hcju/isabelleps huggingface dataset:
induct: fixes x1 :: "nat \<Rightarrow> pdf_type"   and x2 :: "expr"   and x3 :: "pdf_type"   and P :: "(nat \<Rightarrow> pdf_type) \<Rightarrow> expr \<Rightarrow> pdf_type \<Rightarrow> bool" assumes "x1 \<turnstile> x2 : x3"   and "\<And>\<Gamma> x. P \<Gamma> (Var x) (\<Gamma> x)"   and "\<And>\<Gamma> v. P \<Gamma> (Val v) (val_type v)"   and "\<And>\<Gamma> e1 t1 e2 t2. \<lbrakk>\<Gamma> \<turnstile> e1 : t1; P \<Gamma> e1 t1; case_nat t1 \<Gamma> \<turnstile> e2 : t2; P (case_nat t1 \<Gamma>) e2 t2\<rbrakk> \<Longrightarrow> P \<Gamma> (LET e1 IN e2) t2"   and "\<And>\<Gamma> e t oper t'. \<lbrakk>\<Gamma> \<turnstile> e : t; P \<Gamma> e t; op_type oper t = Some t'\<rbrakk> \<Longrightarrow> P \<Gamma> (oper $$ e) t'"   and "\<And>\<Gamma> e1 t1 e2 t2. \<lbrakk>\<Gamma> \<turnstile> e1 : t1; P \<Gamma> e1 t1; \<Gamma> \<turnstile> e2 : t2; P \<Gamma> e2 t2\<rbrakk> \<Longrightarrow> P \<Gamma> <e1 ,  e2> (PRODUCT t1 t2)"   and "\<And>\<Gamma> e dst. \<lbrakk>\<Gamma> \<turnstile> e : dist_param_type dst; P \<Gamma> e (dist_param_type dst)\<rbrakk> \<Longrightarrow> P \<Gamma> (Random dst e) (dist_result_type dst)"   and "\<And>\<Gamma> b e1 t e2. \<lbrakk>\<Gamma> \<turnstile> b : BOOL; P \<Gamma> b BOOL; \<Gamma> \<turnstile> e1 : t; P \<Gamma> e1 t; \<Gamma> \<turnstile> e2 : t; P \<Gamma> e2 t\<rbrakk> \<Longrightarrow> P \<Gamma> (IF b THEN e1 ELSE e2) t"   and "\<And>\<Gamma> t. P \<Gamma> (Fail t) t" shows "P x1 x2 x3"
cases: fixes r :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and a1 :: "'a list"   and a2 :: "'b list" assumes "listrelp r a1 a2" obtains "a1 = []" and "a2 = []"   | x :: "'a" and y :: "'b" and xs :: "'a list" and ys :: "'b list" where "a1 = x # xs" and "a2 = y # ys" and "r x y" and "listrelp r xs ys"
LEN: shows "0 < len \<longrightarrow> di + len \<le> length lsrc \<and> si + len \<le> length lsrc"
preservation_of_binary_predicates_and_closures_2: fixes Rel :: "('c \<times> 'c) set"   and Pred :: "'c \<Rightarrow> 'd \<Rightarrow> bool" assumes "rel_preserves_binary_pred Rel Pred" shows "rel_preserves_binary_pred (Rel\<^sup>+) Pred"
f'_bounded: fixes t :: "real"   and x :: "'a" assumes "t \<in> T"   and "x \<in> X" shows "onorm (f' (t, x)) \<le> B'"
greek_less_nonempty: fixes b :: "(accent \<times> 'a) list"   and a :: "(accent \<times> 'a) list"   and r :: "('a \<times> 'a) set" assumes "b \<noteq> []" shows "((a, b) \<in> greek_less r) = ((a, b) \<in> nest r (greek_less r))"
quicksort_postI: fixes xs' :: "'c list"   and xs :: "'c list"   and R :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and h :: "'c \<Rightarrow> 'd"   and lo :: "nat"   and hi :: "nat" assumes "mset xs' = mset xs"   and "sorted_sublist_map R h xs' lo hi"   and "\<And>i. i < lo \<Longrightarrow> xs' ! i = xs ! i"   and "\<And>j. \<lbrakk>hi < j; j < length xs\<rbrakk> \<Longrightarrow> xs' ! j = xs ! j" shows "quicksort_post R h lo hi xs xs'"
lm_2: fixes n :: "nat" shows "snd (A n) \<in> fmeasurable M"
