Nuprl Lemma : qle-minus
∀[a,b:ℚ]. uiff(a ≤ b;-(b) ≤ -(a))
Proof
Definitions occuring in Statement :
qle: r ≤ s,
qmul: r * s,
rationals: ℚ,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
minus: -n,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
nat_plus: ℕ+,
cand: A c∧ B,
not: ¬A,
implies: P ⇒ Q,
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
and: P ∧ Q,
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T ,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
prop: ℙ,
qmul: r * s,
qle: r ≤ s,
callbyvalueall: callbyvalueall,
evalall: evalall(t),
ifthenelse: if b then t else f fi ,
btrue: tt,
grp_leq: a ≤ b,
qadd_grp: <ℚ+>,
grp_le: ≤b,
pi2: snd(t),
pi1: fst(t),
infix_ap: x f y,
q_le: q_le(r;s),
qeq: qeq(r;s),
qsub: r - s,
qpositive: qpositive(r),
qadd: r + s,
so_lambda: λ2x.t[x],
so_apply: x[s],
has-value: (a)↓,
has-valueall: has-valueall(a),
bfalse: ff,
uiff: uiff(P;Q),
or: P ∨ Q,
decidable: Dec(P),
sq_type: SQType(T),
guard: {T},
band: p ∧b q,
rev_implies: P ⇐ Q
Lemmas referenced :
q-elim,
nat_plus_properties,
iff_weakening_uiff,
assert_wf,
qeq_wf2,
int-subtype-rationals,
equal-wf-base,
rationals_wf,
int_subtype_base,
assert-qeq,
istype-assert,
qdiv-int-elim,
full-omega-unsat,
intformand_wf,
intformeq_wf,
itermVar_wf,
itermConstant_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
nequal_wf,
valueall-type-has-valueall,
product-valueall-type,
int-valueall-type,
evalall-reduce,
uiff_wf,
qle_wf,
qdiv_wf,
qmul_wf,
qle_witness,
decidable__or,
less_than_wf,
decidable__cand,
istype-less_than,
decidable__lt,
decidable__equal_int,
intformnot_wf,
intformor_wf,
itermAdd_wf,
itermMultiply_wf,
int_formula_prop_not_lemma,
int_formula_prop_or_lemma,
int_term_value_add_lemma,
int_term_value_mul_lemma,
bor_wf,
lt_int_wf,
bool_cases,
subtype_base_sq,
bool_wf,
bool_subtype_base,
eqtt_to_assert,
band_wf,
btrue_wf,
assert_of_lt_int,
bfalse_wf,
eq_int_wf,
mul-associates,
mul-commutes,
one-mul,
add-commutes,
iff_transitivity,
assert_of_bor,
assert_of_band,
assert_of_eq_int,
assert_witness
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
productElimination,
isectElimination,
hypothesis,
setElimination,
rename,
lambdaFormation_alt,
independent_functionElimination,
applyEquality,
sqequalRule,
closedConclusion,
natural_numberEquality,
baseClosed,
because_Cache,
dependent_set_memberEquality_alt,
independent_isectElimination,
approximateComputation,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
Error :memTop,
independent_pairFormation,
universeIsType,
voidElimination,
equalityIstype,
inhabitedIsType,
sqequalBase,
equalitySymmetry,
intEquality,
callbyvalueReduce,
sqleReflexivity,
isintReduceTrue,
minusEquality,
productEquality,
independent_pairEquality,
multiplyEquality,
addEquality,
hyp_replacement,
applyLambdaEquality,
isect_memberEquality_alt,
isectIsTypeImplies,
equalityTransitivity,
unionEquality,
baseApply,
unionElimination,
inlFormation_alt,
unionIsType,
productIsType,
instantiate,
cumulativity,
promote_hyp,
inrFormation_alt
Latex:
\mforall{}[a,b:\mBbbQ{}]. uiff(a \mleq{} b;-(b) \mleq{} -(a))
Date html generated:
2020_05_20-AM-09_16_37
Last ObjectModification:
2020_01_25-AM-11_55_28
Theory : rationals
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