hcju/isabelleps huggingface dataset:
finsum_infinite: fixes A :: "'c set"   and f :: "'c \<Rightarrow> 'a vec" assumes "infinite A" shows "finsum V f A = 0\<^sub>v n"
simps_2: fixes v :: "'a"   and va :: "'a list" shows "dense_repr (v # va) = (if lead_coeff (v # va) \<noteq> \<zero> then (lead_coeff (v # va), degree (v # va)) # dense_repr (tl (v # va)) else dense_repr (tl (v # va)))"
hyps: shows "r|\<sigma> @ \<tau>| = r|\<sigma>| + r|\<tau>| -s r \<down>l \<sigma>"
HSem_ignores_fresh_restr': fixes \<Gamma> :: "('b \<times> 'a) list"   and S :: "'b set"   and \<rho> :: "'b \<Rightarrow> 'c" assumes "fv \<Gamma> \<subseteq> S"   and "\<And>x \<rho>. x \<in> domA \<Gamma> \<Longrightarrow> \<lbrakk> the (map_of \<Gamma> x) \<rbrakk>\<^bsub>\<rho>\<^esub> = \<lbrakk> the (map_of \<Gamma> x) \<rbrakk>\<^bsub>\<rho> f|` fv (the (map_of \<Gamma> x))\<^esub>" shows "(\<lbrace> \<Gamma> \<rbrace>\<rho>) f|` S = \<lbrace> \<Gamma> \<rbrace>\<rho> f|` S"
SPEC_eq_is_RETURN_1: fixes x :: "'b" shows "SPEC ((=) x) = RETURN x"
redT_silent_moves_preserves: fixes s :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and s' :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set" assumes "mthr.silent_moves s s'"   and "ts_ok P (thr s) (shr s)" shows "ts_ok P (thr s') (shr s')"
split_asm: fixes P :: "'a \<Rightarrow> bool"   and f1 :: "'b \<Rightarrow> 'a"   and f2 :: "'a"   and f3 :: "'a"   and action :: "'b action" shows "P (case action of NormalAction x \<Rightarrow> f1 x | InitialThreadAction \<Rightarrow> f2 | ThreadFinishAction \<Rightarrow> f3) = (\<not> ((\<exists>x1. action = NormalAction x1 \<and> \<not> P (f1 x1)) \<or> action = InitialThreadAction \<and> \<not> P f2 \<or> action = ThreadFinishAction \<and> \<not> P f3))"
mon_of_entry: fixes fg :: "('a, 'b, 'c, 'd, 'e) flowgraph_rec_scheme"   and p :: "'b" assumes "flowgraph fg" shows "mon_n fg (entry fg p) = mon fg p"
it_paths_SnocE: fixes "is" :: "nat list"   and i :: "nat"   and t :: "'a rose_tree" assumes "is @ [i] \<in> it_paths t" obtains "is \<in> it_paths t \<and> i < length (children (tree_at t is))"
intro: fixes G :: "('c, 'd) monoid_scheme" assumes "finite (carrier G)" shows "finite_group_axioms G"
cycle_decomposition: fixes p :: "'a \<Rightarrow> 'a"   and S :: "'a set" assumes "p permutes S"   and "finite S" shows "cycle_decomp S p"
foldD_equality: fixes B :: "'c set"   and D :: "'d set"   and f :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and A :: "'c set"   and y :: "'d"   and e :: "'d" assumes "LCD B D f"   and "(A, y) \<in> foldSetD D f e"   and "e \<in> D"   and "A \<subseteq> B" shows "foldD D f e A = y"
P2: fixes j :: "nat" assumes "0 < j" shows "foldl1 f2 (replicate j Zero) = Zero"
in_trancl_mono: fixes TI :: "('e \<times> 'e) list"   and TI' :: "('e \<times> 'e) list"   and a :: "'e"   and b :: "'e" assumes "set TI \<subseteq> set TI'"   and "in_trancl TI a b" shows "in_trancl TI' a b"
