Nuprl Lemma : last-decomp2
∀[l:Top List]. (l ~ if (||l|| =z 0) then [] else firstn(||l|| - 1;l) @ [last(l)] fi )
Proof
Definitions occuring in Statement :
firstn: firstn(n;as),
last: last(L),
length: ||as||,
append: as @ bs,
cons: [a / b],
nil: [],
list: T List,
ifthenelse: if b then t else f fi ,
eq_int: (i =z j),
uall: ∀[x:A]. B[x],
top: Top,
subtract: n - m,
natural_number: $n,
sqequal: s ~ t
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
ifthenelse: if b then t else f fi ,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False,
cons: [a / b],
ge: i ≥ j ,
le: A ≤ B,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
prop: ℙ,
decidable: Dec(P),
nequal: a ≠ b ∈ T
Lemmas referenced :
eq_int_wf,
length_wf,
top_wf,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
list_wf,
list-cases,
product_subtype_list,
length_of_cons_lemma,
le_weakening2,
non_neg_length,
decidable__lt,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformeq_wf,
itermAdd_wf,
istype-int,
int_formula_prop_and_lemma,
int_formula_prop_le_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-decomp,
intformnot_wf,
intformless_wf,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesis,
hypothesisEquality,
natural_numberEquality,
inhabitedIsType,
lambdaFormation_alt,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
independent_isectElimination,
sqequalRule,
dependent_pairFormation_alt,
equalityIstype,
promote_hyp,
dependent_functionElimination,
instantiate,
because_Cache,
independent_functionElimination,
voidElimination,
axiomSqEquality,
universeIsType,
hypothesis_subsumption,
Error :memTop,
approximateComputation,
lambdaEquality_alt,
int_eqEquality,
independent_pairFormation
Latex:
\mforall{}[l:Top List]. (l \msim{} if (||l|| =\msubz{} 0) then [] else firstn(||l|| - 1;l) @ [last(l)] fi )
Date html generated:
2020_05_19-PM-09_44_00
Last ObjectModification:
2020_03_09-PM-00_43_04
Theory : list_1
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