Nuprl Lemma : assert-is-qrep
∀p:ℤ × ℕ+. (↑is-qrep(p) ⇐⇒ ∃q:ℚ. (qrep(q) = p ∈ (ℤ × ℕ+)))
Proof
Definitions occuring in Statement :
is-qrep: is-qrep(p),
qrep: qrep(r),
rationals: ℚ,
nat_plus: ℕ+,
assert: ↑b,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
product: x:A × B[x],
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
rev_implies: P ⇐ Q,
exists: ∃x:A. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
mk-rational: mk-rational(a;b),
subtype_rel: A ⊆r B,
nat_plus: ℕ+,
int_nzero: ℤ-o,
uimplies: b supposing a,
nequal: a ≠ b ∈ T ,
not: ¬A,
false: False,
guard: {T},
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
is-qrep: is-qrep(p),
has-value: (a)↓,
uiff: uiff(P;Q),
qrep: qrep(r),
callbyvalueall: callbyvalueall,
has-valueall: has-valueall(a),
ifthenelse: if b then t else f fi ,
bfalse: ff,
spreadn: spread3,
nat: ℕ,
or: P ∨ Q,
sq_type: SQType(T),
assoced: a ~ b,
divides: b | a,
ge: i ≥ j ,
decidable: Dec(P),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
bnot: ¬bb,
assert: ↑b,
pi1: fst(t),
pi2: snd(t)
Lemmas referenced :
assert_wf,
is-qrep_wf,
exists_wf,
rationals_wf,
equal_wf,
nat_plus_wf,
qrep_wf,
mk-rational_wf,
subtype_rel_sets,
less_than_wf,
nequal_wf,
nat_plus_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformeq_wf,
itermVar_wf,
itermConstant_wf,
intformless_wf,
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,
equal-wf-base,
int_subtype_base,
value-type-has-value,
int-value-type,
better-gcd_wf,
bor_wf,
eq_int_wf,
gcd_wf,
or_wf,
equal-wf-T-base,
better-gcd-gcd,
iff_transitivity,
iff_weakening_uiff,
assert_of_bor,
assert_of_eq_int,
valueall-type-has-valueall,
product-valueall-type,
int-valueall-type,
set-valueall-type,
evalall-reduce,
gcd_reduce_property,
gcd_reduce_wf,
nat_wf,
equal-wf-base-T,
coprime_wf,
coprime_elim_a,
subtype_base_sq,
divides_invar_1,
minus-minus,
divides_reflexivity,
one_divs_any,
coprime_elim,
nat_properties,
decidable__equal_int,
intformnot_wf,
itermMultiply_wf,
int_formula_prop_not_lemma,
int_term_value_mul_lemma,
assoced_elim,
le_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_le_int,
eqff_to_assert,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
le_wf,
intformle_wf,
int_formula_prop_le_lemma,
decidable__lt,
product_subtype_base,
set_subtype_base,
qrep-coprime,
absval_wf,
absval_ifthenelse,
lt_int_wf,
bnot_wf,
not_wf,
minus-is-int-iff,
itermMinus_wf,
int_term_value_minus_lemma,
false_wf,
bool_cases,
assert_of_lt_int,
assert_of_bnot
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
independent_pairFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
dependent_functionElimination,
productElimination,
independent_pairEquality,
hypothesisEquality,
hypothesis,
sqequalRule,
lambdaEquality,
productEquality,
intEquality,
dependent_pairFormation,
applyEquality,
because_Cache,
natural_numberEquality,
independent_isectElimination,
setElimination,
rename,
setEquality,
applyLambdaEquality,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
baseClosed,
independent_functionElimination,
callbyvalueReduce,
minusEquality,
equalityTransitivity,
equalitySymmetry,
orFunctionality,
multiplyEquality,
baseApply,
closedConclusion,
unionElimination,
instantiate,
cumulativity,
equalityElimination,
promote_hyp,
dependent_set_memberEquality,
inlFormation,
inrFormation,
pointwiseFunctionality,
impliesFunctionality
Latex:
\mforall{}p:\mBbbZ{} \mtimes{} \mBbbN{}\msupplus{}. (\muparrow{}is-qrep(p) \mLeftarrow{}{}\mRightarrow{} \mexists{}q:\mBbbQ{}. (qrep(q) = p))
Date html generated:
2018_05_21-PM-11_48_55
Last ObjectModification:
2017_07_26-PM-06_43_15
Theory : rationals
Home
Index