hcju/isabelleps huggingface dataset:
M_to_int_mod_ring: fixes m :: "int"   and x :: "'a mod_ring" assumes "poly_mod_type TYPE('a) m" shows "poly_mod.M m (to_int_mod_ring x) = to_int_mod_ring x"
HSigmaD2: fixes a :: "hf"   and b :: "hf"   and A :: "hf"   and B :: "hf \<Rightarrow> hf" assumes "\<langle>a, b\<rangle> \<^bold>\<in> HSigma A B" shows "b \<^bold>\<in> B a"
UNIV_I: fixes x :: "'n" shows "x \<in> UNIV"
greatest_Lower_below: fixes L :: "('a, 'b) gorder_scheme"   and i :: "'a"   and A :: "'a set"   and x :: "'a" assumes "is_glb L i A"   and "x \<in> A"   and "A \<subseteq> carrier L" shows "i \<sqsubseteq>\<^bsub>L\<^esub> x"
match_cong: fixes r :: "'b regex"   and r' :: "'b regex"   and t :: "nat \<Rightarrow> 'b \<Rightarrow> bool"   and t' :: "nat \<Rightarrow> 'b \<Rightarrow> bool" assumes "r = r'"   and "\<And>i z. z \<in> atms r \<Longrightarrow> t i z = t' i z" shows "match t r = match t' r'"
mono_extend_get_offending_flows: fixes M :: "'a SecurityInvariant_configured list"   and V :: "'a set"   and E :: "('a \<times> 'a) set"   and E' :: "('a \<times> 'a) set"   and F' :: "('a \<times> 'a) set" assumes "valid_reqs M"   and "wf_graph \<lparr>nodes = V, edges = E\<rparr>"   and "E' \<subseteq> E"   and "F' \<in> get_offending_flows M \<lparr>nodes = V, edges = E'\<rparr>" shows "\<exists>F\<in>get_offending_flows M \<lparr>nodes = V, edges = E\<rparr>. F' \<subseteq> F"
dvd_div_eq_mult: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and a :: "'c"   and b :: "'c"   and c :: "'c" assumes "class.algebraic_semidom divide plus minus zero times one"   and "a \<noteq> zero"   and "dvd.dvd times a b" shows "(divide b a = c) = (b = times c a)"
fun_relD: fixes f :: "'a \<Rightarrow> 'b"   and f' :: "'c \<Rightarrow> 'f"   and A :: "('a \<times> 'c) set"   and B :: "('b \<times> 'f) set"   and x :: "'a"   and x' :: "'c" assumes "(f, f') \<in> A \<rightarrow> B"   and "(x, x') \<in> A" shows "(f x, f' x') \<in> B"
is_aligned_iff_dvd_nat: fixes ptr :: "'a word"   and n :: "nat" shows "is_aligned ptr n = (2 ^ n dvd unat ptr)"
Embed_def: fixes t :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" shows "Embed t = (\<lambda>ab. t)"
solve: shows "solve_exec_ns cs \<equiv> case ui of [] \<Rightarrow> case assert_all t as of Inl x \<Rightarrow> Inl x | Inr v \<Rightarrow> Inr (trans_v v) | i # x \<Rightarrow> Inl [i]"
assms_2: shows "(i, Store e mo i) \<in> set (node_deliver_messages xs)"
npQring: fixes R :: "('c, 'd) Ring_scheme"   and I :: "'c set"   and a :: "'c"   and n :: "nat" assumes "Ring R"   and "ideal R I"   and "a \<in> carrier R" shows "(a \<uplus>\<^bsub>R\<^esub> I)^\<^bsup>R /\<^sub>r I n\<^esup> = a^\<^bsup>R n\<^esup> \<uplus>\<^bsub>R\<^esub> I"
