Nuprl Lemma : cons_interleaving
∀[T:Type]. ∀x:T. ∀L,L1,L2:T List. (interleaving(T;L1;L2;L) ⇒ interleaving(T;[x / L1];L2;[x / L]))
Proof
Definitions occuring in Statement :
interleaving: interleaving(T;L1;L2;L),
cons: [a / b],
list: T List,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
interleaving: interleaving(T;L1;L2;L),
member: t ∈ T,
top: Top,
and: P ∧ Q,
cand: A c∧ B,
nat: ℕ,
guard: {T},
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
false: False,
uiff: uiff(P;Q),
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
prop: ℙ,
le: A ≤ B,
less_than': less_than'(a;b),
disjoint_sublists: disjoint_sublists(T;L1;L2;L),
int_seg: {i..j-},
lelt: i ≤ j < k,
less_than: a < b,
squash: ↓T,
true: True,
nat_plus: ℕ+,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
cons: [a / b],
colength: colength(L),
nil: [],
it: ⋅,
sq_type: SQType(T),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
increasing: increasing(f;k),
subtract: n - m,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
fadd: fadd(f;g),
select: L[n],
fshift: fshift(f;x),
bool: 𝔹,
unit: Unit,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
bnot: ¬bb,
assert: ↑b,
nequal: a ≠ b ∈ T
Lemmas referenced :
length_of_cons_lemma,
istype-void,
nat_properties,
decidable__equal_int,
length_wf,
add-is-int-iff,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformeq_wf,
itermAdd_wf,
itermVar_wf,
itermConstant_wf,
istype-int,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
false_wf,
add_nat_wf,
istype-false,
le_wf,
decidable__le,
intformle_wf,
int_formula_prop_le_lemma,
fadd_wf,
length_wf_nat,
less_than_wf,
int_seg_wf,
fshift_wf,
add_nat_plus,
nat_plus_properties,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
increasing_wf,
non_neg_length,
select_wf,
cons_wf,
int_seg_properties,
set_subtype_base,
lelt_wf,
int_subtype_base,
interleaving_wf,
list_wf,
istype-universe,
decidable__assert,
null_wf,
assert_of_null,
ge_wf,
list-cases,
product_subtype_list,
colength-cons-not-zero,
colength_wf_list,
subtract-1-ge-0,
subtype_base_sq,
spread_cons_lemma,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
null_nil_lemma,
btrue_wf,
null_cons_lemma,
bfalse_wf,
btrue_neq_bfalse,
nat_wf,
length_of_nil_lemma,
non_nil_length,
iff_weakening_uiff,
assert_wf,
equal-wf-T-base,
fshift_increasing,
fadd_increasing,
const_nondecreasing,
eq_int_wf,
equal-wf-base,
bool_wf,
bnot_wf,
not_wf,
uiff_transitivity,
eqtt_to_assert,
assert_of_eq_int,
iff_transitivity,
eqff_to_assert,
assert_of_bnot,
equal_wf,
iff_weakening_equal,
add-subtract-cancel,
squash_wf,
true_wf,
select_cons_tl,
subtype_rel_self,
int_seg_subtype_nat,
subtract_nat_wf,
subtract-is-int-iff,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
lambdaFormation_alt,
sqequalHypSubstitution,
sqequalRule,
cut,
introduction,
extract_by_obid,
dependent_functionElimination,
thin,
isect_memberEquality_alt,
voidElimination,
hypothesis,
productElimination,
isectElimination,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
setElimination,
rename,
hypothesisEquality,
addEquality,
natural_numberEquality,
unionElimination,
pointwiseFunctionality,
promote_hyp,
baseApply,
closedConclusion,
baseClosed,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
independent_pairFormation,
universeIsType,
dependent_set_memberEquality_alt,
inhabitedIsType,
equalityIsType1,
because_Cache,
imageMemberEquality,
productIsType,
functionExtensionality,
applyEquality,
functionIsType,
imageElimination,
equalityIsType4,
intEquality,
universeEquality,
intWeakElimination,
axiomSqEquality,
functionIsTypeImplies,
hypothesis_subsumption,
instantiate,
equalityIsType3,
functionExtensionality_alt,
equalityElimination,
cumulativity,
equalityIsType2
Latex:
\mforall{}[T:Type]. \mforall{}x:T. \mforall{}L,L1,L2:T List. (interleaving(T;L1;L2;L) {}\mRightarrow{} interleaving(T;[x / L1];L2;[x / L]))
Date html generated:
2019_10_15-AM-10_55_33
Last ObjectModification:
2018_10_09-AM-10_15_59
Theory : list!
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