hcju/isabelleps huggingface dataset:
hyps_6: fixes V :: "char list" assumes "V \<notin> fv (RI (D,throw a) ; Cs \<leftarrow> e\<^sub>1)" shows "sh\<^sub>1 V = l' V"
size_single: fixes b :: "'a" shows "size {#b#} = 1"
eq2: shows "int (\<Sum>hU\<in>set (ps\<^sub>+). Hilbert_fun (cone hU) d) = Hilbert_poly (\<b> ps) (int d)"
Infinitesimal_hcmod_add_diff: fixes u :: "complex star"   and x :: "complex star" assumes "u \<approx> 0" shows "hcmod (x + u) - hcmod x \<in> Infinitesimal"
A: shows "(\<lambda>x. exp (- x / 2)) integrable_on {c..}"
assms_2: fixes m :: "'a measure" assumes "m \<in> M" shows "space m = space N"
swap_def: fixes n :: "nat"   and xs :: "'b list" shows "swap n xs = (if Suc n < length xs then xs[n := xs ! Suc n, Suc n := xs ! n] else xs)"
if_SE_def: fixes c :: "'d \<Rightarrow> bool"   and E :: "'d \<Rightarrow> ('e \<times> 'd) option"   and F :: "'d \<Rightarrow> ('e \<times> 'd) option" shows "_if_SE c E F = (\<lambda>\<sigma>. if c \<sigma> then E \<sigma> else F \<sigma>)"
restrict_space_eq_vimage_algebra: fixes \<Omega> :: "'b set"   and M :: "'b measure" assumes "\<Omega> \<subseteq> space M" shows "sets (restrict_space M \<Omega>) = sets (vimage_algebra \<Omega> (\<lambda>x. x) M)"
bounded_inner_imp_bdd_below: fixes s :: "'b set"   and a :: "'b" assumes "bounded s" shows "bdd_below ((\<lambda>x. x \<bullet> a) ` s)"
R: fixes S :: "'b literal set set"   and C :: "'b literal set"   and D :: "'b literal set"   and k :: "'b" assumes "S \<turnstile> C"   and "S \<turnstile> D"   and "k\<^sup>+ \<in> C"   and "k\<inverse> \<in> D" shows "S \<turnstile> C - {k\<^sup>+} \<union> (D - {k\<inverse>})"
mem_restr_dropD: fixes b :: "nat"   and xs :: "event_data option list"   and R :: "event_data list set" assumes "b \<le> length xs"   and "mem_restr R (drop b xs)" shows "mem_restr (lift_envs' b R) xs"
preserves_arr: fixes f :: "'c" assumes "C.arr f" shows "D.arr (F f)"
subringE_5: fixes H :: "'c set"   and R :: "('c, 'd) ring_scheme"   and h :: "'c" assumes "subring H R"   and "h \<in> H" shows "\<ominus>\<^bsub>R\<^esub> h \<in> H"
better_ImpR_substc: fixes d :: "coname"   and a :: "coname"   and M :: "trm"   and x :: "name"   and z :: "name"   and P :: "trm" assumes "d \<sharp> [a].M" shows "ImpR x.<a>.M d{d:=z.P} = fresh_fun (\<lambda>a'. Cut <a'>.ImpR x.<a>.M a' z.P)"
eq_is_equal: shows "(.=\<^bsub>L\<^esub>) = (=)"
cycle_free_altI: fixes g :: "'b ugraph" assumes "\<And>u v. \<lbrakk>(u, v) \<in> edges g; (u, v) \<in> (edges g - {(u, v), (v, u)})\<^sup>*\<rbrakk> \<Longrightarrow> False" shows "cycle_free g"
semantics_stateful_packet_tagging_intro_start: fixes \<sigma>\<^sub>0 :: "'a"   and \<sigma>' :: "'a"   and ps_processed :: "('b \<times> final_decision) list"   and \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and packet_tagger :: "'a \<Rightarrow> 'b \<Rightarrow> 'd"   and state_upate :: "'a \<Rightarrow> final_decision \<Rightarrow> 'b \<Rightarrow> 'a"   and built_in_chain :: "char list"   and default_policy :: "action"   and ps :: "'b list" assumes "\<sigma>\<^sub>0 = \<sigma>'"   and "ps_processed = []" shows "semantics_stateful_packet_tagging \<Gamma> \<gamma> packet_tagger state_upate \<sigma>\<^sub>0 (built_in_chain, default_policy) ps ps_processed \<sigma>'"
