Nuprl Lemma : decidable__llex-le
∀[A:Type]. ∀[<:A ⟶ A ⟶ ℙ].
((∀a,b:A. (Dec(<[a;b]) ∧ Dec(a = b ∈ A))) ⇒ (∀L1,L2:A List. Dec(L1 llex-le(A;a,b.<[a;b]) L2)))
Proof
Definitions occuring in Statement :
llex-le: llex-le(A;a,b.<[a; b]),
list: T List,
decidable: Dec(P),
uall: ∀[x:A]. B[x],
prop: ℙ,
infix_ap: x f y,
so_apply: x[s1;s2],
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
all: ∀x:A. B[x],
so_lambda: λ2x.t[x],
prop: ℙ,
and: P ∧ Q,
so_apply: x[s1;s2],
so_apply: x[s],
llex-le: llex-le(A;a,b.<[a; b]),
infix_ap: x f y,
decidable: Dec(P),
or: P ∨ Q,
so_lambda: λ2x y.t[x; y],
guard: {T}
Lemmas referenced :
decidable__llex,
list_wf,
all_wf,
decidable_wf,
equal_wf,
not_wf,
or_wf,
llex_wf,
decidable__or,
decidable__equal_list
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
lambdaFormation,
independent_functionElimination,
dependent_functionElimination,
cumulativity,
sqequalRule,
lambdaEquality,
productEquality,
applyEquality,
functionExtensionality,
functionEquality,
because_Cache,
universeEquality,
unionElimination,
inlFormation,
inrFormation,
productElimination
Latex:
\mforall{}[A:Type]. \mforall{}[<:A {}\mrightarrow{} A {}\mrightarrow{} \mBbbP{}].
((\mforall{}a,b:A. (Dec(<[a;b]) \mwedge{} Dec(a = b))) {}\mRightarrow{} (\mforall{}L1,L2:A List. Dec(L1 llex-le(A;a,b.<[a;b]) L2)))
Date html generated:
2017_02_20-AM-10_56_00
Last ObjectModification:
2017_02_02-PM-09_46_22
Theory : general
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