ts_context: fixes freshenLC :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and renameLCs :: "('a \<Rightarrow> 'a) \<Rightarrow> 'b \<Rightarrow> 'b"   and lconsts :: "'b \<Rightarrow> 'a set"   and closed :: "'b \<Rightarrow> bool"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and subst_lconsts :: "'c \<Rightarrow> 'a set"   and subst_renameLCs :: "('a \<Rightarrow> 'a) \<Rightarrow> 'c \<Rightarrow> 'c"   and anyP :: "'b"   and antecedent :: "'d \<Rightarrow> ('b, 'a) antecedent list"   and consequent :: "'d \<Rightarrow> 'b list"   and rules :: "'d stream"   and assumptions :: "'b list"   and conclusions :: "'b list"   and c :: "'b" assumes "Solved_Task freshenLC renameLCs lconsts closed subst subst_lconsts subst_renameLCs anyP antecedent consequent rules assumptions conclusions"   and "c |\<in>| Abstract_Task.conc_forms conclusions" shows "fst (fst (tree.root (Solved_Task.ts freshenLC renameLCs lconsts subst subst_lconsts antecedent consequent rules assumptions c))) |\<subseteq>| Abstract_Task.ass_forms assumptions"
elims_2: fixes x :: "Core.term" assumes "is_app x" obtains uu_ :: "Core.term" and uv_ :: "Core.term" where "x = uu_ $ uv_"
union_left_cancel: fixes K :: "'b multiset"   and M :: "'b multiset"   and N :: "'b multiset" shows "(K + M = K + N) = (M = N)"
OK_def: fixes I :: "'d set"   and F :: "'d \<Rightarrow> 'e program" shows "OK I F = (\<forall>i\<in>I. \<forall>j\<in>I - {i}. Acts (F i) \<subseteq> AllowedActs (F j))"
prod_list_map_hom: fixes xs :: "real poly list" shows "prod_list (map (map_poly complex_of_real) xs) = map_poly complex_of_real (prod_list xs)"
assms_1: shows "filterlim f G (at x within A \<inter> {x. P x})"
eqvt: fixes p :: "perm"   and x :: "'a"   and y :: "'b" shows "p \<bullet> f x y = f (p \<bullet> x) (p \<bullet> y)"
omega_continuous_imp_mono_refl: fixes A :: "'b set"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b" assumes "omega_continuous A less_eq f"   and "x \<in> A"   and "y \<in> A"   and "less_eq x y"   and "less_eq x x"   and "less_eq y y" shows "less_eq (f x) (f y)"
simps_1: fixes f :: "'b \<rightarrow> 'a" shows "smap\<cdot>f\<cdot>[::] = [::]"
monom_zero: fixes a :: "'a" assumes "a \<in> carrier R" shows "Q (monom P a 0)"
var_elist_inj_term: fixes t :: "Term.var"   and w :: "expr list" shows "\<langle>t\<rangle> \<noteq> \<langle>w\<rangle>"
rel_refl_strong: fixes x :: "'b tree"   and Ra :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "\<And>z. z \<in> set_tree x \<Longrightarrow> Ra z z" shows "rel_tree Ra x x"
exhaust: fixes y :: "'a vertex" obtains "y = SOURCE"   | "y = vertex.SINK"   | x3 :: "'a" where "y = \<langle>x3\<rangle>"
filterlim_at_infinity_imp_eventually_ne: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and c :: "'b" assumes "filterlim f at_infinity F" shows "\<forall>\<^sub>F z in F. f z \<noteq> c"
Re_1: shows "(Re (Arccos w))\<^sup>2 \<le> pi\<^sup>2"
cball_empty: fixes e :: "real"   and x :: "'a" assumes "e < 0" shows "cball x e = {}"
intro: fixes ext :: "('b \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'b list \<Rightarrow> 'b list \<Rightarrow> bool" assumes "\<And>gt ys xs x. ext gt ys xs \<Longrightarrow> ext gt (ys @ [x]) (xs @ [x])" shows "ext_compat_snoc_axioms ext"
