hcju/isabelleps huggingface dataset:
s_mul_ext_ne_extend_left: fixes A :: "'a multiset"   and B :: "'a multiset"   and C :: "'a multiset"   and W :: "('a \<times> 'a) set"   and S :: "('a \<times> 'a) set" assumes "A \<noteq> {#}"   and "(B, C) \<in> ns_mul_ext W S" shows "(A + B, C) \<in> s_mul_ext W S"
chine_assoc_in_hom: shows "\<guillemotleft>cods.chine_assoc : Chn ((cod \<mu> \<star> cod \<nu>) \<star> cod \<pi>) \<rightarrow>\<^sub>C Chn (cod \<mu> \<star> cod \<nu> \<star> cod \<pi>)\<guillemotright>"
lc_def: fixes cmp_term :: "'b nat_term_order"   and p :: "'b \<Rightarrow>\<^sub>0 'a" shows "lc_punit cmp_term p = lookup p (lt_punit cmp_term p)"
del_list_sorted4: fixes xs :: "('c \<times> 'd) list"   and a :: "'c"   and a' :: "'d"   and ys :: "('c \<times> 'd) list"   and b :: "'c"   and b' :: "'d"   and zs :: "('c \<times> 'd) list"   and c :: "'c"   and c' :: "'d"   and us :: "('c \<times> 'd) list"   and x :: "'c" assumes "sorted1 (xs @ (a, a') # ys @ (b, b') # zs @ (c, c') # us)"   and "x < c" shows "del_list x (xs @ (a, a') # ys @ (b, b') # zs @ (c, c') # us) = del_list x (xs @ (a, a') # ys @ (b, b') # zs) @ (c, c') # us"
map_last_prefixes: fixes p :: "'b list" shows "map last (tl (prefixes p)) = p"
additive_strength_def: fixes f :: "'b \<Rightarrow> 'b"   and xi :: "'b Gromov_completion" shows "additive_strength f xi = lim (\<lambda>n. Busemann_function_at xi ((f ^^ n) basepoint) basepoint / real n)"
ucycles_distinct_edges: fixes c :: "nat list"   and G :: "nat set \<times> nat set set" assumes "c \<in> ucycles G" shows "distinct (uwalk_edges c)"
EdomArityAnalysis_axioms: shows "EdomArityAnalysis Aexp"
IH: assumes "list_all tfr\<^sub>s\<^sub>t\<^sub>p S"   and "bvars\<^sub>s\<^sub>t S \<inter> range_vars \<theta> = {}" shows "list_all tfr\<^sub>s\<^sub>t\<^sub>p (S \<cdot>\<^sub>s\<^sub>t \<theta>)"
is_segment_def: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and X :: "'b set" assumes "tarski_first3 C" shows "tarski_first3.is_segment X \<equiv> \<exists>x y. X = {x, y}"
ind_eq_fin': fixes a :: "'action"   and v :: "'action list" assumes "Ind {a} (set v)" shows "[a] @ v =\<^sub>F v @ [a]"
HomContraAt: fixes x :: "ZF"   and C :: "('d, 'e, 'f) LSCategory_scheme"   and f :: "'e"   and X :: "'d" assumes "x |\<in>| (Hom\<^bsub>C\<^esub> cod\<^bsub>C\<^esub> f X)" shows "(HomC\<^bsub>C\<^esub>[f,X]) |@| x = m2z\<^bsub>C\<^esub>f ;;\<^bsub>C\<^esub> (z2m\<^bsub>C\<^esub>x)"
lookup_raw_update_by_fun_raw: fixes xs :: "('a \<times> 'c) list \<times> 'o"   and k1 :: "'a"   and f :: "'c \<Rightarrow> 'c"   and k2 :: "'a" assumes "oalist_inv xs" shows "lookup_raw (update_by_fun_raw k1 f xs) k2 = (if k1 = k2 then f else id) (lookup_raw xs k2)"
newton_seq_is_Zp_cauchy: shows "is_Zp_cauchy ns"
distinct_sorted_list_of_set: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and A :: "'c set" assumes "class.linorder less_eq less" shows "distinct (linorder.sorted_list_of_set less_eq A)"
transp_dickson_less: fixes d :: "'a \<Rightarrow> nat"   and m :: "nat" shows "transp (dickson_less d m)"
nop_v11_vdomain: shows "\<D>\<^sub>\<circ> f = A ^\<^sub>\<times> n"
csrp_mem1: fixes I :: "'a set"   and x :: "'a set" assumes "ideal R I"   and "x \<in> carrier (R /\<^sub>r I)" shows "csrp_fn R I x \<in> x"
