hcju/isabelleps huggingface dataset:
iterate_atoms: fixes f :: "'a ltln \<Rightarrow> 'a ltln"   and \<phi> :: "'a ltln"   and n :: "nat" assumes "\<And>\<phi>. atoms_ltln (f \<phi>) \<subseteq> atoms_ltln \<phi>" shows "atoms_ltln (iterate f \<phi> n) \<subseteq> atoms_ltln \<phi>"
exec_call: fixes \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and p :: "'e"   and bdy :: "('d, 'e, 'f) com"   and init :: "'d \<Rightarrow> 'd"   and s :: "'d"   and t :: "'d"   and c :: "'d \<Rightarrow> 'd \<Rightarrow> ('d, 'e, 'f) com"   and return :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and u :: "('d, 'f) xstate" assumes "\<Gamma> p = Some bdy"   and "\<Gamma>\<turnstile> \<langle>bdy,Normal (init s)\<rangle> \<Rightarrow> Normal t"   and "\<Gamma>\<turnstile> \<langle>c s t,Normal (return s t)\<rangle> \<Rightarrow> u" shows "\<Gamma>\<turnstile> \<langle>call init p return c,Normal s\<rangle> \<Rightarrow> u"
ide_dom: fixes f :: "(bool, bool list) concrete_category.arr" assumes "J.arr f" shows "J.ide (J.dom f)"
hd_prog_append_Write\<^sub>s\<^sub>b: fixes p :: "'b"   and sb :: "'b memref list"   and volatile :: "bool"   and a :: "nat"   and sop :: "nat set \<times> ((nat \<Rightarrow> nat option) \<Rightarrow> nat)"   and v :: "nat"   and A :: "nat set"   and L :: "nat set"   and R :: "nat set"   and W :: "nat set" shows "hd_prog p (sb @ [Write\<^sub>s\<^sub>b volatile a sop v A L R W]) = hd_prog p sb"
stream_trans_trans: fixes xs :: "('c \<Rightarrow> 'a) stream"   and i :: "nat" assumes "stream_trans xs" shows "local.trans (xs !! i) (stl xs !! i)"
select_convs_2: fixes p_iiface :: "char list"   and p_oiface :: "char list"   and p_src :: "'a word"   and p_dst :: "'a word"   and p_proto :: "8 word"   and p_sport :: "16 word"   and p_dport :: "16 word"   and p_tcp_flags :: "tcp_flag set"   and p_payload :: "char list"   and more :: "'b" shows "p_oiface \<lparr>p_iiface = p_iiface, p_oiface = p_oiface, p_src = p_src, p_dst = p_dst, p_proto = p_proto, p_sport = p_sport, p_dport = p_dport, p_tcp_flags = p_tcp_flags, p_payload = p_payload, \<dots> = more\<rparr> = p_oiface"
swapAlts_def: fixes a :: "'c"   and b :: "'c"   and u :: "'c" shows "swapAlts a b u \<equiv> if u = a then b else if u = b then a else u"
g': shows "\<guillemotleft>\<sigma>.map\<^sub>0 a' \<star>\<^sub>D g \<star>\<^sub>D \<tau>\<^sub>0 a : F.map\<^sub>0 a \<rightarrow>\<^sub>D F.map\<^sub>0 a'\<guillemotright>"
Ex_less_Suc: fixes n :: "nat"   and P :: "nat \<Rightarrow> bool" shows "(\<exists>i<Suc n. P i) = (P n \<or> (\<exists>i<n. P i))"
leadsto_SF: fixes A :: "state \<times> state \<Rightarrow> bool"   and v :: "state \<Rightarrow> 'a" shows "\<turnstile> (Enabled (<A>_v) \<leadsto> <A>_v) \<longrightarrow> SF(A)_v"
simps_1: fixes p2 :: "'a list" shows "poly_mult [] p2 = []"
assms: shows "(s, None) \<in> terminates_on (maps_raw f g)"
