Nuprl Lemma : general-append-cancellation
∀[T:Type]. ∀[as,bs,cs,ds:T List].
({(as = bs ∈ (T List)) ∧ (cs = ds ∈ (T List))}) supposing
(((||as|| = ||bs|| ∈ ℤ) ∨ (||cs|| = ||ds|| ∈ ℤ)) and
((as @ cs) = (bs @ ds) ∈ (T List)))
Proof
Definitions occuring in Statement :
length: ||as||,
append: as @ bs,
list: T List,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
guard: {T},
or: P ∨ Q,
and: P ∧ Q,
int: ℤ,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
uimplies: b supposing a,
prop: ℙ,
or: P ∨ Q,
guard: {T},
and: P ∧ Q,
so_apply: x[s],
implies: P ⇒ Q,
append: as @ bs,
all: ∀x:A. B[x],
so_lambda: so_lambda(x,y,z.t[x; y; z]),
top: Top,
so_apply: x[s1;s2;s3],
cand: A c∧ B,
cons: [a / b],
false: False,
ge: i ≥ j ,
le: A ≤ B,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
not: ¬A,
subtype_rel: A ⊆r B,
true: True,
squash: ↓T,
iff: P ⇐⇒ Q,
decidable: Dec(P)
Lemmas referenced :
list_induction,
uall_wf,
list_wf,
isect_wf,
equal_wf,
append_wf,
or_wf,
length_wf,
list_ind_nil_lemma,
length_of_nil_lemma,
equal-wf-base-T,
list_ind_cons_lemma,
length_of_cons_lemma,
cons_wf,
list-cases,
nil_wf,
product_subtype_list,
non_neg_length,
satisfiable-full-omega-tt,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformeq_wf,
itermAdd_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
int_formula_prop_wf,
subtype_rel_list,
top_wf,
squash_wf,
true_wf,
add_functionality_wrt_eq,
length_append,
iff_weakening_equal,
reduce_tl_cons_lemma,
and_wf,
tl_wf,
decidable__or,
decidable__equal_int,
intformnot_wf,
intformor_wf,
int_formula_prop_not_lemma,
int_formula_prop_or_lemma,
reduce_hd_cons_lemma,
hd_wf,
ge_wf,
length_cons_ge_one
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
thin,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
sqequalRule,
lambdaEquality,
cumulativity,
hypothesis,
because_Cache,
intEquality,
productEquality,
applyLambdaEquality,
independent_functionElimination,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
productElimination,
independent_pairEquality,
axiomEquality,
baseClosed,
equalityTransitivity,
equalitySymmetry,
lambdaFormation,
rename,
addEquality,
natural_numberEquality,
universeEquality,
unionElimination,
independent_pairFormation,
promote_hyp,
hypothesis_subsumption,
independent_isectElimination,
dependent_pairFormation,
int_eqEquality,
computeAll,
hyp_replacement,
applyEquality,
imageElimination,
imageMemberEquality,
dependent_set_memberEquality,
setElimination
Latex:
\mforall{}[T:Type]. \mforall{}[as,bs,cs,ds:T List].
(\{(as = bs) \mwedge{} (cs = ds)\}) supposing
(((||as|| = ||bs||) \mvee{} (||cs|| = ||ds||)) and
((as @ cs) = (bs @ ds)))
Date html generated:
2017_04_14-AM-09_26_36
Last ObjectModification:
2017_02_27-PM-04_00_50
Theory : list_1
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