Nuprl Lemma : interleaving_of_cons
∀[T:Type]
∀x:T. ∀L,L1,L2:T List.
(interleaving(T;L1;L2;[x / L])
⇐⇒ (0 < ||L1|| c∧ ((L1[0] = x ∈ T) ∧ interleaving(T;tl(L1);L2;L)))
∨ (0 < ||L2|| c∧ ((L2[0] = x ∈ T) ∧ interleaving(T;L1;tl(L2);L))))
Proof
Definitions occuring in Statement :
interleaving: interleaving(T;L1;L2;L),
select: L[n],
length: ||as||,
tl: tl(l),
cons: [a / b],
list: T List,
less_than: a < b,
uall: ∀[x:A]. B[x],
cand: A c∧ B,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
or: P ∨ Q,
and: P ∧ Q,
natural_number: $n,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
or: P ∨ Q,
decidable: Dec(P),
member: t ∈ T,
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
select: L[n],
nil: [],
it: ⋅,
so_lambda: λ2x y.t[x; y],
top: Top,
so_apply: x[s1;s2],
prop: ℙ,
cand: A c∧ B,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
guard: {T},
true: True,
squash: ↓T,
less_than: a < b,
nat_plus: ℕ+,
cons: [a / b],
so_lambda: λ2x.t[x],
so_apply: x[s],
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
bfalse: ff,
disjoint_sublists: disjoint_sublists(T;L1;L2;L),
interleaving: interleaving(T;L1;L2;L),
subtype_rel: A ⊆r B,
lelt: i ≤ j < k,
int_seg: {i..j-},
inject: Inj(A;B;f),
unit: Unit,
bool: 𝔹,
sq_type: SQType(T),
nat: ℕ,
ge: i ≥ j ,
subtract: n - m,
increasing: increasing(f;k),
rev_uimplies: rev_uimplies(P;Q),
colength: colength(L)
Lemmas referenced :
istype-universe,
list_wf,
null_wf,
decidable__assert,
assert_of_null,
length_of_nil_lemma,
stuck-spread,
base_wf,
reduce_tl_nil_lemma,
iff_wf,
interleaving_wf,
cons_wf,
or_wf,
less_than_wf,
length_wf,
equal_wf,
select_wf,
false_wf,
tl_wf,
nil_wf,
equal-wf-base-T,
int_formula_prop_wf,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_less_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
intformeq_wf,
itermAdd_wf,
itermVar_wf,
itermConstant_wf,
intformless_wf,
intformnot_wf,
intformand_wf,
full-omega-unsat,
add-is-int-iff,
decidable__lt,
nat_plus_properties,
nat_plus_wf,
length_wf_nat,
add_nat_plus,
reduce_tl_cons_lemma,
length_of_cons_lemma,
nil_interleaving,
nat_wf,
list_induction,
istype-false,
istype-int,
istype-void,
nil_interleaving2,
not_wf,
assert_wf,
null_nil_lemma,
true_wf,
null_cons_lemma,
iff_weakening_uiff,
non_nil_length,
equal-wf-T-base,
not_functionality_wrt_uiff,
int_subtype_base,
lelt_wf,
set_subtype_base,
le_wf,
decidable__equal_int,
injection_le,
int_seg_wf,
assert_of_le_int,
bnot_of_lt_int,
assert_functionality_wrt_uiff,
eqff_to_assert,
assert_of_lt_int,
eqtt_to_assert,
uiff_transitivity,
bnot_wf,
le_int_wf,
bool_wf,
lt_int_wf,
subtype_base_sq,
int_term_value_subtract_lemma,
int_formula_prop_le_lemma,
itermSubtract_wf,
intformle_wf,
decidable__le,
nat_properties,
int_seg_properties,
non_neg_length,
subtract_wf,
increasing_implies,
increasing_inj,
increasing_implies_le,
istype-less_than,
istype-le,
le-add-cancel,
add-zero,
add-associates,
add_functionality_wrt_le,
add-commutes,
minus-one-mul-top,
zero-add,
minus-one-mul,
minus-add,
condition-implies-le,
not-lt-2,
product_subtype_list,
list-cases,
select-cons-hd,
increasing_wf,
add-member-int_seg2,
length_tl,
subtract_nat_wf,
iff_weakening_equal,
subtype_rel_self,
select_cons_tl,
squash_wf,
select_tl,
length_zero,
spread_cons_lemma,
subtract-1-ge-0,
colength_wf_list,
colength-cons-not-zero,
istype-base,
ge_wf,
cons_interleaving,
cons_interleaving2
Rules used in proof :
universeEquality,
universeIsType,
inhabitedIsType,
unionElimination,
hypothesis,
hypothesisEquality,
isectElimination,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
lambdaFormation_alt,
isect_memberFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
productElimination,
independent_isectElimination,
sqequalRule,
baseClosed,
lambdaFormation,
isect_memberEquality,
voidElimination,
voidEquality,
hyp_replacement,
equalitySymmetry,
applyLambdaEquality,
cumulativity,
because_Cache,
productEquality,
natural_numberEquality,
independent_pairFormation,
equalityTransitivity,
imageElimination,
intEquality,
int_eqEquality,
lambdaEquality,
dependent_pairFormation,
approximateComputation,
closedConclusion,
baseApply,
promote_hyp,
pointwiseFunctionality,
rename,
setElimination,
imageMemberEquality,
dependent_set_memberEquality,
independent_functionElimination,
inrFormation,
functionEquality,
addEquality,
productIsType,
equalityIsType1,
lambdaEquality_alt,
dependent_pairFormation_alt,
dependent_set_memberEquality_alt,
isect_memberEquality_alt,
inlFormation_alt,
unionIsType,
equalityIsType3,
inrFormation_alt,
equalityIsType4,
applyEquality,
equalityElimination,
instantiate,
functionExtensionality,
sqequalBase,
equalityIstype,
Error :memTop,
functionIsType,
inlFormation,
minusEquality,
hypothesis_subsumption,
functionIsTypeImplies,
axiomEquality,
intWeakElimination
Latex:
\mforall{}[T:Type]
\mforall{}x:T. \mforall{}L,L1,L2:T List.
(interleaving(T;L1;L2;[x / L])
\mLeftarrow{}{}\mRightarrow{} (0 < ||L1|| c\mwedge{} ((L1[0] = x) \mwedge{} interleaving(T;tl(L1);L2;L)))
\mvee{} (0 < ||L2|| c\mwedge{} ((L2[0] = x) \mwedge{} interleaving(T;L1;tl(L2);L))))
Date html generated:
2020_05_20-AM-07_48_41
Last ObjectModification:
2020_04_06-PM-06_47_41
Theory : list!
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