hcju/isabelleps huggingface dataset:
coplanar_6: fixes A :: "Points"   and B :: "Points"   and C :: "Points"   and \<alpha> :: "Points"   and \<gamma> :: "Points"   and \<beta> :: "Points" assumes "rk {A, B, C} = 3"   and "rk {B, C, \<alpha>} = 2"   and "rk {A, B, \<gamma>} = 2"   and "rk {A, C, \<beta>} = 2" shows "rk {A, B, C, \<alpha>, \<beta>, \<gamma>} = 3"
lin_b: shows "class.linorder (\<sqsubseteq>\<^sub>b) (\<sqsubset>\<^sub>b)"
simps_9: fixes f :: "'f"   and g :: "'d set"   and c :: "('d, 'e, 'f) com" shows "Language.normalize (Guard f g c) = Guard f g (Language.normalize c)"
assms_2: fixes i :: "V" assumes "i \<in>\<^sub>\<circ> I" shows "A \<subseteq>\<^sub>\<circ> f i"
sset_range: fixes s :: "'d stream" shows "sset s = range ((!!) s)"
stream_from_llist: shows "type_definition llist_of_stream stream_of_llist {xs. \<not> lfinite xs}"
welldefined_subsets2: shows "all_list_constr_R subsets2 (length qs2)"
rcosetGM1g_subset_G: fixes g :: "'a"   and x :: "'a" assumes "g \<in> carrier G"   and "x \<in> M1 #> g" shows "x \<in> carrier G"
ipcidr_tuple_to_wordinterval_def: fixes iprng :: "'b word \<times> nat" shows "ipcidr_tuple_to_wordinterval iprng \<equiv> iprange_interval (ipcidr_to_interval iprng)"
hd_in_awalk_verts_1: fixes u :: "'a"   and p :: "'b list"   and v :: "'a" assumes "H.awalk u p v" shows "u \<in> set (H.awalk_verts u p)"
shift_simps_1: fixes xs :: "'b list"   and s :: "'b stream" shows "shd (xs @- s) = (if xs = [] then shd s else hd xs)"
inner_divide_right: fixes a :: "'c"   and b :: "'c"   and m :: "real" shows "a \<bullet> (b / of_real m) = a \<bullet> b / m"
distinct_11: fixes x1 :: "('c, 'd) Term.term"   and x71 :: "'d list"   and x72 :: "(('c, 'd) Term.term \<times> ('c, 'd) Term.term) list"   and x73 :: "(('c, 'd) Term.term \<times> ('c, 'd) Term.term) list" shows "send\<langle>x1\<rangle> \<noteq> \<forall>x71\<langle>\<or>\<noteq>: x72 \<or>\<notin>: x73\<rangle>"
steps: shows "\<Gamma>\<turnstile> (cs, css, s) \<rightarrow>\<^sup>* ([], [], t)"
InvariantConsistentAfterBacktrack: fixes M :: "(Literal \<times> bool) list"   and M' :: "(Literal \<times> bool) list" assumes "InvariantUniq M"   and "InvariantConsistent M"   and "decisions M \<noteq> []"   and "M' = prefixBeforeLastDecision M @ [(opposite (lastDecision M), False)]" shows "InvariantConsistent M'"
numneg_nz: fixes a :: "Ferrack.num" assumes "nozerocoeff a" shows "nozerocoeff (numneg a)"
residue_holomorphic_over_power': fixes A :: "complex set"   and f :: "complex \<Rightarrow> complex"   and n :: "nat" assumes "open A"   and "0 \<in> A"   and "f holomorphic_on A" shows "residue (\<lambda>z. f z / z ^ Suc n) 0 = (deriv ^^ n) f 0 / fact n"
pt_supp_fun_subset: fixes f :: "'a \<Rightarrow> 'c"   and x :: "'a" assumes "pt TYPE('a) TYPE('b)"   and "pt TYPE('c) TYPE('b)"   and "at TYPE('b)"   and "finite (supp f)"   and "finite (supp x)" shows "supp (f x) \<subseteq> supp f \<union> supp x"
