hcju/isabelleps huggingface dataset:
product_mset_def: fixes A :: "'c multiset"   and B :: "'d multiset" shows "product_mset A B = A \<times># B"
it_to_sorted_list_def: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and s :: "'b set" shows "it_to_sorted_list R s \<equiv> SPEC (\<lambda>l. distinct l \<and> s = set l \<and> sorted_wrt R l)"
conv1: shows "pfinite_gpv \<I>2 (run_converter conv1 a)"
cnj_E1: fixes H :: "fmla set"   and A :: "fmla"   and B :: "fmla" assumes "H \<turnstile> A AND B" shows "H \<turnstile> A"
lt_tail: fixes p :: "'t \<Rightarrow>\<^sub>0 'c" assumes "tail p \<noteq> 0" shows "lt_of_nat_term_order cmp_term (lt (tail p)) (lt p)"
Un_Diff: fixes A :: "'c set"   and B :: "'c set"   and C :: "'c set" shows "A \<union> B - C = A - C \<union> (B - C)"
map_scale_eq_monom_mult: fixes c :: "'b"   and p :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" shows "c \<cdot> p = local.punit.monom_mult c 0 p"
extensible_def: fixes R :: "('a, 'b) labeled_graph \<times> ('a, 'b) labeled_graph"   and G :: "('a, 'c) labeled_graph"   and f :: "('b \<times> 'c) set" shows "extensible R G f \<equiv> \<exists>g. graph_homomorphism (snd R) G g \<and> agree_on (fst R) f g"
nominal_in_names: fixes A :: "'b set"   and H :: "(('c, 'b) fm list \<times> 'b) set"   and i :: "'b" assumes "Hintikka A H"   and "\<exists>block\<in>H. i \<in> block_nominals block" shows "i \<in> names H"
distperm_s'_A: fixes n :: "nat"   and m :: "nat" shows "dist_perm (s'_A n m) init"
surj_on_implies_size_eq_card: fixes A :: "'c set"   and B :: "'d set"   and F :: "('c \<Rightarrow> 'd) set" assumes "finite A"   and "finite B"   and "F \<in> (A \<rightarrow>\<^sub>E B) // domain_and_range_permutation A B"   and "univ (\<lambda>f. f ` A = B) F" shows "size (number_partition_of A B F) = card B"
Cons_4: shows "InvariantWatchListsUniq (getWatchList state)"
C_def: shows "(C::'a::refinement_lattice set) = {c::'a::refinement_lattice}"
assms_2: fixes exp :: "exp"   and var :: "Vars.var" assumes "P exp" shows "P (App exp var)"
at_eq_sup_left_right: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and "open" :: "'a set \<Rightarrow> bool"   and x :: "'a" assumes "class.linorder_topology less_eq less open" shows "topological_space.at_within open x UNIV = sup (topological_space.at_within open x (ord.lessThan less x)) (topological_space.at_within open x (ord.greaterThan less x))"
collapse_2: fixes hd :: "('a, 'b) hd" assumes "is_Sym hd" shows "Sym (hd.sym hd) = hd"
axioms_1: fixes V\<^sub>C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and H\<^sub>C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<a>\<^sub>C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<i>\<^sub>C :: "'c \<Rightarrow> 'c"   and src\<^sub>C :: "'c \<Rightarrow> 'c"   and trg\<^sub>C :: "'c \<Rightarrow> 'c"   and V\<^sub>D :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and H\<^sub>D :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and \<a>\<^sub>D :: "'d \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and \<i>\<^sub>D :: "'d \<Rightarrow> 'd"   and src\<^sub>D :: "'d \<Rightarrow> 'd"   and trg\<^sub>D :: "'d \<Rightarrow> 'd"   and F :: "'c \<Rightarrow> 'd"   and \<Phi>\<^sub>F :: "'c \<times> 'c \<Rightarrow> 'd" assumes "identity_pseudonatural_transformation V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D F \<Phi>\<^sub>F" shows "bicategory V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C"
stop_no_eval: fixes stop :: "'a"   and eval :: "((('a \<times> (Mode \<Rightarrow> 'b set)) \<times> ('b \<Rightarrow> 'c)) \<times> ('a \<times> (Mode \<Rightarrow> 'b set)) \<times> ('b \<Rightarrow> 'c)) set"   and some_val :: "'c"   and some_val' :: "'c"   and mds :: "Mode \<Rightarrow> 'b set"   and mem :: "'b \<Rightarrow> 'c"   and c' :: "'a"   and mds' :: "Mode \<Rightarrow> 'b set"   and mem' :: "'b \<Rightarrow> 'c" assumes "sifum_security stop eval some_val some_val'" shows "(((stop, mds), mem), (c', mds'), mem') \<notin> eval"
swb_def: shows "swb = (\<lambda>A i n. sw A (map (\<lambda>j. (j, Suc j)) [i..<i + n]))"
fin_dg_Arr_vfinite: shows "vfinite (\<CC>\<lparr>Arr\<rparr>)"
msteps_psnoc: fixes \<pi> :: "(char list \<times> 'a list) prefix"   and tdb :: "(char list \<times> 'a list) set \<times> nat"   and st :: "'a mstate" assumes "last_ts \<pi> \<le> snd tdb" shows "msteps (psnoc \<pi> tdb) st = (let (V', st') = msteps \<pi> st; (V'', st'') = mstep tdb st' in (V' \<union> V'', st''))"
cases: fixes a1 :: "nat \<times> nat \<Rightarrow> 'capacity"   and a2 :: "nat \<Rightarrow> nat"   and a3 :: "nat \<times> nat \<Rightarrow> 'capacity"   and a4 :: "nat \<Rightarrow> nat" assumes "((a1, a2), a3, a4) \<in> pr_algo_rel" obtains f :: "nat \<times> nat \<Rightarrow> 'capacity" and l :: "nat \<Rightarrow> nat" and e :: "nat \<times> nat" where "a1 = push_effect fa__ e" and "a2 = la__" and "a3 = fa__" and "a4 = la__" and "Height_Bounded_Labeling c s t fa__ la__" and "push_precond fa__ la__ e"   | f :: "nat \<times> nat \<Rightarrow> 'capacity" and l :: "nat \<Rightarrow> nat" and u :: "nat" where "a1 = f" and "a2 = relabel_effect f l u" and "a3 = f" and "a4 = l" and "Height_Bounded_Labeling c s t f l" and "relabel_precond f l u"
lookup_env_delete_None: fixes x :: "'d"   and m :: "'d \<Rightarrow> 'c" shows "env_delete x m x = \<bottom>"
fv_map_lc: fixes p :: "nat \<Rightarrow> nat"   and f :: "form" shows "fv (map_lc p f) = fv f"
cases: fixes x :: "const" obtains i :: "int" where "x = Const i"   | "x = Any"
Prefixes_def: fixes A :: "'a list set" shows "Prefixes A = {w. \<exists>q. w @ q \<in> A}"
sol_fres: fixes y :: "'a"   and x :: "'a" shows "(y \<cdot> x \<leftarrow> x = y) = (\<exists>z. y = z \<leftarrow> x)"
inv_dghm_ObjMap_vdomain: shows "\<D>\<^sub>\<circ> (inv_dghm \<FF>\<lparr>ObjMap\<rparr>) = \<BB>\<lparr>Obj\<rparr>"
