hcju/isabelleps huggingface dataset:
SUP_cong: fixes A :: "'a set"   and B :: "'a set"   and C :: "'a \<Rightarrow> 'b"   and D :: "'a \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b" assumes "A = B"   and "\<And>x. s2p B x \<Longrightarrow> C x = D x" shows "Sup (\<P> C A) = Sup (\<P> D B)"
choose_pivot3_def: fixes R :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and h :: "'d \<Rightarrow> 'c"   and xs :: "'d list"   and lo :: "nat"   and hi :: "nat" shows "choose_pivot3 R h xs lo hi = ASSERT (lo < length xs) \<bind> (\<lambda>_. ASSERT (hi < length xs) \<bind> (\<lambda>_. let k' = (hi - lo) div 2; k = lo + k' in ASSERT (k < length xs) \<bind> (\<lambda>_. let start = h (xs ! lo); mid = h (xs ! k); end = h (xs ! hi) in if R start mid \<and> R mid end \<or> R end mid \<and> R mid start then RETURN k else if R start end \<and> R end mid \<or> R mid end \<and> R end start then RETURN hi else RETURN lo)))"
lucas_lehmer_mult_assoc: fixes m :: "nat"   and x :: "nat \<times> nat"   and y :: "nat \<times> nat"   and z :: "nat \<times> nat" assumes "0 < m" shows "lucas_lehmer_mult m x (lucas_lehmer_mult m y z) = lucas_lehmer_mult m (lucas_lehmer_mult m x y) z"
sat_\<Phi>: fixes \<xi> :: "Preliminaries.var \<Rightarrow> univ" assumes "wtE \<xi>" shows "satPB \<xi> gPB"
lookupDef_looksup'': fixes n :: "'node"   and g :: "'g"   and v :: "'var" assumes "n \<in> set (\<alpha>n g)" obtains n' :: "'node" and def :: "Def" where "lookupDef g n v = (v, n', def)"
expand_inv_def: fixes \<phi> :: "'c ltlr"   and n_ns :: "('c, 'd) node_scheme \<times> ('c, 'e) node_scheme set" shows "expand_inv \<phi> n_ns \<equiv> case n_ns of (n, ns) \<Rightarrow> expand_inv_node \<phi> n \<and> expand_inv_result \<phi> ns"
item_origin_def: fixes item :: "item" shows "item_origin item = (case item of Item x1 x2 x3 x4 \<Rightarrow> x3)"
Id_on_eqI: fixes a :: "'b"   and b :: "'b"   and A :: "'b set" assumes "a = b"   and "a \<in> A" shows "(a, b) \<in> s2r A"
diff_add_cancel: fixes a :: "'b"   and b :: "'b" shows "a - b + b = a"
fresh_1: fixes a :: "atom"   and trm1 :: "trm"   and trm2 :: "trm" shows "a \<sharp> trm1 EQ trm2 = (a \<sharp> trm1 \<and> a \<sharp> trm2)"
IH2: fixes x :: "'a" assumes "x \<in> carrier G"   and "P x" shows "P (x \<otimes> gen)"
assms_3: shows "\<Theta>,\<Gamma> \<turnstile> Abs T t $ x"
step_redexes_Catch: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'a, 'c) com option"   and r :: "('b, 'a, 'c) com"   and s :: "('b, 'c) xstate"   and r' :: "('b, 'a, 'c) com"   and s' :: "('b, 'c) xstate"   and c\<^sub>2 :: "('b, 'a, 'c) com"   and c :: "('b, 'a, 'c) com" assumes "\<Gamma>\<turnstile> (r, s) \<rightarrow> (r', s')"   and "Catch r c\<^sub>2 \<in> redexes c" shows "\<exists>c'. \<Gamma>\<turnstile> (c, s) \<rightarrow> (c', s') \<and> Catch r' c\<^sub>2 \<in> redexes c'"
lim: shows "AE x in M. ((\<lambda>i. s i x) \<longlongrightarrow> f x) (at_right c)"
C_def: shows "C = (cmod s + r + 1) * (cmod s + r) / x"
FGModuleHom_restrict0_GSubspace: fixes U :: "'v set" assumes "RModule FG (\<cdot>) U \<and> U \<subseteq> V" shows "FGModuleHom G (\<cdot>) U (\<star>) (T \<down> U)"
open_continuous_vimage': fixes s :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and B :: "'c set" assumes "open s"   and "continuous_on s f"   and "open B" shows "open (s \<inter> f -` B)"
W_1: shows "U \<subseteq> f ` (topspace T \<inter> f -` X)"