st_inverse: fixes x :: "real star" assumes "x \<in> HFinite"   and "st x \<noteq> 0" shows "st (inverse x) = inverse (st x)"
f3: shows "\<forall>p ps rss nss m. map ((!) rss) (find_nonzeros_from_input_vec (solve_for_lhs_R p ps nss m)) = get_signs_R (reduce_system_R p (ps, m, nss, rss))"
to_nat_a_not_i: shows "to_nat a \<noteq> i"
i_shrink_eq_NoMsg_iAll_finish_event_conv: fixes k :: "nat"   and event :: "nat \<Rightarrow> bool"   and t0 :: "nat"   and t :: "nat"   and s :: "nat \<Rightarrow> 'a message_af" assumes "1 < k"   and "\<And>t. event t = (t mod k = k - 1)"   and "t0 = t * k" shows "((s \<div> k) t = NoMsg) = (s t0 = NoMsg \<and> (\<circle> t' t0 [0\<dots>]. s t1 = NoMsg. t1 \<U> t2 [0\<dots>] \<oplus> t'. event t2 \<and> s t2 = NoMsg))"
less_gexp_aux_antisym: fixes x :: "'b gexp"   and y :: "'b gexp" shows "GExp_Lexorder.less_gexp_aux x y = (\<not> GExp_Lexorder.less_gexp_aux y x \<and> x \<noteq> y)"
order_equiv_trans: fixes A :: "'b set"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "attractive A less_eq"   and "less_eq x y"   and "equivpartp less_eq y z"   and "x \<in> A"   and "y \<in> A"   and "z \<in> A" shows "less_eq x z"
inf_chain_ldropn_chain: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'b llist"   and n :: "nat" assumes "chain R xs"   and "\<not> lfinite xs" shows "chain R (ldropn n xs)"
eq_ucast_ucast_eq: fixes x :: "'c word"   and y :: "'b word" assumes "LENGTH('b) \<le> LENGTH('c)"   and "x = UCAST('b \<rightarrow> 'c) y" shows "UCAST('c \<rightarrow> 'b) x = y"
sd: shows "\<forall>C\<in>T. S \<turnstile> C"
assms_1: shows "s\<^sub>o \<le> ego.s_stop"
strict_mono_onI: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "\<And>r s. \<lbrakk>r \<in> A; s \<in> A; r < s\<rbrakk> \<Longrightarrow> f r < f s" shows "strict_mono_on f A"
assms_2: fixes w :: "'a stream" assumes "R w" shows "\<exists>u v. w = u @- v \<and> u \<noteq> [] \<and> list_all P u \<and> R v"
other: fixes \<sigma> :: "nat \<Rightarrow> 'a"   and \<sigma>' :: "nat \<Rightarrow> 'a" assumes "other U {i} \<sigma> \<sigma>'" shows "Q \<sigma> \<sigma>'"
hermitian_def: fixes A :: "'c mat" shows "hermitian A = (adjoint A = A)"
full_elims_3: fixes n :: "nat"   and l :: "'b tree23"   and p :: "'b"   and m :: "'b tree23"   and q :: "'b"   and r :: "'b tree23" assumes "full n \<langle>l, p, m, q, r\<rangle>" obtains na :: "nat" where "n = Suc na" and "full na l" and "full na m" and "full na r"
inf_absorb1: fixes x :: "'d upred"   and y :: "'d upred" assumes "y \<sqsubseteq> x" shows "(x \<and> y) = x"
app: fixes u\<^sub>1 :: "'a"   and u\<^sub>2 :: "'a" assumes "t = app u\<^sub>1 u\<^sub>2" shows "thesis"
arr_cod_iff_arr: fixes f :: "'a2" shows "C2.arr (C2.cod f) = C2.arr f"
dickson_grading_varnum: fixes X :: "'c set" assumes "finite X" shows "dickson_grading (varnum X)"
Un_Union_image: fixes A :: "'d \<Rightarrow> 'c set"   and B :: "'d \<Rightarrow> 'c set"   and C :: "'d set" shows "(\<Union>x\<in>C. A x \<union> B x) = \<Union> (A ` C) \<union> \<Union> (B ` C)"
