Nuprl Lemma : mu-ge-property2
∀n:ℤ
  ∀[P:{n...} ⟶ ℙ]
    ∀d:∀n:{n...}. Dec(P[n]). {P[mu-ge(d;n)] ∧ (∀[i:{n..mu-ge(d;n)-}]. (¬P[i]))} supposing ∃m:{n...}. P[m]
Proof
Definitions occuring in Statement : 
mu-ge: mu-ge(f;n)
, 
int_upper: {i...}
, 
int_seg: {i..j-}
, 
decidable: Dec(P)
, 
uimplies: b supposing a
, 
uall: ∀[x:A]. B[x]
, 
prop: ℙ
, 
guard: {T}
, 
so_apply: x[s]
, 
all: ∀x:A. B[x]
, 
exists: ∃x:A. B[x]
, 
not: ¬A
, 
and: P ∧ Q
, 
function: x:A ⟶ B[x]
, 
int: ℤ
Definitions unfolded in proof : 
all: ∀x:A. B[x]
, 
uall: ∀[x:A]. B[x]
, 
uimplies: b supposing a
, 
member: t ∈ T
, 
implies: P 
⇒ Q
, 
decidable: Dec(P)
, 
or: P ∨ Q
, 
isl: isl(x)
, 
assert: ↑b
, 
ifthenelse: if b then t else f fi 
, 
btrue: tt
, 
iff: P 
⇐⇒ Q
, 
and: P ∧ Q
, 
prop: ℙ
, 
rev_implies: P 
⇐ Q
, 
true: True
, 
bfalse: ff
, 
false: False
, 
not: ¬A
, 
so_apply: x[s]
, 
subtype_rel: A ⊆r B
, 
exists: ∃x:A. B[x]
, 
squash: ↓T
, 
top: Top
, 
guard: {T}
, 
mu-ge: mu-ge(f;n)
, 
has-value: (a)↓
, 
strict4: strict4(F)
, 
so_lambda: λ2x.t[x]
, 
so_apply: x[s1;s2;s3;s4]
, 
so_lambda: so_lambda(x,y,z,w.t[x; y; z; w])
, 
int_upper: {i...}
Lemmas referenced : 
int_upper_wf, 
true_wf, 
istype-void, 
subtype_rel_self, 
assert_wf, 
btrue_wf, 
bfalse_wf, 
mu-ge_wf2, 
subtype_rel_union, 
not_wf, 
top_wf, 
decidable_wf, 
istype-int, 
mu-ge-property, 
is-exception_wf, 
base_wf, 
has-value_wf_base, 
equal_wf, 
lifting-strict-decide, 
int_seg_subtype_upper, 
le_reflexive, 
int_seg_wf
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
Error :lambdaFormation_alt, 
Error :isect_memberFormation_alt, 
cut, 
Error :lambdaEquality_alt, 
because_Cache, 
Error :inhabitedIsType, 
hypothesis, 
thin, 
sqequalHypSubstitution, 
unionElimination, 
sqequalRule, 
Error :equalityIsType1, 
equalityTransitivity, 
equalitySymmetry, 
hypothesisEquality, 
dependent_functionElimination, 
independent_functionElimination, 
Error :universeIsType, 
applyEquality, 
functionExtensionality, 
introduction, 
extract_by_obid, 
isectElimination, 
independent_pairFormation, 
natural_numberEquality, 
voidElimination, 
instantiate, 
universeEquality, 
productElimination, 
Error :dependent_pairFormation_alt, 
imageMemberEquality, 
baseClosed, 
imageElimination, 
independent_isectElimination, 
Error :isect_memberEquality_alt, 
Error :unionIsType, 
Error :productIsType, 
Error :functionIsType, 
inlFormation, 
exceptionSqequal, 
inrFormation, 
decideExceptionCases, 
closedConclusion, 
baseApply, 
sqleReflexivity, 
unionEquality, 
callbyvalueDecide, 
lambdaFormation, 
voidEquality, 
isect_memberEquality, 
promote_hyp, 
setElimination, 
rename
Latex:
\mforall{}n:\mBbbZ{}
    \mforall{}[P:\{n...\}  {}\mrightarrow{}  \mBbbP{}]
        \mforall{}d:\mforall{}n:\{n...\}.  Dec(P[n])
            \{P[mu-ge(d;n)]  \mwedge{}  (\mforall{}[i:\{n..mu-ge(d;n)\msupminus{}\}].  (\mneg{}P[i]))\}  supposing  \mexists{}m:\{n...\}.  P[m]
Date html generated:
2019_06_20-PM-01_16_47
Last ObjectModification:
2018_10_06-AM-11_22_01
Theory : int_2
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