chain_subset: fixes ord :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a set"   and B :: "'a set" assumes "Complete_Partial_Order.chain ord A"   and "B \<subseteq> A" shows "Complete_Partial_Order.chain ord B"
open_Diff: fixes "open" :: "'a set \<Rightarrow> bool"   and S :: "'a set"   and T :: "'a set" assumes "class.topological_space open"   and "open S"   and "topological_space.closed open T" shows "open (S - T)"
bst_insert: fixes t :: "'b tree"   and x :: "'b" assumes "bst t" shows "bst (Splay_Tree.insert x t)"
GE_parOf_def: shows "GE_parOf \<equiv> ProblemIkTpartG.GE_parOf parOf"
good_k: fixes x :: "'a"   and k :: "nat" assumes "x \<in> good_set" shows "(T ^^ k) x \<in> good_set \<and> f ((T ^^ k) x) = f x"
P_no_deadends: fixes n :: "nat" assumes "enat (Suc n) < llength P" shows "\<not> deadend (P $ n)"
conditionalize: fixes A :: "'a set"   and g :: "'a \<Rightarrow> 'b" assumes "finite A" shows "prod g A = (\<Prod>a. if a \<in> A then g a else (1::'b))"
h1: shows "(\<exists>a\<in>set (dnf (And \<phi>1 \<phi>2)). f a) = (\<exists>a\<in>set (dnf \<phi>1). \<exists>a'\<in>set (dnf \<phi>2). f a \<and> f a')"
iteration_sep: fixes y :: "'a"   and x :: "'a" assumes "y \<cdot> x \<le> x \<cdot> y" shows "(x + y)\<^sup>\<infinity> = x\<^sup>\<infinity> \<cdot> y\<^sup>\<infinity>"
addpre_r: fixes dsn :: "nat"   and dsk :: "k"   and fl :: "f"   and hops :: "nat"   and nhip :: "nat"   and pre :: "nat set"   and npre :: "nat set" shows "addpre (dsn, dsk, fl, hops, nhip, pre) npre = (dsn, dsk, fl, hops, nhip, pre \<union> npre)"
spr_sim_tFirst_tLast: fixes t :: "('a, 'ePubAct, 'es, 'pPubAct, 'ps) BEState Trace"   and s :: "('a, 'ePubAct, 'es, 'pPubAct, 'ps) SPRstate" assumes "spr_sim t = s"   and "t \<in> SPR.jkbpC" shows "(sprFst s, sprLst s) \<in> sprCRel s"
assms_2: fixes m :: "enat"   and n :: "enat" assumes "(m, n) \<in> X" shows "m = 0 \<and> n = 0 \<or> (\<exists>m' n'. m = eSuc m' \<and> n = eSuc n' \<and> ((m', n') \<in> X \<or> m' = n'))"
absorb1: fixes b :: "'a"   and a :: "'a" assumes "b \<le> a" shows "a + b = a"
gcd_poly_decompose: fixes p :: "'b poly"   and q :: "'b poly" shows "gcd p q = smult (gcd (content p) (content q)) (gcd (primitive_part p) (primitive_part q))"
heap_is_wellformed_children_distinct: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and children :: "(_) node_ptr list" assumes "heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M h"   and "h \<turnstile> Shadow_DOM.get_child_nodes ptr \<rightarrow>\<^sub>r children" shows "distinct children"
sumIdempRight: fixes Rel :: "(pi \<times> pi) set"   and P :: "pi" assumes "Id \<subseteq> Rel" shows "P \<oplus> P \<leadsto>[Rel] P"
eventually_until_eventually: fixes P :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and Q :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" shows "\<diamond>(P \<U> Q) = \<diamond>Q"
