Nuprl Lemma : cons_member!
∀[T:Type]. ∀l:T List. ∀a,x:T. ((x ∈! [a / l]) ⇐⇒ ((x = a ∈ T) ∧ (¬(x ∈ l))) ∨ ((x ∈! l) ∧ (¬(x = a ∈ T))))
Proof
Definitions occuring in Statement :
l_member!: (x ∈! l),
l_member: (x ∈ l),
cons: [a / b],
list: T List,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
not: ¬A,
or: P ∨ Q,
and: P ∧ Q,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
l_member!: (x ∈! l),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
exists: ∃x:A. B[x],
member: t ∈ T,
cand: A c∧ B,
nat: ℕ,
prop: ℙ,
uimplies: b supposing a,
top: Top,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
sq_type: SQType(T),
guard: {T},
nequal: a ≠ b ∈ T ,
select: L[n],
cons: [a / b],
l_member: (x ∈ l),
less_than: a < b,
squash: ↓T,
le: A ≤ B,
less_than': less_than'(a;b),
nat_plus: ℕ+,
true: True
Lemmas referenced :
nat_wf,
less_than_wf,
length_wf,
cons_wf,
select_wf,
length_of_cons_lemma,
istype-void,
nat_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
istype-int,
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_formula_prop_wf,
set_subtype_base,
le_wf,
int_subtype_base,
not_wf,
l_member_wf,
equal_wf,
istype-universe,
list_wf,
decidable__assert,
eq_int_wf,
assert_of_eq_int,
subtype_base_sq,
decidable__equal_int,
intformeq_wf,
int_formula_prop_eq_lemma,
neg_assert_of_eq_int,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
itermAdd_wf,
int_term_value_add_lemma,
select-cons-tl,
add-subtract-cancel,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
non_neg_length,
istype-false,
select-cons-hd,
add_nat_plus,
length_wf_nat,
nat_plus_properties,
add-is-int-iff,
false_wf,
squash_wf,
true_wf,
select_cons_tl,
subtype_rel_self,
iff_weakening_equal,
select_cons_hd
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
Error :isect_memberFormation_alt,
Error :lambdaFormation_alt,
independent_pairFormation,
Error :productIsType,
Error :universeIsType,
cut,
introduction,
extract_by_obid,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
Error :equalityIsType1,
Error :inhabitedIsType,
cumulativity,
because_Cache,
independent_isectElimination,
dependent_functionElimination,
Error :isect_memberEquality_alt,
voidElimination,
natural_numberEquality,
unionElimination,
approximateComputation,
independent_functionElimination,
Error :dependent_pairFormation_alt,
Error :lambdaEquality_alt,
int_eqEquality,
Error :functionIsType,
Error :equalityIsType4,
applyEquality,
intEquality,
Error :unionIsType,
universeEquality,
productElimination,
instantiate,
equalityTransitivity,
equalitySymmetry,
Error :dependent_set_memberEquality_alt,
Error :inlFormation_alt,
addEquality,
imageElimination,
applyLambdaEquality,
Error :inrFormation_alt,
imageMemberEquality,
baseClosed,
pointwiseFunctionality,
promote_hyp,
baseApply,
closedConclusion
Latex:
\mforall{}[T:Type]. \mforall{}l:T List. \mforall{}a,x:T. ((x \mmember{}! [a / l]) \mLeftarrow{}{}\mRightarrow{} ((x = a) \mwedge{} (\mneg{}(x \mmember{} l))) \mvee{} ((x \mmember{}! l) \mwedge{} (\mneg{}(x = a))))
Date html generated:
2019_06_20-PM-01_26_00
Last ObjectModification:
2018_10_05-PM-04_03_57
Theory : list_1
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