IH: fixes bs :: "'b list" shows "length (fst (filter2 P as bs)) = length (snd (filter2 P as bs))"
simps_8: fixes v :: "formula"   and va :: "formula" shows "get_and_list (Or v va) = [Or v va]"
\<alpha>'_simp: fixes f :: "'a"   and g :: "'a"   and h :: "'a" assumes "C.arr f"   and "C.arr g"   and "C.arr h" shows "C.\<alpha>' (f, g, h) = C.tensor (C.tensor f g) h \<cdot> C.assoc' (C.dom f) (C.dom g) (C.dom h)"
wf_mdecl_def: fixes wf_md :: "'a prog \<Rightarrow> char list \<Rightarrow> 'a mdecl \<Rightarrow> bool"   and P :: "'a prog"   and C :: "char list" shows "wf_mdecl wf_md P C \<equiv> \<lambda>(M, b, Ts, T, m). (\<forall>T\<in>set Ts. is_type P T) \<and> is_type P T \<and> wf_md P C (M, b, Ts, T, m)"
finite_inf_insertI: fixes x :: "'a"   and A :: "'a set"   and P :: "'a \<Rightarrow> bool" assumes "\<And>i. is_glb L i (insert x A) \<Longrightarrow> P i"   and "finite A"   and "x \<in> carrier L"   and "A \<subseteq> carrier L" shows "P (\<Sqinter>insert x A)"
wellformed_items_\<I>_k_if_0: fixes k :: "nat" assumes "wellformed_items (\<J> k 0)" shows "wellformed_items (\<I> k)"
path_valid_node_1: fixes sourcenode :: "'g \<Rightarrow> 'h"   and targetnode :: "'g \<Rightarrow> 'h"   and kind :: "'g \<Rightarrow> ('i, 'j, 'k, 'l) edge_kind"   and valid_edge :: "'g \<Rightarrow> bool"   and Entry :: "'h"   and get_proc :: "'h \<Rightarrow> 'l"   and get_return_edges :: "'g \<Rightarrow> 'g set"   and procs :: "('l \<times> 'i list \<times> 'i list) list"   and Main :: "'l"   and n :: "'h"   and as :: "'g list"   and n' :: "'h" assumes "CFG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main"   and "CFG.path sourcenode targetnode valid_edge n as n'" shows "CFG.valid_node sourcenode targetnode valid_edge n"
C: shows "IntAgrK (S (Card 0, 0, run')) = Some d"
cong_mod_left: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and euclidean_size :: "'c \<Rightarrow> nat"   and division_segment :: "'c \<Rightarrow> 'c"   and b :: "'c"   and a :: "'c"   and c :: "'c" assumes "class.unique_euclidean_semiring divide plus minus zero times one modulo euclidean_size division_segment" shows "unique_euclidean_semiring.cong modulo (modulo b a) c a = unique_euclidean_semiring.cong modulo b c a"
Guarantee: fixes f :: "'d"   and F :: "'d set"   and \<Gamma> :: "'e \<Rightarrow> ('f, 'e, 'd) com option"   and \<Theta> :: "('f set \<times> 'e \<times> 'f set \<times> 'f set) set"   and g :: "'f set"   and P :: "'f set"   and c :: "('f, 'e, 'd) com"   and Q :: "'f set"   and A :: "'f set" assumes "f \<in> F"   and "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>(g \<inter> P) c Q,A" shows "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P Guard f g c Q,A"
lossless_funct_14_OT: fixes M :: "'g \<times> 'g \<times> 'g \<times> 'g"   and C :: "bool \<times> bool" shows "lossless_spmf (funct_OT_14 M C)"
strict0: fixes X :: "('\<AA> state \<times> '\<AA> state \<Rightarrow> '\<alpha>1) \<times> ('\<AA> state \<times> '\<AA> state \<Rightarrow> '\<alpha>2)" shows "(case X of (a, b) \<Rightarrow> f invalid a b) = invalid"
Crypt_imp_OR2: fixes B :: "agent"   and NA :: "msg"   and NB :: "msg"   and A :: "agent"   and evs :: "event list" assumes "Crypt (shrK B) \<lbrace>NA, NB, Agent A, Agent B\<rbrace> \<in> parts (knows Spy evs)"   and "B \<notin> bad"   and "evs \<in> otway" shows "\<exists>X. Says B Server \<lbrace>NA, Agent A, Agent B, X, Crypt (shrK B) \<lbrace>NA, NB, Agent A, Agent B\<rbrace>\<rbrace> \<in> set evs"
