Nuprl Lemma : rounded-numerator_wf
∀[r:ℚ]. ∀[k:ℕ+]. (rounded-numerator(r;k) ∈ ℤ)
Proof
Definitions occuring in Statement :
rounded-numerator: rounded-numerator(r;k),
rationals: ℚ,
nat_plus: ℕ+,
uall: ∀[x:A]. B[x],
member: t ∈ T,
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
rationals: ℚ,
quotient: x,y:A//B[x; y],
and: P ∧ Q,
all: ∀x:A. B[x],
implies: P ⇒ Q,
prop: ℙ,
qeq: qeq(r;s),
uimplies: b supposing a,
so_lambda: λ2x.t[x],
so_apply: x[s],
int_nzero: ℤ-o,
callbyvalueall: callbyvalueall,
has-value: (a)↓,
has-valueall: has-valueall(a),
rounded-numerator: rounded-numerator(r;k),
b-union: A ⋃ B,
tunion: ⋃x:A.B[x],
bool: 𝔹,
unit: Unit,
ifthenelse: if b then t else f fi ,
pi2: snd(t),
btrue: tt,
uiff: uiff(P;Q),
nat_plus: ℕ+,
bfalse: ff,
sq_type: SQType(T),
guard: {T},
nequal: a ≠ b ∈ T ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top
Lemmas referenced :
b-union_wf,
int_nzero_wf,
equal-wf-T-base,
bool_wf,
qeq_wf,
equal_wf,
equal-wf-base,
nat_plus_wf,
rationals_wf,
valueall-type-has-valueall,
bunion-valueall-type,
int-valueall-type,
product-valueall-type,
set-valueall-type,
nequal_wf,
evalall-reduce,
eqtt_to_assert,
assert_of_eq_int,
and_wf,
subtype_base_sq,
int_subtype_base,
int_nzero_properties,
nat_plus_properties,
decidable__equal_int,
satisfiable-full-omega-tt,
intformnot_wf,
intformeq_wf,
itermMultiply_wf,
itermVar_wf,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_mul_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
div-cancel,
mul_preserves_eq,
mul_nzero,
intformand_wf,
int_formula_prop_and_lemma,
mul-associates,
mul-commutes,
mul-swap,
div-mul-cancel
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
pointwiseFunctionalityForEquality,
intEquality,
sqequalRule,
pertypeElimination,
productElimination,
thin,
equalityTransitivity,
hypothesis,
equalitySymmetry,
extract_by_obid,
isectElimination,
productEquality,
lambdaFormation,
because_Cache,
hypothesisEquality,
baseClosed,
dependent_functionElimination,
independent_functionElimination,
axiomEquality,
isect_memberEquality,
independent_isectElimination,
lambdaEquality,
natural_numberEquality,
callbyvalueReduce,
imageElimination,
unionElimination,
equalityElimination,
isintReduceTrue,
addLevel,
levelHypothesis,
dependent_set_memberEquality,
independent_pairFormation,
applyLambdaEquality,
setElimination,
rename,
multiplyEquality,
instantiate,
cumulativity,
dependent_pairFormation,
int_eqEquality,
voidElimination,
voidEquality,
computeAll,
divideEquality
Latex:
\mforall{}[r:\mBbbQ{}]. \mforall{}[k:\mBbbN{}\msupplus{}]. (rounded-numerator(r;k) \mmember{} \mBbbZ{})
Date html generated:
2018_05_21-PM-11_44_18
Last ObjectModification:
2017_07_26-PM-06_43_00
Theory : rationals
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