrow> nat list \\<Rightarrow> bool",
"'a list \\<Rightarrow> nat",
"'a option",
"('a \\<Rightarrow> 'b option) \\<Rightarrow> 'a list \\<Rightarrow> 'b list \\<Rightarrow> 'a \\<Rightarrow> 'b option",
"('a \\<Rightarrow> 'b option) \\<Rightarrow> ('a \\<Rightarrow> 'b option) \\<Rightarrow> 'a \\<Rightarrow> 'b option",
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> bool"
] |
[
"primrec substs :: \"(varName \\<rightharpoonup> exp) \\<Rightarrow> exp \\<Rightarrow> exp\"\n and subst_list1 :: \"(varName \\<rightharpoonup> exp) \\<Rightarrow> exp list \\<Rightarrow> exp list\"\n and subst_list2 :: \"(varName \\<rightharpoonup> exp) \\<Rightarrow> exp list \\<Rightarrow> exp list\" where\n \"substs \\<sigma> (Var x) = (case (\\<sigma>(x)) of None \\<Rightarrow> (Var x) | Some p \\<Rightarrow> p)\"\n| \"substs \\<sigma> (FieldProj e f) = FieldProj (substs \\<sigma> e) f\"\n| \"substs \\<sigma> (MethodInvk e m es) = MethodInvk (substs \\<sigma> e) m (subst_list1 \\<sigma> es)\"\n| \"substs \\<sigma> (New C es) = New C (subst_list2 \\<sigma> es)\"\n| \"substs \\<sigma> (Cast C e) = Cast C (substs \\<sigma> e)\"\n| \"subst_list1 \\<sigma> [] = []\"\n| \"subst_list1 \\<sigma> (h # t) = (substs \\<sigma> h) # (subst_list1 \\<sigma> t)\"\n| \"subst_list2 \\<sigma> [] = []\"\n| \"subst_list2 \\<sigma> (h # t) = (substs \\<sigma> h) # (subst_list2 \\<sigma> t)\"",
"datatype (set: 'a) list =\n Nil (\"[]\")\n | Cons (hd: 'a) (tl: \"'a list\") (infixr \"#\" 65)\nfor\n map: map\n rel: list_all2\n pred: list_all\nwhere\n \"tl [] = []\"",
"inductive\n subtypings :: \"[classTable, className list, className list] \\<Rightarrow> bool\" (\"_ \\<turnstile>+ _ <: _\" [80,80,80] 80)\nwhere\n ss_nil : \"CT \\<turnstile>+ [] <: []\"\n| ss_cons : \"\\<lbrakk> CT \\<turnstile> C0 <: D0; CT \\<turnstile>+ Cs <: Ds \\<rbrakk> \\<Longrightarrow> CT \\<turnstile>+ (C0 # Cs) <: (D0 # Ds)\"",
"inductive\n typings :: \"[classTable, varCtx, exp list, className list] \\<Rightarrow> bool\" (\"_;_ \\<turnstile>+ _ : _\" [80,80,80,80] 80)\n and typing :: \"[classTable, varCtx, exp, className] \\<Rightarrow> bool\" (\"_;_ \\<turnstile> _ : _\" [80,80,80,80] 80)\nwhere\n ts_nil : \"CT;\\<Gamma> \\<turnstile>+ [] : []\"\n\n| ts_cons : \n \"\\<lbrakk> CT;\\<Gamma> \\<turnstile> e0 : C0; CT;\\<Gamma> \\<turnstile>+ es : Cs \\<rbrakk> \n \\<Longrightarrow> CT;\\<Gamma> \\<turnstile>+ (e0 # es) : (C0 # Cs)\"\n\n| t_var : \n \"\\<lbrakk> \\<Gamma>(x) = Some C \\<rbrakk> \\<Longrightarrow> CT;\\<Gamma> \\<turnstile> (Var x) : C\"\n\n| t_field : \n \"\\<lbrakk> CT;\\<Gamma> \\<turnstile> e0 : C0;\n fields(CT,C0) = Cf;\n lookup Cf (\\<lambda>fd.(vdName fd = fi)) = Some(fDef);\n vdType fDef = Ci \\<rbrakk>\n \\<Longrightarrow> CT;\\<Gamma> \\<turnstile> FieldProj e0 fi : Ci\"\n\n| t_invk : \n \"\\<lbrakk> CT;\\<Gamma> \\<turnstile> e0 : C0;\n mtype(CT,m,C0) = Ds \\<rightarrow> C;\n CT;\\<Gamma> \\<turnstile>+ es : Cs;\n CT \\<turnstile>+ Cs <: Ds;\n length es = length Ds \\<rbrakk>\n \\<Longrightarrow> CT;\\<Gamma> \\<turnstile> MethodInvk e0 m es : C\"\n\n| t_new : \n \"\\<lbrakk> fields(CT,C) = Df;\n length es = length Df;\n varDefs_types Df = Ds;\n CT;\\<Gamma> \\<turnstile>+ es : Cs;\n CT \\<turnstile>+ Cs <: Ds \\<rbrakk>\n \\<Longrightarrow> CT;\\<Gamma> \\<turnstile> New C es : C\"\n\n| t_ucast : \n \"\\<lbrakk> CT;\\<Gamma> \\<turnstile> e0 : D; \n CT \\<turnstile> D <: C \\<rbrakk>\n \\<Longrightarrow> CT;\\<Gamma> \\<turnstile> Cast C e0 : C\"\n\n| t_dcast : \n \"\\<lbrakk> CT;\\<Gamma> \\<turnstile> e0 : D; \n CT \\<turnstile> C <: D; C \\<noteq> D \\<rbrakk>\n \\<Longrightarrow> CT;\\<Gamma> \\<turnstile> Cast C e0 : C\"\n\n| t_scast : \n \"\\<lbrakk> CT;\\<Gamma> \\<turnstile> e0 : D;\n CT \\<turnstile> C \\<not><: D;\n CT \\<turnstile> D \\<not><: C \\<rbrakk>\n \\<Longrightarrow> CT;\\<Gamma> \\<turnstile> Cast C e0 : C\"",
"abbreviation length :: \"'a list \\<Rightarrow> nat\" where\n\"length \\<equiv> size\"",
"datatype 'a option =\n None\n | Some (the: 'a)",
"definition map_upds :: \"('a \\<rightharpoonup> 'b) \\<Rightarrow> 'a list \\<Rightarrow> 'b list \\<Rightarrow> 'a \\<rightharpoonup> 'b\" where\n\"map_upds m xs ys = m ++ map_of (rev (zip xs ys))\"",
"definition\n map_add :: \"('a \\<rightharpoonup> 'b) \\<Rightarrow> ('a \\<rightharpoonup> 'b) \\<Rightarrow> ('a \\<rightharpoonup> 'b)\" (infixl \"++\" 100) where\n \"m1 ++ m2 = (\\<lambda>x. case m2 x of None \\<Rightarrow> m1 x | Some y \\<Rightarrow> Some y)\"",
"inductive\n ct_typing :: \"classTable \\<Rightarrow> bool\" (\"_ OK\" 80)\nwhere\nct_all_ok: \n \"\\<lbrakk> Object \\<notin> dom(CT); \n \\<forall>C CDef. CT(C) = Some(CDef) \\<longrightarrow> (CT \\<turnstile> CDef OK) \\<and> (cName CDef = C) \\<rbrakk>\n \\<Longrightarrow> CT OK\""
] |
FeatherweightJava/FJSound
|
FJSound.A_1_2(2)
|
lemma A_1_2:
assumes "CT OK"
and "\<Gamma> = \<Gamma>1 ++ \<Gamma>2"
and "\<Gamma>2 = [xs [\<mapsto>] Bs]"
and "length xs = length ds"
and "length Bs = length ds"
and "\<exists>As. CT;\<Gamma>1 \<turnstile>+ ds : As \<and> CT \<turnstile>+ As <: Bs"
shows "CT;\<Gamma> \<turnstile>+ es:Ds \<Longrightarrow> \<exists>Cs. (CT;\<Gamma>1 \<turnstile>+ ([ds/xs]es):Cs \<and> CT \<turnstile>+ Cs <: Ds)" (is "?TYPINGS \<Longrightarrow> ?P1")
and "CT;\<Gamma> \<turnstile> e:D \<Longrightarrow> \<exists>C. (CT;\<Gamma>1 \<turnstile> ((ds/xs)e):C \<and> CT \<turnstile> C <: D)" (is "?TYPING \<Longrightarrow> ?P2")
|
?CT OK \<Longrightarrow> ?\<Gamma> = ?\<Gamma>1.0 ++ ?\<Gamma>2.0 \<Longrightarrow> ?\<Gamma>2.0 = [ ?xs [\<mapsto>] ?Bs] \<Longrightarrow> length ?xs = length ?ds \<Longrightarrow> length ?Bs = length ?ds \<Longrightarrow> \<exists>As. ?CT; ?\<Gamma>1.0 \<turnstile>+ ?ds : As \<and> ?CT \<turnstile>+ As <: ?Bs \<Longrightarrow> ?CT; ?\<Gamma> \<turnstile> ?e : ?D \<Longrightarrow> \<exists>C. ?CT; ?\<Gamma>1.0 \<turnstile> substs [ ?xs [\<mapsto>] ?ds] ?e : C \<and> ?CT \<turnstile> C <: ?D
|
\<lbrakk> ?H1 x_1; x_2 = ?H2 x_3 x_4; x_4 = ?H3 (\<lambda>y_0. ?H4) x_5 x_6; ?H5 x_5 = ?H6 x_7; ?H5 x_6 = ?H6 x_7; \<exists>y_1. ?H7 x_1 x_3 x_7 y_1 \<and> ?H8 x_1 y_1 x_6; ?H9 x_1 x_2 x_8 x_9\<rbrakk> \<Longrightarrow> \<exists>y_2. ?H9 x_1 x_3 (?H10 (?H11 (\<lambda>y_3. ?H12) x_5 x_7) x_8) y_2 \<and> ?H13 x_1 y_2 x_9
|
[
"FJDefs.subtyping",
"FJDefs.substs",
"FJDefs.typing",
"FJDefs.subtypings",
"FJDefs.typings",
"List.length",
"Option.option.None",
"Map.map_upds",
"Map.map_add",
"FJDefs.ct_typing"
] |
[
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> nat \\<Rightarrow> nat \\<Rightarrow> bool",
"(nat \\<Rightarrow> exp option) \\<Rightarrow> exp \\<Rightarrow> exp",
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> (nat \\<Rightarrow> nat option) \\<Rightarrow> exp \\<Rightarrow> nat \\<Rightarrow> bool",
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> nat list \\<Rightarrow> nat list \\<Rightarrow> bool",
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> (nat \\<Rightarrow> nat option) \\<Rightarrow> exp list \\<Rightarrow> nat list \\<Rightarrow> bool",
"'a list \\<Rightarrow> nat",
"'a option",
"('a \\<Rightarrow> 'b option) \\<Rightarrow> 'a list \\<Rightarrow> 'b list \\<Rightarrow> 'a \\<Rightarrow> 'b option",
"('a \\<Rightarrow> 'b option) \\<Rightarrow> ('a \\<Rightarrow> 'b option) \\<Rightarrow> 'a \\<Rightarrow> 'b option",
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> bool"
] |
[
"inductive\n subtyping :: \"[classTable, className, className] \\<Rightarrow> bool\" (\"_ \\<turnstile> _ <: _\" [80,80,80] 80)\nwhere\n s_refl : \"CT \\<turnstile> C <: C\"\n| s_trans : \"\\<lbrakk> CT \\<turnstile> C <: D; CT \\<turnstile> D <: E \\<rbrakk> \\<Longrightarrow> CT \\<turnstile> C <: E\"\n| s_super : \"\\<lbrakk> CT(C) = Some(CDef); cSuper CDef = D \\<rbrakk> \\<Longrightarrow> CT \\<turnstile> C <: D\"",
"primrec substs :: \"(varName \\<rightharpoonup> exp) \\<Rightarrow> exp \\<Rightarrow> exp\"\n and subst_list1 :: \"(varName \\<rightharpoonup> exp) \\<Rightarrow> exp list \\<Rightarrow> exp list\"\n and subst_list2 :: \"(varName \\<rightharpoonup> exp) \\<Rightarrow> exp list \\<Rightarrow> exp list\" where\n \"substs \\<sigma> (Var x) = (case (\\<sigma>(x)) of None \\<Rightarrow> (Var x) | Some p \\<Rightarrow> p)\"\n| \"substs \\<sigma> (FieldProj e f) = FieldProj (substs \\<sigma> e) f\"\n| \"substs \\<sigma> (MethodInvk e m es) = MethodInvk (substs \\<sigma> e) m (subst_list1 \\<sigma> es)\"\n| \"substs \\<sigma> (New C es) = New C (subst_list2 \\<sigma> es)\"\n| \"substs \\<sigma> (Cast C e) = Cast C (substs \\<sigma> e)\"\n| \"subst_list1 \\<sigma> [] = []\"\n| \"subst_list1 \\<sigma> (h # t) = (substs \\<sigma> h) # (subst_list1 \\<sigma> t)\"\n| \"subst_list2 \\<sigma> [] = []\"\n| \"subst_list2 \\<sigma> (h # t) = (substs \\<sigma> h) # (subst_list2 \\<sigma> t)\"",
"inductive\n typings :: \"[classTable, varCtx, exp list, className list] \\<Rightarrow> bool\" (\"_;_ \\<turnstile>+ _ : _\" [80,80,80,80] 80)\n and typing :: \"[classTable, varCtx, exp, className] \\<Rightarrow> bool\" (\"_;_ \\<turnstile> _ : _\" [80,80,80,80] 80)\nwhere\n ts_nil : \"CT;\\<Gamma> \\<turnstile>+ [] : []\"\n\n| ts_cons : \n \"\\<lbrakk> CT;\\<Gamma> \\<turnstile> e0 : C0; CT;\\<Gamma> \\<turnstile>+ es : Cs \\<rbrakk> \n \\<Longrightarrow> CT;\\<Gamma> \\<turnstile>+ (e0 # es) : (C0 # Cs)\"\n\n| t_var : \n \"\\<lbrakk> \\<Gamma>(x) = Some C \\<rbrakk> \\<Longrightarrow> CT;\\<Gamma> \\<turnstile> (Var x) : C\"\n\n| t_field : \n \"\\<lbrakk> CT;\\<Gamma> \\<turnstile> e0 : C0;\n fields(CT,C0) = Cf;\n lookup Cf (\\<lambda>fd.(vdName fd = fi)) = Some(fDef);\n vdType fDef = Ci \\<rbrakk>\n \\<Longrightarrow> CT;\\<Gamma> \\<turnstile> FieldProj e0 fi : Ci\"\n\n| t_invk : \n \"\\<lbrakk> CT;\\<Gamma> \\<turnstile> e0 : C0;\n mtype(CT,m,C0) = Ds \\<rightarrow> C;\n CT;\\<Gamma> \\<turnstile>+ es : Cs;\n CT \\<turnstile>+ Cs <: Ds;\n length es = length Ds \\<rbrakk>\n \\<Longrightarrow> CT;\\<Gamma> \\<turnstile> MethodInvk e0 m es : C\"\n\n| t_new : \n \"\\<lbrakk> fields(CT,C) = Df;\n length es = length Df;\n varDefs_types Df = Ds;\n CT;\\<Gamma> \\<turnstile>+ es : Cs;\n CT \\<turnstile>+ Cs <: Ds \\<rbrakk>\n \\<Longrightarrow> CT;\\<Gamma> \\<turnstile> New C es : C\"\n\n| t_ucast : \n \"\\<lbrakk> CT;\\<Gamma> \\<turnstile> e0 : D; \n CT \\<turnstile> D <: C \\<rbrakk>\n \\<Longrightarrow> CT;\\<Gamma> \\<turnstile> Cast C e0 : C\"\n\n| t_dcast : \n \"\\<lbrakk> CT;\\<Gamma> \\<turnstile> e0 : D; \n CT \\<turnstile> C <: D; C \\<noteq> D \\<rbrakk>\n \\<Longrightarrow> CT;\\<Gamma> \\<turnstile> Cast C e0 : C\"\n\n| t_scast : \n \"\\<lbrakk> CT;\\<Gamma> \\<turnstile> e0 : D;\n CT \\<turnstile> C \\<not><: D;\n CT \\<turnstile> D \\<not><: C \\<rbrakk>\n \\<Longrightarrow> CT;\\<Gamma> \\<turnstile> Cast C e0 : C\"",
"inductive\n subtypings :: \"[classTable, className list, className list] \\<Rightarrow> bool\" (\"_ \\<turnstile>+ _ <: _\" [80,80,80] 80)\nwhere\n ss_nil : \"CT \\<turnstile>+ [] <: []\"\n| ss_cons : \"\\<lbrakk> CT \\<turnstile> C0 <: D0; CT \\<turnstile>+ Cs <: Ds \\<rbrakk> \\<Longrightarrow> CT \\<turnstile>+ (C0 # Cs) <: (D0 # Ds)\"",
"abbreviation length :: \"'a list \\<Rightarrow> nat\" where\n\"length \\<equiv> size\"",
"datatype 'a option =\n None\n | Some (the: 'a)",
"definition map_upds :: \"('a \\<rightharpoonup> 'b) \\<Rightarrow> 'a list \\<Rightarrow> 'b list \\<Rightarrow> 'a \\<rightharpoonup> 'b\" where\n\"map_upds m xs ys = m ++ map_of (rev (zip xs ys))\"",
"definition\n map_add :: \"('a \\<rightharpoonup> 'b) \\<Rightarrow> ('a \\<rightharpoonup> 'b) \\<Rightarrow> ('a \\<rightharpoonup> 'b)\" (infixl \"++\" 100) where\n \"m1 ++ m2 = (\\<lambda>x. case m2 x of None \\<Rightarrow> m1 x | Some y \\<Rightarrow> Some y)\"",
"inductive\n ct_typing :: \"classTable \\<Rightarrow> bool\" (\"_ OK\" 80)\nwhere\nct_all_ok: \n \"\\<lbrakk> Object \\<notin> dom(CT); \n \\<forall>C CDef. CT(C) = Some(CDef) \\<longrightarrow> (CT \\<turnstile> CDef OK) \\<and> (cName CDef = C) \\<rbrakk>\n \\<Longrightarrow> CT OK\""
] |
FeatherweightJava/FJSound
|
FJSound.closed_subterm_split
|
lemma closed_subterm_split:
assumes "CT;\<Gamma> \<turnstile> e : C" and "\<Gamma> = Map.empty"
shows "
((\<exists>C0 es fi. (FieldProj (New C0 es) fi) \<in> subexprs(e))
\<or> (\<exists>C0 es m ds. (MethodInvk (New C0 es) m ds) \<in> subexprs(e))
\<or> (\<exists>C0 D es. (Cast D (New C0 es)) \<in> subexprs(e))
\<or> val(e))" (is "?F e \<or> ?M e \<or> ?C e \<or> ?V e" is "?IH e")
|
?CT; ?\<Gamma> \<turnstile> ?e : ?C \<Longrightarrow> ?\<Gamma> = (\<lambda>x. None) \<Longrightarrow> (\<exists>C0 es fi. FieldProj (New C0 es) fi \<in> subexprs(?e)) \<or> (\<exists>C0 es m ds. MethodInvk (New C0 es) m ds \<in> subexprs(?e)) \<or> (\<exists>C0 D es. Cast D (New C0 es) \<in> subexprs(?e)) \<or> val(?e)
|
\<lbrakk> ?H1 x_1 x_2 x_3 x_4; x_2 = (\<lambda>y_0. ?H2)\<rbrakk> \<Longrightarrow> (\<exists>y_1 y_2 y_3. ?H3 (?H4 (?H5 y_1 y_2) y_3) x_3) \<or> (\<exists>y_4 y_5 y_6 y_7. ?H3 (?H6 (?H5 y_4 y_5) y_6 y_7) x_3) \<or> (\<exists>y_8 y_9 y_10. ?H3 (?H7 y_9 (?H5 y_8 y_10)) x_3) \<or> ?H8 x_3
|
[
"FJDefs.val",
"FJDefs.exp.Cast",
"FJDefs.exp.MethodInvk",
"FJDefs.exp.New",
"FJDefs.exp.FieldProj",
"FJDefs.subexprs",
"Option.option.None",
"FJDefs.typing"
] |
[
"exp \\<Rightarrow> bool",
"nat \\<Rightarrow> exp \\<Rightarrow> exp",
"exp \\<Rightarrow> nat \\<Rightarrow> exp list \\<Rightarrow> exp",
"nat \\<Rightarrow> exp list \\<Rightarrow> exp",
"exp \\<Rightarrow> nat \\<Rightarrow> exp",
"exp \\<Rightarrow> exp \\<Rightarrow> bool",
"'a option",
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> (nat \\<Rightarrow> nat option) \\<Rightarrow> exp \\<Rightarrow> nat \\<Rightarrow> bool"
] |
[
"inductive\n vals :: \"[exp list] \\<Rightarrow> bool\" (\"vals'(_')\" [80] 80)\n and val :: \"[exp] \\<Rightarrow> bool\" (\"val'(_')\" [80] 80)\nwhere\n vals_nil : \"vals([])\"\n | vals_cons : \"\\<lbrakk> val(vh); vals(vt) \\<rbrakk> \\<Longrightarrow> vals((vh # vt))\"\n | val : \"\\<lbrakk> vals(vs) \\<rbrakk> \\<Longrightarrow> val(New C vs)\"",
"datatype exp = \n Var \"varName\"\n | FieldProj \"exp\" \"varName\" \n | MethodInvk \"exp\" \"methodName\" \"exp list\"\n | New \"className\" \"exp list\"\n | Cast \"className\" \"exp\"",
"abbreviation\n subexprs :: \"[exp,exp] \\<Rightarrow> bool\" (\"_ \\<in> subexprs'(_')\" [80,80] 80) where\n \"e' \\<in> subexprs(e) \\<equiv> (e',e) \\<in> isubexprs^*\"",
"datatype 'a option =\n None\n | Some (the: 'a)",
"inductive\n typings :: \"[classTable, varCtx, exp list, className list] \\<Rightarrow> bool\" (\"_;_ \\<turnstile>+ _ : _\" [80,80,80,80] 80)\n and typing :: \"[classTable, varCtx, exp, className] \\<Rightarrow> bool\" (\"_;_ \\<turnstile> _ : _\" [80,80,80,80] 80)\nwhere\n ts_nil : \"CT;\\<Gamma> \\<turnstile>+ [] : []\"\n\n| ts_cons : \n \"\\<lbrakk> CT;\\<Gamma> \\<turnstile> e0 : C0; CT;\\<Gamma> \\<turnstile>+ es : Cs \\<rbrakk> \n \\<Longrightarrow> CT;\\<Gamma> \\<turnstile>+ (e0 # es) : (C0 # Cs)\"\n\n| t_var : \n \"\\<lbrakk> \\<Gamma>(x) = Some C \\<rbrakk> \\<Longrightarrow> CT;\\<Gamma> \\<turnstile> (Var x) : C\"\n\n| t_field : \n \"\\<lbrakk> CT;\\<Gamma> \\<turnstile> e0 : C0;\n fields(CT,C0) = Cf;\n lookup Cf (\\<lambda>fd.(vdName fd = fi)) = Some(fDef);\n vdType fDef = Ci \\<rbrakk>\n \\<Longrightarrow> CT;\\<Gamma> \\<turnstile> FieldProj e0 fi : Ci\"\n\n| t_invk : \n \"\\<lbrakk> CT;\\<Gamma> \\<turnstile> e0 : C0;\n mtype(CT,m,C0) = Ds \\<rightarrow> C;\n CT;\\<Gamma> \\<turnstile>+ es : Cs;\n CT \\<turnstile>+ Cs <: Ds;\n length es = length Ds \\<rbrakk>\n \\<Longrightarrow> CT;\\<Gamma> \\<turnstile> MethodInvk e0 m es : C\"\n\n| t_new : \n \"\\<lbrakk> fields(CT,C) = Df;\n length es = length Df;\n varDefs_types Df = Ds;\n CT;\\<Gamma> \\<turnstile>+ es : Cs;\n CT \\<turnstile>+ Cs <: Ds \\<rbrakk>\n \\<Longrightarrow> CT;\\<Gamma> \\<turnstile> New C es : C\"\n\n| t_ucast : \n \"\\<lbrakk> CT;\\<Gamma> \\<turnstile> e0 : D; \n CT \\<turnstile> D <: C \\<rbrakk>\n \\<Longrightarrow> CT;\\<Gamma> \\<turnstile> Cast C e0 : C\"\n\n| t_dcast : \n \"\\<lbrakk> CT;\\<Gamma> \\<turnstile> e0 : D; \n CT \\<turnstile> C <: D; C \\<noteq> D \\<rbrakk>\n \\<Longrightarrow> CT;\\<Gamma> \\<turnstile> Cast C e0 : C\"\n\n| t_scast : \n \"\\<lbrakk> CT;\\<Gamma> \\<turnstile> e0 : D;\n CT \\<turnstile> C \\<not><: D;\n CT \\<turnstile> D \\<not><: C \\<rbrakk>\n \\<Longrightarrow> CT;\\<Gamma> \\<turnstile> Cast C e0 : C\""
] |
FeatherweightJava/FJSound
|
FJSound.Thm_2_4_3
|
theorem Thm_2_4_3:
assumes e_typ: "CT;Map.empty \<turnstile> e : C"
and ct_ok: "CT OK"
and multisteps: "CT \<turnstile> e \<rightarrow>* e1"
and no_step: "\<not>(\<exists>e2. CT \<turnstile> e1 \<rightarrow> e2)"
shows "(val(e1) \<and> (\<exists>D. CT;Map.empty \<turnstile> e1 : D \<and> CT \<turnstile> D <: C))
\<or> (\<exists>D C es. (Cast D (New C es) \<in> subexprs(e1) \<and> CT \<turnstile> C \<not><: D))"
|
?CT;(\<lambda>x. None) \<turnstile> ?e : ?C \<Longrightarrow> ?CT OK \<Longrightarrow> ?CT \<turnstile> ?e \<rightarrow>* ?e1.0 \<Longrightarrow> \<nexists>e2. ?CT \<turnstile> ?e1.0 \<rightarrow> e2 \<Longrightarrow> val(?e1.0) \<and> (\<exists>D. ?CT;(\<lambda>x. None) \<turnstile> ?e1.0 : D \<and> ?CT \<turnstile> D <: ?C) \<or> (\<exists>D C es. Cast D (New C es) \<in> subexprs(?e1.0) \<and> ?CT \<turnstile> C \<not><: D)
|
\<lbrakk> ?H1 x_1 (\<lambda>y_0. ?H2) x_2 x_3; ?H3 x_1; ?H4 x_1 x_2 x_4; \<nexists>y_1. ?H5 x_1 x_4 y_1\<rbrakk> \<Longrightarrow> ?H6 x_4 \<and> (\<exists>y_2. ?H1 x_1 (\<lambda>y_3. ?H2) x_4 y_2 \<and> ?H7 x_1 y_2 x_3) \<or> (\<exists>y_4 y_5 y_6. ?H8 (?H9 y_4 (?H10 y_5 y_6)) x_4 \<and> ?H11 x_1 y_5 y_4)
|
[
"FJDefs.neg_subtyping",
"FJDefs.exp.New",
"FJDefs.exp.Cast",
"FJDefs.subexprs",
"FJDefs.subtyping",
"FJDefs.val",
"FJDefs.reduction",
"FJDefs.reductions",
"FJDefs.ct_typing",
"Option.option.None",
"FJDefs.typing"
] |
[
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> nat \\<Rightarrow> nat \\<Rightarrow> bool",
"nat \\<Rightarrow> exp list \\<Rightarrow> exp",
"nat \\<Rightarrow> exp \\<Rightarrow> exp",
"exp \\<Rightarrow> exp \\<Rightarrow> bool",
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> nat \\<Rightarrow> nat \\<Rightarrow> bool",
"exp \\<Rightarrow> bool",
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> exp \\<Rightarrow> exp \\<Rightarrow> bool",
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> exp \\<Rightarrow> exp \\<Rightarrow> bool",
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> bool",
"'a option",
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> (nat \\<Rightarrow> nat option) \\<Rightarrow> exp \\<Rightarrow> nat \\<Rightarrow> bool"
] |
[
"abbreviation\n neg_subtyping :: \"[classTable, className, className] \\<Rightarrow> bool\" (\"_ \\<turnstile> _ \\<not><: _\" [80,80,80] 80)\n where \"CT \\<turnstile> S \\<not><: T \\<equiv> \\<not> CT \\<turnstile> S <: T\"",
"datatype exp = \n Var \"varName\"\n | FieldProj \"exp\" \"varName\" \n | MethodInvk \"exp\" \"methodName\" \"exp list\"\n | New \"className\" \"exp list\"\n | Cast \"className\" \"exp\"",
"abbreviation\n subexprs :: \"[exp,exp] \\<Rightarrow> bool\" (\"_ \\<in> subexprs'(_')\" [80,80] 80) where\n \"e' \\<in> subexprs(e) \\<equiv> (e',e) \\<in> isubexprs^*\"",
"inductive\n subtyping :: \"[classTable, className, className] \\<Rightarrow> bool\" (\"_ \\<turnstile> _ <: _\" [80,80,80] 80)\nwhere\n s_refl : \"CT \\<turnstile> C <: C\"\n| s_trans : \"\\<lbrakk> CT \\<turnstile> C <: D; CT \\<turnstile> D <: E \\<rbrakk> \\<Longrightarrow> CT \\<turnstile> C <: E\"\n| s_super : \"\\<lbrakk> CT(C) = Some(CDef); cSuper CDef = D \\<rbrakk> \\<Longrightarrow> CT \\<turnstile> C <: D\"",
"inductive\n vals :: \"[exp list] \\<Rightarrow> bool\" (\"vals'(_')\" [80] 80)\n and val :: \"[exp] \\<Rightarrow> bool\" (\"val'(_')\" [80] 80)\nwhere\n vals_nil : \"vals([])\"\n | vals_cons : \"\\<lbrakk> val(vh); vals(vt) \\<rbrakk> \\<Longrightarrow> vals((vh # vt))\"\n | val : \"\\<lbrakk> vals(vs) \\<rbrakk> \\<Longrightarrow> val(New C vs)\"",
"inductive\n reduction :: \"[classTable, exp, exp] \\<Rightarrow> bool\" (\"_ \\<turnstile> _ \\<rightarrow> _\" [80,80,80] 80)\nwhere\n\n r_field: \n \"\\<lbrakk> fields(CT,C) = Cf; \n lookup2 Cf es (\\<lambda>fd.(vdName fd = fi)) = Some(ei) \\<rbrakk>\n \\<Longrightarrow> CT \\<turnstile> FieldProj (New C es) fi \\<rightarrow> ei\"\n\n| r_invk: \n \"\\<lbrakk> mbody(CT,m,C) = xs . e0;\n substs ((map_upds Map.empty xs ds)(this \\<mapsto> (New C es))) e0 = e0' \\<rbrakk>\n \\<Longrightarrow> CT \\<turnstile> MethodInvk (New C es) m ds \\<rightarrow> e0'\"\n\n| r_cast: \n \"\\<lbrakk> CT \\<turnstile> C <: D \\<rbrakk> \n \\<Longrightarrow> CT \\<turnstile> Cast D (New C es) \\<rightarrow> New C es\"\n\n| rc_field: \n \"\\<lbrakk> CT \\<turnstile> e0 \\<rightarrow> e0' \\<rbrakk> \n \\<Longrightarrow> CT \\<turnstile> FieldProj e0 f \\<rightarrow> FieldProj e0' f\"\n\n| rc_invk_recv: \n \"\\<lbrakk> CT \\<turnstile> e0 \\<rightarrow> e0' \\<rbrakk> \n \\<Longrightarrow> CT \\<turnstile> MethodInvk e0 m es \\<rightarrow> MethodInvk e0' m es\"\n\n| rc_invk_arg: \n \"\\<lbrakk> CT \\<turnstile> ei \\<rightarrow> ei' \\<rbrakk>\n \\<Longrightarrow> CT \\<turnstile> MethodInvk e0 m (el@ei#er) \\<rightarrow> MethodInvk e0 m (el@ei'#er)\"\n\n| rc_new_arg: \n \"\\<lbrakk> CT \\<turnstile> ei \\<rightarrow> ei' \\<rbrakk> \n \\<Longrightarrow> CT \\<turnstile> New C (el@ei#er) \\<rightarrow> New C (el@ei'#er)\"\n\n| rc_cast: \n \"\\<lbrakk> CT \\<turnstile> e0 \\<rightarrow> e0' \\<rbrakk> \n \\<Longrightarrow> CT \\<turnstile> Cast C e0 \\<rightarrow> Cast C e0'\"",
"inductive\n reductions :: \"[classTable, exp, exp] \\<Rightarrow> bool\" (\"_ \\<turnstile> _ \\<rightarrow>* _\" [80,80,80] 80)\nwhere\n rs_refl: \"CT \\<turnstile> e \\<rightarrow>* e\" \n| rs_trans: \"\\<lbrakk> CT \\<turnstile> e \\<rightarrow> e'; CT \\<turnstile> e' \\<rightarrow>* e'' \\<rbrakk> \\<Longrightarrow> CT \\<turnstile> e \\<rightarrow>* e''\"",
"inductive\n ct_typing :: \"classTable \\<Rightarrow> bool\" (\"_ OK\" 80)\nwhere\nct_all_ok: \n \"\\<lbrakk> Object \\<notin> dom(CT); \n \\<forall>C CDef. CT(C) = Some(CDef) \\<longrightarrow> (CT \\<turnstile> CDef OK) \\<and> (cName CDef = C) \\<rbrakk>\n \\<Longrightarrow> CT OK\"",
"datatype 'a option =\n None\n | Some (the: 'a)",
