hcju/isabelleps huggingface dataset:
d_fixpoint: fixes x :: "'a" shows "(\<exists>y. x = ad (ad y)) = (x = ad (ad x))"
inangle_one_side2: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and P :: "'p"   and Q :: "'p" assumes "\<not> Col A B C"   and "\<not> Col A B P"   and "\<not> Col A B Q"   and "\<not> Col C B P"   and "\<not> Col C B Q"   and "P InAngle A B C"   and "Q InAngle A B C" shows "A B OS P Q \<and> C B OS P Q"
simps_4: fixes n :: "nat"   and f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and y :: "'a"   and x :: "'a" assumes "n \<noteq> 1"   and "odd n" shows "efficient_power f y x n = efficient_power f (f x y) (f x x) (n div 2)"
foldr_Nil: fixes f :: "'b \<Rightarrow> 'a \<Rightarrow> 'a" shows "foldr f [] = id"
simps_2: fixes t1 :: "lam"   and t2 :: "lam" shows "height (App t1 t2) = max (height t1) (height t2) + 1"
Auth_Snd: fixes x :: "rat"   and y :: "rat"   and H :: "msg set" assumes "Auth_Data x y \<in> items H"   and "Pri_AgrK x \<in> items H" shows "Pri_AgrK y \<in> items H"
hyps: shows "branch_nominals (mapi_branch (bridge k j xs) branch) \<union> {k, j} = branch_nominals branch \<union> {k, j}"
compl_fam_cong: fixes f :: "'d \<Rightarrow> 'a set"   and g :: "'c \<Rightarrow> 'd"   and A :: "'c set" assumes "compl_fam (f \<circ> g) A"   and "inj_on g A" shows "compl_fam f (g ` A)"
combine_l_distr: fixes Ks :: "'a list"   and Us :: "'a list"   and u :: "'a" assumes "set Ks \<subseteq> carrier R"   and "set Us \<subseteq> carrier R"   and "u \<in> carrier R" shows "combine Ks Us \<otimes> u = combine Ks (map (\<lambda>u'. u' \<otimes> u) Us)"
Bot_prop: fixes x :: "'a" assumes "x \<in> A" shows "(Bot cl, x) \<in> r"
that: fixes x :: "'a"   and xs :: "'a list" assumes "l = x # xs" shows "thesis"
Cons_1: fixes s :: "'a \<Rightarrow> 'b option" assumes "is_valid_problem_sas_plus \<Psi>"   and "\<forall>op'\<in>set ops'. op' \<in> set ((\<phi> \<Psi> )\<^sub>\<O>)" shows "execute_parallel_operator (\<phi>\<^sub>S \<Psi> s) ops' = \<phi>\<^sub>S \<Psi> execute_parallel_operator_sas_plus s (map (strips_op_to_sasp \<Psi>) ops')"
fresh_2: fixes a :: "atom"   and name :: "name" shows "a \<sharp> Var name = a \<sharp> name"
assertion_def: shows "assertion = {x. x \<le> (1::'a) \<and> x * \<top> \<sqinter> x ^ o = x}"
ground_prove: fixes t :: "tm"   and u :: "tm"   and n :: "nat"   and e :: "(name, hf) finfun" assumes "size t + size u < n"   and "ground t"   and "ground u" shows "(\<lbrakk>t\<rbrakk>e \<le> \<lbrakk>u\<rbrakk>e \<longrightarrow> {} \<turnstile> t SUBS u) \<and> (\<lbrakk>t\<rbrakk>e \<^bold>\<in> \<lbrakk>u\<rbrakk>e \<longrightarrow> {} \<turnstile> t IN u)"
conforms_return: fixes x :: "abopt"   and s :: "st"   and G :: "prog"   and L :: "lenv"   and x' :: "abopt"   and s' :: "st"   and L' :: "lenv" assumes "(x, s)\<Colon>\<preceq>(G, L)"   and "(x', s')\<Colon>\<preceq>(G, L')"   and "s\<le>|s'"   and "x' \<noteq> Some (Jump Ret)" shows "(x', set_locals (locals s) s')\<Colon>\<preceq>(G, L)"
