hcju/isabelleps huggingface dataset:
f4: shows "(\<not> (\<forall>i f F. (f \<in> game_sem i \<alpha> F) = (f \<in> game_sem i \<alpha> (selectlike F f (VV (\<lambda>V. \<forall>i f F. (f \<in> game_sem i \<alpha> F) = (f \<in> game_sem i \<alpha> (selectlike F f V))) (\<lambda>V. \<forall>i f F. f \<in> game_sem i \<alpha> F \<longrightarrow> f \<in> game_sem i \<alpha> (selectlike F f V)))))) \<or> (\<forall>i f F. (f \<in> game_sem i \<alpha> F) = (f \<in> game_sem i \<alpha> (selectlike F f (VV (\<lambda>V. \<forall>i f F. (f \<in> game_sem i \<alpha> F) = (f \<in> game_sem i \<alpha> (selectlike F f V))) (\<lambda>V. \<forall>i f F. f \<in> game_sem i \<alpha> F \<longrightarrow> f \<in> game_sem i \<alpha> (selectlike F f V))))))) \<and> ((\<forall>i f F. (f \<in> game_sem i \<alpha> F) = (f \<in> game_sem i \<alpha> (selectlike F f (VV (\<lambda>V. \<forall>i f F. (f \<in> game_sem i \<alpha> F) = (f \<in> game_sem i \<alpha> (selectlike F f V))) (\<lambda>V. \<forall>i f F. f \<in> game_sem i \<alpha> F \<longrightarrow> f \<in> game_sem i \<alpha> (selectlike F f V)))))) \<or> (rr (VV (\<lambda>V. \<forall>i f F. (f \<in> game_sem i \<alpha> F) = (f \<in> game_sem i \<alpha> (selectlike F f V))) (\<lambda>V. \<forall>i f F. f \<in> game_sem i \<alpha> F \<longrightarrow> f \<in> game_sem i \<alpha> (selectlike F f V))) \<alpha> \<notin> game_sem (ii (VV (\<lambda>V. \<forall>i f F. (f \<in> game_sem i \<alpha> F) = (f \<in> game_sem i \<alpha> (selectlike F f V))) (\<lambda>V. \<forall>i f F. f \<in> game_sem i \<alpha> F \<longrightarrow> f \<in> game_sem i \<alpha> (selectlike F f V))) \<alpha>) \<alpha> (FF (VV (\<lambda>V. \<forall>i f F. (f \<in> game_sem i \<alpha> F) = (f \<in> game_sem i \<alpha> (selectlike F f V))) (\<lambda>V. \<forall>i f F. f \<in> game_sem i \<alpha> F \<longrightarrow> f \<in> game_sem i \<alpha> (selectlike F f V))) \<alpha>)) = (rr (VV (\<lambda>V. \<forall>i f F. (f \<in> game_sem i \<alpha> F) = (f \<in> game_sem i \<alpha> (selectlike F f V))) (\<lambda>V. \<forall>i f F. f \<in> game_sem i \<alpha> F \<longrightarrow> f \<in> game_sem i \<alpha> (selectlike F f V))) \<alpha> \<in> game_sem (ii (VV (\<lambda>V. \<forall>i f F. (f \<in> game_sem i \<alpha> F) = (f \<in> game_sem i \<alpha> (selectlike F f V))) (\<lambda>V. \<forall>i f F. f \<in> game_sem i \<alpha> F \<longrightarrow> f \<in> game_sem i \<alpha> (selectlike F f V))) \<alpha>) \<alpha> (selectlike (FF (VV (\<lambda>V. \<forall>i f F. (f \<in> game_sem i \<alpha> F) = (f \<in> game_sem i \<alpha> (selectlike F f V))) (\<lambda>V. \<forall>i f F. f \<in> game_sem i \<alpha> F \<longrightarrow> f \<in> game_sem i \<alpha> (selectlike F f V))) \<alpha>) (rr (VV (\<lambda>V. \<forall>i f F. (f \<in> game_sem i \<alpha> F) = (f \<in> game_sem i \<alpha> (selectlike F f V))) (\<lambda>V. \<forall>i f F. f \<in> game_sem i \<alpha> F \<longrightarrow> f \<in> game_sem i \<alpha> (selectlike F f V))) \<alpha>) (VV (\<lambda>V. \<forall>i f F. (f \<in> game_sem i \<alpha> F) = (f \<in> game_sem i \<alpha> (selectlike F f V))) (\<lambda>V. \<forall>i f F. f \<in> game_sem i \<alpha> F \<longrightarrow> f \<in> game_sem i \<alpha> (selectlike F f V))))))"
