hcju/isabelleps huggingface dataset:
widen_trans: fixes G :: "prog"   and S :: "ty"   and U :: "ty"   and T :: "ty" assumes "G\<turnstile>S\<preceq>U"   and "G\<turnstile>U\<preceq>T"   and "wf_prog G" shows "G\<turnstile>S\<preceq>T"
intro: fixes intT :: "'a \<Rightarrow> univ \<Rightarrow> bool" assumes "M.Tstruct intT" shows "CM.Tstruct intT"
V_def: shows "V = (\<Union>x. U (1 / (real x + 2)))"
simps_3: fixes A :: "'c llist set" assumes "\<not> (\<forall>x\<in>A. lnull x)" shows "lhd (lSup A) = (THE x. x \<in> lhd ` (A \<inter> {xs. \<not> lnull xs}))"
unlex_cases: fixes y :: "'b list" assumes "y \<in> {xs. True}" obtains x :: "'b lexlist" where "y = unlex x"
carr_prodrg_mem_eq: fixes f :: "'d \<Rightarrow> 'e"   and I :: "'d set"   and A :: "'d \<Rightarrow> ('e, 'f) Ring_scheme"   and g :: "'d \<Rightarrow> 'e" assumes "f \<in> carrier (r\<Pi>\<^bsub>I\<^esub> A)"   and "g \<in> carrier (r\<Pi>\<^bsub>I\<^esub> A)"   and "\<forall>i\<in>I. f i = g i" shows "f = g"
baI: fixes \<phi> :: "'BB \<Rightarrow> bool"   and t :: "nat \<Rightarrow> cnf"   and n :: "nat"   and t' :: "nat \<Rightarrow> 'BB" assumes "\<phi> (\<sigma>\<^bsub>the_bb\<^esub>t n)" shows "pb.eval the_bb t t' n [\<phi>]\<^sub>b"
smallD: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'b"   and c :: "real" assumes "f \<in> \<omega>[F](g)"   and "0 < c" shows "\<forall>\<^sub>F x in F. c * norm (g x) \<le> norm (f x)"
Signature_axioms: shows "Sig.Signature TYPE('c) wtFsym wtPsym"
word_rev_gal: fixes w :: "'b word"   and u :: "'b word" assumes "word_reverse w = u" shows "word_reverse u = w"
Max_less_iMin_imp_disjoint: fixes A :: "'b set"   and B :: "'b set" assumes "finite A"   and "Max A < iMin B" shows "A \<inter> B = {}"
run_put_state: fixes s :: "'s"   and m :: "('s, 'm) stateT"   and s' :: "'s" shows "run_state (local.put_state s m) s' = run_state m s"
lm059: fixes G :: "'d set"   and N :: "'c set" shows "allAllocations N G \<subseteq> {a. Domain a \<subseteq> N}"
ss': shows "fst (TSdet init h (as @ [x] @ bs @ [x] @ cs) (length (as @ [x] @ bs @ [x]) + i)) = fst (config\<^sub>p (rTS h) init (as @ [x] @ bs @ [x] @ take i cs))"
f21_split_face: shows "f21 = snd (split_face f v u ws)"
minus: fixes ops :: "'b arith_ops_record"   and R :: "'b \<Rightarrow> 'c \<Rightarrow> bool" assumes "ring_ops ops R" shows "(R ===> R ===> R) (arith_ops_record.minus ops) (-)"
ga_trans: shows "(g, ga) \<in> br g.\<alpha> g.invar"
assms_1: shows "\<not> hb (i, Append i e1) (r, Store e2 mo r)"
closest_pair_rec_code_eq: fixes xs :: "(int \<times> int) list"   and ys :: "(int \<times> int) list"   and c\<^sub>0 :: "int \<times> int"   and c\<^sub>1 :: "int \<times> int"   and ys' :: "(int \<times> int) list"   and \<delta>' :: "int"   and c\<^sub>0' :: "int \<times> int"   and c\<^sub>1' :: "int \<times> int" assumes "1 < length xs"   and "(ys, c\<^sub>0, c\<^sub>1) = closest_pair_rec xs"   and "(ys', \<delta>', c\<^sub>0', c\<^sub>1') = closest_pair_rec_code xs" shows "c\<^sub>0 = c\<^sub>0' \<and> c\<^sub>1 = c\<^sub>1'"
