Nuprl Lemma : CCC-omni-2
∀K:Type
(CCCNSet(K) ⇒ (∀P:K ⟶ K ⟶ ℙ. ((∀k,m:K. Dec(P[k;m])) ⇒ ((∃k:K. ∀m:K. P[k;m]) ∨ (∀k:K. (¬(∀m:K. P[k;m])))))))
Proof
Definitions occuring in Statement :
ccc-nset: CCCNSet(K),
decidable: Dec(P),
prop: ℙ,
so_apply: x[s1;s2],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
or: P ∨ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
false: False,
guard: {T},
subtype_rel: A ⊆r B,
not: ¬A,
decidable: Dec(P),
exists: ∃x:A. B[x],
or: P ∨ Q,
uall: ∀[x:A]. B[x],
so_apply: x[s],
so_apply: x[s1;s2],
prop: ℙ,
so_lambda: λ2x.t[x],
implies: P ⇒ Q,
member: t ∈ T,
all: ∀x:A. B[x]
Lemmas referenced :
istype-void,
subtype_rel_self,
decidable__not,
not_wf,
istype-universe,
ccc-nset_wf,
decidable_wf,
CCC-omni
Rules used in proof :
Error :inlFormation_alt,
voidElimination,
Error :inrFormation_alt,
productElimination,
unionElimination,
Error :inhabitedIsType,
instantiate,
universeEquality,
isectElimination,
Error :functionIsType,
because_Cache,
Error :universeIsType,
applyEquality,
functionEquality,
Error :lambdaEquality_alt,
sqequalRule,
independent_functionElimination,
hypothesisEquality,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
hypothesis,
Error :lambdaFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
extract_by_obid,
introduction,
cut
Latex:
\mforall{}K:Type
(CCCNSet(K)
{}\mRightarrow{} (\mforall{}P:K {}\mrightarrow{} K {}\mrightarrow{} \mBbbP{}
((\mforall{}k,m:K. Dec(P[k;m])) {}\mRightarrow{} ((\mexists{}k:K. \mforall{}m:K. P[k;m]) \mvee{} (\mforall{}k:K. (\mneg{}(\mforall{}m:K. P[k;m])))))))
Date html generated:
2019_06_20-PM-03_02_54
Last ObjectModification:
2019_06_14-AM-10_00_38
Theory : continuity
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