Nuprl Lemma : oalist_pr_length_ind
∀a:LOSet. ∀b:AbDMon. ∀Q:((|a| × |b|) List) ⟶ ((|a| × |b|) List) ⟶ ℙ.
((∀ps,qs:|oal(a;b)|. ((∀us,vs:|oal(a;b)|. (||us|| + ||vs|| < ||ps|| + ||qs|| ⇒ Q[us;vs])) ⇒ Q[ps;qs]))
⇒ {∀ps,qs:|oal(a;b)|. Q[ps;qs]})
Proof
Definitions occuring in Statement :
oalist: oal(a;b),
length: ||as||,
list: T List,
less_than: a < b,
prop: ℙ,
guard: {T},
so_apply: x[s1;s2],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
product: x:A × B[x],
add: n + m,
abdmonoid: AbDMon,
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,
prop: ℙ,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
loset: LOSet,
poset: POSet{i},
qoset: QOSet,
abdmonoid: AbDMon,
dset: DSet,
oalist: oal(a;b),
dset_set: dset_set,
mk_dset: mk_dset(T, eq),
set_car: |p|,
pi1: fst(t),
dset_list: s List,
set_prod: s × t,
dset_of_mon: g↓set,
so_apply: x[s1;s2],
dmon: DMon,
mon: Mon,
so_apply: x[s],
pi2: snd(t),
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
decidable: Dec(P),
or: P ∨ Q,
le: A ≤ B,
less_than': less_than'(a;b),
nat: ℕ,
ge: i ≥ j ,
uiff: uiff(P;Q),
less_than: a < b,
squash: ↓T
Lemmas referenced :
all_wf,
set_car_wf,
oalist_wf,
less_than_wf,
length_wf,
set_prod_wf,
dset_of_mon_wf,
dset_wf,
list_wf,
grp_car_wf,
abdmonoid_wf,
loset_wf,
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,
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,
add_nat_wf,
length_wf_nat,
nat_wf,
nat_properties,
add-is-int-iff,
itermAdd_wf,
int_term_value_add_lemma,
equal_wf,
decidable__lt,
lelt_wf,
set_wf,
primrec-wf2,
pi2_wf,
pi1_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
dependent_functionElimination,
hypothesisEquality,
hypothesis,
applyEquality,
because_Cache,
sqequalRule,
lambdaEquality,
functionEquality,
addEquality,
setElimination,
rename,
functionExtensionality,
productEquality,
universeEquality,
cumulativity,
independent_pairEquality,
natural_numberEquality,
productElimination,
independent_isectElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
unionElimination,
addLevel,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
levelHypothesis,
hypothesis_subsumption,
dependent_set_memberEquality,
pointwiseFunctionality,
promote_hyp,
baseApply,
closedConclusion,
baseClosed,
independent_functionElimination,
imageElimination
Latex:
\mforall{}a:LOSet. \mforall{}b:AbDMon. \mforall{}Q:((|a| \mtimes{} |b|) List) {}\mrightarrow{} ((|a| \mtimes{} |b|) List) {}\mrightarrow{} \mBbbP{}.
((\mforall{}ps,qs:|oal(a;b)|.
((\mforall{}us,vs:|oal(a;b)|. (||us|| + ||vs|| < ||ps|| + ||qs|| {}\mRightarrow{} Q[us;vs])) {}\mRightarrow{} Q[ps;qs]))
{}\mRightarrow{} \{\mforall{}ps,qs:|oal(a;b)|. Q[ps;qs]\})
Date html generated:
2017_10_01-AM-10_02_00
Last ObjectModification:
2017_03_03-PM-01_04_47
Theory : polynom_2
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