hcju/isabelleps huggingface dataset:
MultiSplitAux: fixes M :: "Var list \<times> (int \<times> Instr) list \<times> int"   and l :: "int"   and s :: "Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list"   and k :: "nat"   and ll :: "int"   and t :: "Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list" assumes "(M, l, s, k, ll, t) \<in> MStep"   and "1 \<le> k" shows "\<exists>n m r l1. (M, l, s, n, l1, r) \<in> Step \<and> (M, l1, r, m, ll, t) \<in> MStep \<and> Suc m + n = k"
Abs: fixes b :: "'b set"   and A :: "'b set set"   and a :: "'b set" assumes "b \<in> A" shows "timpls_transformable_to_pred A (Fun (Abs a) []) (Fun (Abs b) [])"
unifiert_unifiers: fixes \<sigma> :: "char list \<Rightarrow> fterm"   and ts :: "fterm set" assumes "unifier\<^sub>t\<^sub>s \<sigma> ts" shows "(\<lambda>x. fterm_to_iterm (\<sigma> x)) \<in> unifiers (fterm_to_iterm ` ts \<times> fterm_to_iterm ` ts)"
\<psi>: fixes x :: "'s \<times> ('c \<Rightarrow> real)"   and y :: "'s \<times> ('c \<Rightarrow> real)" assumes "x \<in> S"   and "x ~ y" shows "\<psi> x = \<psi> y"
integrable_exp_minus_squared_1: fixes A :: "real set" assumes "A \<subseteq> {0..}"   and "A \<in> sets lborel" shows "set_integrable lborel A (\<lambda>t. exp (- t\<^sup>2))"
mb: shows "\<not> is_zero_row_upt_k mb k A"
verts3_K5: fixes G :: "'b pair_pre_digraph" assumes "K\<^bsub>5\<^esub> (with_proj G)" shows "verts3 (with_proj G) = verts (with_proj G)"
Class_widen_Class: fixes P :: "'b prog"   and C :: "char list"   and D :: "char list" shows "subtype P (Class C) (Class D) = P \<turnstile> C \<preceq>\<^sup>* D"
elemDIselemHD: fixes t :: "'a event list"   and P :: "'a process"   and A :: "'a set" assumes "t \<in> D P" shows "trace_hide t (ev ` A) \<in> D (P \ A)"
inj_monoid_add_hom_def: fixes hom :: "'a \<Rightarrow> 'b" shows "inj_monoid_add_hom hom \<equiv> monoid_add_hom hom \<and> inj_semigroup_add_hom hom"
prems: shows "\<forall>x y. x \<le> g x y \<and> y \<le> g x y"
assms: shows "\<exists>y'>r. \<forall>x\<in>{r<..y'}. aEvalUni (EqUni (a, b, c)) x \<or> aEvalUni (LessUni (a, b, c)) x"
cong_exp_mod_1: fixes a :: "'c"   and n :: "'c"   and b :: "nat"   and c :: "'c" shows "[(a mod n) ^ b = c] (mod n) = [a ^ b = c] (mod n)"
finprod_insert: fixes F :: "'c set"   and a :: "'c"   and f :: "'c \<Rightarrow> 'a set" assumes "finite F"   and "a \<notin> F"   and "f \<in> F \<rightarrow> carrier (ideals_set R)"   and "f a \<in> carrier (ideals_set R)" shows "finprod (ideals_set R) f (insert a F) = f a \<otimes>\<^bsub>ideals_set R\<^esub> finprod (ideals_set R) f F"
uniform: fixes point_set :: "'a set"   and block_collection :: "'a set multiset"   and u_block_size :: "int"   and bl :: "'a set" assumes "block_design point_set block_collection u_block_size"   and "bl \<in># block_collection" shows "int (card bl) = u_block_size"