single_rt_fresher: fixes rt1 :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option"   and ip :: "nat"   and rt2 :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option" assumes "the (rt1 ip) \<sqsubseteq> the (rt2 ip)" shows "rt1 \<sqsubseteq>\<^bsub>ip\<^esub> rt2"
ring_hom_def: fixes R :: "('e, 'g) ring_scheme"   and S :: "('f, 'h) ring_scheme" shows "ring_hom R S = {h \<in> carrier R \<rightarrow> carrier S. (\<forall>x y. x \<in> carrier R \<and> y \<in> carrier R \<longrightarrow> h (x \<otimes>\<^bsub>R\<^esub> y) = h x \<otimes>\<^bsub>S\<^esub> h y \<and> h (x \<oplus>\<^bsub>R\<^esub> y) = h x \<oplus>\<^bsub>S\<^esub> h y) \<and> h \<one>\<^bsub>R\<^esub> = \<one>\<^bsub>S\<^esub>}"
sum: fixes ccw :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and S :: "'b set"   and X :: "'c set"   and a :: "'b"   and f :: "'c \<Rightarrow> 'b" assumes "ccw_vector_space ccw S"   and "finite X"   and "X \<noteq> {}"   and "\<And>x. x \<in> X \<Longrightarrow> ccw (0::'b) a (f x)" shows "ccw (0::'b) a (sum f X)"
deg_less2: fixes n :: "nat" assumes "0 < n" shows "degree (hensel_fpxs2 $ n) < degree h"
Branch: fixes c :: "'c"   and rs :: "'d \<Rightarrow> ('a, 'b, 'c, 'd) resumption" assumes "x = Branch c rs" shows "thesis"
mod_mult_div_eq: fixes plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and one :: "'e"   and divide :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and modulo :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and a :: "'e"   and b :: "'e" assumes "class.semiring_modulo plus minus zero times one divide modulo" shows "plus (modulo a b) (times b (divide a b)) = a"
lc_set_iff: fixes c :: "'c"   and F :: "('t \<Rightarrow>\<^sub>0 'c) set" shows "(c \<in> lc_set F) = (\<exists>f\<in>F. f \<noteq> 0 \<and> lc f = c)"
h_def: shows "h \<equiv> \<lambda>x. SOME g. g \<in> R \<and> (\<forall>xa\<in>S. \<bar>f xa - g xa\<bar> < 1 / (1 + real x))"
a_def: shows "a = res_lim (f \<circ> s)"
map_add_upds: fixes m1 :: "'c \<Rightarrow> 'd option"   and m2 :: "'c \<Rightarrow> 'd option"   and xs :: "'c list"   and ys :: "'d list" shows "m1 ++ m2(xs [\<mapsto>] ys) = (m1 ++ m2)(xs [\<mapsto>] ys)"
empty_prob_dom: fixes PROB :: "(('a, 'b) fmap \<times> ('a, 'c) fmap) set" assumes "prob_dom PROB = {}" shows "PROB = {(fmempty, fmempty)} \<or> PROB = {}"
le_1_WtFT: fixes cl :: "('d, 'e, 'f) cmd list" shows "0 \<le> 1 - WtFT cl"
positive: shows "\<forall>n w. 0 < prices Mkt pos_stock n w"
rel_U_witness2: fixes Co1 :: "'n \<Rightarrow> 'o \<Rightarrow> bool"   and Co1' :: "'o \<Rightarrow> 'p \<Rightarrow> bool"   and Co2 :: "'q \<Rightarrow> 'r \<Rightarrow> bool"   and Co2' :: "'r \<Rightarrow> 's \<Rightarrow> bool"   and Contra1 :: "'t \<Rightarrow> 'u \<Rightarrow> bool"   and Contra1' :: "'u \<Rightarrow> 'v \<Rightarrow> bool"   and Contra2 :: "'w \<Rightarrow> 'x \<Rightarrow> bool"   and Contra2' :: "'x \<Rightarrow> 'y \<Rightarrow> bool"   and tytok_G :: "(('z, 'n, 'q, 't, 'w, 'aa) U \<times> (('z, 'n, 'q, 't, 'w, 'aa) U \<times> ('ab, 'p, 's, 'v, 'y, 