Nuprl Lemma : qpositive-qabs
∀[r:ℚ]. ↑qpositive(|r|) supposing ¬(r = 0 ∈ ℚ)
Proof
Definitions occuring in Statement :
qabs: |r|,
qpositive: qpositive(r),
rationals: ℚ,
assert: ↑b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
not: ¬A,
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
qabs: |r|,
callbyvalueall: callbyvalueall,
has-value: (a)↓,
has-valueall: has-valueall(a),
all: ∀x:A. B[x],
or: P ∨ Q,
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 ,
bfalse: ff,
exists: ∃x:A. B[x],
prop: ℙ,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False,
not: ¬A
Lemmas referenced :
valueall-type-has-valueall,
rationals-valueall-type,
evalall-reduce,
rationals_wf,
q_trichotomy,
qpositive_wf,
bool_wf,
eqtt_to_assert,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
assert_witness,
qabs_wf,
not_wf,
equal-wf-T-base
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
because_Cache,
independent_isectElimination,
hypothesis,
callbyvalueReduce,
hypothesisEquality,
dependent_functionElimination,
unionElimination,
lambdaFormation,
equalityElimination,
productElimination,
dependent_pairFormation,
equalityTransitivity,
equalitySymmetry,
promote_hyp,
instantiate,
cumulativity,
independent_functionElimination,
voidElimination,
baseClosed,
isect_memberEquality
Latex:
\mforall{}[r:\mBbbQ{}]. \muparrow{}qpositive(|r|) supposing \mneg{}(r = 0)
Date html generated:
2018_05_21-PM-11_51_35
Last ObjectModification:
2017_07_26-PM-06_44_36
Theory : rationals
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