hcju/isabelleps huggingface dataset:
comm_monoid_axioms: shows "comm_monoid (\<cdot>) 1"
lvl_split: fixes t :: "('c \<times> nat) tree" shows "lvl (split t) = lvl t \<or> lvl (split t) = lvl t + 1 \<and> sngl (split t)"
FF: fixes i :: "'a"   and n :: "nat" assumes "i \<in> K"   and "B / 2 ^ n < r i" shows "\<exists>j. j \<in> F n \<and> \<not> disjnt (cball (a i) (r i)) (cball (a j) (r j)) \<and> cball (a i) (r i) \<subseteq> ball (a j) (5 * r j)"
tab_simps_4: shows "local.dom \<mu> = u\<^sub>1"
IH_4: fixes d :: "nat"   and t' :: "'a tree" assumes "ins n d x r = Unbal t'" shows "height r \<le> height t' \<and> height t' \<le> height r + 1"
pre_between_r12: fixes vs :: "'a list"   and ram1 :: "'a"   and ram2 :: "'a" assumes "pre_between vs ram1 ram2" shows "ram1 \<noteq> ram2"
sWlsNE_def: fixes SEM :: "('g, 'h, 'i, 'j, 'k, 'l, 'm) semDom_scheme" shows "sWlsNE SEM \<equiv> \<forall>s. \<exists>sX. sWls SEM s sX"
aff_dim_subset: fixes S :: "'a set"   and T :: "'a set" assumes "S \<subseteq> T" shows "aff_dim S \<le> aff_dim T"
dirichlet_prod_inverse': fixes f :: "nat \<Rightarrow> 'b"   and i :: "'b" assumes "f 1 * i = (1::'b)" shows "dirichlet_inverse f i \<star> f = (\<lambda>n. if n = 1 then 1::'b else (0::'b))"
jump_poly_mod: fixes q :: "real poly"   and p :: "real poly"   and x :: "real" shows "jump_poly q p x = jump_poly (q mod p) p x"
arr_def: fixes f :: "'a \<times> 'a" shows "VV.arr f \<equiv> VV.domains f \<noteq> {} \<or> VV.codomains f \<noteq> {}"
assms_1: shows "\<not> (x = \<infinity> \<and> y = - \<infinity> \<or> x = - \<infinity> \<and> y = \<infinity>)"
dropWhile_idem: fixes P :: "'b \<Rightarrow> bool"   and xs :: "'b list" shows "dropWhile P (dropWhile P xs) = dropWhile P xs"
Suc: shows "cis (arg (x ^ n)) = cis (arg x * real n)"
derives_induct: fixes a :: "symbol list"   and b :: "symbol list"   and P :: "symbol list \<Rightarrow> bool" assumes "derives a b"   and "P a"   and "\<And>y z. \<lbrakk>derives a y; derives1 y z; P y\<rbrakk> \<Longrightarrow> P z" shows "P b"
M_def: shows "M = (A @\<^sub>c mat_of_col (- b)) @\<^sub>r 0\<^sub>m 1 n @\<^sub>c - 1\<^sub>m 1"
cc'_def: shows "cc' = \<lparr>states = states cc, msgs = msgs cc -# <pJ, inM False> \<union># {# <pJ, inM True> }\<rparr>"
trans: fixes b :: "'b list"   and a :: "'b list"   and c :: "'b list" assumes "prefix b a"   and "prefix c b" shows "prefix c a"
distinct_27: fixes x11 :: "floatarith"   and x12 :: "floatarith"   and x151 :: "floatarith"   and x152 :: "nat" shows "Add x11 x12 \<noteq> x151 ^\<^sub>e x152"
append_Nil2: fixes xs :: "'d list" shows "xs @ [] = xs"
head_if: fixes g :: "nat \<Rightarrow> 'a"   and m :: "nat"   and n :: "nat" shows "prod g {m..n} = (if n < m then 1::'a else prod g {m..<n} * g n)"
additive_wp_PC: fixes a :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and b :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and P :: "'a \<Rightarrow> real" assumes "Transformers.additive (wp a)"   and "Transformers.additive (wp b)" shows "Transformers.additive (wp (a \<^bsub>P\<^esub>\<oplus> b))"
