hcju/isabelleps huggingface dataset:
cl_rank_of: fixes x :: "'a"   and X :: "'a set" assumes "x \<in> X_insert.cl X" shows "X_insert.rank_of (insert x X) = X_insert.rank_of X"
is_MST_def: fixes w :: "'b set \<Rightarrow> nat"   and g :: "'b ugraph"   and t :: "'b ugraph" shows "is_MST w g t \<equiv> is_spanning_tree g t \<and> (\<forall>t'. is_spanning_tree g t' \<longrightarrow> weight w t \<le> weight w t')"
subset_diff: fixes B :: "'k set"   and A :: "'k set"   and g :: "'k \<Rightarrow> 'l" assumes "B \<subseteq> A"   and "finite A" shows "sum g A = sum g (A - B) + sum g B"
not_dbm_lt_le_impl: fixes a :: "'c"   and b :: "'c" assumes "\<not> Lt a \<prec> Le b" shows "b < a"
measurable_g: fixes s :: "'s"   and f :: "'s \<Rightarrow> 'a \<Rightarrow> 'b"   and M :: "'a measure"   and N :: "'b measure"   and g :: "'a \<Rightarrow> 's" assumes "\<And>i. i \<in> acc `` {s} \<Longrightarrow> f i \<in> M \<rightarrow>\<^sub>M N"   and "g \<in> M \<rightarrow>\<^sub>M count_space (acc `` {s})" shows "(\<lambda>x. f (g x) x) \<in> M \<rightarrow>\<^sub>M N"
abs_split_def: fixes xs :: "('b \<times> 'c) list"   and x :: "'c" shows "abs_split xs x = (takeWhile (\<lambda>(uu_, s). s < x) xs, dropWhile (\<lambda>(uu_, s). s < x) xs)"
simps_3: fixes uw :: "'c" shows "typeof (OpUbx2 uw) = Some Ubx2"
subsetI: fixes A :: "'a set"   and B :: "'a set" assumes "\<And>x. x \<in> A \<Longrightarrow> x \<in> B" shows "order.greater_eq B A"
11: fixes i :: "nat" assumes "enat i < llength u"   and "enat i < llength v" shows "u ?! i = v ?! i"
mult_Characters: fixes G :: "('c, 'd) monoid_scheme"   and \<chi>\<^sub>1 :: "'c \<Rightarrow> complex"   and \<chi>\<^sub>2 :: "'c \<Rightarrow> complex" shows "\<chi>\<^sub>1 \<otimes>\<^bsub>Characters G\<^esub> \<chi>\<^sub>2 = (\<lambda>a. \<chi>\<^sub>1 a * \<chi>\<^sub>2 a)"
intros: fixes x :: "'a"   and uu :: "'a list" shows "head_side (x # uu)"
NewC: fixes G :: "prog"   and A :: "'b triple set"   and P :: "'b assn"   and C :: "qtname"   and Q :: "'b assn" assumes "G,A\<turnstile>{Normal P} .Init C. {Alloc G (CInst C) Q}" shows "G,A\<turnstile>{Normal P} NewC C-\<succ> {Q}"
hyps: fixes x :: "'a mod_ring poly"   and y :: "'a mod_ring poly" shows "(x + y) ^ CARD('a) ^ n = x ^ CARD('a) ^ n + y ^ CARD('a) ^ n"
reachable1_awalk: fixes u :: "'a"   and v :: "'a" shows "((u, v) \<in> (parcs (subdivide G e w))\<^sup>+) = (\<exists>p. S.awalk u p v \<and> p \<noteq> [])"
hyps_1: fixes s :: "'a regex" assumes "local.safe_regex Futu Lax r1"   and "s \<in> lpd test i r1" shows "local.safe_regex Futu Lax s"
multiplicative_function_eqI: fixes f :: "nat \<Rightarrow> 'a"   and g :: "nat \<Rightarrow> 'a"   and n :: "nat" assumes "multiplicative_function f"   and "multiplicative_function g"   and "\<And>p k. \<lbrakk>prime p; 0 < k\<rbrakk> \<Longrightarrow> f (p ^ k) = g (p ^ k)" shows "f n = g n"
