hcju/isabelleps huggingface dataset:
is_REL_state_aux2: fixes c :: "nat"   and n :: "nat"   and ts :: "nat \<Rightarrow> thread"   and t :: "nat"   and tk :: "nat" assumes "B.reachable (c, n, ts)"   and "t < N"   and "ts t = REL tk" shows "\<not> is_REL_state (ts(t := INIT))"
le_infI1: fixes a :: "'b"   and x :: "'b"   and b :: "'b" assumes "a \<le> x" shows "a \<sqinter> b \<le> x"
distinct_45: fixes x4 :: "'a OCL_Types.type"   and x6 :: "'a OCL_Types.type" shows "Collection x4 \<noteq> OrderedSet x6"
cardlt: fixes n :: "nat" shows "card {i. i < n} \<le> n"
remdups_adj_rev: fixes xs :: "'g list" shows "remdups_adj (rev xs) = rev (remdups_adj xs)"
map1_rel_witness_generat: fixes A :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and C :: "'i \<Rightarrow> 'j \<Rightarrow> bool"   and R :: "'k \<Rightarrow> 'l \<Rightarrow> bool"   and x :: "('g, 'i, 'k) generat"   and y :: "('h, 'j, 'l) generat" assumes "rel_generat A C R x y" shows "map_generat fst fst fst (rel_witness_generat (x, y)) = x"
summable_in_conv_radius: fixes z :: "'a"   and f :: "nat \<Rightarrow> 'a" assumes "ereal (norm z) < conv_radius f" shows "summable (\<lambda>n. f n * z ^ n)"
circline_set_conjugate_circline: fixes H :: "circline" shows "circline_set (conjugate_circline H) = conjugate ` circline_set H"
name_perm_compose: fixes pi2 :: "(name \<times> name) list"   and pi1 :: "(name \<times> name) list"   and x :: "'a" shows "pi2 \<bullet> pi1 \<bullet> x = (pi2 \<bullet> pi1) \<bullet> pi2 \<bullet> x"
dis_rews_11: shows "is_EQ\<cdot>EQ = TT"
long_division_closed_2: fixes K :: "(('a list \<times> nat) multiset \<Rightarrow> 'a) set"   and p :: "(('a list \<times> nat) multiset \<Rightarrow> 'a) list"   and q :: "(('a list \<times> nat) multiset \<Rightarrow> 'a) list" assumes "subfield K M"   and "p \<in> carrier (K [X]\<^bsub>M\<^esub>)"   and "q \<in> carrier (K [X]\<^bsub>M\<^esub>)" shows "M.pmod p q \<in> carrier (K [X]\<^bsub>M\<^esub>)"
replicate_app_Cons_same: fixes n :: "nat"   and x :: "'c"   and xs :: "'c list" shows "replicate n x @ x # xs = x # replicate n x @ xs"
Max_V_viterbi: fixes s :: "'b"   and os :: "'a list"   and o :: "'a" assumes "s \<in> \<S>" shows "(MAX t\<in>\<S>. viterbi_prob s t os) = Max {emeasure (T (s, o)) {\<omega> \<in> space (stream_space (count_space UNIV)). \<exists>\<omega>'. \<omega> = zip as os @- \<omega>'} |as. length as = length os \<and> set as \<subseteq> \<S>}"
isomorphic_to_identity_is_equivalence: fixes \<phi> :: "'a \<Rightarrow> 'a" assumes "natural_isomorphism (\<cdot>\<^sub>S) (\<cdot>\<^sub>S) L S.map \<phi>" shows "equivalence_functor (\<cdot>\<^sub>S) (\<cdot>\<^sub>S) L"
DynPDG_path_Nil: fixes n :: "'node" assumes "valid_node n" shows "n -[]\<rightarrow>\<^sub>d* n"
\<phi>J: fixes \<psi> :: "'fmla" assumes "\<psi> \<in> fmla"   and "Fvars \<psi> = {inp}" shows "\<phi>J \<psi> \<in> fmla"
assms: shows "(\<box>(\<lambda>s. P s \<longrightarrow> (\<circle>P) s)) \<sigma>"
