Nuprl Lemma : basic_bar_induction
∀[T:Type]. ∀[R,A:n:ℕ ⟶ (ℕn ⟶ T) ⟶ ℙ].
((∀n:ℕ. ∀s:ℕn ⟶ T. Dec(R[n;s]))
⇒ (∀n:ℕ. ∀s:ℕn ⟶ T. (R[n;s] ⇒ A[n;s]))
⇒ (∀n:ℕ. ∀s:ℕn ⟶ T. ((∀t:T. A[n + 1;s++t]) ⇒ A[n;s]))
⇒ (∀alpha:ℕ ⟶ T. (↓∃m:ℕ. R[m;alpha]))
⇒ (∀x:Top. A[0;x]))
Proof
Definitions occuring in Statement :
seq-adjoin: s++t,
int_seg: {i..j-},
nat: ℕ,
decidable: Dec(P),
uall: ∀[x:A]. B[x],
top: Top,
prop: ℙ,
so_apply: x[s1;s2],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
squash: ↓T,
implies: P ⇒ Q,
function: x:A ⟶ B[x],
add: n + m,
natural_number: $n,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
all: ∀x:A. B[x],
member: t ∈ T,
int_seg: {i..j-},
false: False,
lelt: i ≤ j < k,
and: P ∧ Q,
guard: {T},
uimplies: b supposing a,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
nat: ℕ,
le: A ≤ B,
less_than': less_than'(a;b),
not: ¬A,
prop: ℙ,
squash: ↓T,
subtype_rel: A ⊆r B,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
decidable: Dec(P),
or: P ∨ Q,
uiff: uiff(P;Q),
sq_stable: SqStable(P),
subtract: n - m,
top: Top,
exists: ∃x:A. B[x],
seq-append: seq-append(n;m;s1;s2),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
less_than: a < b,
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b
Lemmas referenced :
less_than_transitivity1,
less_than_irreflexivity,
int_seg_wf,
bar_recursion_wf,
false_wf,
le_wf,
nat_wf,
subtype_rel-equal,
equal_wf,
iff_weakening_equal,
top_wf,
all_wf,
squash_wf,
exists_wf,
subtype_rel_dep_function,
int_seg_subtype_nat,
decidable__le,
not-le-2,
sq_stable__le,
condition-implies-le,
minus-add,
minus-one-mul,
zero-add,
minus-one-mul-top,
add-associates,
add-swap,
add-commutes,
add_functionality_wrt_le,
add-zero,
le-add-cancel,
seq-adjoin_wf,
decidable_wf,
minus-zero,
seq-append_wf,
lt_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_lt_int,
less_than_wf,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
iff_transitivity,
assert_wf,
bnot_wf,
not_wf,
iff_weakening_uiff,
assert_of_bnot,
subtract_wf,
not-lt-2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
rename,
introduction,
cut,
functionExtensionality,
sqequalHypSubstitution,
setElimination,
thin,
productElimination,
hypothesis,
extract_by_obid,
isectElimination,
hypothesisEquality,
natural_numberEquality,
independent_isectElimination,
independent_functionElimination,
voidElimination,
because_Cache,
sqequalRule,
dependent_set_memberEquality,
independent_pairFormation,
equalityTransitivity,
equalitySymmetry,
imageElimination,
imageMemberEquality,
baseClosed,
functionEquality,
cumulativity,
applyEquality,
instantiate,
lambdaEquality,
addEquality,
dependent_functionElimination,
unionElimination,
isect_memberEquality,
voidEquality,
intEquality,
minusEquality,
universeEquality,
dependent_pairFormation,
hyp_replacement,
equalityElimination,
lessCases,
sqequalAxiom,
promote_hyp,
impliesFunctionality
Latex:
\mforall{}[T:Type]. \mforall{}[R,A:n:\mBbbN{} {}\mrightarrow{} (\mBbbN{}n {}\mrightarrow{} T) {}\mrightarrow{} \mBbbP{}].
((\mforall{}n:\mBbbN{}. \mforall{}s:\mBbbN{}n {}\mrightarrow{} T. Dec(R[n;s]))
{}\mRightarrow{} (\mforall{}n:\mBbbN{}. \mforall{}s:\mBbbN{}n {}\mrightarrow{} T. (R[n;s] {}\mRightarrow{} A[n;s]))
{}\mRightarrow{} (\mforall{}n:\mBbbN{}. \mforall{}s:\mBbbN{}n {}\mrightarrow{} T. ((\mforall{}t:T. A[n + 1;s++t]) {}\mRightarrow{} A[n;s]))
{}\mRightarrow{} (\mforall{}alpha:\mBbbN{} {}\mrightarrow{} T. (\mdownarrow{}\mexists{}m:\mBbbN{}. R[m;alpha]))
{}\mRightarrow{} (\mforall{}x:Top. A[0;x]))
Date html generated:
2017_04_14-AM-07_27_19
Last ObjectModification:
2017_02_27-PM-02_56_36
Theory : bar-induction
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