pdiv_4: fixes K :: "'a set" assumes "K \<in> p" shows "\<exists>a b. K = cbox a b"
assms: shows "fmdom' (snd h) \<subseteq> fmdom' s"
finite1: fixes A :: "'a set"   and f :: "'v \<Rightarrow> 'a" shows "(\<Sum>\<^sup>+ y. g (y, x) * indicator A (f y)) \<noteq> \<top>"
rot_diamond_eq_diamond_gen: fixes x :: "real"   and y :: "real" assumes "0 \<le> x"   and "x \<le> 1"   and "0 \<le> y"   and "y \<le> 1" shows "rot_diamond_cube (x, y) = diamond_cube_gen (rot_x y x, (y - 1 / 2) / (2 * diamond_y_gen (x_coord (rot_x y x))) + 1 / 2)"
Sim: fixes \<Psi>' :: "'b"   and R :: "('a, 'b, 'c) psi"   and S :: "('a, 'b, 'c) psi" assumes "(\<Psi>', R, S) \<in> Rel" shows "\<Psi>' \<rhd> R \<leadsto><Rel> S"
sat_models_inter_inj_helper: fixes \<phi> :: "'a ltln"   and P :: "'a ltln set"   and \<psi> :: "'a ltln" assumes "prop_atoms \<phi> \<subseteq> P"   and "prop_atoms \<psi> \<subseteq> P"   and "sat_models (abs_ltln\<^sub>P \<phi>) \<inter> Pow P = sat_models (abs_ltln\<^sub>P \<psi>) \<inter> Pow P" shows "\<phi> \<sim>\<^sub>P \<psi>"
elementary_category_with_binary_products_axioms: shows "elementary_category_with_binary_products (\<cdot>) prj0' prj1'"
SeqSubstFormP_imp_LstSeqP: fixes v :: "tm"   and u :: "tm"   and x :: "tm"   and x' :: "tm"   and s :: "tm"   and k :: "tm" shows "{SeqSubstFormP v u x x' s k} \<turnstile> LstSeqP s k (HPair x x')"
add_idem': fixes x :: "'a list set" shows "x + x = x"
lcm_B: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "a * b l\<rightarrow> c = a l\<rightarrow> b l\<rightarrow> c"
jkbpC_tLength_inv: fixes t :: "'s Trace"   and n :: "nat" assumes "t \<in> Clock.jkbpC"   and "tLength t = n" shows "t \<in> Clock.jkbpCn n"
fimage_of_vempty: fixes r :: "V" shows "r `\<^sub>\<bullet> []\<^sub>\<circ> = []\<^sub>\<circ>"
below_bottom_iff: fixes x :: "'e" shows "(x \<sqsubseteq> \<bottom>) = (x = \<bottom>)"
The_cong: fixes P :: "'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool" assumes "\<And>x. P x = Q x" shows "The P = The Q"
sop_closedTr: fixes s :: "'a \<Rightarrow> 'd \<Rightarrow> 'd"   and A :: "'d set"   and i :: "'d \<Rightarrow> 'd"   and z :: "'d"   and f :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and n :: "nat" assumes "ipp_cond1 (s_set R s A) i"   and "ipp_cond2 z (s_set R s A) i f"   and "ipp_cond3 z i"   and "inv_ipp z i f (s_set R s A)"   and "zeroA z i f (s_set R s A) z"   and "sop_distr2 R s f i (s_set R s A) z"   and "sop_assoc R s (aug_pm_set z i (s_set R s A))"   and "sop_inv R s i (s_set R s A)"   and "sop_one R s (aug_pm_set z i (s_set R s A))"   and "sop_z R s z" shows "\<forall>r\<in>carrier R. \<forall>x\<in>add_set f (aug_pm_set z i (s_set R s A)) n. r \<^sub>s\<cdot> x \<in> add_set f (aug_pm_set z i (s_set R s A)) n"
moebius_reciprocal_inv: shows "moebius_inv moebius_reciprocal = moebius_reciprocal"
