Nuprl Lemma : agree_on_common_cons2
∀[T:Type]
∀as,bs:T List. ∀x:T.
(agree_on_common(T;[x / as];bs) ⇐⇒ agree_on_common(T;as;bs) supposing ¬(x ∈ bs)
∧ agree_on_common(T;as;[x / bs]) ⇐⇒ agree_on_common(T;as;bs) supposing ¬(x ∈ as))
Proof
Definitions occuring in Statement :
agree_on_common: agree_on_common(T;as;bs),
l_member: (x ∈ l),
cons: [a / b],
list: T List,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
not: ¬A,
and: P ∧ Q,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
nat: ℕ,
implies: P ⇒ Q,
guard: {T},
ge: i ≥ j ,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
decidable: Dec(P),
or: P ∨ Q,
false: False,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
le: A ≤ B,
agree_on_common: agree_on_common(T;as;bs),
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
cand: A c∧ B,
true: True
Lemmas referenced :
length_wf,
add_nat_wf,
length_wf_nat,
nat_wf,
nat_properties,
decidable__le,
add-is-int-iff,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
itermAdd_wf,
intformeq_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_wf,
false_wf,
equal_wf,
le_wf,
list_wf,
all_wf,
subtract_wf,
not_wf,
l_member_wf,
iff_wf,
agree_on_common_wf,
cons_wf,
set_wf,
less_than_wf,
primrec-wf2,
length_zero,
non_neg_length,
decidable__equal_int,
list_ind_cons_lemma,
list_ind_nil_lemma,
true_wf,
nil_wf,
list_induction,
isect_wf,
agree_on_common_nil,
or_wf,
length_of_cons_lemma,
itermSubtract_wf,
int_term_value_subtract_lemma,
cons_member,
le_weakening2,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
length_of_nil_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
lambdaFormation,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
dependent_set_memberEquality,
addEquality,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
setElimination,
rename,
sqequalRule,
natural_numberEquality,
unionElimination,
pointwiseFunctionality,
promote_hyp,
baseApply,
closedConclusion,
baseClosed,
productElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
because_Cache,
universeIsType,
universeEquality,
functionEquality,
productEquality,
isectEquality,
independent_pairEquality,
axiomEquality,
hyp_replacement,
inlFormation,
inrFormation
Latex:
\mforall{}[T:Type]
\mforall{}as,bs:T List. \mforall{}x:T.
(agree\_on\_common(T;[x / as];bs) \mLeftarrow{}{}\mRightarrow{} agree\_on\_common(T;as;bs) supposing \mneg{}(x \mmember{} bs)
\mwedge{} agree\_on\_common(T;as;[x / bs]) \mLeftarrow{}{}\mRightarrow{} agree\_on\_common(T;as;bs) supposing \mneg{}(x \mmember{} as))
Date html generated:
2019_10_15-AM-10_53_13
Last ObjectModification:
2018_09_27-AM-11_00_36
Theory : list!
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