induct: fixes P :: "Vars.var set \<Rightarrow> stack_elem list \<Rightarrow> bool"   and a0 :: "Vars.var set"   and a1 :: "stack_elem list" assumes "\<And>V. P V []"   and "\<And>V e1 e2 S. P V S \<Longrightarrow> P V (Alts e1 e2 # S)"   and "\<And>V x S. P V S \<Longrightarrow> P V (Arg x # S)"   and "\<And>V x S. \<lbrakk>x \<in> V \<Longrightarrow> P V S; x \<notin> V \<Longrightarrow> P V S\<rbrakk> \<Longrightarrow> P V (Upd x # S)"   and "\<And>V x S. P V S \<Longrightarrow> P V (Dummy x # S)" shows "P a0 a1"
sg1: shows "if \<exists>k<n. nSend k t \<noteq> [] then recv t = nSend (SOME k. k < n \<and> nSend k t \<noteq> []) t else recv t = []"
convex_hull_def: fixes Vs :: "'a vec set" shows "convex_hull Vs = {x. \<exists>Ws c. finite Ws \<and> Ws \<subseteq> Vs \<and> convex_lincomb c Ws x}"
Beta_star_equals_One_star: fixes t1 :: "lam"   and t2 :: "lam" shows " t1 \<longrightarrow>\<^sub>1\<^sup>* t2 =  t1 \<longrightarrow>\<^sub>\<beta>\<^sup>* t2"
simple_path_imp_path: fixes g :: "real \<Rightarrow> 'b" assumes "simple_path g" shows "path g"
index_smult_mat_1: fixes i :: "nat"   and A :: "'c mat"   and j :: "nat"   and a :: "'c" assumes "i < dim_row A"   and "j < dim_col A" shows "(a \<cdot>\<^sub>m A) $$ (i, j) = a * A $$ (i, j)"
append_take_drop_id: fixes n :: "nat"   and xs :: "'k list" shows "take n xs @ drop n xs = xs"
sys_block_sizes_def: shows "ins.sys_block_sizes \<equiv> {int (card bl) |bl. bl \<in># \<A>}"
res_opp_fold_A_chamber_images_2: fixes X :: "'b set set"   and \<A> :: "'b set set set"   and A :: "'b set set"   and B :: "'b set set"   and B' :: "'b set set"   and C :: "'b set"   and D :: "'b set"   and E :: "'b set"   and z :: "'b set" assumes "ChamberComplexApartmentSystemTriangle X \<A> A B B' C D E z" shows "restrict1 (ChamberComplexWithApartmentSystem.canonical_retraction X \<A> A C \<circ> ChamberComplexWithApartmentSystem.canonical_retraction X \<A> B' D) (\<Union> A) ` C = D"
assms: shows "\<forall>\<^sub>F x in at_top. f x - g x = 0"
get_ancestors_obtains_children: fixes h :: "(_) heap"   and ancestor :: "(_) object_ptr"   and ptr :: "(_) object_ptr"   and ancestors :: "(_) object_ptr list" assumes "heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M h"   and "ancestor \<noteq> ptr"   and "ancestor \<in> set ancestors"   and "h \<turnstile> Shadow_DOM.get_ancestors ptr \<rightarrow>\<^sub>r ancestors"   and "ShadowRootClass.type_wf h"   and "ShadowRootClass.known_ptrs h" obtains children :: "(_) node_ptr list" and ancestor_child :: "(_) node_ptr" where "h \<turnstile> Shadow_DOM.get_child_nodes ancestor \<rightarrow>\<^sub>r children" and "ancestor_child \<in> set children" and "cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r ancestor_child \<in> set ancestors"
minus_flow_simps: fixes \<Delta> :: "('a, 'b) network_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal"   and g :: "'a \<times> 'a \<Rightarrow> ennreal"   and x :: "'a"   and y :: "'a" assumes "flow_attainability \<Delta>" shows "flow_attainability.minus_flow \<Delta> f g (x, y) = (if edge \<Delta> x y then f (x, y) - g (x, y) else if edge \<Delta> y x then g (y, x) - f (y, x) else 0)"