intros_9: fixes F :: "'b formula" shows "\<^bold>\<not> F \<^bold>\<rightarrow> F \<^bold>\<rightarrow> \<bottom> \<in> AX10"
contk: shows "continuous_on ({0..1} \<times> {undefined}) k"
emb_part: fixes s :: "char list \<Rightarrow> int"   and q :: "char list \<Rightarrow> int" shows "emb (part s) q = s"
vsv_vid_on: fixes A :: "V" shows "vsv (vid_on A)"
aux: fixes s :: "real" shows "ode (flow0 x0 s) = ode (snd (s, flow0 x0 s))"
Conj_eqvt: fixes xset :: "('a, 'b, 'c, 'd) formula set['a]"   and p :: "perm" assumes "finite (supp xset)" shows "p \<bullet> Conj xset = Conj (p \<bullet> xset)"
closed_w_def: fixes p :: "'b list" shows "closed_w p \<equiv> \<exists>u. awalk u p u \<and> 0 < length p"
all_pairs_list_product: fixes xs :: "'b list" shows "all_pairs xs = List.product xs xs"
IH: fixes X :: "'a set"   and x :: "'a" assumes "bounded_WHILE n cond step m0 = RES X"   and "x \<in> X" shows "rwof m0 cond step x"
part_def: fixes p :: "'a"   and xs :: "'a list" shows "part p xs = part_aux (f p) xs ([], [p], [])"
analz_LtKeys_Tag: fixes NI :: "msg set"   and K :: "msg set" assumes "NI \<subseteq> payload"   and "K \<subseteq> range LtK" shows "analz (NI \<union> K \<union> Tags) \<subseteq> analz NI \<union> K \<union> Tags"
inf_option_def: fixes x :: "'a option"   and y :: "'a option" shows "x \<sqinter> y = (case x of None \<Rightarrow> None | Some x \<Rightarrow> case y of None \<Rightarrow> None | Some y \<Rightarrow> Some (x \<sqinter> y))"
nearly_healthy_wlp_DC: fixes f :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and g :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" assumes "nearly_healthy (wlp f)"   and "nearly_healthy (wlp g)" shows "nearly_healthy (wlp (f \<Sqinter> g))"
RA2_analz_spies: fixes A :: "agent"   and B :: "agent"   and X :: "msg"   and evs :: "event list" assumes "Says A B X \<in> set evs" shows "X \<in> analz (knows Spy evs)"
in_set_impl_in_set_zip1: fixes xs :: "'b list"   and ys :: "'c list"   and x :: "'b" assumes "length xs = length ys"   and "x \<in> set xs" obtains y :: "'c" where "(x, y) \<in> set (zip xs ys)"
ltake_Suc_conv_snoc_lnth: fixes m :: "nat"   and xs :: "'b llist" assumes "enat m < llength xs" shows "ltake (enat (Suc m)) xs = lappend (ltake (enat m) xs) (LCons (xs $ m) LNil)"
assms_1: shows "ys' \<equiv> map (\<lambda>(x, n). x ^ Suc n) ys"
map_simp_ide: fixes a :: "'a" assumes "\<tau>.A.ide a" shows "\<tau>'\<tau>.map a = B (\<tau> a) (\<tau>'.map a)"
mlex_prod_def: fixes f :: "'g \<Rightarrow> nat"   and R :: "('g \<times> 'g) set" shows "f <*mlex*> R = inv_image (less_than <*lex*> R) (\<lambda>x. (f x, x))"
nodes_list_it_correct: fixes g :: "'g" assumes "invar g" shows "set_iterator (foldri (\<alpha>n g)) (nodes (\<alpha> g))"
