hcju/isabelleps huggingface dataset:
extend_in_C_stop: fixes S :: "(nat, 'a) form set"   and C :: "(nat, 'a) form set set"   and f :: "nat \<Rightarrow> (nat, 'a) form"   and n :: "nat" assumes "extend S C f n \<in> C"   and "extend S C f n \<union> {f n} \<notin> C" shows "extend S C f (Suc n) \<in> C"
intervalE: fixes a :: "'b"   and x :: "'b"   and b :: "'b" assumes "a \<le> x \<and> x \<le> b" obtains "x \<in> {a..b}"
Process_eq_spec: fixes P :: "'c process"   and Q :: "'c process" shows "(P = Q) = (F P = F Q \<and> D P = D Q)"
hv_props_4: shows "common_sudiv_exists (two_chain_vertical_boundary two_chain) \<V> \<or> common_reparam_exists \<V> (two_chain_vertical_boundary two_chain)"
positively_invariant_inner: shows "c.positively_invariant {p. 1 / 4 - ((fst p)\<^sup>2 + (snd p)\<^sup>2) \<le> 0}"
prod_emb_PiE_same_index: fixes I :: "'b set"   and E :: "'b \<Rightarrow> 'c set"   and M :: "'b \<Rightarrow> 'c measure" assumes "\<And>i. i \<in> I \<Longrightarrow> E i \<subseteq> space (M i)" shows "prod_emb I M I (Pi\<^sub>E I E) = Pi\<^sub>E I E"
univ_asTerm_alpha: fixes f :: "('c, 'd, 'varSort, 'var, 'e) qTerm \<Rightarrow> 'f"   and X :: "('c, 'd, 'varSort, 'var, 'e) qTerm" assumes "f respectsP (#=)"   and "qGood X" shows "Equiv_Relation2.univ f (asTerm X) = f X"
P6: shows "\<not> Coplanar A B C Q \<and> \<not> Coplanar A B C S"
max2_def: fixes r :: "('a \<times> 'a) set"   and a :: "'a"   and b :: "'a" assumes "wo_rel r" shows "wo_rel.max2 r a b \<equiv> if (a, b) \<in> r then b else a"
xor_left_self: fixes x :: "'b"   and y :: "'b" shows "x XOR x XOR y = y"
abx: shows "prime x \<and> odd x \<and> coprime a b \<and> x dvd a\<^sup>2 + 3 * b\<^sup>2 \<and> \<not> is_qfN x 3"
steps_z_beta_complete: fixes A :: "('d \<times> ('c, real) cconstraint \<times> 'e \<times> 'c list \<times> 'd) set \<times> ('d \<Rightarrow> ('c, real) cconstraint)"   and l :: "'d"   and Z :: "('c \<Rightarrow> real) set"   and l' :: "'d"   and Z' :: "('c \<Rightarrow> real) set" assumes "A \<turnstile> \<langle>l, Z\<rangle> \<leadsto>* \<langle>l', Z'\<rangle>"   and "valid_abstraction A X (\<lambda>x. real (k x))"   and "Z \<subseteq> V" shows "\<exists>Z''. A \<turnstile> \<langle>l, Z\<rangle> \<leadsto>\<^sub>\<beta>* \<langle>l', Z''\<rangle> \<and> Z' \<subseteq> Z''"
none_MT_rulesconc: fixes C :: "'d \<Rightarrow> 'e \<Rightarrow> 'f option"   and a :: "'d list"   and b :: "'d" assumes "none_MT_rules C (a @ [b])" shows "none_MT_rules C a"
assms_1: shows "dbm_entry_val u (Some c1) (Some c2) a"
discrete_ed_ou: fixes X :: "'a set" assumes "discrete X" shows "ed_ou X"
rel_funD: fixes A :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and B :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and f :: "'e \<Rightarrow> 'g"   and g :: "'f \<Rightarrow> 'h"   and x :: "'e"   and y :: "'f" assumes "(A ===> B) f g"   and "A x y" shows "B (f x) (g y)"
dom_char: fixes f :: "'a \<times> 'a \<times> 'a" shows "B.VVV.dom f = (if B.VVV.arr f then B.VxVxV.dom f else B.VxVxV.null)"
obj_simps_4: fixes a :: "('a, 'a set) Maps.arr arrow_of_spans_data" assumes "Span.obj a" shows "Span.dom a = a"
cases: fixes a :: "'c set" assumes "finite a" obtains "a = {}"   | A :: "'c set" and aa :: "'c" where "a = insert aa A" and "finite A"
