Nuprl Lemma : qabs-as-qmax
∀[q:ℚ]. (|q| = qmax(q;-(q)) ∈ ℚ)
Proof
Definitions occuring in Statement :
qabs: |r|,
qmax: qmax(x;y),
qmul: r * s,
rationals: ℚ,
uall: ∀[x:A]. B[x],
minus: -n,
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
qmax: qmax(x;y),
qabs: |r|,
member: t ∈ T,
uimplies: b supposing a,
callbyvalueall: callbyvalueall,
has-value: (a)↓,
has-valueall: has-valueall(a),
subtype_rel: A ⊆r B,
guard: {T},
uiff: uiff(P;Q),
and: P ∧ Q,
true: True,
not: ¬A,
implies: P ⇒ Q,
qeq: qeq(r;s),
evalall: evalall(t),
ifthenelse: if b then t else f fi ,
btrue: tt,
eq_int: (i =z j),
bfalse: ff,
assert: ↑b,
false: False,
prop: ℙ,
qle: r ≤ s,
grp_leq: a ≤ b,
infix_ap: x f y,
grp_le: ≤b,
pi1: fst(t),
pi2: snd(t),
qadd_grp: <ℚ+>,
q_le: q_le(r;s),
qdiv: (r/s),
qmul: r * s,
qinv: 1/r,
bor: p ∨bq,
qpositive: qpositive(r),
qsub: r - s,
qadd: r + s,
band: p ∧b q,
lt_int: i <z j,
qless: r < s,
grp_lt: a < b,
set_lt: a <p b,
set_blt: a <b b,
set_le: ≤b,
oset_of_ocmon: g↓oset,
dset_of_mon: g↓set,
bnot: ¬bb,
all: ∀x:A. B[x],
bool: 𝔹,
unit: Unit,
it: ⋅,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
squash: ↓T
Lemmas referenced :
valueall-type-has-valueall,
rationals_wf,
rationals-valueall-type,
evalall-reduce,
qpositive_wf,
bool_wf,
equal-wf-T-base,
assert_wf,
qless_wf,
int-subtype-rationals,
q_le_wf,
qmul_wf,
qle_wf,
qadd_preserves_qle,
qadd_wf,
qmul_preserves_qle2,
qdiv_wf,
assert-qeq,
equal-wf-base,
qle_witness,
bnot_wf,
not_wf,
qle_complement_qorder,
qless_complement_qorder,
qadd_preserves_qless,
qmul_preserves_qless,
qless_witness,
uiff_transitivity,
eqtt_to_assert,
assert-qpositive,
iff_transitivity,
iff_weakening_uiff,
eqff_to_assert,
assert_of_bnot,
equal_wf,
uiff_transitivity2,
assert-q_le-eq,
squash_wf,
true_wf,
qinverse_q,
qmul_ident,
q_distrib,
iff_weakening_equal,
qmul_zero_qrng,
qmul_ac_1_qrng,
qmul-qdiv-cancel3,
qmul_one_qrng,
qless_transitivity_2_qorder,
qless_irreflexivity
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
cut,
sqequalRule,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesis,
independent_isectElimination,
hypothesisEquality,
callbyvalueReduce,
equalityTransitivity,
equalitySymmetry,
baseClosed,
natural_numberEquality,
applyEquality,
because_Cache,
minusEquality,
productElimination,
lambdaFormation,
independent_pairFormation,
voidElimination,
independent_functionElimination,
unionElimination,
equalityElimination,
impliesFunctionality,
dependent_functionElimination,
lambdaEquality,
imageElimination,
universeEquality,
imageMemberEquality
Latex:
\mforall{}[q:\mBbbQ{}]. (|q| = qmax(q;-(q)))
Date html generated:
2018_05_21-PM-11_57_58
Last ObjectModification:
2017_07_26-PM-06_47_48
Theory : rationals
Home
Index