summable_exp_generic: fixes x :: "'a" shows "summable (\<lambda>n. x ^ n /\<^sub>R fact n)"
induct: fixes x1 :: "com \<times> (char list \<Rightarrow> int)"   and x2 :: "com \<times> (char list \<Rightarrow> int)"   and P :: "com \<times> (char list \<Rightarrow> int) \<Rightarrow> com \<times> (char list \<Rightarrow> int) \<Rightarrow> bool" assumes "x1 \<rightarrow> x2"   and "\<And>x a s. P (x ::= a, s) (SKIP, s(x := aval a s))"   and "\<And>c\<^sub>2 s. P (SKIP;; c\<^sub>2, s) (c\<^sub>2, s)"   and "\<And>c\<^sub>1 s c\<^sub>1' s' c\<^sub>2. \<lbrakk>(c\<^sub>1, s) \<rightarrow> (c\<^sub>1', s'); P (c\<^sub>1, s) (c\<^sub>1', s')\<rbrakk> \<Longrightarrow> P (c\<^sub>1;; c\<^sub>2, s) (c\<^sub>1';; c\<^sub>2, s')"   and "\<And>b s c\<^sub>1 c\<^sub>2. bval b s \<Longrightarrow> P (IF b THEN c\<^sub>1 ELSE c\<^sub>2, s) (c\<^sub>1, s)"   and "\<And>b s c\<^sub>1 c\<^sub>2. \<not> bval b s \<Longrightarrow> P (IF b THEN c\<^sub>1 ELSE c\<^sub>2, s) (c\<^sub>2, s)"   and "\<And>b c s. P (WHILE b DO c, s) (IF b THEN c;; WHILE b DO c ELSE SKIP, s)" shows "P x1 x2"
span_induct: fixes x :: "'b"   and S :: "'b set"   and P :: "'b \<Rightarrow> bool" assumes "x \<in> m1.span S"   and "m1.subspace (Collect P)"   and "\<And>x. x \<in> S \<Longrightarrow> P x" shows "P x"
insert: fixes A :: "'c set"   and x :: "'c"   and g :: "'c \<Rightarrow> 'a" assumes "finite A"   and "x \<notin> A" shows "comm_monoid_add.sum (\<squnion>) bot g (insert x A) = g x \<squnion> comm_monoid_add.sum (\<squnion>) bot g A"
dghm_cn_comp_components_3: fixes \<GG> :: "V"   and \<FF> :: "V" shows "(\<GG> \<^sub>D\<^sub>G\<^sub>H\<^sub>M\<circ> \<FF>)\<lparr>HomDom\<rparr> = op_dg (\<FF>\<lparr>HomDom\<rparr>)"
list_rel_term_poly_list_rel_same_rightD: fixes xa :: "(char list list \<times> int) list"   and y :: "(char list multiset \<times> int) list"   and ya :: "(char list multiset \<times> int) list" assumes "(xa, y) \<in> \<langle>term_poly_list_rel \<times>\<^sub>r int_rel\<rangle>list_rel"   and "(xa, ya) \<in> \<langle>term_poly_list_rel \<times>\<^sub>r int_rel\<rangle>list_rel" shows "y = ya"
True: shows "extended_Gromov_product_at f x y = \<infinity>"
sup_state_opt_def: fixes P :: "'a prog" shows "sup_state_opt P \<equiv> Opt.le (sup_state P)"
RedCondF: fixes e\<^sub>1 :: "char list exp"   and e\<^sub>2 :: "char list exp"   and s :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list" shows "P \<turnstile> \<langle>if (false) e\<^sub>1 else e\<^sub>2,s\<rangle> \<rightarrow> \<langle>e\<^sub>2,s\<rangle>"
simps_2: fixes A :: "'a iarray iarray"   and n :: "nat"   and i :: "nat"   and j :: "nat"   and bezout :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a \<times> 'a \<times> 'a" shows "bezout_iterate_iarrays A (Suc n) i j bezout = (if Suc n \<le> i then A else bezout_iterate_iarrays (bezout_matrix_iarrays A i (Suc n) j bezout **i A) n i j bezout)"
