Nuprl Lemma : sd_ordered_char
∀s:QOSet. ∀us:|s| List. sd_ordered(us) = HTFor{<𝔹,∧b>} v::vs ∈ us. ∀bw(:|s|) ∈ vs. (w <b v)
Proof
Definitions occuring in Statement :
sd_ordered: sd_ordered(as),
ball: ball,
mon_htfor: HTFor{g} h::t ∈ as. f[h; t],
list: T List,
bool: 𝔹,
all: ∀x:A. B[x],
equal: s = t ∈ T,
band_mon: <𝔹,∧b>,
qoset: QOSet,
set_blt: a <b b,
set_car: |p|
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
qoset: QOSet,
dset: DSet,
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
so_lambda: λ2x y.t[x; y],
so_apply: x[s],
band_mon: <𝔹,∧b>,
grp_car: |g|,
pi1: fst(t),
so_apply: x[s1;s2],
implies: P ⇒ Q,
top: Top,
grp_id: e,
pi2: snd(t),
grp_op: *,
infix_ap: x f y,
prop: ℙ,
squash: ↓T,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
band: p ∧b q,
ifthenelse: if b then t else f fi ,
bfalse: ff,
true: True,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
ball: ball,
uiff: uiff(P;Q),
or: P ∨ Q,
assert: ↑b,
cons: [a / b],
cand: A c∧ B,
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
list_induction,
set_car_wf,
equal_wf,
bool_wf,
sd_ordered_wf,
mon_htfor_wf,
band_mon_wf,
ball_wf,
set_blt_wf,
list_wf,
sd_ordered_nil_lemma,
mon_htfor_nil_lemma,
btrue_wf,
sd_ordered_cons_lemma,
mon_htfor_cons_lemma,
qoset_wf,
squash_wf,
true_wf,
before_wf,
band_wf,
iff_weakening_equal,
iff_imp_equal_bool,
assert_wf,
assert_of_band,
iff_wf,
list-cases,
ball_nil_lemma,
nil_wf,
product_subtype_list,
ball_cons_lemma,
cons_wf,
before_cons_lemma,
iff_transitivity,
assert_of_set_lt,
set_lt_wf,
iff_weakening_uiff,
assert_functionality_wrt_uiff,
ball_char,
mem_wf,
set_lt_transitivity_2,
set_leq_weakening_lt,
before_nil_lemma,
mem_cons_lemma,
bor_wf,
set_eq_wf,
or_wf,
assert_of_bor,
assert_of_dset_eq
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
setElimination,
rename,
because_Cache,
hypothesis,
sqequalRule,
lambdaEquality,
dependent_functionElimination,
hypothesisEquality,
applyEquality,
independent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
equalityUniverse,
levelHypothesis,
unionElimination,
equalityElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_isectElimination,
productElimination,
independent_pairFormation,
productEquality,
addLevel,
impliesFunctionality,
promote_hyp,
hypothesis_subsumption,
orFunctionality,
inlFormation
Latex:
\mforall{}s:QOSet. \mforall{}us:|s| List. sd\_ordered(us) = HTFor\{<\mBbbB{},\mwedge{}\msubb{}>\} v::vs \mmember{} us. \mforall{}\msubb{}w(:|s|) \mmember{} vs. (w <\msubb{} v)
Date html generated:
2017_10_01-AM-10_01_32
Last ObjectModification:
2017_03_03-PM-01_04_08
Theory : polynom_2
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