Nuprl Lemma : length_segment
∀[T:Type]. ∀[as:T List]. ∀[i:{0...||as||}]. ∀[j:{i...||as||}]. (||as[i..j-]|| = (j - i) ∈ ℤ)
Proof
Definitions occuring in Statement :
segment: as[m..n-],
length: ||as||,
list: T List,
int_iseg: {i...j},
uall: ∀[x:A]. B[x],
subtract: n - m,
natural_number: $n,
int: ℤ,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
member: t ∈ T,
uall: ∀[x:A]. B[x],
int_iseg: {i...j},
segment: as[m..n-],
and: P ∧ Q,
cand: A c∧ B,
all: ∀x:A. B[x],
implies: P ⇒ Q,
exists: ∃x:A. B[x],
subtype_rel: A ⊆r B,
nat: ℕ,
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
prop: ℙ,
squash: ↓T,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
le: A ≤ B,
subtract: n - m,
top: Top,
uiff: uiff(P;Q),
nat_plus: ℕ+,
less_than: a < b,
less_than': less_than'(a;b),
not: ¬A,
false: False,
decidable: Dec(P),
or: P ∨ Q
Lemmas referenced :
int_iseg_wf,
length_wf,
list_wf,
length_firstn,
nth_tl_wf,
subtract_wf,
non_neg_length,
length_wf_nat,
nat_wf,
set_subtype_base,
le_wf,
int_subtype_base,
equal_wf,
squash_wf,
true_wf,
length_nth_tl,
iff_weakening_equal,
minus-one-mul,
add-commutes,
not-le-2,
add_functionality_wrt_le,
le_reflexive,
minus-one-mul-top,
add-associates,
minus-zero,
add-zero,
one-mul,
zero-add,
add-swap,
add-mul-special,
zero-mul,
two-mul,
mul-distributes-right,
omega-shadow,
less_than_wf,
mul-distributes,
mul-associates,
le-add-cancel,
add-is-int-iff,
minus-add,
mul-commutes,
le-add-cancel-alt,
int_iseg_properties,
nat_properties,
decidable__le
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
hypothesis,
cumulativity,
because_Cache,
natural_numberEquality,
universeEquality,
isect_memberFormation,
sqequalRule,
isect_memberEquality,
axiomEquality,
dependent_set_memberEquality,
lambdaFormation,
dependent_pairFormation,
sqequalIntensionalEquality,
applyEquality,
intEquality,
lambdaEquality,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
independent_functionElimination,
productElimination,
promote_hyp,
independent_pairFormation,
imageElimination,
imageMemberEquality,
baseClosed,
productEquality,
voidElimination,
voidEquality,
addEquality,
multiplyEquality,
minusEquality,
baseApply,
closedConclusion,
unionElimination
Latex:
\mforall{}[T:Type]. \mforall{}[as:T List]. \mforall{}[i:\{0...||as||\}]. \mforall{}[j:\{i...||as||\}]. (||as[i..j\msupminus{}]|| = (j - i))
Date html generated:
2017_04_14-AM-08_47_46
Last ObjectModification:
2017_02_27-PM-03_34_47
Theory : list_0
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