'aa) U) \<times> ('ab, 'p, 's, 'v, 'y, 'aa) U \<times> 'z \<times> ('z \<times> 'ab) \<times> 'ab \<times> 'aa) itself"   and L1 :: "'z \<Rightarrow> 'ab \<Rightarrow> bool"   and x :: "('z, 'n, 'q, 't, 'w, 'aa) U"   and y :: "('ab, 'p, 's, 'v, 'y, 'aa) U" assumes "rel_G_neg_distr_cond Co1 Co1' Co2 Co2' Contra1 Contra1' Contra2 Contra2' tytok_G"   and "rel_U L1 (Co1 OO Co1') (Co2 OO Co2') (Contra1 OO Contra1') (Contra2 OO Contra2') x y" shows "rel_U (\<lambda>(x, y') y. y' = y \<and> L1 x y) Co1' Co2' Contra1' Contra2' (rel_U_witness L1 Co1 Co1' Co2 Co2' Contra1 Contra1' Contra2 Contra2' (x, y)) y"
quot_Mem: fixes x :: "tm"   and y :: "tm" shows "\<guillemotleft>x IN y\<guillemotright> = Q_Mem \<guillemotleft>x\<guillemotright> \<guillemotleft>y\<guillemotright>"
is_parser_error: fixes x :: "char list" shows "is_parser (Parser_Monad.error x)"
T_join_adj: fixes ts :: "'a tree23s" assumes "not_T ts" shows "T_join_adj ts \<le> len ts div 2"
simps: fixes K :: "'a set"   and Us :: "'a list" shows "Span K Us = foldr (line_extension K) Us {\<zero>}"
prod_prime_factors': fixes n :: "nat" assumes "0 < n" shows "f n = (\<Prod>p\<in>prime_factors n. f p ^ multiplicity p n)"
pderivs_simps_1: fixes s :: "'a list" shows "pderivs s Zero = (if s = [] then {Zero} else {})"
wp_g_ode_subset: fixes f :: "'a \<Rightarrow> 'a"   and T :: "real set"   and S :: "'a set"   and \<phi> :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and U :: "'a \<Rightarrow> real set"   and G :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool" assumes "local_flow f T S \<phi>"   and "\<And>s. s2p S s \<Longrightarrow> s2p (U s) 0 \<and> is_interval (U s) \<and> U s \<subseteq> T" shows "wp x\<acute>= \<lambda>t. f & G on U S @ 0 \<lceil>Q\<rceil> = \<lceil>\<lambda>s. s2p S s \<longrightarrow> (\<forall>t\<in>U s. (\<forall>\<tau>\<in>down (U s) t. G (\<phi> \<tau> s)) \<longrightarrow> Q (\<phi> t s))\<rceil>"
seqI': fixes f :: "('e set, 'e \<Rightarrow> 'e) concrete_category.arr"   and a :: "('e set, 'e \<Rightarrow> 'e) concrete_category.arr"   and b :: "('e set, 'e \<Rightarrow> 'e) concrete_category.arr"   and g :: "('e set, 'e \<Rightarrow> 'e) concrete_category.arr"   and c :: "('e set, 'e \<Rightarrow> 'e) concrete_category.arr" assumes "\<guillemotleft>f : a \<rightarrow> b\<guillemotright>"   and "\<guillemotleft>g : b \<rightarrow> c\<guillemotright>" shows "seq g f"
frag_of_in_free_Abelian_group: fixes x :: "'b"   and S :: "'b set" shows "(frag_of x \<in> carrier (free_Abelian_group S)) = (x \<in> S)"
fine_Int: fixes d1 :: "'a \<Rightarrow> 'b set"   and d2 :: "'a \<Rightarrow> 'b set"   and p :: "('a \<times> 'b set) set" shows "((\<lambda>x. d1 x \<inter> d2 x) fine p) = (d1 fine p \<and> d2 fine p)"
inj_on_contraD: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and x :: "'a"   and y :: "'a" assumes "inj_on f A"   and "x \<noteq> y"   and "x \<in> A"   and "y \<in> A" shows "f x \<noteq> f y"
