Nuprl Lemma : qless-minus
∀[a,b:ℚ]. uiff(a < b;-(b) < -(a))
Proof
Definitions occuring in Statement :
qless: r < s,
qmul: r * s,
rationals: ℚ,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
minus: -n,
natural_number: $n
Definitions unfolded in proof :
prop: ℙ,
implies: P ⇒ Q,
and: P ∧ Q,
uiff: uiff(P;Q),
subtype_rel: A ⊆r B,
has-valueall: has-valueall(a),
has-value: (a)↓,
callbyvalueall: callbyvalueall,
uimplies: b supposing a,
q_le: q_le(r;s),
infix_ap: x f y,
pi1: fst(t),
grp_le: ≤b,
qadd_grp: <ℚ+>,
pi2: snd(t),
set_le: ≤b,
set_blt: a <b b,
dset_of_mon: g↓set,
set_lt: a <p b,
oset_of_ocmon: g↓oset,
grp_lt: a < b,
qless: r < s,
member: t ∈ T,
uall: ∀[x:A]. B[x],
qsub: r - s,
all: ∀x:A. B[x],
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
guard: {T},
squash: ↓T,
true: True,
or: P ∨ Q,
false: False,
not: ¬A,
sq_stable: SqStable(P)
Lemmas referenced :
qless_wf,
int-subtype-rationals,
qless_witness,
qmul_wf,
evalall-reduce,
rationals-valueall-type,
rationals_wf,
valueall-type-has-valueall,
qadd_comm_q,
true_wf,
squash_wf,
assert_witness,
assert_of_bnot,
assert-qeq,
assert-qpositive,
assert_of_bor,
assert_of_band,
iff_weakening_uiff,
iff_transitivity,
bnot_wf,
qeq_wf2,
qsub_wf,
qpositive_wf,
bor_wf,
band_wf,
assert_wf,
uiff_wf,
qinv_inv_q,
iff_weakening_equal,
qadd_com,
not_wf,
equal_wf,
qadd_wf,
or_wf,
decidable__equal_rationals,
decidable__qless,
decidable__or,
sq_stable_from_decidable,
istype-universe,
subtype_rel_self
Rules used in proof :
equalitySymmetry,
equalityTransitivity,
independent_functionElimination,
isect_memberEquality,
independent_pairEquality,
productElimination,
applyEquality,
natural_numberEquality,
minusEquality,
because_Cache,
callbyvalueReduce,
hypothesisEquality,
independent_isectElimination,
hypothesis,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
sqequalRule,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
levelHypothesis,
impliesFunctionality,
orFunctionality,
dependent_functionElimination,
addLevel,
universeEquality,
cumulativity,
baseClosed,
imageMemberEquality,
imageElimination,
lambdaEquality,
productEquality,
voidElimination,
lambdaFormation,
independent_pairFormation,
applyLambdaEquality,
hyp_replacement,
inrFormation,
inlFormation,
unionElimination,
lambdaEquality_alt,
universeIsType,
instantiate
Latex:
\mforall{}[a,b:\mBbbQ{}]. uiff(a < b;-(b) < -(a))
Date html generated:
2020_05_20-AM-09_16_31
Last ObjectModification:
2020_02_25-PM-00_11_59
Theory : rationals
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