hcju/isabelleps huggingface dataset:
insert: fixes g :: "'b \<Rightarrow> 'a"   and x :: "'b"   and A :: "'b multiset" shows "\<Sum>\<^sub># (image_mset g (add_mset x A)) = g x + \<Sum>\<^sub># (image_mset g A)"
resolution_rule': fixes C\<^sub>1 :: "fterm literal set"   and Cs :: "fterm literal set set"   and C\<^sub>2 :: "fterm literal set"   and L\<^sub>1 :: "fterm literal set"   and L\<^sub>2 :: "fterm literal set"   and \<sigma> :: "char list \<Rightarrow> fterm"   and C :: "fterm literal set set" assumes "C\<^sub>1 \<in> Cs"   and "C\<^sub>2 \<in> Cs"   and "applicable C\<^sub>1 C\<^sub>2 L\<^sub>1 L\<^sub>2 \<sigma>"   and "C = {resolution C\<^sub>1 C\<^sub>2 L\<^sub>1 L\<^sub>2 \<sigma>}" shows "resolution_step Cs (Cs \<union> C)"
coboundedI2: fixes c :: "'e multiset"   and b :: "'e multiset"   and a :: "'e multiset" assumes "c \<subseteq># b" shows "c \<subseteq># a \<union># b"
is_UP_monomE_3: fixes R :: "('c, 'd) ring_scheme"   and f :: "nat \<Rightarrow> 'c" assumes "UP_ring R"   and "UP_ring.is_UP_monom R f" shows "f = monom (UP R) (f (deg R f)) (deg R f)"
mono_F: shows "mono (\<lambda>F s. \<Sqinter>D\<in>K s. \<integral>\<^sup>+ t. g t (F t) \<partial>measure_pmf D)"
edge_Some1E: fixes x :: "'v"   and y :: "'v option" assumes "edge' \<langle>x\<rangle> y" obtains z :: "'v" where "y = \<langle>z\<rangle>" and "edge \<Gamma> x z \<or> edge \<Gamma> z x"
proj2_set_Col_def: fixes S :: "proj2 set" shows "proj2_set_Col S \<equiv> \<exists>l. \<forall>p\<in>S. proj2_incident p l"
union_code: fixes xs :: "'d list"   and ys :: "'d list" shows "mset xs + mset ys = mset (xs @ ys)"
elims_2: fixes x :: "'a"   and xa :: "'a \<times> 'a" assumes "elem x xa" obtains xb :: "'a" and l :: "'a" and u :: "'a" where "x = xb" and "xa = (l, u)" and "l < xb \<and> xb < u"
offset_poly_single: fixes a :: "'a"   and h :: "'a" shows "offset_poly [:a:] h = [:a:]"
quotient_group_def: fixes H :: "'g set" shows "quotient_group H \<equiv> BinOpSetGroup.Abs_G_perm (G // LCoset_rel H) (+) ` G // LCoset_rel H"
assms_1: shows "\<langle>c \<Leftarrow> t\<rangle>\<^bsub>n\<^esub> \<le> n'"
ereal2_cases: fixes x :: "ereal"   and xa :: "ereal" obtains r :: "real" and ra :: "real" where "x = ereal r__" and "xa = ereal ra"   | r :: "real" where "x = ereal rb__" and "xa = \<infinity>"   | r :: "real" where "x = ereal rc__" and "xa = - \<infinity>"   | r :: "real" where "x = \<infinity>" and "xa = ereal rd__"   | "x = \<infinity>" and "xa = \<infinity>"   | "x = \<infinity>" and "xa = - \<infinity>"   | r :: "real" where "x = - \<infinity>" and "xa = ereal r"   | "x = - \<infinity>" and "xa = \<infinity>"   | "x = - \<infinity>" and "xa = - \<infinity>"
q_def: shows "q = smult s (p \<circ>\<^sub>p [:0::'a, - (1::'a):])"
PLam_component_iff: fixes I :: "nat set"   and F :: "nat \<Rightarrow> ('c \<times> (nat \<Rightarrow> 'c) \<times> 'd) program"   and H :: "((nat \<Rightarrow> 'c) \<times> 'd) program" shows "(PLam I F \<le> H) = (\<forall>i\<in>I. lift i (F i) \<le> H)"
