Nuprl Lemma : power-set-lift-well-founded-implies
∀[T:Type]. ∀[R:T ⟶ T ⟶ ℙ].
((∀x:T. (R x x))
⇒ (∀f:ℕ ⟶ P(T). (↓∃n:ℕ. ((power-set-lift(T;R) (f (n + 1)) (f n)) ⇒ (power-set-lift(T;R) (f n) (f (n + 1))))))
⇒ AFx,y:T.R[x;y])
Proof
Definitions occuring in Statement :
power-set-lift: power-set-lift(T;R),
power-set: P(T),
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,
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],
member: t ∈ T,
implies: P ⇒ Q,
almost-full: AFx,y:T.R[x; y],
all: ∀x:A. B[x],
prop: ℙ,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
nat: ℕ,
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
not: ¬A,
rev_implies: P ⇐ Q,
false: False,
uiff: uiff(P;Q),
uimplies: b supposing a,
sq_stable: SqStable(P),
squash: ↓T,
subtract: n - m,
top: Top,
le: A ≤ B,
less_than': less_than'(a;b),
true: True,
so_apply: x[s],
exists: ∃x:A. B[x],
power-set: P(T),
power-set-lift: power-set-lift(T;R),
set-member: (x ∈ s),
cand: A c∧ B,
guard: {T},
int_upper: {i...},
so_apply: x[s1;s2]
Lemmas referenced :
iff_weakening_equal,
less_than_wf,
not-lt-2,
decidable__lt,
int_subtype_base,
add-is-int-iff,
le_reflexive,
and_wf,
equal_wf,
le-add-cancel2,
zero-mul,
add-mul-special,
int_upper_subtype_int_upper,
int_upper_subtype_nat,
subtype_rel_dep_function,
int_upper_wf,
le_wf,
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,
false_wf,
decidable__le,
power-set-lift_wf,
exists_wf,
squash_wf,
power-set_wf,
all_wf,
nat_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
functionEquality,
lemma_by_obid,
hypothesis,
hypothesisEquality,
thin,
instantiate,
sqequalHypSubstitution,
isectElimination,
applyEquality,
lambdaEquality,
cumulativity,
universeEquality,
sqequalRule,
dependent_set_memberEquality,
addEquality,
setElimination,
rename,
natural_numberEquality,
dependent_functionElimination,
unionElimination,
independent_pairFormation,
voidElimination,
productElimination,
independent_functionElimination,
independent_isectElimination,
imageMemberEquality,
baseClosed,
imageElimination,
isect_memberEquality,
voidEquality,
intEquality,
because_Cache,
minusEquality,
dependent_pairEquality,
dependent_pairFormation,
multiplyEquality,
baseApply,
closedConclusion,
productEquality,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}[T:Type]. \mforall{}[R:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}].
((\mforall{}x:T. (R x x))
{}\mRightarrow{} (\mforall{}f:\mBbbN{} {}\mrightarrow{} P(T)
(\mdownarrow{}\mexists{}n:\mBbbN{}
((power-set-lift(T;R) (f (n + 1)) (f n)) {}\mRightarrow{} (power-set-lift(T;R) (f n) (f (n + 1))))))
{}\mRightarrow{} AFx,y:T.R[x;y])
Date html generated:
2016_05_13-PM-03_51_45
Last ObjectModification:
2016_01_14-PM-07_00_34
Theory : bar-induction
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