Nuprl Lemma : no_repeats-update-alist
∀[A,T:Type]. ∀[eq:EqDecider(T)]. ∀[x:T]. ∀[a:A]. ∀[f:A ⟶ A]. ∀[L:(T × A) List].
no_repeats(T;map(λp.(fst(p));update-alist(eq;L;x;a;n.f[n]))) supposing no_repeats(T;map(λp.(fst(p));L))
Proof
Definitions occuring in Statement :
no_repeats: no_repeats(T;l),
update-alist: update-alist(eq;L;x;z;v.f[v]),
map: map(f;as),
list: T List,
deq: EqDecider(T),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
pi1: fst(t),
lambda: λx.A[x],
function: x:A ⟶ B[x],
product: x:A × B[x],
universe: Type
Definitions unfolded in proof :
deq: EqDecider(T),
subtype_rel: A ⊆r B,
decidable: Dec(P),
so_apply: x[s1;s2],
so_lambda: λ2x y.t[x; y],
squash: ↓T,
less_than: a < b,
sq_type: SQType(T),
it: ⋅,
nil: [],
colength: colength(L),
so_apply: x[s],
so_lambda: λ2x.t[x],
less_than': less_than'(a;b),
le: A ≤ B,
cons: [a / b],
pi1: fst(t),
so_apply: x[s1;s2;s3],
so_lambda: so_lambda3,
update-alist: update-alist(eq;L;x;z;v.f[v]),
or: P ∨ Q,
guard: {T},
prop: ℙ,
and: P ∧ Q,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
uimplies: b supposing a,
ge: i ≥ j ,
false: False,
implies: P ⇒ Q,
nat: ℕ,
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
uiff: uiff(P;Q),
cand: A c∧ B,
bool: 𝔹,
unit: Unit,
btrue: tt,
eqof: eqof(d),
ifthenelse: if b then t else f fi ,
bfalse: ff,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
true: True
Lemmas referenced :
istype-universe,
deq_wf,
istype-nat,
list_wf,
ifthenelse_wf,
list_ind_cons_lemma,
le_wf,
decidable__le,
int_term_value_add_lemma,
int_term_value_subtract_lemma,
int_formula_prop_not_lemma,
itermAdd_wf,
itermSubtract_wf,
intformnot_wf,
subtract_wf,
decidable__equal_int,
spread_cons_lemma,
int_subtype_base,
set_subtype_base,
subtype_base_sq,
subtract-1-ge-0,
update-alist_wf,
pi1_wf,
map_wf,
istype-le,
istype-void,
colength_wf_list,
colength-cons-not-zero,
product_subtype_list,
no_repeats_wf,
nil_wf,
cons_wf,
no_repeats_singleton,
map_cons_lemma,
list_ind_nil_lemma,
map_nil_lemma,
list-cases,
int_formula_prop_eq_lemma,
intformeq_wf,
no_repeats_witness,
istype-less_than,
ge_wf,
int_formula_prop_wf,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_and_lemma,
istype-int,
intformless_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformand_wf,
full-omega-unsat,
nat_properties,
equal-wf-T-base,
bool_wf,
assert_wf,
equal_wf,
no_repeats_cons,
l_member_wf,
bnot_wf,
not_wf,
eqof_wf,
istype-assert,
uiff_transitivity,
eqtt_to_assert,
safe-assert-deq,
iff_transitivity,
iff_weakening_uiff,
eqff_to_assert,
assert_of_bnot,
squash_wf,
true_wf,
member-update-alist1
Rules used in proof :
universeEquality,
functionIsType,
independent_pairEquality,
spreadEquality,
sqequalBase,
intEquality,
baseClosed,
closedConclusion,
baseApply,
imageElimination,
instantiate,
applyEquality,
productIsType,
dependent_set_memberEquality_alt,
because_Cache,
equalityIstype,
productElimination,
hypothesis_subsumption,
promote_hyp,
unionElimination,
productEquality,
functionIsTypeImplies,
inhabitedIsType,
isectIsTypeImplies,
applyLambdaEquality,
equalitySymmetry,
equalityTransitivity,
isect_memberEquality_alt,
voidElimination,
universeIsType,
independent_pairFormation,
sqequalRule,
Error :memTop,
dependent_functionElimination,
int_eqEquality,
lambdaEquality_alt,
dependent_pairFormation_alt,
independent_functionElimination,
approximateComputation,
independent_isectElimination,
natural_numberEquality,
intWeakElimination,
rename,
setElimination,
hypothesis,
hypothesisEquality,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
lambdaFormation_alt,
thin,
cut,
introduction,
isect_memberFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
equalityElimination,
hyp_replacement,
imageMemberEquality,
unionIsType
Latex:
\mforall{}[A,T:Type]. \mforall{}[eq:EqDecider(T)]. \mforall{}[x:T]. \mforall{}[a:A]. \mforall{}[f:A {}\mrightarrow{} A]. \mforall{}[L:(T \mtimes{} A) List].
no\_repeats(T;map(\mlambda{}p.(fst(p));update-alist(eq;L;x;a;n.f[n])))
supposing no\_repeats(T;map(\mlambda{}p.(fst(p));L))
Date html generated:
2020_05_19-PM-09_52_17
Last ObjectModification:
2019_12_26-AM-11_47_52
Theory : decidable!equality
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