hcju/isabelleps huggingface dataset:
a_star_redu_NotR_elim: fixes x :: "name"   and M :: "trm"   and a :: "coname"   and R :: "trm" assumes "NotR x.M a \<longrightarrow>\<^sub>a* R" shows "\<exists>M'. R = NotR x.M' a \<and> M \<longrightarrow>\<^sub>a* M'"
IH: shows "(rGamma has_field_derivative (- 1) ^ n * fact n) (at (- of_nat n))"
WriteVolatile: fixes \<O>s :: "nat set list"   and i :: "nat"   and a :: "nat"   and \<R>s :: "(nat \<Rightarrow> bool option) list"   and A :: "nat set"   and \<S> :: "nat \<Rightarrow> bool option"   and \<O> :: "nat set"   and L :: "nat set"   and R :: "nat set"   and D :: "nat set"   and f :: "(nat \<Rightarrow> nat option) \<Rightarrow> nat"   and W :: "nat set"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and m :: "nat \<Rightarrow> nat"   and \<D> :: "bool" assumes "\<forall>j<length \<O>s. i \<noteq> j \<longrightarrow> a \<notin> \<O>s ! j \<union> dom (\<R>s ! j)"   and "A \<subseteq> dom \<S> \<union> \<O>"   and "L \<subseteq> A"   and "R \<subseteq> \<O>"   and "A \<inter> R = {}"   and "\<forall>j<length \<O>s. i \<noteq> j \<longrightarrow> A \<inter> (\<O>s ! j \<union> dom (\<R>s ! j)) = {}"   and "a \<notin> read_only \<S>" shows "\<O>s,\<R>s,i\<turnstile> (Write True a (D, f) A L R W # is, \<theta>, m, \<D>, \<O>, \<S>) \<surd>"
index_add_mat_3: fixes A :: "'m mat"   and B :: "'m mat" shows "dim_col (A + B) = dim_col B"
le_antisym: fixes L :: "('d, 'e) gorder_scheme"   and x :: "'d"   and y :: "'d" assumes "partial_order L"   and "x \<sqsubseteq>\<^bsub>L\<^esub> y"   and "y \<sqsubseteq>\<^bsub>L\<^esub> x"   and "x \<in> carrier L"   and "y \<in> carrier L" shows "x = y"
nlistListRCons: fixes t :: "('c, 'd) FingerTreeStruc"   and xs :: "('c, 'd) Node list" shows "toList (rconsNlist t xs) = toList t @ nlistToList xs"
or_gt_1: fixes \<G> :: "'a cyclic_group" assumes "chaum_ped_\<Sigma>_base \<G>" shows "1 < order \<G>"
pnop_def: fixes A :: "V"   and n :: "V"   and f :: "V" shows "pnop A n f \<equiv> vsv f \<and> pnop_axioms A n f"
north_not_east_or_west: shows "north \<notin> {east, west}"
\<tau>diverge_neq_wfP_silent_move_from: fixes trsys :: "'a \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> bool"   and \<tau>move :: "'a \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> bool"   and s :: "'a" shows "\<tau>trsys.\<tau>diverge trsys \<tau>move s \<noteq> wfP (flip (\<tau>trsys.silent_move_from trsys \<tau>move s))"
simps_3: fixes x31 :: "'b"   and x32 :: "('a, 'b) form list"   and y31 :: "'b"   and y32 :: "('a, 'b) form list" shows "(Modal x31 x32 = Modal y31 y32) = (x31 = y31 \<and> x32 = y32)"
igSubstIGVar1STR_def: fixes MOD :: "('h, 'i, 'varSort, 'sort, 'j, 'k, 'l, 'm, 'n) model_scheme" shows "igSubstIGVar1STR MOD \<equiv> (\<forall>ys y y1 xs x. ys \<noteq> xs \<or> x \<noteq> y \<longrightarrow> igSubst MOD ys (igVar MOD ys y1) y (igVar MOD xs x) = igVar MOD xs x) \<and> (\<forall>ys y Y xs x. igWls MOD (asSort ys) Y \<longrightarrow> ys \<noteq> xs \<or> y \<noteq> x \<longrightarrow> igSubst MOD ys Y y (igVar MOD xs x) = igVar MOD xs x)"
