Theory Util

theory Util
imports Main
```(*  Title:      HOL/Proofs/Extraction/Util.thy
Author:     Stefan Berghofer, TU Muenchen
*)

section ‹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
then show ?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
then show ?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)
then show ?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
then show ?case by iprover
qed
qed
qed

end
```