unilateral_substitutes_on_bilateral_substitutes_on: fixes A :: "'x set"   and f :: "'x set \<Rightarrow> 'x set" assumes "unilateral_substitutes_on A f" shows "bilateral_substitutes_on A f"
rroots4_def: shows "rroots4 = roots4_generic solve_depressed_quartic_real"
n: fixes F :: "nat \<Rightarrow> 'b poly"   and n :: "nat \<Rightarrow> nat"   and \<delta> :: "nat \<Rightarrow> nat"   and f :: "nat \<Rightarrow> 'b"   and k :: "nat"   and \<beta> :: "nat \<Rightarrow> 'b"   and i :: "nat" assumes "primitive_remainder_sequence F n \<delta> f k \<beta>" shows "n i = degree (F i)"
simps_2: fixes cf :: "('test, 'atom) com \<times> 'state"   and cfl :: "(('test, 'atom) com \<times> 'state) list" shows "parTrace (cf # cfl) = (cfl \<noteq> [] \<longrightarrow> parTrace cfl \<and> cf \<rightarrow>c hd cfl)"
length_bv_prepend_drop: fixes a :: "nat"   and xs :: "'a list"   and b :: "'a" shows "a \<le> length xs \<longrightarrow> length (bv_prepend a b (drop a xs)) = length xs"
hull_hull: fixes S :: "'a set \<Rightarrow> bool"   and s :: "'a set" shows "S hull (S hull s) = S hull s"
Image_SN_on: fixes r :: "('b \<times> 'b) set"   and A :: "'b set" assumes "SN_on r (r `` A)" shows "SN_on r A"
connected_components_of_maximal: fixes C :: "'b set"   and X :: "'b topology"   and S :: "'b set" assumes "C \<in> connected_components_of X"   and "connectedin X S"   and "\<not> disjnt C S" shows "S \<subseteq> C"
permute_fun_def: fixes p :: "'a perm"   and f :: "'b \<Rightarrow> 'c" shows "permute_fun p f = (\<lambda>x. perm2 p (f (perm1 (- p) x)))"
dom_comp: fixes g :: "'a"   and f :: "'a" assumes "S.seq g f" shows "S.dom (S.comp g f) = S.dom f"
image_mset_union: fixes f :: "'c \<Rightarrow> 'b"   and M :: "'c multiset"   and N :: "'c multiset" shows "image_mset f (M + N) = image_mset f M + image_mset f N"
encode_problem_with_operator_interference_exclusion_has_model_then_also_partial_encodings_5: fixes \<A> :: "sat_plan_variable \<Rightarrow> bool"   and \<Pi> :: "'b strips_problem"   and t :: "nat" assumes "\<A> \<Turnstile> \<Phi>\<^sub>\<forall> \<Pi> t" shows "\<A> \<Turnstile> (\<Phi>\<^sub>G \<Pi>) t"
first_correct: fixes q :: "'b queue" assumes "invar q"   and "list q \<noteq> []" shows "first q = hd (list q)"
Ext_method_inheritance: shows "filter_tab (\<lambda>sig m. tprg \<turnstile> Ext inherits method sig m) [fst (case Base_foo of (s, m) \<Rightarrow> (s, Base, m)) \<mapsto> snd (case Base_foo of (s, m) \<Rightarrow> (s, Base, m))] = Map.empty"
assms_1: shows "continuous_on s (apply_chart c \<circ> f)"
zero_less_imp_eq_int: fixes k :: "int" assumes "0 < k" shows "\<exists>n>0. k = int n"
tdropn_TNil: fixes n :: "nat"   and b :: "'b" shows "tdropn n (TNil b) = TNil b"
intros_4: fixes K :: "nat"   and X :: "msg"   and H :: "msg set" assumes "Crypt K X \<in> parts H" shows "X \<in> parts H"
link_tree_invar: fixes t1 :: "('c, 'd) BinomialTree"   and t2 :: "('c, 'd) BinomialTree" assumes "tree_invar t1"   and "tree_invar t2"   and "rank t1 = rank t2" shows "tree_invar (link t1 t2)"
pt_name1: fixes x :: "'a" shows "[] \<bullet> x = x"
schnorr: fixes \<eta> :: "nat" shows "schnorr_sigma_protocol (\<G> \<eta>)"
