hcju/isabelleps huggingface dataset:
terminates_Normal_elim_cases_8: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'a, 'c) com option"   and p :: "'a"   and s :: "'b" assumes "\<Gamma>\<turnstile>Call p \<down> Normal s" obtains bdy :: "('b, 'a, 'c) com" where "\<Gamma> p = Some bdy" and "\<Gamma>\<turnstile>bdy \<down> Normal s"   | "\<Gamma> p = None"
simps: fixes a :: "intv" shows "isConst a = (\<exists>uu_. a = Const uu_)"
cases: fixes x :: "'b match_expr list" obtains "x = []"   | m :: "'b match_expr" and ms :: "'b match_expr list" where "x = m # ms"
no_initial_process_snapshot: fixes channel :: "nat \<Rightarrow> ('d \<times> 'd) option"   and trans :: "'d \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> bool"   and send :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and recv :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and init :: "('d, 'e, 'f) configuration" assumes "cocomputation channel trans send recv init" shows "\<forall>p. \<not> ps init p \<noteq> None"
Iod_empty_Torder: shows "Torder (Iod D {})"
simps_2: fixes a :: "nat" shows "pred (Suc a) = a"
induced_map_def: fixes A :: "'a set" shows "induced_map A = (\<lambda>x. (T ^^ return_time_function A x) x)"
reachable_adjI: fixes G :: "('c, 'd) pre_digraph"   and u :: "'c"   and v :: "'c" assumes "wf_digraph G"   and "u \<rightarrow>\<^bsub>G\<^esub> v" shows "u \<rightarrow>\<^sup>*\<^bsub>G\<^esub> v"
trans: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x .= y"   and "y .= z"   and "x \<in> carrier S"   and "y \<in> carrier S"   and "z \<in> carrier S" shows "x .= z"
simps_1: fixes p :: "Player"   and \<sigma> :: "'a \<Rightarrow> 'a"   and \<sigma>' :: "'a \<Rightarrow> 'a"   and v0 :: "'a" shows "\<not> lnull (greedy_conforming_path p \<sigma> \<sigma>' v0)"
abs_ge_zero: fixes a :: "'a" shows "(0::'a) \<le> \<bar>a\<bar>"
reachable_closed_dfs: fixes xs :: "'rep list" assumes "list_all k_isNode xs" shows "simAbs ` KBPAlg.reachable (set xs) \<subseteq> simAbs ` KBPAlg.set_of (KBPAlg.dfs k_empt xs)"
closure_involutive': fixes Z :: "('c \<Rightarrow> real) set"   and W :: "('c \<Rightarrow> real) set" assumes "Z \<subseteq> Closure\<^sub>\<alpha> W" shows "Closure\<^sub>\<alpha> Z \<subseteq> Closure\<^sub>\<alpha> W"
longest_common_prefix_prefix2: fixes xs :: "'b list"   and ys :: "'b list" shows "prefix (longest_common_prefix xs ys) ys"
mask_AND_NOT_mask: fixes w :: "'a word"   and n :: "nat" shows "(w && ~~ mask n) && mask n = 0"
lens_comp_assoc: fixes X :: "'a \<Longrightarrow> 'c"   and Y :: "'c \<Longrightarrow> 'd"   and Z :: "'d \<Longrightarrow> 'b" shows "X ;\<^sub>L (Y ;\<^sub>L Z) = X ;\<^sub>L Y ;\<^sub>L Z"
assms_1: shows "\<mu> \<in> \<K> (l, R)"
insert_before_list_subset: fixes xs :: "'a list"   and x :: "'a"   and ref :: "'a option" shows "set xs \<subseteq> set (insert_before_list x ref xs)"