list_of_vec_index: fixes v :: "'a vec"   and j :: "nat" shows "list_of_vec v ! j = v $ j"
eq: shows "p \<bullet> bn \<alpha>1 = bn \<alpha>2"
Anno_code_i_i_i_oI: fixes x :: "'b J_prog"   and xa :: "String.literal \<Rightarrow> ty option"   and xb :: "'b expr"   and xc :: "'b expr" assumes "x,xa \<turnstile> xb \<leadsto>' xc" shows "pred.eval (Anno_code_i_i_i_o x xa xb) xc"
fw_spec_def: fixes n :: "nat"   and M :: "nat \<Rightarrow> nat \<Rightarrow> 'b" shows "fw_spec n M \<equiv> SPEC (\<lambda>M'. if \<exists>i\<le>n. M' i i < (0::'b) then \<not> (\<forall>i xs. i \<le> n \<and> set xs \<subseteq> {0..n} \<longrightarrow> (0::'b) \<le> len M i i xs) else \<forall>i\<le>n. \<forall>j\<le>n. M' i j = D M i j n \<and> (\<forall>i xs. i \<le> n \<and> set xs \<subseteq> {0..n} \<longrightarrow> (0::'b) \<le> len M i i xs))"
assms_2: shows "T \<subseteq> U\<^sub>M\<^sub>_\<^sub>2"
bij_betw_byWitness: fixes A :: "'c set"   and f' :: "'d \<Rightarrow> 'c"   and f :: "'c \<Rightarrow> 'd"   and A' :: "'d set" assumes "\<forall>a\<in>A. f' (f a) = a"   and "\<forall>a'\<in>A'. f (f' a') = a'"   and "f ` A \<subseteq> A'"   and "f' ` A' \<subseteq> A" shows "bij_betw f A A'"
rk2_step_flowpipe: fixes X0 :: "'c set"   and h :: "real" assumes "wd TYPE('c)" shows "rk2_step X0 h \<le> SPEC (\<lambda>(h', uu_, RES_ivl, RES). 0 < h' \<and> h' \<le> h \<and> flowpipe0 X0 h' h' RES_ivl RES)"
SD_efficient_def: fixes p :: "'alt pmf" shows "SD_efficient R p = (\<not> (\<exists>q\<in>lotteries_on alts. p \<prec>[Pareto (SD \<circ> R)] q))"
trivial_group_def: fixes G :: "('f, 'g) monoid_scheme" shows "trivial_group G \<equiv> Group.group G \<and> carrier G = {\<one>\<^bsub>G\<^esub>}"
map_vec_vec_of_list_eq_intro: fixes f :: "'b \<Rightarrow> 'a"   and xs :: "'b list"   and g :: "'c \<Rightarrow> 'a"   and ys :: "'c list" assumes "map f xs = map g ys" shows "map_vec f (vec_of_list xs) = map_vec g (vec_of_list ys)"
asem_def: fixes x :: "char list"   and e :: "aexp"   and S :: "(char list \<Rightarrow> 'av) option" shows "asem x e S = (case S of None \<Rightarrow> None | Some S \<Rightarrow> Some (S(x := aval' e S)))"
add_mono: fixes a :: "'c multiset"   and b :: "'c multiset"   and c :: "'c multiset"   and d :: "'c multiset" assumes "a \<subseteq># b"   and "c \<subseteq># d" shows "a + c \<subseteq># b + d"
pred_gpv_lift_resumption: fixes A :: "'d \<Rightarrow> bool"   and C :: "'e \<Rightarrow> bool"   and r :: "('d, 'e, 'f) Resumption.resumption" shows "pred_gpv A C (lift_resumption r) = Resumption.resumption.pred_resumption A C r"
ENC_EXISTS: fixes n :: "nat"   and \<phi> :: "'a formula" assumes "wf_formula n (FEXISTS \<phi>)" shows "lang n (ENC n (FOV (FEXISTS \<phi>))) - {[]} = map \<pi> ` lang (Suc n) (ENC (Suc n) (FOV \<phi>)) - {[]}"
