Nuprl Lemma : assert-deq-f-subset
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[x,y:fset(T)]. uiff(↑(deq-f-subset(eq) x y);x ⊆ y)
Proof
Definitions occuring in Statement :
deq-f-subset: deq-f-subset(eq),
f-subset: xs ⊆ ys,
fset: fset(T),
deq: EqDecider(T),
assert: ↑b,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
apply: f a,
universe: Type
Definitions unfolded in proof :
member: t ∈ T,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
iff: P ⇐⇒ Q,
all: ∀x:A. B[x],
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
and: P ∧ Q,
prop: ℙ,
uiff: uiff(P;Q),
uimplies: b supposing a,
f-subset: xs ⊆ ys,
guard: {T},
subtype_rel: A ⊆r B,
sq_stable: SqStable(P),
squash: ↓T
Lemmas referenced :
decidable__f-subset,
decidable__assert,
sq_stable_from_decidable,
sq_stable__uiff,
deq_wf,
assert_witness,
fset-member_wf,
fset-member_witness,
equal_wf,
assert_wf,
f-subset_wf,
iff_wf,
all_wf,
bool_wf,
fset_wf,
set_wf,
deq-f-subset_wf
Rules used in proof :
cut,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
functionEquality,
cumulativity,
because_Cache,
sqequalRule,
lambdaEquality,
applyEquality,
lambdaFormation,
setElimination,
rename,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
independent_functionElimination,
independent_pairFormation,
isect_memberFormation,
introduction,
isect_memberEquality,
setEquality,
independent_isectElimination,
universeEquality,
productElimination,
independent_pairEquality,
imageMemberEquality,
baseClosed,
imageElimination
Latex:
\mforall{}[T:Type]. \mforall{}[eq:EqDecider(T)]. \mforall{}[x,y:fset(T)]. uiff(\muparrow{}(deq-f-subset(eq) x y);x \msubseteq{} y)
Date html generated:
2016_05_14-PM-03_41_38
Last ObjectModification:
2016_01_14-PM-10_41_05
Theory : finite!sets
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