hcju/isabelleps huggingface dataset:
minimalize_offending_overapprox_sound: fixes G :: "'a graph"   and ff :: "('a \<times> 'a) list"   and nP :: "'a \<Rightarrow> 'b" assumes "wf_graph G"   and "is_offending_flows (set ff) G nP"   and "set ff \<subseteq> edges G"   and "distinct ff" shows "is_offending_flows_min_set (set (local.minimalize_offending_overapprox ff [] G nP)) G nP"
stermsl_choice_disj: fixes p :: "('a, 'b, 'c, 'd) seqp"   and p1 :: "('a, 'b, 'c, 'd) seqp"   and p2 :: "('a, 'b, 'c, 'd) seqp" shows "(p \<in> stermsl (p1 \<oplus> p2)) = (p \<in> stermsl p1 \<or> p \<in> stermsl p2)"
not_vertex: fixes x :: "'v" assumes "x \<notin> A \<Gamma>'"   and "x \<notin> B \<Gamma>'" shows "\<not> vertex \<Gamma>' x"
lookup_combine_with_key: fixes f :: "'b \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a"   and t1 :: "('b, 'a) RBT.rbt"   and t2 :: "('b, 'a) RBT.rbt"   and k :: "'b" shows "RBT.lookup (RBT.combine_with_key f t1 t2) k = combine_options (f k) (RBT.lookup t1 k) (RBT.lookup t2 k)"
append_eq_conv_conj: fixes xs :: "'b list"   and ys :: "'b list"   and zs :: "'b list" shows "(xs @ ys = zs) = (xs = zs \<down> length xs \<and> ys = zs \<up> length xs)"
limitin_atin: fixes Y :: "'b topology"   and f :: "'c \<Rightarrow> 'b"   and y :: "'b"   and X :: "'c topology"   and x :: "'c" shows "limitin Y f y (atin X x) = (y \<in> topspace Y \<and> (x \<in> topspace X \<longrightarrow> (\<forall>V. openin Y V \<and> y \<in> V \<longrightarrow> (\<exists>U. openin X U \<and> x \<in> U \<and> f ` (U - {x}) \<subseteq> V))))"
vsubst_def: fixes ys :: "'varSort"   and y1 :: "'var"   and y2 :: "'var" shows "vsubst ys y1 y2 \<equiv> subst ys (Var ys y1) y2"
order_refl: fixes x :: "('x, nat) pp" shows "ord x x"
frechet_linear: fixes vid1 :: "'a"   and vid2 :: "'a"   and vid3 :: "'a"   and fid1 :: "'b"   and fid2 :: "'b"   and fid3 :: "'b"   and pid1 :: "'c"   and pid2 :: "'c"   and pid3 :: "'c"   and pid4 :: "'c"   and I :: "('d, 'e, 'f) interp"   and \<theta> :: "('d, 'f) trm"   and v :: "(real, 'f) vec" assumes "ids vid1 vid2 vid3 fid1 fid2 fid3 pid1 pid2 pid3 pid4"   and "is_interp I"   and "dfree \<theta>" shows "bounded_linear (frechet I \<theta> v)"
cobounded2: fixes a :: "'d upred"   and b :: "'d upred" shows "b \<sqsubseteq> (a \<and> b)"
Diff_eqvt: fixes p :: "perm"   and A :: "'e set"   and B :: "'e set" shows "p \<bullet> (A - B) = p \<bullet> A - p \<bullet> B"
less_eq_list_drop: fixes xs :: "'a list"   and ys :: "'a list"   and y :: "'a" assumes "xs \<preceq> ys" shows "xs \<preceq> y # ys"
pr_iso_test0_corr: fixes Fs\<^sub>1 :: "'a list list"   and Fs\<^sub>2 :: "'b list list" assumes "\<forall>F\<in>set Fs\<^sub>1. distinct F"   and "\<forall>F\<in>set Fs\<^sub>2. distinct F"   and "distinct Fs\<^sub>1"   and "inj_on (\<lambda>xs. {xs} // {\<cong>}) (set Fs\<^sub>1)"   and "distinct Fs\<^sub>2"   and "inj_on (\<lambda>xs. {xs} // {\<cong>}) (set Fs\<^sub>2)" shows "pr_iso_test0 Map.empty Fs\<^sub>1 Fs\<^sub>2 = (\<exists>\<phi>. is_pr_iso \<phi> Fs\<^sub>1 Fs\<^sub>2)"
