Nuprl Lemma : assert-union-deq
∀[A,B:Type]. ∀[a:EqDecider(A)]. ∀[b:EqDecider(B)]. ∀[x,y:A + B]. uiff(↑(union-deq(A;B;a;b) x y);x = y ∈ (A + B))
Proof
Definitions occuring in Statement :
union-deq: union-deq(A;B;a;b),
deq: EqDecider(T),
assert: ↑b,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
apply: f a,
union: left + right,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
deq: EqDecider(T),
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
equal_wf,
iff_weakening_uiff,
assert_wf,
union-deq_wf,
deq_wf,
assert-deq,
assert_witness,
uiff_wf
Rules used in proof :
cut,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
independent_pairFormation,
isect_memberFormation,
hypothesis,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
unionEquality,
cumulativity,
hypothesisEquality,
because_Cache,
addLevel,
productElimination,
independent_isectElimination,
applyEquality,
lambdaEquality,
setElimination,
rename,
sqequalRule,
independent_functionElimination,
universeEquality,
independent_pairEquality,
isect_memberEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}[A,B:Type]. \mforall{}[a:EqDecider(A)]. \mforall{}[b:EqDecider(B)]. \mforall{}[x,y:A + B].
uiff(\muparrow{}(union-deq(A;B;a;b) x y);x = y)
Date html generated:
2017_04_14-AM-07_39_22
Last ObjectModification:
2017_02_27-PM-03_11_14
Theory : equality!deciders
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