dest_port_int_TCPUDP_def: fixes x :: "int \<times> (int \<times> int \<times> protocol) \<times> (int \<times> int \<times> protocol) \<times> 'a" shows "dest_port x \<equiv> (fst \<circ> snd \<circ> fst \<circ> snd \<circ> snd) x"
USUP_insert: fixes x :: "'c"   and xs :: "'c set"   and P :: "'c \<Rightarrow> ('a, 'b) uexpr" shows "(\<Squnion> i \<in> insert x xs \<bullet> P i) = P x \<squnion> (\<Squnion> i \<in> xs \<bullet> P i)"
free_tv_ML_scheme_list: fixes n :: "nat"   and A :: "type_scheme list" shows "(n \<in> free_tv A) = (n \<in> set (free_tv_ML A))"
r_r_into_trancl: fixes a :: "'b"   and b :: "'b"   and R :: "('b \<times> 'b) set"   and c :: "'b" assumes "(a, b) \<in> R"   and "(b, c) \<in> R" shows "(a, c) \<in> R\<^sup>+"
Ints_of_nat: fixes 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 n :: "nat" assumes "class.ring_1 one times plus zero minus uminus" shows "semiring_1.of_nat one plus zero n \<in> ring_1.Ints one plus zero minus"
tendsto_ine\<^sub>N_def: fixes N :: "'c quasinorm"   and u :: "nat \<Rightarrow> 'c"   and x :: "'c" shows "tendsto_ine\<^sub>N N u x = (\<lambda>n. eNorm N (u n - x)) \<longlonglongrightarrow> 0"
Min_in: fixes A :: "'a word set" assumes "finite A"   and "A \<noteq> {}" shows "signed.Min A \<in> A"
discrete_topology_interior_of: fixes U :: "'c set"   and S :: "'c set" shows "discrete_topology U interior_of S = U \<inter> S"
QAP_Var: fixes \<Gamma> :: "(name \<times> ty) list"   and x :: "name"   and T :: "ty" assumes "valid \<Gamma>"   and "(x, T) \<in> set \<Gamma>" shows "\<Gamma> \<turnstile> Var x \<leftrightarrow> Var x : T"
MatchP: fixes \<sigma> :: "(String.literal \<times> event_data list) trace"   and j :: "nat"   and P :: "String.literal \<Rightarrow> nat option"   and V :: "String.literal \<Rightarrow> (nat \<Rightarrow> event_data list set) option"   and n :: "nat"   and R :: "event_data list set"   and \<phi>s :: "('msaux, 'muaux) mformula list"   and mr :: "mregex"   and r :: "Formula.formula Regex.regex"   and mrs :: "mregex list"   and buf :: "event_data option list set list list"   and nts :: "nat list"   and I :: "\<I>"   and aux :: "(nat \<times> (mregex, event_data option list set) mapping) list" assumes "case to_mregex r of (mr', \<phi>s') \<Rightarrow> list_all2 (wf_mformula \<sigma> j P V n R) \<phi>s \<phi>s' \<and> mr = mr'"   and "mrs = sorted_list_of_set (RPDs mr)"   and "safe_regex Past Strict r"   and "wf_mbufn' \<sigma> P V j n R r buf"   and "wf_ts_regex \<sigma> P j r nts"   and "wf_matchP_aux \<sigma> V n R I r aux (Monitor.progress \<sigma> P (formula.MatchP I r) j)" shows "wf_mformula \<sigma> j P V n R (MMatchP I mr mrs \<phi>s buf nts aux) (formula.MatchP I r)"
Streams_Connected_defs_1: fixes outS :: "'d message_af list"   and inS :: "'d message_af list" shows "f_Streams_Connected outS inS \<equiv> inS = NoMsg # outS"
True_notin_funpow_harry_step: fixes xs :: "bool list" shows "True \<notin> set ((harry_step ^^ harry_meas xs) xs)"
ssufD2: fixes x :: "'b list"   and y :: "'b list" assumes "x <s y" shows "x \<noteq> y"
skl1_refines_init_a0i_i: shows "init skl1 \<subseteq> R0sk1iai `` init a0i"