faithful_functor_axioms_def: fixes A :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and F :: "'a \<Rightarrow> 'b" shows "faithful_functor_axioms A F \<equiv> \<forall>f f'. partial_magma.arr A f \<and> partial_magma.arr A f' \<and> partial_magma.dom A f = partial_magma.dom A f' \<and> partial_magma.cod A f = partial_magma.cod A f' \<longrightarrow> F f = F f' \<longrightarrow> f = f'"
get_local: fixes vi :: "v list"   and j :: "nat"   and s :: "s"   and v :: "v"   and vs :: "v list"   and i :: "nat" assumes "length vi = j" shows "\<lparr>s;vi @ [v] @ vs;[$Get_local j]\<rparr> \<leadsto>_ i \<lparr>s;vi @ [v] @ vs;[$C v]\<rparr>"
ope4: shows "openin (top_of_set (U - C)) (U - C - H4)"
qty_replace_comp_def: fixes pf1 :: "'d \<Rightarrow> nat \<Rightarrow> 'e \<Rightarrow> real"   and x :: "'d"   and pf2 :: "'d \<Rightarrow> nat \<Rightarrow> 'e \<Rightarrow> real" shows "qty_replace_comp pf1 x pf2 = qty_sum (qty_rem_comp pf1 x) (qty_mult_comp pf2 (pf1 x))"
select_convs_4: fixes Obj :: "'e set"   and Mor :: "'d set"   and Dom :: "'d \<Rightarrow> 'e"   and Cod :: "'d \<Rightarrow> 'e"   and Id :: "'e \<Rightarrow> 'd"   and Comp :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and more :: "'f" shows "Cod \<lparr>Obj = Obj, Mor = Mor, Dom = Dom, Cod = Cod, Id = Id, Comp = Comp, \<dots> = more\<rparr> = Cod"
borel_measurable_continuous_on_restrict: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "continuous_on A f" shows "f \<in> borel_measurable (restrict_space borel A)"
conductor_eq_1_iff_principal: shows "(conductor = 1) = (\<chi> = principal_dchar n)"
invar: shows "round_robin_invar (queue, n0) (dom (thr s))"
leadsTo_Un_duplicate: fixes F :: "'c program"   and A :: "'c set"   and A' :: "'c set" assumes "F \<in> A \<longmapsto> A' \<union> A'" shows "F \<in> A \<longmapsto> A'"
logically_equivalent_def: fixes satisfies :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and transition :: "'b \<Rightarrow> ('d, 'b) residual \<Rightarrow> bool"   and P :: "'b"   and Q :: "'b" assumes "indexed_nominal_ts TYPE('a) satisfies transition" shows "indexed_nominal_ts.logically_equivalent TYPE('a) satisfies transition P Q \<equiv> \<forall>x. nominal_ts.valid satisfies transition P x = nominal_ts.valid satisfies transition Q x"
dbl_inc_simps_1: fixes minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and uminus :: "'c \<Rightarrow> 'c"   and one :: "'c"   and k :: "num" assumes "class.neg_numeral minus plus zero uminus" shows "neg_numeral.dbl_inc plus one (uminus (numeral.numeral one plus k)) = uminus (neg_numeral.dbl_dec minus plus one (numeral.numeral one plus k))"
bconf_nonVal: fixes P :: "'a prog"   and sh :: "sheap"   and e :: "'b exp" assumes "P,sh \<turnstile>\<^sub>b (e,True) \<surd>" shows "val_of e = None"
getEnv_preserves_good: fixes rho :: "'d \<Rightarrow> 'e \<Rightarrow> ('f, 'g, 'varSort, 'var, 'h) qTerm set option"   and xs :: "'d"   and x :: "'e"   and X :: "('f, 'g, 'varSort, 'var, 'h) qTerm set" assumes "goodEnv rho"   and "rho xs x = Some X" shows "good X"