vrrestriction_def: fixes D :: "V" shows "vrrestriction D = (\<lambda>\<langle>r, A\<rangle>\<in>\<^sub>\<circ>D. set {\<langle>a, b\<rangle> |a b. b \<in>\<^sub>\<circ> A \<and> \<langle>a, b\<rangle> \<in>\<^sub>\<circ> r})"
Roots_nonempty: fixes p :: "int poly" assumes "p \<in> P" shows "Roots p \<noteq> {}"
basis_enum_of_vec_add: fixes v1 :: "complex vec"   and v2 :: "complex vec" assumes "dim_vec v1 = length canonical_basis"   and "dim_vec v2 = length canonical_basis" shows "basis_enum_of_vec (v1 + v2) = basis_enum_of_vec v1 + basis_enum_of_vec v2"
ring_homomorphism_\<eta>_cong: fixes \<eta> :: "'d \<Rightarrow> 'e"   and R :: "'d set"   and addition :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and multiplication :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and unit :: "'d"   and R' :: "'e set"   and addition' :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and multiplication' :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero' :: "'e"   and unit' :: "'e"   and \<eta>' :: "'d \<Rightarrow> 'e" assumes "ring_homomorphism \<eta> R addition multiplication zero unit R' addition' multiplication' zero' unit'"   and "\<And>x. x \<in> R \<Longrightarrow> \<eta>' x = \<eta> x"   and "\<eta>' \<in> extensional R" shows "ring_homomorphism \<eta>' R addition multiplication zero unit R' addition' multiplication' zero' unit'"
setUp_program_inv: fixes ast :: "AST.module list"   and prog :: "program"   and g :: "gState" assumes "setUp ast = (prog, g)" shows "program_inv prog"
atrm_suc: fixes t :: "'trm" assumes "t \<in> atrm" shows "suc t \<in> atrm"
case_map_generat: fixes io :: "'e \<Rightarrow> 'f \<Rightarrow> 'b"   and a :: "'g \<Rightarrow> 'd"   and b :: "'h \<Rightarrow> 'e"   and d :: "'i \<Rightarrow> 'f"   and r :: "('g, 'h, 'i) generat" shows "(case map_generat a b d r of Pure x \<Rightarrow> pure x | IO x xa \<Rightarrow> io x xa) = (case r of Pure x \<Rightarrow> (pure \<circ> a) x | IO out x \<Rightarrow> (io (b out) \<circ> d) x)"
Arg_times_of_real: fixes r :: "real"   and z :: "complex" assumes "0 < r" shows "Arg (complex_of_real r * z) = Arg z"
join_None2: fixes x :: "'a option" shows "x \<squnion> None = x"
qmpt_factor_iterates: fixes proj :: "(int \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> real) \<Rightarrow> 'b"   and M2 :: "'b measure"   and T2 :: "'b \<Rightarrow> 'b" assumes "I.qmpt_factor proj M2 T2" shows "AE x in MI. \<forall>n. proj ((TI ^^ n) x) = (T2 ^^ n) (proj x)"
prodag_comp_i: fixes a :: "'d \<Rightarrow> 'e"   and I :: "'d set"   and A :: "'d \<Rightarrow> ('e, 'f) aGroup_scheme"   and i :: "'d" assumes "a \<in> carr_prodag I A"   and "i \<in> I" shows "a i \<in> carrier (A i)"
star_n_inverse: fixes X :: "nat \<Rightarrow> 'c" shows "inverse (star_n X) = star_n (\<lambda>n. inverse (X n))"
minus_unique: fixes y :: "nat \<Rightarrow> int"   and x :: "nat \<Rightarrow> int"   and y' :: "nat \<Rightarrow> int" assumes "y \<oplus> x = \<zero>"   and "x \<oplus> y' = \<zero>"   and "x \<in> carrier Zp"   and "y \<in> carrier Zp"   and "y' \<in> carrier Zp" shows "y = y'"