nonzero_inverse_minus_eq: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and one :: "'b"   and times :: "'b \<Rightarrow> '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 a :: "'b" assumes "class.division_ring divide inverse one times plus zero minus uminus"   and "a \<noteq> zero" shows "inverse (uminus a) = uminus (inverse a)"
Infinitesimal_hrabs_iff: fixes x :: "real star" shows "(\<bar>x\<bar> \<in> Infinitesimal) = (x \<in> Infinitesimal)"
assms_2: fixes i :: "'a" assumes "i \<notin> I" shows "x i = undefined"
nat_neg_numeral: fixes k :: "num" shows "nat (- numeral k) = 0"
exhaust: fixes y :: "('c, 'd, 'a, 'b, 'e) qAbs" obtains x1 :: "'a" and x2 :: "'b" and x3 :: "('c, 'd, 'a, 'b, 'e) qTerm" where "y = qAbs x1 x2 x3"
cp\<^sub>P\<^sub>R\<^sub>E': fixes self :: "'\<AA> state \<times> '\<AA> state \<Rightarrow> '\<alpha>0"   and x :: "'b"   and \<tau> :: "'\<AA> state \<times> '\<AA> state" shows "PRE self x \<tau> = PRE (\<lambda>_. self \<tau>) (f_lam x \<tau>) \<tau>"
set_image_mset: fixes f :: "'c \<Rightarrow> 'b"   and M :: "'c multiset" shows "set_mset (image_mset f M) = f ` set_mset M"
Collect_mem_eq: fixes A :: "'e set" shows "{x. x \<in> A} = A"
leg0_simps_4: shows "local.dom s\<^sub>0 = s\<^sub>0"
wp3_SKIP: fixes Q :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool" shows "wp\<^sub>3 SKIP Q = (Q \<and>* $ 1)"
flow_solves_ode: fixes x0 :: "'a" assumes "t0 \<in> T"   and "x0 \<in> X" shows "(flow t0 x0 solves_ode f) (existence_ivl t0 x0) X"
invar: fixes xc :: "'b" assumes "xc \<in> S0" shows "invar xc"
h2b: shows "(\<Sum>i\<le>2. isolate_variable_sparse p var i * Var var ^ i) = (\<Sum>i\<le>1. isolate_variable_sparse p var i * Var var ^ i) + isolate_variable_sparse p var 2 * (Var var)\<^sup>2"
J_3: shows "\<not> vUniform False (initCfgList ! Suc j)"
bounded_convex_hull: fixes s :: "'b set" assumes "bounded s" shows "bounded (convex hull s)"
range_of_not_empty: fixes \<Psi> :: "('d, 'c) sas_plus_problem"   and v :: "'d" shows "(range_of \<Psi> v \<noteq> None \<and> range_of \<Psi> v \<noteq> Some []) = (\<R>\<^sub>+ \<Psi> v \<noteq> {})"
cartesian_monoidal_category_axioms: shows "cartesian_monoidal_category (\<cdot>) local.Prod \<alpha>.map \<iota> dup trm"
eval_cases_17: fixes P :: "(char list list \<times> char list exp) prog"   and e :: "char list exp"   and s :: "heap \<times> sfields \<times> sheap"   and e' :: "char list exp"   and s' :: "heap \<times> sfields \<times> sheap" assumes "P \<turnstile> \<langle>throw e,s\<rangle> \<Rightarrow> \<langle>e',s'\<rangle>" obtains a :: "nat" where "e'a__ = Throw a" and "P \<turnstile> \<langle>e,s\<rangle> \<Rightarrow> \<langle>addr a,s'\<rangle>"   | "e'a__ = THROW NullPointer" and "P \<turnstile> \<langle>e,s\<rangle> \<Rightarrow> \<langle>null,s'\<rangle>"   | e' :: "char list exp" where "e'a__ = throw e'" and "P \<turnstile> \<langle>e,s\<rangle> \<Rightarrow> \<langle>throw e',s'\<rangle>"
signed_lconjaction_rconjaction: fixes s :: "'a"   and x :: "'a \<times> bool" shows "signed_lconjaction s (signed_rconjaction s x) = x"
