hcju/isabelleps huggingface dataset:
simps_19: fixes f1 :: "'a"   and f2 :: "'b \<Rightarrow> 'b \<Rightarrow> 'a"   and f3 :: "'b \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'a"   and f4 :: "('b, 'c) Combinators \<Rightarrow> ('b, 'c) Combinators \<Rightarrow> 'a"   and x41 :: "('b, 'c) Combinators"   and x42 :: "('b, 'c) Combinators" shows "(case x41 \<oplus> x42 of DenyAll \<Rightarrow> f1 | DenyAllFromTo x xa \<Rightarrow> f2 x xa | AllowPortFromTo x xa xb \<Rightarrow> f3 x xa xb | x \<oplus> xa \<Rightarrow> f4 x xa) = f4 x41 x42"
formalSpec: shows "Spec = \<lparr>c_sinvar = \<lambda>G. sinvar_spec G nP, c_offending_flows = \<lambda>G. SecurityInvariant_withOffendingFlows.set_offending_flows sinvar_spec G nP, c_isIFS = nm_receiver_violation m\<rparr>"
B: fixes i :: "'agent"   and A :: "'alt set" assumes "i \<in> agents"   and "A \<subseteq> alts"   and "A \<noteq> {}" shows "Max_wrt_among (R i) A \<noteq> {}"
hp_IdleMarkSweep_locs_def: shows "hp_IdleMarkSweep_locs = {idle_noop_mfence, idle_noop_init_type, mark_end} \<union> sweep_locs \<union> (mark_loop_locs - {mark_loop_get_roots_init_type})"
rel_mapping_def: fixes Q :: "'e \<Rightarrow> 'f \<Rightarrow> bool" shows "rel_mapping Q = rel_fun (=) (rel_option Q)"
simps_2: fixes A :: "agent"   and B :: "agent"   and M :: "msg" shows "sym.implConfid A B M = Enc \<langle>ConfidTag, M\<rangle> (shrK A B)"
splitFace_add_f21': fixes f' :: "face"   and g' :: "graph"   and v :: "nat"   and a :: "nat"   and nvl :: "nat list" assumes "f' \<in> \<F> g'" shows "fst (snd (splitFace g' v a f' nvl)) \<in> \<F> (snd (snd (splitFace g' v a f' nvl)))"
subterms_subst_subset': fixes \<sigma> :: "'e \<Rightarrow> ('c, 'd) Term.term"   and M :: "('c, 'e) Term.term set" shows "subterms\<^sub>s\<^sub>e\<^sub>t M \<cdot>\<^sub>s\<^sub>e\<^sub>t \<sigma> \<subseteq> subterms\<^sub>s\<^sub>e\<^sub>t (M \<cdot>\<^sub>s\<^sub>e\<^sub>t \<sigma>)"
angle_actrans_sem_eq: fixes F :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool"   and v :: "'a \<Rightarrow> 'b" shows "\<langle>F\<rangle>_v = (F \<and> v$ \<noteq> $v)"
assms_4: shows "(r, Expunge e mo r) \<in> set (node_deliver_messages xs)"
is_class_def: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list"   and C :: "char list" shows "is_class P C \<equiv> class P C \<noteq> None"
sel_1: fixes a :: "'b" shows "\<oo> (Atom a) = False"
at_within_open_NO_MATCH: fixes "open" :: "'b set \<Rightarrow> bool"   and a :: "'b"   and s :: "'b set" assumes "class.topological_space open"   and "a \<in> s"   and "open s"   and "NO_MATCH UNIV s" shows "topological_space.at_within open a s = topological_space.at_within open a UNIV"
fcompose_poly: fixes q2 :: "'a poly"   and x :: "'a"   and p :: "'a poly"   and q1 :: "'a poly" assumes "poly q2 x \<noteq> (0::'a)" shows "poly p (poly q1 x / poly q2 x) = poly (fcompose p q1 q2) x / poly (q2 ^ degree p) x"
