Nuprl Lemma : apply-alist-no_repeats
∀[A,T:Type]. ∀[eq:EqDecider(T)]. ∀[L:(T × A) List].
∀[x:T]. ∀[a:A]. apply-alist(eq;L;x) = (inl a) ∈ (A?) supposing (<x, a> ∈ L)
supposing no_repeats(T;map(λp.(fst(p));L))
Proof
Definitions occuring in Statement :
apply-alist: apply-alist(eq;L;x),
no_repeats: no_repeats(T;l),
l_member: (x ∈ l),
map: map(f;as),
list: T List,
deq: EqDecider(T),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
pi1: fst(t),
unit: Unit,
lambda: λx.A[x],
pair: <a, b>,
product: x:A × B[x],
inl: inl x,
union: left + right,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
all: ∀x:A. B[x],
top: Top,
l_member: (x ∈ l),
exists: ∃x:A. B[x],
cand: A c∧ B,
and: P ∧ Q,
nat: ℕ,
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
prop: ℙ,
not: ¬A,
implies: P ⇒ Q,
false: False,
guard: {T},
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
less_than: a < b,
squash: ↓T,
pi1: fst(t),
no_repeats: no_repeats(T;l),
less_than': less_than'(a;b),
true: True,
iff: P ⇐⇒ Q,
pi2: snd(t)
Lemmas referenced :
apply-alist-cases,
subtype_rel_list,
top_wf,
subtype_rel_product,
lelt_wf,
length_wf,
equal_wf,
select_wf,
int_seg_properties,
nat_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_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_formula_prop_wf,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
pi1_wf,
int_seg_wf,
and_wf,
l_member_wf,
no_repeats_wf,
map_wf,
list_wf,
deq_wf,
int_seg_subtype_nat,
false_wf,
map-length,
intformeq_wf,
int_formula_prop_eq_lemma,
nat_wf,
not_wf,
squash_wf,
true_wf,
map_select,
iff_weakening_equal,
pi2_wf,
unit_wf2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
because_Cache,
hypothesisEquality,
applyEquality,
productEquality,
cumulativity,
hypothesis,
independent_isectElimination,
sqequalRule,
lambdaEquality,
lambdaFormation,
isect_memberEquality,
voidElimination,
voidEquality,
productElimination,
setElimination,
rename,
dependent_set_memberEquality,
independent_pairFormation,
independent_functionElimination,
dependent_functionElimination,
unionElimination,
natural_numberEquality,
dependent_pairFormation,
int_eqEquality,
intEquality,
computeAll,
imageElimination,
independent_pairEquality,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
axiomEquality,
universeEquality,
imageMemberEquality,
baseClosed,
hyp_replacement,
inlEquality
Latex:
\mforall{}[A,T:Type]. \mforall{}[eq:EqDecider(T)]. \mforall{}[L:(T \mtimes{} A) List].
\mforall{}[x:T]. \mforall{}[a:A]. apply-alist(eq;L;x) = (inl a) supposing (<x, a> \mmember{} L)
supposing no\_repeats(T;map(\mlambda{}p.(fst(p));L))
Date html generated:
2017_09_29-PM-06_04_26
Last ObjectModification:
2017_07_26-PM-02_53_04
Theory : decidable!equality
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