SHOnextConfig_eq: fixes A :: "('a, 'b, 'c, 'd) CHOAlgorithm_scheme"   and r :: "nat"   and cfg :: "'a \<Rightarrow> 'b"   and HO :: "'a \<Rightarrow> 'a set"   and SHO :: "'a \<Rightarrow> 'a set"   and cfg' :: "'a \<Rightarrow> 'b" shows "SHOnextConfig A r cfg HO SHO cfg' = (\<forall>p. \<exists>\<mu>\<in>SHOmsgVectors A r p cfg (HO p) (SHO p). nextState A r p (cfg p) \<mu> (cfg' p))"
eq_False: fixes P :: "bool" shows "(P = False) = (\<not> P)"
assms_2: shows "P (nsqn\<^sub>r (dsn, dsk, Aodv_Basic.inv, hops, nhip, pre))"
expand: fixes option :: "'b option"   and option' :: "'b option" assumes "(option = None) = (option' = None)"   and "\<lbrakk>option \<noteq> None; option' \<noteq> None\<rbrakk> \<Longrightarrow> the option = the option'" shows "option = option'"
minimal_core: assumes "distinct_indices_state (set_unsat I s')" shows "subsets_sat_core (set_unsat I s')"
equal_zero: fixes emBits :: "nat" assumes "176 + sLen \<le> emBits"   and "roundup emBits 8 * 8 - emBits \<le> roundup emBits 8 * 8 - (176 + sLen)" shows "0 = roundup emBits 8 * 8 - emBits - (roundup emBits 8 * 8 - (176 + sLen))"
lm045: fixes a :: "('c \<times> 'd set) set" shows "\<Union> {{(fst pair, {y}) |y. y \<in> snd pair} |pair. pair \<in> a} = {(fst pair, {y}) |y pair. y \<in> snd pair \<and> pair \<in> a}"
WEAKEN_HNR_POST_def: fixes \<Gamma> :: "assn"   and \<Gamma>' :: "assn"   and \<Gamma>'' :: "assn" shows "WEAKEN_HNR_POST \<Gamma> \<Gamma>' \<Gamma>'' \<equiv> (\<exists>h. h \<Turnstile> \<Gamma>) \<longrightarrow> (\<Gamma>'' \<Longrightarrow>\<^sub>t \<Gamma>')"
transpE: fixes r :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'d"   and y :: "'d"   and z :: "'d" assumes "transp r"   and "r x y"   and "r y z" obtains "r x z"
cinsertI1: fixes a :: "'c"   and B :: "'c cset" shows "a \<in>\<^sub>c cinsert a B"
similar_mat_witD2_7: fixes A :: "'a mat"   and n :: "nat"   and m :: "nat"   and B :: "'a mat"   and P :: "'a mat"   and Q :: "'a mat" assumes "A \<in> carrier_mat n m"   and "similar_mat_wit A B P Q" shows "Q \<in> carrier_mat n n"
act_act': fixes g :: "'a"   and h :: "'a"   and b :: "'b" shows "g \<in> carrier G \<and> h \<in> carrier G \<longrightarrow> b <o g <o h = b <o g \<otimes> h"
closed_lift: fixes e :: "db"   and k :: "nat" assumes "\<forall>v. freedb e v \<longrightarrow> v < k" shows "lift e k = e"
inject_1: fixes x1 :: "b_e"   and y1 :: "b_e" shows "($x1 = $y1) = (x1 = y1)"
tiTable_ack_st_hd: fixes req :: "nat \<Rightarrow> reqType list"   and a1 :: "nat \<Rightarrow> aType list"   and stop :: "nat \<Rightarrow> stopType list"   and lose :: "nat \<Rightarrow> bool list"   and st_in :: "nat \<Rightarrow> GatewayStatus"   and b :: "nat \<Rightarrow> ECall_Info list"   and ack :: "nat \<Rightarrow> GatewayStatus list"   and i1 :: "nat \<Rightarrow> ECall_Info list"   and vc :: "nat \<Rightarrow> vcType list"   and st_out :: "nat \<Rightarrow> GatewayStatus"   and t :: "nat" assumes "tiTable_SampleT req a1 stop lose st_in b ack i1 vc st_out"   and "ts lose"   and "msg (Suc 0) a1"   and "msg (Suc 0) stop" shows "st_out t = hd (ack t)"