EpubKF_synth: fixes K :: "fresh_t"   and H :: "msg set" assumes "epubKF K \<in> synth H" obtains "epubKF K \<in> H"
equivalent_def: fixes S1 :: "'at Literal set set"   and S2 :: "'at Literal set set" shows "equivalent S1 S2 = (entails_formula S1 S2 \<and> entails_formula S2 S1)"
FiniteSingleAgentEnvironment_def: fixes jkbp :: "'a \<Rightarrow> ('a, 'b, 'c) GC list"   and agent :: "'a" shows "FiniteSingleAgentEnvironment jkbp agent \<equiv> FiniteEnvironment jkbp \<and> FiniteSingleAgentEnvironment_axioms agent"
not_None_eq: fixes x :: "'g option" shows "(x \<noteq> None) = (\<exists>y. x = Some y)"
defined_bool_split: fixes \<tau> :: "'a state \<times> 'a state"   and A :: "'a state \<times> 'a state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>" shows "(\<tau> \<Turnstile> \<delta> A) = (\<tau> \<Turnstile> A \<or> \<tau> \<Turnstile> not A)"
Abs_preserves_good: fixes X :: "('b, 'c, 'varSort, 'var, 'd) qTerm set"   and xs :: "'varSort"   and x :: "'var" assumes "good X" shows "goodAbs (Abs xs x X)"
assms_5: fixes x :: "real" assumes "x \<in> {a..b}" shows "(G has_real_derivative g x) (at x within {a..b})"
pseudo_complement: fixes uminus :: "'c \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and x :: "'c"   and y :: "'c" assumes "class.p_algebra uminus inf less_eq less sup bot top" shows "(inf x y = bot) = less_eq x (uminus y)"
mono: shows "(\<forall>y\<in>S. f y \<le> f x) \<or> (\<forall>y\<in>S. f x \<le> f y)"
principal_domain_axioms_def: fixes R :: "('c, 'd) ring_scheme" shows "principal_domain_axioms R \<equiv> \<forall>I. ideal I R \<longrightarrow> (\<exists>a\<in>carrier R. I = PIdl\<^bsub>R\<^esub> a)"
that: assumes "a \<^bold>\<in> A"   and "P a" shows "thesis"
eenv_def: fixes \<xi> :: "Preliminaries.var \<Rightarrow> univ"   and x :: "Preliminaries.var" shows "eenv \<xi> x \<equiv> invt (TE.tpOfV x) (\<xi> x)"
step_Fault_prop: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'a, 'c) com option"   and c :: "('b, 'a, 'c) com"   and s :: "('b, 'c) xstate"   and c' :: "('b, 'a, 'c) com"   and s' :: "('b, 'c) xstate"   and f :: "'c" assumes "\<Gamma>\<turnstile> (c, s) \<rightarrow> (c', s')"   and "s = Fault f" shows "s' = Fault f"
implem_LtK: fixes x :: "chan"   and K :: "ltkey" shows "implem x \<noteq> LtK K"
Mp_dvdm: fixes m :: "int"   and f :: "int poly"   and g :: "int poly" shows "poly_mod.dvdm m (poly_mod.Mp m f) g = poly_mod.dvdm m f g"
fmrel_on_fset_acyclic: fixes xm :: "'f \<rightharpoonup>\<^sub>f 'e"   and R :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and ym :: "'f \<rightharpoonup>\<^sub>f 'e" assumes "acyclicP_on (fmran' xm) R"   and "fmrel_on_fset (fmdom ym) R\<^sup>+\<^sup>+ xm ym"   and "fmrel_on_fset (fmdom xm) R ym xm" shows "xm = ym"
order_asym: fixes R :: "('b \<times> 'b) set" assumes "trans R" shows "asym R = irrefl R"
pcompose_prod: fixes f :: "'c \<Rightarrow> 'b poly"   and A :: "'c set"   and p :: "'b poly" shows "prod f A \<circ>\<^sub>p p = (\<Prod>i\<in>A. f i \<circ>\<^sub>p p)"
trivial_homomorphism_hom_boundary_inclusion: fixes S :: "'b set"   and X :: "'b topology"   and p :: "int" assumes "S = {} \<or> S retract_of_space X" shows "trivial_homomorphism (relative_homology_group p X S) (homology_group (p - 1) (subtopology X S)) (hom_boundary p X S)"