psubst_subst: fixes x :: "name"   and \<theta> :: "(name \<times> trm) list"   and s :: "trm"   and t :: "trm" assumes "x \<sharp> \<theta>" shows "\<theta><t>[x::=s] = ((x, s) # \<theta>)<t>"
AssertAllState''_axioms: shows "AssertAllState'' init assert_bound check"
at_within_interior: fixes x :: "'a"   and S :: "'a set" assumes "x \<in> interior S" shows "at x within S = at x"
fs_sq_norm_Ints: fixes i :: "nat" assumes "i < length fs" shows "\<parallel>fs ! i\<parallel>\<^sup>2 \<in> \<int>"
init_unsat_flag: fixes init :: "(nat \<times> linear_poly) list \<Rightarrow> ('a, 'b) state"   and t :: "(nat \<times> linear_poly) list" assumes "Init init" shows "\<not> \<U> (init t)"
simDef_1: fixes P :: "pi"   and Rs :: "residual"   and C :: "'a"   and Rel :: "(pi \<times> pi) set" shows "weakSimAct P Rs C Rel \<equiv> (\<forall>Q' a x. Rs = a<\<nu>x> \<prec> Q' \<longrightarrow> x \<sharp> C \<longrightarrow> (\<exists>P'. P \<Longrightarrow>\<^sub>l\<^sup>^a<\<nu>x> \<prec> P' \<and> (P', Q') \<in> Rel)) \<and> (\<forall>Q' a x. Rs = a<x> \<prec> Q' \<longrightarrow> x \<sharp> C \<longrightarrow> (\<exists>P''. \<forall>u. \<exists>P'. P \<Longrightarrow>\<^sub>lu in P''\<rightarrow>a<x> \<prec> P' \<and> (P', Q'[x::=u]) \<in> Rel)) \<and> (\<forall>Q' \<alpha>. Rs = \<alpha> \<prec> Q' \<longrightarrow> (\<exists>P'. P \<Longrightarrow>\<^sub>l\<^sup>^\<alpha> \<prec> P' \<and> (P', Q') \<in> Rel))"
lookup_empty: fixes ko :: "'o"   and k :: "'a" shows "lookup (local.empty ko) k = (0::'b)"
execute'_correct: fixes s :: "nat \<Rightarrow> nat option"   and name :: "char list" assumes "s \<in> valid_states"   and "enabled name s" shows "execute' problem name s = execute name s"
composite_cell_in_hom: fixes w :: "'a"   and u :: "'a"   and \<theta> :: "'a" assumes "ide w"   and "\<guillemotleft>w : src u \<rightarrow> src f'\<guillemotright>"   and "\<guillemotleft>\<theta> : f' \<star> w \<Rightarrow> u\<guillemotright>" shows "\<guillemotleft>(r \<star> \<theta>) \<cdot> \<a>[r, f', w] \<cdot> (\<rho>' \<star> w) : g' \<star> w \<Rightarrow> r \<star> u\<guillemotright>"
D_COMP: fixes G :: "((Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool) set"   and c1 :: "OBJ"   and \<Phi> :: "((Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list) \<times> ((Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list) \<times> (Location \<times> Location) set \<Rightarrow> bool"   and c2 :: "OBJ"   and \<Psi> :: "((Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list) \<times> ((Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list) \<times> (Location \<times> Location) set \<Rightarrow> bool" assumes "(G, c1, Sec \<Phi>) \<in> Deriv"   and "(G, c2, Sec \<Psi>) \<in> Deriv" shows "(G, Comp c1 c2, Sec (\<lambda>(s, t, \<beta>). \<exists>r. \<Phi> (r, t, \<beta>) \<and> (\<forall>w \<gamma>. r \<equiv>\<^sub>\<gamma> w \<longrightarrow> \<Psi> (s, w, \<gamma>)))) \<in> Deriv"
