hcju/isabelleps huggingface dataset:
poly_eval_plus: fixes a :: "'b \<Rightarrow> 'a"   and p :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'a"   and q :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'a" shows "poly_eval a (p + q) = poly_eval a p + poly_eval a q"
abs_0: fixes abs :: "'a \<Rightarrow> 'a"   and sgn :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a" assumes "class.idom_abs_sgn abs sgn times one plus zero minus uminus" shows "abs zero = zero"
add_sup_distrib_left: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "a + sup b c = sup (a + b) (a + c)"
Vinv3_invariant: shows "reach v_TS \<subseteq> Vinv3"
2: shows "x \<squnion> z \<in> F \<and> y \<squnion> z \<in> G"
fmempty_lookup: fixes x :: "'d" shows "fmlookup fmempty x = None"
prime: fixes p :: "int" assumes "padic_integers p" shows "prime p"
less_LCons: fixes a :: "'b"   and r :: "'b llist"   and b :: "'b"   and t :: "'b llist" shows "(a ## r < b ## t) = (a = b \<and> r < t)"
lunit_simps_5: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i> :: "'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and f :: "'b" assumes "bicategory V H \<a> \<i> src trg"   and "partial_magma.ide V f" shows "partial_magma.cod V (bicategory.lunit V H \<a> \<i> src trg f) = f"
bit_c'': fixes x :: "'a"   and y :: "'a"   and x0 :: "'a"   and y0 :: "'a"   and s :: "('a list \<times> bool list \<times> 'a list) pmf"   and qs :: "'a list" assumes "x \<noteq> y"   and "{x, y} = {x0, y0}"   and "BIT_inv s x [x0, y0]"   and "set qs \<subseteq> {x, y}"   and "qs \<in> lang (seq [question (Atom x), Atom y, Atom x, Star (Times (Atom y) (Atom x)), Atom x])" shows "T\<^sub>p_on_rand' BIT s qs \<le> 175 / 10\<^sup>2 * real (T\<^sub>p [x, y] qs (OPT2 qs [x, y])) \<and> BIT_inv (Partial_Cost_Model.config'_rand BIT s qs) (last qs) [x0, y0]"
simps_2: fixes x21 :: "'b btree"   and x22 :: "'b"   and x23 :: "'b btree"   and y21 :: "'b btree"   and y22 :: "'b"   and y23 :: "'b btree" shows "(Up\<^sub>i x21 x22 x23 = Up\<^sub>i y21 y22 y23) = (x21 = y21 \<and> x22 = y22 \<and> x23 = y23)"
inject: fixes x1 :: "'a"   and x2 :: "nat"   and x3 :: "'b list"   and y1 :: "'a"   and y2 :: "nat"   and y3 :: "'b list" shows "(Frame x1 x2 x3 = Frame y1 y2 y3) = (x1 = y1 \<and> x2 = y2 \<and> x3 = y3)"
distinct_sorted_list_of_set: fixes A :: "'c set" shows "distinct (sorted_list_of_set A)"
at_within_def: fixes a :: "'a"   and s :: "'a set" shows "at a within s = inf (nhds a) (principal (s - {a}))"
pseudonatural_transformation_axioms: shows "pseudonatural_transformation (\<cdot>\<^sub>C) (\<star>\<^sub>C) \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C (\<cdot>\<^sub>D) (\<star>\<^sub>D) \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D G' \<Phi>\<^sub>G' G \<Phi>\<^sub>G \<tau>'.map\<^sub>0 \<tau>'.map\<^sub>1"
O1_def: shows "O1 = {x \<in> space M. \<forall>B\<ge>N1. real (card ({n. \<exists>l\<in>{1..n}. u n x - u (n - l) x \<le> - c0 * real l} \<inter> {..<B})) < d2 / 2 * real B}"
