Nuprl Lemma : map_is_append
∀[A,B:Type]. ∀[f:A ⟶ B]. ∀[L:A List]. ∀[L1,L2:B List].
{(map(f;firstn(||L1||;L)) = L1 ∈ (B List)) ∧ (map(f;nth_tl(||L1||;L)) = L2 ∈ (B List))}
supposing map(f;L) = (L1 @ L2) ∈ (B List)
Proof
Definitions occuring in Statement :
firstn: firstn(n;as),
length: ||as||,
nth_tl: nth_tl(n;as),
map: map(f;as),
append: as @ bs,
list: T List,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
guard: {T},
and: P ∧ Q,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
guard: {T},
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
not: ¬A,
top: Top,
and: P ∧ Q,
prop: ℙ,
subtype_rel: A ⊆r B,
or: P ∨ Q,
cons: [a / b],
colength: colength(L),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
decidable: Dec(P),
nil: [],
it: ⋅,
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
firstn: firstn(n;as),
cand: A c∧ B,
map: map(f;as),
list_ind: list_ind,
nth_tl: nth_tl(n;as),
le_int: i ≤z j,
lt_int: i <z j,
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
bfalse: ff,
btrue: tt,
le: A ≤ B,
uiff: uiff(P;Q),
bool: 𝔹,
unit: Unit,
true: True
Lemmas referenced :
nat_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
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,
less_than_wf,
equal_wf,
list_wf,
map_wf,
append_wf,
equal-wf-T-base,
nat_wf,
colength_wf_list,
less_than_transitivity1,
less_than_irreflexivity,
list-cases,
product_subtype_list,
spread_cons_lemma,
intformeq_wf,
itermAdd_wf,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
decidable__le,
intformnot_wf,
int_formula_prop_not_lemma,
le_wf,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
subtype_base_sq,
set_subtype_base,
int_subtype_base,
decidable__equal_int,
map_nil_lemma,
list_ind_nil_lemma,
nth_tl_nil,
length_wf,
nil_wf,
equal-wf-base-T,
list_ind_cons_lemma,
cons_wf,
null_nil_lemma,
btrue_wf,
and_wf,
null_wf3,
subtype_rel_list,
top_wf,
null_cons_lemma,
bfalse_wf,
btrue_neq_bfalse,
map_cons_lemma,
length_of_nil_lemma,
length_of_cons_lemma,
first0,
reduce_tl_cons_lemma,
tl_wf,
lt_int_wf,
bool_wf,
assert_wf,
le_int_wf,
add-is-int-iff,
false_wf,
bnot_wf,
add-subtract-cancel,
non_neg_length,
uiff_transitivity,
eqtt_to_assert,
assert_of_lt_int,
eqff_to_assert,
assert_functionality_wrt_uiff,
bnot_of_lt_int,
assert_of_le_int,
bnot_of_le_int,
squash_wf,
true_wf,
reduce_hd_cons_lemma,
hd_wf,
length_cons_ge_one
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
introduction,
cut,
thin,
lambdaFormation,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
independent_functionElimination,
productElimination,
independent_pairEquality,
axiomEquality,
cumulativity,
functionExtensionality,
applyEquality,
equalityTransitivity,
equalitySymmetry,
because_Cache,
unionElimination,
promote_hyp,
hypothesis_subsumption,
applyLambdaEquality,
dependent_set_memberEquality,
addEquality,
baseClosed,
instantiate,
imageElimination,
functionEquality,
universeEquality,
pointwiseFunctionality,
baseApply,
closedConclusion,
equalityElimination,
imageMemberEquality
Latex:
\mforall{}[A,B:Type]. \mforall{}[f:A {}\mrightarrow{} B]. \mforall{}[L:A List]. \mforall{}[L1,L2:B List].
\{(map(f;firstn(||L1||;L)) = L1) \mwedge{} (map(f;nth\_tl(||L1||;L)) = L2)\} supposing map(f;L) = (L1 @ L2)
Date html generated:
2018_05_21-PM-07_20_46
Last ObjectModification:
2017_07_26-PM-05_05_09
Theory : general
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