hcju/isabelleps huggingface dataset:
evaln_elim_cases_30: fixes G :: "prog"   and s :: "st"   and accC :: "qtname"   and statT :: "ref_ty"   and mode :: "inv_mode"   and e :: "expr"   and mn :: "mname"   and pT :: "ty list"   and p :: "expr list"   and n :: "nat"   and v :: "vals"   and s' :: "state" assumes "G\<turnstile>Norm s \<midarrow>In1l ({accC,statT,mode}e\<cdot>mn( {pT}p))\<succ>\<midarrow>n\<rightarrow> (v, s')" obtains a' :: "val" and s1 :: "state" and vs :: "val list" and s2 :: "state" and va :: "val" and s4 :: "state" where "v = In1 va" and "s' = set_lvars (locals (snd s2)) s4" and "G\<turnstile>Norm s \<midarrow>e-\<succ>a'\<midarrow>n\<rightarrow> s1" and "G\<turnstile>s1 \<midarrow>p\<doteq>\<succ>vs\<midarrow>n\<rightarrow> s2" and "G\<turnstile>check_method_access G accC statT mode \<lparr>name = mn, parTs = pT\<rparr> a' (init_lvars G (invocation_declclass G mode (snd s2) a' statT \<lparr>name = mn, parTs = pT\<rparr>) \<lparr>name = mn, parTs = pT\<rparr> mode a' vs s2) \<midarrow>Methd (invocation_declclass G mode (snd s2) a' statT \<lparr>name = mn, parTs = pT\<rparr>) \<lparr>name = mn, parTs = pT\<rparr>-\<succ>va\<midarrow>n\<rightarrow> s4"
domain: shows "Domainp cr_ratfps = (\<lambda>x. constant_term (denominator x) \<noteq> (0::'b))"
widens_trans: fixes P :: "'c prog"   and Ss :: "ty list"   and Ts :: "ty list"   and Us :: "ty list" assumes "P \<turnstile> Ss [\<le>] Ts"   and "P \<turnstile> Ts [\<le>] Us" shows "P \<turnstile> Ss [\<le>] Us"
tabulation_axioms: shows "tabulation (\<cdot>) (\<star>) \<a> \<i> src trg ((f \<star> g) \<star> h) THfgh.\<rho>\<sigma>.tab (tab\<^sub>0 h \<star> THfgh.p\<^sub>0) (tab\<^sub>1 (f \<star> g) \<star> THfgh.p\<^sub>1)"
mon_fs': fixes fa :: "int poly" assumes "fa \<in> set fs" shows "monic fa"
spec\<^sub>3_def: fixes p :: "prog" shows "spec\<^sub>3 p \<equiv> vars p = vars\<^sub>0 \<and> (\<exists>s\<^sub>H. body_split p s\<^sub>L\<^sub>0 s\<^sub>H \<and> wfs \<Gamma>\<^sub>0 s\<^sub>H \<and> high_proc\<^sub>2 s\<^sub>H \<and> high_proc_no_low_output_change s\<^sub>H)"
sum_card_RF: fixes n :: "nat"   and i :: "nat" assumes "n \<le> k"   and "i < m" shows "(\<Sum>j\<le>n. card (RF j i)) = enum (DF k i) n"
map_spmf_conv_ap: fixes f :: "'b \<Rightarrow> 'a"   and p :: "'b spmf" shows "map_spmf f p = ap_spmf (return_spmf f) p"
ladder: shows "LeftDerivationLadder (\<delta> @ \<alpha>) D L \<gamma>"
hom_deduct_if_hom_ik: fixes M :: "('fun, 'var) Term.term set"   and A :: "('lbl strand_label \<times> ('fun, 'var) strand_step) list"   and Sec :: "('fun, 'var) Term.term set"   and m :: "('fun, 'var) Term.term" assumes "\<langle>M;A;Sec\<rangle> \<turnstile>\<^sub>h\<^sub>o\<^sub>m m"   and "\<forall>m\<in>M. homogeneous\<^sub>l\<^sub>s\<^sub>t m A Sec \<and> m \<in> GSMP (trms\<^sub>l\<^sub>s\<^sub>t A)" shows "homogeneous\<^sub>l\<^sub>s\<^sub>t m A Sec \<and> m \<in> GSMP (trms\<^sub>l\<^sub>s\<^sub>t A)"