diff: fixes x :: "'b"   and y :: "'b" shows "a *s (x - y) = a *s x - a *s y"
simps_2: fixes "is" :: "'b set"   and t :: "'b" shows "interrupt_action_ok' is (ClearInterrupt t) = True"
remap_polys_l2_remap_polys_l: fixes spec :: "(char list list \<times> int) list"   and \<V> :: "char list set"   and A :: "(nat, (char list list \<times> int) list) fmap" shows "remap_polys_l2 spec \<V> A \<le> \<Down> Id (remap_polys_l spec \<V> A)"
assms: fixes x :: "'a"   and y :: "'a" assumes "x \<in> set xs"   and "y \<in> set ys" shows "f x \<noteq> f y"
sas_plus_formalism_and_induced_strips_formalism_are_equally_expressive_ii_b: fixes \<Psi> :: "('c, 'd) sas_plus_problem"   and k :: "nat" assumes "is_valid_problem_sas_plus \<Psi>" shows "finite (bounded_solution_set_strips' (\<phi> \<Psi> ) k)"
boundedI: fixes b :: "'b"   and a :: "'b"   and c :: "'b" assumes "b \<le> a"   and "c \<le> a" shows "max b c \<le> a"
matrix_vector_mult_scaleR: fixes A :: "((real, 'b) vec, 'a) vec"   and c :: "real"   and x :: "(real, 'b) vec" shows "A *v c *\<^sub>R x = c *\<^sub>R (A *v x)"
card_0_eq: fixes A :: "'e set" assumes "finite A" shows "(card A = 0) = (A = {})"
is_extensional: fixes f :: "'c local.term set" assumes "\<not> \<F>\<^sub>SC.arr f" shows "DoS.map f = \<F>\<^sub>SC.null"
simps_1: fixes x :: "'d"   and env :: "'d \<Rightarrow> 'c" shows "eval (Variable x) env = env x"
takeWhile_append1: fixes x :: "'a"   and xs :: "'a list"   and P :: "'a \<Rightarrow> bool"   and ys :: "'a list" assumes "x \<in> set xs"   and "\<not> P x" shows "takeWhile P (xs @ ys) = takeWhile P xs"
eq_refl: fixes x :: "'a list"   and y :: "'a list" assumes "x = y" shows "x \<preceq> y"
gcd_neg2: fixes one :: "'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and gcd :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and lcm :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.ring_gcd one zero times gcd lcm divide plus minus unit_factor normalize uminus" shows "gcd a (uminus b) = gcd a b"
Cod_cod: fixes f :: "(bool, 'arr list) concrete_category.arr" assumes "arr f" shows "Cod (cod f) = Cod f"
Mp_ident_iff: fixes f :: "int poly" shows "(q.Mp f = f) = (\<forall>n. poly.coeff f n \<in> {0..<q})"
hyps_3: shows "\<forall>t\<in>{s<..<1}. Re (g t) \<noteq> Re z"
node_deliver_messages_append: fixes xs :: "('a \<times> ('a, 'b) operation) event list"   and ys :: "('a \<times> ('a, 'b) operation) event list" shows "node_deliver_messages (xs @ ys) = node_deliver_messages xs @ node_deliver_messages ys"
mono_neutral_right: fixes T :: "'a set"   and S :: "'a set"   and g :: "'a \<Rightarrow> 'b" assumes "finite T"   and "S \<subseteq> T"   and "\<forall>i\<in>T - S. g i = (1::'b)" shows "prod g T = prod g S"
Rep_regular: fixes x :: "'b regular" shows "Rep_regular x \<in> regular_elements"
