Nuprl Lemma : 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]))
⇒ (∀n:ℕ. ∀s:ℕn ⟶ T. ((∀alpha:ℕ ⟶ T. (↓∃m:ℕ. R[n + m;seq-append(n;m;s;alpha)])) ⇒ A[n;s])))
Proof
Definitions occuring in Statement :
seq-adjoin: s++t,
seq-append: seq-append(n;m;s1;s2),
int_seg: {i..j-},
nat: ℕ,
decidable: Dec(P),
uall: ∀[x:A]. B[x],
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,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
nat: ℕ,
seq-adjoin: s++t,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
prop: ℙ,
so_lambda: λ2x.t[x],
guard: {T},
sq_stable: SqStable(P),
squash: ↓T,
so_apply: x[s],
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
not: ¬A,
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
subtract: n - m,
true: True
Lemmas referenced :
bar_recursion_wf,
int_seg_wf,
subtype_rel-equal,
nat_wf,
seq-normalize_wf,
all_wf,
squash_wf,
exists_wf,
add_nat_wf,
sq_stable__le,
equal_wf,
le_wf,
seq-append_wf,
subtype_rel_dep_function,
int_seg_subtype_nat,
false_wf,
decidable__le,
not-le-2,
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
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
rename,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
sqequalRule,
lambdaEquality,
applyEquality,
functionExtensionality,
because_Cache,
natural_numberEquality,
setElimination,
hypothesis,
functionEquality,
independent_functionElimination,
independent_isectElimination,
dependent_set_memberEquality,
addEquality,
imageMemberEquality,
baseClosed,
imageElimination,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
independent_pairFormation,
unionElimination,
voidElimination,
productElimination,
minusEquality,
universeEquality
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{}n:\mBbbN{}. \mforall{}s:\mBbbN{}n {}\mrightarrow{} T. ((\mforall{}alpha:\mBbbN{} {}\mrightarrow{} T. (\mdownarrow{}\mexists{}m:\mBbbN{}. R[n + m;seq-append(n;m;s;alpha)])) {}\mRightarrow{} A[n;s])))
Date html generated:
2017_04_14-AM-07_27_16
Last ObjectModification:
2017_02_27-PM-02_56_29
Theory : bar-induction
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