Nuprl Lemma : l-last-is-last
∀[l:Top]. (l-last(l) ~ last(l))
Proof
Definitions occuring in Statement :
l-last: l-last(l),
last: last(L),
uall: ∀[x:A]. B[x],
top: Top,
sqequal: s ~ 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,
or: P ∨ Q,
cons: [a / b],
le: A ≤ B,
less_than': less_than'(a;b),
colength: colength(L),
nil: [],
it: ⋅,
guard: {T},
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,
bool: 𝔹,
unit: Unit,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
bnot: ¬bb,
assert: ↑b,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
list_ind: list_ind,
l-last-default: l-last-default(l;d),
l-last: l-last(l),
cand: A c∧ B,
is-exception: is-exception(t),
has-value: (a)↓,
strict4: strict4(F),
so_apply: x[s1;s2;s3;s4],
so_lambda: so_lambda(x,y,z,w.t[x; y; z; w]),
nat_plus: ℕ+,
select: L[n],
last: last(L),
true: True,
label: ...$L... t,
subtract: n - m,
pi2: snd(t),
pi1: fst(t),
length: ||as||,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
null: null(as)
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,
istype-less_than,
top_wf,
list-cases,
l-last-nil,
last_nil,
product_subtype_list,
colength-cons-not-zero,
colength_wf_list,
istype-false,
le_wf,
subtract-1-ge-0,
subtype_base_sq,
intformeq_wf,
int_formula_prop_eq_lemma,
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,
l-last-cons,
last_cons2,
null_wf,
eqtt_to_assert,
assert_of_null,
eqff_to_assert,
bool_subtype_base,
bool_cases_sqequal,
bool_wf,
assert-bnot,
iff_weakening_uiff,
assert_wf,
equal-wf-T-base,
list_wf,
nat_wf,
has-value-l-last-list,
has-value_wf_base,
is-exception_wf,
has-value-last-list,
istype-top,
base_wf,
sqle_wf_base,
has-value-implies-dec-isaxiom-2,
has-value-implies-dec-ispair-2,
exception-not-bottom,
bottom_diverge,
lifting-strict-callbyvalue,
fun_exp_unroll_1,
exception-not-bottom_1,
strictness-apply,
fun_exp0_lemma,
less_than_wf,
satisfiable-full-omega-tt,
equal-wf-base,
equal_wf,
assert_of_lt_int,
lt_int_wf,
length_of_cons_lemma,
int-value-type,
value-type-has-value,
length-nat-if-has-value,
false_wf,
pair-eta,
set-value-type,
exception-not-value,
length_of_nil_lemma,
istype-base,
decidable__lt,
add-associates,
add-swap,
add-commutes,
zero-add,
length-zero-implies-nil,
list_ind_cons_lemma,
list_ind_nil_lemma,
bfalse_wf,
exception-not-value_1
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
sqequalRule,
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,
independent_pairFormation,
Error :universeIsType,
axiomSqEquality,
Error :functionIsTypeImplies,
Error :inhabitedIsType,
unionElimination,
promote_hyp,
hypothesis_subsumption,
productElimination,
Error :equalityIsType1,
because_Cache,
Error :dependent_set_memberEquality_alt,
instantiate,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
imageElimination,
Error :equalityIsType4,
baseApply,
closedConclusion,
baseClosed,
applyEquality,
intEquality,
equalityElimination,
Error :equalityIsType3,
cumulativity,
sqequalSqle,
divergentSqle,
sqleReflexivity,
lambdaFormation,
axiomSqleEquality,
exceptionSqequal,
exceptionCompactness,
callbyvalueReduce,
callbyvalueExceptionCases,
sqleRule,
sqleTransitivity,
inlFormation,
inrFormation,
applyExceptionCases,
callbyvalueApply,
dependent_set_memberEquality,
imageMemberEquality,
computeAll,
voidEquality,
isect_memberEquality,
lambdaEquality,
dependent_pairFormation,
sqequalIntensionalEquality,
productEquality,
spreadExceptionCases,
isect_memberFormation,
lessCases,
addEquality,
callbyvalueAdd,
lessExceptionCases,
addExceptionCases,
minusEquality,
levelHypothesis,
addLevel,
Error :isectIsTypeImplies,
callbyvalueCallbyvalue
Latex:
\mforall{}[l:Top]. (l-last(l) \msim{} last(l))
Date html generated:
2019_06_20-PM-01_20_47
Last ObjectModification:
2018_10_17-AM-11_51_01
Theory : list_1
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