Nuprl Lemma : ratadd_wf
∀[a,b:ℤ × ℕ+]. (ratadd(a;b) ∈ {r:ℤ × ℕ+| ratreal(r) = (ratreal(a) + ratreal(b))} )
Proof
Definitions occuring in Statement :
ratadd: ratadd(x;y),
ratreal: ratreal(r),
req: x = y,
radd: a + b,
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,
ratadd: ratadd(x;y),
nat_plus: ℕ+,
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
exists: ∃x:A. B[x],
has-value: (a)↓,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
top: Top,
prop: ℙ,
squash: ↓T,
nequal: a ≠ b ∈ T ,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
int_nzero: ℤ-o,
sq_type: SQType(T),
so_lambda: λ2x.t[x],
so_apply: x[s],
rneq: x ≠ y,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
better-gcd-gcd,
gcd-properties,
gcd_wf,
value-type-has-value,
int-value-type,
gcd-positive,
nat_plus_subtype_nat,
nat_plus_properties,
decidable__lt,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
istype-int,
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,
decidable__le,
intformle_wf,
int_formula_prop_le_lemma,
subtype_base_sq,
int_subtype_base,
equal_wf,
squash_wf,
true_wf,
istype-universe,
intformeq_wf,
int_formula_prop_eq_lemma,
subtype_rel_self,
iff_weakening_equal,
div-cancel,
nequal_wf,
itermMultiply_wf,
int_term_value_mul_lemma,
mul_positive_iff,
set_subtype_base,
less_than_wf,
nat_plus_wf,
mul_nat_plus,
istype-less_than,
req_functionality,
ratreal_wf,
rdiv_wf,
int-to-real_wf,
rless-int,
mul_bounds_1b,
rless_wf,
radd_wf,
ratreal-req,
radd_functionality,
req_wf,
mul_nzero,
nat_plus_inc_int_nzero,
decidable__equal_int,
req-int-fractions,
uiff_transitivity,
rdiv_functionality,
req_inversion,
radd-int,
req_weakening,
radd-rdiv
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
productElimination,
thin,
sqequalRule,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
setElimination,
rename,
hypothesisEquality,
hypothesis,
dependent_functionElimination,
because_Cache,
inhabitedIsType,
lambdaFormation_alt,
callbyvalueReduce,
intEquality,
independent_isectElimination,
applyEquality,
natural_numberEquality,
unionElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
universeIsType,
instantiate,
cumulativity,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
divideEquality,
equalityIstype,
baseClosed,
sqequalBase,
imageMemberEquality,
dependent_set_memberEquality_alt,
addEquality,
multiplyEquality,
productIsType,
baseApply,
closedConclusion,
independent_pairEquality,
inrFormation_alt,
promote_hyp
Latex:
\mforall{}[a,b:\mBbbZ{} \mtimes{} \mBbbN{}\msupplus{}]. (ratadd(a;b) \mmember{} \{r:\mBbbZ{} \mtimes{} \mBbbN{}\msupplus{}| ratreal(r) = (ratreal(a) + ratreal(b))\} )
Date html generated:
2019_10_30-AM-09_18_01
Last ObjectModification:
2019_01_10-PM-02_51_38
Theory : reals
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