PiE_mem: fixes f :: "'g \<Rightarrow> 'h"   and S :: "'g set"   and T :: "'g \<Rightarrow> 'h set"   and x :: "'g" assumes "f \<in> Pi\<^sub>E S T"   and "x \<in> S" shows "f x \<in> T x"
inf_sup_ord_2: fixes inf :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and less_eq :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and less :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and sup :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and x :: "'k"   and y :: "'k" assumes "class.lattice inf less_eq less sup" shows "less_eq (inf x y) y"
orientation_symmetric: fixes y :: "'a"   and x :: "'a" assumes "y \<squnion> id y = x \<and> y \<sqinter> id y = bot" shows "local.comp_inf.symmetric x"
small_nstep_SucD: fixes small :: "'a \<Rightarrow> 'b \<Rightarrow> 'b set"   and endset :: "'b set"   and \<sigma>' :: "'b"   and P :: "'a"   and \<sigma> :: "'b"   and n :: "nat" assumes "Semantics small endset"   and "\<sigma>' \<in> Semantics.small_nstep small P \<sigma> (Suc n)" shows "\<exists>\<sigma>1. \<sigma>1 \<in> small P \<sigma> \<and> \<sigma>' \<in> Semantics.small_nstep small P \<sigma>1 n"
real_poly_div: fixes p :: "'a poly"   and q :: "'a poly" assumes "set (coeffs p) \<subseteq> \<real>"   and "set (coeffs q) \<subseteq> \<real>" shows "set (coeffs (q div p)) \<subseteq> \<real>"
NSDERIV_NSLIM_iff: fixes f :: "'b \<Rightarrow> 'b"   and x :: "'b"   and D :: "'b" shows "NSDERIV f x :> D = (\<lambda>h. (f (x + h) - f x) / h) \<midarrow>0::'b\<rightarrow>\<^sub>N\<^sub>S D"
code: fixes b :: "bool"   and n :: "real" shows "ln_series_stream_aux b n = MSSCons (if b then - inverse n else inverse n) (ln_series_stream_aux (\<not> b) (n + 1))"
Cd_Above: fixes d :: "'b"   and X :: "'a set" shows "ManiPref'.Cd d X = Above (TruePref.Pd_shuffle_to_top ds' (ManiPref.cop ds) d) (X \<inter> Field (TruePref.Pd_shuffle_to_top ds' (ManiPref.cop ds) d)) \<inter> X"
True_def: shows "True \<equiv> (\<lambda>x. x) = (\<lambda>x. x)"
var_pow_finsum_decomp: fixes K :: "'a set"   and p :: "'a list" assumes "subring K R"   and "p \<in> carrier (K [X])" shows "p = (\<Oplus>\<^bsub>K [X]\<^esub>t\<in>set (dense_repr p). [fst t] \<otimes>\<^bsub>K [X]\<^esub> X [^]\<^bsub>K [X]\<^esub> snd t)"
ras_raw_\<alpha>_def: fixes s :: "'c array \<times> nat" shows "ras_raw_\<alpha> s \<equiv> rev (take (snd s) (list_of_array (fst s)))"
Pair_Enc: fixes X :: "msg"   and Y :: "msg"   and X' :: "msg"   and Y' :: "msg" shows "\<langle>X, Y\<rangle> \<noteq> Enc X' Y'"
siso0_Atm: fixes atm :: "'atom" shows "siso0 (Atm atm) = compatAtm atm"
set_of_elts: fixes x :: "V" shows "ZFC_in_HOL.set (elts x) = x"
valid_frac_def: fixes S :: "'c set"   and R :: "'c set"   and addition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and multiplication :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and unit :: "'c"   and X :: "('c \<times> 'c) set" assumes "Comm_Ring.quotient_ring S R addition multiplication zero unit" shows "quotient_ring.valid_frac S R addition multiplication zero X \<equiv> \<exists>r\<in>R. \<exists>s\<in>S. quotient_ring.frac S R addition multiplication zero r s = X"
