hcju/isabelleps huggingface dataset:
substs_lemma: fixes e :: "nat \<Rightarrow> LambdaType.type"   and u :: "dB"   and T :: "LambdaType.type"   and i :: "nat"   and ts :: "dB list"   and Ts :: "LambdaType.type list" assumes "e \<turnstile> u : T"   and "e\<langle>i:T\<rangle> \<tturnstile> ts : Ts" shows "e \<tturnstile> map (\<lambda>t. t[u/i]) ts : Ts"
position_of_Hd: fixes uv :: "('c, 'd) hd"   and uw :: "dir"   and ux :: "dir list" shows "position_of (Hd uv) (uw # ux) = False"
deltafun_apply_neq: fixes x :: "'c"   and a :: "'c"   and b :: "'d" assumes "x \<noteq> a" shows "(b \<delta> a) x = (0::'d)"
setops\<^sub>s\<^sub>s\<^sub>t\<^sub>p_subst: fixes a :: "('d, 'c) stateful_strand_step"   and \<theta> :: "'c \<Rightarrow> ('d, 'c) Term.term" assumes "set (bvars\<^sub>s\<^sub>s\<^sub>t\<^sub>p a) \<inter> subst_domain \<theta> = {}" shows "setops\<^sub>s\<^sub>s\<^sub>t\<^sub>p (a \<cdot>\<^sub>s\<^sub>s\<^sub>t\<^sub>p \<theta>) = setops\<^sub>s\<^sub>s\<^sub>t\<^sub>p a \<cdot>\<^sub>p\<^sub>s\<^sub>e\<^sub>t \<theta>"
PORI_5: fixes f :: "Three \<Rightarrow> ValD" assumes "f Two = \<bottom>" shows "f \<in> unlr POR.delta"
sel_1: fixes x1 :: "'b literal multiset multiset"   and x2 :: "'b literal multiset"   and x3 :: "'b literal multiset" shows "side_prems_of (Infer x1 x2 x3) = x1"
vertex_subtreeI: fixes G :: "('e, 'f) Graph_scheme"   and T :: "('g, 'h) Graph_scheme"   and bag :: "'g \<Rightarrow> 'e set"   and t :: "'g"   and v :: "'e" assumes "TreeDecomposition G T bag"   and "t \<in> V\<^bsub>T\<^esub>"   and "v \<in> bag t" shows "t \<in> TreeDecomposition.vertex_subtree T bag v"
out_subseqs2_set_def: fixes v :: "'b"   and js :: "nat list" shows "out_subseqs2_set v js \<equiv> {f (tail !! j) v |j. j \<in> set js}"
accumulate_def: shows "accumulate = dds_of (\<lambda>A X. (A |\<union>| X, A |\<union>| X)) {||}"
A92_NotSource_level1: shows "isNotDSource level1 sA92"
assms_5: shows "\<forall>x\<in>set les. case x of (d, e, f) \<Rightarrow> d * ((sqrt (aa\<^sup>2 - 4 * a * ba) - aa) / (2 * a))\<^sup>2 + e * (sqrt (aa\<^sup>2 - 4 * a * ba) - aa) / (2 * a) + f < 0"
ide_apex: fixes J :: "'ag \<Rightarrow> 'ag \<Rightarrow> 'ag"   and C :: "'ah \<Rightarrow> 'ah \<Rightarrow> 'ah"   and D :: "'ag \<Rightarrow> 'ah"   and a :: "'ah"   and \<chi> :: "'ag \<Rightarrow> 'ah" assumes "Limit.cone J C D a \<chi>" shows "partial_magma.ide C a"
that: fixes a :: "'a"   and b :: "'a" assumes "X \<subseteq> cbox a b" shows "thesis"
semClosed_def: fixes \<phi> :: "'aprop dfmla" shows "semClosed \<phi> \<equiv> sem \<phi> any (SOME \<pi>l. pointerOf \<pi>l = s0)"
is_parser_oneof: fixes ts :: "char list list" shows "is_parser (oneof ts)"
assm_5: fixes x1 :: "'b"   and K1 :: "'b set"   and x2 :: "'b"   and K2 :: "'b set" assumes "(x1, K1) \<in> p"   and "(x2, K2) \<in> p"   and "(x1, K1) \<noteq> (x2, K2)" shows "interior K1 \<inter> interior K2 = {}"