GSpan_def: shows "GSpan = R_scalar_mult.RSpan (aezfun_setspan G) fgsmult"
difference_is_product_of_complement: fixes x :: "i"   and y :: "i" assumes "\<exists>w. P w x \<and> \<not> O w y" shows "x \<ominus> y = x \<otimes> \<midarrow> y"
sum_cls_val_process: fixes pf1 :: "'c \<Rightarrow> nat \<Rightarrow> 'd \<Rightarrow> real"   and pf2 :: "'c \<Rightarrow> nat \<Rightarrow> 'd \<Rightarrow> real"   and Mkt :: "('d, 'c) discrete_market" assumes "portfolio pf1"   and "portfolio pf2" shows "\<forall>n w. cls_val_process Mkt (qty_sum pf1 pf2) n w = cls_val_process Mkt pf1 n w + cls_val_process Mkt pf2 n w"
disc_21: fixes x11 :: "'a"   and x12 :: "'b"   and x13 :: "'b" shows "\<not> isRecvMarker (Trans x11 x12 x13)"
integrable_cong_AE_imp: fixes M :: "'a measure"   and g :: "'a \<Rightarrow> 'b"   and f :: "'a \<Rightarrow> 'b" assumes "integrable M g"   and "f \<in> borel_measurable M"   and "AE x in M. g x = f x" shows "integrable M f"
evret_flow: shows "\<forall>\<^sub>F x in at x0 within S. returns_to p2 (flow0 x h)"
assms_2: shows "(I, J) = max_getIJ Q_pos Q_neg V"
smooth_on_compose2: fixes k :: "enat"   and T :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and S :: "'d set"   and g :: "'d \<Rightarrow> 'b"   and U :: "'d set" assumes "k-smooth_on T f"   and "k-smooth_on S g"   and "open U"   and "open T"   and "g ` U \<subseteq> T"   and "U \<subseteq> S" shows "k-smooth_on U (f \<circ> g)"
x_coord_img: shows "x_coord ` {0..1} = {- 1 / 2..1 / 2}"
order_iff_strict: fixes b :: "'k multiset"   and a :: "'k multiset" shows "(b \<subseteq># a) = (b \<subset># a \<or> a = b)"
that: fixes F :: "complex" assumes "(f has_laplace F) s" shows "thesis"
poly_binomial: fixes n :: "nat" shows "[:1::'a, 1::'a:] ^ n = (\<Sum>k\<le>n. monom (of_nat (n choose k)) k)"
gen_sa_impl_rel_eext_def_internal: fixes Rm :: "('g \<times> 'j) set"   and Rl :: "('f \<times> ('h \<Rightarrow> 'i)) set" shows "gen_sa_impl_rel_eext Rm Rl \<equiv> {(\<lparr>sai_L = Li, \<dots> = mi\<rparr>, \<lparr>sa_L = L, \<dots> = m\<rparr>) |Li mi L m. (Li, L) \<in> Rl \<and> (mi, m) \<in> Rm}"
set_integral_finite_Union: fixes I :: "'b set"   and A :: "'b \<Rightarrow> 'c set"   and M :: "'c measure"   and f :: "'c \<Rightarrow> 'd" assumes "finite I"   and "disjoint_family_on A I"   and "\<And>i. i \<in> I \<Longrightarrow> set_integrable M (A i) f"   and "\<And>i. i \<in> I \<Longrightarrow> A i \<in> sets M" shows "set_lebesgue_integral M (\<Union> (A ` I)) f = (\<Sum>i\<in>I. set_lebesgue_integral M (A i) f)"
mult_double_iso: fixes x :: "'a"   and y :: "'a"   and w :: "'a"   and z :: "'a" assumes "x \<le> y" shows "w ; x ; z \<le> w ; y ; z"
simplex_like_special_cosets: fixes X :: "'w set" assumes "X \<in> \<P>" shows "dual_order.simplex_like (dual_order.below_in \<P> X)"
