hcju/isabelleps huggingface dataset:
get_scdom_component_subset_get_dom_component: fixes get_owner_document :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) document_ptr) prog"   and heap_is_wellformed :: "(_) heap \<Rightarrow> bool"   and type_wf :: "(_) heap \<Rightarrow> bool"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and known_ptrs :: "(_) heap \<Rightarrow> bool"   and get_scdom_component :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog"   and get_dom_component :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog"   and h :: "(_) heap"   and ptr :: "(_) object_ptr"   and sc :: "(_) object_ptr list"   and c :: "(_) object_ptr list" assumes "l_get_dom_component_get_scdom_component get_owner_document heap_is_wellformed type_wf known_ptr known_ptrs get_scdom_component get_dom_component"   and "heap_is_wellformed h"   and "type_wf h"   and "known_ptrs h"   and "h \<turnstile> get_scdom_component ptr \<rightarrow>\<^sub>r sc"   and "h \<turnstile> get_dom_component ptr \<rightarrow>\<^sub>r c" shows "set c \<subseteq> set sc"
QmpTr2: fixes H :: "'a set"   and N :: "'a set"   and X :: "'a set" assumes "G \<guillemotright> H "   and "G \<guillemotright> N "   and "H \<subseteq> N"   and "X \<in> carrier (G / H)" shows "(Qm\<^bsub>G H,N\<^esub>) X \<in> carrier (G / N)"
assms_4: shows "\<forall>\<^sub>F t in at_right 0. flow0 x t \<notin> P"
lv_rel_add: shows "(map2 (+), (+)) \<in> lv_rel \<rightarrow> lv_rel \<rightarrow> lv_rel"
discrete_topology_interior_of: fixes U :: "'a set"   and S :: "'a set" shows "discrete_topology U interior_of S = U \<inter> S"
compact_lemma_general: fixes basis :: "'d set"   and proj :: "'f \<Rightarrow> 'd \<Rightarrow> 'e"   and f :: "nat \<Rightarrow> 'f"   and unproj :: "('d \<Rightarrow> 'e) \<Rightarrow> 'f" assumes "finite basis"   and "\<And>k. k \<in> basis \<Longrightarrow> bounded ((\<lambda>x. proj x k) ` range f)"   and "\<And>e k. k \<in> basis \<Longrightarrow> proj (unproj e) k = e k"   and "\<And>x. unproj (proj x) = x" shows "\<forall>d\<subseteq>basis. \<exists>l r. strict_mono r \<and> (\<forall>e>0. \<forall>\<^sub>F n in sequentially. \<forall>i\<in>d. dist (proj (f (r n)) i) (proj l i) < e)"
infinite_set_mset_mset_set: fixes A :: "'d set" assumes "infinite A" shows "set_mset (mset_set A) = {}"
bin_code_code: fixes u :: "'b list"   and v :: "'b list" assumes "u \<cdot> v \<noteq> v \<cdot> u" shows "code {u, v}"
0: fixes i :: "nat"   and g :: "'a"   and h :: "'a" assumes "i < k" shows "(g \<cdot>\<^sub>v gs ! i) \<bullet> (h \<cdot>\<^sub>v gs ! k) = (0::'a)"
real_of_1_0_3: fixes p :: "int poly"   and l :: "rat"   and r :: "rat" assumes "invariant_1 (p, l, r)"   and "r = 0" shows "l = 0"
similar_mat_witD_5: fixes n :: "nat"   and A :: "'a mat"   and B :: "'a mat"   and P :: "'a mat"   and Q :: "'a mat" assumes "n = dim_row A"   and "similar_mat_wit A B P Q" shows "B \<in> carrier_mat n n"
roots_of_real_alg_1: fixes p :: "int poly" assumes "p \<noteq> 0" shows "real_of ` set (roots_of_real_alg p) = {x. poly (real_of_int_poly p) x = 0}"
norm_ket: fixes i :: "'c" shows "norm (ket i) = 1"