imageE: fixes b :: "'g"   and f :: "'h \<Rightarrow> 'g"   and A :: "'h set" assumes "b \<in> f ` A" obtains x :: "'h" where "b = f x" and "x \<in> A"
hm_update_rule': fixes m :: "'a \<Rightarrow> 'b option"   and l :: "('a \<times> 'b) list list"   and ht :: "('a, 'b) hashtable"   and k :: "'a"   and v :: "'b" shows "<is_hashmap' m l ht> hm_update k v ht <is_hashmap (m(k \<mapsto> v))>\<^sub>t"
B_trusts_YM4_shrK: fixes B :: "agent"   and A :: "agent"   and K :: "nat"   and evs :: "event list" assumes "Crypt (shrK B) \<lbrace>Agent A, Key K\<rbrace> \<in> parts (knows Spy evs)"   and "B \<notin> bad"   and "evs \<in> yahalom" shows "\<exists>NA NB. Says Server A \<lbrace>Crypt (shrK A) \<lbrace>Agent B, Key K, Nonce NA, Nonce NB\<rbrace>, Crypt (shrK B) \<lbrace>Agent A, Key K\<rbrace>\<rbrace> \<in> set evs"
right_neutral: fixes a :: "'m" shows "a + (0::'m) = a"
times_fds_def: fixes f :: "'b fds"   and g :: "'b fds" shows "f * g = fds (dirichlet_prod (fds_nth f) (fds_nth g))"
out_subseqs1_set_def: fixes head :: "'a"   and tail :: "'a iarray"   and prevs :: "(nat \<times> 'b) list" shows "out_subseqs1_set head tail prevs \<equiv> (f head \<circ> snd) ` set prevs \<union> \<Union> {out_subseqs2_set tail v [0..<i] |v i. (i, v) \<in> set prevs}"
l_x: shows "s\<turnstile> l reachable_from x"
multiple_block_in: fixes n :: "nat"   and b :: "'a set" assumes "0 < n"   and "b \<in># \<B>" shows "b \<in># multiple_blocks n"
Plus: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and z :: "'a"   and A :: "'b set"   and B :: "'c set"   and g :: "'b + 'c \<Rightarrow> 'a" assumes "comm_monoid_set f z"   and "finite A"   and "finite B" shows "comm_monoid_set.F f z g (A <+> B) = f (comm_monoid_set.F f z (g \<circ> Inl) A) (comm_monoid_set.F f z (g \<circ> Inr) B)"
commute: fixes a :: "'c \<times> 'd"   and b :: "'c \<times> 'd" shows "inf_prod a b = inf_prod b a"
space_closed: shows "sigma_sets \<Omega> M \<subseteq> Pow \<Omega>"
divide_right_mono: fixes divide :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and inverse :: "'d \<Rightarrow> 'd"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sgn :: "'d \<Rightarrow> 'd"   and abs :: "'d \<Rightarrow> 'd"   and a :: "'d"   and b :: "'d"   and c :: "'d" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less_eq a b"   and "less_eq zero c" shows "less_eq (divide a c) (divide b c)"
elims_2: fixes x :: "'b extT"   and xa :: "'b extT" assumes "x < xa" obtains x :: "'b extT" and y :: "'b extT" where "xa__ = x" and "xa = y" and "x \<le> y \<and> \<not> y \<le> x"
in_set_simps_2: fixes x :: "'a"   and y :: "'a" shows "(x \<in> set [y]) = (x = y)"
