Nuprl Lemma : rat-nat-div_wf
∀[a:ℤ × ℕ+]. ∀[n:ℕ+]. (rat-nat-div(a;n) ∈ {r:ℤ × ℕ+| ratreal(r) = (ratreal(a))/n} )
Proof
Definitions occuring in Statement :
rat-nat-div: rat-nat-div(x;n),
ratreal: ratreal(r),
int-rdiv: (a)/k1,
req: x = y,
nat_plus: ℕ+,
uall: ∀[x:A]. B[x],
member: t ∈ T,
set: {x:A| B[x]} ,
product: x:A × B[x],
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
rat-nat-div: rat-nat-div(x;n),
has-value: (a)↓,
uimplies: b supposing a,
nat_plus: ℕ+,
subtype_rel: A ⊆r B,
prop: ℙ,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
decidable: Dec(P),
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
nequal: a ≠ b ∈ T ,
rdiv: (x/y),
req_int_terms: t1 ≡ t2
Lemmas referenced :
value-type-has-value,
int-value-type,
mul_nat_plus,
req_wf,
ratreal_wf,
int-rdiv_wf,
nat_plus_inc_int_nzero,
nat_plus_wf,
istype-int,
rdiv_wf,
int-to-real_wf,
rless-int,
mul_bounds_1b,
rless_wf,
nat_plus_properties,
decidable__lt,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
req_functionality,
ratreal-req,
req_transitivity,
int-rdiv-req,
rdiv_functionality,
req_weakening,
rmul_preserves_req,
rmul_wf,
rinv_wf2,
rneq_functionality,
rmul-int,
rneq-int,
int_entire_a,
intformeq_wf,
int_formula_prop_eq_lemma,
int_subtype_base,
itermSubtract_wf,
itermMultiply_wf,
rmul_functionality,
rinv_functionality2,
req_inversion,
rinv-of-rmul,
rmul-rinv,
req-iff-rsub-is-0,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
productElimination,
thin,
sqequalRule,
callbyvalueReduce,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
intEquality,
independent_isectElimination,
hypothesis,
multiplyEquality,
setElimination,
rename,
hypothesisEquality,
dependent_set_memberEquality_alt,
independent_pairEquality,
universeIsType,
applyEquality,
productIsType,
because_Cache,
inrFormation_alt,
dependent_functionElimination,
independent_functionElimination,
natural_numberEquality,
unionElimination,
approximateComputation,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
lambdaFormation_alt,
equalityIstype,
inhabitedIsType,
baseClosed,
sqequalBase,
equalitySymmetry,
closedConclusion
Latex:
\mforall{}[a:\mBbbZ{} \mtimes{} \mBbbN{}\msupplus{}]. \mforall{}[n:\mBbbN{}\msupplus{}]. (rat-nat-div(a;n) \mmember{} \{r:\mBbbZ{} \mtimes{} \mBbbN{}\msupplus{}| ratreal(r) = (ratreal(a))/n\} )
Date html generated:
2019_10_30-AM-09_26_57
Last ObjectModification:
2019_01_10-PM-02_06_15
Theory : reals
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