hcju/isabelleps huggingface dataset:
prog_unclocked_unchanged: fixes ck :: "bool"   and env :: "v sem_env"   and s :: "'b state"   and prog :: "top0 list"   and res :: "'b state \<times> (v sem_env, v) result" assumes "evaluate_prog ck env s prog res"   and "\<not> ck" shows "snd res \<noteq> Rerr (Rabort Rtimeout_error) \<and> clock (fst res) = clock s"
weak_map_of_SomeI: fixes k :: "'a"   and x :: "'b"   and l :: "('a \<times> 'b) list" assumes "(k, x) \<in> set l" shows "\<exists>x. table_of l k = Some x"
fmap_ext: fixes m :: "'d \<rightharpoonup>\<^sub>f 'e"   and n :: "'d \<rightharpoonup>\<^sub>f 'e" assumes "\<And>x. fmlookup m x = fmlookup n x" shows "m = n"
h2: shows "well_def_signs (length qs) [[1], [- 1]]"
int_induct: fixes P :: "int \<Rightarrow> bool"   and k :: "int"   and i :: "int" assumes "P k"   and "\<And>i. \<lbrakk>k \<le> i; P i\<rbrakk> \<Longrightarrow> P (i + 1)"   and "\<And>i. \<lbrakk>i \<le> k; P i\<rbrakk> \<Longrightarrow> P (i - 1)" shows "P i"
a_all_ptrs_in_heap_def: fixes h :: "(_) heap" shows "local.CD.a_all_ptrs_in_heap h = ((\<forall>ptr\<in>fset (object_ptr_kinds h). set |h \<turnstile> get_child_nodes ptr|\<^sub>r \<subseteq> fset (node_ptr_kinds h)) \<and> (\<forall>document_ptr\<in>fset (document_ptr_kinds h). set |h \<turnstile> get_disconnected_nodes document_ptr|\<^sub>r \<subseteq> fset (node_ptr_kinds h)))"
IH_1: assumes "l \<turnstile> c" shows "s1 = s\<^sub>2_ (< l)"
hyps: assumes "mat (row_length M1) (length M1) M1 \<and> mat (row_length M2) (length M2) M2" shows "mat (row_length M1 * row_length M2) (length M1 * length M2) (M1 \<otimes> M2)"
distproj_def: fixes x :: "'c"   and A :: "'c set" shows "distproj x A = (if proj_set x A \<noteq> {} then SOME y. y \<in> proj_set x A else x)"
mk_eqcl_fixpt_startnode: fixes M :: "nat bdd list \<times> bool list" assumes "0 < length (fst M)" shows "0 < length (snd (mk_eqcl (replicate (length (fst M)) None) [] 0 (fixpt M (init_tr M)))) \<and> fst (mk_eqcl (replicate (length (fst M)) None) [] 0 (fixpt M (init_tr M))) ! 0 = 0 \<and> snd (mk_eqcl (replicate (length (fst M)) None) [] 0 (fixpt M (init_tr M))) ! 0 = 0"
space_lim_sequence: fixes x :: "'a" assumes "x \<in> space M" shows "space (lim_sequence x) = space (Pi\<^sub>M UNIV (\<lambda>i. M))"
Subst_preserves_soundness: fixes X :: "'b list set"   and xrhs :: "'b trm set"   and rhs :: "'b trm set" assumes "X = lang_rhs xrhs"   and "finite rhs" shows "lang_rhs (Subst rhs X xrhs) = lang_rhs rhs"
intro: fixes A :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and F :: "'a \<Rightarrow> 'a" assumes "functor A A F" shows "endofunctor A F"
af_subsequence_append: fixes i :: "nat"   and j :: "nat"   and k :: "nat"   and \<phi> :: "'b ltln"   and w :: "nat \<Rightarrow> 'b set" assumes "i \<le> j"   and "j \<le> k" shows "af (af \<phi> (w [i \<rightarrow> j])) (w [j \<rightarrow> k]) = af \<phi> (w [i \<rightarrow> k])"
