Nuprl Lemma : qless-mk-rational
∀[a,c:ℤ]. ∀[b,d:ℕ+]. uiff(mk-rational(a;b) < mk-rational(c;d);a * d < b * c)
Proof
Definitions occuring in Statement :
qless: r < s,
mk-rational: mk-rational(a;b),
nat_plus: ℕ+,
less_than: a < b,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
multiply: n * m,
int: ℤ
Definitions unfolded in proof :
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
band: p ∧b q,
guard: {T},
sq_type: SQType(T),
false: False,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
or: P ∨ Q,
decidable: Dec(P),
prop: ℙ,
nequal: a ≠ b ∈ T ,
int_nzero: ℤ-o,
bfalse: ff,
btrue: tt,
ifthenelse: if b then t else f fi ,
top: Top,
has-valueall: has-valueall(a),
has-value: (a)↓,
callbyvalueall: callbyvalueall,
all: ∀x:A. B[x],
so_apply: x[s],
so_lambda: λ2x.t[x],
qsub: r - s,
q_le: q_le(r;s),
infix_ap: x f y,
pi1: fst(t),
grp_le: ≤b,
qadd_grp: <ℚ+>,
pi2: snd(t),
set_le: ≤b,
dset_of_mon: g↓set,
oset_of_ocmon: g↓oset,
set_blt: a <b b,
set_lt: a <p b,
grp_lt: a < b,
qless: r < s,
implies: P ⇒ Q,
uimplies: b supposing a,
and: P ∧ Q,
uiff: uiff(P;Q),
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
nat_plus: ℕ+,
member: t ∈ T,
mk-rational: mk-rational(a;b)
Lemmas referenced :
assert_witness,
assert_of_bnot,
assert_of_eq_int,
istype-assert,
assert_of_bor,
or_wf,
assert_of_band,
iff_weakening_uiff,
not_wf,
int_subtype_base,
set_subtype_base,
equal-wf-base,
bnot_wf,
eq_int_wf,
bfalse_wf,
assert_of_lt_int,
btrue_wf,
band_wf,
eqtt_to_assert,
bool_subtype_base,
bool_wf,
subtype_base_sq,
bool_cases,
lt_int_wf,
bor_wf,
assert_wf,
iff_transitivity,
istype-less_than,
int_formula_prop_wf,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
int_term_value_constant_lemma,
int_formula_prop_or_lemma,
int_term_value_var_lemma,
int_term_value_mul_lemma,
int_formula_prop_less_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
intformeq_wf,
itermAdd_wf,
itermConstant_wf,
intformor_wf,
itermVar_wf,
itermMultiply_wf,
intformless_wf,
intformnot_wf,
intformand_wf,
full-omega-unsat,
decidable__lt,
nat_plus_properties,
mul_bounds_1b,
qeq-elim,
nequal_wf,
mul_nzero,
qpositive-elim,
istype-void,
isint-int,
qadd-elim,
int-subtype-rationals,
qmul-elim,
evalall-reduce,
less_than_wf,
set-valueall-type,
int-valueall-type,
product-valueall-type,
valueall-type-has-valueall,
qless_witness,
member-less_than,
istype-int,
nat_plus_wf,
nat_plus_inc_int_nzero,
mk-rational_wf
Rules used in proof :
functionIsType,
inrFormation_alt,
promote_hyp,
baseClosed,
baseApply,
unionEquality,
equalityTransitivity,
cumulativity,
instantiate,
unionIsType,
equalitySymmetry,
sqequalBase,
equalityIstype,
productIsType,
inlFormation_alt,
int_eqEquality,
dependent_pairFormation_alt,
approximateComputation,
unionElimination,
dependent_functionElimination,
independent_pairFormation,
dependent_set_memberEquality_alt,
closedConclusion,
addEquality,
voidElimination,
minusEquality,
callbyvalueReduce,
natural_numberEquality,
lambdaFormation_alt,
lambdaEquality_alt,
intEquality,
productEquality,
independent_functionElimination,
isectIsTypeImplies,
independent_isectElimination,
isect_memberEquality_alt,
independent_pairEquality,
productElimination,
isect_memberFormation_alt,
universeIsType,
inhabitedIsType,
because_Cache,
sqequalRule,
applyEquality,
isectElimination,
extract_by_obid,
introduction,
hypothesis,
cut,
rename,
thin,
setElimination,
sqequalHypSubstitution,
hypothesisEquality,
multiplyEquality,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[a,c:\mBbbZ{}]. \mforall{}[b,d:\mBbbN{}\msupplus{}]. uiff(mk-rational(a;b) < mk-rational(c;d);a * d < b * c)
Date html generated:
2019_10_29-AM-07_46_16
Last ObjectModification:
2019_10_17-AM-09_58_30
Theory : rationals
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