keys_except: fixes p :: "'c \<Rightarrow>\<^sub>0 'd"   and S :: "'c set" shows "keys (except p S) = keys p - S"
weakCongruenceE_2: fixes P :: "ccs"   and Q :: "ccs" assumes "P \<cong> Q" shows "Q \<leadsto><weakBisimulation> P"
exhaust_sel: fixes insert_message :: "('c, 'd) insert_message" obtains "insert_message = InsertMessage (P insert_message) (I insert_message) (S insert_message) (\<Sigma> insert_message)"
Zero_In_SUCC: fixes k :: "tm" shows "{OrdP k} \<turnstile> Zero IN SUCC k"
simps_1: shows "mirror \<langle>\<rangle> = \<langle>\<rangle>"
l_3: shows "(v \<circ> a) \<longlonglongrightarrow> liminf (v \<circ> r \<circ> s)"
fds_abs_converges_norm_iff: fixes f :: "'a fds"   and s :: "'a" shows "fds_abs_converges (fds_norm f) (s \<bullet> (1::'a)) = fds_abs_converges f s"
troeger_2: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(x \<squnion> y)\<^sup>\<circ> * z = x\<^sup>\<circ> * (y * (x \<squnion> y)\<^sup>\<circ> * z \<squnion> z)"
reconstruct_lasso_code_def: fixes node_eq_impl :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and node_hash_impl :: "nat \<Rightarrow> 'c \<Rightarrow> nat"   and node_def_hash_size :: "nat"   and G_impl :: "('c \<Rightarrow> bool, 'c \<Rightarrow> 'c list, 'c list, 'c \<Rightarrow> integer, 'd) gen_igbg_impl_scheme" shows "reconstruct_lasso_code node_eq_impl node_hash_impl node_def_hash_size G_impl \<equiv> \<lambda>x xa. reconstruct_reach_code node_eq_impl node_hash_impl node_def_hash_size G_impl x xa \<bind> (\<lambda>(xb, xc). let y' = atLeastLessThan_tr (bs_empty ()) bs_insert 0 (igbgi_num_acc G_impl); y'a = graph_restrict_right_impl (\<lambda>x f. f x) xa (gi_E G_impl) in WHILE\<^sub>T (\<lambda>(xe, xf, xg). \<not> bs_eq xg y') (\<lambda>(xe, xf, xg). wset_find_path_impl node_hash_impl node_def_hash_size node_eq_impl y'a [xe] (\<lambda>xh. \<not> bs_subset_eq (igbgi_acc G_impl xh) xg) \<bind> (\<lambda>xh. let (xi, xj) = the xh in RETURN (xj, xf @ xi, bs_union xg (igbgi_acc G_impl xj)))) (xc, [], igbgi_acc G_impl xc) \<bind> (\<lambda>(xd, xe, xf). (if is_Nil xe then find_path1_impl node_eq_impl node_hash_impl node_def_hash_size y'a xd (node_eq_impl xc) else wset_find_path_impl node_hash_impl node_def_hash_size node_eq_impl y'a [xd] (node_eq_impl xc)) \<bind> (\<lambda>xg. let (xh, xi) = the xg in RETURN (xb, xe @ xh))))"
pr_sesk_auth: fixes evs :: "event list"   and S :: "agent \<times> nat \<times> nat \<Rightarrow> session"   and A :: "msg set"   and U :: "msg set"   and K :: "rat"   and e :: "rat"   and x :: "rat"   and y :: "rat"   and g :: "rat" assumes "(evs, S, A, U) \<in> protocol"   and "Crypt (sesK K) \<lbrace>pubAK e, Auth_Data x y, pubAK g, Crypt (priSK CA) (Hash (pubAK g))\<rbrace> \<in> parts (A \<union> spies evs)" shows "Key (sesK K) \<in> U"
reindex: fixes h :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and g :: "'b \<Rightarrow> 'c" assumes "inj_on h A" shows "sum g (h ` A) = sum (g \<circ> h) A"
l_i: fixes a :: "'a" assumes "a \<in> carrier G" shows "\<rho> a \<cdot> a = \<one>"