1: fixes k :: "nat" assumes "\<tau> \<sigma> k \<le> \<tau> \<sigma> i + b" shows "k < Monitor.progress \<sigma> \<phi>1 (plen \<pi>)"
assms: shows "\<forall>PROB as s. P PROB \<and> s \<in> valid_states PROB \<and> as \<in> valid_plans PROB \<longrightarrow> (\<exists>as'. exec_plan s as = exec_plan s as' \<and> subseq as' as \<and> length as' < f PROB)"
WTrt_elim_cases_5: fixes P :: "'addr J_prog"   and h :: "'heap"   and E :: "String.literal \<Rightarrow> ty option"   and e :: "'addr expr"   and T :: "ty" assumes "P,E,h \<turnstile> throw e : T" obtains T :: "ty" where "P,E,h \<turnstile> e : T" and "P \<turnstile> T \<le> Class Throwable"
a_star_redu_NotL_elim: fixes a :: "coname"   and M :: "trm"   and x :: "name"   and R :: "trm" assumes "NotL <a>.M x \<longrightarrow>\<^sub>a* R" shows "\<exists>M'. R = NotL <a>.M' x \<and> M \<longrightarrow>\<^sub>a* M'"
rem_def: shows "rem = (\<Prod>p\<in>prime_factors j - {p}. p ^ multiplicity p j)"
extract_None_iff: fixes P :: "'g \<Rightarrow> bool"   and xs :: "'g list" shows "(List.extract P xs = None) = (\<not> (\<exists>x\<in>set xs. P x))"
simps_3: fixes x2 :: "'f" shows "Some x2 \<noteq> None"
length_replicate: fixes n :: "nat"   and x :: "'g" shows "length (replicate n x) = n"
rev_image_eqI: fixes x :: "'b"   and A :: "'b set"   and b :: "'d"   and f :: "'b \<Rightarrow> 'd" assumes "x \<in> A"   and "b = f x" shows "b \<in> f ` A"
unity: fixes G :: "('a, 'b) monoid_scheme" shows "solvable_seq G {\<one>\<^bsub>G\<^esub>}"
f\<^sub>0_def: shows "f\<^sub>0 = \<tau> i(gap i (LEAST t. \<forall>T. gap i T \<le> gap i t) \<mapsto> 0)"
NTC: shows "NTCatCod \<eta> = CatCod G"
numSt_transC: fixes c :: "('a, 'b) com"   and tval :: "'a \<Rightarrow> 'c \<Rightarrow> bool"   and aval :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and s :: "'c"   and c' :: "('a, 'b) com"   and s' :: "'c" assumes "noWhile c"   and "PL.transC tval aval (c, s) (c', s')" shows "numSt c' < numSt c"
map_le_refl: fixes f :: "'c \<Rightarrow> 'd option" shows "f \<subseteq>\<^sub>m f"
mem_eq: shows "\<forall>x. to_total \<Gamma> x = Low \<longrightarrow> mem\<^sub>1 x = mem\<^sub>2 x"
rep_eq: fixes x :: "'b" shows "Rep_formula (Formula.Pred x) = Pred\<^sub>\<alpha> x"
ucast_down_add: fixes a :: "'b word"   and b :: "'b word" assumes "is_down UCAST('b \<rightarrow> 'c)" shows "UCAST('b \<rightarrow> 'c) (a + b) = UCAST('b \<rightarrow> 'c) a + UCAST('b \<rightarrow> 'c) b"
delay_bisimulation_final_base_flip: fixes trsys1 :: "'g \<Rightarrow> 'h \<Rightarrow> 'g \<Rightarrow> bool"   and trsys2 :: "'i \<Rightarrow> 'j \<Rightarrow> 'i \<Rightarrow> bool"   and bisim :: "'g \<Rightarrow> 'i \<Rightarrow> bool"   and \<tau>move1 :: "'g \<Rightarrow> 'h \<Rightarrow> 'g \<Rightarrow> bool"   and \<tau>move2 :: "'i \<Rightarrow> 'j \<Rightarrow> 'i \<Rightarrow> bool"   and final1 :: "'g \<Rightarrow> bool"   and final2 :: "'i \<Rightarrow> bool" assumes "delay_bisimulation_final_base trsys1 trsys2 bisim \<tau>move1 \<tau>move2 final1 final2" shows "delay_bisimulation_final_base trsys2 trsys1 (flip bisim) \<tau>move2 \<tau>move1 final2 final1"
code: fixes xs :: "'d list" shows "cycle xs = hd xs ## cycle (tl xs @ [hd xs])"
Collect_case_prod: fixes P :: "'b \<Rightarrow> bool"   and Q :: "'c \<Rightarrow> bool" shows "{(a, b). P a \<and> Q b} = Collect P \<times> Collect Q"
