Nuprl Lemma : qmin-list-unique
∀[L:ℚ List]. ∀[a:ℚ]. uiff(qmin-list(L) = a ∈ ℚ;(∀b∈L.a ≤ b)) supposing (a ∈ L)
Proof
Definitions occuring in Statement :
qmin-list: qmin-list(L),
qle: r ≤ s,
rationals: ℚ,
l_all: (∀x∈L.P[x]),
l_member: (x ∈ l),
list: T List,
uiff: uiff(P;Q),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
uimplies: b supposing a,
implies: P ⇒ Q,
or: P ∨ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
false: False,
cons: [a / b],
top: Top,
guard: {T},
nat: ℕ,
le: A ≤ B,
decidable: Dec(P),
not: ¬A,
rev_implies: P ⇐ Q,
prop: ℙ,
uiff: uiff(P;Q),
subtract: n - m,
subtype_rel: A ⊆r B,
less_than': less_than'(a;b),
true: True,
l_all: (∀x∈L.P[x]),
int_seg: {i..j-},
lelt: i ≤ j < k,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
less_than: a < b,
squash: ↓T,
l_member: (x ∈ l),
l_exists: (∃x∈L. P[x]),
ge: i ≥ j ,
cand: A c∧ B,
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
qmin-list-bounds,
rationals_wf,
list-cases,
length_of_nil_lemma,
nil_member,
product_subtype_list,
length_of_cons_lemma,
length_wf_nat,
nat_wf,
decidable__lt,
false_wf,
not-lt-2,
condition-implies-le,
minus-add,
minus-one-mul,
zero-add,
minus-one-mul-top,
add-commutes,
add_functionality_wrt_le,
add-associates,
add-zero,
le-add-cancel,
equal_wf,
qle_weakening_eq_qorder,
qle_witness,
select_wf,
int_seg_properties,
length_wf,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
intformless_wf,
int_formula_prop_less_lemma,
int_seg_wf,
qmin-list_wf,
lelt_wf,
qle_wf,
nat_properties,
qle_antisymmetry,
l_all_wf2,
l_member_wf,
squash_wf,
true_wf
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_functionElimination,
isectElimination,
unionElimination,
sqequalRule,
productElimination,
voidElimination,
promote_hyp,
hypothesis_subsumption,
isect_memberEquality,
voidEquality,
lambdaFormation,
setElimination,
rename,
natural_numberEquality,
addEquality,
independent_pairFormation,
independent_isectElimination,
applyEquality,
because_Cache,
minusEquality,
equalityTransitivity,
equalitySymmetry,
lambdaEquality,
dependent_pairFormation,
int_eqEquality,
intEquality,
computeAll,
imageElimination,
dependent_set_memberEquality,
setEquality,
independent_pairEquality,
axiomEquality,
hyp_replacement,
imageMemberEquality,
baseClosed
Latex:
\mforall{}[L:\mBbbQ{} List]. \mforall{}[a:\mBbbQ{}]. uiff(qmin-list(L) = a;(\mforall{}b\mmember{}L.a \mleq{} b)) supposing (a \mmember{} L)
Date html generated:
2018_05_21-PM-11_55_56
Last ObjectModification:
2017_07_26-PM-06_46_33
Theory : rationals
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