mod_smult_left: fixes a :: "'b"   and x :: "'b poly"   and y :: "'b poly" shows "smult a x mod y = smult a (x mod y)"
trans: fixes b :: "'b word"   and a :: "'b word"   and c :: "'b word" assumes "b \<le>s a"   and "c \<le>s b" shows "c \<le>s a"
arrI: fixes f :: "'a arrow_of_spans_data \<times> 'a arrow_of_spans_data" assumes "arr (fst f) \<and> arr (snd f) \<and> src (fst f) = trg (snd f)" shows "VV.arr f"
nat_abs_int_diff: fixes a :: "nat"   and b :: "nat" shows "nat \<bar>int a - int b\<bar> = (if a \<le> b then b - a else a - b)"
type_less_x_Real_intro_2: fixes \<tau> :: "'a basic_type" assumes "\<tau> = Integer" shows "\<tau> < Real"
ideduct_synth_subst_apply: fixes t :: "('fun, 'var) Term.term"   and M :: "('fun, 'var) Term.term set"   and \<theta> :: "'var \<Rightarrow> ('fun, 'var) Term.term" assumes "{} \<turnstile>\<^sub>c t"   and "\<And>v. M \<turnstile>\<^sub>c \<theta> v" shows "M \<turnstile>\<^sub>c t \<cdot> \<theta>"
finite_swap_ne: fixes f :: "'b \<Rightarrow> 'c"   and c :: "'c"   and b :: "'b"   and i :: "'b" assumes "finite {x. f x \<noteq> c}" shows "finite {x. f (swap b i x) \<noteq> c}"
steps_last_step: fixes xs :: "'a list" assumes "steps xs"   and "1 < length xs" shows "\<exists>a. a \<rightarrow> last xs"
iso_lunit: fixes f :: "'a" assumes "ide f" shows "local.iso (PB.lunit f)"
mset_delete_index: fixes i :: "nat"   and xs :: "'b list" assumes "i < length xs" shows "mset (delete_index i xs) = mset xs - {#xs ! i#}"
CFG_SSA_axioms_def: fixes \<alpha>n :: "'a \<Rightarrow> 'b list"   and inEdges' :: "'a \<Rightarrow> 'b \<Rightarrow> ('b \<times> 'c) list"   and defs :: "'a \<Rightarrow> 'b \<Rightarrow> 'd set"   and phis :: "'a \<Rightarrow> 'b \<times> 'd \<Rightarrow> 'd list option" shows "CFG_SSA_axioms \<alpha>n inEdges' defs phis \<equiv> ((\<forall>g. finite (dom (phis g))) \<and> (\<forall>g n v vs. phis g (n, v) = Some vs \<longrightarrow> n \<in> set (\<alpha>n g))) \<and> (\<forall>g n v args. phis g (n, v) = Some args \<longrightarrow> length (graph_path_base.predecessors inEdges' g n) = length args) \<and> (\<forall>n g. n \<in> set (\<alpha>n g) \<longrightarrow> defs g n \<inter> CFG_SSA_base.phiDefs phis g n = {}) \<and> (\<forall>n g m. n \<in> set (\<alpha>n g) \<longrightarrow> m \<in> set (\<alpha>n g) \<longrightarrow> n \<noteq> m \<longrightarrow> CFG_SSA_base.allDefs defs phis g n \<inter> CFG_SSA_base.allDefs defs phis g m = {})"
elims_1: fixes x :: "vname"   and xa :: "vname"   and y :: "bool" assumes "(x < xa) = y" obtains n1 :: "nat" and n2 :: "nat" where "x = I n1__" and "xa = R n2__" and "y"   | n1 :: "nat" and n2 :: "nat" where "x = R n1a__" and "xa = I n2a__" and "\<not> y"   | n1 :: "nat" and n2 :: "nat" where "x = I n1b__" and "xa = I n2b__" and "y = (n1b__ < n2b__)"   | n1 :: "nat" and n2 :: "nat" where "x = R n1" and "xa = R n2" and "y = (n1 < n2)"
c_manifold_order_le: fixes charts :: "('c, 'd) chart set"   and k :: "enat"   and l :: "enat" assumes "c_manifold charts k"   and "l \<le> k" shows "c_manifold charts l"
