header "Synthesis examples, using a crude form of narrowing"
theory Synthesis
imports Arith
begin
text "discovery of predecessor function"
lemma "?a : SUM pred:?A . Eq(N, pred`0, 0)
* (PROD n:N. Eq(N, pred ` succ(n), n))"
apply (tactic "intr_tac []")
apply (tactic eqintr_tac)
apply (rule_tac [3] reduction_rls)
apply (rule_tac [5] comp_rls)
apply (tactic "rew_tac []")
done
text "the function fst as an element of a function type"
lemma [folded basic_defs]:
"A type ==> ?a: SUM f:?B . PROD i:A. PROD j:A. Eq(A, f ` <i,j>, i)"
apply (tactic "intr_tac []")
apply (tactic eqintr_tac)
apply (rule_tac [2] reduction_rls)
apply (rule_tac [4] comp_rls)
apply (tactic "typechk_tac []")
txt "now put in A everywhere"
apply assumption+
done
text "An interesting use of the eliminator, when"
lemma "?a : PROD i:N. Eq(?A, ?b(inl(i)), <0 , i>)
* Eq(?A, ?b(inr(i)), <succ(0), i>)"
apply (tactic "intr_tac []")
apply (tactic eqintr_tac)
apply (rule comp_rls)
apply (tactic "rew_tac []")
done
lemma "?a : PROD 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"
lemma [folded basic_defs]:
"?a : PROD i:N. PROD j:N. Eq(?A, ?b(inl(<i,j>)), i)
* Eq(?A, ?b(inr(<i,j>)), j)"
apply (tactic "intr_tac []")
apply (tactic eqintr_tac)
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 (tactic "typechk_tac []")
done
lemma "?a : PROD i:N. PROD j:N. Eq(?A(i,j), ?b(inl(<i,j>)), i)
* Eq(?A(i,j), ?b(inr(<i,j>)), j)"
oops
lemma "?a : PROD i:N. PROD j:N. Eq(N, ?b(inl(<i,j>)), i)
* Eq(N, ?b(inr(<i,j>)), j)"
oops
text "Deriving the addition operator"
lemma [folded arith_defs]:
"?c : PROD n:N. Eq(N, ?f(0,n), n)
* (PROD m:N. Eq(N, ?f(succ(m), n), succ(?f(m,n))))"
apply (tactic "intr_tac []")
apply (tactic eqintr_tac)
apply (rule comp_rls)
apply (tactic "rew_tac []")
done
text "The addition function -- using explicit lambdas"
lemma [folded arith_defs]:
"?c : SUM plus : ?A .
PROD x:N. Eq(N, plus`0`x, x)
* (PROD y:N. Eq(N, plus`succ(y)`x, succ(plus`y`x)))"
apply (tactic "intr_tac []")
apply (tactic eqintr_tac)
apply (tactic "resolve_tac [TSimp.split_eqn] 3")
apply (tactic "SELECT_GOAL (rew_tac []) 4")
apply (tactic "resolve_tac [TSimp.split_eqn] 3")
apply (tactic "SELECT_GOAL (rew_tac []) 4")
apply (rule_tac [3] p = "y" in NC_succ)
apply (tactic "rew_tac []")
done
end