Nuprl Lemma : int_seg-case
∀i,j:ℤ. ∀[F,G:{i..j-} ⟶ ℙ]. ((∀k:{i..j-}. (F[k] ∨ G[k])) ⇒ ((∃k:{i..j-}. F[k]) ∨ (∀k:{i..j-}. G[k])))
Proof
Definitions occuring in Statement :
int_seg: {i..j-},
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
or: P ∨ Q,
function: x:A ⟶ B[x],
int: ℤ
Definitions unfolded in proof :
prop: ℙ,
subtype_rel: A ⊆r B,
or: P ∨ Q,
so_apply: x[s],
so_lambda: λ2x.t[x],
member: t ∈ T,
implies: P ⇒ Q,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x]
Lemmas referenced :
istype-int,
subtype_rel_self,
int_seg_wf,
int-seg-case_wf
Rules used in proof :
inhabitedIsType,
universeEquality,
instantiate,
unionIsType,
functionIsType,
because_Cache,
hypothesis,
universeIsType,
applyEquality,
lambdaEquality_alt,
sqequalRule,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
cut,
introduction,
rename,
isect_memberFormation_alt,
lambdaFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}i,j:\mBbbZ{}.
\mforall{}[F,G:\{i..j\msupminus{}\} {}\mrightarrow{} \mBbbP{}]. ((\mforall{}k:\{i..j\msupminus{}\}. (F[k] \mvee{} G[k])) {}\mRightarrow{} ((\mexists{}k:\{i..j\msupminus{}\}. F[k]) \mvee{} (\mforall{}k:\{i..j\msupminus{}\}. G[k])))
Date html generated:
2019_10_15-AM-10_19_58
Last ObjectModification:
2019_10_02-PM-06_07_24
Theory : call!by!value_2
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