same_level_path_def: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and as :: "'a list" assumes "CFG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main" shows "CFG.same_level_path kind get_return_edges as \<equiv> CFG.same_level_path_aux kind get_return_edges [] as \<and> CFG.upd_cs kind [] as = []"
intro: fixes Inf :: "'a inference set"   and Red_F :: "'a set \<Rightarrow> 'a set"   and Red_I :: "'a set \<Rightarrow> 'a inference set" assumes "\<And>N. inference_system.Inf_between Inf UNIV (Red_F N) \<subseteq> Red_I N" shows "reduced_calculus_axioms Inf Red_I Red_F"
index_last: fixes xs :: "'b list" assumes "xs \<noteq> []"   and "distinct xs" shows "index xs (last xs) = length xs - 1"
compact_sum: fixes I :: "'b set"   and S :: "'b \<Rightarrow> 'c set"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'd"   and J :: "'b set" assumes "finite I"   and "\<And>i. i \<in> I \<Longrightarrow> compact (S i)"   and "\<And>i. i \<in> I \<Longrightarrow> continuous_on (S i) (f i)"   and "I \<subseteq> J" shows "compact {\<Sum>i\<in>I. f i (x i) |x. x \<in> Pi J S}"
filter_greater: fixes Fa :: "'b set set"   and Ta :: "'b set set"   and F :: "'c set set"   and T :: "'c set set"   and A :: "'c set"   and B :: "'c set" assumes "Topology.filter Fa Ta"   and "F \<in> Filters\<^bsub>T\<^esub>"   and "A \<subseteq> B"   and "B \<subseteq> carrier\<^bsub>T\<^esub>"   and "A \<in> F" shows "B \<in> F"
inv\<^sub>3_def: 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 V :: "'a set" shows "inv\<^sub>3 P\<^sub>1 P\<^sub>2 B\<^sub>1 B\<^sub>2 V = (inv\<^sub>2 P\<^sub>1 P\<^sub>2 B\<^sub>1 B\<^sub>2 V \<and> B\<^sub>2 \<subseteq> S)"
mergesort_eq_val_mergesort_tm: fixes f :: "'a \<Rightarrow> 'b"   and xs :: "'a list" shows "Time_Monad.val (mergesort_tm f xs) = mergesort f xs"
pg: fixes t :: "real" assumes "t \<in> {0..1}" shows "cmod (\<gamma> t - p t) < min d e / 2"
expectation_gpv_mk_lossless_gpv: fixes \<I>' :: "('a, 'b) \<I>"   and gpv :: "('c, 'a, 'b) gpv"   and \<I> :: "('a, 'b) \<I>"   and y :: "ennreal"   and x :: "'c" assumes "\<I>' \<turnstile>g gpv \<surd>"   and "outs_\<I> \<I> = outs_\<I> \<I>'" shows "expectation_gpv 0 \<I>' (\<lambda>_. y) gpv \<le> expectation_gpv 0 \<I> (\<lambda>_. y) (mk_lossless_gpv (responses_\<I> \<I>') x gpv)"
pref_emp: fixes a :: "'b list" assumes "a \<le>p \<epsilon>" shows "a = \<epsilon>"
attract_mono_imp_qfp_complete: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and CC :: "'b set set" assumes "fixed_point_proof A f"   and "attractive A less_eq"   and "CC-complete A less_eq"   and "\<forall>C\<subseteq>A. well_related_set C less_eq \<longrightarrow> C \<in> CC"   and "\<forall>X\<in>CC. \<forall>Y\<in>CC. setwise less_eq X Y \<longrightarrow> X \<union> Y \<in> CC"   and "monotone_on A less_eq less_eq f" shows "CC-complete {p \<in> A. sympartp less_eq (f p) p} less_eq"
