Nuprl Lemma : list_ind_reverse_wf_dependent
∀[A,B:Type].
∀nilcase:B. ∀F:B ⟶ (A List) ⟶ A ⟶ B. ∀P:(A List) ⟶ B ⟶ ℙ.
((P [] nilcase)
⇒ (∀L:A List. ∀x:A. ∀b:B. ((b = list_ind_reverse(L;nilcase;F) ∈ B) ⇒ (P L b) ⇒ (P (L @ [x]) (F b L x))))
⇒ (∀L:A List. (P L list_ind_reverse(L;nilcase;F))))
Proof
Definitions occuring in Statement :
list_ind_reverse: list_ind_reverse(L;nilcase;R),
append: as @ bs,
cons: [a / b],
nil: [],
list: T List,
uall: ∀[x:A]. B[x],
prop: ℙ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
apply: f a,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
so_apply: x[s],
list_ind_reverse: list_ind_reverse(L;nilcase;R),
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,
nequal: a ≠ b ∈ T ,
nat: ℕ,
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
top: Top,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
squash: ↓T,
int_iseg: {i...j},
cand: A c∧ B,
decidable: Dec(P),
true: True,
firstn: firstn(n;as),
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
append: as @ bs,
cons: [a / b]
Lemmas referenced :
nat_wf,
list_wf,
all_wf,
equal_wf,
list_ind_reverse_wf,
append_wf,
cons_wf,
nil_wf,
eq_int_wf,
length_wf,
bool_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,
nat_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformeq_wf,
itermVar_wf,
itermConstant_wf,
intformnot_wf,
int_formula_prop_and_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_not_lemma,
int_formula_prop_wf,
equal-wf-T-base,
length_wf_nat,
intformless_wf,
int_formula_prop_less_lemma,
int_subtype_base,
set_wf,
less_than_wf,
primrec-wf2,
length_zero,
iff_weakening_equal,
firstn_wf,
subtract_wf,
le_wf,
squash_wf,
true_wf,
length_firstn_eq,
decidable__le,
intformle_wf,
itermSubtract_wf,
int_formula_prop_le_lemma,
int_term_value_subtract_lemma,
length_firstn,
last_wf,
non_null_iff_length,
subtype_rel_list,
top_wf,
decidable__lt,
list-cases,
list_ind_nil_lemma,
length_of_nil_lemma,
null_nil_lemma,
product_subtype_list,
length_of_cons_lemma,
null_cons_lemma,
false_wf,
firstn_last
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
introduction,
extract_by_obid,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
sqequalRule,
lambdaEquality,
because_Cache,
functionEquality,
functionExtensionality,
applyEquality,
universeEquality,
natural_numberEquality,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
independent_isectElimination,
dependent_pairFormation,
promote_hyp,
dependent_functionElimination,
instantiate,
independent_functionElimination,
voidElimination,
applyLambdaEquality,
setElimination,
rename,
int_eqEquality,
intEquality,
isect_memberEquality,
voidEquality,
independent_pairFormation,
computeAll,
baseClosed,
baseApply,
closedConclusion,
imageElimination,
dependent_set_memberEquality,
productEquality,
imageMemberEquality,
hypothesis_subsumption,
hyp_replacement
Latex:
\mforall{}[A,B:Type].
\mforall{}nilcase:B. \mforall{}F:B {}\mrightarrow{} (A List) {}\mrightarrow{} A {}\mrightarrow{} B. \mforall{}P:(A List) {}\mrightarrow{} B {}\mrightarrow{} \mBbbP{}.
((P [] nilcase)
{}\mRightarrow{} (\mforall{}L:A List. \mforall{}x:A. \mforall{}b:B.
((b = list\_ind\_reverse(L;nilcase;F)) {}\mRightarrow{} (P L b) {}\mRightarrow{} (P (L @ [x]) (F b L x))))
{}\mRightarrow{} (\mforall{}L:A List. (P L list\_ind\_reverse(L;nilcase;F))))
Date html generated:
2017_04_17-AM-08_44_05
Last ObjectModification:
2017_02_27-PM-05_03_06
Theory : list_1
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