P2_security_rsa: fixes prime_set :: "nat set"   and D :: "bool \<times> (nat \<times> nat) \<times> bool \<times> bool \<Rightarrow> bool spmf"   and B :: "nat \<times> nat \<Rightarrow> nat \<Rightarrow> bool"   and m2 :: "bool"   and HCP_ad :: "real"   and m1 :: "bool \<times> bool" assumes "rsa_base prime_set"   and "\<forall>a. lossless_spmf (D a)"   and "\<And>b\<^sub>\<sigma>. etp.HCP_adv (rsa_base.I prime_set) local.range F\<^sub>i\<^sub>n\<^sub>v B (ETP_base.\<A> local.range B) m2 b\<^sub>\<sigma> D \<le> HCP_ad" shows "sim_det_def.adv_P2 (ETP_base.R2 (rsa_base.I prime_set) local.range B F F\<^sub>i\<^sub>n\<^sub>v) (ETP_base.S2 (rsa_base.I prime_set) local.range B F\<^sub>i\<^sub>n\<^sub>v) funct_OT_12 m1 m2 D \<le> 2 * HCP_ad"
intro: fixes G :: "('c, 'd) pre_digraph" assumes "wf_digraph G"   and "sym_digraph_axioms G" shows "sym_digraph G"
maxR_def: fixes d :: "'b"   and x :: "'a"   and y :: "'a" shows "Bossy''.maxR d x y = (if (x, y) \<in> BPd' d then y else x)"
ltrm_cfs: fixes f :: "nat \<Rightarrow> 'a"   and n :: "nat" assumes "f \<in> carrier P" shows "monom P (f (deg R f)) (deg R f) n = (if n = deg R f then f (deg R f) else \<zero>)"
isn: fixes x :: "'a"   and i :: "'a" assumes "x \<in> Csafe" shows "isnFDERIV DIM('a) ode_e [0..<DIM('a)] [DIM('a)..<2 * DIM('a)] (list_of_eucl x @ list_of_eucl i) (Suc (Suc 0))"
connected_def: fixes S :: "'b set" shows "connected S = (\<nexists>A B. open A \<and> open B \<and> S \<subseteq> A \<union> B \<and> A \<inter> B \<inter> S = {} \<and> A \<inter> S \<noteq> {} \<and> B \<inter> S \<noteq> {})"
I_eps1: fixes f :: "atom fm"   and xs :: "real list" assumes "nqfree f" shows "R.I (qe_eps\<^sub>1 f) xs = (\<exists>x. R.I f (x # xs))"
sets_n: fixes N :: "'a stream measure"   and M :: "'a stream measure"   and y :: "'a" assumes "R N M" shows "sets (n N M y) = sets (stream_space (count_space UNIV))"
range_nth_default: fixes dflt :: "'b"   and xs :: "'b list" shows "range (nth_default dflt xs) = insert dflt (set xs)"
sup_def: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list"   and mxs :: "nat"   and mxl :: "nat" shows "JVM_SemiType.sup P mxs mxl \<equiv> snd (snd (JVM_SemiType.sl P mxs mxl))"
Postdomination_axioms: shows "Postdomination sourcenode targetnode kind valid_edge Entry Exit"
inf_sup_aci_5: fixes x :: "'b \<times> 'c"   and y :: "'b \<times> 'c" shows "sup_prod x y = sup_prod y x"
leadsTo_weaken: fixes F :: "'d program"   and A :: "'d set"   and A' :: "'d set"   and B :: "'d set"   and B' :: "'d set" assumes "F \<in> A \<longmapsto> A'"   and "B \<subseteq> A"   and "A' \<subseteq> B'" shows "F \<in> B \<longmapsto> B'"
Indirect: fixes G :: "prog"   and new :: "qtname \<times> mdecl"   and intr :: "qtname \<times> mdecl"   and old :: "qtname \<times> mdecl" assumes "G \<turnstile> new overrides\<^sub>S intr"   and "G \<turnstile> intr overrides\<^sub>S old" shows "G \<turnstile> new overrides\<^sub>S old"
