Nuprl Lemma : mk_oset_wf
∀[T:Type]. ∀[eq,leq:T ⟶ T ⟶ 𝔹].
(mk_oset(T;eq;leq) ∈ LOSet) supposing (UniformLinorder(T;a,b.↑(a leq b)) and IsEqFun(T;eq))
Proof
Definitions occuring in Statement :
mk_oset: mk_oset(T;eq;leq),
loset: LOSet,
ulinorder: UniformLinorder(T;x,y.R[x; y]),
eqfun_p: IsEqFun(T;eq),
assert: ↑b,
bool: 𝔹,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
infix_ap: x f y,
member: t ∈ T,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
mk_oset: mk_oset(T;eq;leq),
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
prop: ℙ,
so_lambda: λ2x y.t[x; y],
infix_ap: x f y,
so_apply: x[s1;s2],
ulinorder: UniformLinorder(T;x,y.R[x; y]),
and: P ∧ Q,
uorder: UniformOrder(T;x,y.R[x; y]),
loset: LOSet,
poset: POSet{i},
qoset: QOSet,
dset: DSet,
poset_sig: PosetSig,
set_car: |p|,
pi1: fst(t),
set_eq: =b,
pi2: snd(t),
set_leq: a ≤ b,
set_le: ≤b,
upreorder: UniformPreorder(T;x,y.R[x; y])
Lemmas referenced :
ulinorder_wf,
assert_wf,
eqfun_p_wf,
bool_wf,
set_car_wf,
set_eq_wf,
upreorder_wf,
set_leq_wf,
uanti_sym_wf,
connex_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
hypothesis,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
extract_by_obid,
isectElimination,
thin,
hypothesisEquality,
lambdaEquality,
applyEquality,
isect_memberEquality,
because_Cache,
functionEquality,
universeEquality,
productElimination,
dependent_set_memberEquality,
dependent_pairEquality,
productEquality,
independent_pairFormation,
setElimination,
rename
Latex:
\mforall{}[T:Type]. \mforall{}[eq,leq:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbB{}].
(mk\_oset(T;eq;leq) \mmember{} LOSet) supposing (UniformLinorder(T;a,b.\muparrow{}(a leq b)) and IsEqFun(T;eq))
Date html generated:
2018_05_21-PM-03_13_56
Last ObjectModification:
2018_05_19-AM-08_26_35
Theory : sets_1
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