linorder_not_le: fixes x :: "'c"   and y :: "'c" shows "(\<not> x \<le> y) = (y < x)"
cost_splitLeaf: fixes t :: "'a tree"   and a :: "'a"   and w\<^sub>a :: "nat"   and w\<^sub>b :: "nat"   and b :: "'a" assumes "consistent t"   and "a \<in> alphabet t"   and "freq t a = w\<^sub>a + w\<^sub>b" shows "cost (splitLeaf t w\<^sub>a a w\<^sub>b b) = cost t + w\<^sub>a + w\<^sub>b"
wp3'_def: fixes c :: "com"   and Q :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool" shows "wp\<^sub>3\<^sub>' c Q = (\<lambda>(s, n). \<exists>t m. m \<le> n \<and> (c, s) \<Rightarrow>\<^sub>A m \<Down> t \<and> Q (t, n - m))"
subseqs_Cons: fixes ys :: "'d list"   and xs :: "'d list"   and x :: "'d" assumes "ys \<in> set (subseqs xs)" shows "ys \<in> set (subseqs (x # xs))"
r_less_prim: shows "prim_recfn 2 r_less"
2: shows "|\<Union> (range XS)| <o |UNIV|"
fsymsC_def: fixes c :: "('b, 'c) lit list" shows "fsymsC c = \<Union> (set (map fsymsL c))"
l_comb_jointfun_fj: fixes H :: "'a set"   and A :: "'b set"   and s :: "nat \<Rightarrow> 'b"   and n :: "nat"   and m :: "nat"   and f :: "nat \<Rightarrow> 'a"   and g :: "nat \<Rightarrow> 'a" assumes "H \<subseteq> carrier M"   and "ideal R A"   and "s \<in> {j. j \<le> Suc (n + m)} \<rightarrow> A"   and "f \<in> {j. j \<le> n} \<rightarrow> H"   and "g \<in> {j. j \<le> m} \<rightarrow> H" shows "\<Sigma>\<^sub>e M (\<lambda>j. s j \<cdot>\<^sub>s jointfun n f m g j) n = \<Sigma>\<^sub>e M (\<lambda>j. s j \<cdot>\<^sub>s f j) n"
monoid_axioms: shows "monoid (\<squnion>) \<bottom>"
order_gt_0: fixes \<G> :: "'a cyclic_group" assumes "okamoto_base \<G>" shows "0 < order \<G>"
trss_c_no_mon: fixes s :: "'a list"   and c :: "'a list multiset"   and w :: "('b, 'c) label list"   and s' :: "'a list"   and c' :: "'a list multiset" assumes "((s, c), w, s', c') \<in> trcl (trss fg)" shows "mon_c fg c' = mon_c fg c"
mat_add_idem: fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c" shows "f \<oplus> f = f"
ccw_vector_space_axioms_def: fixes ccw :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool" shows "ccw_vector_space_axioms ccw \<equiv> ((\<forall>a x b c. ccw (a + x) (b + x) (c + x) = ccw a b c) \<and> (\<forall>e a b. 0 < e \<longrightarrow> ccw (0::'b) (e *\<^sub>R a) b = ccw (0::'b) a b)) \<and> (\<forall>a b. ccw (0::'b) (- a) b = ccw (0::'b) b a) \<and> (\<forall>a b c. ccw (0::'b) a b \<longrightarrow> ccw (0::'b) c b \<longrightarrow> ccw (0::'b) (a + c) b)"
vconst_on_def: fixes A :: "V"   and c :: "V" shows "vconst_on A c = set {\<langle>a, c\<rangle> |a. a \<in>\<^sub>\<circ> A}"
need_second_arg_def: fixes binop :: "binop"   and v1 :: "val" shows "need_second_arg binop v1 = (\<not> (binop = CondAnd \<and> \<not> the_Bool v1 \<or> binop = CondOr \<and> the_Bool v1))"
