Nuprl Lemma : list_match-aux-cons
∀[A,B:Type]. ∀[R:A ⟶ B ⟶ ℙ].
((∀a:A. ∀b:B. SqStable(R[a;b]))
⇒ (∀bs:B List. ∀u:A. ∀v:A List. ∀used:ℤ List.
(list-match-aux([u / v];bs;used;a,b.R[a;b])
⇐⇒ ∃j:ℕ||bs||. ((¬↑j ∈b used) ∧ R[u;bs[j]] ∧ list-match-aux(v;bs;[j / used];a,b.R[a;b])))))
Proof
Definitions occuring in Statement :
list-match-aux: list-match-aux(L1;L2;used;a,b.R[a; b]),
select: L[n],
length: ||as||,
deq-member: x ∈b L,
cons: [a / b],
list: T List,
int-deq: IntDeq,
int_seg: {i..j-},
assert: ↑b,
sq_stable: SqStable(P),
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
not: ¬A,
implies: P ⇒ Q,
and: P ∧ Q,
function: x:A ⟶ B[x],
natural_number: $n,
int: ℤ,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
all: ∀x:A. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
so_apply: x[s1;s2],
so_apply: x[s],
prop: ℙ,
iff: P ⇐⇒ Q,
and: P ∧ Q,
so_lambda: λ2x y.t[x; y],
rev_implies: P ⇐ Q,
int_seg: {i..j-},
uimplies: b supposing a,
guard: {T},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
less_than: a < b,
squash: ↓T,
subtype_rel: A ⊆r B,
list-match-aux: list-match-aux(L1;L2;used;a,b.R[a; b]),
sq_exists: ∃x:A [B[x]],
le: A ≤ B,
less_than': less_than'(a;b),
nat_plus: ℕ+,
true: True,
uiff: uiff(P;Q),
cand: A c∧ B,
sq_stable: SqStable(P),
select: L[n],
cons: [a / b],
subtract: n - m,
ge: i ≥ j ,
nat: ℕ,
inject: Inj(A;B;f),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
nequal: a ≠ b ∈ T ,
label: ...$L... t
Lemmas referenced :
deq-member_wf,
int-deq_wf,
list_wf,
all_wf,
sq_stable_wf,
list-match-aux_wf,
cons_wf,
exists_wf,
int_seg_wf,
length_wf,
not_wf,
assert_wf,
select_wf,
int_seg_properties,
decidable__le,
full-omega-unsat,
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,
length_of_cons_lemma,
false_wf,
add_nat_plus,
length_wf_nat,
less_than_wf,
nat_plus_wf,
nat_plus_properties,
add-is-int-iff,
itermAdd_wf,
intformeq_wf,
int_term_value_add_lemma,
int_formula_prop_eq_lemma,
equal_wf,
lelt_wf,
assert-deq-member,
l_member_wf,
and_wf,
squash_wf,
le_wf,
add-member-int_seg2,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
non_neg_length,
nat_wf,
nat_properties,
decidable__equal_int,
select-cons-tl,
add-subtract-cancel,
cons_member,
inject_wf,
sq_stable__and,
sq_stable__inject,
sq_stable__all,
sq_stable__not,
eq_int_wf,
bool_wf,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
subtype_rel_self,
eqtt_to_assert,
assert_of_eq_int,
member_wf,
set_subtype_base,
int_subtype_base,
select-cons-hd,
select_cons_tl,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
intEquality,
hypothesis,
hypothesisEquality,
sqequalRule,
lambdaEquality,
applyEquality,
functionEquality,
cumulativity,
universeEquality,
independent_pairFormation,
natural_numberEquality,
productEquality,
dependent_functionElimination,
setElimination,
rename,
functionExtensionality,
because_Cache,
independent_isectElimination,
productElimination,
unionElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
imageElimination,
dependent_set_memberEquality,
imageMemberEquality,
baseClosed,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
pointwiseFunctionality,
promote_hyp,
baseApply,
closedConclusion,
addEquality,
addLevel,
impliesFunctionality,
hyp_replacement,
instantiate,
dependent_set_memberFormation,
equalityElimination,
inlFormation,
levelHypothesis,
impliesLevelFunctionality,
inrFormation
Latex:
\mforall{}[A,B:Type]. \mforall{}[R:A {}\mrightarrow{} B {}\mrightarrow{} \mBbbP{}].
((\mforall{}a:A. \mforall{}b:B. SqStable(R[a;b]))
{}\mRightarrow{} (\mforall{}bs:B List. \mforall{}u:A. \mforall{}v:A List. \mforall{}used:\mBbbZ{} List.
(list-match-aux([u / v];bs;used;a,b.R[a;b])
\mLeftarrow{}{}\mRightarrow{} \mexists{}j:\mBbbN{}||bs||. ((\mneg{}\muparrow{}j \mmember{}\msubb{} used) \mwedge{} R[u;bs[j]] \mwedge{} list-match-aux(v;bs;[j / used];a,b.R[a;b])))))
Date html generated:
2018_05_21-PM-00_47_06
Last ObjectModification:
2018_05_19-AM-06_54_51
Theory : list_1
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