subst_cls_lists_length: fixes Cs :: "'a literal multiset list"   and \<sigma>s :: "'s list" shows "length (Cs \<cdot>\<cdot>cl \<sigma>s) = min (length Cs) (length \<sigma>s)"
that: fixes ms :: "'node list"   and ms'' :: "'node list"   and s'' :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list"   and as' :: "'edge list"   and ms' :: "'node list"   and as'' :: "'edge list" assumes "S,kind \<turnstile> (m # ms,s) =slice_edges S cs as\<Rightarrow>* (ms'',s'')"   and "S,kind \<turnstile> (ms'',s'') =as'\<Rightarrow>\<^sub>\<tau> (m' # ms',s')"   and "ms = targetnodes rs"   and "length ms = length cs"   and "\<forall>i<length cs. call_of_return_node (ms ! i) (sourcenode (cs ! i))"   and "slice_edges S cs as = slice_edges S cs as''"   and "m -as'' @ as'\<rightarrow>* m'"   and "valid_path_aux cs (as'' @ as')" shows "thesis"
simps_2: fixes instr :: "('d, 'c) instr"   and instrs :: "('d, 'c) instr list"   and stack :: "'c list"   and env :: "'d \<Rightarrow> 'c" shows "exec (instr # instrs) stack env = (case instr of Const c \<Rightarrow> exec instrs (c # stack) env | Load x \<Rightarrow> exec instrs (env x # stack) env | Apply f \<Rightarrow> exec instrs (f (hd stack) (hd (tl stack)) # tl (tl stack)) env)"
csol_4: fixes T :: "real set"   and f :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and X :: "'b set"   and t :: "real"   and t0 :: "real"   and x0 :: "'b" assumes "ll_on_open_it T f X"   and "t \<in> ll_on_open.existence_ivl T f X t0 x0" shows "(ll_on_open.csol T f X t0 x0 t solves_ode f) {t0--t} X"
int_shift_bij_2: shows "inv int_left_shift = int_right_shift"
gt_ex: fixes less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and x :: "'e" assumes "class.no_top less_eq less" shows "\<exists>y. less x y"
insert_Diff: fixes a :: "'d"   and A :: "'d set" assumes "a \<in> A" shows "insert a (A - {a}) = A"
a_consistent_stackD: fixes ae :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and a :: "Arity"   and as :: "Arity list"   and \<Gamma> :: "(Vars.var \<times> exp) list"   and e :: "exp"   and S :: "stack_elem list" assumes "a_consistent (ae, a, as) (\<Gamma>, e, S)" shows "Astack S \<sqsubseteq> a"
uniq: fixes \<phi> :: "real" assumes "0 \<le> \<phi>"   and "\<phi> \<le> pi"   and "cos \<phi> = 0" shows "\<phi> = \<theta>"
comp: fixes f :: "'i \<Rightarrow> 'g"   and g :: "'j \<Rightarrow> 'h"   and h :: "'e \<Rightarrow> 'i"   and i :: "'f \<Rightarrow> 'j" shows "map_prod f g \<circ> map_prod h i = map_prod (f \<circ> h) (g \<circ> i)"
sel: fixes callee :: "'e \<Rightarrow> 'a \<Rightarrow> ('b \<times> 'e, 'c, 'd) gpv"   and s :: "'e" shows "run_converter (converter_of_callee callee s) = (\<lambda>a. map_gpv (map_prod id (converter_of_callee callee)) id (callee s a))"