intro: fixes set_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and get_ancestors_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" assumes "\<And>ptr h h'. \<forall>w\<in>set_disconnected_nodes_locs ptr. h \<turnstile> w \<rightarrow>\<^sub>h h' \<longrightarrow> (\<forall>r\<in>get_ancestors_locs. r h h')" shows "l_set_disconnected_nodes_get_ancestors set_disconnected_nodes_locs get_ancestors_locs"
generate_pow_card: fixes a :: "nat \<times> nat" assumes "a \<in> carrier S'" shows "S'.ord a = card (generate S' {a})"
powr_mono2': fixes a :: "real"   and x :: "real"   and y :: "real" assumes "a \<le> 0"   and "0 < x"   and "x \<le> y" shows "y powr a \<le> x powr a"
cnj_x_x: fixes x :: "complex" shows "cnj x * x = \<bar>x\<bar>\<^sup>2"
FGModule_def: fixes G :: "'a set"   and smult :: "('b, 'a) aezfun \<Rightarrow> 'c \<Rightarrow> 'c"   and V :: "'c set" shows "FGModule G smult V \<equiv> Group G \<and> RModule (aezfun_setspan G) smult V"
iMOD_I: fixes x :: "nat"   and m :: "nat"   and r :: "nat" assumes "x mod m = r mod m"   and "r \<le> x" shows "x \<in> [ r, mod m ]"
fm_def: shows "fm = (\<lambda>x. max (- f x) 0)"
that_1: assumes "x = None"   and "run n \<sigma> \<sigma>' r" shows "thesis"
min_dist_is_dist: fixes v :: "nat"   and v' :: "nat" assumes "connected v v'" shows "dist v (min_dist v v') v'"
ssum_map_map: fixes f1 :: "'f \<rightarrow> 'e"   and g1 :: "'h \<rightarrow> 'g"   and f2 :: "'i \<rightarrow> 'f"   and g2 :: "'j \<rightarrow> 'h"   and p :: "'i \<oplus> 'j" assumes "f1\<cdot>\<bottom> = \<bottom>"   and "g1\<cdot>\<bottom> = \<bottom>" shows "ssum_map\<cdot>f1\<cdot>g1\<cdot>(ssum_map\<cdot>f2\<cdot>g2\<cdot>p) = ssum_map\<cdot>(\<Lambda> x. f1\<cdot>(f2\<cdot>x))\<cdot>(\<Lambda> x. g1\<cdot>(g2\<cdot>x))\<cdot>p"
set_plus_image: fixes S :: "'b set"   and T :: "'b set" shows "S + T = (\<lambda>(x, y). x + y) ` (S \<times> T)"
map_prod_def: fixes f :: "'f \<Rightarrow> 'h"   and g :: "'g \<Rightarrow> 'i" shows "map_prod f g = (\<lambda>(x, y). (f x, g y))"
vec_to_iarray_def: fixes A :: "('b, 'c) vec" shows "vec_to_iarray A = IArray.of_fun (\<lambda>i. A $ mod_type_class.from_nat i) CARD('c)"
sorted_wrt_upt_iff: fixes rel :: "nat \<Rightarrow> nat \<Rightarrow> bool"   and a :: "nat"   and b :: "nat" shows "sorted_wrt rel [a..<b] = (\<forall>i j. a \<le> i \<longrightarrow> i < j \<longrightarrow> j < b \<longrightarrow> rel i j)"
bilinear_r_lin1: fixes M1 :: "('g, 'a, 'h) Module_scheme"   and M2 :: "('i, 'a, 'h) Module_scheme"   and N :: "('j, 'a, 'h) Module_scheme"   and m1 :: "'g"   and m2 :: "'i"   and r :: "'a"   and f :: "'g \<times> 'i \<Rightarrow> 'j" assumes "R module M1"   and "R module M2"   and "R module N"   and "m1 \<in> carrier M1"   and "m2 \<in> carrier M2"   and "r \<in> carrier R"   and "bilinear_map f R M1 M2 N" shows "f (m1, r \<cdot>\<^sub>s\<^bsub>M2\<^esub> m2) \<plusminus>\<^bsub>N\<^esub> -\<^sub>a\<^bsub>N\<^esub> r \<cdot>\<^sub>s\<^bsub>N\<^esub> f (m1, m2) = \<zero>\<^bsub>N\<^esub>"
