Nuprl Lemma : list-at-combine-skips
∀[ms,ns:colist(ℕ)]. ∀[k:ℕ]. ∀[T:Type]. ∀[L:colist(T)]. (L@combine-skips(ns;ms;k) = nth_tl(k;L)@ns@ms ∈ colist(T))
Proof
Definitions occuring in Statement :
combine-skips: combine-skips(as;bs;n),
list-at: L1@L2,
nth_tl: nth_tl(n;as),
colist: colist(T),
nat: ℕ,
uall: ∀[x:A]. B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
colist: colist(T),
corec: corec(T.F[T]),
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
all: ∀x:A. B[x],
top: Top,
and: P ∧ Q,
prop: ℙ,
lt_int: i <z j,
subtract: n - m,
ifthenelse: if b then t else f fi ,
btrue: tt,
bool: 𝔹,
unit: Unit,
it: ⋅,
uiff: uiff(P;Q),
bfalse: ff,
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
ext-eq: A ≡ B,
subtype_rel: A ⊆r B,
nil: [],
combine-skips: combine-skips(as;bs;n),
b-union: A ⋃ B,
tunion: ⋃x:A.B[x],
decidable: Dec(P),
pi2: snd(t),
cons: [a / b],
list-at: L1@L2,
null: null(as),
nth_tl: nth_tl(n;as),
le_int: i ≤z j,
squash: ↓T,
true: True,
co-nil: (),
so_lambda: λ2x.t[x],
so_apply: x[s],
tl: tl(l),
co-cons: [x / L]
Lemmas referenced :
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
istype-less_than,
primrec-unroll,
colist_wf,
istype-universe,
istype-nat,
nat_wf,
subtract-1-ge-0,
lt_int_wf,
eqtt_to_assert,
assert_of_lt_int,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
iff_weakening_uiff,
assert_wf,
less_than_wf,
colist-ext,
isaxiom_wf_listunion,
subtype_rel_b-union-left,
unit_wf2,
axiom-listunion,
null_nil_lemma,
reduce_tl_nil_lemma,
list_at_nil2_lemma,
btrue_wf,
it_wf,
ifthenelse_wf,
primrec_wf,
subtract_wf,
decidable__le,
intformnot_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
istype-le,
top_wf,
b-union_wf,
int_seg_wf,
subtype_rel_b-union-right,
non-axiom-listunion,
null_cons_lemma,
reduce_hd_cons_lemma,
reduce_tl_cons_lemma,
intformeq_wf,
int_formula_prop_eq_lemma,
int_subtype_base,
add-zero,
nth_tl_nil,
bfalse_wf,
equal_wf,
squash_wf,
true_wf,
list-at_wf,
subtype_rel_self,
iff_weakening_equal,
le_int_wf,
assert_of_le_int,
le_wf,
co-nil_wf,
itermAdd_wf,
int_term_value_add_lemma,
set_subtype_base,
decidable__equal_int,
nil-at,
co-cons_wf,
subtract-add-cancel
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
Error :isect_memberEquality_alt,
thin,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
Error :lambdaFormation_alt,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
Error :dependent_pairFormation_alt,
Error :lambdaEquality_alt,
int_eqEquality,
dependent_functionElimination,
voidElimination,
sqequalRule,
independent_pairFormation,
Error :universeIsType,
axiomEquality,
Error :functionIsTypeImplies,
Error :inhabitedIsType,
because_Cache,
instantiate,
universeEquality,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
Error :equalityIstype,
promote_hyp,
cumulativity,
hypothesis_subsumption,
applyEquality,
productEquality,
imageMemberEquality,
Error :dependent_pairEquality_alt,
Error :dependent_set_memberEquality_alt,
baseClosed,
sqequalBase,
int_eqReduceFalseSq,
Error :isectIsTypeImplies,
independent_pairEquality,
imageElimination,
Error :isectIsType,
baseApply,
closedConclusion,
intEquality,
axiomSqEquality,
addEquality
Latex:
\mforall{}[ms,ns:colist(\mBbbN{})]. \mforall{}[k:\mBbbN{}]. \mforall{}[T:Type]. \mforall{}[L:colist(T)].
(L@combine-skips(ns;ms;k) = nth\_tl(k;L)@ns@ms)
Date html generated:
2019_06_20-PM-01_21_43
Last ObjectModification:
2018_12_07-PM-06_27_53
Theory : list_1
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