hcju/isabelleps huggingface dataset:
sum_SDG_proc_entry_exit_cdep: fixes a :: "'edge"   and Q :: "('var \<Rightarrow> 'val option) \<times> 'ret \<Rightarrow> bool"   and r :: "'ret"   and p :: "'pname"   and fs :: "(('var \<Rightarrow> 'val option) \<Rightarrow> 'val option) list"   and n :: "'node SDG_node"   and a' :: "'edge"   and n' :: "'node SDG_node" assumes "valid_edge a"   and "kind a = Q:r\<hookrightarrow>\<^bsub>p\<^esub>fs"   and "n = CFG_node (targetnode a)"   and "a' \<in> get_return_edges a"   and "n' = CFG_node (sourcenode a')" shows "n s\<longrightarrow>\<^bsub>cd\<^esub> n'"
Arr_comp: fixes f :: "('obj, 'obj) concrete_category.arr"   and g :: "('obj, 'obj) concrete_category.arr" assumes "arr f"   and "arr g"   and "Dom g = Cod f" shows "g \<cdot> f \<noteq> Null \<and> Dom (g \<cdot> f) \<in> Obj \<and> Cod (g \<cdot> f) \<in> Obj \<and> Map (g \<cdot> f) \<in> (if Dom (g \<cdot> f) = Cod (g \<cdot> f) then {Dom (g \<cdot> f)} else {})"
a_coset_add_inv2: fixes M :: "'a set"   and x :: "'a"   and y :: "'a" assumes "M +> x = M +> y"   and "x \<in> carrier G"   and "y \<in> carrier G"   and "M \<subseteq> carrier G" shows "M +> x \<oplus> \<ominus> y = M"
fold_maybe_or_true_base_never_false: fixes l :: "trilean list" shows "fold (\<or>?) l true \<noteq> false"
set_bd_3: fixes x :: "('l, 'm, 'n, 'k) G" shows "|Gset3 x| \<le>o bd_G"
adm_below: fixes u :: "'b \<Rightarrow> 'c"   and v :: "'b \<Rightarrow> 'c" assumes "cont u"   and "cont v" shows "adm (\<lambda>x. u x \<sqsubseteq> v x)"
neville_aitken: fixes n :: "nat"   and x :: "nat \<Rightarrow> 'b"   and j :: "nat"   and f :: "nat \<Rightarrow> 'b" assumes "\<And>i j. \<lbrakk>i < j; j \<le> n\<rbrakk> \<Longrightarrow> x i \<noteq> x j"   and "j \<le> n" shows "poly (neville_aitken x f n) (x j) = f j"
par_beta_cd: fixes s :: "dB"   and t :: "dB" assumes "s => t" shows "t => cd s"
Qigen_def: fixes Q :: "'v \<Rightarrow> enat"   and \<pi> :: "'v \<Rightarrow> 'a option"   and u :: "'v"   and adjs :: "('v \<times> 'b) list"   and v :: "'v" shows "Qigen Q \<pi> u adjs v = (if v \<notin> fst ` set adjs then Q v else Qinter Q \<pi> u v)"
B: shows "{msg. msg \<in># msgs init \<and> (\<exists>v. <\<bottom>, outM v> = msg)} = {}"
valid_store_sops_nth_update: fixes i :: "nat"   and "is" :: "instr list"   and p :: "'a"   and xs :: "nat \<Rightarrow> nat option"   and sb :: "'a memref list"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option" assumes "i < length ts"   and "\<forall>sop\<in>store_sops is. valid_sop sop" shows "valid_store_sops (ts[i := (p, is, xs, sb, \<D>, \<O>, \<R>)])"
