Nuprl Lemma : rmin_strict_lb
∀x,y,z:ℝ. ((x < z) ∨ (y < z) ⇐⇒ rmin(x;y) < z)
Proof
Definitions occuring in Statement :
rless: x < y,
rmin: rmin(x;y),
real: ℝ,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
or: P ∨ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
or: P ∨ Q,
rless: x < y,
sq_exists: ∃x:{A| B[x]},
member: t ∈ T,
rmin: rmin(x;y),
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
real: ℝ,
prop: ℙ,
rev_implies: P ⇐ Q,
true: True,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
uimplies: b supposing a,
ifthenelse: if b then t else f fi ,
bfalse: ff,
exists: ∃x:A. B[x],
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False,
not: ¬A,
nat_plus: ℕ+,
decidable: Dec(P),
less_than: a < b,
squash: ↓T,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
le: A ≤ B
Lemmas referenced :
less_than_wf,
rmin_wf,
or_wf,
rless_wf,
real_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,
le_wf,
nat_plus_properties,
decidable__lt,
add-is-int-iff,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermAdd_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
false_wf,
squash_wf,
true_wf,
add_functionality_wrt_eq,
imin_unfold,
iff_weakening_equal,
assert_wf,
bnot_wf,
not_wf,
bool_cases,
iff_transitivity,
iff_weakening_uiff,
assert_of_bnot
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
independent_pairFormation,
sqequalHypSubstitution,
unionElimination,
thin,
setElimination,
rename,
introduction,
dependent_set_memberEquality,
hypothesisEquality,
sqequalRule,
cut,
extract_by_obid,
isectElimination,
addEquality,
applyEquality,
hypothesis,
lambdaEquality,
because_Cache,
natural_numberEquality,
intEquality,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
independent_isectElimination,
dependent_pairFormation,
promote_hyp,
dependent_functionElimination,
instantiate,
cumulativity,
independent_functionElimination,
voidElimination,
imageElimination,
pointwiseFunctionality,
baseApply,
closedConclusion,
baseClosed,
int_eqEquality,
isect_memberEquality,
voidEquality,
computeAll,
imageMemberEquality,
universeEquality,
impliesFunctionality,
inlFormation,
dependent_set_memberFormation,
inrFormation
Latex:
\mforall{}x,y,z:\mBbbR{}. ((x < z) \mvee{} (y < z) \mLeftarrow{}{}\mRightarrow{} rmin(x;y) < z)
Date html generated:
2017_10_03-AM-08_30_28
Last ObjectModification:
2017_07_28-AM-07_26_36
Theory : reals
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