finite_cartesian_productD2: fixes A :: "'a set"   and B :: "'b set" assumes "finite (A \<times> B)"   and "A \<noteq> {}" shows "finite B"
mem_notdisjoint2: fixes schedule :: "'b \<Rightarrow> 'a list"   and A :: "'b"   and B :: "'b"   and x :: "'a" assumes "disjoint (schedule A) (schedule B)"   and "x mem schedule A" shows "\<not> x mem schedule B"
IH: fixes mn :: "char list list"   and env :: "v sem_env"   and s :: "'a state" assumes "\<forall>r. \<not> evaluate_decs False mn env s ds r" shows "\<exists>r. evaluate_decs True mn env s ds r \<and> snd r = Rerr (Rabort Rtimeout_error)"
surjective: fixes r :: "('a, 'b, 'c) Signature_scheme" shows "r = \<lparr>BaseTypes = Ty\<^bsub>r\<^esub>, BaseFunctions = Fn\<^bsub>r\<^esub>, SigDom = sDom\<^bsub>r\<^esub>, SigCod = sCod\<^bsub>r\<^esub>, \<dots> = Signature.more r\<rparr>"
extremum: fixes a :: "'c" shows "(0::'c) \<le> a"
inf_right_dist_sup: fixes y :: "'a"   and z :: "'a"   and x :: "'a" shows "(- y \<squnion> - z) \<sqinter> - x = - y \<sqinter> - x \<squnion> - z \<sqinter> - x"
rank: fixes x :: "'a" assumes "x \<in> A" shows "linorder_rank R (set xs) x = index xs x"
mem_existence_ivl_subset: fixes t :: "real"   and x0 :: "'a" assumes "t \<in> existence_ivl0 x0" shows "t \<in> UNIV"
max_min_same_1: fixes x :: "'c"   and y :: "'c" shows "max x (min x y) = x"
dist_tendsto_0_imp_tendsto: fixes f :: "nat \<Rightarrow> real"   and a :: "real" assumes "(\<lambda>i. \<bar>f i - a\<bar>) \<longlonglongrightarrow> 0" shows "f \<longlonglongrightarrow> a"
simps_13: fixes f :: "'b \<Rightarrow> 'a"   and x21 :: "'b"   and x22 :: "'b llist" shows "lmap f (x21 #\<^sub>l x22) = f x21 #\<^sub>l lmap f x22"
tendsto_compose_eventually: fixes g :: "'a \<Rightarrow> 'b"   and m :: "'b"   and l :: "'a"   and f :: "'c \<Rightarrow> 'a"   and F :: "'c filter" assumes "g \<midarrow>l\<rightarrow> m"   and "(f \<longlongrightarrow> l) F"   and "\<forall>\<^sub>F x in F. f x \<noteq> l" shows "((\<lambda>x. g (f x)) \<longlongrightarrow> m) F"
extend_def: fixes forms :: "'a sequent"   and seq :: "'a sequent" shows "extend forms seq \<equiv> if succ seq = Em then  antec forms + antec seq \<Rightarrow>* succ forms else ( antec forms + antec seq \<Rightarrow>* succ seq)"
share_cond: shows "\<not> (\<not> isLeaf_pt node low high \<and> repa (low node) = repa (high node))"
af_letter_build: fixes x :: "'a set"   and w :: "nat \<Rightarrow> 'a set"   and \<phi> :: "'a ltln" shows "(x ## w) \<Turnstile>\<^sub>n \<phi> = w \<Turnstile>\<^sub>n af_letter \<phi> x"
labeling_consistent: fixes R :: "('h \<times> 'i) set"   and A :: "('h, 'j, 'k) sa_rec_scheme"   and B :: "('i, 'j, 'l) sa_rec_scheme"   and a :: "'h"   and b :: "'i" assumes "lsimulation R A B"   and "(a, b) \<in> R" shows "sa_L A a = sa_L B b"
var: fixes \<Gamma> :: "(name, value) fmap"   and name :: "name"   and val :: "value"   and C_info :: "(name, dt_def) fmap" assumes "fmlookup \<Gamma> name = Some val" shows "special_constants.veval' C_info \<Gamma> (Svar name) val"