orthogonal_cut': fixes f :: "'v vertex \<times> 'v vertex \<Rightarrow> ennreal"   and S :: "'v vertex set" assumes "orthogonal \<Delta>'' f S"   and "flow \<Delta>'' f" shows "orthogonal \<Delta> (\<lambda>(x, y). f (Edge x y, Vertex y)) (Vertex -` S)"
intros_3: fixes X :: "msg"   and Y :: "msg"   and H :: "msg set" assumes "\<langle>X, Y\<rangle> \<in> analz H" shows "Y \<in> analz H"
is_sig_red_top_tailI: fixes sing_reg :: "'t \<Rightarrow> 't \<Rightarrow> bool"   and top_tail :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and F :: "('t \<Rightarrow>\<^sub>0 'b) set"   and p :: "'t \<Rightarrow>\<^sub>0 'b" assumes "is_sig_red sing_reg top_tail F p" shows "is_sig_red sing_reg (\<preceq>) F p"
iprev_nth_mono: fixes a :: "nat"   and b :: "nat"   and I :: "nat set" assumes "a \<le> b" shows "I \<leftarrow> b \<le> I \<leftarrow> a"
has_vector_derivative_mult: fixes f :: "real \<Rightarrow> 'a"   and f' :: "'a"   and x :: "real"   and s :: "real set"   and g :: "real \<Rightarrow> 'a"   and g' :: "'a" assumes "(f has_vector_derivative f') (at x within s)"   and "(g has_vector_derivative g') (at x within s)" shows "((\<lambda>x. f x * g x) has_vector_derivative f x * g' + f' * g x) (at x within s)"
iffb_inv2: fixes \<phi> :: "'c set \<Rightarrow> 'd set" assumes "Inf_pres \<phi>" shows "(fb\<^sub>\<F> \<circ> fb\<^sup>-\<^sub>\<F>) \<phi> = \<phi>"
lemma1a: fixes tp :: "bool list"   and tq :: "bool list"   and L :: "bool list \<times> ('b option \<Rightarrow> bool list \<Rightarrow> bool list set) \<times> (bool list \<Rightarrow> bool)"   and R :: "bool list \<times> ('b option \<Rightarrow> bool list \<Rightarrow> bool list set) \<times> (bool list \<Rightarrow> bool)"   and p :: "bool list" assumes "(tp, tq) \<in> (eps (or L R))\<^sup>*"   and "tp = True # p" shows "\<exists>q. (p, q) \<in> (eps L)\<^sup>* \<and> tq = True # q"
Val: fixes P :: "(char list list \<times> char list exp) prog"   and v :: "val"   and s :: "heap \<times> sfields \<times> sheap" shows "P \<turnstile> \<langle>Val v,s\<rangle> \<Rightarrow> \<langle>Val v,s\<rangle>"
assign_simp: fixes x :: "char list"   and a :: "aexp"   and s :: "char list \<Rightarrow> int"   and s' :: "char list \<Rightarrow> int" shows "(x ::= a, s) \<Rightarrow> s' = (s' = s[a/x])"
assms_4: shows "h \<turnstile> get_parent node_ptr \<rightarrow>\<^sub>r None"
prm_set_unit_inaction: fixes a :: "'b"   and S :: "'b set"   and b :: "'b" assumes "a \<notin> S"   and "b \<notin> S" shows "[a \<leftrightarrow> b] {$} S = S"
cl_op_prop: fixes x :: "'b"   and y :: "'b" shows "(cl_op (x \<squnion> y) = cl_op y) = (cl_op x \<le> cl_op y)"
multiset_partition: fixes M :: "'c multiset"   and P :: "'c \<Rightarrow> bool" shows "M = filter_mset P M + {#x \<in># M. \<not> P x#}"
map_add_find_right: fixes n :: "'b \<Rightarrow> 'a option"   and k :: "'b"   and xx :: "'a"   and m :: "'b \<Rightarrow> 'a option" assumes "n k = Some xx" shows "(m ++ n) k = Some xx"
