Nuprl Lemma : bool-bar-induction
∀[T:Type]. ∀[A:(T List) ⟶ ℙ].
∀R:(T List) ⟶ 𝔹
((∀s:{s:T List| ↑R[s]} . A[s])
⇒ (∀s:{s:T List| ¬↑R[s]} . ((∀t:T. A[s @ [t]]) ⇒ A[s]))
⇒ (∀alpha:ℕ ⟶ T. (↓∃n:ℕ. (↑R[map(alpha;upto(n))])))
⇒ A[[]])
Proof
Definitions occuring in Statement :
upto: upto(n),
map: map(f;as),
append: as @ bs,
cons: [a / b],
nil: [],
list: T List,
nat: ℕ,
assert: ↑b,
bool: 𝔹,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
squash: ↓T,
implies: P ⇒ Q,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
universe: Type
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],
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
top: Top,
so_apply: x[s1;s2;s3],
squash: ↓T,
prop: ℙ,
nat: ℕ,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
not: ¬A
Lemmas referenced :
bool_wf,
cons_wf,
append_wf,
not_wf,
list_wf,
upto_wf,
false_wf,
int_seg_subtype_nat,
subtype_rel_dep_function,
int_seg_wf,
map_wf,
assert_wf,
exists_wf,
squash_wf,
all_wf,
nat_wf,
list_ind_nil_lemma,
nil_wf,
bbar-recursion_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
rename,
introduction,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
because_Cache,
hypothesisEquality,
sqequalRule,
lambdaEquality,
applyEquality,
hypothesis,
independent_functionElimination,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
imageElimination,
imageMemberEquality,
baseClosed,
functionEquality,
cumulativity,
natural_numberEquality,
setElimination,
independent_isectElimination,
independent_pairFormation,
setEquality,
universeEquality
Latex:
\mforall{}[T:Type]. \mforall{}[A:(T List) {}\mrightarrow{} \mBbbP{}].
\mforall{}R:(T List) {}\mrightarrow{} \mBbbB{}
((\mforall{}s:\{s:T List| \muparrow{}R[s]\} . A[s])
{}\mRightarrow{} (\mforall{}s:\{s:T List| \mneg{}\muparrow{}R[s]\} . ((\mforall{}t:T. A[s @ [t]]) {}\mRightarrow{} A[s]))
{}\mRightarrow{} (\mforall{}alpha:\mBbbN{} {}\mrightarrow{} T. (\mdownarrow{}\mexists{}n:\mBbbN{}. (\muparrow{}R[map(alpha;upto(n))])))
{}\mRightarrow{} A[[]])
Date html generated:
2016_05_15-PM-10_05_13
Last ObjectModification:
2016_01_16-PM-04_05_39
Theory : bar!induction
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