list_rec_def: fixes l :: "'c SList.list"   and c :: "'b"   and d :: "'c \<Rightarrow> 'c SList.list \<Rightarrow> 'b \<Rightarrow> 'b" shows "list_rec l c d = List_rec (Rep_List l) c (\<lambda>x y. d (inv Leaf x) (Abs_List y))"
pcp_lim_Hintikka: fixes C :: "'b formula set set"   and S :: "'b formula set" assumes "pcp (C::'b formula set set)"   and "subset_closed (C::'b formula set set)"   and "finite_character (C::'b formula set set)"   and "(S::'b formula set) \<in> (C::'b formula set set)" shows "Hintikka (pcp_lim (C::'b formula set set) (S::'b formula set))"
that: fixes f :: "'b \<Rightarrow> 'a" assumes "Y = f ` X"   and "\<forall>x\<in>X. R (f x) x" shows "thesis"
trans: fixes b :: "'d word"   and a :: "'d word"   and c :: "'d word" assumes "b \<le>s a"   and "c \<le>s b" shows "c \<le>s a"
dest_port_ipv4_def: fixes x :: "int \<times> ((int \<times> int \<times> int \<times> int) \<times> int) \<times> ((int \<times> int \<times> int \<times> int) \<times> int) \<times> 'a" shows "dest_port x \<equiv> (snd \<circ> fst \<circ> snd \<circ> snd) x"
env_ext_def: shows "env_ext \<equiv> \<Lambda> v x \<rho> v'. if v = v' then x else \<rho>\<cdot>v'"
summedBidVectorCharacterization: fixes N :: "'a set"   and G :: "'b set"   and bids :: "'a \<times> 'b set \<Rightarrow> 'c" shows "N \<times> (Pow G - {{}}) = Domain (summedBidVectorRel bids N G)"
up_to_def: fixes P :: "nat \<Rightarrow> (nat \<Rightarrow> 'b) \<Rightarrow> 'b measure"   and M :: "nat \<Rightarrow> 'b measure"   and J :: "nat set" assumes "Ionescu_Tulcea P M" shows "Ionescu_Tulcea.up_to J = (LEAST n. \<forall>i\<ge>n. i \<notin> J)"
x_onAxis: shows "xval x = (0::'a) \<and> yval x = (0::'a) \<and> zval x = (0::'a)"
smult_1_left: fixes p :: "'a poly" shows "smult (1::'a) p = p"
idx_bubble_up_fun_correct2: fixes x :: "(nat \<times> 'b) list \<times> nat option list"   and k :: "nat" assumes "index_of_pqueue x" shows "index_of_pqueue (idx_bubble_up_fun x k)"
axioms_2: fixes return :: "'a \<Rightarrow> 'b"   and bind :: "'b \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b"   and get :: "('c \<Rightarrow> 'b) \<Rightarrow> 'b"   and put :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and altc :: "'d cset \<Rightarrow> ('d \<Rightarrow> 'b) \<Rightarrow> 'b" assumes "monad_state_altc return bind get put altc" shows "monad_altc return bind altc"
sp_F2: shows "sp_fundefs (Fubx_get F2)"
ABinds_restrict_below: fixes S :: "Vars.var set"   and \<Gamma> :: "(Vars.var \<times> exp) list"   and ae :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>" shows "ABinds (restrictA S \<Gamma>)\<cdot>ae \<sqsubseteq> ABinds \<Gamma>\<cdot>ae"
nummul: fixes bs :: "real list"   and t :: "Ferrack.num"   and i :: "int" shows "Inum bs (nummul t i) = Inum bs (Mul i t)"
PiM_cong: fixes I :: "'d set"   and J :: "'d set"   and M :: "'d \<Rightarrow> 'e measure"   and N :: "'d \<Rightarrow> 'e measure" assumes "I = J"   and "\<And>x. x \<in> I \<Longrightarrow> M x = N x" shows "Pi\<^sub>M I M = Pi\<^sub>M J N"
AE_contr: fixes P :: "'a \<Rightarrow> bool" assumes "almost_everywhere M P"   and "AE \<omega> in M. \<not> P \<omega>" shows "False"
lazI: fixes a :: "'a list"   and b :: "'b list"   and P :: "'a \<Rightarrow> 'b \<Rightarrow> bool" assumes "length a = length b"   and "\<And>i. i < length b \<Longrightarrow> P (a ! i) (b ! i)" shows "list_all_zip P a b"