"inductive\n typings :: \"[classTable, varCtx, exp list, className list] \\<Rightarrow> bool\" (\"_;_ \\<turnstile>+ _ : _\" [80,80,80,80] 80)\n and typing :: \"[classTable, varCtx, exp, className] \\<Rightarrow> bool\" (\"_;_ \\<turnstile> _ : _\" [80,80,80,80] 80)\nwhere\n ts_nil : \"CT;\\<Gamma> \\<turnstile>+ [] : []\"\n\n| ts_cons : \n \"\\<lbrakk> CT;\\<Gamma> \\<turnstile> e0 : C0; CT;\\<Gamma> \\<turnstile>+ es : Cs \\<rbrakk> \n \\<Longrightarrow> CT;\\<Gamma> \\<turnstile>+ (e0 # es) : (C0 # Cs)\"\n\n| t_var : \n \"\\<lbrakk> \\<Gamma>(x) = Some C \\<rbrakk> \\<Longrightarrow> CT;\\<Gamma> \\<turnstile> (Var x) : C\"\n\n| t_field : \n \"\\<lbrakk> CT;\\<Gamma> \\<turnstile> e0 : C0;\n fields(CT,C0) = Cf;\n lookup Cf (\\<lambda>fd.(vdName fd = fi)) = Some(fDef);\n vdType fDef = Ci \\<rbrakk>\n \\<Longrightarrow> CT;\\<Gamma> \\<turnstile> FieldProj e0 fi : Ci\"\n\n| t_invk : \n \"\\<lbrakk> CT;\\<Gamma> \\<turnstile> e0 : C0;\n mtype(CT,m,C0) = Ds \\<rightarrow> C;\n CT;\\<Gamma> \\<turnstile>+ es : Cs;\n CT \\<turnstile>+ Cs <: Ds;\n length es = length Ds \\<rbrakk>\n \\<Longrightarrow> CT;\\<Gamma> \\<turnstile> MethodInvk e0 m es : C\"\n\n| t_new : \n \"\\<lbrakk> fields(CT,C) = Df;\n length es = length Df;\n varDefs_types Df = Ds;\n CT;\\<Gamma> \\<turnstile>+ es : Cs;\n CT \\<turnstile>+ Cs <: Ds \\<rbrakk>\n \\<Longrightarrow> CT;\\<Gamma> \\<turnstile> New C es : C\"\n\n| t_ucast : \n \"\\<lbrakk> CT;\\<Gamma> \\<turnstile> e0 : D; \n CT \\<turnstile> D <: C \\<rbrakk>\n \\<Longrightarrow> CT;\\<Gamma> \\<turnstile> Cast C e0 : C\"\n\n| t_dcast : \n \"\\<lbrakk> CT;\\<Gamma> \\<turnstile> e0 : D; \n CT \\<turnstile> C <: D; C \\<noteq> D \\<rbrakk>\n \\<Longrightarrow> CT;\\<Gamma> \\<turnstile> Cast C e0 : C\"\n\n| t_scast : \n \"\\<lbrakk> CT;\\<Gamma> \\<turnstile> e0 : D;\n CT \\<turnstile> C \\<not><: D;\n CT \\<turnstile> D \\<not><: C \\<rbrakk>\n \\<Longrightarrow> CT;\\<Gamma> \\<turnstile> Cast C e0 : C\""
] |
FeatherweightJava/FJSound
|
FJSound.A_1_4
|
lemma A_1_4:
assumes ct_ok: "CT OK"
and mb:"mbody(CT,m,C) = xs . e"
and mt:"mtype(CT,m,C) = Ds \<rightarrow> D"
shows "\<exists>D0 C0. (CT \<turnstile> C <: D0) \<and>
(CT \<turnstile> C0 <: D) \<and>
(CT;[xs[\<mapsto>]Ds](this \<mapsto> D0) \<turnstile> e : C0)"
|
?CT OK \<Longrightarrow> mbody(?CT, ?m, ?C) = ?xs . ?e \<Longrightarrow> mtype(?CT, ?m, ?C) = ?Ds \<rightarrow> ?D \<Longrightarrow> \<exists>D0 C0. ?CT \<turnstile> ?C <: D0 \<and> ?CT \<turnstile> C0 <: ?D \<and> ?CT;[ ?xs [\<mapsto>] ?Ds, this \<mapsto> D0] \<turnstile> ?e : C0
|
\<lbrakk> ?H1 x_1; ?H2 x_1 x_2 x_3 x_4 x_5; ?H3 x_1 x_2 x_3 x_6 x_7\<rbrakk> \<Longrightarrow> \<exists>y_0 y_1. ?H4 x_1 x_3 y_0 \<and> ?H4 x_1 y_1 x_7 \<and> ?H5 x_1 (?H6 (?H7 (\<lambda>y_2. ?H8) x_4 x_6) ?H9 (?H10 y_0)) x_5 y_1
|
[
"Option.option.Some",
"FJDefs.this",
"Option.option.None",
"Map.map_upds",
"Fun.fun_upd",
"FJDefs.typing",
"FJDefs.subtyping",
"FJDefs.mtype",
"FJDefs.mbody",
"FJDefs.ct_typing"
] |
[
"'a \\<Rightarrow> 'a option",
"nat",
"'a option",
"('a \\<Rightarrow> 'b option) \\<Rightarrow> 'a list \\<Rightarrow> 'b list \\<Rightarrow> 'a \\<Rightarrow> 'b option",
"('a \\<Rightarrow> 'b) \\<Rightarrow> 'a \\<Rightarrow> 'b \\<Rightarrow> 'a \\<Rightarrow> 'b",
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> (nat \\<Rightarrow> nat option) \\<Rightarrow> exp \\<Rightarrow> nat \\<Rightarrow> bool",
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> nat \\<Rightarrow> nat \\<Rightarrow> bool",
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> nat \\<Rightarrow> nat \\<Rightarrow> nat list \\<Rightarrow> nat \\<Rightarrow> bool",
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> nat \\<Rightarrow> nat \\<Rightarrow> nat list \\<Rightarrow> exp \\<Rightarrow> bool",
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> bool"
] |
[
"datatype 'a option =\n None\n | Some (the: 'a)",
"definition\n this :: \"varName\" where\n \"this == 0\"",
"definition map_upds :: \"('a \\<rightharpoonup> 'b) \\<Rightarrow> 'a list \\<Rightarrow> 'b list \\<Rightarrow> 'a \\<rightharpoonup> 'b\" where\n\"map_upds m xs ys = m ++ map_of (rev (zip xs ys))\"",
"definition fun_upd :: \"('a \\<Rightarrow> 'b) \\<Rightarrow> 'a \\<Rightarrow> 'b \\<Rightarrow> ('a \\<Rightarrow> 'b)\"\n where \"fun_upd f a b = (\\<lambda>x. if x = a then b else f x)\"",
"inductive\n typings :: \"[classTable, varCtx, exp list, className list] \\<Rightarrow> bool\" (\"_;_ \\<turnstile>+ _ : _\" [80,80,80,80] 80)\n and typing :: \"[classTable, varCtx, exp, className] \\<Rightarrow> bool\" (\"_;_ \\<turnstile> _ : _\" [80,80,80,80] 80)\nwhere\n ts_nil : \"CT;\\<Gamma> \\<turnstile>+ [] : []\"\n\n| ts_cons : \n \"\\<lbrakk> CT;\\<Gamma> \\<turnstile> e0 : C0; CT;\\<Gamma> \\<turnstile>+ es : Cs \\<rbrakk> \n \\<Longrightarrow> CT;\\<Gamma> \\<turnstile>+ (e0 # es) : (C0 # Cs)\"\n\n| t_var : \n \"\\<lbrakk> \\<Gamma>(x) = Some C \\<rbrakk> \\<Longrightarrow> CT;\\<Gamma> \\<turnstile> (Var x) : C\"\n\n| t_field : \n \"\\<lbrakk> CT;\\<Gamma> \\<turnstile> e0 : C0;\n fields(CT,C0) = Cf;\n lookup Cf (\\<lambda>fd.(vdName fd = fi)) = Some(fDef);\n vdType fDef = Ci \\<rbrakk>\n \\<Longrightarrow> CT;\\<Gamma> \\<turnstile> FieldProj e0 fi : Ci\"\n\n| t_invk : \n \"\\<lbrakk> CT;\\<Gamma> \\<turnstile> e0 : C0;\n mtype(CT,m,C0) = Ds \\<rightarrow> C;\n CT;\\<Gamma> \\<turnstile>+ es : Cs;\n CT \\<turnstile>+ Cs <: Ds;\n length es = length Ds \\<rbrakk>\n \\<Longrightarrow> CT;\\<Gamma> \\<turnstile> MethodInvk e0 m es : C\"\n\n| t_new : \n \"\\<lbrakk> fields(CT,C) = Df;\n length es = length Df;\n varDefs_types Df = Ds;\n CT;\\<Gamma> \\<turnstile>+ es : Cs;\n CT \\<turnstile>+ Cs <: Ds \\<rbrakk>\n \\<Longrightarrow> CT;\\<Gamma> \\<turnstile> New C es : C\"\n\n| t_ucast : \n \"\\<lbrakk> CT;\\<Gamma> \\<turnstile> e0 : D; \n CT \\<turnstile> D <: C \\<rbrakk>\n \\<Longrightarrow> CT;\\<Gamma> \\<turnstile> Cast C e0 : C\"\n\n| t_dcast : \n \"\\<lbrakk> CT;\\<Gamma> \\<turnstile> e0 : D; \n CT \\<turnstile> C <: D; C \\<noteq> D \\<rbrakk>\n \\<Longrightarrow> CT;\\<Gamma> \\<turnstile> Cast C e0 : C\"\n\n| t_scast : \n \"\\<lbrakk> CT;\\<Gamma> \\<turnstile> e0 : D;\n CT \\<turnstile> C \\<not><: D;\n CT \\<turnstile> D \\<not><: C \\<rbrakk>\n \\<Longrightarrow> CT;\\<Gamma> \\<turnstile> Cast C e0 : C\"",
"inductive\n subtyping :: \"[classTable, className, className] \\<Rightarrow> bool\" (\"_ \\<turnstile> _ <: _\" [80,80,80] 80)\nwhere\n s_refl : \"CT \\<turnstile> C <: C\"\n| s_trans : \"\\<lbrakk> CT \\<turnstile> C <: D; CT \\<turnstile> D <: E \\<rbrakk> \\<Longrightarrow> CT \\<turnstile> C <: E\"\n| s_super : \"\\<lbrakk> CT(C) = Some(CDef); cSuper CDef = D \\<rbrakk> \\<Longrightarrow> CT \\<turnstile> C <: D\"",
"inductive\n mtype :: \"[classTable, methodName, className, className list, className] \\<Rightarrow> bool\" (\"mtype'(_,_,_') = _ \\<rightarrow> _\" [80,80,80,80] 80)\nwhere\n mt_class: \n \"\\<lbrakk> CT(C) = Some(CDef);\n lookup (cMethods CDef) (\\<lambda>md.(mName md = m)) = Some(mDef);\n varDefs_types (mParams mDef) = Bs;\n mReturn mDef = B \\<rbrakk>\n \\<Longrightarrow> mtype(CT,m,C) = Bs \\<rightarrow> B\"\n\n| mt_super: \n \"\\<lbrakk> CT(C) = Some (CDef);\n lookup (cMethods CDef) (\\<lambda>md.(mName md = m)) = None;\n cSuper CDef = D;\n mtype(CT,m,D) = Bs \\<rightarrow> B \\<rbrakk>\n \\<Longrightarrow> mtype(CT,m,C) = Bs \\<rightarrow> B\"",
"inductive\n mbody :: \"[classTable, methodName, className, varName list, exp] \\<Rightarrow> bool\" (\"mbody'(_,_,_') = _ . _\" [80,80,80,80] 80)\nwhere\n mb_class: \n \"\\<lbrakk> CT(C) = Some(CDef);\n lookup (cMethods CDef) (\\<lambda>md.(mName md = m)) = Some(mDef);\n varDefs_names (mParams mDef) = xs;\n mBody mDef = e \\<rbrakk>\n \\<Longrightarrow> mbody(CT,m,C) = xs . e\"\n\n| mb_super:\n \"\\<lbrakk> CT(C) = Some(CDef);\n lookup (cMethods CDef) (\\<lambda>md.(mName md = m)) = None;\n cSuper CDef = D;\n mbody(CT,m,D) = xs . e \\<rbrakk>\n \\<Longrightarrow> mbody(CT,m,C) = xs . e\"",
"inductive\n ct_typing :: \"classTable \\<Rightarrow> bool\" (\"_ OK\" 80)\nwhere\nct_all_ok: \n \"\\<lbrakk> Object \\<notin> dom(CT); \n \\<forall>C CDef. CT(C) = Some(CDef) \\<longrightarrow> (CT \\<turnstile> CDef OK) \\<and> (cName CDef = C) \\<rbrakk>\n \\<Longrightarrow> CT OK\""
] |
FeatherweightJava/FJSound
|
FJSound.Thm_2_4_2_1
|
theorem Thm_2_4_2_1:
assumes "CT;Map.empty \<turnstile> e : C"
and "FieldProj (New C0 es) fi \<in> subexprs(e)"
shows "\<exists>Cf fDef. fields(CT, C0) = Cf \<and> lookup Cf (\<lambda>fd. (vdName fd = fi)) = Some fDef"
|
?CT;(\<lambda>x. None) \<turnstile> ?e : ?C \<Longrightarrow> FieldProj (New ?C0.0 ?es) ?fi \<in> subexprs(?e) \<Longrightarrow> \<exists>Cf fDef. fields(?CT, ?C0.0) = Cf \<and> lookup Cf (\<lambda>fd. vdName fd = ?fi) = Some fDef
|
\<lbrakk> ?H1 x_1 (\<lambda>y_0. ?H2) x_2 x_3; ?H3 (?H4 (?H5 x_4 x_5) x_6) x_2\<rbrakk> \<Longrightarrow> \<exists>y_1 y_2. ?H6 x_1 x_4 y_1 \<and> ?H7 y_1 (\<lambda>y_3. ?H8 y_3 = x_6) = ?H9 y_2
|
[
"Option.option.Some",
"FJDefs.varDef.vdName",
"FJDefs.lookup",
"FJDefs.fields",
"FJDefs.exp.New",
"FJDefs.exp.FieldProj",
"FJDefs.subexprs",
"Option.option.None",
"FJDefs.typing"
] |
[
"'a \\<Rightarrow> 'a option",
"'a varDef_scheme \\<Rightarrow> nat",
"'a list \\<Rightarrow> ('a \\<Rightarrow> bool) \\<Rightarrow> 'a option",
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> nat \\<Rightarrow> varDef list \\<Rightarrow> bool",
"nat \\<Rightarrow> exp list \\<Rightarrow> exp",
"exp \\<Rightarrow> nat \\<Rightarrow> exp",
"exp \\<Rightarrow> exp \\<Rightarrow> bool",
"'a option",
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> (nat \\<Rightarrow> nat option) \\<Rightarrow> exp \\<Rightarrow> nat \\<Rightarrow> bool"
] |
[
"datatype 'a option =\n None\n | Some (the: 'a)",
"record varDef = \n vdName :: \"varName\"\n vdType :: \"className\"",
"primrec lookup :: \"'a list \\<Rightarrow> ('a \\<Rightarrow> bool) \\<Rightarrow> 'a option\"\nwhere\n \"lookup [] P = None\"\n| \"lookup (h#t) P = (if P h then Some h else lookup t P)\"",
"inductive\n fields :: \"[classTable, className, varDef list] \\<Rightarrow> bool\" (\"fields'(_,_') = _\" [80,80,80] 80)\nwhere\n f_obj: \n \"fields(CT,Object) = []\"\n| f_class: \n \"\\<lbrakk> CT(C) = Some(CDef); cSuper CDef = D; cFields CDef = Cf; fields(CT,D) = Dg; DgCf = Dg @ Cf \\<rbrakk> \n \\<Longrightarrow> fields(CT,C) = DgCf\"",
"datatype exp = \n Var \"varName\"\n | FieldProj \"exp\" \"varName\" \n | MethodInvk \"exp\" \"methodName\" \"exp list\"\n | New \"className\" \"exp list\"\n | Cast \"className\" \"exp\"",
"abbreviation\n subexprs :: \"[exp,exp] \\<Rightarrow> bool\" (\"_ \\<in> subexprs'(_')\" [80,80] 80) where\n \"e' \\<in> subexprs(e) \\<equiv> (e',e) \\<in> isubexprs^*\"",
"inductive\n typings :: \"[classTable, varCtx, exp list, className list] \\<Rightarrow> bool\" (\"_;_ \\<turnstile>+ _ : _\" [80,80,80,80] 80)\n and typing :: \"[classTable, varCtx, exp, className] \\<Rightarrow> bool\" (\"_;_ \\<turnstile> _ : _\" [80,80,80,80] 80)\nwhere\n ts_nil : \"CT;\\<Gamma> \\<turnstile>+ [] : []\"\n\n| ts_cons : \n \"\\<lbrakk> CT;\\<Gamma> \\<turnstile> e0 : C0; CT;\\<Gamma> \\<turnstile>+ es : Cs \\<rbrakk> \n \\<Longrightarrow> CT;\\<Gamma> \\<turnstile>+ (e0 # es) : (C0 # Cs)\"\n\n| t_var : \n \"\\<lbrakk> \\<Gamma>(x) = Some C \\<rbrakk> \\<Longrightarrow> CT;\\<Gamma> \\<turnstile> (Var x) : C\"\n\n| t_field : \n \"\\<lbrakk> CT;\\<Gamma> \\<turnstile> e0 : C0;\n fields(CT,C0) = Cf;\n lookup Cf (\\<lambda>fd.(vdName fd = fi)) = Some(fDef);\n vdType fDef = Ci \\<rbrakk>\n \\<Longrightarrow> CT;\\<Gamma> \\<turnstile> FieldProj e0 fi : Ci\"\n\n| t_invk : \n \"\\<lbrakk> CT;\\<Gamma> \\<turnstile> e0 : C0;\n mtype(CT,m,C0) = Ds \\<rightarrow> C;\n CT;\\<Gamma> \\<turnstile>+ es : Cs;\n CT \\<turnstile>+ Cs <: Ds;\n length es = length Ds \\<rbrakk>\n \\<Longrightarrow> CT;\\<Gamma> \\<turnstile> MethodInvk e0 m es : C\"\n\n| t_new : \n \"\\<lbrakk> fields(CT,C) = Df;\n length es = length Df;\n varDefs_types Df = Ds;\n CT;\\<Gamma> \\<turnstile>+ es : Cs;\n CT \\<turnstile>+ Cs <: Ds \\<rbrakk>\n \\<Longrightarrow> CT;\\<Gamma> \\<turnstile> New C es : C\"\n\n| t_ucast : \n \"\\<lbrakk> CT;\\<Gamma> \\<turnstile> e0 : D; \n CT \\<turnstile> D <: C \\<rbrakk>\n \\<Longrightarrow> CT;\\<Gamma> \\<turnstile> Cast C e0 : C\"\n\n| t_dcast : \n \"\\<lbrakk> CT;\\<Gamma> \\<turnstile> e0 : D; \n CT \\<turnstile> C <: D; C \\<noteq> D \\<rbrakk>\n \\<Longrightarrow> CT;\\<Gamma> \\<turnstile> Cast C e0 : C\"\n\n| t_scast : \n \"\\<lbrakk> CT;\\<Gamma> \\<turnstile> e0 : D;\n CT \\<turnstile> C \\<not><: D;\n CT \\<turnstile> D \\<not><: C \\<rbrakk>\n \\<Longrightarrow> CT;\\<Gamma> \\<turnstile> Cast C e0 : C\""
] |
FeatherweightJava/FJSound
|
FJSound.Thm_2_4_2_2
|
lemma Thm_2_4_2_2:
fixes es ds :: "exp list"
assumes "CT;Map.empty \<turnstile> e : C"
and "MethodInvk (New C0 es) m ds \<in> subexprs(e)"
shows "\<exists>xs e0. mbody(CT,m,C0) = xs . e0 \<and> length xs = length ds"
|
?CT;(\<lambda>x. None) \<turnstile> ?e : ?C \<Longrightarrow> MethodInvk (New ?C0.0 ?es) ?m ?ds \<in> subexprs(?e) \<Longrightarrow> \<exists>xs e0. mbody(?CT, ?m, ?C0.0) = xs . e0 \<and> length xs = length ?ds
|
\<lbrakk> ?H1 x_1 (\<lambda>y_0. ?H2) x_2 x_3; ?H3 (?H4 (?H5 x_4 x_5) x_6 x_7) x_2\<rbrakk> \<Longrightarrow> \<exists>y_1 y_2. ?H6 x_1 x_6 x_4 y_1 y_2 \<and> ?H7 y_1 = ?H8 x_7
|
[
"List.length",
"FJDefs.mbody",
"FJDefs.exp.New",
"FJDefs.exp.MethodInvk",
"FJDefs.subexprs",
"Option.option.None",
"FJDefs.typing"
] |
[
"'a list \\<Rightarrow> nat",
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> nat \\<Rightarrow> nat \\<Rightarrow> nat list \\<Rightarrow> exp \\<Rightarrow> bool",
"nat \\<Rightarrow> exp list \\<Rightarrow> exp",
"exp \\<Rightarrow> nat \\<Rightarrow> exp list \\<Rightarrow> exp",
"exp \\<Rightarrow> exp \\<Rightarrow> bool",
"'a option",
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> (nat \\<Rightarrow> nat option) \\<Rightarrow> exp \\<Rightarrow> nat \\<Rightarrow> bool"
] |
[
"abbreviation length :: \"'a list \\<Rightarrow> nat\" where\n\"length \\<equiv> size\"",
"inductive\n mbody :: \"[classTable, methodName, className, varName list, exp] \\<Rightarrow> bool\" (\"mbody'(_,_,_') = _ . _\" [80,80,80,80] 80)\nwhere\n mb_class: \n \"\\<lbrakk> CT(C) = Some(CDef);\n lookup (cMethods CDef) (\\<lambda>md.(mName md = m)) = Some(mDef);\n varDefs_names (mParams mDef) = xs;\n mBody mDef = e \\<rbrakk>\n \\<Longrightarrow> mbody(CT,m,C) = xs . e\"\n\n| mb_super:\n \"\\<lbrakk> CT(C) = Some(CDef);\n lookup (cMethods CDef) (\\<lambda>md.(mName md = m)) = None;\n cSuper CDef = D;\n mbody(CT,m,D) = xs . e \\<rbrakk>\n \\<Longrightarrow> mbody(CT,m,C) = xs . e\"",
"datatype exp = \n Var \"varName\"\n | FieldProj \"exp\" \"varName\" \n | MethodInvk \"exp\" \"methodName\" \"exp list\"\n | New \"className\" \"exp list\"\n | Cast \"className\" \"exp\"",
"abbreviation\n subexprs :: \"[exp,exp] \\<Rightarrow> bool\" (\"_ \\<in> subexprs'(_')\" [80,80] 80) where\n \"e' \\<in> subexprs(e) \\<equiv> (e',e) \\<in> isubexprs^*\"",
"datatype 'a option =\n None\n | Some (the: 'a)",
"inductive\n typings :: \"[classTable, varCtx, exp list, className list] \\<Rightarrow> bool\" (\"_;_ \\<turnstile>+ _ : _\" [80,80,80,80] 80)\n and typing :: \"[classTable, varCtx, exp, className] \\<Rightarrow> bool\" (\"_;_ \\<turnstile> _ : _\" [80,80,80,80] 80)\nwhere\n ts_nil : \"CT;\\<Gamma> \\<turnstile>+ [] : []\"\n\n| ts_cons : \n \"\\<lbrakk> CT;\\<Gamma> \\<turnstile> e0 : C0; CT;\\<Gamma> \\<turnstile>+ es : Cs \\<rbrakk> \n \\<Longrightarrow> CT;\\<Gamma> \\<turnstile>+ (e0 # es) : (C0 # Cs)\"\n\n| t_var : \n \"\\<lbrakk> \\<Gamma>(x) = Some C \\<rbrakk> \\<Longrightarrow> CT;\\<Gamma> \\<turnstile> (Var x) : C\"\n\n| t_field : \n \"\\<lbrakk> CT;\\<Gamma> \\<turnstile> e0 : C0;\n fields(CT,C0) = Cf;\n lookup Cf (\\<lambda>fd.(vdName fd = fi)) = Some(fDef);\n vdType fDef = Ci \\<rbrakk>\n \\<Longrightarrow> CT;\\<Gamma> \\<turnstile> FieldProj e0 fi : Ci\"\n\n| t_invk : \n \"\\<lbrakk> CT;\\<Gamma> \\<turnstile> e0 : C0;\n mtype(CT,m,C0) = Ds \\<rightarrow> C;\n CT;\\<Gamma> \\<turnstile>+ es : Cs;\n CT \\<turnstile>+ Cs <: Ds;\n length es = length Ds \\<rbrakk>\n \\<Longrightarrow> CT;\\<Gamma> \\<turnstile> MethodInvk e0 m es : C\"\n\n| t_new : \n \"\\<lbrakk> fields(CT,C) = Df;\n length es = length Df;\n varDefs_types Df = Ds;\n CT;\\<Gamma> \\<turnstile>+ es : Cs;\n CT \\<turnstile>+ Cs <: Ds \\<rbrakk>\n \\<Longrightarrow> CT;\\<Gamma> \\<turnstile> New C es : C\"\n\n| t_ucast : \n \"\\<lbrakk> CT;\\<Gamma> \\<turnstile> e0 : D; \n CT \\<turnstile> D <: C \\<rbrakk>\n \\<Longrightarrow> CT;\\<Gamma> \\<turnstile> Cast C e0 : C\"\n\n| t_dcast : \n \"\\<lbrakk> CT;\\<Gamma> \\<turnstile> e0 : D; \n CT \\<turnstile> C <: D; C \\<noteq> D \\<rbrakk>\n \\<Longrightarrow> CT;\\<Gamma> \\<turnstile> Cast C e0 : C\"\n\n| t_scast : \n \"\\<lbrakk> CT;\\<Gamma> \\<turnstile> e0 : D;\n CT \\<turnstile> C \\<not><: D;\n CT \\<turnstile> D \\<not><: C \\<rbrakk>\n \\<Longrightarrow> CT;\\<Gamma> \\<turnstile> Cast C e0 : C\""
] |
FeatherweightJava/FJSound
|
FJSound.Cor_2_4_1_multi
| null |
?CT \<turnstile> ?e \<rightarrow>* ?e' \<Longrightarrow> ?CT OK \<Longrightarrow> ?CT; ?\<Gamma> \<turnstile> ?e : ?C \<Longrightarrow> \<exists>C'. ?CT; ?\<Gamma> \<turnstile> ?e' : C' \<and> ?CT \<turnstile> C' <: ?C
|
\<lbrakk> ?H1 x_1 x_2 x_3; ?H2 x_1; ?H3 x_1 x_4 x_2 x_5\<rbrakk> \<Longrightarrow> \<exists>y_0. ?H3 x_1 x_4 x_3 y_0 \<and> ?H4 x_1 y_0 x_5
|
[
"FJDefs.subtyping",
"FJDefs.typing",
"FJDefs.ct_typing",
"FJDefs.reductions"
] |
[
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> nat \\<Rightarrow> nat \\<Rightarrow> bool",
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> (nat \\<Rightarrow> nat option) \\<Rightarrow> exp \\<Rightarrow> nat \\<Rightarrow> bool",
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> bool",
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> exp \\<Rightarrow> exp \\<Rightarrow> bool"
] |
[
"inductive\n subtyping :: \"[classTable, className, className] \\<Rightarrow> bool\" (\"_ \\<turnstile> _ <: _\" [80,80,80] 80)\nwhere\n s_refl : \"CT \\<turnstile> C <: C\"\n| s_trans : \"\\<lbrakk> CT \\<turnstile> C <: D; CT \\<turnstile> D <: E \\<rbrakk> \\<Longrightarrow> CT \\<turnstile> C <: E\"\n| s_super : \"\\<lbrakk> CT(C) = Some(CDef); cSuper CDef = D \\<rbrakk> \\<Longrightarrow> CT \\<turnstile> C <: D\"",
"inductive\n typings :: \"[classTable, varCtx, exp list, className list] \\<Rightarrow> bool\" (\"_;_ \\<turnstile>+ _ : _\" [80,80,80,80] 80)\n and typing :: \"[classTable, varCtx, exp, className] \\<Rightarrow> bool\" (\"_;_ \\<turnstile> _ : _\" [80,80,80,80] 80)\nwhere\n ts_nil : \"CT;\\<Gamma> \\<turnstile>+ [] : []\"\n\n| ts_cons : \n \"\\<lbrakk> CT;\\<Gamma> \\<turnstile> e0 : C0; CT;\\<Gamma> \\<turnstile>+ es : Cs \\<rbrakk> \n \\<Longrightarrow> CT;\\<Gamma> \\<turnstile>+ (e0 # es) : (C0 # Cs)\"\n\n| t_var : \n \"\\<lbrakk> \\<Gamma>(x) = Some C \\<rbrakk> \\<Longrightarrow> CT;\\<Gamma> \\<turnstile> (Var x) : C\"\n\n| t_field : \n \"\\<lbrakk> CT;\\<Gamma> \\<turnstile> e0 : C0;\n fields(CT,C0) = Cf;\n lookup Cf (\\<lambda>fd.(vdName fd = fi)) = Some(fDef);\n vdType fDef = Ci \\<rbrakk>\n \\<Longrightarrow> CT;\\<Gamma> \\<turnstile> FieldProj e0 fi : Ci\"\n\n| t_invk : \n \"\\<lbrakk> CT;\\<Gamma> \\<turnstile> e0 : C0;\n mtype(CT,m,C0) = Ds \\<rightarrow> C;\n CT;\\<Gamma> \\<turnstile>+ es : Cs;\n CT \\<turnstile>+ Cs <: Ds;\n length es = length Ds \\<rbrakk>\n \\<Longrightarrow> CT;\\<Gamma> \\<turnstile> MethodInvk e0 m es : C\"\n\n| t_new : \n \"\\<lbrakk> fields(CT,C) = Df;\n length es = length Df;\n varDefs_types Df = Ds;\n CT;\\<Gamma> \\<turnstile>+ es : Cs;\n CT \\<turnstile>+ Cs <: Ds \\<rbrakk>\n \\<Longrightarrow> CT;\\<Gamma> \\<turnstile> New C es : C\"\n\n| t_ucast : \n \"\\<lbrakk> CT;\\<Gamma> \\<turnstile> e0 : D; \n CT \\<turnstile> D <: C \\<rbrakk>\n \\<Longrightarrow> CT;\\<Gamma> \\<turnstile> Cast C e0 : C\"\n\n| t_dcast : \n \"\\<lbrakk> CT;\\<Gamma> \\<turnstile> e0 : D; \n CT \\<turnstile> C <: D; C \\<noteq> D \\<rbrakk>\n \\<Longrightarrow> CT;\\<Gamma> \\<turnstile> Cast C e0 : C\"\n\n| t_scast : \n \"\\<lbrakk> CT;\\<Gamma> \\<turnstile> e0 : D;\n CT \\<turnstile> C \\<not><: D;\n CT \\<turnstile> D \\<not><: C \\<rbrakk>\n \\<Longrightarrow> CT;\\<Gamma> \\<turnstile> Cast C e0 : C\"",
"inductive\n ct_typing :: \"classTable \\<Rightarrow> bool\" (\"_ OK\" 80)\nwhere\nct_all_ok: \n \"\\<lbrakk> Object \\<notin> dom(CT); \n \\<forall>C CDef. CT(C) = Some(CDef) \\<longrightarrow> (CT \\<turnstile> CDef OK) \\<and> (cName CDef = C) \\<rbrakk>\n \\<Longrightarrow> CT OK\"",
"inductive\n reductions :: \"[classTable, exp, exp] \\<Rightarrow> bool\" (\"_ \\<turnstile> _ \\<rightarrow>* _\" [80,80,80] 80)\nwhere\n rs_refl: \"CT \\<turnstile> e \\<rightarrow>* e\" \n| rs_trans: \"\\<lbrakk> CT \\<turnstile> e \\<rightarrow> e'; CT \\<turnstile> e' \\<rightarrow>* e'' \\<rbrakk> \\<Longrightarrow> CT \\<turnstile> e \\<rightarrow>* e''\""
] |
FeatherweightJava/FJSound
|
FJSound.Thm_2_4_1
|
theorem Thm_2_4_1:
assumes "CT \<turnstile> e \<rightarrow> e'"
and "CT OK"
shows "\<And>C. \<lbrakk> CT;\<Gamma> \<turnstile> e : C \<rbrakk>
\<Longrightarrow> \<exists>C'. (CT;\<Gamma> \<turnstile> e' : C' \<and> CT \<turnstile> C' <: C)"
|
?CT \<turnstile> ?e \<rightarrow> ?e' \<Longrightarrow> ?CT OK \<Longrightarrow> ?CT; ?\<Gamma> \<turnstile> ?e : ?C \<Longrightarrow> \<exists>C'. ?CT; ?\<Gamma> \<turnstile> ?e' : C' \<and> ?CT \<turnstile> C' <: ?C
|
\<lbrakk> ?H1 x_1 x_2 x_3; ?H2 x_1; ?H3 x_1 x_4 x_2 x_5\<rbrakk> \<Longrightarrow> \<exists>y_0. ?H3 x_1 x_4 x_3 y_0 \<and> ?H4 x_1 y_0 x_5
|
[
"FJDefs.subtyping",
"FJDefs.typing",
"FJDefs.ct_typing",
"FJDefs.reduction"
] |
[
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> nat \\<Rightarrow> nat \\<Rightarrow> bool",
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> (nat \\<Rightarrow> nat option) \\<Rightarrow> exp \\<Rightarrow> nat \\<Rightarrow> bool",
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> bool",
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> exp \\<Rightarrow> exp \\<Rightarrow> bool"
] |
[