that: fixes Q' :: "'state" assumes "\<langle>f\<rangle>Q \<rightarrow> \<langle>\<alpha>,Q'\<rangle>"   and "P' \<sim>\<cdot>[L (\<alpha>, F, f)] Q'" shows "thesis"
sync_assoc_oneside_D: fixes P :: "'a process"   and C :: "'a set"   and Q :: "'a process"   and S :: "'a process" shows "D (P \<lbrakk>C\<rbrakk> (Q \<lbrakk>C\<rbrakk> S)) \<subseteq> D (P \<lbrakk>C\<rbrakk> Q \<lbrakk>C\<rbrakk> S)"
conj_cancel_left: fixes x :: "'a" shows "\<sim> x \<^bold>\<sqinter> x = \<zero>"
round_up_def: fixes prec :: "int"   and x :: "real" shows "round_up prec x = real_of_int \<lceil>x * 2 powr real_of_int prec\<rceil> * 2 powr - real_of_int prec"
find_index_less_size_conv: fixes P :: "'c \<Rightarrow> bool"   and xs :: "'c list" shows "(find_index P xs < length xs) = (\<exists>x\<in>set xs. P x)"
envTrans: shows "\<forall>s s' a eact eact' aact aact'. ODList.lookup (ps s) a = ODList.lookup (ps s') a \<and> aact a = aact' a \<longrightarrow> ODList.lookup (ps (envTrans eact aact s)) a = ODList.lookup (ps (envTrans eact' aact' s')) a"
Max_eq_iff: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a set"   and m :: "'a" assumes "class.linorder less_eq less"   and "finite A"   and "A \<noteq> {}" shows "(linorder.Max less_eq A = m) = (m \<in> A \<and> (\<forall>a\<in>A. less_eq a m))"
Ord_Collect_lt: fixes \<alpha> :: "V" assumes "Ord \<alpha>" shows "{\<xi>. Ord \<xi> \<and> \<xi> < \<alpha>} = elts \<alpha>"
tarski_space_axioms_def: fixes B :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool" shows "tarski_space_axioms B \<equiv> \<forall>a b c d t. B a d t \<and> B b d c \<and> a \<noteq> d \<longrightarrow> (\<exists>x y. B a b x \<and> B a c y \<and> B x t y)"
pick_wakeup_spec_via_sel: fixes ws_\<alpha> :: "'a \<Rightarrow> 'b \<Rightarrow> 'c wait_set_status option"   and ws_invar :: "'a \<Rightarrow> bool"   and ws_sel :: "'a \<Rightarrow> ('b \<times> 'c wait_set_status \<Rightarrow> bool) \<Rightarrow> ('b \<times> 'c wait_set_status) option"   and \<sigma>_invar :: "'d \<Rightarrow> 'b set \<Rightarrow> bool" assumes "map_sel' ws_\<alpha> ws_invar ws_sel" shows "pick_wakeup_spec (pick_wakeup_via_sel (\<lambda>s P. ws_sel s (\<lambda>(k, v). P k v))) \<sigma>_invar ws_\<alpha> ws_invar"
cases: fixes a1 :: "('p, 's, 'm) configuration"   and a2 :: "('p, 's, 'm) event llist" assumes "cotrace a1 a2" obtains c :: "('p, 's, 'm) configuration" where "a1 = cb__" and "a2 = LNil"   | c :: "('p, 's, 'm) configuration" and ev :: "('p, 's, 'm) event" and c' :: "('p, 's, 'm) configuration" and t :: "('p, 's, 'm) event llist" where "a1 = c" and "a2 = LCons ev t" and "c \<turnstile> ev \<mapsto> c'" and "cotrace c' t"
append_eq_Cons_conv: fixes ys :: "'k list"   and zs :: "'k list"   and x :: "'k"   and xs :: "'k list" shows "(ys @ zs = x # xs) = (ys = [] \<and> zs = x # xs \<or> (\<exists>ys'. ys = x # ys' \<and> ys' @ zs = xs))"
iT_Plus_neg_iUntil_conv: fixes I :: "nat set"   and k :: "nat"   and P :: "nat \<Rightarrow> bool"   and Q :: "nat \<Rightarrow> bool" shows "(P t1. t1 \<U> t2 I \<oplus>- k. Q t2) = (P (t1 - k). t1 \<U> t2 I \<down>\<ge> k. Q (t2 - k))"
