hcju/isabelleps huggingface dataset:
Throw: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and e :: "char list exp"   and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and a :: "nat"   and s\<^sub>1 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)" assumes "P \<turnstile> \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>addr a,s\<^sub>1\<rangle>" shows "P \<turnstile> \<langle>throw e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>Throw a,s\<^sub>1\<rangle>"
eq: shows "set (map (insertAt i 0) (find_consistent_signs_at_roots (fss ! i) (removeAt i fss))) = insertAt i 0 ` set (characterize_consistent_signs_at_roots (fss ! i) (removeAt i fss))"
compliant_stateful_ACS_only_state_violations_filternew: shows "\<forall>F\<in>get_offending_flows (get_ACS M) (stateful_policy_to_network_graph \<lparr>hosts = hosts \<T>, flows_fix = flows_fix \<T>, flows_state = filternew_flows_state \<T>\<rparr>). F \<subseteq> backflows (filternew_flows_state \<T>)"
state_\<alpha>_conv_2: fixes thr_\<alpha> :: "'f \<Rightarrow> 'a \<Rightarrow> ('b \<times> 'c \<Rightarrow>f nat) option"   and ws_\<alpha> :: "'g \<Rightarrow> 'a \<Rightarrow> 'e wait_set_status option"   and is_\<alpha> :: "'h \<Rightarrow> 'a set"   and s :: "('c, 'a) locks \<times> ('f \<times> 'd) \<times> 'g \<times> 'h" shows "thr (state_refine_base.state_\<alpha> thr_\<alpha> ws_\<alpha> is_\<alpha> s) = thr_\<alpha> (thr s)"
cset_mapM_def: fixes A :: "'c option cset" shows "cset_mapM A = (if None \<in>\<^sub>c A then None else Some (the `\<^sub>c A))"
Rep_mod: fixes x :: "'b bit0" shows "Rep_bit0 x mod int CARD('b bit0) = Rep_bit0 x"
tagged_partial_division_subset: fixes s :: "('a \<times> 'a set) set"   and i :: "'a set"   and t :: "('a \<times> 'a set) set" assumes "s tagged_partial_division_of i"   and "t \<subseteq> s" shows "t tagged_partial_division_of i"
indep_sets_finite_index_sets: fixes F :: "'e \<Rightarrow> 'a set set"   and I :: "'e set" shows "indep_sets F I = (\<forall>J\<subseteq>I. J \<noteq> {} \<longrightarrow> finite J \<longrightarrow> indep_sets F J)"
assms_4: shows "fst ` CX \<union> CY \<union> fst ` P1 \<subseteq> CZ"
elims: fixes x :: "('e \<Rightarrow> 'd) option"   and xa :: "'e option"   and y :: "'d option" assumes "x <*> xa = y" obtains f :: "'e \<Rightarrow> 'd" and xb :: "'e" where "x = Some f" and "xa = Some xb" and "y = Some (f xb)"   | uv_ :: "'e option" where "x = None" and "xa = uv_" and "y = None"   | uu_ :: "('e \<Rightarrow> 'd) option" where "x = uu_" and "xa = None" and "y = None"
order_apply: fixes f :: "'a perm"   and a :: "'a" shows "order f (f \<langle>$\<rangle> a) = order f a"
phils_transitions_take: fixes n :: "nat"   and ps :: "nat list" assumes "0 < n" shows "Tr\<^sub>P n ps = Tr\<^sub>P n (take n ps)"
S: shows "S \<equiv> P\<^sub>n\<^sub>o\<^sub>r\<^sub>m\<lbrakk>Tr,Up\<rbrakk>"
is_weak_ranking_def: fixes xs :: "'c set list" shows "is_weak_ranking xs = ({} \<notin> set xs \<and> (\<forall>i j. i < length xs \<and> j < length xs \<and> i \<noteq> j \<longrightarrow> xs ! i \<inter> xs ! j = {}))"