less_prod_def: fixes x :: "'c \<times> 'd"   and y :: "'c \<times> 'd" shows "less_prod x y \<equiv> fst x \<le> fst y \<and> snd x \<le> snd y \<and> x \<noteq> y"
simps_1: fixes ev_counter :: "nat"   and more :: "'a" shows "ev_counter \<lparr>ev_counter = ev_counter, \<dots> = more\<rparr> = ev_counter"
poly_add_nonzero: fixes p :: "'a poly"   and q :: "'a poly"   and x :: "'a"   and y :: "'a" assumes "p \<noteq> 0"   and "q \<noteq> 0"   and "poly p x = (0::'a)"   and "poly q y = (0::'a)"   and "primitive p" shows "poly_add p q \<noteq> 0"
monoD: fixes f :: "'a \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a" assumes "preord_class.mono f"   and "x \<sqsubseteq> y" shows "f x \<sqsubseteq> f y"
combining_sys_satisfies_properties: fixes p :: "real poly"   and qs1 :: "real poly list"   and qs2 :: "real poly list" assumes "p \<noteq> 0"   and "0 < length qs1"   and "\<forall>q. List.member qs1 q \<longrightarrow> coprime p q"   and "0 < length qs2"   and "\<forall>q. List.member qs2 q \<longrightarrow> coprime p q"   and "satisfies_properties p qs1 (get_subsets (calculate_data p qs1)) (get_signs (calculate_data p qs1)) (get_matrix (calculate_data p qs1))"   and "satisfies_properties p qs2 (get_subsets (calculate_data p qs2)) (get_signs (calculate_data p qs2)) (get_matrix (calculate_data p qs2))" shows "satisfies_properties p (qs1 @ qs2) (get_subsets (snd (combine_systems p (qs1, calculate_data p qs1) (qs2, calculate_data p qs2)))) (get_signs (snd (combine_systems p (qs1, calculate_data p qs1) (qs2, calculate_data p qs2)))) (get_matrix (snd (combine_systems p (qs1, calculate_data p qs1) (qs2, calculate_data p qs2))))"
adjoint_dim: fixes A :: "'a mat"   and n :: "nat" assumes "A \<in> carrier_mat n n" shows "adjoint A \<in> carrier_mat n n"
normalize_idem_imp_is_unit_iff: fixes a :: "'a" assumes "normalize a = a" shows "is_unit a = (a = (1::'a))"
prems_2: fixes i :: "nat"   and j :: "nat" assumes "i < length (x # xs)"   and "j < length ((x # xs) ! i)" shows "P ((x # xs) ! i ! j) ((y # ys) ! i ! j)"
insert_delete_commute: fixes i2 :: "'c"   and e :: "'c \<times> 'd \<times> bool"   and xs :: "('c \<times> 'd \<times> bool) list"   and i1 :: "'c option" assumes "i2 \<noteq> fst e" shows "Ordered_List.insert xs e i1 \<bind> (\<lambda>ys. delete ys i2) = delete xs i2 \<bind> (\<lambda>ys. Ordered_List.insert ys e i1)"
acquired_no_pending_write_in: fixes x :: "nat"   and xs :: "'b memref list"   and A :: "nat set"   and B :: "nat set" assumes "x \<in> acquired False xs A" shows "x \<in> acquired False xs B"
towards_sup_compl: fixes x :: "'a"   and y :: "'a" shows "x \<squnion> - x = y \<squnion> - y"
T8_\<o>: fixes w :: "i"   and \<psi> :: "\<o> \<Rightarrow> \<o>" shows "[\<^bold>\<forall>\<^sub>\<o>x. \<psi> x in w] = (\<forall>x. [\<psi> x in w])"
nres_order_simps_1: fixes M :: "'a nres" shows "SUCCEED \<le> M"
tangent_space_linear_on: fixes X :: "('c \<Rightarrow> real) \<Rightarrow> real"   and p :: "'c" assumes "X \<in> manifold_eucl.dest.tangent_space p" shows "linear_on manifold_eucl.dest.diff_fun_space UNIV (*\<^sub>R) (*\<^sub>R) X"
fresh_swap_compose: fixes X :: "('b, 'c, 'varSort, 'var, 'd) qTerm set"   and xs :: "'varSort"   and y :: "'var"   and z :: "'var"   and x :: "'var" assumes "good X"   and "fresh xs y X"   and "fresh xs z X" shows "X #[y \<and> x]_xs #[z \<and> y]_xs = X #[z \<and> x]_xs"
