hcju/isabelleps huggingface dataset:
assms_4: shows "(ir, Delete is e2) \<in> set (node_deliver_messages xs)"
Quotient_rel_abs: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and Abs :: "'b \<Rightarrow> 'c"   and Rep :: "'c \<Rightarrow> 'b"   and T :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and r :: "'b"   and s :: "'b" assumes "Quotient R Abs Rep T"   and "R r s" shows "Abs r = Abs s"
inf_sup_aci_1: fixes inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and x :: "'c"   and y :: "'c" assumes "class.lattice inf less_eq less sup" shows "inf x y = inf y x"
simps_2: fixes x :: "'b"   and xs :: "'b list" shows "suffixes (x # xs) = suffixes xs @ [x # xs]"
fin_make_untimed_inf_truncate_Nonempty_all0a: fixes z :: "nat \<Rightarrow> 'd list"   and a :: "'d" assumes "z 0 = [a]" shows "\<forall>i. fin_make_untimed (inf_truncate z i) \<noteq> []"
feasible_lfp_loop: shows "feasible (lfp_trans (\<lambda>a. wp (body ;; Embed a \<^bsub>\<guillemotleft> G \<guillemotright>\<^esub>\<oplus> Skip)))"
rbt_lookup_fold_rbt_insert_impl: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and t2 :: "('b, 'c) rbt"   and t1 :: "('b, 'c) rbt" assumes "class.linorder less_eq less"   and "ord.rbt_sorted less t2" shows "ord.rbt_lookup less (RBT_Impl.fold (ord.rbt_insert less) t1 t2) = ord.rbt_lookup less t2 ++ map_of (rev (RBT_Impl.entries t1))"
rt_strictly_fresherI': fixes i :: "nat"   and rt1 :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option"   and rt2 :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option" assumes "rt1 \<sqsubseteq>\<^bsub>i\<^esub> rt2"   and "\<not> rt2 \<sqsubseteq>\<^bsub>i\<^esub> rt1" shows "rt1 \<sqsubset>\<^bsub>i\<^esub> rt2"
exhaust: fixes y :: "('c, 'd) t" obtains x :: "('c \<times> 'd) list" where "y = T x"
h6: shows "tp \<cdot> at r \<cdot> y \<le> at r \<cdot> y \<cdot> tq"
pidgc_\<alpha>_def: fixes pgc :: "('b, 'd, 'e) pid_global_config" shows "pidgc_\<alpha> pgc \<equiv> \<lparr>global_config.processes = mset (pid_global_config.processes pgc), state = pid_global_config.state pgc\<rparr>"
list_slice2_0: fixes xs :: "'b list" shows "list_slice2 xs 0 = (if length xs = 0 then [] else [xs])"
val: fixes n :: "nat" assumes "dec (\<sigma> n p) = Some w" shows "w = v"
rel_inject_2: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and x21 :: "'a"   and x22 :: "'a list"   and y21 :: "'b"   and y22 :: "'b list" shows "list_all2 R (x21 # x22) (y21 # y22) = (R x21 y21 \<and> list_all2 R x22 y22)"
cases: fixes a :: "('test, 'atom, 'choice) cmd \<times> 'state" assumes "discrCf a" obtains cf :: "('test, 'atom, 'choice) cmd \<times> 'state" where "a = cf" and "\<And>i. i < brn (fst cf) \<longrightarrow> snd cf \<approx> snd (cont_eff cf i) \<and> discrCf (cont_eff cf i)"
trans_is_node: fixes A :: "bool list bdd list \<times> bool list"   and n :: "nat"   and q :: "bool list"   and a :: "bool list" assumes "aut_nfa A n"   and "nfa_is_node A q"   and "is_alph n a" shows "nfa_is_node A (nfa_trans A q a)"
