Nuprl Lemma : no_repeats-l_member!
∀[T:Type]. ∀l:T List. ∀x:T. ((x ∈ l) ⇐⇒ (x ∈! l)) supposing no_repeats(T;l)
Proof
Definitions occuring in Statement :
l_member!: (x ∈! l),
no_repeats: no_repeats(T;l),
l_member: (x ∈ l),
list: T List,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
uimplies: b supposing a,
member: t ∈ T,
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
l_member: (x ∈ l),
exists: ∃x:A. B[x],
cand: A c∧ B,
l_member!: (x ∈! l),
nat: ℕ,
no_repeats: no_repeats(T;l),
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
squash: ↓T,
guard: {T},
false: False,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
prop: ℙ,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_stable: SqStable(P),
ge: i ≥ j ,
rev_implies: P ⇐ Q
Lemmas referenced :
no_repeats_witness,
decidable__equal_int,
satisfiable-full-omega-tt,
intformand_wf,
intformeq_wf,
itermVar_wf,
intformnot_wf,
int_formula_prop_and_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_formula_prop_not_lemma,
int_formula_prop_wf,
equal-wf-base,
set_subtype_base,
le_wf,
int_subtype_base,
sq_stable__le,
decidable__le,
intformle_wf,
itermConstant_wf,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
equal_wf,
select_wf,
nat_properties,
less_than_wf,
length_wf,
nat_wf,
all_wf,
l_member_wf,
l_member!_wf,
no_repeats_wf,
list_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
independent_functionElimination,
hypothesis,
rename,
independent_pairFormation,
productElimination,
dependent_pairFormation,
dependent_set_memberEquality,
dependent_functionElimination,
setElimination,
because_Cache,
unionElimination,
independent_isectElimination,
applyLambdaEquality,
sqequalRule,
imageMemberEquality,
baseClosed,
imageElimination,
natural_numberEquality,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
applyEquality,
equalityTransitivity,
equalitySymmetry,
cumulativity,
productEquality,
functionEquality,
universeEquality
Latex:
\mforall{}[T:Type]. \mforall{}l:T List. \mforall{}x:T. ((x \mmember{} l) \mLeftarrow{}{}\mRightarrow{} (x \mmember{}! l)) supposing no\_repeats(T;l)
Date html generated:
2017_04_17-AM-07_29_12
Last ObjectModification:
2017_02_27-PM-04_07_36
Theory : list_1
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