allDefs_var_disjoint: fixes n :: "'a"   and g :: "('a, 'b, 'c) gen_cfg_wf"   and v :: "'b \<times> 'a \<times> Def"   and v' :: "'b \<times> 'a \<times> Def" assumes "n \<in> set (gen_wf_\<alpha>n g)"   and "v \<in> gen_wf.braun_ssa.allDefs g n"   and "v' \<in> gen_wf.braun_ssa.allDefs g n"   and "v \<noteq> v'" shows "gen_wf_var g v' \<noteq> gen_wf_var g v"
elims_3: fixes ty_ent :: "'a \<Rightarrow> PDDL_STRIPS_Semantics.type option"   and x :: "'a ast_effect" assumes "\<not> wf_effect ty_ent x" obtains a :: "'a atom formula list" and d :: "'a atom formula list" where "x = Effect a d" and "\<not> (Ball (set a) (wf_fmla_atom ty_ent) \<and> Ball (set d) (wf_fmla_atom ty_ent))"
hom_mult: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier G"   and "y \<in> carrier G" shows "h' (x \<otimes>\<^bsub>G\<^esub> y) = h' x \<otimes>\<^bsub>H\<^esub> h' y"
cnf_invert_singelton: fixes \<gamma> :: "'b \<Rightarrow> bool"   and a :: "'b negation_type" shows "cnf_to_bool \<gamma> [invert a] = (\<not> cnf_to_bool \<gamma> [a])"
quasi_isometric_def: fixes X :: "'b set"   and Y :: "'c set" shows "quasi_isometric X Y = (\<exists>lambda C f. lambda C -quasi_isometry_between X Y f)"
dom_tensor: fixes f :: "'c local.term set"   and g :: "'c local.term set" assumes "\<F>C.arr f"   and "\<F>C.arr g" shows "\<F>C.dom (f \<otimes> g) = \<F>C.dom f \<otimes> \<F>C.dom g"
moebius_rotation_preserve_cmod: fixes u :: "complex_homo"   and \<phi> :: "real" assumes "u \<noteq> \<infinity>\<^sub>h" shows "cmod (to_complex (moebius_pt (moebius_rotation \<phi>) u)) = cmod (to_complex u)"
typing_alt_simps_9: fixes \<Gamma> :: "String.literal \<rightharpoonup>\<^sub>f 'a OCL_Types.type"   and \<tau> :: "'a OCL_Types.type" shows "(\<Gamma> \<turnstile>\<^sub>E Literal (TupleLiteral []) : \<tau>) = (\<tau> = Tuple fmempty)"
set_help: fixes v :: "'c"   and as :: "'c list"   and bs :: "'c list" assumes "v \<in> set (as @ bs)" shows "v \<in> set as \<or> v \<in> set bs"
cases: fixes x :: "'c formula" obtains "x = \<bottom>"   | "x = \<^bold>\<not> \<bottom>"   | uu_ :: "'c" where "x = Atom uu_"   | uv_ :: "'c" where "x = \<^bold>\<not> (Atom uv_)"   | va :: "'c formula" where "x = \<^bold>\<not> (\<^bold>\<not> va__)"   | va :: "'c formula" and vb :: "'c formula" where "x = \<^bold>\<not> (vaa__ \<^bold>\<and> vb__)"   | va :: "'c formula" and vb :: "'c formula" where "x = \<^bold>\<not> (vab__ \<^bold>\<or> vba__)"   | va :: "'c formula" and vb :: "'c formula" where "x = \<^bold>\<not> (vac__ \<^bold>\<rightarrow> vb)"   | v :: "'c formula" and va :: "'c formula" where "x = vc__ \<^bold>\<and> vad__"   | v :: "'c formula" and va :: "'c formula" where "x = vd__ \<^bold>\<or> vae__"   | v :: "'c formula" and va :: "'c formula" where "x = v \<^bold>\<rightarrow> va"
sorted_wrt_linorder_imp_index_le: fixes A :: "'b set"   and R :: "('b \<times> 'b) set"   and xs :: "'b list"   and x :: "'b"   and y :: "'b" assumes "linorder_on A R"   and "set xs \<subseteq> A"   and "Linorder_Relations.sorted_wrt R xs"   and "x \<in> set xs"   and "y \<in> set xs"   and "(x, y) \<in> R" shows "index xs x \<le> index xs y"
