hcju/isabelleps huggingface dataset:
update_convs_2: fixes user_reg' :: "('b word \<Rightarrow> word5 \<Rightarrow> word32) \<Rightarrow> 'b word \<Rightarrow> word5 \<Rightarrow> word32"   and cpu_reg :: "CPU_register \<Rightarrow> word32"   and user_reg :: "'b word \<Rightarrow> word5 \<Rightarrow> word32"   and sys_reg :: "sys_reg \<Rightarrow> word32"   and mem :: "word8 \<Rightarrow> word36 \<Rightarrow> word8 option"   and mmu :: "MMU_state"   and cache :: "cpu_cache"   and dwrite :: "(int \<times> word32 \<times> CPU_register) list"   and state_var :: "sparc_state_var"   and traps :: "Trap set"   and undef :: "bool"   and more :: "'c" shows "user_reg_update user_reg' \<lparr>cpu_reg = cpu_reg, user_reg = user_reg, sys_reg = sys_reg, mem = mem, mmu = mmu, cache = cache, dwrite = dwrite, state_var = state_var, traps = traps, undef = undef, \<dots> = more\<rparr> = \<lparr>cpu_reg = cpu_reg, user_reg = user_reg' user_reg, sys_reg = sys_reg, mem = mem, mmu = mmu, cache = cache, dwrite = dwrite, state_var = state_var, traps = traps, undef = undef, \<dots> = more\<rparr>"
F_def: shows "F = fds_newman + fds_deriv fds_zeta + fds_const c * fds_zeta"
eqArg: shows "Im (integral {0..t} (\<lambda>x. vector_derivative \<gamma> (at x) / (\<gamma> x - z))) = Arg2pi r"
c1: shows "dim_col (u \<Otimes> v) = 1"
nonneg_irreducible_mat_def: fixes A :: "'b mat" shows "nonneg_irreducible_mat A = (nonneg_mat A \<and> irreducible_mat A)"
psubst_Var: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qTerm set option"   and xs :: "'varSort"   and x :: "'var" assumes "goodEnv rho" shows "Var xs x #[rho] = (case rho xs x of None \<Rightarrow> Var xs x | Some X \<Rightarrow> X)"
hyps: fixes a :: "'a" shows "foldr (foldr f) m a = foldr f (concat m) a"
det3_translate_origin: fixes p :: "real \<times> real"   and q :: "real \<times> real"   and r :: "real \<times> real" assumes "NO_MATCH (0::'a) p" shows "det3 p q r = det3 0 (q - p) (r - p)"
a0: shows "a $ 0 \<noteq> (0::'a)"
simps_2: fixes delta :: "'h"   and inp :: "'d \<Rightarrow> ('d, 'e, 'f, 'g, 'h) qTerm option"   and binp :: "'e \<Rightarrow> ('d, 'e, 'f, 'g, 'h) qAbs option"   and delta' :: "'h"   and inp' :: "'d \<Rightarrow> ('d, 'e, 'f, 'g, 'h) qTerm option"   and binp' :: "'e \<Rightarrow> ('d, 'e, 'f, 'g, 'h) qAbs option" shows "(qOp delta inp binp #= qOp delta' inp' binp') = (delta = delta' \<and> sameDom inp inp' \<and> sameDom binp binp' \<and> liftAll2 (#=) inp inp' \<and> liftAll2 ($=) binp binp')"
zone_delay_mono: fixes A :: "('f \<Rightarrow> 'g) set"   and B :: "('f \<Rightarrow> 'g) set" assumes "A \<subseteq> B" shows "A\<^sup>\<up> \<subseteq> B\<^sup>\<up>"
insert_disjoint_2: fixes a :: "'b"   and A :: "'b set"   and B :: "'b set" shows "({} = insert a A \<inter> B) = (a \<notin> B \<and> {} = A \<inter> B)"
has_sgnx_split: fixes f :: "real \<Rightarrow> real"   and c :: "real"   and x :: "real" shows "(f has_sgnx c) (at x) = ((f has_sgnx c) (at_left x) \<and> (f has_sgnx c) (at_right x))"
