Thm* A:ioa{i:l}(), r:rel(), rho:Decl, de:sig(), e:{[[de]] rho}, a:( [[A.da]] rho), tr:trace_env([[A.da]] rho). tc_ioa(A;de) ![](FONT/eq.png) ioa_mentions_trace(A) ![](FONT/eq.png) trace_consistent_rel(rho;A.da;tr.proj;r) ![](FONT/eq.png) single_valued_decls(A.ds) ![](FONT/eq.png) ( s,x':[[A]] rho de e.state. tc(r;A.ds; < > ;de) ![](FONT/eq.png) closed_rel(r) ![](FONT/eq.png) covers_rel(A;r) ![](FONT/eq.png) [[A]] rho de e.trans(s,a,x') ![](FONT/eq.png) ([[r]] rho A.ds < > de e x' tr ![](FONT/if_big.png) [[wp_rel(A;kind(a);r)]] rho A.ds dec_lookup(A.da;kind(a)) de e s value(a) tr)) | [wp_rel_correctness] |
Thm* A:ioa{i:l}(), de:sig(). tc_ioa(A;de) ![](FONT/eq.png) ioa_mentions_trace(A) ![](FONT/eq.png) ( Q:Fmla, rho:Decl, e:{[[de]] rho}, a:[[A]] rho de e.action, tr:trace_env([[A.da]] rho). tc_ioa(A;de) ![](FONT/eq.png) ioa_mentions_trace(A) ![](FONT/eq.png) trace_consistent_pred(rho;A.da;tr.proj;Q) ![](FONT/eq.png) single_valued_decls(A.ds) ![](FONT/eq.png) ( s,x':[[A]] rho de e.state. tc_pred(Q;A.ds; < > ;de) ![](FONT/eq.png) closed_pred(Q) ![](FONT/eq.png) covers_pred(A;Q) ![](FONT/eq.png) [[A]] rho de e.trans(s,a,x') ![](FONT/eq.png) ([[Q]] rho A.ds < > de e x' tr ![](FONT/if_big.png) [[wp(A;kind(a);Q)]] rho A.ds dec_lookup(A.da;kind(a)) de e s value(a) tr))) | [wp_correctness] |
Thm* A:ioa{i:l}(), de:sig(). tc_ioa(A;de) ![](FONT/eq.png) ioa_mentions_trace(A) ![](FONT/eq.png) ( Q:Fmla, rho:Decl, e:{[[de]] rho}, a:[[A]] rho de e.action, tr:trace_env([[A.da]] rho). trace_consistent_pred(rho;A.da;tr.proj;Q) ![](FONT/eq.png) single_valued_decls(A.ds) ![](FONT/eq.png) ( s,x':[[A]] rho de e.state. tc_pred(Q;A.ds; < > ;de) ![](FONT/eq.png) closed_pred(Q) ![](FONT/eq.png) covers_pred(A;Q) ![](FONT/eq.png) [[A]] rho de e.trans(s,a,x') ![](FONT/eq.png) (pred_mng_2(Q; rho; A.ds; < > ; de; e; s; x'; ; tr) ![](FONT/if_big.png) [[wp2(A;kind(a);Q)]] rho A.ds dec_lookup(A.da;kind(a)) de e s value(a) tr))) | [wp2_correctness] |
Thm* A:ioa{i:l}(), r:rel(), rho:Decl, de:sig(), e:{[[de]] rho}, a:( [[A.da]] rho), tr:trace_env([[A.da]] rho). tc_ioa(A;de) ![](FONT/eq.png) ioa_mentions_trace(A) ![](FONT/eq.png) trace_consistent_rel(rho;A.da;tr.proj;r) ![](FONT/eq.png) single_valued_decls(A.ds) ![](FONT/eq.png) ( s,x':[[A]] rho de e.state. tc(r;A.ds; < > ;de) ![](FONT/eq.png) closed_rel(r) ![](FONT/eq.png) covers_rel(A;r) ![](FONT/eq.png) [[A]] rho de e.trans(s,a,x') ![](FONT/eq.png) (rel_mng_2(r; rho; A.ds; < > ; de; e; s; x'; ; tr) ![](FONT/if_big.png) [[wp2_rel(A;kind(a);r)]] rho A.ds dec_lookup(A.da;kind(a)) de e s value(a) tr)) | [wp2_rel_correctness] |