simps_4: fixes f1 :: "'k"   and f2 :: "'l \<Rightarrow> 'l list \<Rightarrow> 'k" shows "(case [] of [] \<Rightarrow> f1 | x # xa \<Rightarrow> f2 x xa) = f1"
psubstAll_cong: fixes P :: "('d, 'e, 'varSort, 'var, 'f) param"   and X :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and A :: "('d, 'e, 'varSort, 'var, 'f) qAbs set" assumes "goodPar P" shows "(good X \<longrightarrow> (\<forall>rho rho'. {rho, rho'} \<subseteq> envsOf P \<longrightarrow> (\<forall>ys y. fresh ys y X \<or> rho ys y = rho' ys y) \<longrightarrow> X #[rho] = X #[rho'])) \<and> (goodAbs A \<longrightarrow> (\<forall>rho rho'. {rho, rho'} \<subseteq> envsOf P \<longrightarrow> (\<forall>ys y. freshAbs ys y A \<or> rho ys y = rho' ys y) \<longrightarrow> A $[rho] = A $[rho']))"
cases: fixes a :: "'b"   and r :: "('c \<times> 'b) set" assumes "a \<in> Range r" obtains a :: "'c" and b :: "'b" where "aa__ = b" and "(a, b) \<in> r"
dis_log_def: fixes \<A> :: "'grp \<Rightarrow> nat spmf" shows "local.dis_log \<A> = TRY sample_uniform (order \<G>) \<bind> (\<lambda>x. let h = \<^bold>g [^] x in \<A> h \<bind> (\<lambda>x'. return_spmf [x = x'] (mod order \<G>))) ELSE return_spmf False"
In0_mono: fixes M :: "('f, 'g) Old_Datatype.node set"   and N :: "('f, 'g) Old_Datatype.node set" assumes "M \<subseteq> N" shows "In0 M \<subseteq> In0 N"
Cons_3: shows "elementLevel e (m # M') + i \<le> level"
create_character_data_document_in_heap: fixes h :: "(_) heap"   and document_ptr :: "(_) document_ptr"   and "text" :: "char list" assumes "h \<turnstile> ok create_character_data document_ptr text" shows "document_ptr |\<in>| document_ptr_kinds h"
Max_gr_iff: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and A :: "'c set"   and x :: "'c" assumes "class.linorder less_eq less"   and "finite A"   and "A \<noteq> {}" shows "less x (linorder.Max less_eq A) = (\<exists>a\<in>A. less x a)"
prio_pop_min_def: fixes q :: "'a \<Rightarrow> 'b option" shows "prio_pop_min q \<equiv> ASSERT (dom q \<noteq> {}) \<bind> (\<lambda>_. SPEC (\<lambda>(e, w, q'). q' = q(e := None) \<and> q e = Some w \<and> (\<forall>e' w'. q e' = Some w' \<longrightarrow> w \<le> w')))"
put_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_ok: fixes h :: "(_) heap"   and ptr :: "(_) element_ptr"   and setter :: "('m \<Rightarrow> 'm) \<Rightarrow> ('c, 'a, 'b, 'd, 'g option) RElement_scheme \<Rightarrow> ('c, 'a, 'b, 'd, 'g option) RElement_scheme"   and v :: "'m" assumes "ShadowRootClass.type_wf h"   and "ptr |\<in>| element_ptr_kinds h" shows "h \<turnstile> ok put_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t ptr setter v"
range_inter_ts_def: fixes A :: "'b \<Rightarrow> 'a set" shows "\<Rightarrow>\<Inter>A = {x |x. \<forall>k. x \<in> A k}"
random_treap_fold_Leaf: shows "rinss xs \<langle>\<rangle> A = distr (Pi\<^sub>M (set xs) (\<lambda>x. \<U> 0 1)) (tree_sigma (count_space A \<Otimes>\<^sub>M lborel)) (\<lambda>p. foldl (\<lambda>t x. ins x (p x) t) \<langle>\<rangle> xs)"
