Nuprl Lemma : assert-product-deq
∀[A,B:Type]. ∀[a:EqDecider(A)]. ∀[b:EqDecider(B)]. ∀[x,y:A × B]. uiff(↑(product-deq(A;B;a;b) x y);x = y ∈ (A × B))
Proof
Definitions occuring in Statement :
product-deq: product-deq(A;B;a;b),
deq: EqDecider(T),
assert: ↑b,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
apply: f a,
product: x:A × B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
product-deq: product-deq(A;B;a;b),
proddeq: proddeq(a;b),
pi1: fst(t),
pi2: snd(t),
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
deq: EqDecider(T),
cand: A c∧ B,
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
eqof: eqof(d),
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
assert_wf,
band_wf,
iff_transitivity,
eqof_wf,
and_wf,
equal_wf,
iff_weakening_uiff,
assert_of_band,
safe-assert-deq,
assert_witness,
product-deq_wf,
deq_wf,
pi2_wf,
pi1_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
sqequalRule,
productElimination,
thin,
independent_pairFormation,
isect_memberFormation,
cut,
hypothesis,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
applyEquality,
setElimination,
rename,
hypothesisEquality,
introduction,
addLevel,
independent_functionElimination,
because_Cache,
lambdaFormation,
independent_isectElimination,
productEquality,
independent_pairEquality,
lambdaEquality,
universeEquality,
isect_memberEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
promote_hyp
Latex:
\mforall{}[A,B:Type]. \mforall{}[a:EqDecider(A)]. \mforall{}[b:EqDecider(B)]. \mforall{}[x,y:A \mtimes{} B].
uiff(\muparrow{}(product-deq(A;B;a;b) x y);x = y)
Date html generated:
2016_05_14-AM-06_07_27
Last ObjectModification:
2015_12_26-AM-11_46_32
Theory : equality!deciders
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