Nuprl Lemma : repeat-on-success_wf
∀[A,B:Type].
(∀[C:Type]. ∀[f:A ⟶ (C?)]. ∀[g:(ℕ × C) ⟶ (A List) ⟶ (A List)].
∀[h:(ℕ × C) ⟶ B ⟶ B]. ∀[as:A List]. ∀[b:B]. (repeat-on-success(f;g;h;as;b) ∈ A List × B)
supposing ∃m:(A List) ⟶ ℕ
∀as:A List. ∀c:C. ∀j:ℕ.
((first-success(f;as) = (inl <j, c>) ∈ (ℕ × C?)) ⇒ m (g <j, c> as) < m as)) supposing
(value-type(B) and
valueall-type(A))
Proof
Definitions occuring in Statement :
repeat-on-success: repeat-on-success(f;g;h;as;b),
first-success: first-success(f;L),
list: T List,
nat: ℕ,
valueall-type: valueall-type(T),
value-type: value-type(T),
less_than: a < b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
unit: Unit,
member: t ∈ T,
apply: f a,
function: x:A ⟶ B[x],
pair: <a, b>,
product: x:A × B[x],
inl: inl x,
union: left + right,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
exists: ∃x:A. B[x],
all: ∀x:A. B[x],
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
guard: {T},
prop: ℙ,
subtype_rel: A ⊆r B,
squash: ↓T,
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,
top: Top,
le: A ≤ B,
less_than': less_than'(a;b),
true: True,
repeat-on-success: repeat-on-success(f;g;h;as;b),
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_stable: SqStable(P),
has-value: (a)↓
Lemmas referenced :
nat_properties,
less_than_transitivity1,
less_than_irreflexivity,
ge_wf,
less_than_wf,
list_wf,
nat_wf,
subtype_rel-equal,
base_wf,
le_weakening,
equal_wf,
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,
first-success_wf,
int_seg_wf,
length_wf,
unit_wf2,
add_nat_wf,
le_wf,
sq_stable__le,
decidable__lt,
not-lt-2,
add-mul-special,
zero-mul,
exists_wf,
all_wf,
subtype_rel_union,
subtype_rel_product,
int_seg_subtype_nat,
value-type_wf,
valueall-type_wf,
evalall-reduce,
list-valueall-type,
value-type-has-value,
list-value-type,
equal_functionality_wrt_subtype_rel2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
productElimination,
thin,
lambdaFormation,
extract_by_obid,
isectElimination,
hypothesisEquality,
hypothesis,
setElimination,
rename,
sqequalRule,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
independent_functionElimination,
voidElimination,
lambdaEquality,
dependent_functionElimination,
isect_memberEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
applyEquality,
functionExtensionality,
cumulativity,
because_Cache,
dependent_pairFormation,
sqequalIntensionalEquality,
applyLambdaEquality,
imageMemberEquality,
baseClosed,
promote_hyp,
unionElimination,
independent_pairFormation,
addEquality,
minusEquality,
voidEquality,
intEquality,
unionEquality,
productEquality,
dependent_set_memberEquality,
imageElimination,
multiplyEquality,
functionEquality,
inlEquality,
independent_pairEquality,
universeEquality,
callbyvalueReduce
Latex:
\mforall{}[A,B:Type].
(\mforall{}[C:Type]. \mforall{}[f:A {}\mrightarrow{} (C?)]. \mforall{}[g:(\mBbbN{} \mtimes{} C) {}\mrightarrow{} (A List) {}\mrightarrow{} (A List)].
\mforall{}[h:(\mBbbN{} \mtimes{} C) {}\mrightarrow{} B {}\mrightarrow{} B]. \mforall{}[as:A List]. \mforall{}[b:B]. (repeat-on-success(f;g;h;as;b) \mmember{} A List \mtimes{} B)
supposing \mexists{}m:(A List) {}\mrightarrow{} \mBbbN{}
\mforall{}as:A List. \mforall{}c:C. \mforall{}j:\mBbbN{}.
((first-success(f;as) = (inl <j, c>)) {}\mRightarrow{} m (g <j, c> as) < m as)) supposing
(value-type(B) and
valueall-type(A))
Date html generated:
2017_04_14-AM-08_39_46
Last ObjectModification:
2017_02_27-PM-03_30_28
Theory : list_0
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