Nuprl Lemma : append_functionality_wrt_permr
∀T:Type. ∀as,as',bs,bs':T List. ((as ≡(T) as') ⇒ (bs ≡(T) bs') ⇒ ((as @ bs) ≡(T) (as' @ bs')))
Proof
Definitions occuring in Statement :
permr: as ≡(T) bs,
append: as @ bs,
list: T List,
all: ∀x:A. B[x],
implies: P ⇒ Q,
universe: Type
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
permr: as ≡(T) bs,
cand: A c∧ B,
top: Top,
squash: ↓T,
uimplies: b supposing a,
true: True,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
exists: ∃x:A. B[x],
sym_grp: Sym(n),
sq_type: SQType(T),
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
ge: i ≥ j ,
int_seg: {i..j-},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
perm: Perm(T),
nat: ℕ,
less_than: a < b,
app_perm: app_perm(m;n;p;q),
mk_perm: mk_perm(f;b),
perm_f: p.f,
pi1: fst(t),
app_permf: app_permf(m;n;p;q),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
subtract: n - m,
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
permr_wf,
list_wf,
length-append,
istype-void,
equal_wf,
squash_wf,
true_wf,
istype-universe,
add_functionality_wrt_eq,
length_wf,
subtype_rel_self,
iff_weakening_equal,
subtype_rel-equal,
perm_wf,
int_seg_wf,
append_wf,
length_append,
subtype_rel_list,
top_wf,
subtype_base_sq,
int_subtype_base,
int_seg_subtype,
istype-false,
non_neg_length,
int_seg_properties,
decidable__le,
full-omega-unsat,
intformnot_wf,
intformle_wf,
itermAdd_wf,
itermVar_wf,
istype-int,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
select_wf,
perm_f_wf,
le_wf,
less_than_wf,
length_wf_nat,
nat_properties,
intformand_wf,
itermConstant_wf,
int_formula_prop_and_lemma,
int_term_value_constant_lemma,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
intformeq_wf,
int_formula_prop_eq_lemma,
app_perm_wf,
lt_int_wf,
equal-wf-T-base,
bool_wf,
assert_wf,
le_int_wf,
bnot_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,
zero-le-nat,
int_seg_subtype_nat,
select_append_front,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
add-member-int_seg2,
minus-one-mul,
add-mul-special,
zero-mul,
add-associates,
add-commutes,
zero-add,
select_append_back,
add-subtract-cancel,
set_subtype_base,
lelt_wf,
nat_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
universeIsType,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
hypothesis,
inhabitedIsType,
isectElimination,
universeEquality,
productElimination,
independent_pairFormation,
sqequalRule,
isect_memberEquality_alt,
voidElimination,
applyEquality,
lambdaEquality_alt,
imageElimination,
equalityTransitivity,
equalitySymmetry,
intEquality,
because_Cache,
independent_isectElimination,
addEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
instantiate,
independent_functionElimination,
dependent_pairFormation_alt,
dependent_set_memberEquality_alt,
productIsType,
equalityIsType1,
applyLambdaEquality,
setElimination,
rename,
unionElimination,
approximateComputation,
int_eqEquality,
functionIsType,
equalityElimination,
cumulativity
Latex:
\mforall{}T:Type. \mforall{}as,as',bs,bs':T List. ((as \mequiv{}(T) as') {}\mRightarrow{} (bs \mequiv{}(T) bs') {}\mRightarrow{} ((as @ bs) \mequiv{}(T) (as' @ bs')))
Date html generated:
2019_10_16-PM-01_01_01
Last ObjectModification:
2018_10_08-PM-07_04_31
Theory : perms_2
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