ord_eq_le_trans: fixes a :: "'b multiset"   and b :: "'b multiset"   and c :: "'b multiset" assumes "a = b"   and "b \<subseteq># c" shows "a \<subseteq># c"
p_in_reachable: fixes x :: "'s"   and y :: "'s"   and n :: "nat" assumes "(x, y) \<notin> acc" shows "p x y n = 0"
unitary11_sgn_det_orientation': fixes M :: "complex \<times> complex \<times> complex \<times> complex"   and k' :: "real" assumes "congruence M (1, 0, 0, - 1) = cor k' *\<^sub>s\<^sub>m (1, 0, 0, - 1)"   and "k' \<noteq> 0" shows "\<exists>a b k. k \<noteq> 0 \<and> M = k *\<^sub>s\<^sub>m (a, b, cnj b, cnj a) \<and> sgn k' = sgn (Re (mat_det (a, b, cnj b, cnj a)))"
pth_9_2: fixes c :: "real"   and x :: "'c"   and d :: "real"   and z :: "'c" shows "c *\<^sub>R x + (d *\<^sub>R x + z) \<equiv> (c + d) *\<^sub>R x + z"
primitive_matcher_generic_common_matcher: shows "primitive_matcher_generic common_matcher"
conjugate_preserve_poincare_line: fixes u :: "complex_homo"   and v :: "complex_homo" assumes "u \<in> unit_disc"   and "v \<in> unit_disc"   and "u \<noteq> v" shows "poincare_line (conjugate u) (conjugate v) = conjugate_circline (poincare_line u v)"
ipoly_complex_interval_tendsto: fixes C :: "nat \<Rightarrow> complex_interval"   and c :: "complex"   and p :: "int poly" assumes "C \<longlonglongrightarrow>\<^sub>c c" shows "(\<lambda>i. ipoly_complex_interval p (C i)) \<longlonglongrightarrow>\<^sub>c poly (complex_of_int_poly p) c"
induct: fixes P :: "dir list \<Rightarrow> 'c Tree_Relabelling.tree \<Rightarrow> bool"   and a0 :: "dir list"   and a1 :: "'c Tree_Relabelling.tree" assumes "\<And>path l r. P path l \<Longrightarrow> P (dir.Left # path) (Tree_Relabelling.tree.Node l r)"   and "\<And>path l r. P path r \<Longrightarrow> P (dir.Right # path) (Tree_Relabelling.tree.Node l r)"   and "\<And>uu_ x. P uu_ (Tree_Relabelling.tree.Leaf x)"   and "\<And>v va. P [] (Tree_Relabelling.tree.Node v va)" shows "P a0 a1"
SeqThrow: fixes e :: "char list exp"   and e\<^sub>2 :: "char list exp"   and s :: "heap \<times> sfields \<times> sheap"   and b :: "bool"   and P :: "(char list list \<times> char list exp) prog" shows "P \<turnstile> \<langle>throw e;; e\<^sub>2,s,b\<rangle> \<rightarrow> \<langle>throw e,s,b\<rangle>"
finite_UNIV_integer_def: shows "finite_UNIV = Phantom(integer) False"
index_append: fixes xs :: "'b list"   and ys :: "'b list"   and x :: "'b" shows "index (xs @ ys) x = (if x \<in> set xs then index xs x else length xs + index ys x)"
axioms_2: fixes real_resource :: "nat \<Rightarrow> ('i + 'j, 'k + 'l) resource"   and ideal_resource :: "nat \<Rightarrow> ('m + 'j, 'n + 'l) resource"   and sim :: "nat \<Rightarrow> ('i, 'k, 'm, 'n) converter"   and \<I>_real :: "nat \<Rightarrow> ('i, 'k) \<I>"   and \<I>_ideal :: "nat \<Rightarrow> ('m, 'n) \<I>"   and \<I>_common :: "nat \<Rightarrow> ('j, 'l) \<I>"   and bound :: "nat \<Rightarrow> enat"   and lossless :: "bool"   and w :: "nat \<Rightarrow> ('m \<Rightarrow> 'i) \<times> ('k \<Rightarrow> 'n)" assumes "constructive_security2 real_resource ideal_resource sim \<I>_real \<I>_ideal \<I>_common bound lossless w" shows "constructive_security2_axioms sim \<I>_real \<I>_ideal w"
