hcju/isabelleps huggingface dataset:
l2_step1_def: fixes Ra :: "fid_t"   and A :: "agent"   and B :: "agent" shows "l2_step1 Ra A B \<equiv> {(s, s'). Ra \<notin> dom (progress s) \<and> guessed_runs Ra = \<lparr>role = Init, owner = A, partner = B\<rparr> \<and> s' = s\<lparr>progress := progress s(Ra \<mapsto> {xnx, xni, xgnx}), chan := {Confid A B (NonceF (Ra $ ni))} \<union> ({Insec A B (Exp Gen (NonceF (Ra $ nx)))} \<union> chan s)\<rparr>}"
close_consistency: fixes C :: "'b fm set set" assumes "consistency C" shows "consistency (close C)"
filt_equiv_space: fixes F :: "'c \<Rightarrow> 'd measure"   and M :: "'d measure"   and N :: "'d measure" assumes "filt_equiv F M N" shows "space M = space N"
elims_2: fixes satisfies :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and transition :: "'a \<Rightarrow> ('c, 'a) residual \<Rightarrow> bool"   and effect_apply :: "'d \<Rightarrow> 'a \<Rightarrow> 'a"   and L :: "'c \<times> 'd fs_set \<times> 'd \<Rightarrow> 'd fs_set"   and x :: "('a, 'd) L_state"   and xa :: "'b" assumes "effect_nominal_ts satisfies transition effect_apply L"   and "effect_nominal_ts.L_satisfies satisfies x xa" obtains uu_ :: "'d" and uv_ :: "'d fs_set" and P :: "'a" and \<phi> :: "'b" where "x = AC (uu_, uv_, P)" and "xa = \<phi>" and "satisfies P \<phi>"
maxsimpchain_def: fixes xs :: "'b set list" shows "codomain.gallery xs \<equiv> Ball (set xs) codomain.chamber \<and> adjacentchain xs"
pmaxsimpchainD_distinct: fixes X :: "'b set set"   and xs :: "'b set list" assumes "SimplicialComplex X"   and "SimplicialComplex.pmaxsimpchain X xs" shows "distinct xs"
cong_commutativity: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "Cong A B C D" shows "Cong B A D C"
subgroups_Inter: fixes A :: "'a set set" assumes "\<And>Ha. Ha \<in> A \<Longrightarrow> subgroup Ha (G\<lparr>carrier := H\<rparr>)"   and "A \<noteq> {}" shows "subgroup (\<Inter> A) (G\<lparr>carrier := H\<rparr>)"
add_diff_eq: fixes minus :: "'r \<Rightarrow> 'r \<Rightarrow> 'r"   and plus :: "'r \<Rightarrow> 'r \<Rightarrow> 'r"   and zero :: "'r"   and uminus :: "'r \<Rightarrow> 'r"   and a :: "'r"   and b :: "'r"   and c :: "'r" assumes "class.group_add minus plus zero uminus" shows "plus a (minus b c) = minus (plus a b) c"
surj_iff: fixes f :: "'i \<Rightarrow> 'h" shows "surj f = (f \<circ> inv f = id)"
lift_Suc_mono_less_iff: fixes f :: "nat \<Rightarrow> 'c"   and n :: "nat"   and m :: "nat" assumes "\<And>n. f n < f (Suc n)" shows "(f n < f m) = (n < m)"
compose_def: fixes A :: "'a set"   and g :: "'c \<Rightarrow> 'b"   and f :: "'a \<Rightarrow> 'c" shows "g \<circ> f \<down> A = (\<lambda>x\<in>A. g (f x))"
last': shows "last ((C # Cs') @\<^sub>p Ds) = C"
rev_subsetD: fixes c :: "'e"   and A :: "'e set"   and B :: "'e set" assumes "c \<in> A"   and "A \<subseteq> B" shows "c \<in> B"
assms_4: fixes i :: "'b" assumes "i \<in> S" shows "0 < e i"