natural_transformation_axioms: shows "natural_transformation (\<cdot>\<^sub>C) (\<cdot>\<^sub>C) F'G'.GF.map (G' \<circ> (FG.GF.map \<circ> F')) (G' \<circ> (FG.\<eta> \<circ> F'))"
ListAif2: fixes b :: "bexp"   and s :: "char list \<Rightarrow> int"   and upds :: "('a \<times> ((char list \<Rightarrow> int) \<Rightarrow> nat)) list"   and C1 :: "acom"   and C2 :: "acom"   and l :: "'a \<Rightarrow> nat" assumes "\<not> bval b s" shows "preList upds (IF b THEN C1 ELSE C2) l s = preList upds C2 l s"
cone_cone: fixes Vs :: "'a vec set" assumes "Vs \<subseteq> carrier_vec n" shows "cone (cone Vs) = cone Vs"
1: shows "det (mat n n (\<lambda>(i, j). A $$ (p i, j))) = (\<Sum>q | q permutes {0..<n}. signof (q \<circ> p) * (\<Prod>i = 0..<n. A $$ (p i, (q \<circ> p) i)))"
VWO: shows "{(x, y). x \<in> elts y} \<subseteq> VWO \<and> Well_order VWO \<and> Field VWO = UNIV"
rep_eq: fixes x :: "'b fpxs" shows "fpxs_root_order x = nat (LCM r\<in>Formal_Puiseux_Series.supp (fpxs_nth x). snd (quotient_of r))"
W_1: shows "RModule FG (*) W \<and> W \<subseteq> FG"
\<tau>move_heap: fixes t :: "'t"   and x :: "'x1"   and m :: "'m1"   and ta :: "('l, 't, 'x1, 'm1, 'w, 'o) thread_action"   and x' :: "'x1"   and m' :: "'m1" assumes "r1.r_syntax t x m ta x' m'"   and "\<tau>move1 (x, m) ta (x', m')" shows "m = m'"
ex_fresh: fixes T :: "'a"   and A :: "(char list \<times> 'a) fset" shows "\<exists>x. (x, T) |\<notin>| A"
fprod_def: shows "(|\<times>|) \<equiv> map_fun id (map_fun id Abs_fset) (\<lambda>a b. fset a \<times> fset b)"
sum_union_disjoint_finite_set: fixes C :: "nat set"   and g :: "nat \<Rightarrow> int" assumes "finite C" shows "\<forall>A B. A \<inter> B = {} \<and> A \<union> B = C \<longrightarrow> sum g C = sum g A + sum g B"
ipath_iter_conv: fixes p :: "'e list"   and E :: "('e \<times> 'e) set" assumes "p \<noteq> []" shows "ipath E p\<^sup>\<omega> = path E (hd p) p (hd p)"
ZusammenziehungListeA: fixes \<Gamma> :: "'b formula list"   and \<Theta> :: "'b formula list" assumes "\<Gamma> @ \<Gamma> \<Rightarrow> \<Theta>" shows "\<Gamma> \<Rightarrow> \<Theta>"
nth_tabulate: fixes k :: "nat"   and n :: "nat"   and f :: "nat \<Rightarrow> 'b"   and x :: "nat" assumes "k < n" shows "tabulate f x n ! k = f (x + k)"
AboveS_Field: fixes x :: "'d"   and r :: "('d \<times> 'd) set"   and X :: "'d set" assumes "x \<in> AboveS r X" shows "x \<in> Field r"
conseq12': fixes G :: "prog"   and A :: "'b triple set"   and P' :: "'b assn"   and t :: "Term.term"   and Q' :: "'b assn"   and P :: "'b assn"   and Q :: "'b assn" assumes "G,A\<turnstile>{P'} t\<succ> {Q'}"   and "\<forall>s Y' s'. (\<forall>Y Z. P' Y s Z \<longrightarrow> Q' Y' s' Z) \<longrightarrow> (\<forall>Y Z. P Y s Z \<longrightarrow> Q Y' s' Z)" shows "G,A\<turnstile>{P} t\<succ> {Q}"
