Nuprl Lemma : last_lemma
∀[T:Type]. ∀L:T List. ∃L':T List. (L = (L' @ [last(L)]) ∈ (T List)) supposing ¬↑null(L)
Proof
Definitions occuring in Statement :
last: last(L),
null: null(as),
append: as @ bs,
cons: [a / b],
nil: [],
list: T List,
assert: ↑b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
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,
not: ¬A,
implies: P ⇒ Q,
false: False,
exists: ∃x:A. B[x],
prop: ℙ,
top: Top,
nat: ℕ,
int_iseg: {i...j},
and: P ∧ Q,
cand: A c∧ B,
or: P ∨ Q,
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
true: True,
cons: [a / b],
bfalse: ff,
guard: {T},
decidable: Dec(P),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
sq_stable: SqStable(P),
squash: ↓T,
subtract: n - m,
subtype_rel: A ⊆r B,
le: A ≤ B,
less_than': less_than'(a;b),
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
int_seg: {i..j-},
lelt: i ≤ j < k,
less_than: a < b,
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
last: last(L)
Lemmas referenced :
firstn_wf,
subtract_wf,
length_wf,
append_wf,
cons_wf,
last_wf,
nil_wf,
not_wf,
assert_wf,
null_wf,
list_wf,
list_extensionality,
length-append,
istype-void,
length_of_cons_lemma,
length_of_nil_lemma,
less_than_wf,
nat_wf,
length_firstn,
subtract_nat_wf,
list-cases,
null_nil_lemma,
product_subtype_list,
null_cons_lemma,
length_wf_nat,
decidable__le,
istype-false,
not-le-2,
sq_stable__le,
condition-implies-le,
minus-add,
istype-int,
minus-one-mul,
add-swap,
minus-one-mul-top,
add-associates,
add-commutes,
add_functionality_wrt_le,
add-zero,
le-add-cancel2,
nat_properties,
subtract-is-int-iff,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
itermSubtract_wf,
intformeq_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_term_value_subtract_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_wf,
false_wf,
le_wf,
decidable__equal_int,
itermAdd_wf,
int_term_value_add_lemma,
decidable__lt,
select_wf,
intformless_wf,
int_formula_prop_less_lemma,
equal_wf,
squash_wf,
true_wf,
istype-universe,
select_append_front,
select_firstn,
subtype_rel_self,
iff_weakening_equal,
length_firstn_eq,
add_functionality_wrt_eq,
select_append_back,
subtype_base_sq,
set_subtype_base,
int_subtype_base,
select-cons-hd,
minus-minus,
add-mul-special,
zero-add,
zero-mul
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
Error :lambdaFormation_alt,
cut,
introduction,
sqequalRule,
sqequalHypSubstitution,
Error :lambdaEquality_alt,
dependent_functionElimination,
thin,
hypothesisEquality,
voidElimination,
Error :functionIsTypeImplies,
Error :inhabitedIsType,
rename,
Error :dependent_pairFormation_alt,
extract_by_obid,
isectElimination,
hypothesis,
natural_numberEquality,
Error :equalityIsType1,
independent_isectElimination,
Error :universeIsType,
universeEquality,
Error :isect_memberEquality_alt,
setElimination,
Error :dependent_set_memberEquality_alt,
unionElimination,
independent_functionElimination,
promote_hyp,
hypothesis_subsumption,
productElimination,
addEquality,
independent_pairFormation,
imageMemberEquality,
baseClosed,
imageElimination,
applyEquality,
because_Cache,
minusEquality,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
pointwiseFunctionality,
baseApply,
closedConclusion,
approximateComputation,
int_eqEquality,
Error :productIsType,
instantiate,
cumulativity,
intEquality,
multiplyEquality
Latex:
\mforall{}[T:Type]. \mforall{}L:T List. \mexists{}L':T List. (L = (L' @ [last(L)])) supposing \mneg{}\muparrow{}null(L)
Date html generated:
2019_06_20-PM-01_19_42
Last ObjectModification:
2018_10_06-AM-11_23_30
Theory : list_1
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