intruder_deduct_induct: fixes M :: "('fun, 'var) Term.term set"   and t :: "('fun, 'var) Term.term"   and P :: "('fun, 'var) Term.term set \<Rightarrow> ('fun, 'var) Term.term \<Rightarrow> bool" assumes "M \<turnstile> t"   and "\<And>t. t \<in> M \<Longrightarrow> P M t"   and "\<And>T f. \<lbrakk>length T = arity f; public f; \<And>t. t \<in> set T \<Longrightarrow> M \<turnstile> t; \<And>t. t \<in> set T \<Longrightarrow> P M t\<rbrakk> \<Longrightarrow> P M (Fun f T)"   and "\<And>t K T t\<^sub>i. \<lbrakk>M \<turnstile> t; P M t; Ana t = (K, T); \<And>k. k \<in> set K \<Longrightarrow> M \<turnstile> k; \<And>k. k \<in> set K \<Longrightarrow> P M k; t\<^sub>i \<in> set T\<rbrakk> \<Longrightarrow> P M t\<^sub>i" shows "P M t"
Diagonalize_Diag: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and t :: "'a local.term" assumes "monoidal_language C"   and "monoidal_language.Diag C t" shows "monoidal_language.Diagonalize C t = t"
assms_2: fixes res :: "('a, 'b) resource"   and q :: "'a"   and r :: "'b"   and res' :: "('a, 'b) resource" assumes "X res"   and "q \<in> outs_\<I> \<I>"   and "(r, res') \<in> set_spmf (run_resource res q)" shows "r \<in> responses_\<I> \<I> q \<and> (X res' \<or> \<I> \<turnstile>res res' \<surd>)"
broadcasts_unique: fixes i :: "nat"   and j :: "nat"   and m :: "'a" assumes "i \<noteq> j"   and "Broadcast m \<in> set (history i)" shows "Broadcast m \<notin> set (history j)"
that: fixes i :: "nat" assumes "i < max_Var_floatariths fas" shows "vs ! i = ws ! i"
prefix_length_less: fixes xs :: "'e list"   and ys :: "'e list" assumes "strict_prefix xs ys" shows "length xs < length ys"
FF_def: shows "UP.FF \<equiv> \<lambda>\<mu>\<nu>. if VV.arr \<mu>\<nu> then (S.UP (fst \<mu>\<nu>), S.UP (snd \<mu>\<nu>)) else S.VV.null"
adjacentTo_def: fixes v :: "'vertex"   and p :: "('form, 'var) antecedent" shows "adjacentTo v p = (SOME ps. (ps, v, p) \<in> edges)"
assms_3: shows "integral\<^sup>N M h \<noteq> \<infinity>"
fps_deriv_lr_inverse_2: fixes x :: "'a"   and f :: "'a fps"   and y :: "'a" assumes "x * fps_nth f 0 = (1::'a)"   and "fps_nth f 0 * y = (1::'a)" shows "fps_deriv (fps_right_inverse f y) = - fps_right_inverse f y * fps_deriv f * fps_right_inverse f y"
assms: shows "are_operators_interfering op\<^sub>1 op\<^sub>2"
deliver_is_valid: fixes m :: "'a \<times> ('a, 'b) operation"   and i :: "nat" assumes "Deliver m \<in> set (history i)" shows "\<exists>j pre state. pre @ [Broadcast m] prefix of j \<and> apply_operations pre = Some state \<and> valid_behaviours state m"
store_barrier_pending_mod_def: fixes w :: "bool"   and s :: "'b sparc_state" shows "store_barrier_pending_mod w s \<equiv> s\<lparr>state_var := write_store_barrier_pending w (state_var s)\<rparr>"
iT_Div_0: fixes I :: "nat set" assumes "I \<noteq> {}" shows "I \<oslash> 0 = [\<dots>0]"
restrict_map_self: fixes m :: "'a \<Rightarrow> 'b option" shows "m |` dom m = m"
dg_Cod_vrange: shows "\<R>\<^sub>\<circ> (\<CC>\<lparr>Cod\<rparr>) \<subseteq>\<^sub>\<circ> \<CC>\<lparr>Obj\<rparr>"
remove_code_2: fixes x :: "'a"   and xs :: "'a list" shows "Set.remove x (List.coset xs) = List.coset (List.insert x xs)"
assms_2: shows "es \<in> set (sort_by_pri min_pri edges)"