albega0: shows "\<alpha> * \<beta> * \<gamma> \<noteq> 0"
wordinterval_intersection_set_eq: fixes r1 :: "'a wordinterval"   and r2 :: "'a wordinterval" shows "wordinterval_to_set (wordinterval_intersection r1 r2) = wordinterval_to_set r1 \<inter> wordinterval_to_set r2"
test_mult_comp: fixes x :: "'a" shows "!x \<cdot> t x = (0::'a)"
exhaust: fixes y :: "g_level" obtains "y = High"   | "y = Low"
swap_def: fixes i :: "nat"   and x :: "'a"   and a :: "'a array" shows "Array.swap i x a = Heap_Monad.guard (\<lambda>h. i < Array.length h a) (\<lambda>h. (Array.get h a ! i, Array.update a i x h))"
timpl_closure'_param: fixes t :: "(('a, 'b, 'c) prot_fun, 'd) Term.term"   and s :: "(('a, 'b, 'c) prot_fun, 'd) Term.term"   and c :: "('c set \<times> 'c set) set"   and f :: "('a, 'b, 'c) prot_fun"   and g :: "('a, 'b, 'c) prot_fun"   and S :: "(('a, 'b, 'c) prot_fun, 'd) Term.term list"   and T :: "(('a, 'b, 'c) prot_fun, 'd) Term.term list" assumes "(t, s) \<in> timpl_closure' c"   and "f = g \<or> (\<exists>a b. (a, b) \<in> c \<and> f = Abs a \<and> g = Abs b)" shows "(Fun f (S @ t # T), Fun g (S @ s # T)) \<in> timpl_closure' c"
simps_2: fixes t :: "'b tree" shows "merge t \<langle>\<rangle> = t"
rel_set_transfer: fixes A :: "'f \<Rightarrow> 'h \<Rightarrow> bool"   and B :: "'g \<Rightarrow> 'i \<Rightarrow> bool" shows "rel_fun (rel_fun A (rel_fun B (=))) (rel_fun (rel_set A) (rel_fun (rel_set B) (=))) rel_set rel_set"
rbt_comp_minus_def: fixes c :: "'b \<Rightarrow> 'b \<Rightarrow> order"   and t1 :: "('b, 'c) rbt"   and t2 :: "('b, 'c) rbt" shows "rbt_comp_minus c t1 t2 = (let h1 = bheight t1; h2 = bheight t2 in if 2 * h1 \<le> h2 then rbtreeify (fold_rev (\<lambda>k v kvs. if rbt_comp_lookup c t2 k = None then (k, v) # kvs else kvs) t1 []) else RBT_Impl.fold (\<lambda>k v. rbt_comp_delete c k) t2 t1)"
select_convs_2: fixes Pol :: "('c \<times> 'c) set"   and Map :: "'b \<Rightarrow> 'c"   and Doms :: "'c set"   and List :: "'b list"   and ListOp :: "'b list option"   and Set :: "'b set"   and more :: "'d" shows "Map \<lparr>Pol = Pol, Map = Map, Doms = Doms, List = List, ListOp = ListOp, Set = Set, \<dots> = more\<rparr> = Map"
t_2: fixes i :: "nat" shows "f (t i) \<in> A i - f ` C i"
edge_begin_tup: fixes x :: "('d \<times> 'c) \<times> 'd \<times> 'b" shows "edge_begin x = fst (fst x)"
hyps_2: shows "map_poly f (map_poly g p) = map_poly (f \<circ> g) p"
preserves_arr: fixes f :: "'a" assumes "C.arr f" shows "C.arr (trg\<^sub>C f)"
