hcju/isabelleps huggingface dataset:
fold_commute: fixes xs :: "'c list"   and h :: "'d \<Rightarrow> 'e"   and g :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and f :: "'c \<Rightarrow> 'e \<Rightarrow> 'e" assumes "\<And>x. x \<in> set xs \<Longrightarrow> h \<circ> g x = f x \<circ> h" shows "h \<circ> fold g xs = fold f xs \<circ> h"
callee_invariant': fixes y :: "'b"   and s' :: "'s2"   and s :: "'s2"   and x :: "'a" assumes "(y, s') \<in> set_spmf (callee2 s x)"   and "I2 s"   and "x \<in> outs_\<I> \<I>" shows "I2 s' \<and> y \<in> responses_\<I> \<I> x"
implies_intro'_pre: fixes \<Theta> :: "((String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)) \<times> Core.term set"   and \<Gamma> :: "Core.term set"   and B :: "Core.term"   and A :: "Core.term" assumes "wf_theory \<Theta>"   and "\<Theta>,\<Gamma> \<turnstile> B"   and "term_ok \<Theta> A"   and "typ_of A = Some propT"   and "A \<notin> \<Gamma>" shows "\<Theta>,\<Gamma> \<turnstile> A \<longmapsto> B"
simps_2: fixes \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and me :: "'c match_expr"   and p :: "'d" shows "matches \<gamma> (MatchNot me) p = (\<not> matches \<gamma> me p)"
assms: shows "z \<noteq> \<infinity>\<^sub>h"
goedel_numbering_universal: fixes \<psi> :: "nat \<Rightarrow> nat \<Rightarrow> nat option"   and f :: "nat \<Rightarrow> nat option" assumes "goedel_numbering \<psi>"   and "f \<in> \<P>" shows "\<exists>i. \<psi> i = f"
sem_eq1': shows "sterm_sem I (f1 fid2 vid1) (sol 0) = sterm_sem I (f1 fid2 vid1) (fst (mk_v I (OVar vid1) (sol 0, b) (sol 0)))"
update_Gamma_nochange1: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and chain :: "char list"   and rs :: "'c rule list"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and m :: "'c match_expr"   and a :: "action"   and rs' :: "'c rule list"   and s :: "state"   and t :: "state" assumes "\<Gamma>(chain \<mapsto> rs),\<gamma>,p\<turnstile> \<langle>[Rule m a], Undecided\<rangle> \<Rightarrow> Undecided"   and "\<Gamma>(chain \<mapsto> Rule m a # rs),\<gamma>,p\<turnstile> \<langle>rs', s\<rangle> \<Rightarrow> t" shows "\<Gamma>(chain \<mapsto> rs),\<gamma>,p\<turnstile> \<langle>rs', s\<rangle> \<Rightarrow> t"
SUP_cong: fixes A :: "'c set"   and B :: "'c set"   and C :: "'c \<Rightarrow> 'a"   and D :: "'c \<Rightarrow> 'a" assumes "A = B"   and "\<And>x. x \<in> B \<Longrightarrow> C x = D x" shows "Sup (C ` A) = Sup (D ` B)"
nth_list_of_pdevs_pdevs_of_list: fixes xs :: "'b list"   and n :: "nat" assumes "\<And>x. x \<in> set xs \<Longrightarrow> x \<noteq> (0::'b)"   and "n < length xs" shows "list_of_pdevs (pdevs_of_list xs) ! n = (length xs - Suc n, xs ! (length xs - Suc n))"
trace_at_eq: fixes z :: "'a"   and t :: "real"   and x :: "'a"   and \<omega> :: "(real \<times> 'a) stream" assumes "trace_in {z} t x \<omega>" shows "trace_at x \<omega> t = z"
under_def: fixes r :: "('b \<times> 'b) set"   and a :: "'b" shows "under r a \<equiv> {b. (b, a) \<in> r}"