commutative_monoid_axioms: shows "commutative_monoid stfx.carrier_direct_lim stfx.add_rel (stfx.class_of V (zero_str\<^sub>Y V))"
smallI: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b"   and F :: "'a filter" assumes "\<And>c. 0 < c \<Longrightarrow> \<forall>\<^sub>F x in F. R (norm (f x)) (c * norm (g x))" shows "f \<in> l F g"
add_mono_thms_linordered_semiring_1: fixes i :: "'e"   and j :: "'e"   and k :: "'e"   and l :: "'e" assumes "i \<le> j \<and> k \<le> l" shows "i + k \<le> j + l"
divides_prod_l: fixes a :: "'a list"   and b :: "'a list"   and c :: "'a list" assumes "a \<in> carrier (A [X])"   and "b \<in> carrier (A [X])"   and "c \<in> carrier (A [X])"   and "a divides\<^bsub>A [X]\<^esub> b" shows "a divides\<^bsub>A [X]\<^esub> c \<otimes>\<^bsub>A [X]\<^esub> b"
ga_ex_def: shows "ga_ex \<equiv> stutter_extend_ex ga_gex"
intro: fixes \<CC> :: "V"   and \<alpha> :: "V" assumes "vcard \<CC> = 4\<^sub>\<nat>"   and "\<D>\<^sub>\<circ> (\<CC>\<lparr>Dom\<rparr>) = \<CC>\<lparr>Arr\<rparr>"   and "\<R>\<^sub>\<circ> (\<CC>\<lparr>Dom\<rparr>) \<subseteq>\<^sub>\<circ> \<CC>\<lparr>Obj\<rparr>"   and "\<D>\<^sub>\<circ> (\<CC>\<lparr>Cod\<rparr>) = \<CC>\<lparr>Arr\<rparr>"   and "\<R>\<^sub>\<circ> (\<CC>\<lparr>Cod\<rparr>) \<subseteq>\<^sub>\<circ> \<CC>\<lparr>Obj\<rparr>"   and "\<CC>\<lparr>Obj\<rparr> \<in>\<^sub>\<circ> Vset \<alpha>"   and "\<CC>\<lparr>Arr\<rparr> \<in>\<^sub>\<circ> Vset \<alpha>" shows "tiny_digraph_axioms \<alpha> \<CC>"
composite_simps_1: shows "src (\<mu> \<star> \<nu>) = src \<nu>"
concrete_exist: shows "\<lfloor>\<lambda>w. \<forall>x. (Concrete x \<^bold>\<rightarrow> E! x) w\<rfloor>"
assms_1: shows "\<phi> \<equiv> (1 + sqrt 5) / 2"
q63_upto_def: shows "q63 = (\<lambda>k. k\<^sup>2 mod 63) ` {..<63}"
prime_sqrtI: fixes n :: "nat"   and i :: "nat" assumes "2 \<le> n"   and "\<And>j. \<lbrakk>2 \<le> j; j < i\<rbrakk> \<Longrightarrow> \<not> j dvd n"   and "\<not> i * i \<le> n" shows "prime n"
A_def: fixes a :: "'b"   and t :: "'b tree" shows "A a t = real (T_splay a t) + local.\<Phi> (splay a t) - local.\<Phi> t"
conf: shows "Partial_Cost_Model.config' (rTS h0) ([x, y], h) qs = ([x, y], rev qs @ h)"
pos_ord_acyclic: fixes C :: "'a eclause"   and t :: "'a trm" shows "acyclic (pos_ord C t)"
attractor_def: fixes G :: "('c, 'd) ParityGame_scheme"   and p :: "Player"   and W :: "'c set" assumes "ParityGame G" shows "ParityGame.attractor G p W = lfp (\<lambda>S. W \<union> S \<union> ParityGame.directly_attracted G p S)"
C: shows "s\<^sub>0 \<Turnstile> u"
subgraph_isomorphic_def: fixes G' :: "nat set \<times> nat set set"   and G :: "nat set \<times> nat set set" shows "G' \<sqsubseteq> G \<equiv> uwellformed G \<and> (\<exists>G''. G' \<simeq> G'' \<and> subgraph G'' G)"
mod_mult_mult1: 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 c :: "'b"   and a :: "'b"   and b :: "'b" assumes "class.euclidean_semiring_cancel divide plus minus zero times one modulo euclidean_size" shows "modulo (times c a) (times c b) = times c (modulo a b)"