While: fixes b :: "bexp"   and c :: "com" assumes "sec b = 0"   and "0 \<turnstile> c" shows "0 \<turnstile> WHILE b DO c"
borel_integrable_atLeastAtMost': fixes a :: "real"   and b :: "real"   and f :: "real \<Rightarrow> 'c" assumes "continuous_on {a..b} f" shows "set_integrable lborel {a..b} f"
ok_Join_commute_I: fixes F :: "'b program"   and G :: "'b program"   and H :: "'b program" assumes "F ok G"   and "F \<squnion> G ok H" shows "F ok (G \<squnion> H)"
pred_mono_strong: fixes P :: "'c \<Rightarrow> bool"   and x :: "'c stream"   and Pa :: "'c \<Rightarrow> bool" assumes "pred_stream P x"   and "\<And>z. \<lbrakk>z \<in> sset x; P z\<rbrakk> \<Longrightarrow> Pa z" shows "pred_stream Pa x"
redT_updTs_append: fixes ts :: "'f \<Rightarrow> ('g \<times> 'h \<Rightarrow>f nat) option"   and tas :: "('f, 'g, 'i) new_thread_action list"   and tas' :: "('f, 'g, 'i) new_thread_action list" shows "redT_updTs ts (tas @ tas') = redT_updTs (redT_updTs ts tas) tas'"
in_rel: fixes R :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and a :: "'e set['g]"   and b :: "'f set['g]" shows "rel_bset R a b = (\<exists>z. z \<in> {x. set_bset x \<subseteq> {(x, y). R x y}} \<and> map_bset fst z = a \<and> map_bset snd z = b)"
square_bounded_le: fixes a :: "'a"   and b :: "'a" assumes "- a \<le> b \<and> b \<le> a" shows "b * b \<le> a * a"
winding_number_nearby_paths_eq: fixes g :: "real \<Rightarrow> complex"   and h :: "real \<Rightarrow> complex"   and z :: "complex" assumes "path g"   and "path h"   and "pathstart h = pathstart g"   and "pathfinish h = pathfinish g"   and "\<And>t. t \<in> {0..1} \<Longrightarrow> cmod (h t - g t) < cmod (g t - z)" shows "winding_number h z = winding_number g z"
cbox_division_memE: fixes K :: "'c set"   and \<D> :: "'c set set"   and S :: "'c set" assumes "K \<in> \<D>"   and "\<D> division_of S" obtains a :: "'c" and b :: "'c" where "K = cbox a b" and "K \<noteq> {}" and "K \<subseteq> S"
Incremental_NS_Constraint_Ops_axioms: shows "Incremental_NS_Constraint_Ops init_nsc_code assert_nsc_code check_nsc_code solution_nsc_code checkpoint_nsc_code backtrack_nsc_code invariant_nsc checked_nsc"
airr: shows "\<forall>a\<in>set as. irreducible G a"
continuous: fixes prod :: "'e \<Rightarrow> 'f \<Rightarrow> 'g"   and F :: "'h filter"   and f :: "'h \<Rightarrow> 'e"   and g :: "'h \<Rightarrow> 'f" assumes "bounded_bilinear prod"   and "continuous F f"   and "continuous F g" shows "continuous F (\<lambda>x. prod (f x) (g x))"
is_scc_ex1: fixes E :: "('b \<times> 'b) set"   and v :: "'b" shows "\<exists>!scc. is_scc E scc \<and> v \<in> scc"