symclpE: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "symclp r x y" obtains "r x y"   | "r y x"
snil_append_iff: fixes xs :: "[:'r:]"   and ys :: "[:'r:]" shows "(xs :@ ys = [::]) = (xs = [::] \<and> ys = [::])"
insert_Times_insert: fixes a :: "'d"   and A :: "'d set"   and b :: "'e"   and B :: "'e set" shows "insert a A \<times> insert b B = insert (a, b) (A \<times> insert b B \<union> insert a A \<times> B)"
r1: shows "r k (1::'a) ^ k = (1::'a)"
acyclic_star_below_complement: fixes w :: "'a" shows "pd_kleene_allegory_class.acyclic w = (w\<^sup>T\<^sup>\<star> \<le> - w)"
in_set_member: fixes x :: "'d"   and xs :: "'d list" shows "(x \<in> set xs) = List.member xs x"
distinct_61: fixes x4 :: "nat"   and x8 :: "int" shows "Key x4 \<noteq> Pub_AgrK x8"
set: fixes x :: "'a" assumes "x \<in> lset ys" shows "P x = Q x"
subo2: shows "Subobjs P (last Cs) (last Cs # rev Ds)"
ex_min_finite_set: fixes S :: "'a set"   and f :: "'a \<Rightarrow> nat" assumes "finite S"   and "S \<noteq> {}" shows "\<exists>x. x \<in> S \<and> (\<forall>y\<in>S. f x \<le> f y)"
set_pmf_of_set: fixes S :: "'b set" assumes "S \<noteq> {}"   and "finite S" shows "set_pmf (local.pmf_of_set S) = S"
assms: shows "geom_proc n -` {x} \<noteq> {}"
state_update_vs_allInstances_at_post_noincluding': fixes \<sigma>' :: "nat \<Rightarrow> \<langle>'d\<rangle>\<^sub>\<bottom>"   and oid :: "nat"   and Object :: "'d"   and Type :: "'d \<Rightarrow> \<langle>'e\<rangle>\<^sub>\<bottom>"   and \<sigma> :: "'d state"   and A :: "nat \<Rightarrow> \<langle>nat list list list\<rangle>\<^sub>\<bottom>" assumes "\<And>x. \<sigma>' oid = \<lfloor>x\<rfloor> \<Longrightarrow> x = Object"   and "Type Object = None" shows "Type .allInstances() (\<sigma>, \<lparr>heap = \<sigma>'(oid \<mapsto> Object), assocs = A\<rparr>) = Type .allInstances() (\<sigma>, \<lparr>heap = \<sigma>', assocs = A\<rparr>)"
sq_mtx_times_eq: fixes A :: "'a sq_mtx"   and B :: "'a sq_mtx" shows "A * B = to_mtx (\<chi>i j. \<Sum>k\<in>UNIV. A $$ i $ k * B $$ k $ j)"
phis'_codem_def: fixes p :: "'g \<Rightarrow> ('node \<times> 'val, 'val list) mapping"   and g :: "'g"   and "next" :: "'node \<times> 'val"   and next' :: "'val"   and nodes_of_phis :: "('val, ('node \<times> 'val) set) mapping" shows "uninst_code.phis'_codem p g next next' nodes_of_phis = fold (\<lambda>n. Mapping.update n (map (id(snd next := next')) (the (Mapping.lookup (p g) n)))) (sorted_list_of_set (case Mapping.lookup nodes_of_phis (snd next) of None \<Rightarrow> {} | Some x \<Rightarrow> Set.remove next x)) (Mapping.delete next (p g))"
eventually_at: fixes P :: "'c \<Rightarrow> bool"   and a :: "'c"   and S :: "'c set" shows "eventually P (at a within S) = (\<exists>d>0. \<forall>x\<in>S. x \<noteq> a \<and> dist x a < d \<longrightarrow> P x)"
lfilter_code_2: fixes P :: "'a \<Rightarrow> bool"   and x :: "'a"   and xs :: "'a llist" shows "lfilter P (x #\<^sub>l xs) = (if P x then x #\<^sub>l lfilter P xs else lfilter P xs)"
1: shows "\<forall>a\<in>set as. I\<^sub>R a (x # xs)"