diff_self: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and a :: "'b" assumes "class.group_add minus plus zero uminus" shows "minus a a = zero"
joined_is_valid: fixes l :: "(int \<times> (real \<Rightarrow> real \<times> real)) list" assumes "boundary_chain (set l)"   and "\<And>k \<gamma>. (k, \<gamma>) \<in> set l \<Longrightarrow> valid_path \<gamma>"   and "valid_chain_list l" shows "valid_path (rec_join l)"
AE_completion: fixes P :: "'c \<Rightarrow> bool"   and M :: "'c measure" assumes "almost_everywhere M P" shows "almost_everywhere (completion M) P"
eucl_down_def: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and norm :: "'b \<Rightarrow> real"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and sgn :: "'b \<Rightarrow> 'b"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and inner :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and Basis :: "'b set"   and abs :: "'b \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and Basis_list :: "'b list"   and eucl_down :: "int \<Rightarrow> 'b \<Rightarrow> 'b"   and eucl_truncate_down :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and eucl_truncate_up :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and p :: "int"   and b :: "'b" assumes "class.executable_euclidean_space Inf Sup minus dist norm plus zero uminus scaleR sgn uniformity open inner Basis abs inf sup less_eq less Basis_list eucl_down eucl_truncate_down eucl_truncate_up" shows "eucl_down p b = comm_monoid_add.sum plus zero (\<lambda>i. scaleR (round_down p (inner b i)) i) Basis"
intros_9: fixes \<iota> :: "nat"   and V :: "('a, 'b) MWLsCom list"   and m :: "'b \<Rightarrow> 'c"   and E :: "'a \<Rightarrow> ('b \<Rightarrow> 'c) \<Rightarrow> 'c"   and BMap :: "'c \<Rightarrow> bool" shows "((spawn\<^bsub>\<iota>\<^esub> V, m), V, None, m) \<in> MWLs_semantics.MWLsSteps_det E BMap"
col2_cop2__eq: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and U :: "'p"   and V :: "'p"   and P :: "'p"   and Q :: "'p" assumes "\<not> Coplanar A B C U"   and "U \<noteq> V"   and "Coplanar A B C P"   and "Coplanar A B C Q"   and "Col U V P"   and "Col U V Q" shows "P = Q"
monoD: fixes less_eq :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and less :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and f :: "'f \<Rightarrow> 'g"   and x :: "'f"   and y :: "'f" assumes "class.order less_eq less"   and "order.mono less_eq f"   and "less_eq x y" shows "f x \<le> f y"
switch_1: fixes y :: "NominalSequents.var"   and A :: "form"   and x :: "NominalSequents.var" assumes "y \<sharp> A" shows "[y,x]A = [(y, x)] \<bullet> A"
ideal_zero: fixes I :: "'a set" assumes "ideal R I" shows "\<zero> \<in> I"
emeasure_Int_set_pmf: fixes p :: "'c pmf"   and A :: "'c set" shows "emeasure (measure_pmf p) (A \<inter> set_pmf p) = emeasure (measure_pmf p) A"
nms: fixes \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and m :: "'c match_expr"   and p :: "'d"   and \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and rs :: "'c rule list"   and t :: "state"   and a :: "action" assumes "\<not> matches \<gamma> m p"   and "\<Gamma>,\<gamma>,p\<turnstile> rs \<Rightarrow>\<^sub>z t" shows "\<Gamma>,\<gamma>,p\<turnstile> Rule m a # rs \<Rightarrow>\<^sub>z t"