IH_1: fixes x :: "Vars.var"   and r :: "C"   and \<rho> :: "Vars.var \<Rightarrow> C \<rightarrow> CValue" assumes "x \<in> domA \<Gamma>" shows "(\<N>\<lbrakk> the (map_of \<Gamma> x) \<rbrakk>\<^bsub>\<rho>\<^esub>)|\<^bsub>C\<cdot>r\<^esub> = (\<N>\<lbrakk> the (map_of \<Gamma> x) \<rbrakk>\<^bsub>\<rho>|\<^sup>\<circ>\<^bsub>r\<^esub>\<^esub>)|\<^bsub>C\<cdot>r\<^esub>"
simps_2: fixes satisfies :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and transition :: "'b \<Rightarrow> ('d, 'b) residual \<Rightarrow> bool"   and P :: "'b"   and t :: "('e, 'c, 'd) Tree" assumes "nominal_ts satisfies transition" shows "nominal_ts.valid_Tree satisfies transition P (tNot t) = (\<not> nominal_ts.valid_Tree satisfies transition P t)"
phl_export2: fixes z :: "'a"   and w :: "'a"   and x :: "'a"   and y :: "'a" assumes "z \<cdot> w \<le> w \<cdot> z \<cdot> w"   and "x \<cdot> y \<le> y \<cdot> z" shows "x \<cdot> (y \<cdot> w) \<le> y \<cdot> w \<cdot> (z \<cdot> w)"
simps_1: shows "ik\<^sub>s\<^sub>t [] = {}"
IH: shows "(x \<in> set as) = (\<exists>i. mem_idx x as = Some i)"
h3c: shows "\<forall>x. List.member (rows_to_keep (reduce_mat_cols (M_mat signs subsets) (solve_for_lhs p qs subsets (M_mat signs subsets)))) x \<longrightarrow> x < length subsets"
lift_Suc_mono_le: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "nat \<Rightarrow> 'b"   and n :: "nat"   and n' :: "nat" assumes "class.order (less_eq::'b \<Rightarrow> 'b \<Rightarrow> bool) (less::'b \<Rightarrow> 'b \<Rightarrow> bool)"   and "\<And>n::nat. (less_eq::'b \<Rightarrow> 'b \<Rightarrow> bool) ((f::nat \<Rightarrow> 'b) n) (f (Suc n))"   and "(n::nat) \<le> (n'::nat)" shows "(less_eq::'b \<Rightarrow> 'b \<Rightarrow> bool) ((f::nat \<Rightarrow> 'b) (n::nat)) (f (n'::nat))"
set_unsat_simps_5: fixes I :: "'a list"   and s :: "('a, 'b) state" shows "\<U> (set_unsat I s) = True"
nb: shows "valuation_for_operator_variables \<Pi> \<pi> (trace_parallel_plan_strips (\<Pi>\<^sub>I) \<pi>) (Operator k (index (\<Pi>\<^sub>\<O>) op)) = foldr (\<lambda>(k, op) \<A>. \<A>(Operator k (index (\<Pi>\<^sub>\<O>) op) := True)) (concat (map (\<lambda>k. map (Pair k) (\<pi> ! k)) [0..<length (trace_parallel_plan_strips (\<Pi>\<^sub>I) \<pi>) - 1])) \<A>\<^sub>0 (Operator k (index (\<Pi>\<^sub>\<O>) op))"
Solution_imp_subdprodl: fixes a :: "nat list"   and b :: "nat list"   and x :: "nat list"   and y :: "nat list" assumes "hlde a b"   and "(x, y) \<in> Linear_Diophantine_Equations.Solutions a b" shows "hlde.subdprodl a b x y"