hinv_mkt: fixes l :: "'a tree"   and r :: "'a tree"   and x :: "'a" assumes "hinv l"   and "hinv r" shows "hinv (mkt x l r)"
br'_inner_step_proof: fixes \<alpha> :: "'a \<Rightarrow> ('b, 'c) ta_rule set"   and invar :: "'a \<Rightarrow> bool"   and iteratei :: "'a \<Rightarrow> ('d \<Rightarrow> bool) \<Rightarrow> (('b, 'c) ta_rule \<Rightarrow> 'd \<Rightarrow> 'd) \<Rightarrow> 'd \<Rightarrow> 'd"   and cinvar :: "('b, 'c) ta_rule set \<Rightarrow> 'd \<Rightarrow> bool"   and \<delta> :: "('b, 'c) ta_rule set"   and q :: "'b"   and \<Sigma>h :: "'d"   and cstep :: "('b, 'c) ta_rule \<Rightarrow> 'd \<Rightarrow> 'd"   and \<alpha>s :: "'d \<Rightarrow> 'b set \<times> 'b set \<times> (('b, 'c) ta_rule \<Rightarrow> nat option)"   and it_set :: "'a"   and W :: "'b set"   and \<Sigma> :: "'d"   and Q :: "'b set"   and rcm :: "('b, 'c) ta_rule \<Rightarrow> nat option" assumes "set_iteratei \<alpha> invar iteratei"   and "cinvar {r \<in> \<delta>. q \<in> set (rhsq r)} \<Sigma>h"   and "\<And>it r \<Sigma>. \<lbrakk>r \<in> it; it \<subseteq> {r \<in> \<delta>. q \<in> set (rhsq r)}; cinvar it \<Sigma>\<rbrakk> \<Longrightarrow> cinvar (it - {r}) (cstep r \<Sigma>)"   and "\<And>it r \<Sigma>. \<lbrakk>r \<in> it; it \<subseteq> {r \<in> \<delta>. q \<in> set (rhsq r)}; cinvar it \<Sigma>\<rbrakk> \<Longrightarrow> \<alpha>s (cstep r \<Sigma>) = br'_inner_step r (\<alpha>s \<Sigma>)"   and "invar it_set"   and "\<alpha> it_set = {r \<in> \<delta>. q \<in> set (rhsq r)}"   and "q \<in> W"   and "\<alpha>s \<Sigma> = (Q, W, rcm)"   and "\<alpha>s \<Sigma>h = (Q, W - {q}, rcm)" shows "(\<alpha>s \<Sigma>, \<alpha>s (iteratei it_set (\<lambda>_. True) cstep \<Sigma>h)) \<in> br'_step \<delta>"
ostep_invariantI: fixes A :: "('a \<times> 'b, 'c) automaton"   and S :: "'a \<Rightarrow> 'a \<Rightarrow> 'c \<Rightarrow> bool"   and U :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and P :: "('a \<times> 'b) \<times> 'c \<times> 'a \<times> 'b \<Rightarrow> bool" assumes "\<And>\<sigma> s a \<sigma>' s'. \<lbrakk>(\<sigma>, s) \<in> oreachable A S U; ((\<sigma>, s), a, \<sigma>', s') \<in> automaton.trans A; S \<sigma> \<sigma>' a\<rbrakk> \<Longrightarrow> P ((\<sigma>, s), a, \<sigma>', s')" shows "A \<Turnstile>\<^sub>A (S, U \<rightarrow>) P"
append_eq_append_conv: fixes xs :: "'a list"   and ys :: "'a list"   and us :: "'a list"   and vs :: "'a list" assumes "|xs| = |ys| \<or> |us| = |vs|" shows "(xs @ us = ys @ vs) = (xs = ys \<and> us = vs)"
rev_star1: fixes s :: "'b list"   and A :: "'b list set" assumes "s \<in> (rev ` A)\<star>" shows "s \<in> rev ` A\<star>"
rel_cfg_state: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and cfg1 :: "'a cfg"   and cfg2 :: "'b cfg" assumes "rel_cfg P cfg1 cfg2" shows "P (state cfg1) (state cfg2)"
