Nuprl Lemma : decidable__pcw-pp-barred
∀[P:Type]. ∀[A:P ⟶ Type]. ∀[B:p:P ⟶ A[p] ⟶ Type]. ∀[C:p:P ⟶ a:A[p] ⟶ B[p;a] ⟶ P].
∀pp:n:ℕ × (ℕn ⟶ pcw-step(P;p.A[p];p,a.B[p;a];p,a,b.C[p;a;b])). Dec(Barred(pp))
Proof
Definitions occuring in Statement :
pcw-pp-barred: Barred(pp),
pcw-step: pcw-step(P;p.A[p];p,a.B[p; a];p,a,b.C[p; a; b]),
int_seg: {i..j-},
nat: ℕ,
decidable: Dec(P),
uall: ∀[x:A]. B[x],
so_apply: x[s1;s2;s3],
so_apply: x[s1;s2],
so_apply: x[s],
all: ∀x:A. B[x],
function: x:A ⟶ B[x],
product: x:A × B[x],
natural_number: $n,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
nat: ℕ,
pcw-pp-barred: Barred(pp),
decidable: Dec(P),
or: P ∨ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
ge: i ≥ j ,
le: A ≤ B,
and: P ∧ Q,
int_seg: {i..j-},
lelt: i ≤ j < k,
iff: P ⇐⇒ Q,
not: ¬A,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
false: False,
prop: ℙ,
uiff: uiff(P;Q),
uimplies: b supposing a,
subtract: n - m,
subtype_rel: A ⊆r B,
top: Top,
less_than': less_than'(a;b),
true: True,
pcw-step: pcw-step(P;p.A[p];p,a.B[p; a];p,a,b.C[p; a; b]),
spreadn: spread3,
isr: isr(x),
assert: ↑b,
ifthenelse: if b then t else f fi ,
bfalse: ff,
btrue: tt
Lemmas referenced :
nat_properties,
decidable__lt,
nat_wf,
int_seg_wf,
pcw-step_wf,
subtract_wf,
decidable__le,
false_wf,
not-le-2,
less-iff-le,
condition-implies-le,
minus-one-mul,
zero-add,
minus-one-mul-top,
minus-add,
minus-minus,
add-associates,
add-swap,
add-commutes,
add_functionality_wrt_le,
add-zero,
le-add-cancel,
not-lt-2,
add-mul-special,
zero-mul,
le-add-cancel-alt,
lelt_wf,
decidable__and2,
less_than_wf,
decidable__false,
true_wf,
decidable__true,
equal_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
productElimination,
thin,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
setElimination,
rename,
sqequalRule,
dependent_functionElimination,
natural_numberEquality,
unionElimination,
productEquality,
functionEquality,
cumulativity,
lambdaEquality,
applyEquality,
functionExtensionality,
universeEquality,
dependent_set_memberEquality,
independent_pairFormation,
voidElimination,
independent_functionElimination,
independent_isectElimination,
addEquality,
isect_memberEquality,
voidEquality,
intEquality,
minusEquality,
because_Cache,
equalityTransitivity,
equalitySymmetry,
inrFormation
Latex:
\mforall{}[P:Type]. \mforall{}[A:P {}\mrightarrow{} Type]. \mforall{}[B:p:P {}\mrightarrow{} A[p] {}\mrightarrow{} Type]. \mforall{}[C:p:P {}\mrightarrow{} a:A[p] {}\mrightarrow{} B[p;a] {}\mrightarrow{} P].
\mforall{}pp:n:\mBbbN{} \mtimes{} (\mBbbN{}n {}\mrightarrow{} pcw-step(P;p.A[p];p,a.B[p;a];p,a,b.C[p;a;b])). Dec(Barred(pp))
Date html generated:
2017_04_14-AM-07_42_07
Last ObjectModification:
2017_02_27-PM-03_13_51
Theory : co-recursion
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