L3_Controller_One: fixes y :: "nat \<Rightarrow> nat list"   and l :: "nat \<Rightarrow> bit"   and z :: "nat \<Rightarrow> bit list"   and t :: "nat" assumes "Controller_L y (fin_inf_append [Zero] l) l z"   and "l t = One"   and "\<not> hd (y (Suc t)) < 700" shows "z (Suc t) = [Zero]"
preprocess_def: fixes l :: "('a \<times> QDelta ns_constraint) list" shows "preprocess l = (case preprocess_part_1 (map (map_prod id normalize_ns_constraint) l) of (t, as, ui) \<Rightarrow> case preprocess_part_2 as t of (t, tv) \<Rightarrow> (t, as, tv, ui))"
app: shows "app\<^sub>i (i, P, mxs, mpc, rT, \<tau>')"
simps_4: fixes f1 :: "'a"   and f2 :: "color \<Rightarrow> ('b, 'c) rbt \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> ('b, 'c) rbt \<Rightarrow> 'a" shows "(case Empty of Empty \<Rightarrow> f1 | Branch x xa xb xc xd \<Rightarrow> f2 x xa xb xc xd) = f1"
infsetsum'_converges_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'_converges f A = infsetsum'_converges f B"
Ri_subs_Ri_diff_Rf: fixes N :: "'a literal multiset set" shows "Ri N \<subseteq> Ri (N - Rf N)"
vrat_inverse_inverse: fixes x :: "V" assumes "x \<in>\<^sub>\<circ> \<rat>\<^sub>\<circ>" shows "x = (x\<inverse>\<^sub>\<rat>)\<inverse>\<^sub>\<rat>"
remove_child_def: fixes ptr :: "(_) object_ptr"   and child :: "(_) node_ptr" shows "remove_child ptr child = Heap_Error_Monad.bind (get_child_nodes ptr) (\<lambda>children. if child \<notin> set children then error NotFoundError else Heap_Error_Monad.bind (get_owner_document (cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r child)) (\<lambda>owner_document. Heap_Error_Monad.bind (get_disconnected_nodes owner_document) (\<lambda>disc_nodes. Heap_Error_Monad.bind (set_disconnected_nodes owner_document (child # disc_nodes)) (\<lambda>_. set_child_nodes ptr (remove1 child children)))))"
assms_3: assumes "c \<notin>\<^sub>\<circ> A"   and "c \<notin>\<^sub>\<circ> B" shows "P"
neg_numeral_fps_const: fixes k :: "num" shows "- numeral k = fps_const (- numeral k)"
equal_when_halt: fixes s :: "nat"   and lm :: "nat list"   and A :: "abc_inst list"   and na :: "nat"   and lma :: "nat list"   and nb :: "nat"   and lmb :: "nat list" assumes "abc_steps_l (s, lm) A na = (length A, lma)"   and "abc_steps_l (s, lm) A nb = (length A, lmb)" shows "lma = lmb"
cases: fixes \<sigma> :: "('addr, 'thread_id) locks \<times> (('thread_id \<Rightarrow> ((status \<times> 'x) \<times> 'addr \<Rightarrow>f nat) option) \<times> 'heap) \<times> ('thread_id \<Rightarrow> 'addr wait_set_status option) \<times> 'thread_id set"   and a :: "('thread_id \<times> ('addr, 'thread_id) obs_event action) llist" assumes "h.mthr.if.\<E>p \<sigma> a" obtains E' :: "('thread_id \<times> ('addr, 'thread_id, status \<times> 'x, 'heap, 'addr, ('addr, 'thread_id) obs_event action) thread_action) llist" where "a = lconcat (lmap (\<lambda>(t, ta). llist_of (map (Pair t) \<lbrace>ta\<rbrace>\<^bsub>o\<^esub>)) E')" and "h.mthr.if.mthr.Runs \<sigma> E'"
