Nuprl Lemma : isl-apply-alist
∀[A,T:Type].
∀eq:EqDecider(T). ∀x:T. ∀L:(T × A) List.
((↑isl(apply-alist(eq;L;x)) ⇐⇒ (x ∈ map(λp.(fst(p));L)))
∧ (<x, outl(apply-alist(eq;L;x))> ∈ L) supposing ↑isl(apply-alist(eq;L;x)))
Proof
Definitions occuring in Statement :
apply-alist: apply-alist(eq;L;x),
l_member: (x ∈ l),
map: map(f;as),
list: T List,
deq: EqDecider(T),
outl: outl(x),
assert: ↑b,
isl: isl(x),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
pi1: fst(t),
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
lambda: λx.A[x],
pair: <a, b>,
product: x:A × B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
uimplies: b supposing a,
so_lambda: λ2x.t[x],
so_apply: x[s],
top: Top,
and: P ∧ Q,
deq: EqDecider(T),
prop: ℙ,
bfalse: ff,
isl: isl(x),
outl: outl(x),
assert: ↑b,
ifthenelse: if b then t else f fi ,
cand: A c∧ B,
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
false: False,
rev_implies: P ⇐ Q,
not: ¬A,
decidable: Dec(P),
or: P ∨ Q,
exists: ∃x:A. B[x],
l_member: (x ∈ l),
guard: {T},
int_seg: {i..j-},
lelt: i ≤ j < k,
satisfiable_int_formula: satisfiable_int_formula(fmla),
le: A ≤ B,
less_than': less_than'(a;b),
nat: ℕ,
ge: i ≥ j ,
less_than: a < b,
squash: ↓T,
btrue: tt,
true: True
Lemmas referenced :
apply-alist-cases,
subtype_rel_list,
top_wf,
subtype_rel_product,
deq_property,
list_wf,
deq_wf,
decidable__assert,
equal_wf,
assert_wf,
all_wf,
decidable_wf,
map_wf,
pi1_wf,
false_wf,
l_member_wf,
decidable_functionality,
iff_weakening_uiff,
decidable__l_member,
member-map,
int_seg_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformless_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
int_formula_prop_and_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
int_seg_wf,
decidable__equal_int,
subtract_wf,
int_seg_subtype,
decidable__le,
intformnot_wf,
itermSubtract_wf,
intformeq_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
int_formula_prop_eq_lemma,
le_wf,
select_wf,
nat_properties,
less_than_wf,
length_wf,
exists_wf,
decidable__lt,
not_wf,
lelt_wf,
set_wf,
primrec-wf2,
nat_wf,
itermAdd_wf,
int_term_value_add_lemma,
decidable__exists_int_seg,
true_wf,
squash_wf,
pair_eta_rw,
iff_weakening_equal,
select_member,
pi2_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
cut,
introduction,
extract_by_obid,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
lambdaFormation,
applyEquality,
because_Cache,
productEquality,
cumulativity,
independent_isectElimination,
sqequalRule,
lambdaEquality,
isect_memberEquality,
voidElimination,
voidEquality,
universeEquality,
productElimination,
dependent_functionElimination,
setElimination,
rename,
independent_pairEquality,
independent_pairFormation,
independent_functionElimination,
addLevel,
allFunctionality,
unionElimination,
equalitySymmetry,
hyp_replacement,
applyLambdaEquality,
natural_numberEquality,
dependent_pairFormation,
int_eqEquality,
intEquality,
computeAll,
equalityTransitivity,
levelHypothesis,
hypothesis_subsumption,
dependent_set_memberEquality,
functionEquality,
imageElimination,
addEquality,
instantiate,
axiomEquality,
imageMemberEquality,
baseClosed
Latex:
\mforall{}[A,T:Type].
\mforall{}eq:EqDecider(T). \mforall{}x:T. \mforall{}L:(T \mtimes{} A) List.
((\muparrow{}isl(apply-alist(eq;L;x)) \mLeftarrow{}{}\mRightarrow{} (x \mmember{} map(\mlambda{}p.(fst(p));L)))
\mwedge{} (<x, outl(apply-alist(eq;L;x))> \mmember{} L) supposing \muparrow{}isl(apply-alist(eq;L;x)))
Date html generated:
2017_04_14-AM-09_24_08
Last ObjectModification:
2017_02_27-PM-03_59_32
Theory : list_1
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