assms_3: fixes k :: "'a" assumes "k \<in> keys p' \<union> keys q'" shows "f k (lookup p' k) (lookup q' k) = f' k (lookup p' k) (lookup q' k)"
strict_mono_implies_mono: fixes m :: "('d, 'e) state" assumes "strict_mono_state m" shows "mono_state m"
incl: fixes h :: "'a"   and H :: "'a set"   and R :: "('a, 'b) ring_scheme" assumes "h \<in> H" shows "h \<in> generate_ring R H"
FAccNull: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and E :: "char list \<Rightarrow> ty option"   and e :: "expr"   and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and s\<^sub>1 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and F :: "char list"   and Cs :: "char list list" assumes "P,E \<turnstile> \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>null,s\<^sub>1\<rangle>" shows "P,E \<turnstile> \<langle>e\<bullet>F{Cs},s\<^sub>0\<rangle> \<Rightarrow> \<langle>Throw (addr_of_sys_xcpt NullPointer, [NullPointer]),s\<^sub>1\<rangle>"
right_def: fixes tree :: "'b tree" shows "right tree = (case tree of Node x1 x2 x3 \<Rightarrow> x3)"
matches_ipassmt_iface_replace_dstip_mexpr: fixes \<alpha> :: "action \<Rightarrow> ('b, 'c) tagged_packet_scheme \<Rightarrow> bool"   and ipassmt :: "iface \<Rightarrow> ('b word \<times> nat) list option"   and ifce :: "iface"   and a :: "action"   and p :: "('b, 'c) tagged_packet_scheme" shows "matches (common_matcher, \<alpha>) (ipassmt_iface_replace_dstip_mexpr ipassmt ifce) a p = (case ipassmt ifce of None \<Rightarrow> match_iface ifce (p_oiface p) | Some ips \<Rightarrow> p_dst p \<in> ipcidr_union_set (set ips))"
cong: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "linorder.sort_and_count_inversions less_eq less \<equiv> linorder.sort_and_count_inversions less_eq less"
spanE: fixes p :: "'a"   and B :: "'a set" assumes "p \<in> More_Modules.ideal B" obtains A :: "'a set" and q :: "'a \<Rightarrow> 'a" where "finite A" and "A \<subseteq> B" and "p = (\<Sum>b\<in>A. q b * b)"
f_values: shows "(f 0 = 0 \<or> f 0 = 1) \<and> (f 1 = 0 \<or> f 1 = 1)"
prems_4: fixes b :: "bool" shows "X b \<subseteq> ON"
DERIV': fixes f :: "real \<Rightarrow> real"   and D :: "real"   and x :: "real"   and s :: "real set"   and m :: "nat"   and Y :: "real" assumes "(f has_real_derivative D) (at x within s)"   and "m < n \<and> h \<le> f x \<and> f x \<le> 0"   and "diff (Suc m) (f x) * D = Y" shows "((\<lambda>x. diff m (f x)) has_real_derivative Y) (at x within s)"
get_from_list_to_list: fixes e :: "'a" shows "Finca_get (Sinca.Fenv.from_list (Finca_to_list e)) = Finca_get e"
pell_square_solution_nat_aux: fixes x :: "nat"   and y :: "nat" assumes "0 < D"   and "x\<^sup>2 = 1 + D * y\<^sup>2" shows "(x, y) = (1, 0)"
"case": fixes f :: "'n \<Rightarrow> 'o \<Rightarrow> 'm"   and x1 :: "'n"   and x2 :: "'o" shows "(case (x1, x2) of (x, xa) \<Rightarrow> f x xa) = f x1 x2"
tst: shows "thr s t = \<lfloor>(x, no_wait_locks)\<rfloor>"
induct: fixes P :: "Core.term \<Rightarrow> bool"   and "term" :: "Core.term" assumes "\<And>x1 x2. P (Ct x1 x2)"   and "\<And>x1 x2. P (Fv x1 x2)"   and "\<And>x. P (Bv x)"   and "\<And>x1 x2. P x2 \<Longrightarrow> P (Abs x1 x2)"   and "\<And>x1 x2. \<lbrakk>P x1; P x2\<rbrakk> \<Longrightarrow> P (x1 $ x2)" shows "P term"
