Nuprl Lemma : append_overlapping_sublists
∀[T:Type]. ∀L1,L2,L:T List. ∀x:T. L1 @ [x] ⊆ L ⇒ [x / L2] ⊆ L ⇒ L1 @ [x / L2] ⊆ L supposing no_repeats(T;L)
Proof
Definitions occuring in Statement :
sublist: L1 ⊆ L2,
no_repeats: no_repeats(T;l),
append: as @ bs,
cons: [a / b],
nil: [],
list: T List,
uimplies: b supposing a,
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],
uimplies: b supposing a,
member: t ∈ T,
implies: P ⇒ Q,
sublist: L1 ⊆ L2,
no_repeats: no_repeats(T;l),
exists: ∃x:A. B[x],
and: P ∧ Q,
prop: ℙ,
top: Top,
ge: i ≥ j ,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
int_seg: {i..j-},
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
bfalse: ff,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
cand: A c∧ B,
subtype_rel: A ⊆r B,
nat: ℕ,
increasing: increasing(f;k),
less_than: a < b,
squash: ↓T,
subtract: n - m,
true: True,
so_apply: x[s],
so_lambda: λ2x.t[x],
cons: [a / b],
select: L[n]
Lemmas referenced :
no_repeats_witness,
length_of_cons_lemma,
sublist_wf,
cons_wf,
append_wf,
nil_wf,
no_repeats_wf,
list_wf,
istype-universe,
length_wf,
istype-void,
length-append,
length_of_nil_lemma,
istype-false,
le_int_wf,
eqtt_to_assert,
assert_of_le_int,
decidable__lt,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermVar_wf,
itermAdd_wf,
itermConstant_wf,
intformle_wf,
istype-int,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
istype-le,
istype-less_than,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
iff_weakening_uiff,
assert_wf,
le_wf,
subtract_wf,
int_seg_properties,
decidable__le,
add-is-int-iff,
itermSubtract_wf,
int_term_value_subtract_lemma,
false_wf,
intformeq_wf,
int_formula_prop_eq_lemma,
int_seg_wf,
increasing_wf,
length_wf_nat,
select_wf,
length_cons,
non_neg_length,
length_append,
subtype_rel_list,
top_wf,
nat_properties,
add-associates,
minus-one-mul,
add-swap,
add-commutes,
less_than_wf,
squash_wf,
true_wf,
lelt_wf,
decidable__equal_int,
int_subtype_base,
increasing_implies,
nat_wf,
set_subtype_base,
equal_wf,
not_wf,
iff_weakening_equal,
subtype_rel_self,
select_append_back,
add-subtract-cancel,
length_nil,
product_subtype_list,
list-cases,
bnot_wf,
lt_int_wf,
equal-wf-T-base,
uiff_transitivity,
assert_functionality_wrt_uiff,
bnot_of_le_int,
assert_of_lt_int,
add-mul-special,
zero-mul,
select-cons-hd,
select_append_front
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
independent_functionElimination,
hypothesis,
rename,
sqequalRule,
dependent_functionElimination,
Error :memTop,
productElimination,
universeIsType,
inhabitedIsType,
instantiate,
universeEquality,
natural_numberEquality,
addEquality,
voidElimination,
isect_memberEquality_alt,
independent_pairFormation,
dependent_pairFormation_alt,
lambdaEquality_alt,
setElimination,
because_Cache,
unionElimination,
equalityElimination,
independent_isectElimination,
applyEquality,
dependent_set_memberEquality_alt,
approximateComputation,
int_eqEquality,
productIsType,
equalityTransitivity,
equalitySymmetry,
equalityIstype,
promote_hyp,
cumulativity,
pointwiseFunctionality,
baseApply,
closedConclusion,
baseClosed,
functionExtensionality,
functionIsType,
applyLambdaEquality,
imageElimination,
multiplyEquality,
minusEquality,
hyp_replacement,
imageMemberEquality,
equalityIsType1,
intEquality,
voidEquality,
hypothesis_subsumption
Latex:
\mforall{}[T:Type]
\mforall{}L1,L2,L:T List. \mforall{}x:T.
L1 @ [x] \msubseteq{} L {}\mRightarrow{} [x / L2] \msubseteq{} L {}\mRightarrow{} L1 @ [x / L2] \msubseteq{} L supposing no\_repeats(T;L)
Date html generated:
2020_05_19-PM-09_42_19
Last ObjectModification:
2020_02_06-PM-09_36_44
Theory : list_1
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