ord_isom_bij_to: fixes E :: "('e, 'f) Order_scheme"   and f :: "'a \<Rightarrow> 'e" assumes "Order E"   and "ord_isom D E f" shows "bij_to f (carrier D) (carrier E)"
oldDef_defAss: fixes v :: "'val"   and g :: "'g"   and n :: "'node"   and ns :: "'node list" assumes "v \<in> allUses g n"   and "g \<turnstile> Entry g-ns\<rightarrow>n" obtains m :: "'node" where "m \<in> set ns" and "var g v \<in> oldDefs g m"
swap: fixes m :: "'n"   and n :: "'n" assumes "m \<noteq> n" shows "P (\<lambda>x. \<chi>i. x $ Fun.swap m n id i)"
mem: fixes A :: "real mat"   and n :: "nat"   and m :: "nat"   and lam :: "complex"   and n_as :: "(nat \<times> complex) list" assumes "spectral_radius_1_jnf_max A n m lam n_as" shows "(m, lam) \<in> set n_as"
1: shows "\<forall>x\<in>X. n_ivl (lookup (S1 \<triangle> S2) x) \<le> n_ivl (lookup S1 x)"
ntrp_stack_decomp_s: fixes u :: "'a"   and s :: "'a list"   and r :: "'a list"   and c :: "'a list multiset"   and ee :: "('b, 'c) label list el_step"   and s' :: "'a list"   and c' :: "'a list multiset"   and fg :: "('a, 'b, 'c, 'd, 'e) flowgraph_rec_scheme" assumes "((u # s @ r, c), ee, s', c') \<in> ntrp fg" obtains v :: "'a" and rr :: "'a list" where "s' = v # rr @ r" and "((u # s, c), ee, v # rr, c') \<in> ntrp fg"
tensor_add_0_right: fixes A :: "'b tensor" shows "A + tensor0 (dims A) = A"
coreflexive_injective: fixes x :: "'c \<times> 'c \<Rightarrow> 'd" assumes "matrix_idempotent_semiring.coreflexive x" shows "matrix_stone_relation_algebra.injective x"
that: fixes t :: "('a, 'b) Term.term" assumes "t \<in> set ts"   and "t \<unrhd> s" shows "thesis"
small_big_trans: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'b"   and h :: "'a \<Rightarrow> 'b" assumes "f \<in> \<omega>[F](g)"   and "g \<in> \<Omega>[F](h)" shows "f \<in> \<omega>[F](h)"
swap_apply_2: fixes a :: "'c"   and b :: "'c"   and f :: "'c \<Rightarrow> 'b" shows "Fun.swap a b f b = f a"
Gateway_L1: fixes req :: "nat \<Rightarrow> reqType list"   and dt :: "nat \<Rightarrow> ECall_Info list"   and a :: "nat \<Rightarrow> aType list"   and stop :: "nat \<Rightarrow> stopType list"   and lose :: "nat \<Rightarrow> bool list"   and d :: "nat"   and ack :: "nat \<Rightarrow> GatewayStatus list"   and i :: "nat \<Rightarrow> ECall_Info list"   and vc :: "nat \<Rightarrow> vcType list"   and t :: "nat" assumes "Gateway req dt a stop lose d ack i vc"   and "msg (Suc 0) req"   and "msg (Suc 0) a"   and "msg (Suc 0) stop"   and "ts lose"   and "ack t = [init_state]"   and "req (Suc t) = [init]"   and "lose (Suc t) = [False]"   and "lose (Suc (Suc t)) = [False]" shows "ack (Suc (Suc t)) = [connection_ok]"
not_alw_iff: fixes P :: "'c stream \<Rightarrow> bool"   and \<omega> :: "'c stream" shows "(\<not> alw P \<omega>) = ev (\<lambda>xs. \<not> P xs) \<omega>"
sortedP_Cons: fixes xa :: "real \<times> real"   and x :: "real \<times> real"   and xs :: "(real \<times> real) list" assumes "ccw'.sortedP xa (x # xs)" obtains "\<forall>x\<in>set xs. ccw' xa x x" and "ccw'.sortedP xa xs"
