theory Class
imports "../nominal"
begin
section {* Term-Calculus from Urban's PhD *}
atom_decl name coname
nominal_datatype trm =
Ax "name" "coname"
| Cut "\<guillemotleft>coname\<guillemotright>trm" "\<guillemotleft>name\<guillemotright>trm" ("Cut <_>._ (_)._" [100,100,100,100] 100)
| NotR "\<guillemotleft>name\<guillemotright>trm" "coname" ("NotR (_)._ _" [100,100,100] 100)
| NotL "\<guillemotleft>coname\<guillemotright>trm" "name" ("NotL <_>._ _" [100,100,100] 100)
| AndR "\<guillemotleft>coname\<guillemotright>trm" "\<guillemotleft>coname\<guillemotright>trm" "coname" ("AndR <_>._ <_>._ _" [100,100,100,100,100] 100)
| AndL1 "\<guillemotleft>name\<guillemotright>trm" "name" ("AndL1 (_)._ _" [100,100,100] 100)
| AndL2 "\<guillemotleft>name\<guillemotright>trm" "name" ("AndL2 (_)._ _" [100,100,100] 100)
| OrR1 "\<guillemotleft>coname\<guillemotright>trm" "coname" ("OrR1 <_>._ _" [100,100,100] 100)
| OrR2 "\<guillemotleft>coname\<guillemotright>trm" "coname" ("OrR2 <_>._ _" [100,100,100] 100)
| OrL "\<guillemotleft>name\<guillemotright>trm" "\<guillemotleft>name\<guillemotright>trm" "name" ("OrL (_)._ (_)._ _" [100,100,100,100,100] 100)
| ImpR "\<guillemotleft>name\<guillemotright>(\<guillemotleft>coname\<guillemotright>trm)" "coname" ("ImpR (_).<_>._ _" [100,100,100,100] 100)
| ImpL "\<guillemotleft>coname\<guillemotright>trm" "\<guillemotleft>name\<guillemotright>trm" "name" ("ImpL <_>._ (_)._ _" [100,100,100,100,100] 100)
text {* Induction principles *}
thm trm.induct_weak --"weak"
thm trm.induct --"strong"
thm trm.induct' --"strong with explicit context (rarely needed)"
text {* named terms *}
nominal_datatype ntrm = N "\<guillemotleft>name\<guillemotright>trm" ("N (_)._" [100,100] 100)
text {* conamed terms *}
nominal_datatype ctrm = C "\<guillemotleft>coname\<guillemotright>trm" ("C <_>._" [100,100] 100)
text {* We should now define the two forms of substitition :o( *}
consts
substn :: "trm \<Rightarrow> name \<Rightarrow> ctrm \<Rightarrow> trm" ("_[_::=_]" [100,100,100] 100)
substc :: "trm \<Rightarrow> coname \<Rightarrow> ntrm \<Rightarrow> trm" ("_[_::=_]" [100,100,100] 100)
text {* does not work yet *}