case_eq_if: fixes f1 :: "'g option \<Rightarrow> 'f"   and f2 :: "'h \<Rightarrow> ('i \<Rightarrow> ('g, 'h, 'i) Resumption.resumption) \<Rightarrow> 'f"   and resumption :: "('g, 'h, 'i) Resumption.resumption" shows "(case resumption of Resumption.resumption.Done x \<Rightarrow> f1 x | Resumption.resumption.Pause x xa \<Rightarrow> f2 x xa) = (if Resumption.resumption.is_Done resumption then f1 (Resumption.resumption.result resumption) else f2 (Resumption.resumption.output resumption) (Resumption.resumption.resume resumption))"
ctor_rel_induct: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and IR :: "'c F \<Rightarrow> 'd F \<Rightarrow> bool" assumes "\<And>y xa. rel_pre_F R IR y xa \<Longrightarrow> IR (ctor_F y) (ctor_F xa)" shows "rel_F R \<le> IR"
P3: shows "B C Le B' C' \<and> \<not> Cong B C B' C'"
ag_inv_inv: fixes A :: "('d, 'e) aGroup_scheme"   and x :: "'d" assumes "aGroup A"   and "x \<in> carrier A" shows "-\<^sub>a\<^bsub>A\<^esub> (-\<^sub>a\<^bsub>A\<^esub> x) = x"
sorted_insort: fixes x :: "'a"   and xs :: "'a list" shows "sorted (insort x xs) = sorted xs"
l_empty: fixes k :: "'b"   and v :: "real"   and l :: "'b kdt"   and r :: "'b kdt"   and p\<^sub>0 :: "(real, 'b) vec"   and p\<^sub>1 :: "(real, 'b) vec" assumes "invar (Node k v l r)"   and "v < p\<^sub>0 $ k" shows "set_kdt l \<inter> cbox p\<^sub>0 p\<^sub>1 = {}"
hcomp_in_hhom: fixes \<mu> :: "'a"   and a :: "'a"   and b :: "'a"   and \<nu> :: "'a"   and c :: "'a" assumes "\<guillemotleft>\<mu> : a \<rightarrow>\<^sub>C b\<guillemotright>"   and "\<guillemotleft>\<nu> : b \<rightarrow>\<^sub>C c\<guillemotright>" shows "\<guillemotleft>\<nu> \<star>\<^sub>C \<mu> : a \<rightarrow>\<^sub>C c\<guillemotright>"
intro: fixes T :: "real set"   and f :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and X :: "'b set" assumes "interval T"   and "ll_on_open_axioms T f X" shows "ll_on_open T f X"
same_upto_inv_upto_jb: fixes j :: "nat"   and A :: "'a mat"   and B :: "'a mat" assumes "local.same_upto j A B"   and "local.inv_upto local.jb A j" shows "local.inv_upto local.jb B j"
Var_Eq_subst_Iff: fixes i :: "name"   and t :: "tm"   and H :: "fm set"   and A :: "fm" shows "insert (Var i EQ t) H \<turnstile> A(i::=t) IFF A"
nat_le_eq_or_lt: fixes x :: "nat"   and y :: "nat"   and P :: "nat \<Rightarrow> nat \<Rightarrow> bool" assumes "x \<le> y"   and "x = y \<Longrightarrow> P x y"   and "x < y \<Longrightarrow> P x y" shows "P x y"
is_aligned_mask_out_add_eq: fixes p :: "'b word"   and n :: "nat"   and x :: "'b word" assumes "is_aligned p n" shows "p + x AND NOT (mask n) = p + (x AND NOT (mask n))"
load_sub3_low_equal: fixes s1 :: "'b sparc_state"   and s2 :: "'b sparc_state"   and t1 :: "'b sparc_state"   and instr :: "sparc_operation \<times> inst_operand list"   and curr_win :: "'b word"   and rd :: "word5"   and address :: "word32"   and t2 :: "'b sparc_state" assumes "low_equal s1 s2 \<and> t1 = snd (fst (load_sub3 instr curr_win rd 10 address s1)) \<and> t2 = snd (fst (load_sub3 instr curr_win rd 10 address s2))" shows "low_equal t1 t2"
Recv: shows "ev' = Recv i' (occurs_on ev') s u'' u''' m'"
commutes_with_unit: shows "\<eta> \<one> = \<one>'"