map_sel_2: fixes a :: "('a, 'b) tllist"   and f1 :: "'a \<Rightarrow> 'c"   and f2 :: "'b \<Rightarrow> 'd" assumes "\<not> is_TNil a" shows "thd (tmap f1 f2 a) = f1 (thd a)"
fds_nth_deriv: fixes f :: "'b fds"   and n :: "nat" shows "fds_nth (fds_deriv f) n = - ln (real n) *\<^sub>R fds_nth f n"
U_LIMCONS_prefix_complete: fixes vs :: "nat list" assumes "0 < length vs" shows "\<exists>f\<in>U\<^bsub>LIM-CONS\<^esub>. prefix f (length vs - 1) = vs"
le_iff_add: fixes plus :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and zero :: "'f"   and less_eq :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and less :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and a :: "'f"   and b :: "'f" assumes "class.canonically_ordered_monoid_add plus zero less_eq less" shows "less_eq a b = (\<exists>c. b = plus a c)"
IH_1: assumes "braun_indices l = {1..size l}" shows "braun l"
prSwapAbs_def: fixes gA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> ('varSort \<Rightarrow> 'var \<Rightarrow> 'b) \<Rightarrow> 'c"   and SEM :: "('d, 'e, 'f, 'sort, 'g, 'b, 'h) semDom_scheme" shows "prSwapAbs gA SEM \<equiv> \<forall>zs z1 z2 us s A val. wlsAbs (us, s) A \<and> sWlsVal SEM val \<longrightarrow> gA (A $[z1 \<and> z2]_zs) val = gA A (val ^[z1 \<and> z2]_zs)"
inverse_arrow_unique: fixes f :: "'a"   and g :: "'a"   and g' :: "'a" assumes "inverse_arrows f g"   and "inverse_arrows f g'" shows "g = g'"
decompose: shows "set (\<Phi> @ \<Phi>\<^sub>r) = set \<Phi>'"
lexord_partial_trans: fixes xs :: "'g list"   and r :: "('g \<times> 'g) set"   and ys :: "'g list"   and zs :: "'g list" assumes "\<And>x y z. \<lbrakk>x \<in> set xs; (x, y) \<in> r; (y, z) \<in> r\<rbrakk> \<Longrightarrow> (x, z) \<in> r"   and "(xs, ys) \<in> lexord r"   and "(ys, zs) \<in> lexord r" shows "(xs, zs) \<in> lexord r"
PiE: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and B :: "'a \<Rightarrow> 'b set"   and x :: "'a" assumes "f \<in> Pi A B" obtains "f x \<in> B x"   | "x \<notin> A"
Fun_mkArr: fixes A :: "'s set"   and B :: "'s set"   and F :: "'s \<Rightarrow> 's" assumes "arr (mkArr A B F)" shows "Fun (mkArr A B F) = restrict F A"
count_Mn_zero_iff_prim: fixes f :: "recf" shows "(count_Mn f = 0) = Mn_free f"
map_injective: fixes f :: "'c \<Rightarrow> 'b"   and xs :: "'c list"   and ys :: "'c list" assumes "map f xs = map f ys"   and "inj f" shows "xs = ys"
isOK_case_option: fixes P :: "'a + 'b"   and Q :: "'c \<Rightarrow> 'a + 'b"   and x :: "'c option" shows "isOK (case x of None \<Rightarrow> P | Some v \<Rightarrow> Q v) = (case x of None \<Rightarrow> isOK P | Some v \<Rightarrow> isOK (Q v))"
insert_minus_eq: fixes x :: "'d"   and y :: "'d"   and A :: "'d set" assumes "x \<noteq> y" shows "insert x A - {y} = insert x (A - {y})"
Throw: fixes P :: "'d set"   and A :: "'d set"   and \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and \<Theta> :: "('d set \<times> 'e \<times> 'd set \<times> 'd set) set"   and F :: "'f set"   and Q :: "'d set" assumes "P \<subseteq> A" shows "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P Throw Q,A"