case_transfer: fixes S :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and R :: "'e \<Rightarrow> 'f \<Rightarrow> bool" shows "(S ===> (R ===> llist_all2 R ===> S) ===> llist_all2 R ===> S) case_llist case_llist"
closed_segment_line_hyperplanes: fixes a :: "'b"   and b :: "'b" assumes "a \<noteq> b" shows "{a--b} = range (\<lambda>u. a + u *\<^sub>R (b - a)) \<inter> {x. a \<bullet> (b - a) \<le> x \<bullet> (b - a) \<and> x \<bullet> (b - a) \<le> b \<bullet> (b - a)}"
next_in_basics_2: fixes u :: "'a" assumes "u \<in> {a..<b}" shows "u < next_in A u"
funion_absorb: fixes A :: "'c fset" shows "A |\<union>| A = A"
at_within_open_subset: fixes a :: "'a"   and S :: "'a set"   and T :: "'a set" assumes "a \<in> S"   and "open S"   and "S \<subseteq> T" shows "at a within T = at a"
ennreal_pmf_bind: fixes N :: "'d pmf"   and f :: "'d \<Rightarrow> 'c pmf"   and i :: "'c" shows "ennreal (pmf (N \<bind> f) i) = \<integral>\<^sup>+ x. ennreal (pmf (f x) i) \<partial>measure_pmf N"
eta_into_equiv: fixes s :: "dB"   and t :: "dB" assumes "s \<rightarrow>\<^sub>\<eta> t" shows "s \<leftrightarrow> t"
min_elems_le_self: fixes A :: "'c set" shows "min_elems A \<subseteq> A"
\<pi>knipd: shows "(\<pi> @\<^bsup>n\<^esup> \<pi>') k \<noteq> ipd (\<pi> k)"
k: shows "k < ncols (Gram_Schmidt_upt_k A k)"
sum: fixes g :: "'d \<Rightarrow> real"   and A :: "'d set"   and x :: "'c" shows "sum g A *\<^sub>R x = (\<Sum>xa\<in>A. g xa *\<^sub>R x)"
list_vec: fixes xs :: "'a list" shows "list_of_vec (vec_of_list xs) = xs"
lm083: fixes N :: "'a set"   and G :: "'b list"   and bids :: "'a \<times> 'b set \<Rightarrow> 'c" assumes "0 < card N"   and "distinct G" shows "winningAllocationRel N (set G) (\<subseteq>) bids (set (allAllocationsAlg N G))"
simps_2: fixes x :: "'a"   and xs :: "'a list" shows "rotate1 (x # xs) = xs @ [x]"
inv_less_ivl_def: fixes res :: "bool"   and iv1 :: "ivl"   and iv2 :: "ivl" shows "inv_less_ivl res iv1 iv2 = (if res then (iv1 \<sqinter> (below iv2 - [1, 1]), iv2 \<sqinter> (Abs_Int2_ivl.above iv1 + [1, 1])) else (iv1 \<sqinter> Abs_Int2_ivl.above iv2, iv2 \<sqinter> below iv1))"
abd_bcd_abc: fixes a :: "'a"   and b :: "'a"   and d :: "'a"   and c :: "'a" assumes "[[a b d]]"   and "[[b c d]]" shows "[[a b c]]"
\<tau>move1WhileRed: fixes P :: "'a prog"   and h :: "'heap"   and c :: "('b, 'c, 'addr) exp"   and e :: "('b, 'c, 'addr) exp" shows "\<tau>move1 P h (while (c) e)"
strict_order_iff: fixes b :: "'c multiset"   and a :: "'c multiset" shows "(b \<subset># a) = (a = a \<union># b \<and> a \<noteq> b)"
project_constrains: fixes C :: "'c set"   and F :: "'c program"   and A :: "'a set"   and B :: "'a set" shows "(project h C F \<in> A co B) = (F \<in> C \<inter> extend_set h A co extend_set h B \<and> A \<subseteq> B)"
