Nuprl Lemma : sublist_append
∀[T:Type]. ∀L1,L2,L1',L2':T List. (L1 ⊆ L1' ⇒ L2 ⊆ L2' ⇒ L1 @ L2 ⊆ L1' @ L2')
Proof
Definitions occuring in Statement :
sublist: L1 ⊆ L2,
append: as @ bs,
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,
sublist: L1 ⊆ L2,
exists: ∃x:A. B[x],
and: P ∧ Q,
member: t ∈ T,
prop: ℙ,
int_seg: {i..j-},
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
uimplies: b supposing a,
ifthenelse: if b then t else f fi ,
bfalse: ff,
guard: {T},
subtype_rel: A ⊆r B,
lelt: i ≤ j < k,
squash: ↓T,
less_than: a < b,
ge: i ≥ j ,
false: False,
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
le: A ≤ B,
or: P ∨ Q,
decidable: Dec(P),
top: Top,
increasing: increasing(f;k),
cand: A c∧ B,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
nat: ℕ,
true: True,
subtract: n - m
Lemmas referenced :
sublist_wf,
list_wf,
istype-universe,
bnot_wf,
le_wf,
le_int_wf,
less_than_wf,
assert_wf,
equal-wf-T-base,
bool_wf,
length_wf,
lt_int_wf,
uiff_transitivity,
eqtt_to_assert,
assert_of_lt_int,
eqff_to_assert,
assert_functionality_wrt_uiff,
bnot_of_lt_int,
assert_of_le_int,
equal_wf,
lelt_wf,
int_seg_wf,
int_formula_prop_eq_lemma,
intformeq_wf,
add-member-int_seg2,
int_term_value_add_lemma,
int_formula_prop_less_lemma,
itermAdd_wf,
intformless_wf,
decidable__lt,
top_wf,
subtype_rel_list,
length_append,
append_wf,
non_neg_length,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_subtract_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermVar_wf,
itermSubtract_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
full-omega-unsat,
decidable__le,
int_seg_properties,
length-append,
subtract_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
iff_weakening_uiff,
istype-less_than,
istype-int,
istype-le,
length_wf_nat,
nat_properties,
select_append_front,
select_wf,
iff_weakening_equal,
subtract-is-int-iff,
add-is-int-iff,
false_wf,
general_arith_equation1,
minus-one-mul,
add-associates,
add-mul-special,
zero-mul,
add-zero,
itermMultiply_wf,
int_term_value_mul_lemma,
squash_wf,
true_wf,
select_append_back,
subtype_rel_self,
increasing_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
lambdaFormation_alt,
sqequalHypSubstitution,
productElimination,
thin,
dependent_pairFormation_alt,
universeIsType,
cut,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
hypothesis,
inhabitedIsType,
instantiate,
universeEquality,
lambdaEquality,
because_Cache,
baseClosed,
equalitySymmetry,
equalityTransitivity,
rename,
setElimination,
lambdaFormation,
unionElimination,
equalityElimination,
independent_functionElimination,
independent_isectElimination,
sqequalRule,
dependent_functionElimination,
independent_pairFormation,
dependent_set_memberEquality,
cumulativity,
natural_numberEquality,
functionExtensionality,
applyEquality,
imageElimination,
applyLambdaEquality,
intEquality,
int_eqEquality,
dependent_pairFormation,
approximateComputation,
addEquality,
voidEquality,
voidElimination,
isect_memberEquality,
Error :memTop,
equalityIstype,
promote_hyp,
dependent_set_memberEquality_alt,
lambdaEquality_alt,
productIsType,
imageMemberEquality,
pointwiseFunctionality,
baseApply,
closedConclusion,
multiplyEquality,
minusEquality,
functionIsType
Latex:
\mforall{}[T:Type]. \mforall{}L1,L2,L1',L2':T List. (L1 \msubseteq{} L1' {}\mRightarrow{} L2 \msubseteq{} L2' {}\mRightarrow{} L1 @ L2 \msubseteq{} L1' @ L2')
Date html generated:
2020_05_19-PM-09_42_06
Last ObjectModification:
2019_12_31-PM-00_14_53
Theory : list_1
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