ostep_invariant_weakenE: fixes A :: "('a \<times> 'b, 'c) automaton"   and PS :: "'a \<Rightarrow> 'a \<Rightarrow> 'c \<Rightarrow> bool"   and PU :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and P :: "('a \<times> 'b) \<times> 'c \<times> 'a \<times> 'b \<Rightarrow> bool"   and Q :: "('a \<times> 'b) \<times> 'c \<times> 'a \<times> 'b \<Rightarrow> bool"   and QS :: "'a \<Rightarrow> 'a \<Rightarrow> 'c \<Rightarrow> bool"   and QU :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "A \<Turnstile>\<^sub>A (PS, PU \<rightarrow>) P"   and "\<And>t. P t \<Longrightarrow> Q t"   and "\<And>\<sigma> \<sigma>' a. QS \<sigma> \<sigma>' a \<Longrightarrow> PS \<sigma> \<sigma>' a"   and "\<And>\<sigma> \<sigma>'. QU \<sigma> \<sigma>' \<Longrightarrow> PU \<sigma> \<sigma>'" shows "A \<Turnstile>\<^sub>A (QS, QU \<rightarrow>) Q"
intros_1: fixes n :: "nat" shows "[n\<dots>] \<in> i_set_ind"
isSucc_def: fixes r :: "('c \<times> 'c) set"   and i :: "'c" assumes "wo_rel r" shows "isSucc r i \<equiv> \<exists>j. aboveS r j \<noteq> {} \<and> i = succ r j"
IH: fixes stk' :: "'addr val list" assumes "stk = rev vs @ Addr a # stk'"   and "pc < length (compEs2 es)"   and "compEs2 es ! pc = Invoke M (length vs)"   and "n + max_varss es' \<le> length xs"   and "bsoks es n" shows "\<exists>es'a xs'. \<tau>reds1r P t h (es', xs) (es'a, xs') \<and> P,es,h \<turnstile> (es'a, xs') [\<leftrightarrow>] (rev vs @ Addr a # stk', loc, pc, None) \<and> calls1 es'a = \<lfloor>(a, M, vs)\<rfloor>"
map_cong_pred: fixes x :: "'a list"   and ya :: "'a list"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "x = ya"   and "list_all (\<lambda>z. f z = g z) ya" shows "map f x = map g ya"
cases: fixes x :: "'b DBMEntry" obtains d :: "'b" where "x = Le d__"   | d :: "'b" where "x = Lt d"   | "x = \<infinity>"
isoD_2: fixes C1 :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and C2 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'a \<times> 'b" assumes "product_category C1 C2"   and "category.iso (product_category.comp C1 C2) f" shows "category.iso C2 (snd f)"
other: fixes \<sigma> :: "'a"   and p :: "'b"   and \<sigma>' :: "'a" assumes "(\<sigma>, p) \<in> oreachable A S U"   and "P (\<sigma>, p)"   and "U \<sigma> \<sigma>'" shows "P (\<sigma>', p)"
wlsFSb_recAll_termFSbMorph: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) Recursion.model" assumes "wlsFSb MOD" shows "termFSbMorph (rec MOD) (recAbs MOD) MOD"
quotr_unique: fixes x :: "'a"   and y :: "'a" assumes "x \<preceq>\<^sub>R y" shows "\<exists>!z. D x z \<and> y = x \<oplus> z"
zero_on_ennreal: fixes f :: "'c \<Rightarrow> real"   and A :: "'c set" shows "zero_on (\<lambda>x. ennreal (f x)) A = (\<lambda>x. ennreal (zero_on f A x))"
type_less_x_OclSuper_intro: fixes \<tau> :: "'a OCL_Types.type" assumes "\<tau> \<noteq> OclSuper" shows "\<tau> < OclSuper"
cont_memoryless_on: fixes t :: "'a"   and f :: "'a \<Rightarrow> 'a pmf"   and s :: "'a" assumes "t \<in> set_pmf (f s)" shows "cont (memoryless_on f s) t = memoryless_on f t"
exact_seq_mon_eq_triviality: fixes D :: "'b monoid"   and C :: "'b monoid"   and B :: "'b monoid"   and A :: "'b monoid"   and h :: "'b \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'b"   and f :: "'b \<Rightarrow> 'b" assumes "exact_seq ([D, C, B, A], [h, g, f])" shows "inj_on h (carrier C) = trivial_homomorphism B C g"
