Nuprl Lemma : oalist_ind
∀a:LOSet. ∀b:AbDMon. ∀Q:((|a| × |b|) List) ⟶ ℙ.
(Q[[]]
⇒ (∀ws:|oal(a;b)|
(Q[ws] ⇒ (∀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 :
mon: Mon,
dmon: DMon,
abdmonoid: AbDMon,
qoset: QOSet,
poset: POSet{i},
loset: LOSet,
dset_of_mon: g↓set,
set_prod: s × t,
dset_list: s List,
pi1: fst(t),
set_car: |p|,
mk_dset: mk_dset(T, eq),
dset_set: dset_set,
oalist: oal(a;b),
so_apply: x[s],
so_lambda: λ2x.t[x],
prop: ℙ,
dset: DSet,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
all: ∀x:A. B[x],
guard: {T},
squash: ↓T,
less_than: a < b,
ge: i ≥ j ,
nat: ℕ,
less_than': less_than'(a;b),
le: A ≤ B,
or: P ∨ Q,
decidable: Dec(P),
top: Top,
not: ¬A,
false: False,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
uimplies: b supposing a,
and: P ∧ Q,
lelt: i ≤ j < k,
int_seg: {i..j-}
Lemmas referenced :
length_of_cons_lemma,
oalist_cases,
int_seg_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformless_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
int_formula_prop_and_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
int_seg_wf,
decidable__equal_int,
subtract_wf,
int_seg_subtype,
false_wf,
lelt_wf,
decidable__le,
intformnot_wf,
itermSubtract_wf,
intformeq_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
int_formula_prop_eq_lemma,
le_wf,
length_wf,
non_neg_length,
nat_properties,
decidable__lt,
less_than_wf,
set_wf,
primrec-wf2,
nat_wf,
itermAdd_wf,
int_term_value_add_lemma,
length_wf_nat,
set_car_wf,
oalist_wf,
dset_wf,
all_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,
nil_wf,
list_wf,
abdmonoid_wf,
loset_wf
Rules used in proof :
cumulativity,
independent_pairEquality,
productEquality,
productElimination,
universeEquality,
functionEquality,
because_Cache,
rename,
setElimination,
lambdaEquality,
applyEquality,
hypothesis,
hypothesisEquality,
dependent_functionElimination,
thin,
isectElimination,
sqequalHypSubstitution,
lemma_by_obid,
cut,
lambdaFormation,
computationStep,
sqequalTransitivity,
sqequalReflexivity,
sqequalRule,
sqequalSubstitution,
addEquality,
introduction,
independent_functionElimination,
imageElimination,
dependent_set_memberEquality,
hypothesis_subsumption,
levelHypothesis,
setEquality,
equalitySymmetry,
equalityTransitivity,
addLevel,
unionElimination,
computeAll,
independent_pairFormation,
voidEquality,
voidElimination,
isect_memberEquality,
intEquality,
int_eqEquality,
dependent_pairFormation,
independent_isectElimination,
natural_numberEquality
Latex:
\mforall{}a:LOSet. \mforall{}b:AbDMon. \mforall{}Q:((|a| \mtimes{} |b|) List) {}\mrightarrow{} \mBbbP{}.
(Q[[]]
{}\mRightarrow{} (\mforall{}ws:|oal(a;b)|
(Q[ws]
{}\mRightarrow{} (\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:
2016_05_16-AM-08_16_13
Last ObjectModification:
2016_01_16-PM-11_58_49
Theory : polynom_2
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