set_less_eq_aux_code: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'b list"   and ys :: "'b list" assumes "class.linorder less_eq less"   and "linorder.sorted less_eq xs"   and "distinct xs"   and "linorder.sorted less_eq ys"   and "distinct ys" shows "ord.set_less_eq_aux less_eq (set xs) (set ys) = ord.lexordp_eq (\<lambda>x y. less y x) xs ys"
app_Some: fixes i :: "instr"   and 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 T\<^sub>r :: "ty"   and pc :: "nat"   and mpc :: "nat"   and xt :: "(nat \<times> nat \<times> char list \<times> nat \<times> nat) list"   and \<tau> :: "ty list \<times> ty err list" shows "app i P mxs T\<^sub>r pc mpc xt \<lfloor>\<tau>\<rfloor> = (app\<^sub>i (i, P, pc, mxs, T\<^sub>r, \<tau>) \<and> xcpt_app i P pc mxs xt \<tau> \<and> (\<forall>(pc', s')\<in>set (eff i P pc xt \<lfloor>\<tau>\<rfloor>). pc' < mpc))"
dns3: fixes x :: "'a"   and y :: "'a" shows "d (x + y) = d x + d y"
hvzk_h_sub_rewrite: fixes \<G> :: "'a cyclic_group"   and h :: "'a"   and w :: "nat"   and z :: "nat"   and c :: "nat" assumes "chaum_ped_\<Sigma> \<G>"   and "h = \<^bold>g\<^bsub>\<G>\<^esub> [^]\<^bsub>\<G>\<^esub> w"   and "z < order \<G>" shows "\<^bold>g\<^bsub>\<G>\<^esub> [^]\<^bsub>\<G>\<^esub> (z + order \<G> * w * c - w * c) = \<^bold>g\<^bsub>\<G>\<^esub> [^]\<^bsub>\<G>\<^esub> z \<otimes>\<^bsub>\<G>\<^esub> inv\<^bsub>\<G>\<^esub> (h [^]\<^bsub>\<G>\<^esub> c)"
elims: fixes x :: "'a list"   and y :: "'a list" assumes "local.normalize x = y" obtains "x = []" and "y = []"   | v :: "'a" and va :: "'a list" where "x = v # va" and "y = (if lead_coeff (v # va) \<noteq> \<zero> then v # va else local.normalize (tl (v # va)))"
cancels_to_def: shows "cancels_to = cancels_to_1\<^sup>*\<^sup>*"
b_e_check_weaken_type: fixes \<C> :: "t_context"   and es :: "b_e list"   and tn :: "t list"   and tm :: "t list"   and ts :: "t list" assumes "check \<C> es (Type tn) = Type tm" shows "check \<C> es (Type (ts @ tn)) = Type (ts @ tm)"
intruder_synth_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>\<^sub>c 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>\<^sub>c t; \<And>t. t \<in> set T \<Longrightarrow> P M t\<rbrakk> \<Longrightarrow> P M (Fun f T)" shows "P M t"
map_id: fixes t :: "'d option" shows "map_option id t = t"
hd_in_awalk_verts_2: fixes u :: "'a"   and p :: "'b list"   and v :: "'a" assumes "apath u p v" shows "u \<in> set (awalk_verts u p)"
next_suffix_com: fixes n :: "nat"   and s :: "nat \<Rightarrow> 'b" shows "nextsuffix (next n s) = next n (nextsuffix s)"
cmp_fun: fixes f :: "'d \<Rightarrow> 'e"   and A :: "'d set"   and B :: "'e set"   and g :: "'e \<Rightarrow> 'f"   and C :: "'f set" assumes "f \<in> A \<rightarrow> B"   and "g \<in> B \<rightarrow> C" shows "cmp g f \<in> A \<rightarrow> C"
distinct_filter: fixes xs :: "'b list"   and P :: "'b \<Rightarrow> bool" assumes "distinct xs" shows "distinct (filter P xs)"
