Nuprl Lemma : select-append
∀[L1,L2:Top List]. ∀[i:ℕ]. (L1 @ L2[i] ~ if i <z ||L1|| then L1[i] else L2[i - ||L1||] fi )
Proof
Definitions occuring in Statement :
select: L[n]
,
length: ||as||
,
append: as @ bs
,
list: T List
,
nat: ℕ
,
ifthenelse: if b then t else f fi
,
lt_int: i <z j
,
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
,
append: as @ bs
,
so_lambda: so_lambda3,
so_apply: x[s1;s2;s3]
,
select: L[n]
,
nil: []
,
it: ⋅
,
so_lambda: λ2x y.t[x; y]
,
so_apply: x[s1;s2]
,
cons: [a / b]
,
less_than': less_than'(a;b)
,
not: ¬A
,
colength: colength(L)
,
so_lambda: λ2x.t[x]
,
so_apply: x[s]
,
sq_type: SQType(T)
,
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,
stuck-spread,
istype-base,
istype-nat,
product_subtype_list,
colength-cons-not-zero,
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__int_equal,
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,
lt_int_wf,
equal-wf-base,
bool_wf,
istype-int,
assert_wf,
less_than_wf,
le_int_wf,
bnot_wf,
minus-zero,
add-zero,
uiff_transitivity,
eqtt_to_assert,
assert_of_lt_int,
eqff_to_assert,
assert_functionality_wrt_uiff,
bnot_of_lt_int,
assert_of_le_int,
length_wf,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
iff_weakening_uiff,
non_neg_length,
length_wf_nat,
istype-sqequal,
not-lt-2,
le_reflexive,
one-mul,
add-mul-special,
two-mul,
mul-distributes-right,
zero-mul,
omega-shadow,
mul-distributes,
mul-commutes,
mul-associates,
decidable__le,
not-le-2,
less-iff-le,
add-is-int-iff,
select-cons,
bnot_of_le_int
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,
baseClosed,
because_Cache,
promote_hyp,
hypothesis_subsumption,
equalityIstype,
dependent_set_memberEquality_alt,
instantiate,
cumulativity,
intEquality,
equalityTransitivity,
equalitySymmetry,
imageMemberEquality,
applyLambdaEquality,
addEquality,
minusEquality,
baseApply,
closedConclusion,
applyEquality,
sqequalBase,
equalityElimination,
dependent_pairFormation_alt,
multiplyEquality
Latex:
\mforall{}[L1,L2:Top List]. \mforall{}[i:\mBbbN{}]. (L1 @ L2[i] \msim{} if i <z ||L1|| then L1[i] else L2[i - ||L1||] fi )
Date html generated:
2020_05_19-PM-09_37_10
Last ObjectModification:
2020_03_09-PM-01_25_49
Theory : list_0
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