iter_state_iterate_state: fixes cnt :: "'a \<Rightarrow> bool"   and nxt :: "'a \<Rightarrow> 'a"   and sizef :: "'a \<Rightarrow> nat"   and f :: "'a \<Rightarrow> ('b, 'c) state"   and x :: "'a" assumes "iterator cnt nxt sizef" shows "iterator_defs.iter_state cnt nxt f x = iterate_state f (iterator_defs.iterator_to_list cnt nxt x)"
f13: shows "rd = 0 \<longrightarrow> write_reg (if rd = 0 then user_reg_val (fst (fst (get_curr_win () s2))) rd (snd (fst (get_curr_win () s2))) else div_comp_result instr (div_comp_temp_V instr (ucast (div_comp_temp_64bit instr (word_cat (cpu_reg_val Y (snd (fst (get_curr_win () s2)))) (user_reg_val (fst (fst (get_curr_win () s2))) rs1 (snd (fst (get_curr_win () s2))))) operand2 >> 32)) (ucast (div_comp_temp_64bit instr (word_cat (cpu_reg_val Y (snd (fst (get_curr_win () s2)))) (user_reg_val (fst (fst (get_curr_win () s2))) rs1 (snd (fst (get_curr_win () s2))))) operand2 >> 31))) (div_comp_temp_64bit instr (word_cat (cpu_reg_val Y (snd (fst (get_curr_win () s2)))) (user_reg_val (fst (fst (get_curr_win () s2))) rs1 (snd (fst (get_curr_win () s2))))) operand2)) (ucast (get_CWP (cpu_reg_val PSR s1))) 0 (snd (fst (get_curr_win () s1))) = write_reg (if rd = 0 then user_reg_val (fst (fst (get_curr_win () s1))) rd (snd (fst (get_curr_win () s1))) else div_comp_result instr (div_comp_temp_V instr (ucast (div_comp_temp_64bit instr (word_cat (cpu_reg_val Y (snd (fst (get_curr_win () s1)))) (user_reg_val (fst (fst (get_curr_win () s1))) rs1 (snd (fst (get_curr_win () s1))))) operand2 >> 32)) (ucast (div_comp_temp_64bit instr (word_cat (cpu_reg_val Y (snd (fst (get_curr_win () s1)))) (user_reg_val (fst (fst (get_curr_win () s1))) rs1 (snd (fst (get_curr_win () s1))))) operand2 >> 31))) (div_comp_temp_64bit instr (word_cat (cpu_reg_val Y (snd (fst (get_curr_win () s1)))) (user_reg_val (fst (fst (get_curr_win () s1))) rs1 (snd (fst (get_curr_win () s1))))) operand2)) (fst (fst (get_curr_win () s2))) rd (snd (fst (get_curr_win () s1)))"
ide_src: fixes \<mu> :: "'a" assumes "C.arr \<mu>" shows "C.ide (src\<^sub>C \<mu>)"
bottomI: fixes x :: "'b" assumes "x \<sqsubseteq> \<bottom>" shows "x = \<bottom>"
simps_1: fixes a :: "'b" shows "is_Addr (Addr a) = True"
not_all: fixes P :: "'c \<Rightarrow> bool" shows "(\<not> (\<forall>x. P x)) = (\<exists>x. \<not> P x)"
contrapos_corr: fixes I :: "'a set"   and s :: "'a set"   and A :: "'a attree" assumes "\<not> Kripke {s. \<exists>i\<in>I. i \<rightarrow>\<^sub>i* s} I \<turnstile> EF s"   and "attack A = (I, s)" shows "\<not> \<turnstile>A"
indep_iff_subset_basis: fixes X :: "'a set" shows "indep X = (\<exists>B. basis B \<and> X \<subseteq> B)"
equiv: shows "\<forall>\<sigma>. stutter_free \<sigma> \<and> canonical_sequence (atoms_pltl \<phi>) \<sigma> \<longrightarrow> \<sigma> \<Turnstile>\<^sub>p \<psi> = \<sigma> \<Turnstile>\<^sub>p \<phi>"
