Nuprl Lemma : sq_stable_and_decidable
∀[P:ℙ]. (Dec(P) ⇒ (∀[Q:⋂x:P. ℙ]. (SqStable(P) ⇒ (P ⇒ SqStable(Q)) ⇒ SqStable(P ∧ Q))))
Proof
Definitions occuring in Statement :
sq_stable: SqStable(P),
decidable: Dec(P),
uall: ∀[x:A]. B[x],
prop: ℙ,
implies: P ⇒ Q,
and: P ∧ Q,
isect: ⋂x:A. B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
decidable: Dec(P),
or: P ∨ Q,
member: t ∈ T,
prop: ℙ,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
exists: ∃x:A. B[x],
not: ¬A,
false: False,
sq_stable: SqStable(P),
squash: ↓T,
and: P ∧ Q
Lemmas referenced :
decidable_wf,
sq_stable__and,
sq_stable_wf,
isect_subtype_rel_trivial,
subtype_rel_weakening,
ext-eq_weakening,
subtype_rel_wf,
squash_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
sqequalHypSubstitution,
unionElimination,
thin,
cut,
lemma_by_obid,
isectElimination,
hypothesisEquality,
hypothesis,
universeEquality,
independent_functionElimination,
because_Cache,
functionEquality,
applyEquality,
instantiate,
cumulativity,
sqequalRule,
lambdaEquality,
independent_isectElimination,
dependent_pairFormation,
isectEquality,
voidElimination,
imageElimination,
productElimination,
productEquality,
rename,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}[P:\mBbbP{}]. (Dec(P) {}\mRightarrow{} (\mforall{}[Q:\mcap{}x:P. \mBbbP{}]. (SqStable(P) {}\mRightarrow{} (P {}\mRightarrow{} SqStable(Q)) {}\mRightarrow{} SqStable(P \mwedge{} Q))))
Date html generated:
2016_05_15-PM-03_14_22
Last ObjectModification:
2015_12_27-PM-01_02_12
Theory : general
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