jw_App_inv: fixes \<Gamma> :: "(Syntax.var, ty) fmap"   and e :: "Syntax.term"   and e' :: "Syntax.term"   and \<tau>\<^sub>2 :: "ty" assumes "\<Gamma> \<turnstile>\<^sub>W App e e' : \<tau>\<^sub>2" obtains \<tau>\<^sub>1 :: "ty" where "\<Gamma> \<turnstile>\<^sub>W e : Fun \<tau>\<^sub>1 \<tau>\<^sub>2" and "\<Gamma> \<turnstile>\<^sub>W e' : \<tau>\<^sub>1"
fst_hd_half_segments_of_aform: fixes X :: "(real \<times> real) \<times> (real \<times> real) pdevs" assumes "half_segments_of_aform X \<noteq> []" shows "fst (hd (half_segments_of_aform X)) = lowest_vertex (fst X, nlex_pdevs (snd X))"
filter_sup_left_dist_inf: fixes F :: "'a set"   and G :: "'a set"   and H :: "'a set" assumes "ord_class.filter F"   and "ord_class.filter G"   and "ord_class.filter H" shows "filter_sup F (G \<inter> H) = filter_sup F G \<inter> filter_sup F H"
simps_2: fixes l :: "(('c \<times> 'd) \<times> tcolor \<times> 'c \<times> 'd) tree"   and x :: "'c \<times> 'd"   and uu :: "tcolor \<times> 'c \<times> 'd"   and r :: "(('c \<times> 'd) \<times> tcolor \<times> 'c \<times> 'd) tree" shows "invh \<langle>l, (x, uu), r\<rangle> = (invh l \<and> invh r \<and> bheight l = bheight r)"
horizontal_composite: fixes A :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and B :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and F :: "'c \<Rightarrow> 'd"   and G :: "'c \<Rightarrow> 'd"   and \<sigma> :: "'c \<Rightarrow> 'd"   and C :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and H :: "'d \<Rightarrow> 'e"   and K :: "'d \<Rightarrow> 'e"   and \<tau> :: "'d \<Rightarrow> 'e" assumes "natural_transformation A B F G \<sigma>"   and "natural_transformation B C H K \<tau>" shows "natural_transformation A C (H \<circ> F) (K \<circ> G) (\<tau> \<circ> \<sigma>)"
terminates_merge_guards_to_terminates: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "('e, 'd, 'f) com"   and s :: "('e, 'f) xstate" assumes "\<Gamma>\<turnstile>merge_guards c \<down> s" shows "\<Gamma>\<turnstile>c \<down> s"
of_weak_ranking_weak_ranking_aux: shows "of_weak_ranking (R.weak_ranking_aux (inv f -` carrier)) = map_relation (inv f) le"
congruent_triangle_sss_5: fixes a1 :: "'a"   and b1 :: "'a"   and a2 :: "'b"   and b2 :: "'b"   and c1 :: "'a"   and c2 :: "'b" assumes "dist a1 b1 = dist a2 b2"   and "dist b1 c1 = dist b2 c2"   and "dist a1 c1 = dist a2 c2" shows "dist c1 a1 = dist a2 c2"
MtransC_StepC: fixes tval :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and aval :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and cf :: "('a, 'c) com \<times> 'b"   and cf' :: "('a, 'c) com \<times> 'b"   and cf'' :: "('a, 'c) com \<times> 'b" assumes "PL.MtransC tval aval cf cf'"   and "PL.transC tval aval cf' cf''" shows "PL.MtransC tval aval cf cf''"
oGS_\<alpha>_def: fixes I :: "'a \<Rightarrow> node_state option" shows "oGS_\<alpha> I \<equiv> {v. I v = Some DONE}"
in_Vd_conv: fixes v :: "nat"   and c :: "nat \<times> nat \<Rightarrow> 'a"   and src :: "nat"   and d :: "nat" shows "(v \<in> pre_bfs_invar.Vd c src d) = (Graph.connected c src v \<and> Graph.min_dist c src v = d)"
gram_schmidt_fs_int_def: fixes n :: "nat"   and fs :: "'a vec list" shows "gram_schmidt_fs_int n fs \<equiv> gram_schmidt_fs_lin_indpt n fs \<and> gram_schmidt_fs_int_axioms n fs"
inj2: shows "inj_on (\<lambda>xs. {xs} // {\<cong>}) (set Fs\<^sub>2)"