intro: fixes x :: "'a \<Longrightarrow>\<^sub>\<triangle> 'b" assumes "\<And>v. match\<^bsub>x\<^esub> (build\<^bsub>x\<^esub> v) = Some v"   and "\<And>s v. match\<^bsub>x\<^esub> s = Some v \<Longrightarrow> s = build\<^bsub>x\<^esub> v" shows "wb_prism x"
rmshadow: fixes rs :: "'a simple_rule list"   and p :: "'a simple_packet" shows "simple_fw (rmshadow rs UNIV) p = simple_fw rs p"
proj3_update: fixes rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option"   and dip :: "nat"   and dsn :: "nat"   and dsk :: "k"   and flg :: "f"   and hops :: "nat"   and sip :: "nat" assumes "rt \<noteq> update rt dip (dsn, dsk, flg, hops, sip, {})" shows "\<pi>\<^sub>3 (the (update rt dip (dsn, dsk, flg, hops, sip, {}) dip)) = dsk"
func_zero: shows "0 = (\<lambda>x. 0::'b)"
oexp_Sup: fixes \<alpha> :: "V"   and X :: "V set" assumes "\<alpha> \<noteq> 0"   and "Ord \<alpha>"   and "X \<subseteq> ON"   and "small X"   and "X \<noteq> {}" shows "\<alpha> \<up> \<Squnion> X = \<Squnion> ((\<up>) \<alpha> ` X)"
app_vrestriction_def: fixes r :: "V"   and A :: "V" shows "r \<restriction>\<^sub>\<circ> A = set {\<langle>a, b\<rangle> |a b. a \<in>\<^sub>\<circ> A \<and> b \<in>\<^sub>\<circ> A \<and> \<langle>a, b\<rangle> \<in>\<^sub>\<circ> r}"
trans_DT: fixes P :: "'a process"   and Q :: "'a process"   and S :: "'a process" assumes "P \<sqsubseteq>\<^sub>D\<^sub>T Q"   and "Q \<sqsubseteq>\<^sub>D\<^sub>T S" shows "P \<sqsubseteq>\<^sub>D\<^sub>T S"
Nil1: fixes y :: "'b"   and ys :: "'b list" assumes "as = []"   and "bs = y # ys" shows "P"
pd_ret_is_ret: fixes prob :: "('a, 'b, 'c, 'd) ifc_problem"   and x :: "'a" assumes "IFC prob"   and "IFC_def.is_pd prob x (IFC_def.return prob)" shows "x = IFC_def.return prob"
Cons_3: fixes i :: "nat" assumes "i < length (y # ys)" shows "(x # xs) ! i \<le> (y # ys) ! i"
ln_exp_cf5_upper_bound_nonneg: fixes x :: "real" assumes "0 \<le> x"   and "0 < numer_cf5 (- x)" shows "x \<le> ln (exp_cf5 x)"
base: fixes biG :: "('a, 'b) pre_digraph \<times> ('b \<Rightarrow> 'b)" assumes "bidirected_digraph (fst biG) (snd biG)" shows "subdivision biG biG"
dom_iso: fixes x :: "'a"   and y :: "'a" assumes "x \<le> y" shows "ad (ad x) \<le> ad (ad y)"
equator_reflect_K2_isometry: shows "is_K2_isometry equator_reflect"
valid_cfgD: fixes K :: "'a \<Rightarrow> 'a pmf set"   and S :: "'a set"   and cfg :: "'a cfg" assumes "Markov_Decision_Process_Invariant K S"   and "cfg \<in> Markov_Decision_Process_Invariant.valid_cfg K S" shows "cfg \<in> Markov_Decision_Process.cfg_on K (state cfg)"
Instantiation_axioms: shows "Instantiation inPorts outPorts hyps vertices freshenLC renameLCs lconsts closed subst subst_lconsts subst_renameLCs anyP vidx"
map_le_dom_restrict_sub_add: fixes m' :: "'c \<Rightarrow> 'd option"   and m :: "'c \<Rightarrow> 'd option" assumes "m' \<subseteq>\<^sub>m m" shows "m |` (dom m - dom m') ++ m' = m"
