Nuprl Lemma : basic_strong_bar_induction
∀[T:Type]. ∀[R:n:ℕ ⟶ (ℕn ⟶ T) ⟶ T ⟶ ℙ]. ∀[B,A:n:ℕ ⟶ R-consistent-seq(n) ⟶ ℙ].
((∀n:ℕ. ∀s:R-consistent-seq(n). Dec(B[n;s]))
⇒ (∀n:ℕ. ∀s:R-consistent-seq(n). (B[n;s] ⇒ A[n;s]))
⇒ (∀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. A[0;x]))
Proof
Definitions occuring in Statement :
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,
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,
so_lambda: λ2x.t[x],
so_apply: x[s1;s2],
consistent-seq: R-consistent-seq(n),
int_seg: {i..j-},
lelt: i ≤ j < k,
less_than: a < b,
squash: ↓T,
so_apply: x[s],
cand: A c∧ B,
guard: {T},
exists: ∃x:A. B[x],
prop: ℙ,
top: Top,
istype: istype(T),
true: True,
subtract: n - m,
sq_stable: SqStable(P),
uiff: uiff(P;Q),
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
or: P ∨ Q,
decidable: Dec(P)
Lemmas referenced :
istype-top,
subtype_rel_function,
nat_wf,
int_seg_wf,
int_seg_subtype_nat,
istype-false,
subtype_rel_self,
squash_wf,
exists_wf,
subtype_rel_dep_function,
subtype_rel_sets_simple,
and_wf,
le_wf,
less_than_wf,
istype-int,
less_than_transitivity2,
le_weakening2,
istype-le,
istype-less_than,
consistent-seq_wf,
decidable_wf,
istype-nat,
istype-universe,
seq-normalize_wf,
less_than_irreflexivity,
less_than_transitivity1,
subtype_rel-equal,
istype-void,
subtype_rel_sets,
all_wf,
bar_recursion_wf_strong,
seq-add_wf_consistent,
le-add-cancel,
add-zero,
add_functionality_wrt_le,
add-commutes,
add-swap,
add-associates,
minus-one-mul-top,
zero-add,
minus-one-mul,
minus-add,
condition-implies-le,
sq_stable__le,
not-le-2,
decidable__le
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
hypothesis,
sqequalRule,
functionIsType,
setIsType,
because_Cache,
universeIsType,
hypothesisEquality,
applyEquality,
sqequalHypSubstitution,
isectElimination,
thin,
natural_numberEquality,
setElimination,
rename,
independent_isectElimination,
independent_pairFormation,
lambdaEquality_alt,
dependent_set_memberEquality_alt,
productElimination,
imageElimination,
inhabitedIsType,
intEquality,
dependent_functionElimination,
productIsType,
instantiate,
universeEquality,
equalityIsType1,
functionExtensionality_alt,
voidElimination,
isect_memberEquality_alt,
independent_functionElimination,
equalitySymmetry,
equalityTransitivity,
cumulativity,
setEquality,
functionEquality,
functionExtensionality,
minusEquality,
baseClosed,
imageMemberEquality,
unionElimination,
addEquality
Latex:
\mforall{}[T:Type]. \mforall{}[R:n:\mBbbN{} {}\mrightarrow{} (\mBbbN{}n {}\mrightarrow{} T) {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}]. \mforall{}[B,A:n:\mBbbN{} {}\mrightarrow{} R-consistent-seq(n) {}\mrightarrow{} \mBbbP{}].
((\mforall{}n:\mBbbN{}. \mforall{}s:R-consistent-seq(n). Dec(B[n;s]))
{}\mRightarrow{} (\mforall{}n:\mBbbN{}. \mforall{}s:R-consistent-seq(n). (B[n;s] {}\mRightarrow{} A[n;s]))
{}\mRightarrow{} (\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]))
{}\mRightarrow{} (\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. A[0;x]))
Date html generated:
2020_05_19-PM-09_35_52
Last ObjectModification:
2020_01_04-PM-07_56_43
Theory : bar-induction
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