locally_finite_in_def: fixes X :: "'b topology"   and \<A> :: "'b set set" shows "locally_finite_in X \<A> = (\<Union> \<A> \<subseteq> topspace X \<and> (\<forall>x\<in>topspace X. \<exists>V. openin X V \<and> x \<in> V \<and> finite {U \<in> \<A>. U \<inter> V \<noteq> {}}))"
strict_mono_on_subset: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and B :: "'a set" assumes "SetInterval2.strict_mono_on f A"   and "B \<subseteq> A" shows "SetInterval2.strict_mono_on f B"
l_lchild_eq_l_if_well_shaped: fixes t :: "'b tree" assumes "well_shaped t"   and "l t < r t" shows "l (lchild t) = l t"
simps: fixes pi :: "('f \<times> 'f) list"   and x :: "'d"   and y :: "'e" shows "pi \<bullet> (x, y) = (pi \<bullet> x, pi \<bullet> y)"
simps: fixes a1 :: "(name \<times> ty) list"   and a2 :: "lam"   and a3 :: "ty" shows "a1 \<turnstile> a2 : a3 = ((\<exists>\<Gamma> x T. a1 = \<Gamma> \<and> a2 = Var x \<and> a3 = T \<and> valid \<Gamma> \<and> (x, T) \<in> set \<Gamma>) \<or> (\<exists>\<Gamma> t1 T1 T2 t2. a1 = \<Gamma> \<and> a2 = App t1 t2 \<and> a3 = T2 \<and> \<Gamma> \<turnstile> t1 : T1 \<rightarrow> T2 \<and> \<Gamma> \<turnstile> t2 : T1) \<or> (\<exists>x \<Gamma> T1 t T2. a1 = \<Gamma> \<and> a2 = Lam [x].t \<and> a3 = T1 \<rightarrow> T2 \<and> x \<sharp> \<Gamma> \<and> (x, T1) # \<Gamma> \<turnstile> t : T2))"
lower_closure: shows "\<pi>\<^sup>*\<^bsub>F\<^esub> \<in> carrier \<X>\<^bsub>F\<^esub> \<rightarrow> carrier \<Y>\<^bsub>F\<^esub>"
R_linear: fixes x :: "real"   and y :: "real" assumes "\<not> R x y" shows "R y x"
simps_3: fixes indis :: "('a \<times> 'a) set"   and tval :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and aval :: "'c \<Rightarrow> 'a \<Rightarrow> 'a"   and tst :: "'b"   and c1 :: "('b, 'c) com"   and c2 :: "('b, 'c) com" assumes "PL_Indis indis" shows "PL_Indis.SC_ZObis tval aval indis (if tst then c1 else c2) = (if PL_Indis.compatTst tval indis tst then PL_Indis.SC_ZObis tval aval indis c1 \<and> PL_Indis.SC_ZObis tval aval indis c2 else PL_Indis.SC_discr aval indis (if tst then c1 else c2) \<or> PL_Indis.SC_siso tval aval indis (if tst then c1 else c2))"
atU_s_decomp: fixes U :: "'a set"   and s :: "'a list"   and s' :: "'a list" shows "atU_s U (s @ s') = (atU_s U s \<or> s = [] \<and> atU_s U s')"
eqOn_update: fixes S :: "'a set"   and s0 :: "'a"   and \<delta> :: "'a \<Rightarrow> 'a set"   and AP :: "'b set"   and L :: "'a \<Rightarrow> 'b set"   and AP' :: "'b set"   and P :: "pvar set"   and p :: "pvar"   and env :: "pvar \<Rightarrow> nat"   and \<pi>l :: "('a \<times> 'b set) stream list"   and env1 :: "pvar \<Rightarrow> nat"   and \<pi>l1 :: "('a \<times> 'b set) stream list"   and \<pi> :: "('a \<times> 'b set) stream" assumes "Shallow S s0 \<delta> AP L AP'"   and "cpt (P - {p}) env \<pi>l"   and "cpt (P - {p}) env1 \<pi>l1" shows "Shallow.eqOn P (env(p := length \<pi>l)) (\<pi>l @ [\<pi>]) (env1(p := length \<pi>l1)) (\<pi>l1 @ [\<pi>]) = Shallow.eqOn (P - {p}) env \<pi>l env1 \<pi>l1"
