Nuprl Lemma : qpositive_wf
∀[r:ℚ]. (qpositive(r) ∈ 𝔹)
Proof
Definitions occuring in Statement :
qpositive: qpositive(r),
rationals: ℚ,
bool: 𝔹,
uall: ∀[x:A]. B[x],
member: t ∈ T
Definitions unfolded in proof :
btrue: tt,
bfalse: ff,
or: P ∨ Q,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
int_nzero: ℤ-o,
so_apply: x[s],
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
true: True,
squash: ↓T,
uiff: uiff(P;Q),
has-valueall: has-valueall(a),
has-value: (a)↓,
callbyvalueall: callbyvalueall,
uimplies: b supposing a,
prop: ℙ,
qeq: qeq(r;s),
qpositive: qpositive(r),
pi2: snd(t),
ifthenelse: if b then t else f fi ,
unit: Unit,
bool: 𝔹,
tunion: ⋃x:A.B[x],
b-union: A ⋃ B,
implies: P ⇒ Q,
all: ∀x:A. B[x],
and: P ∧ Q,
quotient: x,y:A//B[x; y],
rationals: ℚ,
member: t ∈ T,
uall: ∀[x:A]. B[x],
top: Top,
not: ¬A,
false: False,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
decidable: Dec(P),
nequal: a ≠ b ∈ T ,
gt: i > j,
guard: {T},
sq_type: SQType(T),
cand: A c∧ B
Lemmas referenced :
assert_of_lt_int,
assert_of_band,
assert_of_bor,
iff_weakening_uiff,
iff_transitivity,
iff_wf,
assert_wf,
less_than_wf,
or_wf,
eq_int_wf,
band_wf,
bor_wf,
iff_imp_equal_bool,
nequal_wf,
set-valueall-type,
product-valueall-type,
int_subtype_base,
true_wf,
squash_wf,
lt_int_wf,
assert_of_eq_int,
eqtt_to_assert,
evalall-reduce,
int-valueall-type,
valueall-type-has-valueall,
rationals_wf,
equal-wf-base,
equal_wf,
qeq_wf,
equal-wf-T-base,
int_nzero_wf,
b-union_wf,
bool_wf,
int_formula_prop_wf,
int_formula_prop_or_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_less_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
intformor_wf,
itermVar_wf,
itermConstant_wf,
intformless_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__lt,
int_nzero_properties,
neg_mul_arg_bounds,
gt_wf,
pos_mul_arg_bounds,
subtype_base_sq,
full-omega-unsat,
intformeq_wf,
int_formula_prop_eq_lemma,
decidable__equal_int,
int_entire,
int_term_value_mul_lemma,
itermMultiply_wf,
int_formual_prop_imp_lemma,
intformimplies_wf
Rules used in proof :
orFunctionality,
impliesFunctionality,
addLevel,
isintReduceTrue,
independent_pairFormation,
rename,
setElimination,
multiplyEquality,
independent_pairEquality,
closedConclusion,
baseApply,
imageMemberEquality,
natural_numberEquality,
lambdaEquality,
applyEquality,
callbyvalueReduce,
independent_isectElimination,
axiomEquality,
dependent_functionElimination,
baseClosed,
hypothesisEquality,
independent_functionElimination,
equalityElimination,
unionElimination,
imageElimination,
because_Cache,
lambdaFormation,
productEquality,
intEquality,
isectElimination,
equalitySymmetry,
equalityTransitivity,
thin,
productElimination,
pertypeElimination,
sqequalRule,
hypothesis,
extract_by_obid,
pointwiseFunctionalityForEquality,
sqequalHypSubstitution,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
computeAll,
voidEquality,
voidElimination,
isect_memberEquality,
int_eqEquality,
dependent_pairFormation,
levelHypothesis,
andLevelFunctionality,
cumulativity,
instantiate,
promote_hyp,
inlFormation,
approximateComputation,
inrFormation
Latex:
\mforall{}[r:\mBbbQ{}]. (qpositive(r) \mmember{} \mBbbB{})
Date html generated:
2018_05_21-PM-11_46_06
Last ObjectModification:
2017_07_26-PM-06_43_05
Theory : rationals
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