S1Hist: fixes rmhist :: "state \<Rightarrow> PrIds \<Rightarrow> histState"   and p :: "PrIds" shows "\<turnstile> [HNext rmhist p]_(c p, r p, m p, rmhist!p) \<and> $S1 rmhist p \<longrightarrow> unchanged (rmhist!p)"
product_nn_integral_insert_rev: fixes I :: "'i set"   and i :: "'i"   and f :: "('i \<Rightarrow> 'a) \<Rightarrow> ennreal" assumes "finite I"   and "i \<notin> I"   and "f \<in> borel_measurable (Pi\<^sub>M (insert i I) M)" shows "integral\<^sup>N (Pi\<^sub>M (insert i I) M) f = \<integral>\<^sup>+ y. \<integral>\<^sup>+ x. f (x(i := y)) \<partial>Pi\<^sub>M I M \<partial>M i"
exec_to_exec_restrict: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "('e, 'd, 'f) com"   and s :: "('e, 'f) xstate"   and t :: "('e, 'f) xstate"   and P :: "'d set" assumes "\<Gamma>\<turnstile> \<langle>c,s\<rangle> \<Rightarrow> t" shows "\<exists>t'. \<Gamma>|\<^bsub>P\<^esub>\<turnstile> \<langle>c,s\<rangle> \<Rightarrow> t' \<and> (t = Stuck \<longrightarrow> t' = Stuck) \<and> (\<forall>f. t = Fault f \<longrightarrow> t' \<in> {Fault f, Stuck}) \<and> (t' \<noteq> Stuck \<longrightarrow> t' = t)"
HMA_Diagonal_to_Smith_row_i_PQ: fixes i :: "nat" assumes "i < min CARD('b) CARD('c)" shows "((=) ===> HMA_M3 ===> HMA_M3) (Diagonal_to_Smith_row_i_PQ_JNF i) (Diagonal_to_Smith_row_i_PQ i)"
dividesD: fixes G :: "('c, 'd) monoid_scheme"   and a :: "'c"   and b :: "'c" assumes "a divides\<^bsub>G\<^esub> b" shows "\<exists>c\<in>carrier G. b = a \<otimes>\<^bsub>G\<^esub> c"
group_closure_scalar_mult_left: fixes s :: "'a"   and S :: "'a set"   and n :: "nat" assumes "s \<in> group_closure S" shows "of_nat n * s \<in> group_closure S"
inputInduct: fixes P :: "pi"   and a :: "name"   and x :: "name"   and P' :: "pi"   and F :: "'a \<Rightarrow> pi \<Rightarrow> name \<Rightarrow> name \<Rightarrow> pi \<Rightarrow> bool"   and C :: "'a" assumes "P \<longmapsto> a<x> \<prec> P'"   and "x \<sharp> P"   and "\<And>a x P C. F C (a<x>.P) a x P"   and "\<And>P a x P' b C. \<lbrakk>P \<longmapsto> a<x> \<prec> P'; \<And>C. F C P a x P'\<rbrakk> \<Longrightarrow> F C ([b\<frown>b]P) a x P'"   and "\<And>P a x P' b c C. \<lbrakk>P \<longmapsto> a<x> \<prec> P'; \<And>C. F C P a x P'; b \<noteq> c\<rbrakk> \<Longrightarrow> F C ([b\<noteq>c]P) a x P'"   and "\<And>P Q a x P' C. \<lbrakk>P \<longmapsto> a<x> \<prec> P'; \<And>C. F C P a x P'\<rbrakk> \<Longrightarrow> F C (P \<oplus> Q) a x P'"   and "\<And>P Q a x Q' C. \<lbrakk>Q \<longmapsto> a<x> \<prec> Q'; \<And>C. F C Q a x Q'\<rbrakk> \<Longrightarrow> F C (P \<oplus> Q) a x Q'"   and "\<And>P P' Q a x C. \<lbrakk>P \<longmapsto> a<x> \<prec> P'; x \<sharp> P; x \<sharp> Q; x \<noteq> a; \<And>C. F C P a x P'\<rbrakk> \<Longrightarrow> F C (P \<parallel> Q) a x (P' \<parallel> Q)"   and "\<And>P Q Q' a x C. \<lbrakk>Q \<longmapsto> a<x> \<prec> Q'; x \<sharp> P; x \<sharp> Q; x \<noteq> a; \<And>C. F C Q a x Q'\<rbrakk> \<Longrightarrow> F C (P \<parallel> Q) a x (P \<parallel> Q')"   and "\<And>P P' a x y C. \<lbrakk>P \<longmapsto> a<x> \<prec> P'; y \<noteq> a; y \<noteq> x; y \<sharp> C; \<And>C. F C P a x P'\<rbrakk> \<Longrightarrow> F C (<\<nu>y>P) a x (<\<nu>y>P')"   and "\<And>P a x P' C. \<lbrakk>P \<parallel> !P \<longmapsto> a<x> \<prec> P'; \<And>C. F C (P \<parallel> !P) a x P'\<rbrakk> \<Longrightarrow> F C (!P) a x P'" shows "F C P a x P'"