Some_in_results_gpv_catch_gpv: fixes x :: "'d"   and \<I> :: "('e, 'f) \<I>"   and gpv :: "('d, 'e, 'f) gpv" shows "(Some x \<in> results_gpv \<I> (catch_gpv gpv)) = (x \<in> results_gpv \<I> gpv)"
finite_LPDs: fixes r :: "mregex" shows "finite (LPDs r)"
is_subprob_densityD_4: fixes M :: "'c measure"   and f :: "'c \<Rightarrow> ennreal" assumes "is_subprob_density M f" shows "integral\<^sup>N M f \<le> 1"
map_le_dom_subset_restrict: fixes m' :: "'c \<Rightarrow> 'd option"   and m :: "'c \<Rightarrow> 'd option"   and S :: "'c set" assumes "m' \<subseteq>\<^sub>m m"   and "dom m' \<subseteq> S" shows "m' \<subseteq>\<^sub>m m |` S"
hext_trans: fixes typeof_addr :: "'a \<Rightarrow> 'b \<Rightarrow> htype option"   and h :: "'a"   and h' :: "'a"   and h'' :: "'a" assumes "heap_base.hext typeof_addr h h'"   and "heap_base.hext typeof_addr h' h''" shows "heap_base.hext typeof_addr h h''"
prefixes_length_lower_bound: fixes t :: "nat"   and j :: "nat" shows "Suc t \<le> e_length (local.prefixes j t)"
sep_conj_empty: fixes P :: "'a \<Rightarrow> bool" shows "P ** \<box> = P"
classical_PP: fixes \<phi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "Fvars \<phi> = {}"   and "prv (neg (neg (PP \<langle>\<phi>\<rangle>)))" shows "prv (PP \<langle>\<phi>\<rangle>)"
impl_test_def: fixes ifex_of :: "'d \<Rightarrow> 'e ifex"   and b1 :: "'d"   and b2 :: "'d" shows "impl_test ifex_of b1 b2 = (normif Mapping.empty (ifex_of b1) (ifex_of b2) Trueif = Trueif)"
length_plays_from_strategy: fixes \<sigma> :: "'c list \<Rightarrow> 'c"   and p :: "'c list" shows "llength (strategy_progression \<sigma> p) = \<infinity>"
llist_exhaust: fixes xs :: "'b llist" obtains ys :: "'b list" where "xs = llist_of ys"   | f :: "nat \<Rightarrow> 'b" where "xs = inf_llist f"
is_inf_uniq: fixes x :: "'c"   and y :: "'c"   and inf :: "'c"   and inf' :: "'c" assumes "is_inf x y inf"   and "is_inf x y inf'" shows "inf = inf'"
empty_mgu: fixes L :: "fterm literal set" assumes "unifier\<^sub>l\<^sub>s \<epsilon> L" shows "mgu\<^sub>l\<^sub>s \<epsilon> L"
succs_is_node: fixes ks :: "int list"   and l :: "int"   and x :: "int"   and n :: "nat" assumes "dioph_is_node ks l x" shows "list_all (dioph_is_node ks l) (dioph_ineq_succs n ks x)"
simps_3: fixes x2 :: "'a" shows "\<lfloor>x2\<rfloor> \<noteq> None"
isSng_correct: fixes s :: "'s" assumes "invar s" shows "isSng s = (\<exists>e. \<alpha> s = {e})"
erfc_real_le_0_iff: fixes x :: "real" shows "(erfc x \<le> 1) = (0 \<le> x)"
simps_2: fixes d :: "'b"   and a :: "'b" shows "fromMaybe\<cdot>d\<cdot>(Just\<cdot>a) = a"
inv_comp_right: fixes f :: "'b \<Rightarrow> 'b" assumes "bijection f" shows "f \<circ> inv f = id"
polynomial_of_mset_empty: shows "polynomial_of_mset {#} = 0"
lipschitz_on_add: fixes C :: "real"   and U :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and D :: "real"   and g :: "'b \<Rightarrow> 'c" assumes "C-lipschitz_on U f"   and "D-lipschitz_on U g" shows "(C + D)-lipschitz_on U (\<lambda>x. f x + g x)"