diagonal_to_Smith_PQ_JNF_def: fixes A :: "'b mat"   and bezout :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<times> 'b \<times> 'b \<times> 'b \<times> 'b" shows "diagonal_to_Smith_PQ_JNF A bezout = diagonal_to_Smith_aux_PQ_JNF [0..<min (dim_row A) (dim_col A) - 1] bezout (1\<^sub>m (dim_row A), A, 1\<^sub>m (dim_col A))"
fp_cop_F_preferred: fixes y :: "'x"   and ds :: "'d set"   and X'' :: "'x set"   and x :: "'x" assumes "y \<in> CD_on ds (cop ds \<union> X'')"   and "x \<in> cop ds"   and "Xd x = Xd y" shows "(x, y) \<in> Pd (Xd x)"
zero_order_2: fixes n :: "'b multiset" shows "(n \<subseteq># {#}) = (n = {#})"
simps: fixes k :: "nat"   and ts :: "('a btree \<times> 'a) list"   and t :: "'a btree" shows "split_max k (Node ts t) = (case t of Leaf \<Rightarrow> let (sub, y) = last ts in (Node (butlast ts) sub, y) | Node list btree \<Rightarrow> case split_max k t of (sub, x) \<Rightarrow> (rebalance_last_tree k ts sub, x))"
cong_bigtheta: fixes f :: "'b \<Rightarrow> 'c"   and F :: "'b filter"   and g :: "'b \<Rightarrow> 'c" assumes "f \<in> \<Theta>[F](g)" shows "\<Theta>[F](f) = \<Theta>[F](g)"
strict_iff_order: fixes a :: "'a list"   and b :: "'a list" shows "strict_prefix a b = (prefix a b \<and> a \<noteq> b)"
square: shows "dim_row (submatrix A I J) = dim_col (submatrix A I J)"
distinct_15: fixes x2 :: "nat"   and x3 :: "nat" shows "Number x2 \<noteq> Nonce x3"
dtype2: fixes tds :: "(char list list \<times> char list \<times> (char list \<times> t list) list) list"   and mn :: "char list list"   and s :: "'b state"   and ck :: "bool"   and env :: "v sem_env"   and locs :: "locn \<times> locn" assumes "\<not> check_dup_ctors tds \<or> \<not> disjnt (type_defs_to_new_tdecs mn tds) (defined_types s) \<or> \<not> allDistinct (map (\<lambda>x. case x of (tvs, tn, ctors) \<Rightarrow> tn) tds)" shows "evaluate_dec ck mn env s (Dtype locs tds) (s, Rerr (Rabort Rtype_error))"
ProcRec: fixes Procs :: "'d set"   and \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'g) com option"   and \<Theta> :: "('e set \<times> 'd \<times> 'e set \<times> 'e set) set"   and P :: "'d \<Rightarrow> 'f \<Rightarrow> 'e set"   and r :: "(('e \<times> 'd) \<times> 'e \<times> 'd) set"   and Q :: "'d \<Rightarrow> 'f \<Rightarrow> 'e set"   and A :: "'d \<Rightarrow> 'f \<Rightarrow> 'e set"   and F :: "'g set" assumes "\<forall>p\<in>Procs. \<forall>\<sigma> Z. \<Gamma>,\<Theta> \<union> (\<Union>q\<in>Procs. \<Union>Z. {(P q Z \<inter> {s. ((s, q), \<sigma>, p) \<in> r}, q, Q q Z, A q Z)})\<turnstile>\<^sub>t\<^bsub>/F\<^esub> ({\<sigma>} \<inter> P p Z) the (\<Gamma> p) (Q p Z),(A p Z)"   and "wf r"   and "Procs \<subseteq> dom \<Gamma>" shows "\<forall>p\<in>Procs. \<forall>Z. \<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> (P p Z) Call p (Q p Z),(A p Z)"
cmp_cases: fixes y :: "'a" assumes "y \<in> set (x1 # l')"   and "\<not> cmp e y" shows "cmp y e"
delta_hedging_self_financing: fixes p :: "real"   and M :: "bool stream measure"   and geom_proc :: "nat \<Rightarrow> bool stream \<Rightarrow> real"   and u :: "real"   and d :: "real"   and init :: "real"   and Mkt :: "(bool stream, 'b) discrete_market"   and r :: "real"   and risk_free_asset :: "'b"   and stk :: "'b"   and G :: "nat \<Rightarrow> bool stream measure"   and N :: "bool stream measure"   and der :: "bool stream \<Rightarrow> real"   and matur :: "nat" assumes "CRR_market p M geom_proc u d init Mkt r risk_free_asset stk G" shows "self_financing Mkt (CRR_market.delta_hedging geom_proc Mkt r risk_free_asset stk N der matur)"