old_inferences_between_eq_new_inferences_between: fixes N :: "'a literal multiset set"   and C :: "'a literal multiset" shows "old_concls_of (inference_system.inferences_between (ord_FO_\<Gamma> S) N C) = concl_of ` F.Inf_between N {C}"
mat_extension_def: fixes m :: "'b mat" shows "mat_extension m = tensor_mat m (1\<^sub>m d2)"
tag_name_eq_h3: fixes ptr' :: "(_) element_ptr"   and tag :: "char list" shows "h3 \<turnstile> get_tag_name ptr' \<rightarrow>\<^sub>r tag = h' \<turnstile> get_tag_name ptr' \<rightarrow>\<^sub>r tag"
mat_plusI_def: fixes pl :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and m1 :: "'b list list"   and m2 :: "'b list list" shows "mat_plusI pl m1 m2 = map (\<lambda>uv. vec_plusI pl (fst uv) (snd uv)) (zip m1 m2)"
cases: fixes ground_heads_var :: "'a \<Rightarrow> 'b set"   and gt_sym :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and extf :: "'b \<Rightarrow> (('b, 'a) tm \<Rightarrow> ('b, 'a) tm \<Rightarrow> bool) \<Rightarrow> ('b, 'a) tm list \<Rightarrow> ('b, 'a) tm list \<Rightarrow> bool"   and arity_sym :: "'b \<Rightarrow> enat"   and arity_var :: "'a \<Rightarrow> enat"   and a1 :: "('b, 'a) tm"   and a2 :: "('b, 'a) tm" assumes "rpo_optim ground_heads_var gt_sym extf arity_sym arity_var"   and "rpo_optim.gt ground_heads_var gt_sym extf a1 a2" obtains ti :: "('b, 'a) tm" and t :: "('b, 'a) tm" and s :: "('b, 'a) tm" where "a1 = tb__" and "a2 = sb__" and "ti \<in> set (args tb__)" and "rpo_optim.gt ground_heads_var gt_sym extf ti sb__ \<or> ti = sb__"   | t :: "('b, 'a) tm" and s :: "('b, 'a) tm" where "a1 = tc__" and "a2 = sc__" and "ground_heads.gt_hd gt_sym ground_heads_var (head tc__) (head sc__)" and "rpo_basis.chkvar tc__ sc__" and "rpo_optim.chkargs (rpo_optim.gt ground_heads_var gt_sym extf) tc__ sc__"   | t :: "('b, 'a) tm" and s :: "('b, 'a) tm" where "a1 = t" and "a2 = s" and "head t = head s" and "rpo_optim.chkargs (rpo_optim.gt ground_heads_var gt_sym extf) t s" and "\<forall>f\<in>ground_heads.ground_heads ground_heads_var (head t). extf f (rpo_optim.gt ground_heads_var gt_sym extf) (args t) (args s)"
finite_has_minimal2: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and A :: "'c set"   and a :: "'c" assumes "class.order less_eq less"   and "finite A"   and "a \<in> A" shows "\<exists>m\<in>A. less_eq m a \<and> (\<forall>b\<in>A. less_eq b m \<longrightarrow> m = b)"
carrier_direct_limE: fixes X :: "('c set \<times> 'd) set" assumes "X \<in> dlY.carrier_direct_lim" obtains U :: "'c set" and s :: "'d" where "U \<in> target.neighborhoods (f x)" and "s \<in> \<O>\<^sub>Y U" and "X = dlY.class_of U s"
comb'_repr': fixes \<psi> :: "'b" assumes "\<psi> \<in> span B" shows "comb' (repr' \<psi>) = \<psi>"
asm: shows "\<forall>x\<in>set (Abs_State.dom S2). (x \<in> set (Abs_State.dom S1) \<longrightarrow> fun S1 x \<sqsubseteq> fun S2 x) \<and> (x \<in> set (Abs_State.dom S1) \<or> \<top> \<sqsubseteq> fun S2 x)"
in_outs'_restrict_gpvD: fixes x :: "'out"   and gpv :: "('b, 'out, 'in) gpv" assumes "x \<in> outs'_gpv (local.restrict_gpv gpv)" shows "x \<in> outs_\<I> \<I>"
