| Rank | Theorem | Name |
| 6 | | | 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] |
| cites |
| 1 | | | Thm* r:rel(), as:(LabelTerm) List, ds:Collection(dec()), da:Collection(SimpleType), de:sig(). tc(r;ds;da;de) (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)))) tc(rel_subst2(as;r);ds;da;de) | [rel_subst2_tc] |
| 3 | | | Thm* r:rel(), as:(LabelTerm) 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] |
| 5 | | | 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] |
| 0 | | | Thm* t:Term, e,a,s,tr:Top. [[t]] e s a tr ~ [[unprime(t)]] e s a tr | [term_mng_unprime] |
| 0 | | | Thm* sts:Collection(SimpleType), rho:Decl, v:[[sts]] rho, s:SimpleType. s sts v [[s]] rho | [sts_mng_subtype] |
| 1 | | | Thm* da:Collection(dec()), rho:Decl, tr:trace_env([[da]] rho), y1:Label. trace_consistent(rho;da;tr.proj;trace(y1)) tr.y1 [[lbl_pr( < Trace, y1 > )]] rho | [tproj_w_f2] |
| 1 | | | Thm* ds:Collection(dec()), rho:Decl, s:{[[ds]] rho}, x:Label, t:SimpleType. mk_dec(x, t) ds s.x [[t]] rho | [record_select_wf_decls_mng] |
| 3 | | | Thm* ds,da:Collection(dec()), st1:Collection(SimpleType), de:sig(), rho:Decl, t:Term, s:{[[ds]] rho}, e:{1of([[de]] rho)}, a:SimpleType, v:[[st1]] rho, tr:trace_env([[da]] rho). trace_consistent(rho;da;tr.proj;t) a term_types(ds;st1;de;t) [[t]] e s v tr [[a]] rho | [term_typing] |
| 1 | | | Thm* as:(LabelT) List, d:T, x:Label. (apply_alist(as;x;d) 2of(unzip(as)))  apply_alist(as;x;d) = d | [apply_alist_property] |
| 0 | | | Thm* g:Label, t:Term. term_mentions_guard(g;t) mentions_trace(t) | [mentions_guard_mentions_trace] |