init_fin_simulation_Wakeup1: fixes final1 :: "'a \<Rightarrow> bool"   and r1 :: "('d, 'b, 'a, 'c, 'e, 'f) semantics"   and final2 :: "'g \<Rightarrow> bool"   and r2 :: "('d, 'b, 'g, 'h, 'e, 'f) semantics"   and bisim :: "'b \<Rightarrow> 'a \<times> 'c \<Rightarrow> 'g \<times> 'h \<Rightarrow> bool"   and bisim_wait :: "'a \<Rightarrow> 'g \<Rightarrow> bool"   and \<tau>move1 :: "('d, 'b, 'a, 'c, 'e, 'f) \<tau>moves"   and \<tau>move2 :: "('d, 'b, 'g, 'h, 'e, 'f) \<tau>moves"   and t :: "'b"   and sx1 :: "status \<times> 'a"   and m1 :: "'c"   and sx2 :: "status \<times> 'g"   and m2 :: "'h"   and ta1 :: "('d, 'b, status \<times> 'a, 'c, 'e, 'f action) thread_action"   and sx1' :: "status \<times> 'a"   and m1' :: "'c" assumes "FWdelay_bisimulation_obs final1 r1 final2 r2 bisim bisim_wait \<tau>move1 \<tau>move2"   and "FWbisimulation_base.init_fin_bisim final1 final2 bisim t (sx1, m1) (sx2, m2)"   and "FWbisimulation_base.init_fin_bisim_wait bisim_wait sx1 sx2"   and "multithreaded_base.init_fin final1 r1 t (sx1, m1) ta1 (sx1', m1')"   and "Notified \<in> set \<lbrace>ta1\<rbrace>\<^bsub>w\<^esub> \<or> WokenUp \<in> set \<lbrace>ta1\<rbrace>\<^bsub>w\<^esub>" shows "\<exists>ta2 sx2' m2'. multithreaded_base.init_fin final2 r2 t (sx2, m2) ta2 (sx2', m2') \<and> FWbisimulation_base.init_fin_bisim final1 final2 bisim t (sx1', m1') (sx2', m2') \<and> ta_bisim (FWbisimulation_base.init_fin_bisim final1 final2 bisim) ta1 ta2"
l_known_ptrs_axioms: shows "l_known_ptrs DocumentClass.known_ptr DocumentClass.known_ptrs"
call_return_nodes_in_slice_2: fixes a :: "'edge"   and Q :: "('var \<Rightarrow> 'val option) \<times> 'ret \<Rightarrow> bool"   and p :: "'pname"   and f :: "('var \<Rightarrow> 'val option) \<Rightarrow> ('var \<Rightarrow> 'val option) \<Rightarrow> 'var \<Rightarrow> 'val option"   and a' :: "'edge"   and Q' :: "('var \<Rightarrow> 'val option) \<times> 'ret \<Rightarrow> bool"   and r' :: "'ret"   and fs' :: "(('var \<Rightarrow> 'val option) \<Rightarrow> 'val option) list"   and S :: "'node SDG_node set" assumes "valid_edge a"   and "kind a = Q\<hookleftarrow>\<^bsub>p\<^esub>f"   and "valid_edge a'"   and "kind a' = Q':r'\<hookrightarrow>\<^bsub>p\<^esub>fs'"   and "a \<in> get_return_edges a'"   and "CFG_node (targetnode a) \<in> HRB_slice S" shows "CFG_node (sourcenode a') \<in> HRB_slice S"
UP_def: fixes \<AA> :: "('a \<times> 'a) set"   and x :: "'b" assumes "setcat \<AA>" shows "setcat.UP x \<equiv> concrete_category.MkArr {x} {x} (\<lambda>x\<in>{x}. x)"
inj_on_eq_image: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set"   and X :: "'b set"   and Y :: "'b set" assumes "inj_on f A"   and "order.greater_eq A X"   and "order.greater_eq A Y"   and "f ` X = f ` Y" shows "X = Y"
leg_simps_3: shows "C.arr HfHgh.cod.leg1"
tranclE: fixes a :: "'b"   and b :: "'b"   and r :: "('b \<times> 'b) set" assumes "(a, b) \<in> r\<^sup>+" obtains "(a, b) \<in> r"   | c :: "'b" where "(a, c) \<in> r\<^sup>+" and "(c, b) \<in> r"