eff_def: fixes i :: "'c instr"   and P :: "'d prog"   and pc :: "nat"   and et :: "(nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list"   and t :: "(ty list \<times> ty err list) option" shows "eff i P pc et t \<equiv> case t of None \<Rightarrow> [] | \<lfloor>\<tau>\<rfloor> \<Rightarrow> norm_eff i P pc \<tau> @ xcpt_eff i P pc \<tau> et"
intros_4: fixes KS :: "key list"   and X :: "msg"   and Y :: "msg"   and H :: "msg set" assumes "foldr Crypt KS \<lbrace>X, Y\<rbrace> \<in> crypts H" shows "foldr Crypt KS Y \<in> crypts H"
lunit_hcomp_3: fixes f :: "'d"   and g :: "'d" assumes "D.ide f"   and "D.ide g"   and "src\<^sub>D f = trg\<^sub>D g" shows "\<l>\<^sub>D[f \<star>\<^sub>D g] = (\<l>\<^sub>D[f] \<star>\<^sub>D g) \<cdot>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[trg\<^sub>D f, f, g]"
ord_isom_sym: fixes E :: "('c, 'd) Order_scheme"   and f :: "'a \<Rightarrow> 'c" assumes "Order E"   and "ord_isom D E f" shows "ord_isom E D (f\<inverse>\<^bsub>carrier E,carrier D\<^esub>)"
F3_def: shows "F3 = (\<lambda>x. - c / (x - a) ^ (n - 1) / real (n - 1))"
replicate_mset_Suc: fixes n :: "nat"   and x :: "'a" shows "replicate_mset (Suc n) x = add_mset x (replicate_mset n x)"
point_below_vector: fixes v :: "'a" assumes "is_vector v"   and "v \<noteq> 0" shows "\<exists>x. point x \<and> x \<le> v"
wls_skel_vsubst: fixes s :: "'sort"   and X :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set"   and ys :: "'varSort"   and y1 :: "'var"   and y2 :: "'var" assumes "wls s X" shows "skel (X #[y1 // y2]_ys) = skel X"
map_fst_zip_take: fixes xs :: "'b list"   and ys :: "'c list" shows "map fst (zip xs ys) = take (min (length xs) (length ys)) xs"
a'_def: shows "a' = a - (\<Sum>b\<in>S. (b \<bullet> a / (b \<bullet> b)) *\<^sub>R b)"
cases: fixes x :: "rat \<times> rat" obtains l' :: "rat" and r' :: "rat" where "x = (l', r')"
filter_mset_cong: fixes M :: "'c multiset"   and M' :: "'c multiset"   and P :: "'c \<Rightarrow> bool"   and Q :: "'c \<Rightarrow> bool" assumes "M = M'"   and "\<And>a. a \<in># M \<Longrightarrow> P a = Q a" shows "filter_mset P M = filter_mset Q M"
insert: fixes A :: "'b set"   and x :: "'b" assumes "finite A"   and "x \<notin> A" shows "card (insert x A) = Suc (card A)"
diagonal_to_Smith_row_i_dvd_jj': fixes bezout :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a \<times> 'a \<times> 'a"   and a :: "'b"   and i :: "nat"   and A :: "(('a, 'c) vec, 'b) vec"   and b :: "'c"   and c :: "nat" assumes "is_bezout_ext bezout"   and "to_nat a \<in> {i..<min (nrows A) (ncols A)}"   and "to_nat a = to_nat b"   and "c \<le> i"   and "\<forall>a b. to_nat a \<in> set [i..<min (nrows A) (ncols A)] \<and> to_nat a = to_nat b \<longrightarrow> A $ from_nat c $ from_nat c dvd A $ a $ b" shows "Diagonal_to_Smith_row_i A i bezout $ from_nat c $ from_nat c dvd Diagonal_to_Smith_row_i A i bezout $ a $ b"
