Nuprl Lemma : decidable__equal_compact_domain
∀[T,S:Type]. ((∀a,b:S. Dec(a = b ∈ S)) ⇒ compact-type(T) ⇒ (∀f,g:T ⟶ S. Dec(f = g ∈ (T ⟶ S))))
Proof
Definitions occuring in Statement :
compact-type: compact-type(T),
decidable: Dec(P),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
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,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
compact-type: compact-type(T),
deq: EqDecider(T),
or: P ∨ Q,
decidable: Dec(P),
not: ¬A,
false: False,
exists: ∃x:A. B[x],
uiff: uiff(P;Q),
uimplies: b supposing a,
eqof: eqof(d)
Lemmas referenced :
deq-exists,
compact-type_wf,
all_wf,
decidable_wf,
equal_wf,
equal-wf-T-base,
bool_wf,
eqff_to_assert,
assert_wf,
bnot_wf,
eqof_wf,
not_wf,
iff_transitivity,
iff_weakening_uiff,
assert_of_bnot,
safe-assert-deq,
eqtt_to_assert
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
productElimination,
independent_functionElimination,
hypothesis,
rename,
functionEquality,
cumulativity,
sqequalRule,
lambdaEquality,
universeEquality,
dependent_functionElimination,
applyEquality,
setElimination,
functionExtensionality,
unionElimination,
inrFormation,
because_Cache,
equalitySymmetry,
hyp_replacement,
applyLambdaEquality,
baseClosed,
equalityTransitivity,
independent_isectElimination,
voidElimination,
independent_pairFormation,
impliesFunctionality,
promote_hyp,
inlFormation
Latex:
\mforall{}[T,S:Type]. ((\mforall{}a,b:S. Dec(a = b)) {}\mRightarrow{} compact-type(T) {}\mRightarrow{} (\mforall{}f,g:T {}\mrightarrow{} S. Dec(f = g)))
Date html generated:
2017_10_01-AM-08_29_08
Last ObjectModification:
2017_07_26-PM-04_23_49
Theory : basic
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