distinct_9: fixes x2 :: "('g \<Rightarrow> 'h option) \<Rightarrow> bool"   and x41 :: "('g \<Rightarrow> 'h option) \<times> 'i \<Rightarrow> bool"   and x42 :: "'j"   and x43 :: "('g \<Rightarrow> 'h option) \<Rightarrow> ('g \<Rightarrow> 'h option) \<Rightarrow> 'g \<Rightarrow> 'h option" shows "(x2)\<^sub>\<surd> \<noteq> x41\<hookleftarrow>\<^bsub>x42\<^esub>x43"
invertible: fixes u :: "'a" assumes "u \<in> I" shows "IA.additive.sub.invertible u"
assms_1: shows "finite V \<or> |V| <o |UNIV|"
hyp: shows "(remove_rels ts'', m'', \<S>'') \<Rightarrow>\<^sub>v\<^sup>* (remove_rels ts', m', \<S>')"
cod_char: fixes f :: "(bool, 'arr list) concrete_category.arr" shows "cod f = (if arr f then if f = Zero then Zero else One else null)"
rel_cong: fixes x :: "'c option"   and ya :: "'c option"   and y :: "'d option"   and xa :: "'d option"   and R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and Ra :: "'c \<Rightarrow> 'd \<Rightarrow> bool" assumes "x = ya"   and "y = xa"   and "\<And>z yb. \<lbrakk>z \<in> set_option ya; yb \<in> set_option xa\<rbrakk> \<Longrightarrow> R z yb = Ra z yb" shows "rel_option R x y = rel_option Ra ya xa"
SUP_least: fixes A :: "'c set"   and f :: "'c \<Rightarrow> 'd \<times> 'e"   and u :: "'d \<times> 'e" assumes "\<And>i. i \<in> A \<Longrightarrow> less_eq_prod (f i) u" shows "less_eq_prod (Sup_prod (f ` A)) u"
map2_val_pair_singleton_eq_update_by_fun_pair: fixes xs :: "('a \<times> 'b) list"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'b"   and g :: "('a \<times> 'b) list \<Rightarrow> ('a \<times> 'b) list"   and h :: "('a \<times> 'c) list \<Rightarrow> ('a \<times> 'b) list"   and k :: "'a"   and v :: "'c" assumes "tc.oalist_inv_raw xs"   and "\<And>k x. f k x (0::'c) = x"   and "\<And>zs. tc.oalist_inv_raw zs \<Longrightarrow> g zs = zs"   and "h [(k, v)] = map_val_pair (\<lambda>k. f k (0::'b)) [(k, v)]" shows "map2_val_pair_tc f g h xs [(k, v)] = update_by_fun_pair_tc k (\<lambda>x. f k x v) xs"
sum_var_list_def: fixes m :: "('a \<times> nat) list"   and x :: "'a" shows "sum_var_list m x \<equiv> \<Sum>(y, c)\<leftarrow>m. if x = y then c else 0"
sig_trd_spp_body_alt_Some_1: fixes bs :: "('t \<Rightarrow>\<^sub>0 'b) list"   and v :: "'t"   and p :: "'a \<Rightarrow>\<^sub>0 'b"   and i :: "nat"   and r :: "'a \<Rightarrow>\<^sub>0 'b" assumes "find_sig_reducer (map spp_of bs) v (punit.lt p) 0 = Some i" shows "sig_trd_spp_body (map spp_of bs) v (p, r) = (punit.lower (p - local.punit.monom_mult (punit.lc p / punit.lc (rep_list (bs ! i))) (punit.lt p - punit.lt (rep_list (bs ! i))) (rep_list (bs ! i))) (punit.lt p), r)"
inject: fixes x1 :: "'b"   and x2 :: "'c"   and x3 :: "nat"   and x4 :: "('b, 'c) BinomialTree list"   and y1 :: "'b"   and y2 :: "'c"   and y3 :: "nat"   and y4 :: "('b, 'c) BinomialTree list" shows "(Node x1 x2 x3 x4 = Node y1 y2 y3 y4) = (x1 = y1 \<and> x2 = y2 \<and> x3 = y3 \<and> x4 = y4)"
mono_onE: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and x :: "'a"   and y :: "'a" assumes "Basis.mono_on A f"   and "x \<in> A"   and "y \<in> A"   and "x \<le> y" obtains "f x \<le> f y"