IH: fixes s :: "'a state"   and env :: "v sem_env" assumes "\<And>p e s' env. \<lbrakk>(p, e) \<in> set pes; clock s' \<le> clock s\<rbrakk> \<Longrightarrow> \<exists>s'' r. evaluate True env s' e (s'', r)" shows "\<exists>s' r. evaluate_match True env s v pes v' (s', r)"
h: shows "proots_count (map_poly complex_of_real P) (upper_circle_01 \<union> lower_circle_01) = 1"
Abs_nonempty_list_cases: fixes x :: "'d nonempty_list" obtains y :: "'d list" where "x = Abs_nonempty_list y" and "y \<in> {xs. xs \<noteq> []}"
r_smn_aux_prim: fixes n :: "nat"   and m :: "nat" assumes "0 < n" shows "prim_recfn (Suc m) (r_smn_aux n m)"
DisjI: fixes A0 :: "formula"   and A1 :: "formula"   and Gamma :: "formula list"   and A :: "(formula list \<times> formula list set) set" assumes "A0 # A1 # Gamma \<in> deductions A"   and "Disjs \<subseteq> A" shows "FConj Neg A0 A1 # Gamma \<in> deductions A"
lim: shows "(\<lambda>n. LINT x|M. (F (n + 1) x - F n (T x)) * indicat_real A x) \<longlonglongrightarrow> LINT x|M. f x * indicat_real A x"
simps: fixes cm :: "common_metadata"   and pm :: "participant_metadata"   and views :: "view list" shows "transaction_merkle.the_Transaction (Transaction cm pm views) = ((cm, pm), views)"
QR_decomposition_iarrays_efficient_def: fixes A :: "real iarray iarray" shows "QR_decomposition_iarrays_efficient A = (let Q = divide_by_norm_iarray (Gram_Schmidt_matrix_iarrays_efficient A) in (Q, transpose_iarray Q **i A))"
Hermite_ring_def: shows "class.Hermite_ring TYPE('b) \<equiv> \<forall>A. admits_triangular_reduction A"
tNDComm: fixes a :: "'b set set"   and b :: "'c set set"   and c :: "'b set set"   and d :: "'c set set" assumes "twoNetsDistinct a b c d" shows "twoNetsDistinct c d a b"
v_eq: fixes cfg :: "'s cfg" assumes "cfg \<in> valid_cfg" shows "v cfg = emeasure (T cfg) {x \<in> space St. (HLD S1 suntil HLD S2) (state cfg ## x)}"
that: fixes \<F>' :: "'a set set" assumes "finite \<F>'"   and "\<Union> \<F>' = \<Union> \<F>"   and "\<And>X. X \<in> \<F>' \<Longrightarrow> diameter X < e"   and "\<And>X. X \<in> \<F>' \<Longrightarrow> polytope X"   and "\<And>X. X \<in> \<F>' \<Longrightarrow> aff_dim X \<le> d"   and "\<And>X Y. \<lbrakk>X \<in> \<F>'; Y \<in> \<F>'\<rbrakk> \<Longrightarrow> X \<inter> Y face_of X"   and "\<And>C. C \<in> \<F>' \<Longrightarrow> \<exists>D. D \<in> \<F> \<and> C \<subseteq> D"   and "\<And>C x. C \<in> \<F> \<and> x \<in> C \<Longrightarrow> \<exists>D. D \<in> \<F>' \<and> x \<in> D \<and> D \<subseteq> C" shows "thesis"
