Nuprl Lemma : firstn_last
∀[T:Type]. ∀[L:T List]. L = (firstn(||L|| - 1;L) @ [last(L)]) ∈ (T List) supposing ¬↑null(L)
Proof
Definitions occuring in Statement :
firstn: firstn(n;as),
last: last(L),
length: ||as||,
null: null(as),
append: as @ bs,
cons: [a / b],
nil: [],
list: T List,
assert: ↑b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
not: ¬A,
subtract: n - m,
natural_number: $n,
universe: Type,
equal: s = t ∈ 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,
ge: i ≥ j ,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
and: P ∧ Q,
prop: ℙ,
guard: {T},
or: P ∨ Q,
firstn: firstn(n;as),
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
append: as @ bs,
true: True,
cons: [a / b],
le: A ≤ B,
less_than': less_than'(a;b),
colength: colength(L),
nil: [],
it: ⋅,
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
less_than: a < b,
squash: ↓T,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
decidable: Dec(P),
subtype_rel: A ⊆r B,
bfalse: ff,
bool: 𝔹,
unit: Unit,
uiff: uiff(P;Q),
subtract: n - m,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
rev_uimplies: rev_uimplies(P;Q)
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,
less_than_wf,
intformeq_wf,
int_formula_prop_eq_lemma,
list-cases,
null_nil_lemma,
list_ind_nil_lemma,
not_wf,
true_wf,
product_subtype_list,
colength-cons-not-zero,
colength_wf_list,
istype-false,
le_wf,
assert_wf,
null_wf,
subtract-1-ge-0,
subtype_base_sq,
set_subtype_base,
int_subtype_base,
spread_cons_lemma,
decidable__equal_int,
subtract_wf,
intformnot_wf,
itermSubtract_wf,
itermAdd_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
int_term_value_add_lemma,
decidable__le,
null_cons_lemma,
length_of_cons_lemma,
false_wf,
nat_wf,
list_wf,
lt_int_wf,
length_wf,
equal-wf-T-base,
bool_wf,
list_ind_cons_lemma,
le_int_wf,
bnot_wf,
uiff_transitivity,
eqtt_to_assert,
assert_of_lt_int,
eqff_to_assert,
assert_functionality_wrt_uiff,
bnot_of_lt_int,
assert_of_le_int,
cons_wf,
squash_wf,
istype-universe,
equal_wf,
length_of_nil_lemma,
append_wf,
firstn_wf,
last_wf,
bfalse_wf,
assert_elim,
btrue_neq_bfalse,
nil_wf,
subtype_rel_self,
iff_weakening_equal,
last_cons,
add-subtract-cancel,
last_singleton,
length_zero,
non_neg_length
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
thin,
Error :lambdaFormation_alt,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
Error :dependent_pairFormation_alt,
Error :lambdaEquality_alt,
int_eqEquality,
dependent_functionElimination,
Error :isect_memberEquality_alt,
voidElimination,
sqequalRule,
independent_pairFormation,
Error :universeIsType,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
Error :functionIsTypeImplies,
Error :inhabitedIsType,
unionElimination,
promote_hyp,
hypothesis_subsumption,
productElimination,
Error :equalityIsType1,
because_Cache,
Error :dependent_set_memberEquality_alt,
instantiate,
imageElimination,
Error :equalityIsType4,
baseApply,
closedConclusion,
baseClosed,
applyEquality,
intEquality,
universeEquality,
addEquality,
equalityElimination,
imageMemberEquality,
hyp_replacement
Latex:
\mforall{}[T:Type]. \mforall{}[L:T List]. L = (firstn(||L|| - 1;L) @ [last(L)]) supposing \mneg{}\muparrow{}null(L)
Date html generated:
2019_06_20-PM-01_34_42
Last ObjectModification:
2018_10_06-AM-11_23_14
Theory : list_1
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