Nuprl Lemma : rmin-rmax-real-decomp
∀[r:ℝ]. ((rmin(|r|;rmax(r0;r)) + rmax(-(|r|);rmin(r0;r))) = r)
Proof
Definitions occuring in Statement :
rabs: |x|,
rmin: rmin(x;y),
rmax: rmax(x;y),
req: x = y,
rminus: -(x),
radd: a + b,
int-to-real: r(n),
real: ℝ,
uall: ∀[x:A]. B[x],
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
real: ℝ,
bdd-diff: bdd-diff(f;g),
exists: ∃x:A. B[x],
nat: ℕ,
le: A ≤ B,
less_than': less_than'(a;b),
not: ¬A,
false: False,
all: ∀x:A. B[x],
int-to-real: r(n),
rmin: rmin(x;y),
rabs: |x|,
rminus: -(x),
rmax: rmax(x;y),
nat_plus: ℕ+,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
prop: ℙ,
sq_type: SQType(T),
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
less_than: a < b,
true: True,
squash: ↓T,
bfalse: ff,
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
absval: |i|
Lemmas referenced :
req_witness,
radd_wf,
rmin_wf,
rabs_wf,
rmax_wf,
int-to-real_wf,
rminus_wf,
real_wf,
req-iff-bdd-diff,
nat_plus_wf,
istype-void,
istype-le,
subtype_base_sq,
int_subtype_base,
nat_plus_properties,
decidable__equal_int,
full-omega-unsat,
intformnot_wf,
intformeq_wf,
itermMultiply_wf,
itermConstant_wf,
itermVar_wf,
istype-int,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_mul_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
absval_wf,
subtract_wf,
bdd-diff_functionality,
radd-bdd-diff,
bdd-diff_weakening,
lt_int_wf,
eqtt_to_assert,
assert_of_lt_int,
istype-top,
eqff_to_assert,
bool_cases_sqequal,
bool_wf,
bool_subtype_base,
assert-bnot,
iff_weakening_uiff,
assert_wf,
less_than_wf,
istype-less_than,
imin_wf,
imax_wf,
ifthenelse_wf,
le_int_wf,
assert_of_le_int,
le_wf,
intformle_wf,
int_formula_prop_le_lemma,
intformand_wf,
itermMinus_wf,
int_formula_prop_and_lemma,
int_term_value_minus_lemma,
add-zero,
intformless_wf,
int_formula_prop_less_lemma,
zero-add,
absval_unfold,
add_functionality_wrt_eq,
imin_unfold,
squash_wf,
true_wf,
istype-universe,
imax_unfold,
iff_weakening_equal,
itermSubtract_wf,
int_term_value_subtract_lemma,
istype-false
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
natural_numberEquality,
independent_functionElimination,
universeIsType,
productElimination,
independent_isectElimination,
applyEquality,
lambdaEquality_alt,
setElimination,
rename,
inhabitedIsType,
because_Cache,
sqequalRule,
addEquality,
equalityTransitivity,
equalitySymmetry,
dependent_pairFormation_alt,
dependent_set_memberEquality_alt,
independent_pairFormation,
lambdaFormation_alt,
voidElimination,
instantiate,
cumulativity,
intEquality,
dependent_functionElimination,
unionElimination,
approximateComputation,
int_eqEquality,
isect_memberEquality_alt,
equalityIstype,
functionIsType,
minusEquality,
equalityElimination,
lessCases,
axiomSqEquality,
isectIsTypeImplies,
imageMemberEquality,
baseClosed,
imageElimination,
promote_hyp,
sqequalIntensionalEquality,
universeEquality
Latex:
\mforall{}[r:\mBbbR{}]. ((rmin(|r|;rmax(r0;r)) + rmax(-(|r|);rmin(r0;r))) = r)
Date html generated:
2019_10_29-AM-09_34_45
Last ObjectModification:
2019_04_12-AM-11_18_48
Theory : reals
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