induct: fixes P :: "'c rexp \<Rightarrow> bool"   and a0 :: "'c rexp" assumes "\<And>r s. \<lbrakk>P r; P s\<rbrakk> \<Longrightarrow> P (Plus r s)"   and "\<And>r s. \<lbrakk>P r; P s\<rbrakk> \<Longrightarrow> P (rexp.Inter r s)"   and "P Full"   and "P Zero"   and "\<And>r. P (rexp.Not r)"   and "P One"   and "\<And>v. P (Atom v)"   and "\<And>v va. P (Times v va)"   and "\<And>v. P (Star v)"   and "\<And>v. P (Pr v)" shows "P a0"
finite_UNIV_unit_def: shows "finite_UNIV = Phantom(unit) True"
update_Some_unfold: fixes k :: "'d"   and v :: "'c"   and al :: "('d \<times> 'c) list"   and x :: "'d"   and y :: "'c" shows "(map_of (AList.update k v al) x = Some y) = (x = k \<and> v = y \<or> x \<noteq> k \<and> map_of al x = Some y)"
smult_dvd_iff: fixes a :: "'a"   and p :: "'a poly"   and q :: "'a poly" shows "(smult a p dvd q) = (if a = (0::'a) then q = 0 else p dvd q)"
\<I>\<^sub>\<tau>_5: shows "(\<forall>n. \<I>\<^sub>\<tau> \<Turnstile> \<langle>proj_unl n \<A>'\<rangle>) \<or> (\<exists>\<A>''. prefix \<A>'' \<A>' \<and> strand_leaks\<^sub>l\<^sub>s\<^sub>t \<A>'' Sec \<I>\<^sub>\<tau>)"
powser_ms_aux_MSLNil_iff: fixes cs :: "real msllist"   and f :: "('a \<times> real) msllist" shows "(powser_ms_aux cs f = MSLNil) = (cs = MSLNil)"
nearly_healthy_unitaryD: fixes t :: "('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real"   and P :: "'c \<Rightarrow> real" assumes "nearly_healthy t"   and "unitary P" shows "unitary (t P)"
lower: fixes N :: "nat" assumes "n \<le> N" shows "x < u N + ereal r"
assigns_comp: fixes f :: "'a \<Rightarrow> 'a"   and g :: "'a \<Rightarrow> 'a" shows "\<langle>f\<rangle>\<^sub>a ;; \<langle>g\<rangle>\<^sub>a = \<langle>g \<circ> f\<rangle>\<^sub>a"
RETURN_ref_RETURND: fixes c :: "'c"   and R :: "('c \<times> 'd) set"   and a :: "'d" assumes "RETURN c \<le> \<Down> R (RETURN a)" shows "(c, a) \<in> R"
absorb2: fixes b :: "'a"   and a :: "'a" assumes "b \<le> a" shows "inf a b = b"
coeff_all_0: fixes p :: "'a mpoly" assumes "\<And>m. MPoly_Type.coeff p m = (0::'a)" shows "p = 0"
axiom9_F: fixes x :: "('o, 'p, 'n) F" shows "|F'set4 x| \<le>o F'bd"
less_eq: fixes a :: "'a operation"   and b :: "'a operation" shows "(a \<le> b) = (if pos a \<noteq> pos b then pos a < pos b else if is_INS a \<noteq> is_INS b then is_INS a \<and> \<not> is_INS b else if op_idx a \<noteq> op_idx b then op_idx a < op_idx b else op_int a \<le> op_int b)"
square_free_factorization_int_def: fixes f :: "int poly" shows "square_free_factorization_int f = (case x_split f of (n, g) \<Rightarrow> case square_free_factorization_int' g of (d, fs) \<Rightarrow> if n = 0 then (d, fs) else (d, (Polynomial.monom 1 1, n - 1) # fs))"
monad_left_unit: fixes x :: "'f"   and f :: "'f \<rightarrow> 'd\<cdot>'e" shows "return\<cdot>x \<bind> f = f\<cdot>x"