sdrop_bottom_iff: fixes i :: "Integer"   and xs :: "[:'a:]" shows "(sdrop\<cdot>i\<cdot>xs = \<bottom>) = (i = \<bottom> \<or> xs = \<bottom>)"
prems_2: shows "\<forall>ws. length ws = n \<longrightarrow> P (bdd_lookup (Branch l r) ws)"
infsI_sset: fixes w :: "'b stream"   and A :: "'b set" assumes "sset w \<subseteq> A" shows "infs A w"
Inf_greatest: fixes A :: "'a set"   and z :: "'a" assumes "\<And>x. x \<in> A \<Longrightarrow> z \<le> x" shows "z \<le> \<Sqinter> A"
divide_eq_eq: fixes b :: "'a"   and c :: "'a"   and a :: "'a" shows "(b / c = a) = (if c \<noteq> (0::'a) then b = a * c else a = (0::'a))"
set_merge_wrt: fixes rel :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and xs :: "'e list"   and ys :: "'e list" shows "set (merge_wrt rel xs ys) = set xs \<union> set ys"
ex_ncol_cop2: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" shows "\<exists>E F. Coplanar A B C E \<and> Coplanar A B C F \<and> \<not> Col D E F"
step_IH_cong: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and rs1 :: "'c rule list"   and t :: "state"   and rs2 :: "'c rule list"   and r :: "'c rule"   and s :: "state" assumes "\<And>s. \<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>rs1, s\<rangle> \<Rightarrow> t = \<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>rs2, s\<rangle> \<Rightarrow> t" shows "\<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>r # rs1, s\<rangle> \<Rightarrow> t = \<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>r # rs2, s\<rangle> \<Rightarrow> t"
dtor2_def: fixes x :: "'b JF2" shows "dtor2 x = F2map id Abs_JF1 Abs_JF2 (str_final2 (Rep_JF2 x))"
prems_1: fixes ns :: "'node list" assumes "g \<turnstile> Entry g-ns\<rightarrow>n"   and "EntryPath g ns" shows "thesis"
CHAOS\<^sub>S\<^sub>K\<^sub>I\<^sub>P_DF\<^sub>S\<^sub>K\<^sub>I\<^sub>P_refine_F: fixes A :: "'a set" shows "CHAOS\<^sub>S\<^sub>K\<^sub>I\<^sub>P A \<sqsubseteq>\<^sub>F DF\<^sub>S\<^sub>K\<^sub>I\<^sub>P A"
pos_mod_conj: fixes b :: "int"   and a :: "int" assumes "0 < b" shows "0 \<le> a mod b \<and> a mod b < b"
simps: fixes a1 :: "'b"   and a2 :: "'b"   and r :: "('b \<times> 'b) set" shows "((a1, a2) \<in> r\<^sup>*) = ((\<exists>a. a1 = a \<and> a2 = a) \<or> (\<exists>a b c. a1 = a \<and> a2 = c \<and> (a, b) \<in> r\<^sup>* \<and> (b, c) \<in> r))"
split_aforms_lemma: fixes x :: "(real \<times> real pdevs) list"   and y :: "real list set"   and z :: "real list set"   and l :: "nat"   and i :: "nat" assumes "(x, y) \<in> aforms_rel"   and "z \<subseteq> y"   and "l = length x" shows "\<exists>a b. (split_aforms x i, a, b) \<in> aforms_rel \<times>\<^sub>r aforms_rel \<and> env_len a l \<and> env_len b l \<and> z \<subseteq> a \<union> b"
span_induct: fixes scale :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and x :: "'b"   and S :: "'b set"   and P :: "'b \<Rightarrow> bool" assumes "module scale"   and "x \<in> module.span scale S"   and "module.subspace scale (Collect P)"   and "\<And>x. x \<in> S \<Longrightarrow> P x" shows "P x"