induct: fixes x1 :: "(('sf, 'sc, 'sz) formula list \<times> ('sf, 'sc, 'sz) formula list) list \<times> ('sf, 'sc, 'sz) formula list \<times> ('sf, 'sc, 'sz) formula list"   and x2 :: "nat"   and x3 :: "('sf, 'sc, 'sz) step"   and P :: "(('sf, 'sc, 'sz) formula list \<times> ('sf, 'sc, 'sz) formula list) list \<times> ('sf, 'sc, 'sz) formula list \<times> ('sf, 'sc, 'sz) formula list \<Rightarrow> nat \<Rightarrow> ('sf, 'sc, 'sz) step \<Rightarrow> bool" assumes "step_ok x1 x2 x3"   and "\<And>SG i a C. SG ! i = ([], [get_axiom a]) \<Longrightarrow> P (SG, C) i (Axiom a)"   and "\<And>SG i a \<sigma> C. \<lbrakk>SG ! i = ([], [Fsubst (get_axiom a) \<sigma>]); Fadmit \<sigma> (get_axiom a); ssafe \<sigma>\<rbrakk> \<Longrightarrow> P (SG, C) i (AxSubst a \<sigma>)"   and "\<And>SG C i j L. \<lbrakk>lrule_ok SG C i j L; j < length (fst (SG ! i))\<rbrakk> \<Longrightarrow> P (SG, C) i (Lrule L j)"   and "\<And>SG C i j L. \<lbrakk>rrule_ok SG C i j L; j < length (snd (SG ! i))\<rbrakk> \<Longrightarrow> P (SG, C) i (Rrule L j)"   and "\<And>\<phi> i SG C. \<lbrakk>fsafe \<phi>; i < length SG\<rbrakk> \<Longrightarrow> P (SG, C) i (Cut \<phi>)"   and "\<And>SG i j k C. \<lbrakk>fst (SG ! i) ! j = snd (SG ! i) ! k; j < length (fst (SG ! i)); k < length (snd (SG ! i))\<rbrakk> \<Longrightarrow> P (SG, C) i (CloseId j k)"   and "\<And>SG i C a p. SG ! i = ([], [[[a]]p]) \<Longrightarrow> P (SG, C) i G"   and "\<And>SG i ODE \<sigma> C. \<lbrakk>SG ! i = ([], [Fsubst ([[EvolveODE (OProd (OSing vid1 (f1 fid1 vid1)) ODE) (p1 vid2 vid1)]]P pid1 \<leftrightarrow> ([[EvolveODE (OProd (OSing vid1 (f1 fid1 vid1)) ODE) (p1 vid2 vid1)]][[DiffAssign vid1 (f1 fid1 vid1)]]P pid1)) \<sigma>]); ssafe \<sigma>; osafe ODE; {Inl vid1, Inr vid1} \<inter> BVO ODE = {}; Fadmit \<sigma> ([[EvolveODE (OProd (OSing vid1 (f1 fid1 vid1)) ODE) (p1 vid2 vid1)]]P pid1 \<leftrightarrow> ([[EvolveODE (OProd (OSing vid1 (f1 fid1 vid1)) ODE) (p1 vid2 vid1)]][[DiffAssign vid1 (f1 fid1 vid1)]]P pid1))\<rbrakk> \<Longrightarrow> P (SG, C) i (DEAxiomSchema ODE \<sigma>)"   and "\<And>SG i \<phi> \<psi> \<sigma> C. \<lbrakk>SG ! i = ([], [Fsubst (InContext pid1 \<phi> \<leftrightarrow> InContext pid1 \<psi>) \<sigma>]); valid (\<phi> \<leftrightarrow> \<psi>); fsafe \<phi>; fsafe \<psi>; ssafe \<sigma>; Fadmit \<sigma> (InContext pid1 \<phi> \<leftrightarrow> InContext pid1 \<psi>)\<rbrakk> \<Longrightarrow> P (SG, C) i (CE \<phi> \<psi> \<sigma>)"   and "\<And>SG i p \<theta> \<theta>' \<sigma> C. \<lbrakk>SG ! i = ([], [Fsubst ($\<phi> p (singleton \<theta>) \<leftrightarrow> $\<phi> p (singleton \<theta>')) \<sigma>]); valid (Equals \<theta> \<theta>'); dsafe \<theta>; dsafe \<theta>'; ssafe \<sigma>; Fadmit \<sigma> ($\<phi> p (singleton \<theta>) \<leftrightarrow> $\<phi> p (singleton \<theta>'))\<rbrakk> \<Longrightarrow> P (SG, C) i (CQ \<theta> \<theta>' \<sigma>)" shows "P x1 x2 x3"
Nil_is_map_conv: fixes f :: "'e \<Rightarrow> 'd"   and xs :: "'e list" shows "([] = map f xs) = (xs = [])"