assms: shows "<\<nu>x>a{x}.P \<longmapsto> b[y] \<prec> P'"
image_sub: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and B :: "'d set"   and A1 :: "'c set" assumes "f \<in> A \<rightarrow> B"   and "A1 \<subseteq> A" shows "f ` A1 \<subseteq> B"
sup_bot_left: fixes sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and bot :: "'a"   and x :: "'a" assumes "bounded_semilattice_sup_bot sup less_eq less bot" shows "sup bot x = x"
assms_3: shows "K \<subseteq>\<^sub>\<circ> I"
max_Var_floatarith_le_max_Var_floatariths_nth: fixes n :: "nat"   and fas :: "floatarith list" assumes "n < length fas" shows "max_Var_floatarith (fas ! n) \<le> max_Var_floatariths fas"
var_tys_def: fixes G :: "prog"   and oi :: "obj_tag"   and r :: "oref" shows "var_tys G oi r = (case r of Inl a \<Rightarrow> case oi of CInst C \<Rightarrow> fields_table G C (\<lambda>n f. \<not> static f) (+) Map.empty | Arr T k \<Rightarrow> Map.empty (+) arr_comps T k | Inr C \<Rightarrow> fields_table G C (\<lambda>fn f. declclassf fn = C \<and> static f) (+) Map.empty)"
tagged_fun_relD_none: fixes f :: "'e \<Rightarrow> 'f"   and f' :: "'g \<Rightarrow> 'h"   and A :: "('e \<times> 'g) set"   and B :: "('f \<times> 'h) set"   and x :: "'e"   and x' :: "'g" assumes "(f, f') \<in> A \<rightarrow> B"   and "(x, x') \<in> A" shows "(f x, f' x') \<in> B"
sep_conj_sep_impl: fixes P :: "'a \<Rightarrow> bool"   and s :: "'a"   and Q :: "'a \<Rightarrow> bool"   and R :: "'a \<Rightarrow> bool" assumes "P s"   and "\<And>s. (P ** Q) s \<Longrightarrow> R s" shows "(Q \<longrightarrow>\<^sup>* R) s"
parallelD2: fixes x :: "'b list"   and y :: "'b list" assumes "x \<parallel> y" shows "\<not> prefix y x"
p': shows "y\<^sub>1 [^] 2 = x\<^sub>1 [^] 3 \<oplus> a \<otimes> x\<^sub>1 \<oplus> b"
map_of_concat: fixes k :: "'a"   and l :: "('a \<times> 'b) list list" assumes "k \<in> dom (map_of (concat l))" shows "\<exists>i. k \<in> dom (map_of (l ! i)) \<and> i < length l"
divide_left: fixes f :: "'b \<Rightarrow> 'c"   and F :: "'b filter"   and g :: "'b \<Rightarrow> 'c"   and h :: "'b \<Rightarrow> 'c" assumes "\<forall>\<^sub>F x in F. f x \<noteq> (0::'c)"   and "\<forall>\<^sub>F x in F. g x \<noteq> (0::'c)"   and "g \<in> \<Theta>[F](f)" shows "(\<lambda>x. h x / f x) \<in> \<Theta>[F](\<lambda>x. h x / g x)"
X: fixes v :: "'a"   and x :: "'a set" assumes "v \<notin> proj_verts_H ` x" shows "v = w \<or> (\<exists>y. v = proj_verts_H y \<and> y \<notin> x)"
le_cases3: fixes x :: "'b word"   and y :: "'b word"   and z :: "'b word" obtains "x \<le>s y" and "y \<le>s z"   | "y \<le>s x" and "x \<le>s z"   | "x \<le>s z" and "z \<le>s y"   | "z \<le>s y" and "y \<le>s x"   | "y \<le>s z" and "z \<le>s x"   | "z \<le>s x" and "x \<le>s y"