that: fixes c :: "recf" assumes "recfn 1 c"   and "Partial_Recursive.total c"   and "\<forall>i x. eval psi [i, x] = eval r_phi [the (eval c [i]), x]" shows "thesis"
onll_def: fixes \<Gamma> :: "'k \<Rightarrow> ('i, 'j, 'k, 'l) seqp"   and P :: "('h \<times> 'l) \<times> 'm \<times> 'h \<times> 'l \<Rightarrow> bool" shows "onll \<Gamma> P \<equiv> \<lambda>((\<xi>, p), a, \<xi>', p'). \<forall>l\<in>labels \<Gamma> p. \<forall>l'\<in>labels \<Gamma> p'. P ((\<xi>, l), a, \<xi>', l')"
gen_akiyama_tanigawa_n_0: fixes f :: "nat \<Rightarrow> real"   and n :: "nat" shows "gen_akiyama_tanigawa f n 0 = (\<Sum>k\<le>n. (- 1) ^ k * fact k * real (Stirling (Suc n) (Suc k)) * f k)"
map_iterator_linord_foldli_conv: fixes iti :: "('c \<Rightarrow> bool) \<Rightarrow> ('a \<times> 'd \<Rightarrow> 'c \<Rightarrow> 'c) \<Rightarrow> 'c \<Rightarrow> 'c"   and m :: "'a \<Rightarrow> 'd option" shows "map_iterator_linord iti m = (\<exists>l0. distinct (map fst l0) \<and> m = map_of l0 \<and> sorted (map fst l0) \<and> iti = foldli l0)"
embedded_serial_strips_plan_operator_structure: fixes ops' :: "('c \<times> 'd) strips_operator list"   and \<pi> :: "('c \<times> 'd) strips_operator list"   and \<Pi> :: "('c, 'd) sas_plus_problem" assumes "ops' \<in> set (List_Supplement.embed \<pi>)" obtains op :: "('c \<times> 'd) strips_operator" where "op \<in> set \<pi>" and "map (strips_op_to_sasp \<Pi>) ops' = [\<phi>\<^sub>O\<inverse> \<Pi> op]"
length_brauns: fixes k :: "nat"   and xs :: "'b list" shows "length (brauns k xs) = min (length xs) (2 ^ k)"
simps: fixes a1 :: "('a \<times> 'a) set"   and a2 :: "('a \<times> 'a) set"   and a3 :: "nat"   and a4 :: "nat \<Rightarrow> 'a"   and a5 :: "nat \<Rightarrow> bool"   and a6 :: "nat"   and a7 :: "'a \<times> 'a" shows "relto_fun a1 a2 a3 a4 a5 a6 a7 = (\<exists>as a m b sel A B n. a1 = A \<and> a2 = B \<and> a3 = n \<and> a4 = as \<and> a5 = sel \<and> a6 = m \<and> a7 = (a, b) \<and> as 0 = a \<and> as m = b \<and> (\<forall>x<m. (sel x \<longrightarrow> (as x, as (Suc x)) \<in> A) \<and> (\<not> sel x \<longrightarrow> (as x, as (Suc x)) \<in> B)) \<and> n = card {i. i < m \<and> sel i} \<and> (n = 0) = (m = 0))"
intro: fixes getIJ :: "(nat set \<times> 'b option list set) set \<Rightarrow> (nat set \<times> 'b option list set) set \<Rightarrow> nat set \<Rightarrow> nat set \<times> nat set" assumes "\<And>V Q_pos Q_neg I J. \<lbrakk>2 \<le> card V; getIJ Q_pos Q_neg V = (I, J)\<rbrakk> \<Longrightarrow> 1 \<le> card I \<and> 1 \<le> card J \<and> V = I \<union> J \<and> I \<inter> J = {}" shows "getIJ getIJ"
simps_2: fixes t :: "('d, 'c) Term.term" shows "fv\<^sub>s\<^sub>s\<^sub>t\<^sub>p (receive\<langle>t\<rangle>) = fv t"
assms_1: shows "\<lbrace>p \<and> b\<rbrace> S \<lbrace>p\<rbrace>\<^sub>u"
full_imp_height: fixes n :: "nat"   and t :: "'a tree23" assumes "full n t" shows "height t = n"