3: shows "Max (lift (liset visible (llist_of_stream u))) < i"
mat_assoc_test_3: fixes A :: "complex mat"   and n :: "nat"   and B :: "complex mat"   and C :: "complex mat"   and D :: "complex mat" assumes "A \<in> carrier_mat n n"   and "B \<in> carrier_mat n n"   and "C \<in> carrier_mat n n"   and "D \<in> carrier_mat n n" shows "A * 1\<^sub>m n * 1\<^sub>m n * B * 1\<^sub>m n = A * B"
deleteMin_correct_2: fixes q :: "('a, 'b) BsSkewBinomialHeapStruc.BsSkewBinomialTree list" assumes "SkewBinomialHeapStruc.invar (map BsSkewBinomialHeapStruc.bsmapt q)"   and "q \<noteq> []" shows "image_mset fst (SkewBinomialHeapStruc.queue_to_multiset (map BsSkewBinomialHeapStruc.bsmapt (BsSkewBinomialHeapStruc.deleteMin q))) = image_mset fst (SkewBinomialHeapStruc.queue_to_multiset (map BsSkewBinomialHeapStruc.bsmapt q)) - {#BsSkewBinomialHeapStruc.findMin q#}"
cases: fixes x :: "'a option" obtains "x = None"   | xa :: "'a" where "x = Some xa"
bel: shows "spick w n True \<in> geom_proc n -` {x} \<and> spick w n False \<in> geom_proc n -` {x}"
finite_measure_cut_measurable: fixes M :: "'c measure"   and Q :: "('d \<times> 'c) set"   and N :: "'d measure" assumes "finite_measure M"   and "Q \<in> sets (N \<Otimes>\<^sub>M M)" shows "(\<lambda>x. emeasure M (Pair x -` Q)) \<in> borel_measurable N"
Cons_1: fixes P1 :: "'b \<Rightarrow> bool"   and l :: "'b list"   and PProbs :: "'a \<Rightarrow> bool"   and s :: "'a"   and PProbl :: "'b list \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'b list \<Rightarrow> 'a"   and P2 :: "'b \<Rightarrow> bool"   and k1 :: "nat"   and P3 :: "'b \<Rightarrow> bool" assumes "x = filter (\<lambda>x. \<not> P1 x) l"   and "PProbs s"   and "PProbl l"   and "\<forall>l s. PProbs s \<and> PProbl l \<and> list_all P1 l \<longrightarrow> (\<exists>l'. f s l' = f s l \<and> length (filter P2 l') \<le> k1 \<and> length (filter P3 l') \<le> length (filter P3 l) \<and> list_all P1 l' \<and> subseq l' l)"   and "\<forall>s l1 l2. f (f s l1) l2 = f s (l1 @ l2)"   and "\<forall>s l. PProbs s \<and> PProbl l \<longrightarrow> PProbs (f s l)"   and "\<forall>l1 l2. subseq l1 l2 \<and> PProbl l2 \<longrightarrow> PProbl l1"   and "\<forall>l1 l2. PProbl (l1 @ l2) = (PProbl l1 \<and> PProbl l2)" shows "\<exists>l'. f s l' = f s l \<and> length (filter P3 l') \<le> length (filter P3 l) \<and> (\<forall>l''. sublist l'' l' \<and> list_all P1 l'' \<longrightarrow> length (filter P2 l'') \<le> k1) \<and> subseq l' l"
empty_neutral_2: fixes x :: "'c multiset" shows "x + {#} = x"
disj_of_clause_is: fixes R :: "'b literal list" shows "is_disj (disj_of_clause R)"
T1: fixes F :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" shows "\<turnstile> \<box>\<box>F = \<box>F"
bind_def: fixes A :: "'m set"   and f :: "'m \<Rightarrow> 'l set" shows "Set.bind A f = {x. \<exists>B\<in>f ` A. x \<in> B}"