IH: fixes i :: "nat"   and j :: "nat" assumes "xs ! i = x"   and "xs ! j = x"   and "i < j"   and "j < length xs"   and "distinct xs" shows "False"
1: fixes s :: "'a list"   and "is" :: "'b"   and q :: "'a" shows "\<forall>((free, paid), uu_)\<in>set_pmf (snd A (s, is) q). paid = []"
rem_effectless_works_4': fixes as :: "('d \<times> ('e, 'f) fmap) list"   and A :: "('d \<times> ('e, 'f) fmap) set" assumes "as \<in> valid_plans A" shows "rem_effectless_act as \<in> valid_plans A"
functions_of_def: fixes A :: "'c set"   and B :: "'d set"   and N :: "nat multiset" shows "functions_of A B N = {f \<in> A \<rightarrow>\<^sub>E B. image_mset card (mset_set ((\<lambda>b. {x \<in> A. f x = b}) ` B - {{}})) = N}"
std_apart_instance_of\<^sub>l\<^sub>s1: fixes C\<^sub>1 :: "fterm literal set" shows "instance_of\<^sub>l\<^sub>s C\<^sub>1 (C\<^sub>1 \<cdot>\<^sub>l\<^sub>s (\<lambda>x. \<epsilon> (''1'' @ x)))"
is_trail_rev: fixes G :: "('a, 'b) graph"   and v' :: "'a"   and ps :: "('a \<times> 'b \<times> 'a) list"   and v :: "'a" assumes "valid_unMultigraph G" shows "valid_unMultigraph.is_trail G v' (rev_path ps) v = valid_unMultigraph.is_trail G v ps v'"
index_map_inj_on: fixes f :: "'a \<Rightarrow> 'b"   and S :: "'a set"   and y :: "'a"   and xs :: "'a list" assumes "inj_on f S"   and "y \<in> S"   and "set xs \<subseteq> S" shows "index (map f xs) (f y) = index xs y"
rel_nabla_bin_d_closed: fixes x :: "('a \<times> 'a) set"   and q :: "('a \<times> 'a) set" shows "rdom (rel_nabla_bin x q) = rel_nabla_bin x q"
range_eqI: fixes b :: "'a"   and f :: "'b \<Rightarrow> 'a"   and x :: "'b" assumes "b = f x" shows "b \<in> range f"
uminus_uminus_image: fixes S :: "'b set" shows "uminus ` uminus ` S = S"
one_dim_inverse: fixes scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and scaleC :: "complex \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and norm :: "'b \<Rightarrow> real"   and sgn :: "'b \<Rightarrow> 'b"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and cinner :: "'b \<Rightarrow> 'b \<Rightarrow> complex"   and canonical_basis :: "'b list"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "complex" assumes "class.one_dim scaleR scaleC plus zero minus uminus dist norm sgn uniformity open cinner canonical_basis divide inverse one times" shows "inverse (scaleC a one) = scaleC (inverse a) one"
is_empty_def: fixes A :: "'a set" shows "Set.is_empty A = (A = {})"
ctxt_subst_basecase_1: fixes c :: "cmd" shows "\<forall>n. c[n=n \<diamond>]\<^sup>C = c"
Rep_lifted_distrib_lattice_top_induct: fixes y :: "('c, 'd) phi \<Rightarrow> 'd"   and P :: "(('c, 'd) phi \<Rightarrow> 'd) \<Rightarrow> bool" assumes "y \<in> {xf. True}"   and "\<And>x. P (Rep_lifted_distrib_lattice_top x)" shows "P y"
iso_in_homI: fixes f :: "'a \<times> 'a"   and a :: "'a \<times> 'a"   and b :: "'a \<times> 'a" assumes "B.VV.in_hom f a b"   and "B.VV.iso f" shows "B.VV.iso_in_hom f a b"