Ostrowski_base_npow_not_one: fixes N :: "nat"   and j :: "nat"   and n :: "nat"   and vv :: "nat \<Rightarrow> 'a \<Rightarrow> ant" assumes "0 < N"   and "j \<le> Suc n"   and "vals_nonequiv K (Suc n) vv" shows "1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup> \<noteq> \<zero>"
map_return_pmf: fixes f :: "'b \<Rightarrow> 'a"   and x :: "'b" shows "map_pmf f (return_pmf x) = return_pmf (f x)"
differing_vars_neg_intro: 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 mem\<^sub>1 :: "'b \<Rightarrow> 'c"   and x :: "'b"   and mems :: "(('b \<Rightarrow> 'c) \<times> ('b \<Rightarrow> 'c)) list"   and i :: "nat"   and mem\<^sub>2 :: "'b \<Rightarrow> 'c" assumes "sifum_security stop eval some_val some_val'"   and "mem\<^sub>1 x = fst (mems ! i) x"   and "mem\<^sub>2 x = snd (mems ! i) x" shows "x \<notin> sifum_security.differing_vars_lists mem\<^sub>1 mem\<^sub>2 mems i"
float_round_down: fixes prec :: "nat"   and x :: "float" shows "real_of_float (float_round_down prec x) \<le> real_of_float x"
mapl_F_def: fixes l1 :: "'q \<Rightarrow> 'aa"   and l2 :: "'r \<Rightarrow> 'ab"   and l3 :: "'s \<Rightarrow> 'ac" shows "mapl_F l1 l2 l3 = map_F l1 l2 l3 id id id id id id"
add_initial_modes_def: fixes cmds :: "'Com list" shows "add_initial_modes cmds = zip cmds (replicate (length cmds) mds\<^sub>s)"
msubstpos_I: fixes p :: "fm"   and vs :: "'b list"   and c :: "poly"   and x :: "'b"   and bs :: "'b list"   and t :: "tm" assumes "islin p"   and "(0::'b) < \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup>" shows "Ifm vs (x # bs) (msubstpos p c t) = Ifm vs (Itm vs (x # bs) t / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup> # bs) p"
entries_rbtreeify: fixes kvs :: "('a \<times> 'b) list" shows "entries (rbtreeify kvs) = kvs"
invertible_det_nz: fixes A :: "(('a, 'b) vec, 'b) vec" shows "invertible A = (det A \<noteq> (0::'a))"
nth_sshiftr: fixes w :: "'b word"   and m :: "nat"   and n :: "nat" shows "(w >>> m) !! n = (n < size w \<and> (if size w \<le> n + m then w !! (size w - 1) else w !! (n + m)))"
abs_div_pos: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inverse :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sgn :: "'c \<Rightarrow> 'c"   and abs :: "'c \<Rightarrow> 'c"   and y :: "'c"   and x :: "'c" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less zero y" shows "divide (abs x) y = abs (divide x y)"
scale_eq_0_iff: fixes a :: "real"   and x :: "'d" shows "(a *\<^sub>R x = (0::'d)) = (a = 0 \<or> x = (0::'d))"
hseqI': fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and \<mu> :: "'b"   and \<nu> :: "'b" assumes "horizontal_composition V H src trg"   and "partial_magma.arr V \<mu>"   and "partial_magma.arr V \<nu>"   and "src \<nu> = trg \<mu>" shows "partial_magma.arr V (H \<nu> \<mu>)"
