Nuprl Lemma : l_all_exists_injection
∀[A,B:Type]. ∀[R:A ⟶ B ⟶ ℙ]. ∀[P:B ⟶ ℙ].
∀L:A List
((∀x∈L.∃y:B. (R[x;y] ∧ P[y])) ⇒ (∃f:ℕ||L|| ⟶ {y:B| P[y]} . Inj(ℕ||L||;{y:B| P[y]} ;f))) supposing
(no_repeats(A;L) and
(∀x1,x2:A. ∀y:B. (R[x1;y] ⇒ R[x2;y] ⇒ (x1 = x2 ∈ A))))
Proof
Definitions occuring in Statement :
l_all: (∀x∈L.P[x]),
no_repeats: no_repeats(T;l),
length: ||as||,
list: T List,
inject: Inj(A;B;f),
int_seg: {i..j-},
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
so_apply: x[s],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
natural_number: $n,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
uimplies: b supposing a,
member: t ∈ T,
implies: P ⇒ Q,
so_apply: x[s1;s2],
subtype_rel: A ⊆r B,
prop: ℙ,
l_all: (∀x∈L.P[x]),
so_lambda: λ2x.t[x],
and: P ∧ Q,
so_apply: x[s],
exists: ∃x:A. B[x],
int_seg: {i..j-},
guard: {T},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
top: Top,
less_than: a < b,
squash: ↓T,
pi1: fst(t),
inject: Inj(A;B;f),
no_repeats: no_repeats(T;l),
le: A ≤ B,
less_than': less_than'(a;b),
nat: ℕ,
ge: i ≥ j
Lemmas referenced :
no_repeats_witness,
l_all_wf,
exists_wf,
l_member_wf,
no_repeats_wf,
all_wf,
equal_wf,
list_wf,
int_seg_wf,
length_wf,
select_wf,
int_seg_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_and_lemma,
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,
inject_wf,
decidable__equal_int_seg,
int_seg_subtype_nat,
false_wf,
nat_properties,
set_wf,
le_wf,
intformeq_wf,
int_formula_prop_eq_lemma,
lelt_wf,
nat_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
introduction,
sqequalRule,
sqequalHypSubstitution,
lambdaEquality,
dependent_functionElimination,
thin,
hypothesisEquality,
axiomEquality,
hypothesis,
applyEquality,
functionExtensionality,
cumulativity,
universeEquality,
because_Cache,
rename,
extract_by_obid,
isectElimination,
independent_functionElimination,
productEquality,
setElimination,
setEquality,
functionEquality,
promote_hyp,
productElimination,
dependent_set_memberEquality,
natural_numberEquality,
independent_isectElimination,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
imageElimination,
equalityTransitivity,
equalitySymmetry,
hyp_replacement
Latex:
\mforall{}[A,B:Type]. \mforall{}[R:A {}\mrightarrow{} B {}\mrightarrow{} \mBbbP{}]. \mforall{}[P:B {}\mrightarrow{} \mBbbP{}].
\mforall{}L:A List
((\mforall{}x\mmember{}L.\mexists{}y:B. (R[x;y] \mwedge{} P[y])) {}\mRightarrow{} (\mexists{}f:\mBbbN{}||L|| {}\mrightarrow{} \{y:B| P[y]\} . Inj(\mBbbN{}||L||;\{y:B| P[y]\} ;f))) suppos\000Cing
(no\_repeats(A;L) and
(\mforall{}x1,x2:A. \mforall{}y:B. (R[x1;y] {}\mRightarrow{} R[x2;y] {}\mRightarrow{} (x1 = x2))))
Date html generated:
2016_10_21-AM-10_27_12
Last ObjectModification:
2016_07_12-AM-05_40_33
Theory : list_1
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