Nuprl Lemma : add_nat_plus
∀[i:ℕ]. ∀[j:ℕ+]. (i + j ∈ ℕ+)
Proof
Definitions occuring in Statement :
nat_plus: ℕ+,
nat: ℕ,
uall: ∀[x:A]. B[x],
member: t ∈ T,
add: n + m
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat_plus: ℕ+,
nat: ℕ,
prop: ℙ,
all: ∀x:A. B[x],
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
top: Top,
subtract: n - m,
ge: i ≥ j ,
subtype_rel: A ⊆r B,
le: A ≤ B,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
implies: P ⇒ Q,
not: ¬A,
false: False,
decidable: Dec(P),
or: P ∨ Q
Lemmas referenced :
decidable__lt,
nat_properties,
nat_plus_properties,
mul-swap,
mul-associates,
mul-commutes,
omega-shadow,
minus-zero,
minus-one-mul-top,
not-lt-2,
add-zero,
zero-mul,
mul-distributes-right,
add-commutes,
two-mul,
add-mul-special,
one-mul,
zero-add,
minus-one-mul,
add-associates,
le_reflexive,
subtract_wf,
add_functionality_wrt_le,
less-iff-le,
nat_wf,
nat_plus_wf,
less_than_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
dependent_set_memberEquality,
addEquality,
sqequalHypSubstitution,
setElimination,
thin,
rename,
hypothesisEquality,
lemma_by_obid,
isectElimination,
natural_numberEquality,
hypothesis,
sqequalRule,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
isect_memberEquality,
because_Cache,
dependent_functionElimination,
productElimination,
independent_isectElimination,
multiplyEquality,
voidElimination,
voidEquality,
minusEquality,
applyEquality,
lambdaEquality,
intEquality,
independent_pairFormation,
imageMemberEquality,
baseClosed,
independent_functionElimination,
unionElimination
Latex:
\mforall{}[i:\mBbbN{}]. \mforall{}[j:\mBbbN{}\msupplus{}]. (i + j \mmember{} \mBbbN{}\msupplus{})
Date html generated:
2016_05_13-PM-03_39_23
Last ObjectModification:
2016_01_14-PM-06_38_31
Theory : arithmetic
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