observable_move_vpa_path: fixes S :: "'node SDG_node set"   and f :: "'edge \<Rightarrow> ('var, 'val, 'ret, 'pname) edge_kind"   and m :: "'node"   and ms :: "'node list"   and s :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list"   and a :: "'edge"   and m' :: "'node"   and ms' :: "'node list"   and s' :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list"   and rs :: "'edge list"   and cs :: "'edge list" assumes "S,f \<turnstile> (m # ms,s) -a\<rightarrow> (m' # ms',s')"   and "valid_node m"   and "\<forall>i<length rs. rs ! i \<in> get_return_edges (cs ! i)"   and "ms = targetnodes rs"   and "valid_return_list rs m"   and "length rs = length cs" shows "valid_path_aux cs [a]"
stmt: fixes m :: "'a \<times> 'a set" assumes "m \<in># N \<times># {#b#}" shows "(fst m \<in> snd m) = (fst m \<in> b)"
split_at_it_def: fixes n :: "nat"   and xs :: "'a list" shows "split_at_it n xs = split_at_it' [] n xs"
eqset_imp_iff: fixes A :: "'b set"   and B :: "'b set"   and x :: "'b" assumes "A = B" shows "(x \<in> A) = (x \<in> B)"
res: shows "yun_gcd.yun_monic_factorization gcd p = bs"
aperiodic_def: fixes K :: "'a \<Rightarrow> 'a pmf"   and C :: "'a set" shows "MC_syntax.aperiodic K C = (C \<in> UNIV // MC_syntax.communicating K \<and> MC_syntax.period K C = 1)"
responses_\<I>_mono: fixes \<I> :: "('d, 'e) \<I>"   and \<I>' :: "('d, 'e) \<I>"   and x :: "'d" assumes "\<I> \<le> \<I>'"   and "x \<in> outs_\<I> \<I>" shows "responses_\<I> \<I>' x \<subseteq> responses_\<I> \<I> x"
cis_diff_cis_opposite: fixes \<phi> :: "real" shows "cis \<phi> - cis (- \<phi>) = 2 * \<i> * cor (sin \<phi>)"
group_action_def: fixes G :: "('d, 'e) monoid_scheme"   and \<phi> :: "'d \<Rightarrow> 'f \<Rightarrow> 'f"   and M :: "'f set" shows "group_action G \<phi> M \<equiv> Group.group G \<and> group_action_axioms G \<phi> M"
mem_is_intervalI: fixes s :: "'b set"   and a :: "'b"   and b :: "'b"   and x :: "'b" assumes "is_interval s"   and "a \<in> s"   and "b \<in> s"   and "\<And>i. i \<in> Basis \<Longrightarrow> a \<bullet> i \<le> x \<bullet> i \<and> x \<bullet> i \<le> b \<bullet> i \<or> b \<bullet> i \<le> x \<bullet> i \<and> x \<bullet> i \<le> a \<bullet> i" shows "x \<in> s"
Dag_is_DAG: fixes p :: "ref"   and l :: "ref \<Rightarrow> ref"   and r :: "ref \<Rightarrow> ref"   and t :: "dag" assumes "Dag p l r t" shows "DAG t"
mat_mult: fixes nr :: "nat"   and n :: "nat"   and m1 :: "'b list list"   and nc :: "nat"   and m2 :: "'b list list"   and ze :: "'b"   and pl :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and ti :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" assumes "mat nr n m1"   and "mat n nc m2" shows "mat nr nc (mat_multI ze pl ti nr m1 m2)"
Class_right_inverse: fixes G :: "'c set"   and composition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit :: "'c"   and E :: "('c \<times> 'c) set"   and a :: "'c" assumes "group_congruence G composition unit E"   and "a \<in> G" shows "monoid_congruence.quotient_composition G composition E (equivalence.Class G E a) (equivalence.Class G E (monoid.inverse G composition unit a)) = equivalence.Class G E unit"
invertible_eq_bij: fixes m :: "(('a, 'b) vec, 'c) vec" shows "invertible m = bij ((*v) m)"