progress_b_e: fixes \<C> :: "t_context"   and b_es :: "b_e list"   and ts :: "t list"   and ts' :: "t list"   and \<S> :: "s_context"   and cs :: "e list"   and i :: "nat"   and vs :: "v list"   and s :: "s" assumes "\<C> \<turnstile> b_es : ts _> ts'"   and "\<S>\<bullet>\<C> \<turnstile> cs : [] _> ts"   and "\<And>lholed. \<not> Lfilled 0 lholed [$Return] (cs @ ($* b_es))"   and "\<And>i lholed. \<not> Lfilled 0 lholed [$Br i] (cs @ ($* b_es))"   and "const_list cs"   and "\<not> const_list ($* b_es)"   and "i < length (s_inst \<S>)"   and "length (local \<C>) = length vs"   and "Option.is_none (memory \<C>) = Option.is_none (inst.mem (inst s ! i))" shows "\<exists>a s' vs' es'. \<lparr>s;vs;cs @ ($* b_es)\<rparr> \<leadsto>_ i \<lparr>s';vs';es'\<rparr>"
I_in_cb_def: fixes s :: "'a list"   and j :: "nat" shows "I_in_cb s j \<equiv> \<lambda>i. if j = 1 then i = 0 else strict_border (take (i - 1) s) (take (j - 1) s) \<and> \<ff> s j \<le> i + 1"
map_bset_permute: fixes p :: "perm"   and B :: "'a set['b]" shows "p \<bullet> B = map_bset (permute p) B"
IH: fixes ys :: "['a]" shows "eq\<cdot>xs\<cdot>(take\<cdot>(length\<cdot>xs)\<cdot>ys) \<sqsubseteq> isPrefixOf\<cdot>xs\<cdot>ys"
snoc_2: shows "(m, l') = foldl f (m', t') (xs @ [x])"
HFail_HInv4b_p: fixes s :: "state"   and s' :: "state"   and p :: "Proc" assumes "HFail s s' p" shows "HInv4b s' p"
0: shows "\<not> real_of_int (T_A' n) \<le> real (n * length init)"
B_def: fixes n :: "nat" shows "B n = cbox (- real n *\<^sub>R One) (real n *\<^sub>R One)"
J: fixes k :: "real" shows "((\<lambda>t. exp (t *\<^sub>R - s) * g t) has_integral J k) {0..k}"
max_hi_exists: fixes t :: "('b ivl \<times> 'b) tree" assumes "inv_max_hi t"   and "t \<noteq> \<langle>\<rangle>" shows "\<exists>a\<in>Tree2.set_tree t. high a = max_hi t"
mute_preserves: fixes q :: "'a"   and k :: "nat"   and l :: "nat"   and r :: "'a"   and A :: "'a mat"   and r' :: "'a"   and A' :: "'a mat"   and i :: "nat"   and j :: "nat" assumes "local.mute q k l (r, A) = (r', A')"   and "i < dim_row A"   and "j < dim_col A"   and "l < dim_row A"   and "k < dim_row A"   and "i \<noteq> k" shows "A' $$ (i, j) = A $$ (i, j)"
assms_4: shows "cs\<^bsup>\<pi> \<guillemotleft> k\<^esup> n = cs\<^bsup>\<pi>' \<guillemotleft> k'\<^esup> n'"
directly_attracted_def: fixes p :: "Player"   and S :: "'a set" shows "directly_attracted p S \<equiv> {v \<in> V - S. \<not> deadend v \<and> (v \<in> VV p \<longrightarrow> (\<exists>w. v \<rightarrow> w \<and> w \<in> S)) \<and> (v \<in> VV p** \<longrightarrow> (\<forall>w. v \<rightarrow> w \<longrightarrow> w \<in> S))}"
icard_atLeast: fixes u :: "nat" shows "icard {u..} = \<infinity>"
ifFalse: fixes subst :: "('a, 'b, 'c) expr \<Rightarrow> 'c \<Rightarrow> ('a, 'b, 'c) expr \<Rightarrow> ('a, 'b, 'c) expr"   and s :: "'a \<Rightarrow> (('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('a, 'b, 'c) expr) option"   and r :: "'a"   and \<sigma> :: "'b \<Rightarrow> ('a, 'b, 'c) val option"   and \<tau> :: "'b \<Rightarrow> ('a, 'b, 'c) val option"   and \<E> :: "('a, 'b, 'c) cntxt"   and e1 :: "('a, 'b, 'c) expr"   and e2 :: "('a, 'b, 'c) expr" assumes "substitution subst"   and "s r = Some (\<sigma>, \<tau>, \<E> [Ite (VE (CV F)) e1 e2])" shows "substitution.revision_step subst r s (s(r \<mapsto> (\<sigma>, \<tau>, \<E> [e2])))"
dcd_4: shows "cs\<^bsup>path \<sigma>\<^esup> n = cs\<^bsup>path \<sigma>'\<^esup> n'"
tauChainFresh: fixes P :: "('a, 'b, 'c) psi"   and P' :: "('a, 'b, 'c) psi"   and \<Psi> :: "'b"   and x :: "name" assumes "\<Psi> \<rhd> P \<Longrightarrow>\<^sup>^\<^sub>\<tau> P'"   and "x \<sharp> P" shows "x \<sharp> P'"