discrete_eqs_2: fixes X :: "'a set" assumes "discrete X" shows "K = id"
elim_eq: fixes \<sigma> :: "('b \<times> 'b trm) list"   and \<theta> :: "('b \<times> 'b trm) list"   and x :: "'b" assumes "\<sigma> \<doteq> \<theta>" shows "elim \<sigma> x = elim \<theta> x"
t_stop_pos: fixes a :: "real"   and v :: "real" assumes "braking_movement a v"   and "v \<noteq> 0" shows "0 < movement.t_stop a v"
simps_4: fixes uu :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and uv :: "'d" shows "Semantics.matches uu MatchAny uv = True"
card_UN_disjoint: fixes I :: "'b set"   and A :: "'b \<Rightarrow> 'c set" assumes "finite I"   and "\<forall>i\<in>I. finite (A i)"   and "\<forall>i\<in>I. \<forall>j\<in>I. i \<noteq> j \<longrightarrow> A i \<inter> A j = {}" shows "card (\<Union> (A ` I)) = (\<Sum>i\<in>I. card (A i))"
cnorm_le_square: fixes x :: "'a"   and a :: "real" shows "(norm x \<le> a) = (0 \<le> a \<and> x \<bullet>\<^sub>C x \<le> complex_of_real (a\<^sup>2))"
f_Exec_Stream_Init_append: fixes trans_fun :: "'e \<Rightarrow> 'd \<Rightarrow> 'd"   and xs :: "'e list"   and ys :: "'e list"   and c :: "'d" shows "f_Exec_Comp_Stream_Init trans_fun (xs @ ys) c = f_Exec_Comp_Stream_Init trans_fun xs c @ tl (f_Exec_Comp_Stream_Init trans_fun ys (f_Exec_Comp trans_fun xs c))"
rel_witness_generat2: fixes A :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and C :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and R :: "'h \<Rightarrow> 'i \<Rightarrow> bool"   and x :: "('d, 'f, 'h) generat"   and y :: "('e, 'g, 'i) generat" assumes "rel_generat A C R x y" shows "rel_generat (\<lambda>(a, b') b. b = b' \<and> A a b') (\<lambda>(c, d') d. d = d' \<and> C c d') (\<lambda>(r, s') s. s = s' \<and> R r s') (rel_witness_generat (x, y)) y"
span_redundant: fixes x :: "'b"   and S :: "'b set" assumes "x \<in> span S" shows "span (insert x S) = span S"
hs_dfs_invar_final: fixes \<Sigma>i :: "('a, unit) HashMap.hashmap"   and post :: "'a \<Rightarrow> 'a dlist" shows "\<forall>sc. sc \<in> wa_invar (det_wa_wa (hs_dfs_dwa \<Sigma>i post)) \<and> sc \<notin> wa_cond (det_wa_wa (hs_dfs_dwa \<Sigma>i post)) \<longrightarrow> fst (dfs_\<alpha> (hs_dfs_\<alpha> sc)) = (hs_R post)\<^sup>* `` hs.\<alpha> \<Sigma>i"
valid_K_cfg: fixes cfg :: "('s \<times> ('c \<Rightarrow> real) set) cfg"   and cfg' :: "('s \<times> ('c \<Rightarrow> real) set) cfg" assumes "cfg \<in> R_G.valid_cfg"   and "cfg' \<in> set_pmf (K_cfg cfg)" shows "cfg' \<in> R_G.valid_cfg"
u_def: shows "u \<equiv> up \<otimes>\<^bsub>relative_homology_group (int n) (nsphere n) (equator n)\<^esub> inv\<^bsub>relative_homology_group (int n) (nsphere n) (equator n)\<^esub> un"
dperm_inv: fixes as :: "(nat \<times> nat list) list"   and rs :: "'b list"   and s0 :: "'b list"   and n :: "nat" assumes "MTF_Off as rs s0" shows "dist_perm (MTF_Off.s_A as rs s0 n) (MTF_Off.s_mtf rs s0 n)"
prime_ge_2_nat: fixes p :: "nat" assumes "normalization_semidom_class.prime p" shows "2 \<le> p"
