Nuprl Lemma : nth_tl_is_fseg
∀[T:Type]. ∀L1,L2:T List. (fseg(T;L1;L2) ⇐⇒ ∃n:ℕ||L2|| + 1. (L1 = nth_tl(n;L2) ∈ (T List)))
Proof
Definitions occuring in Statement :
fseg: fseg(T;L1;L2),
length: ||as||,
nth_tl: nth_tl(n;as),
list: T List,
int_seg: {i..j-},
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
add: n + m,
natural_number: $n,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
fseg: fseg(T;L1;L2),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
exists: ∃x:A. B[x],
member: t ∈ T,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
rev_implies: P ⇐ Q,
int_seg: {i..j-},
lelt: i ≤ j < k,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
le: A ≤ B,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
top: Top,
subtype_rel: A ⊆r B,
less_than': less_than'(a;b),
squash: ↓T,
true: True,
guard: {T},
less_than: a < b,
uiff: uiff(P;Q),
int_iseg: {i...j}
Lemmas referenced :
exists_wf,
list_wf,
equal_wf,
append_wf,
int_seg_wf,
length_wf,
nth_tl_wf,
non_neg_length,
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,
decidable__lt,
intformless_wf,
itermAdd_wf,
int_formula_prop_less_lemma,
int_term_value_add_lemma,
lelt_wf,
nth_tl_append,
int_seg_subtype,
false_wf,
le_wf,
squash_wf,
true_wf,
add_functionality_wrt_eq,
length_append,
subtype_rel_list,
top_wf,
iff_weakening_equal,
int_seg_properties,
add-is-int-iff,
firstn_wf,
append_firstn_lastn,
subtype_rel_sets
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
lambdaFormation,
independent_pairFormation,
sqequalHypSubstitution,
productElimination,
thin,
cut,
introduction,
extract_by_obid,
isectElimination,
cumulativity,
hypothesisEquality,
hypothesis,
lambdaEquality,
natural_numberEquality,
addEquality,
setElimination,
rename,
universeEquality,
dependent_pairFormation,
dependent_set_memberEquality,
dependent_functionElimination,
because_Cache,
unionElimination,
independent_isectElimination,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
addLevel,
applyEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
imageMemberEquality,
baseClosed,
independent_functionElimination,
applyLambdaEquality,
pointwiseFunctionality,
promote_hyp,
baseApply,
closedConclusion,
hyp_replacement,
equalityUniverse,
levelHypothesis,
productEquality,
setEquality
Latex:
\mforall{}[T:Type]. \mforall{}L1,L2:T List. (fseg(T;L1;L2) \mLeftarrow{}{}\mRightarrow{} \mexists{}n:\mBbbN{}||L2|| + 1. (L1 = nth\_tl(n;L2)))
Date html generated:
2017_04_17-AM-07_33_05
Last ObjectModification:
2017_02_27-PM-04_09_23
Theory : list_1
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