domdConcStart: fixes x :: "int \<times> (int \<times> int) \<times> (int \<times> int) \<times> DummyContent"   and a :: "((int \<times> int) set set, int) Combinators"   and b :: "((int \<times> int) set set, int) Combinators list" assumes "x \<in> dom (C (list2FWpolicy (a # b)))"   and "x \<notin> dom (C (list2FWpolicy b))" shows "x \<in> dom (C a)"
simps_6: fixes \<phi> :: "'b ltln"   and \<psi> :: "'b ltln"   and N :: "'b ltln set" shows "\<phi> or\<^sub>n \<psi>[N]\<^sub>\<Pi>\<^sub>2 = \<phi>[N]\<^sub>\<Pi>\<^sub>2 or\<^sub>n \<psi>[N]\<^sub>\<Pi>\<^sub>2"
ext_inject: fixes nodes :: "'c set"   and edges :: "('c \<times> 'c) set"   and more :: "'d"   and nodes' :: "'c set"   and edges' :: "('c \<times> 'c) set"   and more' :: "'d" shows "(\<lparr>nodes = nodes, edges = edges, \<dots> = more\<rparr> = \<lparr>nodes = nodes', edges = edges', \<dots> = more'\<rparr>) = (nodes = nodes' \<and> edges = edges' \<and> more = more')"
tranclp_unfold: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b" shows "r\<^sup>+\<^sup>+ a b \<equiv> (a, b) \<in> {(x, y). r x y}\<^sup>+"
br'_rcm_aux': fixes Q :: "'d set"   and W :: "'d set"   and rcm :: "('d, 'e) ta_rule \<Rightarrow> nat option"   and \<delta> :: "('d, 'e) ta_rule set"   and q :: "'d" assumes "(Q, W, rcm) \<in> br'_invar \<delta>"   and "q \<in> W" shows "{r \<in> \<delta>. q \<in> set (rhsq r) \<and> the (rcm r) \<le> Suc 0} = {r \<in> \<delta>. q \<in> set (rhsq r) \<and> set (rhsq r) \<subseteq> Q - (W - {q})}"
servK_authentic_bis: fixes authK :: "nat"   and servK :: "nat"   and B :: "agent"   and Ts :: "nat"   and servTicket :: "msg"   and evs :: "event list" assumes "Crypt authK \<lbrace>Key servK, Agent B, Number Ts, servTicket\<rbrace> \<in> parts (knows Spy evs)"   and "Key authK \<notin> analz (knows Spy evs)"   and "B \<noteq> Tgs"   and "evs \<in> kerbIV" shows "\<exists>A. Says Tgs A (Crypt authK \<lbrace>Key servK, Agent B, Number Ts, servTicket\<rbrace>) \<in> set evs"
assms_1: shows "b \<in> sphere (0::'a) 1"
kernel_zero: fixes i :: "'a"   and R :: "('a, 'e) ring_scheme"   and S :: "('c, 'f) ring_scheme" assumes "i \<in> a_kernel R S h" shows "h i = \<zero>\<^bsub>S\<^esub>"
i_Exec_Stream_Init_nth_Suc_eq_i_Exec_Stream_nth: fixes trans_fun :: "'b \<Rightarrow> 'a \<Rightarrow> 'a"   and input :: "nat \<Rightarrow> 'b"   and c :: "'a"   and n :: "nat" shows "i_Exec_Comp_Stream_Init trans_fun input c (Suc n) = i_Exec_Comp_Stream trans_fun input c n"
times_fun_def: shows "(\<oplus>) \<equiv> rel_partial_semigroup.times_rel_fun"
prj_simps_1: shows "C.seq gf.prj\<^sub>1 \<p>\<^sub>1[f.leg0 \<cdot> gf.prj\<^sub>0, g.leg1]"
