Nuprl Lemma : wqo-less_wf
∀[T:Type]. ∀[R:T ⟶ T ⟶ ℙ]. ∀[bs,as:n:ℕ × (ℕn ⟶ T)]. (wqo-less(T;x,y.R[x;y];bs;as) ∈ ℙ)
Proof
Definitions occuring in Statement :
wqo-less: wqo-less(T;x,y.R[x; y];bs;as)
,
int_seg: {i..j-}
,
nat: ℕ
,
uall: ∀[x:A]. B[x]
,
prop: ℙ
,
so_apply: x[s1;s2]
,
member: t ∈ T
,
function: x:A ⟶ B[x]
,
product: x:A × B[x]
,
natural_number: $n
,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x]
,
member: t ∈ T
,
wqo-less: wqo-less(T;x,y.R[x; y];bs;as)
,
so_lambda: λ2x.t[x]
,
prop: ℙ
,
and: P ∧ Q
,
nat: ℕ
,
int_seg: {i..j-}
,
lelt: i ≤ j < k
,
le: A ≤ B
,
all: ∀x:A. B[x]
,
decidable: Dec(P)
,
or: P ∨ Q
,
iff: P
⇐⇒ Q
,
not: ¬A
,
rev_implies: P
⇐ Q
,
implies: P
⇒ Q
,
false: False
,
uiff: uiff(P;Q)
,
uimplies: b supposing a
,
guard: {T}
,
subtract: n - m
,
subtype_rel: A ⊆r B
,
top: Top
,
less_than': less_than'(a;b)
,
true: True
,
sq_type: SQType(T)
,
so_apply: x[s1;s2]
,
so_apply: x[s]
Lemmas referenced :
exists_wf,
equal_wf,
int_seg_wf,
seq-add_wf,
decidable__lt,
false_wf,
not-lt-2,
le_antisymmetry_iff,
condition-implies-le,
minus-add,
minus-one-mul,
add-swap,
minus-one-mul-top,
add-commutes,
add_functionality_wrt_le,
nat_wf,
le-add-cancel,
and_wf,
le_wf,
less_than_wf,
subtype_base_sq,
int_subtype_base,
subtype_rel_self,
all_wf,
not_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
lambdaEquality,
productElimination,
productEquality,
intEquality,
setElimination,
rename,
because_Cache,
hypothesis,
addEquality,
natural_numberEquality,
functionEquality,
functionExtensionality,
applyEquality,
dependent_set_memberEquality,
independent_pairFormation,
dependent_functionElimination,
unionElimination,
lambdaFormation,
voidElimination,
independent_functionElimination,
independent_isectElimination,
isect_memberEquality,
voidEquality,
minusEquality,
instantiate,
equalityTransitivity,
equalitySymmetry,
axiomEquality,
universeEquality
Latex:
\mforall{}[T:Type]. \mforall{}[R:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}]. \mforall{}[bs,as:n:\mBbbN{} \mtimes{} (\mBbbN{}n {}\mrightarrow{} T)]. (wqo-less(T;x,y.R[x;y];bs;as) \mmember{} \mBbbP{})
Date html generated:
2017_04_14-AM-07_29_07
Last ObjectModification:
2017_02_27-PM-02_57_06
Theory : bar-induction
Home
Index