t_stop_def: fixes a :: "real"   and v :: "real" shows "movement.t_stop a v = - v / a"
rec_eq_def: shows "rec_eq = Cn 2 rec_minus [Cn 2 (constn 1) [recf.id 2 0], Cn 2 rec_add [Cn 2 rec_minus [recf.id 2 0, recf.id 2 1], Cn 2 rec_minus [recf.id 2 1, recf.id 2 0]]]"
Cons: fixes i :: "nat"   and j :: "nat" shows "(i < row_length M * row_length M2 \<and> j < length M * length M2) \<and> mat (row_length M) (length M) M \<and> mat (row_length M2) (length M2) M2 \<longrightarrow> (M \<otimes> M2) ! j ! i = M ! (j div length M2) ! (i div row_length M2) * M2 ! (j mod length M2) ! (i mod row_length M2)"
append_eq_conv_conj: fixes xs :: "'g list"   and ys :: "'g list"   and zs :: "'g list" shows "(xs @ ys = zs) = (xs = take (length xs) zs \<and> ys = drop (length xs) zs)"
measure_space_def: fixes \<Omega> :: "'b set"   and A :: "'b set set"   and \<mu> :: "'b set \<Rightarrow> ennreal" shows "measure_space \<Omega> A \<mu> = (sigma_algebra \<Omega> A \<and> positive A \<mu> \<and> countably_additive A \<mu>)"
invalidate_other: fixes dip :: "nat"   and dests :: "nat \<Rightarrow> nat option"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option" assumes "dip \<notin> dom dests" shows "invalidate rt dests dip = rt dip"
gim: shows "g ` \<Union> \<F> \<subseteq> T"
r_inv_ex: fixes prod :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and x :: "'a" assumes "Dgrp prod one" shows "\<exists>y. prod x y = one"
QuoteP_Zero: shows "{} \<turnstile> QuoteP Zero Zero"
pull_up_configuration_state_rank: fixes q :: "'a ltl_prop_equiv_quotient"   and n :: "nat" assumes "configuration q n = {}" shows "state_rank q n = None"
AE_prob_1: fixes M :: "'c measure"   and A :: "'c set" assumes "prob_space M"   and "Sigma_Algebra.measure M A = 1" shows "AE x in M. x \<in> A"
utp_defs_11: fixes P :: "'b \<times> 'd \<Rightarrow> bool"   and Q :: "'d \<times> 'c \<Rightarrow> bool" shows "P ;; Q \<equiv> \<lambda>r. r \<in> {p. P p} O {q. Q q}"
continuous_openin_preimage_eq: fixes S :: "'b set"   and f :: "'b \<Rightarrow> 'c" shows "continuous_on S f = (\<forall>T. open T \<longrightarrow> openin (top_of_set S) (S \<inter> f -` T))"
eval_fds_mult: fixes f :: "'a fds"   and s :: "'a"   and g :: "'a fds" assumes "fds_abs_converges f s"   and "fds_abs_converges g s" shows "eval_fds (f * g) s = eval_fds f s * eval_fds g s"
indRelRTPO_is_preorder: fixes TRel :: "('procT \<times> 'procT) set" assumes "refl TRel" shows "preorder (indRelRTPO TRel)"
cod_simp: fixes f :: "'c \<times> 'c" assumes "C'.VV.arr f" shows "C'.VV.cod f = C.VxV.cod f"
im_subgroup: fixes H :: "'m set" assumes "VecGroup.Subgroup H" shows "Group (T ` H) \<and> T ` H \<subseteq> ImG"
iffs: fixes lens_get :: "'b \<Rightarrow> 'a"   and lens_put :: "'b \<Rightarrow> 'a \<Rightarrow> 'b"   and more :: "'c"   and lens_get' :: "'b \<Rightarrow> 'a"   and lens_put' :: "'b \<Rightarrow> 'a \<Rightarrow> 'b"   and more' :: "'c" shows "(\<lparr>lens_get = lens_get, lens_put = lens_put, \<dots> = more\<rparr> = \<lparr>lens_get = lens_get', lens_put = lens_put', \<dots> = more'\<rparr>) = (lens_get = lens_get' \<and> lens_put = lens_put' \<and> more = more')"
test_bit_word_transfer: shows "rel_fun pcr_word (=) (\<lambda>k n. n < LENGTH('b) \<and> bit k n) (!!)"