I'_def: shows "I' \<equiv> \<lambda>x. if x \<notin> M then I x else Const (intv_const (I x) + 1)"
all_after_eq_6: fixes P :: "'b \<Rightarrow> 'a \<Rightarrow> bool" shows "(\<forall>x. P x)$ \<equiv> \<forall>x. P x$"
emeasure_space: fixes M :: "'c measure"   and A :: "'c set" shows "emeasure M A \<le> emeasure M (space M)"
rep_eq: fixes x :: "'k word" shows "the_signed_int x = signed_take_bit (LENGTH('k) - Suc 0) (Word.rep x)"
complex_of_real_of: shows "complex_of \<circ> real_of = id"
lateEarlyTauChain: fixes P :: "pi"   and Q :: "pi" assumes "P \<Longrightarrow>\<^sub>\<tau>\<^sub>l Q" shows "(P, Q) \<in> {(P, P') |P P'. P \<longmapsto>\<^sub>e \<tau> \<prec>\<^sub>e P'}\<^sup>*"
perm_bottom: fixes \<pi> :: "perm" shows "\<pi> \<bullet> \<bottom> = \<bottom>"
natUnion_def: fixes f :: "nat \<Rightarrow> 'b set" shows "natUnion f = \<Union> {f n |n. True}"
in_red_subsets: shows "List.member (reduction_subsets_R p qs signs subsets (M_mat_R signs subsets)) (a, b)"
psubst_Op_simp: fixes inp :: "'d \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm set option"   and binp :: "'e \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qAbs set option"   and rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm set option"   and delta :: "'f" assumes "goodInp inp"   and "goodBinp binp"   and "goodEnv rho" shows "Op delta inp binp #[rho] = Op delta inp %[rho] binp %%[rho]"
mat_of_col_dim_3: fixes v :: "'b vec" shows "mat_of_col v \<in> carrier_mat (dim_vec v) 1"
gen_span_def: shows "gen_span \<equiv> \<lambda>G gens. {x. gen_spanp G (\<lambda>x. x \<in> gens) x}"
sf_c_sum_plus_c_fst: fixes u :: "nat" shows "sf (c_sum u) + c_fst u = u"
sel_1: fixes x1 :: "'a"   and x2 :: "'b"   and x3 :: "nat"   and x4 :: "('a, 'b) BinomialHeapStruc.BinomialTree list" shows "BinomialHeapStruc.val (BinomialHeapStruc.Node x1 x2 x3 x4) = x1"
map_lift_spmf: fixes f :: "'k \<Rightarrow> 'h"   and g :: "'l \<Rightarrow> 'i"   and p :: "'k spmf" shows "map_gpv f g (lift_spmf p) = lift_spmf (map_spmf f p)"
distinct_119: fixes x10 :: "nat" shows "DBtt \<noteq> DBNum x10"
n0': fixes n :: "nat" assumes "n0' \<le> n" shows "norm (f n) \<le> c * norm (g n)"
ctr_1: fixes x :: "('f, 'g, 'h) Resumption.resumption"   and f :: "'f \<Rightarrow> ('i, 'g, 'h) Resumption.resumption" assumes "Resumption.resumption.is_Done x"   and "Resumption.resumption.result x \<noteq> None \<longrightarrow> Resumption.resumption.is_Done (f (the (Resumption.resumption.result x)))" shows "x \<bind> f = Resumption.resumption.Done (Resumption.resumption.result x \<bind> Resumption.resumption.result \<circ> f)"
Keys_init_syzygy_list: fixes bs :: "('t \<Rightarrow>\<^sub>0 'c) list" shows "Keys (set (init_syzygy_list bs)) = map_component (\<lambda>k. k + length bs) ` Keys (set bs) \<union> (\<lambda>i. term_of_pair (0::'a, i)) ` {0..<length bs}"