compute_SCC_impl_def: shows "compute_SCC_impl \<equiv> stat_start_nres \<bind> (\<lambda>_. let so = ([], Map.empty) in FOREACH\<^bsup>\<lambda>it (l, s). cscc_outer_invar it (l, oGS_\<alpha> s)\<^esup> V0 (\<lambda>v0 (l, I0). if \<not> is_done_oimpl v0 I0 then let ls = (l, initial_impl v0 I0) in WHILE\<^sub>T\<^bsup>\<lambda>(l, s). cscc_invar v0 (oGS_\<alpha> I0) (l, GS.\<alpha> s)\<^esup> (\<lambda>(l, s). \<not> path_is_empty_impl s) (\<lambda>(l, s). select_edge_impl s \<bind> (\<lambda>(vo, s). case vo of None \<Rightarrow> last_seg_impl s \<bind> (\<lambda>scc. pop_impl s \<bind> (\<lambda>s. let l = scc # l in RETURN (l, s))) | Some v \<Rightarrow> if is_on_stack_impl v s then collapse_impl v s \<bind> (\<lambda>s. RETURN (l, s)) else if \<not> is_done_impl v s then RETURN (l, push_impl v s) else RETURN (l, s))) ls \<bind> (\<lambda>(l, S, B, I, P). RETURN (l, I)) else RETURN (l, I0)) so \<bind> (\<lambda>(l, D). stat_stop_nres \<bind> (\<lambda>_. RETURN l)))"
assert_all_state_sat: fixes t :: "(nat \<times> linear_poly) list"   and as :: "('i \<times> 'a atom) list"   and s' :: "('i, 'a) state" assumes "\<triangle> t"   and "assert_all_state t as = s'"   and "\<not> \<U> s'" shows "\<Turnstile> s'"
dghm_length: shows "vcard \<FF> = 4\<^sub>\<nat>"
primeI: fixes x :: "'a" assumes "prime_elem x"   and "normalize x = x" shows "prime x"
finite_enumerate_mono_iff: fixes S :: "'b set"   and m :: "nat"   and n :: "nat" assumes "finite S"   and "m < card S"   and "n < card S" shows "(wellorder_class.enumerate S m < wellorder_class.enumerate S n) = (m < n)"
assms: shows "P \<Longrightarrow>\<^sub>l a<x> \<prec> P'"
Tarjan_invar_eq: fixes s :: "'v tarjan_state" shows "DFS_invar G tarjan_params s = Tarjan_invar G s"
init_a_def: fixes r :: "'b rexp" shows "init_a r = (True, empty_mrexp r)"
strict_trans2: fixes b :: "'b list"   and a :: "'b list"   and c :: "'b list" assumes "b <p a"   and "c \<le>p b" shows "c <p a"
whisker_left: fixes \<nu> :: "'a"   and \<mu> :: "'a"   and f :: "'a" assumes "seq \<nu> \<mu>"   and "ide f" shows "f \<star> \<nu> \<cdot> \<mu> = (f \<star> \<nu>) \<cdot> (f \<star> \<mu>)"
igSubstAbsIPresIGWlsAbs_fromMOD: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) Recursion.model" assumes "gSubstAbsPresGWlsAbs MOD" shows "igSubstAbsIPresIGWlsAbs (fromMOD MOD)"
empty_state_list_lemma: fixes s :: "('c, 'd) fmap"   and as :: "(('c, 'd) fmap \<times> ('c, 'd) fmap) list" shows "[] \<noteq> state_list s as"
antisym: fixes b :: "'a \<times> 'b"   and a :: "'a \<times> 'b" assumes "less_eq_prod b a"   and "less_eq_prod a b" shows "a = b"
prems_6: shows "older_seniors y n \<subseteq> older_seniors x n"
T_on'_embed: fixes A :: "('a list \<Rightarrow> 'b) \<times> ('a list \<times> 'b \<Rightarrow> 'a \<Rightarrow> (nat \<times> nat list) \<times> 'b)"   and s0 :: "'a list"   and x :: "'b"   and qs :: "'a list" shows "real (Move_to_Front.T_on' A (s0, x) qs) = Move_to_Front.T_on_rand' (Partial_Cost_Model.embed A) (return_pmf (s0, x)) qs"
pair_bidirected_digraph_slim: shows "pair_bidirected_digraph H.slim"
simps_2: fixes e :: "'b equation" shows "subterms_of_lit (Neg e) = subterms_of_eq e"
pprt_def: fixes x :: "'a" shows "pprt x = sup x (0::'a)"