Thm* A:ioa{i:l}(), r:rel(), rho:Decl, de:sig(), e:{[[de]] rho}, a:( [[A.da]] rho), tr:trace_env([[A.da]] rho). tc_ioa(A;de) ![](FONT/eq.png) ioa_mentions_trace(A) ![](FONT/eq.png) trace_consistent_rel(rho;A.da;tr.proj;r) ![](FONT/eq.png) single_valued_decls(A.ds) ![](FONT/eq.png) ( s,x':[[A]] rho de e.state. tc(r;A.ds;dec_lookup(A.da;kind(a));de) ![](FONT/eq.png) covers_rel(A;r) ![](FONT/eq.png) [[A]] rho de e.trans(s,a,x') ![](FONT/eq.png) (rel_mng_2(r; rho; A.ds; dec_lookup(A.da;kind(a)); de; e; s; x'; value(a); tr) ![](FONT/if_big.png) [[wp2_rel(A;kind(a);r)]] rho A.ds dec_lookup(A.da;kind(a)) de e s value(a) tr)) | [wp2_rel_correct] |
Thm* p:Fmla, ds,daa:Collection(dec()), da:Collection(SimpleType), de:sig(), rho:Decl, e:{[[de]] rho}, s,s':{[[ds]] rho}, a:[[da]] rho, tr:trace_env([[daa]] rho). trace_consistent_pred(rho;daa;tr.proj;p) ![](FONT/eq.png) tc_pred(p;ds;da;de) ![](FONT/eq.png) (pred_mng_2(pred_unprime(p); rho; ds; da; de; e; s; s'; a; tr) ![](FONT/if_big.png) [[p]] rho ds da de e s a tr) | [pred_mng2_unprime] |
Thm* p:Fmla, ds,daa:Collection(dec()), da:Collection(SimpleType), de:sig(), rho:Decl, e:{[[de]] rho}, s,s':{[[ds]] rho}, a:[[da]] rho, tr:trace_env([[daa]] rho). trace_consistent_pred(rho;daa;tr.proj;p) ![](FONT/eq.png) tc_pred(p;ds;da;de) ![](FONT/eq.png) (pred_mng_2((p)'; rho; ds; da; de; e; s; s'; a; tr) ![](FONT/if_big.png) [[p]] rho ds da de e s' a tr) | [pred_mng2_addprime] |
Thm* p:Fmla, rho:Decl, ds,daa:Collection(dec()), da1,da2:Collection(SimpleType), de:sig(), s:{[[ds]] rho}, e:{[[de]] rho}, a1,a2:Top, tr:trace_env([[daa]] rho). trace_consistent_pred(rho;daa;tr.proj;p) ![](FONT/eq.png) tc_pred(p;ds;da1;de) ![](FONT/eq.png) closed_pred(p) ![](FONT/eq.png) ([[p]] rho ds da1 de e s a1 tr ![](FONT/if_big.png) [[p]] rho ds da2 de e s a2 tr) | [closed_pred_mng] |
Thm* p1,p2:Fmla, ds1,ds2,daa:Collection(dec()), da1,da2:Collection(SimpleType), de:sig(), rho:Decl, e:{[[de]] rho}, s:{[[ds1]] rho}, a:[[da1]] rho, tr:trace_env([[daa]] rho). trace_consistent_pred(rho;daa;tr.proj;p1) ![](FONT/eq.png) tc_pred(p1;ds1;da1;de) ![](FONT/eq.png) p1 = p2 ![](FONT/eq.png) ds1 = ds2 ![](FONT/eq.png) da1 = da2 ![](FONT/eq.png) ([[p1]] rho ds1 da1 de e s a tr ![](FONT/if_big.png) [[p2]] rho ds2 da2 de e s a tr) | [pred_mng_functionality] |
Thm* p,q:Fmla, rho:Decl, da:Collection(dec()), R:(Label![](FONT/dash.png) Label![](FONT/dash.png) ![](FONT/then_med.png) ). trace_consistent_pred(rho;da;R;p q) ![](FONT/if_big.png) trace_consistent_pred(rho;da;R;p) & trace_consistent_pred(rho;da;R;q) | [trace_consistent_pred_and] |
Thm* r:rel(), as:(Label Term) List, ds,daa:Collection(dec()), da:Collection(SimpleType), de:sig(), rho:Decl, e:{[[de]] rho}, s,s':{[[ds]] rho}, k:Label, a:[[da]] rho, tr:trace_env([[daa]] rho). trace_consistent_rel(rho;daa;tr.proj;r) ![](FONT/eq.png) tc(r;ds;da;de) ![](FONT/eq.png) subst_mentions_trace(as) ![](FONT/eq.png) ( x:Label. (x rel_primed_vars(r)) ![](FONT/eq.png) ( t:SimpleType. mk_dec(x, t) ds ![](FONT/eq.png) t term_types(ds;da;de;apply_alist(as;x;x)) & s'.x = [[apply_alist(as;x;x)]] 1of(e) s a tr [[t]] rho)) ![](FONT/eq.png) tc(rel_subst2(as;r);ds;da;de) & trace_consistent_rel(rho;daa;tr.proj;rel_subst2(as;r)) & ([[rel_subst2(as;r)]] rho ds da de e s a tr ![](FONT/if_big.png) rel_mng_2(r; rho; ds; da; de; e; s; s'; a; tr)) | [rel_subst_mng_2_iff] |
