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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