CONS_inj: fixes x :: "bool list \<times> bool list"   and \<AA> :: "interp"   and y :: "bool list \<times> bool list"   and \<BB> :: "interp" assumes "size_atom x = #\<^sub>V \<AA>"   and "size_atom y = #\<^sub>V \<AA>"   and "#\<^sub>V \<AA> = #\<^sub>V \<BB>" shows "(CONS x \<AA> = CONS y \<BB>) = (x = y \<and> \<AA> = \<BB>)"
indexToVertexList_def: fixes f :: "face"   and v :: "nat"   and "is" :: "nat list" shows "indexToVertexList f v is \<equiv> hideDups (map (\<lambda>k. f\<^bsup>k\<^esup> \<bullet> v) is)"
interval_eq_iff: fixes a :: "'b interval"   and b :: "'b interval" shows "(a = b) = (lower a = lower b \<and> upper a = upper b)"
5: assumes "hom_decomp ps" shows "hom_decomp qs"
inject_1: fixes x1 :: "('e, 'f) atm"   and y1 :: "('e, 'f) atm" shows "(Pos x1 = Pos y1) = (x1 = y1)"
assms_5: shows "D.internal_equivalence (F f) (F g) (D.inv (\<Phi> (g, f)) \<cdot>\<^sub>D F \<eta> \<cdot>\<^sub>D unit (src\<^sub>C f)) (D.inv (unit (trg\<^sub>C f)) \<cdot>\<^sub>D F \<epsilon> \<cdot>\<^sub>D \<Phi> (f, g))"
empty_correct: fixes q :: "('c, 'd) SkewBinomialHeap" shows "(to_mset q = {#}) = (q = local.empty)"
keys_eq_empty_iff: fixes c :: "'a \<Rightarrow>\<^sub>0 'b" shows "(keys c = {}) = (c = 0)"
simps_1: fixes c :: "qtname" shows "constVal (NewC c) = None"
mono': fixes m :: "nat"   and n :: "nat" assumes "m \<le> n"   and "0 < m" shows "f n \<le> f m"
frechet_derivative_pair: fixes f :: "'b \<Rightarrow> 'c"   and x :: "'b"   and g :: "'b \<Rightarrow> 'd" assumes "f differentiable at x"   and "g differentiable at x" shows "frechet_derivative (\<lambda>x. (f x, g x)) (at x) = (\<lambda>v. (frechet_derivative f (at x) v, frechet_derivative g (at x) v))"
m_in_mr_coset: fixes H :: "'a set"   and m :: "'a" assumes "submodule R M H"   and "m \<in> carrier M" shows "m \<in> m \<uplus>\<^bsub>M\<^esub> H"
wo_rel_def: fixes r :: "('c \<times> 'c) set" shows "wo_rel r \<equiv> Well_order r"
mru_guard_def: fixes s :: "v_state"   and Q :: "process set"   and v :: "val" shows "mru_guard s Q v \<equiv> Q \<in> Quorum \<and> (let mru = mru_of_set (votes s) Q in mru = None \<or> (\<exists>r. mru = Some (r, v)))"
sw_def: fixes ys :: "'e"   and y1 :: "'d"   and y2 :: "'d"   and xs :: "'e"   and x :: "'d" shows "x @xs[y1 \<and> y2]_ys \<equiv> if ys = xs then if x = y1 then y2 else if x = y2 then y1 else x else x"
bigtheta_inverse: fixes f :: "'b \<Rightarrow> 'c"   and F :: "'b filter"   and g :: "'b \<Rightarrow> 'c" shows "((\<lambda>x. inverse (f x)) \<in> \<Theta>[F](\<lambda>x. inverse (g x))) = (f \<in> \<Theta>[F](g))"
SUP_LE_P_upper: fixes pair :: "'b \<Rightarrow> 'c \<Rightarrow> 'a"   and v :: "'b"   and i :: "'c"   and u :: "'a"   and P :: "'b \<Rightarrow> 'c \<Rightarrow> 'd" assumes "pair v i \<sqsubseteq> u" shows "P v i \<sqsubseteq> DiagramTermination.SUP_LE_P pair P u i"
null_part_sets_2: fixes S :: "'b set"   and M :: "'b measure" assumes "S \<in> sets M" shows "emeasure M (null_part M S) = 0"