exists_unique_gen: fixes K :: "'a list set"   and I :: "'a list list set" assumes "subfield K (K [X])"   and "ideal I (K [X]\<^bsub>K [X]\<^esub>)"   and "I \<noteq> {[]}" shows "\<exists>!p. p \<in> carrier (K [X]\<^bsub>K [X]\<^esub>) \<and> lead_coeff p = \<one>\<^bsub>K [X]\<^esub> \<and> I = PIdl\<^bsub>K [X]\<^bsub>K [X]\<^esub>\<^esub> p"
cases: fixes x :: "cell" obtains "x = Bk"   | "x = Oc"
simps_1: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and \<alpha> :: "'c \<Rightarrow> 'd" assumes "class.linorder less_eq less" shows "linorder.eval_monom_list \<alpha> [] = (1::'d)"
repv_selectlike_other: fixes x :: "variable"   and y :: "variable"   and \<omega> :: "variable \<Rightarrow> real"   and d :: "real"   and X :: "(variable \<Rightarrow> real) set" assumes "x \<noteq> y" shows "(repv \<omega> x d \<in> selectlike X \<omega> {y}) = (repv \<omega> x d \<in> X)"
semiring_norm_85: fixes m :: "num" shows "(num.Bit0 m = num.One) = False"
poset_simplex_map_def: fixes P :: "'a set"   and x :: "'a" shows "poset_simplex_map P x = {y. pseudominimal_in (P .\<^bold>\<le> x) y}"
XsubY: shows "bangRel weakBisim \<subseteq> Strong_Early_Bisim.bisim O bangRel weakBisim O Strong_Early_Bisim.bisim"
INF_lower2: fixes Inf :: "'h set \<Rightarrow> 'h"   and Sup :: "'h set \<Rightarrow> 'h"   and inf :: "'h \<Rightarrow> 'h \<Rightarrow> 'h"   and less_eq :: "'h \<Rightarrow> 'h \<Rightarrow> bool"   and less :: "'h \<Rightarrow> 'h \<Rightarrow> bool"   and sup :: "'h \<Rightarrow> 'h \<Rightarrow> 'h"   and bot :: "'h"   and top :: "'h"   and i :: "'i"   and A :: "'i set"   and f :: "'i \<Rightarrow> 'h"   and u :: "'h" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "i \<in> A"   and "less_eq (f i) u" shows "less_eq (Inf (f ` A)) u"
seq_inf_par: fixes a :: "'c"   and b :: "'c" shows "a ; b \<le>\<^sub>\<K> a * b"
set_solutions_iff: fixes a :: "nat list"   and b :: "nat list" shows "set (solutions a b) = {(x, y). (x, y) \<in> set (check_generate a b) \<and> \<not> (\<exists>(u, v)\<in>set (check_generate a b). u @ v <\<^sub>v x @ y)}"
image_int_atLeastAtMost: fixes a :: "nat"   and b :: "nat" shows "int ` {a..b} = {int a..int b}"
finite_out_arcs: fixes G :: "('c, 'd) pre_digraph"   and v :: "'c" assumes "fin_digraph G" shows "finite (out_arcs G v)"
not_less_iff_gr_or_eq: fixes x :: "'a word"   and y :: "'a word" shows "(\<not> x <s y) = (y <s x \<or> x = y)"
minus_floatarith_def: fixes f :: "floatarith"   and g :: "floatarith" shows "f - g = floatarith.Add f (floatarith.Minus g)"
nstut_not: fixes F :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool" assumes "NSTUTINV F" shows "NSTUTINV \<not> F"
A_3: shows "(\<Sigma>, \<Sigma>') \<in> br_step \<delta>"
len_stable_down: fixes v :: "view"   and v1 :: "view"   and v2 :: "view"   and ts :: "traffic"   and c :: "cars" shows "v=v1--v2 \<longrightarrow> len v ts c = len v1 ts c"