nprovRulesp_nprovRules_eq: shows "nprovRulesp = (\<lambda>x xa. (x, xa) \<in> nprovRules)"
subst_compose_def: fixes \<sigma> :: "'c \<Rightarrow> ('d, 'f) Term.term"   and \<tau> :: "'f \<Rightarrow> ('d, 'e) Term.term" shows "\<sigma> \<circ>\<^sub>s \<tau> = (\<lambda>x. \<sigma> x \<cdot> \<tau>)"
height_balR: fixes l :: "('b \<times> nat) tree"   and r :: "('b \<times> nat) tree"   and a :: "'b" assumes "avl l"   and "avl r"   and "height r = height l + 2" shows "height (balR l a r) \<in> {height l + 2, height l + 3}"
asIMOD_igSwapBinp: fixes SEM :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'b) semDom"   and val :: "'varSort \<Rightarrow> 'var \<Rightarrow> 'b"   and zs :: "'varSort"   and z1 :: "'var"   and z2 :: "'var"   and binp :: "'c \<Rightarrow> (('varSort \<Rightarrow> 'var \<Rightarrow> 'b) \<Rightarrow> ('varSort, 'b) sAbs) option" assumes "sWlsVal SEM val" shows "lift (\<lambda>A. A val) (igSwapBinp (asIMOD SEM) zs z1 z2 binp) = lift (\<lambda>A. A (val ^[z1 \<and> z2]_zs)) binp"
fundchamber_S_image_neq_fundchamber: fixes s :: "'a permutation" assumes "s \<in> (\<Union>(f, g)\<in>fundfoldpairs. {Abs_induced_automorph f g})" shows "s `\<rightarrow> C0 \<noteq> C0"
blinding_respects_hashes_parametric_aux: fixes A1 :: "('e \<Rightarrow> 'e \<Rightarrow> bool) \<Rightarrow> ('f \<Rightarrow> 'f \<Rightarrow> bool) \<Rightarrow> bool"   and A2 :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and A4 :: "bool \<Rightarrow> bool \<Rightarrow> bool"   and A3 :: "'g \<Rightarrow> 'h \<Rightarrow> bool" assumes "(A1 ===> (A2 ===> A2 ===> (=)) ===> A4) (\<le>) (\<le>)"   and "bi_unique A3" shows "((A2 ===> A3) ===> A1 ===> A4) blinding_respects_hashes blinding_respects_hashes"
body': shows "P \<turnstile> \<langle>body,(h\<^sub>2, l\<^sub>2(this \<mapsto> Addr a, pns [\<mapsto>] vs))\<rangle> \<rightarrow>* \<langle>ef,(h\<^sub>3, l\<^sub>2 ++ l\<^sub>3)\<rangle>"
K2_isometry_swap: fixes a :: "proj2"   and b :: "proj2" assumes "a \<in> hyp2"   and "b \<in> hyp2" shows "\<exists>J. is_K2_isometry J \<and> apply_cltn2 a J = b \<and> apply_cltn2 b J = a"
l6_4_1: fixes P :: "'p"   and A :: "'p"   and B :: "'p" assumes "P Out A B"   and "Col A P B" shows "\<not> Bet A P B"
mset_heap_rev_eq: fixes ts :: "'b tree list" shows "mset_heap (rev ts) = mset_heap ts"
path_from_toE: fixes G :: "('e, 'f) Graph_scheme"   and v :: "'e"   and xs :: "'e list"   and w :: "'e" assumes "Graph G"   and "v \<leadsto>xs\<leadsto>\<^bsub>G\<^esub> w" shows "(Graph.walk G xs \<and> distinct xs) \<and> xs \<noteq> [] \<and> hd xs = v \<and> last xs = w"
operat: shows "(if g integrable_on cbox a b then Some (integral (cbox a b) g) else None) = F (\<lambda>i. if g integrable_on i then Some (integral i g) else None) p"
Approximation1_5P: fixes n :: "nat"   and vv :: "nat \<Rightarrow> 'a \<Rightarrow> ant" assumes "vals_nonequiv K (Suc n) vv"   and "n_val K (vv 0) = vv 0" shows "\<exists>x\<in>carrier K. vv 0 x = 1 \<and> (\<forall>j\<in>nset (Suc 0) (Suc n). vv j x = 0)"
