# Theory Uprod

```(* Title: HOL/Library/Uprod.thy
Author: Andreas Lochbihler, ETH Zurich *)

section ‹Unordered pairs›

theory Uprod imports Main begin

typedef ('a, 'b) commute = "{f :: 'a ⇒ 'a ⇒ 'b. ∀x y. f x y = f y x}"
morphisms apply_commute Abs_commute
by auto

setup_lifting type_definition_commute

lemma apply_commute_commute: "apply_commute f x y = apply_commute f y x"
by(transfer) simp

context includes lifting_syntax begin

lift_definition rel_commute :: "('a ⇒ 'b ⇒ bool) ⇒ ('c ⇒ 'd ⇒ bool) ⇒ ('a, 'c) commute ⇒ ('b, 'd) commute ⇒ bool"
is "λA B. A ===> A ===> B" .

end

definition eq_upair :: "('a × 'a) ⇒ ('a × 'a) ⇒ bool"
where "eq_upair = (λ(a, b) (c, d). a = c ∧ b = d ∨ a = d ∧ b = c)"

lemma eq_upair_simps [simp]:
"eq_upair (a, b) (c, d) ⟷ a = c ∧ b = d ∨ a = d ∧ b = c"

lemma equivp_eq_upair: "equivp eq_upair"

quotient_type 'a uprod = "'a × 'a" / eq_upair by(rule equivp_eq_upair)

lift_definition Upair :: "'a ⇒ 'a ⇒ 'a uprod" is Pair parametric Pair_transfer[of "A" "A" for A] .

lemma uprod_exhaust [case_names Upair, cases type: uprod]:
obtains a b where "x = Upair a b"
by transfer fastforce

lemma Upair_inject [simp]: "Upair a b = Upair c d ⟷ a = c ∧ b = d ∨ a = d ∧ b = c"
by transfer auto

code_datatype Upair

lift_definition case_uprod :: "('a, 'b) commute ⇒ 'a uprod ⇒ 'b" is case_prod
parametric case_prod_transfer[of A A for A] by auto

lemma case_uprod_simps [simp, code]: "case_uprod f (Upair x y) = apply_commute f x y"
by transfer auto

lemma uprod_split: "P (case_uprod f x) ⟷ (∀a b. x = Upair a b ⟶ P (apply_commute f a b))"
by transfer auto

lemma uprod_split_asm: "P (case_uprod f x) ⟷ ¬ (∃a b. x = Upair a b ∧ ¬ P (apply_commute f a b))"
by transfer auto

lift_definition not_equal :: "('a, bool) commute" is "(≠)" by auto

lemma apply_not_equal [simp]: "apply_commute not_equal x y ⟷ x ≠ y"
by transfer simp

definition proper_uprod :: "'a uprod ⇒ bool"
where "proper_uprod = case_uprod not_equal"

lemma proper_uprod_simps [simp, code]: "proper_uprod (Upair x y) ⟷ x ≠ y"

context includes lifting_syntax begin

private lemma set_uprod_parametric':
"(rel_prod A A ===> rel_set A) (λ(a, b). {a, b}) (λ(a, b). {a, b})"
by transfer_prover

lift_definition set_uprod :: "'a uprod ⇒ 'a set" is "λ(a, b). {a, b}"
parametric set_uprod_parametric' by auto

lemma set_uprod_simps [simp, code]: "set_uprod (Upair x y) = {x, y}"
by transfer simp

lemma finite_set_uprod [simp]: "finite (set_uprod x)"
by(cases x) simp

private lemma map_uprod_parametric':
"((A ===> B) ===> rel_prod A A ===> rel_prod B B) (λf. map_prod f f) (λf. map_prod f f)"
by transfer_prover

lift_definition map_uprod :: "('a ⇒ 'b) ⇒ 'a uprod ⇒ 'b uprod" is "λf. map_prod f f"
parametric map_uprod_parametric' by auto

lemma map_uprod_simps [simp, code]: "map_uprod f (Upair x y) = Upair (f x) (f y)"
by transfer simp

private lemma rel_uprod_transfer':
"((A ===> B ===> (=)) ===> rel_prod A A ===> rel_prod B B ===> (=))
(λR (a, b) (c, d). R a c ∧ R b d ∨ R a d ∧ R b c) (λR (a, b) (c, d). R a c ∧ R b d ∨ R a d ∧ R b c)"
by transfer_prover

lift_definition rel_uprod :: "('a ⇒ 'b ⇒ bool) ⇒ 'a uprod ⇒ 'b uprod ⇒ bool"
is "λR (a, b) (c, d). R a c ∧ R b d ∨ R a d ∧ R b c" parametric rel_uprod_transfer'
by auto

lemma rel_uprod_simps [simp, code]:
"rel_uprod R (Upair a b) (Upair c d) ⟷ R a c ∧ R b d ∨ R a d ∧ R b c"
by transfer auto

lemma Upair_parametric [transfer_rule]: "(A ===> A ===> rel_uprod A) Upair Upair"
unfolding rel_fun_def by transfer auto

lemma case_uprod_parametric [transfer_rule]:
"(rel_commute A B ===> rel_uprod A ===> B) case_uprod case_uprod"
unfolding rel_fun_def by transfer(force dest: rel_funD)

end

bnf uprod: "'a uprod"
map: map_uprod
sets: set_uprod
bd: natLeq
rel: rel_uprod
proof -
show "map_uprod id = id" unfolding fun_eq_iff by transfer auto