fully_abstract_wrt_equivalences_impl_symcl_source_target_relation_is_preorder: fixes SRel :: "('procS \<times> 'procS) set"   and TRel :: "('procT \<times> 'procT) set" assumes "\<forall>S1 S2. ((S1, S2) \<in> SRel) = ((\<lbrakk>S1\<rbrakk>, \<lbrakk>S2\<rbrakk>) \<in> TRel)"   and "refl TRel"   and "sym TRel"   and "trans TRel" shows "\<exists>Rel. (\<forall>S. (SourceTerm S, TargetTerm (\<lbrakk>S\<rbrakk>)) \<in> Rel) \<and> SRel = _Collect (S1, S2) ((SourceTerm S1, SourceTerm S2) \<in> symcl Rel) \<and> TRel = _Collect (T1, T2) ((TargetTerm T1, TargetTerm T2) \<in> symcl Rel) \<and> preorder (symcl Rel)"
equals_rsp: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and Abs :: "'a \<Rightarrow> 'b"   and Rep :: "'b \<Rightarrow> 'a"   and xa :: "'a"   and xb :: "'a"   and ya :: "'a"   and yb :: "'a" assumes "Quotient3 R Abs Rep"   and "R xa xb"   and "R ya yb" shows "R xa ya = R xb yb"
small_vrat_le: shows "small {[a, b]\<^sub>\<circ> |a b. [a, b]\<^sub>\<circ> \<in>\<^sub>\<circ> \<rat>\<^sub>\<circ> ^\<^sub>\<times> 2\<^sub>\<nat> \<and> rat_of_vrat a \<le> rat_of_vrat b}"
alpha_Tree_valid_Tree: fixes satisfies :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and transition :: "'b \<Rightarrow> ('d, 'b) residual \<Rightarrow> bool"   and t1 :: "('e, 'c, 'd) Tree"   and t2 :: "('e, 'c, 'd) Tree"   and P :: "'b" assumes "nominal_ts satisfies transition"   and "t1 =\<^sub>\<alpha> t2" shows "nominal_ts.valid_Tree satisfies transition P t1 = nominal_ts.valid_Tree satisfies transition P t2"
gt_sym_wf: shows "wfP (\<lambda>f g. g >\<^sub>s f)"
galois_aux3: fixes x :: "'a"   and y :: "'a" shows "(x + y = 1) = (- x \<le> y)"
subset_insertI2: fixes A :: "'a set"   and B :: "'a set"   and b :: "'a" assumes "A \<subseteq> B" shows "A \<subseteq> insert b B"
R'_def: shows "R' = (\<Sum>(p, i)\<in>{(p, i). prime p \<and> 1 < i \<and> p ^ i \<le> n}. ln (real p) / real (p ^ i))"
A_1: shows "set \<phi>2s' \<subseteq> set \<phi>1s"
is_dense_def: fixes T :: "'b set set"   and A :: "'b set" shows "(A dense\<^bsub>T\<^esub>) = (A densein\<^bsub>T\<^esub> carrier\<^bsub>T\<^esub>)"
zero_reorient: fixes x :: "'a" shows "((0::'a) = x) = (x = (0::'a))"
NotDSource_NotSource: fixes i :: "AbstrLevelsID"   and S :: "CSet" assumes "\<forall>C\<in>AbstrLevel i. S \<notin> DSources i C" shows "\<forall>Z\<in>AbstrLevel i. S \<notin> Sources i Z"
assms: shows "externals (ioa.asig A) = externals (ioa.asig B)"
sup_idemp: fixes a :: "'a" shows "a \<squnion> a = a"
dom_lhs: shows "\<D>\<^sub>\<circ> \<FF> = 4\<^sub>\<nat>"
is_processT: fixes P :: "'c process" shows "([], {}) \<in> F P \<and> (\<forall>s X. (s, X) \<in> F P \<longrightarrow> front_tickFree s) \<and> (\<forall>s t. (s @ t, {}) \<in> F P \<longrightarrow> (s, {}) \<in> F P) \<and> (\<forall>s X Y. (s, Y) \<in> F P \<and> X \<subseteq> Y \<longrightarrow> (s, X) \<in> F P) \<and> (\<forall>s X Y. (s, X) \<in> F P \<and> (\<forall>c. c \<in> Y \<longrightarrow> (s @ [c], {}) \<notin> F P) \<longrightarrow> (s, X \<union> Y) \<in> F P) \<and> (\<forall>s X. (s @ [tick], {}) \<in> F P \<longrightarrow> (s, X - {tick}) \<in> F P) \<and> (\<forall>s t. s \<in> D P \<and> tickFree s \<and> front_tickFree t \<longrightarrow> s @ t \<in> D P) \<and> (\<forall>s X. s \<in> D P \<longrightarrow> (s, X) \<in> F P) \<and> (\<forall>s. s @ [tick] \<in> D P \<longrightarrow> s \<in> D P)"