wf: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "well_founded R" shows "wfP R"
simps_1: fixes I :: "'f \<Rightarrow> 'g option"   and P :: "'g \<Rightarrow> 'f \<Rightarrow> 'h \<Rightarrow> 'i \<Rightarrow> bool" shows "upd_invs I P [] = I"
lossless_gpv_induct_strong: fixes \<I> :: "('d, 'e) \<I>"   and gpv :: "('f, 'd, 'e) gpv"   and P :: "('f, 'd, 'e) gpv \<Rightarrow> bool" assumes "lossless_gpv \<I> gpv"   and "\<And>p. \<lbrakk>lossless_spmf p; \<And>gpv. gpv \<in> sub_gpvs \<I> (GPV p) \<Longrightarrow> lossless_gpv \<I> gpv; \<And>gpv. gpv \<in> sub_gpvs \<I> (GPV p) \<Longrightarrow> P gpv\<rbrakk> \<Longrightarrow> P (GPV p)" shows "P gpv"
dotprod_gt0: fixes x :: "nat list"   and y :: "nat list" assumes "length x = length y"   and "\<exists>i<length y. 0 < x ! i \<and> 0 < y ! i" shows "0 < x \<bullet> y"
s_1: shows "s \<in>\<^sub>A path_summary loc' loc"
mult_scalar_monomial: fixes t :: "'c"   and c :: "'d"   and p :: "'c \<Rightarrow>\<^sub>0 'd" shows "MPoly_Type_Class.punit.mult_scalar (monomial c t) p = MPoly_Type_Class.punit.monom_mult c t p"
Block: fixes P :: "'d set"   and init :: "'d \<Rightarrow> 'd"   and P' :: "'d \<Rightarrow> 'd 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"   and bdy :: "('d, 'e, 'f) com"   and return :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and R :: "'d \<Rightarrow> 'd \<Rightarrow> 'd set"   and A :: "'d set"   and c :: "'d \<Rightarrow> 'd \<Rightarrow> ('d, 'e, 'f) com"   and Q :: "'d set" assumes "P \<subseteq> {s. init s \<in> P' s}"   and "\<forall>s. \<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> (P' s) bdy {t. return s t \<in> R s t},{t. return s t \<in> A}"   and "\<forall>s t. \<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> (R s t) c s t Q,A" shows "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P block init bdy return c Q,A"
nat_pow_Suc2: fixes x :: "'a vec"   and n :: "nat" assumes "x \<in> carrier_vec n" shows "[Suc n] \<cdot>\<^bsub>V\<^esub> x = x + [n] \<cdot>\<^bsub>V\<^esub> x"
vfsequence_Limit_vcons_in_VsetI: fixes xs :: "V"   and \<alpha> :: "V"   and x :: "V" assumes "vfsequence xs"   and "Limit \<alpha>"   and "x \<in>\<^sub>\<circ> Vset \<alpha>"   and "xs \<in>\<^sub>\<circ> Vset \<alpha>" shows "xs #\<^sub>\<circ> x \<in>\<^sub>\<circ> Vset \<alpha>"
power_add_numeral2: fixes a :: "'a"   and m :: "num"   and n :: "num"   and b :: "'a" shows "a ^ numeral m * (a ^ numeral n * b) = a ^ numeral (m + n) * b"
extend_def: fixes h :: "'d \<times> 'e \<Rightarrow> 'c"   and F :: "'d program" shows "extend h F = mk_program (extend_set h (Init F), extend_act h ` Acts F, project_act h -` AllowedActs F)"
upset_set_to_downset_set2: fixes X :: "'a set" shows "\<Up> X = \<partial> ` \<Down> (\<partial> ` X)"
syzygy_module_def: fixes B :: "('t \<Rightarrow>\<^sub>0 'c) set" shows "pmdl.syzygy_module B = {s. pmdl.represents B s 0}"
