Nuprl Lemma : cWO-rel-path-barred
∀[T:Type]
∀[R:T ⟶ T ⟶ ℙ]
(∀f:ℕ ⟶ T. (↓∃m:ℕ. ∃n:ℕm. (¬R[f n;f m])))
⇒ (∀alpha:{f:ℕ ⟶ (T?)| ∀x:ℕ. (cWO-rel(R) x f (f x))} . (↓∃m:ℕ. (cWObar() m alpha)))
supposing ∀a,b,c:T. (R[a;b] ⇒ R[b;c] ⇒ R[a;c])
supposing T
Proof
Definitions occuring in Statement :
cWObar: cWObar(),
cWO-rel: cWO-rel(R),
int_seg: {i..j-},
nat: ℕ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
squash: ↓T,
implies: P ⇒ Q,
unit: Unit,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x],
union: left + right,
natural_number: $n,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
implies: P ⇒ Q,
all: ∀x:A. B[x],
cWO-rel: cWO-rel(R),
cWObar: cWObar(),
squash: ↓T,
prop: ℙ,
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
nat: ℕ,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
not: ¬A,
so_apply: x[s],
so_apply: x[s1;s2],
int_seg: {i..j-},
lelt: i ≤ j < k,
exists: ∃x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
subtract: n - m,
top: Top,
true: True,
cand: A c∧ B,
isl: isl(x),
isr: isr(x),
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
bfalse: ff,
guard: {T},
sq_stable: SqStable(P),
nat_plus: ℕ+,
less_than: a < b,
sq_type: SQType(T),
outl: outl(x)
Lemmas referenced :
set_wf,
nat_wf,
unit_wf2,
all_wf,
cWO-rel_wf,
subtype_rel_function,
int_seg_wf,
int_seg_subtype_nat,
false_wf,
subtype_rel_self,
squash_wf,
exists_wf,
not_wf,
le_wf,
equal_wf,
less_than_wf,
assert_wf,
isr_wf,
subtract_wf,
decidable__le,
not-le-2,
less-iff-le,
condition-implies-le,
minus-one-mul,
zero-add,
minus-one-mul-top,
minus-add,
minus-minus,
add-associates,
add-swap,
add-commutes,
add_functionality_wrt_le,
add-zero,
le-add-cancel,
decidable__exists_int_seg,
decidable__and2,
decidable__lt,
decidable__assert,
isl_wf,
true_wf,
add-member-int_seg2,
le-add-cancel2,
and_wf,
not-lt-2,
add-subtract-cancel,
outl_wf,
primrec-wf2,
add-mul-special,
zero-mul,
sq_stable__and,
sq_stable__le,
sq_stable__less_than,
member-less_than,
add-is-int-iff,
int_subtype_base,
le_reflexive,
one-mul,
two-mul,
mul-distributes-right,
minus-zero,
omega-shadow,
mul-distributes,
mul-associates,
nat_properties,
le-add-cancel-alt,
le_weakening2,
subtype_base_sq,
set_subtype_base,
decidable__int_equal,
not-equal-2,
less_than_transitivity2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
sqequalHypSubstitution,
sqequalRule,
hypothesis,
imageElimination,
imageMemberEquality,
hypothesisEquality,
thin,
baseClosed,
extract_by_obid,
isectElimination,
functionEquality,
unionEquality,
lambdaEquality,
applyEquality,
natural_numberEquality,
setElimination,
rename,
because_Cache,
independent_isectElimination,
independent_pairFormation,
dependent_set_memberEquality,
productElimination,
dependent_functionElimination,
setEquality,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
cumulativity,
universeEquality,
unionElimination,
independent_functionElimination,
addEquality,
productEquality,
functionExtensionality,
voidElimination,
voidEquality,
minusEquality,
intEquality,
instantiate,
dependent_pairFormation,
multiplyEquality,
baseApply,
closedConclusion,
addLevel,
levelHypothesis,
hyp_replacement,
inlEquality,
applyLambdaEquality,
promote_hyp
Latex:
\mforall{}[T:Type]
\mforall{}[R:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}]
(\mforall{}f:\mBbbN{} {}\mrightarrow{} T. (\mdownarrow{}\mexists{}m:\mBbbN{}. \mexists{}n:\mBbbN{}m. (\mneg{}R[f n;f m])))
{}\mRightarrow{} (\mforall{}alpha:\{f:\mBbbN{} {}\mrightarrow{} (T?)| \mforall{}x:\mBbbN{}. (cWO-rel(R) x f (f x))\} . (\mdownarrow{}\mexists{}m:\mBbbN{}. (cWObar() m alpha)))
supposing \mforall{}a,b,c:T. (R[a;b] {}\mRightarrow{} R[b;c] {}\mRightarrow{} R[a;c])
supposing T
Date html generated:
2019_06_20-AM-11_29_36
Last ObjectModification:
2018_08_21-PM-01_53_39
Theory : bar-induction
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