mset_unplusm_dist_cases2: fixes B :: "'a multiset"   and C :: "'a multiset"   and s :: "'a"   and A :: "'a multiset" assumes "B + C = {#s#} + A" obtains "B = {#s#} + (B - {#s#})" and "A = B - {#s#} + C"   | "C = {#s#} + (C - {#s#})" and "A = B + (C - {#s#})"
Sr_pOp_closed: fixes S :: "'a set"   and x :: "'a"   and y :: "'a" assumes "sr R S"   and "x \<in> S"   and "y \<in> S" shows "x \<plusminus> y \<in> S"
eigenvalue_root_char_poly: fixes A :: "'a mat"   and n :: "nat"   and k :: "'a" assumes "A \<in> carrier_mat n n" shows "eigenvalue A k = (poly (char_poly A) k = (0::'a))"
asm: shows "(ftv T - ftv \<Gamma>) \<sharp>* \<theta>"
a: shows "limsup (\<lambda>x. ereal (real (card (A \<inter> {..<x})) / real x)) \<le> 1"
order_refl: fixes x :: "('d, 'e) Term.term" shows "x \<unrhd> x"
assms_2: shows "f \<in>\<^sub>\<circ> \<AA>\<lparr>Arr\<rparr>"
pr_analz_used: fixes evs :: "event list"   and S :: "agent \<times> nat \<times> nat \<Rightarrow> session"   and A :: "msg set"   and U :: "msg set" assumes "(evs, S, A, U) \<in> protocol" shows "A \<subseteq> U"
stmt_step_data_dependency_consistent_instrs: fixes \<theta> :: "nat \<Rightarrow> nat option"   and s :: "stmt"   and t :: "nat"   and s' :: "stmt"   and t' :: "nat"   and "is" :: "instr list" assumes "\<theta>\<turnstile> (s, t) \<rightarrow>\<^sub>s ((s', t'), is)"   and "valid_sops_stmt t s" shows "data_dependency_consistent_instrs {i. i < t} is"
in_set_zipE: fixes x :: "'g"   and y :: "'h"   and xs :: "'g list"   and ys :: "'h list" assumes "(x, y) \<in> set (zip xs ys)" obtains "x \<in> set xs" and "y \<in> set ys"
crdt_ops_distinct: fixes xs :: "('a \<times> 'b) list"   and deps :: "'b \<Rightarrow> 'a set" assumes "crdt_ops xs deps" shows "distinct xs"
ivl_rep_of_set_coll_def: fixes X :: "'c set" shows "ivl_rep_of_set_coll X = sets_of_coll (X ::: clw_rel appr_rel) \<bind> op_ivl_rep_of_sets"
stream_append_Cons: fixes x :: "'b stream"   and ys :: "'b list stream"   and zs :: "'b list stream" shows "smap2 (@) (smap2 (#) x ys) zs = smap2 (#) x (smap2 (@) ys zs)"
card_of_Sigma_cong1: fixes I :: "'c set"   and A :: "'c \<Rightarrow> 'd set"   and B :: "'c \<Rightarrow> 'e set" assumes "\<forall>i\<in>I. |A i| =o |B i|" shows "|Sigma I A| =o |Sigma I B|"
fg: shows "homotopic_with (\<lambda>k. True) Y Y (f \<circ> g) id"
compress_normalize_output_interfaces_nnf: fixes m :: "'b common_primitive match_expr"   and m' :: "'b common_primitive match_expr" assumes "normalized_nnf_match m"   and "compress_normalize_output_interfaces m = Some m'" shows "normalized_nnf_match m'"
stream_space_single_set: fixes p :: "real"   and M :: "bool stream measure"   and F :: "nat \<Rightarrow> bool stream measure"   and X :: "nat \<Rightarrow> bool stream \<Rightarrow> 'b"   and r :: "'b stream"   and n :: "nat" assumes "infinite_cts_filtration p M F"   and "borel_adapt_stoch_proc F X"   and "r \<in> range (proj_stoch_proc X n)" shows "infinite_cts_filtration.stream_space_single (proj_stoch_proc X n) r \<in> sets (stream_space borel)"
