header {* Auxiliary lemmas used in program extraction examples *}
theory Util
imports Main
begin
text {*
Decidability of equality on natural numbers.
*}
lemma nat_eq_dec: "!!n::nat. m = n ∨ m ≠ n"
apply (induct m)
apply (case_tac n)
apply (case_tac [3] n)
apply (simp only: nat.simps, iprover?)+
done
text {*
Well-founded induction on natural numbers, derived using the standard
structural induction rule.
*}
lemma nat_wf_ind:
assumes R: "!!x::nat. (!!y. y < x ==> P y) ==> P x"
shows "P z"
proof (rule R)
show "!!y. y < z ==> P y"
proof (induct z)
case 0
thus ?case by simp
next
case (Suc n y)
from nat_eq_dec show ?case
proof
assume ny: "n = y"
have "P n"
by (rule R) (rule Suc)
with ny show ?case by simp
next
assume "n ≠ y"
with Suc have "y < n" by simp
thus ?case by (rule Suc)
qed
qed
qed
text {*
Bounded search for a natural number satisfying a decidable predicate.
*}
lemma search:
assumes dec: "!!x::nat. P x ∨ ¬ P x"
shows "(∃x<y. P x) ∨ ¬ (∃x<y. P x)"
proof (induct y)
case 0 show ?case by simp
next
case (Suc z)
thus ?case
proof
assume "∃x<z. P x"
then obtain x where le: "x < z" and P: "P x" by iprover
from le have "x < Suc z" by simp
with P show ?case by iprover
next
assume nex: "¬ (∃x<z. P x)"
from dec show ?case
proof
assume P: "P z"
have "z < Suc z" by simp
with P show ?thesis by iprover
next
assume nP: "¬ P z"
have "¬ (∃x<Suc z. P x)"
proof
assume "∃x<Suc z. P x"
then obtain x where le: "x < Suc z" and P: "P x" by iprover
have "x < z"
proof (cases "x = z")
case True
with nP and P show ?thesis by simp
next
case False
with le show ?thesis by simp
qed
with P have "∃x<z. P x" by iprover
with nex show False ..
qed
thus ?case by iprover
qed
qed
qed
end