Nuprl Lemma : rat2real-qadd
∀[a,b:ℚ]. (rat2real(a + b) = (rat2real(a) + rat2real(b)))
Proof
Definitions occuring in Statement :
rat2real: rat2real(q),
req: x = y,
radd: a + b,
uall: ∀[x:A]. B[x],
qadd: r + s,
rationals: ℚ
Definitions unfolded in proof :
rev_uimplies: rev_uimplies(P;Q),
uiff: uiff(P;Q),
decidable: Dec(P),
rev_implies: P ⇐ Q,
or: P ∨ Q,
guard: {T},
rneq: x ≠ y,
top: Top,
false: False,
satisfiable_int_formula: satisfiable_int_formula(fmla),
nequal: a ≠ b ∈ T ,
int_nzero: ℤ-o,
rat2real: rat2real(q),
bfalse: ff,
ifthenelse: if b then t else f fi ,
has-valueall: has-valueall(a),
has-value: (a)↓,
callbyvalueall: callbyvalueall,
so_apply: x[s],
so_lambda: λ2x.t[x],
qadd: r + s,
mk-rational: mk-rational(a;b),
uimplies: b supposing a,
prop: ℙ,
and: P ∧ Q,
iff: P ⇐⇒ Q,
subtype_rel: A ⊆r B,
implies: P ⇒ Q,
not: ¬A,
cand: A c∧ B,
nat_plus: ℕ+,
exists: ∃x:A. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
radd-int-fractions,
req_weakening,
radd_functionality,
int-rdiv-req,
rless_wf,
istype-less_than,
int_formula_prop_not_lemma,
intformnot_wf,
decidable__lt,
mul_bounds_1b,
rless-int,
rdiv_wf,
int-to-real_wf,
nequal_wf,
int_formula_prop_wf,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_eq_lemma,
istype-void,
int_formula_prop_and_lemma,
istype-int,
intformless_wf,
itermConstant_wf,
itermVar_wf,
intformeq_wf,
intformand_wf,
full-omega-unsat,
mul_nzero,
int-rdiv_wf,
req_functionality,
mk-rational-qdiv,
evalall-reduce,
int-valueall-type,
product-valueall-type,
valueall-type-has-valueall,
req_witness,
radd_wf,
qdiv_wf,
qadd_wf,
rat2real_wf,
req_wf,
istype-assert,
assert-qeq,
int_subtype_base,
rationals_wf,
equal-wf-base,
int-subtype-rationals,
qeq_wf2,
assert_wf,
iff_weakening_uiff,
nat_plus_properties,
q-elim
Rules used in proof :
unionElimination,
inrFormation_alt,
addEquality,
sqequalBase,
equalityIstype,
independent_pairFormation,
voidElimination,
int_eqEquality,
dependent_pairFormation_alt,
approximateComputation,
multiplyEquality,
dependent_set_memberEquality_alt,
callbyvalueReduce,
independent_pairEquality,
lambdaEquality_alt,
intEquality,
productEquality,
universeIsType,
isectIsTypeImplies,
isect_memberEquality_alt,
inhabitedIsType,
independent_isectElimination,
applyLambdaEquality,
equalitySymmetry,
hyp_replacement,
baseClosed,
because_Cache,
natural_numberEquality,
sqequalRule,
applyEquality,
independent_functionElimination,
lambdaFormation_alt,
rename,
setElimination,
hypothesis,
isectElimination,
productElimination,
hypothesisEquality,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
extract_by_obid,
cut,
introduction,
isect_memberFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[a,b:\mBbbQ{}]. (rat2real(a + b) = (rat2real(a) + rat2real(b)))
Date html generated:
2019_10_31-AM-05_57_04
Last ObjectModification:
2019_10_30-PM-02_52_55
Theory : reals
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