Nuprl Lemma : firstn_decomp
∀[T:Type]. ∀[j:ℕ]. ∀[l:T List]. (firstn(j - 1;l) @ [l[j - 1]] ~ firstn(j;l)) supposing (j - 1 < ||l|| and 0 < j)
Proof
Definitions occuring in Statement :
firstn: firstn(n;as),
select: L[n],
length: ||as||,
append: as @ bs,
cons: [a / b],
nil: [],
list: T List,
nat: ℕ,
less_than: a < b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
subtract: n - m,
natural_number: $n,
universe: Type,
sqequal: s ~ t
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
guard: {T},
uimplies: b supposing a,
prop: ℙ,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
and: P ∧ Q,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
not: ¬A,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
subtract: n - m,
subtype_rel: A ⊆r B,
top: Top,
le: A ≤ B,
true: True,
sq_type: SQType(T),
firstn: firstn(n;as),
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
lt_int: i <z j,
select: L[n],
cons: [a / b],
ifthenelse: if b then t else f fi ,
bfalse: ff,
btrue: tt,
append: as @ bs,
bool: 𝔹,
unit: Unit,
it: ⋅,
exists: ∃x:A. B[x],
bnot: ¬bb,
assert: ↑b
Lemmas referenced :
nat_properties,
less_than_transitivity1,
less_than_irreflexivity,
ge_wf,
less_than_wf,
subtract_wf,
length_wf,
list_wf,
decidable__le,
false_wf,
not-ge-2,
less-iff-le,
condition-implies-le,
minus-one-mul,
zero-add,
minus-one-mul-top,
minus-add,
minus-minus,
add-associates,
add-swap,
add-commutes,
add_functionality_wrt_le,
add-zero,
le-add-cancel,
nat_wf,
decidable__equal_int,
subtype_base_sq,
int_subtype_base,
list_decomp,
decidable__lt,
not-lt-2,
le_antisymmetry_iff,
list_ind_cons_lemma,
list_ind_nil_lemma,
first0,
tl_wf,
subtype_rel_list,
top_wf,
not-equal-2,
le-add-cancel2,
squash_wf,
true_wf,
length_tl,
iff_weakening_equal,
lt_int_wf,
bool_wf,
equal-wf-base,
assert_wf,
eqtt_to_assert,
assert_of_lt_int,
select-cons-tl,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
le_int_wf,
le_wf,
bnot_wf,
uiff_transitivity,
assert_functionality_wrt_uiff,
bnot_of_lt_int,
assert_of_le_int
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
lambdaFormation,
natural_numberEquality,
independent_isectElimination,
independent_functionElimination,
voidElimination,
sqequalRule,
lambdaEquality,
dependent_functionElimination,
isect_memberEquality,
sqequalAxiom,
cumulativity,
equalityTransitivity,
equalitySymmetry,
imageElimination,
productElimination,
because_Cache,
unionElimination,
independent_pairFormation,
addEquality,
applyEquality,
voidEquality,
intEquality,
minusEquality,
universeEquality,
instantiate,
imageMemberEquality,
baseClosed,
baseApply,
closedConclusion,
equalityElimination,
dependent_pairFormation,
promote_hyp
Latex:
\mforall{}[T:Type]. \mforall{}[j:\mBbbN{}]. \mforall{}[l:T List].
(firstn(j - 1;l) @ [l[j - 1]] \msim{} firstn(j;l)) supposing (j - 1 < ||l|| and 0 < j)
Date html generated:
2017_04_14-AM-08_48_03
Last ObjectModification:
2017_02_27-PM-03_35_09
Theory : list_0
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