set_shadow_root_get_tag_name: fixes ptr :: "(_) element_ptr"   and h :: "(_) heap"   and h' :: "(_) heap"   and ptr' :: "(_) element_ptr" shows "\<forall>w\<in>set_shadow_root_locs ptr. h \<turnstile> w \<rightarrow>\<^sub>h h' \<longrightarrow> (\<forall>r\<in>get_tag_name_locs ptr'. r h h')"
min_maxsimpchainD_maxsimpchain: fixes xs :: "'a set list" assumes "min_gallery xs" shows "gallery xs"
simps_3: fixes x :: "'e"   and p :: "'f \<Rightarrow> 'e"   and q :: "'g \<Rightarrow> 'e"   and D :: "'f \<Rightarrow> 'h"   and E :: "'g \<Rightarrow> 'h" assumes "x \<notin> range p"   and "x \<notin> range q" shows "con_comp_map D E p q x = None"
vars_def: shows "vars \<equiv> (x, y, sem, pc1, pc2)"
injection_defined_rev: fixes f :: "'b \<rightarrow> 'a"   and g :: "'a \<rightarrow> 'b"   and z :: "'a" assumes "\<forall>x. f\<cdot>(g\<cdot>x) = x"   and "g\<cdot>z = \<bottom>" shows "z = \<bottom>"
bounded_byI2: fixes P :: "'a \<Rightarrow> real"   and b :: "real" assumes "P \<tturnstile> \<lambda>s. b" shows "bounded_by b P"
assms_2: fixes n :: "nat" assumes "dimension n K E" shows "P"
eval_poly_sum: fixes p :: "'b \<Rightarrow> rat poly"   and A :: "'b set"   and x :: "'a" shows "eval_poly of_rat (sum p A) x = (\<Sum>k\<in>A. eval_poly of_rat (p k) x)"
Suc_1: fixes j :: "nat"   and i :: "nat" assumes "k = j - i"   and "i \<le> j"   and "j < length \<GG>" shows "\<GG> ! i \<subseteq> \<GG> ! j"
ccBind_def: fixes v :: "Vars.var"   and e :: "exp" shows "ccBind v e = (\<Lambda> (ae, G). if v--v\<notin>G \<or> \<not> isVal e then \<G>\<^sup>\<bottom>\<^bsub>ae v\<^esub> e G|` fv e else fv e\<^sup>2)"
group_comm_groupI: fixes G :: "('c, 'd) monoid_scheme" assumes "Group.group G"   and "\<And>x y. \<lbrakk>x \<in> carrier G; y \<in> carrier G\<rbrakk> \<Longrightarrow> x \<otimes>\<^bsub>G\<^esub> y = y \<otimes>\<^bsub>G\<^esub> x" shows "comm_group G"
rs_def: shows "set rs = {row A i |i. i < nr} \<union> set Ns \<union> uminus ` set Ns"
unique_lemma: fixes PB :: "msg"   and RB :: "msg"   and KXY :: "nat"   and evs :: "event list"   and A :: "agent"   and K :: "nat"   and B :: "agent"   and N :: "msg"   and A' :: "agent"   and B' :: "agent"   and N' :: "msg" assumes "(PB, RB, KXY) \<in> respond evs"   and "Crypt (shrK A) \<lbrace>Key K, Agent B, N\<rbrace> \<in> parts {RB}"   and "Crypt (shrK A') \<lbrace>Key K, Agent B', N'\<rbrace> \<in> parts {RB}" shows "A' = A \<and> B' = B \<or> A' = B \<and> B' = A"
ID_ceq_neq_None: shows "ID CEQ('a) \<noteq> None"
wt_jvm_prog_impl_wt_instr: fixes \<Phi> :: "String.literal \<Rightarrow> String.literal \<Rightarrow> (ty list \<times> ty err list) option list"   and P :: "(nat \<times> nat \<times> 'a instr list \<times> (nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list) prog"   and C :: "String.literal"   and M :: "String.literal"   and Ts :: "ty list"   and T :: "ty"   and mxs :: "nat"   and mxl\<^sub>0 :: "nat"   and ins :: "'a instr list"   and xt :: "(nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list"   and pc :: "nat" assumes "wf_jvm_prog\<^bsub>\<Phi>\<^esub> P"   and "P \<turnstile> C sees M: Ts\<rightarrow>T = \<lfloor>(mxs, mxl\<^sub>0, ins, xt)\<rfloor> in C"   and "pc < length ins" shows "P,T,mxs,length ins,xt \<turnstile> ins ! pc,pc :: \<Phi> C M"