wf\<^sub>t\<^sub>r\<^sub>m_def: fixes t :: "('fun, 'a) Term.term" shows "wf\<^sub>t\<^sub>r\<^sub>m t \<equiv> \<forall>f T. Fun f T \<sqsubseteq> t \<longrightarrow> length T = arity f"
exec: shows "steps0 (1, l, r) ap stp = (0, l', r')"
gt_iff_wt_unary_diff_same: fixes t :: "('s, 'v) tm"   and s :: "('s, 'v) tm" shows "local.kbo_std.gt t s = (local.kbo_std.gt_wt t s \<or> local.kbo_std.gt_unary t s \<or> local.kbo_std.gt_diff t s \<or> local.kbo_std.gt_same t s)"
one_in_complex_cone: fixes a :: "real"   and b :: "real" assumes "a \<le> 0"   and "0 \<le> b" shows "1 \<in> complex_cone a b"
abs_eq: fixes x :: "int" shows "to_bl (Word.Word x) = bin_to_bl LENGTH('b) x"
shadow_root_ptr_kinds_commutes: fixes shadow_root_ptr :: "(_) shadow_root_ptr"   and h :: "(_) heap" shows "(cast\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r shadow_root_ptr |\<in>| document_ptr_kinds h) = (shadow_root_ptr |\<in>| shadow_root_ptr_kinds h)"
rtrancl_fold_product: fixes f :: "'c \<Rightarrow> 'a \<Rightarrow> 'a"   and g :: "'d \<Rightarrow> 'b \<Rightarrow> 'b"   and A :: "'c set"   and B :: "'d set"   and R :: "'c \<Rightarrow> 'd \<Rightarrow> bool" shows "{((r, s), f a r, g b s) |a b r s. a \<in> A \<and> b \<in> B \<and> R a b}\<^sup>* = {((r, s), fold f w1 r, fold g w2 s) |w1 w2 r s. w1 \<in> lists A \<and> w2 \<in> lists B \<and> list_all2 R w1 w2}"
node_deliverTE': fixes s :: "'c net_state"   and i :: "nat"   and d :: "nat"   and s' :: "'c net_state"   and S :: "('c \<times> 'd seq_action \<times> 'c) set" assumes "(s, i:deliver(d), s') \<in> node_sos S" obtains sa :: "'c" and s'a :: "'c" and R :: "nat set" where "s = NodeS i sa R" and "s' = NodeS i s'a R" and "(sa, deliver d, s'a) \<in> S"
paste_subrel: fixes P :: "('a \<times> 'b) set"   and Q :: "('a \<times> 'b) set" assumes "P || Domain Q \<subseteq> Q" shows "P +< Q = P \<union> Q"
continuous_at_open: fixes x :: "'b"   and f :: "'b \<Rightarrow> 'c" shows "isCont f x = (\<forall>t. open t \<and> f x \<in> t \<longrightarrow> (\<exists>s. open s \<and> x \<in> s \<and> (\<forall>x'\<in>s. f x' \<in> t)))"
sorted_inl: fixes X :: "'a \<times> (real \<times> real) pdevs" shows "ccw.sortedP 0 (ccw.selsort 0 (inl (snd X)))"
vwb_lens_def: fixes x :: "'a \<Longrightarrow> 'b" shows "vwb_lens x \<equiv> wb_lens x \<and> mwb_lens x"
classes_coverture: shows "\<Union> classes\<^bsub>thy_order\<^esub> = carrier thy_order"
rep_eq: shows "Rep_scene \<bottom>\<^sub>S = (\<lambda>x y. x)"
assms: shows "\<Gamma> : e \<Down>\<^bsub>L\<^esub> \<Delta> : z"
sorted_list_of_fempty: shows "sorted_list_of_fset {||} = []"
dickson_gradingD1: fixes d :: "'b \<Rightarrow> nat"   and s :: "'b"   and t :: "'b" assumes "dickson_grading d" shows "d (s + t) = ord_class.max (d s) (d t)"