usubst_ode_ext2: fixes \<sigma> :: "(char \<Rightarrow> trm option) \<times> (char \<Rightarrow> trm option) \<times> (char \<Rightarrow> fml option) \<times> (char \<Rightarrow> game option)"   and U :: "variable set"   and x :: "char"   and \<theta> :: "trm"   and F :: "real \<Rightarrow> variable \<Rightarrow> real"   and \<omega> :: "variable \<Rightarrow> real"   and I :: "interp" assumes "usubstappt \<sigma> (U \<union> {RVar x, DVar x}) \<theta> \<noteq> undeft"   and "Uvariation (F 0) \<omega> (U \<union> {RVar x, DVar x})" shows "solves_ODE I F x (the (usubstappt \<sigma> (U \<union> {RVar x, DVar x}) \<theta>)) = solves_ODE (USubst.adjoint \<sigma> I \<omega>) F x \<theta>"
allocationPowerset: fixes a :: "('c \<times> 'd set) set"   and G :: "'d set"   and N :: "'c set" assumes "a \<in> allAllocations N G" shows "a \<subseteq> N \<times> (Pow G - {{}})"
awalk_append_iff: fixes u :: "'a"   and p :: "'b list"   and q :: "'b list"   and v :: "'a" shows "awalk u (p @ q) v = (awalk u p (awlast u p) \<and> awalk (awlast u p) q v)"
div_smult_right: fixes a :: "'a"   and x :: "'a poly"   and y :: "'a poly" assumes "a \<noteq> (0::'a)" shows "x div Polynomial.smult a y = Polynomial.smult (inverse a) (x div y)"
fps_Lcm_def: shows "Lcm = normalization_euclidean_semiring_class.Lcm"
sorted_antimono_prefix: fixes xs :: "'b list"   and ys :: "'b list" assumes "prefix xs ys"   and "sorted ys" shows "sorted xs"
run_get_env: fixes get :: "('c \<Rightarrow> 'a) \<Rightarrow> 'a"   and f :: "'c \<Rightarrow> ('b, 'a) envT"   and r :: "'b" shows "run_env (get_env get f) r = get (\<lambda>s. run_env (f s) r)"
f_the_inv_into_f_bij_betw: fixes f :: "'d \<Rightarrow> 'e"   and A :: "'d set"   and B :: "'e set"   and x :: "'e" assumes "bij_betw f A B"   and "bij_betw f A B \<Longrightarrow> x \<in> B" shows "f (the_inv_into A f x) = x"
\<Phi>a_def: shows "\<Psi>a = (\<lambda>f. \<lambda>x'\<in>\<psi> ` S'.set (S'.dom f). \<psi> (S'.Fun f (INV.\<psi> x')))"
next_rbt_entries_generator: shows "list.next rbt_entries_generator = rbt_entries_next"
dot_cross7_self_1: fixes x :: "(real, 7) vec"   and y :: "(real, 7) vec" shows "x \<bullet> x \<times>\<^sub>7 y = 0"
left_unique_rel_gpv'': fixes A :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and C :: "'i \<Rightarrow> 'j \<Rightarrow> bool"   and R :: "'k \<Rightarrow> 'l \<Rightarrow> bool" assumes "left_unique A"   and "left_unique C"   and "left_total R" shows "left_unique (rel_gpv'' A C R)"
preorder_ow_axioms: shows "preorder_ow U (\<le>\<^sub>o\<^sub>w) (<\<^sub>o\<^sub>w)"
simpleDefs_phiDefs_var_disjoint: fixes v :: "'a"   and g :: "'g"   and n :: "'node" assumes "v \<in> other.phiDefs g n"   and "n \<in> set (\<alpha>n g)" shows "var' g v \<notin> oldDefs g n"
ident_rel_refl: fixes G :: "('b, 'c) labeled_graph"   and idt :: "'b" assumes "graph G"   and "ident_rel idt G" shows "maintained (reflexivity_rule idt) G"
at_fin_set_supports: fixes X :: "'a set" assumes "at TYPE('a)" shows "X supports X"
