Nuprl Lemma : sq_stable__ex_int_upper
∀n:ℤ. ∀[P:{n...} ⟶ ℙ]. ((∀m:{n...}. Dec(P[m])) ⇒ SqStable(∃m:{n...}. P[m]))
Proof
Definitions occuring in Statement :
int_upper: {i...},
sq_stable: SqStable(P),
decidable: Dec(P),
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
int: ℤ
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
sq_stable: SqStable(P),
member: t ∈ T,
squash: ↓T,
uimplies: b supposing a,
guard: {T},
prop: ℙ,
exists: ∃x:A. B[x],
so_apply: x[s],
and: P ∧ Q,
subtype_rel: A ⊆r B,
decidable: Dec(P),
or: P ∨ Q,
top: Top,
not: ¬A,
false: False,
isl: isl(x),
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
true: True,
bfalse: ff,
outl: outl(x)
Lemmas referenced :
mu-ge-property2,
squash_wf,
int_upper_wf,
istype-int_upper,
decidable_wf,
istype-int,
mu-ge_wf2,
subtype_rel_union,
subtype_rel_self,
not_wf,
top_wf,
istype-void,
istype-assert,
btrue_wf,
bfalse_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :lambdaFormation_alt,
Error :isect_memberFormation_alt,
rename,
introduction,
sqequalHypSubstitution,
imageElimination,
cut,
extract_by_obid,
dependent_functionElimination,
thin,
because_Cache,
isectElimination,
hypothesisEquality,
independent_isectElimination,
hypothesis,
Error :universeIsType,
sqequalRule,
productEquality,
applyEquality,
Error :functionIsType,
universeEquality,
productElimination,
functionExtensionality,
Error :lambdaEquality_alt,
instantiate,
Error :isect_memberEquality_alt,
voidElimination,
Error :unionIsType,
Error :dependent_pairFormation_alt,
Error :inhabitedIsType,
unionElimination,
Error :equalityIstype,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
natural_numberEquality,
Error :dependent_pairEquality_alt
Latex:
\mforall{}n:\mBbbZ{}. \mforall{}[P:\{n...\} {}\mrightarrow{} \mBbbP{}]. ((\mforall{}m:\{n...\}. Dec(P[m])) {}\mRightarrow{} SqStable(\mexists{}m:\{n...\}. P[m]))
Date html generated:
2019_06_20-PM-01_16_59
Last ObjectModification:
2019_01_09-PM-03_00_42
Theory : int_2
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