abs_2: shows "\<Theta>,\<Gamma> \<turnstile> Abs T t $ x"
MPoly_inverse: fixes y :: "(nat \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'a" assumes "y \<in> UNIV" shows "mapping_of (MPoly y) = y"
simps_1: fixes n :: "n" shows "mul2 Z n = Z"
hom_add: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier G"   and "y \<in> carrier G" shows "h (x \<oplus> y) = h x \<oplus>\<^bsub>H\<^esub> h y"
replicate_mset_eq_empty_iff: fixes n :: "nat"   and a :: "'k" shows "(replicate_mset n a = {#}) = (n = 0)"
assms_5: shows "h \<turnstile> get_root_node ptr \<rightarrow>\<^sub>h h'"
f_Exec_Stream_Acc_Output__State_Idle_eq: fixes l :: "nat"   and k :: "nat"   and localState :: "'d \<Rightarrow> 'e"   and trans_fun :: "'f message_af \<Rightarrow> 'd \<Rightarrow> 'd"   and xs :: "'f message_af list"   and output_fun :: "'d \<Rightarrow> 'g message_af"   and c :: "'d" assumes "0 < l"   and "l \<le> k"   and "Exec_Equal_State localState trans_fun"   and "\<And>n. n < length xs \<Longrightarrow> State_Idle localState output_fun trans_fun (f_Exec_Comp_Stream_Acc_LocalState l localState trans_fun xs c ! n)" shows "f_Exec_Comp_Stream_Acc_Output k output_fun trans_fun xs c = f_Exec_Comp_Stream_Acc_Output l output_fun trans_fun xs c"
assms_7: fixes scrut :: "exp"   and e1 :: "exp"   and e2 :: "exp"   and c :: "'a" assumes "\<And>c. P c scrut"   and "\<And>c. P c e1"   and "\<And>c. P c e2" shows "P c (scrut ? e1 : e2)"
subst_eq_cl: fixes \<sigma> :: "('b \<times> 'b trm) list"   and \<eta> :: "('b \<times> 'b trm) list"   and C :: "'b literal set" assumes "\<sigma> \<doteq> \<eta>" shows "subst_cl C \<sigma> = subst_cl C \<eta>"
lossless_return_spmf: fixes x :: "'a" shows "lossless_spmf (return_spmf x)"
witness_weights: shows "witness_l rs = insert_weights shared_weights (deep_model_l rs) witness_weights"
Pi_iff: fixes f :: "'h \<Rightarrow> 'i"   and I :: "'h set"   and X :: "'h \<Rightarrow> 'i set" shows "(f \<in> Pi I X) = (\<forall>i\<in>I. f i \<in> X i)"
span_add_eq: fixes x :: "'b"   and S :: "'b set"   and y :: "'b" assumes "x \<in> ideal S" shows "(x + y \<in> ideal S) = (y \<in> ideal S)"
pn_props_correct: fixes ps :: "'a ltlr set" assumes "finite ps" shows "pn_props ps \<le> SPEC (\<lambda>r. r = ({p. prop\<^sub>r(p) \<in> ps}, {p. nprop\<^sub>r(p) \<in> ps}))"
path_defs_2: fixes g :: "real \<Rightarrow> 'b" shows "pathstart g = g 0"
mk_mat_def: fixes nr :: "nat"   and nc :: "nat"   and f :: "nat \<times> nat \<Rightarrow> 'a" shows "mk_mat nr nc f \<equiv> \<lambda>(i, j). if i < nr \<and> j < nc then f (i, j) else undef_mat nr nc f (i, j)"
leg1_simps_2: shows "hseq t\<^sub>1 t\<^sub>0u\<^sub>1.p\<^sub>1"