map_fst_zip_take: fixes xs :: "'d list"   and ys :: "'e list" shows "map fst (xs || ys) = take (min (length xs) (length ys)) xs"
cases: fixes x :: "('e, 'f, 'g, 'h) SLin_state_scheme \<times> ('i, 'j, 'k, 'l) SLin_state_scheme" obtains s1 :: "('e, 'f, 'g, 'h) SLin_state_scheme" and s2 :: "('i, 'j, 'k, 'l) SLin_state_scheme" where "x = (s1, s2)"
phi': fixes n :: "nat"   and z :: "'a list \<times> bool list \<times> 'a list" shows "\<phi> n z = (\<Sum>(x, y)\<in>Inv (fst z) (s_A n). if fst (snd z) ! index init y then 2 else 1)"
c: shows "c \<in> \<real>\<^sub>\<ge>\<^sub>0"
SeqReds: fixes e :: "expr"   and s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and e' :: "expr"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and E :: "char list \<Rightarrow> ty option"   and e\<^sub>2 :: "expr" assumes "P,E \<turnstile> \<langle>e,s\<rangle> \<rightarrow>* \<langle>e',s'\<rangle>" shows "P,E \<turnstile> \<langle>e;; e\<^sub>2,s\<rangle> \<rightarrow>* \<langle>e';; e\<^sub>2,s'\<rangle>"
map_key_PP_times: fixes p :: "('d, 'e) pp \<Rightarrow>\<^sub>0 'f"   and q :: "('d, 'e) pp \<Rightarrow>\<^sub>0 'f" shows "Poly_Mapping.map_key PP (p * q) = Poly_Mapping.map_key PP p * Poly_Mapping.map_key PP q"
trans: fixes a :: "'a multiset"   and b :: "'a multiset"   and c :: "'a multiset" assumes "a \<subseteq># b"   and "b \<subseteq># c" shows "a \<subseteq># c"
Basis_list_R3: shows "Basis_list = [(1, 0, 0), (0, 1, 0), (0, 0, 1)]"
der_gS': fixes x :: "(real, 'n) vec" assumes "x \<in> S'" shows "(g has_derivative g' x) (at x within S')"
simps_15: fixes x2 :: "'b" shows "set_err (OK x2) = {x2}"
map_of_heap_def: fixes lookup :: "'d \<Rightarrow> 'c option Heap"   and heap :: "heap"   and k :: "'d" shows "heap_mem_defs.map_of_heap lookup heap k = fst (the (execute (lookup k) heap))"
length'_def: fixes as :: "'b iarray" shows "length' as = integer_of_nat (List.length (list_of as))"
bind_returns_error_I: fixes h :: "'l"   and f :: "('l, 'm, 'n) prog"   and e :: "'m"   and g :: "'n \<Rightarrow> ('l, 'm, 'o) prog" assumes "h \<turnstile> f \<rightarrow>\<^sub>e e" shows "h \<turnstile> Heap_Error_Monad.bind f g \<rightarrow>\<^sub>e e"
Zleast_LI: fixes n :: "nat"   and P :: "nat \<Rightarrow> ('a \<Rightarrow> ant) set"   and I :: "'a set"   and j :: "nat"   and x :: "'a" assumes "distinct_pds K n P"   and "ideal (O\<^bsub>K P n\<^esub>) I"   and "j \<le> n"   and "I \<noteq> {\<zero>\<^bsub>O\<^bsub>K P n\<^esub>\<^esub>}"   and "x \<in> I" shows "LI K (\<nu>\<^bsub>K P j\<^esub>) I \<le> (\<nu>\<^bsub>K P j\<^esub>) x"
hmem_succ_self: fixes k :: "hf" shows "k \<^bold>\<in> succ k"
proj_verts_H_def: fixes x :: "'a" shows "proj_verts_H x \<equiv> if x = w then u else x"