flatten_pi_2_monotone: fixes w :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltln"   and N :: "'b ltln set"   and N' :: "'b ltln set" assumes "w \<Turnstile>\<^sub>n \<phi>[N]\<^sub>\<Pi>\<^sub>2"   and "N \<subseteq> N'" shows "w \<Turnstile>\<^sub>n \<phi>[N']\<^sub>\<Pi>\<^sub>2"
interpret_floatarith_FDERIV_floatariths_append: fixes fas :: "floatarith list"   and x :: "'a"   and ramsch :: "real list" assumes "isFDERIV DIM('a) [0..<DIM('a)] fas (list_of_eucl x @ ramsch)"   and "max_Var_floatariths fas \<le> DIM('a) + length ramsch"   and "length fas = DIM('a)" shows "((\<lambda>x. einterpret fas (list_of_eucl x @ ramsch)) has_derivative (\<lambda>d. einterpret (FDERIV_floatariths fas [0..<DIM('a)] (map Var [DIM('a) + length ramsch..<2 * DIM('a) + length ramsch])) (list_of_eucl x @ ramsch @ list_of_eucl d))) (at x)"
Category_def: fixes C :: "('a, 'b, 'c) Category_scheme" shows "Category C \<equiv> ExtCategory C \<and> Category_axioms C"
assigned_nodes_def: shows "assigned_nodes = (\<lambda>slot. Heap_Error_Monad.bind (get_tag_name slot) (\<lambda>tag. Heap_Error_Monad.bind (if tag \<noteq> ''slot'' then error HierarchyRequestError else return ()) (\<lambda>_. Heap_Error_Monad.bind (get_root_node (cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r slot)) (\<lambda>root. if is_shadow_root_ptr_kind\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r root then Heap_Error_Monad.bind (get_host (the (cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r root))) (\<lambda>host. Heap_Error_Monad.bind (get_child_nodes (cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r host)) (filter_M (\<lambda>slotable. Heap_Error_Monad.bind (find_slot False slotable) (\<lambda>found_slot. return (found_slot = Some slot))))) else return []))))"
swapSyms_def: fixes t :: "'a tree"   and a :: "'a"   and b :: "'a" shows "swapSyms t a b = swapLeaves t (freq t a) a (freq t b) b"
d1_4: fixes K :: "'a set" assumes "K \<in> p1" shows "\<exists>a b. K = cbox a b"
per_init_self: fixes a :: "'c"   and b :: "'c"   and D :: "'c set" assumes "(a, b) \<in> per_init D" shows "a = b"
openin_Inter: fixes \<F> :: "'c set set"   and T :: "'c topology" assumes "finite \<F>"   and "\<F> \<noteq> {}"   and "\<And>X. X \<in> \<F> \<Longrightarrow> openin T X" shows "openin T (\<Inter> \<F>)"
while_option_stop: fixes b :: "'b \<Rightarrow> bool"   and c :: "'b \<Rightarrow> 'b"   and s :: "'b"   and t :: "'b" assumes "while_option b c s = Some t" shows "\<not> b t"
N_saturated: shows "\<forall>N. saturated N \<and> (\<forall>B\<in>Bot. B \<notin> N) \<longrightarrow> (\<exists>B\<in>Bot. \<exists>qi\<in>Q. \<not> entails_q qi N {B})"
\<section>: fixes ka :: "nat"   and kb :: "nat"   and zs :: "nat list" shows "\<not> Form_Body ka kb ys xs zs"
left_tree_disjoint: fixes G :: "('c, 'd) Graph_scheme"   and s :: "'c"   and t :: "'c" assumes "Tree G"   and "s \<rightarrow>\<^bsub>G\<^esub> t" shows "Tree.left_tree G s t \<inter> Tree.left_tree G t s = {}"