dvd_imp_le: fixes k :: "nat"   and n :: "nat" assumes "k dvd n"   and "0 < n" shows "k \<le> n"
fixp_lowerbound: fixes Sup :: "'g set \<Rightarrow> 'g"   and less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and f :: "'g \<Rightarrow> 'g"   and z :: "'g" assumes "class.ccpo Sup less_eq less"   and "monotone less_eq less_eq f"   and "less_eq (f z) z" shows "less_eq (ccpo.fixp Sup less_eq f) z"
reflects_equivalence_map: fixes f :: "'a" assumes "C.ide f"   and "D.equivalence_map (F f)" shows "C.equivalence_map f"
split: fixes P :: "'b \<Rightarrow> bool"   and f1 :: "'b"   and f2 :: "nat \<Rightarrow> 'b"   and nat :: "nat" shows "P (case nat of 0 \<Rightarrow> f1 | Suc x \<Rightarrow> f2 x) = ((nat = 0 \<longrightarrow> P f1) \<and> (\<forall>x2. nat = Suc x2 \<longrightarrow> P (f2 x2)))"
OMRU_inv1_def: shows "OMRU_inv1 = {s. \<forall>p. case mru_vote s p of None \<Rightarrow> True | Some (r, x) \<Rightarrow> r < opt_mru_state.next_round s}"
succ_ss_V: fixes c :: "nat \<times> nat \<Rightarrow> 'c"   and u :: "nat" shows "Graph.E c `` {u} \<subseteq> Graph.V c"
cinsert_cinter_cinsert: fixes a :: "'f"   and A :: "'f cset"   and B :: "'f cset" shows "cInt (cinsert a A) (cinsert a B) = cinsert a (cInt A B)"
Cons: fixes xs :: "expr list"   and ys :: "val list"   and y :: "val" assumes "map_val xs ys" shows "map_val (Val y # xs) (y # ys)"
simps_9: fixes i :: "int"   and c :: "int"   and e :: "Cooper.num" shows "iszlfm (Dvd i (CN 0 c e)) = (0 < c \<and> 0 < i \<and> numbound0 e)"
homotopic_paths_sym: fixes s :: "'b set"   and p :: "real \<Rightarrow> 'b"   and q :: "real \<Rightarrow> 'b" assumes "homotopic_paths s p q" shows "homotopic_paths s q p"
oghoare_Parallel: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "'d \<Rightarrow> ('e, 'd, 'f) ann list option"   and F :: "'f set"   and P :: "('e, 'd, 'f) ann"   and cs :: "('e, 'd, 'f) com list"   and Q :: "'e set"   and A :: "'e set" assumes "\<Gamma>, \<Theta> \<turnstile>\<^bsub>/F\<^esub> P (Parallel cs) Q, A" shows "\<exists>as. P = AnnPar as \<and> length as = length cs \<and> \<Inter> (set (map (\<lambda>a. fst (snd a)) as)) \<subseteq> Q \<and> \<Union> (set (map (\<lambda>a. snd (snd a)) as)) \<subseteq> A \<and> (\<forall>i<length cs. \<exists>Q' A'. \<Gamma>, \<Theta> \<turnstile>\<^bsub>/F\<^esub> (fst (as ! i)) (cs ! i) Q', A' \<and> Q' \<subseteq> fst (snd (as ! i)) \<and> A' \<subseteq> snd (snd (as ! i))) \<and> interfree \<Gamma> \<Theta> F as cs"
sel_8: fixes x11 :: "'c"   and x12 :: "'c"   and x13 :: "'c"   and x14 :: "'a" shows "pac_res (Add x11 x12 x13 x14) = x14"
phiDefNodes_aux_pruned: fixes m :: "'node"   and g :: "'g"   and v :: "'var"   and n :: "'node" assumes "m \<in> phiDefNodes_aux g v (\<alpha>n g) n"   and "n \<in> set (\<alpha>n g)"   and "v \<in> uses g n" shows "braun_ssa.liveVal g (v, m, PhiDef)"