nat_term_comp_def: fixes cmp :: "'b \<Rightarrow> 'b \<Rightarrow> order" shows "nat_term_comp cmp = ((\<forall>u v. snd (rep_nat_term u) = snd (rep_nat_term v) \<longrightarrow> fst (rep_nat_term u) = 0 \<longrightarrow> cmp u v \<noteq> Gt) \<and> (\<forall>u v. fst (rep_nat_term u) = fst (rep_nat_term v) \<longrightarrow> snd (rep_nat_term u) < snd (rep_nat_term v) \<longrightarrow> cmp u v = Lt) \<and> (\<forall>t u v. cmp u v = Lt \<longrightarrow> cmp (splus t u) (splus t v) = Lt) \<and> (\<forall>u v a b. fst (rep_nat_term u) = fst (rep_nat_term a) \<longrightarrow> fst (rep_nat_term v) = fst (rep_nat_term b) \<longrightarrow> snd (rep_nat_term u) = snd (rep_nat_term v) \<longrightarrow> snd (rep_nat_term a) = snd (rep_nat_term b) \<longrightarrow> cmp a b = Lt \<longrightarrow> cmp u v = Lt))"
HEndPhase0_HInv4b_p_blocksOf: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and bk :: "DiskBlock" assumes "HEndPhase0 s s' p"   and "HInv4d s p"   and "Inv2c_inner s p"   and "bk \<in> blocksOf s p" shows "bal bk < mbal (dblock s' p)"
conjugation_multr: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(x \<cdot> y \<sqinter> z = \<bottom>) = (y \<sqinter> (x \<rhd> z) = \<bottom>)"
order_iff: fixes a :: "'b"   and b :: "'b" shows "order.greater_eq b a = (a = inf a b)"
wfG': shows "wf_graph \<lparr>nodes = V, edges = E - E'\<rparr>"
to_poly_a_inv: fixes a :: "'a" assumes "a \<in> carrier R" shows "to_polynomial R (\<ominus> a) = \<ominus>\<^bsub>P\<^esub> to_polynomial R a"
SAs_def: shows "SAs = fst \<circ> snd \<circ> Rep_hierauto"
i_set_ind_def: shows "i_set_ind \<equiv> {x. i_set_indp x}"
oneof_def: fixes xs :: "char list list" shows "oneof xs = ??.Parser_Monad.oneof_aux xs xs"
step_elim_cases_5: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "('e, 'd, 'f) com"   and cs :: "('e, 'd, 'f) com list"   and css :: "(('e, 'd, 'f) com list \<times> ('e, 'd, 'f) com list) list"   and s :: "'e"   and u :: "('e, 'd, 'f) com list \<times> (('e, 'd, 'f) com list \<times> ('e, 'd, 'f) com list) list \<times> ('e, 'f) xstate" assumes "\<Gamma>\<turnstile> (c # cs, css, Abrupt s) \<rightarrow> u" obtains "u = (cs, css, Abrupt s)"
simps_3: fixes x21 :: "'a"   and x22 :: "'a list" shows "x21 # x22 \<noteq> []"
i2: fixes M2 :: "lam" assumes " P \<longrightarrow>\<^sub>1 M2" shows "\<exists>M3.  P' \<longrightarrow>\<^sub>1 M3 \<and>  M2 \<longrightarrow>\<^sub>1 M3"
except_minus: fixes p :: "'c \<Rightarrow>\<^sub>0 'd"   and q :: "'c \<Rightarrow>\<^sub>0 'd"   and S :: "'c set" shows "except (p - q) S = except p S - except q S"
write0_def: fixes c :: "'c"   and P :: "('c, 'd) action" shows "c \<rightarrow> P \<equiv> Prefix0 c P"
strict_implies_not_eq: fixes b :: "'b list"   and a :: "'b list" assumes "b <p a" shows "a \<noteq> b"