exec_blockStuck: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and bdy :: "('e, 'd, 'f) com"   and init :: "'e \<Rightarrow> 'e"   and s :: "'e"   and return :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and c :: "'e \<Rightarrow> 'e \<Rightarrow> ('e, 'd, 'f) com" assumes "\<Gamma>\<turnstile> \<langle>bdy,Normal (init s)\<rangle> \<Rightarrow> Stuck" shows "\<Gamma>\<turnstile> \<langle>block init bdy return c,Normal s\<rangle> \<Rightarrow> Stuck"
ope: fixes U :: "'a set" assumes "openin (top_of_set S) U" shows "openin (top_of_set T) (f ` U)"
DiamondOne: fixes trans :: "'a \<Rightarrow> 'b \<Rightarrow> 'c messageValue \<Rightarrow> 'b"   and sends :: "'a \<Rightarrow> 'b \<Rightarrow> 'c messageValue \<Rightarrow> ('a, 'c) message \<Rightarrow> nat"   and cfg :: "('a, 'c, 'b) configuration"   and m :: "('a, 'c) message"   and cfg1 :: "('a, 'c, 'b) configuration"   and p :: "'a"   and q :: "'a"   and m' :: "('a, 'c) message"   and cfg2 :: "('a, 'c, 'b) configuration" assumes "asynchronousSystem.steps trans sends cfg m cfg1"   and "p \<noteq> q"   and "isReceiverOf p m"   and "\<not> isReceiverOf p m'"   and "isReceiverOf q m'"   and "\<not> isReceiverOf q m"   and "asynchronousSystem.steps trans sends cfg m' cfg2" shows "\<exists>cfg'. asynchronousSystem.steps trans sends cfg1 m' cfg' \<and> asynchronousSystem.steps trans sends cfg2 m cfg'"
diagram_def: fixes J :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and D :: "'a \<Rightarrow> 'b" shows "diagram J C D \<equiv> category C \<and> category J \<and> functor J C D"
S3: shows "A = {x. \<exists>y. pA x y = 0}"
helper_cases_call_subchain_defined_or_return: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and rs_called :: "'c rule list"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d" assumes "\<forall>x\<in>ran \<Gamma>. wf_chain \<Gamma> x"   and "\<forall>rsg\<in>ran \<Gamma>. \<forall>r\<in>set rsg. (\<forall>chain. get_action r \<noteq> Goto chain) \<and> get_action r \<noteq> Unknown"   and "\<forall>y m. \<forall>r\<in>set rs_called. r = Rule m (Call y) \<longrightarrow> (\<exists>t. \<Gamma>,\<gamma>,p\<turnstile> \<langle>[Rule m (Call y)], Undecided\<rangle> \<Rightarrow> t)"   and "wf_chain \<Gamma> rs_called"   and "\<forall>r\<in>set rs_called. (\<forall>chain. get_action r \<noteq> Goto chain) \<and> get_action r \<noteq> Unknown" shows "(\<exists>t. \<Gamma>,\<gamma>,p\<turnstile> \<langle>rs_called, Undecided\<rangle> \<Rightarrow> t) \<or> (\<exists>rs_called1 rs_called2 m'. rs_called = rs_called1 @ [Rule m' Return] @ rs_called2 \<and> matches \<gamma> m' p \<and> \<Gamma>,\<gamma>,p\<turnstile> \<langle>rs_called1, Undecided\<rangle> \<Rightarrow> Undecided)"
bigo_add_commute_imp: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd"   and h :: "'c \<Rightarrow> 'd" assumes "f \<in> g +o O(h)" shows "g \<in> f +o O(h)"
zmod_eq_0D: fixes m :: "int"   and d :: "int" assumes "m mod d = 0" shows "\<exists>q. m = d * q"
