Nuprl Lemma : remove-repeats-length-no-repeats-iff
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[L:T List]. uiff(no_repeats(T;L);||remove-repeats(eq;L)|| = ||L|| ∈ ℤ)
Proof
Definitions occuring in Statement :
remove-repeats: remove-repeats(eq;L),
no_repeats: no_repeats(T;l),
length: ||as||,
list: T List,
deq: EqDecider(T),
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
int: ℤ,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uiff: uiff(P;Q),
and: P ∧ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
prop: ℙ,
implies: P ⇒ Q,
squash: ↓T,
true: True,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
all: ∀x:A. B[x],
top: Top,
cand: A c∧ B,
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
or: P ∨ Q,
le: A ≤ B,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
decidable: Dec(P)
Lemmas referenced :
no_repeats_wf,
equal_wf,
length_wf,
remove-repeats_wf,
list_wf,
deq_wf,
no_repeats_witness,
squash_wf,
true_wf,
remove-repeats-length-no-repeats,
iff_weakening_equal,
list_induction,
remove_repeats_nil_lemma,
length_of_nil_lemma,
no_repeats_nil,
equal-wf-base,
no_repeats_cons,
list_ind_cons_lemma,
list_ind_nil_lemma,
remove-repeats-append,
cons_wf,
nil_wf,
length-append,
length_of_cons_lemma,
remove-repeats-append-one-member,
remove-repeats-set-equal,
set-equal-remove-repeats,
remove-repeats-length-leq,
satisfiable-full-omega-tt,
intformand_wf,
intformle_wf,
itermVar_wf,
intformeq_wf,
itermAdd_wf,
itermConstant_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_var_lemma,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
decidable__equal_int,
add-is-int-iff,
intformnot_wf,
int_formula_prop_not_lemma,
false_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
independent_pairFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
hypothesis,
intEquality,
because_Cache,
universeEquality,
isect_memberFormation,
sqequalRule,
productElimination,
independent_pairEquality,
isect_memberEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
applyEquality,
lambdaEquality,
imageElimination,
independent_isectElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
functionEquality,
lambdaFormation,
rename,
dependent_functionElimination,
voidElimination,
voidEquality,
equalityUniverse,
levelHypothesis,
unionElimination,
dependent_pairFormation,
int_eqEquality,
computeAll,
pointwiseFunctionality,
promote_hyp,
baseApply,
closedConclusion
Latex:
\mforall{}[T:Type]. \mforall{}[eq:EqDecider(T)]. \mforall{}[L:T List]. uiff(no\_repeats(T;L);||remove-repeats(eq;L)|| = ||L||)
Date html generated:
2017_04_17-AM-09_12_47
Last ObjectModification:
2017_02_27-PM-05_19_44
Theory : decidable!equality
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