eventually_frequently_const_simps_5: fixes P :: "'b \<Rightarrow> bool"   and C :: "bool"   and F :: "'b filter" shows "(\<forall>\<^sub>F x in F. P x \<longrightarrow> C) = (frequently P F \<longrightarrow> C)"
tinres_commute: fixes r :: "cell list"   and r' :: "cell list" assumes "tinres r r'" shows "tinres r' r"
sort_oalist_id: fixes comp :: "'a \<Rightarrow> 'a \<Rightarrow> order"   and xs :: "('a \<times> 'b) list" assumes "comparator comp"   and "comparator.oalist_inv_raw comp xs" shows "comparator.sort_oalist comp xs = xs"
sample_get: fixes p :: "'p pmf"   and f :: "'p \<Rightarrow> 's \<Rightarrow> 'm" shows "sample p (\<lambda>x. get (f x)) = get (\<lambda>s. sample p (\<lambda>x. f x s))"
hyps: fixes cs :: "'edge list" assumes "upd_cs cs as' = c' # cs'" shows "upd_cs cs (as' @ [a]) = cs'"
case_eq_if: fixes f :: "'d \<Rightarrow> 'e \<Rightarrow> 'c"   and prod :: "'d \<times> 'e" shows "(case prod of (x, xa) \<Rightarrow> f x xa) = f (fst prod) (snd prod)"
assms_1: fixes i :: "'a" assumes "i \<in> A" shows "\<exists>j\<in>B. f i = g j"
finfun_Diag_update1: fixes f :: "('a, 'b) finfun"   and a :: "'a"   and b :: "'b"   and g :: "('a, 'c) finfun" shows "finfun_Diag (finfun_update f a b) g = finfun_update (finfun_Diag f g) a (b, finfun_apply g a)"
coinduct: fixes X :: "'a list set \<Rightarrow> 'a list set \<Rightarrow> bool"   and x1 :: "'a list set"   and x2 :: "'a list set" assumes "X x1 x2"   and "\<And>x1 x2. X x1 x2 \<Longrightarrow> \<exists>K L. x1 = K \<and> x2 = L \<and> K \<subseteq> lists S \<and> L \<subseteq> lists S \<and> ([] \<in> K) = ([] \<in> L) \<and> (\<forall>x. x \<in> S \<longrightarrow> X (Deriv x K) (Deriv x L) \<or> local.bisimilar (Deriv x K) (Deriv x L))" shows "local.bisimilar x1 x2"
def: shows "roots4_generic depressed_solver p = map (\<lambda>y. y - b / (4::'a)) (depressed_solver (c - (3::'a) / (8::'a) * b\<^sup>2) ((b ^ 3 - (4::'a) * b * c + (8::'a) * d) / (8::'a)) ((- (3::'a) * b ^ 4 + (256::'a) * e - (64::'a) * b * d + (16::'a) * b\<^sup>2 * c) / (256::'a)))"
mat_of_cols_index: fixes i :: "nat"   and n :: "nat"   and j :: "nat"   and ws :: "'b vec list" assumes "i < n"   and "j < length ws" shows "mat_of_cols n ws $$ (i, j) = ws ! j $ i"
Cons: fixes ys :: "'d list" shows "zip (map f xs) (map g ys) = map2 (\<lambda>x y. (f x, g y)) xs ys"
no_matching_Goto_Cons: fixes \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and r :: "'c rule"   and rs :: "'c rule list" assumes "no_matching_Goto \<gamma> p [r]"   and "no_matching_Goto \<gamma> p rs" shows "no_matching_Goto \<gamma> p (r # rs)"
induct: fixes P :: "('a \<times> QDelta ns_constraint) list \<Rightarrow> nat \<Rightarrow> bool"   and a0 :: "('a \<times> QDelta ns_constraint) list"   and a1 :: "nat" assumes "\<And>v. P [] v"   and "\<And>i h t v. P t v \<Longrightarrow> P ((i, h) # t) v" shows "P a0 a1"
\<alpha>0_2: shows "\<alpha> \<noteq> 0"
