Nuprl Lemma : int_seg_subtype_nat_plus
∀[a,b:ℤ]. {a..b-} ⊆r ℕ+ supposing 1 ≤ a
Proof
Definitions occuring in Statement :
int_seg: {i..j-},
nat_plus: ℕ+,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
le: A ≤ B,
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-},
nat_plus: ℕ+,
so_lambda: λ2x.t[x],
lelt: i ≤ j < k,
so_apply: x[s],
subtype_rel: A ⊆r B,
and: P ∧ Q,
prop: ℙ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
le: A ≤ B,
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
not: ¬A,
rev_implies: P ⇐ Q,
false: False,
uiff: uiff(P;Q),
top: Top,
less_than': less_than'(a;b),
true: True
Lemmas referenced :
subtype_rel_sets,
and_wf,
le_wf,
less_than_wf,
decidable__lt,
false_wf,
not-lt-2,
add_functionality_wrt_le,
add-associates,
add-swap,
add-commutes,
zero-add,
le-add-cancel
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
intEquality,
because_Cache,
lambdaEquality,
hypothesisEquality,
hypothesis,
natural_numberEquality,
independent_isectElimination,
setElimination,
rename,
setEquality,
lambdaFormation,
productElimination,
dependent_functionElimination,
unionElimination,
independent_pairFormation,
voidElimination,
independent_functionElimination,
addEquality,
applyEquality,
isect_memberEquality,
voidEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}[a,b:\mBbbZ{}]. \{a..b\msupminus{}\} \msubseteq{}r \mBbbN{}\msupplus{} supposing 1 \mleq{} a
Date html generated:
2016_05_13-PM-03_33_12
Last ObjectModification:
2015_12_26-AM-09_45_03
Theory : arithmetic
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