weight_finite: fixes x :: "'v" shows "weight (quotient_web \<Gamma> f) x \<noteq> \<top>"
trans: fixes A :: "wall"   and B :: "wall"   and C :: "wall" assumes "A ~f B"   and "B ~f C" shows "A ~f C"
subset_insert_iff: fixes A :: "'b set"   and x :: "'b"   and B :: "'b set" shows "((A::'b set) \<subseteq> (x::'b) \<triangleright> (B::'b set)) = (if x \<in> A then A - {x} \<subseteq> B else A \<subseteq> B)"
circ_rtc_2: fixes x :: "'a" shows "Sf x (x\<^sup>\<circ>) = x\<^sup>\<circ>"
ladder_stepdown_prepare: fixes L :: "(nat \<times> nat \<times> nat) list" assumes "1 < length L" shows "L = (ladder_n L 0, ladder_j L 0, ladder_i L 0) # (ladder_n L 1, ladder_j L 1, ladder_ix L 1) # drop 2 L"
map_add_left_eq: fixes h\<^sub>0 :: "'c \<Rightarrow> 'd option"   and h :: "'c \<Rightarrow> 'd option"   and h\<^sub>1 :: "'c \<Rightarrow> 'd option" assumes "h\<^sub>0 ++ h = h\<^sub>1 ++ h"   and "h\<^sub>0 \<bottom> h"   and "h\<^sub>1 \<bottom> h" shows "h\<^sub>0 = h\<^sub>1"
app_equality: fixes f :: "hf"   and x :: "hf"   and y :: "hf" assumes "hfunction f"   and "\<langle>x, y\<rangle> \<^bold>\<in> f" shows "happ f x = y"
stieltjes_gamma_def: fixes n :: "nat" shows "stieltjes_gamma n = of_real (\<Sum>k. ln (real (k + 1)) ^ n / real (k + 1) - (ln (real (k + 2)) ^ (n + 1) - ln (real (k + 1)) ^ (n + 1)) / (real n + 1))"
raw_converter_invariant_run_pfinite_converter: fixes \<I> :: "('h, 'i) \<I>"   and \<I>' :: "('j, 'k) \<I>" shows "raw_converter_invariant \<I> \<I>' run_converter (\<lambda>conv. pfinite_converter \<I> \<I>' conv \<and> \<I>, \<I>' \<turnstile>\<^sub>C conv \<surd>)"
leq_refl: fixes x :: "('f, 'g, 'h) Resumption.resumption" shows "resumption_ord x x"
consequence_relation_family_axioms: shows "consequence_relation_family Bot_F Q (\<lambda>q N1 N2. entails_q q (\<G>_Fset_q q N1) (\<G>_Fset_q q N2))"
nan_not_zero: fixes x :: "('a, 'b) IEEE.float" assumes "IEEE.is_nan x" shows "\<not> IEEE.is_zero x"
prime_factors_product: 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 x :: "'b"   and y :: "'b" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize"   and "x \<noteq> zero"   and "y \<noteq> zero" shows "set_mset (factorial_semiring.prime_factorization zero times one normalize (times x y)) = set_mset (factorial_semiring.prime_factorization zero times one normalize x) \<union> set_mset (factorial_semiring.prime_factorization zero times one normalize y)"
absorb_Some_NoneD: fixes j :: "jump"   and abr :: "abrupt" assumes "absorb j (Some abr) = None" shows "abr = Jump j"
predCC_def: fixes S :: "Vars.var set"   and f :: "Arity \<rightarrow> CoCalls" shows "predCC S f = (\<Lambda> a. if a \<noteq> 0 then f\<cdot>(pred\<cdot>a) G|` S else S\<^sup>2)"
fset_intro: fixes x :: "'b fset"   and y :: "'b fset" assumes "\<langle>x\<rangle>\<^sub>f = \<langle>y\<rangle>\<^sub>f" shows "x = y"
handshake_in_syncD: fixes s :: "'e process_name \<Rightarrow> ('f, 'e, 'g, 'h) Model.local_state" assumes "All (ghost_hs_in_sync (s sys))"   and "handshake_phase_inv s" shows "\<forall>m'. ghost_hs_phase (s (mutator m')) = sys_ghost_hs_phase s"
