Nuprl Lemma : not-not-all-int_seg-xmiddle
∀a,b:ℤ. ∀P:{a..b-} ⟶ ℙ. (¬¬(∀i:{a..b-}. (P[i] ∨ (¬P[i]))))
Proof
Definitions occuring in Statement :
int_seg: {i..j-},
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
not: ¬A,
or: P ∨ Q,
function: x:A ⟶ B[x],
int: ℤ
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
implies: P ⇒ Q,
false: False,
and: P ∧ Q,
ge: i ≥ j ,
le: A ≤ B,
cand: A c∧ B,
less_than: a < b,
squash: ↓T,
guard: {T},
uimplies: b supposing a,
prop: ℙ,
not: ¬A,
int_seg: {i..j-},
sq_stable: SqStable(P),
lelt: i ≤ j < k,
uiff: uiff(P;Q),
subtract: n - m,
top: Top,
less_than': less_than'(a;b),
true: True,
or: P ∨ Q,
so_apply: x[s],
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
decidable: Dec(P),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
nat_plus: ℕ+,
sq_type: SQType(T)
Lemmas referenced :
nat_properties,
less_than_transitivity1,
less_than_irreflexivity,
ge_wf,
istype-less_than,
sq_stable__le,
less-iff-le,
condition-implies-le,
minus-add,
istype-void,
minus-one-mul,
add-swap,
minus-one-mul-top,
add-associates,
zero-add,
add_functionality_wrt_le,
add-commutes,
le-add-cancel,
int_seg_wf,
subtype_rel_self,
subtract-1-ge-0,
istype-nat,
add-is-int-iff,
int_subtype_base,
decidable__lt,
istype-false,
not-lt-2,
subtract_wf,
le-add-cancel2,
istype-le,
not-not-1-xmiddle,
add-mul-special,
zero-mul,
add-zero,
le-add-cancel-alt,
le_reflexive,
one-mul,
two-mul,
mul-distributes-right,
not-le-2,
omega-shadow,
decidable__le,
double-negation-hyp-elim,
all_wf,
or_wf,
not_wf,
subtype_base_sq,
set_subtype_base,
lelt_wf,
decidable__int_equal,
not-equal-2,
minus-minus,
subtract_nat_wf,
istype-int
Rules used in proof :
cut,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
independent_pairFormation,
productElimination,
imageElimination,
natural_numberEquality,
independent_isectElimination,
independent_functionElimination,
voidElimination,
universeIsType,
sqequalRule,
lambdaEquality_alt,
dependent_functionElimination,
functionIsTypeImplies,
inhabitedIsType,
addEquality,
isect_memberEquality_alt,
minusEquality,
imageMemberEquality,
baseClosed,
functionIsType,
unionIsType,
applyEquality,
instantiate,
universeEquality,
because_Cache,
dependent_set_memberEquality_alt,
baseApply,
closedConclusion,
unionElimination,
productIsType,
multiplyEquality,
functionEquality,
unionEquality,
cumulativity,
intEquality,
equalityTransitivity,
equalitySymmetry,
equalityIstype
Latex:
\mforall{}a,b:\mBbbZ{}. \mforall{}P:\{a..b\msupminus{}\} {}\mrightarrow{} \mBbbP{}. (\mneg{}\mneg{}(\mforall{}i:\{a..b\msupminus{}\}. (P[i] \mvee{} (\mneg{}P[i]))))
Date html generated:
2020_05_19-PM-09_36_10
Last ObjectModification:
2019_10_17-PM-02_49_36
Theory : int_1
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