Nuprl Lemma : pcw-pp-barred-W-decidable
∀[A:Type]. ∀[B:A ⟶ Type]. ∀n:ℕ. ∀s:ℕn ⟶ cw-step(A;a.B[a]). (Barred(<n, s>) ∨ (¬Barred(<n, s>)))
Proof
Definitions occuring in Statement :
cw-step: cw-step(A;a.B[a]),
pcw-pp-barred: Barred(pp),
int_seg: {i..j-},
nat: ℕ,
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
not: ¬A,
or: P ∨ Q,
function: x:A ⟶ B[x],
pair: <a, b>,
natural_number: $n,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
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],
cw-step: cw-step(A;a.B[a]),
nat: ℕ,
decidable: Dec(P)
Lemmas referenced :
decidable__pcw-pp-barred,
unit_wf2,
it_wf,
int_seg_wf,
pcw-step_wf,
cw-step_wf,
nat_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesis,
sqequalRule,
lambdaEquality,
hypothesisEquality,
applyEquality,
dependent_functionElimination,
dependent_pairEquality,
functionEquality,
natural_numberEquality,
setElimination,
rename,
cumulativity,
universeEquality
Latex:
\mforall{}[A:Type]. \mforall{}[B:A {}\mrightarrow{} Type]. \mforall{}n:\mBbbN{}. \mforall{}s:\mBbbN{}n {}\mrightarrow{} cw-step(A;a.B[a]). (Barred(<n, s>) \mvee{} (\mneg{}Barred(<n, s>)))
Date html generated:
2016_05_14-AM-06_15_10
Last ObjectModification:
2015_12_26-PM-00_05_05
Theory : co-recursion
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