borel_measurable_LIMSEQ_real: fixes M :: "'b measure"   and u :: "nat \<Rightarrow> 'b \<Rightarrow> real"   and u' :: "'b \<Rightarrow> real" assumes "\<And>x. x \<in> space M \<Longrightarrow> (\<lambda>i. u i x) \<longlonglongrightarrow> u' x"   and "\<And>i. u i \<in> borel_measurable M" shows "u' \<in> borel_measurable M"
le_succ_iff: fixes i :: "V"   and j :: "V" assumes "Ord i"   and "Ord j" shows "(ZFC_in_HOL.succ i \<subseteq>\<^sub>\<circ> ZFC_in_HOL.succ j) = (i \<subseteq>\<^sub>\<circ> j)"
path_hd: fixes p :: "'e list"   and E :: "('e \<times> 'e) set"   and v :: "'e"   and w :: "'e" assumes "p \<noteq> []"   and "path E v p w" shows "hd p = v"
acyclic_6_implies_3a: fixes x :: "'a" assumes "acyclic_6 x" shows "acyclic_3a x"
assms_5: shows "A $$ (k, k) = A $$ (i, i)"
euclidean_space_basis_vector_def: shows "local.euclidean_space_basis_vector \<equiv> map_fun id Abs_euclidean_space (\<lambda>x. indicat_real {x})"
basevars_def: fixes vs :: "state \<Rightarrow> 'a" shows "basevars vs \<equiv> surj vs"
map_closed: fixes a :: "'a" assumes "a \<in> G" shows "(a)\<^sub>R \<in> Translations"
greaterThan_def: fixes less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and l :: "'d" shows "ord.greaterThan less l \<equiv> {x. less l x}"
decseq_bounded: fixes X :: "nat \<Rightarrow> real"   and B :: "real" assumes "decseq X"   and "\<forall>i. B \<le> X i" shows "Bseq X"
simps_1: fixes S :: "'a"   and S' :: "'a" shows "(SKIP {S} \<sqsubseteq> SKIP {S'}) = (S \<sqsubseteq> S')"
cdisi: shows "d\<inverse> O s\<inverse> \<subseteq> d\<inverse>"
as_S_not_empty: shows "{x \<in> I. \<nexists>n. I \<leftarrow> n = x} \<noteq> {}"
continuous_on_imp_continuous_within: fixes s :: "'d set"   and f :: "'d \<Rightarrow> 'e"   and t :: "'d set"   and x :: "'d" assumes "continuous_on s f"   and "t \<subseteq> s"   and "x \<in> s" shows "continuous (at x within t) f"
float_neg_sign: fixes a :: "('a, 'b) IEEE.float" shows "sign a \<noteq> sign (- a)"
insert: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and z :: "'a"   and g :: "'b \<Rightarrow> 'a"   and x :: "'b"   and A :: "'b multiset" assumes "comm_monoid_mset f z" shows "comm_monoid_mset.F f z (image_mset g (add_mset x A)) = f (g x) (comm_monoid_mset.F f z (image_mset g A))"
sat_inf_imp_ground_red_fam_inter: fixes N :: "'f set"   and \<iota>' :: "'f inference"   and q :: "'q"   and \<iota> :: "'g inference" assumes "empty_ord.saturated N"   and "\<iota>' \<in> Inf_from N"   and "q \<in> Q"   and "\<G>_I_q q \<iota>' \<noteq> None \<and> \<iota> \<in> the (\<G>_I_q q \<iota>')" shows "\<iota> \<in> Red_I_q q (\<G>_Fset_q q N)"
Func_singleton: fixes A :: "'c set"   and x :: "'d" shows "|Func A {x}| =o |{x}|"
irrefl: fixes a :: "'b list" shows "\<not> strict_prefix a a"
gcsrp_one_def: fixes G :: "('e, 'f) Group_scheme"   and H :: "'e set" shows "gcsrp_one G H = gcsrp_map G H H"
length_induct: fixes P :: "'e list \<Rightarrow> bool"   and xs :: "'e list" assumes "\<And>xs. \<forall>ys. length ys < length xs \<longrightarrow> P ys \<Longrightarrow> P xs" shows "P xs"