cases: fixes Aexp :: "exp \<Rightarrow> Arity \<rightarrow> Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and Aheap :: "(Vars.var \<times> exp) list \<Rightarrow> exp \<Rightarrow> Arity \<rightarrow> Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and a1 :: "(Vars.var \<Rightarrow> Arity\<^sub>\<bottom>) \<times> Arity \<times> Arity list"   and a2 :: "(Vars.var \<times> exp) list \<times> exp \<times> stack_elem list" assumes "ArityAnalysisLetSafe Aexp Aheap"   and "ArityAnalysisLetSafe.a_consistent Aexp a1 a2" obtains ae :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>" and \<Gamma> :: "(Vars.var \<times> exp) list" and S :: "stack_elem list" and a :: "Arity" and e :: "exp" and as :: "Arity list" where "a1 = (ae, a, as)" and "a2 = (\<Gamma>, e, S)" and "edom ae \<subseteq> domA \<Gamma> \<union> upds S" and "Astack S \<sqsubseteq> a" and "(ArityAnalysis.ABinds Aexp \<Gamma>\<cdot>ae \<squnion> Aexp e\<cdot>a \<squnion> ArityAnalysis.AEstack Aexp as S) f|` (domA \<Gamma> \<union> upds S) \<sqsubseteq> ae" and "stack_consistent as S"
hequiv_def: fixes H :: "(('c, 'd) fm list \<times> 'd) set"   and i :: "'d"   and j :: "'d" shows "hequiv H i j \<equiv> Nom j at i in' H"
mcont_ltake2: fixes n :: "enat" shows "mcont lSup lprefix lSup lprefix (ltake n)"
that: assumes "finite (UNIV // eq_app_right L)"   and "card (UNIV // eq_app_right L) \<le> card (UNIV // R)" shows "thesis"
ltree_merge: fixes l :: "('b \<times> nat) tree"   and r :: "('b \<times> nat) tree" assumes "ltree l"   and "ltree r" shows "ltree (merge l r)"
timpl_closure_FunI: fixes T :: "(('a, 'b, 'c) prot_fun, 'd) Term.term list"   and S :: "(('a, 'b, 'c) prot_fun, 'd) Term.term list"   and c :: "('c set \<times> 'c set) set"   and f :: "('a, 'b, 'c) prot_fun"   and g :: "('a, 'b, 'c) prot_fun" assumes "\<And>i. i < length T \<Longrightarrow> (T ! i, S ! i) \<in> timpl_closure' c"   and "length T = length S"   and "f = g \<or> (\<exists>a b. (a, b) \<in> c\<^sup>+ \<and> f = Abs a \<and> g = Abs b)" shows "(Fun f T, Fun g S) \<in> timpl_closure' c"
augment_list_Suc_prefix: fixes n :: "nat"   and f :: "'b list \<Rightarrow> 'b"   and p :: "'b list" shows "prefix ((augment_list f ^^ n) p) ((augment_list f ^^ Suc n) p)"
ennreal_SUP_const_minus: fixes I :: "'a set"   and c :: "ennreal"   and f :: "'a \<Rightarrow> ennreal" assumes "I \<noteq> {}"   and "c < \<top>" shows "(\<Sqinter>x\<in>I. c - f x) = c - \<Squnion> (f ` I)"
m_comm: fixes x :: "(((nat \<Rightarrow> real) \<Rightarrow> 'a) \<Rightarrow>\<^sub>0 int) set"   and y :: "(((nat \<Rightarrow> real) \<Rightarrow> 'a) \<Rightarrow>\<^sub>0 int) set" assumes "x \<in> carrier (homology_group p X)"   and "y \<in> carrier (homology_group p X)" shows "x \<otimes>\<^bsub>homology_group p X\<^esub> y = y \<otimes>\<^bsub>homology_group p X\<^esub> x"
collect_locksE: fixes l :: "'c"   and las :: "'c lock_actions" assumes "l \<in> collect_locks las" obtains "Lock \<in> set (las $ l)"
a: shows " App t1 t2 \<longrightarrow>\<^sub>1 t'"
ordinal_exp_eq_0: fixes x :: "ordinal"   and y :: "ordinal" shows "(x ** y = 0) = (x = 0 \<and> 0 < y)"