pre_ranks_pre_oldest_token_Min_state_special: fixes q :: "'b"   and n :: "nat" assumes "\<not> sink q"   and "configuration q (Suc n) \<noteq> {}" shows "(Min (pre_ranks (\<lambda>q. state_rank q n) (w n) q) = max_rank) = (Min (pre_oldest_tokens q n) = Suc n)"
rbd_comp_pres: fixes R :: "('c \<times> 'e) set"   and S :: "('e \<times> 'd) set" shows "bd\<^sub>\<R> (R ; S) = bd\<^sub>\<R> S \<circ> bd\<^sub>\<R> R"
arg_eqD: fixes x :: "real"   and y :: "real" assumes "arg (cis x) = arg (cis y)"   and "- pi < x"   and "x \<le> pi"   and "- pi < y"   and "y \<le> pi" shows "x = y"
wf_progress: fixes s :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and t :: "'t"   and x :: "'x" assumes "s \<in> wf_state"   and "thr s t = \<lfloor>(x, no_wait_locks)\<rfloor>"   and "\<not> final x" shows "\<exists>taa x' m'. t \<turnstile> \<langle>x, shr s\<rangle> -taa\<rightarrow> \<langle>x', m'\<rangle>"
state_succ_def: fixes s :: "('b, 'c) fmap"   and a :: "('b, 'c) fmap \<times> ('b, 'c) fmap" shows "state_succ s a \<equiv> if fst a \<subseteq>\<^sub>f s then snd a ++ s else s"
dom_closed: fixes f :: "'a" assumes "left a f" shows "left a (local.dom f)"
permute_minus_cancel_2: fixes permute :: "perm \<Rightarrow> 'b \<Rightarrow> 'b"   and p :: "perm"   and x :: "'b" assumes "class.pt permute" shows "permute (- p) (permute p x) = x"
m_sup: shows "old_concls_of (inference_system.inferences_between (ord_FO_\<Gamma> S) (fst ` FL.active_subset N) C) \<subseteq> fst ` M"
ext_induct: fixes P :: "'a heap_scheme \<Rightarrow> bool"   and r :: "'a heap_scheme" assumes "\<And>arrays refs lim more. P \<lparr>arrays = arrays, refs = refs, lim = lim, \<dots> = more\<rparr>" shows "P r"
rtrancl_eq_or_trancl: fixes x :: "'b"   and y :: "'b"   and R :: "('b \<times> 'b) set" shows "((x, y) \<in> R\<^sup>*) = (x = y \<or> x \<noteq> y \<and> (x, y) \<in> R\<^sup>+)"
return_def: fixes x :: "'b" shows "return x = Heap_Monad.heap (Pair x)"
assms: shows "compile_env \<A> \<O> e = Ok e'"
sup_absorb: fixes x :: "'a"   and y :: "'a" shows "- x \<squnion> - x \<sqinter> - y = - x"
strict_order_N_def: fixes a :: "'a N"   and b :: "'a N" shows "a < b \<equiv> a \<le> b \<and> \<not> b \<le> a"
SPLIT_1: shows "(([u], {#}), eel, sh, ch) \<in> ntrp fg"
interrupt_actions_ok_nthD: fixes "is" :: "'b set"   and ias :: "'b interrupt_action list"   and n :: "nat" assumes "interrupt_actions_ok is ias"   and "n < length ias" shows "interrupt_action_ok (redT_updIs is (take n ias)) (ias ! n)"
rtrancl_reflcl_absorb: fixes R :: "('b \<times> 'b) set" shows "(R\<^sup>*)\<^sup>= = R\<^sup>*"
prop_make_vert_equiv_2: fixes n :: "nat" shows "length (make_vert_equiv n) = 2 ^ n"
new_asms_only_for_priv_ignores_old_vars: shows "ignores_old_vars new_asms_only_for_priv"
assms_3: shows "S \<inter> - T \<noteq> {}"
mem_CD_on_Cd: fixes x :: "'a"   and ds :: "'b set"   and X :: "'a set" shows "(x \<in> ManiPref.CD_on ds X) = (x \<in> ManiPref.Cd (Xd x) X \<and> Xd x \<in> ds)"