length_coeffs: fixes p :: "'b poly" assumes "p \<noteq> 0" shows "length (coeffs p) = degree p + 1"
eval_monom: fixes r :: "int"   and s :: "int"   and n :: "nat" assumes "r \<in> carrier INTEG"   and "s \<in> carrier INTEG" shows "eval INTEG INTEG id s (monom (UP INTEG) r n) = id r \<otimes>\<^bsub>INTEG\<^esub> s [^]\<^bsub>INTEG\<^esub> n"
fold_mset_add_mset: fixes s :: "'a list"   and x :: "'a"   and M :: "'a multiset" shows "fold_mset insort s (add_mset x M) = insort x (fold_mset insort s M)"
i'_bot: shows "(i - 1) * 2 ^ (l' - l) < i'"
exact_seq_eq_triviality: fixes E :: "'b monoid"   and D :: "'b monoid"   and C :: "'b monoid"   and B :: "'b monoid"   and A :: "'b monoid"   and k :: "'b \<Rightarrow> 'b"   and h :: "'b \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'b"   and f :: "'b \<Rightarrow> 'b" assumes "exact_seq ([E, D, C, B, A], [k, h, g, f])" shows "trivial_group C = (f ` carrier A = carrier B \<and> inj_on k (carrier D))"
monoid_cancel_def: fixes G :: "('c, 'd) monoid_scheme" shows "monoid_cancel G \<equiv> Group.monoid G \<and> monoid_cancel_axioms G"
abs_ge_self: fixes abs :: "'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b" assumes "class.ordered_ab_group_add_abs abs plus zero minus uminus less_eq less" shows "less_eq a (abs a)"
All_less_Suc2: fixes n :: "nat"   and P :: "nat \<Rightarrow> bool" shows "(\<forall>i<Suc n. P i) = (P 0 \<and> (\<forall>i<n. P (Suc i)))"
assms_5: shows "VS\<^sub>2.span T \<subseteq> f ` VS\<^sub>1.span S"
upd_set'_lookup: fixes m :: "('b, 'a) mapping"   and d :: "'a"   and f :: "'a \<Rightarrow> 'a"   and X :: "'b set"   and a :: "'b" shows "Mapping.lookup (upd_set' m d f X) a = (if a \<in> X then case Mapping.lookup m a of None \<Rightarrow> Some d | Some b \<Rightarrow> Some (f b) else Mapping.lookup m a)"
times_rel_fun_def: fixes f :: "'c \<times> 'd \<Rightarrow> 'e"   and g :: "'c \<times> 'd \<Rightarrow> 'e" shows "ps_prod.rel_partial_semigroup.times_rel_fun f g = \<otimes> (\<lambda>x y z. ps_prod_D y z \<and> x = ps_prod_times y z) f g"
sum: fixes g :: "'c \<Rightarrow> 'b"   and S :: "'c set" shows "g__ (sum g S) = (\<Sum>a\<in>S. g__ (g a))"
arr_cod_iff_arr: fixes f :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" shows "arr (cod f) = arr f"
1: shows "homotopic_paths s p (p +++ linepath (pathfinish p) (pathfinish p))"
assms_3: shows "\<I>' \<turnstile>g gpv \<surd>"
bind_right_unit': fixes m :: "'a \<Rightarrow> (unit \<times> 'a) option" shows "m ;- unit\<^sub>S\<^sub>E () = m"
emb_sprod_def: shows "EMB('e \<otimes> 'f) = sprod_emb oo sprod_map\<cdot>EMB('e)\<cdot>EMB('f)"
cInf_eq_minimum: fixes z :: "'b multiset"   and X :: "'b multiset set" assumes "z \<in> X"   and "\<And>x. x \<in> X \<Longrightarrow> z \<subseteq># x" shows "Inf X = z"
conc_UNION_distrib_1: fixes A :: "'a list set"   and M :: "'b \<Rightarrow> 'a list set"   and I :: "'b set" shows "A @@ \<Union> (M ` I) = (\<Union>i\<in>I. A @@ M i)"