Thm* r:rel(), ds1,ds2,da:Collection(dec()), da1,da2:Collection(SimpleType), de:sig(), rho:Decl, e:{[[de]] rho}, s:{[[ds1]] rho}, a:[[da1]] rho, tr:trace_env([[da]] rho). trace_consistent_rel(rho;da;tr.proj;r) ![](FONT/eq.png) tc(r;ds1;da1;de) ![](FONT/eq.png) ds1 = ds2 ![](FONT/eq.png) da1 = da2 ![](FONT/eq.png) ([[r]] rho ds1 da1 de e s a tr ![](FONT/if_big.png) [[r]] rho ds2 da2 de e s a tr) | [rel_mng_functionality] |
Thm* p:Collection(rel()), ds:Collection(dec()), da:Collection(SimpleType), de:sig(). tc_pred(p;ds;da;de) ![](FONT/if_big.png) ( r p.tc(r;ds;da;de)) | [tc_pred_col_all] |
Thm* r:rel(), rho:Decl, ds,daa:Collection(dec()), da1,da2:Collection(SimpleType), de:sig(), s:{[[ds]] rho}, e:{[[de]] rho}, a1,a2:Top, tr:trace_env([[daa]] rho). trace_consistent_rel(rho;daa;tr.proj;r) ![](FONT/eq.png) tc(r;ds;da1;de) ![](FONT/eq.png) closed_rel(r) ![](FONT/eq.png) ([[r]] rho ds da1 de e s a1 tr ![](FONT/if_big.png) [[r]] rho ds da2 de e s a2 tr) | [closed_rel_mng_2] |
Thm* r1,r2:rel(), ds,daa:Collection(dec()), da:Collection(SimpleType), de:sig(), rho:Decl, e:{[[de]] rho}, s1,s2:{[[ds]] rho}, a:[[da]] rho, tr:trace_env([[daa]] rho). trace_consistent_rel(rho;daa;tr.proj;r1) ![](FONT/eq.png) trace_consistent_rel(rho;daa;tr.proj;r2) ![](FONT/eq.png) tc(r1;ds;da;de) ![](FONT/eq.png) tc(r2;ds;da;de) ![](FONT/eq.png) r1.name = r2.name ![](FONT/eq.png) ||r1.args|| = ||r2.args|| ![](FONT/eq.png) ( i: . i < ||r1.args|| ![](FONT/eq.png) [[r1.args[i]]] 1of(e) s1 a tr = [[r2.args[i]]] 1of(e) s1 s2 a tr [[rel_arg_typ(r1.name;i;de)]] rho) ![](FONT/eq.png) ([[r1]] rho ds da de e s1 a tr ![](FONT/if_big.png) rel_mng_2(r2; rho; ds; da; de; e; s1; s2; a; tr)) | [rel_mng_2_iff] |
Thm* P:Fmla, ds:Collection(dec()), da:Collection(SimpleType), de:sig(). tc_pred(P;ds;da;de) ![](FONT/if_big.png) tc_pred(pred_unprime(P);ds;da;de) | [tc_pred_unprime] |
Thm* P:Fmla, ds:Collection(dec()), da:Collection(SimpleType), de:sig(). tc_pred(P;ds;da;de) ![](FONT/if_big.png) tc_pred((P)';ds;da;de) | [tc_pred_addprime] |
Thm* p1,p2:Fmla, ds1,ds2:Collection(dec()), da1,da2:Collection(SimpleType), de:sig(). p1 = p2 ![](FONT/eq.png) ds1 = ds2 ![](FONT/eq.png) da1 = da2 ![](FONT/eq.png) (tc_pred(p1;ds1;da1;de) ![](FONT/if_big.png) tc_pred(p2;ds2;da2;de)) | [tc_pred_functionality] |
Thm* r:rel(), ds:Collection(dec()), da:Collection(SimpleType), de:sig(). tc_pred( < r > ;ds;da;de) ![](FONT/if_big.png) tc(r;ds;da;de) | [tc_pred_singleton] |
Thm* r:rel(), ds:Collection(dec()), da:Collection(SimpleType), de:sig(). tc_pred(r;ds;da;de) ![](FONT/if_big.png) tc(r;ds;da;de) | [tc_pred_pred_rel] |