vconn_implies_hpar_1: fixes \<mu> :: "'c"   and f :: "'c"   and g :: "'c" assumes "\<guillemotleft>\<mu> : f \<rightarrow>\<^sub>C g\<guillemotright>" shows "src\<^sub>C \<mu> = src\<^sub>C f"
ran_m_fmdrop_notin: fixes C :: "'c"   and N :: "('c, 'd) fmap" assumes "C \<notin># dom_m N" shows "ran_m (fmdrop C N) = ran_m N"
tllist_all2_tfinite2_terminalD: fixes P :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and Q :: "'i \<Rightarrow> 'j \<Rightarrow> bool"   and xs :: "('g, 'i) tllist"   and ys :: "('h, 'j) tllist" assumes "tllist_all2 P Q xs ys"   and "tfinite ys" shows "Q (terminal xs) (terminal ys)"
scaleR_left_distrib: fixes a :: "real"   and b :: "real"   and x :: "'a" shows "(a + b) *\<^sub>R x = a *\<^sub>R x + b *\<^sub>R x"
\<alpha>_no_x: shows "\<alpha> \<upharpoonleft> X = []"
int_card: shows "int (CARD('a) - 1) = int CARD('a) - 1"
2: fixes S :: "'a set" assumes "open S"   and "x \<in> S" shows "\<exists>A\<in>\<A>. A \<subseteq> S"
set_bits_bit_eq: fixes 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 divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bit :: "'a \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and drop_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and take_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and "and" :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and or :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and xor :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and mask :: "nat \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and not :: "'a \<Rightarrow> 'a"   and set_bits :: "(nat \<Rightarrow> bool) \<Rightarrow> 'a"   and a :: "'a" assumes "class.bit_comprehension plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask uminus not set_bits" shows "set_bits (bit a) = a"
add_mset: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and z :: "'a"   and x :: "'a"   and N :: "'a multiset" assumes "comm_monoid_mset f z" shows "comm_monoid_mset.F f z (add_mset x N) = f x (comm_monoid_mset.F f z N)"
mono_gamma_o: fixes sa :: "(char list \<Rightarrow> 'av) option"   and sa' :: "(char list \<Rightarrow> 'av) option" assumes "sa \<sqsubseteq> sa'" shows "\<gamma>\<^sub>o sa \<subseteq> \<gamma>\<^sub>o sa'"
irreflexive_axioms: shows "irreflexive A (\<bowtie>)"
aux: fixes x :: "'b"   and A :: "'b list set" shows "set ` (#) x ` A = insert x ` set ` A"
arr_char: fixes \<mu> :: "('a, 'a set) Maps.arr arrow_of_spans_data" shows "Span.arr \<mu> = arrow_of_spans (\<odot>) \<mu>"
asm_3: shows "circlepath z r x = circlepath z r y"
find_the_pair: fixes pairs :: "('a \<times> 'b) list"   and P :: "'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'b" assumes "distinct (map fst pairs)"   and "\<And>x y. \<lbrakk>x \<in> set (map fst pairs); y \<in> set (map fst pairs); P x; P y\<rbrakk> \<Longrightarrow> x = y"   and "(x, y) \<in> set pairs"   and "P x" shows "List.find (\<lambda>(x, uu_). P x) pairs = Some (x, y)"