approx_closed_set_with_set_zero_measure_boundary: fixes S :: "real set"   and epsilon :: "real" assumes "closed S"   and "0 < epsilon"   and "S \<noteq> {}" shows "\<exists>r<epsilon. 0 < r \<and> prob {x. infdist x S = r} = 0 \<and> prob {x. infdist x S \<le> r} < prob S + epsilon"
mult_eq_0_iff: fixes plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and a :: "'e"   and b :: "'e" assumes "class.semiring_no_zero_divisors plus zero times" shows "(times a b = zero) = (a = zero \<or> b = zero)"
advantage_nonneg: fixes \<A> :: "('m, 'n) distinguisher"   and res1 :: "('m, 'n) resource"   and res2 :: "('m, 'n) resource" shows "0 \<le> advantage \<A> res1 res2"
inject_1: fixes x1 :: "'a"   and y1 :: "'a" shows "(LOC x1 = LOC y1) = (x1 = y1)"
reach_loc: shows "s\<turnstile> l reachable_from (s@@arrLoc T a i)"
mult_implies_one_step: fixes r :: "('k \<times> 'k) set"   and M :: "'k multiset"   and N :: "'k multiset" assumes "trans r"   and "(M, N) \<in> mult r" shows "\<exists>I J K. N = I + J \<and> M = I + K \<and> J \<noteq> {#} \<and> (\<forall>k\<in>#K. \<exists>j\<in>#J. (k, j) \<in> r)"
l'_def_2: shows "les' = vs @ [Label n es' lfilledk'] @ es''"
map_\<I>_id0: shows "map_\<I> id id = id"
rep_eq: fixes x :: "'a"   and xa :: "'b"   and xb :: "('a, 'b) alist" shows "impl_of (update x xa xb) = AList.update x xa (impl_of xb)"
Cons_path: fixes x :: "'b"   and xs :: "'b list"   and t :: "'b ttree" shows "(x # xs \<in> paths t) = (possible t x \<and> xs \<in> paths (nxt t x))"
succs_2: shows "succs i \<tau>' pc \<noteq> []"
exec_wfs_no_error: fixes \<Gamma> :: "char list set"   and s :: "stmt"   and \<sigma> :: "char list \<Rightarrow> nat option" assumes "wfs \<Gamma> s"   and "match \<sigma> \<Gamma>" shows "\<not> s \<rhd> \<sigma> \<leadsto> None"
gbm_edges_list_it_impl: shows "graph_edges_it gbm_\<alpha> gbm_invar gbm_edges_list_it"
unstream_concatMapS: fixes f :: "'e \<rightarrow> ('b, 'd) Stream"   and a :: "('e, 'c) Stream" shows "unstream\<cdot>(concatMapS\<cdot>f\<cdot>a) = concatMapL\<cdot>(unstream oo f)\<cdot>(unstream\<cdot>a)"
encode_operators_def: fixes \<Pi> :: "'a strips_problem"   and t :: "nat" shows "encode_operators \<Pi> t \<equiv> let ops = \<Pi>\<^sub>\<O> in encode_all_operator_preconditions \<Pi> ops t \<^bold>\<and> encode_all_operator_effects \<Pi> ops t"
continuous_ident: fixes x :: "'b"   and S :: "'b set" shows "continuous (at x within S) (\<lambda>x. x)"
assms: fixes r :: "char list"   and s :: "char list" shows "\<forall>x\<in>set xs. elt x r @ s = elt x (r @ s)"
eq_neg_iff_add_eq_0: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.group_add minus plus zero uminus" shows "(a = uminus b) = (plus a b = zero)"
summable_strong: fixes z :: "real" assumes "norm z < 1" shows "summable (\<lambda>n. (a gchoose n) * z ^ n)"
c3_def: shows "c3 = Min {c1 * ((1 - b) / 2) / max 1 (max ((b / 2) powr (p + 1)) ((b * 3 / 2) powr (p + 1))) |b. b \<in> set bs}"