LPD_safe: fixes g :: "safety"   and mr :: "mregex"   and \<phi>s :: "Formula.formula list"   and ms :: "mregex" assumes "safe_regex Futu g (from_mregex mr \<phi>s)"   and "ms \<in> LPD mr" shows "safe_regex Futu g (from_mregex ms \<phi>s)"
measure_eqI_finite: fixes M :: "'c measure"   and A :: "'c set"   and N :: "'c measure" assumes "sets M = Pow A"   and "sets N = Pow A"   and "finite A"   and "\<And>a. a \<in> A \<Longrightarrow> emeasure M {a} = emeasure N {a}" shows "M = N"
mat_assoc_test_5: fixes A :: "complex mat"   and n :: "nat"   and B :: "complex mat"   and C :: "complex mat"   and D :: "complex mat" assumes "A \<in> carrier_mat n n"   and "B \<in> carrier_mat n n"   and "C \<in> carrier_mat n n"   and "D \<in> carrier_mat n n" shows "A + (B - C) = A + B - C"
lfp_subset_iter: fixes f :: "'a set \<Rightarrow> 'a set"   and f' :: "'a set \<Rightarrow> 'a set"   and D :: "'a set"   and n :: "nat"   and A :: "'a set" assumes "mono f"   and "\<And>X. f X \<subseteq> f' X"   and "lfp f \<subseteq> D" shows "lfp f \<subseteq> iter f' n A D"
abs_eq: shows "1 = Interval (1::'a, 1::'a)"
at_within_cbox_finite: fixes x :: "'b"   and a :: "'b"   and b :: "'b"   and S :: "'b set" assumes "x \<in> box a b"   and "x \<notin> S"   and "finite S" shows "at x within cbox a b - S = at x"
AS_prec_after_step: shows "\<forall>s a a'. (\<exists>aseq\<in>AS_set. is_sub_seq a a' aseq) \<and> invariant s \<and> AS_precondition s (current s) a \<and> \<not> aborting s (current s) a \<and> \<not> waiting s (current s) a \<longrightarrow> AS_precondition (kstep s a) (current s) a'"
simps_1: fixes \<sigma> :: "'a"   and y :: "'b"   and \<tau> :: "'a" shows "cSnoc \<lbrace> \<sigma> \<rbrace> y \<tau> = \<lbrace> \<sigma> \<rbrace> \<cdot> y \<cdot> \<lbrace> \<tau> \<rbrace>"
F_inj_composition_permutation: fixes \<phi> :: "nat \<Rightarrow> nat"   and g :: "nat \<Rightarrow> nat" assumes "\<phi> permutes {0..<n}"   and "g \<in> local.F_inj" shows "g \<circ> \<phi> \<in> local.F_inj"
Lyndon_mono_sing: fixes u :: "'a list" assumes "Lyndon u" shows "Lyndon_mono [u]"
induct: fixes P :: "int \<Rightarrow> int list \<Rightarrow> atom \<Rightarrow> bool"   and a0 :: "int"   and a1 :: "int list"   and a2 :: "atom" assumes "\<And>i' ks' i k ks. P i' ks' (Le i (k # ks))"   and "\<And>i' ks' d i k ks. P i' ks' (Dvd d i (k # ks))"   and "\<And>i' ks' d i k ks. P i' ks' (NDvd d i (k # ks))"   and "\<And>i' ks' v. P i' ks' (Le v [])"   and "\<And>i' ks' v va. P i' ks' (Dvd v va [])"   and "\<And>i' ks' v va. P i' ks' (NDvd v va [])" shows "P a0 a1 a2"
simps: fixes prob :: "('a, 'b, 'c, 'd) ifc_problem"   and \<pi> :: "nat \<Rightarrow> 'a"   and n :: "nat" shows "IFC_def.cs prob \<pi> n = (if Ex (IFC_def.is_icdi prob n \<pi>) then IFC_def.cs prob \<pi> (The (IFC_def.is_icdi prob n \<pi>)) @ [\<pi> n] else [\<pi> n])"