range_to_int_mod_ring: shows "range to_int_mod_ring = {0..<int CARD('b)}"
fsubset_antisym: fixes A :: "'x fset"   and B :: "'x fset" assumes "A |\<subseteq>| B"   and "B |\<subseteq>| A" shows "A = B"
domo_bfilter: fixes b :: "bexp"   and X :: "char list set"   and S :: "'av Abs_State.st option"   and bv :: "bool" assumes "vars b \<subseteq> X"   and "domo S \<subseteq> X" shows "domo (bfilter b bv S) \<subseteq> X"
mset_neg_empty: shows "mset_neg {#}\<^sub>z = {#}"
unat_plus_simple: fixes x :: "'b word"   and y :: "'b word" shows "(x \<le> x + y) = (unat (x + y) = unat x + unat y)"
perm_set_def: fixes pi :: "('b \<times> 'b) list"   and X :: "'a set" shows "pi \<bullet> X = {pi \<bullet> x |x. x \<in> X}"
Dev_Lam: fixes x :: "name"   and M :: "lam"   and N :: "lam" assumes " Lam [x].M \<longrightarrow>\<^sub>d N" shows "\<exists>N'. N = Lam [x].N' \<and>  M \<longrightarrow>\<^sub>d N'"
app_vlrestrictionD_1: fixes a :: "V"   and b :: "V"   and r :: "V"   and A :: "V" assumes "\<langle>a, b\<rangle> \<in>\<^sub>\<circ> r \<restriction>\<^sup>l\<^sub>\<circ> A" shows "a \<in>\<^sub>\<circ> A"
splitAtRec_conv: fixes x :: "'c"   and bs :: "'c list"   and xs :: "'c list" shows "splitAtRec x bs xs = (bs @ takeWhile (\<lambda>y. y \<noteq> x) xs, tl (dropWhile (\<lambda>y. y \<noteq> x) xs))"
semilattice_sup_ow_axioms: shows "semilattice_sup_ow U\<^sub>2 sup\<^sub>2 (\<le>\<^sub>o\<^sub>w\<^sub>.\<^sub>2) (<\<^sub>o\<^sub>w\<^sub>.\<^sub>2)"
pos_divide_less_eq: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'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 sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and c :: "'b"   and b :: "'b"   and a :: "'b" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less zero c" shows "less (divide b c) a = less b (times a c)"
valid_path_aux_split_2: fixes cs :: "'edge list"   and as :: "'edge list"   and as' :: "'edge list" assumes "valid_path_aux cs (as @ as')" shows "valid_path_aux (upd_cs cs as) as'"
mult_right_less_imp_less: 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 a :: "'c"   and c :: "'c"   and b :: "'c" assumes "class.linordered_semiring plus minus less_eq less zero times"   and "less (times a c) (times b c)"   and "less_eq zero c" shows "less a b"
preserves_arr: fixes f :: "('a \<Rightarrow> 'b, 'a \<Rightarrow> 'b) concrete_category.arr \<times> 'a" assumes "A_BxA.arr f" shows "B.arr (E.map f)"
interp_factorization_def: shows "interp_factorization = (\<lambda>(a, cs). smult a (\<Prod>(c, n)\<leftarrow>cs. [:- c, 1::'a:] ^ Suc n))"
distinct_1: fixes x11 :: "CAttId"   and x12 :: "nat"   and x2 :: "AttId" shows "objLoc x11 x12 \<noteq> staticLoc x2"
inject: fixes x2 :: "'f"   and y2 :: "'f" shows "(\<lfloor>x2\<rfloor> = \<lfloor>y2\<rfloor>) = (x2 = y2)"
t18_18_aux: fixes A :: "'p"   and B :: "'p"   and D :: "'p"   and E :: "'p"   and C :: "'p"   and F :: "'p" assumes "Cong A B D E"   and "Cong A C D F"   and "F D E LtA C A B"   and "\<not> Col A B C"   and "\<not> Col D E F"   and "D F Le D E" shows "E F Lt B C"
map_poly_smult: fixes f :: "'b \<Rightarrow> 'a"   and c :: "'b"   and p :: "'b poly" assumes "f (0::'b) = (0::'a)"   and "\<And>c x. f (c * x) = f c * f x" shows "map_poly f (Polynomial.smult c p) = Polynomial.smult (f c) (map_poly f p)"