upset_set_anti: fixes dual :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "class.preorder_with_dual dual less_eq less" shows "mono (ord.upset_set less_eq)"
ev_1: shows "\<forall>\<^sub>F x in F. 0 \<le> f x"
intros: fixes A :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and B :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and res1 :: "('c, 'e) resource"   and res2 :: "('d, 'f) resource" assumes "rel_fun A (rel_spmf (rel_prod B (rel_resource A B))) (run_resource res1) (run_resource res2)" shows "rel_resource A B res1 res2"
sas_plus_problem_to_strips_problem_variable_set_element_iff: fixes \<Psi> :: "('c, 'd) sas_plus_problem"   and v :: "'c"   and a :: "'d" assumes "is_valid_problem_sas_plus \<Psi>" shows "((v, a) \<in> set ((\<phi> \<Psi> )\<^sub>\<V>)) = (v \<in> set (\<Psi>\<^sub>\<V>\<^sub>+) \<and> a \<in> \<R>\<^sub>+ \<Psi> v)"
Plus_comm: fixes r :: "'a language"   and s :: "'a language" shows "Plus r s = Plus s r"
lookup_raw_tl_raw': fixes xs :: "('a \<times> 'c) list \<times> 'o"   and k :: "'a" assumes "oalist_inv xs" shows "lookup_raw (tl_raw xs) k = (if k = fst (list.hd (fst xs)) then 0::'c else lookup_raw xs k)"
cod_in_codomains: fixes f :: "'c" assumes "C.codomains f \<noteq> {}" shows "C.cod f \<in> C.codomains f"
cases: fixes E :: "('b \<times> 'b) set"   and a1 :: "'b"   and a2 :: "'b list"   and a3 :: "'b" assumes "path E a1 a2 a3" obtains u :: "'b" where "a1 = u__" and "a2 = []" and "a3 = u__"   | u :: "'b" and v :: "'b" and l :: "'b list" and w :: "'b" where "a1 = u" and "a2 = u # l" and "a3 = w" and "(u, v) \<in> E" and "path E v l w"
assign_supd: fixes f :: "'a \<Rightarrow> st \<Rightarrow> st"   and v :: "'a"   and x :: "abopt"   and s :: "st" shows "assign (\<lambda>v. supd (f v)) v (x, s) = (x, if x = None then f v s else s)"
all_set_conv_all_nth: fixes xs :: "'c list"   and P :: "'c \<Rightarrow> bool" shows "(\<forall>x\<in>set xs. P x) = (\<forall>i<length xs. P (xs ! i))"
subst_cls_mset_add_mset: fixes C :: "'a literal multiset"   and CC :: "'a literal multiset multiset"   and \<sigma> :: "'s" shows "add_mset C CC \<cdot>cm \<sigma> = add_mset (C \<cdot> \<sigma>) (CC \<cdot>cm \<sigma>)"
simps_2: fixes x2 :: "'e"   and y2 :: "'e" shows "(Inr x2 = Inr y2) = (x2 = y2)"
monofun_cfun_arg: fixes x :: "'b"   and y :: "'b"   and f :: "'b \<rightarrow> 'c" assumes "x \<sqsubseteq> y" shows "f\<cdot>x \<sqsubseteq> f\<cdot>y"
parallel_fixp_induct_strong: fixes luba :: "'c set \<Rightarrow> 'c"   and orda :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and lubb :: "'d set \<Rightarrow> 'd"   and ordb :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and P :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'c"   and g :: "'d \<Rightarrow> 'd" assumes "class.ccpo luba orda (mk_less orda)"   and "class.ccpo lubb ordb (mk_less ordb)"   and "ccpo.admissible (prod_lub luba lubb) (rel_prod orda ordb) (\<lambda>x. P (fst x) (snd x))"   and "monotone orda orda f"   and "monotone ordb ordb g"   and "P (luba {}) (lubb {})"   and "\<And>x y. \<lbrakk>orda x (ccpo.fixp luba orda f); ordb y (ccpo.fixp lubb ordb g); P x y\<rbrakk> \<Longrightarrow> P (f x) (g y)" shows "P (ccpo.fixp luba orda f) (ccpo.fixp lubb ordb g)"