arctan_add_raw: fixes x :: "real"   and y :: "real" assumes "\<bar>arctan x + arctan y\<bar> < pi / 2" shows "arctan x + arctan y = arctan ((x + y) / (1 - x * y))"
is_left_adjoint_char_1: fixes f :: "'a arrow_of_spans_data" shows "is_left_adjoint f = (ide f \<and> C.iso (Leg0 (Dom f)))"
l3_refines_l2: shows "refines (R23s \<inter> reach l2 \<times> (l3_inv1 \<inter> l3_inv2 \<inter> l3_inv3 \<inter> l3_inv4)) med23s l2 l3"
trg_dom: fixes \<mu> :: "'a arrow_of_spans_data" assumes "arr \<mu>" shows "trg (local.dom \<mu>) = trg \<mu>"
pairwise_empty: fixes P :: "'c \<Rightarrow> 'c \<Rightarrow> bool" shows "pairwise P {}"
set_tl_is_subset: fixes x :: "'a list" shows "set (tl x) \<subseteq> set x"
indep_system_subset: fixes \<E> :: "'a set" assumes "\<E> \<subseteq> \<E>" shows "indep_system \<E> (\<E>.indep_in \<E>)"
simps_56: fixes x7 :: "'a OCL_Types.type"   and x4 :: "'a OCL_Types.type" shows "Bag x7 \<noteq> Collection x4"
nat_pow_0: fixes x :: "int" shows "x [^]\<^bsub>INTEG\<^esub> 0 = \<one>\<^bsub>INTEG\<^esub>"
ocrit_spec_chain_ocrit: shows "punit.ocrit_spec punit.chain_ocrit"
vsv_vrange_vsubset: fixes A :: "V" assumes "\<And>x. x \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> vint_uminus \<Longrightarrow> -\<^sub>\<int> x \<in>\<^sub>\<circ> A" shows "\<R>\<^sub>\<circ> vint_uminus \<subseteq>\<^sub>\<circ> A"
intro: fixes Xd :: "'d \<Rightarrow> 'e"   and Xh :: "'d \<Rightarrow> 'f"   and Pd :: "'e \<Rightarrow> ('d \<times> 'd) set"   and Ch :: "'f \<Rightarrow> 'd set \<Rightarrow> 'd set"   and x :: "'d"   and X :: "'d set"   and ds :: "'e set"   and Pd' :: "'e \<Rightarrow> ('d \<times> 'd) set" assumes "ContractsWithUnilateralSubstitutesAndIRCAndLAD Xd Xh Pd Ch"   and "TruePrefs Xd Xh Pd Ch x X ds Pd'" shows "ContractsWithUnilateralSubstitutesAndIRCAndLADAndTruePrefs Xd Xh Pd Ch x X ds Pd'"
conf_clinit_nProc_dist: fixes P :: "jvm_prog"   and sh :: "sheap"   and frs :: "frame list"   and C :: "char list" assumes "conf_clinit P sh frs"   and "\<forall>sfs. sh C \<noteq> \<lfloor>(sfs, Processing)\<rfloor>" shows "distinct (C # clinit_classes frs)"
SEQ_iff: fixes f :: "nat \<Rightarrow> 'a"   and A :: "'a set" shows "(f \<in> SEQ A) = (\<forall>i. f i \<in> A)"
inj_list_succ: fixes xs :: "'b list" assumes "distinct xs" shows "inj (list_succ xs)"
delete_edges_wf: fixes G :: "'a graph"   and E :: "('a \<times> 'a) set" assumes "wf_graph G" shows "wf_graph (delete_edges G E)"
bconf_Cond: fixes P :: "'a prog"   and sh :: "sheap"   and b :: "'b exp"   and e\<^sub>1 :: "'b exp"   and e\<^sub>2 :: "'b exp"   and b' :: "bool" shows "(P,sh \<turnstile>\<^sub>b (if (b) e\<^sub>1 else e\<^sub>2,b') \<surd>) = (P,sh \<turnstile>\<^sub>b (b,b') \<surd>)"
nsgPr2: fixes N :: "'a set"   and a :: "'a"   and h :: "'a" assumes "G \<triangleright> N"   and "a \<in> carrier G"   and "h \<in> N" shows "\<rho> a \<cdot> (h \<cdot> a) \<in> N"
of_nat_ge_1_iff: fixes x :: "nat" shows "((1::'a) \<le> of_nat x) = (1 \<le> x)"