exists_min_ground_head: fixes ground_heads_var :: "'a \<Rightarrow> 'b set"   and gt_sym :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and \<epsilon> :: "nat"   and \<delta> :: "nat"   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 wt_sym :: "'b \<Rightarrow> hmultiset"   and coef_sym :: "'b \<Rightarrow> nat \<Rightarrow> hmultiset"   and \<zeta> :: "('b, 'a) hd" assumes "tkbo_coefs ground_heads_var gt_sym \<epsilon> \<delta> extf arity_sym arity_var wt_sym coef_sym" shows "\<exists>f. f \<in> ground_heads.ground_heads ground_heads_var \<zeta> \<and> (\<forall>g\<in>ground_heads.ground_heads ground_heads_var \<zeta>. wt_sym f + of_nat \<delta> * hmset_of_enat (arity_sym f) \<le> wt_sym g + of_nat \<delta> * hmset_of_enat (arity_sym g))"
pos_zmod_mult_2: fixes a :: "int"   and b :: "int" assumes "0 \<le> a" shows "(1 + 2 * b) mod (2 * a) = 1 + 2 * (b mod a)"
is_tangle_left_compose: fixes x :: "wall"   and y :: "wall" assumes "is_tangle_diagram (x \<circ> y)" shows "is_tangle_diagram x"
unifiers_insert_VarD_1: fixes \<sigma> :: "'a \<Rightarrow> ('b, 'a) Term.term"   and x :: "'a"   and t :: "('b, 'a) Term.term"   and E :: "(('b, 'a) Term.term \<times> ('b, 'a) Term.term) set" assumes "\<sigma> \<in> unifiers (insert (Var x, t) E)" shows "subst x t \<circ>\<^sub>s \<sigma> = \<sigma>"
HF_def: fixes A :: "hf set" shows "HF.HF A = Abs_hf (set_encode (Rep_hf ` A))"
PfP_implies_ModPon_PfP_quot: fixes H :: "fm set"   and \<alpha> :: "fm"   and \<beta> :: "fm" assumes "H \<turnstile> PfP \<guillemotleft>\<alpha> IMP \<beta>\<guillemotright>"   and "H \<turnstile> PfP \<guillemotleft>\<alpha>\<guillemotright>" shows "H \<turnstile> PfP \<guillemotleft>\<beta>\<guillemotright>"
eq: fixes p :: "nat" assumes "prime p" shows "complex_of_real (ln (real p) / real p) * hurwitz_zeta (real p) s = 1 / (s - 1) * (complex_of_real (ln (real p)) / (of_nat p powr s - 1) + B p s)"
compat: fixes i :: "nat"   and j :: "nat" shows "compatible (Y i) (Z j)"
pr_1_eq_1': shows "PrimRec1 = PrimRec1'"
a1: shows "\<forall>va. virt_to_phys va (mmu s2) (mem s1) = virt_to_phys va (mmu s2) (mem s2)"
induct: fixes P :: "'b formula \<Rightarrow> bool"   and a0 :: "'b formula" assumes "\<And>k. P (Atom k)"   and "\<And>k. P (\<^bold>\<not> (Atom k))"   and "P \<bottom>"   and "P (\<^bold>\<not> \<bottom>)"   and "\<And>F G. \<lbrakk>P F; P G\<rbrakk> \<Longrightarrow> P (F \<^bold>\<and> G)"   and "\<And>F G. \<lbrakk>\<And>x xa xb. P F; \<And>x xa xb. P G\<rbrakk> \<Longrightarrow> P (F \<^bold>\<or> G)"   and "\<And>va. P (\<^bold>\<not> (\<^bold>\<not> va))"   and "\<And>va vb. P (\<^bold>\<not> (va \<^bold>\<and> vb))"   and "\<And>va vb. P (\<^bold>\<not> (va \<^bold>\<or> vb))"   and "\<And>va vb. P (\<^bold>\<not> (va \<^bold>\<rightarrow> vb))"   and "\<And>v va. P (v \<^bold>\<rightarrow> va)" shows "P a0"
is_extensional: fixes f :: "'b \<times> 'b \<times> 'b" assumes "\<not> VVV.arr f" shows "\<a>[fst f, fst (snd f), snd (snd f)] = null"
valid_decompD_2: fixes X :: "'x set"   and ps :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list"   and h :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and U :: "'x set" assumes "valid_decomp X ps"   and "(h, U) \<in> set ps" shows "h \<noteq> 0"
assms_2: shows "\<forall>f\<in>M. mcont \<Or> (\<sqsubseteq>) Sup (\<le>) f"
v_2: fixes l :: "nat" shows "v \<bullet> (T ^^ l) (0::'a) \<le> - A * real l"
gc_to_red: fixes Ns :: "('a literal multiset \<times> label) set llist" assumes "chain (\<leadsto>GC) Ns" shows "chain (\<rhd>L) Ns"