dvd_normalize_div: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and b :: "'a"   and a :: "'a" assumes "class.normalization_semidom_multiplicative divide plus minus zero times one unit_factor normalize"   and "dvd.dvd times b a" shows "normalize (divide a b) = divide (normalize a) (normalize b)"
distinct_26: fixes name' :: "name"   and ccs' :: "ccs"   and ccs1 :: "ccs"   and ccs2 :: "ccs" shows "\<lparr>\<nu>name'\<rparr>ccs' \<noteq> ccs1 \<parallel> ccs2"
Gamma_semilattice_axioms: shows "Gamma_semilattice num' plus' \<gamma>"
compact_LConsI: fixes xs :: "'c llist"   and x :: "'c" assumes "ccpo.compact lSup (\<sqsubseteq>) xs" shows "ccpo.compact lSup (\<sqsubseteq>) (LCons x xs)"
monic: shows "monic (\<Prod>(a, i)\<in>as. a ^ i)"
Seq: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c\<^sub>1 :: "('e, 'd, 'f) com"   and s :: "'e"   and s' :: "('e, 'f) xstate"   and c\<^sub>2 :: "('e, 'd, 'f) com"   and t :: "('e, 'f) xstate" assumes "\<Gamma>\<turnstile> \<langle>c\<^sub>1,Normal s\<rangle> \<Rightarrow> s'"   and "\<Gamma>\<turnstile> \<langle>c\<^sub>2,s'\<rangle> \<Rightarrow> t" shows "\<Gamma>\<turnstile> \<langle>Seq c\<^sub>1 c\<^sub>2,Normal s\<rangle> \<Rightarrow> t"
r_step_Mn: fixes e :: "nat" shows "eval r_step_Mn [e] \<down>= estep_Mn e"
globally_consistent_def: fixes A :: "'Var \<Rightarrow> ('Val \<times> 'Val) option"   and mds :: "Mode \<Rightarrow> 'Var set" shows "globally_consistent A mds \<equiv> finite (dom A) \<and> (\<forall>x\<in>dom A. globally_consistent_var A mds x)"
m_closed: fixes x :: "'a vec"   and y :: "'a vec" assumes "x \<in> carrier_vec n"   and "y \<in> carrier_vec n" shows "x + y \<in> carrier_vec n"
det_transpose: fixes A :: "'b mat"   and n :: "nat" assumes "A \<in> carrier_mat n n" shows "Determinant.det A\<^sup>T = Determinant.det A"
aux: shows "int ((x ^ a) ^ q - 1) = int ((x ^ a - 1) * sum ((^) (x ^ a)) {..<q})"
inject_2: fixes x31 :: "'c"   and x32 :: "('c \<Rightarrow> 'd) \<Rightarrow> 'd"   and y31 :: "'c"   and y32 :: "('c \<Rightarrow> 'd) \<Rightarrow> 'd" shows "(Assign x31 x32 = Assign y31 y32) = (x31 = y31 \<and> x32 = y32)"
tcopy_def: shows "tcopy \<equiv> tcopy_begin |+| tcopy_loop |+| tcopy_end"
negligible_const_iff: fixes c :: "real" shows "negligible (\<lambda>_. c) = (c = 0)"
filter_def: fixes P :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and t :: "('c, 'd) RBT.rbt" shows "RBT.filter P t = RBT.fold (\<lambda>k v t. if P k v then RBT.insert k v t else t) t RBT.empty"
unrest_usubst_def: fixes x :: "'e \<Longrightarrow> 'f"   and \<sigma> :: "'f \<Rightarrow> 'f" shows "(x \<sharp> \<sigma>) = (\<forall>\<rho> v. \<sigma> (put\<^bsub>x\<^esub> \<rho> v) = put\<^bsub>x\<^esub> (\<sigma> \<rho>) v)"
ssortedD: fixes s :: "'b stream"   and i :: "nat" assumes "ssorted s" shows "s !! i \<le> stl s !! i"
1: shows "OFCLASS('a, Hermite_ring_class) &&& \<forall>a b c. ideal_generated {a, b, c} = ideal_generated {1::'a} \<longrightarrow> (\<exists>p q. ideal_generated {p * a, p * b + q * c} = ideal_generated {1::'a})"
