Nuprl Lemma : list_decomp_last
∀[T:Type]. ∀L:T List. ∃L':T List. (L = (L' @ [last(L)]) ∈ (T List)) supposing 0 < ||L||
Proof
Definitions occuring in Statement :
last: last(L),
length: ||as||,
append: as @ bs,
cons: [a / b],
nil: [],
list: T List,
less_than: a < b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
natural_number: $n,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
uimplies: b supposing a,
member: t ∈ T,
so_lambda: λ2x.t[x],
implies: P ⇒ Q,
prop: ℙ,
or: P ∨ Q,
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
false: False,
and: P ∧ Q,
cons: [a / b],
top: Top,
bfalse: ff,
not: ¬A,
so_apply: x[s],
exists: ∃x:A. B[x],
decidable: Dec(P),
subtype_rel: A ⊆r B,
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
guard: {T},
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
satisfiable_int_formula: satisfiable_int_formula(fmla)
Lemmas referenced :
member-less_than,
length_wf,
list_induction,
less_than_wf,
exists_wf,
list_wf,
equal_wf,
append_wf,
cons_wf,
last_wf,
list-cases,
null_nil_lemma,
length_of_nil_lemma,
product_subtype_list,
null_cons_lemma,
length_of_cons_lemma,
false_wf,
nil_wf,
decidable__equal_int,
length-zero-implies-sq-nil,
subtype_rel_list,
top_wf,
list_ind_nil_lemma,
squash_wf,
true_wf,
last_singleton,
iff_weakening_equal,
bfalse_wf,
assert_elim,
btrue_neq_bfalse,
assert_wf,
null_wf,
decidable__lt,
add-is-int-iff,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
intformeq_wf,
itermAdd_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
int_formula_prop_wf,
last_cons,
list_ind_cons_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
natural_numberEquality,
cumulativity,
hypothesisEquality,
hypothesis,
independent_isectElimination,
rename,
sqequalRule,
lambdaEquality,
functionEquality,
because_Cache,
dependent_functionElimination,
unionElimination,
imageElimination,
productElimination,
voidElimination,
promote_hyp,
hypothesis_subsumption,
isect_memberEquality,
voidEquality,
independent_functionElimination,
addEquality,
universeEquality,
applyEquality,
dependent_pairFormation,
equalityTransitivity,
equalitySymmetry,
imageMemberEquality,
baseClosed,
addLevel,
levelHypothesis,
pointwiseFunctionality,
baseApply,
closedConclusion,
int_eqEquality,
intEquality,
independent_pairFormation,
computeAll,
equalityUniverse
Latex:
\mforall{}[T:Type]. \mforall{}L:T List. \mexists{}L':T List. (L = (L' @ [last(L)])) supposing 0 < ||L||
Date html generated:
2017_04_17-AM-08_44_46
Last ObjectModification:
2017_02_27-PM-05_02_38
Theory : list_1
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