(* 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)"
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>)
\<times> 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 \<times> N.*)
schematic_goal "?a : \i:N. Eq(?A(i), ?b(inl(i)), <0 , i>)
\<times> 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)
\<times> 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)
\<times> 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)
\<times> 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)
\<times> (\<Prod>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 .
\<Prod>x:N. Eq(N, plus`0`x, x)
\<times> (\<Prod>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
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