M6: fixes F :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and G :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and v :: "'b \<Rightarrow> 'c" shows "\<turnstile> \<box>[F \<and> G]_v \<longrightarrow> \<box>[F]_v \<and> \<box>[G]_v"
integral_measure_pmf_real: fixes A :: "'c set"   and M :: "'c pmf"   and f :: "'c \<Rightarrow> real" assumes "finite A"   and "\<And>a. \<lbrakk>a \<in> set_pmf M; f a \<noteq> 0\<rbrakk> \<Longrightarrow> a \<in> A" shows "measure_pmf.expectation M f = (\<Sum>a\<in>A. f a * pmf M a)"
finite_channels: shows "finite {i. \<exists>p q. channel i = Some (p, q)}"
vimage_algebra_vimage_algebra_eq: fixes f :: "'c \<Rightarrow> 'd"   and X :: "'c set"   and Y :: "'d set"   and g :: "'d \<Rightarrow> 'e"   and M :: "'e measure" assumes "f \<in> X \<rightarrow> Y"   and "g \<in> Y \<rightarrow> space M" shows "vimage_algebra X f (vimage_algebra Y g M) = vimage_algebra X (\<lambda>x. g (f x)) M"
False: shows "\<not> (n = 0 \<or> k = 0)"
ccBind_cong: fixes cccexp1 :: "exp \<Rightarrow> Arity \<rightarrow> CoCalls"   and e :: "exp"   and cccexp2 :: "exp \<Rightarrow> Arity \<rightarrow> CoCalls"   and x :: "Vars.var" assumes "cccexp1 e = cccexp2 e" shows "CoCallAnalysis.ccBind cccexp1 x e = CoCallAnalysis.ccBind cccexp2 x e"
2: fixes k :: "nat"   and m :: "'b \<Rightarrow> nat option" shows "dom (snd (fold (\<lambda>x (k, m). (Suc k, m(x \<mapsto> k))) tsl' (k, m))) = dom m \<union> set tsl'"
deletion_reduce: fixes ss :: "'w list" assumes "ss \<in> lists S" shows "\<exists>ts. ts \<in> ssubseqs ss \<inter> reduced_words_for S (sum_list ss)"
project_stable_mono: fixes D :: "'c set"   and C :: "'c set"   and F :: "'c program"   and A :: "'a set" assumes "D \<subseteq> C"   and "project h C F \<in> stable A" shows "project h D F \<in> stable A"
induct: fixes x1 :: "'b \<Rightarrow> ('c, 'd) trm"   and x2 :: "('c + 'b, 'd) trm"   and P :: "('b \<Rightarrow> ('c, 'd) trm) \<Rightarrow> ('c + 'b, 'd) trm \<Rightarrow> bool" assumes "TadmitFO x1 x2"   and "\<And>\<sigma> \<theta>. \<lbrakk>TadmitFFO \<sigma> \<theta>; NTUadmit \<sigma> \<theta> UNIV; dfree (TsubstFO \<theta> \<sigma>)\<rbrakk> \<Longrightarrow> P \<sigma> (Differential \<theta>)"   and "\<And>\<sigma> args f. \<lbrakk>\<And>i. TadmitFO \<sigma> (args i); \<And>i. P \<sigma> (args i)\<rbrakk> \<Longrightarrow> P \<sigma> ($f f args)"   and "\<And>\<sigma> \<theta>1 \<theta>2. \<lbrakk>TadmitFO \<sigma> \<theta>1; P \<sigma> \<theta>1; TadmitFO \<sigma> \<theta>2; P \<sigma> \<theta>2\<rbrakk> \<Longrightarrow> P \<sigma> (Plus \<theta>1 \<theta>2)"   and "\<And>\<sigma> \<theta>1 \<theta>2. \<lbrakk>TadmitFO \<sigma> \<theta>1; P \<sigma> \<theta>1; TadmitFO \<sigma> \<theta>2; P \<sigma> \<theta>2\<rbrakk> \<Longrightarrow> P \<sigma> (Times \<theta>1 \<theta>2)"   and "\<And>\<sigma> x. P \<sigma> ($' x)"   and "\<And>\<sigma> x. P \<sigma> (Var x)"   and "\<And>\<sigma> r. P \<sigma> (Const r)" shows "P x1 x2"