Ord_linear_le: fixes k :: "hf"   and l :: "hf" assumes "Ord k"   and "Ord l" obtains "k \<le> l"   | "l \<le> k"
line_eq_endpoint_iff_1: fixes a :: "'b"   and b :: "'b"   and i :: "real" shows "({a -- b}\<^bsub>i\<^esub> = b) = (a = b \<or> i = 1)"
fds_nth_one: shows "($) 1 = (\<lambda>n. if n = 1 then 1::'b else (0::'b))"
path_connected_space_iff_components_subset_singleton: fixes X :: "'a topology" shows "path_connected_space X = (\<exists>a. path_components_of X \<subseteq> {a})"
that: assumes "\<sigma>' = Some (snd (Ref.alloc x (the_state \<sigma>)))"   and "r = fst (Ref.alloc x (the_state \<sigma>))"   and "Ref.get (the_state \<sigma>') r = x" shows "thesis"
inf_left_commute: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "inf x (inf y z) = inf y (inf x z)"
conseqPost: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "('e set \<times> 'd \<times> 'e set \<times> 'e set) set"   and F :: "'f set"   and P :: "'e set"   and c :: "('e, 'd, 'f) com"   and Q' :: "'e set"   and A' :: "'e set"   and Q :: "'e set"   and A :: "'e set" assumes "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P c Q',A'"   and "Q' \<subseteq> Q"   and "A' \<subseteq> A" shows "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P c Q,A"
inducts: fixes x :: "nat \<times> nat"   and P :: "nat \<times> nat \<Rightarrow> bool" assumes "zerozero x"   and "P (0, 0)" shows "P x"
assms_1: shows "A \<or> B \<or> C \<or> D"
intro: fixes h :: "'e \<Rightarrow> 'f"   and G :: "('e, 'g) monoid_scheme"   and H :: "('f, 'h) monoid_scheme" assumes "h \<in> hom G H" shows "group_hom_axioms G H h"
onl_invariantD: fixes A :: "('e \<times> ('f, 'g, 'h, 'i) seqp, 'j) automaton"   and I :: "'j \<Rightarrow> bool"   and \<Gamma> :: "'h \<Rightarrow> ('f, 'g, 'h, 'i) seqp"   and P :: "'e \<times> 'i \<Rightarrow> bool"   and \<xi> :: "'e"   and p :: "('f, 'g, 'h, 'i) seqp"   and l :: "'i" assumes "A \<TTurnstile> (I \<rightarrow>) onl \<Gamma> P"   and "(\<xi>, p) \<in> reachable A I"   and "l \<in> labels \<Gamma> p" shows "P (\<xi>, l)"
prog_not_eq_in_par_ctran_aux: fixes P :: "'b com option list"   and s :: "'b"   and Q :: "'b com option list"   and t :: "'b" assumes "(P, s) -pc\<rightarrow> (Q, t)" shows "P \<noteq> Q"
fresh_fv: fixes e :: "'c"   and x :: "'b" assumes "finite (fv e)" shows "atom x \<sharp> fv e = atom x \<sharp> e"
PK: shows "(P, K) = local.reduce_column_aux div_op xs H (addrow_mat (dim_row H) (- div_op (H $$ (i, 0)) (H $$ (0, 0))) i 0 * P_init, addrow (- div_op (H $$ (i, 0)) (H $$ (0, 0))) i 0 K_init)"
abs_lattice: fixes a :: "'a" shows "\<bar>a\<bar> = sup a (- a)"
Rep_process_inverse: fixes x :: "'b process" shows "Abs_process (Rep_process x) = x"
merge_multiset: fixes l :: "'c Tree"   and g :: "'c Tree" shows "Heap.multiset l + Heap.multiset g = Heap.multiset (merge l g)"