"inductive\n subtyping :: \"[classTable, className, className] \\<Rightarrow> bool\" (\"_ \\<turnstile> _ <: _\" [80,80,80] 80)\nwhere\n s_refl : \"CT \\<turnstile> C <: C\"\n| s_trans : \"\\<lbrakk> CT \\<turnstile> C <: D; CT \\<turnstile> D <: E \\<rbrakk> \\<Longrightarrow> CT \\<turnstile> C <: E\"\n| s_super : \"\\<lbrakk> CT(C) = Some(CDef); cSuper CDef = D \\<rbrakk> \\<Longrightarrow> CT \\<turnstile> C <: D\"",
"inductive\n typings :: \"[classTable, varCtx, exp list, className list] \\<Rightarrow> bool\" (\"_;_ \\<turnstile>+ _ : _\" [80,80,80,80] 80)\n and typing :: \"[classTable, varCtx, exp, className] \\<Rightarrow> bool\" (\"_;_ \\<turnstile> _ : _\" [80,80,80,80] 80)\nwhere\n ts_nil : \"CT;\\<Gamma> \\<turnstile>+ [] : []\"\n\n| ts_cons : \n \"\\<lbrakk> CT;\\<Gamma> \\<turnstile> e0 : C0; CT;\\<Gamma> \\<turnstile>+ es : Cs \\<rbrakk> \n \\<Longrightarrow> CT;\\<Gamma> \\<turnstile>+ (e0 # es) : (C0 # Cs)\"\n\n| t_var : \n \"\\<lbrakk> \\<Gamma>(x) = Some C \\<rbrakk> \\<Longrightarrow> CT;\\<Gamma> \\<turnstile> (Var x) : C\"\n\n| t_field : \n \"\\<lbrakk> CT;\\<Gamma> \\<turnstile> e0 : C0;\n fields(CT,C0) = Cf;\n lookup Cf (\\<lambda>fd.(vdName fd = fi)) = Some(fDef);\n vdType fDef = Ci \\<rbrakk>\n \\<Longrightarrow> CT;\\<Gamma> \\<turnstile> FieldProj e0 fi : Ci\"\n\n| t_invk : \n \"\\<lbrakk> CT;\\<Gamma> \\<turnstile> e0 : C0;\n mtype(CT,m,C0) = Ds \\<rightarrow> C;\n CT;\\<Gamma> \\<turnstile>+ es : Cs;\n CT \\<turnstile>+ Cs <: Ds;\n length es = length Ds \\<rbrakk>\n \\<Longrightarrow> CT;\\<Gamma> \\<turnstile> MethodInvk e0 m es : C\"\n\n| t_new : \n \"\\<lbrakk> fields(CT,C) = Df;\n length es = length Df;\n varDefs_types Df = Ds;\n CT;\\<Gamma> \\<turnstile>+ es : Cs;\n CT \\<turnstile>+ Cs <: Ds \\<rbrakk>\n \\<Longrightarrow> CT;\\<Gamma> \\<turnstile> New C es : C\"\n\n| t_ucast : \n \"\\<lbrakk> CT;\\<Gamma> \\<turnstile> e0 : D; \n CT \\<turnstile> D <: C \\<rbrakk>\n \\<Longrightarrow> CT;\\<Gamma> \\<turnstile> Cast C e0 : C\"\n\n| t_dcast : \n \"\\<lbrakk> CT;\\<Gamma> \\<turnstile> e0 : D; \n CT \\<turnstile> C <: D; C \\<noteq> D \\<rbrakk>\n \\<Longrightarrow> CT;\\<Gamma> \\<turnstile> Cast C e0 : C\"\n\n| t_scast : \n \"\\<lbrakk> CT;\\<Gamma> \\<turnstile> e0 : D;\n CT \\<turnstile> C \\<not><: D;\n CT \\<turnstile> D \\<not><: C \\<rbrakk>\n \\<Longrightarrow> CT;\\<Gamma> \\<turnstile> Cast C e0 : C\"",
"inductive\n ct_typing :: \"classTable \\<Rightarrow> bool\" (\"_ OK\" 80)\nwhere\nct_all_ok: \n \"\\<lbrakk> Object \\<notin> dom(CT); \n \\<forall>C CDef. CT(C) = Some(CDef) \\<longrightarrow> (CT \\<turnstile> CDef OK) \\<and> (cName CDef = C) \\<rbrakk>\n \\<Longrightarrow> CT OK\"",
"inductive\n reduction :: \"[classTable, exp, exp] \\<Rightarrow> bool\" (\"_ \\<turnstile> _ \\<rightarrow> _\" [80,80,80] 80)\nwhere\n\n r_field: \n \"\\<lbrakk> fields(CT,C) = Cf; \n lookup2 Cf es (\\<lambda>fd.(vdName fd = fi)) = Some(ei) \\<rbrakk>\n \\<Longrightarrow> CT \\<turnstile> FieldProj (New C es) fi \\<rightarrow> ei\"\n\n| r_invk: \n \"\\<lbrakk> mbody(CT,m,C) = xs . e0;\n substs ((map_upds Map.empty xs ds)(this \\<mapsto> (New C es))) e0 = e0' \\<rbrakk>\n \\<Longrightarrow> CT \\<turnstile> MethodInvk (New C es) m ds \\<rightarrow> e0'\"\n\n| r_cast: \n \"\\<lbrakk> CT \\<turnstile> C <: D \\<rbrakk> \n \\<Longrightarrow> CT \\<turnstile> Cast D (New C es) \\<rightarrow> New C es\"\n\n| rc_field: \n \"\\<lbrakk> CT \\<turnstile> e0 \\<rightarrow> e0' \\<rbrakk> \n \\<Longrightarrow> CT \\<turnstile> FieldProj e0 f \\<rightarrow> FieldProj e0' f\"\n\n| rc_invk_recv: \n \"\\<lbrakk> CT \\<turnstile> e0 \\<rightarrow> e0' \\<rbrakk> \n \\<Longrightarrow> CT \\<turnstile> MethodInvk e0 m es \\<rightarrow> MethodInvk e0' m es\"\n\n| rc_invk_arg: \n \"\\<lbrakk> CT \\<turnstile> ei \\<rightarrow> ei' \\<rbrakk>\n \\<Longrightarrow> CT \\<turnstile> MethodInvk e0 m (el@ei#er) \\<rightarrow> MethodInvk e0 m (el@ei'#er)\"\n\n| rc_new_arg: \n \"\\<lbrakk> CT \\<turnstile> ei \\<rightarrow> ei' \\<rbrakk> \n \\<Longrightarrow> CT \\<turnstile> New C (el@ei#er) \\<rightarrow> New C (el@ei'#er)\"\n\n| rc_cast: \n \"\\<lbrakk> CT \\<turnstile> e0 \\<rightarrow> e0' \\<rbrakk> \n \\<Longrightarrow> CT \\<turnstile> Cast C e0 \\<rightarrow> Cast C e0'\""
] |
FeatherweightJava/FJSound
|
FJSound.A_1_1
|
lemma A_1_1:
assumes "CT \<turnstile> C <: D" and "CT OK"
shows "(mtype(CT,m,D) = Cs \<rightarrow> C0) \<Longrightarrow> (mtype(CT,m,C) = Cs \<rightarrow> C0)"
|
?CT \<turnstile> ?C <: ?D \<Longrightarrow> ?CT OK \<Longrightarrow> mtype(?CT, ?m, ?D) = ?Cs \<rightarrow> ?C0.0 \<Longrightarrow> mtype(?CT, ?m, ?C) = ?Cs \<rightarrow> ?C0.0
|
\<lbrakk> ?H1 x_1 x_2 x_3; ?H2 x_1; ?H3 x_1 x_4 x_3 x_5 x_6\<rbrakk> \<Longrightarrow> ?H3 x_1 x_4 x_2 x_5 x_6
|
[
"FJDefs.mtype",
"FJDefs.ct_typing",
"FJDefs.subtyping"
] |
[
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> nat \\<Rightarrow> nat \\<Rightarrow> nat list \\<Rightarrow> nat \\<Rightarrow> bool",
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> bool",
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> nat \\<Rightarrow> nat \\<Rightarrow> bool"
] |
[
"inductive\n mtype :: \"[classTable, methodName, className, className list, className] \\<Rightarrow> bool\" (\"mtype'(_,_,_') = _ \\<rightarrow> _\" [80,80,80,80] 80)\nwhere\n mt_class: \n \"\\<lbrakk> CT(C) = Some(CDef);\n lookup (cMethods CDef) (\\<lambda>md.(mName md = m)) = Some(mDef);\n varDefs_types (mParams mDef) = Bs;\n mReturn mDef = B \\<rbrakk>\n \\<Longrightarrow> mtype(CT,m,C) = Bs \\<rightarrow> B\"\n\n| mt_super: \n \"\\<lbrakk> CT(C) = Some (CDef);\n lookup (cMethods CDef) (\\<lambda>md.(mName md = m)) = None;\n cSuper CDef = D;\n mtype(CT,m,D) = Bs \\<rightarrow> B \\<rbrakk>\n \\<Longrightarrow> mtype(CT,m,C) = Bs \\<rightarrow> B\"",
"inductive\n ct_typing :: \"classTable \\<Rightarrow> bool\" (\"_ OK\" 80)\nwhere\nct_all_ok: \n \"\\<lbrakk> Object \\<notin> dom(CT); \n \\<forall>C CDef. CT(C) = Some(CDef) \\<longrightarrow> (CT \\<turnstile> CDef OK) \\<and> (cName CDef = C) \\<rbrakk>\n \\<Longrightarrow> CT OK\"",
"inductive\n subtyping :: \"[classTable, className, className] \\<Rightarrow> bool\" (\"_ \\<turnstile> _ <: _\" [80,80,80] 80)\nwhere\n s_refl : \"CT \\<turnstile> C <: C\"\n| s_trans : \"\\<lbrakk> CT \\<turnstile> C <: D; CT \\<turnstile> D <: E \\<rbrakk> \\<Longrightarrow> CT \\<turnstile> C <: E\"\n| s_super : \"\\<lbrakk> CT(C) = Some(CDef); cSuper CDef = D \\<rbrakk> \\<Longrightarrow> CT \\<turnstile> C <: D\""
] |
FeatherweightJava/FJSound
|
FJSound.mtype_mbody
|
lemma mtype_mbody:
fixes Cs :: "nat list"
assumes "mtype(CT,m,C) = Cs \<rightarrow> C0"
shows "\<exists>xs e. mbody(CT,m,C) = xs . e \<and> length xs = length Cs"
|
mtype(?CT, ?m, ?C) = ?Cs \<rightarrow> ?C0.0 \<Longrightarrow> \<exists>xs e. mbody(?CT, ?m, ?C) = xs . e \<and> length xs = length ?Cs
|
?H1 x_1 x_2 x_3 x_4 x_5 \<Longrightarrow> \<exists>y_0 y_1. ?H2 x_1 x_2 x_3 y_0 y_1 \<and> ?H3 y_0 = ?H3 x_4
|
[
"List.length",
"FJDefs.mbody",
"FJDefs.mtype"
] |
[
"'a list \\<Rightarrow> nat",
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> nat \\<Rightarrow> nat \\<Rightarrow> nat list \\<Rightarrow> exp \\<Rightarrow> bool",
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> nat \\<Rightarrow> nat \\<Rightarrow> nat list \\<Rightarrow> nat \\<Rightarrow> bool"
] |
[
"abbreviation length :: \"'a list \\<Rightarrow> nat\" where\n\"length \\<equiv> size\"",
"inductive\n mbody :: \"[classTable, methodName, className, varName list, exp] \\<Rightarrow> bool\" (\"mbody'(_,_,_') = _ . _\" [80,80,80,80] 80)\nwhere\n mb_class: \n \"\\<lbrakk> CT(C) = Some(CDef);\n lookup (cMethods CDef) (\\<lambda>md.(mName md = m)) = Some(mDef);\n varDefs_names (mParams mDef) = xs;\n mBody mDef = e \\<rbrakk>\n \\<Longrightarrow> mbody(CT,m,C) = xs . e\"\n\n| mb_super:\n \"\\<lbrakk> CT(C) = Some(CDef);\n lookup (cMethods CDef) (\\<lambda>md.(mName md = m)) = None;\n cSuper CDef = D;\n mbody(CT,m,D) = xs . e \\<rbrakk>\n \\<Longrightarrow> mbody(CT,m,C) = xs . e\"",
"inductive\n mtype :: \"[classTable, methodName, className, className list, className] \\<Rightarrow> bool\" (\"mtype'(_,_,_') = _ \\<rightarrow> _\" [80,80,80,80] 80)\nwhere\n mt_class: \n \"\\<lbrakk> CT(C) = Some(CDef);\n lookup (cMethods CDef) (\\<lambda>md.(mName md = m)) = Some(mDef);\n varDefs_types (mParams mDef) = Bs;\n mReturn mDef = B \\<rbrakk>\n \\<Longrightarrow> mtype(CT,m,C) = Bs \\<rightarrow> B\"\n\n| mt_super: \n \"\\<lbrakk> CT(C) = Some (CDef);\n lookup (cMethods CDef) (\\<lambda>md.(mName md = m)) = None;\n cSuper CDef = D;\n mtype(CT,m,D) = Bs \\<rightarrow> B \\<rbrakk>\n \\<Longrightarrow> mtype(CT,m,C) = Bs \\<rightarrow> B\""
] |
FeatherweightJava/FJSound
|
FJSound.mtype_mbody_length
|
lemma mtype_mbody_length:
assumes mt:"mtype(CT,m,C) = Cs \<rightarrow> C0"
and mb:"mbody(CT,m,C) = xs . e"
shows "length xs = length Cs"
|
mtype(?CT, ?m, ?C) = ?Cs \<rightarrow> ?C0.0 \<Longrightarrow> mbody(?CT, ?m, ?C) = ?xs . ?e \<Longrightarrow> length ?xs = length ?Cs
|
\<lbrakk> ?H1 x_1 x_2 x_3 x_4 x_5; ?H2 x_1 x_2 x_3 x_6 x_7\<rbrakk> \<Longrightarrow> ?H3 x_6 = ?H3 x_4
|
[
"List.length",
"FJDefs.mbody",
"FJDefs.mtype"
] |
[
"'a list \\<Rightarrow> nat",
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> nat \\<Rightarrow> nat \\<Rightarrow> nat list \\<Rightarrow> exp \\<Rightarrow> bool",
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> nat \\<Rightarrow> nat \\<Rightarrow> nat list \\<Rightarrow> nat \\<Rightarrow> bool"
] |
[
"abbreviation length :: \"'a list \\<Rightarrow> nat\" where\n\"length \\<equiv> size\"",
"inductive\n mbody :: \"[classTable, methodName, className, varName list, exp] \\<Rightarrow> bool\" (\"mbody'(_,_,_') = _ . _\" [80,80,80,80] 80)\nwhere\n mb_class: \n \"\\<lbrakk> CT(C) = Some(CDef);\n lookup (cMethods CDef) (\\<lambda>md.(mName md = m)) = Some(mDef);\n varDefs_names (mParams mDef) = xs;\n mBody mDef = e \\<rbrakk>\n \\<Longrightarrow> mbody(CT,m,C) = xs . e\"\n\n| mb_super:\n \"\\<lbrakk> CT(C) = Some(CDef);\n lookup (cMethods CDef) (\\<lambda>md.(mName md = m)) = None;\n cSuper CDef = D;\n mbody(CT,m,D) = xs . e \\<rbrakk>\n \\<Longrightarrow> mbody(CT,m,C) = xs . e\"",
"inductive\n mtype :: \"[classTable, methodName, className, className list, className] \\<Rightarrow> bool\" (\"mtype'(_,_,_') = _ \\<rightarrow> _\" [80,80,80,80] 80)\nwhere\n mt_class: \n \"\\<lbrakk> CT(C) = Some(CDef);\n lookup (cMethods CDef) (\\<lambda>md.(mName md = m)) = Some(mDef);\n varDefs_types (mParams mDef) = Bs;\n mReturn mDef = B \\<rbrakk>\n \\<Longrightarrow> mtype(CT,m,C) = Bs \\<rightarrow> B\"\n\n| mt_super: \n \"\\<lbrakk> CT(C) = Some (CDef);\n lookup (cMethods CDef) (\\<lambda>md.(mName md = m)) = None;\n cSuper CDef = D;\n mtype(CT,m,D) = Bs \\<rightarrow> B \\<rbrakk>\n \\<Longrightarrow> mtype(CT,m,C) = Bs \\<rightarrow> B\""
] |
FeatherweightJava/FJSound
|
FJSound.sub_fields
|
lemma sub_fields:
assumes "CT \<turnstile> C <: D"
shows "\<And>Dg. fields(CT,D) = Dg \<Longrightarrow> \<exists>Cf. fields(CT,C) = (Dg@Cf)"
|
?CT \<turnstile> ?C <: ?D \<Longrightarrow> fields(?CT, ?D) = ?Dg \<Longrightarrow> \<exists>Cf. fields(?CT, ?C) = (?Dg @ Cf)
|
\<lbrakk> ?H1 x_1 x_2 x_3; ?H2 x_1 x_3 x_4\<rbrakk> \<Longrightarrow> \<exists>y_0. ?H2 x_1 x_2 (?H3 x_4 y_0)
|
[
"List.append",
"FJDefs.fields",
"FJDefs.subtyping"
] |
[
"'a list \\<Rightarrow> 'a list \\<Rightarrow> 'a list",
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> nat \\<Rightarrow> varDef list \\<Rightarrow> bool",
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> nat \\<Rightarrow> nat \\<Rightarrow> bool"
] |
[
"primrec append :: \"'a list \\<Rightarrow> 'a list \\<Rightarrow> 'a list\" (infixr \"@\" 65) where\nappend_Nil: \"[] @ ys = ys\" |\nappend_Cons: \"(x#xs) @ ys = x # xs @ ys\"",
"inductive\n fields :: \"[classTable, className, varDef list] \\<Rightarrow> bool\" (\"fields'(_,_') = _\" [80,80,80] 80)\nwhere\n f_obj: \n \"fields(CT,Object) = []\"\n| f_class: \n \"\\<lbrakk> CT(C) = Some(CDef); cSuper CDef = D; cFields CDef = Cf; fields(CT,D) = Dg; DgCf = Dg @ Cf \\<rbrakk> \n \\<Longrightarrow> fields(CT,C) = DgCf\"",
"inductive\n subtyping :: \"[classTable, className, className] \\<Rightarrow> bool\" (\"_ \\<turnstile> _ <: _\" [80,80,80] 80)\nwhere\n s_refl : \"CT \\<turnstile> C <: C\"\n| s_trans : \"\\<lbrakk> CT \\<turnstile> C <: D; CT \\<turnstile> D <: E \\<rbrakk> \\<Longrightarrow> CT \\<turnstile> C <: E\"\n| s_super : \"\\<lbrakk> CT(C) = Some(CDef); cSuper CDef = D \\<rbrakk> \\<Longrightarrow> CT \\<turnstile> C <: D\""
] |
FeatherweightJava/FJSound
|
FJSound.A_1_3(1)
|
lemma A_1_3:
shows "(CT;\<Gamma>2 \<turnstile>+ es : Cs) \<Longrightarrow> (CT;\<Gamma>1++\<Gamma>2 \<turnstile>+ es : Cs)" (is "?P1 \<Longrightarrow> ?P2")
and "CT;\<Gamma>2 \<turnstile> e : C \<Longrightarrow> CT;\<Gamma>1++\<Gamma>2 \<turnstile> e : C" (is "?Q1 \<Longrightarrow> ?Q2")
|
?CT; ?\<Gamma>2.0 \<turnstile>+ ?es : ?Cs \<Longrightarrow> ?CT; ?\<Gamma>1.0 ++ ?\<Gamma>2.0 \<turnstile>+ ?es : ?Cs
|
?H1 x_1 x_2 x_3 x_4 \<Longrightarrow> ?H1 x_1 (?H2 x_5 x_2) x_3 x_4
|
[
"Map.map_add",
"FJDefs.typings"
] |
[
"('a \\<Rightarrow> 'b option) \\<Rightarrow> ('a \\<Rightarrow> 'b option) \\<Rightarrow> 'a \\<Rightarrow> 'b option",
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> (nat \\<Rightarrow> nat option) \\<Rightarrow> exp list \\<Rightarrow> nat list \\<Rightarrow> bool"
] |
[
"definition\n map_add :: \"('a \\<rightharpoonup> 'b) \\<Rightarrow> ('a \\<rightharpoonup> 'b) \\<Rightarrow> ('a \\<rightharpoonup> 'b)\" (infixl \"++\" 100) where\n \"m1 ++ m2 = (\\<lambda>x. case m2 x of None \\<Rightarrow> m1 x | Some y \\<Rightarrow> Some y)\"",
"inductive\n typings :: \"[classTable, varCtx, exp list, className list] \\<Rightarrow> bool\" (\"_;_ \\<turnstile>+ _ : _\" [80,80,80,80] 80)\n and typing :: \"[classTable, varCtx, exp, className] \\<Rightarrow> bool\" (\"_;_ \\<turnstile> _ : _\" [80,80,80,80] 80)\nwhere\n ts_nil : \"CT;\\<Gamma> \\<turnstile>+ [] : []\"\n\n| ts_cons : \n \"\\<lbrakk> CT;\\<Gamma> \\<turnstile> e0 : C0; CT;\\<Gamma> \\<turnstile>+ es : Cs \\<rbrakk> \n \\<Longrightarrow> CT;\\<Gamma> \\<turnstile>+ (e0 # es) : (C0 # Cs)\"\n\n| t_var : \n \"\\<lbrakk> \\<Gamma>(x) = Some C \\<rbrakk> \\<Longrightarrow> CT;\\<Gamma> \\<turnstile> (Var x) : C\"\n\n| t_field : \n \"\\<lbrakk> CT;\\<Gamma> \\<turnstile> e0 : C0;\n fields(CT,C0) = Cf;\n lookup Cf (\\<lambda>fd.(vdName fd = fi)) = Some(fDef);\n vdType fDef = Ci \\<rbrakk>\n \\<Longrightarrow> CT;\\<Gamma> \\<turnstile> FieldProj e0 fi : Ci\"\n\n| t_invk : \n \"\\<lbrakk> CT;\\<Gamma> \\<turnstile> e0 : C0;\n mtype(CT,m,C0) = Ds \\<rightarrow> C;\n CT;\\<Gamma> \\<turnstile>+ es : Cs;\n CT \\<turnstile>+ Cs <: Ds;\n length es = length Ds \\<rbrakk>\n \\<Longrightarrow> CT;\\<Gamma> \\<turnstile> MethodInvk e0 m es : C\"\n\n| t_new : \n \"\\<lbrakk> fields(CT,C) = Df;\n length es = length Df;\n varDefs_types Df = Ds;\n CT;\\<Gamma> \\<turnstile>+ es : Cs;\n CT \\<turnstile>+ Cs <: Ds \\<rbrakk>\n \\<Longrightarrow> CT;\\<Gamma> \\<turnstile> New C es : C\"\n\n| t_ucast : \n \"\\<lbrakk> CT;\\<Gamma> \\<turnstile> e0 : D; \n CT \\<turnstile> D <: C \\<rbrakk>\n \\<Longrightarrow> CT;\\<Gamma> \\<turnstile> Cast C e0 : C\"\n\n| t_dcast : \n \"\\<lbrakk> CT;\\<Gamma> \\<turnstile> e0 : D; \n CT \\<turnstile> C <: D; C \\<noteq> D \\<rbrakk>\n \\<Longrightarrow> CT;\\<Gamma> \\<turnstile> Cast C e0 : C\"\n\n| t_scast : \n \"\\<lbrakk> CT;\\<Gamma> \\<turnstile> e0 : D;\n CT \\<turnstile> C \\<not><: D;\n CT \\<turnstile> D \\<not><: C \\<rbrakk>\n \\<Longrightarrow> CT;\\<Gamma> \\<turnstile> Cast C e0 : C\""
] |
FeatherweightJava/FJSound
|
FJSound.A_1_3(2)
|
lemma A_1_3:
shows "(CT;\<Gamma>2 \<turnstile>+ es : Cs) \<Longrightarrow> (CT;\<Gamma>1++\<Gamma>2 \<turnstile>+ es : Cs)" (is "?P1 \<Longrightarrow> ?P2")
and "CT;\<Gamma>2 \<turnstile> e : C \<Longrightarrow> CT;\<Gamma>1++\<Gamma>2 \<turnstile> e : C" (is "?Q1 \<Longrightarrow> ?Q2")
|
?CT; ?\<Gamma>2.0 \<turnstile> ?e : ?C \<Longrightarrow> ?CT; ?\<Gamma>1.0 ++ ?\<Gamma>2.0 \<turnstile> ?e : ?C
|
?H1 x_1 x_2 x_3 x_4 \<Longrightarrow> ?H1 x_1 (?H2 x_5 x_2) x_3 x_4
|
[
"Map.map_add",
"FJDefs.typing"
] |
[
"('a \\<Rightarrow> 'b option) \\<Rightarrow> ('a \\<Rightarrow> 'b option) \\<Rightarrow> 'a \\<Rightarrow> 'b option",
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> (nat \\<Rightarrow> nat option) \\<Rightarrow> exp \\<Rightarrow> nat \\<Rightarrow> bool"
] |
[
"definition\n map_add :: \"('a \\<rightharpoonup> 'b) \\<Rightarrow> ('a \\<rightharpoonup> 'b) \\<Rightarrow> ('a \\<rightharpoonup> 'b)\" (infixl \"++\" 100) where\n \"m1 ++ m2 = (\\<lambda>x. case m2 x of None \\<Rightarrow> m1 x | Some y \\<Rightarrow> Some y)\"",
"inductive\n typings :: \"[classTable, varCtx, exp list, className list] \\<Rightarrow> bool\" (\"_;_ \\<turnstile>+ _ : _\" [80,80,80,80] 80)\n and typing :: \"[classTable, varCtx, exp, className] \\<Rightarrow> bool\" (\"_;_ \\<turnstile> _ : _\" [80,80,80,80] 80)\nwhere\n ts_nil : \"CT;\\<Gamma> \\<turnstile>+ [] : []\"\n\n| ts_cons : \n \"\\<lbrakk> CT;\\<Gamma> \\<turnstile> e0 : C0; CT;\\<Gamma> \\<turnstile>+ es : Cs \\<rbrakk> \n \\<Longrightarrow> CT;\\<Gamma> \\<turnstile>+ (e0 # es) : (C0 # Cs)\"\n\n| t_var : \n \"\\<lbrakk> \\<Gamma>(x) = Some C \\<rbrakk> \\<Longrightarrow> CT;\\<Gamma> \\<turnstile> (Var x) : C\"\n\n| t_field : \n \"\\<lbrakk> CT;\\<Gamma> \\<turnstile> e0 : C0;\n fields(CT,C0) = Cf;\n lookup Cf (\\<lambda>fd.(vdName fd = fi)) = Some(fDef);\n vdType fDef = Ci \\<rbrakk>\n \\<Longrightarrow> CT;\\<Gamma> \\<turnstile> FieldProj e0 fi : Ci\"\n\n| t_invk : \n \"\\<lbrakk> CT;\\<Gamma> \\<turnstile> e0 : C0;\n mtype(CT,m,C0) = Ds \\<rightarrow> C;\n CT;\\<Gamma> \\<turnstile>+ es : Cs;\n CT \\<turnstile>+ Cs <: Ds;\n length es = length Ds \\<rbrakk>\n \\<Longrightarrow> CT;\\<Gamma> \\<turnstile> MethodInvk e0 m es : C\"\n\n| t_new : \n \"\\<lbrakk> fields(CT,C) = Df;\n length es = length Df;\n varDefs_types Df = Ds;\n CT;\\<Gamma> \\<turnstile>+ es : Cs;\n CT \\<turnstile>+ Cs <: Ds \\<rbrakk>\n \\<Longrightarrow> CT;\\<Gamma> \\<turnstile> New C es : C\"\n\n| t_ucast : \n \"\\<lbrakk> CT;\\<Gamma> \\<turnstile> e0 : D; \n CT \\<turnstile> D <: C \\<rbrakk>\n \\<Longrightarrow> CT;\\<Gamma> \\<turnstile> Cast C e0 : C\"\n\n| t_dcast : \n \"\\<lbrakk> CT;\\<Gamma> \\<turnstile> e0 : D; \n CT \\<turnstile> C <: D; C \\<noteq> D \\<rbrakk>\n \\<Longrightarrow> CT;\\<Gamma> \\<turnstile> Cast C e0 : C\"\n\n| t_scast : \n \"\\<lbrakk> CT;\\<Gamma> \\<turnstile> e0 : D;\n CT \\<turnstile> C \\<not><: D;\n CT \\<turnstile> D \\<not><: C \\<rbrakk>\n \\<Longrightarrow> CT;\\<Gamma> \\<turnstile> Cast C e0 : C\""
] |
FeatherweightJava/FJDefs
|
FJDefs.typing_induct
|
lemma typing_induct:
assumes "CT;\<Gamma> \<turnstile> e : C" (is ?T)
and "\<And>C CT \<Gamma> x. \<Gamma> x = Some C \<Longrightarrow> P CT \<Gamma> (Var x) C"
and "\<And>C0 CT Cf Ci \<Gamma> e0 fDef fi. \<lbrakk>CT;\<Gamma> \<turnstile> e0 : C0; P CT \<Gamma> e0 C0; fields(CT,C0) = Cf; lookup Cf (\<lambda>fd. vdName fd = fi) = Some fDef; vdType fDef = Ci\<rbrakk> \<Longrightarrow> P CT \<Gamma> (FieldProj e0 fi) Ci"
and "\<And>C C0 CT Cs Ds \<Gamma> e0 es m. \<lbrakk>CT;\<Gamma> \<turnstile> e0 : C0; P CT \<Gamma> e0 C0; mtype(CT,m,C0) = Ds \<rightarrow> C; CT;\<Gamma> \<turnstile>+ es : Cs; \<And>i . \<lbrakk> i < length es \<rbrakk> \<Longrightarrow> P CT \<Gamma> (es!i) (Cs!i); CT \<turnstile>+ Cs <: Ds; length es = length Ds\<rbrakk> \<Longrightarrow> P CT \<Gamma> (MethodInvk e0 m es) C"
and "\<And>C CT Cs Df Ds \<Gamma> es. \<lbrakk>fields(CT,C) = Df; length es = length Df; varDefs_types Df = Ds; CT;\<Gamma> \<turnstile>+ es : Cs; \<And>i. \<lbrakk> i < length es \<rbrakk> \<Longrightarrow> P CT \<Gamma> (es!i) (Cs!i); CT \<turnstile>+ Cs <: Ds\<rbrakk> \<Longrightarrow> P CT \<Gamma> (New C es) C"
and "\<And>C CT D \<Gamma> e0. \<lbrakk>CT;\<Gamma> \<turnstile> e0 : D; P CT \<Gamma> e0 D; CT \<turnstile> D <: C\<rbrakk> \<Longrightarrow> P CT \<Gamma> (Cast C e0) C"
and "\<And>C CT D \<Gamma> e0. \<lbrakk>CT;\<Gamma> \<turnstile> e0 : D; P CT \<Gamma> e0 D; CT \<turnstile> C <: D; C \<noteq> D\<rbrakk> \<Longrightarrow> P CT \<Gamma> (Cast C e0) C"
and "\<And>C CT D \<Gamma> e0. \<lbrakk>CT;\<Gamma> \<turnstile> e0 : D; P CT \<Gamma> e0 D; CT \<turnstile> C \<not><: D; CT \<turnstile> D \<not><: C\<rbrakk> \<Longrightarrow> P CT \<Gamma> (Cast C e0) C"
shows "P CT \<Gamma> e C" (is ?P)
|
?CT; ?\<Gamma> \<turnstile> ?e : ?C \<Longrightarrow> (\<And>C CT \<Gamma> x. \<Gamma> x = Some C \<Longrightarrow> ?P CT \<Gamma> (Var x) C) \<Longrightarrow> (\<And>C0 CT Cf Ci \<Gamma> e0 fDef fi. CT;\<Gamma> \<turnstile> e0 : C0 \<Longrightarrow> ?P CT \<Gamma> e0 C0 \<Longrightarrow> fields(CT,C0) = Cf \<Longrightarrow> lookup Cf (\<lambda>fd. vdName fd = fi) = Some fDef \<Longrightarrow> vdType fDef = Ci \<Longrightarrow> ?P CT \<Gamma> (FieldProj e0 fi) Ci) \<Longrightarrow> (\<And>C C0 CT Cs Ds \<Gamma> e0 es m. CT;\<Gamma> \<turnstile> e0 : C0 \<Longrightarrow> ?P CT \<Gamma> e0 C0 \<Longrightarrow> mtype(CT,m,C0) = Ds \<rightarrow> C \<Longrightarrow> CT;\<Gamma> \<turnstile>+ es : Cs \<Longrightarrow> (\<And>i. i < length es \<Longrightarrow> ?P CT \<Gamma> (es ! i) (Cs ! i)) \<Longrightarrow> CT \<turnstile>+ Cs <: Ds \<Longrightarrow> length es = length Ds \<Longrightarrow> ?P CT \<Gamma> (MethodInvk e0 m es) C) \<Longrightarrow> (\<And>C CT Cs Df Ds \<Gamma> es. fields(CT,C) = Df \<Longrightarrow> length es = length Df \<Longrightarrow> varDefs_types Df = Ds \<Longrightarrow> CT;\<Gamma> \<turnstile>+ es : Cs \<Longrightarrow> (\<And>i. i < length es \<Longrightarrow> ?P CT \<Gamma> (es ! i) (Cs ! i)) \<Longrightarrow> CT \<turnstile>+ Cs <: Ds \<Longrightarrow> ?P CT \<Gamma> (New C es) C) \<Longrightarrow> (\<And>C CT D \<Gamma> e0. CT;\<Gamma> \<turnstile> e0 : D \<Longrightarrow> ?P CT \<Gamma> e0 D \<Longrightarrow> CT \<turnstile> D <: C \<Longrightarrow> ?P CT \<Gamma> (Cast C e0) C) \<Longrightarrow> (\<And>C CT D \<Gamma> e0. CT;\<Gamma> \<turnstile> e0 : D \<Longrightarrow> ?P CT \<Gamma> e0 D \<Longrightarrow> CT \<turnstile> C <: D \<Longrightarrow> C \<noteq> D \<Longrightarrow> ?P CT \<Gamma> (Cast C e0) C) \<Longrightarrow> (\<And>C CT D \<Gamma> e0. CT;\<Gamma> \<turnstile> e0 : D \<Longrightarrow> ?P CT \<Gamma> e0 D \<Longrightarrow> CT \<turnstile> C \<not><: D \<Longrightarrow> CT \<turnstile> D \<not><: C \<Longrightarrow> ?P CT \<Gamma> (Cast C e0) C) \<Longrightarrow> ?P ?CT ?\<Gamma> ?e ?C