pr_auth_key_used: fixes evs :: "event list"   and S :: "agent \<times> nat \<times> nat \<Rightarrow> session"   and A :: "msg set"   and U :: "msg set"   and n :: "nat" assumes "(evs, S, A, U) \<in> protocol" shows "Pri_AgrK (priAK n) \<in> U"
not_final_thread_existsE: fixes s :: "('a, 'b) locks \<times> (('b \<Rightarrow> ('x1 \<times> 'a \<Rightarrow>f nat) option) \<times> 'c) \<times> ('b \<Rightarrow> 'd wait_set_status option) \<times> 'b set"   and t :: "'b" assumes "r1.not_final_thread s t" obtains x :: "'x1" and ln :: "'a \<Rightarrow>f nat" where "thr s t = \<lfloor>(x, ln)\<rfloor>"
deriv_const: fixes a :: "'a"   and b :: "'a" assumes "a \<in> carrier R"   and "b \<in> carrier R" shows "deriv (monom P a 0) b = \<zero>"
rank_invar_hd_cons: fixes bq :: "('c, 'd) SkewBinomialTree list"   and t :: "('c, 'd) SkewBinomialTree" assumes "rank_invar bq"   and "rank t < rank (hd bq)" shows "rank_invar (t # bq)"
sorted_repeat_poly_list_rel_ConsD: fixes ys :: "char list list"   and n :: "int"   and p :: "(char list list \<times> int) list"   and a :: "(char list multiset \<times> int) multiset"   and S :: "char list list \<Rightarrow> char list list \<Rightarrow> bool" assumes "((ys, n) # p, a) \<in> sorted_repeat_poly_list_rel S" shows "(p, remove1_mset (mset ys, n) a) \<in> sorted_repeat_poly_list_rel S \<and> (mset ys, n) \<in># a \<and> (\<forall>x\<in>set p. S ys (fst x)) \<and> sorted_wrt var_order ys \<and> distinct ys \<and> n \<noteq> 0 \<and> nonzero_coeffs a"
wsumSS_1: shows "(\<Sum>t\<in>S. cc t *\<^sub>R t) = x - c a *\<^sub>R a"
plus_adds: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and s :: "'a"   and t :: "'a"   and u :: "'a" assumes "class.comm_powerprod plus minus zero" shows "plus.adds plus (plus s t) u = (plus.adds plus t u \<and> plus.adds plus s (minus u t))"
weakPsiCongParSym: fixes P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi"   and \<Psi> :: "'b"   and R :: "('a, 'b, 'c) psi" assumes "\<forall>\<Psi>. \<Psi> \<rhd> P \<doteq> Q" shows "\<Psi> \<rhd> R \<parallel> P \<doteq> R \<parallel> Q"
emeasure_distr: fixes f :: "'b \<Rightarrow> 'c"   and M :: "'b measure"   and N :: "'c measure"   and A :: "'c set" assumes "f \<in> M \<rightarrow>\<^sub>M N"   and "A \<in> sets N" shows "emeasure (distr M N f) A = emeasure M (f -` A \<inter> space M)"
induced_closed: fixes A :: "'a set" assumes "A \<in> contained.Partition" shows "induced A \<in> G'"
Levellist_length: fixes hds :: "ref list"   and p :: "ref \<Rightarrow> ref"   and ll :: "ref list list" assumes "Levellist hds p ll" shows "length ll = length hds"
assms_6: shows "\<forall>n. \<forall>asset\<in>support_set pf. finite (prices Mkt asset n ` space M)"
indexed_const_index_free: fixes k :: "'a"   and i :: "'c" shows "index_free (indexed_const k) i"
left_idem: fixes a :: "'d upred"   and b :: "'d upred" shows "(a \<and> a \<and> b) = (a \<and> b)"
