Nuprl Lemma : rminus-nonneg
∀[x:ℝ]. (x = r0) supposing (rnonneg(-(x)) and rnonneg(x))
Proof
Definitions occuring in Statement :
rnonneg: rnonneg(x),
req: x = y,
rminus: -(x),
int-to-real: r(n),
real: ℝ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
rminus: -(x),
rnonneg: rnonneg(x),
req: x = y,
all: ∀x:A. B[x],
int-to-real: r(n),
implies: P ⇒ Q,
prop: ℙ,
subtype_rel: A ⊆r B,
real: ℝ,
nat_plus: ℕ+,
top: Top,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
less_than: a < b,
less_than': less_than'(a;b),
true: True,
squash: ↓T,
not: ¬A,
false: False,
le: A ≤ B,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
bfalse: ff,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
subtract: n - m
Lemmas referenced :
nat_plus_wf,
req_witness,
int-to-real_wf,
rnonneg_wf,
rminus_wf,
real_wf,
absval_unfold,
lt_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_lt_int,
top_wf,
less_than_wf,
nat_plus_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermVar_wf,
itermConstant_wf,
itermMinus_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_term_value_minus_lemma,
int_formula_prop_wf,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
minus-one-mul,
mul-associates,
mul-commutes,
zero-mul,
add-zero
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
sqequalRule,
lambdaFormation,
extract_by_obid,
hypothesis,
isectElimination,
thin,
hypothesisEquality,
natural_numberEquality,
independent_functionElimination,
applyEquality,
lambdaEquality,
setElimination,
rename,
isect_memberEquality,
because_Cache,
equalityTransitivity,
equalitySymmetry,
multiplyEquality,
voidElimination,
voidEquality,
dependent_functionElimination,
minusEquality,
unionElimination,
equalityElimination,
productElimination,
independent_isectElimination,
lessCases,
sqequalAxiom,
independent_pairFormation,
imageMemberEquality,
baseClosed,
imageElimination,
dependent_set_memberEquality,
dependent_pairFormation,
int_eqEquality,
intEquality,
computeAll,
promote_hyp,
instantiate,
cumulativity
Latex:
\mforall{}[x:\mBbbR{}]. (x = r0) supposing (rnonneg(-(x)) and rnonneg(x))
Date html generated:
2017_10_03-AM-08_24_15
Last ObjectModification:
2017_07_28-AM-07_23_13
Theory : reals
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