Nuprl Lemma : bar_recursion_wf_strong
∀[T:Type]. ∀[R:n:ℕ ⟶ (ℕn ⟶ T) ⟶ T ⟶ ℙ]. ∀[B:n:ℕ ⟶ {s:ℕn ⟶ T| ∀x:ℕn. (R x s (s x))} ⟶ ℙ].
∀[A:n:ℕ ⟶ R-consistent-seq(n) ⟶ ℙ]. ∀[d:∀n:ℕ. ∀s:R-consistent-seq(n). Dec(B[n;s])]. ∀[b:∀n:ℕ. ∀s:R-consistent-seq(n).
(B[n;s] ⇒ A[n;s])].
∀[i:∀n:ℕ. ∀s:R-consistent-seq(n). ((∀t:{t:T| R n s t} . A[n + 1;s.t@n]) ⇒ A[n;s])].
((∀alpha:{f:ℕ ⟶ T| ∀x:ℕ. (R x f (f x))} . (↓∃m:ℕ. B[m;alpha])) ⇒ (∀x:Top. (bar_recursion(d;b;i;0;x) ∈ A[0;x])))
Proof
Definitions occuring in Statement :
bar_recursion: bar_recursion,
consistent-seq: R-consistent-seq(n),
seq-add: s.x@n,
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,
member: t ∈ T,
set: {x:A| B[x]} ,
apply: f a,
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,
subtype_rel: A ⊆r B,
nat: ℕ,
uimplies: b supposing a,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
not: ¬A,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s1;s2],
so_apply: x[s],
int_seg: {i..j-},
lelt: i ≤ j < k,
guard: {T},
exists: ∃x:A. B[x],
consistent-seq: R-consistent-seq(n),
decidable: Dec(P),
bar_recursion: bar_recursion,
or: P ∨ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
sq_stable: SqStable(P),
squash: ↓T,
subtract: n - m,
true: True,
seq-add: s.x@n,
top: Top
Lemmas referenced :
top_wf,
subtype_rel_function,
int_seg_wf,
int_seg_subtype_nat,
false_wf,
subtype_rel_self,
squash_wf,
exists_wf,
nat_wf,
subtype_rel_sets,
all_wf,
le_wf,
subtype_rel_dep_function,
and_wf,
less_than_wf,
less_than_transitivity2,
le_weakening2,
subtype_rel_set,
consistent-seq_wf,
decidable_wf,
seq-add_wf_consistent,
equal_wf,
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
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
Error :lambdaFormation_alt,
Error :universeIsType,
cut,
introduction,
extract_by_obid,
hypothesis,
sqequalRule,
Error :functionIsType,
Error :setIsType,
because_Cache,
hypothesisEquality,
applyEquality,
sqequalHypSubstitution,
isectElimination,
thin,
natural_numberEquality,
setElimination,
rename,
independent_isectElimination,
independent_pairFormation,
lambdaFormation,
lambdaEquality,
functionEquality,
dependent_set_memberEquality,
productElimination,
intEquality,
setEquality,
dependent_functionElimination,
universeEquality,
instantiate,
strong_bar_Induction,
unionElimination,
functionExtensionality,
independent_functionElimination,
voidElimination,
cumulativity,
equalityTransitivity,
equalitySymmetry,
addEquality,
imageMemberEquality,
baseClosed,
imageElimination,
minusEquality,
isect_memberEquality,
voidEquality
Latex:
\mforall{}[T:Type]. \mforall{}[R:n:\mBbbN{} {}\mrightarrow{} (\mBbbN{}n {}\mrightarrow{} T) {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}]. \mforall{}[B:n:\mBbbN{} {}\mrightarrow{} \{s:\mBbbN{}n {}\mrightarrow{} T| \mforall{}x:\mBbbN{}n. (R x s (s x))\} {}\mrightarrow{} \mBbbP{}].
\mforall{}[A:n:\mBbbN{} {}\mrightarrow{} R-consistent-seq(n) {}\mrightarrow{} \mBbbP{}]. \mforall{}[d:\mforall{}n:\mBbbN{}. \mforall{}s:R-consistent-seq(n). Dec(B[n;s])].
\mforall{}[b:\mforall{}n:\mBbbN{}. \mforall{}s:R-consistent-seq(n). (B[n;s] {}\mRightarrow{} A[n;s])]. \mforall{}[i:\mforall{}n:\mBbbN{}. \mforall{}s:R-consistent-seq(n).
((\mforall{}t:\{t:T| R n s t\} . A[n + 1;s.t@n])
{}\mRightarrow{} A[n;s])].
((\mforall{}alpha:\{f:\mBbbN{} {}\mrightarrow{} T| \mforall{}x:\mBbbN{}. (R x f (f x))\} . (\mdownarrow{}\mexists{}m:\mBbbN{}. B[m;alpha]))
{}\mRightarrow{} (\mforall{}x:Top. (bar\_recursion(d;b;i;0;x) \mmember{} A[0;x])))
Date html generated:
2019_06_20-AM-11_28_54
Last ObjectModification:
2018_09_26-AM-11_13_41
Theory : bar-induction
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