llist_inf_simps_3: fixes x :: "'b"   and xs :: "'b llist"   and y :: "'b"   and ys :: "'b llist" shows "LCons x xs \<sqinter> LCons y ys = (if x = y then LCons x (xs \<sqinter> ys) else LNil)"
oalist_inv_raw_induct: fixes xs :: "('a \<times> 'e) list"   and P :: "('a \<times> 'e) list \<Rightarrow> bool" assumes "oalist_inv_raw xs"   and "P []"   and "\<And>k v xs. \<lbrakk>oalist_inv_raw ((k, v) # xs); oalist_inv_raw xs; v \<noteq> (0::'e); \<And>k'. k' \<in> fst ` set xs \<Longrightarrow> lt k k'; P xs\<rbrakk> \<Longrightarrow> P ((k, v) # xs)" shows "P xs"
assms: shows "u \<noteq> 0\<^sub>v (dim_vec u)"
take_drop: fixes n :: "nat"   and m :: "nat"   and xs :: "'g list" shows "take n (drop m xs) = drop m (take (n + m) xs)"
minus_bit1_def: fixes x :: "'b bit1"   and y :: "'b bit1" shows "x - y = Abs_bit1' (Rep_bit1 x - Rep_bit1 y)"
times_def_2: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and conv :: "'c \<Rightarrow> 'c"   and x :: "'c"   and y :: "'c" assumes "class.linear_aggregation_algebra one times uminus inf less_eq less sup bot top plus conv" shows "times x y = inf x y"
\<gamma>: fixes \<D> :: "('m \<times> 'm set) set" assumes "\<D> tagged_division_of cbox a b"   and "\<gamma> fine \<D>" shows "norm ((\<Sum>(x, K)\<in>\<D>. content K *\<^sub>R f x) - integral (cbox a b) f) < e / (2 * (real DIM('n) + 1))"
INFM_mono: fixes P :: "'c \<Rightarrow> bool"   and Q :: "'c \<Rightarrow> bool" assumes "\<exists>\<^sub>\<infinity>x. P x"   and "\<And>x. P x \<Longrightarrow> Q x" shows "\<exists>\<^sub>\<infinity>x. Q x"
higherPowers_eq: fixes b :: "nat"   and i :: "nat"   and p :: "nat \<Rightarrow>\<^sub>0 'b"   and x :: "nat" shows "lookup (higherPowers b i p) x = lookup p (inc_above b i x)"
simps_2: fixes n :: "nat" shows "mk_tree (Suc n) = (let t = mk_tree n in \<triangle> \<^bold>l: t \<^bold>r: t)"
is_pure_def: fixes P :: "'c \<Rightarrow> 'd \<Rightarrow> assn" shows "is_pure P \<equiv> \<exists>P'. \<forall>x x'. P x x' = \<up> (P' x x')"
geodesic_segment_dist'_1: fixes x :: "'a"   and z :: "'a"   and y :: "'a" assumes "dist x z = dist x y + dist y z" shows "y \<in> {x--z}"
projection_fixes_image: fixes M :: "'b set"   and x :: "'b" assumes "closed_csubspace M"   and "x \<in> M" shows "projection M x = x"
bisim1_imp_bisim1': fixes P :: "'a prog"   and h :: "'heap"   and e :: "'addr expr1"   and exs :: "'addr expr1 \<times> 'addr val list"   and s :: "'addr val list \<times> 'addr val list \<times> nat \<times> 'addr option"   and n :: "nat" assumes "P,e,h \<turnstile> exs \<leftrightarrow> s" shows "P,e,n,h \<turnstile>' exs \<leftrightarrow> s"
intro: fixes G :: "('c, 'd) monoid_scheme" assumes "monoid_cancel G"   and "Group.comm_monoid G" shows "comm_monoid_cancel G"
addpre_simps_3: fixes v :: "nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set"   and pre :: "nat set" shows "\<pi>\<^sub>4 (addpre v pre) = \<pi>\<^sub>4 v"
