Nuprl Lemma : oal_merge_sd_ordered
∀a:LOSet. ∀b:AbDMon. ∀ps,qs:(|a| × |b|) List.
((↑sd_ordered(map(λx.(fst(x));ps))) ⇒ (↑sd_ordered(map(λx.(fst(x));qs))) ⇒ (↑sd_ordered(map(λx.(fst(x));ps ++ qs))))
Proof
Definitions occuring in Statement :
oal_merge: ps ++ qs,
sd_ordered: sd_ordered(as),
map: map(f;as),
list: T List,
assert: ↑b,
pi1: fst(t),
all: ∀x:A. B[x],
implies: P ⇒ Q,
lambda: λx.A[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],
prop: ℙ,
implies: P ⇒ Q,
so_apply: x[s1;s2],
guard: {T},
or: P ∨ Q,
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
true: True,
cons: [a / b],
uimplies: b supposing a,
sq_type: SQType(T),
uiff: uiff(P;Q),
and: P ∧ Q,
bfalse: ff,
band: p ∧b q,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
bool: 𝔹,
grp_car: |g|,
pi1: fst(t),
band_mon: <𝔹,∧b>,
rev_uimplies: rev_uimplies(P;Q),
grp_op: *,
pi2: snd(t),
infix_ap: x f y,
unit: Unit,
it: ⋅,
iff: P ⇐⇒ Q,
not: ¬A,
rev_implies: P ⇐ Q,
false: False,
cand: A c∧ B,
ball: ball,
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,
sd_ordered_wf,
map_wf,
pi1_wf_top,
oal_merge_wf,
list_wf,
abdmonoid_wf,
loset_wf,
list-cases,
length_of_nil_lemma,
map_nil_lemma,
oal_merge_left_nil_lemma,
sd_ordered_nil_lemma,
istype-true,
istype-less_than,
length_wf,
istype-assert,
product_subtype_list,
length_of_cons_lemma,
map_cons_lemma,
oal_merge_right_nil_lemma,
sd_ordered_cons_lemma,
before_wf,
bool_cases,
subtype_base_sq,
bool_wf,
bool_subtype_base,
eqtt_to_assert,
band_wf,
btrue_wf,
bfalse_wf,
cons_wf,
assert_functionality_wrt_uiff,
mon_htfor_wf,
band_mon_wf,
iabmonoid_subtype_imon,
abmonoid_subtype_iabmonoid,
subtype_rel_transitivity,
abmonoid_wf,
iabmonoid_wf,
imon_wf,
ball_wf,
set_blt_wf,
subtype_rel_self,
mon_subtype_grp_sig,
abmonoid_subtype_mon,
mon_wf,
grp_sig_wf,
sd_ordered_char,
mon_htfor_cons_lemma,
assert_of_band,
oal_merge_conses_lemma,
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,
istype-void,
grp_eq_wf,
grp_op_wf,
grp_id_wf,
equal_wf,
assert_of_mon_eq,
oal_merge_dom_pred,
ball_cons_lemma,
decidable__lt,
full-omega-unsat,
intformnot_wf,
intformless_wf,
itermAdd_wf,
itermVar_wf,
itermConstant_wf,
istype-int,
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,
assert_functionality_wrt_bimplies,
ball_functionality_wrt_bimplies,
set_blt_functionality_wrt_set_lt_r,
set_lt_complement,
set_leq_antisymmetry
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
productEquality,
isectElimination,
setElimination,
rename,
hypothesisEquality,
hypothesis,
sqequalRule,
lambdaEquality_alt,
functionEquality,
because_Cache,
productElimination,
independent_pairEquality,
Error :memTop,
productIsType,
universeIsType,
inhabitedIsType,
independent_functionElimination,
unionElimination,
natural_numberEquality,
functionIsType,
addEquality,
promote_hyp,
hypothesis_subsumption,
instantiate,
cumulativity,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
applyEquality,
isect_memberEquality_alt,
equalityElimination,
baseClosed,
independent_pairFormation,
voidElimination,
equalityIstype,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
hyp_replacement,
applyLambdaEquality
Latex:
\mforall{}a:LOSet. \mforall{}b:AbDMon. \mforall{}ps,qs:(|a| \mtimes{} |b|) List.
((\muparrow{}sd\_ordered(map(\mlambda{}x.(fst(x));ps)))
{}\mRightarrow{} (\muparrow{}sd\_ordered(map(\mlambda{}x.(fst(x));qs)))
{}\mRightarrow{} (\muparrow{}sd\_ordered(map(\mlambda{}x.(fst(x));ps ++ qs))))
Date html generated:
2020_05_20-AM-09_35_53
Last ObjectModification:
2020_01_08-PM-06_17_02
Theory : polynom_2
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