trans: fixes f :: "'b \<times> nat"   and g :: "'b \<times> nat"   and h :: "'b \<times> nat" assumes "f \<in> F"   and "g \<in> F"   and "h \<in> F"   and "P f g"   and "P g h" shows "P f h"
in_pmdlE_dgrad_p_set: fixes d :: "'a \<Rightarrow> nat"   and B :: "('a \<Rightarrow>\<^sub>0 'c) set"   and m :: "nat"   and p :: "'a \<Rightarrow>\<^sub>0 'c" assumes "hom_grading d"   and "B \<subseteq> punit.dgrad_p_set d m"   and "p \<in> punit.dgrad_p_set d m"   and "p \<in> local.punit.pmdl B" obtains A :: "('a \<Rightarrow>\<^sub>0 'c) set" and q :: "('a \<Rightarrow>\<^sub>0 'c) \<Rightarrow> 'a \<Rightarrow>\<^sub>0 'c" where "finite A" and "A \<subseteq> B" and "\<And>b. q b \<in> punit.dgrad_p_set d m" and "p = (\<Sum>b\<in>A. local.punit.mult_scalar (q b) b)"
cod_char: fixes f :: "('a, 'a set) Maps.arr arrow_of_spans_data \<times> ('a, 'a set) Maps.arr arrow_of_spans_data" shows "Span.VV.cod f = (if Span.VV.arr f then Span.VxV.cod f else Span.VxV.null)"
lemma7: fixes r :: "dB"   and r' :: "dB" assumes "r \<rightarrow>\<^sub>s r'"   and "NF r'" shows "r \<rightarrow>\<^sub>l r'"
subst_atm_mset_lists_id_subst: fixes AAs :: "'a multiset list" shows "AAs \<cdot>\<cdot>aml replicate (length AAs) id_subst = AAs"
update_by_fun_pair_tc_def: shows "update_by_fun_pair_tc \<equiv> comparator.update_by_fun_pair comparator_of"
correctness: fixes G :: "('a, 'b) form list"   and p :: "('a, 'b) form" assumes "G \<turnstile> p" shows "\<forall>e f g. e,f,g,G \<Turnstile> p"
uniformity_euclidean_space_def: shows "uniformity = (INF e\<in>{0<..}. principal {(x, y). dist x y < e})"
t: shows "t \<in> {t \<in> \<Union>w\<in>W. lconjby w ` S. \<not> gcd_nat.greater_eq (count_list (lconjseq ss) t) 2}"
consec_trans_lesser: fixes i :: "nat_int"   and j :: "nat_int"   and k :: "nat_int" shows "consec i j \<and> consec j k \<longrightarrow> (\<forall>n m. n \<^bold>\<in> i \<and> m \<^bold>\<in> k \<longrightarrow> n < m)"
rqp_then_div_int: fixes r :: "int poly"   and q :: "int poly"   and p :: "int poly" assumes "r * q = p"   and "q \<noteq> 0" shows "div_int_poly p q = Some r"
M\<^sub>R_1: shows "[M\<^sub>R]\<^bsub>v,n\<^esub> = R"
IH2: assumes "\<exists>pn. p2 \<in> subterms (\<Gamma> pn)"   and "((\<xi>, p2), a, \<xi>', p') \<in> seqp_sos \<Gamma>" shows "\<exists>pn. p' \<in> subterms (\<Gamma> pn)"
f_shrink_def: fixes xs :: "'b message_af list"   and k :: "nat" shows "xs \<div> k \<equiv> f_aggregate xs k last_message"
simplify_pathE: fixes g :: "'a ugraph"   and u :: "'a"   and p :: "('a \<times> 'a) list"   and v :: "'a" assumes "path g u p v" obtains p' :: "('a \<times> 'a) list" where "path g u p' v" and "simple p'" and "set p' \<subseteq> set p"