in_eq_out_degree: fixes G :: "('a, 'b) pre_digraph"   and v :: "'a" assumes "graph G" shows "in_degree G v = out_degree G v"
assms_2: shows "p - p' \<in> More_Modules.ideal polynomial_bool"
vector_one_nth: fixes x :: "('c, 1) vec" shows "vec (x $ 1) = x"
134: fixes x :: "'a"   and y :: "'a" shows "- x \<squnion> - (- - x \<sqinter> y) = - (- - x \<sqinter> y)"
cond_spmf_def: fixes p :: "'c spmf"   and A :: "'c set" shows "cond_spmf p A = (if set_spmf p \<inter> A = {} then return_pmf None else cond_pmf p (Some ` A))"
enum_ex_Agent_def: fixes P :: "Agent \<Rightarrow> bool" shows "enum_class.enum_ex P = (P Child0 \<or> P Child1 \<or> P Child2)"
shconf_hupd_obj: fixes P :: "'a prog"   and h :: "heap"   and sh :: "sheap"   and a :: "nat"   and C :: "char list"   and fs :: "Objects.fields"   and fs' :: "Objects.fields" assumes "P,h \<turnstile>\<^sub>s sh \<surd>"   and "h a = \<lfloor>(C, fs)\<rfloor>" shows "P,h(a \<mapsto> (C, fs')) \<turnstile>\<^sub>s sh \<surd>"
pr_rec1: fixes h :: "nat \<Rightarrow> nat \<Rightarrow> nat"   and a :: "nat" assumes "h \<in> PrimRec2" shows "PrimRecOp1 a h \<in> PrimRec1"
add_less_zeroD: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and x :: "'b"   and y :: "'b" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one"   and "less (plus x y) zero" shows "less x zero \<or> less y zero"
coalg_F1set3: fixes B1 :: "'d set"   and B2 :: "'e set"   and s1 :: "'d \<Rightarrow> ('f, 'd, 'e) F1"   and s2 :: "'e \<Rightarrow> ('f, 'd, 'e) F2"   and a :: "'d" assumes "coalg B1 B2 s1 s2"   and "a \<in> B1" shows "F1set3 (s1 a) \<subseteq> B2"
rel_cfg_cont: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and cfg1 :: "'a cfg"   and cfg2 :: "'b cfg" assumes "rel_cfg P cfg1 cfg2" shows "rel_pmf (\<lambda>s t. rel_cfg P (cont cfg1 s) (cont cfg2 t)) (action cfg1) (action cfg2)"
bisim1TryCatch1: fixes P :: "'a prog"   and h :: "'heap"   and e :: "'addr expr1"   and a :: "'addr"   and xs :: "'addr val list"   and stk :: "'addr val list"   and loc :: "'addr val list"   and pc :: "nat"   and C' :: "String.literal"   and C :: "String.literal"   and V :: "nat"   and e2 :: "'addr expr1" assumes "P,e,h \<turnstile> (Throw a, xs) \<leftrightarrow> (stk, loc, pc, \<lfloor>a\<rfloor>)"   and "typeof_addr h a = \<lfloor>Class_type C'\<rfloor>"   and "P \<turnstile> C' \<preceq>\<^sup>* C" shows "P,try e catch(C V) e2,h \<turnstile> ({V:Class C=None; e2}, xs[V := Addr a]) \<leftrightarrow> ([Addr a], loc, Suc (length (compE2 e)), None)"
poly_mult_const_1: fixes K :: "'a set"   and p :: "'a list"   and a :: "'a" assumes "subring K R"   and "polynomial K p"   and "a \<in> K - {\<zero>}" shows "poly_mult [a] p = map ((\<otimes>) a) p"
Deriv_lists: fixes c :: "'a"   and S :: "'a set" assumes "c \<in> S" shows "Deriv c (lists S) = lists S"
2: fixes n :: "nat" assumes "n < (LEAST n. xs !! n = x)" shows "xs !! n \<noteq> x"
