Nuprl Lemma : rless-iff-rpositive
∀x,y:ℝ. (x < y ⇐⇒ rpositive(y - x))
Proof
Definitions occuring in Statement :
rless: x < y,
rpositive: rpositive(x),
rsub: x - y,
real: ℝ,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
rev_implies: P ⇐ Q,
subtype_rel: A ⊆r B,
real: ℝ,
nat_plus: ℕ+,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
exists: ∃x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
top: Top,
rpositive: rpositive(x),
sq_exists: ∃x:{A| B[x]},
rsub: x - y,
rminus: -(x),
radd: a + b,
accelerate: accelerate(k;f),
has-value: (a)↓,
nequal: a ≠ b ∈ T ,
sq_type: SQType(T),
guard: {T},
int_nzero: ℤ-o,
nat: ℕ,
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_stable: SqStable(P),
le: A ≤ B,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
btrue: tt,
bfalse: ff,
rless: x < y
Lemmas referenced :
minus-is-int-iff,
add-is-int-iff,
mul_preserves_lt,
assert_of_bnot,
iff_weakening_uiff,
iff_transitivity,
eqff_to_assert,
assert_of_lt_int,
eqtt_to_assert,
bool_subtype_base,
bool_wf,
bool_cases,
not_wf,
bnot_wf,
assert_wf,
lt_int_wf,
false_wf,
int_term_value_minus_lemma,
int_term_value_add_lemma,
int_term_value_subtract_lemma,
itermMinus_wf,
itermAdd_wf,
itermSubtract_wf,
subtract-is-int-iff,
subtract_wf,
absval_ifthenelse,
sq_stable__le,
set_wf,
nat_wf,
absval_wf,
rem_bounds_absval,
iff_weakening_equal,
reg-seq-list-add-as-l_sum,
squash_wf,
nequal_wf,
div_rem_sum2,
mul_cancel_in_lt,
l_sum_nil_lemma,
l_sum_cons_lemma,
map_nil_lemma,
map_cons_lemma,
true_wf,
equal_wf,
int_subtype_base,
subtype_base_sq,
decidable__lt,
int_formula_prop_eq_lemma,
intformeq_wf,
decidable__equal_int,
nil_wf,
nat_plus_wf,
cons_wf,
int-value-type,
value-type-has-value,
int_formula_prop_wf,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_mul_lemma,
int_term_value_var_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
intformless_wf,
itermConstant_wf,
itermMultiply_wf,
itermVar_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__le,
nat_plus_properties,
mul_nat_plus,
less_than_wf,
rless-iff-large-diff,
real_wf,
rsub_wf,
rpositive_wf,
rless_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
independent_pairFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
applyEquality,
lambdaEquality,
setElimination,
rename,
sqequalRule,
dependent_functionElimination,
productElimination,
independent_functionElimination,
dependent_set_memberEquality,
natural_numberEquality,
introduction,
imageMemberEquality,
baseClosed,
multiplyEquality,
unionElimination,
independent_isectElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
dependent_set_memberFormation,
callbyvalueReduce,
sqleReflexivity,
functionEquality,
minusEquality,
because_Cache,
equalityTransitivity,
equalitySymmetry,
addLevel,
instantiate,
cumulativity,
divideEquality,
addEquality,
imageElimination,
universeEquality,
remainderEquality,
pointwiseFunctionality,
promote_hyp,
baseApply,
closedConclusion,
impliesFunctionality
Latex:
\mforall{}x,y:\mBbbR{}. (x < y \mLeftarrow{}{}\mRightarrow{} rpositive(y - x))
Date html generated:
2016_05_18-AM-07_03_48
Last ObjectModification:
2016_01_17-AM-01_50_32
Theory : reals
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