extended_simps_8: fixes x :: "'b extended" shows "x \<le> \<stileturn>"
b'_insertion'': fixes x :: "real" shows "insertion (nth_default 0 (xs' @ x # xs)) (isolate_variable_sparse p var (Suc 0)) = b'"
pi_approx': fixes n :: "nat"   and k :: "nat" assumes "even n"   and "0 < n"   and "k \<le> 13 * n - 2" shows "\<bar>pi - pi_approx n\<bar> \<le> inverse (2 ^ k)"
\<epsilon>o_in_hom: fixes x :: "'a" assumes "Adj.C.ide x" shows "\<guillemotleft>Adj'.\<epsilon>o x : F (G' x) \<rightarrow>\<^sub>C x\<guillemotright>"
select_convs_1: fixes prg :: "prog"   and cls :: "qtname"   and lcl :: "lenv"   and more :: "'a" shows "prg \<lparr>prg = prg, cls = cls, lcl = lcl, \<dots> = more\<rparr> = prg"
topspace_subtopology: fixes U :: "'a topology"   and V :: "'a set" shows "topspace (subtopology U V) = topspace U \<inter> V"
not_eq_order_implies_strict: fixes a :: "'b list"   and b :: "'b list" assumes "a \<noteq> b"   and "b \<le>p a" shows "b <p a"
that: fixes \<alpha> :: "'a \<Rightarrow> 'b"   and I :: "'a \<Rightarrow> bool" assumes "R = brp \<alpha> I" shows "thesis"
homeomorphic_imp_homotopy_equivalent_space: fixes X :: "'b topology"   and Y :: "'c topology" assumes "X homeomorphic_space Y" shows "X homotopy_equivalent_space Y"
norm_diff_triangle_ineq: fixes minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and dist :: "'e \<Rightarrow> 'e \<Rightarrow> real"   and norm :: "'e \<Rightarrow> real"   and plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and uminus :: "'e \<Rightarrow> 'e"   and scaleR :: "real \<Rightarrow> 'e \<Rightarrow> 'e"   and sgn :: "'e \<Rightarrow> 'e"   and uniformity :: "('e \<times> 'e) filter"   and "open" :: "'e set \<Rightarrow> bool"   and a :: "'e"   and b :: "'e"   and c :: "'e"   and d :: "'e" assumes "class.real_normed_vector minus dist norm plus zero uminus scaleR sgn uniformity open" shows "norm (minus (plus a b) (plus c d)) \<le> norm (minus a c) + norm (minus b d)"
Int_atLeastAtMost: fixes a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" shows "{a..b} \<inter> {c..d} = {max a c..min b d}"
intros: fixes rel :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and I :: "'b \<Rightarrow> bool" assumes "\<And>x y. \<lbrakk>rel x y; I x\<rbrakk> \<Longrightarrow> I y" shows "invariant rel I"
tendsto_bcontfun_uniform_limit: fixes f :: "'c \<Rightarrow> 'd \<Rightarrow>\<^sub>C 'e"   and l :: "'d \<Rightarrow>\<^sub>C 'e"   and F :: "'c filter" assumes "(f \<longlongrightarrow> l) F" shows "uniform_limit UNIV (\<lambda>x. apply_bcontfun (f x)) (apply_bcontfun l) F"
order_greater_eq_two_iff: fixes f :: "'a perm"   and a :: "'a" shows "(2 \<le> order f a) = (a \<in> affected f)"
max_fun_diff_max: fixes f :: "'b \<Rightarrow> 'a"   and g :: "'b \<Rightarrow> 'a"   and x :: "'b" assumes "f \<noteq> g"   and "f \<in> FINFUNC"   and "g \<in> FINFUNC"   and "x \<in> {a \<in> Field s. f a \<noteq> g a}" shows "(x, s.max_fun_diff f g) \<in> s"