step_Cons: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "('e, 'd, 'f) com"   and cs :: "('e, 'd, 'f) com list"   and css :: "(('e, 'd, 'f) com list \<times> ('e, 'd, 'f) com list) list"   and s :: "('e, 'f) xstate"   and cs' :: "('e, 'd, 'f) com list"   and css' :: "(('e, 'd, 'f) com list \<times> ('e, 'd, 'f) com list) list"   and t :: "('e, 'f) xstate" assumes "\<Gamma>\<turnstile> (c # cs, css, s) \<rightarrow> (cs', css', t)" shows "\<exists>pcss. css' = pcss @ css \<and> (if pcss = [] then \<exists>ps. cs' = ps @ cs else \<exists>pcs_normal pcs_abrupt. pcss = [(pcs_normal @ cs, pcs_abrupt @ cs)])"
preserves_hom: fixes f :: "'b"   and a :: "'b"   and b :: "'b" assumes "F.B.in_hom f a b" shows "\<guillemotleft>G f : G a \<rightarrow> G b\<guillemotright>"
TAbs_preservation: fixes \<Gamma> :: "binding list"   and T\<^sub>1\<^sub>1 :: "POPLmarkRecord.type"   and t\<^sub>1\<^sub>2 :: "trm"   and T\<^sub>2 :: "POPLmarkRecord.type"   and T :: "POPLmarkRecord.type" assumes "\<Gamma> \<turnstile> (\<lambda><:T\<^sub>1\<^sub>1. t\<^sub>1\<^sub>2) \<bullet>\<^sub>\<tau> T\<^sub>2 : T" shows "\<Gamma> \<turnstile> t\<^sub>1\<^sub>2[0 \<mapsto>\<^sub>\<tau> T\<^sub>2] : T"
sorted_rem: fixes t :: "tree"   and x :: "nat" assumes "t \<noteq> Tip"   and "x \<in> set_of (rem t)"   and "BinaryTree_TacticStyle.sorted t" shows "x < rm t"
fmran_fmmap: fixes f :: "'c \<Rightarrow> 'a"   and m :: "('b, 'c) fmap" shows "fmran (fmmap f m) = f |`| fmran m"
False: shows "z \<notin> \<int>\<^sub>\<le>\<^sub>0"
SN_imp_acyclic: fixes R :: "('a \<times> 'a) set" assumes "SN R" shows "acyclic R"
in_fset_mset_mset: fixes x :: "'c"   and N :: "'c multiset" shows "(x |\<in>| fset_mset N) = (x \<in># N)"
select_convs_2: fixes pverts :: "'b set"   and parcs :: "('b \<times> 'b) set"   and more :: "'c" shows "parcs \<lparr>pverts = pverts, parcs = parcs, \<dots> = more\<rparr> = parcs"
zero: fixes U :: "'c set"   and V :: "'c set" assumes "Subspace.subspace U V"   and "vectorspace V" shows "(0::'c) \<in> U"
open_Diff: fixes "open" :: "'c set \<Rightarrow> bool"   and S :: "'c set"   and T :: "'c set" assumes "class.topological_space open"   and "open S"   and "topological_space.closed open T" shows "open (S - T)"
xs_1: shows "Polynomial.smult (Polynomial.lead_coeff p) (\<Prod>x\<leftarrow>xs. [:- x, 1:]) = p"
muaux_axioms: shows "muaux valid_muaux init_muaux add_new_muaux length_muaux eval_muaux"
rep_metric_completion_def: shows "rep_metric_completion \<equiv> quot_type.rep Rep_metric_completion"
preorder_on_restrict_subset: fixes A :: "'a set" assumes "A \<subseteq> Aa__ - M" shows "Order_Predicates.preorder_on A (restrict_relation A (of_weak_ranking (weak_ranking_aux (Aa__ - M))))"
simps_2: fixes c :: "'a"   and s :: "'a FOL_Fitting.term" shows "subc c s TT = TT"
simp'_2: fixes x :: "Core.term" shows "subst_bv x g = g"
w\<^sub>h'_is_map: shows "is_left_adjoint w\<^sub>h'"
lnth_llist_of: fixes xs :: "'a list" shows "($) (llist_of xs) = (!) xs"