res: fixes r :: "'c"   and s' :: "'a"   and s :: "'a"   and q :: "'b" assumes "(r, s') \<in> results_gpv \<I>' (callee s q)"   and "q \<in> outs_\<I> \<I>"   and "I s" shows "r \<in> responses_\<I> \<I> q \<and> I s'"
D_def: shows "D = mat_of_rows (dim_col A) [Matrix.row C 0, Matrix.row C 1]"
zbeforey_ind: shows "index (fst (TSdet init h (as @ [x] @ bs @ [x] @ cs) (length (as @ [x] @ bs @ [x]) + i))) z < index (fst (TSdet init h (as @ [x] @ bs @ [x] @ cs) (length (as @ [x] @ bs @ [x]) + i))) (cs ! i)"
less_vcard_VPow: fixes A :: "V" shows "vcard A < vcard (VPow A)"
Domain_O: fixes a :: "'e set"   and x :: "('e \<times> 'f) set"   and y :: "('f \<times> 'g) set" assumes "a \<subseteq> Domain x"   and "x `` a \<subseteq> Domain y" shows "a \<subseteq> Domain (x O y)"
dyn_standard_control_dependence_def: fixes n :: "'node"   and n' :: "'node"   and as :: "'edge list" shows "n controls\<^sub>s n' via as \<equiv> \<exists>a a' as'. as = a # as' \<and> n' \<notin> set (sourcenodes as) \<and> n -as\<rightarrow>* n' \<and> n' postdominates targetnode a \<and> valid_edge a' \<and> sourcenode a' = n \<and> \<not> n' postdominates targetnode a'"
simps_12: fixes C :: "char list"   and M :: "char list"   and es :: "'a exp list" shows "max_vars (C\<bullet>\<^sub>sM(es)) = max_varss es"
split_maxD: fixes t :: "('c \<times> nat) tree"   and t' :: "('c \<times> nat) tree"   and x :: "'c" assumes "split_max t = (t', x)"   and "t \<noteq> \<langle>\<rangle>"   and "invar t" shows "Tree2.inorder t' @ [x] = Tree2.inorder t"
fusion_spec: fixes p :: "'e \<Rightarrow> bool"   and r :: "'e \<Rightarrow> 'd \<Rightarrow> bool"   and p' :: "'e \<Rightarrow> bool"   and r' :: "'e \<Rightarrow> 'd \<Rightarrow> bool" shows "({. p .} \<circ> [: r :]) \<parallel> ({. p' .} \<circ> [: r' :]) = {. p \<sqinter> p' .} \<circ> [: r \<sqinter> r' :]"
MkIde_Dom: fixes a :: "('o, 'a) arr" assumes "arr a" shows "MkIde (Dom a) = local.dom a"
cobounded1: fixes a :: "'g multiset"   and b :: "'g multiset" shows "a \<subseteq># a \<union># b"
assms_1: shows "(\<sigma>, \<zeta>) \<in> oreachable (A \<langle>\<langle>\<^bsub>i\<^esub> qmsg) (otherwith S {i} (orecvmsg R)) (other U {i})"
assms_1: fixes i :: "nat" assumes "i < a" shows "length (f i) = d"
Sbis_imp_ZObis: fixes c :: "('test, 'atom, 'choice) cmd" assumes "c \<approx>s c" shows "c \<approx>01 c"
l_set_disconnected_nodes_axioms: shows "l_set_disconnected_nodes type_wf set_disconnected_nodes set_disconnected_nodes_locs"
vinsert_compr: fixes y :: "V"   and A :: "V" shows "vinsert y A = set {x. x = y \<or> x \<in>\<^sub>\<circ> A}"
complete_development_exists: fixes A :: "'a trm" shows "\<exists>X. A >>> X"
relcompp_distrib2: fixes S :: "'a \<Rightarrow> 'c \<Rightarrow> bool"   and T :: "'a \<Rightarrow> 'c \<Rightarrow> bool"   and R :: "'c \<Rightarrow> 'b \<Rightarrow> bool" shows "sup S T OO R = sup (S OO R) (T OO R)"
fact_real_float_equiv: fixes n :: "nat" shows "real_of_float (fact n) = fact n"
one_extB_def: shows "1 \<equiv> extB.Val top"