comp_assoc: fixes h :: "'s"   and g :: "'s"   and f :: "'s" shows "(h \<cdot>\<^sub>S g) \<cdot>\<^sub>S f = h \<cdot>\<^sub>S g \<cdot>\<^sub>S f"
config'_rand_append: fixes step :: "'a \<Rightarrow> 'c \<Rightarrow> 'd \<Rightarrow> 'a"   and A :: "('a \<Rightarrow> 'b pmf) \<times> ('a \<times> 'b \<Rightarrow> 'c \<Rightarrow> ('d \<times> 'b) pmf)"   and s :: "('a \<times> 'b) pmf"   and xs :: "'c list"   and ys :: "'c list" shows "On_Off.config'_rand step A s (xs @ ys) = On_Off.config'_rand step A (On_Off.config'_rand step A s xs) ys"
disc2_noC: shows "\<forall>a. \<not> disc2 (Dst_Ports a)"
mHom_tOp_assoc: fixes N :: "('g, 'b, 'h) Module_scheme"   and f :: "'a \<Rightarrow> 'g"   and g :: "'a \<Rightarrow> 'g"   and h :: "'a \<Rightarrow> 'g" assumes "R module N"   and "f \<in> mHom R M N"   and "g \<in> mHom R M N"   and "h \<in> mHom R M N" shows "tOp_mHom R M N (tOp_mHom R M N f g) h = tOp_mHom R M N f (tOp_mHom R M N g h)"
\<Delta>_naturality: shows "(\<mu> \<star> r\<^sub>0) \<cdot> \<rho> = (s \<star> \<mu>.the_\<theta>) \<cdot> \<a>[s, s\<^sub>0, \<mu>.chine] \<cdot> (\<sigma> \<star> \<mu>.chine) \<cdot> \<mu>.the_\<nu>"
fps_const_1_eq_1: shows "fps_const (1::'b) = 1"
mconn_def: fixes E :: "('b \<times> 'b) set" shows "mconn E = E\<^sup>* \<inter> (E\<inverse>)\<^sup>*"
inv_start: shows "inv_end x (Suc 0, l, r)"
one_is_normal: shows "{\<one>\<^bsub>H\<^esub>} \<lhd> H"
rem_comps_spec_struct: fixes sel :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> nat \<times> 'g \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and ap :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> nat \<times> 'g \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and ab :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> nat \<times> 'g \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and compl :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> nat \<times> 'g \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> 'f) list \<times> 'g"   and gs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and bs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and data :: "nat \<times> nat \<times> 'g"   and ps :: "((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and sps :: "((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and aux :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> 'f) list \<times> 'g"   and hs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and data' :: "nat \<times> 'g" assumes "struct_spec sel ap ab compl"   and "rem_comps_spec (gs @ bs) data"   and "ps \<noteq> []"   and "set ps \<subseteq> set bs \<times> (set gs \<union> set bs)"   and "sps = sel gs bs ps (snd data)"   and "aux = compl gs bs (ps -- sps) sps (snd data)"   and "(hs, data') = add_indices aux (snd data)" shows "rem_comps_spec (gs @ ab gs bs hs data') (fst data - count_const_lt_components (fst aux), data')"
mylem2: fixes w2 :: "32 Word.word"   and w1 :: "32 Word.word" assumes "w2 < w1" shows "uint w2 < uint w1"
inv\<^sub>3E_2: fixes P\<^sub>1 :: "'a set set"   and P\<^sub>2 :: "'a set set"   and B\<^sub>1 :: "'a set"   and B\<^sub>2 :: "'a set"   and S :: "'a set"   and L :: "'a set" assumes "inv\<^sub>3 P\<^sub>1 P\<^sub>2 B\<^sub>1 B\<^sub>2 S L" shows "B\<^sub>2 \<subseteq> {u \<in> U. w u \<le> real c / 2}"
