Nuprl Lemma : subtract_nat_wf
∀[i,j:ℤ]. i - j ∈ ℕ supposing j ≤ i
Proof
Definitions occuring in Statement :
nat: ℕ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
le: A ≤ B,
member: t ∈ T,
subtract: n - m,
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
nat: ℕ,
le: A ≤ B,
and: P ∧ Q,
subtract: n - m,
prop: ℙ,
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
top: Top,
uiff: uiff(P;Q),
nat_plus: ℕ+,
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 :
subtract_wf,
minus-one-mul,
le_wf,
istype-int,
add_functionality_wrt_le,
le_reflexive,
minus-one-mul-top,
istype-void,
one-mul,
add-swap,
add-commutes,
add-associates,
add-mul-special,
two-mul,
mul-distributes-right,
zero-mul,
zero-add,
not-le-2,
minus-zero,
add-zero,
omega-shadow,
less_than_wf,
mul-distributes,
mul-associates,
mul-commutes,
le-add-cancel-alt,
decidable__le
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
Error :dependent_set_memberEquality_alt,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
productElimination,
sqequalRule,
Error :universeIsType,
natural_numberEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
Error :isect_memberEquality_alt,
because_Cache,
Error :inhabitedIsType,
multiplyEquality,
independent_isectElimination,
dependent_functionElimination,
applyEquality,
Error :lambdaEquality_alt,
voidElimination,
addEquality,
minusEquality,
independent_pairFormation,
imageMemberEquality,
baseClosed,
independent_functionElimination,
unionElimination
Latex:
\mforall{}[i,j:\mBbbZ{}]. i - j \mmember{} \mBbbN{} supposing j \mleq{} i
Date html generated:
2019_06_20-AM-11_26_25
Last ObjectModification:
2018_10_03-AM-10_13_23
Theory : arithmetic
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