intros: fixes A :: "'a \<Rightarrow> 'd \<Rightarrow> bool"   and \<I> :: "('b, 'c) \<I>"   and gpv :: "('a, 'b, 'c) gpv"   and gpv' :: "('d, 'b, 'c) gpv" assumes "rel_spmf (eq_\<I>_generat A \<I> (eq_\<I>_gpv A \<I>)) (the_gpv gpv) (the_gpv gpv')" shows "eq_\<I>_gpv A \<I> gpv gpv'"
sorted_sorted_list_of_set: fixes A :: "'a set" shows "sorted (sorted_list_of_set A)"
JOIN_VARS_def: fixes f :: "'b"   and g :: "'c"   and P :: "'b \<Rightarrow> 'c \<Rightarrow> 'a" shows "JOIN_VARS f g P \<equiv> P f g"
assms_3: fixes T' :: "'a set" assumes "S \<subseteq> T'"   and "closed T'" shows "T \<subseteq> T'"
exp_minus_inverse: fixes x :: "'a" shows "exp x * exp (- x) = (1::'a)"
expand_term_prop: fixes \<phi> :: "'b ltlr"   and n_ns :: "'b node \<times> 'b node set" assumes "expand_inv \<phi> n_ns" shows "expand\<^sub>T n_ns \<le> SPEC (\<lambda>(uu_, nds). old_next_pair ` snd n_ns \<subseteq> old_next_pair ` nds \<and> expand_inv_result \<phi> nds)"
mem_grow_def: fixes m :: "mem"   and n :: "nat" shows "mem_grow m n = mem_append m (bytes_replicate (n * 64000) 0)"
assms_2: shows "x1 \<in> carrier (\<G> n)"
assms_1: shows "ideal \<aa> R (+) (\<cdot>) \<zero> \<one>"
bounded_region: fixes X :: "'a set"   and k :: "'a \<Rightarrow> nat"   and I :: "'a \<Rightarrow> Regions.intv"   and r :: "('a \<times> 'a) set"   and u :: "'a \<Rightarrow> real" assumes "finite X"   and "Regions.valid_region X k I r"   and "\<not> (\<forall>t\<ge>0. \<forall>u\<in>Regions.region X I r. u \<oplus> t \<in> Regions.region X I r)"   and "u \<in> Regions.region X I r" shows "u \<oplus> 1 \<notin> Regions.region X I r"
rank_ins: fixes bq :: "('c, 'd) SkewBinomialTree list"   and t :: "('c, 'd) SkewBinomialTree" assumes "rank_invar bq" shows "rank_invar (ins t bq)"
sep_conj_sep_impl2: fixes P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool"   and h :: "'a"   and R :: "'a \<Rightarrow> bool" assumes "(P \<and>* Q) h"   and "\<And>h. P h \<Longrightarrow> (Q \<longrightarrow>* R) h" shows "R h"
atms_empty_progress: fixes m :: "modality"   and g :: "safety"   and r :: "Formula.formula Regex.regex"   and \<sigma> :: "(String.literal \<times> event_data list) trace"   and P :: "String.literal \<Rightarrow> nat option"   and j :: "nat" assumes "safe_regex m g r"   and "atms r = {}" shows "progress_regex \<sigma> P r j = j"
comp_tuple_arr: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and prj0 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and prj1 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b"   and g :: "'b"   and h :: "'b"   and k :: "'b"   and l :: "'b" assumes "elementary_category_with_pullbacks C prj0 prj1"   and "category.commutative_square C f g h k"   and "partial_magma.arr C (C h l)" shows "C (elementary_category_with_pullbacks.tuple C prj0 prj1 h f g k) l = elementary_category_with_pullbacks.tuple C prj0 prj1 (C h l) f g (C k l)"
targetnodes_def: fixes xs :: "'edge list" shows "targetnodes xs \<equiv> map targetnode xs"
comp_null_2: fixes f :: "'a \<times> 'a" shows "B.VV.comp f B.VV.null = B.VV.null"
