# Theory Intf_Hash

```section ‹\isaheader{Hashable Interface}›
theory Intf_Hash
imports
Main
"../../Lib/HashCode"
"../../Lib/Code_Target_ICF"
Automatic_Refinement.Automatic_Refinement
begin

type_synonym 'a eq = "'a ⇒ 'a ⇒ bool"
type_synonym 'k bhc = "nat ⇒ 'k ⇒ nat"

subsection ‹Abstract and concrete hash functions›

definition is_bounded_hashcode :: "('c×'a) set ⇒ 'c eq ⇒ 'c bhc ⇒ bool"
where "is_bounded_hashcode R eq bhc ≡
((eq,(=)) ∈ R → R → bool_rel) ∧
(∀n. ∀ x ∈ Domain R. ∀ y ∈ Domain R. eq x y ⟶ bhc n x = bhc n y) ∧
(∀n x. 1 < n ⟶ bhc n x < n)"
definition abstract_bounded_hashcode :: "('c×'a) set ⇒ 'c bhc ⇒ 'a bhc"
where "abstract_bounded_hashcode Rk bhc n x' ≡
if x' ∈ Range Rk
then THE c. ∃x. (x,x') ∈ Rk ∧ bhc n x = c
else 0"

lemma is_bounded_hashcodeI[intro]:
"((eq,(=)) ∈ R → R → bool_rel) ⟹
(⋀x y n. x ∈ Domain R ⟹ y ∈ Domain R ⟹ eq x y ⟹ bhc n x = bhc n y) ⟹
(⋀x n. 1 < n ⟹ bhc n x < n) ⟹ is_bounded_hashcode R eq bhc"
unfolding is_bounded_hashcode_def by force

lemma is_bounded_hashcodeD[dest]:
assumes "is_bounded_hashcode R eq bhc"
shows "(eq,(=)) ∈ R → R → bool_rel" and
"⋀n x y. x ∈ Domain R ⟹ y ∈ Domain R ⟹ eq x y ⟹ bhc n x = bhc n y" and
"⋀n x. 1 < n ⟹ bhc n x < n"
using assms unfolding is_bounded_hashcode_def by simp_all

lemma bounded_hashcode_welldefined:
assumes BHC: "is_bounded_hashcode Rk eq bhc" and
R1: "(x1,x') ∈ Rk" and R2: "(x2,x') ∈ Rk"
shows "bhc n x1 = bhc n x2"
proof-
from is_bounded_hashcodeD[OF BHC] have "(eq,(=)) ∈ Rk → Rk → bool_rel" by simp
with R1 R2 have "eq x1 x2" by (force dest: fun_relD)
thus ?thesis using R1 R2 BHC by blast
qed

lemma abstract_bhc_correct[intro]:
assumes "is_bounded_hashcode Rk eq bhc"
shows "(bhc, abstract_bounded_hashcode Rk bhc) ∈
nat_rel → Rk → nat_rel" (is "(bhc, ?bhc') ∈ _")
proof (intro fun_relI)
fix n n' x x'
assume A: "(n,n') ∈ nat_rel" and B: "(x,x') ∈ Rk"
hence C: "n = n'" and D: "x' ∈ Range Rk" by auto
have "?bhc' n' x' = bhc n x"
unfolding abstract_bounded_hashcode_def
apply (simp add: C D, rule)
apply (intro exI conjI, fact B, rule refl)
apply (elim exE conjE, hypsubst,
erule bounded_hashcode_welldefined[OF assms _ B])
done
thus "(bhc n x, ?bhc' n' x') ∈ nat_rel" by simp
qed

lemma abstract_bhc_is_bhc[intro]:
fixes Rk :: "('c×'a) set"
assumes bhc: "is_bounded_hashcode Rk eq bhc"
shows "is_bounded_hashcode Id (=) (abstract_bounded_hashcode Rk bhc)"
(is "is_bounded_hashcode _ (=) ?bhc'")
proof
fix x'::'a and y'::'a and n'::nat assume "x' = y'"
thus "?bhc' n' x' = ?bhc' n' y'" by simp
next
fix x'::'a and n'::nat assume "1 < n'"
from abstract_bhc_correct[OF bhc] show "?bhc' n' x' < n'"
proof (cases "x' ∈ Range Rk")
case False
with ‹1 < n'› show ?thesis
unfolding abstract_bounded_hashcode_def by simp
next
case True
then obtain x where "(x,x') ∈ Rk" ..
have "(n',n') ∈ nat_rel" ..
from abstract_bhc_correct[OF assms] have "?bhc' n' x' = bhc n' x"
apply -
apply (drule fun_relD[OF _ ‹(n',n') ∈ nat_rel›],
drule fun_relD[OF _ ‹(x,x') ∈ Rk›], simp)
done
also from ‹1 < n'› and bhc have "... < n'" by blast
finally show "?bhc' n' x' < n'" .
qed
qed simp

(*lemma hashable_bhc_is_bhc[autoref_ga_rules]:
"⟦STRUCT_EQ_tag eq (=;) REL_IS_ID R⟧ ⟹ is_bounded_hashcode R eq bounded_hashcode"
unfolding is_bounded_hashcode_def

(* TODO: This is a hack that causes the relation to be instantiated to Id, if it is not
yet fixed! *)
lemma hashable_bhc_is_bhc[autoref_ga_rules]:
"⟦STRUCT_EQ_tag eq (=); REL_FORCE_ID R⟧ ⟹ is_bounded_hashcode R eq bounded_hashcode_nat"
unfolding is_bounded_hashcode_def