no: fixes x :: "'a" assumes "norm x = 1"   and "b \<bullet> x \<noteq> 1" shows "(norm (f x))\<^sup>2 = 4 * (1 + b \<bullet> x) / (1 - b \<bullet> x)"
cindex_polyE_rec: fixes a :: "real"   and b :: "real"   and p :: "real poly"   and q :: "real poly" assumes "a < b"   and "coprime p q" shows "cindex_polyE a b q p = real_of_int (cross_alt q p a b) / 2 + cindex_polyE a b (- (p mod q)) q"
invar_step: fixes r :: "('Q, 'L) ta_rule"   and it :: "('Q, 'L) ta_rule set"   and \<Sigma> :: "'\<Sigma>" assumes "r \<in> it"   and "it \<subseteq> {r \<in> \<delta>. q \<in> set (rhsq r)}"   and "cinvar it \<Sigma>" shows "cinvar (it - {r}) (cstep r \<Sigma>)"
Eq_subst_fm_Iff: fixes t :: "tm"   and u :: "tm"   and H :: "fm set"   and A :: "fm"   and i :: "name" shows "insert (t EQ u) H \<turnstile> A(i::=t) IFF A(i::=u)"
gdpr_scenario'_def: shows "gdpr_scenario' \<equiv> Infrastructure ex_graph' local_policies"
dfs_algo_def: fixes \<Sigma>i :: "'d set"   and R :: "('d \<times> 'd) set" shows "dfs_algo \<Sigma>i R \<equiv> \<lparr>wa_cond = dfs_cond, wa_step = dfs_step R, wa_initial = dfs_initial \<Sigma>i, wa_invar = dfs_invar \<Sigma>i R\<rparr>"
inverse_in_subgroup_generated: fixes a :: "'a"   and H :: "'a set" assumes "a \<in> subgroup_generated H" shows "local.inverse a \<in> subgroup_generated H"
lec_of_constraint_real: fixes c :: "constraint"   and v :: "nat \<Rightarrow> real" assumes "is_le c" shows "v \<Turnstile>\<^sub>l\<^sub>e of_rat_lec (lec_of_constraint c) = v \<Turnstile>\<^sub>c c"
assms_2: shows "\<forall>\<^sub>F n in sequentially. 0 < f n"
simps_2: fixes l :: "'b tree23"   and uu :: "'b"   and r :: "'b tree23" shows "complete \<langle>l, uu, r\<rangle> = (height l = height r \<and> complete l \<and> complete r)"
tuple_in'_def: shows "tuple_in' \<equiv> fold (\<lambda>(t, X) tuple_in. upd_set tuple_in (\<lambda>_. t) {as \<in> X. valid_tuple tuple_since (t, as)}) move tuple_in"
echelon_form_echelon_form_of: fixes bezout :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<times> 'b \<times> 'b \<times> 'b \<times> 'b"   and A :: "(('b, 'c) vec, 'd) vec" assumes "is_bezout_ext bezout" shows "echelon_form (echelon_form_of A bezout)"
admissible_def: fixes w :: "face \<Rightarrow> nat"   and g :: "graph" shows "admissible w g \<equiv> admissible\<^sub>1 w g \<and> admissible\<^sub>2 w g \<and> admissible\<^sub>3 w g"
represents_add_rat: fixes p :: "int poly"   and x :: "'a"   and r :: "rat" assumes "p represents x" shows "poly_add_rat r p represents of_rat r + x"
main_lemma_2: fixes \<Gamma> :: "(name \<times> ty) list"   and p :: "trm"   and q :: "trm"   and T :: "ty" assumes "\<Gamma> \<turnstile> p \<leftrightarrow> q : T" shows "\<Gamma> \<turnstile> p is q : T"
pow_generator_mod_int: fixes k :: "int" shows "\<^bold>g [^] (k mod int (order \<G>)) = \<^bold>g [^] k"
evContainsAtom: fixes fs :: "formula list"   and f :: "nat \<Rightarrow> (signs \<times> predicate \<times> vbl list) list \<times> (nat \<times> formula) list"   and i :: "nat"   and z :: "signs"   and P :: "predicate"   and vs :: "vbl list" assumes "branch subs (pseq fs) f"   and "\<forall>n. \<not> proofTree (tree subs (f n))"   and "EV (contains f (i, FAtom z P vs))" shows "\<exists>n. \<forall>m. FAtom z P vs \<in> set (sequent (f (n + m)))"
