Nuprl Lemma : add-monotonic
∀a,b,c,d:ℤ. (a < b ⇒ ((c = d ∈ ℤ) ∨ c < d) ⇒ a + c < b + d)
Proof
Definitions occuring in Statement :
less_than: a < b,
all: ∀x:A. B[x],
implies: P ⇒ Q,
or: P ∨ Q,
add: n + m,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
prop: ℙ,
squash: ↓T,
member: t ∈ T,
less_than': less_than'(a;b),
cand: A c∧ B,
and: P ∧ Q,
less_than: a < b,
implies: P ⇒ Q,
all: ∀x:A. B[x],
not: ¬A,
false: False,
or: P ∨ Q,
decidable: Dec(P),
sq_type: SQType(T),
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
guard: {T},
uimplies: b supposing a,
top: Top,
true: True
Lemmas referenced :
less_than_wf,
int_subtype_base,
equal-wf-base,
or_wf,
decidable__less_than',
subtype_base_sq,
iff_weakening_equal,
false_wf,
true_wf,
top_wf,
add-comm
Rules used in proof :
applyEquality,
intEquality,
isectElimination,
extract_by_obid,
sqequalHypSubstitution,
baseClosed,
thin,
imageMemberEquality,
sqequalRule,
introduction,
hypothesisEquality,
addEquality,
hypothesis,
independent_pairFormation,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
addMonotonic,
voidElimination,
independent_functionElimination,
productElimination,
imageElimination,
unionElimination,
dependent_functionElimination,
cumulativity,
instantiate,
independent_isectElimination,
voidEquality,
promote_hyp,
because_Cache,
isect_memberEquality,
axiomSqEquality,
isect_memberFormation,
natural_numberEquality,
equalityTransitivity,
lessCases,
sqequalBase,
equalitySymmetry,
baseInt,
Error :lessTransitive
Latex:
\mforall{}a,b,c,d:\mBbbZ{}. (a < b {}\mRightarrow{} ((c = d) \mvee{} c < d) {}\mRightarrow{} a + c < b + d)
Date html generated:
2019_06_20-AM-11_19_49
Last ObjectModification:
2018_10_15-AM-08_55_51
Theory : sqequal_1
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