Theory Synthesis

(*  Title:      CTT/ex/Synthesis.thy
    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
    Copyright   1991  University of Cambridge
*)

section ‹Synthesis examples, using a crude form of narrowing›

theory Synthesis
  imports "../CTT"
begin

text ‹discovery of predecessor function›
schematic_goal "?a : ∑pred:?A . Eq(N, pred`0, 0) × (∏n:N. Eq(N, pred ` succ(n), n))"
  apply intr
    apply eqintr
    apply (rule_tac [3] reduction_rls)
      apply (rule_tac [5] comp_rls)
        apply rew
  done

text ‹the function fst as an element of a function type›
schematic_goal [folded basic_defs]:
  "A type ⟹ ?a: ∑f:?B . ∏i:A. ∏j:A. Eq(A, f ` <i,j>, i)"
  apply intr
   apply eqintr
   apply (rule_tac [2] reduction_rls)
     apply (rule_tac [4] comp_rls)
       apply typechk
  txt "now put in A everywhere"
   apply assumption+
  done

text ‹An interesting use of the eliminator, when›
  (*The early implementation of unification caused non-rigid path in occur check
  See following example.*)
schematic_goal "?a : ∏i:N. Eq(?A, ?b(inl(i)), <0    ,   i>)
                   × Eq(?A, ?b(inr(i)), <succ(0), i>)"
  apply intr
   apply eqintr
   apply (rule comp_rls)
     apply rew
  done

(*Here we allow the type to depend on i.
 This prevents the cycle in the first unification (no longer needed).
 Requires flex-flex to preserve the dependence.
 Simpler still: make ?A into a constant type N × N.*)
schematic_goal "?a : ∏i:N. Eq(?A(i), ?b(inl(i)), <0   ,   i>)
                  ×  Eq(?A(i), ?b(inr(i)), <succ(0),i>)"
  oops

  text ‹A tricky combination of when and split›
    (*Now handled easily, but caused great problems once*)
schematic_goal [folded basic_defs]:
  "?a : ∏i:N. ∏j:N. Eq(?A, ?b(inl(<i,j>)), i)
                           ×  Eq(?A, ?b(inr(<i,j>)), j)"
  apply intr
   apply eqintr
   apply (rule PlusC_inl [THEN trans_elem])
      apply (rule_tac [4] comp_rls)
        apply (rule_tac [7] reduction_rls)
           apply (rule_tac [10] comp_rls)
             apply typechk
  done

(*similar but allows the type to depend on i and j*)
schematic_goal "?a : ∏i:N. ∏j:N. Eq(?A(i,j), ?b(inl(<i,j>)), i)
                          ×   Eq(?A(i,j), ?b(inr(<i,j>)), j)"
  oops

(*similar but specifying the type N simplifies the unification problems*)
schematic_goal "?a : ∏i:N. ∏j:N. Eq(N, ?b(inl(<i,j>)), i)
                          ×   Eq(N, ?b(inr(<i,j>)), j)"
  oops


  text ‹Deriving the addition operator›
schematic_goal [folded arith_defs]:
  "?c : ∏n:N. Eq(N, ?f(0,n), n)
                  ×  (∏m:N. Eq(N, ?f(succ(m), n), succ(?f(m,n))))"
  apply intr
   apply eqintr
   apply (rule comp_rls)
    apply rew
  done

text ‹The addition function -- using explicit lambdas›
schematic_goal [folded arith_defs]:
  "?c : ∑plus : ?A .
         ∏x:N. Eq(N, plus`0`x, x)
                ×  (∏y:N. Eq(N, plus`succ(y)`x, succ(plus`y`x)))"
  apply intr
    apply eqintr
    apply (tactic "resolve_tac \<^context> [TSimp.split_eqn] 3")
     apply (tactic "SELECT_GOAL (rew_tac \<^context> []) 4")
         apply (tactic "resolve_tac \<^context> [TSimp.split_eqn] 3")
          apply (tactic "SELECT_GOAL (rew_tac \<^context> []) 4")
              apply (rule_tac [3] p = "y" in NC_succ)
    (**  by (resolve_tac @{context} comp_rls 3);  caused excessive branching  **)
                apply rew
  done

end