Nuprl Lemma : isl-first-success
∀[T:Type]. ∀[A:T ⟶ Type].
∀f:x:T ⟶ (A[x]?). ∀L:T List.
((↑isl(first-success(f;L)))
⇒ (fst(outl(first-success(f;L))) < ||L||
∧ ((f L[fst(outl(first-success(f;L)))])
= (inl (snd(outl(first-success(f;L)))))
∈ (A[L[fst(outl(first-success(f;L)))]]?))
∧ (∀x∈firstn(fst(outl(first-success(f;L)));L).↑isr(f x))))
Proof
Definitions occuring in Statement :
firstn: firstn(n;as),
l_all: (∀x∈L.P[x]),
first-success: first-success(f;L),
select: L[n],
length: ||as||,
list: T List,
outl: outl(x),
assert: ↑b,
isr: isr(x),
isl: isl(x),
less_than: a < b,
uall: ∀[x:A]. B[x],
so_apply: x[s],
pi1: fst(t),
pi2: snd(t),
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
unit: Unit,
apply: f a,
function: 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],
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
guard: {T},
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
prop: ℙ,
less_than: a < b,
squash: ↓T,
isl: isl(x),
outl: outl(x),
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
pi1: fst(t),
pi2: snd(t),
iff: P ⇐⇒ Q,
cand: A c∧ B,
unit: Unit,
bfalse: ff
Lemmas referenced :
first-success_wf,
select_wf,
int_seg_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
first-success-is-inl,
true_wf,
false_wf,
equal_wf,
assert_wf,
isl_wf,
int_seg_wf,
length_wf,
unit_wf2,
list_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
cumulativity,
hypothesisEquality,
sqequalRule,
lambdaEquality,
applyEquality,
functionExtensionality,
hypothesis,
unionEquality,
productEquality,
because_Cache,
independent_isectElimination,
setElimination,
rename,
productElimination,
dependent_functionElimination,
unionElimination,
natural_numberEquality,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
imageElimination,
independent_functionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
functionEquality,
universeEquality
Latex:
\mforall{}[T:Type]. \mforall{}[A:T {}\mrightarrow{} Type].
\mforall{}f:x:T {}\mrightarrow{} (A[x]?). \mforall{}L:T List.
((\muparrow{}isl(first-success(f;L)))
{}\mRightarrow{} (fst(outl(first-success(f;L))) < ||L||
\mwedge{} ((f L[fst(outl(first-success(f;L)))]) = (inl (snd(outl(first-success(f;L))))))
\mwedge{} (\mforall{}x\mmember{}firstn(fst(outl(first-success(f;L)));L).\muparrow{}isr(f x))))
Date html generated:
2017_04_14-AM-09_23_42
Last ObjectModification:
2017_02_27-PM-03_58_26
Theory : list_1
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