Nuprl Lemma : single_iseg
∀[T:Type]. ∀L:T List. ∀x:T. ([x] ≤ L ⇐⇒ 0 < ||L|| ∧ (L[0] = x ∈ T))
Proof
Definitions occuring in Statement :
iseg: l1 ≤ l2,
select: L[n],
length: ||as||,
cons: [a / b],
nil: [],
list: T List,
less_than: a < b,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
natural_number: $n,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
or: P ∨ Q,
cons: [a / b],
select: L[n],
uimplies: b supposing a,
nil: [],
it: ⋅,
so_lambda: λ2x y.t[x; y],
top: Top,
so_apply: x[s1;s2],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
not: ¬A,
false: False,
prop: ℙ,
rev_implies: P ⇐ Q,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
nat_plus: ℕ+,
true: True,
guard: {T},
decidable: Dec(P),
uiff: uiff(P;Q),
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x]
Lemmas referenced :
list-cases,
product_subtype_list,
list_wf,
length_of_nil_lemma,
stuck-spread,
base_wf,
iseg_nil,
cons_wf,
nil_wf,
assert_elim,
null_wf,
null_cons_lemma,
bfalse_wf,
btrue_neq_bfalse,
iseg_wf,
less_than_wf,
equal-wf-base-T,
length_of_cons_lemma,
add_nat_plus,
length_wf_nat,
nat_plus_wf,
nat_plus_properties,
decidable__lt,
add-is-int-iff,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
itermAdd_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_term_value_add_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_wf,
false_wf,
equal_wf,
nil_iseg,
length_wf,
cons_iseg,
iff_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
hypothesisEquality,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesis,
dependent_functionElimination,
unionElimination,
promote_hyp,
hypothesis_subsumption,
productElimination,
sqequalRule,
cumulativity,
universeEquality,
baseClosed,
independent_isectElimination,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
independent_functionElimination,
equalityTransitivity,
equalitySymmetry,
because_Cache,
imageElimination,
productEquality,
natural_numberEquality,
dependent_set_memberEquality,
imageMemberEquality,
applyLambdaEquality,
setElimination,
rename,
pointwiseFunctionality,
baseApply,
closedConclusion,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
computeAll,
addEquality,
addLevel,
impliesFunctionality
Latex:
\mforall{}[T:Type]. \mforall{}L:T List. \mforall{}x:T. ([x] \mleq{} L \mLeftarrow{}{}\mRightarrow{} 0 < ||L|| \mwedge{} (L[0] = x))
Date html generated:
2017_04_17-AM-08_46_22
Last ObjectModification:
2017_02_27-PM-05_04_04
Theory : list_1
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