not_P: fixes t :: "nat" assumes "t < t\<^sub>0" shows "\<not> (local.prefixes t j \<down> \<and> x < e_length (the (local.prefixes t j)))"
fds_integral_add: fixes a :: "'b"   and b :: "'b"   and f :: "'b fds"   and g :: "'b fds" shows "fds_integral (a + b) (f + g) = fds_integral a f + fds_integral b g"
per_def: fixes r :: "rule" shows "per r \<equiv> \<forall>s r1 sl' s'. s \<in> UNIV \<and> enabled r s \<and> r1 \<in> R - {r} \<and> eff' r1 s = Some sl' \<and> s' |\<in>| sl' \<longrightarrow> enabled r s'"
prems_2: shows "subst_domain \<theta> \<inter> bvars\<^sub>s\<^sub>s\<^sub>t (s # S) = {}"
ts_ok_init_fin_lift_init_fin_lift_state: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool"   and s :: "status"   and \<sigma> :: "('d, 'a) locks \<times> (('a \<Rightarrow> ('b \<times> 'd \<Rightarrow>f nat) option) \<times> 'c) \<times> ('a \<Rightarrow> 'e wait_set_status option) \<times> 'a set" shows "ts_ok (init_fin_lift P) (thr (init_fin_lift_state s \<sigma>)) (shr (init_fin_lift_state s \<sigma>)) = ts_ok P (thr \<sigma>) (shr \<sigma>)"
Zp_def: shows "Zp \<equiv> padic_int p"
csi: shows "cs\<^bsup>\<pi>\<^esup> i = cs\<^bsup>\<pi>'\<^esup> i'"
map_is_rbt: fixes less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'c \<Rightarrow> 'b"   and t :: "('a, 'c) RBT_Impl.rbt" shows "ord.is_rbt less (RBT_Impl.map f t) = ord.is_rbt less t"
lt_vec_of: fixes sp :: "'t \<times> ('a \<Rightarrow>\<^sub>0 'b)" assumes "spp_inv sp" shows "lt (vec_of sp) = fst sp"
vector_scalar_mult_def: fixes c :: "'a"   and x :: "('a, 'b) Finite_Cartesian_Product.vec" shows "c *s x = (\<chi>i. c * x $h i)"
fmadd_empty_1: fixes m :: "'c \<rightharpoonup>\<^sub>f 'd" shows "fmempty ++\<^sub>f m = m"
inv_solve_left: fixes a :: "(((nat \<Rightarrow> real) \<Rightarrow> nat \<Rightarrow> real) \<Rightarrow>\<^sub>0 int) set"   and b :: "(((nat \<Rightarrow> real) \<Rightarrow> nat \<Rightarrow> real) \<Rightarrow>\<^sub>0 int) set"   and c :: "(((nat \<Rightarrow> real) \<Rightarrow> nat \<Rightarrow> real) \<Rightarrow>\<^sub>0 int) set" assumes "a \<in> carrier (relative_homology_group (int n) (nsphere n) (equator n))"   and "b \<in> carrier (relative_homology_group (int n) (nsphere n) (equator n))"   and "c \<in> carrier (relative_homology_group (int n) (nsphere n) (equator n))" shows "(a = inv\<^bsub>relative_homology_group (int n) (nsphere n) (equator n)\<^esub> b \<otimes>\<^bsub>relative_homology_group (int n) (nsphere n) (equator n)\<^esub> c) = (c = b \<otimes>\<^bsub>relative_homology_group (int n) (nsphere n) (equator n)\<^esub> a)"
fresh_14: fixes a :: "atom"   and "term" :: "Syntax.term" shows "a \<sharp> Unroll term = a \<sharp> term"
cut_greater_not_empty_iff: fixes I :: "'a set"   and t :: "'a" shows "(I \<down>> t \<noteq> {}) = (\<exists>x\<in>I. t < x)"
Rep_unit: fixes x :: "unit" shows "Rep_unit x \<in> {True}"