Lim_transform_within_openin: fixes f :: "'c \<Rightarrow> 'd"   and l :: "'d"   and a :: "'c"   and T :: "'c set"   and S :: "'c set"   and g :: "'c \<Rightarrow> 'd" assumes "(f \<longlongrightarrow> l) (at a within T)"   and "openin (top_of_set T) S"   and "a \<in> S"   and "\<And>x. \<lbrakk>x \<in> S; x \<noteq> a\<rbrakk> \<Longrightarrow> f x = g x" shows "(g \<longlongrightarrow> l) (at a within T)"
diff_mult_distrib: fixes m :: "nat"   and n :: "nat"   and k :: "nat" shows "(m - n) * k = m * k - n * k"
eventually_at_ball: fixes d :: "real"   and z :: "'a"   and A :: "'a set" assumes "0 < d" shows "\<forall>\<^sub>F t in at z within A. t \<in> ball z d \<and> t \<in> A"
simps_7: fixes t :: "'c local.term" shows "\<^bold>\<lfloor>\<^bold>\<r>\<^bold>[t\<^bold>]\<^bold>\<rfloor> = \<^bold>\<lfloor>t\<^bold>\<rfloor>"
distinct_1: fixes x1 :: "'g" shows "Node x1 \<noteq> NewEntry"
right_neutral: fixes a :: "'a" shows "a XOR (0::'a) = a"
frameResChainFresh': fixes xvec :: "name list"   and yvec :: "name list"   and F :: "'b frame" shows "xvec \<sharp>* (\<lparr>\<nu>*yvec\<rparr>F) = (\<forall>x\<in>set xvec. x mem yvec \<or> x \<sharp> F)"
f_def: fixes y :: "real" shows "f y = {z. (y, z) \<in> cone hull {1} \<times> S}"
prems: shows "\<turnstile> (\<Delta> @ B # \<Gamma>) ok"
f_def: fixes x :: "int" shows "local.f x = {x, local.f_1 x}"
tau_transition_trans: fixes satisfies :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and transition :: "'a \<Rightarrow> ('c, 'a) residual \<Rightarrow> bool"   and \<tau> :: "'c"   and P :: "'a"   and Q :: "'a"   and R :: "'a" assumes "weak_nominal_ts satisfies transition \<tau>"   and "weak_nominal_ts.tau_transition transition \<tau> P Q"   and "weak_nominal_ts.tau_transition transition \<tau> Q R" shows "weak_nominal_ts.tau_transition transition \<tau> P R"
rel_refl_strong: fixes x :: "('b, 'd, 'e) gpv"   and R1a :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and R2a :: "'d \<Rightarrow> 'd \<Rightarrow> bool" assumes "\<And>z1. z1 \<in> results'_gpv x \<Longrightarrow> R1a z1 z1"   and "\<And>z2. z2 \<in> outs'_gpv x \<Longrightarrow> R2a z2 z2" shows "rel_gpv R1a R2a x x"
mangoldt_primepow: fixes p :: "nat"   and k :: "nat" assumes "prime p" shows "mangoldt (p ^ k) = (if 0 < k then of_real (ln (real p)) else (0::'a))"
ss_pos: fixes x :: "'a" assumes "x \<in> S" shows "0 < supp_sum (\<lambda>V. setdist {x} (S - V)) \<C>"
scale_zero_right: fixes scale :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and a :: "'c" assumes "module scale" shows "scale a (0::'d) = (0::'d)"
to_fun_eval: fixes f :: "nat \<Rightarrow> 'a"   and x :: "'a" assumes "f \<in> carrier P"   and "x \<in> carrier R" shows "to_fun f x = eval R R (\<lambda>x. x) x f"
nc_sccs_eq_reachable: assumes "\<not> param_DFS_defs.cond G (Tarjan_def.tarjan_params G) s" shows "(g_E G)\<^sup>* `` g_V0 G = \<Union> (sccs s)"
m_comm: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier R"   and "y \<in> carrier R" shows "x \<otimes> y = y \<otimes> x"