keys_restrict_indets: fixes p :: "('c option \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd" assumes "None \<notin> indets p" shows "keys (restrict_indets p) = restrict_indets_pp ` keys p"
ppoint_mult_closed: fixes a :: "int"   and b :: "int"   and p :: "int \<times> int \<times> int"   and n :: "nat" assumes "a \<in> carrier R"   and "b \<in> carrier R"   and "on_curvep a b p" shows "on_curvep a b (ppoint_mult a n p)"
subdigraph_axioms: shows "\<AA> \<subseteq>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<CC>"
sys_block_sizes_in: fixes bl :: "'a set" assumes "bl \<in># \<B>" shows "int (card bl) \<in> sys_block_sizes"
path_20: fixes pth :: "((nat \<times> (form, nat) out_port) \<times> nat \<times> (form, nat) antecedent) list" assumes "task.path 2 0 pth" shows "(1, allI_input) \<in> snd ` set pth"
assms_3: fixes y :: "nat" assumes "y < z" shows "eval f (y # xs) \<down>\<noteq> 0"
nonword_leftmost_exists: fixes a :: "symbol list" assumes "is_sentence a"   and "\<not> is_word a" shows "\<exists>i. leftmost i a"
fst_mismatch_def: fixes u\<^sub>0 :: "'b list"   and u\<^sub>1 :: "'b list" assumes "binary_code u\<^sub>0 u\<^sub>1" shows "binary_code.c\<^sub>0 u\<^sub>0 u\<^sub>1 = (u\<^sub>0 \<cdot> u\<^sub>1) ! \<^bold>|binary_code.\<alpha> u\<^sub>0 u\<^sub>1\<^bold>|"
DIAndCutP12_def: fixes vid1 :: "'a"   and vid2 :: "'a"   and vid3 :: "'a"   and fid1 :: "'b"   and fid2 :: "'b"   and fid3 :: "'b"   and pid1 :: "'c"   and pid2 :: "'c"   and pid3 :: "'c"   and pid4 :: "'c" assumes "ids vid1 vid2 vid3 fid1 fid2 fid3 pid1 pid2 pid3 pid4" shows "ids.DIAndCutP12 vid1 pid1 pid2 = (([[$\<alpha> vid1]]Pc pid1 \<rightarrow> Pc pid2 \<rightarrow> (Pc pid1 && Pc pid2)) \<rightarrow> ([[$\<alpha> vid1]]Pc pid1) \<rightarrow> [[$\<alpha> vid1]]Pc pid2 \<rightarrow> (Pc pid1 && Pc pid2))"
s_1: shows "\<forall>i. flow0 x (s i) \<notin> A \<and> flow0 x (s i) \<notin> B"
pp_of_term_def: fixes v :: "'t" shows "tp.pp_of_term v = fst (pair_of_term v)"
rel_FGcontra_neg_distr_imp: fixes Co1 :: "'p \<Rightarrow> 'q \<Rightarrow> bool"   and Co1' :: "'q \<Rightarrow> 'r \<Rightarrow> bool"   and Co4 :: "'s \<Rightarrow> 't \<Rightarrow> bool"   and Co4' :: "'t \<Rightarrow> 'u \<Rightarrow> bool"   and Co5 :: "'v \<Rightarrow> 'w \<Rightarrow> bool"   and Co5' :: "'w \<Rightarrow> 'x \<Rightarrow> bool"   and Contra1 :: "'y \<Rightarrow> 'ae \<Rightarrow> bool"   and Contra2 :: "'z \<Rightarrow> 'af \<Rightarrow> bool"   and Contra3 :: "'aa \<Rightarrow> 'ag \<Rightarrow> bool"   and Contra4 :: "'ab \<Rightarrow> 'ah \<Rightarrow> bool"   and Co2 :: "'ac \<Rightarrow> 'ai \<Rightarrow> bool"   and Contra1' :: "'ae \<Rightarrow> 'aj \<Rightarrow> bool"   and Contra2' :: "'af \<Rightarrow> 'ak \<Rightarrow> bool"   and Contra3' :: "'ag \<Rightarrow> 'al \<Rightarrow> bool"   and Contra4' :: "'ah \<Rightarrow> 'am \<Rightarrow> bool"   and Co2' :: "'ai \<Rightarrow> 'an \<Rightarrow> bool"   and Contra5 :: "'ao \<Rightarrow> 'ap \<Rightarrow> bool"   and Contra5' :: "'ap \<Rightarrow> 'aq \<Rightarrow> bool"   and tytok_F :: "('ar \<times> 'as \<times> 'at \<times> 'p \<times> 'q \<times> 'r \<times> 'au \<times> 'av \<times> 'aw \<times> 'ax) itself"   and tytok_G :: "('y \<times> 'ae \<times> 'aj \<times> 'z \<times> 'af \<times> 'ak \<times> 'ad) itself"   and Co3 :: "'au \<Rightarrow> 'av \<Rightarrow> bool"   and Co3' :: "'av \<Rightarrow> 'aw \<Rightarrow> bool"   and tytok_FGcontra :: "('ar \<times> 'as \<times> 'at \<times> 'ad \<times> 'ax) itself" assumes "rel_F_neg_distr_cond Co1 Co1' Co4 Co4' Co5 Co5' (rel_G Contra1 Contra2 Contra3 Contra4 Co1 Co2) (rel_G Contra1' Contra2' Contra3' Contra4' Co1' Co2') Contra1 Contra1' Contra5 Contra5' tytok_F"   and "rel_G_pos_distr_cond Contra3 Contra3' Contra4 Contra4' Co1 Co1' Co2 Co2' tytok_G" shows "rel_FGcontra_neg_distr_cond Co1 Co1' Co2 Co2' Co3 Co3' Co4 Co4' Co5 Co5' Contra1 Contra1' Contra2 Contra2' Contra3 Contra3' Contra4 Contra4' Contra5 Contra5' tytok_FGcontra"
IsLegalRcvArg_def: fixes ra :: "rpcOp" shows "IsLegalRcvArg ra \<equiv> case ra of memcall m \<Rightarrow> True | othercall v \<Rightarrow> False"
mult_pos_neg2: fixes a :: "'b"   and b :: "'b" assumes "(0::'b) < a"   and "b < (0::'b)" shows "b * a < (0::'b)"
nprv_clear4_4: fixes var :: "'a set"   and trm :: "'b set"   and fmla :: "'c set"   and Var :: "'a \<Rightarrow> 'b"   and FvarsT :: "'b \<Rightarrow> 'a set"   and substT :: "'b \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b"   and Fvars :: "'c \<Rightarrow> 'a set"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'c"   and eql :: "'b \<Rightarrow> 'b \<Rightarrow> 'c"   and cnj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and imp :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and all :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and exi :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and fls :: "'c"   and dsj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and num :: "'b set"   and prv :: "'c \<Rightarrow> bool"   and \<phi>1 :: "'c"   and \<phi>2 :: "'c"   and \<phi>3 :: "'c"   and \<psi> :: "'c"   and \<phi>4 :: "'c" assumes "Deduct_with_False_Disj var trm fmla Var FvarsT substT Fvars subst eql cnj imp all exi fls dsj num prv"   and "Deduct_with_False_Disj.nprv cnj imp fls prv {\<phi>1, \<phi>2, \<phi>3} \<psi>"   and "\<phi>1 \<in> fmla"   and "\<phi>2 \<in> fmla"   and "\<phi>3 \<in> fmla"   and "\<phi>4 \<in> fmla"   and "\<psi> \<in> fmla" shows "Deduct_with_False_Disj.nprv cnj imp fls prv {\<phi>1, \<phi>2, \<phi>3, \<phi>4} \<psi>"
map_simp_1: shows "D.map J.Zero = C.dom f0"
preserves_reflects_arr: fixes f :: "'a" shows "S.arr (L f) = S.arr f"
abs_convergent_euler_product: fixes f :: "nat \<Rightarrow> 'a" assumes "summable (\<lambda>n. norm (f n))"   and "multiplicative_function f" shows "abs_convergent_prod (\<lambda>p. if prime p then \<Sum>n. f (p ^ n) else (1::'a))"
