Nuprl Lemma : decidable__equal_fset
∀[T:Type]. ((∀x,y:T. Dec(x = y ∈ T)) ⇒ (∀xs,ys:fset(T). Dec(xs = ys ∈ fset(T))))
Proof
Definitions occuring in Statement :
fset: fset(T),
decidable: Dec(P),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
all: ∀x:A. B[x],
member: t ∈ T,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
iff: P ⇐⇒ Q,
and: P ∧ Q,
uimplies: b supposing a,
f-subset: xs ⊆ ys,
rev_implies: P ⇐ Q
Lemmas referenced :
fset_wf,
all_wf,
decidable_wf,
equal_wf,
mk_deq_wf,
f-subset_weakening,
fset-member_witness,
fset-member_wf,
f-subset_antisymmetry,
and_wf,
f-subset_wf,
decidable_functionality,
decidable__and2,
decidable__f-subset
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
sqequalRule,
lambdaEquality,
universeEquality,
rename,
introduction,
independent_pairFormation,
independent_isectElimination,
dependent_functionElimination,
isect_memberEquality,
independent_functionElimination,
equalityTransitivity,
equalitySymmetry,
because_Cache,
productElimination
Latex:
\mforall{}[T:Type]. ((\mforall{}x,y:T. Dec(x = y)) {}\mRightarrow{} (\mforall{}xs,ys:fset(T). Dec(xs = ys)))
Date html generated:
2016_05_14-PM-03_41_41
Last ObjectModification:
2015_12_26-PM-06_40_20
Theory : finite!sets
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