Units_one_closed: shows "\<one>\<^bsub>ideals_set R\<^esub> \<in> Units (ideals_set R)"
sources_aux_single_dom: fixes I :: "('c \<times> 'c) set"   and D :: "'d \<Rightarrow> 'c"   and u :: "'c"   and xs :: "'d list" shows "sources_aux I D {u} xs = insert u (sources I D u xs)"
select_convs_5: fixes Pol :: "('c \<times> 'c) set"   and Map :: "'b \<Rightarrow> 'c"   and Doms :: "'c set"   and List :: "'b list"   and ListOp :: "'b list option"   and Set :: "'b set"   and more :: "'d" shows "ListOp \<lparr>Pol = Pol, Map = Map, Doms = Doms, List = List, ListOp = ListOp, Set = Set, \<dots> = more\<rparr> = ListOp"
interact_sing: fixes x :: "'a list"   and ys :: "'a list list" shows "interact [x] ys = x @ concat ys"
fbd_sup_pres: shows "sup_pres bd\<^sub>\<F>"
powers_3: fixes b :: "'a"   and a :: "'a"   and c :: "'a" shows "b + (a + c) = a + (b + c)"
assms_2: shows "Counterclockwise_2D_Arbitrary.lex p r"
iso_inv_iso: fixes f :: "'a" assumes "B.iso f" shows "B.iso (B.inv f)"
AllN_7: shows "\<forall>x. insertion (nth_default 0 ((drop z L)[var := x])) b = Bv"
is_preserved_by_base_iso: fixes \<phi> :: "'a"   and r' :: "'a" assumes "\<guillemotleft>\<phi> : g \<star> f\<^sup>* \<Rightarrow> r'\<guillemotright>"   and "local.iso \<phi>" shows "tabulation (\<cdot>) (\<star>) \<a> \<i> src trg r' ((\<phi> \<star> f) \<cdot> \<rho>) f g"
relators_1: fixes rs :: "'g list" assumes "rs \<in> R" shows "rs \<in> lists (S \<union> uminus ` S)"
rev_conv_fold: fixes xs :: "'a list" shows "rev xs = fold (#) xs []"
run_reachable: fixes s\<^sub>0 :: "'a"   and lstep :: "'b \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and s :: "nat \<Rightarrow> 'a"   and i :: "nat" assumes "DF_System.run s\<^sub>0 lstep s" shows "system.reachable s\<^sub>0 lstep (s i)"
Int_commute: fixes A :: "'k set"   and B :: "'k set" shows "A \<inter> B = B \<inter> A"
edge'_def: fixes xo :: "'v vertex"   and yo :: "'v vertex" shows "edge' xo yo = (case (xo, yo) of (SOURCE, \<langle>y\<rangle>) \<Rightarrow> y = b | (SOURCE, _) \<Rightarrow> False | (vertex.SINK, \<langle>x\<rangle>) \<Rightarrow> x \<in> A \<Gamma> | (vertex.SINK, _) \<Rightarrow> False | (\<langle>x\<rangle>, SOURCE) \<Rightarrow> False | (\<langle>x\<rangle>, vertex.SINK) \<Rightarrow> x \<in> A \<Gamma> | (\<langle>x\<rangle>, \<langle>y\<rangle>) \<Rightarrow> edge \<Gamma> x y \<or> edge \<Gamma> y x)"
list_all2_Nil: fixes P :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and ys :: "'h list" shows "list_all2 P [] ys = (ys = [])"
diff_rat: fixes b :: "int"   and d :: "int"   and a :: "int"   and c :: "int" assumes "b \<noteq> 0"   and "d \<noteq> 0" shows "Fract a b - Fract c d = Fract (a * d - c * b) (b * d)"
set_intros_1: fixes x31 :: "'d"   and x32 :: "'d ifex"   and x33 :: "'d ifex" shows "x31 \<in> set_ifex (IF x31 x32 x33)"
lub_below_iff: fixes below :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and S :: "nat \<Rightarrow> 'c"   and x :: "'c" assumes "class.cpo below"   and "po.chain below S" shows "below (po.lub below (range S)) x = (\<forall>i. below (S i) x)"