Same: assumes "lvl (update x y t) = lvl t" shows "thesis"
of_nat_le_of_nat_power_cancel_iff: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and b :: "nat"   and w :: "nat"   and x :: "nat" assumes "class.linordered_semidom plus minus less_eq less zero times one" shows "less_eq (power.power one times (semiring_1.of_nat one plus zero b) w) (semiring_1.of_nat one plus zero x) = (b ^ w \<le> x)"
terminates_Normal_elim_cases_2: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'a, 'c) com option"   and f :: "'c"   and g :: "'b set"   and c :: "('b, 'a, 'c) com"   and s :: "'b" assumes "\<Gamma>\<turnstile>Guard f g c \<down> Normal s" obtains "s \<in> g" and "\<Gamma>\<turnstile>c \<down> Normal s"   | "s \<notin> g"
order_le_less_trans: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<le> y"   and "y < z" shows "x < z"
iTILL_Min: fixes n :: "nat" shows "iMin [\<dots>n] = 0"
simps_2: fixes uu :: "('b \<times> nat) tree"   and uv :: "'b"   and lv :: "nat"   and uw :: "('b \<times> nat) tree" shows "lvl \<langle>uu, (uv, lv), uw\<rangle> = lv"
plus_subset1: fixes f :: "'b \<Rightarrow> 'c"   and F :: "'b filter"   and g :: "'b \<Rightarrow> 'c" assumes "f \<in> o[F](g)" shows "o[F](g) \<subseteq> o[F](\<lambda>x. f x + g x)"
hta_add_qi_def: fixes qi :: "'a"   and H :: "('a, 'b) hashedTa" shows "hta_add_qi qi H \<equiv> init_hta (hs.ins qi (hta_Qi H)) (hta_\<delta> H)"
dghm_ObjMap_app_in_HomCod_Obj: fixes a :: "V" assumes "a \<in>\<^sub>\<circ> \<CC>\<lparr>Obj\<rparr>" shows "\<FF>\<lparr>ObjMap\<rparr>\<lparr>a\<rparr> \<in>\<^sub>\<circ> dg_prod I \<AA>\<lparr>Obj\<rparr>"
\<zeta>_def: shows "\<zeta> \<equiv> Re (zeta (complex_of_real s))"
opp_fold_A_chamber_system_image: fixes X :: "'b set set"   and \<A> :: "'b set set set"   and A :: "'b set set"   and B :: "'b set set"   and B' :: "'b set set"   and C :: "'b set"   and D :: "'b set"   and E :: "'b set"   and z :: "'b set" assumes "ChamberComplexApartmentSystemTriangle X \<A> A B B' C D E z" shows "(ChamberComplexWithApartmentSystem.canonical_retraction X \<A> A C \<circ> ChamberComplexWithApartmentSystem.canonical_retraction X \<A> B' D) \<turnstile> ChamberComplex.chamber_system A = {F \<in> ChamberComplex.chamber_system A. ChamberComplex.face_distance X z F = ChamberComplex.chamber_distance X D F}"
implAuth_synth_analz: fixes H :: "msg set"   and A :: "agent"   and B :: "agent"   and M :: "msg" assumes "H \<subseteq> payload \<union> sym.valid \<union> range LtK \<union> Tags"   and "sym.implAuth A B M \<in> synth (analz H)" shows "sym.implAuth A B M \<in> H \<or> M \<in> synth (analz H) \<and> (A, B) \<in> broken H"