refine: shows "(uncurry2 full_checker_l_impl, uncurry2 full_checker_l) \<in> poly_assn\<^sup>k *\<^sub>a polys_assn_input\<^sup>d *\<^sub>a (list_assn (pac_step_rel_assn uint64_nat_assn poly_assn string_assn))\<^sup>k \<rightarrow>\<^sub>a status_assn raw_string_assn \<times>\<^sub>a hs.assn string_assn \<times>\<^sub>a polys_assn"
sum_sum_distr_fun: fixes Y :: "'c set"   and h :: "'c \<Rightarrow> 'c set"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "finite Y"   and "\<And>y. finite (h y)" shows "\<Sum> ((\<lambda>y. \<Sum> (f ` h y) \<cdot> g y) ` Y) = \<Sum> {\<Sum> {f x \<cdot> g y |x. x \<in> h y} |y. y \<in> Y}"
vec_nth_inject: fixes x :: "('e, 'd) Finite_Cartesian_Product.vec"   and y :: "('e, 'd) Finite_Cartesian_Product.vec" shows "(($h) x = ($h) y) = (x = y)"
apex_def: shows "assoc\<mu>\<nu>\<pi>.dom.apex \<equiv> C.dom assoc\<mu>\<nu>\<pi>.dom.leg0"
inverse_arrowsE: fixes f :: "'a1"   and g :: "'a1" assumes "C1.inverse_arrows f g" obtains "C1.ide (g \<cdot>\<^sub>1 f)" and "C1.ide (f \<cdot>\<^sub>1 g)"
s_antimono: fixes x :: "real"   and y :: "real" assumes "x \<le> y"   and "ego2.t_stop \<le> x" shows "ego2.s y \<le> ego2.s x"
pr0_def: fixes a :: "'a"   and b :: "'a" shows "pr0 a b \<equiv> if ide a \<and> ide b then snd (SOME x. has_as_binary_product a b (fst x) (snd x)) else null"
hyps_4: fixes b :: "name"   and ba :: "trm"   and bb :: "(name \<times> trm) list" assumes "b \<sharp> t" shows "((b, ba) # bb)<t> = bb<t>"
CHAOS_has_all_tickFree_failures: fixes a :: "'b event list"   and A :: "'b set"   and b :: "'b event set" assumes "tickFree a"   and "{x. ev x \<in> set a} \<subseteq> A" shows "(a, b) \<in> F (CHAOS A)"
refine: assumes "R = Id" shows "(cr_rename_gba_impl, cr_rename_gba) \<in> \<langle>\<langle>Rm, R\<rangle>node_rel\<rangle>list_set_rel \<rightarrow> \<langle>R\<rangle>ltlr_rel \<rightarrow> \<langle>gbav_impl_rel_ext unit_rel nat_rel (\<langle>Id\<rangle>fun_set_rel)\<rangle>nres_rel"
assms: shows "h \<turnstile> find_slot open_flag slotable \<rightarrow>\<^sub>r Some slot"
hyp_c2: shows "\<forall>Z. \<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> {s. s = Z \<and> \<Gamma>\<turnstile> \<langle>c\<^sub>2,Normal s\<rangle> \<Rightarrow>\<notin>{Stuck} \<union> Fault ` (- F) \<and> \<Gamma>\<turnstile>the (\<Gamma> p) \<down> Normal \<sigma> \<and> (\<exists>cs css. \<Gamma>\<turnstile> ([the (\<Gamma> p)], [], Normal \<sigma>) \<rightarrow>\<^sup>* (c\<^sub>2 # cs, css, Normal s))} c\<^sub>2 {t. \<Gamma>\<turnstile> \<langle>c\<^sub>2,Normal Z\<rangle> \<Rightarrow> Normal t},{t. \<Gamma>\<turnstile> \<langle>c\<^sub>2,Normal Z\<rangle> \<Rightarrow> Abrupt t}"
sum_mset_distrib_left: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and c :: "'b"   and f :: "'c \<Rightarrow> 'b"   and M :: "'c multiset" assumes "class.semiring_0 plus zero times" shows "times c (comm_monoid_add.sum_mset plus zero (image_mset f M)) = comm_monoid_add.sum_mset plus zero {#times c (f x). x \<in># M#}"
sort_by_pri_edges: fixes edges :: "'a edge_scheme list"   and A :: "'a edge_scheme set"   and es :: "'a edge_scheme list"   and min_pri :: "nat" assumes "set edges \<subseteq> A"   and "es \<in> set (sort_by_pri min_pri edges)" shows "set es \<subseteq> A"