fBexI: fixes P :: "'a \<Rightarrow> bool"   and x :: "'a"   and A :: "'a fset" assumes "P x"   and "x |\<in>| A" shows "fBex A P"
infinite_Func: fixes A :: "'c set"   and b1 :: "'d"   and b2 :: "'d"   and B :: "'d set" assumes "infinite A"   and "{b1, b2} \<subseteq> B"   and "b1 \<noteq> b2" shows "infinite (Func A B)"
RSpan_contains_spanset_single: fixes R :: "'a set"   and smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and M :: "'b set"   and m :: "'b" assumes "RModule R smult M"   and "m \<in> M" shows "m \<in> R_scalar_mult.RSpan R smult [m]"
STEP: shows "(\<Sigma>h, \<Sigma>') \<in> brw_inner_step r"
less_eq_V_0_iff: fixes x :: "V" shows "(x \<le> 0) = (x = 0)"
order_char_poly_smult: fixes A :: "complex mat"   and n :: "nat"   and k :: "complex"   and x :: "complex" assumes "A \<in> carrier_mat n n"   and "k \<noteq> 0" shows "order x (char_poly (k \<cdot>\<^sub>m A)) = order (x / k) (char_poly A)"
timpl_closure_Var_inv: fixes t :: "(('e, 'f, 'g) prot_fun, 'h) Term.term"   and x :: "'h"   and TI :: "('g set \<times> 'g set) set" assumes "t \<in> timpl_closure (Var x) TI" shows "t = Var x"
equivalent_on_term: fixes I :: "'b trm \<Rightarrow> 'b trm \<Rightarrow> bool"   and \<sigma> :: "('b \<times> 'b trm) list"   and \<eta> :: "('b \<times> 'b trm) list"   and t :: "'b trm" assumes "congruence I"   and "equivalent_on \<sigma> \<eta> (vars_of t) I" shows "I (t \<lhd> \<sigma>) (t \<lhd> \<eta>)"
assms_3: shows "\<forall>z\<in>V. \<forall>zs\<in>fZs z. qFreshAbs zs z A"
mod0_uint64_def: fixes x :: "uint64" shows "mod0_uint64 x = undefined (mod) x 0"
ary_\<rho>x: fixes x :: "'v" shows "arity_var x \<le> arity (\<rho> x) \<and> local.ground_heads (head (\<rho> x)) \<subseteq> ground_heads_var x"
split: fixes P :: "'a \<Rightarrow> bool"   and f1 :: "'a"   and f2 :: "'a"   and f3 :: "'a"   and f4 :: "expr \<Rightarrow> 'a"   and f5 :: "varRef \<Rightarrow> expr \<Rightarrow> 'a"   and f6 :: "procVarDecl \<Rightarrow> 'a"   and f7 :: "String.literal \<Rightarrow> expr list \<Rightarrow> 'a"   and f8 :: "chanRef \<Rightarrow> expr list \<Rightarrow> bool \<Rightarrow> 'a"   and f9 :: "chanRef \<Rightarrow> recvArg list \<Rightarrow> bool \<Rightarrow> bool \<Rightarrow> 'a"   and edgeEffect :: "edgeEffect" shows "P (case edgeEffect of EEEnd \<Rightarrow> f1 | EEId \<Rightarrow> f2 | EEGoto \<Rightarrow> f3 | EEAssert x \<Rightarrow> f4 x | EEAssign x xa \<Rightarrow> f5 x xa | EEDecl x \<Rightarrow> f6 x | EERun x xa \<Rightarrow> f7 x xa | EESend x xa xb \<Rightarrow> f8 x xa xb | EERecv x xa xb xc \<Rightarrow> f9 x xa xb xc) = ((edgeEffect = EEEnd \<longrightarrow> P f1) \<and> (edgeEffect = EEId \<longrightarrow> P f2) \<and> (edgeEffect = EEGoto \<longrightarrow> P f3) \<and> (\<forall>x4. edgeEffect = EEAssert x4 \<longrightarrow> P (f4 x4)) \<and> (\<forall>x51 x52. edgeEffect = EEAssign x51 x52 \<longrightarrow> P (f5 x51 x52)) \<and> (\<forall>x6. edgeEffect = EEDecl x6 \<longrightarrow> P (f6 x6)) \<and> (\<forall>x71 x72. edgeEffect = EERun x71 x72 \<longrightarrow> P (f7 x71 x72)) \<and> (\<forall>x81 x82 x83. edgeEffect = EESend x81 x82 x83 \<longrightarrow> P (f8 x81 x82 x83)) \<and> (\<forall>x91 x92 x93 x94. edgeEffect = EERecv x91 x92 x93 x94 \<longrightarrow> P (f9 x91 x92 x93 x94)))"