|
\<lbrakk> ?H1 x_1 x_2 x_3 x_4; \<And>y_0 y_1 y_2 y_3. y_2 y_3 = ?H2 y_0 \<Longrightarrow> x_5 y_1 y_2 (?H3 y_3) y_0; \<And>y_4 y_5 y_6 y_7 y_8 y_9 y_10 y_11. \<lbrakk> ?H1 y_5 y_8 y_9 y_4; x_5 y_5 y_8 y_9 y_4; ?H4 y_5 y_4 y_6; ?H5 y_6 (\<lambda>y_12. ?H6 y_12 = y_11) = ?H7 y_10; ?H8 y_10 = y_7\<rbrakk> \<Longrightarrow> x_5 y_5 y_8 (?H9 y_9 y_11) y_7; \<And>y_13 y_14 y_15 y_16 y_17 y_18 y_19 y_20 y_21. \<lbrakk> ?H1 y_15 y_18 y_19 y_14; x_5 y_15 y_18 y_19 y_14; ?H10 y_15 y_21 y_14 y_17 y_13; ?H11 y_15 y_18 y_20 y_16; \<And>y_22. y_22 < ?H12 y_20 \<Longrightarrow> x_5 y_15 y_18 (?H13 y_20 y_22) (?H14 y_16 y_22); ?H15 y_15 y_16 y_17; ?H12 y_20 = ?H16 y_17\<rbrakk> \<Longrightarrow> x_5 y_15 y_18 (?H17 y_19 y_21 y_20) y_13; \<And>y_23 y_24 y_25 y_26 y_27 y_28 y_29. \<lbrakk> ?H4 y_24 y_23 y_26; ?H12 y_29 = ?H18 y_26; ?H19 y_26 = y_27; ?H11 y_24 y_28 y_29 y_25; \<And>y_30. y_30 < ?H12 y_29 \<Longrightarrow> x_5 y_24 y_28 (?H13 y_29 y_30) (?H14 y_25 y_30); ?H15 y_24 y_25 y_27\<rbrakk> \<Longrightarrow> x_5 y_24 y_28 (?H20 y_23 y_29) y_23; \<And>y_31 y_32 y_33 y_34 y_35. \<lbrakk> ?H1 y_32 y_34 y_35 y_33; x_5 y_32 y_34 y_35 y_33; ?H21 y_32 y_33 y_31\<rbrakk> \<Longrightarrow> x_5 y_32 y_34 (?H22 y_31 y_35) y_31; \<And>y_36 y_37 y_38 y_39 y_40. \<lbrakk> ?H1 y_37 y_39 y_40 y_38; x_5 y_37 y_39 y_40 y_38; ?H21 y_37 y_36 y_38; y_36 \<noteq> y_38\<rbrakk> \<Longrightarrow> x_5 y_37 y_39 (?H22 y_36 y_40) y_36; \<And>y_41 y_42 y_43 y_44 y_45. \<lbrakk> ?H1 y_42 y_44 y_45 y_43; x_5 y_42 y_44 y_45 y_43; ?H23 y_42 y_41 y_43; ?H23 y_42 y_43 y_41\<rbrakk> \<Longrightarrow> x_5 y_42 y_44 (?H22 y_41 y_45) y_41\<rbrakk> \<Longrightarrow> x_5 x_1 x_2 x_3 x_4
|
[
"FJDefs.neg_subtyping",
"FJDefs.exp.Cast",
"FJDefs.subtyping",
"FJDefs.exp.New",
"FJDefs.varDefs_types",
"FJDefs.exp.MethodInvk",
"FJDefs.subtypings",
"List.nth",
"List.length",
"FJDefs.typings",
"FJDefs.mtype",
"FJDefs.exp.FieldProj",
"FJDefs.varDef.vdType",
"FJDefs.varDef.vdName",
"FJDefs.lookup",
"FJDefs.fields",
"FJDefs.exp.Var",
"Option.option.Some",
"FJDefs.typing"
] |
[
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> nat \\<Rightarrow> nat \\<Rightarrow> bool",
"nat \\<Rightarrow> exp \\<Rightarrow> exp",
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> nat \\<Rightarrow> nat \\<Rightarrow> bool",
"nat \\<Rightarrow> exp list \\<Rightarrow> exp",
"varDef list \\<Rightarrow> nat list",
"exp \\<Rightarrow> nat \\<Rightarrow> exp list \\<Rightarrow> exp",
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> nat list \\<Rightarrow> nat list \\<Rightarrow> bool",
"'a list \\<Rightarrow> nat \\<Rightarrow> 'a",
"'a list \\<Rightarrow> nat",
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> (nat \\<Rightarrow> nat option) \\<Rightarrow> exp list \\<Rightarrow> nat list \\<Rightarrow> bool",
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> nat \\<Rightarrow> nat \\<Rightarrow> nat list \\<Rightarrow> nat \\<Rightarrow> bool",
"exp \\<Rightarrow> nat \\<Rightarrow> exp",
"'a varDef_scheme \\<Rightarrow> nat",
"'a varDef_scheme \\<Rightarrow> nat",
"'a list \\<Rightarrow> ('a \\<Rightarrow> bool) \\<Rightarrow> 'a option",
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> nat \\<Rightarrow> varDef list \\<Rightarrow> bool",
"nat \\<Rightarrow> exp",
"'a \\<Rightarrow> 'a option",
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> (nat \\<Rightarrow> nat option) \\<Rightarrow> exp \\<Rightarrow> nat \\<Rightarrow> bool"
] |
[
"abbreviation\n neg_subtyping :: \"[classTable, className, className] \\<Rightarrow> bool\" (\"_ \\<turnstile> _ \\<not><: _\" [80,80,80] 80)\n where \"CT \\<turnstile> S \\<not><: T \\<equiv> \\<not> CT \\<turnstile> S <: T\"",
"datatype exp = \n Var \"varName\"\n | FieldProj \"exp\" \"varName\" \n | MethodInvk \"exp\" \"methodName\" \"exp list\"\n | New \"className\" \"exp list\"\n | Cast \"className\" \"exp\"",
"inductive\n subtyping :: \"[classTable, className, className] \\<Rightarrow> bool\" (\"_ \\<turnstile> _ <: _\" [80,80,80] 80)\nwhere\n s_refl : \"CT \\<turnstile> C <: C\"\n| s_trans : \"\\<lbrakk> CT \\<turnstile> C <: D; CT \\<turnstile> D <: E \\<rbrakk> \\<Longrightarrow> CT \\<turnstile> C <: E\"\n| s_super : \"\\<lbrakk> CT(C) = Some(CDef); cSuper CDef = D \\<rbrakk> \\<Longrightarrow> CT \\<turnstile> C <: D\"",
"definition\n varDefs_types :: \"varDef list \\<Rightarrow> className list\" where\n \"varDefs_types = map vdType\"",
"inductive\n subtypings :: \"[classTable, className list, className list] \\<Rightarrow> bool\" (\"_ \\<turnstile>+ _ <: _\" [80,80,80] 80)\nwhere\n ss_nil : \"CT \\<turnstile>+ [] <: []\"\n| ss_cons : \"\\<lbrakk> CT \\<turnstile> C0 <: D0; CT \\<turnstile>+ Cs <: Ds \\<rbrakk> \\<Longrightarrow> CT \\<turnstile>+ (C0 # Cs) <: (D0 # Ds)\"",
"primrec (nonexhaustive) nth :: \"'a list => nat => 'a\" (infixl \"!\" 100) where\nnth_Cons: \"(x # xs) ! n = (case n of 0 \\<Rightarrow> x | Suc k \\<Rightarrow> xs ! k)\"\n \\<comment> \\<open>Warning: simpset does not contain this definition, but separate\n theorems for \\<open>n = 0\\<close> and \\<open>n = Suc k\\<close>\\<close>",
"abbreviation length :: \"'a list \\<Rightarrow> nat\" where\n\"length \\<equiv> size\"",
"inductive\n typings :: \"[classTable, varCtx, exp list, className list] \\<Rightarrow> bool\" (\"_;_ \\<turnstile>+ _ : _\" [80,80,80,80] 80)\n and typing :: \"[classTable, varCtx, exp, className] \\<Rightarrow> bool\" (\"_;_ \\<turnstile> _ : _\" [80,80,80,80] 80)\nwhere\n ts_nil : \"CT;\\<Gamma> \\<turnstile>+ [] : []\"\n\n| ts_cons : \n \"\\<lbrakk> CT;\\<Gamma> \\<turnstile> e0 : C0; CT;\\<Gamma> \\<turnstile>+ es : Cs \\<rbrakk> \n \\<Longrightarrow> CT;\\<Gamma> \\<turnstile>+ (e0 # es) : (C0 # Cs)\"\n\n| t_var : \n \"\\<lbrakk> \\<Gamma>(x) = Some C \\<rbrakk> \\<Longrightarrow> CT;\\<Gamma> \\<turnstile> (Var x) : C\"\n\n| t_field : \n \"\\<lbrakk> CT;\\<Gamma> \\<turnstile> e0 : C0;\n fields(CT,C0) = Cf;\n lookup Cf (\\<lambda>fd.(vdName fd = fi)) = Some(fDef);\n vdType fDef = Ci \\<rbrakk>\n \\<Longrightarrow> CT;\\<Gamma> \\<turnstile> FieldProj e0 fi : Ci\"\n\n| t_invk : \n \"\\<lbrakk> CT;\\<Gamma> \\<turnstile> e0 : C0;\n mtype(CT,m,C0) = Ds \\<rightarrow> C;\n CT;\\<Gamma> \\<turnstile>+ es : Cs;\n CT \\<turnstile>+ Cs <: Ds;\n length es = length Ds \\<rbrakk>\n \\<Longrightarrow> CT;\\<Gamma> \\<turnstile> MethodInvk e0 m es : C\"\n\n| t_new : \n \"\\<lbrakk> fields(CT,C) = Df;\n length es = length Df;\n varDefs_types Df = Ds;\n CT;\\<Gamma> \\<turnstile>+ es : Cs;\n CT \\<turnstile>+ Cs <: Ds \\<rbrakk>\n \\<Longrightarrow> CT;\\<Gamma> \\<turnstile> New C es : C\"\n\n| t_ucast : \n \"\\<lbrakk> CT;\\<Gamma> \\<turnstile> e0 : D; \n CT \\<turnstile> D <: C \\<rbrakk>\n \\<Longrightarrow> CT;\\<Gamma> \\<turnstile> Cast C e0 : C\"\n\n| t_dcast : \n \"\\<lbrakk> CT;\\<Gamma> \\<turnstile> e0 : D; \n CT \\<turnstile> C <: D; C \\<noteq> D \\<rbrakk>\n \\<Longrightarrow> CT;\\<Gamma> \\<turnstile> Cast C e0 : C\"\n\n| t_scast : \n \"\\<lbrakk> CT;\\<Gamma> \\<turnstile> e0 : D;\n CT \\<turnstile> C \\<not><: D;\n CT \\<turnstile> D \\<not><: C \\<rbrakk>\n \\<Longrightarrow> CT;\\<Gamma> \\<turnstile> Cast C e0 : C\"",
"inductive\n mtype :: \"[classTable, methodName, className, className list, className] \\<Rightarrow> bool\" (\"mtype'(_,_,_') = _ \\<rightarrow> _\" [80,80,80,80] 80)\nwhere\n mt_class: \n \"\\<lbrakk> CT(C) = Some(CDef);\n lookup (cMethods CDef) (\\<lambda>md.(mName md = m)) = Some(mDef);\n varDefs_types (mParams mDef) = Bs;\n mReturn mDef = B \\<rbrakk>\n \\<Longrightarrow> mtype(CT,m,C) = Bs \\<rightarrow> B\"\n\n| mt_super: \n \"\\<lbrakk> CT(C) = Some (CDef);\n lookup (cMethods CDef) (\\<lambda>md.(mName md = m)) = None;\n cSuper CDef = D;\n mtype(CT,m,D) = Bs \\<rightarrow> B \\<rbrakk>\n \\<Longrightarrow> mtype(CT,m,C) = Bs \\<rightarrow> B\"",
"record varDef = \n vdName :: \"varName\"\n vdType :: \"className\"",
"primrec lookup :: \"'a list \\<Rightarrow> ('a \\<Rightarrow> bool) \\<Rightarrow> 'a option\"\nwhere\n \"lookup [] P = None\"\n| \"lookup (h#t) P = (if P h then Some h else lookup t P)\"",
"inductive\n fields :: \"[classTable, className, varDef list] \\<Rightarrow> bool\" (\"fields'(_,_') = _\" [80,80,80] 80)\nwhere\n f_obj: \n \"fields(CT,Object) = []\"\n| f_class: \n \"\\<lbrakk> CT(C) = Some(CDef); cSuper CDef = D; cFields CDef = Cf; fields(CT,D) = Dg; DgCf = Dg @ Cf \\<rbrakk> \n \\<Longrightarrow> fields(CT,C) = DgCf\"",
"datatype 'a option =\n None\n | Some (the: 'a)"
] |
FeatherweightJava/Execute
|
Execute.rc_invk_arg'
|
lemma rc_invk_arg': "CT \<turnstile> ei \<rightarrow> ei' \<Longrightarrow> append el (ei # er) e' \<Longrightarrow> append el (ei' # er) e'' \<Longrightarrow>
CT \<turnstile> MethodInvk e m e' \<rightarrow> MethodInvk e m e''"
|
?CT \<turnstile> ?ei \<rightarrow> ?ei' \<Longrightarrow> Execute.append ?el (?ei # ?er) ?e' \<Longrightarrow> Execute.append ?el (?ei' # ?er) ?e'' \<Longrightarrow> ?CT \<turnstile> MethodInvk ?e ?m ?e' \<rightarrow> MethodInvk ?e ?m ?e''
|
\<lbrakk> ?H1 x_1 x_2 x_3; ?H2 x_4 (?H3 x_2 x_5) x_6; ?H2 x_4 (?H3 x_3 x_5) x_7\<rbrakk> \<Longrightarrow> ?H1 x_1 (?H4 x_8 x_9 x_6) (?H4 x_8 x_9 x_7)
|
[
"FJDefs.exp.MethodInvk",
"List.list.Cons",
"Execute.append",
"FJDefs.reduction"
] |
[
"exp \\<Rightarrow> nat \\<Rightarrow> exp list \\<Rightarrow> exp",
"'a \\<Rightarrow> 'a list \\<Rightarrow> 'a list",
"'a list \\<Rightarrow> 'a list \\<Rightarrow> 'a list \\<Rightarrow> bool",
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> exp \\<Rightarrow> exp \\<Rightarrow> bool"
] |
[
"datatype exp = \n Var \"varName\"\n | FieldProj \"exp\" \"varName\" \n | MethodInvk \"exp\" \"methodName\" \"exp list\"\n | New \"className\" \"exp list\"\n | Cast \"className\" \"exp\"",
"datatype (set: 'a) list =\n Nil (\"[]\")\n | Cons (hd: 'a) (tl: \"'a list\") (infixr \"#\" 65)\nfor\n map: map\n rel: list_all2\n pred: list_all\nwhere\n \"tl [] = []\"",
"definition append where \"append xs ys zs = (zs = xs @ ys)\"",
"inductive\n reduction :: \"[classTable, exp, exp] \\<Rightarrow> bool\" (\"_ \\<turnstile> _ \\<rightarrow> _\" [80,80,80] 80)\nwhere\n\n r_field: \n \"\\<lbrakk> fields(CT,C) = Cf; \n lookup2 Cf es (\\<lambda>fd.(vdName fd = fi)) = Some(ei) \\<rbrakk>\n \\<Longrightarrow> CT \\<turnstile> FieldProj (New C es) fi \\<rightarrow> ei\"\n\n| r_invk: \n \"\\<lbrakk> mbody(CT,m,C) = xs . e0;\n substs ((map_upds Map.empty xs ds)(this \\<mapsto> (New C es))) e0 = e0' \\<rbrakk>\n \\<Longrightarrow> CT \\<turnstile> MethodInvk (New C es) m ds \\<rightarrow> e0'\"\n\n| r_cast: \n \"\\<lbrakk> CT \\<turnstile> C <: D \\<rbrakk> \n \\<Longrightarrow> CT \\<turnstile> Cast D (New C es) \\<rightarrow> New C es\"\n\n| rc_field: \n \"\\<lbrakk> CT \\<turnstile> e0 \\<rightarrow> e0' \\<rbrakk> \n \\<Longrightarrow> CT \\<turnstile> FieldProj e0 f \\<rightarrow> FieldProj e0' f\"\n\n| rc_invk_recv: \n \"\\<lbrakk> CT \\<turnstile> e0 \\<rightarrow> e0' \\<rbrakk> \n \\<Longrightarrow> CT \\<turnstile> MethodInvk e0 m es \\<rightarrow> MethodInvk e0' m es\"\n\n| rc_invk_arg: \n \"\\<lbrakk> CT \\<turnstile> ei \\<rightarrow> ei' \\<rbrakk>\n \\<Longrightarrow> CT \\<turnstile> MethodInvk e0 m (el@ei#er) \\<rightarrow> MethodInvk e0 m (el@ei'#er)\"\n\n| rc_new_arg: \n \"\\<lbrakk> CT \\<turnstile> ei \\<rightarrow> ei' \\<rbrakk> \n \\<Longrightarrow> CT \\<turnstile> New C (el@ei#er) \\<rightarrow> New C (el@ei'#er)\"\n\n| rc_cast: \n \"\\<lbrakk> CT \\<turnstile> e0 \\<rightarrow> e0' \\<rbrakk> \n \\<Longrightarrow> CT \\<turnstile> Cast C e0 \\<rightarrow> Cast C e0'\""
] |
FeatherweightJava/Execute
|
Execute.rc_new_arg'
|
lemma rc_new_arg': "CT \<turnstile> ei \<rightarrow> ei' \<Longrightarrow> append el (ei # er) e \<Longrightarrow> append el (ei' # er) e'
==> CT \<turnstile> New C e \<rightarrow> New C e'"
|
?CT \<turnstile> ?ei \<rightarrow> ?ei' \<Longrightarrow> Execute.append ?el (?ei # ?er) ?e \<Longrightarrow> Execute.append ?el (?ei' # ?er) ?e' \<Longrightarrow> ?CT \<turnstile> New ?C ?e \<rightarrow> New ?C ?e'
|
\<lbrakk> ?H1 x_1 x_2 x_3; ?H2 x_4 (?H3 x_2 x_5) x_6; ?H2 x_4 (?H3 x_3 x_5) x_7\<rbrakk> \<Longrightarrow> ?H1 x_1 (?H4 x_8 x_6) (?H4 x_8 x_7)
|
[
"FJDefs.exp.New",
"List.list.Cons",
"Execute.append",
"FJDefs.reduction"
] |
[
"nat \\<Rightarrow> exp list \\<Rightarrow> exp",
"'a \\<Rightarrow> 'a list \\<Rightarrow> 'a list",
"'a list \\<Rightarrow> 'a list \\<Rightarrow> 'a list \\<Rightarrow> bool",
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> exp \\<Rightarrow> exp \\<Rightarrow> bool"
] |
[
"datatype exp = \n Var \"varName\"\n | FieldProj \"exp\" \"varName\" \n | MethodInvk \"exp\" \"methodName\" \"exp list\"\n | New \"className\" \"exp list\"\n | Cast \"className\" \"exp\"",
"datatype (set: 'a) list =\n Nil (\"[]\")\n | Cons (hd: 'a) (tl: \"'a list\") (infixr \"#\" 65)\nfor\n map: map\n rel: list_all2\n pred: list_all\nwhere\n \"tl [] = []\"",
"definition append where \"append xs ys zs = (zs = xs @ ys)\"",
"inductive\n reduction :: \"[classTable, exp, exp] \\<Rightarrow> bool\" (\"_ \\<turnstile> _ \\<rightarrow> _\" [80,80,80] 80)\nwhere\n\n r_field: \n \"\\<lbrakk> fields(CT,C) = Cf; \n lookup2 Cf es (\\<lambda>fd.(vdName fd = fi)) = Some(ei) \\<rbrakk>\n \\<Longrightarrow> CT \\<turnstile> FieldProj (New C es) fi \\<rightarrow> ei\"\n\n| r_invk: \n \"\\<lbrakk> mbody(CT,m,C) = xs . e0;\n substs ((map_upds Map.empty xs ds)(this \\<mapsto> (New C es))) e0 = e0' \\<rbrakk>\n \\<Longrightarrow> CT \\<turnstile> MethodInvk (New C es) m ds \\<rightarrow> e0'\"\n\n| r_cast: \n \"\\<lbrakk> CT \\<turnstile> C <: D \\<rbrakk> \n \\<Longrightarrow> CT \\<turnstile> Cast D (New C es) \\<rightarrow> New C es\"\n\n| rc_field: \n \"\\<lbrakk> CT \\<turnstile> e0 \\<rightarrow> e0' \\<rbrakk> \n \\<Longrightarrow> CT \\<turnstile> FieldProj e0 f \\<rightarrow> FieldProj e0' f\"\n\n| rc_invk_recv: \n \"\\<lbrakk> CT \\<turnstile> e0 \\<rightarrow> e0' \\<rbrakk> \n \\<Longrightarrow> CT \\<turnstile> MethodInvk e0 m es \\<rightarrow> MethodInvk e0' m es\"\n\n| rc_invk_arg: \n \"\\<lbrakk> CT \\<turnstile> ei \\<rightarrow> ei' \\<rbrakk>\n \\<Longrightarrow> CT \\<turnstile> MethodInvk e0 m (el@ei#er) \\<rightarrow> MethodInvk e0 m (el@ei'#er)\"\n\n| rc_new_arg: \n \"\\<lbrakk> CT \\<turnstile> ei \\<rightarrow> ei' \\<rbrakk> \n \\<Longrightarrow> CT \\<turnstile> New C (el@ei#er) \\<rightarrow> New C (el@ei'#er)\"\n\n| rc_cast: \n \"\\<lbrakk> CT \\<turnstile> e0 \\<rightarrow> e0' \\<rbrakk> \n \\<Longrightarrow> CT \\<turnstile> Cast C e0 \\<rightarrow> Cast C e0'\""
] |
FeatherweightJava/Execute
|
Execute.method_typing_aux
|
lemma method_typing_aux:
"(\<forall>Ds D0. mtype(CT,m,D) = Ds \<rightarrow> D0 \<longrightarrow> Cs = Ds \<and> C = D0) = (\<not> method_typing_aux CT m D Cs C)"
|
(\<forall>Ds D0. mtype(?CT, ?m, ?D) = Ds \<rightarrow> D0 \<longrightarrow> ?Cs = Ds \<and> ?C = D0) = (\<not> method_typing_aux ?CT ?m ?D ?Cs ?C)
|
(\<forall>y_0 y_1. ?H1 x_1 x_2 x_3 y_0 y_1 \<longrightarrow> x_4 = y_0 \<and> x_5 = y_1) = (\<not> ?H2 x_1 x_2 x_3 x_4 x_5)
|
[
"Execute.method_typing_aux",
"FJDefs.mtype"
] |
[
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> nat \\<Rightarrow> nat \\<Rightarrow> nat list \\<Rightarrow> nat \\<Rightarrow> bool",
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> nat \\<Rightarrow> nat \\<Rightarrow> nat list \\<Rightarrow> nat \\<Rightarrow> bool"
] |
[
"definition method_typing_aux\nwhere\n \"method_typing_aux CT m D Cs C = (\\<not> (\\<forall>Ds D0. mtype(CT,m,D) = Ds \\<rightarrow> D0 \\<longrightarrow> Cs = Ds \\<and> C = D0))\"",
"inductive\n mtype :: \"[classTable, methodName, className, className list, className] \\<Rightarrow> bool\" (\"mtype'(_,_,_') = _ \\<rightarrow> _\" [80,80,80,80] 80)\nwhere\n mt_class: \n \"\\<lbrakk> CT(C) = Some(CDef);\n lookup (cMethods CDef) (\\<lambda>md.(mName md = m)) = Some(mDef);\n varDefs_types (mParams mDef) = Bs;\n mReturn mDef = B \\<rbrakk>\n \\<Longrightarrow> mtype(CT,m,C) = Bs \\<rightarrow> B\"\n\n| mt_super: \n \"\\<lbrakk> CT(C) = Some (CDef);\n lookup (cMethods CDef) (\\<lambda>md.(mName md = m)) = None;\n cSuper CDef = D;\n mtype(CT,m,D) = Bs \\<rightarrow> B \\<rbrakk>\n \\<Longrightarrow> mtype(CT,m,C) = Bs \\<rightarrow> B\""
] |
FeatherweightJava/Execute
|
Execute.mtype_i_i_i_i_iE
| null |
pred.eval (mtype_i_i_i_i_i ?x ?xa ?xb ?xc ?xd) ?y \<Longrightarrow> (mtype(?x, ?xa, ?xb) = ?xc \<rightarrow> ?xd \<Longrightarrow> ?P) \<Longrightarrow> ?P
|
\<lbrakk> ?H1 (?H2 x_1 x_2 x_3 x_4 x_5) x_6; ?H3 x_1 x_2 x_3 x_4 x_5 \<Longrightarrow> x_7\<rbrakk> \<Longrightarrow> x_7
|
[
"FJDefs.mtype",
"Execute.mtype_i_i_i_i_i",
"Predicate.pred.eval"
] |
[
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> nat \\<Rightarrow> nat \\<Rightarrow> nat list \\<Rightarrow> nat \\<Rightarrow> bool",
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> nat \\<Rightarrow> nat \\<Rightarrow> nat list \\<Rightarrow> nat \\<Rightarrow> unit Predicate.pred",
"'a Predicate.pred \\<Rightarrow> 'a \\<Rightarrow> bool"
] |
[
"inductive\n mtype :: \"[classTable, methodName, className, className list, className] \\<Rightarrow> bool\" (\"mtype'(_,_,_') = _ \\<rightarrow> _\" [80,80,80,80] 80)\nwhere\n mt_class: \n \"\\<lbrakk> CT(C) = Some(CDef);\n lookup (cMethods CDef) (\\<lambda>md.(mName md = m)) = Some(mDef);\n varDefs_types (mParams mDef) = Bs;\n mReturn mDef = B \\<rbrakk>\n \\<Longrightarrow> mtype(CT,m,C) = Bs \\<rightarrow> B\"\n\n| mt_super: \n \"\\<lbrakk> CT(C) = Some (CDef);\n lookup (cMethods CDef) (\\<lambda>md.(mName md = m)) = None;\n cSuper CDef = D;\n mtype(CT,m,D) = Bs \\<rightarrow> B \\<rbrakk>\n \\<Longrightarrow> mtype(CT,m,C) = Bs \\<rightarrow> B\"",
"datatype (plugins only: extraction) (dead 'a) pred = Pred (eval: \"'a \\<Rightarrow> bool\")"
] |
FeatherweightJava/Execute
|
Execute.method_typing_aux_i_i_i_i_iE
| null |
pred.eval (method_typing_aux_i_i_i_i_i ?x ?xa ?xb ?xc ?xd) ?y \<Longrightarrow> (method_typing_aux ?x ?xa ?xb ?xc ?xd \<Longrightarrow> ?P) \<Longrightarrow> ?P
|
\<lbrakk> ?H1 (?H2 x_1 x_2 x_3 x_4 x_5) x_6; ?H3 x_1 x_2 x_3 x_4 x_5 \<Longrightarrow> x_7\<rbrakk> \<Longrightarrow> x_7
|
[
"Execute.method_typing_aux",
"Execute.method_typing_aux_i_i_i_i_i",
"Predicate.pred.eval"
] |
[
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> nat \\<Rightarrow> nat \\<Rightarrow> nat list \\<Rightarrow> nat \\<Rightarrow> bool",
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> nat \\<Rightarrow> nat \\<Rightarrow> nat list \\<Rightarrow> nat \\<Rightarrow> unit Predicate.pred",
"'a Predicate.pred \\<Rightarrow> 'a \\<Rightarrow> bool"
] |
[
"definition method_typing_aux\nwhere\n \"method_typing_aux CT m D Cs C = (\\<not> (\\<forall>Ds D0. mtype(CT,m,D) = Ds \\<rightarrow> D0 \\<longrightarrow> Cs = Ds \\<and> C = D0))\"",
"datatype (plugins only: extraction) (dead 'a) pred = Pred (eval: \"'a \\<Rightarrow> bool\")"
] |
FeatherweightJava/Execute
|
Execute.mbody_i_i_i_i_iE
| null |
pred.eval (mbody_i_i_i_i_i ?x ?xa ?xb ?xc ?xd) ?y \<Longrightarrow> (mbody(?x, ?xa, ?xb) = ?xc . ?xd \<Longrightarrow> ?P) \<Longrightarrow> ?P
|
\<lbrakk> ?H1 (?H2 x_1 x_2 x_3 x_4 x_5) x_6; ?H3 x_1 x_2 x_3 x_4 x_5 \<Longrightarrow> x_7\<rbrakk> \<Longrightarrow> x_7
|
[
"FJDefs.mbody",
"Execute.mbody_i_i_i_i_i",
"Predicate.pred.eval"
] |
[
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> nat \\<Rightarrow> nat \\<Rightarrow> nat list \\<Rightarrow> exp \\<Rightarrow> bool",
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> nat \\<Rightarrow> nat \\<Rightarrow> nat list \\<Rightarrow> exp \\<Rightarrow> unit Predicate.pred",
"'a Predicate.pred \\<Rightarrow> 'a \\<Rightarrow> bool"
] |
[
"inductive\n mbody :: \"[classTable, methodName, className, varName list, exp] \\<Rightarrow> bool\" (\"mbody'(_,_,_') = _ . _\" [80,80,80,80] 80)\nwhere\n mb_class: \n \"\\<lbrakk> CT(C) = Some(CDef);\n lookup (cMethods CDef) (\\<lambda>md.(mName md = m)) = Some(mDef);\n varDefs_names (mParams mDef) = xs;\n mBody mDef = e \\<rbrakk>\n \\<Longrightarrow> mbody(CT,m,C) = xs . e\"\n\n| mb_super:\n \"\\<lbrakk> CT(C) = Some(CDef);\n lookup (cMethods CDef) (\\<lambda>md.(mName md = m)) = None;\n cSuper CDef = D;\n mbody(CT,m,D) = xs . e \\<rbrakk>\n \\<Longrightarrow> mbody(CT,m,C) = xs . e\"",
"datatype (plugins only: extraction) (dead 'a) pred = Pred (eval: \"'a \\<Rightarrow> bool\")"
] |
FeatherweightJava/Execute
|
Execute.mbody_i_i_i_o_oE
| null |
pred.eval (mbody_i_i_i_o_o ?x ?xa ?xb) (?xc, ?xd) \<Longrightarrow> (mbody(?x, ?xa, ?xb) = ?xc . ?xd \<Longrightarrow> ?P) \<Longrightarrow> ?P
|
\<lbrakk> ?H1 (?H2 x_1 x_2 x_3) (x_4, x_5); ?H3 x_1 x_2 x_3 x_4 x_5 \<Longrightarrow> x_6\<rbrakk> \<Longrightarrow> x_6
|
[
"FJDefs.mbody",
"Execute.mbody_i_i_i_o_o",
"Predicate.pred.eval"
] |
[
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> nat \\<Rightarrow> nat \\<Rightarrow> nat list \\<Rightarrow> exp \\<Rightarrow> bool",
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> nat \\<Rightarrow> nat \\<Rightarrow> (nat list \\<times> exp) Predicate.pred",
"'a Predicate.pred \\<Rightarrow> 'a \\<Rightarrow> bool"
] |
[
"inductive\n mbody :: \"[classTable, methodName, className, varName list, exp] \\<Rightarrow> bool\" (\"mbody'(_,_,_') = _ . _\" [80,80,80,80] 80)\nwhere\n mb_class: \n \"\\<lbrakk> CT(C) = Some(CDef);\n lookup (cMethods CDef) (\\<lambda>md.(mName md = m)) = Some(mDef);\n varDefs_names (mParams mDef) = xs;\n mBody mDef = e \\<rbrakk>\n \\<Longrightarrow> mbody(CT,m,C) = xs . e\"\n\n| mb_super:\n \"\\<lbrakk> CT(C) = Some(CDef);\n lookup (cMethods CDef) (\\<lambda>md.(mName md = m)) = None;\n cSuper CDef = D;\n mbody(CT,m,D) = xs . e \\<rbrakk>\n \\<Longrightarrow> mbody(CT,m,C) = xs . e\"",
"datatype (plugins only: extraction) (dead 'a) pred = Pred (eval: \"'a \\<Rightarrow> bool\")"
] |
FeatherweightJava/Execute
|
Execute.mtype_i_i_i_o_oE
| null |
pred.eval (mtype_i_i_i_o_o ?x ?xa ?xb) (?xc, ?xd) \<Longrightarrow> (mtype(?x, ?xa, ?xb) = ?xc \<rightarrow> ?xd \<Longrightarrow> ?P) \<Longrightarrow> ?P
|
\<lbrakk> ?H1 (?H2 x_1 x_2 x_3) (x_4, x_5); ?H3 x_1 x_2 x_3 x_4 x_5 \<Longrightarrow> x_6\<rbrakk> \<Longrightarrow> x_6
|
[
"FJDefs.mtype",
"Execute.mtype_i_i_i_o_o",
"Predicate.pred.eval"
] |
[
