Nuprl Lemma : no_repeats-length-le-by-relation
∀[A,B:Type]. ∀[R:A ⟶ B ⟶ ℙ].
∀[as:A List]. ∀[bs:B List]. (||as|| ≤ ||bs||) supposing ((∀a∈as.(∃b∈bs. R a b)) and no_repeats(A;as))
supposing ∀a1,a2:A. ∀b:B. ((R a1 b) ⇒ (R a2 b) ⇒ (a1 = a2 ∈ A))
Proof
Definitions occuring in Statement :
l_exists: (∃x∈L. P[x]),
l_all: (∀x∈L.P[x]),
no_repeats: no_repeats(T;l),
length: ||as||,
list: T List,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
prop: ℙ,
le: A ≤ B,
all: ∀x:A. B[x],
implies: P ⇒ Q,
apply: f a,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
l_exists: (∃x∈L. P[x]),
l_all: (∀x∈L.P[x]),
exists: ∃x:A. B[x],
inject: Inj(A;B;f),
all: ∀x:A. B[x],
implies: P ⇒ Q,
int_seg: {i..j-},
guard: {T},
lelt: i ≤ j < k,
and: P ∧ Q,
le: A ≤ B,
less_than: a < b,
squash: ↓T,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
top: Top,
prop: ℙ,
subtype_rel: A ⊆r B,
ge: i ≥ j ,
so_lambda: λ2x.t[x],
so_apply: x[s],
nat: ℕ,
cand: A c∧ B,
pi1: fst(t),
no_repeats: no_repeats(T;l),
less_than': less_than'(a;b),
sq_type: SQType(T)
Lemmas referenced :
injection_le,
length_wf_nat,
select_wf,
int_seg_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
non_neg_length,
length_wf,
int_seg_wf,
set_subtype_base,
lelt_wf,
int_subtype_base,
inject_wf,
le_witness_for_triv,
l_all_wf,
l_exists_wf,
l_member_wf,
no_repeats_wf,
list_wf,
subtype_rel_self,
istype-universe,
istype-le,
istype-less_than,
nat_properties,
intformeq_wf,
int_formula_prop_eq_lemma,
squash_wf,
le_wf,
less_than_wf,
decidable__equal_int_seg,
int_seg_subtype_nat,
istype-false,
subtype_base_sq,
istype-nat
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalHypSubstitution,
sqequalRule,
hypothesis,
promote_hyp,
thin,
productElimination,
extract_by_obid,
isectElimination,
hypothesisEquality,
independent_isectElimination,
dependent_pairFormation_alt,
lambdaFormation_alt,
dependent_functionElimination,
setElimination,
rename,
because_Cache,
natural_numberEquality,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
imageElimination,
unionElimination,
approximateComputation,
independent_functionElimination,
lambdaEquality_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
universeIsType,
applyEquality,
inhabitedIsType,
equalityIstype,
intEquality,
sqequalBase,
setIsType,
isectIsTypeImplies,
functionIsType,
instantiate,
universeEquality,
dependent_set_memberEquality_alt,
productIsType,
functionExtensionality,
hyp_replacement,
productEquality,
imageMemberEquality,
baseClosed,
cumulativity
Latex:
\mforall{}[A,B:Type]. \mforall{}[R:A {}\mrightarrow{} B {}\mrightarrow{} \mBbbP{}].
\mforall{}[as:A List]. \mforall{}[bs:B List].
(||as|| \mleq{} ||bs||) supposing ((\mforall{}a\mmember{}as.(\mexists{}b\mmember{}bs. R a b)) and no\_repeats(A;as))
supposing \mforall{}a1,a2:A. \mforall{}b:B. ((R a1 b) {}\mRightarrow{} (R a2 b) {}\mRightarrow{} (a1 = a2))
Date html generated:
2020_05_19-PM-09_42_52
Last ObjectModification:
2019_10_23-PM-03_44_02
Theory : list_1
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