Nuprl Lemma : q-ineq-test
∀[a,b,c:ℚ]. (False) supposing (0 < c and ((b + ((1/3) * c)) ≤ a) and ((a + c + c) ≤ b))
Proof
Definitions occuring in Statement :
qle: r ≤ s,
qless: r < s,
qdiv: (r/s),
qmul: r * s,
qadd: r + s,
rationals: ℚ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
false: False,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
false: False,
prop: ℙ,
subtype_rel: A ⊆r B,
not: ¬A,
implies: P ⇒ Q,
uiff: uiff(P;Q),
and: P ∧ Q,
qeq: qeq(r;s),
callbyvalueall: callbyvalueall,
evalall: evalall(t),
ifthenelse: if b then t else f fi ,
btrue: tt,
eq_int: (i =z j),
bfalse: ff,
assert: ↑b,
guard: {T},
qless: r < s,
grp_lt: a < b,
set_lt: a <p b,
set_blt: a <b b,
band: p ∧b q,
infix_ap: x f y,
set_le: ≤b,
pi2: snd(t),
oset_of_ocmon: g↓oset,
dset_of_mon: g↓set,
grp_le: ≤b,
pi1: fst(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,
lt_int: i <z j,
bnot: ¬bb,
true: True,
squash: ↓T
Lemmas referenced :
qmul-ident-div,
qmul_preserves_qle,
uiff_transitivity,
qmul_zero_qrng,
q_distrib,
qmul_ident,
qmul_assoc,
qadd_inv_assoc_q,
qadd_ac_1_q,
mon_assoc_q,
qmul_over_plus_qrng,
qadd_preserves_qle,
qinv_inv_q,
mon_ident_q,
qinverse_q,
qadd_comm_q,
true_wf,
squash_wf,
qadd_preserves_qless,
uiff_transitivity2,
qle_witness,
equal-wf-base,
qless_transitivity_2_qorder,
qle_transitivity_qorder,
rationals_wf,
equal_wf,
assert-qeq,
qdiv_wf,
qmul_wf,
qadd_wf,
qle_wf,
int-subtype-rationals,
qless_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
hypothesis,
sqequalRule,
sqequalHypSubstitution,
because_Cache,
lemma_by_obid,
isectElimination,
thin,
natural_numberEquality,
applyEquality,
hypothesisEquality,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
lambdaFormation,
productElimination,
independent_pairFormation,
voidElimination,
minusEquality,
baseClosed,
independent_functionElimination,
lambdaEquality,
imageElimination,
imageMemberEquality
Latex:
\mforall{}[a,b,c:\mBbbQ{}]. (False) supposing (0 < c and ((b + ((1/3) * c)) \mleq{} a) and ((a + c + c) \mleq{} b))
Date html generated:
2016_05_15-PM-11_04_46
Last ObjectModification:
2016_01_16-PM-09_31_01
Theory : rationals
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