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
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