Nuprl Lemma : round-robin-list-index
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[x:T]. ∀[L:T List]. (round-robin(L) outl(list-index(eq;L;x))) = x ∈ T supposing (x ∈ L)
Proof
Definitions occuring in Statement :
round-robin: round-robin(L),
list-index: list-index(d;L;x),
l_member: (x ∈ l),
list: T List,
deq: EqDecider(T),
outl: outl(x),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
apply: f a,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
round-robin: round-robin(L),
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
nat_plus: ℕ+,
or: P ∨ Q,
false: False,
cons: [a / b],
top: Top,
guard: {T},
nat: ℕ,
le: A ≤ B,
cand: A c∧ B,
decidable: Dec(P),
not: ¬A,
uiff: uiff(P;Q),
subtract: n - m,
less_than': less_than'(a;b),
true: True,
ge: i ≥ j ,
outl: outl(x),
int_seg: {i..j-},
isl: isl(x),
lelt: i ≤ j < k,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
prop: ℙ,
sq_type: SQType(T)
Lemmas referenced :
list-index-property,
subtype_base_sq,
int_subtype_base,
rem_base_case,
outl_wf,
isl-list-index,
length_wf,
list-cases,
length_of_nil_lemma,
nil_member,
product_subtype_list,
length_of_cons_lemma,
istype-void,
length_wf_nat,
decidable__lt,
istype-false,
not-lt-2,
condition-implies-le,
minus-add,
minus-one-mul,
zero-add,
minus-one-mul-top,
add-commutes,
add_functionality_wrt_le,
add-associates,
add-zero,
le-add-cancel,
istype-less_than,
non_neg_length,
assert_elim,
btrue_wf,
bfalse_wf,
btrue_neq_bfalse,
nat_properties,
int_seg_properties,
full-omega-unsat,
intformand_wf,
intformless_wf,
itermVar_wf,
intformnot_wf,
istype-int,
int_formula_prop_and_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_formula_prop_not_lemma,
int_formula_prop_wf,
l_member_wf,
list_wf,
deq_wf,
istype-universe
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
independent_isectElimination,
sqequalRule,
instantiate,
cumulativity,
intEquality,
because_Cache,
dependent_functionElimination,
productElimination,
independent_functionElimination,
Error :dependent_set_memberEquality_alt,
unionElimination,
voidElimination,
promote_hyp,
hypothesis_subsumption,
Error :isect_memberEquality_alt,
Error :inhabitedIsType,
Error :lambdaFormation_alt,
setElimination,
rename,
independent_pairFormation,
natural_numberEquality,
addEquality,
minusEquality,
Error :equalityIstype,
equalityTransitivity,
equalitySymmetry,
Error :productIsType,
applyLambdaEquality,
approximateComputation,
Error :dependent_pairFormation_alt,
Error :lambdaEquality_alt,
int_eqEquality,
Error :universeIsType,
universeEquality
Latex:
\mforall{}[T:Type]. \mforall{}[eq:EqDecider(T)]. \mforall{}[x:T]. \mforall{}[L:T List].
(round-robin(L) outl(list-index(eq;L;x))) = x supposing (x \mmember{} L)
Date html generated:
2019_06_20-PM-01_56_52
Last ObjectModification:
2019_03_06-AM-10_52_27
Theory : decidable!equality
Home
Index