112: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "- x \<sqinter> y \<squnion> (- - x \<sqinter> y \<squnion> z) = y \<squnion> z"
assms_3: shows "ipresIGFreshAbs hA' MOD' MOD''"
next_action_after_next_state: shows "\<forall>s execs d. current s \<noteq> d \<longrightarrow> fst (CISK_control (int.next_state s execs) d (execs d)) = None \<or> fst (CISK_control (int.next_state s execs) d (execs d)) = fst (CISK_control s d (execs d))"
finite_partitions_k_parts: fixes n :: "nat"   and k :: "nat" shows "finite {p. p partitions n \<and> sum p {..n} = k}"
lderivs_final: fixes n :: "nat"   and r :: "'b rexp"   and ws :: "'a list" assumes "local.wf n r"   and "wf_word n ws" shows "final (lderivs ws r) = (ws \<in> lang n r)"
LDF': shows "is_sentence (b1 @ [X] @ b2) \<and> is_sentence c \<and> LeftDerivation (b1 @ [X] @ b2) D c \<and> length b1 < length (b1 @ [X] @ b2) \<and> j < length c \<and> (b1 @ [X] @ b2) ! length b1 = c ! j \<and> (\<exists>E F. D = E @ derivation_shift F 0 (Suc j) \<and> LeftDerivation (take (length b1) (b1 @ [X] @ b2)) E (take j c) \<and> LeftDerivation (drop (Suc (length b1)) (b1 @ [X] @ b2)) F (drop (Suc j) c))"
assms_2: shows "strategy_winning_by_Odd \<sigma> p"
matches_op: fixes pat :: "[:'r:]" shows "matches\<cdot>pat = smap\<cdot>cfst oo sfilter\<cdot>(snull oo csnd oo csnd) oo sscanl\<cdot>(\<Lambda> (n, usvs) x. (n + 1, op\<cdot>pat\<cdot>usvs\<cdot>x))\<cdot>(0, [::], pat)"
simps: fixes a :: "nat"   and E :: "('a \<times> ('b, 'a) obs_event action) llist" shows "(a \<in> read_actions E) = (\<exists>aa ad al v. a = aa \<and> aa \<in> actions E \<and> action_obs E aa = NormalAction (ReadMem ad al v))"
sum_mset_distrib_left: fixes plus :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and zero :: "'k"   and times :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and c :: "'k"   and f :: "'l \<Rightarrow> 'k"   and M :: "'l multiset" assumes "class.semiring_0 plus zero times" shows "times c (comm_monoid_add.sum_mset plus zero (image_mset f M)) = comm_monoid_add.sum_mset plus zero {#times c (f x). x \<in># M#}"
nonabort_par_top: fixes c :: "'a::refinement_lattice" assumes "chaos \<sqsubseteq> (c::'a::refinement_lattice)" shows "(c::'a::refinement_lattice) \<parallel> \<top> = \<top>"
elims_2: fixes x :: "(('Var, 'AExp, 'BExp) Stmt \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)"   and xa :: "(('Var, 'AExp, 'BExp) Stmt \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)" assumes "bisim_helper x xa" obtains c\<^sub>1 :: "('Var, 'AExp, 'BExp) Stmt" and mds :: "Mode \<Rightarrow> 'Var set" and mem\<^sub>1 :: "'Var \<Rightarrow> 'Val" and c\<^sub>2 :: "('Var, 'AExp, 'BExp) Stmt" and mds\<^sub>2 :: "Mode \<Rightarrow> 'Var set" and mem\<^sub>2 :: "'Var \<Rightarrow> 'Val" where "x = \<langle>c\<^sub>1, mds, mem\<^sub>1\<rangle>" and "xa = \<langle>c\<^sub>2, mds\<^sub>2, mem\<^sub>2\<rangle>" and "mem\<^sub>1 =\<^bsub>mds\<^esub>\<^sup>l mem\<^sub>2"
compl_inf: fixes x :: "'a"   and y :: "'a" shows "- (x \<sqinter> y) = - x \<squnion> - y"
