Nuprl Lemma : listify_length
∀[T:Type]. ∀m,n:ℤ. ∀f:{m..n-} ⟶ T. (n < m ∨ (||listify(f;m;n)|| = (n - m) ∈ ℤ))
Proof
Definitions occuring in Statement :
length: ||as||,
listify: listify(f;m;n),
int_seg: {i..j-},
less_than: a < b,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
or: P ∨ Q,
function: x:A ⟶ B[x],
subtract: n - m,
int: ℤ,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
decidable: Dec(P),
or: P ∨ Q,
prop: ℙ,
iff: P ⇐⇒ Q,
and: P ∧ Q,
not: ¬A,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
false: False,
uiff: uiff(P;Q),
uimplies: b supposing a,
guard: {T},
subtract: n - m,
subtype_rel: A ⊆r B,
top: Top,
le: A ≤ B,
less_than': less_than'(a;b),
true: True,
so_lambda: λ2x.t[x],
int_lower: {...i},
so_apply: x[s],
listify: listify(f;m;n),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff
Lemmas referenced :
int_seg_wf,
decidable__le,
decidable__lt,
equal_wf,
length_wf,
listify_wf,
subtract_wf,
false_wf,
not-lt-2,
decidable__int_equal,
not-equal-2,
not-le-2,
condition-implies-le,
minus-add,
minus-one-mul,
add-swap,
minus-one-mul-top,
add-commutes,
add_functionality_wrt_le,
le-add-cancel,
or_wf,
less_than_wf,
int_lower_ind,
all_wf,
int_lower_wf,
le_int_wf,
bool_wf,
equal-wf-base,
int_subtype_base,
assert_wf,
le_wf,
add-mul-special,
zero-mul,
lt_int_wf,
bnot_wf,
less_than_irreflexivity,
uiff_transitivity,
eqtt_to_assert,
assert_of_le_int,
length_of_nil_lemma,
eqff_to_assert,
assert_functionality_wrt_uiff,
bnot_of_le_int,
assert_of_lt_int,
length_of_cons_lemma,
equal-wf-T-base,
add-associates,
le-add-cancel-alt,
subtype_rel_dep_function,
int_seg_subtype,
add-zero,
le_reflexive,
le_antisymmetry_iff,
minus-minus,
zero-add
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
functionEquality,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
cumulativity,
intEquality,
universeEquality,
dependent_functionElimination,
unionElimination,
inlFormation,
independent_pairFormation,
voidElimination,
productElimination,
independent_isectElimination,
sqequalRule,
inrFormation,
addEquality,
natural_numberEquality,
applyEquality,
lambdaEquality,
isect_memberEquality,
voidEquality,
because_Cache,
minusEquality,
independent_functionElimination,
addLevel,
orFunctionality,
instantiate,
setElimination,
rename,
functionExtensionality,
baseApply,
closedConclusion,
baseClosed,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
multiplyEquality,
dependent_set_memberEquality
Latex:
\mforall{}[T:Type]. \mforall{}m,n:\mBbbZ{}. \mforall{}f:\{m..n\msupminus{}\} {}\mrightarrow{} T. (n < m \mvee{} (||listify(f;m;n)|| = (n - m)))
Date html generated:
2017_04_14-AM-08_36_04
Last ObjectModification:
2017_02_27-PM-03_28_19
Theory : list_0
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