simps: fixes a1 :: "(att \<times> char list, att, nat) env"   and a2 :: "operation list"   and a3 :: "(att \<times> char list, att, nat) env" shows "a1 \<Midarrow>a2\<Rightarrow> a3 = ((\<exists>root. a1 = root \<and> a2 = [] \<and> a3 = root) \<or> (\<exists>root x root' xs root''. a1 = root \<and> a2 = x # xs \<and> a3 = root'' \<and> root \<midarrow>x\<rightarrow> root' \<and> root' \<Midarrow>xs\<Rightarrow> root''))"
single_code: fixes x :: "'a" shows "single x = Seq (\<lambda>u. Insert x \<bottom>)"
bounded_Union: fixes F :: "'b set set" assumes "finite F"   and "\<forall>S\<in>F. bounded S" shows "bounded (\<Union> F)"
toTree_def: fixes t :: "('c \<times> 'd) list" shows "toTree t \<equiv> Abs_FingerTree (FingerTreeStruc.toTree t)"
max_points_square: fixes ps :: "(int \<times> int) set"   and x :: "real"   and y :: "real"   and \<delta> :: "real" assumes "\<forall>p\<in>ps. (case p of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) \<in> cbox (x, y) (x + \<delta>, y + \<delta>)"   and "sparse \<delta> ps"   and "0 \<le> \<delta>" shows "card ps \<le> 4"
npow_exp_minusTr2: fixes x :: "'a"   and i :: "int"   and j :: "int" assumes "x \<in> carrier K"   and "x \<noteq> \<zero>"   and "0 \<le> i"   and "0 \<le> j"   and "0 \<le> i - j" shows "x\<^bsub>K\<^esub>\<^bsup>(i - j)\<^esup> = x^\<^bsup>K nat i\<^esup> \<cdot>\<^sub>r x\<^bsup>\<hyphen> K\<^esup>^\<^bsup>K nat j\<^esup>"
sum_constant_scale: fixes y :: "'b"   and A :: "'c set" shows "(\<Sum>x\<in>A. y) = of_nat (card A) * y"
Step_Cut: fixes SG :: "(('sf, 'sc, 'sz) formula list \<times> ('sf, 'sc, 'sz) formula list) list"   and C :: "('sf, 'sc, 'sz) formula list \<times> ('sf, 'sc, 'sz) formula list"   and i :: "nat"   and \<phi> :: "('sf, 'sc, 'sz) formula" shows "step_result (SG, C) (i, Cut \<phi>) = (let (A, S) = SG ! i in ((\<phi> # A, S) # (A, \<phi> # S) # closeI SG i, C))"
sym: fixes comp :: "'a \<Rightarrow> 'a \<Rightarrow> order"   and x :: "'a"   and y :: "'a" assumes "comparator comp" shows "invert_order (comp x y) = comp y x"
dim_le_card': fixes s :: "('a, 'b) vec set" assumes "finite s" shows "vec.dim s \<le> card s"
GrpE: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and x :: "'b"   and y :: "'c" assumes "Grp A f x y" obtains "f x = y" and "x \<in> A"
maxsimp_vertices: fixes X :: "'b set set"   and x :: "'b set" assumes "SimplicialComplex X"   and "SimplicialComplex.maxsimp X x" shows "order.greater_eq (\<Union> X) x"
fds_truncate_cong: fixes m :: "nat"   and f :: "'b fds"   and f' :: "'b fds" assumes "\<And>n. \<lbrakk>0 < m; 0 < n; n \<le> m\<rbrakk> \<Longrightarrow> fds_nth f n = fds_nth f' n" shows "fds_truncate m f = fds_truncate m f'"
h3: shows "\<forall>x. insertion (nth_default 0 (xs @ x # \<Gamma>)) (isolate_variable_sparse p var 2) \<noteq> 0 \<or> insertion (nth_default 0 (xs @ x # \<Gamma>)) (isolate_variable_sparse p var 1) \<noteq> 0 \<or> insertion (nth_default 0 (xs @ x # \<Gamma>)) (isolate_variable_sparse p var 0) \<noteq> 0"
sublens_def: fixes X :: "'e \<Longrightarrow> 'f"   and Y :: "'g \<Longrightarrow> 'f" shows "X \<subseteq>\<^sub>L Y = (\<exists>Z. vwb_lens Z \<and> X = Z ;\<^sub>L Y)"