natural_isomorphism_axioms: shows "natural_isomorphism A B F F F"
Le_LCons: fixes xs :: "'c llist"   and ys :: "'c llist"   and x :: "'c" assumes "xs \<sqsubseteq> ys" shows "LCons x xs \<sqsubseteq> LCons x ys"
weakCongWeakEq: fixes P :: "pi"   and Q :: "pi" assumes "P \<simeq>\<^sup>s Q" shows "P \<simeq> Q"
totient_conv_moebius_mu_of_nat: fixes n :: "nat" shows "of_nat (totient n) = dirichlet_prod moebius_mu of_nat n"
adopt_node_preserves_wellformedness: fixes h :: "(_) heap"   and document_ptr :: "(_) document_ptr"   and child :: "(_) node_ptr"   and h' :: "(_) heap" assumes "heap_is_wellformed h"   and "h \<turnstile> adopt_node document_ptr child \<rightarrow>\<^sub>h h'"   and "known_ptrs h"   and "type_wf h" shows "heap_is_wellformed h'"
3: fixes n :: "nat" shows "\<Union> (S ` {..n}) \<subseteq> \<Union> (S ` {..Suc n})"
rep_eq: fixes x :: "'b fset" shows "fBex x = Bex (fset x)"
that: fixes w :: "'action list" assumes "path w p"   and "target w p = q"   and "set w \<inter> cuts = {}"   and "w \<noteq> []" shows "thesis"
that: fixes c :: "'c" assumes "homotopic_with (\<lambda>h. True) X Z (g \<circ> f) (\<lambda>x. c)" shows "thesis"
Complex_mult_complex_of_real: fixes x :: "real"   and y :: "real"   and r :: "real" shows "Complex x y * complex_of_real r = Complex (x * r) (y * r)"
dim_eq_span: fixes S :: "('a, 'b) vec set"   and T :: "('a, 'b) vec set" assumes "S \<subseteq> T"   and "vec.dim T \<le> vec.dim S" shows "vec.span S = vec.span T"
int_clset_is_a_model: fixes S :: "'a eclause set"   and pair :: "'a eclause \<times> ('a \<times> 'a trm) list" assumes "ground_inference_saturated S"   and "\<forall>x\<in>S. finite (cl_ecl x)"   and "Ball S well_constrained"   and "\<forall>x\<in>S. cl_ecl x \<noteq> {}"   and "closed_under_renaming S" shows "\<forall>C \<sigma>. fst pair = C \<longrightarrow> \<sigma> = snd pair \<longrightarrow> C \<in> S \<longrightarrow> ground_clause (subst_cl (cl_ecl C) \<sigma>) \<longrightarrow> all_trms_irreducible (subst_set (trms_ecl C) \<sigma>) (\<lambda>t. trm_rep t S) \<longrightarrow> validate_ground_clause (same_values (\<lambda>t. trm_rep t S)) (subst_cl (cl_ecl C) \<sigma>)"
primmultgreater: fixes p :: "nat"   and q :: "nat" assumes "prime p"   and "prime q"   and "p \<noteq> 2"   and "q \<noteq> 2" shows "2 < p * q"
le_iff_add: fixes a :: "'f multiset"   and b :: "'f multiset" shows "(a \<subseteq># b) = (\<exists>c. b = a + c)"
comp_char: fixes g :: "'a arrow_of_spans_data \<times> 'a arrow_of_spans_data"   and f :: "'a arrow_of_spans_data \<times> 'a arrow_of_spans_data" shows "VxV.comp g f = (if seq (fst g) (fst f) \<and> seq (snd g) (snd f) then (fst g \<bullet> fst f, snd g \<bullet> snd f) else (null, null))"
W_cmpser_def: fixes G :: "('e, 'f) Group_scheme"   and n :: "nat"   and g :: "nat \<Rightarrow> 'e set" shows "W_cmpser G n g = (if n = 0 then d_gchain G 0 g else D_gchain G n g \<and> (\<forall>l\<le>n - 1. \<natural>\<^bsub>G\<^esub>g l \<triangleright> g (Suc l)))"