intro: fixes H :: "'e set"   and M :: "('f, 'e, 'g) module_scheme"   and R :: "('f, 'h) ring_scheme" assumes "subgroup H (add_monoid M)"   and "submodule_axioms H R M" shows "submodule H R M"
match_iface_case_nowildcard: fixes i :: "char list"   and p_i :: "char list" assumes "\<not> iface_name_is_wildcard i" shows "match_iface (Iface i) p_i = (i = p_i)"
surj_hom_subgroup: fixes F :: "('e, 'f) monoid_scheme"   and \<phi> :: "'a \<Rightarrow> 'e" assumes "group_hom G F \<phi>"   and "\<phi> ` carrier G = carrier F" shows "subgroup (\<phi> ` H) F"
disjoint_sum: fixes A :: "'d set"   and B :: "'d set set"   and f :: "'d \<Rightarrow> 'e" assumes "Congruence.partition A B"   and "finite A" shows "sum (sum f) B = sum f A"
cases: fixes x :: "('c, 'd) SkewBinomialTree" obtains e :: "'c" and a :: "'d" and r :: "nat" and ts :: "('c, 'd) SkewBinomialTree list" where "x = Node e a r ts"
ld_\<gamma>: shows "LeftDerivationFix (\<alpha>' @ \<delta>) (i + ix) D j \<gamma>"
rivest_hiding: fixes \<A> :: "(nat \<times> nat \<Rightarrow> ((nat \<times> nat) \<times> 'b) spmf) \<times> (nat \<Rightarrow> 'b \<Rightarrow> bool spmf)" shows "spmf (rivest_commit.hiding_game_ind_cpa \<A>) True - 1 / 2 = 0"
Int_insert_right_if0: fixes a :: "'d"   and A :: "'d set"   and B :: "'d set" assumes "a \<notin> A" shows "A \<inter> insert a B = A \<inter> B"
subdivision_genus: fixes G :: "('c, 'd) pre_digraph"   and rev_G :: "'d \<Rightarrow> 'd"   and H :: "('c, 'd) pre_digraph"   and rev_H :: "'d \<Rightarrow> 'd"   and HM :: "'d pre_map"   and m :: "int" assumes "subdivision (G, rev_G) (H, rev_H)"   and "digraph_map H HM"   and "pre_digraph_map.euler_genus H HM = m" shows "\<exists>GM. digraph_map G GM \<and> pre_digraph_map.euler_genus G GM = m"
nn_integral_count_space_finite: fixes A :: "'c set"   and f :: "'c \<Rightarrow> ennreal" assumes "finite A" shows "integral\<^sup>N (count_space A) f = sum f A"
ran_map_add: fixes m1 :: "'c \<Rightarrow> 'd option"   and m2 :: "'c \<Rightarrow> 'd option" assumes "dom m1 \<inter> dom m2 = {}" shows "ran (m1 ++ m2) = ran m1 \<union> ran m2"
foldl_conv_fold: fixes f :: "'e \<Rightarrow> 'f \<Rightarrow> 'e"   and s :: "'e"   and xs :: "'f list" shows "foldl f s xs = fold (\<lambda>x s. f s x) xs s"
is_sig_red_singletonI: fixes sing_reg :: "'t \<Rightarrow> 't \<Rightarrow> bool"   and top_tail :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and F :: "('t \<Rightarrow>\<^sub>0 'b) set"   and r :: "'t \<Rightarrow>\<^sub>0 'b" assumes "is_sig_red sing_reg top_tail F r" obtains f :: "'t \<Rightarrow>\<^sub>0 'b" where "f \<in> F" and "is_sig_red sing_reg top_tail {f} r"
IH: fixes \<xi> :: "V" assumes "\<xi> \<in> elts \<mu>" shows "\<xi> \<le> \<alpha> + \<xi>"
WN_onD: fixes r :: "('d \<times> 'd) set"   and A :: "'d set"   and a :: "'d" assumes "WN_on r A"   and "a \<in> A" shows "\<exists>b. (a, b) \<in> r\<^sup>!"