arrE: fixes f :: "'a" assumes "S.arr f" shows "right a f"
spec\<^sub>5_def: fixes p :: "prog" shows "spec\<^sub>5 p \<equiv> para p = [''x'', ''y''] \<and> (\<exists>e\<^sub>2. body p = Add (Var ''x'') e\<^sub>2 \<and> wfe \<Gamma>\<^sub>x\<^sub>y e\<^sub>2 \<and> (\<forall>x y. eval (\<E>\<^sub>x\<^sub>y x y) e\<^sub>2 = Some (2 * x + 2 * y)) \<and> coste e\<^sub>2 \<le> 2)"
hom_graph_var: fixes f :: "'a \<Rightarrow> 'b"   and i :: "'a" shows "hom_graph f {i} (var i) (f i)"
hext_dom_typeof_addr_subset: fixes h :: "'heap"   and h' :: "'heap" assumes "h \<unlhd> h'" shows "dom (typeof_addr h) \<subseteq> dom (typeof_addr h')"
ennreal_of_enat_Sup: fixes X :: "enat set" shows "ennreal_of_enat (\<Squnion> X) = \<Squnion> (ennreal_of_enat ` X)"
thetaIfWT_WbisT: shows "thetaIfWT \<subseteq> WbisT"
open_except_set: fixes A :: "'b set"   and x :: "'b" assumes "finite A"   and "x \<notin> A" shows "\<exists>U. open U \<and> x \<in> U \<and> U \<inter> A = {}"
\<G>_subset: fixes Inf_F :: "'a inference set"   and Bot_G :: "'b set"   and Inf_G :: "'b inference set"   and entails_G :: "'b set \<Rightarrow> 'b set \<Rightarrow> bool"   and Red_I_G :: "'b set \<Rightarrow> 'b inference set"   and Red_F_G :: "'b set \<Rightarrow> 'b set"   and Bot_F :: "'a set"   and \<G>_F :: "'a \<Rightarrow> 'b set"   and \<G>_I :: "'a inference \<Rightarrow> 'b inference set option"   and N1 :: "'a set"   and N2 :: "'a set" assumes "standard_lifting Inf_F Bot_G Inf_G entails_G Red_I_G Red_F_G Bot_F \<G>_F \<G>_I"   and "N1 \<subseteq> N2" shows "\<Union> (\<G>_F ` N1) \<subseteq> \<Union> (\<G>_F ` N2)"
AbstTermP_imp_VarP: fixes v :: "tm"   and i :: "tm"   and t :: "tm"   and u :: "tm" shows "{AbstTermP v i t u} \<turnstile> VarP v"
le: shows "le_vec (A1n *v y) (cc *s y)"
map_le_restrict: fixes m :: "'c \<Rightarrow> 'd option"   and m' :: "'c \<Rightarrow> 'd option" assumes "m \<subseteq>\<^sub>m m'" shows "m = m' |` dom m"
integrable_subinterval: fixes f :: "real \<Rightarrow> real"   and a :: "real"   and b :: "real"   and a' :: "real"   and b' :: "real" assumes "integrable f a b"   and "a \<le> a'"   and "b' \<le> b" shows "integrable f a' b'"
p2: fixes z :: "'a" shows "p2 z \<in> T'"
abs_insert_impl: fixes M :: "msg"   and x :: "tag"   and A :: "agent"   and B :: "agent"   and S :: "msg set" assumes "M \<in> payload" shows "local.abs (insert (implem (Chan x A B M)) S) = insert (Chan x A B M) (local.abs S)"
1630: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(x \<squnion> y) \<sqinter> - z \<le> x \<sqinter> - z \<squnion> y"
Carmichael_number_imp_squarefree_alt: fixes n :: "nat" assumes "Carmichael_number n" shows "\<exists>p l. n = p * l \<and> prime p \<and> \<not> p dvd l \<and> 2 < l"
preserves_reflects_arr: fixes f :: "'c" shows "S.arr (\<Psi>'.map f) = Cop.arr f"
next1_subset_frmls: fixes \<phi> :: "'d ltlr"   and \<psi> :: "'d ltlr" assumes "\<phi> \<in> next1 \<psi>" shows "\<phi> \<in> subfrmlsr \<psi>"