numneg_def: fixes t :: "MIR.num" shows "numneg t \<equiv> nummul t (- 1)"
finite_almost_full_on: fixes A :: "'a set"   and P :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "finite A"   and "reflp_on P A" shows "almost_full_on P A"
Step1_5_1a: fixes p :: "PrIds" shows "\<turnstile> IPImp p \<longrightarrow> (\<forall>l. \<box>$MemInv mm l)"
dom_fun_upd: fixes f :: "'a \<Rightarrow> 'b option"   and x :: "'a"   and y :: "'b option" shows "dom (f(x := y)) = (if y = None then dom f - {x} else insert x (dom f))"
simps_3: fixes bs :: "int list"   and n :: "nat"   and c :: "int"   and a :: "Cooper.num" shows "Inum bs (CN n c a) = c * bs ! n + Inum bs a"
Fun_incl: fixes f :: "'s" assumes "incl f" shows "Fun f = (\<lambda>x\<in>Dom f. x)"
assms_2: fixes x :: "real" assumes "a < ereal x"   and "ereal x < b" shows "(g has_real_derivative g' x) (at x)"
power4_eq: fixes x :: "real" shows "x ^ 4 = x * x * x * x"
symmetric_reachable': fixes v :: "'a"   and w :: "'a" assumes "v \<rightarrow>\<^sup>* w" shows "w \<rightarrow>\<^sup>* v"
all_conj_distrib: fixes P :: "'e \<Rightarrow> bool"   and Q :: "'e \<Rightarrow> bool" shows "(\<forall>x. P x \<and> Q x) = ((\<forall>x. P x) \<and> (\<forall>x. Q x))"
nex: shows "\<not> (\<exists>(a, b, c)\<in>les. a = 0 \<and> b = 0 \<and> c = 0)"
simps_18: fixes r :: "'a rexp"   and s :: "'a rexp"   and v :: "'a rexp"   and va :: "'a rexp" shows "nTimes (Times r s) (Times v va) = Times r (nTimes s (Times v va))"
mult_strict_mono': fixes a :: "'c"   and b :: "'c"   and c :: "'c"   and d :: "'c" assumes "a < b"   and "c < d"   and "(0::'c) \<le> a"   and "(0::'c) \<le> c" shows "a * c < b * d"
dvd: fixes k :: "int" assumes "k dvd p" shows "\<bar>k\<bar> = 1 \<or> \<bar>k\<bar> = p"
simps: fixes R :: "('b \<times> 'b) set"   and xs :: "'b list" shows "rqs'_cost R xs = (if xs = [] then return_pmf 0 else pmf_of_set {..<length xs} \<bind> (\<lambda>i. let x = xs ! i in map_pmf (\<lambda>(n1, y). length xs - 1 + n1 + y) (pair_pmf (rqs'_cost R (lesss R x xs)) (rqs'_cost R (greaters R x xs)))))"
b0: shows "0 < m div 2 ^ multiplicity 2 m div (2 ^ (multiplicity 2 m + 1) - 1)"
whenE_def: fixes P :: "bool"   and f :: "'c \<Rightarrow> (('d + unit) \<times> 'c) set \<times> bool" shows "whenE P f \<equiv> if P then f else returnOk ()"
ereal_ex_split: fixes P :: "ereal \<Rightarrow> bool" shows "(\<exists>x. P x) = (P \<infinity> \<or> (\<exists>x. P (ereal x)) \<or> P (- \<infinity>))"
AE_discrete_difference: fixes X :: "'b set"   and M :: "'b measure" assumes "countable X"   and "\<And>x. x \<in> X \<Longrightarrow> emeasure M {x} = 0"   and "\<And>x. x \<in> X \<Longrightarrow> {x} \<in> sets M" shows "AE x in M. x \<notin> X"
2: shows "(\<Union>i. A i \<inter> b) \<in> smallest_ccdi_sets \<Omega> M"
set_list_of_vec: fixes v :: "'b Matrix.vec" shows "set (list_of_vec v) = set\<^sub>v v"
dagger_annil: fixes x :: "'a" shows "(1::'a) + x \<cdot> (0::'a) = (x \<cdot> (0::'a))\<^sup>\<star>"