prob_of_id_def: fixes m :: "'b Monomorphic_Monad.id" shows "prob_of_id m = return_pmf (extract m)"
z: shows "\<forall>x\<in>S. x \<le> z"
13: shows "vector r \<and> injective r \<and> v\<^sup>T = r\<^sup>T * t\<^sup>\<star>"
lookup_action_Some_in_\<delta>: fixes op :: "(nat, nat) sas_plus_operator"   and ast_op :: "char list \<times> (nat \<times> nat) list \<times> ((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list \<times> nat" assumes "lookup_action op = Some ast_op" shows "ast_op \<in> set ast\<delta>"
lcm_eq_0_iff: fixes one :: "'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize" shows "(lcm a b = zero) = (a = zero \<or> b = zero)"
dual_inf: fixes x :: "'b"   and y :: "'b"   and sup :: "'b" shows "is_inf (dual x) (dual y) (dual sup) = is_sup x y sup"
Preds_mkinterp: fixes C :: "char \<Rightarrow> real"   and F :: "char \<Rightarrow> real \<Rightarrow> real"   and P :: "char \<Rightarrow> real \<Rightarrow> bool"   and G :: "char \<Rightarrow> (variable \<Rightarrow> real) set \<Rightarrow> (variable \<Rightarrow> real) set" shows "Preds (mkinterp (C, F, P, G)) = P"
UP_l_one: fixes p :: "nat \<Rightarrow> 'a" assumes "p \<in> carrier P" shows "\<one>\<^bsub>P\<^esub> \<otimes>\<^bsub>P\<^esub> p = p"
138: fixes x :: "'a" shows "- x \<sqinter> - x = - x"
lookupNotIn: fixes x :: "'e"   and m :: "'e \<Rightarrow>\<^sub>0 'f" assumes "x \<notin> keys m" shows "lookup m x = (0::'f)"
SPEC_rule_param2: fixes f :: "'d \<Rightarrow> ('b \<times> 'c) nres"   and x :: "'d"   and P :: "'d \<Rightarrow> 'b \<times> 'c \<Rightarrow> bool"   and x' :: "'d" assumes "f x \<le> SPEC (P x)"   and "\<And>r1 r2. P x (r1, r2) \<Longrightarrow> P x' (r1, r2)" shows "f x \<le> SPEC (P x')"
cs_sub_cas: shows "\<Sum>\<^sub># (mset Cs) \<subseteq># \<Sum>\<^sub># (mset CAs)"
Red1TryCatch: fixes h :: "'heap"   and a :: "'addr"   and D :: "String.literal"   and P :: "'addr J1_prog"   and C :: "String.literal"   and V :: "nat"   and x :: "'addr val list"   and uf :: "bool"   and t :: "'thread_id"   and e2 :: "'addr expr1" assumes "typeof_addr h a = \<lfloor>Class_type D\<rfloor>"   and "P \<turnstile> D \<preceq>\<^sup>* C"   and "V < length x" shows "uf,P,t \<turnstile>1 \<langle>try Throw a catch(C V) e2,(h, x)\<rangle> -\<lbrace>\<rbrace>\<rightarrow> \<langle>{V:Class C=None; e2},(h, x[V := Addr a])\<rangle>"
card_insert_disjoint: fixes A :: "'c set"   and x :: "'c" assumes "finite A"   and "x \<notin> A" shows "card (Set.insert x A) = Suc (card A)"
orthogonal_commute: fixes x :: "'b"   and y :: "'b" shows "orthogonal x y = orthogonal y x"
transpose_matrix_vector: fixes A :: "(('b, 'c) vec, 'd) vec"   and x :: "('b, 'd) vec" shows "Finite_Cartesian_Product.transpose A *v x = x v* A"
arrow_of_spans_axioms: shows "arrow_of_spans (\<cdot>) \<lparr>Chn = fgh.chine_assoc, Dom = Dom ((f \<star> g) \<star> h), Cod = Cod (f \<star> g \<star> h)\<rparr>"