lsol_ode: shows "(solution solves_ode f) (cball t et) (cball (flow t0 x0 t) ex)"
complete_def: fixes dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and S :: "'b set" assumes "class.metric_space dist uniformity open" shows "uniform_space.complete open uniformity S = (\<forall>f. (\<forall>n. f n \<in> S) \<and> uniform_space.Cauchy uniformity f \<longrightarrow> (\<exists>l\<in>S. filterlim f (topological_space.nhds open l) sequentially))"
Lower_dual: fixes L :: "('a, 'b) gorder_scheme"   and A :: "'a set" shows "Lower (inv_gorder L) A = Upper L A"
igConsIPresIGWls_def: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'b, 'c, 'd, 'e) model_scheme" shows "igConsIPresIGWls MOD \<equiv> igVarIPresIGWls MOD \<and> igAbsIPresIGWls MOD \<and> igOpIPresIGWls MOD"
Dom_g: shows "\<AA>\<lparr>Dom\<rparr>\<lparr>g\<rparr> = a"
pre_mono: fixes P :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and P' :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and C :: "acom"   and l :: "char list \<Rightarrow> nat"   and s :: "char list \<Rightarrow> int" assumes "\<forall>l s. P l s \<longrightarrow> P' l s"   and "pre C P l s" shows "pre C P' l s"
simps_1: fixes q :: "'b fm" shows "conjoin [] q = q"
\<beta>3: shows "\<beta> 3 = (- 1) ^ (\<delta> 1 + 1)"
rel_fundefs_def: fixes f :: "'a \<Rightarrow> ('b, 'c, 'd, 'op) Std.instr fundef option"   and g :: "'a \<Rightarrow> ('b, 'c, 'd, 'op, 'opinl) Inca.instr fundef option" shows "rel_fundefs f g = (\<forall>x. rel_option (rel_fundef (\<lambda>x y. x = norm_instr y)) (f x) (g x))"
prm_unit_inaction: fixes x :: "'b"   and a :: "'b"   and b :: "'b" assumes "x \<noteq> a"   and "x \<noteq> b" shows "[a \<leftrightarrow> b] $ x = x"
assms_1: shows "\<forall>x y. x \<le> y \<longrightarrow> f x \<le> f y"
simps: fixes a :: "'a multiset" shows "local.well_base a = (\<exists>M. a = M \<and> (\<forall>n. count M n < base))"
map_is_cnf: fixes f :: "'a formula"   and F :: "'a \<Rightarrow> 'b" assumes "is_cnf f" shows "is_cnf (map_formula F f)"
that: fixes i :: "nat" assumes "xs ! i = x"   and "ys ! i = y"   and "i < length xs"   and "i < length ys" shows "thesis"
exi_E: fixes A :: "fmla"   and H :: "fmla set"   and B :: "fmla"   and i :: "name" assumes "insert A H \<turnstile> B"   and "atom i \<sharp> B"   and "\<forall>C\<in>H. atom i \<sharp> C" shows "insert (exi i A) H \<turnstile> B"
single_valued_UN_chain: fixes S :: "nat \<Rightarrow> ('a \<times> 'b) set" assumes "chain S"   and "\<And>n. single_valued (S n)" shows "single_valued (\<Union> (range S))"
simps_23: fixes vc :: "'b acom"   and vd :: "'b acom"   and v :: "char list"   and va :: "aexp"   and vb :: "'b" shows "(vc;; vd \<sqsubseteq> v ::= va {vb}) = False"
wellformed_policy1_charn: fixes p :: "('b, 'c) Combinators list" assumes "wellformed_policy1 p"   and "DenyAll \<in> set p" shows "\<exists>p'. p = DenyAll # p' \<and> DenyAll \<notin> set p'"