filter_plus': fixes a :: "'av"   and a1 :: "'av"   and a2 :: "'av"   and b1 :: "'av"   and b2 :: "'av"   and n1 :: "int"   and n2 :: "int" assumes "filter_plus' a a1 a2 = (b1, b2)"   and "n1 \<in> \<gamma> a1"   and "n2 \<in> \<gamma> a2"   and "n1 + n2 \<in> \<gamma> a" shows "n1 \<in> \<gamma> b1 \<and> n2 \<in> \<gamma> b2"
closed_segment_commute: fixes a :: "'c"   and b :: "'c" shows "{a--b} = {b--a}"
case_1: fixes f1 :: "nat \<Rightarrow> nat \<Rightarrow> 'a"   and f2 :: "nat \<Rightarrow> nat \<Rightarrow> 'a"   and x11 :: "nat"   and x12 :: "nat" shows "(case Pkt x11 x12 of Pkt x xa \<Rightarrow> f1 x xa | Newpkt x xa \<Rightarrow> f2 x xa) = f1 x11 x12"
f_def: shows "f = (\<lambda>s. complex_of_real (2 * pi) powr (s - 1) * \<i> * Gamma (1 - s) * (\<i> powr - s * hurwitz_zeta q (1 - s) - \<i> powr s * hurwitz_zeta (1 - q) (1 - s)))"
disconnected_nodes_eq_h2: fixes doc_ptr :: "(_) document_ptr"   and disc_nodes :: "(_) node_ptr list" shows "h2 \<turnstile> get_disconnected_nodes doc_ptr \<rightarrow>\<^sub>r disc_nodes = h' \<turnstile> get_disconnected_nodes doc_ptr \<rightarrow>\<^sub>r disc_nodes"
const_def: fixes X :: "'a \<Rightarrow> 'b" shows "const X \<equiv> \<forall>\<tau> \<tau>'. X \<tau> = X \<tau>'"
antisym_iff_exe_antisym: fixes l :: "('a \<times> 'a) list" shows "antisym (set l) = exe_antisym l"
local: fixes p :: "('b, 'c, 'a, 'd) seqp"   and l :: "'d"   and \<sigma> :: "nat \<Rightarrow> 'b"   and a :: "'c seq_action"   and \<sigma>' :: "nat \<Rightarrow> 'b"   and q :: "('b, 'c, 'a, 'd) seqp"   and l' :: "'d"   and pp :: "('b, 'c, 'a, 'd) seqp" assumes "p \<in> cterms \<Gamma>"   and "l \<in> labels \<Gamma> p"   and "((\<sigma>, p), a, \<sigma>', q) \<in> oseqp_sos \<Gamma> i"   and "((\<sigma>, p), a, \<sigma>', q) \<in> automaton.trans A"   and "l' \<in> labels \<Gamma> q"   and "(\<sigma>, pp) \<in> oreachable A S U"   and "p \<in> sterms \<Gamma> pp"   and "(\<sigma>', q) \<in> oreachable A S U"   and "S \<sigma> \<sigma>' a" shows "P ((\<sigma>, l), a, \<sigma>', l')"
hom_power: fixes x :: "'b poly"   and n :: "nat" shows "map_poly to_fract (x ^ n) = map_poly to_fract x ^ n"
B_def: shows "B = {x \<in> A. \<forall>n\<in>{1..}. (T ^^ n) x \<in> space M - A}"
flat_codom: fixes f :: "'b \<rightarrow> 'a"   and x :: "'b"   and c :: "'a" assumes "f\<cdot>x = c" shows "f\<cdot>\<bottom> = \<bottom> \<or> (\<forall>z. f\<cdot>z = c)"
isAbrE: fixes S :: "('a, 'b) xstate" assumes "isAbr S" obtains s :: "'a" where "S = Abrupt s"
cross_components_1: fixes x :: "(real, 3) vec"   and y :: "(real, 3) vec" shows "cross3 x y $ 1 = x $ 2 * y $ 3 - y $ 2 * x $ 3"
pq_order_unique_subgrp: fixes q :: "nat"   and p :: "nat" assumes "finite (carrier (G\<lparr>carrier := H\<rparr>))"   and "order (G\<lparr>carrier := H\<rparr>) = q * p"   and "prime p"   and "prime q"   and "p < q" shows "\<exists>!Q. Q \<in> Hgroup.subgroups_of_size q"
