Nuprl Lemma : set-equal-no_repeats-length
∀[T:Type]. ∀[as,bs:T List].
(||as|| = ||bs|| ∈ ℤ) supposing (set-equal(T;as;bs) and no_repeats(T;bs) and no_repeats(T;as))
Proof
Definitions occuring in Statement :
set-equal: set-equal(T;x;y),
no_repeats: no_repeats(T;l),
length: ||as||,
list: T List,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
int: ℤ,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
cand: A c∧ B,
implies: P ⇒ Q,
decidable: Dec(P),
or: P ∨ Q,
le: A ≤ B,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
prop: ℙ
Lemmas referenced :
no_repeats_wf,
set-equal_wf,
int_formula_prop_wf,
int_formula_prop_le_lemma,
int_term_value_var_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
intformle_wf,
itermVar_wf,
intformeq_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__equal_int,
l_contains-no_repeats-length,
set-equal-l_contains
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
dependent_functionElimination,
productElimination,
independent_functionElimination,
hypothesis,
independent_pairFormation,
because_Cache,
independent_isectElimination,
unionElimination,
natural_numberEquality,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
computeAll,
axiomEquality,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}[T:Type]. \mforall{}[as,bs:T List].
(||as|| = ||bs||) supposing (set-equal(T;as;bs) and no\_repeats(T;bs) and no\_repeats(T;as))
Date html generated:
2016_05_14-PM-01_39_25
Last ObjectModification:
2016_01_15-AM-08_25_04
Theory : list_1
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