wordinterval_sort: fixes w :: "'a wordinterval" shows "wordinterval_to_set (wordinterval_sort w) = wordinterval_to_set w"
equiv_domain_permutation: fixes A :: "'a set"   and B :: "'b set" shows "equiv (A \<rightarrow>\<^sub>E B) (domain_permutation A B)"
22: shows "prv (imp (PP \<langle>neg \<phi>\<rangle>) \<phi>)"
cong_r_symmetric: fixes F :: "'a set"   and a :: "'a"   and b :: "'a" assumes "F \<in> filters" shows "cong_r F a b = cong_r F b a"
is_glbI: fixes S :: "'a set"   and x :: "'a" assumes "S >| x"   and "\<And>u. S >| u \<Longrightarrow> u \<sqsubseteq> x" shows "S >>| x"
Icc_subset_Iic_iff: fixes l :: "'a multiset"   and h :: "'a multiset"   and h' :: "'a multiset" shows "(subset_mset.atLeastAtMost l h \<subseteq> subset_mset.atMost h') = (\<not> l \<subseteq># h \<or> h \<subseteq># h')"
verified_def: fixes G :: "((Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool) set" shows "verified G = (\<forall>A. A \<in> G \<longrightarrow> G \<rhd> body : A)"
bound: fixes t :: "'a" assumes "t \<in> T" shows "f t \<noteq> l"
section_char: fixes f :: "'s" shows "section f = (arr f \<and> (Dom f = {} \<longrightarrow> Cod f = {}) \<and> inj_on (Fun f) (Dom f))"
Range_Id_on: fixes A :: "'g set" shows "Range (Id_on A) = A"
inner_diff_left: fixes x :: "'b"   and y :: "'b"   and z :: "'b" shows "inner (x - y) z = inner x z - inner y z"
PMS_lemma: fixes PMS :: "nat"   and evs :: "event list"   and K :: "nat"   and Na :: "nat"   and Nb :: "nat"   and NA :: "nat"   and NB :: "nat"   and role :: "role" assumes "Nonce PMS \<notin> parts (knows Spy evs)"   and "K = sessionK ((Na, Nb, PRF (PMS, NA, NB)), role)"   and "evs \<in> tls" shows "Key K \<notin> parts (knows Spy evs) \<and> (\<forall>Y. Crypt K Y \<notin> parts (knows Spy evs))"
inv_init_imp_prop_safe_def: fixes summary :: "'a \<Rightarrow> 'a \<Rightarrow> 'b antichain"   and results_in :: "'c \<Rightarrow> 'b \<Rightarrow> 'c"   and c :: "('d, 'c, 'a, 'e) Combined.configuration_scheme" assumes "dataflow_topology summary results_in" shows "dataflow_topology.inv_init_imp_prop_safe summary results_in c = (\<forall>p. init c p \<longrightarrow> dataflow_topology.impl_safe summary results_in (prop_config c p) \<and> dataflow_topology.safe summary results_in (prop_config c p))"
acquire: fixes s :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and t :: "'t"   and x :: "'x"   and ln :: "'l \<Rightarrow>f nat" assumes "thr s t = \<lfloor>(x, ln)\<rfloor>"   and "ln \<noteq> no_wait_locks"   and "\<not> waiting (wset s t)"   and "may_acquire_all (locks s) t ln" shows "t \<in> active_threads s"
case_3: fixes f1 :: "'a"   and f2 :: "'a"   and f3 :: "'a"   and f4 :: "'a" shows "(case C of A \<Rightarrow> f1 | B \<Rightarrow> f2 | C \<Rightarrow> f3 | D \<Rightarrow> f4) = f3"
smod_int_compares_2: fixes a :: "int"   and b :: "int" assumes "0 \<le> a"   and "0 < b" shows "0 \<le> a smod b"
ordst_ordstNS_trans: fixes X :: "char list"   and Y :: "char list"   and Z :: "char list" assumes "ordst X Y"   and "ordstNS Y Z" shows "ordst X Z"