restrict_linorder: fixes D :: "'a set"   and cmp :: "'a \<Rightarrow> 'a \<Rightarrow> comp_res"   and D' :: "'a set" assumes "linorder_on D cmp"   and "D' \<subseteq> D" shows "linorder_on D' cmp"
elims_2: fixes x :: "'a ifex" assumes "ifex_no_twice x" obtains v :: "'a" and t :: "'a ifex" and e :: "'a ifex" where "x = IF v t e" and "v \<notin> ifex_var_set t \<union> ifex_var_set e \<and> ifex_no_twice t \<and> ifex_no_twice e"   | "x = Trueif"   | "x = Falseif"
convex_imp_locally_path_connected: fixes S :: "'a set" assumes "convex S" shows "locally path_connected S"
execute: fixes p :: "nat \<times> 'c"   and A :: "('d, 'c) nba"   and w :: "'d stream"   and V :: "(nat \<times> 'c) set"   and a :: "nat \<times> 'c" assumes "p \<in> gnodes A w V"   and "case p of (k, p) \<Rightarrow> w !! k \<in> alphabet A \<and> a \<in> {Suc k} \<times> transition A (w !! k) p \<inter> V" shows "a \<in> gnodes A w V"
assn_aci_10: fixes a :: "assn"   and b :: "assn"   and c :: "assn" shows "a * b * c = a * c * b"
mapi_branch_id: fixes f :: "('b, 'c) fm \<Rightarrow> ('b, 'c) fm"   and branch :: "(('b, 'c) fm list \<times> 'c) list" shows "mapi_branch (mapper f {}) branch = branch"
distinct_36: fixes x81 :: "pat"   and x82 :: "trm"   and x83 :: "trm"   and x31 :: "POPLmarkRecord.type"   and x32 :: "trm" shows "(LET x81 = x82 IN x83) \<noteq> (\<lambda><:x31. x32)"
preserves_hom: fixes f :: "'c"   and a :: "'c"   and b :: "'c" assumes "\<guillemotleft>f : a \<rightarrow>\<^sub>C b\<guillemotright>" shows "\<guillemotleft>G f : G a \<rightarrow>\<^sub>D G b\<guillemotright>"
2: shows "singular_face (Suc p) 0 (simplex_cone p v (oriented_simplex p l)) = oriented_simplex p l"
induct: fixes x1 :: "'a OCL_Types.type"   and x2 :: "'a OCL_Types.type"   and P :: "'a OCL_Types.type \<Rightarrow> 'a OCL_Types.type \<Rightarrow> bool" assumes "to_nonunique_collection x1 x2"   and "\<And>\<tau>. P (Collection \<tau>) (Bag \<tau>)"   and "\<And>\<tau>. P (Set \<tau>) (Bag \<tau>)"   and "\<And>\<tau>. P (OrderedSet \<tau>) (Sequence \<tau>)"   and "\<And>\<tau>. P (Bag \<tau>) (Bag \<tau>)"   and "\<And>\<tau>. P (Sequence \<tau>) (Sequence \<tau>)" shows "P x1 x2"
preprm_compose_def: fixes f :: "('b \<times> 'b) list"   and g :: "('b \<times> 'b) list" shows "preprm_compose f g \<equiv> f @ g"
alignUp_def: fixes x :: "'b word"   and n :: "nat" shows "alignUp x n \<equiv> x + 2 ^ n - 1 AND NOT (2 ^ n - 1)"
initial_no_mon: fixes fg :: "('c, 'd, 'e, 'f, 'g) flowgraph_rec_scheme"   and p :: "'d" assumes "flowgraph fg"   and "initialproc fg p" shows "mon fg p = {}"
wit_LNil: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "wit R LNil = LNil"
Sigma_Image: fixes A :: "'d set"   and B :: "'d \<Rightarrow> 'c set"   and X :: "'d set" shows "Sigma A B `` X = \<Union> (B ` (X \<inter> A))"
P2: shows "Bet C B C0 \<and> Cong B C0 C B"
neg_eq_FF_iff: fixes x :: "tr" shows "(neg\<cdot>x = FF) = (x = TT)"
SUP_mult_left_ennreal: fixes c :: "ennreal"   and f :: "'c \<Rightarrow> ennreal"   and I :: "'c set" shows "c * \<Squnion> (f ` I) = (\<Squnion>i\<in>I. c * f i)"