restrict_to_halfspaces_def: fixes sctns :: "'b sctn set"   and X :: "'b set" shows "restrict_to_halfspaces sctns X = SPEC (\<lambda>R. R = X \<inter> below_halfspaces sctns)"
rcons_def: fixes t :: "('a, 'b) FingerTree"   and a :: "'a \<times> 'b" shows "FingerTree.rcons t a \<equiv> Abs_FingerTree (FingerTreeStruc.rcons (Rep_FingerTree t) a)"
rename_image_preserves: fixes h :: "'c \<Rightarrow> 'd"   and v :: "'c \<Rightarrow> 'e" assumes "bij h" shows "rename h ` preserves v = preserves (v \<circ> inv h)"
necessary_condition_one: shows "\<Lambda> * (\<v> - 1) div (\<k> - 1) * (\<k> - 1) = \<Lambda> * (\<v> - 1)"
shift_append_preserves_T: fixes \<tau> :: "ty"   and \<rho>3 :: "val set list"   and \<rho>2 :: "val set list" shows "T \<tau> \<rho>3 = T (shift (length \<rho>2) 0 \<tau>) (\<rho>2 @ \<rho>3)"
lcs_comm: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and lcs :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and s :: "'c"   and t :: "'c" assumes "class.lcs_powerprod plus minus zero lcs" shows "lcs s t = lcs t s"
UP_tuple_DOWN_membI: fixes f :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and g :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and x :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "span f g"   and "x \<in> Dom f" shows "UP \<langle>DOWN (Fun f x), DOWN (Fun g x)\<rangle> \<in> local.set (local.prod (cod f) (cod g))"
assms: shows "Gromov_completion_rel (\<lambda>n. x) u"
min_rvar_incdec_eq_None: fixes dir :: "('i, 'a) Direction"   and s :: "('i, 'a) state"   and eq :: "nat \<times> linear_poly"   and "is" :: "'i list" assumes "min_rvar_incdec_eq dir s eq = Inl is" shows "(\<forall>x\<in>rvars_eq eq. \<not> (0 < coeff (rhs eq) x \<and> \<lhd>\<^sub>u\<^sub>b (Direction.lt dir) (\<langle>\<V> s\<rangle> x) (Direction.UB dir s x) \<or> coeff (rhs eq) x < 0 \<and> \<rhd>\<^sub>l\<^sub>b (Direction.lt dir) (\<langle>\<V> s\<rangle> x) (Direction.LB dir s x))) \<and> set is = {Direction.LI dir s (lhs eq)} \<union> {Direction.LI dir s x |x. x \<in> rvars_eq eq \<and> coeff (rhs eq) x < 0} \<union> {Direction.UI dir s x |x. x \<in> rvars_eq eq \<and> 0 < coeff (rhs eq) x} \<and> (dir = Positive \<or> dir = Negative \<longrightarrow> Direction.LI dir s (lhs eq) \<in> indices_state s \<longrightarrow> set is \<subseteq> indices_state s)"
containsTrueNotUnit: fixes l :: "Literal"   and c :: "Literal list"   and v :: "Literal list" assumes "l el c"   and "literalTrue l v"   and "consistent v" shows "\<nexists>ul. isUnitClause c ul v"
extremum: fixes a :: "'a" shows "\<bottom> \<sqsubseteq> a"
mat_cons3: fixes env :: "v sem_env"   and s :: "'b state"   and p :: "pat"   and v1 :: "v"   and ck :: "bool"   and e :: "exp0"   and pes :: "(pat \<times> exp0) list"   and err_v :: "v" assumes "pmatch (c env) (refs s) p v1 [] = Match_type_error" shows "evaluate_match ck env s v1 ((p, e) # pes) err_v (s, Rerr (Rabort Rtype_error))"
cod_char: fixes f :: "'c local.term set \<times> 'c local.term set" shows "CC.cod f = (if arr (fst f) \<and> arr (snd f) then (cod (fst f), cod (snd f)) else (null, null))"