nn_comparable: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and one :: "'a"   and sgn :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.nice_ordered_field plus minus less_eq less zero times abs uminus divide inverse one sgn"   and "less_eq zero a"   and "less_eq zero b" shows "less_eq a b \<or> less_eq b a"
verts3_subdivide: fixes e :: "'a \<times> 'a"   and w :: "'a" assumes "e \<in> parcs H"   and "w \<notin> pverts H" shows "verts3 (with_proj (subdivide H e w)) = verts3 (with_proj H)"
final_2: fixes channel :: "nat \<Rightarrow> ('d \<times> 'd) option"   and trans :: "'d \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> bool"   and send :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and recv :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and init :: "('d, 'e, 'f) configuration"   and t :: "('d, 'e, 'f) event llist" assumes "cocomputation channel trans send recv init"   and "distributed_system.cotrace channel trans send recv init t" shows "\<forall>p. ps (cocomputation.final channel trans send recv init t) p \<noteq> None"
fin: fixes A :: "'c set" assumes "V.fin_dim"   and "V.module.lin_indpt A"   and "A \<subseteq> carrier V" shows "finite A"
assms: assumes "vassn_tag (hn_ctxt A x y)" shows "P' x"
cSup_eq_maximum: fixes z :: "'g multiset"   and X :: "'g multiset set" assumes "z \<in> X"   and "\<And>x. x \<in> X \<Longrightarrow> x \<subseteq># z" shows "Sup X = z"
prob_space_stream_space: shows "prob_space (stream_space (stream_space M))"
from_digits_less: fixes n :: "nat"   and d :: "nat \<Rightarrow> nat" assumes "\<forall>i<n. d i < base" shows "from_digits n d < base ^ n"
check_neq_bot_snoc: fixes \<C> :: "t_context"   and es :: "b_e list"   and e :: "b_e"   and ts :: "checker_type" assumes "check \<C> (es @ [e]) ts \<noteq> Bot" shows "check \<C> es ts \<noteq> Bot"
gcd_dvd_1: fixes a :: "'d"   and b :: "'d" shows "(some_gcd a b dvd (1::'d)) = comm_monoid_mult_class.coprime a b"
step_1: assumes "j + 2 < card X"   and "0 < j + 1"   and "j + 1 + 1 \<le> card X" shows "aa (j + 1) + bb (j + 1) \<le> Dube_aux (card X) (poly_deg f) (j + 1)"
multiset_filter_subset: fixes f :: "'k \<Rightarrow> bool"   and M :: "'k multiset" shows "filter_mset f M \<subseteq># M"
Iff_refl: fixes H :: "fmla set"   and A :: "fmla" shows "H \<turnstile> A IFF A"
drop_norm_stack: fixes n :: "nat"   and xs :: "('dyn, 'ubx1, 'ubx2) unboxed list" shows "drop n (norm_stack xs) = norm_stack (drop n xs)"
rf: fixes i :: "nat"   and j :: "nat" shows "eval rf [i, j] = eval (r_smn 1 2) [encode r, i, j]"
eval_Z: fixes x :: "nat" shows "eval Z [x] \<down>= 0"
has_integral_spike_interior: fixes f :: "'a \<Rightarrow> 'b"   and y :: "'b"   and a :: "'a"   and b :: "'a"   and g :: "'a \<Rightarrow> 'b" assumes "(f has_integral y) (cbox a b)"   and "\<And>x. x \<in> box a b \<Longrightarrow> g x = f x" shows "(g has_integral y) (cbox a b)"
dom_inv: fixes f :: "'a" assumes "S.iso f" shows "S.dom (S.inv f) = S.cod f"