errMOD_def: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) model" shows "errMOD MOD \<equiv> \<lparr>igWls = \<lambda>s eX. case eX of ERR \<Rightarrow> False | OK X \<Rightarrow> igWls MOD s X, igWlsAbs = \<lambda>(us, s) eA. case eA of ERR \<Rightarrow> False | OK A \<Rightarrow> igWlsAbs MOD (us, s) A, igVar = \<lambda>xs x. OK (igVar MOD xs x), igAbs = \<lambda>xs x eX. if eX \<noteq> ERR \<and> (\<exists>s. isInBar (xs, s) \<and> igWls MOD s (check eX)) then OK (igAbs MOD xs x (check eX)) else ERR, igOp = \<lambda>delta einp ebinp. if liftAll (\<lambda>X. X \<noteq> ERR) einp \<and> liftAll (\<lambda>A. A \<noteq> ERR) ebinp \<and> igWlsInp MOD delta (checkI einp) \<and> igWlsBinp MOD delta (checkI ebinp) then OK (igOp MOD delta (checkI einp) (checkI ebinp)) else ERR, igFresh = \<lambda>ys y eX. if eX \<noteq> ERR \<and> (\<exists>s. igWls MOD s (check eX)) then igFresh MOD ys y (check eX) else True, igFreshAbs = \<lambda>ys y eA. if eA \<noteq> ERR \<and> (\<exists>us s. igWlsAbs MOD (us, s) (check eA)) then igFreshAbs MOD ys y (check eA) else True, igSwap = \<lambda>zs z1 z2 eX. if eX \<noteq> ERR \<and> (\<exists>s. igWls MOD s (check eX)) then OK (igSwap MOD zs z1 z2 (check eX)) else ERR, igSwapAbs = \<lambda>zs z1 z2 eA. if eA \<noteq> ERR \<and> (\<exists>us s. igWlsAbs MOD (us, s) (check eA)) then OK (igSwapAbs MOD zs z1 z2 (check eA)) else ERR, igSubst = \<lambda>ys eY y eX. if eY \<noteq> ERR \<and> igWls MOD (asSort ys) (check eY) \<and> eX \<noteq> ERR \<and> (\<exists>s. igWls MOD s (check eX)) then OK (igSubst MOD ys (check eY) y (check eX)) else ERR, igSubstAbs = \<lambda>ys eY y eA. if eY \<noteq> ERR \<and> igWls MOD (asSort ys) (check eY) \<and> eA \<noteq> ERR \<and> (\<exists>us s. igWlsAbs MOD (us, s) (check eA)) then OK (igSubstAbs MOD ys (check eY) y (check eA)) else ERR\<rparr>"
substEnv_compose2: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qTerm set option"   and Y :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and Z :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and ys :: "'varSort"   and zs :: "'varSort"   and y :: "'var"   and z :: "'var" assumes "goodEnv rho"   and "good Y"   and "good Z"   and "ys \<noteq> zs \<or> y \<noteq> z"   and "fresh ys y Z" shows "rho &[Y / y]_ys &[Z / z]_zs = rho &[Z / z]_zs &[Y #[Z / z]_zs / y]_ys"
D: shows "offs_num (length ns) (x # xs) index key mi ma j \<le> offs_next ns ub (x # xs) index key mi ma j - ns ! j"
happens_before_change_prefix: fixes P :: "'a prog"   and E :: "('b \<times> ('c, 'b) obs_event action) llist"   and a :: "nat"   and a' :: "nat"   and E' :: "('b \<times> ('c, 'b) obs_event action) llist"   and n :: "enat" assumes "P,E \<turnstile> a \<le>hb a'"   and "thread_start_actions_ok E'"   and "ltake n E [\<approx>] ltake n E'"   and "enat a < n"   and "enat a' < n" shows "P,E' \<turnstile> a \<le>hb a'"
Union_path_components_of: fixes X :: "'b topology" shows "\<Union> (path_components_of X) = topspace X"
digraph_op: fixes \<alpha> :: "V"   and \<CC> :: "V" assumes "digraph \<alpha> \<CC>" shows "digraph \<alpha> (op_dg \<CC>)"
