Nuprl Lemma : subtract-1-ge-0
∀z:ℤ. (0 < z ⇒ ((z - 1) ≥ 0 ))
Proof
Definitions occuring in Statement :
less_than: a < b,
ge: i ≥ j ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
subtract: n - m,
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
not: ¬A,
rev_implies: P ⇐ Q,
false: False,
uiff: uiff(P;Q),
uimplies: b supposing a,
subtract: n - m,
subtype_rel: A ⊆r B,
top: Top,
le: A ≤ B,
less_than': less_than'(a;b),
true: True,
prop: ℙ
Lemmas referenced :
decidable__le,
subtract_wf,
istype-false,
not-ge-2,
less-iff-le,
condition-implies-le,
minus-one-mul,
zero-add,
minus-one-mul-top,
istype-void,
minus-add,
minus-minus,
add-associates,
add-swap,
add-commutes,
add_functionality_wrt_le,
add-zero,
le-add-cancel,
less_than_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
natural_numberEquality,
isectElimination,
hypothesisEquality,
hypothesis,
unionElimination,
independent_pairFormation,
voidElimination,
productElimination,
independent_functionElimination,
independent_isectElimination,
addEquality,
sqequalRule,
applyEquality,
Error :lambdaEquality_alt,
Error :isect_memberEquality_alt,
Error :universeIsType,
intEquality,
minusEquality,
Error :inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
because_Cache
Latex:
\mforall{}z:\mBbbZ{}. (0 < z {}\mRightarrow{} ((z - 1) \mgeq{} 0 ))
Date html generated:
2019_06_20-AM-11_23_07
Last ObjectModification:
2018_09_29-AM-11_35_51
Theory : arithmetic
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