mono_invE: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'c"   and x :: "'b"   and y :: "'b" assumes "class.linorder less_eq less"   and "order.mono less_eq f"   and "f x < f y" obtains "less_eq x y"
ccompatible_def: fixes \<R> :: "('a \<Rightarrow> 'b) set set"   and cc :: "('a, 'b) cconstraint" shows "ccompatible \<R> cc \<equiv> \<forall>R\<in>\<R>. R \<subseteq> {v. v \<turnstile> cc} \<or> {v. v \<turnstile> cc} \<inter> R = {}"
cmult_in_iff: fixes c :: "'b"   and f :: "'c \<Rightarrow> 'b"   and F :: "'c filter"   and g :: "'c \<Rightarrow> 'b" assumes "c \<noteq> (0::'b)" shows "((\<lambda>x. c * f x) \<in> O[F](g)) = (f \<in> O[F](g))"
assms_3: shows "c \<in>\<^sub>\<circ> dg_prod K \<AA>\<lparr>Obj\<rparr>"
cases: fixes x :: "nat" obtains "x = 0"   | c :: "nat" where "x = Suc c"
T6: fixes w :: "i"   and \<psi> :: "\<o>" shows "[\<^bold>\<box>\<psi> in w] = (\<forall>v. [\<psi> in v])"
unrestrictn_edges: fixes g :: "'b ugraph"   and V :: "'b set" shows "edges (restrict_nodes g V) \<subseteq> edges g"
br'_invar_def: fixes \<delta> :: "('a, 'b) ta_rule set" shows "br'_invar \<delta> \<equiv> br'_invar_add \<delta> \<inter> {\<Sigma>. br'_\<alpha> \<Sigma> \<in> br_invar \<delta>}"
intro: fixes \<SS> :: "V"   and \<alpha> :: "V" assumes "vcard \<SS> = 3\<^sub>\<nat>"   and "\<Z>_csgrp \<alpha> [\<SS>\<lparr>\<A>\<rparr>, \<SS>\<lparr>vplus\<rparr>]\<^sub>\<circ>"   and "\<Z>_sgrp \<alpha> [\<SS>\<lparr>\<A>\<rparr>, \<SS>\<lparr>vmult\<rparr>]\<^sub>\<circ>"   and "\<And>a b c. \<lbrakk>a \<in>\<^sub>\<circ> \<SS>\<lparr>\<A>\<rparr>; b \<in>\<^sub>\<circ> \<SS>\<lparr>\<A>\<rparr>; c \<in>\<^sub>\<circ> \<SS>\<lparr>\<A>\<rparr>\<rbrakk> \<Longrightarrow> (a +\<^sub>\<circ>\<^bsub>\<SS>\<^esub> b) *\<^sub>\<circ>\<^bsub>\<SS>\<^esub> c = a *\<^sub>\<circ>\<^bsub>\<SS>\<^esub> c +\<^sub>\<circ>\<^bsub>\<SS>\<^esub> b *\<^sub>\<circ>\<^bsub>\<SS>\<^esub> c"   and "\<And>a b c. \<lbrakk>a \<in>\<^sub>\<circ> \<SS>\<lparr>\<A>\<rparr>; b \<in>\<^sub>\<circ> \<SS>\<lparr>\<A>\<rparr>; c \<in>\<^sub>\<circ> \<SS>\<lparr>\<A>\<rparr>\<rbrakk> \<Longrightarrow> a *\<^sub>\<circ>\<^bsub>\<SS>\<^esub> (b +\<^sub>\<circ>\<^bsub>\<SS>\<^esub> c) = a *\<^sub>\<circ>\<^bsub>\<SS>\<^esub> b +\<^sub>\<circ>\<^bsub>\<SS>\<^esub> a *\<^sub>\<circ>\<^bsub>\<SS>\<^esub> c" shows "\<Z>_srng_axioms \<alpha> \<SS>"
filtercomap_top: fixes f :: "'a \<Rightarrow> 'b" shows "filtercomap f top = top"
assms_2: fixes x :: "'a acom" assumes "P x" shows "P (f x)"
distinct_290: fixes x16 :: "Syntax.term"   and x14 :: "Syntax.term" shows "Unauth x16 \<noteq> Unroll x14"
filter_False: fixes xs :: "'d list"   and P :: "'d \<Rightarrow> bool" assumes "\<forall>x\<in>set xs. \<not> P x" shows "filter P xs = []"
msubstle_nb: fixes a :: "poly"   and r :: "tm"   and t :: "tm"   and s :: "tm"   and c :: "poly"   and d :: "poly" assumes "islin (Le (CNP 0 a r))"   and "tmbound0 t"   and "tmbound0 s" shows "bound0 (msubstle c t d s a r)"