bound: fixes \<eta> :: "nat" shows "interaction_bounded_by' isl (\<A> \<eta>) (q \<eta>)"
IntI: fixes c :: "'c"   and A :: "'c set"   and B :: "'c set" assumes "c \<in> A"   and "c \<in> B" shows "c \<in> A \<inter> B"
induct: fixes E :: "('b \<times> 'b) set"   and x1 :: "'b"   and x2 :: "'b list"   and x3 :: "'b"   and P :: "'b \<Rightarrow> 'b list \<Rightarrow> 'b \<Rightarrow> bool" assumes "path E x1 x2 x3"   and "\<And>u. P u [] u"   and "\<And>u v l w. \<lbrakk>(u, v) \<in> E; path E v l w; P v l w\<rbrakk> \<Longrightarrow> P u (u # l) w" shows "P x1 x2 x3"
process_prop_6_def: fixes P :: "('c list \<times> 'c set) set \<times> 'c list set" shows "process_prop_6 P \<equiv> \<forall>xs X. xs \<in> snd P \<longrightarrow> (xs, X) \<in> fst P"
optimal_set_def: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and S :: "'a set" shows "optimal_set f S = {x \<in> S. \<forall>y\<in>S. f x y}"
le_cSup_iff: fixes A :: "'a set"   and x :: "'a" assumes "A \<noteq> {}"   and "bdd_above A" shows "(x \<le> Sup A) = (\<forall>y<x. \<exists>a\<in>A. y < a)"
fx: fixes x :: "real"   and t :: "'b" assumes "x \<in> U"   and "t \<in> cbox a b" shows "((\<lambda>x. f x t) has_vector_derivative fx x t) (at x within U)"
epath_filter_selfloop: fixes x :: "'b"   and F :: "'b uprod set"   and a :: "'b"   and p :: "'b uprod list"   and b :: "'b" assumes "epath (insert (Upair x x) F) a p b" shows "\<exists>p. epath F a p b"
rel_filter_neg_distr_cond'_stronger: fixes C :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and C' :: "'e \<Rightarrow> 'f \<Rightarrow> bool" assumes "rel_filter_neg_distr_cond' C C'" shows "rel_pred2_neg_distr_cond C C'"
exec_move_CASI3: fixes ci :: "('addr, 'heap) check_instr"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and e3 :: "'addr expr1"   and h :: "'heap"   and stk :: "'addr val list"   and loc :: "'addr val list"   and pc :: "nat"   and xcp :: "'addr option"   and ta :: "('addr, 'thread_id, 'heap) jvm_thread_action"   and h' :: "'heap"   and stk' :: "'addr val list"   and loc' :: "'addr val list"   and pc' :: "nat"   and xcp' :: "'addr option"   and e1 :: "'addr expr1"   and D :: "String.literal"   and F :: "String.literal"   and e2 :: "'addr expr1"   and v' :: "'addr val"   and v :: "'addr val" assumes "exec_move ci P t e3 h (stk, loc, pc, xcp) ta h' (stk', loc', pc', xcp')" shows "exec_move ci P t (e1\<bullet>compareAndSwap(D\<bullet>F, e2, e3)) h (stk @ [v', v], loc, length (compE2 e1) + length (compE2 e2) + pc, xcp) ta h' (stk' @ [v', v], loc', length (compE2 e1) + length (compE2 e2) + pc', xcp')"
domdConcStart: fixes x :: "int \<times> (int \<times> int \<times> protocol) \<times> (int \<times> int \<times> protocol) \<times> DummyContent"   and a :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators"   and b :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators list" assumes "x \<in> dom (Cp (list2FWpolicy (a # b)))"   and "x \<notin> dom (Cp (list2FWpolicy b))" shows "x \<in> dom (Cp a)"