fun_chain_at_top_at_top: fixes f :: "'a \<Rightarrow> 'a"   and n :: "nat" assumes "filterlim f at_top at_top" shows "filterlim (f ^^ n) at_top at_top"
abs_eq: fixes x :: "'e" shows "return_pmf x = Abs_pmf (return (count_space UNIV) x)"
take_Suc_eq: fixes n :: "nat"   and xs :: "'a list" shows "take (Suc n) xs = (if n < length xs then take n xs @ [xs ! n] else xs)"
push_bit_eq_0_iff: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and division_segment :: "'a \<Rightarrow> 'a"   and bit :: "'a \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and drop_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and take_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and n :: "nat"   and a :: "'a" assumes "class.unique_euclidean_semiring_with_bit_shifts divide plus minus zero times one modulo euclidean_size division_segment bit push_bit drop_bit take_bit" shows "(push_bit n a = zero) = (a = zero)"
tensor_one_from_lookup: shows "1 = tensor_from_lookup [] (\<lambda>_. 1::'b)"
inj_onD: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and x :: "'a"   and y :: "'a" assumes "inj_on f A"   and "f x = f y"   and "x \<in> A"   and "y \<in> A" shows "x = y"
length_filter_sort_nth: fixes xs :: "'a list"   and n :: "nat" assumes "distinct xs"   and "n < length xs" shows "length (filter (\<lambda>x. x \<le> sort xs ! n) xs) = n + 1"
relto_fun_into_reltos: fixes A :: "('a \<times> 'a) set"   and B :: "('a \<times> 'a) set"   and n :: "nat"   and as :: "nat \<Rightarrow> 'a"   and sel :: "nat \<Rightarrow> bool"   and m :: "nat"   and a :: "'a"   and b :: "'a" assumes "relto_fun A B n as sel m (a, b)" shows "(a, b) \<in> (B\<^sup>* O A O B\<^sup>*) ^^ n"
assms_2: shows "approx_floatariths_aformerr p ra VS = Some X"
Di'_drop_subset: fixes i :: "nat"   and l :: "nat"   and y :: "nat list" shows "Di' i (drop l y) \<subseteq> Di' i y"
simps_1: fixes xs :: "'b list" shows "remove_all_cycles [] xs = xs"
distinct_25: fixes x2 :: "nat"   and x81 :: "nat"   and x82 :: "msg" shows "Number x2 \<noteq> Crypt x81 x82"
graph_more_rel_internal_def: fixes Rm :: "('i \<times> 'l) set"   and Rv :: "('g \<times> 'j) set"   and Rw :: "('h \<times> 'k) set" shows "graph_more_rel Rm Rv Rw \<equiv> {(g, g'). (nodes g, nodes g') \<in> \<langle>Rv\<rangle>set_rel \<and> (edges g, edges g') \<in> \<langle>Rv \<times>\<^sub>r Rw \<times>\<^sub>r Rv\<rangle>set_rel \<and> (graph.more g, graph.more g') \<in> Rm}"
RSsecure_def: fixes R :: "(Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool"   and S :: "(Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool"   and c :: "IMP" shows "RSsecure R S c = (\<forall>s t ss tt. R s t \<longrightarrow> s , c \<Down> ss  \<longrightarrow> t , c \<Down> tt  \<longrightarrow> S ss tt)"
F: shows "\<forall>A\<in>F. A \<subseteq> E \<and> s \<le> card A \<and> card A \<le> k"
prems_1: shows "v0 \<leadsto>xs\<leadsto> w"