P_subterm: fixes t :: "('fun, 'var) Term.term"   and t' :: "('fun, 'var) Term.term" assumes "homogeneous\<^sub>l\<^sub>s\<^sub>t t \<A> Sec \<and> t \<in> GSMP (trms\<^sub>l\<^sub>s\<^sub>t \<A>)"   and "t' \<sqsubseteq> t" shows "homogeneous\<^sub>l\<^sub>s\<^sub>t t' \<A> Sec \<and> t' \<in> GSMP (trms\<^sub>l\<^sub>s\<^sub>t \<A>)"
small_trans: fixes f :: "'b \<Rightarrow> 'c"   and F :: "'b filter"   and g :: "'b \<Rightarrow> 'c"   and h :: "'b \<Rightarrow> 'c" assumes "f \<in> o[F](g)"   and "g \<in> o[F](h)" shows "f \<in> o[F](h)"
assms_3: shows "bvars\<^sub>s\<^sub>t S \<inter> (subst_domain \<delta> \<union> range_vars \<delta>) = {}"
u_def: shows "u = (\<lambda>x. - 2 / 3 * (4 - x) powr (3 / 2))"
trancl_into_rtrancl: fixes a :: "'e"   and b :: "'e"   and r :: "('e \<times> 'e) set" assumes "(a, b) \<in> r\<^sup>+" shows "(a, b) \<in> r\<^sup>*"
norm_le_zero_iff: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and norm :: "'b \<Rightarrow> real"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and sgn :: "'b \<Rightarrow> 'b"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and x :: "'b" assumes "class.real_normed_vector minus dist norm plus zero uminus scaleR sgn uniformity open" shows "(norm x \<le> 0) = (x = zero)"
execute_bind_eq_SomeI: fixes f :: "'a Heap"   and h :: "heap"   and x :: "'a"   and h' :: "heap"   and g :: "'a \<Rightarrow> 'b Heap"   and y :: "'b"   and h'' :: "heap" assumes "execute f h = Some (x, h')"   and "execute (g x) h' = Some (y, h'')" shows "execute (f \<bind> g) h = Some (y, h'')"
assms_4: shows "\<forall>s\<in>pair ` setops\<^sub>s\<^sub>s\<^sub>t S. \<forall>t\<in>pair ` setops\<^sub>s\<^sub>s\<^sub>t S. (\<exists>\<sigma> \<theta> \<rho>. wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<sigma> \<and> wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<theta> \<and> wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (subst_range \<sigma>) \<and> wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (subst_range \<theta>) \<and> Unifier \<rho> (s \<cdot> \<sigma>) (t \<cdot> \<theta>)) \<longrightarrow> (\<exists>\<delta>. Unifier \<delta> s t)"
lift_subc_1: fixes c :: "'a"   and s :: "'a FOL_Fitting.term"   and t :: "'a FOL_Fitting.term" shows "liftt (subc_term c s t) = subc_term c (liftt s) (liftt t)"
f9: shows "\<not> (\<exists>(a', b', c')\<in>set c. a' \<noteq> 0 \<and> - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and> (\<forall>(d, e, f)\<in>set a. d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 + e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) + f = 0) \<and> (\<forall>(d, e, f)\<in>set b. d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 + e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) + f < 0) \<and> (\<forall>(d, e, f)\<in>set c. d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 + e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) + f \<le> 0) \<and> (\<forall>(d, e, f)\<in>set d. d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 + e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) + f \<noteq> 0))"
mult_assoc: fixes a :: "'a \<Rightarrow> ('b, 'a) Term.term"   and b :: "'a \<Rightarrow> ('b, 'a) Term.term"   and c :: "'a \<Rightarrow> ('b, 'a) Term.term" shows "a \<circ>\<^sub>s b \<circ>\<^sub>s c = a \<circ>\<^sub>s (b \<circ>\<^sub>s c)"
prems_1: shows "tfr\<^sub>s\<^sub>s\<^sub>t\<^sub>p (snd a)"