butlast_subset: fixes xs :: "'a list"   and A :: "'a set" assumes "xs \<noteq> []"   and "set xs \<subseteq> A" shows "set (butlast xs) \<subseteq> A"
assms_3: shows "CX \<union> CY \<union> fst ` P1 \<subseteq> CZ"
R_H2_commute2: fixes P :: "('c, 'd) alpha_rp_scheme \<times> ('c, 'd) alpha_rp_scheme \<Rightarrow> bool" shows "R (H2 P) = H2 (R P)"
supp_def: fixes permute :: "perm \<Rightarrow> 'c \<Rightarrow> 'c"   and x :: "'c" assumes "class.pt permute" shows "pt.supp permute x = {a. infinite {b. permute (a \<rightleftharpoons> b) x \<noteq> x}}"
min_distI2: fixes c :: "nat \<times> nat \<Rightarrow> 'b"   and v :: "nat"   and v' :: "nat"   and Q :: "nat \<Rightarrow> bool" assumes "Graph.connected c v v'"   and "\<And>d. \<lbrakk>Graph.dist c v d v'; \<And>d'. Graph.dist c v d' v' \<Longrightarrow> d \<le> d'\<rbrakk> \<Longrightarrow> Q d" shows "Q (Graph.min_dist c v v')"
rulesInPCs_16: shows "Perms \<subseteq> CutFreePC"
cases: fixes x :: "'a ifex" obtains "x = Trueif"   | "x = Falseif"   | v :: "'a" and t :: "'a ifex" and e :: "'a ifex" where "x = IF v t e"
transfer_rule_numeral_uint: shows "rel_fun (=) cr_uint64 numeral numeral"
in_read_only_convs_1: fixes a :: "nat"   and \<S> :: "nat \<Rightarrow> bool option"   and A :: "nat set"   and L :: "nat set" shows "(a \<in> read_only (\<S> \<ominus>\<^bsub>A\<^esub> L)) = (a \<in> read_only \<S> \<and> a \<notin> L \<and> a \<notin> A)"
measure_Int_set_pmf: fixes p :: "'b pmf"   and A :: "'b set" shows "measure_pmf.prob p (A \<inter> set_pmf p) = measure_pmf.prob p A"
isSPath_nt_parallel_pf: fixes s :: "nat"   and p :: "(nat \<times> nat) list"   and t :: "nat" assumes "isSimplePath s p t" shows "set p \<inter> (set p)\<inverse> = {}"
UnionI: fixes X :: "'g set"   and C :: "'g set set"   and A :: "'g" assumes "X \<in> C"   and "A \<in> X" shows "A \<in> \<Union> C"
rel_interior_closure_convex_segment: fixes S :: "'b set"   and a :: "'b"   and b :: "'b" assumes "convex S"   and "a \<in> rel_interior S"   and "b \<in> closure S" shows "open_segment a b \<subseteq> rel_interior S"
special_points_on_y_axis''_3: shows "\<infinity>\<^sub>h \<in> circline_set y_axis"
subadditive: fixes x :: "'a"   and y :: "'a" assumes "x \<in> V"   and "y \<in> V" shows "\<parallel>x + y\<parallel> \<le> \<parallel>x\<parallel> + \<parallel>y\<parallel>"
fresh_19: fixes a :: "name"   and x4 :: "name"   and x5 :: "trm"   and x2 :: "name"   and x3 :: "trm"   and x1 :: "name" shows "a \<sharp> OrL x4.x5 x2.x3 x1 = (a \<sharp> [x4].x5 \<and> a \<sharp> [x2].x3 \<and> a \<sharp> x1)"
weak_least_unique: fixes L :: "('a, 'b) gorder_scheme"   and x :: "'a"   and A :: "'a set"   and y :: "'a" assumes "weak_partial_order L"   and "least L x A"   and "least L y A" shows "x .=\<^bsub>L\<^esub> y"
convergent_Cauchy: fixes "open" :: "'a set \<Rightarrow> bool"   and uniformity :: "('a \<times> 'a) filter"   and X :: "nat \<Rightarrow> 'a" assumes "class.uniform_space open uniformity"   and "topological_space.convergent open X" shows "uniform_space.Cauchy uniformity X"
