Nuprl Lemma : oalist_cases
∀a:LOSet. ∀b:AbDMon. ∀Q:((|a| × |b|) List) ⟶ 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: [],
list: T List,
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>,
product: x:A × B[x],
equal: s = t ∈ T,
abdmonoid: AbDMon,
grp_id: e,
grp_car: |g|,
loset: LOSet,
set_car: |p|
Definitions unfolded in proof :
guard: {T},
all: ∀x:A. B[x],
implies: P ⇒ Q,
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),
oalist: oal(a;b),
dset_set: dset_set,
dset_list: s List,
dset_of_mon: g↓set,
prop: ℙ,
so_apply: x[s],
and: P ∧ Q,
pi2: snd(t),
sq_stable: SqStable(P),
not: ¬A,
false: False,
squash: ↓T,
or: P ∨ Q,
top: Top,
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
bfalse: ff,
cons: [a / b],
set_eq: =b,
bool: 𝔹,
unit: Unit,
it: ⋅,
uiff: uiff(P;Q),
uimplies: b supposing a,
band: p ∧b q,
exists: ∃x:A. B[x],
sq_type: SQType(T),
bnot: ¬bb,
grp_car: |g|,
infix_ap: x f y,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
cand: A c∧ B,
true: True
Lemmas referenced :
set_car_wf,
oalist_wf,
grp_car_wf,
assert_wf,
before_wf,
map_wf,
set_prod_wf,
dset_of_mon_wf,
not_wf,
equal_wf,
grp_id_wf,
cons_wf,
subtype_rel_self,
nil_wf,
list_wf,
abdmonoid_wf,
loset_wf,
sq_stable__and,
sd_ordered_wf,
mem_wf,
dset_of_mon_wf0,
sq_stable__assert,
sq_stable__not,
list-cases,
map_nil_lemma,
istype-void,
sd_ordered_nil_lemma,
mem_nil_lemma,
true_wf,
false_wf,
product_subtype_list,
map_cons_lemma,
sd_ordered_cons_lemma,
mem_cons_lemma,
eqtt_to_assert,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
bfalse_wf,
bor_wf,
infix_ap_wf,
bool_wf,
grp_eq_wf,
iff_transitivity,
or_wf,
iff_weakening_uiff,
assert_of_bor,
assert_of_mon_eq,
assert_of_band,
squash_wf,
dset_wf,
assert_elim
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
lambdaFormation_alt,
universeIsType,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
dependent_functionElimination,
hypothesisEquality,
hypothesis,
applyEquality,
lambdaEquality_alt,
setElimination,
rename,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
functionIsType,
because_Cache,
productElimination,
productEquality,
independent_pairEquality,
instantiate,
universeEquality,
isect_memberEquality_alt,
independent_functionElimination,
voidElimination,
functionIsTypeImplies,
imageMemberEquality,
baseClosed,
imageElimination,
unionElimination,
productIsType,
promote_hyp,
hypothesis_subsumption,
equalityElimination,
independent_isectElimination,
dependent_pairFormation_alt,
equalityIsType1,
functionEquality,
independent_pairFormation,
inlFormation_alt,
inrFormation_alt,
unionIsType,
dependent_set_memberEquality_alt,
natural_numberEquality,
cumulativity
Latex:
\mforall{}a:LOSet. \mforall{}b:AbDMon. \mforall{}Q:((|a| \mtimes{} |b|) List) {}\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_15
Last ObjectModification:
2018_10_08-PM-00_30_43
Theory : polynom_2
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