ext: fixes g :: "'a"   and f :: "'a" assumes "A g f \<noteq> A.null" shows "A.seq g f"
And_propos': fixes x :: "'a ltl" assumes "x \<in> set xs" shows "propos x \<subseteq> propos (And xs)"
ide_cod: fixes f :: "'b" assumes "B.arr f" shows "B.ide (B.cod f)"
bigtheta_const_ln_powr: fixes a :: "real"   and d :: "real" assumes "0 < a" shows "(\<lambda>x. ln (a * x) powr d) \<in> \<Theta>(\<lambda>x. ln x powr d)"
Above_def: fixes r :: "('a \<times> 'a) set"   and A :: "'a set" shows "Above r A \<equiv> {b \<in> Field r. \<forall>a\<in>A. (a, b) \<in> r}"
count: fixes x :: "'a multiset" shows "count x \<in> multiset"
fps_nonzeroI: fixes f :: "'b fps"   and n :: "nat" assumes "f $ n \<noteq> (0::'b)" shows "f \<noteq> 0"
closure_lessThan: fixes b :: "'b" shows "closure {..<b} = {..b}"
eint_defs_5: fixes m :: "eint"   and n :: "eint" shows "(m < n) = (case m of eint m1 \<Rightarrow> case n of eint n1 \<Rightarrow> m1 < n1 | \<infinity> \<Rightarrow> True | \<infinity> \<Rightarrow> False)"
vallam: fixes x :: "nat"   and e :: "exp" shows "isval (ELam x e)"
left_unique_def: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> bool" shows "left_unique R = (\<forall>x y z. R x z \<longrightarrow> R y z \<longrightarrow> x = y)"
linmap_im_lincomb: fixes A :: "'b set"   and N :: "('g, 'b, 'h) Module_scheme"   and f :: "'a \<Rightarrow> 'g"   and H :: "'a set"   and s :: "nat \<Rightarrow> 'b"   and n :: "nat"   and g :: "nat \<Rightarrow> 'a" assumes "ideal R A"   and "R module N"   and "f \<in> mHom R M N"   and "H \<subseteq> carrier M"   and "s \<in> {j. j \<le> n} \<rightarrow> A"   and "g \<in> {j. j \<le> n} \<rightarrow> H" shows "f (l_comb R M n s g) = l_comb R N n s (cmp f g)"
simps_3: fixes x :: "'c event"   and s :: "'c event list"   and X :: "'c event set" shows "setinterleaving (x # s, X, []) = (if x \<in> X then {} else {x # u |u. u setinterleaves ((s, []), X)})"
reachable_induce_subgraphD: fixes S :: "'a set"   and u :: "'a"   and v :: "'a" assumes "u \<rightarrow>\<^sup>*\<^bsub>(G \<restriction> S) \<restriction> S\<^esub> v"   and "S \<subseteq> verts (G \<restriction> S)" shows "u \<rightarrow>\<^sup>*\<^bsub>G \<restriction> S\<^esub> v"
bezout_ring_imp_diagonal_admits_SNF_JNF: assumes "class.bezout_ring (*) (1::'b) (+) (0::'b) (-) uminus" shows "\<forall>A. admits_SNF_JNF A"
lfp_induct: fixes f :: "'g \<Rightarrow> 'g"   and P :: "'g" assumes "mono f"   and "f (inf (lfp f) P) \<le> P" shows "lfp f \<le> P"
walk_last_edge: fixes xs :: "'a list"   and ys :: "'a list" assumes "H.walk (xs @ ys)"   and "xs \<noteq> []"   and "ys \<noteq> []" shows "last xs \<rightarrow>\<^bsub>H\<^esub> hd ys"
limit_mat_scale: fixes X :: "nat \<Rightarrow> complex mat"   and A :: "complex mat"   and m :: "nat"   and c :: "complex" assumes "limit_mat X A m" shows "limit_mat (\<lambda>n. c \<cdot>\<^sub>m X n) (c \<cdot>\<^sub>m A) m"
