Nuprl Lemma : no_repeats-append
∀[T:Type]. ∀[L1,L2:T List]. uiff(no_repeats(T;L1 @ L2);{no_repeats(T;L1) ∧ no_repeats(T;L2) ∧ l_disjoint(T;L1;L2)})
Proof
Definitions occuring in Statement :
l_disjoint: l_disjoint(T;l1;l2),
no_repeats: no_repeats(T;l),
append: as @ bs,
list: T List,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
guard: {T},
and: P ∧ Q,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
and: P ∧ Q,
prop: ℙ,
uiff: uiff(P;Q),
guard: {T},
l_disjoint: l_disjoint(T;l1;l2),
or: P ∨ Q,
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
cons: [a / b],
decidable: Dec(P),
colength: colength(L),
nil: [],
it: ⋅,
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
subtype_rel: A ⊆r B,
cand: A c∧ B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
istype-less_than,
no_repeats_witness,
intformeq_wf,
int_formula_prop_eq_lemma,
list-cases,
list_ind_nil_lemma,
no_repeats_nil,
l_disjoint_nil,
nil_wf,
no_repeats_wf,
l_disjoint_wf,
product_subtype_list,
colength-cons-not-zero,
colength_wf_list,
decidable__le,
intformnot_wf,
int_formula_prop_not_lemma,
istype-le,
append_wf,
subtract-1-ge-0,
subtype_base_sq,
set_subtype_base,
int_subtype_base,
spread_cons_lemma,
decidable__equal_int,
subtract_wf,
itermSubtract_wf,
itermAdd_wf,
int_term_value_subtract_lemma,
int_term_value_add_lemma,
le_wf,
list_ind_cons_lemma,
no_repeats_cons,
cons_wf,
l_disjoint_cons2,
l_member_wf,
member_append,
istype-nat,
list_wf,
istype-universe,
not_wf,
iff_transitivity,
iff_weakening_uiff
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
thin,
Error :lambdaFormation_alt,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
Error :dependent_pairFormation_alt,
Error :lambdaEquality_alt,
int_eqEquality,
dependent_functionElimination,
Error :isect_memberEquality_alt,
voidElimination,
sqequalRule,
independent_pairFormation,
Error :universeIsType,
productElimination,
independent_pairEquality,
because_Cache,
Error :functionIsTypeImplies,
Error :inhabitedIsType,
Error :isectIsTypeImplies,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
unionElimination,
Error :productIsType,
promote_hyp,
hypothesis_subsumption,
Error :equalityIstype,
Error :dependent_set_memberEquality_alt,
instantiate,
imageElimination,
baseApply,
closedConclusion,
baseClosed,
applyEquality,
intEquality,
sqequalBase,
Error :unionIsType,
universeEquality,
productEquality,
unionEquality,
Error :functionIsType,
Error :inlFormation_alt,
Error :inrFormation_alt
Latex:
\mforall{}[T:Type]. \mforall{}[L1,L2:T List].
uiff(no\_repeats(T;L1 @ L2);\{no\_repeats(T;L1) \mwedge{} no\_repeats(T;L2) \mwedge{} l\_disjoint(T;L1;L2)\})
Date html generated:
2019_06_20-PM-01_27_27
Last ObjectModification:
2019_01_15-PM-02_18_12
Theory : list_1
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