lunit_simps_1: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i> :: "'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and f :: "'b" assumes "bicategory V H \<a> \<i> src trg"   and "partial_magma.ide V f" shows "partial_magma.arr V (bicategory.lunit V H \<a> \<i> src trg f)"
ign_res_def: fixes P :: "'b assn" shows "P\<down> = (\<lambda>Y s Z. \<exists>Y. P Y s Z)"
restrict_merge_2: fixes I :: "'d set"   and J :: "'d set"   and x :: "'d \<Rightarrow> 'e"   and y :: "'d \<Rightarrow> 'e" assumes "I \<inter> J = {}" shows "restrict (merge I J (x, y)) J = restrict y J"
mult_reals: fixes a :: "complex"   and b :: "complex" assumes "is_real a"   and "is_real b" shows "is_real (a * b)"
balL_def: fixes m :: "nat"   and AB :: "('b \<times> nat) tree"   and b :: "'b"   and C :: "('b \<times> nat) tree" assumes "HBT m" shows "HBT.balL m AB b C = (if ht AB = ht C + m + 1 then case AB of \<langle>A, (a, x), B\<rangle> \<Rightarrow> if ht B \<le> ht A then node A a (node B b C) else case B of \<langle>B\<^sub>1, (ab, x), B\<^sub>2\<rangle> \<Rightarrow> node (node A a B\<^sub>1) ab (node B\<^sub>2 b C) else node AB b C)"
siso_Atm: fixes cval :: "'a \<Rightarrow> 'b \<Rightarrow> real"   and indis :: "('b \<times> 'b) set"   and aval :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and tval :: "'d \<Rightarrow> 'b \<Rightarrow> bool"   and atm :: "'c" assumes "PL_Indis cval indis" shows "PL_Indis.siso aval tval cval indis (Atm atm) = PL_Indis.compatAtm aval indis atm"
lift_pref_profile_def: fixes agents :: "'a set"   and alts :: "'b set"   and agents' :: "'a set"   and alts' :: "'b set"   and R :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool" shows "lift_pref_profile agents alts agents' alts' R = (\<lambda>i x y. x \<in> alts' \<and> y \<in> alts' \<and> i \<in> agents' \<and> (x = y \<or> x \<notin> alts \<or> i \<notin> agents \<or> y \<in> alts \<and> R i x y))"
run_one_step_basic_tee_local_result: fixes d :: "nat"   and i :: "nat"   and s :: "s"   and vs :: "v list"   and ves :: "v list"   and x16 :: "nat"   and s' :: "s"   and vs' :: "v list"   and res :: "res_step" assumes "run_one_step d i (s, vs, ves, $Tee_local x16) = (s', vs', res)" shows "(\<exists>r. res = RSNormal r) \<or> (\<exists>e. res = RSCrash e)"
qtable1: shows "qtable n (fv \<phi>) (mem_restr R) (\<lambda>v. MFOTL.sat \<sigma> (map the v) ne \<phi>) rel1"
a_ptr_disconnected_node_rel_def: fixes h :: "(_) heap" shows "local.a_ptr_disconnected_node_rel h = (\<lambda>(x, y). (cast\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^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 x, cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^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 y)) ` {(document_ptr, disconnected_node). document_ptr |\<in>| document_ptr_kinds h \<and> disconnected_node \<in> set |h \<turnstile> get_disconnected_nodes document_ptr|\<^sub>r}"
cone_def: fixes n :: "nat"   and Vs :: "'b vec set" shows "gram_schmidt.cone n Vs = {x. \<exists>Ws. finite Ws \<and> Ws \<subseteq> Vs \<and> x \<in> gram_schmidt.finite_cone n Ws}"
firstPosDomainForElements: fixes a :: "'b"   and l :: "'b list" shows "(0 \<le> firstPos a l \<and> firstPos a l < length l) = (a \<in> set l)"
outside_inside: fixes S :: "'a set" shows "outside S = - (S \<union> inside S)"
fset_to_fset: fixes A :: "'b set" assumes "finite A" shows "fset (the_inv fset A) = A"