pts_symbolic_c_induct: fixes \<S> :: "('fun, 'var) strand_step list set"   and \<A> :: "('fun, 'var) extstrand_step list"   and \<S>' :: "('fun, 'var) strand_step list set"   and \<A>' :: "('fun, 'var) extstrand_step list" assumes "(\<S>, \<A>) \<Rightarrow>\<^sup>\<bullet>\<^sub>c (\<S>', \<A>')" obtains "[] \<in> \<S>" and "\<S>' = update\<^sub>s\<^sub>t \<S> []" and "\<A>' = \<A>"   | t :: "('fun, 'var) Term.term" and S :: "('fun, 'var) strand_step list" where "send\<langle>t__\<rangle>\<^sub>s\<^sub>t # S__ \<in> \<S>" and "\<S>' = update\<^sub>s\<^sub>t \<S> (send\<langle>t__\<rangle>\<^sub>s\<^sub>t # S__)" and "\<A>' = \<A> @ [Step (receive\<langle>t__\<rangle>\<^sub>s\<^sub>t)]"   | t :: "('fun, 'var) Term.term" and S :: "('fun, 'var) strand_step list" where "receive\<langle>ta__\<rangle>\<^sub>s\<^sub>t # Sa__ \<in> \<S>" and "\<S>' = update\<^sub>s\<^sub>t \<S> (receive\<langle>ta__\<rangle>\<^sub>s\<^sub>t # Sa__)" and "\<A>' = \<A> @ [Step (send\<langle>ta__\<rangle>\<^sub>s\<^sub>t)]"   | a :: "poscheckvariant" and t :: "('fun, 'var) Term.term" and t' :: "('fun, 'var) Term.term" and S :: "('fun, 'var) strand_step list" where "\<langle>a: t \<doteq> t'\<rangle>\<^sub>s\<^sub>t # Sb__ \<in> \<S>" and "\<S>' = update\<^sub>s\<^sub>t \<S> (\<langle>a: t \<doteq> t'\<rangle>\<^sub>s\<^sub>t # Sb__)" and "\<A>' = \<A> @ [Step \<langle>a: t \<doteq> t'\<rangle>\<^sub>s\<^sub>t]"   | X :: "'var list" and F :: "(('fun, 'var) Term.term \<times> ('fun, 'var) Term.term) list" and S :: "('fun, 'var) strand_step list" where "\<forall>X\<langle>\<or>\<noteq>: F\<rangle>\<^sub>s\<^sub>t # S \<in> \<S>" and "\<S>' = update\<^sub>s\<^sub>t \<S> (\<forall>X\<langle>\<or>\<noteq>: F\<rangle>\<^sub>s\<^sub>t # S)" and "\<A>' = \<A> @ [Step \<forall>X\<langle>\<or>\<noteq>: F\<rangle>\<^sub>s\<^sub>t]"   | f :: "'fun" and T :: "('fun, 'var) Term.term list" where "Fun f T \<sqsubseteq>\<^sub>s\<^sub>e\<^sub>t ik\<^sub>e\<^sub>s\<^sub>t \<A> \<union> assignment_rhs\<^sub>e\<^sub>s\<^sub>t \<A>" and "\<S>' = \<S>" and "\<A>' = \<A> @ [Decomp (Fun f T)]"
run_bind_state: fixes bind :: "'a \<Rightarrow> ('c \<times> 'b \<Rightarrow> 'a) \<Rightarrow> 'a"   and x :: "('b, 'a) stateT"   and f :: "'c \<Rightarrow> ('b, 'a) stateT"   and s :: "'b" shows "run_state (bind_state bind x f) s = bind (run_state x s) (\<lambda>(a, s'). run_state (f a) s')"
trivial_stateful_policy_compliance: fixes V :: "'b set"   and E :: "('b \<times> 'b) set"   and M :: "'b SecurityInvariant_configured list" assumes "wf_graph \<lparr>nodes = V, edges = E\<rparr>"   and "valid_reqs M"   and "all_security_requirements_fulfilled M \<lparr>nodes = V, edges = E\<rparr>" shows "stateful_policy_compliance \<lparr>hosts = V, flows_fix = E, flows_state = {}\<rparr> \<lparr>nodes = V, edges = E\<rparr> M"
finite_arcs: fixes G :: "('c, 'd) pre_digraph" assumes "fin_digraph G" shows "finite (arcs G)"
set_plus_rearrange3: fixes a :: "'b"   and B :: "'b set"   and C :: "'b set" shows "a +o B + C = a +o (B + C)"
chine_assoc_props_3: shows "doms.Prj\<^sub>1\<^sub>0 \<cdot> doms.chine_assoc = doms.Prj\<^sub>0\<^sub>1"
in_homI: fixes f :: "'a \<times> 'a"   and a :: "'a \<times> 'a"   and b :: "'a \<times> 'a" assumes "VoV.arr f"   and "VoV.dom f = a"   and "VoV.cod f = b" shows "VoV.in_hom f a b"
