Nuprl Lemma : add-member-int_seg1
∀[k,m,n,x:ℤ]. uiff(k + x ∈ {m..n-};x ∈ {m - k..n - k-})
Proof
Definitions occuring in Statement :
int_seg: {i..j-},
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtract: n - m,
add: n + m,
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
int_seg: {i..j-},
lelt: i ≤ j < k,
squash: ↓T,
prop: ℙ,
implies: P ⇒ Q,
all: ∀x:A. B[x],
sq_stable: SqStable(P),
subtract: n - m,
top: Top,
le: A ≤ B,
subtype_rel: A ⊆r B,
nat_plus: ℕ+,
less_than: a < b,
less_than': less_than'(a;b),
true: True,
not: ¬A,
false: False,
decidable: Dec(P),
or: P ∨ Q
Lemmas referenced :
decidable__lt,
less-iff-le,
not-lt-2,
le-add-cancel-alt,
mul-commutes,
mul-swap,
mul-distributes,
omega-shadow,
mul-distributes-right,
two-mul,
mul-associates,
not-le-2,
add-zero,
zero-add,
zero-mul,
add-mul-special,
add-swap,
one-mul,
add-associates,
minus-add,
minus-one-mul-top,
le_reflexive,
add_functionality_wrt_le,
add-is-int-iff,
int_subtype_base,
int_seg_wf,
equal-wf-base,
subtract_wf,
add-commutes,
minus-one-mul,
squash_wf,
member-less_than,
sq_stable__less_than,
decidable__le,
sq_stable_from_decidable,
sq_stable__and,
less_than_wf,
le_wf,
and_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
independent_pairFormation,
dependent_set_memberEquality,
hypothesisEquality,
sqequalHypSubstitution,
applyEquality,
lambdaEquality,
setElimination,
thin,
rename,
hypothesis,
sqequalRule,
imageMemberEquality,
baseClosed,
setEquality,
intEquality,
lemma_by_obid,
isectElimination,
because_Cache,
imageElimination,
isect_memberEquality,
independent_functionElimination,
dependent_functionElimination,
equalityTransitivity,
equalitySymmetry,
lambdaFormation,
independent_isectElimination,
productElimination,
voidElimination,
voidEquality,
axiomEquality,
baseApply,
closedConclusion,
addEquality,
independent_pairEquality,
multiplyEquality,
natural_numberEquality,
minusEquality,
unionElimination
Latex:
\mforall{}[k,m,n,x:\mBbbZ{}]. uiff(k + x \mmember{} \{m..n\msupminus{}\};x \mmember{} \{m - k..n - k\msupminus{}\})
Date html generated:
2016_05_13-PM-03_38_53
Last ObjectModification:
2016_01_14-PM-06_39_31
Theory : arithmetic
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