distinct_Ex1: fixes xs :: "'g list"   and x :: "'g" assumes "distinct xs"   and "x \<in> set xs" shows "\<exists>!i. i < length xs \<and> xs ! i = x"
qp_post: shows "is_quantum_predicate (tensor_P post (1\<^sub>m K))"
subgroup_def: fixes H :: "'e set"   and G :: "('e, 'f) monoid_scheme" shows "subgroup H G \<equiv> (H \<subseteq> carrier G \<and> (\<forall>x y. x \<in> H \<longrightarrow> y \<in> H \<longrightarrow> x \<otimes>\<^bsub>G\<^esub> y \<in> H)) \<and> \<one>\<^bsub>G\<^esub> \<in> H \<and> (\<forall>x. x \<in> H \<longrightarrow> inv\<^bsub>G\<^esub> x \<in> H)"
interior_rel_interior: fixes S :: "'b set" assumes "aff_dim S = int DIM('b)" shows "rel_interior S = interior S"
False: shows "\<not> (q = 0 \<or> degree p = 0)"
antisym_conv: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and y :: "'b"   and x :: "'b" assumes "class.order less_eq less"   and "less_eq y x" shows "less_eq x y = (x = y)"
mat_inverse_def: fixes A :: "'b mat" shows "mat_inverse A = (if dim_row A = dim_col A then let one = 1\<^sub>m (dim_row A) in case gauss_jordan A one of (B, C) \<Rightarrow> if B = one then Some C else None else None)"
elementLevelPrefixElement: fixes e :: "'a"   and level :: "nat"   and M :: "('a \<times> bool) list" assumes "e \<in> set (elements (prefixToLevel level M))" shows "elementLevel e (prefixToLevel level M) = elementLevel e M"
simps_11: fixes f1 :: "'e"   and f2 :: "'f \<Rightarrow> 'e"   and f3 :: "'g \<Rightarrow> 'f \<Rightarrow> 'e"   and x31 :: "'g"   and x32 :: "'f" shows "(case Yield x31 x32 of Done \<Rightarrow> f1 | Skip x \<Rightarrow> f2 x | Yield x xa \<Rightarrow> f3 x xa) = f3 x31 x32"
rbt_lookup_ins: fixes t :: "('a, 'c) rbt"   and f :: "'a \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and k :: "'a"   and v :: "'c"   and x :: "'a" assumes "rbt_sorted t" shows "rbt_lookup (rbt_ins f k v t) x = (rbt_lookup t(k \<mapsto> case rbt_lookup t k of None \<Rightarrow> v | Some w \<Rightarrow> f k w v)) x"
coeffM_zero_row_def: fixes A :: "(('c poly, 'd) vec, 'e) vec"   and i :: "'e" shows "coeffM_zero_row A i = (\<chi>j. coeff (A $ i $ j) 0)"
bounded_integrals_over_subintervals: fixes f :: "'a \<Rightarrow> 'b"   and a :: "'a"   and b :: "'a" assumes "f integrable_on cbox a b" shows "bounded {integral (cbox c d) f |c d. cbox c d \<subseteq> cbox a b}"
bin_sign_simps_3: shows "bin_sign (- 1) = - 1"
gcd_exp_weak: fixes a :: "'a"   and n :: "nat"   and b :: "'a" shows "gcd (a ^ n) (b ^ n) = normalize (gcd a b ^ n)"
primitive_dchar_iff: shows "primitive_dchar n (principal_dchar n) = (\<not> (\<exists>d<n. principal.induced_modulus d))"
ev_smap: fixes P :: "'c stream \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'c"   and s :: "'d stream" shows "ev P (smap f s) = ev (\<lambda>x. P (smap f x)) s"