show_r_zero: fixes x :: "'a"   and a :: "'a" assumes "x \<in> carrier R"   and "a \<in> carrier R" shows "(x \<oplus> a = x) = (a = \<zero>)"
flow_initial_time_if: fixes x0 :: "'a"   and t0 :: "real" shows "flow0 x0 (t0 - t0) = (if t0 \<in> UNIV \<and> x0 \<in> X then x0 else (0::'a))"
assms: shows "\<forall>x y. x \<noteq> y \<longrightarrow> f x \<noteq> f y"
keys_phidefNodes: fixes g :: "'g" assumes "finite (dom (Mapping.lookup (phis g)))" shows "Mapping.keys (uninst_code.ssa.phidefNodes phis g) = snd ` dom (Mapping.lookup (phis g))"
finite_h': fixes e :: "'v \<times> 'v" shows "h' e \<noteq> \<top>"
get_heap_returns_result: fixes h :: "'x"   and f :: "'x \<Rightarrow> ('x, 'y, 'z) prog"   and x :: "'z" shows "h \<turnstile> Heap_Error_Monad.bind get_heap f \<rightarrow>\<^sub>r x = h \<turnstile> f h \<rightarrow>\<^sub>r x"
case_resumption_def: fixes t :: "'g \<Rightarrow> 'f"   and l :: "'h \<Rightarrow> ('g, 'h, 'i, 'j) resumption \<Rightarrow> 'f"   and br :: "'i \<Rightarrow> ('j \<Rightarrow> ('g, 'h, 'i, 'j) resumption) \<Rightarrow> 'f"   and r :: "('g, 'h, 'i, 'j) resumption" shows "(case r of Terminal x \<Rightarrow> t x | Linear x xa \<Rightarrow> l x xa | Branch x xa \<Rightarrow> br x xa) = case_RESUMPTION t (\<lambda>b r. l b (Abs_resumption r)) (\<lambda>c rs. br c (\<lambda>d. Abs_resumption (rs d))) (Rep_resumption r)"
Map_hcomp: fixes \<mu> :: "('a E.term, 'a) arr"   and \<nu> :: "('a E.term, 'a) arr" assumes "arr \<mu>"   and "arr \<nu>"   and "src \<mu> = trg \<nu>" shows "Map (\<mu> \<star> \<nu>) = B.can (Cod \<mu> \<^bold>\<lfloor>\<^bold>\<star>\<^bold>\<rfloor> Cod \<nu>) (Cod \<mu> \<^bold>\<star> Cod \<nu>) \<cdot>\<^sub>B (Map \<mu> \<star>\<^sub>B Map \<nu>) \<cdot>\<^sub>B B.can (Dom \<mu> \<^bold>\<star> Dom \<nu>) (Dom \<mu> \<^bold>\<lfloor>\<^bold>\<star>\<^bold>\<rfloor> Dom \<nu>)"
Left_Coset_def: fixes x :: "'a"   and H :: "'b set" shows "x \<cdot>| H = {x \<cdot> h |h. h \<in> H}"
indexToVertexList_natToVertexList_eq: fixes f :: "face"   and v :: "nat"   and "is" :: "nat list" assumes "distinct (vertices f)"   and "v \<in> \<V> f"   and "\<forall>i\<in>set is. i < |vertices f|"   and "is \<noteq> []"   and "hd is = 0" shows "indexToVertexList f v is = natToVertexList v f is"
list_gmean_gt_iff: fixes one :: "real list"   and two :: "real list" assumes "0 < prod_list one"   and "0 < prod_list two"   and "one \<noteq> []"   and "two \<noteq> []"   and "prod_list two < prod_list one"   and "length one = length two" shows "gmean two < gmean one"
C_subset: fixes M2 :: "('d, 'e, 'f) FSM"   and M1 :: "('d, 'e, 'g) FSM"   and \<Omega> :: "('d, 'e) ATC set"   and V :: "'d list set"   and m :: "nat"   and i :: "nat" shows "C M2 M1 \<Omega> V m i \<subseteq> TS M2 M1 \<Omega> V m i"
module_vec_simps_4: fixes ty :: "'b itself"   and n :: "nat" shows "(\<odot>\<^bsub>module_vec ty n\<^esub>) = (\<cdot>\<^sub>v)"
simps_8: fixes \<sigma> :: "(char \<Rightarrow> trm option) \<times> (char \<Rightarrow> trm option) \<times> (char \<Rightarrow> fml option) \<times> (char \<Rightarrow> game option)"   and U :: "variable set"   and x :: "char"   and \<theta> :: "trm" shows "usubstappp \<sigma> U (ODE x \<theta>) = (U \<union> {RVar x, DVar x}, ODEo x (usubstappt \<sigma> (U \<union> {RVar x, DVar x}) \<theta>))"