wf_idecl_supD: fixes G :: "prog"   and I :: "qtname"   and i :: "iface"   and J :: "qtname" assumes "wf_idecl G (I, i)"   and "J \<in> set (isuperIfs i)" shows "is_acc_iface G (pid I) J \<and> (J, I) \<notin> (subint1 G)\<^sup>+"
terminate_F_lemma: fixes m :: "nat"   and r :: "nat" assumes "terminate rec_halt [m, r]" shows "terminate rec_F [m, r]"
mult_zero_left: fixes x :: "V" shows "0 * x = 0"
wary_\<rho>x: fixes x :: "'v" shows "wary (\<rho> x)"
zero_eq_add_iff_both_eq_0: fixes x :: "'a multiset"   and y :: "'a multiset" shows "({#} = x + y) = (x = {#} \<and> y = {#})"
axioms_1: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<omega> :: "'b" assumes "right_hom V H \<omega>" shows "weak_composition V H"
tensor_mat_mult: fixes m1 :: "'a mat"   and m2 :: "'a mat"   and m3 :: "'a mat"   and m4 :: "'a mat" assumes "m1 \<in> carrier_mat d1 d1"   and "m2 \<in> carrier_mat d1 d1"   and "m3 \<in> carrier_mat d2 d2"   and "m4 \<in> carrier_mat d2 d2" shows "tensor_mat (m1 * m2) (m3 * m4) = tensor_mat m1 m3 * tensor_mat m2 m4"
assms_2: shows "A2 = B2 \<union> C2"
set_1: fixes x1 :: "'b" shows "set_flat_complete_lattice (flat_complete_lattice.Value x1) = {x1}"
z_gt: shows "\<forall>y. y \<in># Ya \<longrightarrow> y \<in> A \<and> z \<in> A \<and> R y z"
subgroupE_1: fixes H :: "'a set" assumes "subgroup H (G\<lparr>carrier := H\<rparr>)" shows "H \<subseteq> carrier (G\<lparr>carrier := H\<rparr>)"
InitBlockReds: fixes e :: "char list exp"   and h :: "heap"   and l :: "sfields"   and V :: "char list"   and v :: "val"   and sh :: "sheap"   and b :: "bool"   and e' :: "char list exp"   and h' :: "heap"   and l' :: "sfields"   and sh' :: "sheap"   and b' :: "bool"   and P :: "(char list list \<times> char list exp) prog"   and T :: "ty" assumes "P \<turnstile> \<langle>e,(h, l(V \<mapsto> v), sh),b\<rangle> \<rightarrow>* \<langle>e',(h', l', sh'),b'\<rangle>" shows "P \<turnstile> \<langle>{V:T; V:=Val v;; e},(h, l, sh),b\<rangle> \<rightarrow>* \<langle>{V:T; V:=Val (the (l' V));; e'},(h', l'(V := l V), sh'),b'\<rangle>"
cmod_complex_polar: fixes r :: "real"   and a :: "real" shows "cmod (cor r * (cor (cos a) + \<i> * cor (sin a))) = \<bar>r\<bar>"
oppOrdering_def: shows "oppOrdering = (\<Lambda> x. case x of LT \<Rightarrow> GT | EQ \<Rightarrow> EQ | GT \<Rightarrow> LT)"
integral_measurable_subprob_algebra: fixes f :: "'c \<Rightarrow> 'd"   and N :: "'c measure" assumes "f \<in> borel_measurable N" shows "(\<lambda>M. integral\<^sup>L M f) \<in> borel_measurable (subprob_algebra N)"
gs_dominate: fixes g1 :: "'a \<Rightarrow> real"   and g2 :: "'a \<Rightarrow> real"   and h1 :: "'a \<Rightarrow> real"   and h2 :: "'a \<Rightarrow> real" assumes "g1 \<in> G"   and "g2 \<in> G"   and "h1 \<in> H"   and "h2 \<in> H"   and "g1 \<in> o[F](g2)" shows "(\<lambda>x. g1 x * h1 x) \<in> o[F](\<lambda>x. g2 x * h2 x)"
simps_1: fixes x :: "octo"   and y :: "octo" shows "Ree (x + y) = Ree x + Ree y"
rsp: shows "eq_onp (\<lambda>f. \<forall>\<^sub>\<infinity>n. f (- int n) = (0::'a)) (\<lambda>k. if k = 0 then 1::'a else (0::'a)) (\<lambda>k. if k = 0 then 1::'a else (0::'a))"