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> nat \\<Rightarrow> nat \\<Rightarrow> nat list \\<Rightarrow> nat \\<Rightarrow> bool",
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> nat \\<Rightarrow> nat \\<Rightarrow> (nat list \\<times> nat) Predicate.pred",
"'a Predicate.pred \\<Rightarrow> 'a \\<Rightarrow> bool"
] |
[
"inductive\n mtype :: \"[classTable, methodName, className, className list, className] \\<Rightarrow> bool\" (\"mtype'(_,_,_') = _ \\<rightarrow> _\" [80,80,80,80] 80)\nwhere\n mt_class: \n \"\\<lbrakk> CT(C) = Some(CDef);\n lookup (cMethods CDef) (\\<lambda>md.(mName md = m)) = Some(mDef);\n varDefs_types (mParams mDef) = Bs;\n mReturn mDef = B \\<rbrakk>\n \\<Longrightarrow> mtype(CT,m,C) = Bs \\<rightarrow> B\"\n\n| mt_super: \n \"\\<lbrakk> CT(C) = Some (CDef);\n lookup (cMethods CDef) (\\<lambda>md.(mName md = m)) = None;\n cSuper CDef = D;\n mtype(CT,m,D) = Bs \\<rightarrow> B \\<rbrakk>\n \\<Longrightarrow> mtype(CT,m,C) = Bs \\<rightarrow> B\"",
"datatype (plugins only: extraction) (dead 'a) pred = Pred (eval: \"'a \\<Rightarrow> bool\")"
] |
FeatherweightJava/Execute
|
Execute.mbody_i_i_i_i_oE
| null |
pred.eval (mbody_i_i_i_i_o ?x ?xa ?xb ?xc) ?xd \<Longrightarrow> (mbody(?x, ?xa, ?xb) = ?xc . ?xd \<Longrightarrow> ?P) \<Longrightarrow> ?P
|
\<lbrakk> ?H1 (?H2 x_1 x_2 x_3 x_4) x_5; ?H3 x_1 x_2 x_3 x_4 x_5 \<Longrightarrow> x_6\<rbrakk> \<Longrightarrow> x_6
|
[
"FJDefs.mbody",
"Execute.mbody_i_i_i_i_o",
"Predicate.pred.eval"
] |
[
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> nat \\<Rightarrow> nat \\<Rightarrow> nat list \\<Rightarrow> exp \\<Rightarrow> bool",
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> nat \\<Rightarrow> nat \\<Rightarrow> nat list \\<Rightarrow> exp Predicate.pred",
"'a Predicate.pred \\<Rightarrow> 'a \\<Rightarrow> bool"
] |
[
"inductive\n mbody :: \"[classTable, methodName, className, varName list, exp] \\<Rightarrow> bool\" (\"mbody'(_,_,_') = _ . _\" [80,80,80,80] 80)\nwhere\n mb_class: \n \"\\<lbrakk> CT(C) = Some(CDef);\n lookup (cMethods CDef) (\\<lambda>md.(mName md = m)) = Some(mDef);\n varDefs_names (mParams mDef) = xs;\n mBody mDef = e \\<rbrakk>\n \\<Longrightarrow> mbody(CT,m,C) = xs . e\"\n\n| mb_super:\n \"\\<lbrakk> CT(C) = Some(CDef);\n lookup (cMethods CDef) (\\<lambda>md.(mName md = m)) = None;\n cSuper CDef = D;\n mbody(CT,m,D) = xs . e \\<rbrakk>\n \\<Longrightarrow> mbody(CT,m,C) = xs . e\"",
"datatype (plugins only: extraction) (dead 'a) pred = Pred (eval: \"'a \\<Rightarrow> bool\")"
] |
FeatherweightJava/Execute
|
Execute.mtype_i_i_i_o_iE
| null |
pred.eval (mtype_i_i_i_o_i ?x ?xa ?xb ?xd) ?xc \<Longrightarrow> (mtype(?x, ?xa, ?xb) = ?xc \<rightarrow> ?xd \<Longrightarrow> ?P) \<Longrightarrow> ?P
|
\<lbrakk> ?H1 (?H2 x_1 x_2 x_3 x_4) x_5; ?H3 x_1 x_2 x_3 x_5 x_4 \<Longrightarrow> x_6\<rbrakk> \<Longrightarrow> x_6
|
[
"FJDefs.mtype",
"Execute.mtype_i_i_i_o_i",
"Predicate.pred.eval"
] |
[
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> nat \\<Rightarrow> nat \\<Rightarrow> nat list \\<Rightarrow> nat \\<Rightarrow> bool",
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> nat \\<Rightarrow> nat \\<Rightarrow> nat \\<Rightarrow> nat list Predicate.pred",
"'a Predicate.pred \\<Rightarrow> 'a \\<Rightarrow> bool"
] |
[
"inductive\n mtype :: \"[classTable, methodName, className, className list, className] \\<Rightarrow> bool\" (\"mtype'(_,_,_') = _ \\<rightarrow> _\" [80,80,80,80] 80)\nwhere\n mt_class: \n \"\\<lbrakk> CT(C) = Some(CDef);\n lookup (cMethods CDef) (\\<lambda>md.(mName md = m)) = Some(mDef);\n varDefs_types (mParams mDef) = Bs;\n mReturn mDef = B \\<rbrakk>\n \\<Longrightarrow> mtype(CT,m,C) = Bs \\<rightarrow> B\"\n\n| mt_super: \n \"\\<lbrakk> CT(C) = Some (CDef);\n lookup (cMethods CDef) (\\<lambda>md.(mName md = m)) = None;\n cSuper CDef = D;\n mtype(CT,m,D) = Bs \\<rightarrow> B \\<rbrakk>\n \\<Longrightarrow> mtype(CT,m,C) = Bs \\<rightarrow> B\"",
"datatype (plugins only: extraction) (dead 'a) pred = Pred (eval: \"'a \\<Rightarrow> bool\")"
] |
FeatherweightJava/Execute
|
Execute.mbody_i_i_i_o_iE
| null |
pred.eval (mbody_i_i_i_o_i ?x ?xa ?xb ?xd) ?xc \<Longrightarrow> (mbody(?x, ?xa, ?xb) = ?xc . ?xd \<Longrightarrow> ?P) \<Longrightarrow> ?P
|
\<lbrakk> ?H1 (?H2 x_1 x_2 x_3 x_4) x_5; ?H3 x_1 x_2 x_3 x_5 x_4 \<Longrightarrow> x_6\<rbrakk> \<Longrightarrow> x_6
|
[
"FJDefs.mbody",
"Execute.mbody_i_i_i_o_i",
"Predicate.pred.eval"
] |
[
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> nat \\<Rightarrow> nat \\<Rightarrow> nat list \\<Rightarrow> exp \\<Rightarrow> bool",
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> nat \\<Rightarrow> nat \\<Rightarrow> exp \\<Rightarrow> nat list Predicate.pred",
"'a Predicate.pred \\<Rightarrow> 'a \\<Rightarrow> bool"
] |
[
"inductive\n mbody :: \"[classTable, methodName, className, varName list, exp] \\<Rightarrow> bool\" (\"mbody'(_,_,_') = _ . _\" [80,80,80,80] 80)\nwhere\n mb_class: \n \"\\<lbrakk> CT(C) = Some(CDef);\n lookup (cMethods CDef) (\\<lambda>md.(mName md = m)) = Some(mDef);\n varDefs_names (mParams mDef) = xs;\n mBody mDef = e \\<rbrakk>\n \\<Longrightarrow> mbody(CT,m,C) = xs . e\"\n\n| mb_super:\n \"\\<lbrakk> CT(C) = Some(CDef);\n lookup (cMethods CDef) (\\<lambda>md.(mName md = m)) = None;\n cSuper CDef = D;\n mbody(CT,m,D) = xs . e \\<rbrakk>\n \\<Longrightarrow> mbody(CT,m,C) = xs . e\"",
"datatype (plugins only: extraction) (dead 'a) pred = Pred (eval: \"'a \\<Rightarrow> bool\")"
] |
FeatherweightJava/Execute
|
Execute.mtype_i_i_i_i_oE
| null |
pred.eval (mtype_i_i_i_i_o ?x ?xa ?xb ?xc) ?xd \<Longrightarrow> (mtype(?x, ?xa, ?xb) = ?xc \<rightarrow> ?xd \<Longrightarrow> ?P) \<Longrightarrow> ?P
|
\<lbrakk> ?H1 (?H2 x_1 x_2 x_3 x_4) x_5; ?H3 x_1 x_2 x_3 x_4 x_5 \<Longrightarrow> x_6\<rbrakk> \<Longrightarrow> x_6
|
[
"FJDefs.mtype",
"Execute.mtype_i_i_i_i_o",
"Predicate.pred.eval"
] |
[
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> nat \\<Rightarrow> nat \\<Rightarrow> nat list \\<Rightarrow> nat \\<Rightarrow> bool",
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> nat \\<Rightarrow> nat \\<Rightarrow> nat list \\<Rightarrow> nat Predicate.pred",
"'a Predicate.pred \\<Rightarrow> 'a \\<Rightarrow> bool"
] |
[
"inductive\n mtype :: \"[classTable, methodName, className, className list, className] \\<Rightarrow> bool\" (\"mtype'(_,_,_') = _ \\<rightarrow> _\" [80,80,80,80] 80)\nwhere\n mt_class: \n \"\\<lbrakk> CT(C) = Some(CDef);\n lookup (cMethods CDef) (\\<lambda>md.(mName md = m)) = Some(mDef);\n varDefs_types (mParams mDef) = Bs;\n mReturn mDef = B \\<rbrakk>\n \\<Longrightarrow> mtype(CT,m,C) = Bs \\<rightarrow> B\"\n\n| mt_super: \n \"\\<lbrakk> CT(C) = Some (CDef);\n lookup (cMethods CDef) (\\<lambda>md.(mName md = m)) = None;\n cSuper CDef = D;\n mtype(CT,m,D) = Bs \\<rightarrow> B \\<rbrakk>\n \\<Longrightarrow> mtype(CT,m,C) = Bs \\<rightarrow> B\"",
"datatype (plugins only: extraction) (dead 'a) pred = Pred (eval: \"'a \\<Rightarrow> bool\")"
] |
FeatherweightJava/Execute
|
Execute.typings_i_i_i_iE
| null |
pred.eval (typings_i_i_i_i ?x ?xa ?xb ?xc) ?y \<Longrightarrow> (?x; ?xa \<turnstile>+ ?xb : ?xc \<Longrightarrow> ?P) \<Longrightarrow> ?P
|
\<lbrakk> ?H1 (?H2 x_1 x_2 x_3 x_4) x_5; ?H3 x_1 x_2 x_3 x_4 \<Longrightarrow> x_6\<rbrakk> \<Longrightarrow> x_6
|
[
"FJDefs.typings",
"Execute.typings_i_i_i_i",
"Predicate.pred.eval"
] |
[
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> (nat \\<Rightarrow> nat option) \\<Rightarrow> exp list \\<Rightarrow> nat list \\<Rightarrow> bool",
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> (nat \\<Rightarrow> nat option) \\<Rightarrow> exp list \\<Rightarrow> nat list \\<Rightarrow> unit Predicate.pred",
"'a Predicate.pred \\<Rightarrow> 'a \\<Rightarrow> bool"
] |
[
"inductive\n typings :: \"[classTable, varCtx, exp list, className list] \\<Rightarrow> bool\" (\"_;_ \\<turnstile>+ _ : _\" [80,80,80,80] 80)\n and typing :: \"[classTable, varCtx, exp, className] \\<Rightarrow> bool\" (\"_;_ \\<turnstile> _ : _\" [80,80,80,80] 80)\nwhere\n ts_nil : \"CT;\\<Gamma> \\<turnstile>+ [] : []\"\n\n| ts_cons : \n \"\\<lbrakk> CT;\\<Gamma> \\<turnstile> e0 : C0; CT;\\<Gamma> \\<turnstile>+ es : Cs \\<rbrakk> \n \\<Longrightarrow> CT;\\<Gamma> \\<turnstile>+ (e0 # es) : (C0 # Cs)\"\n\n| t_var : \n \"\\<lbrakk> \\<Gamma>(x) = Some C \\<rbrakk> \\<Longrightarrow> CT;\\<Gamma> \\<turnstile> (Var x) : C\"\n\n| t_field : \n \"\\<lbrakk> CT;\\<Gamma> \\<turnstile> e0 : C0;\n fields(CT,C0) = Cf;\n lookup Cf (\\<lambda>fd.(vdName fd = fi)) = Some(fDef);\n vdType fDef = Ci \\<rbrakk>\n \\<Longrightarrow> CT;\\<Gamma> \\<turnstile> FieldProj e0 fi : Ci\"\n\n| t_invk : \n \"\\<lbrakk> CT;\\<Gamma> \\<turnstile> e0 : C0;\n mtype(CT,m,C0) = Ds \\<rightarrow> C;\n CT;\\<Gamma> \\<turnstile>+ es : Cs;\n CT \\<turnstile>+ Cs <: Ds;\n length es = length Ds \\<rbrakk>\n \\<Longrightarrow> CT;\\<Gamma> \\<turnstile> MethodInvk e0 m es : C\"\n\n| t_new : \n \"\\<lbrakk> fields(CT,C) = Df;\n length es = length Df;\n varDefs_types Df = Ds;\n CT;\\<Gamma> \\<turnstile>+ es : Cs;\n CT \\<turnstile>+ Cs <: Ds \\<rbrakk>\n \\<Longrightarrow> CT;\\<Gamma> \\<turnstile> New C es : C\"\n\n| t_ucast : \n \"\\<lbrakk> CT;\\<Gamma> \\<turnstile> e0 : D; \n CT \\<turnstile> D <: C \\<rbrakk>\n \\<Longrightarrow> CT;\\<Gamma> \\<turnstile> Cast C e0 : C\"\n\n| t_dcast : \n \"\\<lbrakk> CT;\\<Gamma> \\<turnstile> e0 : D; \n CT \\<turnstile> C <: D; C \\<noteq> D \\<rbrakk>\n \\<Longrightarrow> CT;\\<Gamma> \\<turnstile> Cast C e0 : C\"\n\n| t_scast : \n \"\\<lbrakk> CT;\\<Gamma> \\<turnstile> e0 : D;\n CT \\<turnstile> C \\<not><: D;\n CT \\<turnstile> D \\<not><: C \\<rbrakk>\n \\<Longrightarrow> CT;\\<Gamma> \\<turnstile> Cast C e0 : C\"",
"datatype (plugins only: extraction) (dead 'a) pred = Pred (eval: \"'a \\<Rightarrow> bool\")"
] |
FeatherweightJava/Execute
|
Execute.typing_i_i_i_iE
| null |
pred.eval (typing_i_i_i_i ?x ?xa ?xb ?xc) ?y \<Longrightarrow> (?x; ?xa \<turnstile> ?xb : ?xc \<Longrightarrow> ?P) \<Longrightarrow> ?P
|
\<lbrakk> ?H1 (?H2 x_1 x_2 x_3 x_4) x_5; ?H3 x_1 x_2 x_3 x_4 \<Longrightarrow> x_6\<rbrakk> \<Longrightarrow> x_6
|
[
"FJDefs.typing",
"Execute.typing_i_i_i_i",
"Predicate.pred.eval"
] |
[
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> (nat \\<Rightarrow> nat option) \\<Rightarrow> exp \\<Rightarrow> nat \\<Rightarrow> bool",
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> (nat \\<Rightarrow> nat option) \\<Rightarrow> exp \\<Rightarrow> nat \\<Rightarrow> unit Predicate.pred",
"'a Predicate.pred \\<Rightarrow> 'a \\<Rightarrow> bool"
] |
[
"inductive\n typings :: \"[classTable, varCtx, exp list, className list] \\<Rightarrow> bool\" (\"_;_ \\<turnstile>+ _ : _\" [80,80,80,80] 80)\n and typing :: \"[classTable, varCtx, exp, className] \\<Rightarrow> bool\" (\"_;_ \\<turnstile> _ : _\" [80,80,80,80] 80)\nwhere\n ts_nil : \"CT;\\<Gamma> \\<turnstile>+ [] : []\"\n\n| ts_cons : \n \"\\<lbrakk> CT;\\<Gamma> \\<turnstile> e0 : C0; CT;\\<Gamma> \\<turnstile>+ es : Cs \\<rbrakk> \n \\<Longrightarrow> CT;\\<Gamma> \\<turnstile>+ (e0 # es) : (C0 # Cs)\"\n\n| t_var : \n \"\\<lbrakk> \\<Gamma>(x) = Some C \\<rbrakk> \\<Longrightarrow> CT;\\<Gamma> \\<turnstile> (Var x) : C\"\n\n| t_field : \n \"\\<lbrakk> CT;\\<Gamma> \\<turnstile> e0 : C0;\n fields(CT,C0) = Cf;\n lookup Cf (\\<lambda>fd.(vdName fd = fi)) = Some(fDef);\n vdType fDef = Ci \\<rbrakk>\n \\<Longrightarrow> CT;\\<Gamma> \\<turnstile> FieldProj e0 fi : Ci\"\n\n| t_invk : \n \"\\<lbrakk> CT;\\<Gamma> \\<turnstile> e0 : C0;\n mtype(CT,m,C0) = Ds \\<rightarrow> C;\n CT;\\<Gamma> \\<turnstile>+ es : Cs;\n CT \\<turnstile>+ Cs <: Ds;\n length es = length Ds \\<rbrakk>\n \\<Longrightarrow> CT;\\<Gamma> \\<turnstile> MethodInvk e0 m es : C\"\n\n| t_new : \n \"\\<lbrakk> fields(CT,C) = Df;\n length es = length Df;\n varDefs_types Df = Ds;\n CT;\\<Gamma> \\<turnstile>+ es : Cs;\n CT \\<turnstile>+ Cs <: Ds \\<rbrakk>\n \\<Longrightarrow> CT;\\<Gamma> \\<turnstile> New C es : C\"\n\n| t_ucast : \n \"\\<lbrakk> CT;\\<Gamma> \\<turnstile> e0 : D; \n CT \\<turnstile> D <: C \\<rbrakk>\n \\<Longrightarrow> CT;\\<Gamma> \\<turnstile> Cast C e0 : C\"\n\n| t_dcast : \n \"\\<lbrakk> CT;\\<Gamma> \\<turnstile> e0 : D; \n CT \\<turnstile> C <: D; C \\<noteq> D \\<rbrakk>\n \\<Longrightarrow> CT;\\<Gamma> \\<turnstile> Cast C e0 : C\"\n\n| t_scast : \n \"\\<lbrakk> CT;\\<Gamma> \\<turnstile> e0 : D;\n CT \\<turnstile> C \\<not><: D;\n CT \\<turnstile> D \\<not><: C \\<rbrakk>\n \\<Longrightarrow> CT;\\<Gamma> \\<turnstile> Cast C e0 : C\"",
"datatype (plugins only: extraction) (dead 'a) pred = Pred (eval: \"'a \\<Rightarrow> bool\")"
] |
FeatherweightJava/Execute
|
Execute.typings_i_i_i_oE
| null |
pred.eval (typings_i_i_i_o ?x ?xa ?xb) ?xc \<Longrightarrow> (?x; ?xa \<turnstile>+ ?xb : ?xc \<Longrightarrow> ?P) \<Longrightarrow> ?P
|
\<lbrakk> ?H1 (?H2 x_1 x_2 x_3) x_4; ?H3 x_1 x_2 x_3 x_4 \<Longrightarrow> x_5\<rbrakk> \<Longrightarrow> x_5
|
[
"FJDefs.typings",
"Execute.typings_i_i_i_o",
"Predicate.pred.eval"
] |
[
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> (nat \\<Rightarrow> nat option) \\<Rightarrow> exp list \\<Rightarrow> nat list \\<Rightarrow> bool",
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> (nat \\<Rightarrow> nat option) \\<Rightarrow> exp list \\<Rightarrow> nat list Predicate.pred",
"'a Predicate.pred \\<Rightarrow> 'a \\<Rightarrow> bool"
] |
[
"inductive\n typings :: \"[classTable, varCtx, exp list, className list] \\<Rightarrow> bool\" (\"_;_ \\<turnstile>+ _ : _\" [80,80,80,80] 80)\n and typing :: \"[classTable, varCtx, exp, className] \\<Rightarrow> bool\" (\"_;_ \\<turnstile> _ : _\" [80,80,80,80] 80)\nwhere\n ts_nil : \"CT;\\<Gamma> \\<turnstile>+ [] : []\"\n\n| ts_cons : \n \"\\<lbrakk> CT;\\<Gamma> \\<turnstile> e0 : C0; CT;\\<Gamma> \\<turnstile>+ es : Cs \\<rbrakk> \n \\<Longrightarrow> CT;\\<Gamma> \\<turnstile>+ (e0 # es) : (C0 # Cs)\"\n\n| t_var : \n \"\\<lbrakk> \\<Gamma>(x) = Some C \\<rbrakk> \\<Longrightarrow> CT;\\<Gamma> \\<turnstile> (Var x) : C\"\n\n| t_field : \n \"\\<lbrakk> CT;\\<Gamma> \\<turnstile> e0 : C0;\n fields(CT,C0) = Cf;\n lookup Cf (\\<lambda>fd.(vdName fd = fi)) = Some(fDef);\n vdType fDef = Ci \\<rbrakk>\n \\<Longrightarrow> CT;\\<Gamma> \\<turnstile> FieldProj e0 fi : Ci\"\n\n| t_invk : \n \"\\<lbrakk> CT;\\<Gamma> \\<turnstile> e0 : C0;\n mtype(CT,m,C0) = Ds \\<rightarrow> C;\n CT;\\<Gamma> \\<turnstile>+ es : Cs;\n CT \\<turnstile>+ Cs <: Ds;\n length es = length Ds \\<rbrakk>\n \\<Longrightarrow> CT;\\<Gamma> \\<turnstile> MethodInvk e0 m es : C\"\n\n| t_new : \n \"\\<lbrakk> fields(CT,C) = Df;\n length es = length Df;\n varDefs_types Df = Ds;\n CT;\\<Gamma> \\<turnstile>+ es : Cs;\n CT \\<turnstile>+ Cs <: Ds \\<rbrakk>\n \\<Longrightarrow> CT;\\<Gamma> \\<turnstile> New C es : C\"\n\n| t_ucast : \n \"\\<lbrakk> CT;\\<Gamma> \\<turnstile> e0 : D; \n CT \\<turnstile> D <: C \\<rbrakk>\n \\<Longrightarrow> CT;\\<Gamma> \\<turnstile> Cast C e0 : C\"\n\n| t_dcast : \n \"\\<lbrakk> CT;\\<Gamma> \\<turnstile> e0 : D; \n CT \\<turnstile> C <: D; C \\<noteq> D \\<rbrakk>\n \\<Longrightarrow> CT;\\<Gamma> \\<turnstile> Cast C e0 : C\"\n\n| t_scast : \n \"\\<lbrakk> CT;\\<Gamma> \\<turnstile> e0 : D;\n CT \\<turnstile> C \\<not><: D;\n CT \\<turnstile> D \\<not><: C \\<rbrakk>\n \\<Longrightarrow> CT;\\<Gamma> \\<turnstile> Cast C e0 : C\"",
"datatype (plugins only: extraction) (dead 'a) pred = Pred (eval: \"'a \\<Rightarrow> bool\")"
] |
FeatherweightJava/Execute
|
Execute.typing_i_i_i_oE
| null |
pred.eval (typing_i_i_i_o ?x ?xa ?xb) ?xc \<Longrightarrow> (?x; ?xa \<turnstile> ?xb : ?xc \<Longrightarrow> ?P) \<Longrightarrow> ?P
|
\<lbrakk> ?H1 (?H2 x_1 x_2 x_3) x_4; ?H3 x_1 x_2 x_3 x_4 \<Longrightarrow> x_5\<rbrakk> \<Longrightarrow> x_5
|
[
"FJDefs.typing",
"Execute.typing_i_i_i_o",
"Predicate.pred.eval"
] |
[
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> (nat \\<Rightarrow> nat option) \\<Rightarrow> exp \\<Rightarrow> nat \\<Rightarrow> bool",
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> (nat \\<Rightarrow> nat option) \\<Rightarrow> exp \\<Rightarrow> nat Predicate.pred",
"'a Predicate.pred \\<Rightarrow> 'a \\<Rightarrow> bool"
] |
[
"inductive\n typings :: \"[classTable, varCtx, exp list, className list] \\<Rightarrow> bool\" (\"_;_ \\<turnstile>+ _ : _\" [80,80,80,80] 80)\n and typing :: \"[classTable, varCtx, exp, className] \\<Rightarrow> bool\" (\"_;_ \\<turnstile> _ : _\" [80,80,80,80] 80)\nwhere\n ts_nil : \"CT;\\<Gamma> \\<turnstile>+ [] : []\"\n\n| ts_cons : \n \"\\<lbrakk> CT;\\<Gamma> \\<turnstile> e0 : C0; CT;\\<Gamma> \\<turnstile>+ es : Cs \\<rbrakk> \n \\<Longrightarrow> CT;\\<Gamma> \\<turnstile>+ (e0 # es) : (C0 # Cs)\"\n\n| t_var : \n \"\\<lbrakk> \\<Gamma>(x) = Some C \\<rbrakk> \\<Longrightarrow> CT;\\<Gamma> \\<turnstile> (Var x) : C\"\n\n| t_field : \n \"\\<lbrakk> CT;\\<Gamma> \\<turnstile> e0 : C0;\n fields(CT,C0) = Cf;\n lookup Cf (\\<lambda>fd.(vdName fd = fi)) = Some(fDef);\n vdType fDef = Ci \\<rbrakk>\n \\<Longrightarrow> CT;\\<Gamma> \\<turnstile> FieldProj e0 fi : Ci\"\n\n| t_invk : \n \"\\<lbrakk> CT;\\<Gamma> \\<turnstile> e0 : C0;\n mtype(CT,m,C0) = Ds \\<rightarrow> C;\n CT;\\<Gamma> \\<turnstile>+ es : Cs;\n CT \\<turnstile>+ Cs <: Ds;\n length es = length Ds \\<rbrakk>\n \\<Longrightarrow> CT;\\<Gamma> \\<turnstile> MethodInvk e0 m es : C\"\n\n| t_new : \n \"\\<lbrakk> fields(CT,C) = Df;\n length es = length Df;\n varDefs_types Df = Ds;\n CT;\\<Gamma> \\<turnstile>+ es : Cs;\n CT \\<turnstile>+ Cs <: Ds \\<rbrakk>\n \\<Longrightarrow> CT;\\<Gamma> \\<turnstile> New C es : C\"\n\n| t_ucast : \n \"\\<lbrakk> CT;\\<Gamma> \\<turnstile> e0 : D; \n CT \\<turnstile> D <: C \\<rbrakk>\n \\<Longrightarrow> CT;\\<Gamma> \\<turnstile> Cast C e0 : C\"\n\n| t_dcast : \n \"\\<lbrakk> CT;\\<Gamma> \\<turnstile> e0 : D; \n CT \\<turnstile> C <: D; C \\<noteq> D \\<rbrakk>\n \\<Longrightarrow> CT;\\<Gamma> \\<turnstile> Cast C e0 : C\"\n\n| t_scast : \n \"\\<lbrakk> CT;\\<Gamma> \\<turnstile> e0 : D;\n CT \\<turnstile> C \\<not><: D;\n CT \\<turnstile> D \\<not><: C \\<rbrakk>\n \\<Longrightarrow> CT;\\<Gamma> \\<turnstile> Cast C e0 : C\"",
"datatype (plugins only: extraction) (dead 'a) pred = Pred (eval: \"'a \\<Rightarrow> bool\")"
] |
FeatherweightJava/Execute
|
Execute.subtypings_i_i_iE
| null |
pred.eval (subtypings_i_i_i ?x ?xa ?xb) ?y \<Longrightarrow> (?x \<turnstile>+ ?xa <: ?xb \<Longrightarrow> ?P) \<Longrightarrow> ?P
|
\<lbrakk> ?H1 (?H2 x_1 x_2 x_3) x_4; ?H3 x_1 x_2 x_3 \<Longrightarrow> x_5\<rbrakk> \<Longrightarrow> x_5
|
[
"FJDefs.subtypings",
"Execute.subtypings_i_i_i",
"Predicate.pred.eval"
] |
[
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> nat list \\<Rightarrow> nat list \\<Rightarrow> bool",
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> nat list \\<Rightarrow> nat list \\<Rightarrow> unit Predicate.pred",
"'a Predicate.pred \\<Rightarrow> 'a \\<Rightarrow> bool"
] |
[
"inductive\n subtypings :: \"[classTable, className list, className list] \\<Rightarrow> bool\" (\"_ \\<turnstile>+ _ <: _\" [80,80,80] 80)\nwhere\n ss_nil : \"CT \\<turnstile>+ [] <: []\"\n| ss_cons : \"\\<lbrakk> CT \\<turnstile> C0 <: D0; CT \\<turnstile>+ Cs <: Ds \\<rbrakk> \\<Longrightarrow> CT \\<turnstile>+ (C0 # Cs) <: (D0 # Ds)\"",
"datatype (plugins only: extraction) (dead 'a) pred = Pred (eval: \"'a \\<Rightarrow> bool\")"
] |
FeatherweightJava/Execute
|
Execute.subtyping_i_i_iE
| null |
pred.eval (subtyping_i_i_i ?x ?xa ?xb) ?y \<Longrightarrow> (?x \<turnstile> ?xa <: ?xb \<Longrightarrow> ?P) \<Longrightarrow> ?P
|
\<lbrakk> ?H1 (?H2 x_1 x_2 x_3) x_4; ?H3 x_1 x_2 x_3 \<Longrightarrow> x_5\<rbrakk> \<Longrightarrow> x_5
|
[
"FJDefs.subtyping",
"Execute.subtyping_i_i_i",
"Predicate.pred.eval"
] |
[
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> nat \\<Rightarrow> nat \\<Rightarrow> bool",
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> nat \\<Rightarrow> nat \\<Rightarrow> unit Predicate.pred",
"'a Predicate.pred \\<Rightarrow> 'a \\<Rightarrow> bool"
] |
[
"inductive\n subtyping :: \"[classTable, className, className] \\<Rightarrow> bool\" (\"_ \\<turnstile> _ <: _\" [80,80,80] 80)\nwhere\n s_refl : \"CT \\<turnstile> C <: C\"\n| s_trans : \"\\<lbrakk> CT \\<turnstile> C <: D; CT \\<turnstile> D <: E \\<rbrakk> \\<Longrightarrow> CT \\<turnstile> C <: E\"\n| s_super : \"\\<lbrakk> CT(C) = Some(CDef); cSuper CDef = D \\<rbrakk> \\<Longrightarrow> CT \\<turnstile> C <: D\"",
"datatype (plugins only: extraction) (dead 'a) pred = Pred (eval: \"'a \\<Rightarrow> bool\")"
] |
FeatherweightJava/Execute
|
Execute.reductions_i_i_iE
| null |
pred.eval (reductions_i_i_i ?x ?xa ?xb) ?y \<Longrightarrow> (?x \<turnstile> ?xa \<rightarrow>* ?xb \<Longrightarrow> ?P) \<Longrightarrow> ?P
|
\<lbrakk> ?H1 (?H2 x_1 x_2 x_3) x_4; ?H3 x_1 x_2 x_3 \<Longrightarrow> x_5\<rbrakk> \<Longrightarrow> x_5
|
[
"FJDefs.reductions",
"Execute.reductions_i_i_i",
"Predicate.pred.eval"
] |
[
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> exp \\<Rightarrow> exp \\<Rightarrow> bool",
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> exp \\<Rightarrow> exp \\<Rightarrow> unit Predicate.pred",
"'a Predicate.pred \\<Rightarrow> 'a \\<Rightarrow> bool"
] |
[
"inductive\n reductions :: \"[classTable, exp, exp] \\<Rightarrow> bool\" (\"_ \\<turnstile> _ \\<rightarrow>* _\" [80,80,80] 80)\nwhere\n rs_refl: \"CT \\<turnstile> e \\<rightarrow>* e\" \n| rs_trans: \"\\<lbrakk> CT \\<turnstile> e \\<rightarrow> e'; CT \\<turnstile> e' \\<rightarrow>* e'' \\<rbrakk> \\<Longrightarrow> CT \\<turnstile> e \\<rightarrow>* e''\"",
"datatype (plugins only: extraction) (dead 'a) pred = Pred (eval: \"'a \\<Rightarrow> bool\")"
] |
FeatherweightJava/Execute
|
Execute.reduction_i_i_iE
| null |
pred.eval (reduction_i_i_i ?x ?xa ?xb) ?y \<Longrightarrow> (?x \<turnstile> ?xa \<rightarrow> ?xb \<Longrightarrow> ?P) \<Longrightarrow> ?P
|
\<lbrakk> ?H1 (?H2 x_1 x_2 x_3) x_4; ?H3 x_1 x_2 x_3 \<Longrightarrow> x_5\<rbrakk> \<Longrightarrow> x_5
|
[
"FJDefs.reduction",
"Execute.reduction_i_i_i",
"Predicate.pred.eval"
] |
[
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> exp \\<Rightarrow> exp \\<Rightarrow> bool",
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> exp \\<Rightarrow> exp \\<Rightarrow> unit Predicate.pred",