wtl_Nil: fixes A :: "'c set"   and r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and T :: "'c"   and B :: "'c"   and c :: "'c list"   and step :: "nat \<Rightarrow> 'c \<Rightarrow> (nat \<times> 'c) list"   and pc :: "nat"   and s :: "'c" assumes "lbv A r f T B" shows "wtl_inst_list [] c f r T B step pc s = s"
case_nat_update_0: fixes x :: "'a"   and \<sigma> :: "nat \<Rightarrow> 'a"   and y :: "'a" shows "(case_nat x \<sigma>)(0 := y) = case_nat y \<sigma>"
wtTE_intT: fixes \<xi> :: "Preliminaries.var \<Rightarrow> univ"   and x :: "Preliminaries.var" assumes "I.wtE \<xi>" shows "I_intT (tpOfV x) (\<xi> x)"
sel: fixes x2 :: "'c" shows "the \<lfloor>x2\<rfloor> = x2"
NoMessageLoss: fixes trans :: "'a \<Rightarrow> 'b \<Rightarrow> 'c messageValue \<Rightarrow> 'b"   and sends :: "'a \<Rightarrow> 'b \<Rightarrow> 'c messageValue \<Rightarrow> ('a, 'c) message \<Rightarrow> nat"   and cfg1 :: "('a, 'c, 'b) configuration"   and Q :: "'a set"   and cfg2 :: "('a, 'c, 'b) configuration"   and p :: "'a"   and m' :: "('a, 'c) message" assumes "asynchronousSystem.qReachable trans sends cfg1 (UNIV - Q) cfg2"   and "p \<in> Q"   and "isReceiverOf p m'" shows "msgs cfg1 m' \<le> msgs cfg2 m'"
greaterThanAtMost_def: fixes less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and l :: "'e"   and u :: "'e" shows "ord.greaterThanAtMost less_eq less l u \<equiv> ord.greaterThan less l \<inter> ord.atMost less_eq u"
half_negative_int_iff: fixes k :: "int" shows "(k div 2 < 0) = (k < 0)"
heapToAssn_eqvt: fixes p :: "perm"   and \<Gamma> :: "(Vars.var \<times> exp) list" shows "p \<bullet> heapToAssn \<Gamma> = heapToAssn (p \<bullet> \<Gamma>)"
monog: fixes x :: "real"   and y :: "real" assumes "a \<le> x"   and "x \<le> y"   and "y \<le> b" shows "g x \<le> g y"
single_valuedI: fixes r :: "('a \<times> 'b) set" assumes "\<And>x y z. \<lbrakk>(x, y) \<in> r; (x, z) \<in> r\<rbrakk> \<Longrightarrow> y = z" shows "single_valued r"
trg_trg: fixes \<mu> :: "'a" assumes "C.arr \<mu>" shows "trg\<^sub>C (trg\<^sub>C \<mu>) = trg\<^sub>C \<mu>"
finprod_closed: fixes f :: "'c \<Rightarrow> 'a"   and A :: "'c set" assumes "f \<in> A \<rightarrow> carrier G" shows "finprod G f A \<in> carrier G"
circline_eq_cmat_hermitean_nonzero: fixes H :: "complex \<times> complex \<times> complex \<times> complex"   and H' :: "complex \<times> complex \<times> complex \<times> complex" assumes "hermitean H \<and> H \<noteq> mat_zero"   and "circline_eq_cmat H H'" shows "hermitean H' \<and> H' \<noteq> mat_zero"
reachable_weaken_TT: fixes s :: "'a"   and A :: "('a, 'b) automaton"   and I :: "'b \<Rightarrow> bool" assumes "s \<in> reachable A I" shows "s \<in> reachable A TT"