map_of_map_to_set: fixes l :: "('f \<times> 'g) list"   and m :: "'f \<Rightarrow> 'g option" assumes "distinct (map fst l)" shows "(map_of l = m) = (set l = map_to_set m)"
IllegalMonitorState_def: shows "IllegalMonitorState = STR ''java/lang/IllegalMonitorStateException''"
univ_poly_is_domain: fixes R :: "('c, 'd) ring_scheme"   and K :: "'c set" assumes "domain R"   and "subring K R" shows "domain (K [X]\<^bsub>R\<^esub>)"
False: shows "\<langle>vcard xs', x\<rangle> \<notin>\<^sub>\<circ> xs'"
subcls1_single_valued: fixes P :: "'b prog" shows "single_valued (subcls1 P)"
prod_rev_ffact_nat: fixes n :: "nat"   and m :: "nat" assumes "n \<le> m" shows "prod ((+) (m - n)) {1..n} = ffact n m"
map_project_def: fixes f :: "'d \<Rightarrow> 'c option"   and A :: "'d set" shows "List.map_project f A = {b. \<exists>a\<in>A. f a = Some b}"
LocalSecrets_L1: fixes key :: "Keys"   and P :: "specID" assumes "kKS key \<in> LocalSecrets P"   and "kKS key \<notin> \<Union> (LocalSecrets ` subcomponents P)" shows "kKS key \<notin> specKeysSecrets P"
id_Rel_components_1: fixes A :: "V" shows "id_Rel A\<lparr>ArrVal\<rparr> = vid_on A"
append_set_prefix: fixes xs :: "'e list"   and T :: "'e list set"   and X :: "'e set" assumes "xs \<in> append_set T X" shows "butlast xs \<in> T"
startsby_power: fixes a :: "'a fps"   and v :: "'a"   and n :: "nat" assumes "fps_nth a 0 = v" shows "fps_nth (a ^ n) 0 = v ^ n"
Comm2: fixes P :: "pi"   and a :: "name"   and b :: "name"   and P' :: "pi"   and Q :: "pi"   and Q' :: "pi" assumes "P \<Longrightarrow>a[b] \<prec> P'"   and "Q \<Longrightarrow>a<b> \<prec> Q'" shows "P \<parallel> Q \<Longrightarrow>\<tau> \<prec> P' \<parallel> Q'"
subset_antisym: fixes A :: "'d set"   and B :: "'d set" assumes "A \<subseteq> B"   and "B \<subseteq> A" shows "A = B"
divisionTr4: fixes g :: "'a"   and f :: "'a" assumes "Corps S"   and "g \<in> carrier R"   and "g \<noteq> \<zero>"   and "0 < deg_n R S X g"   and "f \<in> carrier R" shows "\<exists>q\<in>carrier R. f = q \<cdot>\<^sub>r g \<or> (\<exists>r\<in>carrier R. r \<noteq> \<zero> \<and> f = q \<cdot>\<^sub>r g \<plusminus> r \<and> deg_n R S X r < deg_n R S X g)"
compositor_is_natural_isomorphism: shows "natural_isomorphism VV.comp (\<bullet>) HoSPN_SPN.map SPNoH.map \<Phi>.map"
g_def: shows "g \<equiv> \<lambda>k. moebius_mu k * h k"
span_mono: fixes scale :: "'d \<Rightarrow> 'e \<Rightarrow> 'e"   and A :: "'e set"   and B :: "'e set" assumes "module scale"   and "A \<subseteq> B" shows "module.span scale A \<subseteq> module.span scale B"
conj_zero_left: fixes x :: "'c" shows "(0::'c) AND x = (0::'c)"
dim_col_of_dagger: fixes M :: "complex mat" shows "dim_col M\<^sup>\<dagger> = dim_row M"
ordLeq_refl: fixes r :: "('d \<times> 'd) set" assumes "Card_order r" shows "ordLeq3 r r"
iNext_iEx_iff: fixes t0 :: "nat"   and I :: "nat set"   and P :: "nat \<Rightarrow> bool" shows "iNext t0 I P = (\<diamond> t [\<dots>0] \<oplus> inext t0 I. P t)"