select_convs_1: fixes lasso_reach :: "'e list"   and lasso_va :: "'e"   and lasso_cysfx :: "'e list"   and more :: "'f" shows "lasso_reach \<lparr>lasso_reach = lasso_reach, lasso_va = lasso_va, lasso_cysfx = lasso_cysfx, \<dots> = more\<rparr> = lasso_reach"
bij_imp_permutes: fixes p :: "'b \<Rightarrow> 'b"   and S :: "'b set" assumes "bij_betw p S S"   and "\<And>x. x \<notin> S \<Longrightarrow> p x = x" shows "p permutes S"
split: fixes P :: "'a \<Rightarrow> bool"   and f1 :: "'a"   and f2 :: "'b \<Rightarrow> 'b pred \<Rightarrow> 'a"   and f3 :: "'b pred \<Rightarrow> 'b seq \<Rightarrow> 'a"   and seq :: "'b seq" shows "P (case seq of Empty \<Rightarrow> f1 | Insert x xa \<Rightarrow> f2 x xa | Join x xa \<Rightarrow> f3 x xa) = ((seq = Empty \<longrightarrow> P f1) \<and> (\<forall>x21 x22. seq = Insert x21 x22 \<longrightarrow> P (f2 x21 x22)) \<and> (\<forall>x31 x32. seq = Join x31 x32 \<longrightarrow> P (f3 x31 x32)))"
nsum_memTr: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'a" shows "(\<forall>j\<le>n. f j \<in> carrier A) \<longrightarrow> \<Sigma>\<^sub>e A f n \<in> carrier A"
cut_less_Min_not_empty: fixes I :: "'a set"   and t :: "'a" assumes "I \<noteq> {}"   and "iMin I < t" shows "I \<down>< t \<noteq> {}"
Tensor_right_id: fixes xs :: "'a list list" shows "xs \<otimes> [[id]] = xs"
nat_pow_mult: fixes G :: "('a, 'b) monoid_scheme"   and x :: "'a"   and n :: "nat"   and m :: "nat" assumes "Group.monoid G"   and "x \<in> carrier G" shows "x [^]\<^bsub>G\<^esub> n \<otimes>\<^bsub>G\<^esub> x [^]\<^bsub>G\<^esub> m = x [^]\<^bsub>G\<^esub> (n + m)"
rel_eq: shows "tllist_all2 (=) (=) = (=)"
divide_nonzero: fixes x :: "nat \<Rightarrow> int"   and y :: "nat \<Rightarrow> int" assumes "x \<in> nonzero Zp"   and "y \<in> nonzero Zp"   and "val_Zp y \<le> val_Zp x" shows "local.divide x y \<in> nonzero Zp"
snd: shows "q \<approx>[\<R>] mix_pmf Uq B W"
compet_def: fixes step :: "'b \<Rightarrow> 'c \<Rightarrow> 'd \<Rightarrow> 'b"   and t :: "'b \<Rightarrow> 'c \<Rightarrow> 'd \<Rightarrow> nat"   and wf :: "'b \<Rightarrow> 'c list \<Rightarrow> bool"   and A :: "('b \<Rightarrow> 'e) \<times> ('b \<times> 'e \<Rightarrow> 'c \<Rightarrow> 'd \<times> 'e)"   and c :: "real"   and S :: "'b set" shows "On_Off.compet step t wf A c S = (\<forall>s\<in>S. \<exists>b\<ge>0. \<forall>rs. wf s rs \<longrightarrow> real (On_Off.T step t s rs (On_Off.off2 step A (s, fst A s) rs)) \<le> c * real (On_Off.T_opt step t s rs) + b)"
card_le_if_inj_on_rel: fixes B :: "'a set"   and A :: "'b set"   and r :: "'b \<Rightarrow> 'a \<Rightarrow> bool" assumes "finite B"   and "\<And>a. a \<in> A \<Longrightarrow> \<exists>b. b \<in> B \<and> r a b"   and "\<And>a1 a2 b. \<lbrakk>a1 \<in> A; a2 \<in> A; b \<in> B; r a1 b; r a2 b\<rbrakk> \<Longrightarrow> a1 = a2" shows "card A \<le> card B"
set_map: fixes f :: "'d \<Rightarrow> 'c"   and v :: "'d option" shows "set_option (map_option f v) = f ` set_option v"