AUX: assumes "v0 \<in> it"   and "it \<subseteq> V0"   and "V0 \<subseteq> U"   and "E `` U \<subseteq> U" shows "E\<^sup>* `` {v0} \<subseteq> U"
oconf_def: fixes P :: "'b prog"   and h :: "heap"   and obj :: "obj" shows "P,h \<turnstile> obj \<surd> \<equiv> let (C, fs) = obj in \<forall>F D T. P \<turnstile> C has F,NonStatic:T in D \<longrightarrow> (\<exists>v. fs (F, D) = \<lfloor>v\<rfloor> \<and> P,h \<turnstile> v :\<le> T)"
start_fin_in_steps_atom: fixes a :: "'b"   and w :: "'b list" shows "((start (atom a), [False]) \<in> NAe.steps (atom a) w) = (w = [a])"
parallel_state_iso_def: shows "parallel_state_iso = (\<lambda>((s11, s12), s21, s22). ((s11, s21), s12, s22), \<lambda>((s11, s21), s12, s22). ((s11, s12), s21, s22))"
ET_Source_Conf: fixes TS :: "('d \<times> (('d, 'e) expr \<times> ('f data \<Rightarrow> bool) \<times> 'e set \<times> 'f pupdate) \<times> 'd) set"   and ST :: "('d, 'e, 'f) status" assumes "TS \<subseteq> ET ST" shows "Source TS \<subseteq> Conf ST"
Mp_surjec: fixes H :: "'a set"   and N :: "'a set" assumes "submodule R M H"   and "submodule R M N"   and "H \<subseteq> N" shows "surjec\<^bsub>M /\<^sub>m H,M /\<^sub>m N\<^esub> (Mp\<^bsub>M  H,N\<^esub>)"
a_incr: fixes bs :: "bool list" shows "real (t\<^sub>i\<^sub>n\<^sub>c\<^sub>r bs) + \<Phi> (incr bs) - \<Phi> bs = 2"
take_Suc_prefix: fixes n :: "nat"   and p :: "'c list" shows "prefix (take n p) (take (Suc n) p)"
l2_inv4'_init: shows "init l2 \<subseteq> l2_inv4'"
uw\<theta>: shows "ide w \<and> \<guillemotleft>\<theta>' : f \<star> w \<Rightarrow> src r\<guillemotright>"
simps_1: fixes n :: "nat" shows "unbox_num (DNum n) = Some n"
init_wf_state: fixes \<sigma> :: "nat \<Rightarrow> 'c list"   and wf_atom :: "nat \<Rightarrow> 'd \<Rightarrow> bool"   and project :: "'c \<Rightarrow> 'c"   and lookup :: "'d \<Rightarrow> 'c \<Rightarrow> bool"   and init :: "'d rexp \<Rightarrow> 'e"   and delta :: "'c \<Rightarrow> 'e \<Rightarrow> 'e"   and final :: "'e \<Rightarrow> bool"   and wf_state :: "'e \<Rightarrow> bool"   and post :: "'e \<Rightarrow> 'e"   and L :: "'e \<Rightarrow> 'c list set"   and n :: "nat"   and r :: "'d rexp" assumes "rexp_DA \<sigma> wf_atom project lookup init delta final wf_state post L n"   and "alphabet.wf wf_atom n r" shows "wf_state (init r)"
order_tendstoD_1: fixes f :: "'c \<Rightarrow> 'a"   and y :: "'a"   and F :: "'c filter"   and a :: "'a" assumes "(f \<longlongrightarrow> y) F"   and "a < y" shows "\<forall>\<^sub>F x in F. a < f x"
Cart_lambda_cong: fixes f :: "'d \<Rightarrow> 'e"   and g :: "'d \<Rightarrow> 'e" assumes "\<And>x. f x = g x" shows "vec_lambda f = vec_lambda g"
ZFfunComp: fixes A :: "ZF"   and f :: "ZF \<Rightarrow> ZF"   and B :: "ZF"   and C :: "ZF"   and g :: "ZF \<Rightarrow> ZF" assumes "\<forall>x. x |\<in>| A \<longrightarrow> f x |\<in>| B" shows "ZFfun A B f |o| ZFfun B C g = ZFfun A C (g \<circ> f)"