vconverse_atD: fixes r :: "V"   and b :: "V"   and a :: "V" assumes "vsv r"   and "\<langle>b, a\<rangle> \<in>\<^sub>\<circ> r\<inverse>\<^sub>\<circ>" shows "r\<lparr>a\<rparr> = b"
is_exec_of_def: fixes A :: "('c, 'd) ioa"   and e :: "'c \<times> ('d \<times> 'c) list" shows "is_exec_of A e \<equiv> fst e \<in> start A \<and> is_exec_frag_of A e"
mset_set_Union: fixes A :: "'b set"   and B :: "'b set" assumes "finite A"   and "finite B"   and "A \<inter> B = {}" shows "mset_set (A \<union> B) = mset_set A + mset_set B"
edges_add_edge: fixes v :: "'c"   and e :: "'d"   and v' :: "'c"   and g :: "('c, 'd, 'e) graph_scheme" shows "edges (add_edge v e v' g) = insert (v, e, v') (edges g)"
is_InfI: fixes A :: "'b set"   and inf :: "'b" assumes "\<And>x. x \<in> A \<Longrightarrow> inf \<sqsubseteq> x"   and "\<And>z. \<forall>x\<in>A. z \<sqsubseteq> x \<Longrightarrow> z \<sqsubseteq> inf" shows "is_Inf A inf"
Collect_ball_eq: fixes A :: "'c set"   and P :: "'b \<Rightarrow> 'c \<Rightarrow> bool" shows "{x. \<forall>y\<in>A. P x y} = (\<Inter>y\<in>A. {x. P x y})"
span_is_monotone: fixes S :: "'a vec set"   and T :: "'a vec set" assumes "S \<subseteq> T" shows "local.span S \<subseteq> local.span T"
CAND_AndR_elim: fixes a :: "coname"   and b :: "coname"   and M :: "trm"   and c :: "coname"   and N :: "trm"   and B :: "ty" assumes "<a>:AndR <b>.M <c>.N a \<in> \<parallel><B>\<parallel>"   and "<a>:AndR <b>.M <c>.N a \<notin> BINDINGc B (\<parallel>(B)\<parallel>)" shows "\<exists>B1 B2. B = B1 AND B2 \<and> <b>:M \<in> \<parallel><B1>\<parallel> \<and> <c>:N \<in> \<parallel><B2>\<parallel>"
deduct2: fixes p :: "('a, 'b) fm"   and ps :: "('a, 'b) fm list"   and q :: "('a, 'b) fm" assumes "p # ps \<turnstile> q" shows "ps \<turnstile> p \<^bold>\<longrightarrow> q"
oa_contingent_3: fixes v :: "i"   and x :: "\<nu>" shows "[\<lparr>A!,x\<^sup>P\<rparr> \<^bold>\<equiv> \<^bold>\<not>\<lparr>O!,x\<^sup>P\<rparr> in v]"
skip_doesnt_write: fixes annos :: "'Var ModeUpd list"   and x :: "'Var" shows "doesnt_modify (Skip \<otimes> annos) x"
no_outstanding_write_to_read_only_memory_def: fixes \<S> :: "nat \<Rightarrow> bool option"   and ts :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'b memref list \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list" shows "no_outstanding_write_to_read_only_memory \<S> ts \<equiv> \<forall>i p is \<O> \<R> \<D> \<theta> sb. i < length ts \<longrightarrow> ts ! i = (p, is, \<theta>, sb, \<D>, \<O>, \<R>) \<longrightarrow> no_write_to_read_only_memory \<S> sb"
Abs_binding_inverse: fixes y :: "binding_Rep" assumes "y \<in> Collect binding_Rep_set" shows "Rep_binding (Abs_binding y) = y"
monad_altc_envT: fixes return :: "'a \<Rightarrow> 'b"   and bind :: "'b \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b"   and altc :: "'c cset \<Rightarrow> ('c \<Rightarrow> 'b) \<Rightarrow> 'b" assumes "monad_altc return bind altc" shows "monad_altc (return_env return) (bind_env bind) (altc_env altc)"
