Nuprl Lemma : easy-member-int_seg
∀[i,j,a:ℤ]. (j - a ∈ {i..j-}) supposing (((i + a) ≤ j) and 0 < a)
Proof
Definitions occuring in Statement :
int_seg: {i..j-},
less_than: a < b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
le: A ≤ B,
member: t ∈ T,
subtract: n - m,
add: n + m,
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
le: A ≤ B,
cand: A c∧ B,
less_than: a < b,
squash: ↓T,
top: Top,
subtract: n - m,
all: ∀x:A. B[x],
uiff: uiff(P;Q),
nat_plus: ℕ+,
less_than': less_than'(a;b),
true: True,
implies: P ⇒ Q,
not: ¬A,
false: False,
decidable: Dec(P),
or: P ∨ Q
Lemmas referenced :
subtract_wf,
add-commutes,
istype-void,
minus-one-mul,
istype-le,
istype-less_than,
istype-int,
not-le-2,
add_functionality_wrt_le,
le_reflexive,
minus-one-mul-top,
add-associates,
one-mul,
add-swap,
add-mul-special,
zero-mul,
zero-add,
add-zero,
two-mul,
mul-distributes-right,
omega-shadow,
mul-distributes,
mul-swap,
mul-associates,
le-add-cancel,
less-iff-le,
not-lt-2,
minus-add,
decidable__le,
decidable__lt
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
dependent_set_memberEquality_alt,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
independent_pairFormation,
productElimination,
imageElimination,
sqequalRule,
because_Cache,
isect_memberEquality_alt,
voidElimination,
productIsType,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
addEquality,
isectIsTypeImplies,
inhabitedIsType,
natural_numberEquality,
dependent_functionElimination,
independent_isectElimination,
multiplyEquality,
minusEquality,
imageMemberEquality,
baseClosed,
independent_functionElimination,
unionElimination
Latex:
\mforall{}[i,j,a:\mBbbZ{}]. (j - a \mmember{} \{i..j\msupminus{}\}) supposing (((i + a) \mleq{} j) and 0 < a)
Date html generated:
2020_05_19-PM-09_35_43
Last ObjectModification:
2019_12_08-PM-06_18_58
Theory : arithmetic
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