Nuprl Lemma : squash-list-exists
∀[A,B:Type]. ∀[R:A ⟶ B ⟶ ℙ].
∀as:A List
((∀i:ℕ||as||. (↓∃b:B. R[as[i];b])) ⇒ (↓∃bs:B List. ((||bs|| = ||as|| ∈ ℤ) ∧ (∀i:ℕ||as||. R[as[i];bs[i]]))))
Proof
Definitions occuring in Statement :
select: L[n],
length: ||as||,
list: T List,
int_seg: {i..j-},
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
squash: ↓T,
implies: P ⇒ Q,
and: P ∧ Q,
function: x:A ⟶ B[x],
natural_number: $n,
int: ℤ,
universe: Type,
equal: s = t ∈ T
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,
squash: ↓T,
or: P ∨ Q,
select: L[n],
nil: [],
it: ⋅,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
so_lambda: λ2x.t[x],
guard: {T},
int_seg: {i..j-},
lelt: i ≤ j < k,
so_apply: x[s],
cons: [a / b],
le: A ≤ B,
less_than': less_than'(a;b),
colength: colength(L),
sq_type: SQType(T),
less_than: a < b,
decidable: Dec(P),
subtype_rel: A ⊆r B,
cand: A c∧ B,
uiff: uiff(P;Q),
subtract: n - m,
nat_plus: ℕ+,
true: True,
iff: P ⇐⇒ Q,
rev_implies: 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,
less_than_wf,
list-cases,
length_of_nil_lemma,
stuck-spread,
istype-base,
int_seg_wf,
squash_wf,
exists_wf,
int_seg_properties,
product_subtype_list,
colength-cons-not-zero,
colength_wf_list,
istype-false,
le_wf,
subtract-1-ge-0,
subtype_base_sq,
nat_wf,
set_subtype_base,
int_subtype_base,
spread_cons_lemma,
decidable__equal_int,
subtract_wf,
intformnot_wf,
intformeq_wf,
itermSubtract_wf,
itermAdd_wf,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_subtract_lemma,
int_term_value_add_lemma,
decidable__le,
length_of_cons_lemma,
length_wf,
select_wf,
cons_wf,
non_neg_length,
decidable__lt,
list_wf,
nil_wf,
length_nil,
add-member-int_seg2,
select-cons-tl,
add-subtract-cancel,
add_nat_plus,
length_wf_nat,
nat_plus_properties,
equal_wf,
true_wf,
add_functionality_wrt_eq,
iff_weakening_equal,
select_cons_tl,
subtype_rel_self
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
Error :lambdaFormation_alt,
thin,
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,
imageElimination,
imageMemberEquality,
baseClosed,
Error :functionIsTypeImplies,
Error :inhabitedIsType,
unionElimination,
Error :functionIsType,
cumulativity,
applyEquality,
functionExtensionality,
because_Cache,
productElimination,
promote_hyp,
hypothesis_subsumption,
Error :equalityIsType1,
Error :dependent_set_memberEquality_alt,
instantiate,
intEquality,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
Error :equalityIsType4,
addEquality,
universeEquality,
Error :productIsType,
Error :equalityIsType3
Latex:
\mforall{}[A,B:Type]. \mforall{}[R:A {}\mrightarrow{} B {}\mrightarrow{} \mBbbP{}].
\mforall{}as:A List
((\mforall{}i:\mBbbN{}||as||. (\mdownarrow{}\mexists{}b:B. R[as[i];b]))
{}\mRightarrow{} (\mdownarrow{}\mexists{}bs:B List. ((||bs|| = ||as||) \mwedge{} (\mforall{}i:\mBbbN{}||as||. R[as[i];bs[i]]))))
Date html generated:
2019_06_20-PM-01_48_38
Last ObjectModification:
2018_10_04-PM-02_28_58
Theory : list_1
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