less_HMSet_iff_less_multiset_ext\<^sub>D\<^sub>M: fixes M :: "hmultiset multiset"   and N :: "hmultiset multiset" shows "(HMSet M < HMSet N) = less_multiset_ext\<^sub>D\<^sub>M (<) M N"
size_multiset_overloaded_eq: fixes x :: "'b multiset" shows "size x = sum (count x) (set_mset x)"
Rep_ell2_cases: fixes y :: "'c \<Rightarrow> complex" assumes "y \<in> {x. has_ell2_norm x}" obtains x :: "'c ell2" where "y = Rep_ell2 x"
tr2_def: shows "path M2 (io_init || tr2) (initial M2) \<and> length tr2 = length io_init"
hseq_leg\<^sub>0: shows "hseq (tab\<^sub>0 (cod \<tau>)) \<tau>.chine"
SC_def: fixes l :: "nat list" shows "SC l = Suc (hd0 l)"
Sup_MPLS_s_lt_if: fixes PROB :: "(('c, 'd) fmap \<times> ('c, 'd) fmap) set"   and s :: "('c, 'd) fmap"   and k :: "nat" assumes "\<forall>x\<in>MPLS_s PROB s. x < k" shows "Sup (MPLS_s PROB s) < k"
asymp_equiv_eventually_zeros: fixes f :: "'b \<Rightarrow> 'c"   and F :: "'b filter"   and g :: "'b \<Rightarrow> 'c" assumes "f \<sim>[F] g" shows "\<forall>\<^sub>F x in F. (f x = (0::'c)) = (g x = (0::'c))"
spec2: fixes P :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'c"   and y :: "'d" assumes "\<forall>x y. P x y" shows "P x y"
arcs_app_inv_iso_subgraph: fixes hom :: "('a, 'b, 'e, 'f) digraph_isomorphism"   and A :: "'b set" assumes "digraph_isomorphism hom"   and "A \<subseteq> arcs G" shows "iso_arcs (inv_iso hom) ` iso_arcs hom ` A = A"
qFreshAll_simps_1: fixes xs :: "'d"   and x :: "'e"   and ys :: "'d"   and y :: "'e" shows "qFresh xs x (qVar ys y) = (xs \<noteq> ys \<or> x \<noteq> y)"
bvars\<^sub>s\<^sub>t_split: fixes S :: "('b, 'a) strand_step list"   and S' :: "('b, 'a) strand_step list" shows "bvars\<^sub>s\<^sub>t (S @ S') = bvars\<^sub>s\<^sub>t S \<union> bvars\<^sub>s\<^sub>t S'"
list_rel_simp_1: fixes l' :: "'b list"   and R :: "('a \<times> 'b) set" shows "(([], l') \<in> \<langle>R\<rangle>list_rel) = (l' = [])"
ord_le_eq_trans: fixes a :: "'b"   and b :: "'b"   and c :: "'b" assumes "a \<le> b"   and "b = c" shows "a \<le> c"
n_bigger: shows "\<forall>nj\<in>nj_set. nj \<le> n"
length_I: shows "length (sorted_list_of_set I) = card I"
cEnum_finite_5_def: shows "CENUM(Enum.finite_5) = Some (enum_class.enum, enum_class.enum_all, enum_class.enum_ex)"
Un_absorb: fixes A :: "'d set" shows "A \<union> A = A"
nth_non_equal_first_eq: fixes x :: "'g"   and y :: "'g"   and xs :: "'g list"   and n :: "nat" assumes "x \<noteq> y" shows "((x # xs) ! n = y) = (xs ! (n - 1) = y \<and> 0 < n)"
hyps_1: shows "x \<in>\<^sub>\<circ> \<omega>"
step_le_x: fixes i :: "nat"   and x :: "real" assumes "i < k"   and "x\<^sub>1 \<le> x" shows "bs ! i * x + (hs ! i) x \<le> x"
cyclic_on_edge_succ: fixes x :: "'a" assumes "x \<in> verts H"   and "out_arcs H x \<noteq> {}" shows "cyclic_on (edge_succ HM) (out_arcs H x)"
convex_hull_linear_image: fixes f :: "'b \<Rightarrow> 'c"   and s :: "'b set" assumes "linear f" shows "f ` (convex hull s) = convex hull f ` s"
