Nuprl Lemma : int_seg_cases
∀[m,n:ℤ]. ∀[x:{m..n-}]. x ∈ {m + 1..n-} supposing ¬(x = m ∈ ℤ)
Proof
Definitions occuring in Statement :
int_seg: {i..j-},
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
not: ¬A,
member: t ∈ T,
add: n + m,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
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,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
not: ¬A,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
false: False,
prop: ℙ,
uiff: uiff(P;Q),
subtype_rel: A ⊆r B,
top: Top,
less_than': less_than'(a;b),
true: True,
subtract: n - m
Lemmas referenced :
decidable__le,
false_wf,
not-le-2,
not-equal-2,
add_functionality_wrt_le,
add-associates,
add-commutes,
le-add-cancel,
condition-implies-le,
minus-add,
minus-one-mul,
add-swap,
minus-one-mul-top,
zero-add,
le-add-cancel2,
and_wf,
le_wf,
less_than_wf,
not_wf,
equal_wf,
int_seg_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
setElimination,
thin,
rename,
dependent_set_memberEquality,
hypothesisEquality,
productElimination,
independent_pairFormation,
hypothesis,
lemma_by_obid,
dependent_functionElimination,
addEquality,
natural_numberEquality,
unionElimination,
lambdaFormation,
voidElimination,
independent_functionElimination,
independent_isectElimination,
isectElimination,
sqequalRule,
applyEquality,
lambdaEquality,
isect_memberEquality,
voidEquality,
intEquality,
because_Cache,
minusEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}[m,n:\mBbbZ{}]. \mforall{}[x:\{m..n\msupminus{}\}]. x \mmember{} \{m + 1..n\msupminus{}\} supposing \mneg{}(x = m)
Date html generated:
2016_05_13-PM-03_32_53
Last ObjectModification:
2015_12_26-AM-09_45_27
Theory : arithmetic
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