{-# OPTIONS --allow-unsolved-metas #-} open import Level open import Ordinals module OPair {n : Level } (O : Ordinals {n}) where open import zf open import logic import OD open import Relation.Nullary open import Relation.Binary open import Data.Empty open import Relation.Binary open import Relation.Binary.Core open import Relation.Binary.PropositionalEquality open import Data.Nat renaming ( zero to Zero ; suc to Suc ; ℕ to Nat ; _⊔_ to _n⊔_ ) open inOrdinal O open OD O open OD.OD open OD.HOD open ODAxiom odAxiom open _∧_ open _∨_ open Bool open _==_ <_,_> : (x y : HOD) → HOD < x , y > = (x , x ) , (x , y ) exg-pair : { x y : HOD } → (x , y ) =h= ( y , x ) exg-pair {x} {y} = record { eq→ = left ; eq← = right } where left : {z : Ordinal} → odef (x , y) z → odef (y , x) z left (case1 t) = case2 t left (case2 t) = case1 t right : {z : Ordinal} → odef (y , x) z → odef (x , y) z right (case1 t) = case2 t right (case2 t) = case1 t ord≡→≡ : { x y : HOD } → od→ord x ≡ od→ord y → x ≡ y ord≡→≡ eq = subst₂ (λ j k → j ≡ k ) oiso oiso ( cong ( λ k → ord→od k ) eq ) od≡→≡ : { x y : Ordinal } → ord→od x ≡ ord→od y → x ≡ y od≡→≡ eq = subst₂ (λ j k → j ≡ k ) diso diso ( cong ( λ k → od→ord k ) eq ) eq-prod : { x x' y y' : HOD } → x ≡ x' → y ≡ y' → < x , y > ≡ < x' , y' > eq-prod refl refl = refl prod-eq : { x x' y y' : HOD } → < x , y > =h= < x' , y' > → (x ≡ x' ) ∧ ( y ≡ y' ) prod-eq {x} {x'} {y} {y'} eq = record { proj1 = lemmax ; proj2 = lemmay } where lemma0 : {x y z : HOD } → ( x , x ) =h= ( z , y ) → x ≡ y lemma0 {x} {y} eq with trio< (od→ord x) (od→ord y) lemma0 {x} {y} eq | tri< a ¬b ¬c with eq← eq {od→ord y} (case2 refl) lemma0 {x} {y} eq | tri< a ¬b ¬c | case1 s = ⊥-elim ( o<¬≡ (sym s) a ) lemma0 {x} {y} eq | tri< a ¬b ¬c | case2 s = ⊥-elim ( o<¬≡ (sym s) a ) lemma0 {x} {y} eq | tri≈ ¬a b ¬c = ord≡→≡ b lemma0 {x} {y} eq | tri> ¬a ¬b c with eq← eq {od→ord y} (case2 refl) lemma0 {x} {y} eq | tri> ¬a ¬b c | case1 s = ⊥-elim ( o<¬≡ s c ) lemma0 {x} {y} eq | tri> ¬a ¬b c | case2 s = ⊥-elim ( o<¬≡ s c ) lemma2 : {x y z : HOD } → ( x , x ) =h= ( z , y ) → z ≡ y lemma2 {x} {y} {z} eq = trans (sym (lemma0 lemma3 )) ( lemma0 eq ) where lemma3 : ( x , x ) =h= ( y , z ) lemma3 = ==-trans eq exg-pair lemma1 : {x y : HOD } → ( x , x ) =h= ( y , y ) → x ≡ y lemma1 {x} {y} eq with eq← eq {od→ord y} (case2 refl) lemma1 {x} {y} eq | case1 s = ord≡→≡ (sym s) lemma1 {x} {y} eq | case2 s = ord≡→≡ (sym s) lemma4 : {x y z : HOD } → ( x , y ) =h= ( x , z ) → y ≡ z lemma4 {x} {y} {z} eq with eq← eq {od→ord z} (case2 refl) lemma4 {x} {y} {z} eq | case1 s with ord≡→≡ s -- x ≡ z ... | refl with lemma2 (==-sym eq ) ... | refl = refl lemma4 {x} {y} {z} eq | case2 s = ord≡→≡ (sym s) -- y ≡ z lemmax : x ≡ x' lemmax with eq→ eq {od→ord (x , x)} (case1 refl) lemmax | case1 s = lemma1 (ord→== s ) -- (x,x)≡(x',x') lemmax | case2 s with lemma2 (ord→== s ) -- (x,x)≡(x',y') with x'≡y' ... | refl = lemma1 (ord→== s ) lemmay : y ≡ y' lemmay with lemmax ... | refl with lemma4 eq -- with (x,y)≡(x,y') ... | eq1 = lemma4 (ord→== (cong (λ k → od→ord k ) eq1 )) -- -- unlike ordered pair, ZFProduct is not a HOD data ord-pair : (p : Ordinal) → Set n where pair : (x y : Ordinal ) → ord-pair ( od→ord ( < ord→od x , ord→od y > ) ) ZFProduct : OD ZFProduct = record { def = λ x → ord-pair x } -- open import Relation.Binary.HeterogeneousEquality as HE using (_≅_ ) -- eq-pair : { x x' y y' : Ordinal } → x ≡ x' → y ≡ y' → pair x y ≅ pair x' y' -- eq-pair refl refl = HE.refl pi1 : { p : Ordinal } → ord-pair p → Ordinal pi1 ( pair x y) = x π1 : { p : HOD } → def ZFProduct (od→ord p) → HOD π1 lt = ord→od (pi1 lt ) pi2 : { p : Ordinal } → ord-pair p → Ordinal pi2 ( pair x y ) = y π2 : { p : HOD } → def ZFProduct (od→ord p) → HOD π2 lt = ord→od (pi2 lt ) op-cons : { ox oy : Ordinal } → def ZFProduct (od→ord ( < ord→od ox , ord→od oy > )) op-cons {ox} {oy} = pair ox oy def-subst : {Z : OD } {X : Ordinal }{z : OD } {x : Ordinal }→ def Z X → Z ≡ z → X ≡ x → def z x def-subst df refl refl = df p-cons : ( x y : HOD ) → def ZFProduct (od→ord ( < x , y >)) p-cons x y = def-subst {_} {_} {ZFProduct} {od→ord (< x , y >)} (pair (od→ord x) ( od→ord y )) refl ( let open ≡-Reasoning in begin od→ord < ord→od (od→ord x) , ord→od (od→ord y) > ≡⟨ cong₂ (λ j k → od→ord < j , k >) oiso oiso ⟩ od→ord < x , y > ∎ ) op-iso : { op : Ordinal } → (q : ord-pair op ) → od→ord < ord→od (pi1 q) , ord→od (pi2 q) > ≡ op op-iso (pair ox oy) = refl p-iso : { x : HOD } → (p : def ZFProduct (od→ord x) ) → < π1 p , π2 p > ≡ x p-iso {x} p = ord≡→≡ (op-iso p) p-pi1 : { x y : HOD } → (p : def ZFProduct (od→ord < x , y >) ) → π1 p ≡ x p-pi1 {x} {y} p = proj1 ( prod-eq ( ord→== (op-iso p) )) p-pi2 : { x y : HOD } → (p : def ZFProduct (od→ord < x , y >) ) → π2 p ≡ y p-pi2 {x} {y} p = proj2 ( prod-eq ( ord→== (op-iso p))) ω-pair : {x y : HOD} → infinite ∋ x → infinite ∋ y → od→ord < x , y > o< next o∅ ω-pair {x} {y} lx ly = lemma where lemma1 : od→ord x o< od→ord y → od→ord ( x , x ) o< od→ord ( x , y ) lemma1 = {!!} lemma0 : od→ord x o< od→ord y → od→ord < x , y > o< osuc (next (od→ord (x , y))) lemma0 x <⟨ ho< ⟩ next (omax (od→ord (x , x)) (od→ord (x , y))) ≡⟨ cong (λ k → next k ) (sym (omax< _ _ (lemma1 x o< next o∅ lemma with trio< (od→ord x) (od→ord y) lemma | tri< a ¬b ¬c = {!!} lemma | tri≈ ¬a b ¬c = next< {!!} {!!} lemma | tri> ¬a ¬b c = {!!} _⊗_ : (A B : HOD) → HOD A ⊗ B = Union ( Replace B (λ b → Replace A (λ a → < a , b > ) )) product→ : {A B a b : HOD} → A ∋ a → B ∋ b → ( A ⊗ B ) ∋ < a , b > product→ {A} {B} {a} {b} A∋a B∋b = {!!} record IsProduct (A B p : HOD) (A⊗B∋p : (A ⊗ B ) ∋ p ) : Set (suc n) where field is-pair : def ZFProduct (od→ord p) π1A : A ∋ π1 is-pair π2B : B ∋ π2 is-pair product← : {A B a b p : HOD} → (lt : (A ⊗ B ) ∋ p ) → IsProduct A B p lt product← lt = record { is-pair = {!!} ; π1A = {!!} ; π2B = {!!} } ZFP : (A B : HOD) → HOD ZFP A B = record { od = record { def = λ x → def ZFProduct x ∧ ( { x : Ordinal } → (p : def ZFProduct x ) → checkAB p ) } ; odmax = {!!} ;