conseq: fixes P :: "(char list \<Rightarrow> int) \<Rightarrow> enat"   and c :: "com"   and Q :: "(char list \<Rightarrow> int) \<Rightarrow> enat"   and k :: "nat"   and P' :: "(char list \<Rightarrow> int) \<Rightarrow> enat"   and Q' :: "(char list \<Rightarrow> int) \<Rightarrow> enat" assumes "\<turnstile>\<^sub>2\<^sub>' {P} c {Q}"   and "\<And>s. P s \<le> enat k * P' s"   and "\<And>s. enat k * Q' s \<le> Q s"   and "0 < k" shows "\<turnstile>\<^sub>2\<^sub>' {P'} c {Q'}"
cd_3: shows "(\<lambda>x. real x powr p * (1 + integral {a..real x} (\<lambda>u. u powr p * ln u powr (p' - 1) / u powr (p + 1)))) \<in> \<Theta>(\<lambda>x. c * real x powr p + d * real x powr p * ln (real x) powr p')"
proj_iff: fixes A :: "'a set"   and r :: "('a \<times> 'a) set"   and x :: "'a"   and y :: "'a" assumes "equiv A r"   and "{x, y} \<subseteq> A" shows "(proj r x = proj r y) = ((x, y) \<in> r)"
assms_2: shows "prim_recfn (Suc n) h"
elims: fixes x :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and xa :: "'c list"   and xb :: "'b"   and y :: "'b list" assumes "scan x xa xb = y" obtains f :: "'c \<Rightarrow> 'b \<Rightarrow> 'b" and a :: "'b" where "x = f__" and "xa = []" and "xb = a__" and "y = []"   | f :: "'c \<Rightarrow> 'b \<Rightarrow> 'b" and xc :: "'c" and xs :: "'c list" and a :: "'b" where "x = f" and "xa = xc # xs" and "xb = a" and "y = f xc a # scan f xs (f xc a)"
is_ub_def: fixes below :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and S :: "'a set"   and x :: "'a" assumes "class.po below" shows "po.is_ub below S x = (\<forall>y\<in>S. below y x)"
unfoldE_2: fixes P :: "pi"   and Q :: "pi"   and s :: "(name \<times> name) list" assumes "P \<simeq>\<^sup>s Q" shows "Q[
] \<leadsto><weakBisim> P[
]"
ale_zle: fixes i :: "int"   and j :: "int" shows "(ant i \<le> ant j) = (i \<le> j)"
case_prod_curry: fixes f :: "'a \<times> 'b \<Rightarrow> 'c" shows "(\<lambda>(x, y). curry f x y) = f"
assms: shows "A \<inter>\<^sub>\<circ> B \<noteq> 0"
irreducible\<^sub>d_dvd_smult: fixes p :: "'b poly"   and q :: "'b poly" assumes "0 < degree p"   and "irreducible\<^sub>d q"   and "p dvd q" shows "\<exists>c. c \<noteq> (0::'b) \<and> q = smult c p"
of_nat_0_le_iff: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and n :: "nat" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "less_eq zero (semiring_1.of_nat one plus zero n)"
mult_cancel1_gr0: fixes k :: "nat"   and m :: "nat"   and n :: "nat" assumes "0 < k" shows "(k * m = k * n) = (m = n)"
qsend_snth: fixes \<omega> :: "(('test, 'atom, 'choice) cmd \<times> 'state) stream"   and n :: "nat" assumes "qsend \<omega> = enat n" shows "discrCf (\<omega> !! n)"
sign_simps_36: fixes am :: "'b"   and bm :: "'b"   and ck :: "'b" shows "am * (bm + ck) = am * bm + am * ck"
convergent_prod_to_zero_iff: fixes f :: "nat \<Rightarrow> 'a" assumes "convergent_prod f" shows "(\<lambda>n. prod f {..n}) \<longlonglongrightarrow> (0::'a) = (\<exists>i. f i = (0::'a))"
interp_alt_def: fixes head :: "'b"   and ops :: "('b \<times> 'b option) list" shows "interp_alt head ops \<equiv> foldl insert_alt {(head, None)} (map (\<lambda>x. case x of (oid, None) \<Rightarrow> (oid, head) | (oid, Some ref) \<Rightarrow> (oid, ref)) ops)"
isolated_other_da: fixes u :: "'a" assumes "u \<notin> {tail G a, head G a}" shows "(u \<in> H.isolated_verts) = (u \<in> G.isolated_verts)"
tc_lower: fixes p :: "'a \<Rightarrow>\<^sub>0 'c"   and v :: "'a" assumes "punit.tt p \<prec> v" shows "punit.tc (punit.lower p v) = punit.tc p"
