Nuprl Lemma : decidable__cand
∀[P:ℙ]. ∀[Q:⋂x:P. ℙ]. (Dec(P) ⇒ (P ⇒ Dec(Q)) ⇒ Dec(P c∧ Q))
Proof
Definitions occuring in Statement :
decidable: Dec(P),
uall: ∀[x:A]. B[x],
cand: A c∧ B,
prop: ℙ,
implies: P ⇒ Q,
isect: ⋂x:A. B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
cand: A c∧ B,
and: P ∧ Q,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
exists: ∃x:A. B[x]
Lemmas referenced :
decidable__and2,
decidable_wf,
isect_subtype_rel_trivial,
subtype_rel_weakening,
ext-eq_weakening,
subtype_rel_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
independent_functionElimination,
hypothesis,
because_Cache,
functionEquality,
applyEquality,
instantiate,
cumulativity,
universeEquality,
sqequalRule,
lambdaEquality,
independent_isectElimination,
dependent_pairFormation,
isectIsType,
universeIsType,
inhabitedIsType
Latex:
\mforall{}[P:\mBbbP{}]. \mforall{}[Q:\mcap{}x:P. \mBbbP{}]. (Dec(P) {}\mRightarrow{} (P {}\mRightarrow{} Dec(Q)) {}\mRightarrow{} Dec(P c\mwedge{} Q))
Date html generated:
2019_10_15-AM-10_47_01
Last ObjectModification:
2018_09_27-AM-09_35_29
Theory : basic
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