image_cong: fixes M :: "'a set"   and N :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "M = N"   and "\<And>x. s2p N x \<Longrightarrow> f x = g x" shows "\<P> f M = \<P> g N"
linear_surjective_right_inverse: fixes f :: "'b \<Rightarrow> 'c" assumes "linear (*a) (*b) f"   and "surj f" shows "\<exists>g. linear (*b) (*a) g \<and> f \<circ> g = id"
crename_NotL: fixes R :: "trm"   and a :: "coname"   and b :: "coname"   and c :: "coname"   and N :: "trm"   and y :: "name" assumes "R[a\<turnstile>c>b] = NotL <c>.N y"   and "c \<sharp> (R, a, b)" shows "\<exists>N'. R = NotL <c>.N' y \<and> N'[a\<turnstile>c>b] = N"
inputCases': fixes a :: "name"   and b :: "name"   and P :: "pi"   and Rs :: "residual" assumes "a<b>.P \<longmapsto> Rs" obtains x :: "name" and aa :: "name" and P :: "pi" where "a = aa \<and> [b].Pa__ = [x].P" and "Rs = aa<x> \<prec> P" and "x \<noteq> aa"
hyps: fixes used :: "'b set" assumes "p \<in>. ps" shows "\<exists>i. {\<^bold>@ i q, \<^bold>\<diamond> Nom i} \<subseteq> set (witness_list ps used)"
integrable_fst: fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c" assumes "integrable (M1 \<Otimes>\<^sub>M M2) (\<lambda>(x, y). f x y)" shows "integrable M1 (\<lambda>x. integral\<^sup>L M2 (f x))"
symmetric: fixes P :: "ccs"   and Q :: "ccs" assumes "P \<sim> Q" shows "Q \<sim> P"
assms_2: shows "VLambda I A \<in>\<^sub>\<circ> Vset \<alpha>"
matrix_to_iarray_eq_of_fun: fixes A :: "(('d, 'e) vec, 'c) vec"   and f :: "nat \<Rightarrow> 'd iarray"   and n :: "nat" assumes "\<forall>i. vec_to_iarray (A $ i) = f (mod_type_class.to_nat i)"   and "n = IArray.length (matrix_to_iarray A)" shows "matrix_to_iarray A = IArray.of_fun f n"
stop: fixes g :: "'f \<Rightarrow> ('e, 'f) step"   and s :: "'f" assumes "g s = Done" shows "s \<in> terminates_on g"
is_functor: shows "functor (\<cdot>\<^sub>J) (\<cdot>) \<chi>.A.map"
WTrtFAccNT: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and E :: "char list \<Rightarrow> ty option"   and e :: "char list exp"   and F :: "char list"   and D :: "char list"   and T :: "ty" assumes "P,E,h \<turnstile> e : NT" shows "P,E,h \<turnstile> e\<bullet>F{D} : T"
fun_lub_def: fixes L :: "'i set \<Rightarrow> 'h"   and A :: "('g \<Rightarrow> 'i) set" shows "fun_lub L A = (\<lambda>x. L {y. \<exists>f\<in>A. y = f x})"
mset: shows "mset_set {0..<length us} = {#i#} + {#j#} + mset_set ({0..<length us} - {i, j})"
curry_conv: fixes f :: "'c \<times> 'd \<Rightarrow> 'b"   and a :: "'c"   and b :: "'d" shows "curry f a b = f (a, b)"
X: shows "X = Re (2 * z / (1 + z * cnj z))"
ord_isom_surj: fixes E :: "('e, 'f) Order_scheme"   and f :: "'a \<Rightarrow> 'e"   and b :: "'e" assumes "Order E"   and "ord_isom D E f"   and "b \<in> carrier E" shows "\<exists>a\<in>carrier D. b = f a"
Err_le_conv: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and e :: "'b err" shows "(Err \<sqsubseteq>\<^bsub>Err.le r\<^esub> e) = (e = Err)"