set_integrable_abs_iff: fixes M :: "'b measure"   and A :: "'b set"   and f :: "'b \<Rightarrow> real" assumes "set_borel_measurable M A f" shows "set_integrable M A (\<lambda>x. \<bar>f x\<bar>) = set_integrable M A f"
ResPres: fixes \<Psi>' :: "'b"   and S :: "('a, 'b, 'c) psi"   and T :: "('a, 'b, 'c) psi"   and xvec :: "name list" assumes "(\<Psi>', S, T) \<in> Rel"   and "xvec \<sharp>* \<Psi>'" shows "(\<Psi>', \<lparr>\<nu>*xvec\<rparr>S, \<lparr>\<nu>*xvec\<rparr>T) \<in> Rel"
sel_4: fixes x1 :: "'g" shows "ttl (TNil x1) = TNil x1"
sin_paired: fixes x :: "real" shows "(\<lambda>n. (- 1) ^ n / fact (2 * n + 1) * x ^ (2 * n + 1)) sums sin x"
assms_2: assumes "nsqn\<^sub>r r < nsqn\<^sub>r r'" shows "P r r'"
list_update_append: fixes xs :: "'b list"   and ys :: "'b list"   and n :: "nat"   and x :: "'b" shows "(xs @ ys)[n := x] = (if n < length xs then xs[n := x] @ ys else xs @ ys[n - length xs := x])"
y: shows "invariant_1_2 (p2, l2, r2)"
surjective: fixes r :: "('b, 'c) node_scheme" shows "r = \<lparr>name = name r, incoming = incoming r, new = new r, old = old r, next = next r, \<dots> = node.more r\<rparr>"
hyps_2: fixes d' :: "('p, 's, 'm) configuration" assumes "trace c t d'" shows "c' = d'"
except_eq_zero_iff: fixes p :: "'c \<Rightarrow>\<^sub>0 'd"   and S :: "'c set" shows "(except p S = 0) = (keys p \<subseteq> S)"
iff: shows "(winding_number \<gamma> z \<in> \<int>) = (exp (contour_integral p (\<lambda>w. 1 / (w - z))) = 1)"
unit_eq_div2: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and b :: "'b"   and a :: "'b"   and c :: "'b" assumes "class.algebraic_semidom divide plus minus zero times one"   and "dvd.dvd times b one" shows "(a = divide c b) = (times a b = c)"
dd_true: shows "INTERP N \<Turnstile>m mset CAs"
ThrowRedsThrow: fixes e :: "char list exp"   and s :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and a :: "char list exp"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list" assumes "P \<turnstile> \<langle>e,s\<rangle> \<rightarrow>* \<langle>throw a,s'\<rangle>" shows "P \<turnstile> \<langle>throw e,s\<rangle> \<rightarrow>* \<langle>throw a,s'\<rangle>"
dynmethd_declclass: fixes G :: "prog"   and statC :: "qtname"   and dynC :: "qtname"   and sig :: "sig"   and m :: "qtname \<times> methd" assumes "dynmethd G statC dynC sig = Some m"   and "wf_prog G"   and "is_class G statC" shows "methd G (declclass m) sig = Some m"
ruler_pref: fixes w :: "'b list"   and v :: "'b list"   and z :: "'b list" assumes "w \<le>p v \<cdot> z" shows "w \<bowtie> v"
callee: fixes x :: "'a"   and s :: "'d" assumes "x \<in> outs_\<I> \<I>" shows "plossless_gpv \<I>' (callee s x)"
equation_minus_iff: fixes minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and uminus :: "'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c" assumes "class.group_add minus plus zero uminus" shows "(a = uminus b) = (b = uminus a)"
cases: fixes a :: "'a itrm" assumes "a \<in> CF" obtains x :: "dB" where "a = Pure xa__"   | f :: "'a itrm" and x :: "'a" where "a = f \<diamondop> Opaque x" and "f \<in> CF"