monoWS: fixes P :: "complex mat"   and Q :: "complex mat" assumes "is_quantum_predicate P"   and "is_quantum_predicate Q"   and "P \<le>\<^sub>L Q" shows "wlp S P \<le>\<^sub>L wlp S Q"
bidel_arc_same_face_def: fixes G :: "('c, 'd) pre_digraph"   and M :: "'d pre_map"   and a :: "'d" shows "bidel_arc_same_face G M a \<equiv> bidel_arc G M a \<and> bidel_arc_same_face_axioms M a"
inf_sup_ord_1: fixes x :: "'b"   and y :: "'b" shows "inf x y \<le> x"
ENC_FEXISTS: fixes n :: "nat"   and \<phi> :: "'a formula" assumes "wf_formula n (FEXISTS \<phi>)" shows "lang n (ENC n (FOV (FEXISTS \<phi>))) = SAMEQUOT (any, replicate n False) (map \<pi> ` lang (Suc n) (ENC (Suc n) (FOV \<phi>)))"
order_refl: fixes x :: "'a" shows "x \<sqsubseteq> x"
assms_1: shows "wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<delta>"
good_def: fixes eff :: "'d \<Rightarrow> 'e \<Rightarrow> 'e fset \<Rightarrow> bool"   and rules :: "'d stream"   and "structure" :: "'f set"   and sat :: "'f \<Rightarrow> 'e \<Rightarrow> bool"   and \<delta> :: "'e \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> ('g \<times> bool \<times> 'g) set"   and Ord :: "('h \<times> 'h) set"   and \<sigma> :: "'g \<times> 'f \<Rightarrow> 'h"   and t :: "('e \<times> 'd) tree" assumes "Infinite_Soundness eff rules structure sat \<delta> Ord \<sigma>" shows "Infinite_Soundness.good \<delta> t \<equiv> \<forall>steps. ipath t steps \<longrightarrow> ev (\<lambda>steps'. \<exists>bs Ms. Infinite_Soundness.follow \<delta> bs Ms steps' \<and> Infinite_Soundness.infDecr bs) steps"
replicating_portfolio_def: fixes pf :: "'a \<Rightarrow> nat \<Rightarrow> bool stream \<Rightarrow> real"   and der :: "bool stream \<Rightarrow> real"   and matur :: "nat" shows "replicating_portfolio pf der matur = (stock_portfolio Mkt pf \<and> trading_strategy pf \<and> self_financing Mkt pf \<and> AEeq M (cls_val_process Mkt pf matur) der)"
saturate_mono: fixes C :: "'a literal multiset"   and CC :: "'a literal multiset set"   and DD :: "'a literal multiset set" assumes "C \<in> saturate CC"   and "CC \<subseteq> DD" shows "C \<in> saturate DD"
hfref_compI_PRE_aux: fixes f :: "'a \<Rightarrow> 'b Heap"   and g :: "'c \<Rightarrow> 'd nres"   and P :: "'c \<Rightarrow> bool"   and RR' :: "('c \<Rightarrow> 'a \<Rightarrow> assn) \<times> ('c \<Rightarrow> 'a \<Rightarrow> assn)"   and S :: "'d \<Rightarrow> 'b \<Rightarrow> assn"   and h :: "'e \<Rightarrow> 'f nres"   and Q :: "'e \<Rightarrow> bool"   and T :: "('c \<times> 'e) set"   and U :: "('d \<times> 'f) set" assumes "(f, g) \<in> [P]\<^sub>a RR' \<rightarrow> S"   and "(g, h) \<in> [Q]\<^sub>f T \<rightarrow> \<langle>U\<rangle>nres_rel" shows "(f, h) \<in> [comp_PRE T Q (\<lambda>_. P) (\<lambda>_. True)]\<^sub>a hrp_comp RR' T \<rightarrow> hr_comp S U"
perm_contr: fixes P :: "'b list \<Rightarrow> 'c"   and xs :: "'b list" assumes "\<forall>xs ys. xs <~~> ys \<longrightarrow> P xs = P ys"   and "\<forall>x xs. P (x # x # xs) = P (x # xs)" shows "P xs = P (remdups xs)"