exec_Seq': fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'a, 'c) com option"   and c1 :: "('b, 'a, 'c) com"   and s :: "('b, 'c) xstate"   and s' :: "('b, 'c) xstate"   and c2 :: "('b, 'a, 'c) com"   and s'' :: "('b, 'c) xstate" assumes "\<Gamma>\<turnstile> \<langle>c1,s\<rangle> \<Rightarrow> s'"   and "\<Gamma>\<turnstile> \<langle>c2,s'\<rangle> \<Rightarrow> s''" shows "\<Gamma>\<turnstile> \<langle>Seq c1 c2,s\<rangle> \<Rightarrow> s''"
rotate_id: fixes n :: "nat"   and xs :: "'b list" assumes "n mod \<^bold>|xs\<^bold>| = 0" shows "rotate n xs = xs"
listset_Group_plus_closed: fixes Gs :: "'a set list"   and as :: "'a list"   and bs :: "'a list" assumes "\<forall>G\<in>set Gs. Group G"   and "as \<in> listset Gs"   and "bs \<in> listset Gs" shows "map2 (+) as bs \<in> listset Gs"
assms: shows "Q \<Longrightarrow>\<^sub>\<tau> Q'"
restrict_relp_cong: fixes P :: "'c \<Rightarrow> bool"   and P' :: "'c \<Rightarrow> bool"   and Q :: "'d \<Rightarrow> bool"   and Q' :: "'d \<Rightarrow> bool"   and R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and R' :: "'c \<Rightarrow> 'd \<Rightarrow> bool" assumes "P = P'"   and "Q = Q'"   and "\<And>x y. \<lbrakk>P x; Q y\<rbrakk> \<Longrightarrow> R x y = R' x y" shows "R \<upharpoonleft> P \<otimes> Q = R' \<upharpoonleft> P' \<otimes> Q'"
canonical_embedding_is_hom: fixes K :: "'a set" assumes "subring K R" shows "poly_of_const \<in> ring_hom (R\<lparr>carrier := K\<rparr>) (K [X])"
dens_ctxt_measure_nonempty: fixes \<Y> :: "nat set \<times> nat set \<times> (nat \<Rightarrow> pdf_type) \<times> ((nat \<Rightarrow> val) \<Rightarrow> ennreal)"   and \<rho> :: "nat \<Rightarrow> val" shows "space (dens_ctxt_measure \<Y> \<rho>) \<noteq> {}"
cases: fixes a :: "('fun, 'var) extstrand_step list" assumes "well_analyzed a" obtains "a = []"   | A :: "('fun, 'var) extstrand_step list" and x :: "('fun, 'var) strand_step" where "a = Aa__ @ [Step x]" and "well_analyzed Aa__"   | A :: "('fun, 'var) extstrand_step list" and t :: "('fun, 'var) Term.term" where "a = A @ [Decomp t]" and "well_analyzed A" and "t \<in> subterms\<^sub>s\<^sub>e\<^sub>t (ik\<^sub>e\<^sub>s\<^sub>t A \<union> assignment_rhs\<^sub>e\<^sub>s\<^sub>t A) - range Var"
rel_envT_simps: fixes R :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and M :: "'i \<Rightarrow> 'j \<Rightarrow> bool"   and f :: "'g \<Rightarrow> 'i"   and g :: "'h \<Rightarrow> 'j" shows "local.rel_envT R M (EnvT f) (EnvT g) = (R ===> M) f g"
msort_poly_impl_def: shows "msort_poly_impl = msort (\<lambda>a b. fst a \<le> fst b)"
discr_Wbis: fixes c :: "('test, 'atom) com"   and d :: "('test, 'atom) com" assumes "discr c"   and "discr d" shows "c \<approx>w d"