Par_def: fixes A1 :: "('c, 'd) action"   and ns1 :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and cs :: "'c set"   and ns2 :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and A2 :: "('c, 'd) action" shows "A1 \<lbrakk> ns1 | cs | ns2 \<rbrakk> A2 \<equiv> action_of (R (\<lambda>(S, S'). (\<nexists>tr1 tr2. (action.relation_of A1\<^sup>f\<^sub>f ;; (\<lambda>(S, S'). tr1 = tr S)) (S, S') \<and> (spec False (wait S) (action.relation_of A2) ;; (\<lambda>(S, uu_). tr2 = tr S)) (S, S') \<and> tr_filter tr1 cs = tr_filter tr2 cs) \<and> (\<nexists>tr1 tr2. (spec False (wait S) (action.relation_of A1) ;; (\<lambda>(S, uu_). tr1 = tr S)) (S, S') \<and> (action.relation_of A2\<^sup>f\<^sub>f ;; (\<lambda>(S, S'). tr2 = tr S)) (S, S') \<and> tr_filter tr1 cs = tr_filter tr2 cs) \<turnstile> \<lambda>(S, S'). \<exists>s1 s2. ((\<lambda>(A, A'). action.relation_of A1\<^sup>t\<^sub>f (A, s1) \<and> action.relation_of A2\<^sup>t\<^sub>f (A, s2)) ;; M_par s1 ns1 s2 ns2 cs) (S, S')))"
inv_t2: shows "tree_invar (Node e2 a2 r ts2)"
assms: shows "inv\<^sub>2 P\<^sub>1 P\<^sub>2 B\<^sub>1 B\<^sub>2 V"
m_o2: fixes X :: "char list set"   and o1 :: "(char list \<Rightarrow> 'av) option"   and o2 :: "(char list \<Rightarrow> 'av) option" assumes "finite X"   and "top_on\<^sub>o o1 (- X)"   and "top_on\<^sub>o o2 (- X)"   and "o1 < o2" shows "m\<^sub>o o2 X < m\<^sub>o o1 X"
update_left: fixes y :: "nat"   and t :: "(nat \<times> linear_poly) list"   and s :: "('c, 'a) state"   and x :: "nat"   and v :: "'a" assumes "y \<in> lvars t" shows "\<exists>eqa\<in>set t. lhs eqa = y \<and> Mapping.lookup (foldl (\<lambda>v' e. Mapping.update (lhs e) (rhs_eq_val (\<V> s) x v e) v') (\<V> s) t) y = Some (rhs_eq_val (\<V> s) x v eqa)"
trivial_homology_group: fixes p :: "int"   and X :: "'a topology" assumes "p < 0" shows "trivial_group (homology_group p X)"
normalize_replicate_zero: fixes n :: "nat"   and p :: "'a list list" shows "UP.normalize (replicate n \<zero>\<^bsub>K [X]\<^esub> @ p) = UP.normalize p"
diff_numeral_special_10: shows "- (1::'a) - (1::'a) = - (2::'a)"
configuration_finite: fixes \<Sigma> :: "'c set"   and \<delta> :: "'d \<Rightarrow> 'c \<Rightarrow> 'd"   and q\<^sub>0 :: "'d"   and w :: "nat \<Rightarrow> 'c"   and q :: "'d"   and n :: "nat" assumes "semi_mojmir \<Sigma> \<delta> q\<^sub>0 w" shows "finite (semi_mojmir_def.configuration \<delta> q\<^sub>0 w q n)"
blinding_of_T_mono: fixes bo :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and bo' :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and rh :: "'e \<Rightarrow> 'f" assumes "bo \<le> bo'" shows "blinding_of_T rh bo \<le> blinding_of_T rh bo'"
add_mset_remove_trivial_eq: fixes N :: "'c multiset"   and a :: "'c" shows "(N = add_mset a (remove1_mset a N)) = (a \<in># N)"
p_def: fixes b :: "'b set" shows "p b \<equiv> independent b \<and> span b = span V"
val_update_rightmost_neq_None: fixes t :: "'b tree"   and f :: "'b \<Rightarrow> 'b" assumes "val t \<noteq> None" shows "val (update_rightmost f t) \<noteq> None"