"'a Predicate.pred \\<Rightarrow> 'a \\<Rightarrow> bool"
] |
[
"inductive\n reduction :: \"[classTable, exp, exp] \\<Rightarrow> bool\" (\"_ \\<turnstile> _ \\<rightarrow> _\" [80,80,80] 80)\nwhere\n\n r_field: \n \"\\<lbrakk> fields(CT,C) = Cf; \n lookup2 Cf es (\\<lambda>fd.(vdName fd = fi)) = Some(ei) \\<rbrakk>\n \\<Longrightarrow> CT \\<turnstile> FieldProj (New C es) fi \\<rightarrow> ei\"\n\n| r_invk: \n \"\\<lbrakk> mbody(CT,m,C) = xs . e0;\n substs ((map_upds Map.empty xs ds)(this \\<mapsto> (New C es))) e0 = e0' \\<rbrakk>\n \\<Longrightarrow> CT \\<turnstile> MethodInvk (New C es) m ds \\<rightarrow> e0'\"\n\n| r_cast: \n \"\\<lbrakk> CT \\<turnstile> C <: D \\<rbrakk> \n \\<Longrightarrow> CT \\<turnstile> Cast D (New C es) \\<rightarrow> New C es\"\n\n| rc_field: \n \"\\<lbrakk> CT \\<turnstile> e0 \\<rightarrow> e0' \\<rbrakk> \n \\<Longrightarrow> CT \\<turnstile> FieldProj e0 f \\<rightarrow> FieldProj e0' f\"\n\n| rc_invk_recv: \n \"\\<lbrakk> CT \\<turnstile> e0 \\<rightarrow> e0' \\<rbrakk> \n \\<Longrightarrow> CT \\<turnstile> MethodInvk e0 m es \\<rightarrow> MethodInvk e0' m es\"\n\n| rc_invk_arg: \n \"\\<lbrakk> CT \\<turnstile> ei \\<rightarrow> ei' \\<rbrakk>\n \\<Longrightarrow> CT \\<turnstile> MethodInvk e0 m (el@ei#er) \\<rightarrow> MethodInvk e0 m (el@ei'#er)\"\n\n| rc_new_arg: \n \"\\<lbrakk> CT \\<turnstile> ei \\<rightarrow> ei' \\<rbrakk> \n \\<Longrightarrow> CT \\<turnstile> New C (el@ei#er) \\<rightarrow> New C (el@ei'#er)\"\n\n| rc_cast: \n \"\\<lbrakk> CT \\<turnstile> e0 \\<rightarrow> e0' \\<rbrakk> \n \\<Longrightarrow> CT \\<turnstile> Cast C e0 \\<rightarrow> Cast C e0'\"",
"datatype (plugins only: extraction) (dead 'a) pred = Pred (eval: \"'a \\<Rightarrow> bool\")"
] |
FeatherweightJava/Execute
|
Execute.method_typings_i_i_iE
| null |
pred.eval (method_typings_i_i_i ?x ?xa ?xb) ?y \<Longrightarrow> (?x \<turnstile>+ ?xa OK IN ?xb \<Longrightarrow> ?P) \<Longrightarrow> ?P
|
\<lbrakk> ?H1 (?H2 x_1 x_2 x_3) x_4; ?H3 x_1 x_2 x_3 \<Longrightarrow> x_5\<rbrakk> \<Longrightarrow> x_5
|
[
"FJDefs.method_typings",
"Execute.method_typings_i_i_i",
"Predicate.pred.eval"
] |
[
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> methodDef list \\<Rightarrow> nat \\<Rightarrow> bool",
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> methodDef list \\<Rightarrow> nat \\<Rightarrow> unit Predicate.pred",
"'a Predicate.pred \\<Rightarrow> 'a \\<Rightarrow> bool"
] |
[
"inductive\n method_typings :: \"[classTable, methodDef list, className] \\<Rightarrow> bool\" (\"_ \\<turnstile>+ _ OK IN _\" [80,80,80] 80)\nwhere\n ms_nil : \n \"CT \\<turnstile>+ [] OK IN C\"\n\n| ms_cons : \n \"\\<lbrakk> CT \\<turnstile> m OK IN C; \n CT \\<turnstile>+ ms OK IN C \\<rbrakk>\n \\<Longrightarrow> CT \\<turnstile>+ (m # ms) OK IN C\"",
"datatype (plugins only: extraction) (dead 'a) pred = Pred (eval: \"'a \\<Rightarrow> bool\")"
] |
FeatherweightJava/Execute
|
Execute.method_typing_i_i_iE
| null |
pred.eval (method_typing_i_i_i ?x ?xa ?xb) ?y \<Longrightarrow> (?x \<turnstile> ?xa OK IN ?xb \<Longrightarrow> ?P) \<Longrightarrow> ?P
|
\<lbrakk> ?H1 (?H2 x_1 x_2 x_3) x_4; ?H3 x_1 x_2 x_3 \<Longrightarrow> x_5\<rbrakk> \<Longrightarrow> x_5
|
[
"FJDefs.method_typing",
"Execute.method_typing_i_i_i",
"Predicate.pred.eval"
] |
[
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> methodDef \\<Rightarrow> nat \\<Rightarrow> bool",
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> methodDef \\<Rightarrow> nat \\<Rightarrow> unit Predicate.pred",
"'a Predicate.pred \\<Rightarrow> 'a \\<Rightarrow> bool"
] |
[
"inductive\n method_typing :: \"[classTable, methodDef, className] \\<Rightarrow> bool\" (\"_ \\<turnstile> _ OK IN _\" [80,80,80] 80)\nwhere\nm_typing:\n \"\\<lbrakk> CT(C) = Some(CDef);\n cName CDef = C;\n cSuper CDef = D;\n mName mDef = m;\n lookup (cMethods CDef) (\\<lambda>md.(mName md = m)) = Some(mDef);\n mReturn mDef = C0; mParams mDef = Cxs; mBody mDef = e0;\n varDefs_types Cxs = Cs;\n varDefs_names Cxs = xs;\n \\<Gamma> = (map_upds Map.empty xs Cs)(this \\<mapsto> C); \n CT;\\<Gamma> \\<turnstile> e0 : E0;\n CT \\<turnstile> E0 <: C0;\n \\<forall>Ds D0. (mtype(CT,m,D) = Ds \\<rightarrow> D0) \\<longrightarrow> (Cs=Ds \\<and> C0=D0) \\<rbrakk>\n \\<Longrightarrow> CT \\<turnstile> mDef OK IN C\"",
"datatype (plugins only: extraction) (dead 'a) pred = Pred (eval: \"'a \\<Rightarrow> bool\")"
] |
FeatherweightJava/Execute
|
Execute.fields_i_i_iE
| null |
pred.eval (fields_i_i_i ?x ?xa ?xb) ?y \<Longrightarrow> (fields(?x, ?xa) = ?xb \<Longrightarrow> ?P) \<Longrightarrow> ?P
|
\<lbrakk> ?H1 (?H2 x_1 x_2 x_3) x_4; ?H3 x_1 x_2 x_3 \<Longrightarrow> x_5\<rbrakk> \<Longrightarrow> x_5
|
[
"FJDefs.fields",
"Execute.fields_i_i_i",
"Predicate.pred.eval"
] |
[
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> nat \\<Rightarrow> varDef list \\<Rightarrow> bool",
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> nat \\<Rightarrow> varDef list \\<Rightarrow> unit Predicate.pred",
"'a Predicate.pred \\<Rightarrow> 'a \\<Rightarrow> bool"
] |
[
"inductive\n fields :: \"[classTable, className, varDef list] \\<Rightarrow> bool\" (\"fields'(_,_') = _\" [80,80,80] 80)\nwhere\n f_obj: \n \"fields(CT,Object) = []\"\n| f_class: \n \"\\<lbrakk> CT(C) = Some(CDef); cSuper CDef = D; cFields CDef = Cf; fields(CT,D) = Dg; DgCf = Dg @ Cf \\<rbrakk> \n \\<Longrightarrow> fields(CT,C) = DgCf\"",
"datatype (plugins only: extraction) (dead 'a) pred = Pred (eval: \"'a \\<Rightarrow> bool\")"
] |
FeatherweightJava/Execute
|
Execute.append_i_i_iE
| null |
pred.eval (append_i_i_i ?x ?xa ?xb) ?y \<Longrightarrow> (Execute.append ?x ?xa ?xb \<Longrightarrow> ?P) \<Longrightarrow> ?P
|
\<lbrakk> ?H1 (?H2 x_1 x_2 x_3) x_4; ?H3 x_1 x_2 x_3 \<Longrightarrow> x_5\<rbrakk> \<Longrightarrow> x_5
|
[
"Execute.append",
"Execute.append_i_i_i",
"Predicate.pred.eval"
] |
[
"'a list \\<Rightarrow> 'a list \\<Rightarrow> 'a list \\<Rightarrow> bool",
"'a list \\<Rightarrow> 'a list \\<Rightarrow> 'a list \\<Rightarrow> unit Predicate.pred",
"'a Predicate.pred \\<Rightarrow> 'a \\<Rightarrow> bool"
] |
[
"definition append where \"append xs ys zs = (zs = xs @ ys)\"",
"datatype (plugins only: extraction) (dead 'a) pred = Pred (eval: \"'a \\<Rightarrow> bool\")"
] |
FeatherweightJava/Execute
|
Execute.append_o_o_iE
| null |
pred.eval (append_o_o_i ?xb) (?x, ?xa) \<Longrightarrow> (Execute.append ?x ?xa ?xb \<Longrightarrow> ?P) \<Longrightarrow> ?P
|
\<lbrakk> ?H1 (?H2 x_1) (x_2, x_3); ?H3 x_2 x_3 x_1 \<Longrightarrow> x_4\<rbrakk> \<Longrightarrow> x_4
|
[
"Execute.append",
"Execute.append_o_o_i",
"Predicate.pred.eval"
] |
[
"'a list \\<Rightarrow> 'a list \\<Rightarrow> 'a list \\<Rightarrow> bool",
"'a list \\<Rightarrow> ('a list \\<times> 'a list) Predicate.pred",
"'a Predicate.pred \\<Rightarrow> 'a \\<Rightarrow> bool"
] |
[
"definition append where \"append xs ys zs = (zs = xs @ ys)\"",
"datatype (plugins only: extraction) (dead 'a) pred = Pred (eval: \"'a \\<Rightarrow> bool\")"
] |
FeatherweightJava/Execute
|
Execute.reductions_i_i_oE
| null |
pred.eval (reductions_i_i_o ?x ?xa) ?xb \<Longrightarrow> (?x \<turnstile> ?xa \<rightarrow>* ?xb \<Longrightarrow> ?P) \<Longrightarrow> ?P
|
\<lbrakk> ?H1 (?H2 x_1 x_2) x_3; ?H3 x_1 x_2 x_3 \<Longrightarrow> x_4\<rbrakk> \<Longrightarrow> x_4
|
[
"FJDefs.reductions",
"Execute.reductions_i_i_o",
"Predicate.pred.eval"
] |
[
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> exp \\<Rightarrow> exp \\<Rightarrow> bool",
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> exp \\<Rightarrow> exp Predicate.pred",
"'a Predicate.pred \\<Rightarrow> 'a \\<Rightarrow> bool"
] |
[
"inductive\n reductions :: \"[classTable, exp, exp] \\<Rightarrow> bool\" (\"_ \\<turnstile> _ \\<rightarrow>* _\" [80,80,80] 80)\nwhere\n rs_refl: \"CT \\<turnstile> e \\<rightarrow>* e\" \n| rs_trans: \"\\<lbrakk> CT \\<turnstile> e \\<rightarrow> e'; CT \\<turnstile> e' \\<rightarrow>* e'' \\<rbrakk> \\<Longrightarrow> CT \\<turnstile> e \\<rightarrow>* e''\"",
"datatype (plugins only: extraction) (dead 'a) pred = Pred (eval: \"'a \\<Rightarrow> bool\")"
] |
FeatherweightJava/Execute
|
Execute.reduceE
| null |
pred.eval (reduce ?x ?xa) ?xb \<Longrightarrow> (?x \<turnstile> ?xa \<rightarrow> ?xb \<Longrightarrow> ?P) \<Longrightarrow> ?P
|
\<lbrakk> ?H1 (?H2 x_1 x_2) x_3; ?H3 x_1 x_2 x_3 \<Longrightarrow> x_4\<rbrakk> \<Longrightarrow> x_4
|
[
"FJDefs.reduction",
"Execute.reduce",
"Predicate.pred.eval"
] |
[
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> exp \\<Rightarrow> exp \\<Rightarrow> bool",
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> exp \\<Rightarrow> exp Predicate.pred",
"'a Predicate.pred \\<Rightarrow> 'a \\<Rightarrow> bool"
] |
[
"inductive\n reduction :: \"[classTable, exp, exp] \\<Rightarrow> bool\" (\"_ \\<turnstile> _ \\<rightarrow> _\" [80,80,80] 80)\nwhere\n\n r_field: \n \"\\<lbrakk> fields(CT,C) = Cf; \n lookup2 Cf es (\\<lambda>fd.(vdName fd = fi)) = Some(ei) \\<rbrakk>\n \\<Longrightarrow> CT \\<turnstile> FieldProj (New C es) fi \\<rightarrow> ei\"\n\n| r_invk: \n \"\\<lbrakk> mbody(CT,m,C) = xs . e0;\n substs ((map_upds Map.empty xs ds)(this \\<mapsto> (New C es))) e0 = e0' \\<rbrakk>\n \\<Longrightarrow> CT \\<turnstile> MethodInvk (New C es) m ds \\<rightarrow> e0'\"\n\n| r_cast: \n \"\\<lbrakk> CT \\<turnstile> C <: D \\<rbrakk> \n \\<Longrightarrow> CT \\<turnstile> Cast D (New C es) \\<rightarrow> New C es\"\n\n| rc_field: \n \"\\<lbrakk> CT \\<turnstile> e0 \\<rightarrow> e0' \\<rbrakk> \n \\<Longrightarrow> CT \\<turnstile> FieldProj e0 f \\<rightarrow> FieldProj e0' f\"\n\n| rc_invk_recv: \n \"\\<lbrakk> CT \\<turnstile> e0 \\<rightarrow> e0' \\<rbrakk> \n \\<Longrightarrow> CT \\<turnstile> MethodInvk e0 m es \\<rightarrow> MethodInvk e0' m es\"\n\n| rc_invk_arg: \n \"\\<lbrakk> CT \\<turnstile> ei \\<rightarrow> ei' \\<rbrakk>\n \\<Longrightarrow> CT \\<turnstile> MethodInvk e0 m (el@ei#er) \\<rightarrow> MethodInvk e0 m (el@ei'#er)\"\n\n| rc_new_arg: \n \"\\<lbrakk> CT \\<turnstile> ei \\<rightarrow> ei' \\<rbrakk> \n \\<Longrightarrow> CT \\<turnstile> New C (el@ei#er) \\<rightarrow> New C (el@ei'#er)\"\n\n| rc_cast: \n \"\\<lbrakk> CT \\<turnstile> e0 \\<rightarrow> e0' \\<rbrakk> \n \\<Longrightarrow> CT \\<turnstile> Cast C e0 \\<rightarrow> Cast C e0'\"",
"datatype (plugins only: extraction) (dead 'a) pred = Pred (eval: \"'a \\<Rightarrow> bool\")"
] |
FeatherweightJava/Execute
|
Execute.fields_i_i_oE
| null |
pred.eval (fields_i_i_o ?x ?xa) ?xb \<Longrightarrow> (fields(?x, ?xa) = ?xb \<Longrightarrow> ?P) \<Longrightarrow> ?P
|
\<lbrakk> ?H1 (?H2 x_1 x_2) x_3; ?H3 x_1 x_2 x_3 \<Longrightarrow> x_4\<rbrakk> \<Longrightarrow> x_4
|
[
"FJDefs.fields",
"Execute.fields_i_i_o",
"Predicate.pred.eval"
] |
[
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> nat \\<Rightarrow> varDef list \\<Rightarrow> bool",
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> nat \\<Rightarrow> varDef list Predicate.pred",
"'a Predicate.pred \\<Rightarrow> 'a \\<Rightarrow> bool"
] |
[
"inductive\n fields :: \"[classTable, className, varDef list] \\<Rightarrow> bool\" (\"fields'(_,_') = _\" [80,80,80] 80)\nwhere\n f_obj: \n \"fields(CT,Object) = []\"\n| f_class: \n \"\\<lbrakk> CT(C) = Some(CDef); cSuper CDef = D; cFields CDef = Cf; fields(CT,D) = Dg; DgCf = Dg @ Cf \\<rbrakk> \n \\<Longrightarrow> fields(CT,C) = DgCf\"",
"datatype (plugins only: extraction) (dead 'a) pred = Pred (eval: \"'a \\<Rightarrow> bool\")"
] |
FeatherweightJava/Execute
|
Execute.subtypings_i_i_oE
| null |
pred.eval (subtypings_i_i_o ?x ?xa) ?xb \<Longrightarrow> (?x \<turnstile>+ ?xa <: ?xb \<Longrightarrow> ?P) \<Longrightarrow> ?P
|
\<lbrakk> ?H1 (?H2 x_1 x_2) x_3; ?H3 x_1 x_2 x_3 \<Longrightarrow> x_4\<rbrakk> \<Longrightarrow> x_4
|
[
"FJDefs.subtypings",
"Execute.subtypings_i_i_o",
"Predicate.pred.eval"
] |
[
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> nat list \\<Rightarrow> nat list \\<Rightarrow> bool",
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> nat list \\<Rightarrow> nat list Predicate.pred",
"'a Predicate.pred \\<Rightarrow> 'a \\<Rightarrow> bool"
] |
[
"inductive\n subtypings :: \"[classTable, className list, className list] \\<Rightarrow> bool\" (\"_ \\<turnstile>+ _ <: _\" [80,80,80] 80)\nwhere\n ss_nil : \"CT \\<turnstile>+ [] <: []\"\n| ss_cons : \"\\<lbrakk> CT \\<turnstile> C0 <: D0; CT \\<turnstile>+ Cs <: Ds \\<rbrakk> \\<Longrightarrow> CT \\<turnstile>+ (C0 # Cs) <: (D0 # Ds)\"",
"datatype (plugins only: extraction) (dead 'a) pred = Pred (eval: \"'a \\<Rightarrow> bool\")"
] |
FeatherweightJava/Execute
|
Execute.append_o_i_iE
| null |
pred.eval (append_o_i_i ?xa ?xb) ?x \<Longrightarrow> (Execute.append ?x ?xa ?xb \<Longrightarrow> ?P) \<Longrightarrow> ?P
|
\<lbrakk> ?H1 (?H2 x_1 x_2) x_3; ?H3 x_3 x_1 x_2 \<Longrightarrow> x_4\<rbrakk> \<Longrightarrow> x_4
|
[
"Execute.append",
"Execute.append_o_i_i",
"Predicate.pred.eval"
] |
[
"'a list \\<Rightarrow> 'a list \\<Rightarrow> 'a list \\<Rightarrow> bool",
"'a list \\<Rightarrow> 'a list \\<Rightarrow> 'a list Predicate.pred",
"'a Predicate.pred \\<Rightarrow> 'a \\<Rightarrow> bool"
] |
[
"definition append where \"append xs ys zs = (zs = xs @ ys)\"",
"datatype (plugins only: extraction) (dead 'a) pred = Pred (eval: \"'a \\<Rightarrow> bool\")"
] |
FeatherweightJava/Execute
|
Execute.append_i_o_iE
| null |
pred.eval (append_i_o_i ?x ?xb) ?xa \<Longrightarrow> (Execute.append ?x ?xa ?xb \<Longrightarrow> ?P) \<Longrightarrow> ?P
|
\<lbrakk> ?H1 (?H2 x_1 x_2) x_3; ?H3 x_1 x_3 x_2 \<Longrightarrow> x_4\<rbrakk> \<Longrightarrow> x_4
|
[
"Execute.append",
"Execute.append_i_o_i",
"Predicate.pred.eval"
] |
[
"'a list \\<Rightarrow> 'a list \\<Rightarrow> 'a list \\<Rightarrow> bool",
"'a list \\<Rightarrow> 'a list \\<Rightarrow> 'a list Predicate.pred",
"'a Predicate.pred \\<Rightarrow> 'a \\<Rightarrow> bool"
] |
[
"definition append where \"append xs ys zs = (zs = xs @ ys)\"",
"datatype (plugins only: extraction) (dead 'a) pred = Pred (eval: \"'a \\<Rightarrow> bool\")"
] |
FeatherweightJava/Execute
|
Execute.append_i_i_oE
| null |
pred.eval (append_i_i_o ?x ?xa) ?xb \<Longrightarrow> (Execute.append ?x ?xa ?xb \<Longrightarrow> ?P) \<Longrightarrow> ?P
|
\<lbrakk> ?H1 (?H2 x_1 x_2) x_3; ?H3 x_1 x_2 x_3 \<Longrightarrow> x_4\<rbrakk> \<Longrightarrow> x_4
|
[
"Execute.append",
"Execute.append_i_i_o",
"Predicate.pred.eval"
] |
[
"'a list \\<Rightarrow> 'a list \\<Rightarrow> 'a list \\<Rightarrow> bool",
"'a list \\<Rightarrow> 'a list \\<Rightarrow> 'a list Predicate.pred",
"'a Predicate.pred \\<Rightarrow> 'a \\<Rightarrow> bool"
] |
[
"definition append where \"append xs ys zs = (zs = xs @ ys)\"",
"datatype (plugins only: extraction) (dead 'a) pred = Pred (eval: \"'a \\<Rightarrow> bool\")"
] |
FeatherweightJava/Execute
|
Execute.supertypes_ofE
| null |
pred.eval (supertypes_of ?x ?xa) ?xb \<Longrightarrow> (?x \<turnstile> ?xa <: ?xb \<Longrightarrow> ?P) \<Longrightarrow> ?P
|
\<lbrakk> ?H1 (?H2 x_1 x_2) x_3; ?H3 x_1 x_2 x_3 \<Longrightarrow> x_4\<rbrakk> \<Longrightarrow> x_4
|
[
"FJDefs.subtyping",
"Execute.supertypes_of",
"Predicate.pred.eval"
] |
[
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> nat \\<Rightarrow> nat \\<Rightarrow> bool",
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> nat \\<Rightarrow> nat Predicate.pred",
"'a Predicate.pred \\<Rightarrow> 'a \\<Rightarrow> bool"
] |
[
"inductive\n subtyping :: \"[classTable, className, className] \\<Rightarrow> bool\" (\"_ \\<turnstile> _ <: _\" [80,80,80] 80)\nwhere\n s_refl : \"CT \\<turnstile> C <: C\"\n| s_trans : \"\\<lbrakk> CT \\<turnstile> C <: D; CT \\<turnstile> D <: E \\<rbrakk> \\<Longrightarrow> CT \\<turnstile> C <: E\"\n| s_super : \"\\<lbrakk> CT(C) = Some(CDef); cSuper CDef = D \\<rbrakk> \\<Longrightarrow> CT \\<turnstile> C <: D\"",
"datatype (plugins only: extraction) (dead 'a) pred = Pred (eval: \"'a \\<Rightarrow> bool\")"
] |
FeatherweightJava/Execute
|
Execute.eq_i_iE
| null |
pred.eval (eq_i_i ?x ?xa) ?y \<Longrightarrow> (Predicate.eq ?x ?xa \<Longrightarrow> ?P) \<Longrightarrow> ?P
|
\<lbrakk> ?H1 (?H2 x_1 x_2) x_3; ?H3 x_1 x_2 \<Longrightarrow> x_4\<rbrakk> \<Longrightarrow> x_4
|
[
"Predicate.eq",
"Execute.eq_i_i",
"Predicate.pred.eval"
] |
[
"'a \\<Rightarrow> 'a \\<Rightarrow> bool",
"'a \\<Rightarrow> 'a \\<Rightarrow> unit Predicate.pred",
"'a Predicate.pred \\<Rightarrow> 'a \\<Rightarrow> bool"
] |
[
"inductive eq :: \"'a \\<Rightarrow> 'a \\<Rightarrow> bool\" where \"eq x x\"",
"datatype (plugins only: extraction) (dead 'a) pred = Pred (eval: \"'a \\<Rightarrow> bool\")"
] |
FeatherweightJava/Execute
|
Execute.class_typing_i_iE
| null |
pred.eval (class_typing_i_i ?x ?xa) ?y \<Longrightarrow> (?x \<turnstile> ?xa OK \<Longrightarrow> ?P) \<Longrightarrow> ?P
|
\<lbrakk> ?H1 (?H2 x_1 x_2) x_3; ?H3 x_1 x_2 \<Longrightarrow> x_4\<rbrakk> \<Longrightarrow> x_4
|
[
"FJDefs.class_typing",
"Execute.class_typing_i_i",
"Predicate.pred.eval"
] |
[
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> classDef \\<Rightarrow> bool",
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> classDef \\<Rightarrow> unit Predicate.pred",
"'a Predicate.pred \\<Rightarrow> 'a \\<Rightarrow> bool"
] |
[
"inductive\n class_typing :: \"[classTable, classDef] \\<Rightarrow> bool\" (\"_ \\<turnstile> _ OK\" [80,80] 80)\nwhere\nt_class: \"\\<lbrakk> cName CDef = C; \n cSuper CDef = D;\n cConstructor CDef = KDef;\n cMethods CDef = M;\n kName KDef = C;\n kParams KDef = (Dg@Cf);\n kSuper KDef = varDefs_names Dg;\n kInits KDef = varDefs_names Cf;\n fields(CT,D) = Dg;\n CT \\<turnstile>+ M OK IN C \\<rbrakk>\n \\<Longrightarrow> CT \\<turnstile> CDef OK\"",
"datatype (plugins only: extraction) (dead 'a) pred = Pred (eval: \"'a \\<Rightarrow> bool\")"
] |
FeatherweightJava/Execute
|
Execute.mtype_i_i_i_i_iI
| null |
mtype(?x, ?xa, ?xb) = ?xc \<rightarrow> ?xd \<Longrightarrow> pred.eval (mtype_i_i_i_i_i ?x ?xa ?xb ?xc ?xd) ()
|
?H1 x_1 x_2 x_3 x_4 x_5 \<Longrightarrow> ?H2 (?H3 x_1 x_2 x_3 x_4 x_5) ?H4
|
[
"Product_Type.Unity",
"Execute.mtype_i_i_i_i_i",
"Predicate.pred.eval",
"FJDefs.mtype"
] |
[
"unit",
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> nat \\<Rightarrow> nat \\<Rightarrow> nat list \\<Rightarrow> nat \\<Rightarrow> unit Predicate.pred",
"'a Predicate.pred \\<Rightarrow> 'a \\<Rightarrow> bool",
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> nat \\<Rightarrow> nat \\<Rightarrow> nat list \\<Rightarrow> nat \\<Rightarrow> bool"
] |
[
"definition Unity :: unit (\"'(')\")\n where \"() = Abs_unit True\"",
"datatype (plugins only: extraction) (dead 'a) pred = Pred (eval: \"'a \\<Rightarrow> bool\")",
"inductive\n mtype :: \"[classTable, methodName, className, className list, className] \\<Rightarrow> bool\" (\"mtype'(_,_,_') = _ \\<rightarrow> _\" [80,80,80,80] 80)\nwhere\n mt_class: \n \"\\<lbrakk> CT(C) = Some(CDef);\n lookup (cMethods CDef) (\\<lambda>md.(mName md = m)) = Some(mDef);\n varDefs_types (mParams mDef) = Bs;\n mReturn mDef = B \\<rbrakk>\n \\<Longrightarrow> mtype(CT,m,C) = Bs \\<rightarrow> B\"\n\n| mt_super: \n \"\\<lbrakk> CT(C) = Some (CDef);\n lookup (cMethods CDef) (\\<lambda>md.(mName md = m)) = None;\n cSuper CDef = D;\n mtype(CT,m,D) = Bs \\<rightarrow> B \\<rbrakk>\n \\<Longrightarrow> mtype(CT,m,C) = Bs \\<rightarrow> B\""
] |
FeatherweightJava/Execute
|
Execute.mtype_i_i_i_o_oI
| null |
mtype(?x, ?xa, ?xb) = ?xc \<rightarrow> ?xd \<Longrightarrow> pred.eval (mtype_i_i_i_o_o ?x ?xa ?xb) (?xc, ?xd)
|
?H1 x_1 x_2 x_3 x_4 x_5 \<Longrightarrow> ?H2 (?H3 x_1 x_2 x_3) (x_4, x_5)
|
[
"Execute.mtype_i_i_i_o_o",
"Predicate.pred.eval",
"FJDefs.mtype"
] |
[
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> nat \\<Rightarrow> nat \\<Rightarrow> (nat list \\<times> nat) Predicate.pred",
"'a Predicate.pred \\<Rightarrow> 'a \\<Rightarrow> bool",
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> nat \\<Rightarrow> nat \\<Rightarrow> nat list \\<Rightarrow> nat \\<Rightarrow> bool"
] |
[
"datatype (plugins only: extraction) (dead 'a) pred = Pred (eval: \"'a \\<Rightarrow> bool\")",
"inductive\n mtype :: \"[classTable, methodName, className, className list, className] \\<Rightarrow> bool\" (\"mtype'(_,_,_') = _ \\<rightarrow> _\" [80,80,80,80] 80)\nwhere\n mt_class: \n \"\\<lbrakk> CT(C) = Some(CDef);\n lookup (cMethods CDef) (\\<lambda>md.(mName md = m)) = Some(mDef);\n varDefs_types (mParams mDef) = Bs;\n mReturn mDef = B \\<rbrakk>\n \\<Longrightarrow> mtype(CT,m,C) = Bs \\<rightarrow> B\"\n\n| mt_super: \n \"\\<lbrakk> CT(C) = Some (CDef);\n lookup (cMethods CDef) (\\<lambda>md.(mName md = m)) = None;\n cSuper CDef = D;\n mtype(CT,m,D) = Bs \\<rightarrow> B \\<rbrakk>\n \\<Longrightarrow> mtype(CT,m,C) = Bs \\<rightarrow> B\""
] |
FeatherweightJava/Execute
|
Execute.mbody_i_i_i_i_iI
| null |
mbody(?x, ?xa, ?xb) = ?xc . ?xd \<Longrightarrow> pred.eval (mbody_i_i_i_i_i ?x ?xa ?xb ?xc ?xd) ()
|
?H1 x_1 x_2 x_3 x_4 x_5 \<Longrightarrow> ?H2 (?H3 x_1 x_2 x_3 x_4 x_5) ?H4
|
[
"Product_Type.Unity",
"Execute.mbody_i_i_i_i_i",
"Predicate.pred.eval",
"FJDefs.mbody"
] |
[
"unit",
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> nat \\<Rightarrow> nat \\<Rightarrow> nat list \\<Rightarrow> exp \\<Rightarrow> unit Predicate.pred",
"'a Predicate.pred \\<Rightarrow> 'a \\<Rightarrow> bool",
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> nat \\<Rightarrow> nat \\<Rightarrow> nat list \\<Rightarrow> exp \\<Rightarrow> bool"
] |
[
"definition Unity :: unit (\"'(')\")\n where \"() = Abs_unit True\"",
"datatype (plugins only: extraction) (dead 'a) pred = Pred (eval: \"'a \\<Rightarrow> bool\")",
"inductive\n mbody :: \"[classTable, methodName, className, varName list, exp] \\<Rightarrow> bool\" (\"mbody'(_,_,_') = _ . _\" [80,80,80,80] 80)\nwhere\n mb_class: \n \"\\<lbrakk> CT(C) = Some(CDef);\n lookup (cMethods CDef) (\\<lambda>md.(mName md = m)) = Some(mDef);\n varDefs_names (mParams mDef) = xs;\n mBody mDef = e \\<rbrakk>\n \\<Longrightarrow> mbody(CT,m,C) = xs . e\"\n\n| mb_super:\n \"\\<lbrakk> CT(C) = Some(CDef);\n lookup (cMethods CDef) (\\<lambda>md.(mName md = m)) = None;\n cSuper CDef = D;\n mbody(CT,m,D) = xs . e \\<rbrakk>\n \\<Longrightarrow> mbody(CT,m,C) = xs . e\""
] |
FeatherweightJava/Execute
|
Execute.mbody_i_i_i_o_oI
| null |