summable_powser_split_head: fixes f :: "nat \<Rightarrow> 'b"   and z :: "'b" shows "summable (\<lambda>n. f (Suc n) * z ^ n) = summable (\<lambda>n. f n * z ^ n)"
t_heap_of_B_def: fixes xs :: "'b list" shows "t_heap_of_B xs = length xs + t_heapify (length xs) xs"
assms_4: shows "row_to_poly ts r1 = row_to_poly ts r2"
alist_pqueue: fixes xs :: "('d, 'c) bintree option list" assumes "distinct (vals xs)" shows "set (elements xs) = set (pq.alist_of (pqueue xs))"
ordinal_util_imp_rat_prefs: fixes carrier :: "'b set"   and relation :: "('b \<times> 'b) set"   and u :: "'b \<Rightarrow> real" assumes "ordinal_utility carrier relation u" shows "rational_preference carrier relation"
wf_stateD: 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" assumes "s \<in> wf_state" shows "lock_thread_ok (locks s) (thr s) \<and> wset_final_ok (wset s) (thr s)"
inj: fixes z :: "real" assumes "\<bar>z - x\<bar> \<le> d" shows "g (f z) = z"
bdd_below_Ico: fixes a :: "'a"   and b :: "'a" shows "bdd_below {a..<b}"
rel_refl: fixes Ra :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'e \<rightharpoonup>\<^sub>f 'd" assumes "\<And>x. Ra x x" shows "fmrel Ra x x"
ttm_children: fixes t :: "('c, 'd) SkewBinomialTree" shows "tree_to_multiset t = {#(val t, prio t)#} + queue_to_multiset (children t)"
locally_path_connected_space: fixes X :: "'c topology" shows "locally_path_connected_space X = (\<forall>V x. openin X V \<and> x \<in> V \<longrightarrow> (\<exists>U. openin X U \<and> path_connectedin X U \<and> x \<in> U \<and> U \<subseteq> V))"
unity_root_eq_1_iff_int: fixes k :: "nat"   and n :: "int" assumes "0 < k" shows "(unity_root k n = 1) = (int k dvd n)"
idealI: fixes R :: "('e, 'f) ring_scheme"   and I :: "'e set" assumes "ring R"   and "subgroup I (add_monoid R)"   and "\<And>a x. \<lbrakk>a \<in> I; x \<in> carrier R\<rbrakk> \<Longrightarrow> x \<otimes>\<^bsub>R\<^esub> a \<in> I"   and "\<And>a x. \<lbrakk>a \<in> I; x \<in> carrier R\<rbrakk> \<Longrightarrow> a \<otimes>\<^bsub>R\<^esub> x \<in> I" shows "ideal I R"
eq_upto_0: fixes B :: "(nat \<Rightarrow> 'b) set"   and f :: "nat \<Rightarrow> 'b" shows "eq_upto B f 0 = B"
non_empty_dbm_diag_set: fixes v :: "'b \<Rightarrow> nat"   and n :: "nat"   and M :: "nat \<Rightarrow> nat \<Rightarrow> 'c DBMEntry" assumes "\<forall>c. 0 < v c \<and> (\<forall>x y. v x \<le> n \<and> v y \<le> n \<and> v x = v y \<longrightarrow> x = y)"   and "[M]\<^bsub>v,n\<^esub> \<noteq> {}" shows "[M]\<^bsub>v,n\<^esub> = [(\<lambda>i j. if i = j then \<one> else M i j)]\<^bsub>v,n\<^esub>"
insertI1: fixes a :: "'c"   and B :: "'c set" shows "a \<in> {a} \<union> B"
