Nuprl Lemma : values-for-distinct-property
∀[A,V:Type].
∀eq:EqDecider(A). ∀L:(A × V) List.
((||values-for-distinct(eq;L)|| = ||remove-repeats(eq;map(λp.(fst(p));L))|| ∈ ℤ)
∧ (∀i:ℕ||remove-repeats(eq;map(λp.(fst(p));L))||
(<remove-repeats(eq;map(λp.(fst(p));L))[i], values-for-distinct(eq;L)[i]> ∈ L)))
Proof
Definitions occuring in Statement :
values-for-distinct: values-for-distinct(eq;L),
remove-repeats: remove-repeats(eq;L),
l_member: (x ∈ l),
select: L[n],
length: ||as||,
map: map(f;as),
list: T List,
deq: EqDecider(T),
int_seg: {i..j-},
uall: ∀[x:A]. B[x],
pi1: fst(t),
all: ∀x:A. B[x],
and: P ∧ Q,
lambda: λx.A[x],
pair: <a, b>,
product: x:A × B[x],
natural_number: $n,
int: ℤ,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
so_apply: x[s],
so_lambda: λ2x.t[x],
member: t ∈ T,
cand: A c∧ B,
and: P ∧ Q,
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
top: Top,
values-for-distinct: values-for-distinct(eq;L),
iff: P ⇐⇒ Q,
prop: ℙ,
false: False,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
implies: P ⇒ Q,
not: ¬A,
or: P ∨ Q,
decidable: Dec(P),
le: A ≤ B,
lelt: i ≤ j < k,
uimplies: b supposing a,
int_seg: {i..j-},
pi1: fst(t),
rev_implies: P ⇐ Q,
squash: ↓T,
true: True,
subtype_rel: A ⊆r B,
outl: outl(x),
isl: isl(x)
Lemmas referenced :
deq_wf,
list_wf,
pi1_wf,
map_wf,
remove-repeats_wf,
length_wf,
int_seg_wf,
istype-void,
length-map,
select_member,
int_formula_prop_less_lemma,
intformless_wf,
decidable__lt,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
istype-int,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
full-omega-unsat,
decidable__le,
int_seg_properties,
select_wf,
member-remove-repeats,
isl-apply-alist,
l_member_wf,
squash_wf,
true_wf,
istype-universe,
select-map,
subtype_rel_list,
top_wf,
apply-alist_wf,
assert_elim,
btrue_wf,
bfalse_wf,
btrue_neq_bfalse
Rules used in proof :
because_Cache,
inhabitedIsType,
productIsType,
sqequalRule,
lambdaEquality_alt,
productEquality,
hypothesisEquality,
natural_numberEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
universeIsType,
hypothesis,
independent_pairFormation,
cut,
lambdaFormation_alt,
isect_memberFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
voidElimination,
isect_memberEquality_alt,
dependent_functionElimination,
int_eqEquality,
dependent_pairFormation_alt,
independent_functionElimination,
approximateComputation,
unionElimination,
equalitySymmetry,
equalityTransitivity,
independent_isectElimination,
rename,
setElimination,
productElimination,
hyp_replacement,
applyEquality,
imageElimination,
instantiate,
universeEquality,
independent_pairEquality,
imageMemberEquality,
baseClosed,
equalityIstype,
dependent_set_memberEquality_alt,
applyLambdaEquality
Latex:
\mforall{}[A,V:Type].
\mforall{}eq:EqDecider(A). \mforall{}L:(A \mtimes{} V) List.
((||values-for-distinct(eq;L)|| = ||remove-repeats(eq;map(\mlambda{}p.(fst(p));L))||)
\mwedge{} (\mforall{}i:\mBbbN{}||remove-repeats(eq;map(\mlambda{}p.(fst(p));L))||
(<remove-repeats(eq;map(\mlambda{}p.(fst(p));L))[i], values-for-distinct(eq;L)[i]> \mmember{} L)))
Date html generated:
2019_10_15-AM-10_24_16
Last ObjectModification:
2019_08_05-PM-02_03_49
Theory : decidable!equality
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