ccHeap_simp1: fixes \<Gamma> :: "(Vars.var \<times> exp) list"   and e :: "exp"   and a :: "Arity" assumes "\<not> nonrec \<Gamma>" shows "ccHeap \<Gamma> e\<cdot>a = CCfix \<Gamma>\<cdot>(Afix \<Gamma>\<cdot>(Aexp_syn' a e \<squnion> (\<lambda>_. up\<cdot>0) f|` thunks \<Gamma>), ccExp_syn a e)"
RCALL_def: fixes D :: "'j" shows "RCALL D \<equiv> D"
path_unique: shows "P \<turnstile> Path last Cs to C unique"
val_of_nonzero_3: fixes f :: "nat \<Rightarrow> int"   and p :: "int" assumes "f \<in> padic_set p"   and "f \<noteq> padic_zero p"   and "prime p" shows "f (nat (padic_val p f) + 1) \<noteq> 0"
Par_perm: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "A B Par C D" shows "A B Par C D \<and> B A Par C D \<and> A B Par D C \<and> B A Par D C \<and> C D Par A B \<and> C D Par B A \<and> D C Par A B \<and> D C Par B A"
assms_2: fixes k :: "nat" assumes "k \<in> {nat \<lfloor>a\<rfloor><..nat \<lfloor>b\<rfloor>}" shows "\<not> prime k"
conjugate_char_1_pp: fixes f :: "'a \<Rightarrow> 'a"   and g :: "'a \<Rightarrow> 'a" shows "bounded_lattice.conjugate (\<squnion>) top f g = (\<forall>x y. - - f x \<squnion> - y \<le> f (x \<squnion> - g y) \<and> - - g y \<squnion> - x \<le> g (y \<squnion> - f x))"
assm_bound_funct_OT_12_collapse: fixes \<sigma> :: "bool"   and D :: "bool \<times> 'index \<times> bool \<times> bool \<Rightarrow> bool spmf"   and HCP_ad :: "real"   and m1 :: "bool \<times> bool" assumes "\<forall>b\<^sub>\<sigma>. local.etp.HCP_adv \<A> \<sigma> b\<^sub>\<sigma> D \<le> HCP_ad" shows "\<bar>spmf (funct_OT_12 m1 \<sigma> \<bind> (\<lambda>(out1, out2). local.etp.HCP_game \<A> \<sigma> out2 D)) True - 1 / 2\<bar> \<le> HCP_ad"
CollectD: fixes a :: "'b::type"   and P :: "'b::type \<Rightarrow> bool" assumes "(a::'b::type) \<in> {x::'b::type. (P::'b::type \<Rightarrow> bool) x}" shows "(P::'b::type \<Rightarrow> bool) (a::'b::type)"
finite_Inter: fixes M :: "'c set set" assumes "\<exists>A\<in>M. finite A" shows "finite (\<Inter> M)"
norm_Snorm: fixes s :: "'a" assumes "s \<in> set Snorm" shows "norm s = 1"
Class_is_Left_Coset: fixes g :: "'a" assumes "g \<in> G" shows "contained.Class g = g \<cdot>| L"
not_subtypes: fixes CT :: "nat \<Rightarrow> classDef option"   and A :: "nat"   and C :: "nat"   and D :: "nat" assumes "CT \<turnstile> A <: C"   and "CT \<turnstile> D \<not><: C"   and "CT \<turnstile> C \<not><: D" shows "CT \<turnstile> A \<not><: D"
globally_weak_until: fixes w :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltln"   and \<psi> :: "'b ltln" shows "w \<Turnstile>\<^sub>n G\<^sub>n (\<phi> W\<^sub>n \<psi>) = w \<Turnstile>\<^sub>n G\<^sub>n (\<phi> or\<^sub>n \<psi>)"
constant_function_eq: fixes a :: "'a"   and b :: "'a" assumes "a \<in> carrier R"   and "b \<in> carrier R" shows "constant_function (carrier R) a b = a"
monom_Suc: fixes a :: "'a"   and n :: "nat" shows "monom a (Suc n) = pCons (0::'a) (monom a n)"