assms: shows "t \<in> ik\<^sub>s\<^sub>t S"
l_set_shadow_root_get_child_nodes_axioms: shows "l_set_shadow_root_get_child_nodes type_wf set_shadow_root set_shadow_root_locs known_ptr get_child_nodes get_child_nodes_locs"
real_hyp2_C_cosh_dist: fixes a :: "hyp2"   and b :: "hyp2"   and c :: "hyp2"   and d :: "hyp2" assumes "a b \<congruent>\<^sub>K c d" shows "cosh_dist (Rep_hyp2 a) (Rep_hyp2 b) = cosh_dist (Rep_hyp2 c) (Rep_hyp2 d)"
prems_1: fixes i :: "nat" assumes "i \<noteq> j"   and "i < length (A # As)" shows "(A # As) ! i = tensor0 ds"
measure_space_inter: fixes s :: "nat set set set"   and t :: "nat set set set" assumes "s \<in> events"   and "t \<in> events"   and "prob t = prob (space P)" shows "prob (s \<inter> t) = prob s"
le_less_trans: fixes D :: "('e, 'f) Order_scheme"   and a :: "'e"   and b :: "'e"   and c :: "'e" assumes "Order D"   and "a \<in> carrier D"   and "b \<in> carrier D"   and "c \<in> carrier D"   and "a \<preceq>\<^bsub>D\<^esub> b"   and "b \<prec>\<^bsub>D\<^esub> c" shows "a \<prec>\<^bsub>D\<^esub> c"
normF: shows "1 \<le> \<parallel>F\<parallel>\<^sup>2"
IH: fixes s :: "('a, 'b) fmap"   and PROB :: "(('a, 'b) fmap \<times> ('a, 'b) fmap) set" assumes "s \<in> valid_states PROB"   and "as \<in> valid_plans PROB" shows "last ` state_set (state_list s as) \<subseteq> valid_states PROB"
open_dist: fixes S :: "'c set" shows "open S = (\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S)"
rel_flip: fixes R1 :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and R2 :: "'d \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "('e, 'a, 'c) sp\<^sub>\<mu>"   and b :: "('e, 'b, 'd) sp\<^sub>\<mu>" shows "rel_sp\<^sub>\<mu> R1\<inverse>\<inverse> R2\<inverse>\<inverse> a b = rel_sp\<^sub>\<mu> R1 R2 b a"
pre_Less: fixes x :: "'a language"   and y :: "'a language"   and R :: "'a language \<Rightarrow> 'a language \<Rightarrow> bool" assumes "x \<le> y" shows "Plus_Times_pre_cong R x y"
dim_gen_eigenspace_def: fixes A :: "'a mat"   and ev :: "'a"   and k :: "nat" shows "local.dim_gen_eigenspace A ev k = kernel_dim (char_matrix A ev ^\<^sub>m k)"
exhaust: fixes y :: "net_tree" obtains x11 :: "nat" and x12 :: "nat set" where "y = \<langle>x11; x12\<rangle>"   | x21 :: "net_tree" and x22 :: "net_tree" where "y = x21 \<parallel> x22"
assms_3: fixes i :: "'a" assumes "i \<in> A"   and "i \<noteq> j" shows "g i = (0::'b)"
wit_LCons2: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b"   and xs :: "'b llist" shows "wit R (LCons x (LCons y xs)) = LCons x (prepend (pick R x y) (wit R (LCons y xs)))"
extremum_unique: fixes a :: "'a multiset" shows "(a \<subseteq># {#}) = (a = {#})"
simps_2: fixes x21 :: "'a"   and x22 :: "'a"   and x23 :: "'b" shows "Null \<noteq> MkArr x21 x22 x23"
leg_simps_1: shows "C.arr Cod\<mu>_Cod\<nu>.leg0"
prm_disagreement_ext: fixes x :: "'b"   and \<pi> :: "'b prm"   and \<sigma> :: "'b prm" shows "x \<in> ds \<pi> \<sigma> \<equiv> \<pi> $ x \<noteq> \<sigma> $ x"