fdia_def: fixes x :: "'a"   and y :: "'a" shows "antidomain_semiringl.fdia antirange_op (\<lambda>x y. y \<cdot> x) x y = ar ar (y \<cdot> x)"
FinFunc_osum: fixes fg :: "'b + 'c \<Rightarrow> 'a" shows "(fg \<in> rst.FINFUNC) = (fg \<circ> Inl \<in> FinFunc r s \<and> fg \<circ> Inr \<in> FinFunc r t)"
cong_4321: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "Cong A B C D" shows "Cong D C B A"
prj0_simps_1: fixes f :: "'a"   and g :: "'a" assumes "cospan f g" shows "arr \<p>\<^sub>0[f, g]"
density_context_empty: shows "density_context {} (set vs \<union> set vs') \<Gamma> (\<lambda>_. 1)"
sep_conj_true: shows "((\<lambda>s. True) \<and>* (\<lambda>s. True)) = (\<lambda>s. True)"
BIT_step_def: fixes s :: "'b list \<times> bool list \<times> 'b list"   and q :: "'b" shows "BIT_step s q = (let a = (if fst (snd s) ! index (snd (snd s)) q then 0 else length (fst s), []) in return_pmf (a, flip (index (snd (snd s)) q) (fst (snd s)), snd (snd s)))"
inj_on_def: fixes f :: "'f \<Rightarrow> 'g"   and A :: "'f set" shows "inj_on f A = (\<forall>x\<in>A. \<forall>y\<in>A. f x = f y \<longrightarrow> x = y)"
continuous_const: fixes F :: "'b filter"   and c :: "'c" shows "continuous F (\<lambda>x. c)"
eq_iff: fixes x :: "'b"   and d :: "nat"   and y :: "'b" shows "(monom x d = monom y d) = (x = y)"
small_nstep_base: fixes P :: "'prog"   and \<sigma> :: "'state" shows "small_nstep P \<sigma> 0 = {\<sigma>}"
intro: fixes R :: "('c, 'd) ring_scheme"   and S :: "('e, 'f) ring_scheme"   and h :: "'c \<Rightarrow> 'e" assumes "ring R"   and "ring S"   and "ring_hom_ring_axioms R S h" shows "ring_hom_ring R S h"
eps_iff: fixes q :: "hf"   and Q :: "hf set" shows "(q \<in> ST.epsclo Q) = (q \<in> Q \<inter> insert q0 (dfa.states MS) \<or> q = q0 \<and> dfa.final MS \<inter> Q \<noteq> {} \<or> q = dfa.init MS \<and> insert q0 (dfa.final MS) \<inter> Q \<noteq> {})"
AB: shows "A \<inter> B \<in> null_sets lborel"
witness_propD: fixes \<delta> :: "('a, 'b) ta_rule set"   and m :: "'a \<Rightarrow> 'b tree option"   and q :: "'a"   and t :: "'b tree" assumes "witness_prop \<delta> m"   and "m q = Some t" shows "accs \<delta> t q"
load_sub3_def: fixes instr :: "sparc_operation \<times> inst_operand list"   and curr_win :: "'b word"   and rd :: "word5"   and asi :: "word8"   and address :: "word32" shows "load_sub3 instr curr_win rd asi address \<equiv> do (result, new_state) \<leftarrow> gets (memory_read asi address); if result = None then do raise_trap data_access_exception; return () od else do data_word \<leftarrow> gets (\<lambda>s. case result of Some v \<Rightarrow> v); modify (\<lambda>s. new_state); word0 \<leftarrow> gets (\<lambda>s. ld_word0 instr data_word address); if rd \<noteq> 0 \<and> fst instr \<in> {load_store_type LD, load_store_type LDA, load_store_type LDUH, load_store_type LDSB, load_store_type LDUB, load_store_type LDUBA, load_store_type LDSH, load_store_type LDSHA, load_store_type LDUHA, load_store_type LDSBA} then do write_reg word0 curr_win rd; return () od else load_sub2 address asi rd curr_win word0 od od"