Nuprl Lemma : rmin_strict_ub
∀x,y,z:ℝ. ((z < x) ∧ (z < y) ⇐⇒ z < rmin(x;y))
Proof
Definitions occuring in Statement :
rless: x < y,
rmin: rmin(x;y),
real: ℝ,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
rev_implies: P ⇐ Q,
exists: ∃x:A. B[x],
nat_plus: ℕ+,
guard: {T},
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
top: Top,
rmin: rmin(x;y),
squash: ↓T,
real: ℝ,
int_upper: {i...},
le: A ≤ B,
true: True,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
uiff: uiff(P;Q),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b
Lemmas referenced :
rless_wf,
rmin_wf,
real_wf,
rless-iff4,
imax_nat_plus,
nat_plus_wf,
nat_plus_properties,
decidable__lt,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
intformeq_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_wf,
equal_wf,
imax_wf,
less_than_wf,
squash_wf,
true_wf,
less_than_transitivity1,
imin_unfold,
iff_weakening_equal,
int_upper_wf,
all_wf,
int_upper_properties,
decidable__le,
intformle_wf,
int_formula_prop_le_lemma,
imax_lb,
le_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_le_int,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
le_wf,
int_upper_subtype_int_upper,
imax_ub,
rmin-rleq,
rless_transitivity1
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
independent_pairFormation,
sqequalHypSubstitution,
productElimination,
thin,
productEquality,
cut,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
hypothesis,
dependent_functionElimination,
independent_functionElimination,
because_Cache,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
setElimination,
rename,
unionElimination,
natural_numberEquality,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
computeAll,
dependent_set_memberEquality,
applyEquality,
imageElimination,
addEquality,
imageMemberEquality,
baseClosed,
universeEquality,
equalityElimination,
promote_hyp,
instantiate,
cumulativity,
inrFormation,
inlFormation
Latex:
\mforall{}x,y,z:\mBbbR{}. ((z < x) \mwedge{} (z < y) \mLeftarrow{}{}\mRightarrow{} z < rmin(x;y))
Date html generated:
2017_10_03-AM-08_30_20
Last ObjectModification:
2017_07_28-AM-07_26_31
Theory : reals
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