Nuprl Lemma : qbetween-qdist
∀[a,b,r,s:ℚ]. (qdist(a;b) ≤ (s - r)) supposing (r ≤ b ≤ s and r ≤ a ≤ s)
Proof
Definitions occuring in Statement :
qbetween: a ≤ b ≤ c,
qdist: qdist(r;s),
qle: r ≤ s,
qsub: r - s,
rationals: ℚ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x]
Definitions unfolded in proof :
qdist: qdist(r;s),
qbetween: a ≤ b ≤ c,
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
and: P ∧ Q,
qabs: |r|,
callbyvalueall: callbyvalueall,
has-value: (a)↓,
has-valueall: has-valueall(a),
implies: P ⇒ Q,
prop: ℙ,
true: True,
subtype_rel: A ⊆r B,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
squash: ↓T,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
all: ∀x:A. B[x],
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
not: ¬A,
qsub: r - s
Lemmas referenced :
valueall-type-has-valueall,
rationals_wf,
rationals-valueall-type,
qsub_wf,
evalall-reduce,
qle_witness,
qabs_wf,
qle_wf,
qpositive_wf,
bool_wf,
equal-wf-T-base,
assert_wf,
qless_wf,
int-subtype-rationals,
qle-normalize,
bnot_wf,
not_wf,
squash_wf,
true_wf,
ifthenelse_wf,
qminus-qsub,
iff_weakening_equal,
uiff_transitivity,
eqtt_to_assert,
assert-qpositive,
iff_transitivity,
iff_weakening_uiff,
eqff_to_assert,
assert_of_bnot,
equal_wf,
qadd_wf,
qmul_wf,
mon_assoc_q,
qadd_com,
qadd_assoc,
qadd-non-neg
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
productElimination,
thin,
extract_by_obid,
isectElimination,
hypothesis,
independent_isectElimination,
hypothesisEquality,
callbyvalueReduce,
because_Cache,
independent_functionElimination,
productEquality,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
natural_numberEquality,
baseClosed,
applyEquality,
lambdaEquality,
imageElimination,
universeEquality,
imageMemberEquality,
lambdaFormation,
unionElimination,
equalityElimination,
independent_pairFormation,
impliesFunctionality,
dependent_functionElimination,
hyp_replacement,
applyLambdaEquality,
minusEquality
Latex:
\mforall{}[a,b,r,s:\mBbbQ{}]. (qdist(a;b) \mleq{} (s - r)) supposing (r \mleq{} b \mleq{} s and r \mleq{} a \mleq{} s)
Date html generated:
2018_05_21-PM-11_58_55
Last ObjectModification:
2017_07_26-PM-06_48_24
Theory : rationals
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