Nuprl Lemma : rmax-minus-rmin
∀a,b:ℝ. (|a - b| = (rmax(a;b) - rmin(a;b)))
Proof
Definitions occuring in Statement :
rabs: |x|,
rmin: rmin(x;y),
rmax: rmax(x;y),
rsub: x - y,
req: x = y,
real: ℝ,
all: ∀x:A. B[x]
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
top: Top,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
rsub: x - y,
subtype_rel: A ⊆r B,
real: ℝ,
rminus: -(x),
rmax: rmax(x;y),
rmin: rmin(x;y),
bdd-diff: bdd-diff(f;g),
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
exists: ∃x:A. B[x],
nat: ℕ,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
true: True,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
nat_plus: ℕ+,
satisfiable_int_formula: satisfiable_int_formula(fmla),
less_than: a < b,
squash: ↓T,
subtract: n - m
Lemmas referenced :
rabs-as-rmax,
real_wf,
req-iff-bdd-diff,
rmax_wf,
rsub_wf,
rminus_wf,
rmin_wf,
radd_wf,
imax_wf,
nat_plus_wf,
bdd-diff_functionality,
rmax_functionality_wrt_bdd-diff,
rminus_functionality_wrt_bdd-diff,
radd-bdd-diff,
false_wf,
le_wf,
all_wf,
absval_wf,
subtract_wf,
imin_wf,
ifthenelse_wf,
le_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_le_int,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
nat_plus_properties,
add-is-int-iff,
minus-is-int-iff,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermAdd_wf,
itermVar_wf,
itermMinus_wf,
intformnot_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_term_value_minus_lemma,
int_formula_prop_not_lemma,
int_formula_prop_wf,
absval_unfold,
top_wf,
less_than_wf,
squash_wf,
true_wf,
imax_unfold,
add_functionality_wrt_eq,
minus_functionality_wrt_eq,
imin_unfold,
nat_wf,
subtype_rel_self,
iff_weakening_equal,
minus-add,
minus-minus,
add-associates,
minus-one-mul,
add-mul-special,
add-commutes,
add-swap,
zero-mul,
zero-add
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalRule,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
isect_memberEquality,
voidElimination,
voidEquality,
hypothesis,
hypothesisEquality,
productElimination,
independent_isectElimination,
applyEquality,
lambdaEquality,
setElimination,
rename,
addEquality,
because_Cache,
minusEquality,
dependent_functionElimination,
independent_functionElimination,
dependent_pairFormation,
dependent_set_memberEquality,
natural_numberEquality,
independent_pairFormation,
intEquality,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
promote_hyp,
instantiate,
cumulativity,
pointwiseFunctionality,
baseApply,
closedConclusion,
baseClosed,
approximateComputation,
int_eqEquality,
multiplyEquality,
lessCases,
imageMemberEquality,
isect_memberFormation,
axiomSqEquality,
imageElimination,
universeEquality
Latex:
\mforall{}a,b:\mBbbR{}. (|a - b| = (rmax(a;b) - rmin(a;b)))
Date html generated:
2019_10_29-AM-09_39_00
Last ObjectModification:
2018_08_20-PM-09_46_50
Theory : reals
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