Nuprl Lemma : list_ind-as-fix
∀[L:Top List]. ∀[x,F:Top].
(rec-case(L) of
[] => x
b::bs =>
r.F[r;b;bs] ~ fix((λR,L. if null(L) then x else F (R tl(L)) hd(L) tl(L) fi )) L)
Proof
Definitions occuring in Statement :
hd: hd(l),
null: null(as),
tl: tl(l),
list_ind: list_ind,
list: T List,
ifthenelse: if b then t else f fi ,
uall: ∀[x:A]. B[x],
top: Top,
so_apply: x[s1;s2;s3],
apply: f a,
fix: fix(F),
lambda: λx.A[x],
sqequal: s ~ t
Definitions unfolded in proof :
so_apply: x[s1;s2;s3],
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,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
not: ¬A,
top: Top,
and: P ∧ Q,
prop: ℙ,
subtype_rel: A ⊆r B,
guard: {T},
or: P ∨ Q,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
ifthenelse: if b then t else f fi ,
btrue: tt,
cons: [a / b],
colength: colength(L),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
decidable: Dec(P),
nil: [],
it: ⋅,
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
bfalse: ff
Lemmas referenced :
nat_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
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,
less_than_wf,
top_wf,
equal-wf-T-base,
nat_wf,
colength_wf_list,
less_than_transitivity1,
less_than_irreflexivity,
list_wf,
list-cases,
list_ind_nil_lemma,
null_nil_lemma,
reduce_tl_nil_lemma,
product_subtype_list,
spread_cons_lemma,
intformeq_wf,
itermAdd_wf,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
decidable__le,
intformnot_wf,
int_formula_prop_not_lemma,
le_wf,
equal_wf,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
subtype_base_sq,
set_subtype_base,
int_subtype_base,
decidable__equal_int,
list_ind_cons_lemma,
null_cons_lemma,
reduce_tl_cons_lemma,
reduce_hd_cons_lemma
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
introduction,
cut,
thin,
lambdaFormation,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
independent_functionElimination,
sqequalAxiom,
applyEquality,
because_Cache,
unionElimination,
promote_hyp,
hypothesis_subsumption,
productElimination,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
dependent_set_memberEquality,
addEquality,
baseClosed,
instantiate,
cumulativity,
imageElimination
Latex:
\mforall{}[L:Top List]. \mforall{}[x,F:Top].
(rec-case(L) of
[] => x
b::bs =>
r.F[r;b;bs] \msim{} fix((\mlambda{}R,L. if null(L) then x else F (R tl(L)) hd(L) tl(L) fi )) L)
Date html generated:
2017_10_01-AM-09_10_59
Last ObjectModification:
2017_07_26-PM-04_47_13
Theory : general
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