intros_2: fixes \<phi>s :: "Formula.formula list"   and \<kappa> :: "mregex \<Rightarrow> mregex"   and \<kappa>' :: "Formula.formula Regex.regex \<Rightarrow> Formula.formula Regex.regex"   and mr :: "mregex" assumes "ok_lctxt \<phi>s \<kappa> \<kappa>'" shows "ok_lctxt \<phi>s (\<lambda>t. \<kappa> (MTimes t mr)) (\<lambda>t. \<kappa>' (regex.Times t (from_mregex mr \<phi>s)))"
csi: shows "cs\<^bsup>\<pi> @\<^bsup>k\<^esup> \<pi>l\<^esup> i = cs\<^bsup>\<pi>'\<^esup> i'"
lspasl_starl_der: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and sep_disj :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a \<Rightarrow> bool"   and B :: "'a \<Rightarrow> bool"   and h0 :: "'a" assumes "class.heap_sep_algebra plus zero sep_disj"   and "sep_algebra.sep_conj plus sep_disj A B h0" shows "\<exists>h1 h2. heap_sep_algebra.tern_rel plus sep_disj h1 h2 h0 \<and> A h1 \<and> B h2"
induct: fixes G :: "prog"   and accclass :: "qtname"   and x1 :: "qtname \<times> memberdecl"   and x2 :: "qtname"   and P :: "qtname \<times> memberdecl \<Rightarrow> qtname \<Rightarrow> bool" assumes "G \<turnstile> x1 in x2 dyn_accessible_from accclass"   and "\<And>membr class. \<lbrakk>G \<turnstile> membr member_in class; G \<turnstile> membr in class permits_acc_from accclass\<rbrakk> \<Longrightarrow> P membr class"   and "\<And>membr C new old supr class. \<lbrakk>G \<turnstile> membr member_in class; membr = (C, mdecl new); G \<turnstile> (C, new) overrides old; G\<turnstile>class\<prec>\<^sub>C supr; G \<turnstile>Method old in supr dyn_accessible_from accclass; P (methdMembr old) supr\<rbrakk> \<Longrightarrow> P membr class" shows "P x1 x2"
findnth_state_all_le1: fixes k :: "nat"   and ap :: "abc_inst list"   and n :: "nat"   and a :: "action"   and b :: "nat" assumes "k < length ap"   and "ap ! k = Inc n"   and "(a, b) \<in> set (shift (findnth n) (start_of (layout_of ap) k - Suc 0))" shows "b \<le> start_of (layout_of ap) (length ap)"
Units_bijective: shows "Units = {\<alpha> \<in> S \<rightarrow>\<^sub>E S. bij_betw \<alpha> S S}"
arr_dom: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and f :: "'a" assumes "category C"   and "partial_magma.arr C f" shows "partial_magma.arr C (partial_magma.dom C f)"
vars_mset_arg: fixes t :: "('b, 'a) tm" shows "vars_mset (arg t) \<subseteq># vars_mset t"
that: fixes ns :: "nat list" assumes "\<gamma> = omega_sum ns"   and "normal ns" shows "thesis"
congs_map: fixes F\<^sub>1 :: "'a list"   and F\<^sub>2 :: "'a list"   and f :: "'a \<Rightarrow> 'b" assumes "F\<^sub>1 \<cong> F\<^sub>2" shows "map f F\<^sub>1 \<cong> map f F\<^sub>2"
map: fixes f1 :: "'c \<Rightarrow> 'a"   and f2 :: "'d \<Rightarrow> 'b"   and x1 :: "('c, 'd) form multiset"   and x2 :: "('c, 'd) form multiset" shows "map_sequent f1 f2 ( x1 \<Rightarrow>* x2) = ( image_mset (map_form f1 f2) x1 \<Rightarrow>* image_mset (map_form f1 f2) x2)"
healthy_scalingD: fixes t :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" assumes "healthy t" shows "scaling t"
\<mu>': fixes C :: "'a set set" assumes "C \<subseteq> M"   and "finite C"   and "disjoint C" shows "\<mu>' (\<Union> C) = sum \<mu> C"
simps_2: fixes M2 :: "('e, 'f, 'g) FSM"   and M1 :: "('e, 'f, 'h) FSM"   and \<Omega> :: "('e, 'f) ATC set"   and V :: "'e list set"   and m :: "nat" shows "TS M2 M1 \<Omega> V m (Suc 0) = V"