simps: fixes callee :: "'i \<Rightarrow> 'j \<Rightarrow> ('k \<times> 'i) spmf"   and c :: "('h, 'j, 'k) gpv"   and s :: "'i" shows "exec_gpv callee c s = the_gpv c \<bind> case_generat (\<lambda>x. return_spmf (x, s)) (\<lambda>out c. callee s out \<bind> (\<lambda>(x, y). exec_gpv callee (c x) y))"
prems: fixes i :: "nat" assumes "i \<le> Suc n" shows "ideal (I i) R"
iso_iff: fixes h :: "'e \<Rightarrow> 'f"   and G :: "('e, 'g) monoid_scheme"   and H :: "('f, 'h) monoid_scheme" shows "(h \<in> Group.iso G H) = (h \<in> hom G H \<and> h ` carrier G = carrier H \<and> inj_on h (carrier G))"
dis_rews_2: shows "is_CB\<cdot>\<bottom> = \<bottom>"
nu_exchange_2: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and l :: "'b \<Rightarrow> 'b"   and u :: "'b \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'b"   and h :: "'b \<Rightarrow> 'b" assumes "class.order less_eq less"   and "ord.galois less_eq l u"   and "ord.isotone less_eq g"   and "ord.isotone less_eq h"   and "order.has_greatest_postfixpoint less_eq (u \<circ> h)"   and "order.has_greatest_postfixpoint less_eq (h \<circ> u)"   and "order.has_greatest_postfixpoint less_eq (h \<circ> g)"   and "order.has_greatest_fixpoint less_eq (g \<circ> h)"   and "order.has_greatest_fixpoint less_eq (h \<circ> g)"   and "ord.lifted_less_eq less_eq (g \<circ> h \<circ> u) (u \<circ> h \<circ> g)" shows "less_eq (order.the_greatest_fixpoint less_eq (h \<circ> g)) (order.the_greatest_fixpoint less_eq (h \<circ> u))"
infinite_remove: fixes S :: "'e set"   and a :: "'e" assumes "infinite S" shows "infinite (S - {a})"
continuous_on_f: fixes S :: "'b set"   and g :: "'b \<Rightarrow> real"   and h :: "'b \<Rightarrow> 'a" assumes "continuous_on S g"   and "continuous_on S h"   and "g ` S \<subseteq> T"   and "h ` S \<subseteq> X" shows "continuous_on S (\<lambda>x. f (g x) (h x))"
atLeastAtMost_subseteq_atLeastLessThan_iff: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" assumes "class.preorder less_eq less" shows "(ord.atLeastAtMost less_eq a b \<subseteq> ord.atLeastLessThan less_eq less c d) = (less_eq a b \<longrightarrow> less_eq c a \<and> less b d)"
transfer: shows "(pcr_CoCalls ===> pcr_CoCalls ===> pcr_CoCalls) (\<union>) coCallsJoin"
hyps_2: shows "Suc n = (LEAST n. \<exists>xs\<in>L. n = length xs)"
vpair_nonzero: fixes x :: "V"   and y :: "V" shows "\<langle>x, y\<rangle> \<noteq> 0"
e'_eq: fixes i :: "nat" assumes "i < min d d'" shows "e' i = e i"
h2: shows "\<forall>I \<omega> X. (\<omega> \<in> game_sem I \<alpha> X) = (\<omega> \<in> game_sem I \<alpha> (selectlike X \<omega> N))"
assms_3: shows "Env\<turnstile> dom (locals (snd s0)) \<guillemotright>t\<guillemotright> A"
point_minus: fixes xA :: "real"   and yA :: "real" shows "- Point xA yA = Point (- xA) (- yA)"
\<rho>: shows "\<sigma> \<in> space (state_measure V \<Gamma>)"