CallRec: fixes P :: "'d set"   and p :: "'e"   and Q :: "'d set"   and A :: "'d set"   and Specs :: "('d set \<times> 'e \<times> 'd set \<times> 'd set) set"   and r :: "(('d \<times> 'e) \<times> 'd \<times> 'e) set"   and Specs_wf :: "'e \<Rightarrow> 'd \<Rightarrow> ('d set \<times> 'e \<times> 'd set \<times> 'd set) 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" assumes "(P, p, Q, A) \<in> Specs"   and "wf r"   and "Specs_wf = (\<lambda>p \<sigma>. (\<lambda>(P, q, Q, A). (P \<inter> {s. ((s, q), \<sigma>, p) \<in> r}, q, Q, A)) ` Specs)"   and "\<forall>(P, p, Q, A)\<in>Specs. p \<in> dom \<Gamma> \<and> (\<forall>\<sigma>. \<Gamma>,\<Theta> \<union> Specs_wf p \<sigma>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> ({\<sigma>} \<inter> P) the (\<Gamma> p) Q,A)" shows "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P Call p Q,A"
ivl_rel_def: fixes S :: "('a \<times> 'c) set" shows "\<langle>S\<rangle>ivl_rel = (S \<times>\<^sub>r S) O br set_of_ivl top"
pmdl_image_monic: fixes B :: "('a \<Rightarrow>\<^sub>0 'c) set" shows "local.punit.pmdl (punit.monic ` B) = local.punit.pmdl B"
fun_add_closed: fixes R :: "('c, 'd) ring_scheme"   and f :: "'e \<Rightarrow> 'c"   and S :: "'e set"   and g :: "'e \<Rightarrow> 'c" assumes "ring_functions R"   and "f \<in> carrier (function_ring S R)"   and "g \<in> carrier (function_ring S R)" shows "f \<oplus>\<^bsub>function_ring S R\<^esub> g \<in> carrier (function_ring S R)"
bb: shows "Xs = butlast (butlast (D # E # Es))"
id_take_nth_drop: fixes i :: "nat"   and xs :: "'g list" assumes "i < length xs" shows "xs = take i xs @ xs ! i # drop (Suc i) xs"
InfCard_csucc: fixes \<kappa> :: "V" assumes "InfCard \<kappa>" shows "InfCard (csucc \<kappa>)"
not_new: fixes \<sigma> :: "'l \<Rightarrow> ('r, 'l, 'v) val option"   and \<tau> :: "'l \<Rightarrow> ('r, 'l, 'v) val option"   and \<E> :: "('r, 'l, 'v) cntxt"   and v :: "('r, 'l, 'v) val" shows "s r \<noteq> Some (\<sigma>, \<tau>, \<E> [Ref (VE v)])"
assms_3: shows "ipresIGSwap h' MOD' MOD''"
bl_sshiftr: fixes n :: "nat"   and w :: "'a word" assumes "n \<le> size w" shows "to_bl (w >>> n) = replicate n (hd (to_bl w)) @ take (size w - n) (to_bl w)"
PRE: shows "RETURN x \<le> lfp (\<lambda>x. sup m (x \<bind> a))"
stutter_selection_lproject_visible: fixes u :: "'action stream"   and p :: "'state" assumes "run u p" shows "stutter_selection (lift (liset visible (llist_of_stream u))) (llist_of_stream (smap int (p ## trace u p)))"
p_0: fixes x :: "'s"   and y :: "'s" shows "p x y 0 = (if x = y then 1 else 0)"
Eats_EQ_Zero_E: fixes t :: "tm"   and u :: "tm"   and H :: "fm set"   and A :: "fm" shows "insert (Eats t u EQ Zero) H \<turnstile> A"
nth_map_matrix: fixes f :: "'d \<Rightarrow> 'a"   and x :: "(('d, 'b) Finite_Cartesian_Product.vec, 'c) Finite_Cartesian_Product.vec"   and i :: "'c"   and j :: "'b" shows "Finite_Cartesian_Product.map_matrix f x $h i $h j = f (x $h i $h j)"
thunks_def: fixes \<Gamma> :: "(Vars.var \<times> exp) list" shows "thunks \<Gamma> = {x. case map_of \<Gamma> x of None \<Rightarrow> False | Some e \<Rightarrow> \<not> isVal e}"