weak_bisim_def: fixes ev\<^sub>A :: "('a \<Rightarrow> 'b) \<Rightarrow> 'c \<Rightarrow> 'b"   and ev\<^sub>B :: "('a \<Rightarrow> 'b) \<Rightarrow> 'd \<Rightarrow> bool"   and aexp_vars :: "'c \<Rightarrow> 'a set"   and bexp_vars :: "'d \<Rightarrow> 'a set"   and dma :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> Sec"   and \<C>_vars :: "'a \<Rightarrow> 'a set"   and bexp_neg :: "'d \<Rightarrow> 'd"   and assign_post :: "'d set \<Rightarrow> 'a \<Rightarrow> 'c \<Rightarrow> 'd set"   and dma_type :: "'a \<Rightarrow> 'd set"   and pred_False :: "'d"   and \<T>\<^sub>1 :: "(((('a, 'c, 'd) Stmt \<times> (Mode \<Rightarrow> 'a set)) \<times> ('a \<Rightarrow> 'b)) \<times> (('a, 'c, 'd) Stmt \<times> (Mode \<Rightarrow> 'a set)) \<times> ('a \<Rightarrow> 'b)) set"   and \<T> :: "(((('a, 'c, 'd) Stmt \<times> (Mode \<Rightarrow> 'a set)) \<times> ('a \<Rightarrow> 'b)) \<times> (('a, 'c, 'd) Stmt \<times> (Mode \<Rightarrow> 'a set)) \<times> ('a \<Rightarrow> 'b)) set" assumes "sifum_types ev\<^sub>A ev\<^sub>B aexp_vars bexp_vars dma \<C>_vars bexp_neg assign_post dma_type pred_False" shows "sifum_types.weak_bisim ev\<^sub>A ev\<^sub>B \<T>\<^sub>1 \<T> \<equiv> \<forall>c\<^sub>1 c\<^sub>2 mds mem\<^sub>1 mem\<^sub>2 c\<^sub>1' mds' mem\<^sub>1'. (((c\<^sub>1, mds), mem\<^sub>1), (c\<^sub>2, mds), mem\<^sub>2) \<in> \<T>\<^sub>1 \<and> (((c\<^sub>1, mds), mem\<^sub>1), (c\<^sub>1', mds'), mem\<^sub>1') \<in> sifum_lang_no_dma.eval\<^sub>w ev\<^sub>A ev\<^sub>B \<longrightarrow> (\<exists>c\<^sub>2' mem\<^sub>2'. (((c\<^sub>2, mds), mem\<^sub>2), (c\<^sub>2', mds'), mem\<^sub>2') \<in> sifum_lang_no_dma.eval\<^sub>w ev\<^sub>A ev\<^sub>B \<and> (((c\<^sub>1', mds'), mem\<^sub>1'), (c\<^sub>2', mds'), mem\<^sub>2') \<in> \<T>)"
RModHom_idhom: shows "RModuleHom FG (\<cdot>) V (\<cdot>) (id \<down> V)"
closure_involutive_aux: fixes U :: "('c \<Rightarrow> real) set set" assumes "U \<subseteq> \<R>" shows "Closure\<^sub>\<alpha> \<Union> U = \<Union> U"
powerset_lift_def: fixes f :: "'c \<rightarrow> 'd set" shows "\<^ps> f = (\<Lambda> S. \<Union> (Rep_cfun f ` S))"
preservation1_eq: fixes x :: "'a"   and y :: "'a" assumes "x ; y \<le> x ; y ; x"   and "y ; x \<le> y" shows "(x ; y)\<^sup>\<star> ; x = x ; y\<^sup>\<star>"
\<I>_1: shows "\<lbrakk>{}; {}; unlabel \<A>\<rbrakk>\<^sub>s \<I>"
assms_3: shows "\<forall>\<Sigma>. sp_instr F (xs ! pc) \<Sigma> = sp_instr F instr \<Sigma>"
card_of_underS: fixes r :: "('a \<times> 'a) set"   and a :: "'a" assumes "Card_order r"   and "a \<in> Field r" shows "|underS r a| <o r"
simps_7: fixes I1 :: "WS1S_Nameful.fo \<Rightarrow> nat"   and I2 :: "WS1S_Nameful.so \<Rightarrow> nat fset"   and X :: "WS1S_Nameful.so"   and \<phi> :: "WS1S_Nameful.ws1s" shows "satisfies I1 I2 (Ex2 X \<phi>) = (\<exists>N. satisfies I1 (I2(X := N)) \<phi>)"
expand_new_name_expand_init: fixes nm :: "nat" shows "expand_init < expand_new_name nm"
hop2: shows "local.prf.game_1 (prf'_reduction \<A>) = map_spmf fst game_semi_forgetful"