Nuprl Lemma : rleq-rmax
∀[x,y:ℝ]. ((x ≤ rmax(x;y)) ∧ (y ≤ rmax(x;y)))
Proof
Definitions occuring in Statement :
rleq: x ≤ y,
rmax: rmax(x;y),
real: ℝ,
uall: ∀[x:A]. B[x],
and: P ∧ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
rleq: x ≤ y,
rnonneg: rnonneg(x),
rmax: rmax(x;y),
rsub: x - y,
rminus: -(x),
radd: a + b,
accelerate: accelerate(k;f),
member: t ∈ T,
uall: ∀[x:A]. B[x],
nat_plus: ℕ+,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
and: P ∧ Q,
prop: ℙ,
implies: P ⇒ Q,
has-value: (a)↓,
uimplies: b supposing a,
so_lambda: λ2x.t[x],
so_apply: x[s],
real: ℝ,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
or: P ∨ Q,
guard: {T},
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
nequal: a ≠ b ∈ T ,
sq_type: SQType(T),
subtype_rel: A ⊆r B,
le: A ≤ B,
nat: ℕ,
rev_uimplies: rev_uimplies(P;Q),
ge: i ≥ j ,
cand: A c∧ B,
uiff: uiff(P;Q)
Lemmas referenced :
nat_plus_wf,
real_wf,
mul_nat_plus,
less_than_wf,
value-type-has-value,
set-value-type,
int-value-type,
equal_wf,
imax_ub,
nat_plus_properties,
decidable__le,
satisfiable-full-omega-tt,
intformnot_wf,
intformle_wf,
itermVar_wf,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
le_wf,
squash_wf,
true_wf,
reg-seq-list-add-as-l_sum,
cons_wf,
imax_wf,
nil_wf,
subtype_base_sq,
int_subtype_base,
equal-wf-base,
iff_weakening_equal,
map_cons_lemma,
map_nil_lemma,
l_sum_cons_lemma,
l_sum_nil_lemma,
intformand_wf,
itermConstant_wf,
itermAdd_wf,
itermMinus_wf,
int_formula_prop_and_lemma,
int_term_value_constant_lemma,
int_term_value_add_lemma,
int_term_value_minus_lemma,
false_wf,
le_functionality,
le_weakening,
div_bounds_1,
less_than'_wf,
rsub_wf,
rmax_wf,
rleq_functionality,
req_weakening,
rmax-com
Rules used in proof :
cut,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalRule,
callbyvalueReduce,
sqleReflexivity,
introduction,
extract_by_obid,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
dependent_set_memberEquality,
natural_numberEquality,
independent_pairFormation,
imageMemberEquality,
hypothesisEquality,
baseClosed,
independent_isectElimination,
intEquality,
lambdaEquality,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
independent_functionElimination,
setElimination,
rename,
applyEquality,
functionExtensionality,
because_Cache,
productElimination,
inlFormation,
applyLambdaEquality,
unionElimination,
dependent_pairFormation,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
imageElimination,
minusEquality,
divideEquality,
functionEquality,
addLevel,
instantiate,
cumulativity,
universeEquality,
addEquality,
isect_memberFormation,
independent_pairEquality,
axiomEquality
Latex:
\mforall{}[x,y:\mBbbR{}]. ((x \mleq{} rmax(x;y)) \mwedge{} (y \mleq{} rmax(x;y)))
Date html generated:
2017_10_03-AM-08_29_58
Last ObjectModification:
2017_07_28-AM-07_26_16
Theory : reals
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