w_1: shows "((\<lambda>x. if x \<in> i then f x else (0::'a)) has_integral w) (cbox a b)"
assms_2: shows "(_High_) \<notin> backward_slice S"
algebra_axioms: fixes M :: "'c measure" shows "algebra (space M) (sets M)"
produces_imp_Max_atom: fixes S :: "'b literal multiset \<Rightarrow> 'b literal multiset"   and N :: "'b literal multiset set"   and C :: "'b literal multiset"   and A :: "'b" assumes "ground_resolution_with_selection S"   and "ground_resolution_with_selection.production S N C = {A}" shows "A = Max (atms_of C)"
equals0D: fixes A :: "'g set"   and a :: "'g" assumes "A = {}" shows "a \<notin> A"
elims_2: fixes x :: "'d \<times> 'b" assumes "valid_in_port x" obtains v :: "'d" and p :: "'b" where "x = (v, p)" and "v |\<in>| vertices \<and> p |\<in>| inPorts (nodeOf v)"
normalize_1_iff: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and a :: "'b" assumes "class.normalization_semidom divide plus minus zero times one unit_factor normalize" shows "(normalize a = one) = dvd.dvd times a one"
g_def: shows "g \<equiv> \<lambda>z. if z \<in> \<int> then complex_of_real pi else Gamma z * Gamma (1 - z) * sin (complex_of_real pi * z)"
sel_1: fixes x1 :: "'c"   and x2 :: "real" shows "normal (Sctn x1 x2) = x1"
Unifier_comp: fixes t :: "('c, 'd) Term.term"   and \<theta> :: "'d \<Rightarrow> ('c, 'd) Term.term"   and \<delta> :: "'d \<Rightarrow> ('c, 'd) Term.term"   and u :: "('c, 'd) Term.term" assumes "Unifier (\<theta> \<circ>\<^sub>s \<delta>) t u" shows "Unifier \<delta> (t \<cdot> \<theta>) (u \<cdot> \<theta>)"
presFresh_fromIMor: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<times> 'c"   and hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<times> 'd"   and MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) Recursion.model" assumes "ipresCons h hA (fromMOD MOD)"   and "ipresFresh h (fromMOD MOD)" shows "presFresh (fromIMor h) MOD"
restr_eqE: fixes F :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and F' :: "'a \<Rightarrow> 'b"   and x :: "'a" assumes "restrict F A = restrict F' A"   and "x \<in> A" shows "F x = F' x"
select_def: fixes xs :: "'a list" shows "Random.select xs = scomp (Random.range (natural_of_nat (length xs))) (\<lambda>k. Pair (xs ! nat_of_natural k))"
Composition': fixes SCompose :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and SImp :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and SBottom :: "'d"   and SChanEq :: "'f \<Rightarrow> 'f \<Rightarrow> 'e"   and \<Psi> :: "'d"   and \<Psi>' :: "'d"   and \<Psi>'' :: "'d"   and \<Psi>''' :: "'d" assumes "assertion SCompose SImp SBottom SChanEq"   and "assertionAux.AssertionStatEq SImp \<Psi> \<Psi>'"   and "assertionAux.AssertionStatEq SImp \<Psi>'' \<Psi>'''" shows "assertionAux.AssertionStatEq SImp (SCompose \<Psi> \<Psi>'') (SCompose \<Psi>' \<Psi>''')"
