Nuprl Lemma : oalist_cases_a
∀a:LOSet. ∀b:AbDMon. ∀Q:|oal(a;b)| ⟶ ℙ.
(Q[[]]
⇒ (∀ws:|oal(a;b)|. ∀x:|a|. ∀y:|b|. ((↑before(x;map(λx.(fst(x));ws))) ⇒ (¬(y = e ∈ |b|)) ⇒ Q[[<x, y> / ws]]))
⇒ {∀ws:|oal(a;b)|. Q[ws]})
Proof
Definitions occuring in Statement :
oalist: oal(a;b),
before: before(u;ps),
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_car: |p|
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
guard: {T},
member: t ∈ T,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
dset: DSet,
loset: LOSet,
poset: POSet{i},
qoset: QOSet,
abdmonoid: AbDMon,
dmon: DMon,
mon: Mon,
set_prod: s × t,
mk_dset: mk_dset(T, eq),
set_car: |p|,
pi1: fst(t),
dset_of_mon: g↓set,
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
top: Top,
oalist: oal(a;b),
dset_set: dset_set,
dset_list: s List,
not: ¬A,
false: False,
prop: ℙ,
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,
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,
grp_car: |g|,
squash: ↓T,
sq_stable: SqStable(P),
infix_ap: x f y,
bnot: ¬bb,
sq_type: SQType(T),
exists: ∃x:A. B[x],
band: p ∧b q,
uiff: uiff(P;Q),
unit: Unit,
bool: 𝔹,
set_eq: =b,
cons: [a / b],
or: P ∨ Q,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
list: T List
Lemmas referenced :
set_car_wf,
oalist_wf,
grp_car_wf,
istype-assert,
before_wf,
map_wf,
set_prod_wf,
dset_of_mon_wf,
pi1_wf_top,
subtype_rel_product,
top_wf,
istype-void,
grp_id_wf,
subtype_rel_self,
mem_wf,
nil_wf,
dset_of_mon_wf0,
sd_ordered_wf,
pi2_wf,
abdmonoid_wf,
loset_wf,
sq_stable__not,
sq_stable__assert,
not_wf,
assert_wf,
sq_stable__and,
grp_eq_wf,
bool_wf,
infix_ap_wf,
bor_wf,
bfalse_wf,
assert-bnot,
bool_subtype_base,
subtype_base_sq,
bool_cases_sqequal,
eqff_to_assert,
eqtt_to_assert,
mem_cons_lemma,
sd_ordered_cons_lemma,
map_cons_lemma,
product_subtype_list,
false_wf,
true_wf,
mem_nil_lemma,
sd_ordered_nil_lemma,
map_nil_lemma,
list-cases,
assert_of_mon_eq,
assert_of_bor,
equal_wf,
or_wf,
iff_transitivity,
assert_of_band,
iff_weakening_uiff,
list_wf,
cons_in_oalist
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
sqequalRule,
functionIsType,
universeIsType,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
dependent_functionElimination,
hypothesisEquality,
hypothesis,
applyEquality,
lambdaEquality_alt,
setElimination,
rename,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
because_Cache,
independent_isectElimination,
isect_memberEquality_alt,
voidElimination,
equalityIstype,
instantiate,
universeEquality,
natural_numberEquality,
independent_pairFormation,
dependent_set_memberEquality_alt,
productEquality,
productIsType,
productElimination,
imageElimination,
baseClosed,
imageMemberEquality,
functionIsTypeImplies,
independent_functionElimination,
functionEquality,
equalityIsType1,
dependent_pairFormation_alt,
equalityElimination,
hypothesis_subsumption,
promote_hyp,
unionElimination,
unionIsType,
inrFormation_alt,
inlFormation_alt,
setEquality
Latex:
\mforall{}a:LOSet. \mforall{}b:AbDMon. \mforall{}Q:|oal(a;b)| {}\mrightarrow{} \mBbbP{}.
(Q[[]]
{}\mRightarrow{} (\mforall{}ws:|oal(a;b)|. \mforall{}x:|a|. \mforall{}y:|b|.
((\muparrow{}before(x;map(\mlambda{}x.(fst(x));ws))) {}\mRightarrow{} (\mneg{}(y = e)) {}\mRightarrow{} Q[[<x, y> / ws]]))
{}\mRightarrow{} \{\mforall{}ws:|oal(a;b)|. Q[ws]\})
Date html generated:
2019_10_16-PM-01_07_19
Last ObjectModification:
2018_12_08-AM-11_57_16
Theory : polynom_2
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