red_arg_implies_red_trm: fixes S :: "'a eclause set"   and t1 :: "'a trm"   and t :: "'a trm"   and t2 :: "'a trm" assumes "st_red S t1"   and "t = t1 \<cdot> t2 \<or> t = t2 \<cdot> t1" shows "st_red S t"
extended_Gromov_distance_PInf_boundary_1: fixes x :: "'a Gromov_completion"   and y :: "'a Gromov_completion" assumes "x \<in> Gromov_boundary" shows "extended_Gromov_distance x y = \<infinity>"
2202: fixes x :: "'a"   and y :: "'a" shows "- x \<squnion> - - (y \<squnion> x) = top"
v2: fixes \<Gamma> :: "(name \<times> ty) list"   and x :: "name"   and T :: "ty" assumes "valid \<Gamma>"   and "x \<sharp> \<Gamma>" shows "valid ((x, T) # \<Gamma>)"
assms_1: shows "(s, s') \<in> \<langle>R\<rangle>set_rel"
if_zero: fixes thn :: "exp"   and els :: "exp" shows "EIf (ENat 0) thn els \<longrightarrow> els"
degree_add_leq: fixes a :: "'a mpoly"   and var :: "nat"   and x :: "nat"   and b :: "'a mpoly" assumes "MPoly_Type.degree a var \<le> x"   and "MPoly_Type.degree b var \<le> x" shows "MPoly_Type.degree (a + b) var \<le> x"
Q_def: shows "Q \<equiv> ts_process {[], [Some b]}"
mat_swapcols_def: fixes k :: "nat"   and l :: "nat"   and A :: "'a mat" shows "swapcols k l A = mat (dim_row A) (dim_col A) (\<lambda>(i, j). if k = j then A $$ (i, l) else if l = j then A $$ (i, k) else A $$ (i, j))"
comp_def: fixes f :: "'c local.term set"   and g :: "'c local.term set" shows "f \<cdot> g \<equiv> if ARR f \<and> ARR g \<and> DOM f = COD g then mkarr (rep f \<^bold>\<cdot> rep g) else {}"
tranclD2: fixes x :: "'c"   and y :: "'c"   and R :: "('c \<times> 'c) set" assumes "(x, y) \<in> R\<^sup>+" shows "\<exists>z. (x, z) \<in> R\<^sup>* \<and> (z, y) \<in> R"
healthy_wp_DC: fixes f :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and g :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" assumes "healthy (wp f)"   and "healthy (wp g)" shows "healthy (wp (f \<Sqinter> g))"
uniformity_refl: fixes "open" :: "'b set \<Rightarrow> bool"   and uniformity :: "('b \<times> 'b) filter"   and E :: "'b \<times> 'b \<Rightarrow> bool"   and x :: "'b" assumes "class.uniform_space open uniformity"   and "eventually E uniformity" shows "E (x, x)"
lookup_pp_of_fun: fixes f :: "'a \<Rightarrow> 'b" assumes "finite {x. f x \<noteq> (0::'b)}" shows "lookup_pp (pp_of_fun f) = f"
L4_9: fixes \<phi> :: "'b ltlr"   and qs :: "'b node set"   and \<xi> :: "nat \<Rightarrow> 'b set" assumes "create_gba_from_nodes_precond \<phi> qs"   and "gba.accept (create_gba_from_nodes \<phi> qs) \<xi>" shows "\<xi> \<Turnstile>\<^sub>r \<phi>"
tendsto_zero: fixes g :: "'c \<Rightarrow> 'a"   and F :: "'c filter" assumes "(g \<longlongrightarrow> (0::'a)) F" shows "((\<lambda>x. f (g x)) \<longlongrightarrow> (0::'b)) F"
Gauss_Jordan_column_k_def: fixes A' :: "nat \<times> (('b, 'c) vec, 'd) vec"   and k :: "nat" shows "Gauss_Jordan_column_k A' k = (let i = fst A'; A = snd A'; from_nat_i = mod_type_class.from_nat i; from_nat_k = mod_type_class.from_nat k in if (\<forall>m\<ge>from_nat_i. A $ m $ from_nat_k = (0::'b)) \<or> i = nrows A then (i, A) else (i + 1, Gauss_Jordan_in_ij A from_nat_i from_nat_k))"
Rep_bit0_inject: fixes x :: "'a bit0"   and y :: "'a bit0" shows "(Rep_bit0 x = Rep_bit0 y) = (x = y)"