Nuprl Lemma : list_ind-general-wf
∀[A:Type]. ∀[B:(A List) ⟶ ℙ]. ∀[x:B[[]]]. ∀[F:∀a:A. ∀L:A List. (B[L] ⇒ B[[a / L]])]. ∀[L:A List].
(rec-case(L) of
[] => x
h::t =>
r.F[h;t;r] ∈ B[L])
Proof
Definitions occuring in Statement :
list_ind: list_ind,
cons: [a / b],
nil: [],
list: T List,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2;s3],
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
prop: ℙ,
so_lambda: λ2x.t[x],
implies: P ⇒ Q,
so_apply: x[s],
nat: ℕ,
false: False,
ge: i ≥ j ,
guard: {T},
uimplies: b supposing a,
subtype_rel: A ⊆r B,
top: Top,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
not: ¬A,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
subtract: n - m,
le: A ≤ B,
less_than': less_than'(a;b),
true: True,
pi1: fst(t),
nil: [],
cons: [a / b],
pi2: snd(t),
sq_type: SQType(T)
Lemmas referenced :
list_wf,
all_wf,
cons_wf,
nil_wf,
nat_properties,
less_than_transitivity1,
less_than_irreflexivity,
ge_wf,
less_than_wf,
equal-wf-T-base,
colength_wf_list,
colength-zero,
subtype_rel_list,
top_wf,
list_ind_nil_lemma,
decidable__le,
subtract_wf,
false_wf,
not-ge-2,
less-iff-le,
condition-implies-le,
minus-one-mul,
zero-add,
minus-one-mul-top,
minus-add,
minus-minus,
add-associates,
add-swap,
add-commutes,
add_functionality_wrt_le,
add-zero,
le-add-cancel,
colength-positive2,
le_weakening2,
le_wf,
list_ind_cons_lemma,
list-cases,
product_subtype_list,
subtype_base_sq,
int_subtype_base,
nat_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
thin,
sqequalHypSubstitution,
dependent_functionElimination,
hypothesisEquality,
equalityTransitivity,
hypothesis,
equalitySymmetry,
sqequalRule,
axiomEquality,
extract_by_obid,
isectElimination,
cumulativity,
isect_memberEquality,
because_Cache,
lambdaEquality,
functionEquality,
applyEquality,
functionExtensionality,
universeEquality,
lambdaFormation,
setElimination,
rename,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
independent_functionElimination,
voidElimination,
intEquality,
voidEquality,
baseClosed,
unionElimination,
independent_pairFormation,
productElimination,
addEquality,
minusEquality,
dependent_set_memberEquality,
promote_hyp,
hypothesis_subsumption,
instantiate
Latex:
\mforall{}[A:Type]. \mforall{}[B:(A List) {}\mrightarrow{} \mBbbP{}]. \mforall{}[x:B[[]]]. \mforall{}[F:\mforall{}a:A. \mforall{}L:A List. (B[L] {}\mRightarrow{} B[[a / L]])]. \mforall{}[L:A List].
(rec-case(L) of
[] => x
h::t =>
r.F[h;t;r] \mmember{} B[L])
Date html generated:
2017_04_14-AM-07_54_41
Last ObjectModification:
2017_02_27-PM-03_21_30
Theory : list_0
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