pm_of_poly_pCons: fixes x :: "'a"   and c :: "'b"   and p :: "'b poly" shows "pm_of_poly x (pCons c p) = monomial c 0 + punit.monom_mult (1::'b) (monomial 1 x) (pm_of_poly x p)"
countable_oexp: fixes \<alpha> :: "V" assumes "\<alpha> \<in> elts \<omega>1" shows "\<omega> \<up> \<alpha> \<in> elts \<omega>1"
enc_atom_lang_AQ: fixes r :: "nat"   and I :: "('b + 'b set) list"   and p :: "'b"   and n :: "nat"   and a :: "'a" assumes "r < length I"   and "case I ! r of Inl p' \<Rightarrow> p = p' | Inr P \<Rightarrow> p \<in> P"   and "length I = n"   and "a \<in> set \<Sigma>" shows "[enc_atom I p a] \<in> lang n (Atom (AQ r a))"
a_notin_n_pow2: fixes a :: "'a"   and n :: "nat" assumes "a \<in> carrier R"   and "\<not> Unit R a"   and "a \<noteq> \<zero>"   and "0 < n" shows "a^\<^bsup>R n\<^esup> \<notin> R \<diamondsuit>\<^sub>p a \<^bsup>\<diamondsuit>R Suc n\<^esup>"
subspaceI: fixes S :: "'c set" assumes "(0::'c) \<in> S"   and "\<And>x y. \<lbrakk>x \<in> S; y \<in> S\<rbrakk> \<Longrightarrow> x + y \<in> S"   and "\<And>c x. x \<in> S \<Longrightarrow> c * x \<in> S" shows "ideal.subspace S"
assms_2: shows "es\<^sub>1 = map (\<lambda>v. (v, True)) (add_effects_of op)"
bi_unique_def: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool" shows "bi_unique R = ((\<forall>x y z. R x y \<longrightarrow> R x z \<longrightarrow> y = z) \<and> (\<forall>x y z. R x z \<longrightarrow> R y z \<longrightarrow> x = y))"
imp_qf: fixes p :: "fm"   and q :: "fm" assumes "qfree p"   and "qfree q" shows "qfree (imp p q)"
s: fixes k :: "'a \<Rightarrow> nat"   and v :: "'a \<Rightarrow> nat"   and n :: "nat"   and not_in_X :: "'a"   and A :: "('b \<times> ('a, real) cconstraint \<times> ('a set \<times> 'b) pmf) set \<times> ('b \<Rightarrow> ('a, real) cconstraint)"   and \<phi> :: "'b \<times> ('a \<Rightarrow> real) \<Rightarrow> bool"   and \<psi> :: "'b \<times> ('a \<Rightarrow> real) \<Rightarrow> bool"   and s :: "'b \<times> ('a \<Rightarrow> real)" assumes "Probabilistic_Timed_Automaton_Regions_Reachability k v n not_in_X A \<phi> \<psi> s" shows "s \<in> Probabilistic_Timed_Automaton.S A"
ipv6_preferred_to_compressed_pull_out_if: fixes a :: "16 word"   and b :: "16 word"   and c :: "16 word"   and d :: "16 word"   and e :: "16 word"   and f :: "16 word"   and g :: "16 word"   and h :: "16 word" shows "ipv6_preferred_to_compressed (IPv6AddrPreferred a b c d e f g h) = (if 1 < max_zero_streak [a, b, c, d, e, f, g, h] then List_explode (List_replace1 (replicate (max_zero_streak [a, b, c, d, e, f, g, h]) 0) [] (goup_by_zeros [a, b, c, d, e, f, g, h])) else map Some [a, b, c, d, e, f, g, h])"
rts2b: shows "\<forall>p q i. correct p (te p i) \<and> correct q (te q i) \<longrightarrow> \<bar>te p i - te q i\<bar> \<le> \<beta>"
