Nuprl Lemma : list_ind_reverse_unfold
∀[A,B:Type].
∀nilcase:B. ∀F:B ⟶ (A List) ⟶ A ⟶ B. ∀L:A List.
((||L|| > 0)
⇒ (list_ind_reverse(L;nilcase;F) ~ F list_ind_reverse(firstn(||L|| - 1;L);nilcase;F) firstn(||L|| - 1;L) last(L)))
Proof
Definitions occuring in Statement :
list_ind_reverse: list_ind_reverse(L;nilcase;R),
firstn: firstn(n;as),
last: last(L),
length: ||as||,
list: T List,
uall: ∀[x:A]. B[x],
gt: i > j,
all: ∀x:A. B[x],
implies: P ⇒ Q,
apply: f a,
function: x:A ⟶ B[x],
subtract: n - m,
natural_number: $n,
universe: Type,
sqequal: s ~ t
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
implies: P ⇒ Q,
list_ind_reverse: list_ind_reverse(L;nilcase;R),
prop: ℙ,
gt: i > j,
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 ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
bfalse: ff,
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b
Lemmas referenced :
gt_wf,
length_wf,
list_wf,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
satisfiable-full-omega-tt,
intformand_wf,
intformeq_wf,
itermVar_wf,
itermConstant_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
sqequalRule,
hypothesis,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
natural_numberEquality,
functionEquality,
lambdaEquality,
dependent_functionElimination,
sqequalAxiom,
because_Cache,
universeEquality,
isect_memberEquality,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
independent_isectElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
promote_hyp,
instantiate,
independent_functionElimination
Latex:
\mforall{}[A,B:Type].
\mforall{}nilcase:B. \mforall{}F:B {}\mrightarrow{} (A List) {}\mrightarrow{} A {}\mrightarrow{} B. \mforall{}L:A List.
((||L|| > 0)
{}\mRightarrow{} (list\_ind\_reverse(L;nilcase;F) \msim{} F list\_ind\_reverse(firstn(||L|| - 1;L);nilcase;F)
firstn(||L|| - 1;L)
last(L)))
Date html generated:
2017_04_17-AM-08_44_17
Last ObjectModification:
2017_02_27-PM-05_01_25
Theory : list_1
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