vlrestriction_vinsert_nin: fixes a :: "V"   and A :: "V"   and b :: "V"   and r :: "V" assumes "a \<notin>\<^sub>\<circ> A" shows "vinsert \<langle>a, b\<rangle> r \<restriction>\<^sup>l\<^sub>\<circ> A = r \<restriction>\<^sup>l\<^sub>\<circ> A"
mat1_vert_wall_left: fixes b :: "wall" assumes "is_tangle_diagram b" shows "rat_poly.matrix_mult (blockmat (make_vert_block (nat (domain_wall b)))) (kauff_mat b) = kauff_mat b"
inclusion_of_generators_def: shows "\<F>\<^sub>SC.inclusion_of_generators \<equiv> \<F>C.inclusion_of_generators"
AUX: fixes a :: "'a"   and b :: "'a" assumes "(a, b) \<in> R'\<^sup>*"   and "a \<noteq> x"   and "b \<noteq> x" shows "(a, b) \<in> R\<^sup>*"
surj_f: shows "\<forall>x\<in>X. \<exists>n<card X. f n = x"
Cons_3: shows "vars_constraints (map snd (hh # t)) \<subseteq> V"
A'_wf': shows "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (set A')"
trms\<^sub>s\<^sub>t_def: fixes S :: "('a, 'b) strand_step list" shows "trms\<^sub>s\<^sub>t S \<equiv> \<Union> (trms\<^sub>s\<^sub>t\<^sub>p ` set S)"
fold_Option_bind_eq_Some_at_each_point_Some: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'a \<Rightarrow> 'a option"   and l1 :: "('b \<times> 'c) list"   and l2 :: "('b \<times> 'c) list"   and start :: "'a option"   and res :: "'a" assumes "fold (\<lambda>(x, y) option. Option.bind option (f x y)) (l1 @ l2) start = Some res" shows "\<exists>point. fold (\<lambda>(x, y) option. Option.bind option (f x y)) l1 start = Some point \<and> fold (\<lambda>(x, y) option. Option.bind option (f x y)) l2 (Some point) = Some res"
R1_def: shows "R1 = {(w1, w2). \<exists>\<iota> \<iota>' b b' c1 c1' c2 c2'. w1 = [c1;while\<^bsub>\<iota>\<^esub> b do c2 od] \<and> w2 = [c1';while\<^bsub>\<iota>'\<^esub> b' do c2' od] \<and> \<iota> \<notin> set (PPc c) \<and> \<iota>' \<notin> set (PPc c) \<and> ([c1], [c1']) \<in> R \<and> ([c2], [c2']) \<in> R \<and> b \<equiv>\<^bsub>d\<^esub> b'}"
finprod_swap: fixes A :: "'e set"   and B :: "'f set"   and g :: "'e \<Rightarrow> 'f \<Rightarrow> 'a" assumes "finite A"   and "finite B"   and "\<And>a b. \<lbrakk>a \<in> A; b \<in> B\<rbrakk> \<Longrightarrow> g a b \<in> carrier G" shows "(\<Otimes>(b, a)\<in>B \<times> A. g a b) = (\<Otimes>(a, b)\<in>A \<times> B. g a b)"
map_index_eq_imp_length_eq: fixes f :: "nat \<Rightarrow> 'c \<Rightarrow> 'b"   and xs :: "'c list"   and g :: "nat \<Rightarrow> 'd \<Rightarrow> 'b"   and ys :: "'d list" assumes "map_index f xs = map_index g ys" shows "length xs = length ys"
Rep_resumption_Linear: fixes b :: "'g"   and r :: "('h, 'g, 'f, 'i) resumption" shows "Rep_resumption (Linear b r) = LINEAR b (Rep_resumption r)"
bounded_iff: fixes b :: "'a"   and c :: "'a"   and a :: "'a" shows "(b + c \<le> a) = (b \<le> a \<and> c \<le> a)"
proj_pick: fixes P :: "'c \<Rightarrow> bool"   and \<phi> :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and X :: "'c set" assumes "equivP P \<phi>"   and "(P /// \<phi>) X" shows "proj \<phi> (pick X) = X"