sum_coefs_ge_num_args_if_\<delta>\<^sub>h_eq_0: fixes ground_heads_var :: "'a \<Rightarrow> 'b set"   and gt_sym :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and \<epsilon> :: "nat"   and \<delta> :: "nat"   and extf :: "'b \<Rightarrow> (('b, 'a) tm \<Rightarrow> ('b, 'a) tm \<Rightarrow> bool) \<Rightarrow> ('b, 'a) tm list \<Rightarrow> ('b, 'a) tm list \<Rightarrow> bool"   and arity_sym :: "'b \<Rightarrow> enat"   and arity_var :: "'a \<Rightarrow> enat"   and wt_sym :: "'b \<Rightarrow> hmultiset"   and coef_sym :: "'b \<Rightarrow> nat \<Rightarrow> hmultiset"   and A :: "'a pvar \<Rightarrow> hmultiset"   and s :: "('b, 'a) tm" assumes "tkbo_coefs ground_heads_var gt_sym \<epsilon> \<delta> extf arity_sym arity_var wt_sym coef_sym"   and "tkbo_coefs.legal_passign ground_heads_var \<delta> arity_sym wt_sym A"   and "of_nat \<delta> = 0"   and "arity.wary arity_sym arity_var s" shows "num_args s \<le> sum_coefs (eval_tpoly A (tkbo_coefs.wt ground_heads_var \<delta> arity_sym wt_sym coef_sym s))"
simps: fixes f :: "'b \<Rightarrow> 'b"   and s :: "'b"   and acc :: "'b list"   and x :: "'b" shows "orbit_list_impl f s acc x = (let x' = f x in if x' = s then rev (x # acc) else orbit_list_impl f s (x # acc) x')"
int_of_fbint_inverse: fixes x :: "fbint" shows "fbint (int_of_fbint x) = x"
inj_img_insertE: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and x :: "'d"   and B :: "'d set" assumes "inj_on f A"   and "x \<notin> B"   and "insert x B = f ` A" obtains x' :: "'c" and A' :: "'c set" where "x' \<notin> A'" and "A = insert x' A'" and "x = f x'" and "B = f ` A'"
bounded_iff: fixes b :: "'c"   and c :: "'c"   and a :: "'c" shows "(b \<squnion> c \<le> a) = (b \<le> a \<and> c \<le> a)"
spr_commonKnowledge_def: fixes envObs :: "'e \<Rightarrow> ('e, 'f, 'g) BEState \<Rightarrow> 'h \<times> 'g option"   and cec :: "(('e, 'f, 'g) BEState \<times> ('e, 'f, 'g) BEState) odlist" shows "spr_commonKnowledge envObs cec \<equiv> \<lambda>as s. let r = \<lambda>a. fromList (concat (map (\<lambda>s'. concat (map (\<lambda>s''. if (s', s'') \<in> spr_repRels envObs a then [(s', s'')] else []) (toList cec))) (toList cec))); R = toList (big_union r as) in fromList (memo_list_trancl R s)"
Gets_imp_Says: fixes B :: "agent"   and X :: "msg"   and evs :: "event list" assumes "Gets B X \<in> set evs"   and "evs \<in> yahalom" shows "\<exists>A. Says A B X \<in> set evs"
lm102: fixes l :: "'c set list"   and G :: "'c list" assumes "l \<in> set (all_partitions_list G)"   and "distinct G" shows "distinct l"
ln_Gamma_series_complex_converges_aux: fixes z :: "complex"   and k :: "nat" assumes "z \<noteq> 0"   and "2 * cmod z \<le> real k"   and "2 \<le> k" shows "cmod (z * Ln (1 - 1 / of_nat k) + Ln (z / of_nat k + 1)) \<le> 2 * (cmod z + (cmod z)\<^sup>2) / (real k)\<^sup>2"
RBTid1_def: fixes bal_i :: "nat \<Rightarrow> nat \<Rightarrow> bool" shows "RBTid1 TYPE('b) bal_i \<equiv> RBTi1 TYPE('b) bal_i"