equivalence_refines_to_adjoint_equivalence: fixes f :: "'a"   and g :: "'a"   and \<eta> :: "'a" assumes "B.equivalence_map f"   and "\<guillemotleft>g : trg f \<rightarrow>\<^sub>B src f\<guillemotright>"   and "B.ide g"   and "\<guillemotleft>\<eta> : src f \<Rightarrow>\<^sub>B g \<star> f\<guillemotright>"   and "B.iso \<eta>" shows "\<exists>!\<epsilon>. adjoint_equivalence_in_bicategory (\<cdot>) (\<star>) \<a> \<i> src trg f g \<eta> \<epsilon>"
M: shows "ExtAgrK (S (Card n, n, run)) = Some (c * (s' + a' * b'))"
exc2: shows "abc_steps_l (s, lm) A nb = (length A, lmb)"
matrix_to_iarray_bezout_iterate: fixes n :: "nat"   and A :: "(('a, 'b) vec, 'c) vec"   and i :: "'c"   and j :: "'b"   and bezout :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a \<times> 'a \<times> 'a" assumes "n < nrows A" shows "matrix_to_iarray (bezout_iterate A n i j bezout) = bezout_iterate_iarrays (matrix_to_iarray A) n (mod_type_class.to_nat i) (mod_type_class.to_nat j) bezout"
lm047: fixes P :: "('d \<times> 'c) set"   and Q :: "('d \<times> 'c) set" assumes "Range P \<inter> Range Q = {}"   and "runiq (P\<inverse>)"   and "runiq (Q\<inverse>)" shows "runiq ((P \<union> Q)\<inverse>)"
mk_acc_impl_correct: fixes Flist' :: "'a set list"   and Flist :: "'a set list" assumes "(Flist', Flist) \<in> Id"   and "\<forall>A\<in>set Flist. finite A" shows "mk_acc_impl Flist' \<le> \<Down> Id (RETURN (\<lambda>v. {i. i < length Flist \<and> v \<in> Flist ! i}))"
simps_2: fixes uu :: "'b" shows "bn (S_action.Pred uu) = {}"
octo_distrib_right: fixes b :: "octo"   and c :: "octo"   and a :: "octo" shows "(b + c) * a = b * a + c * a"
r2: shows "R2 = T2 OO T2\<inverse>\<inverse>"
Says_Tgs_AKcryptSK: fixes A :: "agent"   and authK :: "nat"   and servK :: "nat"   and B :: "agent"   and tt :: "msg"   and X :: "msg"   and evs :: "event list"   and authK' :: "nat" assumes "Says Tgs A \<lbrace>Crypt authK \<lbrace>Key servK, Agent B, tt\<rbrace>, X\<rbrace> \<in> set evs"   and "authK' \<noteq> authK"   and "evs \<in> kerbV" shows "\<not> AKcryptSK authK' servK evs"
that: fixes p :: "'b list" assumes "trail (tail G a0) p (head G an)"   and "p \<noteq> []"   and "hd p = a0"   and "last p = an"   and "set p \<subseteq> face_cycle_set a" shows "thesis"
c_prod_assoc: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<parallel> y \<parallel> z = x \<parallel> (y \<parallel> z)"
assms_3: shows "\<guillemotleft>f : F.map\<^sub>0 b \<rightarrow>\<^sub>C F.map\<^sub>0 b'\<guillemotright>"
prefix_fininf_append: fixes u :: "'a list"   and v :: "'a list"   and w :: "'a stream" assumes "u \<le>\<^sub>F\<^sub>I v @- w" obtains "u \<le> v"   | z :: "'a list" where "u = v @ z" and "z \<le>\<^sub>F\<^sub>I w"
B: shows "(g, h) \<in> [Q]\<^sub>f T \<rightarrow> \<langle>U\<rangle>nres_rel"
part_def: fixes J :: "'b set"   and P :: "'b set set" shows "part J P \<equiv> \<Union> P = J \<and> (\<forall>J1 J2. J1 \<in> P \<and> J2 \<in> P \<and> J1 \<noteq> J2 \<longrightarrow> J1 \<inter> J2 = {})"
