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(), as:(Label Term) List, daa:Collection(dec()), rho:Decl, te:(Label![](FONT/dash.png) Label![](FONT/dash.png) ![](FONT/then_med.png) ). trace_consistent_rel(rho;daa;te;r) ![](FONT/eq.png) subst_mentions_trace(as) ![](FONT/eq.png) trace_consistent_rel(rho;daa;te;rel_subst2(as;r)) | [trace_consistent_rel_subst2] |
Thm* r:rel(), as:(Label Term) List, daa:Collection(dec()), rho:Decl, te:(Label![](FONT/dash.png) Label![](FONT/dash.png) ![](FONT/then_med.png) ). trace_consistent_rel(rho;daa;te;r) ![](FONT/eq.png) subst_mentions_trace(as) ![](FONT/eq.png) trace_consistent_rel(rho;daa;te;rel_subst(as;r)) | [trace_consistent_rel_subst] |