Nuprl Lemma : pair-order
∀[A,B:Type]. ∀[Ra:A ⟶ A ⟶ ℙ]. ∀[Rb:B ⟶ B ⟶ ℙ].
(Order(A;a,a'.Ra[a;a'])
⇒ Order(B;b,b'.Rb[b;b'])
⇒ Order(A × B;x,y.Ra[fst(x);fst(y)] ∧ ((¬((fst(x)) = (fst(y)) ∈ A)) ∨ Rb[snd(x);snd(y)])))
Proof
Definitions occuring in Statement :
order: Order(T;x,y.R[x; y]),
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
pi1: fst(t),
pi2: snd(t),
not: ¬A,
implies: P ⇒ Q,
or: P ∨ Q,
and: P ∧ Q,
function: x:A ⟶ B[x],
product: x:A × B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
order: Order(T;x,y.R[x; y]),
and: P ∧ Q,
cand: A c∧ B,
refl: Refl(T;x,y.E[x; y]),
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
pi1: fst(t),
member: t ∈ T,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
guard: {T},
so_apply: x[s1;s2],
trans: Trans(T;x,y.E[x; y]),
pi2: snd(t),
subtype_rel: A ⊆r B,
anti_sym: AntiSym(T;x,y.R[x; y]),
so_lambda: λ2x y.t[x; y],
not: ¬A,
false: False
Lemmas referenced :
not_wf,
equal_wf,
pi1_wf,
pi2_wf,
or_wf,
order_wf,
and_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
cut,
independent_pairFormation,
hypothesis,
inlFormation,
sqequalRule,
hypothesisEquality,
introduction,
extract_by_obid,
isectElimination,
cumulativity,
lambdaEquality,
independent_pairEquality,
unionElimination,
inrFormation,
applyEquality,
functionExtensionality,
productEquality,
because_Cache,
functionEquality,
universeEquality,
dependent_functionElimination,
independent_functionElimination,
equalityTransitivity,
equalitySymmetry,
hyp_replacement,
dependent_set_memberEquality,
applyLambdaEquality,
setElimination,
rename,
voidElimination
Latex:
\mforall{}[A,B:Type]. \mforall{}[Ra:A {}\mrightarrow{} A {}\mrightarrow{} \mBbbP{}]. \mforall{}[Rb:B {}\mrightarrow{} B {}\mrightarrow{} \mBbbP{}].
(Order(A;a,a'.Ra[a;a'])
{}\mRightarrow{} Order(B;b,b'.Rb[b;b'])
{}\mRightarrow{} Order(A \mtimes{} B;x,y.Ra[fst(x);fst(y)] \mwedge{} ((\mneg{}((fst(x)) = (fst(y)))) \mvee{} Rb[snd(x);snd(y)])))
Date html generated:
2017_02_20-AM-10_47_10
Last ObjectModification:
2017_02_02-PM-10_10_45
Theory : rel_1
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