Status_Step: 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"   and U :: "'f update" assumes "TS \<in> HPT ST"   and "U \<in> ResolveRacing TS" shows "(HA ST, StepConf (HA ST) (Conf ST) TS, StepActEvent TS, U !!! Value ST) \<in> status"
pivot_funD_1: fixes A :: "'b mat"   and nr :: "nat"   and f :: "nat \<Rightarrow> nat"   and nc :: "nat"   and i :: "nat" assumes "dim_row A = nr"   and "pivot_fun A f nc"   and "i < nr" shows "f i \<le> nc"
fct_gen_subalgebra_eq_sets: fixes M :: "'b measure"   and P :: "'b measure"   and N :: "'c measure"   and X :: "'b \<Rightarrow> 'c" assumes "sets M = sets P" shows "fct_gen_subalgebra M N X = fct_gen_subalgebra P N X"
skip: fixes mem :: "'Var \<Rightarrow> 'Val" shows "eval\<^sub>w_simple_abv (Skip, mem) (Stop, mem)"
Union_in_chain: fixes \<B> :: "'b set set"   and \<A> :: "'b set set" assumes "finite \<B>"   and "\<B> \<noteq> {}"   and "subset.chain \<A> \<B>" shows "\<Union> \<B> \<in> \<B>"
swapEnvDom_preserves_wls: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set option"   and xs :: "'varSort"   and x :: "'var"   and y :: "'var" assumes "wlsEnv rho" shows "wlsEnv (swapEnvDom xs x y rho)"
assms: shows "NormalAction (WriteMem ad al v') \<in> set obs"
execute_parallel_operator_negative_effect_if: fixes s :: "'b \<Rightarrow> bool option"   and ops :: "'b strips_operator list"   and op :: "'b strips_operator"   and v :: "'b" assumes "are_all_operators_applicable s ops"   and "are_all_operator_effects_consistent ops"   and "op \<in> set ops"   and "v \<in> set (delete_effects_of op)" shows "execute_parallel_operator s ops v = Some False"
coeff_monom: fixes a :: "'b"   and m :: "nat"   and n :: "nat" shows "coeff (monom a m) n = (if m = n then a else (0::'b))"
prod_mset_prime_factorization: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and x :: "'a" assumes "class.factorial_semiring_multiplicative divide plus minus zero times one unit_factor normalize"   and "x \<noteq> zero" shows "comm_monoid_mult.prod_mset times one (factorial_semiring.prime_factorization zero times one normalize x) = normalize x"
surj_card_le: fixes A :: "'b set"   and B :: "'c set"   and f :: "'b \<Rightarrow> 'c" assumes "finite A"   and "B \<subseteq> f ` A" shows "card B \<le> card A"
alpha_tConj: fixes tset1 :: "('a, 'b, 'c, 'd) FL_Formula.Tree set['a]"   and tset2 :: "('a, 'b, 'c, 'd) FL_Formula.Tree set['a]" shows "(FL_Formula.Tree.tConj tset1 =\<^sub>\<alpha> FL_Formula.Tree.tConj tset2) = rel_bset (=\<^sub>\<alpha>) tset1 tset2"