Qg_tassocTr: fixes N :: "'a set"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "G \<triangleright> N"   and "a \<in> carrier G"   and "b \<in> carrier G"   and "c \<in> carrier G" shows "c_top G N (N \<bullet> a) (c_top G N (N \<bullet> b) (N \<bullet> c)) = c_top G N (c_top G N (N \<bullet> a) (N \<bullet> b)) (N \<bullet> c)"
dlex_pm_plus_monotone: fixes s :: "'a \<Rightarrow>\<^sub>0 'b"   and t :: "'a \<Rightarrow>\<^sub>0 'b"   and u :: "'a \<Rightarrow>\<^sub>0 'b" assumes "dlex_pm s t" shows "dlex_pm (s + u) (t + u)"
sum_shift_lb_Suc0_0_upt: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and f :: "nat \<Rightarrow> 'a"   and k :: "nat" assumes "class.comm_monoid_add plus zero"   and "f 0 = zero" shows "comm_monoid_add.sum plus zero f {Suc 0..<k} = comm_monoid_add.sum plus zero f {0..<k}"
assms_5: shows "valid_unMultigraph.is_Eulerian_trail G1 v1 ps1 v1'"
det_def: shows "Square_Matrix.det \<equiv> map_fun to_fun id (\<lambda>A. \<Sum>p | p permutes UNIV. of_int (sign p) * (\<Prod>i\<in>UNIV. A i (p i)))"
random_scheduler_def: fixes seed :: "natural \<times> natural"   and s :: "('l, 't) locks \<times> ('m_t \<times> 'm) \<times> 'm_w \<times> 's_i" shows "local.random_scheduler seed s = next_thread seed (active_threads s) \<bind> (\<lambda>(t, seed'). step_thread (\<lambda>ta. seed') s t)"
inverse: fixes f :: "'b \<Rightarrow> 'c"   and F :: "'b filter"   and g :: "'b \<Rightarrow> 'c" assumes "\<forall>\<^sub>F x in F. f x \<noteq> (0::'c)"   and "\<forall>\<^sub>F x in F. g x \<noteq> (0::'c)"   and "f \<in> \<Omega>[F](g)" shows "(\<lambda>x. inverse (g x)) \<in> \<Omega>[F](\<lambda>x. inverse (f x))"
le_iff_sup: fixes x :: "'b"   and y :: "'b" shows "(x \<le> y) = (x + y = y)"
rep_eq: fixes x :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and xa :: "('c, 'd) fmap" shows "fmpred x xa = map_pred x (fmlookup xa)"
2: shows "{k. (T ^^ k) ((T ^^ n) x) \<in> A} \<noteq> {}"
simps_1: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and E :: "char list \<Rightarrow> ty option"   and h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option" shows "(P,E,h \<turnstile> [] [:]\<^bsub>NT\<^esub> []) = True"
bilinear_bounded_pos: fixes h :: "'a \<Rightarrow> 'b \<Rightarrow> 'c" assumes "bilinear h" shows "\<exists>B>0. \<forall>x y. norm (h x y) \<le> B * norm x * norm y"
selectlike_Vagree: fixes \<nu> :: "variable \<Rightarrow> real"   and \<omega> :: "variable \<Rightarrow> real"   and V :: "variable set"   and X :: "(variable \<Rightarrow> real) set" assumes "Vagree \<nu> \<omega> V" shows "selectlike X \<nu> V = selectlike X \<omega> V"
rcos_disjoint: fixes G :: "('a, 'b) monoid_scheme"   and H :: "'a set" assumes "Group.group G"   and "subgroup H G" shows "disjoint (rcosets\<^bsub>G\<^esub> H)"
Inf_set_def: fixes A :: "'b set set" shows "\<Inter> A = {x. Inf ((\<in>) x ` A)}"