# Theory Synthesis

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

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

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

```