rsemimodule2: fixes next_behaviour :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and next_stimulus :: "'c \<times> 'd \<Rightarrow> 'c"   and s :: "'c"   and a :: "'d"   and b :: "'d" assumes "c2ka next_behaviour next_stimulus" shows "next_stimulus (s, a + b) = next_stimulus (s, a) \<oplus> next_stimulus (s, b)"
diff_eq_diff_less_eq: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c"   and b :: "'c"   and c :: "'c"   and d :: "'c" assumes "class.ordered_ab_group_add plus zero minus uminus less_eq less"   and "minus a b = minus c d" shows "less_eq a b = less_eq c d"
D: shows "(Owner n, Crypt (SesK SK) (Pwd n)) \<in> s"
proper_compact_cball: fixes x :: "'c"   and r :: "real" assumes "proper UNIV" shows "compact (cball x r)"
simps_1: fixes step :: "nat \<Rightarrow> atom list \<Rightarrow> atom fm list \<Rightarrow> atom fm"   and F :: "(atom list \<times> atom fm list \<times> nat) list" shows "the_real_step_augment step 0 F = list_disj (map (\<lambda>(L, F, n). ExN n (list_conj (map fm.Atom L @ F))) F)"
W_AIB: shows "induced_map A --` WA \<inter> induced_map A --` WB = {}"
map_simp_ide: fixes a :: "'a arrow_of_spans_data" assumes "ide a" shows "\<ll>.map a = \<l> a"
assms: shows "f \<in> carrier T \<rightarrow> carrier S"
units_power_order_eq_one: fixes R :: "('a, 'b) ring_scheme"   and a :: "'a" assumes "cring R"   and "finite (Units R)"   and "a \<in> Units R" shows "a [^]\<^bsub>R\<^esub> card (Units R) = \<one>\<^bsub>R\<^esub>"
add_sign_intros_4: fixes a :: "'c multiset"   and b :: "'c multiset" assumes "{#} \<subseteq># a"   and "{#} \<subseteq># b" shows "{#} \<subseteq># a + b"
prefix_def: fixes xs :: "'c list"   and ys :: "'c list" shows "prefix xs ys = (\<exists>zs. ys = xs @ zs)"
UP_r_distr: fixes R :: "('c, 'd) ring_scheme"   and p :: "nat \<Rightarrow> 'c"   and q :: "nat \<Rightarrow> 'c"   and r :: "nat \<Rightarrow> 'c" assumes "UP_ring R"   and "p \<in> carrier (UP R)"   and "q \<in> carrier (UP R)"   and "r \<in> carrier (UP R)" shows "r \<otimes>\<^bsub>UP R\<^esub> (p \<oplus>\<^bsub>UP R\<^esub> q) = r \<otimes>\<^bsub>UP R\<^esub> p \<oplus>\<^bsub>UP R\<^esub> r \<otimes>\<^bsub>UP R\<^esub> q"
mod_mod_trivial: fixes a :: "'a"   and b :: "'a" shows "a mod b mod b = a mod b"
pr0_in_hom': fixes a :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and b :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "ide a"   and "ide b" shows "\<guillemotleft>pr0 a b : local.dom (pr0 a b) \<rightarrow> b\<guillemotright>"
exec_move_newArray: fixes pc :: "nat"   and e :: "'addr expr1"   and ci :: "('addr, 'heap) check_instr"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and T :: "ty"   and h :: "'heap"   and stk :: "'addr val list"   and loc :: "'addr val list"   and xcp :: "'addr option" assumes "pc < length (compE2 e)" shows "exec_move ci P t (newA T\<lfloor>e\<rceil>) h (stk, loc, pc, xcp) = exec_move ci P t e h (stk, loc, pc, xcp)"
conjug_prim_root: fixes u :: "'b list"   and v :: "'b list" assumes "u \<sim> v"   and "u \<noteq> \<epsilon>" shows "\<rho> u \<sim> \<rho> v"