mult_smult_assoc_mat: fixes A :: "'c mat"   and nr :: "nat"   and n :: "nat"   and B :: "'c mat"   and nc :: "nat"   and k :: "'c" assumes "A \<in> carrier_mat nr n"   and "B \<in> carrier_mat n nc" shows "k \<cdot>\<^sub>m A * B = k \<cdot>\<^sub>m (A * B)"
scale_eq_0_iff: fixes a :: "complex"   and x :: "'b" shows "(a *\<^sub>C x = (0::'b)) = (a = 0 \<or> x = (0::'b))"
f_def: shows "f \<equiv> \<lambda>x. \<integral>\<^sup>+ t. (if g t = x then 1 else 0) * f' t \<partial>M"
assms: fixes n :: "nat" shows "u n \<in> space\<^sub>N N"
rev_unsimp: fixes xs :: "'a list"   and r :: "'a"   and rs :: "'a list" shows "rev xs @ r # rs = rev (r # xs) @ rs"
select_convs_3: fixes states :: "hf set"   and init :: "hf"   and final :: "hf set"   and nxt :: "hf \<Rightarrow> 'b \<Rightarrow> hf"   and more :: "'c" shows "dfa.final \<lparr>dfa.states = states, init = init, final = final, nxt = nxt, \<dots> = more\<rparr> = final"
group_by_drop_foldr: fixes x :: "nat"   and l :: "'b list"   and f :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "drop x l = foldr (@) (group_by f (drop x l)) []"
lens_override_put_right_in: fixes A :: "'e \<Longrightarrow> 'f"   and X :: "'g \<Longrightarrow> 'f"   and S\<^sub>1 :: "'f"   and S\<^sub>2 :: "'f"   and v :: "'g" assumes "vwb_lens A"   and "X \<subseteq>\<^sub>L A" shows "S\<^sub>1 \<triangleleft>\<^bsub>A\<^esub> put\<^bsub>X\<^esub> S\<^sub>2 v = put\<^bsub>X\<^esub> (S\<^sub>1 \<triangleleft>\<^bsub>A\<^esub> S\<^sub>2) v"
lc_approx_def: fixes prog :: "'a program_scheme" shows "lc_approx prog \<equiv> {\<lparr>command = c, state = \<lparr>local_state.variables = v\<rparr>\<rparr> |c v. c \<in> cfg_approx prog \<and> dom v \<subseteq> local_var_approx prog}"
simps_1: fixes n :: "lname" shows "assignsV (LVar n) = {}"
Implies_def: fixes P :: "('e, 'f, 'g) formula"   and Q :: "('e, 'f, 'g) formula" shows "(P \<rightarrow> Q) = (Q || ! P)"
index_hma_def: fixes A :: "(('d, 'e) Finite_Cartesian_Product.vec, 'f) Finite_Cartesian_Product.vec"   and i :: "'f"   and j :: "'e" shows "index_hma A i j \<equiv> A $h i $h j"
compare_pow_le_iff: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and p :: "nat"   and x :: "'a"   and y :: "'a" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one"   and "0 < p"   and "less_eq zero x"   and "less_eq zero y" shows "less_eq (power.power one times x p) (power.power one times y p) = less_eq x y"
symE: fixes r :: "('b \<times> 'b) set"   and b :: "'b"   and a :: "'b" assumes "sym r"   and "(b, a) \<in> r" obtains "(a, b) \<in> r"
wlp_loop_mono: fixes t :: "('s \<Rightarrow> real) \<Rightarrow> 's \<Rightarrow> real"   and u :: "('s \<Rightarrow> real) \<Rightarrow> 's \<Rightarrow> real" assumes "le_utrans t u"   and "\<And>P. unitary P \<Longrightarrow> unitary (t P)"   and "\<And>P. unitary P \<Longrightarrow> unitary (u P)" shows "le_utrans (wlp (body ;; Embed t \<^bsub>\<guillemotleft> G \<guillemotright>\<^esub>\<oplus> Skip)) (wlp (body ;; Embed u \<^bsub>\<guillemotleft> G \<guillemotright>\<^esub>\<oplus> Skip))"
