Nuprl Lemma : oal_merge_dom_pred
∀a:LOSet. ∀b:AbDMon. ∀Q:|a| ⟶ 𝔹. ∀ps,qs:(|a| × |b|) List.
((↑(∀bx(:|a|) ∈ map(λx.(fst(x));ps)
Q[x]))
⇒ (↑(∀bx(:|a|) ∈ map(λx.(fst(x));qs)
Q[x]))
⇒ (↑(∀bx(:|a|) ∈ map(λx.(fst(x));ps ++ qs)
Q[x])))
Proof
Definitions occuring in Statement :
oal_merge: ps ++ qs,
ball: ball,
map: map(f;as),
list: T List,
assert: ↑b,
bool: 𝔹,
so_apply: x[s],
pi1: fst(t),
all: ∀x:A. B[x],
implies: P ⇒ Q,
lambda: λx.A[x],
function: x:A ⟶ B[x],
product: x:A × B[x],
abdmonoid: AbDMon,
grp_car: |g|,
loset: LOSet,
set_car: |p|
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
loset: LOSet,
poset: POSet{i},
qoset: QOSet,
dset: DSet,
abdmonoid: AbDMon,
dmon: DMon,
mon: Mon,
so_lambda: λ2x y.t[x; y],
implies: P ⇒ Q,
prop: ℙ,
pi1: fst(t),
so_lambda: λ2x.t[x],
so_apply: x[s],
ball: ball,
so_apply: x[s1;s2],
guard: {T},
or: P ∨ Q,
top: Top,
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
true: True,
cons: [a / b],
bool: 𝔹,
unit: Unit,
it: ⋅,
band: p ∧b q,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
bfalse: ff,
false: False,
iff: P ⇐⇒ Q,
not: ¬A,
rev_implies: P ⇐ Q,
infix_ap: x f y,
rev_uimplies: rev_uimplies(P;Q),
cand: A c∧ B,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x]
Lemmas referenced :
list_pr_length_ind,
set_car_wf,
grp_car_wf,
assert_wf,
ball_wf,
map_wf,
oal_merge_wf,
list_wf,
bool_wf,
abdmonoid_wf,
loset_wf,
list-cases,
length_of_nil_lemma,
map_nil_lemma,
oal_merge_left_nil_lemma,
ball_nil_lemma,
true_wf,
all_wf,
less_than_wf,
length_wf,
product_subtype_list,
length_of_cons_lemma,
map_cons_lemma,
oal_merge_right_nil_lemma,
ball_cons_lemma,
eqtt_to_assert,
equal_wf,
bool_cases_sqequal,
pi1_wf,
oal_merge_conses_lemma,
set_blt_wf,
uiff_transitivity,
equal-wf-T-base,
set_lt_wf,
assert_of_set_lt,
iff_transitivity,
bnot_wf,
not_wf,
iff_weakening_uiff,
eqff_to_assert,
assert_of_bnot,
grp_eq_wf,
grp_op_wf,
grp_id_wf,
assert_of_mon_eq,
assert_of_band,
cons_wf,
decidable__lt,
satisfiable-full-omega-tt,
intformnot_wf,
intformless_wf,
itermAdd_wf,
itermVar_wf,
itermConstant_wf,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
productEquality,
isectElimination,
setElimination,
rename,
hypothesisEquality,
hypothesis,
sqequalRule,
lambdaEquality,
functionEquality,
because_Cache,
productElimination,
applyEquality,
functionExtensionality,
independent_functionElimination,
unionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
natural_numberEquality,
addEquality,
promote_hyp,
hypothesis_subsumption,
equalityElimination,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
independent_pairEquality,
baseClosed,
independent_pairFormation,
impliesFunctionality,
dependent_pairFormation,
int_eqEquality,
intEquality,
computeAll
Latex:
\mforall{}a:LOSet. \mforall{}b:AbDMon. \mforall{}Q:|a| {}\mrightarrow{} \mBbbB{}. \mforall{}ps,qs:(|a| \mtimes{} |b|) List.
((\muparrow{}(\mforall{}\msubb{}x(:|a|) \mmember{} map(\mlambda{}x.(fst(x));ps)
Q[x]))
{}\mRightarrow{} (\muparrow{}(\mforall{}\msubb{}x(:|a|) \mmember{} map(\mlambda{}x.(fst(x));qs)
Q[x]))
{}\mRightarrow{} (\muparrow{}(\mforall{}\msubb{}x(:|a|) \mmember{} map(\mlambda{}x.(fst(x));ps ++ qs)
Q[x])))
Date html generated:
2017_10_01-AM-10_02_32
Last ObjectModification:
2017_03_03-PM-01_05_17
Theory : polynom_2
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