add_diff_cancel_right': fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.cancel_ab_semigroup_add plus minus" shows "minus (plus a b) b = a"
trivial_mute_fun: shows "mute_fun trivial_mute_fun"
not_in_closure_trivial_limitI: fixes x :: "'b"   and s :: "'b set" assumes "x \<notin> closure s" shows "at x within s = bot"
trcl_uncons: fixes c :: "'f"   and a :: "'g"   and w :: "'g list"   and c' :: "'f"   and t :: "('f \<times> 'g \<times> 'f) set" assumes "(c, a # w, c') \<in> trcl t" shows "\<exists>ch. (c, a, ch) \<in> t \<and> (ch, w, c') \<in> trcl t"
assms_4: shows "finite (\<Union> (block_nominals ` S))"
finite_image_iff: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set" assumes "inj_on f A" shows "finite (f ` A) = finite A"
stabiliser_cosets_equivalent: fixes G :: "('d, 'e) monoid_scheme"   and action :: "'d \<Rightarrow> 'f \<Rightarrow> 'f"   and g :: "'d"   and h :: "'d"   and x :: "'f" assumes "orbit_stabiliser G action"   and "g \<in> carrier G"   and "h \<in> carrier G" shows "(action g x = action h x) = (g <#\<^bsub>G\<^esub> orbit_stabiliser.stabiliser G action x = h <#\<^bsub>G\<^esub> orbit_stabiliser.stabiliser G action x)"
simps_3: fixes ref :: "'c" shows "list_op_deps (Delete ref) = {ref}"
encL: fixes S :: "'procS" shows "indRelp (TargetTerm (\<lbrakk>S\<rbrakk>)) (SourceTerm S)"
simulation1_rtrancl: 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 tlsim :: "'h \<Rightarrow> 'j \<Rightarrow> bool"   and s1 :: "'g"   and tls1 :: "'h list"   and s1' :: "'g"   and s2 :: "'i" assumes "bisimulation trsys1 trsys2 bisim tlsim"   and "rtrancl3p trsys1 s1 tls1 s1'"   and "bisim s1 s2" shows "\<exists>s2' tls2. rtrancl3p trsys2 s2 tls2 s2' \<and> bisim s1' s2' \<and> list_all2 tlsim tls1 tls2"
snd_sum_list: fixes xs :: "('b \<times> 'a) list" shows "snd (sum_list xs) = sum_list (map snd xs)"
WHILEI_tailrec_conv: fixes I :: "'b \<Rightarrow> bool"   and c :: "'b \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'b nres" shows "WHILE\<^bsup>I\<^esup> c f = REC (tailrec_body (\<lambda>s. ASSERT (I s) \<bind> (\<lambda>_. ASSUME (c s) \<bind> (\<lambda>_. f s))) (\<lambda>s. ASSERT (I s) \<bind> (\<lambda>_. ASSUME (\<not> c s) \<bind> (\<lambda>_. RETURN s))))"
inject_3: fixes x3 :: "agent"   and y3 :: "agent" shows "(privK x3 = privK y3) = (x3 = y3)"