linf_norm_vec_def: fixes v :: "'a vec" shows "\<parallel>v\<parallel>\<^sub>\<infinity> \<equiv> Missing_Lemmas.max_list (map abs (list_of_vec v) @ [0::'a])"
induct: fixes P :: "'b ptrm \<Rightarrow> bool"   and ptrm :: "'b ptrm" assumes "P PUnit"   and "\<And>x. P (PVar x)"   and "\<And>x1 x2a. \<lbrakk>P x1; P x2a\<rbrakk> \<Longrightarrow> P (PApp x1 x2a)"   and "\<And>x1 x2a x3. P x3 \<Longrightarrow> P (PFn x1 x2a x3)"   and "\<And>x1 x2a. \<lbrakk>P x1; P x2a\<rbrakk> \<Longrightarrow> P (PPair x1 x2a)"   and "\<And>x. P x \<Longrightarrow> P (PFst x)"   and "\<And>x. P x \<Longrightarrow> P (PSnd x)" shows "P ptrm"
call_eq: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and c :: "char list"   and rs :: "'c rule list"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and m :: "'c match_expr"   and p :: "'d"   and s :: "state"   and t :: "state" assumes "\<Gamma> c = Some rs"   and "matches \<gamma> m p"   and "\<forall>r\<in>set rs. get_action r \<noteq> Return" shows "\<Gamma>,\<gamma>,p\<turnstile> \<langle>[Rule m (Call c)], s\<rangle> \<Rightarrow> t = \<Gamma>,\<gamma>,p\<turnstile> \<langle>rs, s\<rangle> \<Rightarrow> t"
zip_append1: fixes xs :: "'g list"   and ys :: "'g list"   and zs :: "'h list" shows "zip (xs @ ys) zs = zip xs (take (length xs) zs) @ zip ys (drop (length xs) zs)"
1: shows "{(x, y). y \<in> s} = UNIV \<times> s"
trancl_unfold_right: fixes r :: "('a \<times> 'a) set" shows "r\<^sup>+ = r\<^sup>* ; r"
lookup_inverse: fixes x :: "'c \<Rightarrow>\<^sub>0 'd" shows "Abs_poly_mapping (lookup x) = x"
word_reverse_eq_of_bl_rev_to_bl: fixes w :: "'a word" shows "word_reverse w = of_bl (rev (to_bl w))"
smult_scalar_prod_distrib: fixes v\<^sub>1 :: "'a vec"   and n :: "nat"   and v\<^sub>2 :: "'a vec"   and a :: "'a" assumes "v\<^sub>1 \<in> carrier_vec n"   and "v\<^sub>2 \<in> carrier_vec n" shows "(a \<cdot>\<^sub>v v\<^sub>1) \<bullet> v\<^sub>2 = a * (v\<^sub>1 \<bullet> v\<^sub>2)"
sel_2: fixes x1 :: "'c"   and x2 :: "'b" shows "final_b (x1, x2) = x2"
beta: fixes rs :: "(Term_Class.term \<times> Term_Class.term) fset"   and t :: "Term_Class.term"   and t' :: "Term_Class.term" shows "rs \<turnstile> \<Lambda> t $ t' \<longrightarrow> t [t']\<^sub>\<beta>"
proots_within_union: fixes p :: "'b poly"   and A :: "'b set"   and B :: "'b set" shows "proots_within p A \<union> proots_within p B = proots_within p (A \<union> B)"
hnorm_minus_cancel: fixes x :: "'a star" shows "hnorm (- x) = hnorm x"
fold_mset_union: fixes s :: "'b"   and M :: "'a multiset"   and N :: "'a multiset" shows "fold_mset f s (M + N) = fold_mset f (fold_mset f s M) N"
pedersen: fixes \<eta> :: "nat" shows "pedersen (\<G> \<eta>)"
authKeys_def: fixes evs :: "event list" shows "authKeys evs = {authK. \<exists>A Peer Ta. Says Kas A (Crypt (shrK A) \<lbrace>Key authK, Agent Peer, Number Ta, Crypt (shrK Peer) \<lbrace>Agent A, Agent Peer, Key authK, Number Ta\<rbrace>\<rbrace>) \<in> set evs}"