\require preamble

\* Intuitionistic Predicate Logic with Equality *\

\open elements \* [1] 2.1. 2.2. 3.1 *\

   \decl "logical false" False: *Prop

   \decl "logical conjunction" And: *Prop => *Prop -> *Prop

   \decl "logical disjunction" Or: *Prop => *Prop -> *Prop

\* implication and non-dependent abstraction are isomorphic *\
   \def "logical implication" 
      Imp = [p:*Prop, q:*Prop] p -> q : *Prop => *Prop -> *Prop

\* comprehension and dependent abstraction are isomorphic *\
   \def "unrestricted logical comprehension"
      All = [q:*Obj->*Prop] [x:*Obj] q(x) : (*Obj -> *Prop) -> *Prop

   \decl "unrestricted logical existence" Ex: (*Obj -> *Prop) -> *Prop

   \decl "syntactical identity" Id: *Obj => *Obj -> *Prop

\close

\open abbreviations \* [1] 2.3. *\

   \def "logical negation" 
      Not = [p:*Prop] p -> False : *Prop -> *Prop

   \def "logical equivalence"
      Iff = [p:*Prop, q:*Prop] And(p -> q, q -> p) : *Prop => *Prop -> *Prop

   \def "unrestricted strict logical existence"
      EX = [p:*Obj->*Prop] Ex([x:*Obj] And(p(x), [y:*Obj] p(y) -> Id(x, y)))
         : (*Obj -> *Prop) -> *Prop

   \def "negated syntactical identity"
      NId = [x:*Obj, y:*Obj] Not(Id(x, y)) : *Obj => *Obj -> *Prop

\close