locate_pred_unique: fixes al :: "'b list"   and f :: "'b \<Rightarrow> nat"   and i :: "nat"   and n1_j1 :: "nat \<times> nat"   and n2_j2 :: "nat \<times> nat" assumes "\<And>a. a \<in> set al \<Longrightarrow> 0 < f a"   and "locate_pred f al i n1_j1"   and "locate_pred f al i n2_j2" shows "n1_j1 = n2_j2"
pre_zeta_aux_eq': fixes a :: "real"   and m :: "nat"   and s :: "complex"   and n :: "nat" assumes "0 < a"   and "- 2 * real m < Re s"   and "- 2 * real n < Re s" shows "pre_zeta_aux m a s = pre_zeta_aux n a s"
simps_2: fixes m :: "'b match_expr" shows "matcheq_matchAny (MatchNot m) = (\<not> matcheq_matchAny m)"
graph_plan_card_state_set: fixes vs :: "'d set"   and s :: "('d, 'e) fmap" assumes "finite vs" shows "card (fmdom' (fmrestrict_set vs s)) \<le> card vs"
encode_negative_transition_frame_axiom_valid: fixes sv :: "sat_plan_variable"   and t :: "nat"   and v :: "nat \<times> nat" assumes "sv \<in> atoms (encode_negative_transition_frame_axiom (\<phi> prob_with_noop abs_prob ) t v)"   and "t < h" shows "valid_state_var sv"
execl_all_sub2: fixes P :: "'a \<Rightarrow> instr list"   and x :: "'a"   and P' :: "'b \<Rightarrow> instr list"   and x' :: "'b"   and P'' :: "instr list"   and cfs :: "(int \<times> (char list \<Rightarrow> int) \<times> int list) list"   and I :: "(char list \<Rightarrow> int) \<Rightarrow> int"   and Q :: "(char list \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and F :: "(char list \<Rightarrow> int) \<Rightarrow> int list \<Rightarrow> int list"   and I' :: "(char list \<Rightarrow> int) \<Rightarrow> int"   and Q' :: "(char list \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and F' :: "(char list \<Rightarrow> int) \<Rightarrow> int list \<Rightarrow> int list" assumes "P x @ P' x' @ P'' \<Turnstile> cfs\<box>"   and "\<And>cfs. P x \<Turnstile> cfs\<box> \<Longrightarrow> (case cfs ! 0 of (pc, s, stk) \<Rightarrow> \<lambda>(pc', s', stk'). pc' = pc + size (P x) + I s \<and> Q s s' \<and> stk' = F s stk) (cfs ! (length cfs - 1))"   and "\<And>cfs. P' x' \<Turnstile> cfs\<box> \<Longrightarrow> (case cfs ! 0 of (pc, s, stk) \<Rightarrow> \<lambda>(pc', s', stk'). pc' = pc + size (P' x') + I' s \<and> Q' s s' \<and> stk' = F' s stk) (cfs ! (length cfs - 1))"   and "I (fst (snd (cfs ! 0))) = 0" shows "\<exists>k<length cfs. \<exists>t. (case cfs ! 0 of (pc, s, stk) \<Rightarrow> \<lambda>(pc', s', stk'). pc = 0 \<and> pc' = size (P x) + size (P' x') + I' t \<and> Q s t \<and> Q' t s' \<and> stk' = F' t (F s stk)) (cfs ! k)"
assms_10: shows "pathstart g $ 1 < pathfinish f $ 1"
constant_is_Zp_cauchy: fixes s :: "nat \<Rightarrow> nat \<Rightarrow> int" assumes "is_constant_seq Zp s" shows "is_Zp_cauchy s"
L1: shows "\<forall>u. \<exists>P'. P \<Longrightarrow>\<^sub>lu in P''\<rightarrow>a<c> \<prec> P' \<and> (P', ([(x, c)] \<bullet> Q')[c::=u]) \<in> Rel"
WHILET_eq_WHILE: fixes b :: "'b \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'b nres"   and s0 :: "'b" assumes "WHILE\<^sub>T b f s0 \<noteq> FAIL" shows "WHILE\<^sub>T b f s0 = WHILE b f s0"