kcomp_def: fixes f :: "'a \<Rightarrow> 'c set"   and g :: "'c \<Rightarrow> 'b set" shows "f \<circ>\<^sub>K g = \<mu> \<circ> \<P> g \<circ> f"
inj_on_singl: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and a0 :: "'a" assumes "inj_on f A"   and "a0 \<notin> A"   and "\<And>a. a \<in> A \<Longrightarrow> f a \<noteq> f a0" shows "inj_on f ({a0} \<union> A)"
prod_neutral_const: fixes f :: "nat \<Rightarrow> 'a" shows "prod f {1..1} = f 1"
Gp_sg_subset: fixes H :: "'a set"   and K :: "'a set" assumes "G \<guillemotright> H "   and "\<natural>H \<guillemotright> K " shows "K \<subseteq> H"
Alm_all_FG_G: fixes w :: "nat \<Rightarrow> 'a set"   and \<psi> :: "'a ltln" shows "\<forall>\<^sub>\<infinity>i. suffix i w \<Turnstile>\<^sub>n F\<^sub>n (G\<^sub>n \<psi>) = suffix i w \<Turnstile>\<^sub>n G\<^sub>n \<psi>"
splus_adds_term_mono: fixes s :: "'a"   and t :: "'a"   and u :: "'t"   and v :: "'t" assumes "s adds t"   and "u adds\<^sub>t v" shows "s \<oplus> u adds\<^sub>t t \<oplus> v"
diff_right_mono: 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" assumes "class.ordered_ab_group_add plus zero minus uminus less_eq less"   and "less_eq a b" shows "less_eq (minus a c) (minus b c)"
monoid_mult_hom_1_def: fixes hom :: "'a \<Rightarrow> 'b" shows "monoid_mult_hom_1 hom \<equiv> monoid_mult_hom hom \<and> one_hom_1 hom"
simps: fixes r :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and a :: "'e" shows "Rangep r a = (\<exists>aa b. a = b \<and> r aa b)"
k: shows "\<forall>x. x \<in> set (i # xs) \<longrightarrow> x < min (nrows A) (ncols A)"
assms_2: fixes n :: "nat" shows "f n \<in> space\<^sub>N (\<LL> p M)"
a_known_ptr_def: fixes ptr :: "(_) object_ptr" shows "ElementClass.known_ptr ptr = (NodeClass.known_ptr ptr \<or> is_element_ptr\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr)"
affected_inverse: fixes f :: "'b perm" shows "affected (inverse f) = affected f"
reducing_def: fixes splitf :: "'b list \<Rightarrow> 'b list \<times> 'b list" shows "reducing splitf = (\<forall>xs ys zs. splitf xs = (ys, zs) \<and> ys \<noteq> [] \<longrightarrow> length zs < length xs)"
rtrancl_induct: fixes a :: "'a"   and b :: "'a"   and r :: "('a \<times> 'a) set"   and P :: "'a \<Rightarrow> bool" assumes "(a, b) \<in> r\<^sup>*"   and "P a"   and "\<And>y z. \<lbrakk>(a, y) \<in> r\<^sup>*; (y, z) \<in> r; P y\<rbrakk> \<Longrightarrow> P z" shows "P b"
1: shows "\<forall>a\<in>set as. I\<^sub>d\<^sub>l\<^sub>o a (x # xs)"
rel_FGf_def: fixes L1 :: "'ac \<Rightarrow> 'am \<Rightarrow> bool"   and L2 :: "'ad \<Rightarrow> 'an \<Rightarrow> bool"   and Co1 :: "'af \<Rightarrow> 'ao \<Rightarrow> bool"   and Co2 :: "'ag \<Rightarrow> 'ap \<Rightarrow> bool"   and Contra1 :: "'ai \<Rightarrow> 'aq \<Rightarrow> bool"   and Contra2 :: "'aj \<Rightarrow> 'ar \<Rightarrow> bool" shows "rel_FGf L1 L2 Co1 Co2 Contra1 Contra2 = rel_F L1 L2 (=) Co1 Co2 (=) Contra1 Contra2 (=)"