lift_append: fixes f :: "'e \<Rightarrow> 'a"   and gens :: "'e set"   and x :: "(bool \<times> 'e) list"   and y :: "(bool \<times> 'e) list" assumes "f \<in> gens \<rightarrow> carrier G"   and "x \<in> lists (UNIV \<times> gens)"   and "y \<in> lists (UNIV \<times> gens)" shows "G.lift f (x @ y) = G.lift f x \<otimes>\<^bsub>G\<^esub> G.lift f y"
assms_2: shows "\<HH> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<BB>"
dgrad_p_set_closed_minus: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and d :: "'a \<Rightarrow> nat"   and m :: "nat"   and q :: "'t \<Rightarrow>\<^sub>0 'c" assumes "p \<in> dgrad_p_set d m"   and "q \<in> dgrad_p_set d m" shows "p - q \<in> dgrad_p_set d m"
simps_39: fixes x61 :: "'c"   and x62 :: "('a, 'b, 'c) Stmt"   and x41 :: "('a, 'b, 'c) Stmt"   and x42 :: "('a, 'b, 'c) Stmt" shows "While x61 x62 \<noteq> x41 ;; x42"
wtm: shows "wt_method P C b Ts T\<^sub>r mxs mxl\<^sub>0 is xt \<tau>s"
is_selection: fixes K :: "'a set"   and A :: "'a set"   and \<phi> :: "'a" assumes "K = Cn A"   and "K .\<bottom>. \<phi> \<noteq> {}" shows "\<gamma>\<^sub>A\<^sub>G\<^sub>M K \<phi> \<subseteq> K .\<bottom>. \<phi>"
dvd_square_part_iff: fixes x :: "'a"   and n :: "'a" shows "(x dvd square_part n) = (x\<^sup>2 dvd n)"
deg_pm_restrict_indets_pp: fixes t :: "'b option \<Rightarrow>\<^sub>0 nat" shows "deg_pm (restrict_indets_pp t) + lookup t None = deg_pm t"
intros_2: fixes i :: "nat"   and b :: "nat"   and n :: "C"   and ms :: "C list" assumes "i \<noteq> 0"   and "i < b"   and "n \<in> hbase b"   and "C ms \<in> hbase b"   and "\<And>m'. m' \<in> set ms \<Longrightarrow> evalC b n < evalC b m'" shows "C (replicate i n @ ms) \<in> hbase b"
map_poly_pderiv: fixes p :: "rat poly" shows "map_poly of_rat (pderiv p) = pderiv (map_poly of_rat p)"
ba_iso_right_invertible: fixes xf :: "('c, 'd) lifted_boolean_algebra" shows "ba_iso (ba_iso_inv xf) = xf"
rh: fixes U :: "'a set" assumes "is_open U" shows "ring_homomorphism (identity (\<FF> U)) (\<FF> U) +\<^bsub>U\<^esub> \<cdot>\<^bsub>U\<^esub> \<zero>\<^bsub>U\<^esub> \<one>\<^bsub>U\<^esub> (imS.im_sheaf U) (imS.add_im_sheaf U) (imS.mult_im_sheaf U) (imS.zero_im_sheaf U) (imS.one_im_sheaf U)"
assms_3: shows "a \<rightarrow>\<^sup>* b"
ref_pres_att: fixes A :: "'a attree"   and A' :: "'a attree" assumes "A \<sqsubseteq> A'" shows "attack A = attack A'"
residual_r_set_def: fixes X :: "'a set"   and Z :: "'a set" shows "X \<rightarrow> Z = \<Union> {Y. X \<cdot> Y \<subseteq> Z}"
aforms_of_ivls_refine: fixes xrs :: "real list"   and yrs :: "real list"   and xri :: "real list"   and yri :: "real list" assumes "list_all2 (\<le>) xrs yrs"   and "(xri, xrs) \<in> rl_rel"   and "(yri, yrs) \<in> rl_rel" shows "(aforms_of_ivls xri yri, lv_ivl xrs yrs) \<in> aforms_rel"
smaps_over_def: fixes A :: "'c set"   and B :: "'d set" shows "smaps_over A B = {m. sdom m \<subseteq> A \<and> sran m \<subseteq> B}"