k_incr: fixes k :: "int" shows "f (of_int (k + 1)) - g (of_int (k + 1)) = f (of_int k) - g (of_int k)"
Span_smult_closed: fixes R :: "('a, 'b) ring_scheme"   and K :: "'a set"   and Us :: "'a list"   and k :: "'a"   and v :: "'a" assumes "ring R"   and "subfield K R"   and "set Us \<subseteq> carrier R"   and "k \<in> K"   and "v \<in> ring.Span R K Us" shows "k \<otimes>\<^bsub>R\<^esub> v \<in> ring.Span R K Us"
elemins: shows "z \<in> set (fst (TSdet init h (as @ [x] @ bs @ [x] @ cs) (length (as @ [x] @ bs @ [x]) + i)))"
nospoof: fixes rs :: "'b common_primitive rule list"   and ipassmt :: "iface \<Rightarrow> ('b word \<times> nat) list option" assumes "simple_ruleset rs" shows "(\<forall>iface\<in>dom ipassmt. nospoof TYPE('c) iface ipassmt rs) = no_spoofing TYPE('c) ipassmt rs"
positive_part_def: fixes C :: "'at Literal set" shows "positive_part C = {L. (\<exists>A. L = Pos A) \<and> L \<in> C}"
simps_6: fixes e\<^sub>1 :: "('f, 'g, 'h) exp"   and bop :: "bop"   and e\<^sub>2 :: "('f, 'g, 'h) exp"   and A :: "'f set option" shows "\<D> (e\<^sub>1 \<guillemotleft>bop\<guillemotright> e\<^sub>2) A = (\<D> e\<^sub>1 A \<and> \<D> e\<^sub>2 (A \<squnion> \<A> e\<^sub>1))"
zero_eq_0: fixes n :: "nat"   and \<chi> :: "nat \<Rightarrow> complex" assumes "dcharacter n \<chi>" shows "\<chi> 0 = 0"
replicate_0: fixes x :: "'a" shows "replicate 0 x = []"
is_of_type_def: fixes ty_ent :: "'a \<Rightarrow> PDDL_STRIPS_Semantics.type option"   and v :: "'a"   and T :: "PDDL_STRIPS_Semantics.type" shows "is_of_type ty_ent v T = (case ty_ent v of None \<Rightarrow> False | Some vT \<Rightarrow> of_type vT T)"
then_: shows "eval (Cn 1 r_ifeq [Cn 1 r_sigma [Cn 1 r_hd [r_pdec1], Cn 1 r_update [r_pdec1, r_pdec2, r_const 1]], Cn 1 r_sigma [Cn 1 r_hd [r_pdec1], Cn 1 r_take [r_pdec2, r_pdec1]], Cn 1 r_prod_encode [Cn 1 r_snoc [r_pdec1, Z], r_pdec2], Cn 1 r_prod_encode [Cn 1 r_snoc [Cn 1 r_update [r_pdec1, r_pdec2, r_const 1], Z], Cn 1 r_length [r_pdec1]]]) [arg] \<down>= (if the (\<sigma> (e_hd b) (e_update b k 1)) = the (\<sigma> (e_hd b) (e_take k b)) then prod_encode (e_snoc b 0, k) else prod_encode (e_snoc (e_update b k 1) 0, e_length b))"
index_zero_mat_3: fixes nr :: "nat"   and nc :: "nat" shows "dim_col (0\<^sub>m nr nc) = nc"
assms_4: shows "eint 2 * val_Zp ([n] \<cdot> \<one>) < val_Zp (\<one> \<ominus> a)"
simps_22: fixes e1 :: "'b aexp"   and e2 :: "'b aexp"   and v :: "value" shows "AExp_Lexorder.less_aexp_aux (Times e1 e2) (L v) = False"
im_bottom: assumes "domain.has_bottom P" shows "f dombot = codbot"
pr_passwd_secrecy: fixes evs :: "event list"   and S :: "agent \<times> nat \<times> nat \<Rightarrow> session"   and A :: "msg set"   and U :: "msg set"   and m :: "nat" assumes "(evs, S, A, U) \<in> protocol"   and "User m \<noteq> Spy" shows "Passwd m \<notin> analz (A \<union> spies evs)"
complex_cnj_complex_of_real: fixes x :: "real" shows "cnj (complex_of_real x) = complex_of_real x"