Nuprl Lemma : l_disjoint_from-upto
∀[L:ℤ List]. ∀n1:ℤ. ∀[n2:ℤ]. uiff(∀x:ℤ. ((x ∈ L) ⇒ (x < n1 ∨ (x ≥ n2 )));l_disjoint(ℤ;L;[n1, n2)))
Proof
Definitions occuring in Statement :
from-upto: [n, m),
l_disjoint: l_disjoint(T;l1;l2),
l_member: (x ∈ l),
list: T List,
less_than: a < b,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
ge: i ≥ j ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
or: P ∨ Q,
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
member: t ∈ T,
l_disjoint: l_disjoint(T;l1;l2),
not: ¬A,
implies: P ⇒ Q,
false: False,
subtype_rel: A ⊆r B,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
or: P ∨ Q,
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
decidable: Dec(P),
iff: P ⇐⇒ Q,
rev_uimplies: rev_uimplies(P;Q),
guard: {T},
sq_stable: SqStable(P),
squash: ↓T
Lemmas referenced :
sq_stable__le,
int_formula_prop_not_lemma,
intformnot_wf,
decidable__lt,
decidable__le,
not_wf,
not_over_and,
int_formula_prop_wf,
int_formula_prop_le_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_and_lemma,
intformle_wf,
itermVar_wf,
intformless_wf,
intformand_wf,
satisfiable-full-omega-tt,
from-upto-member,
list_wf,
l_disjoint_wf,
ge_wf,
or_wf,
all_wf,
less_than_wf,
le_wf,
subtype_rel_list,
from-upto_wf,
l_member_wf,
and_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
independent_pairFormation,
introduction,
cut,
sqequalRule,
sqequalHypSubstitution,
lambdaEquality,
dependent_functionElimination,
thin,
hypothesisEquality,
voidElimination,
lemma_by_obid,
isectElimination,
intEquality,
hypothesis,
applyEquality,
setEquality,
because_Cache,
independent_isectElimination,
setElimination,
rename,
functionEquality,
productElimination,
independent_functionElimination,
unionElimination,
natural_numberEquality,
dependent_pairFormation,
int_eqEquality,
isect_memberEquality,
voidEquality,
computeAll,
inlFormation,
inrFormation,
imageMemberEquality,
baseClosed,
imageElimination
Latex:
\mforall{}[L:\mBbbZ{} List]. \mforall{}n1:\mBbbZ{}. \mforall{}[n2:\mBbbZ{}]. uiff(\mforall{}x:\mBbbZ{}. ((x \mmember{} L) {}\mRightarrow{} (x < n1 \mvee{} (x \mgeq{} n2 )));l\_disjoint(\mBbbZ{};L;[n1, n2)))
Date html generated:
2016_05_14-PM-02_01_50
Last ObjectModification:
2016_01_15-AM-08_08_55
Theory : list_1
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