Nuprl Lemma : descending-chain-barred-implies-wellfounded
∀[T:Type]. ∀[<,B:T ⟶ T ⟶ ℙ].
((∀x,y,z:T. ((x < y) ⇒ (y < z) ⇒ (x < z)))
⇒ (∀x,y:T. Dec(x B y))
⇒ (∀x,y:T. ((x B y) ⇒ (¬(x < y))))
⇒ (∀f:ℕ ⟶ T. (↓∃j:ℕ. ∃i:ℕj. ((f j) B (f i))))
⇒ WellFnd{i}(T;x,y.x < y))
Proof
Definitions occuring in Statement :
int_seg: {i..j-},
nat: ℕ,
wellfounded: WellFnd{i}(A;x,y.R[x; y]),
decidable: Dec(P),
uall: ∀[x:A]. B[x],
prop: ℙ,
infix_ap: x f y,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
squash: ↓T,
implies: P ⇒ Q,
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n,
universe: Type
Definitions unfolded in proof :
label: ...$L... t,
assert: ↑b,
ifthenelse: if b then t else f fi ,
bnot: ¬bb,
sq_type: SQType(T),
bfalse: ff,
less_than: a < b,
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
seq-append: seq-append(n;m;s1;s2),
seq-adjoin: s++t,
sq_stable: SqStable(P),
exists: ∃x:A. B[x],
squash: ↓T,
true: True,
less_than': less_than'(a;b),
top: Top,
subtype_rel: A ⊆r B,
subtract: n - m,
uiff: uiff(P;Q),
false: False,
rev_implies: P ⇐ Q,
not: ¬A,
iff: P ⇐⇒ Q,
or: P ∨ Q,
decidable: Dec(P),
infix_ap: x f y,
so_apply: x[s1;s2],
so_apply: x[s],
all: ∀x:A. B[x],
uimplies: b supposing a,
guard: {T},
le: A ≤ B,
and: P ∧ Q,
lelt: i ≤ j < k,
prop: ℙ,
int_seg: {i..j-},
so_lambda: λ2x.t[x],
nat: ℕ,
so_lambda: λ2x y.t[x; y],
member: t ∈ T,
wellfounded: WellFnd{i}(A;x,y.R[x; y]),
implies: P ⇒ Q,
uall: ∀[x:A]. B[x]
Lemmas referenced :
less_than_irreflexivity,
not-equal-2,
le_antisymmetry_iff,
decidable__int_equal,
le-add-cancel2,
assert_of_bnot,
iff_weakening_uiff,
bnot_wf,
assert_wf,
iff_transitivity,
bool_subtype_base,
subtype_base_sq,
bool_cases_sqequal,
eqff_to_assert,
top_wf,
assert_of_lt_int,
eqtt_to_assert,
bool_wf,
lt_int_wf,
le_reflexive,
sq_stable__le,
iff_weakening_equal,
decidable_wf,
not_wf,
int_seg_subtype_nat,
squash_wf,
seq-adjoin_wf,
int_subtype_base,
set_subtype_base,
add-is-int-iff,
decidable__exists_int_seg,
le-add-cancel-alt,
zero-mul,
add-mul-special,
not-lt-2,
decidable__lt,
le-add-cancel,
add-zero,
add_functionality_wrt_le,
less-iff-le,
add-commutes,
add-swap,
add-associates,
minus-minus,
minus-add,
nat_wf,
minus-one-mul-top,
zero-add,
minus-one-mul,
condition-implies-le,
not-le-2,
false_wf,
decidable__le,
subtract_wf,
less_than_transitivity1,
equal_wf,
all_wf,
less_than_wf,
le_wf,
and_wf,
le_weakening2,
less_than_transitivity2,
infix_ap_wf,
int_seg_wf,
exists_wf,
basic_bar_induction
Rules used in proof :
multiplyEquality,
hyp_replacement,
impliesFunctionality,
promote_hyp,
dependent_pairFormation,
sqequalAxiom,
lessCases,
equalityElimination,
equalitySymmetry,
equalityTransitivity,
imageMemberEquality,
imageElimination,
baseClosed,
closedConclusion,
baseApply,
intEquality,
minusEquality,
voidEquality,
isect_memberEquality,
addEquality,
independent_functionElimination,
voidElimination,
unionElimination,
functionEquality,
dependent_functionElimination,
independent_isectElimination,
independent_pairFormation,
productElimination,
dependent_set_memberEquality,
applyEquality,
functionExtensionality,
universeEquality,
cumulativity,
instantiate,
hypothesis,
because_Cache,
rename,
setElimination,
natural_numberEquality,
lambdaEquality,
sqequalRule,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
lambdaFormation,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[T:Type]. \mforall{}[<,B:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}].
((\mforall{}x,y,z:T. ((x < y) {}\mRightarrow{} (y < z) {}\mRightarrow{} (x < z)))
{}\mRightarrow{} (\mforall{}x,y:T. Dec(x B y))
{}\mRightarrow{} (\mforall{}x,y:T. ((x B y) {}\mRightarrow{} (\mneg{}(x < y))))
{}\mRightarrow{} (\mforall{}f:\mBbbN{} {}\mrightarrow{} T. (\mdownarrow{}\mexists{}j:\mBbbN{}. \mexists{}i:\mBbbN{}j. ((f j) B (f i))))
{}\mRightarrow{} WellFnd\{i\}(T;x,y.x < y))
Date html generated:
2017_09_29-PM-05_47_54
Last ObjectModification:
2017_09_22-AM-08_34_20
Theory : bar-induction
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