pow_mono_exp: fixes a :: "'d"   and m :: "nat"   and n :: "nat" assumes "(1::'d) \<le> a"   and "m \<le> n" shows "a ^ m \<le> a ^ n"
mayDiverge_not_transT: fixes tval :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and aval :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and c :: "('a, 'c) com"   and s :: "'b"   and s' :: "'b" assumes "PL.mayDiverge tval aval c s" shows "\<not> PL.transT tval aval (c, s) s'"
fst_divmod: 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 modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and euclidean_size :: "'b \<Rightarrow> nat"   and division_segment :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and divmod :: "num \<Rightarrow> num \<Rightarrow> 'b \<times> 'b"   and divmod_step :: "num \<Rightarrow> 'b \<times> 'b \<Rightarrow> 'b \<times> 'b"   and m :: "num"   and n :: "num" assumes "class.unique_euclidean_semiring_numeral divide plus minus zero times one modulo euclidean_size division_segment less_eq less divmod divmod_step" shows "fst (divmod m n) = divide (numeral.numeral one plus m) (numeral.numeral one plus n)"
image_mset_remove1_mset_if: fixes f :: "'d \<Rightarrow> 'c"   and M :: "'d multiset"   and a :: "'d" shows "image_mset f (remove1_mset a M) = (if a \<in># M then remove1_mset (f a) (image_mset f M) else image_mset f M)"
Spy_know_private_keys_iff: fixes evs :: "event list"   and b :: "keymode"   and A :: "agent" assumes "evs \<in> certified_mail" shows "(Key (privateKey b A) \<in> parts (knows Spy evs)) = (A \<in> bad)"
ltl_lang_equiv_def: fixes \<phi> :: "'a ltln"   and \<psi> :: "'a ltln" shows "\<phi> \<sim>\<^sub>L \<psi> \<equiv> \<forall>w. w \<Turnstile>\<^sub>n \<phi> = w \<Turnstile>\<^sub>n \<psi>"
Empty_is_rbt: fixes less :: "'a \<Rightarrow> 'a \<Rightarrow> bool" shows "ord.is_rbt less Empty"
timedelayed_e1: fixes \<Gamma> :: "'b constr list"   and n :: "nat"   and K\<^sub>1 :: "clock"   and \<delta>\<tau> :: "'b tag_const"   and K\<^sub>2 :: "clock"   and K\<^sub>3 :: "clock"   and \<Psi> :: "'b TESL_atomic list"   and \<Phi> :: "'b TESL_atomic list" shows "\<Gamma>, n \<turnstile> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^sub>e K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Phi>"
comp_vwb_lens: fixes x :: "'a \<Longrightarrow> 'b"   and y :: "'b \<Longrightarrow> 'c" assumes "vwb_lens x"   and "vwb_lens y" shows "vwb_lens (x ;\<^sub>L y)"
deflation_below_comp1: fixes f :: "'a \<rightarrow> 'a"   and g :: "'a \<rightarrow> 'a"   and x :: "'a" assumes "deflation f"   and "deflation g"   and "f \<sqsubseteq> g" shows "f\<cdot>(g\<cdot>x) = f\<cdot>x"
poly_root_factor_3: fixes r :: "'b"   and k :: "'b" shows "(poly [:r, 1::'b:] k = (0::'b)) = (k = - r)"
insert_if: fixes A :: "'b set"   and g :: "'b \<Rightarrow> 'a"   and x :: "'b" assumes "finite A" shows "sum g (insert x A) = (if x \<in> A then sum g A else g x + sum g A)"
S9: shows "nat_to_list u = c_fst (c_snd (u - 1)) # c_unfold (c_len u - 1) (c_snd (c_snd (u - 1)))"
lt_Integer_numeral_simps_5: fixes k :: "num" shows "lt\<cdot>0\<cdot>(numeral k) = TT"