mult_le_0_iff: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and abs :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.linordered_ring_strict plus zero minus uminus less_eq less abs times" shows "less_eq (times a b) zero = (less_eq zero a \<and> less_eq b zero \<or> less_eq a zero \<and> less_eq zero b)"
scaleC_adj: fixes a :: "complex"   and A :: "'d \<Rightarrow>\<^sub>C\<^sub>L 'c" shows "(a *\<^sub>C A)* = cnj a *\<^sub>C A*"
Dp_Mp_eq: fixes f :: "int poly"   and m :: "int" shows "f = poly_mod.Mp m f + smult m (poly_mod.Dp m f)"
confluent_beta_eta_reds: shows "Joinable.confluent {(s, t). s \<rightarrow>\<^sub>\<beta>\<^sub>\<eta>\<^sup>* t}"
mask_eq_decr_exp: fixes n :: "nat" shows "mask n = 2 ^ n - 1"
unique_circline_set: fixes A :: "complex_homo"   and B :: "complex_homo"   and C :: "complex_homo" assumes "A \<noteq> B"   and "A \<noteq> C"   and "B \<noteq> C" shows "\<exists>!H. A \<in> circline_set H \<and> B \<in> circline_set H \<and> C \<in> circline_set H"
Image_def: fixes r :: "('c \<times> 'b) set"   and s :: "'c set" shows "r `` s = {y. \<exists>x\<in>s. (x, y) \<in> r}"
top_comp: fixes x :: "'a" shows "\<top> * x = \<top>"
lookup_env_restr_not_there: fixes x :: "'d"   and S :: "'d set"   and m :: "'d \<Rightarrow> 'e" assumes "x \<notin> S" shows "(m f|` S) x = \<bottom>"
bot_eq_sup_iff: fixes x :: "'g multiset"   and y :: "'g multiset" shows "({#} = x \<union># y) = (x = {#} \<and> y = {#})"
similar_sq_mtx_def: fixes A :: "'b sq_mtx"   and B :: "'b sq_mtx" shows "(A \<sim> B) = (\<exists>P. mtx_invertible P \<and> A = P\<^sup>-\<^sup>1 * B * P)"
dom: shows "f \<in> {0, 1} \<rightarrow>\<^sub>E {0, 1}"
ereal_le_add_mono2: fixes x :: "ereal"   and z :: "ereal"   and y :: "ereal" assumes "x \<le> z"   and "0 \<le> y" shows "x \<le> y + z"
Dom_in_Obj: fixes f :: "('o, 'a) arr" assumes "arr f" shows "Dom f \<in> Obj"
simps_3: fixes v :: "String.literal"   and va :: "Formula.formula"   and vb :: "Formula.formula" shows "remove_neg (formula.Let v va vb) = formula.Let v va vb"
SR1Subset: fixes p :: "('b, 'c) Combinators list" shows "set (removeShadowRules1 p) \<subseteq> set p"
p_1: fixes z :: "nat" shows "p z tagged_division_of cbox a b"
mult_mono_nonpos_nonpos: fixes c :: "'a"   and a :: "'a"   and d :: "'a"   and b :: "'a" assumes "c \<le> a"   and "d \<le> b"   and "c \<le> (0::'a)"   and "b \<le> (0::'a)" shows "a * b \<le> c * d"
dual_neg_top: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and dual :: "'b \<Rightarrow> 'b"   and dual_star :: "'b \<Rightarrow> 'b"   and omega :: "'b \<Rightarrow> 'b"   and star :: "'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and x :: "'b" assumes "class.mbt_algebra one times inf less_eq less sup dual dual_star omega star bot top" shows "sup (times (dual x) bot) (times x top) = top"
bop_ueval: fixes f :: "'e \<Rightarrow> 'f \<Rightarrow> 'c"   and x :: "('e, 'd) uexpr"   and y :: "('f, 'd) uexpr"   and b :: "'d" shows "\<lbrakk>bop f x y\<rbrakk>\<^sub>e b = f (\<lbrakk>x\<rbrakk>\<^sub>e b) (\<lbrakk>y\<rbrakk>\<^sub>e b)"