mbody(?x, ?xa, ?xb) = ?xc . ?xd \<Longrightarrow> pred.eval (mbody_i_i_i_o_o ?x ?xa ?xb) (?xc, ?xd)
|
?H1 x_1 x_2 x_3 x_4 x_5 \<Longrightarrow> ?H2 (?H3 x_1 x_2 x_3) (x_4, x_5)
|
[
"Execute.mbody_i_i_i_o_o",
"Predicate.pred.eval",
"FJDefs.mbody"
] |
[
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> nat \\<Rightarrow> nat \\<Rightarrow> (nat list \\<times> exp) Predicate.pred",
"'a Predicate.pred \\<Rightarrow> 'a \\<Rightarrow> bool",
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> nat \\<Rightarrow> nat \\<Rightarrow> nat list \\<Rightarrow> exp \\<Rightarrow> bool"
] |
[
"datatype (plugins only: extraction) (dead 'a) pred = Pred (eval: \"'a \\<Rightarrow> bool\")",
"inductive\n mbody :: \"[classTable, methodName, className, varName list, exp] \\<Rightarrow> bool\" (\"mbody'(_,_,_') = _ . _\" [80,80,80,80] 80)\nwhere\n mb_class: \n \"\\<lbrakk> CT(C) = Some(CDef);\n lookup (cMethods CDef) (\\<lambda>md.(mName md = m)) = Some(mDef);\n varDefs_names (mParams mDef) = xs;\n mBody mDef = e \\<rbrakk>\n \\<Longrightarrow> mbody(CT,m,C) = xs . e\"\n\n| mb_super:\n \"\\<lbrakk> CT(C) = Some(CDef);\n lookup (cMethods CDef) (\\<lambda>md.(mName md = m)) = None;\n cSuper CDef = D;\n mbody(CT,m,D) = xs . e \\<rbrakk>\n \\<Longrightarrow> mbody(CT,m,C) = xs . e\""
] |
FeatherweightJava/Execute
|
Execute.method_typing_aux_i_i_i_i_iI
| null |
method_typing_aux ?x ?xa ?xb ?xc ?xd \<Longrightarrow> pred.eval (method_typing_aux_i_i_i_i_i ?x ?xa ?xb ?xc ?xd) ()
|
?H1 x_1 x_2 x_3 x_4 x_5 \<Longrightarrow> ?H2 (?H3 x_1 x_2 x_3 x_4 x_5) ?H4
|
[
"Product_Type.Unity",
"Execute.method_typing_aux_i_i_i_i_i",
"Predicate.pred.eval",
"Execute.method_typing_aux"
] |
[
"unit",
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> nat \\<Rightarrow> nat \\<Rightarrow> nat list \\<Rightarrow> nat \\<Rightarrow> unit Predicate.pred",
"'a Predicate.pred \\<Rightarrow> 'a \\<Rightarrow> bool",
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> nat \\<Rightarrow> nat \\<Rightarrow> nat list \\<Rightarrow> nat \\<Rightarrow> bool"
] |
[
"definition Unity :: unit (\"'(')\")\n where \"() = Abs_unit True\"",
"datatype (plugins only: extraction) (dead 'a) pred = Pred (eval: \"'a \\<Rightarrow> bool\")",
"definition method_typing_aux\nwhere\n \"method_typing_aux CT m D Cs C = (\\<not> (\\<forall>Ds D0. mtype(CT,m,D) = Ds \\<rightarrow> D0 \\<longrightarrow> Cs = Ds \\<and> C = D0))\""
] |
FeatherweightJava/Execute
|
Execute.eq_o_iE
| null |
pred.eval (eq_o_i ?xa) ?x \<Longrightarrow> (Predicate.eq ?x ?xa \<Longrightarrow> ?P) \<Longrightarrow> ?P
|
\<lbrakk> ?H1 (?H2 x_1) x_2; ?H3 x_2 x_1 \<Longrightarrow> x_3\<rbrakk> \<Longrightarrow> x_3
|
[
"Predicate.eq",
"Execute.eq_o_i",
"Predicate.pred.eval"
] |
[
"'a \\<Rightarrow> 'a \\<Rightarrow> bool",
"'a \\<Rightarrow> 'a Predicate.pred",
"'a Predicate.pred \\<Rightarrow> 'a \\<Rightarrow> bool"
] |
[
"inductive eq :: \"'a \\<Rightarrow> 'a \\<Rightarrow> bool\" where \"eq x x\"",
"datatype (plugins only: extraction) (dead 'a) pred = Pred (eval: \"'a \\<Rightarrow> bool\")"
] |
FeatherweightJava/Execute
|
Execute.eq_i_oE
| null |
pred.eval (eq_i_o ?x) ?xa \<Longrightarrow> (Predicate.eq ?x ?xa \<Longrightarrow> ?P) \<Longrightarrow> ?P
|
\<lbrakk> ?H1 (?H2 x_1) x_2; ?H3 x_1 x_2 \<Longrightarrow> x_3\<rbrakk> \<Longrightarrow> x_3
|
[
"Predicate.eq",
"Execute.eq_i_o",
"Predicate.pred.eval"
] |
[
"'a \\<Rightarrow> 'a \\<Rightarrow> bool",
"'a \\<Rightarrow> 'a Predicate.pred",
"'a Predicate.pred \\<Rightarrow> 'a \\<Rightarrow> bool"
] |
[
"inductive eq :: \"'a \\<Rightarrow> 'a \\<Rightarrow> bool\" where \"eq x x\"",
"datatype (plugins only: extraction) (dead 'a) pred = Pred (eval: \"'a \\<Rightarrow> bool\")"
] |
FeatherweightJava/Execute
|
Execute.mtype_i_i_i_o_iI
| null |
mtype(?x, ?xa, ?xb) = ?xc \<rightarrow> ?xd \<Longrightarrow> pred.eval (mtype_i_i_i_o_i ?x ?xa ?xb ?xd) ?xc
|
?H1 x_1 x_2 x_3 x_4 x_5 \<Longrightarrow> ?H2 (?H3 x_1 x_2 x_3 x_5) x_4
|
[
"Execute.mtype_i_i_i_o_i",
"Predicate.pred.eval",
"FJDefs.mtype"
] |
[
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> nat \\<Rightarrow> nat \\<Rightarrow> nat \\<Rightarrow> nat list Predicate.pred",
"'a Predicate.pred \\<Rightarrow> 'a \\<Rightarrow> bool",
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> nat \\<Rightarrow> nat \\<Rightarrow> nat list \\<Rightarrow> nat \\<Rightarrow> bool"
] |
[
"datatype (plugins only: extraction) (dead 'a) pred = Pred (eval: \"'a \\<Rightarrow> bool\")",
"inductive\n mtype :: \"[classTable, methodName, className, className list, className] \\<Rightarrow> bool\" (\"mtype'(_,_,_') = _ \\<rightarrow> _\" [80,80,80,80] 80)\nwhere\n mt_class: \n \"\\<lbrakk> CT(C) = Some(CDef);\n lookup (cMethods CDef) (\\<lambda>md.(mName md = m)) = Some(mDef);\n varDefs_types (mParams mDef) = Bs;\n mReturn mDef = B \\<rbrakk>\n \\<Longrightarrow> mtype(CT,m,C) = Bs \\<rightarrow> B\"\n\n| mt_super: \n \"\\<lbrakk> CT(C) = Some (CDef);\n lookup (cMethods CDef) (\\<lambda>md.(mName md = m)) = None;\n cSuper CDef = D;\n mtype(CT,m,D) = Bs \\<rightarrow> B \\<rbrakk>\n \\<Longrightarrow> mtype(CT,m,C) = Bs \\<rightarrow> B\""
] |
FeatherweightJava/Execute
|
Execute.mtype_i_i_i_i_oI
| null |
mtype(?x, ?xa, ?xb) = ?xc \<rightarrow> ?xd \<Longrightarrow> pred.eval (mtype_i_i_i_i_o ?x ?xa ?xb ?xc) ?xd
|
?H1 x_1 x_2 x_3 x_4 x_5 \<Longrightarrow> ?H2 (?H3 x_1 x_2 x_3 x_4) x_5
|
[
"Execute.mtype_i_i_i_i_o",
"Predicate.pred.eval",
"FJDefs.mtype"
] |
[
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> nat \\<Rightarrow> nat \\<Rightarrow> nat list \\<Rightarrow> nat Predicate.pred",
"'a Predicate.pred \\<Rightarrow> 'a \\<Rightarrow> bool",
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> nat \\<Rightarrow> nat \\<Rightarrow> nat list \\<Rightarrow> nat \\<Rightarrow> bool"
] |
[
"datatype (plugins only: extraction) (dead 'a) pred = Pred (eval: \"'a \\<Rightarrow> bool\")",
"inductive\n mtype :: \"[classTable, methodName, className, className list, className] \\<Rightarrow> bool\" (\"mtype'(_,_,_') = _ \\<rightarrow> _\" [80,80,80,80] 80)\nwhere\n mt_class: \n \"\\<lbrakk> CT(C) = Some(CDef);\n lookup (cMethods CDef) (\\<lambda>md.(mName md = m)) = Some(mDef);\n varDefs_types (mParams mDef) = Bs;\n mReturn mDef = B \\<rbrakk>\n \\<Longrightarrow> mtype(CT,m,C) = Bs \\<rightarrow> B\"\n\n| mt_super: \n \"\\<lbrakk> CT(C) = Some (CDef);\n lookup (cMethods CDef) (\\<lambda>md.(mName md = m)) = None;\n cSuper CDef = D;\n mtype(CT,m,D) = Bs \\<rightarrow> B \\<rbrakk>\n \\<Longrightarrow> mtype(CT,m,C) = Bs \\<rightarrow> B\""
] |
FeatherweightJava/Execute
|
Execute.mbody_i_i_i_i_oI
| null |
mbody(?x, ?xa, ?xb) = ?xc . ?xd \<Longrightarrow> pred.eval (mbody_i_i_i_i_o ?x ?xa ?xb ?xc) ?xd
|
?H1 x_1 x_2 x_3 x_4 x_5 \<Longrightarrow> ?H2 (?H3 x_1 x_2 x_3 x_4) x_5
|
[
"Execute.mbody_i_i_i_i_o",
"Predicate.pred.eval",
"FJDefs.mbody"
] |
[
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> nat \\<Rightarrow> nat \\<Rightarrow> nat list \\<Rightarrow> exp Predicate.pred",
"'a Predicate.pred \\<Rightarrow> 'a \\<Rightarrow> bool",
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> nat \\<Rightarrow> nat \\<Rightarrow> nat list \\<Rightarrow> exp \\<Rightarrow> bool"
] |
[
"datatype (plugins only: extraction) (dead 'a) pred = Pred (eval: \"'a \\<Rightarrow> bool\")",
"inductive\n mbody :: \"[classTable, methodName, className, varName list, exp] \\<Rightarrow> bool\" (\"mbody'(_,_,_') = _ . _\" [80,80,80,80] 80)\nwhere\n mb_class: \n \"\\<lbrakk> CT(C) = Some(CDef);\n lookup (cMethods CDef) (\\<lambda>md.(mName md = m)) = Some(mDef);\n varDefs_names (mParams mDef) = xs;\n mBody mDef = e \\<rbrakk>\n \\<Longrightarrow> mbody(CT,m,C) = xs . e\"\n\n| mb_super:\n \"\\<lbrakk> CT(C) = Some(CDef);\n lookup (cMethods CDef) (\\<lambda>md.(mName md = m)) = None;\n cSuper CDef = D;\n mbody(CT,m,D) = xs . e \\<rbrakk>\n \\<Longrightarrow> mbody(CT,m,C) = xs . e\""
] |
FeatherweightJava/Execute
|
Execute.mbody_i_i_i_o_iI
| null |
mbody(?x, ?xa, ?xb) = ?xc . ?xd \<Longrightarrow> pred.eval (mbody_i_i_i_o_i ?x ?xa ?xb ?xd) ?xc
|
?H1 x_1 x_2 x_3 x_4 x_5 \<Longrightarrow> ?H2 (?H3 x_1 x_2 x_3 x_5) x_4
|
[
"Execute.mbody_i_i_i_o_i",
"Predicate.pred.eval",
"FJDefs.mbody"
] |
[
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> nat \\<Rightarrow> nat \\<Rightarrow> exp \\<Rightarrow> nat list Predicate.pred",
"'a Predicate.pred \\<Rightarrow> 'a \\<Rightarrow> bool",
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> nat \\<Rightarrow> nat \\<Rightarrow> nat list \\<Rightarrow> exp \\<Rightarrow> bool"
] |
[
"datatype (plugins only: extraction) (dead 'a) pred = Pred (eval: \"'a \\<Rightarrow> bool\")",
"inductive\n mbody :: \"[classTable, methodName, className, varName list, exp] \\<Rightarrow> bool\" (\"mbody'(_,_,_') = _ . _\" [80,80,80,80] 80)\nwhere\n mb_class: \n \"\\<lbrakk> CT(C) = Some(CDef);\n lookup (cMethods CDef) (\\<lambda>md.(mName md = m)) = Some(mDef);\n varDefs_names (mParams mDef) = xs;\n mBody mDef = e \\<rbrakk>\n \\<Longrightarrow> mbody(CT,m,C) = xs . e\"\n\n| mb_super:\n \"\\<lbrakk> CT(C) = Some(CDef);\n lookup (cMethods CDef) (\\<lambda>md.(mName md = m)) = None;\n cSuper CDef = D;\n mbody(CT,m,D) = xs . e \\<rbrakk>\n \\<Longrightarrow> mbody(CT,m,C) = xs . e\""
] |
FeatherweightJava/Execute
|
Execute.typings_i_i_i_iI
| null |
?x; ?xa \<turnstile>+ ?xb : ?xc \<Longrightarrow> pred.eval (typings_i_i_i_i ?x ?xa ?xb ?xc) ()
|
?H1 x_1 x_2 x_3 x_4 \<Longrightarrow> ?H2 (?H3 x_1 x_2 x_3 x_4) ?H4
|
[
"Product_Type.Unity",
"Execute.typings_i_i_i_i",
"Predicate.pred.eval",
"FJDefs.typings"
] |
[
"unit",
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> (nat \\<Rightarrow> nat option) \\<Rightarrow> exp list \\<Rightarrow> nat list \\<Rightarrow> unit Predicate.pred",
"'a Predicate.pred \\<Rightarrow> 'a \\<Rightarrow> bool",
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> (nat \\<Rightarrow> nat option) \\<Rightarrow> exp list \\<Rightarrow> nat list \\<Rightarrow> bool"
] |
[
"definition Unity :: unit (\"'(')\")\n where \"() = Abs_unit True\"",
"datatype (plugins only: extraction) (dead 'a) pred = Pred (eval: \"'a \\<Rightarrow> bool\")",
"inductive\n typings :: \"[classTable, varCtx, exp list, className list] \\<Rightarrow> bool\" (\"_;_ \\<turnstile>+ _ : _\" [80,80,80,80] 80)\n and typing :: \"[classTable, varCtx, exp, className] \\<Rightarrow> bool\" (\"_;_ \\<turnstile> _ : _\" [80,80,80,80] 80)\nwhere\n ts_nil : \"CT;\\<Gamma> \\<turnstile>+ [] : []\"\n\n| ts_cons : \n \"\\<lbrakk> CT;\\<Gamma> \\<turnstile> e0 : C0; CT;\\<Gamma> \\<turnstile>+ es : Cs \\<rbrakk> \n \\<Longrightarrow> CT;\\<Gamma> \\<turnstile>+ (e0 # es) : (C0 # Cs)\"\n\n| t_var : \n \"\\<lbrakk> \\<Gamma>(x) = Some C \\<rbrakk> \\<Longrightarrow> CT;\\<Gamma> \\<turnstile> (Var x) : C\"\n\n| t_field : \n \"\\<lbrakk> CT;\\<Gamma> \\<turnstile> e0 : C0;\n fields(CT,C0) = Cf;\n lookup Cf (\\<lambda>fd.(vdName fd = fi)) = Some(fDef);\n vdType fDef = Ci \\<rbrakk>\n \\<Longrightarrow> CT;\\<Gamma> \\<turnstile> FieldProj e0 fi : Ci\"\n\n| t_invk : \n \"\\<lbrakk> CT;\\<Gamma> \\<turnstile> e0 : C0;\n mtype(CT,m,C0) = Ds \\<rightarrow> C;\n CT;\\<Gamma> \\<turnstile>+ es : Cs;\n CT \\<turnstile>+ Cs <: Ds;\n length es = length Ds \\<rbrakk>\n \\<Longrightarrow> CT;\\<Gamma> \\<turnstile> MethodInvk e0 m es : C\"\n\n| t_new : \n \"\\<lbrakk> fields(CT,C) = Df;\n length es = length Df;\n varDefs_types Df = Ds;\n CT;\\<Gamma> \\<turnstile>+ es : Cs;\n CT \\<turnstile>+ Cs <: Ds \\<rbrakk>\n \\<Longrightarrow> CT;\\<Gamma> \\<turnstile> New C es : C\"\n\n| t_ucast : \n \"\\<lbrakk> CT;\\<Gamma> \\<turnstile> e0 : D; \n CT \\<turnstile> D <: C \\<rbrakk>\n \\<Longrightarrow> CT;\\<Gamma> \\<turnstile> Cast C e0 : C\"\n\n| t_dcast : \n \"\\<lbrakk> CT;\\<Gamma> \\<turnstile> e0 : D; \n CT \\<turnstile> C <: D; C \\<noteq> D \\<rbrakk>\n \\<Longrightarrow> CT;\\<Gamma> \\<turnstile> Cast C e0 : C\"\n\n| t_scast : \n \"\\<lbrakk> CT;\\<Gamma> \\<turnstile> e0 : D;\n CT \\<turnstile> C \\<not><: D;\n CT \\<turnstile> D \\<not><: C \\<rbrakk>\n \\<Longrightarrow> CT;\\<Gamma> \\<turnstile> Cast C e0 : C\""
] |
FeatherweightJava/Execute
|
Execute.typing_i_i_i_iI
| null |
?x; ?xa \<turnstile> ?xb : ?xc \<Longrightarrow> pred.eval (typing_i_i_i_i ?x ?xa ?xb ?xc) ()
|
?H1 x_1 x_2 x_3 x_4 \<Longrightarrow> ?H2 (?H3 x_1 x_2 x_3 x_4) ?H4
|
[
"Product_Type.Unity",
"Execute.typing_i_i_i_i",
"Predicate.pred.eval",
"FJDefs.typing"
] |
[
"unit",
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> (nat \\<Rightarrow> nat option) \\<Rightarrow> exp \\<Rightarrow> nat \\<Rightarrow> unit Predicate.pred",
"'a Predicate.pred \\<Rightarrow> 'a \\<Rightarrow> bool",
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> (nat \\<Rightarrow> nat option) \\<Rightarrow> exp \\<Rightarrow> nat \\<Rightarrow> bool"
] |
[
"definition Unity :: unit (\"'(')\")\n where \"() = Abs_unit True\"",
"datatype (plugins only: extraction) (dead 'a) pred = Pred (eval: \"'a \\<Rightarrow> bool\")",
"inductive\n typings :: \"[classTable, varCtx, exp list, className list] \\<Rightarrow> bool\" (\"_;_ \\<turnstile>+ _ : _\" [80,80,80,80] 80)\n and typing :: \"[classTable, varCtx, exp, className] \\<Rightarrow> bool\" (\"_;_ \\<turnstile> _ : _\" [80,80,80,80] 80)\nwhere\n ts_nil : \"CT;\\<Gamma> \\<turnstile>+ [] : []\"\n\n| ts_cons : \n \"\\<lbrakk> CT;\\<Gamma> \\<turnstile> e0 : C0; CT;\\<Gamma> \\<turnstile>+ es : Cs \\<rbrakk> \n \\<Longrightarrow> CT;\\<Gamma> \\<turnstile>+ (e0 # es) : (C0 # Cs)\"\n\n| t_var : \n \"\\<lbrakk> \\<Gamma>(x) = Some C \\<rbrakk> \\<Longrightarrow> CT;\\<Gamma> \\<turnstile> (Var x) : C\"\n\n| t_field : \n \"\\<lbrakk> CT;\\<Gamma> \\<turnstile> e0 : C0;\n fields(CT,C0) = Cf;\n lookup Cf (\\<lambda>fd.(vdName fd = fi)) = Some(fDef);\n vdType fDef = Ci \\<rbrakk>\n \\<Longrightarrow> CT;\\<Gamma> \\<turnstile> FieldProj e0 fi : Ci\"\n\n| t_invk : \n \"\\<lbrakk> CT;\\<Gamma> \\<turnstile> e0 : C0;\n mtype(CT,m,C0) = Ds \\<rightarrow> C;\n CT;\\<Gamma> \\<turnstile>+ es : Cs;\n CT \\<turnstile>+ Cs <: Ds;\n length es = length Ds \\<rbrakk>\n \\<Longrightarrow> CT;\\<Gamma> \\<turnstile> MethodInvk e0 m es : C\"\n\n| t_new : \n \"\\<lbrakk> fields(CT,C) = Df;\n length es = length Df;\n varDefs_types Df = Ds;\n CT;\\<Gamma> \\<turnstile>+ es : Cs;\n CT \\<turnstile>+ Cs <: Ds \\<rbrakk>\n \\<Longrightarrow> CT;\\<Gamma> \\<turnstile> New C es : C\"\n\n| t_ucast : \n \"\\<lbrakk> CT;\\<Gamma> \\<turnstile> e0 : D; \n CT \\<turnstile> D <: C \\<rbrakk>\n \\<Longrightarrow> CT;\\<Gamma> \\<turnstile> Cast C e0 : C\"\n\n| t_dcast : \n \"\\<lbrakk> CT;\\<Gamma> \\<turnstile> e0 : D; \n CT \\<turnstile> C <: D; C \\<noteq> D \\<rbrakk>\n \\<Longrightarrow> CT;\\<Gamma> \\<turnstile> Cast C e0 : C\"\n\n| t_scast : \n \"\\<lbrakk> CT;\\<Gamma> \\<turnstile> e0 : D;\n CT \\<turnstile> C \\<not><: D;\n CT \\<turnstile> D \\<not><: C \\<rbrakk>\n \\<Longrightarrow> CT;\\<Gamma> \\<turnstile> Cast C e0 : C\""
] |
FeatherweightJava/Execute
|
Execute.typings_i_i_i_oI
| null |
?x; ?xa \<turnstile>+ ?xb : ?xc \<Longrightarrow> pred.eval (typings_i_i_i_o ?x ?xa ?xb) ?xc
|
?H1 x_1 x_2 x_3 x_4 \<Longrightarrow> ?H2 (?H3 x_1 x_2 x_3) x_4
|
[
"Execute.typings_i_i_i_o",
"Predicate.pred.eval",
"FJDefs.typings"
] |
[
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> (nat \\<Rightarrow> nat option) \\<Rightarrow> exp list \\<Rightarrow> nat list Predicate.pred",
"'a Predicate.pred \\<Rightarrow> 'a \\<Rightarrow> bool",
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> (nat \\<Rightarrow> nat option) \\<Rightarrow> exp list \\<Rightarrow> nat list \\<Rightarrow> bool"
] |
[
"datatype (plugins only: extraction) (dead 'a) pred = Pred (eval: \"'a \\<Rightarrow> bool\")",
"inductive\n typings :: \"[classTable, varCtx, exp list, className list] \\<Rightarrow> bool\" (\"_;_ \\<turnstile>+ _ : _\" [80,80,80,80] 80)\n and typing :: \"[classTable, varCtx, exp, className] \\<Rightarrow> bool\" (\"_;_ \\<turnstile> _ : _\" [80,80,80,80] 80)\nwhere\n ts_nil : \"CT;\\<Gamma> \\<turnstile>+ [] : []\"\n\n| ts_cons : \n \"\\<lbrakk> CT;\\<Gamma> \\<turnstile> e0 : C0; CT;\\<Gamma> \\<turnstile>+ es : Cs \\<rbrakk> \n \\<Longrightarrow> CT;\\<Gamma> \\<turnstile>+ (e0 # es) : (C0 # Cs)\"\n\n| t_var : \n \"\\<lbrakk> \\<Gamma>(x) = Some C \\<rbrakk> \\<Longrightarrow> CT;\\<Gamma> \\<turnstile> (Var x) : C\"\n\n| t_field : \n \"\\<lbrakk> CT;\\<Gamma> \\<turnstile> e0 : C0;\n fields(CT,C0) = Cf;\n lookup Cf (\\<lambda>fd.(vdName fd = fi)) = Some(fDef);\n vdType fDef = Ci \\<rbrakk>\n \\<Longrightarrow> CT;\\<Gamma> \\<turnstile> FieldProj e0 fi : Ci\"\n\n| t_invk : \n \"\\<lbrakk> CT;\\<Gamma> \\<turnstile> e0 : C0;\n mtype(CT,m,C0) = Ds \\<rightarrow> C;\n CT;\\<Gamma> \\<turnstile>+ es : Cs;\n CT \\<turnstile>+ Cs <: Ds;\n length es = length Ds \\<rbrakk>\n \\<Longrightarrow> CT;\\<Gamma> \\<turnstile> MethodInvk e0 m es : C\"\n\n| t_new : \n \"\\<lbrakk> fields(CT,C) = Df;\n length es = length Df;\n varDefs_types Df = Ds;\n CT;\\<Gamma> \\<turnstile>+ es : Cs;\n CT \\<turnstile>+ Cs <: Ds \\<rbrakk>\n \\<Longrightarrow> CT;\\<Gamma> \\<turnstile> New C es : C\"\n\n| t_ucast : \n \"\\<lbrakk> CT;\\<Gamma> \\<turnstile> e0 : D; \n CT \\<turnstile> D <: C \\<rbrakk>\n \\<Longrightarrow> CT;\\<Gamma> \\<turnstile> Cast C e0 : C\"\n\n| t_dcast : \n \"\\<lbrakk> CT;\\<Gamma> \\<turnstile> e0 : D; \n CT \\<turnstile> C <: D; C \\<noteq> D \\<rbrakk>\n \\<Longrightarrow> CT;\\<Gamma> \\<turnstile> Cast C e0 : C\"\n\n| t_scast : \n \"\\<lbrakk> CT;\\<Gamma> \\<turnstile> e0 : D;\n CT \\<turnstile> C \\<not><: D;\n CT \\<turnstile> D \\<not><: C \\<rbrakk>\n \\<Longrightarrow> CT;\\<Gamma> \\<turnstile> Cast C e0 : C\""
] |
FeatherweightJava/Execute
|
Execute.typing_i_i_i_oI
| null |
?x; ?xa \<turnstile> ?xb : ?xc \<Longrightarrow> pred.eval (typing_i_i_i_o ?x ?xa ?xb) ?xc
|
?H1 x_1 x_2 x_3 x_4 \<Longrightarrow> ?H2 (?H3 x_1 x_2 x_3) x_4
|
[
"Execute.typing_i_i_i_o",
"Predicate.pred.eval",
"FJDefs.typing"
] |
[
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> (nat \\<Rightarrow> nat option) \\<Rightarrow> exp \\<Rightarrow> nat Predicate.pred",
"'a Predicate.pred \\<Rightarrow> 'a \\<Rightarrow> bool",
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> (nat \\<Rightarrow> nat option) \\<Rightarrow> exp \\<Rightarrow> nat \\<Rightarrow> bool"
] |
[
"datatype (plugins only: extraction) (dead 'a) pred = Pred (eval: \"'a \\<Rightarrow> bool\")",
"inductive\n typings :: \"[classTable, varCtx, exp list, className list] \\<Rightarrow> bool\" (\"_;_ \\<turnstile>+ _ : _\" [80,80,80,80] 80)\n and typing :: \"[classTable, varCtx, exp, className] \\<Rightarrow> bool\" (\"_;_ \\<turnstile> _ : _\" [80,80,80,80] 80)\nwhere\n ts_nil : \"CT;\\<Gamma> \\<turnstile>+ [] : []\"\n\n| ts_cons : \n \"\\<lbrakk> CT;\\<Gamma> \\<turnstile> e0 : C0; CT;\\<Gamma> \\<turnstile>+ es : Cs \\<rbrakk> \n \\<Longrightarrow> CT;\\<Gamma> \\<turnstile>+ (e0 # es) : (C0 # Cs)\"\n\n| t_var : \n \"\\<lbrakk> \\<Gamma>(x) = Some C \\<rbrakk> \\<Longrightarrow> CT;\\<Gamma> \\<turnstile> (Var x) : C\"\n\n| t_field : \n \"\\<lbrakk> CT;\\<Gamma> \\<turnstile> e0 : C0;\n fields(CT,C0) = Cf;\n lookup Cf (\\<lambda>fd.(vdName fd = fi)) = Some(fDef);\n vdType fDef = Ci \\<rbrakk>\n \\<Longrightarrow> CT;\\<Gamma> \\<turnstile> FieldProj e0 fi : Ci\"\n\n| t_invk : \n \"\\<lbrakk> CT;\\<Gamma> \\<turnstile> e0 : C0;\n mtype(CT,m,C0) = Ds \\<rightarrow> C;\n CT;\\<Gamma> \\<turnstile>+ es : Cs;\n CT \\<turnstile>+ Cs <: Ds;\n length es = length Ds \\<rbrakk>\n \\<Longrightarrow> CT;\\<Gamma> \\<turnstile> MethodInvk e0 m es : C\"\n\n| t_new : \n \"\\<lbrakk> fields(CT,C) = Df;\n length es = length Df;\n varDefs_types Df = Ds;\n CT;\\<Gamma> \\<turnstile>+ es : Cs;\n CT \\<turnstile>+ Cs <: Ds \\<rbrakk>\n \\<Longrightarrow> CT;\\<Gamma> \\<turnstile> New C es : C\"\n\n| t_ucast : \n \"\\<lbrakk> CT;\\<Gamma> \\<turnstile> e0 : D; \n CT \\<turnstile> D <: C \\<rbrakk>\n \\<Longrightarrow> CT;\\<Gamma> \\<turnstile> Cast C e0 : C\"\n\n| t_dcast : \n \"\\<lbrakk> CT;\\<Gamma> \\<turnstile> e0 : D; \n CT \\<turnstile> C <: D; C \\<noteq> D \\<rbrakk>\n \\<Longrightarrow> CT;\\<Gamma> \\<turnstile> Cast C e0 : C\"\n\n| t_scast : \n \"\\<lbrakk> CT;\\<Gamma> \\<turnstile> e0 : D;\n CT \\<turnstile> C \\<not><: D;\n CT \\<turnstile> D \\<not><: C \\<rbrakk>\n \\<Longrightarrow> CT;\\<Gamma> \\<turnstile> Cast C e0 : C\""
] |
FeatherweightJava/Execute
|
Execute.subtypings_i_i_iI
| null |
?x \<turnstile>+ ?xa <: ?xb \<Longrightarrow> pred.eval (subtypings_i_i_i ?x ?xa ?xb) ()
|
?H1 x_1 x_2 x_3 \<Longrightarrow> ?H2 (?H3 x_1 x_2 x_3) ?H4
|
[
"Product_Type.Unity",
"Execute.subtypings_i_i_i",
"Predicate.pred.eval",
"FJDefs.subtypings"
] |
[
"unit",
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> nat list \\<Rightarrow> nat list \\<Rightarrow> unit Predicate.pred",
"'a Predicate.pred \\<Rightarrow> 'a \\<Rightarrow> bool",
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> nat list \\<Rightarrow> nat list \\<Rightarrow> bool"
] |
[
"definition Unity :: unit (\"'(')\")\n where \"() = Abs_unit True\"",
"datatype (plugins only: extraction) (dead 'a) pred = Pred (eval: \"'a \\<Rightarrow> bool\")",
"inductive\n subtypings :: \"[classTable, className list, className list] \\<Rightarrow> bool\" (\"_ \\<turnstile>+ _ <: _\" [80,80,80] 80)\nwhere\n ss_nil : \"CT \\<turnstile>+ [] <: []\"\n| ss_cons : \"\\<lbrakk> CT \\<turnstile> C0 <: D0; CT \\<turnstile>+ Cs <: Ds \\<rbrakk> \\<Longrightarrow> CT \\<turnstile>+ (C0 # Cs) <: (D0 # Ds)\""
] |
FeatherweightJava/Execute
|
Execute.subtyping_i_i_iI
| null |
?x \<turnstile> ?xa <: ?xb \<Longrightarrow> pred.eval (subtyping_i_i_i ?x ?xa ?xb) ()
|
?H1 x_1 x_2 x_3 \<Longrightarrow> ?H2 (?H3 x_1 x_2 x_3) ?H4
|
[
"Product_Type.Unity",
"Execute.subtyping_i_i_i",
"Predicate.pred.eval",
"FJDefs.subtyping"
] |
[
"unit",
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> nat \\<Rightarrow> nat \\<Rightarrow> unit Predicate.pred",
"'a Predicate.pred \\<Rightarrow> 'a \\<Rightarrow> bool",
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> nat \\<Rightarrow> nat \\<Rightarrow> bool"
] |
[
"definition Unity :: unit (\"'(')\")\n where \"() = Abs_unit True\"",
"datatype (plugins only: extraction) (dead 'a) pred = Pred (eval: \"'a \\<Rightarrow> bool\")",