until_not_relesased_now: fixes \<phi> :: "'c stream \<Rightarrow> bool"   and \<psi> :: "'c stream \<Rightarrow> bool"   and \<omega> :: "'c stream" assumes "(\<phi> until \<psi>) \<omega>"   and "\<not> \<psi> \<omega>" shows "\<phi> \<omega>"
HDomain_Incl_Zero: fixes H :: "fm set"   and r :: "tm" shows "H \<turnstile> HDomain_Incl r Zero"
pow_mat_dim_1: fixes A :: "'a mat"   and k :: "nat" shows "dim_row (A ^\<^sub>m k) = dim_row A"
f_Exec_Stream_Init_eq_f_Exec_Stream_Cons: fixes trans_fun :: "'e \<Rightarrow> 'd \<Rightarrow> 'd"   and xs :: "'e list"   and c :: "'d" shows "f_Exec_Comp_Stream_Init trans_fun xs c = c # f_Exec_Comp_Stream trans_fun xs c"
domains_comp: fixes g :: "'c"   and f :: "'c" assumes "C.seq g f" shows "C.domains (C g f) = C.domains f"
gcd_mult_right_left_cancel: fixes a :: "'a"   and c :: "'a"   and b :: "'a" assumes "coprime a c" shows "gcd a (c * b) = gcd a b"
pdevs_val_uminus_pdevs: fixes e :: "nat \<Rightarrow> real"   and x :: "'a pdevs" shows "pdevs_val e (uminus_pdevs x) = - pdevs_val e x"
bounded_iff: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "(a \<sqsubseteq> b \<sqinter> c) = (a \<sqsubseteq> b \<and> a \<sqsubseteq> c)"
pairs_def: fixes phi :: "'c \<Rightarrow> 'd Filters.filter" assumes "triple phi" shows "triple.pairs phi = {(x, y). \<exists>z. y = phi (- x) \<squnion> up_filter z}"
case_3: fixes f1 :: "'a"   and f2 :: "'a"   and f3 :: "'a"   and f4 :: "'a"   and f5 :: "char list \<Rightarrow> 'a" shows "(case Integer of Void \<Rightarrow> f1 | Boolean \<Rightarrow> f2 | Integer \<Rightarrow> f3 | NT \<Rightarrow> f4 | Class x \<Rightarrow> f5 x) = f3"
floor_add_int: fixes x :: "'a"   and z :: "int" shows "\<lfloor>x\<rfloor> + z = \<lfloor>x + of_int z\<rfloor>"
swap_preserves_fresh: fixes X :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and xs :: "'varSort"   and ys :: "'varSort"   and y1 :: "'var"   and y2 :: "'var"   and x :: "'var" assumes "good X" shows "fresh xs (x @xs[y1 \<and> y2]_ys) (X #[y1 \<and> y2]_ys) = fresh xs x X"
frameFresh: fixes x :: "name"   and A\<^sub>F :: "name list"   and \<Psi>\<^sub>F :: "'b" assumes "x \<sharp> A\<^sub>F"   and "x \<sharp> \<langle>A\<^sub>F, \<Psi>\<^sub>F\<rangle>" shows "x \<sharp> \<Psi>\<^sub>F"
mem_affine: fixes S :: "'a set"   and x :: "'a"   and y :: "'a"   and u :: "real"   and v :: "real" assumes "affine S"   and "x \<in> S"   and "y \<in> S"   and "u + v = 1" shows "u *\<^sub>R x + v *\<^sub>R y \<in> S"
l2: shows "(abscissa A - abscissa X) * (ordinate A - ordinate B) = (ordinate A - ordinate X) * (abscissa A - abscissa B) \<and> (abscissa C - abscissa X) * (ordinate C - ordinate D) = (ordinate C - ordinate X) * (abscissa C - abscissa D)"
wb_lens_def: fixes x :: "'c \<Longrightarrow> 'd" shows "wb_lens x \<equiv> weak_lens x \<and> wb_lens_axioms x"
lc_eq: shows "lc p + lc q = (0::'b)"
plus_right_distributivity: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "a + b * c = a * c + (b * c)"