Nuprl Lemma : zero-qle-qabs
∀[r:ℚ]. (0 ≤ |r|)
Proof
Definitions occuring in Statement :
qabs: |r|,
qle: r ≤ s,
rationals: ℚ,
uall: ∀[x:A]. B[x],
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
qabs: |r|,
uimplies: b supposing a,
callbyvalueall: callbyvalueall,
has-value: (a)↓,
has-valueall: has-valueall(a),
all: ∀x:A. B[x],
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
ifthenelse: if b then t else f fi ,
guard: {T},
subtype_rel: A ⊆r B,
bfalse: ff,
exists: ∃x:A. B[x],
prop: ℙ,
or: P ∨ Q,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
false: False,
not: ¬A,
squash: ↓T,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
valueall-type-has-valueall,
rationals_wf,
rationals-valueall-type,
evalall-reduce,
qpositive_wf,
bool_wf,
eqtt_to_assert,
assert-qpositive,
qle_weakening_lt_qorder,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
qless_wf,
int-subtype-rationals,
qle_witness,
qabs_wf,
qless_trichot_qorder,
qle-iff,
qmul_wf,
qminus-positive,
or_wf,
equal-wf-base-T,
squash_wf,
true_wf,
qinv_id_q,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesis,
independent_isectElimination,
hypothesisEquality,
callbyvalueReduce,
lambdaFormation,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
natural_numberEquality,
applyEquality,
because_Cache,
dependent_pairFormation,
promote_hyp,
dependent_functionElimination,
instantiate,
cumulativity,
independent_functionElimination,
voidElimination,
minusEquality,
addLevel,
orFunctionality,
baseClosed,
inlFormation,
inrFormation,
lambdaEquality,
imageElimination,
universeEquality,
imageMemberEquality,
hyp_replacement,
applyLambdaEquality
Latex:
\mforall{}[r:\mBbbQ{}]. (0 \mleq{} |r|)
Date html generated:
2018_05_21-PM-11_52_34
Last ObjectModification:
2017_07_26-PM-06_45_06
Theory : rationals
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