permutes_pair_eq: fixes p :: "'b \<Rightarrow> 'b"   and S :: "'b set" assumes "p permutes S" shows "{(p s, s) |s. s \<in> S} = {(s, inv p s) |s. s \<in> S}"
non_empty_terminating_path_root_end_iff: fixes x :: "'a" shows "terminating_path_root_end (start_points x) x (end_points x) = (terminating_path x \<and> x \<noteq> 0)"
HStartBallot_HInv4d_q: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and q :: "Proc" assumes "HStartBallot s s' p"   and "HInv4d s q"   and "p \<noteq> q" shows "HInv4d s' q"
rel_fun_neg_distr_imp: fixes A :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and A' :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and tytok :: "('h \<times> 'i \<times> 'j) itself" assumes "left_unique A"   and "right_total A"   and "right_unique A'"   and "left_total A'" shows "rel_fun_neg_distr_cond A A' tytok"
\<psi>_condition_def: fixes T :: "'v \<Rightarrow> 'w"   and S :: "(('f, 'g) aezfun \<Rightarrow> 'v) \<Rightarrow> 'w" shows "\<psi>_condition T S \<equiv> VectorSpaceHom (\<currency>\<currency>) indV (\<sharp>\<star>) S \<and> map (map S) indVfbasis = negHorbit_homVfbasis T"
divide_right_mono: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<le> b"   and "(0::'a) \<le> c" shows "a / c \<le> b / c"
comp_prefs_comp: fixes u :: "'b list"   and z :: "'b list"   and v :: "'b list"   and w :: "'b list" assumes "u \<cdot> z \<bowtie> v \<cdot> w" shows "u \<bowtie> v"
1: shows "\<forall>PROB s as. finite PROB \<and> P PROB \<and> as \<in> valid_plans PROB \<and> s \<in> valid_states PROB \<longrightarrow> (\<exists>x. x \<in> PLS s as \<and> x < f PROB s)"
size_if_splay: fixes a :: "'b"   and t :: "'b tree"   and l :: "'b tree"   and u :: "'b"   and r :: "'b tree" assumes "splay a t = \<langle>l, u, r\<rangle>" shows "size t = size l + size r + 1"
v_2: shows "[u]\<^sub>\<R> = region X I' r'"
restrict_indets_reduced_GB_1: fixes X :: "'a set"   and F :: "(('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c) set" assumes "finite X"   and "F \<subseteq> P[X]" shows "punit.is_Groebner_basis (restrict_indets ` extended_ord.punit.reduced_GB (homogenize None ` extend_indets ` F))"
less_le_not_le: fixes ko :: "'a key_order"   and x :: "'a"   and y :: "'a" shows "ko.lt ko x y = strict (ko.le ko) x y"
sets_Sup_measure'2: fixes M :: "'a measure set" shows "sets (local.Sup_measure' M) = sigma_sets (\<Union> (space ` M)) (\<Union> (sets ` M))"
real_euclid_B_disjunction: fixes l :: "real"   and b :: "(real, 'b) vec"   and a :: "(real, 'b) vec"   and c :: "(real, 'b) vec" assumes "0 \<le> l"   and "b - a = l *\<^sub>R (c - a)" shows "B\<^sub>\<real> a b c \<or> B\<^sub>\<real> a c b"
path_defs_4: fixes g :: "real \<Rightarrow> 'c" shows "path_image g = g ` {0..1}"
iIN_geD: fixes x :: "nat"   and n :: "nat"   and d :: "nat" assumes "x \<in> [n\<dots>,d]" shows "n \<le> x"
assms_2: shows "h = trdsp (map fst bs) (p, q)"