nth_default_beyond: fixes xs :: "'a list"   and n :: "nat"   and dflt :: "'a" assumes "length xs \<le> n" shows "nth_default dflt xs n = dflt"
LEX_imp_less: fixes A :: "'b set"   and P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "nat \<Rightarrow> 'b"   and g :: "nat \<Rightarrow> 'b"   and k :: "nat" assumes "\<forall>x\<in>A. \<not> P x x"   and "f \<in> SEQ A \<or> g \<in> SEQ A"   and "LEX P f g"   and "\<forall>i<k. f i = g i"   and "f k \<noteq> g k" shows "P (f k) (g k)"
icdf: shows "\<forall>i'. f (Suc i') icd\<^bsup>\<pi>'\<^esup>\<rightarrow> f i'"
Disjunctive_def: shows "Apply.Disjunctive = {x. \<forall>X. x (\<Squnion>X) = \<Squnion>(x ` X)}"
simps: fixes a1 :: "'a set"   and a2 :: "nat"   and a3 :: "'a \<Rightarrow> 'a" shows "swapidseq_ext a1 a2 a3 = (a1 = {} \<and> a2 = 0 \<and> a3 = id \<or> (\<exists>S n p a. a1 = insert a S \<and> a2 = n \<and> a3 = p \<and> swapidseq_ext S n p \<and> a \<notin> S) \<or> (\<exists>S n p a b. a1 = insert a (insert b S) \<and> a2 = Suc n \<and> a3 = Fun.swap a b id \<circ> p \<and> swapidseq_ext S n p \<and> a \<noteq> b))"
finter_fempty_right: fixes A :: "'b fset" shows "A |\<inter>| {||} = {||}"
axioms_1: fixes A :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and B :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and F :: "'d \<Rightarrow> 'e"   and G :: "'d \<Rightarrow> 'e"   and \<tau> :: "'d \<Rightarrow> 'e" assumes "inverse_transformation A B F G \<tau>" shows "category A"
NF_def: fixes A :: "('b \<times> 'b) set" shows "NF A = {a. A `` {a} = {}}"
Suc_2: shows "Suc n = card {x. \<exists>p\<in>set (pders p). poly p x = 0 \<and> a < x \<and> x < b}"
loopsup: fixes M :: "(variable \<Rightarrow> real) set set" assumes "\<forall>Z\<in>M. P Z" shows "P (\<Union> M)"
ctr: fixes x :: "octo"   and y :: "octo" shows "x - y = Octo (Ree x - Ree y) (Im1 x - Im1 y) (Im2 x - Im2 y) (Im3 x - Im3 y) (Im4 x - Im4 y) (Im5 x - Im5 y) (Im6 x - Im6 y) (Im7 x - Im7 y)"
convex_hull_bound: fixes X :: "'a vec set"   and Bnd :: "'a" assumes "X \<subseteq> Bounded_vec Bnd"   and "X \<subseteq> carrier_vec n" shows "convex_hull X \<subseteq> Bounded_vec Bnd"
MapsToCompI: fixes C :: "('a, 'b, 'c) Category_scheme"   and f :: "'b"   and g :: "'b" assumes "Category C"   and "f \<approx>>\<^bsub>C\<^esub> g" shows "f ;;\<^bsub>C\<^esub> g maps\<^bsub>C\<^esub> dom\<^bsub>C\<^esub> f to cod\<^bsub>C\<^esub> g"
kleisli_def: fixes f :: "'e \<Rightarrow> 'e option"   and g :: "'e \<Rightarrow> 'e option" shows "f \<rhd> g \<equiv> \<lambda>x. f x \<bind> g"
order_trans_rules_22: fixes x :: "'b"   and y :: "'b"   and z :: "'b" assumes "x < y"   and "y \<le> z" shows "x < z"
lepoll_lists: fixes A :: "'b set" shows "A \<lesssim> lists A"