l_set_tag_name_get_tag_name\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) heap \<Rightarrow> bool"   and get_tag_name :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, char list) prog"   and get_tag_name_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and set_tag_name :: "(_) element_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_tag_name_locs :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, unit) prog set" shows "l_set_tag_name_get_tag_name\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_tag_name get_tag_name_locs set_tag_name set_tag_name_locs \<equiv> l_get_tag_name\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_tag_name get_tag_name_locs \<and> l_set_tag_name\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M set_tag_name set_tag_name_locs"
path1: fixes u :: "'a"   and v :: "'a"   and E :: "('a \<times> 'a) set" assumes "(u, v) \<in> E" shows "path E u [u] v"
app_vrrestrictionE: fixes x :: "V"   and r :: "V"   and A :: "V" assumes "x \<in>\<^sub>\<circ> r \<restriction>\<^sup>r\<^sub>\<circ> A" obtains a :: "V" and b :: "V" where "x = \<langle>a, b\<rangle>" and "b \<in>\<^sub>\<circ> A" and "\<langle>a, b\<rangle> \<in>\<^sub>\<circ> r"
o1: shows "t1 = s \<or> (t1, s) \<in> trm_ord"
ord_spmf_bind_reflI: fixes p :: "'a spmf"   and R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'b spmf"   and g :: "'a \<Rightarrow> 'b spmf" assumes "\<And>x. x \<in> set_spmf p \<Longrightarrow> ord_spmf R (f x) (g x)" shows "ord_spmf R (bind_spmf p f) (bind_spmf p g)"
SUP_eq_iff: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and I :: "'c set"   and c :: "'b"   and f :: "'c \<Rightarrow> 'b" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "I \<noteq> {}"   and "\<And>i. i \<in> I \<Longrightarrow> less_eq c (f i)" shows "(Sup (f ` I) = c) = (\<forall>i\<in>I. f i = c)"
assms_3: fixes vs :: "nat \<Rightarrow> nat"   and vs' :: "nat \<Rightarrow> nat"   and vs'' :: "nat \<Rightarrow> nat" assumes "P vs vs'"   and "Q vs' vs''" shows "R vs vs''"
exec_dynCall: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and init :: "'e \<Rightarrow> 'e"   and p :: "'e \<Rightarrow> 'd"   and s :: "'e"   and return :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and c :: "'e \<Rightarrow> 'e \<Rightarrow> ('e, 'd, 'f) com"   and t :: "('e, 'f) xstate" assumes "\<Gamma>\<turnstile> \<langle>call init (p s) return c,Normal s\<rangle> \<Rightarrow> t" shows "\<Gamma>\<turnstile> \<langle>dynCall init p return c,Normal s\<rangle> \<Rightarrow> t"
of_real_eq_iff: fixes x :: "real"   and y :: "real" shows "(of_real x = of_real y) = (x = y)"
congBisim: fixes P :: "pi"   and Q :: "pi" assumes "P \<approx>\<^sup>s Q" shows "P \<approx> Q"
wfvarsoccs\<^sub>s\<^sub>s\<^sub>t_def: fixes S :: "('d, 'c) stateful_strand_step list" shows "wfvarsoccs\<^sub>s\<^sub>s\<^sub>t S \<equiv> \<Union> (set (map wfvarsoccs\<^sub>s\<^sub>s\<^sub>t\<^sub>p S))"
poly_IVT: fixes a :: "real"   and b :: "real"   and p :: "real poly" assumes "a < b"   and "poly p a * poly p b < 0" shows "\<exists>x>a. x < b \<and> poly p x = 0"