show "map_uprod (g ∘ f) = map_uprod g ∘ map_uprod f" for f :: "'a ⇒ 'b" and g :: "'b ⇒ 'c"
unfolding fun_eq_iff by transfer auto
show "map_uprod f x = map_uprod g x" if "⋀z. z ∈ set_uprod x ⟹ f z = g z"
for f :: "'a ⇒ 'b" and g x using that by transfer auto
show "set_uprod ∘ map_uprod f = (`) f ∘ set_uprod" for f :: "'a ⇒ 'b" by transfer auto
show "card_order natLeq" by(rule natLeq_card_order)
show "BNF_Cardinal_Arithmetic.cinfinite natLeq" by(rule natLeq_cinfinite)
show "regularCard natLeq" by(rule regularCard_natLeq)
show "ordLess2 (card_of (set_uprod x)) natLeq" for x :: "'a uprod"
by (auto simp flip: finite_iff_ordLess_natLeq)
show "rel_uprod R OO rel_uprod S ≤ rel_uprod (R OO S)"
for R :: "'a ⇒ 'b ⇒ bool" and S :: "'b ⇒ 'c ⇒ bool" by(rule predicate2I)(transfer; auto)
show "rel_uprod R = (λx y. ∃z. set_uprod z ⊆ {(x, y). R x y} ∧ map_uprod fst z = x ∧ map_uprod snd z = y)"
for R :: "'a ⇒ 'b ⇒ bool" by transfer(auto simp add: fun_eq_iff)
qed

lemma pred_uprod_code [simp, code]: "pred_uprod P (Upair x y) ⟷ P x ∧ P y"

instantiation uprod :: (equal) equal begin

definition equal_uprod :: "'a uprod ⇒ 'a uprod ⇒ bool"
where "equal_uprod = (=)"

lemma equal_uprod_code [code]:
"HOL.equal (Upair x y) (Upair z u) ⟷ x = z ∧ y = u ∨ x = u ∧ y = z"
unfolding equal_uprod_def by simp

end

quickcheck_generator uprod constructors: Upair

lemma UNIV_uprod: "UNIV = (λx. Upair x x) ` UNIV ∪ (λ(x, y). Upair x y) ` Sigma UNIV (λx. UNIV - {x})"
apply(rule set_eqI)
subgoal for x by(cases x) auto
done

context begin
private lift_definition upair_inv :: "'a uprod ⇒ 'a"
is "λ(x, y). if x = y then x else undefined" by auto

lemma finite_UNIV_prod [simp]:
"finite (UNIV :: 'a uprod set) ⟷ finite (UNIV :: 'a set)" (is "?lhs = ?rhs")
proof
assume ?lhs
hence "finite (range (λx :: 'a. Upair x x))" by(rule finite_subset[rotated]) simp
hence "finite (upair_inv ` range (λx :: 'a. Upair x x))" by(rule finite_imageI)
also have "upair_inv (Upair x x) = x" for x :: 'a by transfer simp
then have "upair_inv ` range (λx :: 'a. Upair x x) = UNIV" by(auto simp add: image_image)
finally show ?rhs .

end

lemma card_UNIV_uprod:
"card (UNIV :: 'a uprod set) = card (UNIV :: 'a set) * (card (UNIV :: 'a set) + 1) div 2"
(is "?UPROD = ?A * _ div _")
proof(cases "finite (UNIV :: 'a set)")
case True
from True obtain f :: "nat ⇒ 'a" where bij: "bij_betw f {0..<?A} UNIV"
by (blast dest: ex_bij_betw_nat_finite)
hence [simp]: "f ` {0..<?A} = UNIV" by(rule bij_betw_imp_surj_on)
have "UNIV = (λ(x, y). Upair (f x) (f y)) ` (SIGMA x:{0..<?A}. {..x})"
apply(rule set_eqI)
subgoal for x
apply(cases x)
apply(clarsimp)
subgoal for a b
apply(cases "inv_into {0..<?A} f a ≤ inv_into {0..<?A} f b")
subgoal by(rule rev_image_eqI[where x="(inv_into {0..<?A} f _, inv_into {0..<?A} f _)"])
(auto simp add: inv_into_into[where A="{0..<?A}" and f=f, simplified] intro: f_inv_into_f[where f=f, symmetric])
subgoal
apply(simp only: not_le)
apply(drule less_imp_le)
apply(rule rev_image_eqI[where x="(inv_into {0..<?A} f _, inv_into {0..<?A} f _)"])
apply(auto simp add: inv_into_into[where A="{0..<?A}" and f=f, simplified] intro: f_inv_into_f[where f=f, symmetric])
done
done
done
done
hence "?UPROD = card …" by simp
also have "… = card (SIGMA x:{0..<?A}. {..x})"
apply(rule card_image)
using bij[THEN bij_betw_imp_inj_on]
by(simp add: inj_on_def Ball_def)(metis leD le_eq_less_or_eq le_less_trans)
also have "… = sum Suc {0..<?A}"
by (subst card_SigmaI) simp_all
also have "… = sum of_nat {Suc 0..?A}"
using sum.atLeastLessThan_reindex [symmetric, of Suc 0 ?A id]
by (simp del: sum.op_ivl_Suc add: atLeastLessThanSuc_atLeastAtMost)
also have "… = ?A * (?A + 1) div 2"
using gauss_sum_from_Suc_0 [of ?A, where ?'a = nat] by simp
finally show ?thesis .
qed simp

end
```