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)  ioa_mentions_trace(A)  trace_consistent_rel(rho;A.da;tr.proj;r)  single_valued_decls(A.ds)  ( s,x':[[A]] rho de e.state. tc(r;A.ds; < > ;de)  closed_rel(r)  covers_rel(A;r)  [[A]] rho de e.trans(s,a,x')  ([[r]] rho A.ds < > de e x' tr  [[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}(), r:rel(), rho:Decl, de:sig(), e:{[[de]] rho}, a:( [[A.da]] rho), tr:trace_env([[A.da]] rho). tc_ioa(A;de)  ioa_mentions_trace(A)  trace_consistent_rel(rho;A.da;tr.proj;r)  single_valued_decls(A.ds)  ( s,x':[[A]] rho de e.state. tc(r;A.ds; < > ;de)  closed_rel(r)  covers_rel(A;r)  [[A]] rho de e.trans(s,a,x')  (rel_mng_2(r; rho; A.ds; < > ; de; e; s; x'; ; tr)  [[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)  ioa_mentions_trace(A)  trace_consistent_rel(rho;A.da;tr.proj;r)  single_valued_decls(A.ds)  ( s,x':[[A]] rho de e.state. tc(r;A.ds;dec_lookup(A.da;kind(a));de)  covers_rel(A;r)  [[A]] rho de e.trans(s,a,x')  (rel_mng_2(r; rho; A.ds; dec_lookup(A.da;kind(a)); de; e; s; x'; value(a); tr)  [[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* A:ioa{i:l}(), rho:Decl, r:rel(), R:(Label Label  ), k:Label. ioa_mentions_trace(A)  trace_consistent_rel(rho;A.da;R;r)  trace_consistent_pred(rho;A.da;R;wp2_rel(A;k;r)) | [trace_consistent_wp2_rel] |
Thm* A:ioa{i:l}(), rho:Decl, r:rel(), R:(Label Label  ), k:Label. ioa_mentions_trace(A)  trace_consistent_rel(rho;A.da;R;r)  trace_consistent_pred(rho;A.da;R;wp_rel(A;k;r)) | [trace_consistent_wp_rel] |
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)  tc(r;ds;da;de)  subst_mentions_trace(as)  ( x:Label. (x rel_primed_vars(r))  ( t:SimpleType. mk_dec(x, t) ds  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))  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  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)  tc(r;ds1;da1;de)  ds1 = ds2  da1 = da2  ([[r]] rho ds1 da1 de e s a tr  [[r]] rho ds2 da2 de e s a tr) | [rel_mng_functionality] |
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)  tc(r;ds;da1;de)  closed_rel(r)  ([[r]] rho ds da1 de e s a1 tr  [[r]] rho ds da2 de e s a2 tr) | [closed_rel_mng_2] |
Thm* rho:Decl, ds,daa:Collection(dec()), da1:Collection(SimpleType), de:sig(), s:{[[ds]] rho}, e:{[[de]] rho}, tr:trace_env([[daa]] rho), r:rel(). closed_rel(r)  tc(r;ds;da1;de)  trace_consistent_rel(rho;daa;tr.proj;r)  [[r]] rho ds da1 de e s tr Prop | [rel_mng_wf_closed] |
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)  trace_consistent_rel(rho;daa;tr.proj;r2)  tc(r1;ds;da;de)  tc(r2;ds;da;de)  r1.name = r2.name  ||r1.args|| = ||r2.args||  ( i: . i < ||r1.args||  [[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)  ([[r1]] rho ds da de e s1 a tr  rel_mng_2(r2; rho; ds; da; de; e; s1; s2; a; tr)) | [rel_mng_2_iff] |
Thm* r:rel(), ds,da:Collection(dec()), de:sig(), rho:Decl, st1:Collection(SimpleType), e:{[[de]] rho}, s:{[[ds]] rho}, a:[[st1]] rho, tr:trace_env([[da]] rho). trace_consistent_rel(rho;da;tr.proj;r)  tc(r;ds;st1;de)  [[r]] rho ds st1 de e s a tr Prop | [rel_mng_wf] |
Thm* r:rel(), as:(Label Term) List, daa:Collection(dec()), rho:Decl, te:(Label Label  ). trace_consistent_rel(rho;daa;te;r)  subst_mentions_trace(as)  trace_consistent_rel(rho;daa;te;rel_subst2(as;r)) | [trace_consistent_rel_subst2] |
Thm* r:rel(), ds,da:Collection(dec()), de:sig(), rho:Decl, st1:Collection(SimpleType), e:{[[de]] rho}, s,s':{[[ds]] rho}, a:[[st1]] rho, tr:trace_env([[da]] rho). trace_consistent_rel(rho;da;tr.proj;r)  tc(r;ds;st1;de)  rel_mng_2(r; rho; ds; st1; de; e; s; s'; a; tr) Prop | [rel_mng_2_wf] |
Thm* r:rel(), i: , ds,da:Collection(dec()), st1:Collection(SimpleType), de:sig(), rho:Decl, s:{[[ds]] rho}, e:{1of([[de]] rho)}, a:[[st1]] rho, tr:trace_env([[da]] rho). trace_consistent_rel(rho;da;tr.proj;r)  tc(r;ds;st1;de)  i < ||r.args||  [[r.args[i]]] e s a tr [[rel_arg_typ(r.name;i;de)]] rho | [rel_arg_typing] |
Thm* r:rel(), as:(Label Term) List, daa:Collection(dec()), rho:Decl, te:(Label Label  ). trace_consistent_rel(rho;daa;te;r)  subst_mentions_trace(as)  trace_consistent_rel(rho;daa;te;rel_subst(as;r)) | [trace_consistent_rel_subst] |
Thm* r:rel(), i: , ds,da:Collection(dec()), st1:Collection(SimpleType), de:sig(), rho:Decl, s,s':{[[ds]] rho}, e:{1of([[de]] rho)}, a:[[st1]] rho, tr:trace_env([[da]] rho). trace_consistent_rel(rho;da;tr.proj;r)  tc(r;ds;st1;de)  i < ||r.args||  [[r.args[i]]] e s s' a tr [[rel_arg_typ(r.name;i;de)]] rho | [rel_arg_typing2] |
Def trace_consistent_pred(rho;da;R;p) == ( r p.trace_consistent_rel(rho;da;R;r)) | [trace_consistent_pred] |