vconn_implies_hpar_1: fixes \<mu> :: "'d"   and f :: "'d"   and g :: "'d" assumes "\<guillemotleft>\<mu> : f \<Rightarrow>\<^sub>D g\<guillemotright>" shows "src\<^sub>D \<mu> = src\<^sub>D f"
s_not_ex: fixes x :: "nat" assumes "x \<in> I"   and "\<nexists>n. I \<leftarrow> n = x" shows "x \<in> S"
v11_vconverse_app: fixes a :: "V"   and b :: "V" assumes "r\<lparr>a\<rparr> = b"   and "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> r" shows "r\<inverse>\<^sub>\<circ>\<lparr>b\<rparr> = a"
get_document_ptr_simp5: fixes character_data_ptr :: "(_) character_data_ptr"   and document_ptr :: "(_) document_ptr"   and f :: "('p, 'o, 'n, 'v option) RDocument_scheme"   and h :: "(_) heap" shows "get character_data_ptr (put document_ptr f h) = get character_data_ptr h"
monomial_decomp_def: fixes ps :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list" shows "monomial_decomp ps = (\<forall>hU\<in>set ps. is_monomial (fst hU) \<and> lcf (fst hU) = (1::'b))"
simps_16: fixes f1 :: "ipc_direction_t \<Rightarrow> ipc_stage_t \<Rightarrow> thread_id_t \<Rightarrow> page_t \<Rightarrow> 'a"   and f2 :: "ev_wait_stage_t \<Rightarrow> ev_consume_t \<Rightarrow> 'a"   and f3 :: "ev_signal_stage_t \<Rightarrow> thread_id_t \<Rightarrow> 'a"   and f4 :: "'a"   and x11 :: "ipc_direction_t"   and x12 :: "ipc_stage_t"   and x13 :: "thread_id_t"   and x14 :: "page_t" shows "(case SK_IPC x11 x12 x13 x14 of SK_IPC x xa xb xc \<Rightarrow> f1 x xa xb xc | SK_EV_WAIT x xa \<Rightarrow> f2 x xa | SK_EV_SIGNAL x xa \<Rightarrow> f3 x xa | NONE \<Rightarrow> f4) = f1 x11 x12 x13 x14"
map_fst_pair_pmf: fixes A :: "'d pmf"   and B :: "'e pmf" shows "map_pmf fst (pair_pmf A B) = A"
not_less_real: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and a :: "'d"   and b :: "'d" assumes "class.semiring_real_line plus zero minus less_eq less times"   and "less a zero \<or> a = zero \<or> less zero a"   and "less b zero \<or> b = zero \<or> less zero b" shows "(\<not> less b a) = less_eq a b"
abs_eq: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'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"   and x :: "int \<times> int" assumes "class.field_char_0 divide inverse times one plus zero minus uminus"   and "ratrel x x" shows "field_char_0.of_rat divide one plus zero minus (Abs_Rat x) = divide (ring_1.of_int one plus zero minus (fst x)) (ring_1.of_int one plus zero minus (snd x))"
induct: fixes P :: "('c, 'd) BinomialTree list \<Rightarrow> bool"   and a0 :: "('c, 'd) BinomialTree list" assumes "\<And>t. P [t]"   and "\<And>t v va. \<lbrakk>P (v # va); \<not> prio t \<le> prio (getMinTree (v # va)) \<Longrightarrow> P (v # va)\<rbrakk> \<Longrightarrow> P (t # v # va)"   and "P []" shows "P a0"
msg_ok_Pkt: fixes \<sigma> :: "nat \<Rightarrow> state"   and data :: "nat"   and src :: "nat" shows "msg_ok \<sigma> (Pkt data src) = (data \<le> no (\<sigma> src))"