assms_3: shows "\<forall>f\<in>Pi I S. finprod G f I = \<one> \<longrightarrow> (\<forall>i\<in>I. f i = \<one>)"
bot_filter_def: shows "bot = Abs_filter (\<lambda>P. True)"
inv_list_rel_eq: fixes R :: "('b \<times> 'a) set" shows "(\<langle>R\<rangle>list_rel)\<inverse> = \<langle>R\<inverse>\<rangle>list_rel"
simps_1: fixes S :: "'b set" shows "scylinder S [] = streams S"
mt_sym: fixes a :: "'a \<Rightarrow> 'b option"   and b :: "'a \<Rightarrow> 'c option" assumes "dom a \<inter> dom b = {}" shows "dom b \<inter> dom a = {}"
finite_intersection_of_Inter: fixes \<U> :: "'c set set"   and P :: "'c set \<Rightarrow> bool" assumes "finite \<U>"   and "\<And>S. S \<in> \<U> \<Longrightarrow> (finite intersection_of P) S" shows "(finite intersection_of P) (\<Inter> \<U>)"
low_def: fixes p :: "'b ivl" shows "low p = fst (Rep_ivl p)"
exe_mode_val_def: fixes state :: "'b sparc_state" shows "exe_mode_val state \<equiv> get_exe_mode (state_var state)"
imp_graph_def: fixes cnf :: "lit uprod set" shows "imp_graph cnf = \<Union> (edges_of_clause ` cnf)"
good_item_simps_81: fixes A :: "('a, 'b, 'varSort, 'var, 'c) qAbs set"   and xsa :: "'varSort"   and xa :: "'var"   and x1 :: "'var" assumes "goodAbs A"   and "freshAbs xsa xa A" shows "A $[x1 // xa]_xsa = A"
cases_2: fixes t :: "('b, 'c) xstate" assumes "\<Gamma>\<turnstile> \<langle>c,s\<rangle> \<Rightarrow> t"   and "AlternativeSmallStep.inf \<Gamma> cs css t" shows "P"
lm030: fixes a :: "('c \<times> 'd) set"   and b :: "('c \<times> 'd) set" assumes "a \<in> injectionsUniverse" shows "a - b \<in> injectionsUniverse"
X_3: assumes "list_all2 (\<lambda>x ys. x \<in> set ys) yss (map cyc_permutations xss)" shows "map set xss = map set yss"
concat_eq_concat_iff: fixes xs :: "'g list list"   and ys :: "'g list list" assumes "\<forall>(x, y)\<in>set (zip xs ys). length x = length y"   and "length xs = length ys" shows "(concat xs = concat ys) = (xs = ys)"
simps_2: fixes G :: "'b list_graph"   and M :: "'b SecurityInvariant list"   and accu :: "('b \<times> 'b) list"   and e :: "'b \<times> 'b"   and Es :: "('b \<times> 'b) list" shows "TopoS_Stateful_Policy_impl.filter_compliant_stateful_ACS_accu G M accu (e # Es) = (if e \<notin> set (backlinks (edgesL G)) \<and> (\<forall>F\<in>set (implc_get_offending_flows (TopoS_Composition_Theory_impl.get_ACS M) (stateful_list_policy_to_list_graph \<lparr>hostsL = nodesL G, flows_fixL = edgesL G, flows_stateL = e # accu\<rparr>)). set F \<subseteq> set (backlinks (e # accu))) then TopoS_Stateful_Policy_impl.filter_compliant_stateful_ACS_accu G M (e # accu) Es else TopoS_Stateful_Policy_impl.filter_compliant_stateful_ACS_accu G M accu Es)"
closedin_closed: fixes U :: "'b set"   and S :: "'b set" shows "closedin (top_of_set U) S = (\<exists>T. closed T \<and> S = U \<inter> T)"
tvs_bind_fv: fixes v :: "variable"   and T :: "typ"   and t :: "Core.term" shows "tvs (bind_fv (v, T) t) \<union> tvsT T = tvs t \<union> tvsT T"
