Nuprl Lemma : AF-path-barred
∀[T:Type]. ∀[R:T ⟶ T ⟶ ℙ].
(AFx,y:T.R[x;y] ⇒ (∀alpha:{f:ℕ ⟶ (T?)| ∀x:ℕ. (AF-spread-law(x,y.R[x;y]) x f (f x))} . (↓∃m:ℕ. (AFbar() m alpha))))
Proof
Definitions occuring in Statement :
AFbar: AFbar(),
AF-spread-law: AF-spread-law(x,y.R[x; y]),
almost-full: AFx,y:T.R[x; y],
nat: ℕ,
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,
unit: Unit,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x],
union: left + right,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
all: ∀x:A. B[x],
AFbar: AFbar(),
nat: ℕ,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
not: ¬A,
prop: ℙ,
decidable: Dec(P),
or: P ∨ Q,
exists: ∃x:A. B[x],
cand: A c∧ B,
less_than: a < b,
squash: ↓T,
true: True,
subtract: n - m,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
uimplies: b supposing a,
subtype_rel: A ⊆r B,
top: Top,
AF-spread-law: AF-spread-law(x,y.R[x; y]),
almost-full: AFx,y:T.R[x; y],
so_lambda: λ2x.t[x],
so_apply: x[s],
so_apply: x[s1;s2],
so_lambda: λ2x y.t[x; y],
int_seg: {i..j-},
assert: ↑b,
ifthenelse: if b then t else f fi ,
isl: isl(x),
btrue: tt,
lelt: i ≤ j < k,
outl: outl(x),
sq_type: SQType(T),
guard: {T},
bfalse: ff,
sq_stable: SqStable(P)
Lemmas referenced :
AF-spread-law_wf,
decidable__assert,
isr_wf,
unit_wf2,
nat_wf,
false_wf,
le_wf,
less_than_wf,
assert_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,
outl_wf,
not-isr-assert-isl,
equal_wf,
set_wf,
all_wf,
subtype_rel_function,
int_seg_wf,
int_seg_subtype_nat,
subtype_rel_self,
almost-full_wf,
decidable__exists_int_seg,
decidable__and2,
decidable__lt,
isl_wf,
not-isl-assert-isr,
and_wf,
subtype_base_sq,
int_subtype_base,
sq_stable__le,
not-lt-2,
add-mul-special,
zero-mul,
le-add-cancel2
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
lambdaFormation,
setElimination,
rename,
sqequalRule,
dependent_functionElimination,
cumulativity,
applyEquality,
functionExtensionality,
dependent_set_memberEquality,
natural_numberEquality,
independent_pairFormation,
unionElimination,
dependent_pairFormation,
because_Cache,
imageMemberEquality,
baseClosed,
productEquality,
voidElimination,
productElimination,
independent_functionElimination,
independent_isectElimination,
addEquality,
lambdaEquality,
isect_memberEquality,
voidEquality,
minusEquality,
intEquality,
promote_hyp,
unionEquality,
equalityTransitivity,
equalitySymmetry,
imageElimination,
functionEquality,
universeEquality,
instantiate,
hyp_replacement,
applyLambdaEquality,
addLevel,
levelHypothesis,
inlEquality,
multiplyEquality
Latex:
\mforall{}[T:Type]. \mforall{}[R:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}].
(AFx,y:T.R[x;y]
{}\mRightarrow{} (\mforall{}alpha:\{f:\mBbbN{} {}\mrightarrow{} (T?)| \mforall{}x:\mBbbN{}. (AF-spread-law(x,y.R[x;y]) x f (f x))\}
(\mdownarrow{}\mexists{}m:\mBbbN{}. (AFbar() m alpha))))
Date html generated:
2019_06_20-AM-11_29_23
Last ObjectModification:
2018_08_21-PM-01_52_50
Theory : bar-induction
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