minus_diff_commute: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and b :: "'c"   and a :: "'c" assumes "class.ab_group_add plus zero minus uminus" shows "minus (uminus b) a = minus (uminus a) b"
sort_mset_cong: fixes xs :: "'a list"   and ys :: "'a list" assumes "mset xs = mset ys" shows "sort xs = sort ys"
A: shows "\<forall>a b. r (a, b\<lparr>ok := True\<rparr>) \<longrightarrow> r (a, b\<lparr>ok := False\<rparr>)"
ln_aform_err: fixes x :: "real"   and e :: "nat \<Rightarrow> real"   and X :: "(real \<times> real pdevs) \<times> real"   and p :: "nat"   and Y :: "(real \<times> real pdevs) \<times> real" assumes "x \<in> aform_err e X"   and "ln_aform_err p X = Some Y"   and "e \<in> UNIV \<rightarrow> {- 1..1}" shows "ln x \<in> aform_err e Y"
wls_swapAbs_involutive: fixes us :: "'varSort"   and s :: "'sort"   and A :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set"   and zs :: "'varSort"   and x :: "'var"   and y :: "'var" assumes "wlsAbs (us, s) A" shows "A $[x \<and> y]_zs $[x \<and> y]_zs = A"
m_closed: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier (Ring_Divisibility.mult_of R)"   and "y \<in> carrier (Ring_Divisibility.mult_of R)" shows "x \<otimes> y \<in> carrier (Ring_Divisibility.mult_of R)"
minus_minus: fixes x :: "'c \<Rightarrow> 'a" assumes "x \<in> carrier F" shows "\<ominus>\<^bsub>F\<^esub> (\<ominus>\<^bsub>F\<^esub> x) = x"
two_cong_0_iff_nat: fixes p :: "nat" shows "[2 = 0] (mod int p) = (p = 1 \<or> p = 2)"
one_subgroup_generated: fixes G :: "('b, 'c) monoid_scheme"   and S :: "'b set" shows "\<one>\<^bsub>subgroup_generated G S\<^esub> = \<one>\<^bsub>G\<^esub>"
aa: shows "AAA = {#L \<in># D. L = Pos A#}"
get_local_def: fixes s :: "state"   and x :: "vname" shows "s<x> \<equiv> the (locals s x)"
simps_1: fixes r :: "'a"   and uu :: "'c" shows "|Inr (r, uu)|\<^sub>r = r"
v_def: shows "rat_of_int v = rat_of_int x / rat_of_int y"
kinematic_triangle_def: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "\<triangle> a b c \<equiv> a \<in> \<E> \<and> b \<in> \<E> \<and> c \<in> \<E> \<and> a \<noteq> b \<and> a \<noteq> c \<and> b \<noteq> c \<and> (\<exists>Q\<in>\<P>. \<exists>R\<in>\<P>. Q \<noteq> R \<and> (\<exists>S\<in>\<P>. Q \<noteq> S \<and> R \<noteq> S \<and> a \<in> Q \<and> b \<in> Q \<and> a \<in> R \<and> c \<in> R \<and> b \<in> S \<and> c \<in> S))"
of_real_erf_numeral: fixes n :: "num" shows "of_real (erf (numeral n)) = erf (numeral n)"
unit_disc_fix_preserves_poincare_in_angle: fixes M :: "moebius"   and p :: "complex_homo"   and a :: "complex_homo"   and b :: "complex_homo"   and c :: "complex_homo" assumes "unit_disc_fix M"   and "p \<in> unit_disc"   and "a \<in> unit_disc"   and "b \<in> unit_disc"   and "c \<in> unit_disc" shows "poincare_in_angle (moebius_pt M p) (moebius_pt M a) (moebius_pt M b) (moebius_pt M c) = poincare_in_angle p a b c"