is_unifier_alt: fixes AA :: "'a set"   and \<sigma> :: "'s" assumes "finite AA" shows "is_unifier \<sigma> AA = (\<forall>A\<in>AA. \<forall>B\<in>AA. A \<cdot>a \<sigma> = B \<cdot>a \<sigma>)"
exception_\<I>_def: fixes \<I> :: "('a, 'b) \<I>" shows "exception_\<I> \<I> = map_\<I> id exception_of_option (stop_\<I> \<I>)"
YFtorNtMapsTo: fixes C :: "('d, 'e, 'f) LSCategory_scheme"   and f :: "'e" assumes "LSCategory C"   and "f \<in> mor\<^bsub>C\<^esub>" shows "YFtorNT C f maps\<^bsub>CatExp (Op C) SET\<^esub> Hom\<^bsub>C\<^esub>[\<midarrow>,dom\<^bsub>C\<^esub> f] to Hom\<^bsub>C\<^esub>[\<midarrow>,cod\<^bsub>C\<^esub> f]"
generate_idem_Un: fixes A :: "'a set" assumes "A \<subseteq> carrier G" shows "generate G (\<Union>x\<in>A. generate G {x}) = generate G A"
tendsto_prod: fixes S :: "'b set"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'd"   and L :: "'b \<Rightarrow> 'd"   and F :: "'c filter" assumes "\<And>i. i \<in> S \<Longrightarrow> (f i \<longlongrightarrow> L i) F" shows "((\<lambda>x. \<Prod>i\<in>S. f i x) \<longlongrightarrow> prod L S) F"
is_derivation_is_sentence: fixes s :: "symbol list" assumes "is_derivation s" shows "is_sentence s"
poly_subst_id: fixes p :: "('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and f :: "'a \<Rightarrow> ('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" assumes "\<And>x. x \<in> indets p \<Longrightarrow> f x = monomial (1::'b) (monomial 1 x)" shows "poly_subst f p = p"
Sbis_trace: fixes cf1 :: "('test, 'atom, 'choice) cmd \<times> 'state"   and cf2 :: "('test, 'atom, 'choice) cmd \<times> 'state"   and n :: "nat"   and s' :: "'state" assumes "proper (fst cf1)"   and "proper (fst cf2)"   and "fst cf1 \<approx>s fst cf2"   and "snd cf1 \<approx> snd cf2" shows "T.prob cf1 {cfT \<in> space (T cf1). snd ((cf1 ## cfT) !! n) \<approx> s'} = T.prob cf2 {cfT \<in> space (T cf2). snd ((cf2 ## cfT) !! n) \<approx> s'}"
smult: fixes a :: "'a"   and p :: "nat \<Rightarrow> 'a" assumes "a \<in> carrier R"   and "p \<in> carrier P"   and "Q p" shows "Q (a \<odot>\<^bsub>P\<^esub> p)"
runit_char_1: fixes f :: "'a" assumes "ide f" shows "arr (PB.runit f) \<and> src f \<in> sources (PB.runit f) \<and> trg f \<in> targets (PB.runit f)"
effective_well_defined_Tensor: fixes M1 :: "'a list list"   and M2 :: "'a list list" assumes "mat (row_length M1) (length M1) M1"   and "mat (row_length M2) (length M2) M2" shows "mat (row_length M1 * row_length M2) (length M1 * length M2) (M1 \<otimes> M2)"
gc_system_init_def: shows "gc_system_init = (\<lambda>s. gc_initial_state (s gc) \<and> (\<forall>m. mut_initial_state (s (mutator m))) \<and> sys_initial_state (s sys) \<and> valid_refs s)"
path_from_toI: fixes xs :: "'a list"   and v :: "'a"   and w :: "'a" assumes "path xs"   and "xs \<noteq> []"   and "hd xs = v"   and "last xs = w" shows "v \<leadsto>xs\<leadsto> w"
sig_trd_def: fixes bs :: "('t \<Rightarrow>\<^sub>0 'b) list"   and p :: "'t \<Rightarrow>\<^sub>0 'b" shows "sig_trd bs p = (if rep_list p = 0 then p else sig_trd_aux bs (punit.lt (rep_list p), p))"
assms_4: fixes x :: "'b" assumes "inv0 x" shows "I (f x)"
val_cong: fixes x :: "'b tm"   and y :: "'b tm" assumes "x = y" shows "Time_Monad.val x = Time_Monad.val y"