power_less_imp_less_exp: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and a :: "'a"   and m :: "nat"   and n :: "nat" assumes "class.linordered_semidom plus minus less_eq less zero times one"   and "less one a"   and "less (power.power one times a m) (power.power one times a n)" shows "m < n"
adh_imp_closure: fixes x :: "'a"   and A :: "'a set" assumes "x \<in> carrier"   and "x adh A" shows "x \<in> closure A"
elims: fixes x :: "('sf, 'sc, 'sz) interp"   and xa :: "('sf, 'sc, 'sz) formula list \<times> ('sf, 'sc, 'sz) formula list"   and y :: "((real, 'sz) vec \<times> (real, 'sz) vec) set" assumes "seq_sem x xa = y" obtains I :: "('sf, 'sc, 'sz) interp" and S :: "('sf, 'sc, 'sz) formula list \<times> ('sf, 'sc, 'sz) formula list" where "x = I" and "xa = S" and "y = fml_sem I (seq2fml S)"
closure_Un_frontier: fixes S :: "'c set" shows "closure S = S \<union> frontier S"
Vinv1D: fixes x :: "'a v_state_scheme"   and r :: "nat" assumes "x \<in> Vinv1"   and "next_round x \<le> r" shows "votes x r = Map.empty"
pseudo_proj_False_finite_image: fixes n :: "nat" shows "finite (range (pseudo_proj_False n))"
distinct_mset_union: fixes A :: "'a multiset"   and B :: "'a multiset" assumes "distinct_mset (A + B)" shows "distinct_mset A"
\<b>_le: fixes X :: "'x set"   and ps :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list"   and d :: "nat"   and i :: "nat" assumes "finite X"   and "\<a> ps \<le> d"   and "\<And>h' U'. \<lbrakk>(h', U') \<in> set ps; i \<le> card U'\<rbrakk> \<Longrightarrow> poly_deg h' < d" shows "\<b> ps i \<le> d"
continuous_map_pullback: fixes T1 :: "'d topology"   and T2 :: "'e topology"   and g :: "'d \<Rightarrow> 'e"   and A :: "'f set"   and f :: "'f \<Rightarrow> 'd" assumes "continuous_map T1 T2 g" shows "continuous_map (pullback_topology A f T1) T2 (g \<circ> f)"
sortedP_append_iff: fixes xs :: "'a list"   and ys :: "'a list" shows "sortedP (xs @ ys) = (sortedP xs \<and> sortedP ys \<and> (\<forall>x\<in>set xs. \<forall>y\<in>set ys. le x y))"
lincomb_list_carrier: fixes vs :: "'a vec list"   and c :: "nat \<Rightarrow> 'a" assumes "set vs \<subseteq> carrier_vec nr" shows "nr.lincomb_list c vs \<in> carrier_vec nr"
r_coset_hom_Mod: fixes H :: "'c set"   and G :: "('c, 'd) monoid_scheme" assumes "H \<lhd> G" shows "(#>\<^bsub>G\<^esub>) H \<in> hom G (G Mod H)"
add_set_single2: fixes f :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and z :: "'e"   and i :: "'e \<Rightarrow> 'e"   and a :: "'e"   and x :: "'e"   and n :: "nat"   and y :: "'e" assumes "commute_bpp f (aug_pm_set z i {a})"   and "assoc_bpp (aug_pm_set z i {a}) f"   and "ipp_cond1 {a} i"   and "ipp_cond2 z {a} i f"   and "ipp_cond3 z i"   and "inv_ipp z i f {a}"   and "commute_bpp f (aug_pm_set z i {a})"   and "zeroA z i f {a} z"   and "x \<in> add_set f {\<^sub>i- a} n"   and "y \<in> add_set f {\<^sub>i- a} n" shows "x = y"
differentiable_sqnorm_at: fixes a :: "'a" shows "(\<lambda>x. (norm x)\<^sup>2) differentiable at a"
distrib_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"   and z :: "'c" assumes "class.distrib_lattice inf less_eq less sup" shows "sup x (inf y z) = inf (sup x y) (sup x z)"