"inductive\n subtyping :: \"[classTable, className, className] \\<Rightarrow> bool\" (\"_ \\<turnstile> _ <: _\" [80,80,80] 80)\nwhere\n s_refl : \"CT \\<turnstile> C <: C\"\n| s_trans : \"\\<lbrakk> CT \\<turnstile> C <: D; CT \\<turnstile> D <: E \\<rbrakk> \\<Longrightarrow> CT \\<turnstile> C <: E\"\n| s_super : \"\\<lbrakk> CT(C) = Some(CDef); cSuper CDef = D \\<rbrakk> \\<Longrightarrow> CT \\<turnstile> C <: D\""
] |
FeatherweightJava/Execute
|
Execute.reductions_i_i_iI
| null |
?x \<turnstile> ?xa \<rightarrow>* ?xb \<Longrightarrow> pred.eval (reductions_i_i_i ?x ?xa ?xb) ()
|
?H1 x_1 x_2 x_3 \<Longrightarrow> ?H2 (?H3 x_1 x_2 x_3) ?H4
|
[
"Product_Type.Unity",
"Execute.reductions_i_i_i",
"Predicate.pred.eval",
"FJDefs.reductions"
] |
[
"unit",
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> exp \\<Rightarrow> exp \\<Rightarrow> unit Predicate.pred",
"'a Predicate.pred \\<Rightarrow> 'a \\<Rightarrow> bool",
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> exp \\<Rightarrow> exp \\<Rightarrow> bool"
] |
[
"definition Unity :: unit (\"'(')\")\n where \"() = Abs_unit True\"",
"datatype (plugins only: extraction) (dead 'a) pred = Pred (eval: \"'a \\<Rightarrow> bool\")",
"inductive\n reductions :: \"[classTable, exp, exp] \\<Rightarrow> bool\" (\"_ \\<turnstile> _ \\<rightarrow>* _\" [80,80,80] 80)\nwhere\n rs_refl: \"CT \\<turnstile> e \\<rightarrow>* e\" \n| rs_trans: \"\\<lbrakk> CT \\<turnstile> e \\<rightarrow> e'; CT \\<turnstile> e' \\<rightarrow>* e'' \\<rbrakk> \\<Longrightarrow> CT \\<turnstile> e \\<rightarrow>* e''\""
] |
FeatherweightJava/Execute
|
Execute.reduction_i_i_iI
| null |
?x \<turnstile> ?xa \<rightarrow> ?xb \<Longrightarrow> pred.eval (reduction_i_i_i ?x ?xa ?xb) ()
|
?H1 x_1 x_2 x_3 \<Longrightarrow> ?H2 (?H3 x_1 x_2 x_3) ?H4
|
[
"Product_Type.Unity",
"Execute.reduction_i_i_i",
"Predicate.pred.eval",
"FJDefs.reduction"
] |
[
"unit",
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> exp \\<Rightarrow> exp \\<Rightarrow> unit Predicate.pred",
"'a Predicate.pred \\<Rightarrow> 'a \\<Rightarrow> bool",
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> exp \\<Rightarrow> exp \\<Rightarrow> bool"
] |
[
"definition Unity :: unit (\"'(')\")\n where \"() = Abs_unit True\"",
"datatype (plugins only: extraction) (dead 'a) pred = Pred (eval: \"'a \\<Rightarrow> bool\")",
"inductive\n reduction :: \"[classTable, exp, exp] \\<Rightarrow> bool\" (\"_ \\<turnstile> _ \\<rightarrow> _\" [80,80,80] 80)\nwhere\n\n r_field: \n \"\\<lbrakk> fields(CT,C) = Cf; \n lookup2 Cf es (\\<lambda>fd.(vdName fd = fi)) = Some(ei) \\<rbrakk>\n \\<Longrightarrow> CT \\<turnstile> FieldProj (New C es) fi \\<rightarrow> ei\"\n\n| r_invk: \n \"\\<lbrakk> mbody(CT,m,C) = xs . e0;\n substs ((map_upds Map.empty xs ds)(this \\<mapsto> (New C es))) e0 = e0' \\<rbrakk>\n \\<Longrightarrow> CT \\<turnstile> MethodInvk (New C es) m ds \\<rightarrow> e0'\"\n\n| r_cast: \n \"\\<lbrakk> CT \\<turnstile> C <: D \\<rbrakk> \n \\<Longrightarrow> CT \\<turnstile> Cast D (New C es) \\<rightarrow> New C es\"\n\n| rc_field: \n \"\\<lbrakk> CT \\<turnstile> e0 \\<rightarrow> e0' \\<rbrakk> \n \\<Longrightarrow> CT \\<turnstile> FieldProj e0 f \\<rightarrow> FieldProj e0' f\"\n\n| rc_invk_recv: \n \"\\<lbrakk> CT \\<turnstile> e0 \\<rightarrow> e0' \\<rbrakk> \n \\<Longrightarrow> CT \\<turnstile> MethodInvk e0 m es \\<rightarrow> MethodInvk e0' m es\"\n\n| rc_invk_arg: \n \"\\<lbrakk> CT \\<turnstile> ei \\<rightarrow> ei' \\<rbrakk>\n \\<Longrightarrow> CT \\<turnstile> MethodInvk e0 m (el@ei#er) \\<rightarrow> MethodInvk e0 m (el@ei'#er)\"\n\n| rc_new_arg: \n \"\\<lbrakk> CT \\<turnstile> ei \\<rightarrow> ei' \\<rbrakk> \n \\<Longrightarrow> CT \\<turnstile> New C (el@ei#er) \\<rightarrow> New C (el@ei'#er)\"\n\n| rc_cast: \n \"\\<lbrakk> CT \\<turnstile> e0 \\<rightarrow> e0' \\<rbrakk> \n \\<Longrightarrow> CT \\<turnstile> Cast C e0 \\<rightarrow> Cast C e0'\""
] |
FeatherweightJava/Execute
|
Execute.method_typings_i_i_iI
| null |
?x \<turnstile>+ ?xa OK IN ?xb \<Longrightarrow> pred.eval (method_typings_i_i_i ?x ?xa ?xb) ()
|
?H1 x_1 x_2 x_3 \<Longrightarrow> ?H2 (?H3 x_1 x_2 x_3) ?H4
|
[
"Product_Type.Unity",
"Execute.method_typings_i_i_i",
"Predicate.pred.eval",
"FJDefs.method_typings"
] |
[
"unit",
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> methodDef list \\<Rightarrow> nat \\<Rightarrow> unit Predicate.pred",
"'a Predicate.pred \\<Rightarrow> 'a \\<Rightarrow> bool",
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> methodDef list \\<Rightarrow> nat \\<Rightarrow> bool"
] |
[
"definition Unity :: unit (\"'(')\")\n where \"() = Abs_unit True\"",
"datatype (plugins only: extraction) (dead 'a) pred = Pred (eval: \"'a \\<Rightarrow> bool\")",
"inductive\n method_typings :: \"[classTable, methodDef list, className] \\<Rightarrow> bool\" (\"_ \\<turnstile>+ _ OK IN _\" [80,80,80] 80)\nwhere\n ms_nil : \n \"CT \\<turnstile>+ [] OK IN C\"\n\n| ms_cons : \n \"\\<lbrakk> CT \\<turnstile> m OK IN C; \n CT \\<turnstile>+ ms OK IN C \\<rbrakk>\n \\<Longrightarrow> CT \\<turnstile>+ (m # ms) OK IN C\""
] |
FeatherweightJava/Execute
|
Execute.method_typing_i_i_iI
| null |
?x \<turnstile> ?xa OK IN ?xb \<Longrightarrow> pred.eval (method_typing_i_i_i ?x ?xa ?xb) ()
|
?H1 x_1 x_2 x_3 \<Longrightarrow> ?H2 (?H3 x_1 x_2 x_3) ?H4
|
[
"Product_Type.Unity",
"Execute.method_typing_i_i_i",
"Predicate.pred.eval",
"FJDefs.method_typing"
] |
[
"unit",
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> methodDef \\<Rightarrow> nat \\<Rightarrow> unit Predicate.pred",
"'a Predicate.pred \\<Rightarrow> 'a \\<Rightarrow> bool",
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> methodDef \\<Rightarrow> nat \\<Rightarrow> bool"
] |
[
"definition Unity :: unit (\"'(')\")\n where \"() = Abs_unit True\"",
"datatype (plugins only: extraction) (dead 'a) pred = Pred (eval: \"'a \\<Rightarrow> bool\")",
"inductive\n method_typing :: \"[classTable, methodDef, className] \\<Rightarrow> bool\" (\"_ \\<turnstile> _ OK IN _\" [80,80,80] 80)\nwhere\nm_typing:\n \"\\<lbrakk> CT(C) = Some(CDef);\n cName CDef = C;\n cSuper CDef = D;\n mName mDef = m;\n lookup (cMethods CDef) (\\<lambda>md.(mName md = m)) = Some(mDef);\n mReturn mDef = C0; mParams mDef = Cxs; mBody mDef = e0;\n varDefs_types Cxs = Cs;\n varDefs_names Cxs = xs;\n \\<Gamma> = (map_upds Map.empty xs Cs)(this \\<mapsto> C); \n CT;\\<Gamma> \\<turnstile> e0 : E0;\n CT \\<turnstile> E0 <: C0;\n \\<forall>Ds D0. (mtype(CT,m,D) = Ds \\<rightarrow> D0) \\<longrightarrow> (Cs=Ds \\<and> C0=D0) \\<rbrakk>\n \\<Longrightarrow> CT \\<turnstile> mDef OK IN C\""
] |
FeatherweightJava/Execute
|
Execute.fields_i_i_iI
| null |
fields(?x, ?xa) = ?xb \<Longrightarrow> pred.eval (fields_i_i_i ?x ?xa ?xb) ()
|
?H1 x_1 x_2 x_3 \<Longrightarrow> ?H2 (?H3 x_1 x_2 x_3) ?H4
|
[
"Product_Type.Unity",
"Execute.fields_i_i_i",
"Predicate.pred.eval",
"FJDefs.fields"
] |
[
"unit",
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> nat \\<Rightarrow> varDef list \\<Rightarrow> unit Predicate.pred",
"'a Predicate.pred \\<Rightarrow> 'a \\<Rightarrow> bool",
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> nat \\<Rightarrow> varDef list \\<Rightarrow> bool"
] |
[
"definition Unity :: unit (\"'(')\")\n where \"() = Abs_unit True\"",
"datatype (plugins only: extraction) (dead 'a) pred = Pred (eval: \"'a \\<Rightarrow> bool\")",
"inductive\n fields :: \"[classTable, className, varDef list] \\<Rightarrow> bool\" (\"fields'(_,_') = _\" [80,80,80] 80)\nwhere\n f_obj: \n \"fields(CT,Object) = []\"\n| f_class: \n \"\\<lbrakk> CT(C) = Some(CDef); cSuper CDef = D; cFields CDef = Cf; fields(CT,D) = Dg; DgCf = Dg @ Cf \\<rbrakk> \n \\<Longrightarrow> fields(CT,C) = DgCf\""
] |
FeatherweightJava/Execute
|
Execute.append_i_i_iI
| null |
Execute.append ?x ?xa ?xb \<Longrightarrow> pred.eval (append_i_i_i ?x ?xa ?xb) ()
|
?H1 x_1 x_2 x_3 \<Longrightarrow> ?H2 (?H3 x_1 x_2 x_3) ?H4
|
[
"Product_Type.Unity",
"Execute.append_i_i_i",
"Predicate.pred.eval",
"Execute.append"
] |
[
"unit",
"'a list \\<Rightarrow> 'a list \\<Rightarrow> 'a list \\<Rightarrow> unit Predicate.pred",
"'a Predicate.pred \\<Rightarrow> 'a \\<Rightarrow> bool",
"'a list \\<Rightarrow> 'a list \\<Rightarrow> 'a list \\<Rightarrow> bool"
] |
[
"definition Unity :: unit (\"'(')\")\n where \"() = Abs_unit True\"",
"datatype (plugins only: extraction) (dead 'a) pred = Pred (eval: \"'a \\<Rightarrow> bool\")",
"definition append where \"append xs ys zs = (zs = xs @ ys)\""
] |
FeatherweightJava/Execute
|
Execute.append_o_o_iI
| null |
Execute.append ?x ?xa ?xb \<Longrightarrow> pred.eval (append_o_o_i ?xb) (?x, ?xa)
|
?H1 x_1 x_2 x_3 \<Longrightarrow> ?H2 (?H3 x_3) (x_1, x_2)
|
[
"Execute.append_o_o_i",
"Predicate.pred.eval",
"Execute.append"
] |
[
"'a list \\<Rightarrow> ('a list \\<times> 'a list) Predicate.pred",
"'a Predicate.pred \\<Rightarrow> 'a \\<Rightarrow> bool",
"'a list \\<Rightarrow> 'a list \\<Rightarrow> 'a list \\<Rightarrow> bool"
] |
[
"datatype (plugins only: extraction) (dead 'a) pred = Pred (eval: \"'a \\<Rightarrow> bool\")",
"definition append where \"append xs ys zs = (zs = xs @ ys)\""
] |
FeatherweightJava/Execute
|
Execute.subtypings_i_i_oI
| null |
?x \<turnstile>+ ?xa <: ?xb \<Longrightarrow> pred.eval (subtypings_i_i_o ?x ?xa) ?xb
|
?H1 x_1 x_2 x_3 \<Longrightarrow> ?H2 (?H3 x_1 x_2) x_3
|
[
"Execute.subtypings_i_i_o",
"Predicate.pred.eval",
"FJDefs.subtypings"
] |
[
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> nat list \\<Rightarrow> nat list Predicate.pred",
"'a Predicate.pred \\<Rightarrow> 'a \\<Rightarrow> bool",
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> nat list \\<Rightarrow> nat list \\<Rightarrow> bool"
] |
[
"datatype (plugins only: extraction) (dead 'a) pred = Pred (eval: \"'a \\<Rightarrow> bool\")",
"inductive\n subtypings :: \"[classTable, className list, className list] \\<Rightarrow> bool\" (\"_ \\<turnstile>+ _ <: _\" [80,80,80] 80)\nwhere\n ss_nil : \"CT \\<turnstile>+ [] <: []\"\n| ss_cons : \"\\<lbrakk> CT \\<turnstile> C0 <: D0; CT \\<turnstile>+ Cs <: Ds \\<rbrakk> \\<Longrightarrow> CT \\<turnstile>+ (C0 # Cs) <: (D0 # Ds)\""
] |
FeatherweightJava/Execute
|
Execute.supertypes_ofI
| null |
?x \<turnstile> ?xa <: ?xb \<Longrightarrow> pred.eval (supertypes_of ?x ?xa) ?xb
|
?H1 x_1 x_2 x_3 \<Longrightarrow> ?H2 (?H3 x_1 x_2) x_3
|
[
"Execute.supertypes_of",
"Predicate.pred.eval",
"FJDefs.subtyping"
] |
[
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> nat \\<Rightarrow> nat Predicate.pred",
"'a Predicate.pred \\<Rightarrow> 'a \\<Rightarrow> bool",
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> nat \\<Rightarrow> nat \\<Rightarrow> bool"
] |
[
"datatype (plugins only: extraction) (dead 'a) pred = Pred (eval: \"'a \\<Rightarrow> bool\")",
"inductive\n subtyping :: \"[classTable, className, className] \\<Rightarrow> bool\" (\"_ \\<turnstile> _ <: _\" [80,80,80] 80)\nwhere\n s_refl : \"CT \\<turnstile> C <: C\"\n| s_trans : \"\\<lbrakk> CT \\<turnstile> C <: D; CT \\<turnstile> D <: E \\<rbrakk> \\<Longrightarrow> CT \\<turnstile> C <: E\"\n| s_super : \"\\<lbrakk> CT(C) = Some(CDef); cSuper CDef = D \\<rbrakk> \\<Longrightarrow> CT \\<turnstile> C <: D\""
] |
FeatherweightJava/Execute
|
Execute.reductions_i_i_oI
| null |
?x \<turnstile> ?xa \<rightarrow>* ?xb \<Longrightarrow> pred.eval (reductions_i_i_o ?x ?xa) ?xb
|
?H1 x_1 x_2 x_3 \<Longrightarrow> ?H2 (?H3 x_1 x_2) x_3
|
[
"Execute.reductions_i_i_o",
"Predicate.pred.eval",
"FJDefs.reductions"
] |
[
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> exp \\<Rightarrow> exp Predicate.pred",
"'a Predicate.pred \\<Rightarrow> 'a \\<Rightarrow> bool",
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> exp \\<Rightarrow> exp \\<Rightarrow> bool"
] |
[
"datatype (plugins only: extraction) (dead 'a) pred = Pred (eval: \"'a \\<Rightarrow> bool\")",
"inductive\n reductions :: \"[classTable, exp, exp] \\<Rightarrow> bool\" (\"_ \\<turnstile> _ \\<rightarrow>* _\" [80,80,80] 80)\nwhere\n rs_refl: \"CT \\<turnstile> e \\<rightarrow>* e\" \n| rs_trans: \"\\<lbrakk> CT \\<turnstile> e \\<rightarrow> e'; CT \\<turnstile> e' \\<rightarrow>* e'' \\<rbrakk> \\<Longrightarrow> CT \\<turnstile> e \\<rightarrow>* e''\""
] |
FeatherweightJava/Execute
|
Execute.reduceI
| null |
?x \<turnstile> ?xa \<rightarrow> ?xb \<Longrightarrow> pred.eval (reduce ?x ?xa) ?xb
|
?H1 x_1 x_2 x_3 \<Longrightarrow> ?H2 (?H3 x_1 x_2) x_3
|
[
"Execute.reduce",
"Predicate.pred.eval",
"FJDefs.reduction"
] |
[
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> exp \\<Rightarrow> exp Predicate.pred",
"'a Predicate.pred \\<Rightarrow> 'a \\<Rightarrow> bool",
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> exp \\<Rightarrow> exp \\<Rightarrow> bool"
] |
[
"datatype (plugins only: extraction) (dead 'a) pred = Pred (eval: \"'a \\<Rightarrow> bool\")",
"inductive\n reduction :: \"[classTable, exp, exp] \\<Rightarrow> bool\" (\"_ \\<turnstile> _ \\<rightarrow> _\" [80,80,80] 80)\nwhere\n\n r_field: \n \"\\<lbrakk> fields(CT,C) = Cf; \n lookup2 Cf es (\\<lambda>fd.(vdName fd = fi)) = Some(ei) \\<rbrakk>\n \\<Longrightarrow> CT \\<turnstile> FieldProj (New C es) fi \\<rightarrow> ei\"\n\n| r_invk: \n \"\\<lbrakk> mbody(CT,m,C) = xs . e0;\n substs ((map_upds Map.empty xs ds)(this \\<mapsto> (New C es))) e0 = e0' \\<rbrakk>\n \\<Longrightarrow> CT \\<turnstile> MethodInvk (New C es) m ds \\<rightarrow> e0'\"\n\n| r_cast: \n \"\\<lbrakk> CT \\<turnstile> C <: D \\<rbrakk> \n \\<Longrightarrow> CT \\<turnstile> Cast D (New C es) \\<rightarrow> New C es\"\n\n| rc_field: \n \"\\<lbrakk> CT \\<turnstile> e0 \\<rightarrow> e0' \\<rbrakk> \n \\<Longrightarrow> CT \\<turnstile> FieldProj e0 f \\<rightarrow> FieldProj e0' f\"\n\n| rc_invk_recv: \n \"\\<lbrakk> CT \\<turnstile> e0 \\<rightarrow> e0' \\<rbrakk> \n \\<Longrightarrow> CT \\<turnstile> MethodInvk e0 m es \\<rightarrow> MethodInvk e0' m es\"\n\n| rc_invk_arg: \n \"\\<lbrakk> CT \\<turnstile> ei \\<rightarrow> ei' \\<rbrakk>\n \\<Longrightarrow> CT \\<turnstile> MethodInvk e0 m (el@ei#er) \\<rightarrow> MethodInvk e0 m (el@ei'#er)\"\n\n| rc_new_arg: \n \"\\<lbrakk> CT \\<turnstile> ei \\<rightarrow> ei' \\<rbrakk> \n \\<Longrightarrow> CT \\<turnstile> New C (el@ei#er) \\<rightarrow> New C (el@ei'#er)\"\n\n| rc_cast: \n \"\\<lbrakk> CT \\<turnstile> e0 \\<rightarrow> e0' \\<rbrakk> \n \\<Longrightarrow> CT \\<turnstile> Cast C e0 \\<rightarrow> Cast C e0'\""
] |
FeatherweightJava/Execute
|
Execute.fields_i_i_oI
| null |
fields(?x, ?xa) = ?xb \<Longrightarrow> pred.eval (fields_i_i_o ?x ?xa) ?xb
|
?H1 x_1 x_2 x_3 \<Longrightarrow> ?H2 (?H3 x_1 x_2) x_3
|
[
"Execute.fields_i_i_o",
"Predicate.pred.eval",
"FJDefs.fields"
] |
[
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> nat \\<Rightarrow> varDef list Predicate.pred",
"'a Predicate.pred \\<Rightarrow> 'a \\<Rightarrow> bool",
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> nat \\<Rightarrow> varDef list \\<Rightarrow> bool"
] |
[
"datatype (plugins only: extraction) (dead 'a) pred = Pred (eval: \"'a \\<Rightarrow> bool\")",
"inductive\n fields :: \"[classTable, className, varDef list] \\<Rightarrow> bool\" (\"fields'(_,_') = _\" [80,80,80] 80)\nwhere\n f_obj: \n \"fields(CT,Object) = []\"\n| f_class: \n \"\\<lbrakk> CT(C) = Some(CDef); cSuper CDef = D; cFields CDef = Cf; fields(CT,D) = Dg; DgCf = Dg @ Cf \\<rbrakk> \n \\<Longrightarrow> fields(CT,C) = DgCf\""
] |
FeatherweightJava/Execute
|
Execute.append_o_i_iI
| null |
Execute.append ?x ?xa ?xb \<Longrightarrow> pred.eval (append_o_i_i ?xa ?xb) ?x
|
?H1 x_1 x_2 x_3 \<Longrightarrow> ?H2 (?H3 x_2 x_3) x_1
|
[
"Execute.append_o_i_i",
"Predicate.pred.eval",
"Execute.append"
] |
[
"'a list \\<Rightarrow> 'a list \\<Rightarrow> 'a list Predicate.pred",
"'a Predicate.pred \\<Rightarrow> 'a \\<Rightarrow> bool",
"'a list \\<Rightarrow> 'a list \\<Rightarrow> 'a list \\<Rightarrow> bool"
] |
[
"datatype (plugins only: extraction) (dead 'a) pred = Pred (eval: \"'a \\<Rightarrow> bool\")",
"definition append where \"append xs ys zs = (zs = xs @ ys)\""
] |
FeatherweightJava/Execute
|
Execute.append_i_o_iI
| null |
Execute.append ?x ?xa ?xb \<Longrightarrow> pred.eval (append_i_o_i ?x ?xb) ?xa
|
?H1 x_1 x_2 x_3 \<Longrightarrow> ?H2 (?H3 x_1 x_3) x_2
|
[
"Execute.append_i_o_i",
"Predicate.pred.eval",
"Execute.append"
] |
[
"'a list \\<Rightarrow> 'a list \\<Rightarrow> 'a list Predicate.pred",
"'a Predicate.pred \\<Rightarrow> 'a \\<Rightarrow> bool",
"'a list \\<Rightarrow> 'a list \\<Rightarrow> 'a list \\<Rightarrow> bool"
] |
[
"datatype (plugins only: extraction) (dead 'a) pred = Pred (eval: \"'a \\<Rightarrow> bool\")",
"definition append where \"append xs ys zs = (zs = xs @ ys)\""
] |
FeatherweightJava/Execute
|
Execute.append_i_i_oI
| null |
Execute.append ?x ?xa ?xb \<Longrightarrow> pred.eval (append_i_i_o ?x ?xa) ?xb
|
?H1 x_1 x_2 x_3 \<Longrightarrow> ?H2 (?H3 x_1 x_2) x_3
|
[
"Execute.append_i_i_o",
"Predicate.pred.eval",
"Execute.append"
] |
[
"'a list \\<Rightarrow> 'a list \\<Rightarrow> 'a list Predicate.pred",
"'a Predicate.pred \\<Rightarrow> 'a \\<Rightarrow> bool",
"'a list \\<Rightarrow> 'a list \\<Rightarrow> 'a list \\<Rightarrow> bool"
] |
[
"datatype (plugins only: extraction) (dead 'a) pred = Pred (eval: \"'a \\<Rightarrow> bool\")",
"definition append where \"append xs ys zs = (zs = xs @ ys)\""
] |
FeatherweightJava/Execute
|
Execute.eq_i_iI
| null |
Predicate.eq ?x ?xa \<Longrightarrow> pred.eval (eq_i_i ?x ?xa) ()
|
?H1 x_1 x_2 \<Longrightarrow> ?H2 (?H3 x_1 x_2) ?H4
|
[
"Product_Type.Unity",
"Execute.eq_i_i",
"Predicate.pred.eval",
"Predicate.eq"
] |
[
"unit",
"'a \\<Rightarrow> 'a \\<Rightarrow> unit Predicate.pred",
"'a Predicate.pred \\<Rightarrow> 'a \\<Rightarrow> bool",
"'a \\<Rightarrow> 'a \\<Rightarrow> bool"
] |
[
"definition Unity :: unit (\"'(')\")\n where \"() = Abs_unit True\"",
"datatype (plugins only: extraction) (dead 'a) pred = Pred (eval: \"'a \\<Rightarrow> bool\")",
"inductive eq :: \"'a \\<Rightarrow> 'a \\<Rightarrow> bool\" where \"eq x x\""
] |
FeatherweightJava/Execute
|
Execute.class_typing_i_iI
| null |
?x \<turnstile> ?xa OK \<Longrightarrow> pred.eval (class_typing_i_i ?x ?xa) ()
|
?H1 x_1 x_2 \<Longrightarrow> ?H2 (?H3 x_1 x_2) ?H4
|
[
"Product_Type.Unity",
"Execute.class_typing_i_i",
"Predicate.pred.eval",
"FJDefs.class_typing"
] |
[
"unit",
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> classDef \\<Rightarrow> unit Predicate.pred",
"'a Predicate.pred \\<Rightarrow> 'a \\<Rightarrow> bool",
"(nat \\<Rightarrow> classDef option) \\<Rightarrow> classDef \\<Rightarrow> bool"
] |
[
"definition Unity :: unit (\"'(')\")\n where \"() = Abs_unit True\"",
"datatype (plugins only: extraction) (dead 'a) pred = Pred (eval: \"'a \\<Rightarrow> bool\")",
"inductive\n class_typing :: \"[classTable, classDef] \\<Rightarrow> bool\" (\"_ \\<turnstile> _ OK\" [80,80] 80)\nwhere\nt_class: \"\\<lbrakk> cName CDef = C; \n cSuper CDef = D;\n cConstructor CDef = KDef;\n cMethods CDef = M;\n kName KDef = C;\n kParams KDef = (Dg@Cf);\n kSuper KDef = varDefs_names Dg;\n kInits KDef = varDefs_names Cf;\n fields(CT,D) = Dg;\n CT \\<turnstile>+ M OK IN C \\<rbrakk>\n \\<Longrightarrow> CT \\<turnstile> CDef OK\""
] |
FeatherweightJava/Execute
|
Execute.eq_o_iI
| null |
Predicate.eq ?x ?xa \<Longrightarrow> pred.eval (eq_o_i ?xa) ?x
|
?H1 x_1 x_2 \<Longrightarrow> ?H2 (?H3 x_2) x_1
|
[
"Execute.eq_o_i",
"Predicate.pred.eval",
"Predicate.eq"
] |
[
"'a \\<Rightarrow> 'a Predicate.pred",
"'a Predicate.pred \\<Rightarrow> 'a \\<Rightarrow> bool",
"'a \\<Rightarrow> 'a \\<Rightarrow> bool"
] |
[
"datatype (plugins only: extraction) (dead 'a) pred = Pred (eval: \"'a \\<Rightarrow> bool\")",
"inductive eq :: \"'a \\<Rightarrow> 'a \\<Rightarrow> bool\" where \"eq x x\""
] |
FeatherweightJava/Execute
|
Execute.eq_i_oI
| null |
Predicate.eq ?x ?xa \<Longrightarrow> pred.eval (eq_i_o ?x) ?xa
|
?H1 x_1 x_2 \<Longrightarrow> ?H2 (?H3 x_1) x_2
|
[
"Execute.eq_i_o",
"Predicate.pred.eval",
"Predicate.eq"
] |
[
"'a \\<Rightarrow> 'a Predicate.pred",
"'a Predicate.pred \\<Rightarrow> 'a \\<Rightarrow> bool",
"'a \\<Rightarrow> 'a \\<Rightarrow> bool"
] |
[
"datatype (plugins only: extraction) (dead 'a) pred = Pred (eval: \"'a \\<Rightarrow> bool\")",
"inductive eq :: \"'a \\<Rightarrow> 'a \\<Rightarrow> bool\" where \"eq x x\""
] |
FeatherweightJava/FJAux
|
FJAux.map_upds_index
|
lemma map_upds_index:
assumes "length xs = length As"
and "[xs[\<mapsto>]As]x = Some Ai"
shows "\<exists>i.(As!i = Ai)
\<and> (i < length As)
\<and> (\<forall>(Bs::'c list).((length Bs = length As)
\<longrightarrow> ([xs[\<mapsto>]Bs] x = Some (Bs !i))))"
(is "\<exists>i. ?P i xs As"
is "\<exists>i.(?P1 i As) \<and> (?P2 i As) \<and> (\<forall>Bs::('c list).(?P3 i xs As Bs))")
|
length ?xs = length ?As \<Longrightarrow> [ ?xs [\<mapsto>] ?As] ?x = Some ?Ai \<Longrightarrow> \<exists>i. ?As ! i = ?Ai \<and> i < length ?As \<and> (\<forall>Bs. length Bs = length ?As \<longrightarrow> [ ?xs [\<mapsto>] Bs] ?x = Some (Bs ! i))
|
\<lbrakk> ?H1 x_1 = ?H2 x_2; ?H3 (\<lambda>y_0. ?H4) x_1 x_2 x_3 = ?H5 x_4\<rbrakk> \<Longrightarrow> \<exists>y_1. ?H6 x_2 y_1 = x_4 \<and> y_1 < ?H2 x_2 \<and> (\<forall>y_2. ?H7 y_2 = ?H2 x_2 \<longrightarrow> ?H8 (\<lambda>y_3. ?H9) x_1 y_2 x_3 = ?H10 (?H11 y_2 y_1))
|
[
"List.nth",
"Option.option.Some",
"Option.option.None",
"Map.map_upds",
"List.length"
] |
[
"'a list \\<Rightarrow> nat \\<Rightarrow> 'a",
"'a \\<Rightarrow> 'a option",
"'a option",
"('a \\<Rightarrow> 'b option) \\<Rightarrow> 'a list \\<Rightarrow> 'b list \\<Rightarrow> 'a \\<Rightarrow> 'b option",
"'a list \\<Rightarrow> nat"
] |
[
"primrec (nonexhaustive) nth :: \"'a list => nat => 'a\" (infixl \"!\" 100) where\nnth_Cons: \"(x # xs) ! n = (case n of 0 \\<Rightarrow> x | Suc k \\<Rightarrow> xs ! k)\"\n \\<comment> \\<open>Warning: simpset does not contain this definition, but separate\n theorems for \\<open>n = 0\\<close> and \\<open>n = Suc k\\<close>\\<close>",
"datatype 'a option =\n None\n | Some (the: 'a)",
"definition map_upds :: \"('a \\<rightharpoonup> 'b) \\<Rightarrow> 'a list \\<Rightarrow> 'b list \\<Rightarrow> 'a \\<rightharpoonup> 'b\" where\n\"map_upds m xs ys = m ++ map_of (rev (zip xs ys))\"",
"abbreviation length :: \"'a list \\<Rightarrow> nat\" where\n\"length \\<equiv> size\""
] |
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