Nuprl Lemma : oalist_cases_b
∀a:LOSet. ∀b:AbDMon. ∀Q:|oal(a;b)| ⟶ ℙ.
(Q[[]]
⇒ (∀ws:|oal(a;b)|. ∀k:|a|. ∀v:|b|.
((↑(∀bx(:|a|) ∈ map(λz.(fst(z));ws). (x <b k))) ⇒ (¬(v = e ∈ |b|)) ⇒ Q[[<k, v> / ws]]))
⇒ {∀ws:|oal(a;b)|. Q[ws]})
Proof
Definitions occuring in Statement :
oalist: oal(a;b),
ball: ball,
map: map(f;as),
cons: [a / b],
nil: [],
assert: ↑b,
prop: ℙ,
guard: {T},
so_apply: x[s],
pi1: fst(t),
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
lambda: λx.A[x],
function: x:A ⟶ B[x],
pair: <a, b>,
equal: s = t ∈ T,
abdmonoid: AbDMon,
grp_id: e,
grp_car: |g|,
loset: LOSet,
set_blt: a <b b,
set_car: |p|
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
guard: {T},
member: t ∈ T,
so_lambda: λ2x.t[x],
so_apply: x[s],
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
dset: DSet,
prop: ℙ,
abdmonoid: AbDMon,
dmon: DMon,
mon: Mon,
loset: LOSet,
poset: POSet{i},
qoset: QOSet,
set_prod: s × t,
mk_dset: mk_dset(T, eq),
set_car: |p|,
pi1: fst(t),
oalist: oal(a;b),
dset_set: dset_set,
dset_list: s List,
dset_of_mon: g↓set,
ball: ball,
and: P ∧ Q,
cand: A c∧ B,
assert: ↑b,
ifthenelse: if b then t else f fi ,
sd_ordered: sd_ordered(as),
ycomb: Y,
list_ind: list_ind,
map: map(f;as),
nil: [],
it: ⋅,
btrue: tt,
true: True,
not: ¬A,
false: False,
mem: a ∈b as,
mon_for: For{g} x ∈ as. f[x],
for: For{T,op,id} x ∈ as. f[x],
reduce: reduce(f;k;as),
grp_id: e,
pi2: snd(t),
bor_mon: <𝔹,∨b>,
bfalse: ff
Lemmas referenced :
oalist_cases_a,
set_car_wf,
oalist_wf,
dset_wf,
not_wf,
equal_wf,
grp_car_wf,
grp_id_wf,
assert_wf,
before_wf,
map_wf,
set_prod_wf,
dset_of_mon_wf,
all_wf,
ball_wf,
set_blt_wf,
mem_wf,
nil_wf,
dset_of_mon_wf0,
sd_ordered_wf,
abdmonoid_wf,
loset_wf,
before_imp_before_all,
cons_in_oalist,
before_all_imp_before,
abdmonoid_abmonoid
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
sqequalRule,
lambdaEquality,
applyEquality,
isectElimination,
hypothesis,
setElimination,
rename,
independent_functionElimination,
because_Cache,
productElimination,
functionEquality,
universeEquality,
natural_numberEquality,
independent_pairFormation,
dependent_set_memberEquality,
productEquality,
cumulativity
Latex:
\mforall{}a:LOSet. \mforall{}b:AbDMon. \mforall{}Q:|oal(a;b)| {}\mrightarrow{} \mBbbP{}.
(Q[[]]
{}\mRightarrow{} (\mforall{}ws:|oal(a;b)|. \mforall{}k:|a|. \mforall{}v:|b|.
((\muparrow{}(\mforall{}\msubb{}x(:|a|) \mmember{} map(\mlambda{}z.(fst(z));ws). (x <\msubb{} k))) {}\mRightarrow{} (\mneg{}(v = e)) {}\mRightarrow{} Q[[<k, v> / ws]]))
{}\mRightarrow{} \{\mforall{}ws:|oal(a;b)|. Q[ws]\})
Date html generated:
2016_05_16-AM-08_16_09
Last ObjectModification:
2015_12_28-PM-06_29_13
Theory : polynom_2
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