is_cdi_def: fixes prob :: "('a, 'b, 'c, 'd) ifc_problem"   and i :: "nat"   and \<pi> :: "nat \<Rightarrow> 'a"   and k :: "nat" shows "IFC_def.is_cdi prob i \<pi> k = (IFC_def.is_path prob \<pi> \<and> k < i \<and> \<pi> i \<noteq> IFC_def.return prob \<and> (\<forall>j\<in>{k..i}. \<pi> j \<noteq> IFC_def.ipd prob (\<pi> k)))"
T8_2: fixes w :: "i"   and \<psi> :: "\<Pi>\<^sub>2 \<Rightarrow> \<o>" shows "[\<^bold>\<forall>\<^sub>2x. \<psi> x in w] = (\<forall>x. [\<psi> x in w])"
residual_abs_rep: fixes res :: "('a, 'b) residual" shows "abs_residual (rep_residual res) = res"
Pair_eqD1: fixes x :: "'a"   and y :: "'b"   and x' :: "'a"   and y' :: "'b" assumes "(x, y) = (x', y')" shows "x = x'"
bit_signed_drop_bit_iff: fixes m :: "nat"   and w :: "'b word"   and n :: "nat" shows "bit (signed_drop_bit m w) n = bit w (if LENGTH('b) - m \<le> n \<and> n < LENGTH('b) then LENGTH('b) - 1 else m + n)"
elims: fixes x :: "nat"   and y :: "'a fract poly" assumes "local.F x = y" obtains i :: "nat" where "x = i" and "y = (if i = 0 then 1 else if i = 1 then s.ffp G1 else if i = 2 then s.ffp G2 else let G = pseudo_mod (local.F (i - 2)) (local.F (i - 1)) in if local.F (i - 1) = 0 \<or> G = 0 then 0 else Polynomial.smult (inverse (local.b i)) G)"
map_simp: fixes f :: "'a1" assumes "A1.arr f" shows "A1.map f = f"
r_delay_aux_prim: shows "prim_recfn 3 r_delay_aux"
repeat_Cons_right: fixes n :: "nat"   and xs :: "'a list" shows "repeat (Suc n) xs = repeat n xs @ xs"
card_partition': fixes A :: "'a set"   and R :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and k :: "nat" assumes "\<forall>v\<in>A. card {n. R v n} = k"   and "0 < k"   and "finite A"   and "\<forall>v1 v2. v1 \<noteq> v2 \<longrightarrow> {n. R v1 n} \<inter> {n. R v2 n} = {}" shows "card (\<Union>v\<in>A. {n. R v n}) = k * card A"
vec_to_lin_poly_coeff_access: fixes x :: "nat"   and y :: "rat vec" assumes "x < dim_vec y" shows "y $ x = Abstract_Linear_Poly.coeff (vec_to_lpoly y) x"
\<mu>': shows "\<guillemotleft>\<mu>' : C.cod \<mu> \<Rightarrow>\<^sub>C C.dom \<mu>\<guillemotright> \<and> F \<mu>' = D.inv (F \<mu>)"
local_lipschitz_add: fixes T :: "real set"   and S :: "'b set"   and f1 :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and f2 :: "real \<Rightarrow> 'b \<Rightarrow> 'b" assumes "local_lipschitz T S f1"   and "local_lipschitz T S f2" shows "local_lipschitz T S (\<lambda>t s. f1 t s + f2 t s)"
relpow_Suc_E2: fixes x :: "'b"   and z :: "'b"   and n :: "nat"   and R :: "('b \<times> 'b) set" assumes "(x, z) \<in> R ^^ Suc n" obtains y :: "'b" where "(x, y) \<in> R" and "(y, z) \<in> R ^^ n"
app_def: fixes f :: "V"   and x :: "V" shows "app f x \<equiv> THE y. \<langle>x, y\<rangle> \<in> elts f"
simps: fixes v :: "letter list"   and a :: "letter list list" shows "L v a = ((\<exists>w ws. a = w # ws \<and> emb w v) \<or> (\<exists>ws w. a = w # ws \<and> L v ws))"