right_total_fun: fixes A :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and B :: "'g \<Rightarrow> 'h \<Rightarrow> bool" assumes "right_unique A"   and "right_total B" shows "right_total (A ===> B)"
insert_eqpoll_insert_iff: fixes a :: "'a"   and A :: "'a set"   and b :: "'b"   and B :: "'b set" assumes "a \<notin> A"   and "b \<notin> B" shows "(insert a A \<approx> insert b B) = (A \<approx> B)"
Suc_1: fixes I :: "nat \<Rightarrow> 'a set" assumes "\<And>i. i \<le> n \<Longrightarrow> ideal (I i) R" shows "canonical_proj_ext I n \<in> ring_hom R (RDirProd_list (map (\<lambda>i. R Quot I i) [0..<Suc n]))"
infinite_inj_imageE: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and B :: "'b set" assumes "infinite A"   and "inj_on f A"   and "f ` A \<subseteq> B" shows "infinite B"
in_hom_char: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and Arr :: "'b \<Rightarrow> bool"   and f :: "'b"   and a :: "'b"   and b :: "'b" assumes "subcategory C Arr" shows "partial_magma.in_hom (subcategory.comp C Arr) f a b = (partial_magma.arr (subcategory.comp C Arr) a \<and> partial_magma.arr (subcategory.comp C Arr) b \<and> partial_magma.arr (subcategory.comp C Arr) f \<and> partial_magma.in_hom C f a b)"
simps_2: fixes f :: "'f"   and ts :: "('f, 'b) Term.term list" shows "weight (Fun f ts) = (let n = length ts; scff = scf (f, n) in w (f, n) + sum_list (map2 (\<lambda>ti i. weight ti * scff i) ts [0..<n]))"
sint_uint: fixes w :: "'a Word.word" shows "sint w = signed_take_bit (LENGTH('a) - Suc 0) (uint w)"
sorted_list_of_fset_def: shows "sorted_list_of_fset \<equiv> map_fun fset id sorted_list_of_set"
add: fixes b1 :: "complex"   and b2 :: "complex" shows "cnj (b1 + b2) = cnj b1 + cnj b2"
abs_summable_on_normI: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set" assumes "f abs_summable_on A" shows "(\<lambda>x. norm (f x)) abs_summable_on A"
A: shows "((F \<circ> real_of_ereal) \<longlongrightarrow> A) (at_right a)"
wf_fmla_atom_constT_imp_objT: fixes Q :: "variable \<Rightarrow> PDDL_STRIPS_Semantics.type option"   and a :: "PDDL_STRIPS_Semantics.term atom formula" assumes "wf_fmla_atom (ty_term Q constT) a" shows "wf_fmla_atom (ty_term Q objT) a"
jkbpCn_jkbpC_subset: fixes n :: "nat" shows "jkbpC\<^bsub>n\<^esub> \<subseteq> jkbpC"
is_unit_prod_mset_primes_iff: fixes A :: "'a multiset" assumes "\<And>x. x \<in># A \<Longrightarrow> prime x" shows "is_unit (\<Prod>\<^sub># A) = (A = {#})"
agreement: fixes t :: "'b trm"   and r :: "('b \<times> 'b trm) list"   and s :: "('b \<times> 'b trm) list" shows "(t \<lhd> r = t \<lhd> s) = (\<forall>v\<in>vars_of t. Var v \<lhd> r = Var v \<lhd> s)"
new_element_ptr_not_in_heap: fixes h :: "(_) heap"   and h' :: "(_) heap"   and new_element_ptr :: "(_) element_ptr" assumes "h \<turnstile> new_element \<rightarrow>\<^sub>h h'"   and "h \<turnstile> new_element \<rightarrow>\<^sub>r new_element_ptr" shows "new_element_ptr |\<notin>| element_ptr_kinds h"