intro: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and J :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and D :: "'b \<Rightarrow> 'a" assumes "category C"   and "category J"   and "functor J C D" shows "diagram J C D"
restrict_disjoint_right: fixes ls :: "'c list"   and ns :: "'c list"   and ms :: "'c list"   and ks :: "'c list"   and f :: "'c \<times> 'c \<Rightarrow> 'd" assumes "disjoint ls ns" shows "ms \<langle>ks \<langle>f\<rangle> ls\<rangle> ns = mbot"
infsetsum'_subset_zero: fixes A :: "'c set"   and B :: "'c set"   and f :: "'c \<Rightarrow> 'd" assumes "\<And>x. x \<in> A - B \<union> (B - A) \<Longrightarrow> f x = (0::'d)" shows "infsetsum' f A = infsetsum' f B"
lfinite_lrel: fixes xs :: "'c llist"   and R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and ys :: "'d llist" assumes "lfinite xs"   and "lrel R xs ys" shows "lfinite ys"
tllist_all2_flip: fixes P :: "'h \<Rightarrow> 'g \<Rightarrow> bool"   and Q :: "'j \<Rightarrow> 'i \<Rightarrow> bool"   and xs :: "('g, 'i) tllist"   and ys :: "('h, 'j) tllist" shows "tllist_all2 (flip P) (flip Q) xs ys = tllist_all2 P Q ys xs"
cnf2bhv_def: fixes tCMP :: "'a \<Rightarrow> cnf \<Rightarrow> 'b"   and active :: "'a \<Rightarrow> cnf \<Rightarrow> bool"   and c :: "'a"   and t :: "nat \<Rightarrow> cnf"   and n :: "nat" shows "dynamic_component.cnf2bhv tCMP active c t n \<equiv> the_enat (llength (dynamic_component.proj tCMP active c (inf_llist t))) - 1 + (n - dynamic_component.lActive active c t)"
that: fixes n :: "nat" assumes "\<And>z. \<lbrakk>z \<in> lang r; n \<le> length z\<rbrakk> \<Longrightarrow> \<exists>u v w. z = u @ v @ w \<and> length (u @ v) \<le> n \<and> v \<noteq> [] \<and> (\<forall>i. u @ repeat i v @ w \<in> lang r)" shows "thesis"
simps: fixes M :: "('e, 'f, 'g) FSM" shows "well_formed M = (finite_FSM M \<and> (\<forall>s1 x y. x \<notin> inputs M \<or> y \<notin> outputs M \<longrightarrow> succ M (x, y) s1 = {}) \<and> inputs M \<noteq> {} \<and> outputs M \<noteq> {})"
Lower_memI: fixes A :: "'a set"   and L :: "('a, 'b) gorder_scheme"   and x :: "'a" assumes "\<And>y. y \<in> A \<Longrightarrow> x \<sqsubseteq>\<^bsub>L\<^esub> y"   and "x \<in> carrier L" shows "x \<in> Lower L A"
intros_1: fixes con :: "(char list, prop_funs) pform"   and \<Gamma> :: "(char list, prop_funs) pform fset" assumes "con |\<in>| \<Gamma>" shows "task1_1.eff Axiom (\<Gamma> \<turnstile> con) {||}"
consume_top_leq: fixes ts :: "ct list"   and ts' :: "ct list"   and c_t :: "checker_type" assumes "consume (TopType ts) ts' = c_t"   and "length ts \<le> length ts'"   and "c_t \<noteq> Bot" shows "c_t = TopType []"
diamond_def: fixes r :: "('b \<times> 'b) set" shows "\<diamond> r = (r\<inverse> O r \<subseteq> r O r\<inverse>)"
simps_2: fixes s :: "'b set"   and t :: "'b set"   and ts :: "'b set list" shows "difference_list s (t # ts) = (t - s) # difference_list s ts"
comm_ring_axioms: shows "comm_ring R (+) (\<cdot>) \<zero> \<one>"
nump_num_eq: shows "nump = (\<lambda>x. x \<in> num)"
