Nuprl Lemma : oalist_ind_a
∀a:LOSet. ∀b:AbDMon. ∀Q:|oal(a;b)| ⟶ 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: [],
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 :
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,
so_apply: x[s],
prop: ℙ,
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],
uimplies: b supposing a,
top: Top,
oalist: oal(a;b),
dset_set: dset_set,
dset_list: s List,
not: ¬A,
false: False,
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|,
int_seg: {i..j-},
lelt: i ≤ j < k,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
sq_type: SQType(T),
nat: ℕ,
ge: i ≥ j ,
le: A ≤ B,
less_than: a < b,
squash: ↓T,
list: T List
Lemmas referenced :
set_car_wf,
oalist_wf,
subtype_rel_self,
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,
mem_wf,
nil_wf,
dset_of_mon_wf0,
sd_ordered_wf,
pi2_wf,
abdmonoid_wf,
loset_wf,
int_seg_properties,
full-omega-unsat,
intformand_wf,
intformless_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
istype-int,
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,
subtype_base_sq,
set_subtype_base,
lelt_wf,
int_subtype_base,
intformnot_wf,
intformeq_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_subtract_lemma,
decidable__le,
decidable__lt,
istype-le,
istype-less_than,
non_neg_length,
nat_properties,
length_wf,
guard_wf,
all_wf,
le_wf,
primrec-wf2,
itermAdd_wf,
int_term_value_add_lemma,
istype-nat,
length_wf_nat,
oalist_cases_a,
less_than_wf,
cons_wf,
length_of_cons_lemma,
cons_in_oalist,
list_wf,
assert_wf,
not_wf
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,
instantiate,
universeEquality,
independent_isectElimination,
isect_memberEquality_alt,
voidElimination,
equalityIstype,
natural_numberEquality,
independent_pairFormation,
dependent_set_memberEquality_alt,
productEquality,
productIsType,
productElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
int_eqEquality,
unionElimination,
cumulativity,
intEquality,
applyLambdaEquality,
hypothesis_subsumption,
imageElimination,
functionEquality,
setIsType,
addEquality,
voidEquality,
independent_pairEquality,
setEquality
Latex:
\mforall{}a:LOSet. \mforall{}b:AbDMon. \mforall{}Q:|oal(a;b)| {}\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:
2019_10_16-PM-01_07_23
Last ObjectModification:
2018_12_08-AM-11_57_18
Theory : polynom_2
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