Nuprl Lemma : firstn-append
∀[L1,L2:Top List]. ∀[n:ℕ]. (firstn(n;L1 @ L2) ~ if n ≤z ||L1|| then firstn(n;L1) else L1 @ firstn(n - ||L1||;L2) fi )
Proof
Definitions occuring in Statement :
firstn: firstn(n;as),
length: ||as||,
append: as @ bs,
list: T List,
nat: ℕ,
le_int: i ≤z j,
ifthenelse: if b then t else f fi ,
uall: ∀[x:A]. B[x],
top: Top,
subtract: n - m,
sqequal: s ~ t
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
nat: ℕ,
implies: P ⇒ Q,
false: False,
and: P ∧ Q,
ge: i ≥ j ,
le: A ≤ B,
cand: A c∧ B,
less_than: a < b,
squash: ↓T,
guard: {T},
uimplies: b supposing a,
prop: ℙ,
or: P ∨ Q,
firstn: firstn(n;as),
append: as @ bs,
so_lambda: so_lambda3,
so_apply: x[s1;s2;s3],
cons: [a / b],
less_than': less_than'(a;b),
not: ¬A,
colength: colength(L),
nil: [],
it: ⋅,
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
sq_stable: SqStable(P),
decidable: Dec(P),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
subtract: n - m,
true: True,
subtype_rel: A ⊆r B,
bool: 𝔹,
unit: Unit,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
exists: ∃x:A. B[x],
bnot: ¬bb,
assert: ↑b,
nat_plus: ℕ+
Lemmas referenced :
nat_properties,
less_than_transitivity1,
less_than_irreflexivity,
ge_wf,
istype-less_than,
top_wf,
list-cases,
list_ind_nil_lemma,
length_of_nil_lemma,
product_subtype_list,
colength-cons-not-zero,
istype-nat,
colength_wf_list,
istype-void,
istype-le,
list_wf,
subtract-1-ge-0,
subtype_base_sq,
nat_wf,
set_subtype_base,
le_wf,
int_subtype_base,
spread_cons_lemma,
sq_stable__le,
decidable__equal_int,
subtract_wf,
istype-false,
not-equal-2,
condition-implies-le,
add-associates,
minus-add,
minus-one-mul,
add-swap,
minus-one-mul-top,
le_antisymmetry_iff,
add_functionality_wrt_le,
add-commutes,
zero-add,
le-add-cancel,
minus-minus,
list_ind_cons_lemma,
length_of_cons_lemma,
le_weakening2,
le_int_wf,
equal-wf-base,
bool_wf,
istype-int,
assert_wf,
lt_int_wf,
less_than_wf,
bnot_wf,
eqtt_to_assert,
assert_of_le_int,
eqff_to_assert,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
iff_weakening_uiff,
assert_of_lt_int,
minus-zero,
not-lt-2,
add-zero,
less-iff-le,
uiff_transitivity,
assert_functionality_wrt_uiff,
bnot_of_le_int,
bnot_of_lt_int,
length_wf,
equal-wf-T-base,
add-is-int-iff,
length_wf_nat,
decidable__le,
not-le-2,
non_neg_length,
istype-sqequal,
le_reflexive,
one-mul,
add-mul-special,
two-mul,
mul-distributes-right,
zero-mul,
omega-shadow,
mul-distributes,
mul-commutes,
mul-associates
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
thin,
lambdaFormation_alt,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
independent_pairFormation,
productElimination,
imageElimination,
natural_numberEquality,
independent_isectElimination,
independent_functionElimination,
voidElimination,
universeIsType,
sqequalRule,
lambdaEquality_alt,
dependent_functionElimination,
isect_memberEquality_alt,
axiomSqEquality,
isectIsTypeImplies,
inhabitedIsType,
functionIsTypeImplies,
unionElimination,
Error :memTop,
promote_hyp,
hypothesis_subsumption,
equalityIstype,
dependent_set_memberEquality_alt,
because_Cache,
instantiate,
cumulativity,
intEquality,
equalityTransitivity,
equalitySymmetry,
imageMemberEquality,
baseClosed,
applyLambdaEquality,
addEquality,
minusEquality,
baseApply,
closedConclusion,
applyEquality,
sqequalBase,
equalityElimination,
dependent_pairFormation_alt,
multiplyEquality
Latex:
\mforall{}[L1,L2:Top List]. \mforall{}[n:\mBbbN{}].
(firstn(n;L1 @ L2) \msim{} if n \mleq{}z ||L1|| then firstn(n;L1) else L1 @ firstn(n - ||L1||;L2) fi )
Date html generated:
2020_05_19-PM-09_37_57
Last ObjectModification:
2020_03_09-PM-01_32_32
Theory : list_0
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