V_arrow_elim_weak: fixes u :: "trm"   and T\<^sub>1 :: "ty"   and T\<^sub>2 :: "ty" assumes "u \<in> V (T\<^sub>1\<rightarrow>T\<^sub>2)" obtains a :: "name" and t :: "trm" where "u = Lam [a].t" and "\<forall>v\<in>V T\<^sub>1. \<exists>v'. t[a::=v] \<Down> v' \<and> v' \<in> V T\<^sub>2"
prefs_undefined': fixes i :: "'a" assumes "i \<notin> agents" shows "R i = (\<lambda>_ _. False)"
less_def: fixes x :: "nat"   and y :: "nat" shows "(x dvd y \<and> x \<noteq> y) = (x dvd y \<and> x \<noteq> y)"
inject_5: fixes x51 :: "nat"   and x52 :: "'a"   and x53 :: "('a, 'b) MWLsCom"   and y51 :: "nat"   and y52 :: "'a"   and y53 :: "('a, 'b) MWLsCom" shows "(while\<^bsub>x51\<^esub> x52 do x53 od = while\<^bsub>y51\<^esub> y52 do y53 od) = (x51 = y51 \<and> x52 = y52 \<and> x53 = y53)"
simps: fixes a :: "'b"   and b :: "'b"   and x :: "'b" shows "swp_apply (a, b) x = (if x = a then b else if x = b then a else x)"
Cauchy_Im: fixes X :: "nat \<Rightarrow> complex" assumes "Cauchy X" shows "Cauchy (\<lambda>n. Im (X n))"
intro: fixes G :: "'b pair_pre_digraph" assumes "\<And>e. e \<in> parcs G \<Longrightarrow> fst e \<in> pverts G"   and "\<And>e. e \<in> parcs G \<Longrightarrow> snd e \<in> pverts G" shows "pair_wf_digraph G"
dres_nres_rel_internal_def: fixes R :: "('a \<times> 'b) set" shows "dres_nres_rel R \<equiv> {(c, a). nres_of c \<le> \<Down> R a}"
that: fixes a :: "V"   and f :: "V" assumes "a \<in>\<^sub>\<circ> Vset \<alpha>"   and "f \<in>\<^sub>\<circ> Vset \<alpha>"   and "\<BB> = smc_1 a f" shows "thesis"
assms_4: shows "\<forall>\<^sub>F i in I. \<forall>x\<in>X. f i x \<le> f' i x"
inv_closed: fixes x :: "'e" assumes "x \<in> carrier N" shows "\<ominus>\<^bsub>N\<^esub> x \<in> carrier N"
lm145: fixes A :: "'a set" shows "finite A = (card (Pow A) \<noteq> 0)"
find_base_vectors_1: fixes A :: "'a mat"   and nr :: "nat"   and nc :: "nat" assumes "row_echelon_form A"   and "A \<in> carrier_mat nr nc" shows "set (find_base_vectors A) \<subseteq> mat_kernel A"
switch_always_exists: fixes v :: "view" shows "\<forall>c. \<exists>u. v = c > u"
vassn_tag_def: fixes \<Gamma> :: "assn" shows "vassn_tag \<Gamma> \<equiv> \<exists>h. h \<Turnstile> \<Gamma>"
select_convs_1: fixes ip :: "nat"   and sn :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option"   and rreqs :: "(nat \<times> nat) set"   and store :: "nat \<Rightarrow> (p \<times> nat list) option"   and msg :: "msg"   and data :: "nat"   and dests :: "nat \<Rightarrow> nat option"   and rreqid :: "nat"   and dip :: "nat"   and oip :: "nat"   and hops :: "nat"   and dsn :: "nat"   and dsk :: "k"   and osn :: "nat"   and sip :: "nat"   and more :: "'a" shows "ip \<lparr>ip = ip, sn = sn, rt = rt, rreqs = rreqs, store = store, msg = msg, data = data, dests = dests, rreqid = rreqid, dip = dip, oip = oip, hops = hops, dsn = dsn, dsk = dsk, osn = osn, sip = sip, \<dots> = more\<rparr> = ip"