berlekamp_resulting_mat_closed_1: fixes u :: "'c mod_ring poly" shows "berlekamp_resulting_mat u \<in> carrier_mat (degree u) (degree u)"
prems: shows "ves = map (Basic \<circ> EConst) (a # vs)"
coinduct: fixes X :: "('a, 'b, 'c) psi \<Rightarrow> bool"   and x :: "('a, 'b, 'c) psi" assumes "X x"   and "\<And>x. X x \<Longrightarrow> \<exists>P. x = P \<and> (\<forall>\<Psi>. insertAssertion (extractFrame P) \<Psi> \<simeq>\<^sub>F \<langle>\<epsilon>, \<Psi>\<rangle> \<and> (\<forall>Rs. \<Psi> \<rhd> P \<longmapsto> Rs \<longrightarrow> (\<exists>P'. Rs = \<tau> \<prec> P' \<and> (X P' \<or> quiet P'))))" shows "quiet x"
simps_4: fixes c :: "('p, 's, 'm) configuration"   and i :: "nat"   and p :: "'p"   and q :: "'p"   and s :: "'s"   and s' :: "'s"   and msg :: "'m"   and c' :: "('p, 's, 'm) configuration" shows "c \<turnstile> Send i p q s s' msg \<mapsto> c' = (can_occur (Send i p q s s' msg) c \<and> states c' p = s' \<and> (\<forall>r. r \<noteq> p \<longrightarrow> states c' r = states c r) \<and> msgs c' i = msgs c i @ [Msg msg] \<and> (\<forall>i'. i \<noteq> i' \<longrightarrow> msgs c' i' = msgs c i') \<and> no_snapshot_change c c')"
am_assn_def: fixes N :: "nat" shows "??.Network_Impl.am_assn N \<equiv> is_nf N []"
uminus_Assertion_def: fixes p :: "'c Assertion" shows "- p = abs_wpt (neg_assert {\<cdot> p })"
cod_cod: fixes f :: "('a S.E.term, 'a) S.arr" assumes "S.arr f" shows "S.cod (S.cod f) = S.cod f"
cases: fixes a1 :: "string_ternop"   and a2 :: "'a OCL_Types.type"   and a3 :: "'b OCL_Types.type"   and a4 :: "'c OCL_Types.type"   and a5 :: "'d OCL_Types.type" assumes "string_ternop_type a1 a2 a3 a4 a5" obtains \<sigma> :: "'b OCL_Types.type" and \<rho> :: "'c OCL_Types.type" where "a1 = SubstringOp" and "a2 = String[1]" and "a3 = \<sigma>" and "a4 = \<rho>" and "a5 = String[1]" and "\<sigma> = UnlimitedNatural[1]\<midarrow>Integer[1]" and "\<rho> = UnlimitedNatural[1]\<midarrow>Integer[1]"
defUse_path_ex: fixes v :: "'val"   and g :: "'g"   and m :: "'node" assumes "v \<in> allUses g m"   and "m \<in> set (\<alpha>n g)" obtains ns :: "'node list" where "g \<turnstile> defNode g v-ns\<rightarrow>m" and "old.EntryPath g ns"
nextFreeBlockInvariant_def: fixes state :: "CFile" shows "nextFreeBlockInvariant state \<equiv> (fileSize state + blockSize - 1) div blockSize = nextFreeBlock state"
simulation_refl: shows "simulation Id (stutter_extend A) (stutter_extend A)"
flatten_sigma_1_length: fixes \<phi> :: "'b ltln"   and N :: "'b ltln set" shows "size (\<phi>[N]\<^sub>\<Sigma>\<^sub>1) \<le> size \<phi>"
continuous_map_closure_preimage_subset: fixes X :: "'b topology"   and Y :: "'c topology"   and f :: "'b \<Rightarrow> 'c"   and T :: "'c set" assumes "continuous_map X Y f" shows "X closure_of {x \<in> topspace X. f x \<in> T} \<subseteq> {x \<in> topspace X. f x \<in> Y closure_of T}"
span_in_category_axioms: shows "span_in_category (\<odot>) (Cod (SPN \<mu>))"