trans_less_equiv: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "cmp a b = Less"   and "cmp b c = Equiv" shows "cmp a c = Less"
subtopology_restrict: fixes X :: "'a topology"   and S :: "'a set" shows "subtopology X (topspace X \<inter> S) = subtopology X S"
rel_setD2: fixes R :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and A :: "'g set"   and B :: "'h set"   and y :: "'h" assumes "rel_set R A B"   and "y \<in> B" shows "\<exists>x\<in>A. R x y"
sel_1: fixes x21 :: "'d"   and x22 :: "'d list" shows "hd (x21 # x22) = x21"
sep_add_disj: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and sep_disj :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a" assumes "class.heap_sep_algebra plus zero sep_disj"   and "sep_disj x x" shows "x = zero"
h2a: shows "MPoly_Type.degree p var = 2 \<longrightarrow> p = (\<Sum>i\<le>2. isolate_variable_sparse p var i * Var var ^ i)"
power_transfer: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> bool" assumes "R (1::'b) (1::'c)"   and "(R ===> R ===> R) (*) (*)" shows "(R ===> (=) ===> R) (^) (^)"
is_OpUbx2_def: fixes unboxed :: "('a, 'b, 'c) unboxed" shows "is_OpUbx2 unboxed = (\<exists>x3. unboxed = OpUbx2 x3)"
rl: shows "(c::'a::refinement_lattice)\<^sup>\<omega> ; \<top> \<sqsubseteq> c\<^sup>\<infinity>"
i_satisfies_atom_set'_stronger: fixes Iv :: "'a set \<times> (nat \<Rightarrow> 'b)"   and as :: "('a \<times> 'b atom) set" assumes "Iv \<Turnstile>\<^sub>i\<^sub>a\<^sub>e\<^sub>s as" shows "Iv \<Turnstile>\<^sub>i\<^sub>a\<^sub>s as"
sin_def: shows "sin = (\<lambda>x. \<Sum>n. sin_coeff n *\<^sub>R x ^ n)"
fps_mult_assoc: fixes f :: "('c, 'd) fps"   and g :: "('c, 'd) fps"   and h :: "('c, 'd) fps" shows "f \<cdot> (g \<cdot> h) = f \<cdot> g \<cdot> h"
eq_minus_divide_eq: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inverse :: "'c \<Rightarrow> 'c"   and one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c"   and c :: "'c" assumes "class.division_ring divide inverse one times plus zero minus uminus" shows "(a = uminus (divide b c)) = (if c \<noteq> zero then times a c = uminus b else a = zero)"
b: shows "(\<Sum>b2\<in>#remove1_mset b1 \<B>. int (nat (b1 |\<inter>| b2) choose 2)) = int (\<Sum>b2\<in>#remove1_mset b1 \<B>. nat (b1 |\<inter>| b2) choose 2)"
C: shows "x * \<top> \<sqinter> x ^ o = x"
perm_restrict_id: fixes f :: "'c \<Rightarrow> 'c"   and S :: "'c set" assumes "f permutes S" shows "perm_restrict f S = f"
exists_some_channel: shows "\<exists>i p q. channel i = Some (p, q)"
terminal_def: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and b :: "'b" assumes "category C" shows "category.terminal C b \<equiv> partial_magma.ide C b \<and> (\<forall>a. partial_magma.ide C a \<longrightarrow> (\<exists>!f. partial_magma.in_hom C f a b))"
realistic_executions_def: fixes execs :: "'dom_t \<Rightarrow> 'action_t list list" shows "int.realistic_executions execs \<equiv> \<forall>d. realistic_execution (execs d)"