Nuprl Lemma : qadd_positive
∀[r,s:ℚ]. (↑qpositive(r + s)) supposing ((↑qpositive(s)) and (↑qpositive(r)))
Proof
Definitions occuring in Statement :
qpositive: qpositive(r),
qadd: r + s,
rationals: ℚ,
assert: ↑b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
nat_plus: ℕ+,
cand: A c∧ B,
not: ¬A,
implies: P ⇒ Q,
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
and: P ∧ Q,
qdiv: (r/s),
top: Top,
ifthenelse: if b then t else f fi ,
btrue: tt,
mk-rational: mk-rational(a;b),
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T ,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
prop: ℙ,
bfalse: ff,
or: P ∨ Q,
nat: ℕ,
decidable: Dec(P),
guard: {T},
sq_type: SQType(T),
uiff: uiff(P;Q),
band: p ∧b q,
rev_implies: P ⇐ Q
Lemmas referenced :
q-elim,
nat_plus_properties,
iff_weakening_uiff,
assert_wf,
qeq_wf2,
int-subtype-rationals,
equal-wf-base,
rationals_wf,
int_subtype_base,
assert-qeq,
istype-assert,
qinv-elim,
qmul-elim,
isint-int,
istype-void,
mk-rational_wf,
full-omega-unsat,
intformand_wf,
intformeq_wf,
itermVar_wf,
itermConstant_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
nequal_wf,
qpositive-elim,
qadd-elim,
mul_nzero,
mul-associates,
mul-commutes,
mul-swap,
one-mul,
add-commutes,
add_nat_plus,
multiply_nat_wf,
decidable__le,
intformnot_wf,
intformle_wf,
intformor_wf,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_formula_prop_or_lemma,
istype-le,
multiply_nat_plus,
decidable__lt,
istype-less_than,
itermAdd_wf,
itermMultiply_wf,
int_term_value_add_lemma,
int_term_value_mul_lemma,
mul_bounds_1b,
iff_transitivity,
bor_wf,
lt_int_wf,
bool_cases,
subtype_base_sq,
bool_wf,
bool_subtype_base,
eqtt_to_assert,
band_wf,
btrue_wf,
assert_of_lt_int,
bfalse_wf,
less_than_wf,
assert_of_bor,
assert_of_band,
qpositive_wf,
qdiv_wf,
qadd_wf,
assert_witness
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
productElimination,
isectElimination,
hypothesis,
setElimination,
rename,
lambdaFormation_alt,
independent_functionElimination,
applyEquality,
sqequalRule,
natural_numberEquality,
because_Cache,
baseClosed,
isect_memberEquality_alt,
voidElimination,
closedConclusion,
dependent_set_memberEquality_alt,
independent_isectElimination,
approximateComputation,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
independent_pairFormation,
universeIsType,
equalityIstype,
inhabitedIsType,
sqequalBase,
equalitySymmetry,
intEquality,
multiplyEquality,
addEquality,
inlFormation_alt,
unionElimination,
equalityTransitivity,
applyLambdaEquality,
productIsType,
unionIsType,
instantiate,
cumulativity,
unionEquality,
productEquality,
promote_hyp,
inrFormation_alt,
hyp_replacement,
isectEquality,
isectIsTypeImplies
Latex:
\mforall{}[r,s:\mBbbQ{}]. (\muparrow{}qpositive(r + s)) supposing ((\muparrow{}qpositive(s)) and (\muparrow{}qpositive(r)))
Date html generated:
2019_10_16-AM-11_47_56
Last ObjectModification:
2019_06_25-PM-00_20_47
Theory : rationals
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