augment_edge_same: fixes e :: "nat \<times> nat"   and f :: "nat \<times> nat \<Rightarrow> 'capacity"   and \<Delta> :: "'capacity" assumes "e \<in> E" shows "augment_edge f e \<Delta> e = f e + \<Delta>"
coeff_simp: fixes R :: "('c, 'd) ring_scheme"   and f :: "nat \<Rightarrow> 'c" assumes "UP_ring R"   and "f \<in> carrier (UP R)" shows "coeff (UP R) f = f"
smaps_over_finite: fixes A :: "'a set"   and B :: "'b set" assumes "finite A"   and "finite B" shows "finite (smaps_over A B)"
clw_rel_def: fixes A :: "('a \<times> 'b set) set" shows "clw_rel A = lw_rel O \<langle>A\<rangle>set_rel O br \<Union> top"
final_threadI: fixes s :: "('e, 'f) locks \<times> (('f \<Rightarrow> ('x \<times> 'e \<Rightarrow>f nat) option) \<times> 'g) \<times> ('f \<Rightarrow> 'h wait_set_status option) \<times> 'f set"   and t :: "'f"   and x :: "'x" assumes "thr s t = \<lfloor>(x, no_wait_locks)\<rfloor>"   and "final x"   and "wset s t = None" shows "final_thread s t"
sn_forget': fixes x :: "name"   and p :: "trm"   and t :: "trm"   and k :: "stack" assumes "SN (t[x::=p] \<star> k)"   and "x \<sharp> k" shows "SN (t \<star> k)"
assms_2: shows "\<forall>q. List.member index_list q \<longrightarrow> q < length arb_list"
a4: shows "More_Modules.ideal (F \<union> B) \<subseteq> More_Modules.ideal (A \<union> B)"
LAss_\<tau>red1r: fixes uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and h :: "'heap"   and e :: "'addr expr1"   and xs :: "'addr val list"   and e' :: "'addr expr1"   and xs' :: "'addr val list"   and V :: "nat" assumes "\<tau>red1gr uf P t h (e, xs) (e', xs')" shows "\<tau>red1gr uf P t h (V:=e, xs) (V:=e', xs')"
pi_def: shows "pi = 2 * (THE x. 0 \<le> x \<and> x \<le> 2 \<and> cos x = 0)"
C_5: shows "\<forall>l s. P l s \<longrightarrow> time C s \<le> k * e s"
simps_1: fixes m :: "'node"   and m' :: "'node"   and S :: "'node SDG_node set"   and f :: "('var \<Rightarrow> 'val option) \<Rightarrow> 'var \<Rightarrow> 'val option" shows "slice_kind_aux m m' S \<Up>f = (if m \<in> \<lfloor>S\<rfloor>\<^bsub>CFG\<^esub> then \<Up>f else \<Up>id)"
inf_compl_bot_right: fixes x :: "'a"   and y :: "'a" shows "x \<cdot> (y \<cdot> - x) = 0"
fix_ind2: fixes P :: "'d \<Rightarrow> bool"   and F :: "'d \<rightarrow> 'd" assumes "adm P"   and "P \<bottom>"   and "P (F\<cdot>\<bottom>)"   and "\<And>x. \<lbrakk>P x; P (F\<cdot>x)\<rbrakk> \<Longrightarrow> P (F\<cdot>(F\<cdot>x))" shows "P (fix\<cdot>F)"
monomial_functions: fixes R :: "('c, 'd) ring_scheme"   and c :: "'c"   and n :: "nat" assumes "U_function_ring R"   and "c \<in> carrier R" shows "monomial_function R c n \<in> carrier (function_ring (carrier R) R)"
Rep_scene_cases: fixes y :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" assumes "y \<in> {F. overrider F}" obtains x :: "'a scene" where "y = Rep_scene x"
h: shows "- BVG \<alpha> = \<Union> {{x} |x. x \<notin> BVG \<alpha>}"