\<BB>_def: shows "\<BB> \<equiv> {(fold lQuot w (lang n r), fold lQuot w (lang n s)) |w. w \<in> lists (\<Sigma> n)}"
ht_hash_ls_rehash: fixes l :: "('a \<times> 'b) list list" shows "ht_hash (ls_rehash l)"
code: fixes xs :: "'b llist" shows "lalternate xs = (case xs of LNil \<Rightarrow> LNil | LCons x xs \<Rightarrow> LCons x (lalternate (ltl xs)))"
less_linear: fixes D :: "('d, 'e) Order_scheme"   and a :: "'d"   and b :: "'d" assumes "Torder D"   and "a \<in> carrier D"   and "b \<in> carrier D" shows "a \<prec>\<^bsub>D\<^esub> b \<or> a = b \<or> b \<prec>\<^bsub>D\<^esub> a"
seqsim_empty_empty: fixes s :: "nat \<Rightarrow> 'b"   and t :: "nat \<Rightarrow> 'b" assumes "s \<approx> t"   and "emptyseq s" shows "emptyseq t"
map\<^sub>0_simps_1: fixes a :: "'b" assumes "B.obj a" shows "C.obj (\<tau>.G.map\<^sub>0 a)"
simps_2: fixes n :: "nat"   and l :: "real" shows "nxt Del (n, l) = (n - 1, if f1 * l \<le> real (n - 1) then l else if l0 \<le> l / c then l / c else l)"
typing_alt_simps_38: fixes \<Gamma> :: "String.literal \<rightharpoonup>\<^sub>f 'a OCL_Types.type"   and a :: "'a expr"   and \<tau> :: "'a OCL_Types.type" shows "(\<Gamma> \<turnstile>\<^sub>P CollectionItem a : \<tau>) = (\<Gamma> \<turnstile>\<^sub>E a : \<tau>)"
index_order3: fixes x :: "'a"   and y :: "'a"   and z :: "'a"   and f :: "nat \<Rightarrow> 'a"   and a :: "nat"   and b :: "nat"   and c :: "nat"   and X :: "'a set" assumes "[[x y z]]"   and "f a = x"   and "f b = y"   and "f c = z"   and "long_ch_by_ord f X"   and "finite X \<longrightarrow> a < card X"   and "finite X \<longrightarrow> b < card X"   and "finite X \<longrightarrow> c < card X" shows "a < b \<and> b < c \<or> c < b \<and> b < a"
integrable: shows "complex_set_integrable lebesgue {0<..} (\<lambda>t. complex_of_real t powr (z - 1) / complex_of_real (exp t))"
assms_2: shows "A * D \<le> (cmod BC)\<^sup>2"
nth_butlast: fixes n :: "nat"   and xs :: "'c list" assumes "n < length (butlast xs)" shows "butlast xs ! n = xs ! n"
less_3: fixes k' :: "nat"   and i' :: "nat" assumes "k' cd\<^bsup>\<pi>\<^esup>\<rightarrow> k"   and "P k'"   and "i' icd\<^bsup>\<pi>\<^esup>\<rightarrow> k'" shows "P i'"
assms_6: fixes t :: "real" assumes "t \<in> {a..b}" shows "D \<le> infdist (f t) G"
Replacement_def: fixes A :: "ZF"   and f :: "ZF \<Rightarrow> ZF option" shows "Replacement A f \<equiv> Repl (Sep A (\<lambda>a. f a \<noteq> None)) (the \<circ> f)"
std_simplex: shows "convex hull insert (0::'a) Basis = {x. (\<forall>i\<in>Basis. 0 \<le> x \<bullet> i) \<and> sum ((\<bullet>) x) Basis \<le> 1}"
ccw'_not_coll_1: fixes a :: "real \<times> real"   and b :: "real \<times> real"   and c :: "real \<times> real" assumes "ccw' a b c" shows "det3 a b c \<noteq> 0"
assoc: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "(a AND b) AND c = a AND b AND c"
funToCone_def: fixes F :: "'j \<Rightarrow> 's" shows "funToCone F \<equiv> \<lambda>j. if J.arr j then S.mkPoint (D j) (F j) else S.null"