l_new_document_get_disconnected_nodes_def: fixes get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_new_document_get_disconnected_nodes get_disconnected_nodes get_disconnected_nodes_locs \<equiv> (\<forall>new_document_ptr ptr' h h' r. new_document_ptr \<noteq> ptr' \<longrightarrow> h \<turnstile> new_document \<rightarrow>\<^sub>r new_document_ptr \<longrightarrow> h \<turnstile> new_document \<rightarrow>\<^sub>h h' \<longrightarrow> r \<in> get_disconnected_nodes_locs ptr' \<longrightarrow> r h h') \<and> (\<forall>h new_document_ptr h'. h \<turnstile> new_document \<rightarrow>\<^sub>r new_document_ptr \<longrightarrow> h \<turnstile> new_document \<rightarrow>\<^sub>h h' \<longrightarrow> h' \<turnstile> get_disconnected_nodes new_document_ptr \<rightarrow>\<^sub>r [])"
iwlsFSwSTR_iterSTR_ipresOp: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) model" assumes "iwlsFSwSTR MOD" shows "ipresOp (iterSTR MOD) (iterAbsSTR MOD) MOD"
length_awalk_verts: fixes u :: "'a"   and p :: "'b list" shows "length (awalk_verts u p) = Suc (length p)"
that: fixes a :: "V"   and b :: "V" assumes "x = \<langle>a, b\<rangle>"   and "a \<in>\<^sub>\<circ> A"   and "b \<in>\<^sub>\<circ> A"   and "\<langle>a, b\<rangle> \<in>\<^sub>\<circ> r" shows "thesis"
poly_at_bot_or_top_at_bot: fixes p :: "real poly" assumes "1 \<le> degree p"   and "lead_coeff p < 0" shows "filterlim (poly p) (if even (degree p) then at_bot else at_top) at_bot"
implies_below_eq: fixes y :: "'a"   and x :: "'a" shows "y \<sqinter> (x \<leadsto> y) = y"
timpl_closure'_def: fixes TI :: "('c set \<times> 'c set) set" shows "timpl_closure' TI \<equiv> (timpl_closure'_step TI)\<^sup>*"
lincomb_Cons: fixes r :: "'r"   and rs :: "'r list"   and m :: "'m"   and ms :: "'m list" shows "(r # rs) \<bullet>\<cdot> (m # ms) = r \<cdot> m + rs \<bullet>\<cdot> ms"
discI_1: fixes resumption :: "('f, 'g, 'h) Resumption.resumption"   and x1 :: "'f option" assumes "resumption = Resumption.resumption.Done x1" shows "Resumption.resumption.is_Done resumption"
dvdm_imp_div_mod: fixes u :: "int poly"   and g :: "int poly" assumes "p.dvdm u g" shows "\<exists>q r. g = q * u + smult k r"
bind_rpv_apply: fixes rpv :: "('g, 'e, 'f) rpv"   and f :: "'g \<Rightarrow> ('d, 'e, 'f) gpv"   and input :: "'f" shows "(rpv \<bind> f) input = rpv input \<bind> f"
finite_set_antichain: fixes A :: "'a antichain" shows "finite (set_antichain A)"
divide_leq: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and scaleRat :: "rat \<Rightarrow> 'a \<Rightarrow> 'a"   and c :: "rat"   and a :: "'a"   and b :: "'a" assumes "class.linordered_rational_vector less_eq less plus zero minus uminus scaleRat"   and "less_eq (scaleRat c a) b"   and "0 < c" shows "less_eq a (scaleRat (1 / c) b)"
simps_2: fixes \<pi> :: "char list \<Rightarrow> com option"   and x :: "char list"   and y :: "char list"   and s :: "char list \<Rightarrow> int \<Rightarrow> int" shows "small_step \<pi> (x[] ::= y, s) = Some (SKIP, s(x := s y))"