induct: fixes P :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list \<Rightarrow> bool"   and a0 :: "'a list"   and a1 :: "'a list"   and a2 :: "'a list" assumes "\<And>a as b bs accs. \<lbrakk>a < b \<Longrightarrow> P as (b # bs) (a # accs); \<lbrakk>\<not> a < b; a = b\<rbrakk> \<Longrightarrow> P as bs (a # accs); \<lbrakk>\<not> a < b; a \<noteq> b\<rbrakk> \<Longrightarrow> P (a # as) bs (b # accs)\<rbrakk> \<Longrightarrow> P (a # as) (b # bs) accs"   and "\<And>bs accs. P [] bs accs"   and "\<And>v va accs. P (v # va) [] accs" shows "P a0 a1 a2"
nary_operations_def: fixes \<OO>\<pp> :: "'a \<Rightarrow> 'b list \<Rightarrow> 'b"   and \<AA>\<rr>\<ii>\<tt>\<yy> :: "'a \<Rightarrow> nat" shows "nary_operations \<OO>\<pp> \<AA>\<rr>\<ii>\<tt>\<yy> \<equiv> \<forall>xs op. length xs = \<AA>\<rr>\<ii>\<tt>\<yy> op \<longrightarrow> (\<exists>y. \<OO>\<pp> op xs = y)"
S_def: shows "S = four_block_mat (Matrix.mat 1 1 (\<lambda>(a, b). H $$ (0, 0))) (0\<^sub>m 1 (dim_col A - 1)) (0\<^sub>m 1 1) H_1xn"
wf_trancl: fixes r :: "('a \<times> 'a) set" assumes "wf r" shows "wf (r\<^sup>+)"
T7: fixes w :: "i"   and \<psi> :: "\<o>" shows "[\<^bold>\<A>\<psi> in w] = [\<psi> in dw]"
nn_integral_lborel_prod: fixes f :: "'b \<Rightarrow> real \<Rightarrow> ennreal" assumes "\<And>b. b \<in> Basis \<Longrightarrow> f b \<in> borel_measurable borel"   and "\<And>b x. b \<in> Basis \<Longrightarrow> 0 \<le> f b x" shows "\<integral>\<^sup>+ x. (\<Prod>b\<in>Basis. f b (x \<bullet> b)) \<partial>lborel = (\<Prod>b\<in>Basis. integral\<^sup>N lborel (f b))"
items_le_def: fixes k :: "nat"   and I :: "item set" shows "items_le k I = {x \<in> I. item_end x \<le> k}"
ideriv_def: fixes R :: "('b \<times> 'b) set"   and S :: "('b \<times> 'b) set"   and as :: "nat \<Rightarrow> 'b" shows "ideriv R S as = ((\<forall>i. (as i, as (Suc i)) \<in> R \<union> S) \<and> (\<exists>\<^sub>\<infinity>i. (as i, as (Suc i)) \<in> R))"
ex_ex1I: fixes P :: "'e \<Rightarrow> bool" assumes "\<exists>x. P x"   and "\<And>x y. \<lbrakk>P x; P y\<rbrakk> \<Longrightarrow> x = y" shows "\<exists>!x. P x"
one_dim_iso_def: fixes a :: "'b" shows "one_dim_iso a = of_complex ((1::'b) \<bullet>\<^sub>C a)"
assms_7: shows "A1 \<noteq> [] \<and> A2 \<noteq> [] \<and> B1 \<noteq> [] \<and> B2 \<noteq> []"
commutative: fixes M :: "'c set"   and composition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit :: "'c"   and x :: "'c"   and y :: "'c" assumes "commutative_monoid M composition unit"   and "x \<in> M"   and "y \<in> M" shows "composition x y = composition y x"
iadd_Suc_right: fixes m :: "enat"   and n :: "enat" shows "m + eSuc n = eSuc (m + n)"
ide_mkIde: fixes A :: "(('c + 'd) set, 'c + 'd \<Rightarrow> 'c + 'd) concrete_category.arr set" assumes "A \<subseteq> S.Univ" shows "S.ide (S.mkIde A)"
program_typedef_5: fixes F :: "'b program" shows "Init F \<equiv> case Rep_Program F of (init, acts, allowed) \<Rightarrow> init"
append_eq_map_conv: fixes ys :: "'a list"   and zs :: "'a list"   and f :: "'b \<Rightarrow> 'a"   and xs :: "'b list" shows "(ys @ zs = map f xs) = (\<exists>us vs. xs = us @ vs \<and> ys = map f us \<and> zs = map f vs)"