matrix_to_iarray_third_Gauss_Jordan_column_k_PA: fixes i :: "nat"   and A :: "(('b, 'c) vec, 'd) vec"   and k :: "nat"   and P :: "(('b, 'd) vec, 'd) vec" assumes "i \<le> nrows A"   and "k < ncols A" shows "matrix_to_iarray (snd (snd (Gauss_Jordan_column_k_PA (P, i, A) k))) = snd (snd (Gauss_Jordan_column_k_iarrays_PA (matrix_to_iarray P, i, matrix_to_iarray A) k))"
get_proc_Exit: shows "lift_get_proc get_proc Main NewExit = Main"
index_smult_mat_2: fixes a :: "'b"   and A :: "'b mat" shows "dim_row (a \<cdot>\<^sub>m A) = dim_row A"
space_stat: fixes K :: "'a \<Rightarrow> 'a pmf"   and C :: "'a set" shows "space (MC_syntax.stat K C) = UNIV"
simps_2: fixes t :: "('a, 'b) Term.term"   and \<theta> :: "'b \<Rightarrow> ('a, 'b) Term.term" shows "Decomp t \<cdot>\<^sub>e\<^sub>s\<^sub>t\<^sub>p \<theta> = Decomp (t \<cdot> \<theta>)"
nonneg_Reals_of_real_iff: fixes r :: "real" shows "(of_real r \<in> \<real>\<^sub>\<ge>\<^sub>0) = (0 \<le> r)"
hcomp_isomorphic_ide: fixes f :: "'b"   and g :: "'b"   and h :: "'b" assumes "f \<cong>\<^sub>D g"   and "D.ide h"   and "src\<^sub>D f = trg\<^sub>D h" shows "f \<star>\<^sub>D h \<cong>\<^sub>D g \<star>\<^sub>D h"
fim: shows "f ` rel_interior S \<subseteq> rel_interior T"
function_of_in_extensional_funcset: fixes enum :: "nat \<Rightarrow> 'b"   and A :: "'b set"   and xs :: "'c list"   and B :: "'c set" assumes "bij_betw enum {0..<card A} A"   and "set xs \<subseteq> B"   and "length xs = card A" shows "function_of A enum xs \<in> A \<rightarrow>\<^sub>E B"
distinct_13: fixes x1 :: "('a, 'b, 'c) val"   and x8 :: "('a, 'b, 'c) expr" shows "VE x1 \<noteq> Rjoin x8"
mult_right_mono: fixes a :: "'c"   and b :: "'c"   and c :: "'c" assumes "a \<le> b"   and "(0::'c) \<le> c" shows "a * c \<le> b * c"
r2: shows "r2 division_of \<Union> (r1 - p) \<inter> \<Union> q"
frameIntCommutativity: fixes A\<^sub>F :: "name list"   and \<Psi> :: "'b"   and \<Psi>' :: "'b" shows "\<langle>A\<^sub>F, \<Psi> \<otimes> \<Psi>'\<rangle> \<simeq>\<^sub>F \<langle>A\<^sub>F, \<Psi>' \<otimes> \<Psi>\<rangle>"
numeral_neq_neg_numeral: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and m :: "num"   and n :: "num" assumes "class.ring_char_0 one times plus zero minus uminus" shows "numeral.numeral one plus m \<noteq> uminus (numeral.numeral one plus n)"
qSwapAll_preserves_qGoodAll1: fixes X :: "('d, 'e, 'varSort, 'var, 'f) qTerm"   and zs :: "'varSort"   and x :: "'var"   and y :: "'var"   and A :: "('d, 'e, 'varSort, 'var, 'f) qAbs" shows "(qGood X \<longrightarrow> qGood (X #[[x \<and> y]]_zs)) \<and> (qGoodAbs A \<longrightarrow> qGoodAbs (A $[[x \<and> y]]_zs))"
cod_char: fixes f :: "'a \<times> 'a" shows "B.VV.cod f = (if B.VV.arr f then B.VxV.cod f else B.VxV.null)"
pff: shows "\<forall>(a, w, b)\<in>E - edges H. \<not> forest (add_edge a w b H)"
gpv_stop_map': fixes f :: "'d \<Rightarrow> 'a"   and g :: "'e \<Rightarrow> 'b"   and h :: "'c \<Rightarrow> 'f"   and gpv :: "('d, 'e, 'f) gpv" shows "gpv_stop (map_gpv' f g h gpv) = map_gpv' (map_option f) g (map_option h) (gpv_stop gpv)"