Nuprl Lemma : oal_neg_non_id_vals
∀a:LOSet. ∀b:AbDGrp. ∀ps:(|a| × |b|) List. ((¬↑(e ∈b map(λx.(snd(x));ps))) ⇒ (¬↑(e ∈b map(λx.(snd(x));--ps))))
Proof
Definitions occuring in Statement :
oal_neg: --ps,
mem: a ∈b as,
map: map(f;as),
list: T List,
assert: ↑b,
pi2: snd(t),
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
lambda: λx.A[x],
product: x:A × B[x],
dset_of_mon: g↓set,
abdgrp: AbDGrp,
grp_id: e,
grp_car: |g|,
loset: LOSet,
set_car: |p|
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
not: ¬A,
false: False,
member: t ∈ T,
loset: LOSet,
poset: POSet{i},
qoset: QOSet,
dset: DSet,
abdgrp: AbDGrp,
abgrp: AbGrp,
grp: Group{i},
mon: Mon,
oal_neg: --ps,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
dmon: DMon,
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
pi2: snd(t),
dset_of_mon: g↓set,
set_car: |p|,
pi1: fst(t),
prop: ℙ,
top: Top,
compose: f o g,
nat: ℕ,
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
and: P ∧ Q,
guard: {T},
or: P ∨ Q,
assert: ↑b,
ifthenelse: if b then t else f fi ,
bfalse: ff,
cons: [a / b],
le: A ≤ B,
less_than': less_than'(a;b),
colength: colength(L),
nil: [],
it: ⋅,
sq_type: SQType(T),
less_than: a < b,
squash: ↓T,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
decidable: Dec(P),
set_eq: =b,
infix_ap: x f y,
grp_car: |g|,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
true: True
Lemmas referenced :
oal_neg_wf,
assert_wf,
mem_wf,
dset_of_mon_wf,
subtype_rel_sets,
grp_id_wf,
map_wf,
set_car_wf,
grp_car_wf,
dset_of_mon_wf0,
not_wf,
list_wf,
abdgrp_wf,
loset_wf,
map_map,
istype-void,
mon_wf,
inverse_wf,
grp_op_wf,
grp_inv_wf,
comm_wf,
eqfun_p_wf,
grp_eq_wf,
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
less_than_wf,
assert_witness,
intformeq_wf,
int_formula_prop_eq_lemma,
list-cases,
map_nil_lemma,
mem_nil_lemma,
product_subtype_list,
colength-cons-not-zero,
colength_wf_list,
istype-false,
le_wf,
subtract-1-ge-0,
subtype_base_sq,
set_subtype_base,
int_subtype_base,
spread_cons_lemma,
decidable__equal_int,
subtract_wf,
intformnot_wf,
itermSubtract_wf,
itermAdd_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
int_term_value_add_lemma,
decidable__le,
map_cons_lemma,
mem_cons_lemma,
nat_wf,
bor_wf,
infix_ap_wf,
bool_wf,
or_wf,
equal_wf,
subtype_rel_self,
iff_transitivity,
iff_weakening_uiff,
assert_of_bor,
assert_of_mon_eq,
grp_subtype_igrp,
abgrp_subtype_grp,
abdgrp_subtype_abgrp,
subtype_rel_transitivity,
abgrp_wf,
grp_wf,
igrp_wf,
squash_wf,
true_wf,
istype-universe,
grp_inv_inv,
grp_inv_id,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
setElimination,
rename,
because_Cache,
hypothesis,
hypothesisEquality,
independent_functionElimination,
voidElimination,
universeIsType,
isectElimination,
applyEquality,
sqequalRule,
instantiate,
independent_isectElimination,
lambdaEquality_alt,
setIsType,
productEquality,
productElimination,
productIsType,
isect_memberEquality_alt,
setEquality,
cumulativity,
intWeakElimination,
natural_numberEquality,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
independent_pairFormation,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
functionIsTypeImplies,
inhabitedIsType,
unionElimination,
promote_hyp,
hypothesis_subsumption,
equalityIsType1,
dependent_set_memberEquality_alt,
imageElimination,
equalityIsType4,
baseApply,
closedConclusion,
baseClosed,
intEquality,
unionIsType,
inlFormation_alt,
inrFormation_alt,
universeEquality,
imageMemberEquality
Latex:
\mforall{}a:LOSet. \mforall{}b:AbDGrp. \mforall{}ps:(|a| \mtimes{} |b|) List.
((\mneg{}\muparrow{}(e \mmember{}\msubb{} map(\mlambda{}x.(snd(x));ps))) {}\mRightarrow{} (\mneg{}\muparrow{}(e \mmember{}\msubb{} map(\mlambda{}x.(snd(x));--ps))))
Date html generated:
2019_10_16-PM-01_07_46
Last ObjectModification:
2018_10_08-PM-05_27_24
Theory : polynom_2
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