rel_gpv''_lossless_gpvD1: fixes A :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and C :: "'h \<Rightarrow> 'i \<Rightarrow> bool"   and R :: "'j \<Rightarrow> 'k \<Rightarrow> bool"   and gpv :: "('f, 'h, 'j) gpv"   and gpv' :: "('g, 'i, 'k) gpv"   and \<I> :: "('h, 'j) \<I>"   and \<I>' :: "('i, 'k) \<I>" assumes "rel_gpv'' A C R gpv gpv'"   and "lossless_gpv \<I> gpv"   and "rel_\<I> C R \<I> \<I>'" shows "lossless_gpv \<I>' gpv'"
abs_prod_mset: fixes A :: "'a multiset" shows "\<bar>\<Prod>\<^sub># A\<bar> = \<Prod>\<^sub># (image_mset abs A)"
nxtActive_lactive: fixes n :: "nat"   and c :: "'id"   and t :: "nat \<Rightarrow> cnf" assumes "\<exists>i\<ge>n. \<parallel>c\<parallel>\<^bsub>t i\<^esub>"   and "\<not> (\<exists>i>\<langle>c \<rightarrow> t\<rangle>\<^bsub>n\<^esub>. \<parallel>c\<parallel>\<^bsub>t i\<^esub>)" shows "\<langle>c \<rightarrow> t\<rangle>\<^bsub>n\<^esub> = \<langle>c \<and> t\<rangle>"
poly_mult_append_zero: fixes p :: "'a list"   and q :: "'a list" assumes "set p \<subseteq> carrier R"   and "set q \<subseteq> carrier R" shows "poly_mult (p @ [\<zero>]) q = local.normalize (poly_mult p q @ [\<zero>])"
zero_le_one: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "less_eq zero one"
current: fixes f :: "'a \<times> 'a \<Rightarrow> ennreal"   and \<Gamma> :: "('a, 'b) web_scheme" assumes "\<And>x. d_OUT f x \<le> weight \<Gamma> x"   and "\<And>x. d_IN f x \<le> weight \<Gamma> x"   and "\<And>x. x \<notin> A \<Gamma> \<Longrightarrow> d_OUT f x \<le> d_IN f x"   and "\<And>a. a \<in> A \<Gamma> \<Longrightarrow> d_IN f a = 0"   and "\<And>b. b \<in> B \<Gamma> \<Longrightarrow> d_OUT f b = 0"   and "\<And>e. e \<notin> \<^bold>E\<^bsub>\<Gamma>\<^esub> \<Longrightarrow> f e = 0" shows "current \<Gamma> f"
comm_ring_1: shows "OFCLASS('a bit1, comm_ring_1_class)"
abs_minus_le_zero: fixes abs :: "'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a" assumes "class.ordered_ab_group_add_abs abs plus zero minus uminus less_eq less" shows "less_eq (uminus (abs a)) zero"
is_Prog\<^sub>s\<^sub>b_simps_3: fixes p :: "'b"   and p' :: "'b"   and "is" :: "instr list" shows "is_Prog\<^sub>s\<^sub>b (Prog\<^sub>s\<^sub>b p p' is) = True"
assms: shows "P = (\<lambda>x y. f x \<le> f y)"
map_entry_def: fixes i :: "nat"   and f :: "'a \<Rightarrow> 'a"   and a :: "'a array" shows "Array.map_entry i f a = Heap_Monad.guard (\<lambda>h. i < Array.length h a) (\<lambda>h. (a, Array.update a i (f (Array.get h a ! i)) h))"
AE_alw_ev_same_loc_iff: fixes cfg :: "('s \<times> ('c \<Rightarrow> real)) cfg" assumes "cfg \<in> MDP.valid_cfg" shows "alternating cfg = alternating (absc cfg)"
hbt_balL: fixes l :: "('b \<times> nat) tree"   and r :: "('b \<times> nat) tree"   and a :: "'b" assumes "hbt l"   and "hbt r"   and "height r - m \<le> height l \<and> height l \<le> height r + m + 1" shows "hbt (balL l a r)"