in_set_zipE: fixes x :: "'b"   and y :: "'c"   and xs :: "'b list"   and ys :: "'c list" assumes "(x, y) \<in> set (xs || ys)" obtains "x \<in> set xs" and "y \<in> set ys"
assms: shows "subst_domain \<theta> \<inter> set (bvars\<^sub>s\<^sub>t\<^sub>p a) = {}"
input_sizes_deep_model: fixes rs :: "nat list" assumes "2 \<le> length rs" shows "input_sizes (deep_model_l rs) = replicate (2 ^ (length rs - 2)) (last rs)"
assms: fixes c :: "'a" assumes "c \<in> set (coeffs p)" shows "a dvd c"
ag_pOp_add_r: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<in> carrier M"   and "b \<in> carrier M"   and "c \<in> carrier M"   and "a = b" shows "a \<plusminus> c = b \<plusminus> c"
comp_weak_lens: fixes x :: "'b \<Longrightarrow> 'c"   and y :: "'c \<Longrightarrow> 'd" assumes "weak_lens x"   and "weak_lens y" shows "weak_lens (x ;\<^sub>L y)"
assms: shows "\<forall>i\<in>Field r. local.aboveS i \<noteq> {} \<and> succ i \<in> Field r"
is_solution_solve: shows "is_solution x (Gauss_Jordan A) (P_Gauss_Jordan A *v b)"
assms_5: fixes it :: "'u set"   and \<sigma> :: "'\<sigma>" assumes "it \<subseteq> dom (\<alpha> m)"   and "it \<noteq> {}"   and "\<not> c \<sigma>"   and "I it \<sigma>"   and "\<forall>k\<in>it. \<forall>j\<in>dom (\<alpha> m) - it. j \<le> k" shows "P \<sigma>"
acts_MapOps_update_def: shows "acts_MapOps_update \<equiv> \<lambda>k v m. trie_odlist_update_with (fst k) m empty_trie (trie_odlist_update (snd k) v)"
d1_eq: fixes x :: "'a" shows "x = d x \<sqinter> x"
NewInitThrow: fixes sh :: "sheap"   and C :: "char list"   and P :: "(char list list \<times> char list exp) prog"   and h :: "heap"   and l :: "sfields"   and a :: "char list exp"   and s' :: "heap \<times> sfields \<times> sheap" assumes "\<nexists>sfs. sh C = \<lfloor>(sfs, Done)\<rfloor>"   and "P \<turnstile> \<langle>INIT C ([C],False) \<leftarrow> unit,(h, l, sh)\<rangle> \<Rightarrow> \<langle>throw a,s'\<rangle>"   and "is_class P C" shows "P \<turnstile> \<langle>new C,(h, l, sh)\<rangle> \<Rightarrow> \<langle>throw a,s'\<rangle>"
bij_lens_inv_left: fixes X :: "'d \<Longrightarrow> 'e" assumes "bij_lens X" shows "inv\<^sub>L X ;\<^sub>L X = 1\<^sub>L"
dvd_monic: fixes p :: "'a poly"   and c :: "'a"   and q :: "'a poly" assumes "lead_coeff p = (1::'a)"   and "p dvd smult c q"   and "c \<noteq> (0::'a)" shows "p dvd q"
simps_1: fixes v :: "bool"   and s :: "char list \<Rightarrow> int option" shows "pbval (Bc v) s = v"
insert_union_subset_iff: fixes a :: "'b"   and A :: "'b multiset"   and B :: "'b multiset" shows "(add_mset a A \<subset># B) = (a \<in># B \<and> A \<subset># B - {#a#})"
subdivision_base: fixes G :: "('a, 'b) pre_digraph"   and rev_G :: "'b \<Rightarrow> 'b" assumes "bidirected_digraph G rev_G" shows "subdivision (G, rev_G) (G, rev_G)"