Nuprl Lemma : q-rel-decider_wf
∀r:ℤ. ∀x:ℚ. (q-rel-decider(r;x) ∈ Dec(q-rel(r;x)))
Proof
Definitions occuring in Statement :
q-rel-decider: q-rel-decider(r;x),
q-rel: q-rel(r;x),
rationals: ℚ,
decidable: Dec(P),
all: ∀x:A. B[x],
member: t ∈ T,
int: ℤ
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
q-rel-decider: q-rel-decider(r;x),
q-rel: q-rel(r;x),
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
exposed-btrue: exposed-btrue,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
ifthenelse: if b then t else f fi ,
band: p ∧b q,
bnot: ¬bb,
bfalse: ff,
decidable: Dec(P),
not: ¬A,
or: P ∨ Q,
subtype_rel: A ⊆r B,
exposed-it: exposed-it,
bor: p ∨bq,
prop: ℙ,
exists: ∃x:A. B[x],
sq_type: SQType(T),
guard: {T},
assert: ↑b,
false: False,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top
Lemmas referenced :
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
qeq_wf2,
assert-qeq,
equal-wf-base-T,
rationals_wf,
false_wf,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
bor_wf,
qpositive_wf,
bfalse_wf,
assert-qpositive,
satisfiable-full-omega-tt,
intformand_wf,
intformeq_wf,
itermVar_wf,
itermConstant_wf,
int_formula_prop_and_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
qless_wf,
int-subtype-rationals,
neg_assert_of_eq_int,
qle_witness,
qle_weakening_eq_qorder,
qle_weakening_lt_qorder,
qless_complement_qorder,
qle_antisymmetry,
qless_witness
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
natural_numberEquality,
hypothesis,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
independent_isectElimination,
sqequalRule,
because_Cache,
applyEquality,
inlEquality,
axiomEquality,
functionEquality,
baseClosed,
dependent_pairFormation,
promote_hyp,
dependent_functionElimination,
instantiate,
cumulativity,
independent_functionElimination,
voidElimination,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidEquality,
independent_pairFormation,
computeAll,
inrEquality
Latex:
\mforall{}r:\mBbbZ{}. \mforall{}x:\mBbbQ{}. (q-rel-decider(r;x) \mmember{} Dec(q-rel(r;x)))
Date html generated:
2018_05_22-AM-00_19_02
Last ObjectModification:
2017_07_26-PM-06_53_48
Theory : rationals
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