f_Exec_State_Idle_append_replicate_NoMsg_ge_state: fixes localState :: "'d \<Rightarrow> 'e"   and output_fun :: "'d \<Rightarrow> 'f message_af"   and trans_fun :: "'g message_af \<Rightarrow> 'd \<Rightarrow> 'd"   and xs :: "'g message_af list"   and m :: "nat"   and c :: "'d"   and n :: "nat" assumes "State_Idle localState output_fun trans_fun (localState (f_Exec_Comp trans_fun (xs @ NoMsg\<^bsup>m\<^esup>) c))"   and "m \<le> n" shows "localState (f_Exec_Comp trans_fun (xs @ NoMsg\<^bsup>n\<^esup>) c) = localState (f_Exec_Comp trans_fun (xs @ NoMsg\<^bsup>m\<^esup>) c)"
satA_subst: fixes \<xi> :: "Preliminaries.var \<Rightarrow> 'univ"   and \<rho> :: "Preliminaries.var \<Rightarrow> 'fsym trm"   and at :: "('fsym, 'psym) atm" shows "satA \<xi> (substA \<rho> at) = satA (int \<xi> \<circ> \<rho>) at"
subdegree_minus_cutoff_ge: fixes f :: "'b fps"   and n :: "nat" assumes "f - fps_cutoff n f \<noteq> 0" shows "n \<le> subdegree (f - fps_cutoff n f)"
Brouwer_degree2_retraction_map: fixes p :: "nat"   and f :: "(nat \<Rightarrow> real) \<Rightarrow> nat \<Rightarrow> real" assumes "retraction_map (nsphere p) (nsphere p) f" shows "\<bar>Brouwer_degree2 p f\<bar> = 1"
return_time_eqI: fixes f :: "'b \<Rightarrow> 'b"   and X :: "'b set"   and P :: "'b set"   and t :: "real"   and x :: "'b" assumes "auto_ll_on_open f X"   and "closed P"   and "0 < t"   and "t \<in> auto_ll_on_open.existence_ivl0 f X x"   and "auto_ll_on_open.flow0 f X x t \<in> P"   and "\<And>s. \<lbrakk>0 < s; s < t\<rbrakk> \<Longrightarrow> auto_ll_on_open.flow0 f X x s \<notin> P" shows "auto_ll_on_open.return_time f X P x = t"
arcs_G: shows "arcs G = arcs H \<union> {uv, rev_G uv} - {uw, rev_H uw, vw, rev_H vw}"
HComplex_inject: fixes x :: "real star"   and y :: "real star"   and x' :: "real star"   and y' :: "real star" shows "(HComplex x y = HComplex x' y') = (x = x' \<and> y = y')"
assms: shows "s \<in> timpl_closure (Fun f T) TI"
fSuc: fixes n :: "nat" assumes "f n \<in> deriv s \<and> fst (f n) = n \<and> infinite (deriv (snd (f n))) \<and> \<not> is_axiom (s_of_ns (snd (f n)))" shows "f (Suc n) \<in> deriv s \<and> fst (f (Suc n)) = Suc n \<and> snd (f (Suc n)) \<in> set (subs (snd (f n))) \<and> infinite (deriv (snd (f (Suc n)))) \<and> \<not> is_axiom (s_of_ns (snd (f (Suc n))))"
not_less_zero: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and n :: "'b" assumes "class.canonically_ordered_monoid_add plus zero less_eq less" shows "\<not> less n zero"
simps_1: fixes atom_ordering :: "('a \<times> 'a) set"   and A :: "'b" assumes "propositional_atoms atom_ordering" shows "complement (Pos A) = Neg A"
that: fixes f :: "'a \<Rightarrow> 'a"   and g :: "'a \<Rightarrow> 'a" assumes "homeomorphism (cball a r \<inter> T) (cball a r \<inter> T) f g"   and "f u = v"   and "\<And>x. \<lbrakk>x \<in> sphere a r; x \<in> T\<rbrakk> \<Longrightarrow> f x = x" shows "thesis"