Nuprl Lemma : qmin-list-bounds
∀L:ℚ List
(0 < ||L||
⇒ (∀x:ℚ
((x ≤ qmin-list(L) ⇐⇒ (∀y∈L.x ≤ y))
∧ (qmin-list(L) ≤ x ⇐⇒ (∃y∈L. y ≤ x))
∧ (x < qmin-list(L) ⇐⇒ (∀y∈L.x < y))
∧ (qmin-list(L) < x ⇐⇒ (∃y∈L. y < x)))))
Proof
Definitions occuring in Statement :
qmin-list: qmin-list(L),
qle: r ≤ s,
qless: r < s,
rationals: ℚ,
l_exists: (∃x∈L. P[x]),
l_all: (∀x∈L.P[x]),
length: ||as||,
list: T List,
less_than: a < b,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
and: P ∧ Q,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
iff: P ⇐⇒ Q,
qmin: qmin(x;y),
prop: ℙ,
rev_implies: P ⇐ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
uimplies: b supposing a,
guard: {T},
ifthenelse: if b then t else f fi ,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
false: False,
not: ¬A,
cand: A c∧ B,
qmin-list: qmin-list(L)
Lemmas referenced :
rationals_wf,
istype-less_than,
length_wf,
list_wf,
combine-list-rel-and,
qmin_wf,
qle_wf,
q_le_wf,
eqtt_to_assert,
assert-q_le-eq,
iff_weakening_equal,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
qmin-assoc,
assert_wf,
bnot_wf,
not_wf,
istype-assert,
istype-void,
qle_complement_qorder,
qless_transitivity_1_qorder,
qle_weakening_lt_qorder,
qle_transitivity_qorder,
bool_cases,
iff_transitivity,
iff_weakening_uiff,
assert_of_bnot,
combine-list-rel-or,
uiff_transitivity2,
equal-wf-T-base,
qless_transitivity_2_qorder,
qless_wf,
qless_transitivity
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
independent_pairFormation,
universeIsType,
cut,
introduction,
extract_by_obid,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
natural_numberEquality,
hypothesisEquality,
dependent_functionElimination,
sqequalRule,
lambdaEquality_alt,
inhabitedIsType,
independent_functionElimination,
productElimination,
unionElimination,
equalityElimination,
because_Cache,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
dependent_pairFormation_alt,
equalityIstype,
promote_hyp,
instantiate,
cumulativity,
voidElimination,
productIsType,
functionIsType,
baseClosed,
unionIsType,
inlFormation_alt,
inrFormation_alt
Latex:
\mforall{}L:\mBbbQ{} List
(0 < ||L||
{}\mRightarrow{} (\mforall{}x:\mBbbQ{}
((x \mleq{} qmin-list(L) \mLeftarrow{}{}\mRightarrow{} (\mforall{}y\mmember{}L.x \mleq{} y))
\mwedge{} (qmin-list(L) \mleq{} x \mLeftarrow{}{}\mRightarrow{} (\mexists{}y\mmember{}L. y \mleq{} x))
\mwedge{} (x < qmin-list(L) \mLeftarrow{}{}\mRightarrow{} (\mforall{}y\mmember{}L.x < y))
\mwedge{} (qmin-list(L) < x \mLeftarrow{}{}\mRightarrow{} (\mexists{}y\mmember{}L. y < x)))))
Date html generated:
2020_05_20-AM-09_16_05
Last ObjectModification:
2020_01_06-PM-05_24_48
Theory : rationals
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