Cond_\<tau>execI2: fixes ci :: "('addr, 'heap) check_instr"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and e' :: "'addr expr1"   and h :: "'heap"   and stk :: "'addr val list"   and loc :: "'addr val list"   and pc :: "nat"   and xcp :: "'addr option"   and stk' :: "'addr val list"   and loc' :: "'addr val list"   and pc' :: "nat"   and xcp' :: "'addr option"   and e :: "'addr expr1"   and e'' :: "'addr expr1" assumes "\<tau>exec_move ci P t e' h (stk, loc, pc, xcp) (stk', loc', pc', xcp')" shows "\<tau>exec_move ci P t (if (e) e' else e'') h (stk, loc, Suc (length (compE2 e) + pc), xcp) (stk', loc', Suc (length (compE2 e) + pc'), xcp')"
sappend_snil_id_left: shows "sappend\<cdot>[::] = ID"
filterlim_at_bot_imp_at_infinity: fixes f :: "'a \<Rightarrow> real"   and F :: "'a filter" assumes "filterlim f at_bot F" shows "filterlim f at_infinity F"
point_swap: fixes p :: "'a"   and q :: "'a"   and x :: "'a" assumes "point p"   and "point q" shows "(p \<le> x ; q) = (q \<le> x\<^sup>T ; p)"
sourcenodes_def: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and xs :: "'a list" assumes "CFG sourcenode targetnode valid_edge Entry" shows "CFG.sourcenodes sourcenode xs \<equiv> map sourcenode xs"
finite_set_strict_sorted: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a set" assumes "class.linorder less_eq less"   and "finite A" obtains l :: "'a list" where "linorder.strict_sorted less l" and "set l = A" and "length l = card A"
tt_min_keys: fixes v :: "'a"   and p :: "'a \<Rightarrow>\<^sub>0 'c" assumes "v \<in> keys p" shows "punit.tt p \<preceq> v"
product_path: fixes w :: "('d \<times> 'e) list"   and r\<^sub>1 :: "'f list"   and r\<^sub>2 :: "'g list"   and A :: "('d, 'e, 'f) FSM"   and B :: "('d, 'e, 'g) FSM"   and p\<^sub>1 :: "'f"   and p\<^sub>2 :: "'g" assumes "length w = length r\<^sub>1"   and "length r\<^sub>1 = length r\<^sub>2" shows "path (product A B) (w || r\<^sub>1 || r\<^sub>2) (p\<^sub>1, p\<^sub>2) = (path A (w || r\<^sub>1) p\<^sub>1 \<and> path B (w || r\<^sub>2) p\<^sub>2)"
tendsto_top_iff_ennreal: fixes f :: "'b \<Rightarrow> ennreal"   and F :: "'b filter" shows "(f \<longlongrightarrow> top) F = (\<forall>l\<ge>0. \<forall>\<^sub>F x in F. ennreal l < f x)"
split_H2: shows "(H2_UL, H2_UR, H2_DL, H2_DR) = split_block H2 1 1"
functionEquivalenceOnSets: fixes X :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "\<forall>x\<in>X. f x = g x" shows "f ` X = g ` X"
proper_it'I: fixes it :: "'a \<Rightarrow> ('b \<Rightarrow> bool) \<Rightarrow> ('c \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'b"   and it' :: "'a \<Rightarrow> ('d \<Rightarrow> bool) \<Rightarrow> ('c \<Rightarrow> 'd \<Rightarrow> 'd) \<Rightarrow> 'd \<Rightarrow> 'd" assumes "\<And>s. proper_it (it s) (it' s)" shows "proper_it' it it'"
Int_insert_right_if0: fixes a :: "'a"   and A :: "'a set"   and B :: "'a set" assumes "a \<notin> A" shows "A \<inter> insert a B = A \<inter> B"
aff_dim_translation_eq_subtract: fixes a :: "'c"   and S :: "'c set" shows "aff_dim ((\<lambda>x. x - a) ` S) = aff_dim S"