orbit_apply_inverse: fixes f :: "'a perm"   and a :: "'a" shows "orbit f (inverse f \<langle>$\<rangle> a) = orbit f a"
phl_cons2: fixes w :: "'a"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "w \<le> x"   and "y \<cdot> z \<le> w \<cdot> y" shows "y \<cdot> z \<le> x \<cdot> y"
path_pd_ipd: fixes \<pi> :: "nat \<Rightarrow> 'n"   and k :: "nat"   and n :: "nat" assumes "is_path \<pi>"   and "\<pi> k \<noteq> local.return"   and "\<pi> n \<noteq> \<pi> k"   and "\<pi> n pd\<rightarrow> \<pi> k"   and "k < n" obtains l :: "nat" where "k < l" and "l \<le> n" and "\<pi> l = ipd (\<pi> k)"
ltl\<^sub>P_abs_rep: fixes \<phi> :: "'a ltl_prop_equiv_quotient" shows "Abs (Rep \<phi>) = \<phi>"
3: assumes "P \<turnstile> typeof\<^bsub>h\<^esub> v = \<lfloor>Class C'\<rfloor>" shows "P \<turnstile> typeof\<^bsub>h(a \<mapsto> (C, S'))\<^esub> v = \<lfloor>Class C'\<rfloor>"
simps_5: fixes v :: "nat"   and va :: "real mpoly"   and L :: "atom list" shows "find_lucky_eq v (Neq va # L) = find_lucky_eq v L"
numeral_uint64: fixes n :: "num" shows "numeral n = Uint64 (numeral n)"
if_split: fixes P :: "'e \<Rightarrow> bool"   and Q :: "bool"   and x :: "'e"   and y :: "'e" shows "P (if Q then x else y) = ((Q \<longrightarrow> P x) \<and> (\<not> Q \<longrightarrow> P y))"
assms_2: shows "theta\<inverse> \<subseteq> WretrT (theta\<inverse> \<union> WbisT)"
seqSubstMismatch: fixes a :: "name"   and b :: "name"   and P :: "pi"   and \<sigma> :: "(name \<times> name) list" shows "[a\<noteq>b]P[<\<sigma>>] = [seq_subst_name a \<sigma>\<noteq>seq_subst_name b \<sigma>](P[<\<sigma>>])"
select_convs_2: fixes g_V :: "'e set"   and g_E :: "('e \<times> 'e) set"   and g_V0 :: "'e set"   and more :: "'f" shows "g_E \<lparr>g_V = g_V, g_E = g_E, g_V0 = g_V0, \<dots> = more\<rparr> = g_E"
assms_3: shows "f \<cdot> \<iota> = \<iota> \<cdot> (f \<otimes> f)"
Union: fixes W :: "'b set set"   and F :: "'b program"   and B :: "'b set" assumes "W \<noteq> {}"   and "\<forall>U\<in>W. U \<in> wens_set F B" shows "\<Union> W \<in> wens_set F B"
is_trace_of_def: fixes A :: "('c, 'd) ioa"   and tr :: "'d list" shows "is_trace_of A tr \<equiv> \<exists>sch. is_schedule_of A sch \<and> tr = filter (\<lambda>a. a \<in> externals (ioa.asig A)) sch"
foundation: fixes z :: "hf" assumes "z \<noteq> 0" shows "\<exists>w. w \<^bold>\<in> z \<and> w \<sqinter> z = 0"
inverts_mat_def: fixes A :: "'d mat"   and B :: "'d mat" shows "inverts_mat A B \<equiv> A * B = 1\<^sub>m (dim_row A)"
Interleaves_ipurge_tr: fixes D :: "'a \<Rightarrow> 'b"   and I :: "('b \<times> 'b) set"   and u :: "'b"   and xs :: "'a list" shows "xs \<cong> {ipurge_tr_rev I D u xs, rev (ipurge_tr (I\<inverse>) D u (rev xs)), \<lambda>y ys. D y \<in> sinks (I\<inverse>) D u (rev (y # ys))}"
less_irrefl: fixes D :: "('c, 'd) Order_scheme"   and a :: "'c"   and C :: "bool" assumes "Order D"   and "a \<in> carrier D"   and "a \<prec>\<^bsub>D\<^esub> a" shows "C"