singleton_iff: fixes x :: "'a" shows "singleton x = (x \<noteq> 0 \<and> x\<^sup>T ; 1 ; x + x ; 1 ; x\<^sup>T \<le> 1')"
intros_2: fixes P2 :: "'a \<Rightarrow> bool"   and b :: "'a"   and P1 :: "'b \<Rightarrow> bool" assumes "P2 b" shows "pred_sum P1 P2 (Inr b)"
len_eq: shows "length (fst gc\<^sub>A) = length cms"
list_all2_takeI: fixes P :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and xs :: "'c list"   and ys :: "'d list"   and n :: "nat" assumes "list_all2 P xs ys" shows "list_all2 P (take n xs) (take n ys)"
supt_var: fixes x :: "'e"   and u :: "('d, 'e) Term.term" shows "(Var x, u) \<notin> {\<rhd>}"
lookup_map_of: fixes t :: "('c \<times> 'd) bro"   and h :: "nat"   and x :: "'c" assumes "t \<in> T h"   and "sorted1 (inorder t)" shows "lookup t x = map_of (inorder t) x"
intro: fixes lift :: "'a \<Rightarrow> 'b" assumes "\<And>a b. lift (a + b) = lift a + lift b"   and "\<And>a b. lift (a * b) = lift a * lift b"   and "lift (0::'a) = (0::'b)"   and "lift (1::'a) = (1::'b)" shows "pfd_homomorphism lift"
assms_1: shows "valid_unMultigraph.is_Eulerian_trail (del_unEdge v0 w v1 G) v1 ps v2"
unit_closed: shows "zero_local_ring_at \<in> carrier_local_ring_at"
transC_induct: fixes tval :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and aval :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and x1a :: "('a, 'c) com"   and x1b :: "'b"   and x2a :: "('a, 'c) com"   and x2b :: "'b"   and P1 :: "('a, 'c) com \<Rightarrow> 'b \<Rightarrow> ('a, 'c) com \<Rightarrow> 'b \<Rightarrow> bool" assumes "PL.transC tval aval (x1a, x1b) (x2a, x2b)"   and "\<And>c1 s c1' s' c2. \<lbrakk>PL.transC tval aval (c1, s) (c1', s'); P1 c1 s c1' s'\<rbrakk> \<Longrightarrow> P1 (c1 ;; c2) s (c1' ;; c2) s'"   and "\<And>c1 s s' c2. PL.transT tval aval (c1, s) s' \<Longrightarrow> P1 (c1 ;; c2) s c2 s'"   and "\<And>tst s c1 c2. tval tst s \<Longrightarrow> P1 (if tst then c1 else c2) s c1 s"   and "\<And>tst s c1 c2. \<not> tval tst s \<Longrightarrow> P1 (if tst then c1 else c2) s c2 s"   and "\<And>tst s c. tval tst s \<Longrightarrow> P1 (while tst do c) s (c ;; while tst do c) s"   and "\<And>c1 s c1' s' c2. \<lbrakk>PL.transC tval aval (c1, s) (c1', s'); P1 c1 s c1' s'\<rbrakk> \<Longrightarrow> P1 (c1 | c2) s (c1' | c2) s'"   and "\<And>c2 s c2' s' c1. \<lbrakk>PL.transC tval aval (c2, s) (c2', s'); P1 c2 s c2' s'\<rbrakk> \<Longrightarrow> P1 (c1 | c2) s (c1 | c2') s'"   and "\<And>c1 s s' c2. PL.transT tval aval (c1, s) s' \<Longrightarrow> P1 (c1 | c2) s c2 s'"   and "\<And>c2 s s' c1. PL.transT tval aval (c2, s) s' \<Longrightarrow> P1 (c1 | c2) s c1 s'"   and "\<And>c s. True"   and "\<And>c s c' s' c'' s''. \<lbrakk>PL.MtransC tval aval (c, s) (c', s'); True; PL.transC tval aval (c', s') (c'', s''); P1 c' s' c'' s''\<rbrakk> \<Longrightarrow> True" shows "P1 x1a x1b x2a x2b"