Nuprl Lemma : last-lemma-sq
∀[T:Type]. ∀[L:T List]. L ~ firstn(||L|| - 1;L) @ [last(L)] 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,
sqequal: s ~ t
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
prop: ℙ,
not: ¬A,
implies: P ⇒ Q,
uiff: uiff(P;Q),
and: P ∧ Q,
gt: i > j,
all: ∀x:A. B[x],
or: P ∨ Q,
false: False,
cons: [a / b],
top: Top,
guard: {T},
nat: ℕ,
le: A ≤ B,
decidable: Dec(P),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
subtract: n - m,
subtype_rel: A ⊆r B,
less_than': less_than'(a;b),
true: True,
squash: ↓T,
int_iseg: {i...j},
cand: A c∧ B,
less_than: a < b,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
last: last(L),
ge: i ≥ j ,
int_seg: {i..j-},
lelt: i ≤ j < k
Lemmas referenced :
not_wf,
assert_wf,
null_wf,
list_wf,
assert_of_null,
equal-wf-T-base,
list-cases,
length_of_nil_lemma,
nil_wf,
product_subtype_list,
length_of_cons_lemma,
length_wf_nat,
nat_wf,
decidable__lt,
false_wf,
not-lt-2,
condition-implies-le,
minus-add,
minus-one-mul,
zero-add,
minus-one-mul-top,
add-commutes,
add_functionality_wrt_le,
add-associates,
add-zero,
le-add-cancel,
equal_wf,
list_decomp,
nth_tl_wf,
subtract_wf,
length_wf,
less_than_wf,
squash_wf,
true_wf,
length_nth_tl,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermSubtract_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_subtract_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
le_wf,
iff_weakening_equal,
select-nthtl,
subtype_rel_list,
top_wf,
decidable__equal_int,
intformeq_wf,
int_formula_prop_eq_lemma,
reduce_tl_nil_lemma,
reduce_tl_cons_lemma,
equal-wf-base,
non_neg_length,
nat_properties,
itermAdd_wf,
int_term_value_add_lemma,
append_firstn_lastn_sq,
lelt_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
hypothesis,
sqequalAxiom,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
sqequalRule,
isect_memberEquality,
because_Cache,
equalityTransitivity,
equalitySymmetry,
universeEquality,
lambdaFormation,
independent_functionElimination,
productElimination,
independent_isectElimination,
baseClosed,
promote_hyp,
dependent_functionElimination,
unionElimination,
voidElimination,
hypothesis_subsumption,
voidEquality,
setElimination,
rename,
natural_numberEquality,
addEquality,
independent_pairFormation,
applyEquality,
lambdaEquality,
intEquality,
minusEquality,
imageElimination,
dependent_set_memberEquality,
dependent_pairFormation,
int_eqEquality,
computeAll,
productEquality,
imageMemberEquality,
applyLambdaEquality
Latex:
\mforall{}[T:Type]. \mforall{}[L:T List]. L \msim{} firstn(||L|| - 1;L) @ [last(L)] supposing \mneg{}\muparrow{}null(L)
Date html generated:
2017_04_17-AM-07_33_17
Last ObjectModification:
2017_02_27-PM-04_11_03
Theory : list_1
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