MkArr_Map: fixes f :: "('obj, 'obj) concrete_category.arr" assumes "arr f" shows "MkArr (Dom f) (Cod f) (Map f) = f"
run_owner_iii_rel: fixes s :: "(agent \<times> msg) set"   and n :: "nat" assumes "s\<^sub>0 \<Turnstile> s" shows "s \<Turnstile> fst (run_owner_iii n s)"
meyer_1: fixes x :: "'b" shows "x\<^sup>\<star> = ((1::'b) + x) \<cdot> (x \<cdot> x)\<^sup>\<star>"
summable_zeta: fixes s :: "complex" assumes "1 < Re s" shows "summable (\<lambda>n. of_nat (Suc n) powr - s)"
"3_5": fixes t :: "('d, 'c) Term.term"   and s :: "('d, 'c) Term.term"   and \<theta> :: "'c \<Rightarrow> ('d, 'c) Term.term" shows "fv\<^sub>s\<^sub>s\<^sub>t\<^sub>p (delete\<langle>t,s\<rangle> \<cdot>\<^sub>s\<^sub>s\<^sub>t\<^sub>p \<theta>) = fv (t \<cdot> \<theta>) \<union> fv (s \<cdot> \<theta>)"
sound_modes_1: shows "sound_mode_use (cms\<^sub>1, mem\<^sub>1)"
hom_0_iff: fixes x :: "'a" shows "([:x:] = 0) = (x = (0::'a))"
twise_set_pair_index: fixes point_set :: "'b set"   and block_collection :: "'b set multiset"   and grouping :: "int"   and index :: "int"   and ps :: "'b set"   and ps2 :: "'b set" assumes "t_wise_balance point_set block_collection grouping index"   and "ps \<subseteq> point_set"   and "ps2 \<subseteq> point_set"   and "ps \<noteq> ps2"   and "int (card ps) = grouping"   and "int (card ps2) = grouping" shows "block_collection index ps = block_collection index ps2"
simps_2: fixes e :: "expr" shows "\<lbrace>Negation e\<rbrace> = - \<lbrace>e\<rbrace>"
update_preserves_le: fixes mds\<^sub>1 :: "Mode \<Rightarrow> 'Var set"   and mds\<^sub>2 :: "Mode \<Rightarrow> 'Var set"   and annos :: "'Var ModeUpd list" assumes "mds\<^sub>1 \<le> mds\<^sub>2" shows "mds\<^sub>1 \<oplus> annos \<le> mds\<^sub>2 \<oplus> annos"
lfp_def: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and f :: "'b \<Rightarrow> 'b" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "complete_lattice.lfp Inf less_eq f = Inf {u. less_eq (f u) u}"
dim_vec_of_list_h: fixes h :: "'d poly"   and u :: "'e poly" assumes "degree h < degree u" shows "dim_vec (vec_of_list (coeffs h @ replicate (degree u - length (coeffs h)) (0::'d))) = degree u"
is_ext_def: fixes y' :: "'c"   and g :: "'d"   and f :: "'c" shows "y\<eta>.is_ext y' g f \<equiv> \<guillemotleft>f : F y \<rightarrow>\<^sub>C y'\<guillemotright> \<and> g = G f \<cdot>\<^sub>D \<eta>.map y"
fold_append: fixes f :: "'d \<Rightarrow> 'c \<Rightarrow> 'c"   and xs :: "'d list"   and ys :: "'d list" shows "fold f (xs @ ys) = fold f ys \<circ> fold f xs"
supp_PTuple: fixes p :: "pat"   and q :: "pat" shows "supp \<langle>\<langle>p, q\<rangle>\<rangle> = supp p \<union> supp q"
ring_hom_cring_axioms_def: fixes R :: "('a, 'b) ring_scheme"   and S :: "('c, 'd) ring_scheme"   and h :: "'a \<Rightarrow> 'c" shows "ring_hom_cring_axioms R S h \<equiv> h \<in> ring_hom R S"