Nuprl Lemma : locally-ranked-is-well-founded
∀[T:Type]. ∀[R:T ⟶ T ⟶ ℙ].
(Trans(T;x,y.R[y;x])
⇒ (∀k:ℕ. ∀rank:T ⟶ ℕ. ∀l:T ⟶ ℕk.
((∀x,y:T. (((l x) = (l y) ∈ ℤ) ⇒ R[x;y] ⇒ rank x < rank y)) ⇒ tcWO(T;x,y.R[y;x]))))
Proof
Definitions occuring in Statement :
trans: Trans(T;x,y.E[x; y]),
tcWO: tcWO(T;x,y.>[x; y]),
int_seg: {i..j-},
nat: ℕ,
less_than: a < b,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
all: ∀x:A. B[x],
implies: P ⇒ Q,
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n,
int: ℤ,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
all: ∀x:A. B[x],
tcWO: tcWO(T;x,y.>[x; y]),
and: P ∧ Q,
member: t ∈ T,
so_apply: x[s1;s2],
subtype_rel: A ⊆r B,
squash: ↓T,
int_seg: {i..j-},
so_lambda: λ2x.t[x],
nat: ℕ,
so_apply: x[s],
uimplies: b supposing a,
prop: ℙ,
so_lambda: λ2x y.t[x; y],
guard: {T},
trans: Trans(T;x,y.E[x; y]),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
sq_stable: SqStable(P),
lelt: i ≤ j < k,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
subtract: n - m,
top: Top,
true: True,
ge: i ≥ j ,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
less_than: a < b,
cand: A c∧ B
Lemmas referenced :
istype-int,
set_subtype_base,
lelt_wf,
int_subtype_base,
subtype_rel_self,
istype-less_than,
istype-nat,
int_seg_wf,
trans_wf,
istype-universe,
Dickson's lemma,
eq_int_wf,
eqtt_to_assert,
assert_of_eq_int,
add_nat_wf,
istype-false,
istype-le,
sq_stable__le,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
int_seg_subtype_nat,
assert_wf,
bnot_wf,
not_wf,
equal-wf-base,
istype-assert,
istype-void,
istype-sqequal,
add-is-int-iff,
le_wf,
le_antisymmetry_iff,
condition-implies-le,
minus-add,
minus-one-mul,
zero-add,
minus-one-mul-top,
add-commutes,
add_functionality_wrt_le,
add-associates,
add-zero,
le-add-cancel,
subtract_wf,
bool_cases,
iff_transitivity,
iff_weakening_uiff,
assert_of_bnot,
int_seg_properties,
nat_properties,
add-swap,
less-iff-le,
le-add-cancel2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
Error :lambdaFormation_alt,
independent_pairFormation,
Error :universeIsType,
cut,
applyEquality,
hypothesisEquality,
hypothesis,
thin,
because_Cache,
sqequalHypSubstitution,
sqequalRule,
introduction,
imageElimination,
imageMemberEquality,
baseClosed,
Error :functionIsType,
Error :equalityIstype,
extract_by_obid,
isectElimination,
intEquality,
Error :lambdaEquality_alt,
natural_numberEquality,
setElimination,
rename,
independent_isectElimination,
sqequalBase,
equalitySymmetry,
instantiate,
universeEquality,
Error :inhabitedIsType,
dependent_functionElimination,
independent_functionElimination,
unionElimination,
equalityElimination,
productElimination,
Error :dependent_set_memberEquality_alt,
addEquality,
equalityTransitivity,
Error :dependent_pairFormation_alt,
promote_hyp,
cumulativity,
voidElimination,
applyLambdaEquality,
Error :isect_memberEquality_alt,
minusEquality,
baseApply,
closedConclusion,
Error :productIsType
Latex:
\mforall{}[T:Type]. \mforall{}[R:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}].
(Trans(T;x,y.R[y;x])
{}\mRightarrow{} (\mforall{}k:\mBbbN{}. \mforall{}rank:T {}\mrightarrow{} \mBbbN{}. \mforall{}l:T {}\mrightarrow{} \mBbbN{}k.
((\mforall{}x,y:T. (((l x) = (l y)) {}\mRightarrow{} R[x;y] {}\mRightarrow{} rank x < rank y)) {}\mRightarrow{} tcWO(T;x,y.R[y;x]))))
Date html generated:
2019_06_20-PM-00_30_10
Last ObjectModification:
2019_01_04-AM-11_46_07
Theory : rel_1
Home
Index