Nuprl Lemma : rminus-rmax
∀[x,y:ℝ]. (-(rmax(x;y)) = rmin(-(x);-(y)))
Proof
Definitions occuring in Statement :
rmin: rmin(x;y),
rmax: rmax(x;y),
req: x = y,
rminus: -(x),
real: ℝ,
uall: ∀[x:A]. B[x]
Definitions unfolded in proof :
subtract: n - m,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
nat: ℕ,
subtype_rel: A ⊆r B,
absval: |i|,
decidable: Dec(P),
le: A ≤ B,
squash: ↓T,
less_than': less_than'(a;b),
less_than: a < b,
top: Top,
satisfiable_int_formula: satisfiable_int_formula(fmla),
nat_plus: ℕ+,
not: ¬A,
false: False,
assert: ↑b,
bnot: ¬bb,
guard: {T},
sq_type: SQType(T),
or: P ∨ Q,
prop: ℙ,
exists: ∃x:A. B[x],
bfalse: ff,
ifthenelse: if b then t else f fi ,
uimplies: b supposing a,
and: P ∧ Q,
uiff: uiff(P;Q),
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
true: True,
real: ℝ,
implies: P ⇒ Q,
rmax: rmax(x;y),
rmin: rmin(x;y),
rminus: -(x),
all: ∀x:A. B[x],
req: x = y,
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
zero-mul,
add-mul-special,
add-commutes,
minus-minus,
minus-one-mul,
iff_weakening_equal,
subtype_rel_self,
imin_unfold,
imax_unfold,
minus_functionality_wrt_eq,
subtract_wf,
true_wf,
squash_wf,
nat_wf,
absval_wf,
false_wf,
int_formula_prop_less_lemma,
intformless_wf,
int_term_value_constant_lemma,
int_term_value_mul_lemma,
int_term_value_add_lemma,
itermConstant_wf,
itermMultiply_wf,
itermAdd_wf,
decidable__le,
top_wf,
assert_of_lt_int,
lt_int_wf,
absval_unfold,
int_formula_prop_wf,
int_term_value_minus_lemma,
int_term_value_var_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermMinus_wf,
itermVar_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
full-omega-unsat,
less_than_wf,
nat_plus_properties,
le_wf,
assert-bnot,
bool_subtype_base,
subtype_base_sq,
bool_cases_sqequal,
equal_wf,
eqff_to_assert,
assert_of_le_int,
eqtt_to_assert,
bool_wf,
le_int_wf,
ifthenelse_wf,
imax_wf,
real_wf,
rmin_wf,
rmax_wf,
rminus_wf,
req_witness,
nat_plus_wf
Rules used in proof :
universeEquality,
imageElimination,
baseClosed,
imageMemberEquality,
sqequalAxiom,
lessCases,
multiplyEquality,
addEquality,
independent_pairFormation,
voidEquality,
int_eqEquality,
lambdaEquality,
approximateComputation,
dependent_set_memberEquality,
voidElimination,
cumulativity,
instantiate,
dependent_functionElimination,
promote_hyp,
dependent_pairFormation,
independent_isectElimination,
productElimination,
equalitySymmetry,
equalityTransitivity,
equalityElimination,
unionElimination,
natural_numberEquality,
minusEquality,
intEquality,
rename,
setElimination,
applyEquality,
because_Cache,
isect_memberEquality,
independent_functionElimination,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
hypothesis,
extract_by_obid,
sqequalRule,
lambdaFormation,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[x,y:\mBbbR{}]. (-(rmax(x;y)) = rmin(-(x);-(y)))
Date html generated:
2018_05_22-PM-01_21_00
Last ObjectModification:
2018_05_21-AM-00_05_08
Theory : reals
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