simps_2: fixes f :: "nat \<Rightarrow> 'a list"   and n :: "nat" shows "??.Relative_Rewriting.choice f (Suc n) = (let (i, j) = ??.Relative_Rewriting.choice f n in if Suc j < length (f i) then (i, Suc j) else (Suc i, 0))"
induct: fixes P :: "'a \<Rightarrow> 'a \<times> nat \<Rightarrow> bool"   and a0 :: "'a"   and a1 :: "'a \<times> nat" assumes "\<And>p x acc. (\<not> (x = (0::'a) \<or> is_unit p \<or> \<not> p dvd x) \<Longrightarrow> P p (x div p, acc + 1)) \<Longrightarrow> P p (x, acc)" shows "P a0 a1"
divide_divide_eq_left: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "class.field divide inverse times one plus zero minus uminus" shows "divide (divide a b) c = divide a (times b c)"
adjoint: shows "dist_over_inf (F::'a::refinement_lattice \<Rightarrow> 'a::refinement_lattice)"
lnth_lmap: fixes n :: "nat"   and xs :: "'a llist"   and f :: "'a \<Rightarrow> 'b" assumes "enat n < llength xs" shows "lnth (lmap f xs) n = f (lnth xs n)"
fmdom'_notI: fixes m :: "'f \<rightharpoonup>\<^sub>f 'e"   and x :: "'f" assumes "fmlookup m x = None" shows "x \<notin> fmdom' m"
map_custom_empty_axioms: fixes maxsize :: "nat" shows "map_custom_empty (PR_CONST (op_hm_empty_sz maxsize))"
fBall_ffilter2: fixes X :: "'a fset"   and Y :: "'a fset"   and f :: "'a \<Rightarrow> bool" assumes "X = Y"   and "fBall X f" shows "ffilter f X = Y"
power_impl: fixes b :: "'a"   and a :: "'a"   and n :: "nat" assumes "b l\<rightarrow> a = a" shows "b ^ n l\<rightarrow> a = a"
proc_ord2a: fixes P :: "'a process"   and Q :: "'a process"   and s :: "'a event list"   and X :: "'a event set" assumes "P \<sqsubseteq> Q"   and "s \<notin> D P" shows "((s, X) \<in> F P) = ((s, X) \<in> F Q)"
Cachera_def: fixes n :: "nat" shows "Cachera n = (\<lambda>s0 s. True, \<lambda>s0 (ops, s, h) (k, v). |k| \<le> |h| + n, \<lambda>s0 (ops, s, h) k. |k| \<le> |h| + n)"
class_subcls1: fixes P :: "'b prog"   and y :: "char list"   and P' :: "'b prog"   and z :: "char list" assumes "class P y = class P' y"   and "P \<turnstile> y \<prec>\<^sup>1 z" shows "P' \<turnstile> y \<prec>\<^sup>1 z"
least_prefixpoint: fixes f :: "'a \<Rightarrow> 'a" assumes "has_least_prefixpoint f" shows "is_least_prefixpoint f (p\<mu> f)"
backward_finite_path_root_acyclic_terminating_aux: fixes r :: "'a"   and x :: "'a" assumes "backward_finite_path_root r x"   and "is_acyclic x" shows "x ; r = 0"
satisfies_T0: shows "is_left_adjoint (tab\<^sub>0 r)"
assms_2: fixes n :: "nat" shows "P (\<psi>\<^sup>D\<^bsub>n\<^esub>\<cdot>x) (\<psi>\<^sup>A\<^bsub>n\<^esub>\<cdot>y)"
Top_def: fixes S :: "alt set"   and L :: "(alt \<times> alt) set" shows "Top S L \<equiv> {(a, b). a <\<^bsub>L\<^esub> b \<and> (a \<in> S \<and> b \<in> S \<or> a \<notin> S \<and> b \<notin> S)} \<union> {(a, b). a \<notin> S \<and> b \<in> S}"
witness: shows "Min {h. h \<le> k \<and> P h} \<le> k \<and> P (Min {h. h \<le> k \<and> P h})"