hoareQ_sound: fixes P :: "(char list \<Rightarrow> int) \<Rightarrow> enat"   and c :: "com"   and Q :: "(char list \<Rightarrow> int) \<Rightarrow> enat" assumes "\<turnstile>\<^sub>2\<^sub>' {P} c {Q}" shows "\<Turnstile>\<^sub>2\<^sub>' {P} c {Q}"
ord_eq_le_trans: fixes a :: "'b"   and b :: "'b"   and c :: "'b" assumes "a = b"   and "b \<le> c" shows "a \<le> c"
StepConf_def: fixes A :: "('d, 'e, 'f) hierauto"   and C :: "'d set"   and TS :: "('d \<times> (('d, 'e) expr \<times> ('f data \<Rightarrow> bool) \<times> 'e set \<times> 'f pupdate) \<times> 'd) set" shows "StepConf A C TS \<equiv> C - ChiStar A `` Source TS \<union> Target TS \<union> ChiRel A `` Target TS \<inter> HAInitStates A \<union> (Restr (ChiRel A) (HAInitStates A))\<^sup>+ `` (ChiRel A `` Target TS \<inter> HAInitStates A)"
impl_refines_spec': shows "COPY' \<sqsubseteq> SYSTEM'"
step: shows "(is, \<theta>, sb, m, \<D>, \<O>, \<R>, \<S>) \<rightarrow>\<^sub>s\<^sub>b\<^sub>h (is', \<theta>', sb', m', \<D>', \<O>', \<R>', \<S>')"
idiff_eint_eint: fixes a :: "int"   and b :: "int" shows "eint a - eint b = eint (a - b)"
chamber_system_into: fixes X :: "'b set set"   and f :: "'b \<Rightarrow> 'b" assumes "ChamberComplexFolding X f" shows "order.greater_eq (ChamberComplex.chamber_system X) (f \<turnstile> ChamberComplex.chamber_system X)"
cut_less_finite: fixes I :: "'a set"   and t :: "'a" assumes "finite I" shows "finite (I \<down>< t)"
is_ground_cls_def: fixes subst_atm :: "'b \<Rightarrow> 'c \<Rightarrow> 'b"   and C :: "'b literal multiset" shows "substitution_ops.is_ground_cls subst_atm C = (\<forall>L. L \<in># C \<longrightarrow> substitution_ops.is_ground_lit subst_atm L)"
continuous_on_const: fixes s :: "'c set"   and c :: "'d" shows "continuous_on s (\<lambda>x. c)"
simps_2: fixes Pi :: "(('c, 'd) fmap \<times> ('c, 'd) fmap) set"   and s :: "('c, 'd) fmap" shows "valid_path Pi [s] = (s \<in> valid_states Pi)"
de_Morgan_disj: fixes x :: "'b"   and y :: "'b" shows "NOT (x OR y) = NOT x AND NOT y"
Abs_eq_iff_3: fixes cs :: "atom list"   and x :: "'a"   and cs' :: "atom list"   and y :: "'a" shows "([cs]lst. x = [cs']lst. y) = (\<exists>p. (cs, x) \<approx>lst (=) supp p (cs', y))"
bex_empty: fixes P :: "'a \<Rightarrow> bool" shows "Bex {} P = False"
freshFrame: fixes F :: "'e frame"   and C :: "'f" obtains A\<^sub>F :: "name list" and \<Psi>\<^sub>F :: "'e" where "F = \<langle>A\<^sub>F, \<Psi>\<^sub>F\<rangle>" and "distinct A\<^sub>F" and "A\<^sub>F \<sharp>* C"
permute_Tree_tConj: fixes p :: "perm"   and tset :: "('e, 'f, 'g) Tree set['e]" shows "p \<bullet> tConj tset = tConj (p \<bullet> tset)"
simps_3: fixes c1 :: "com"   and c2 :: "com" shows "D (c1;; c2) = D c1 O D c2"
ordinal_inc_self: fixes S :: "'c Order" assumes "Worder S" shows "S \<in> ordinal_number S"
exexH_k_dim: fixes k :: "nat" assumes "k < n" shows "exexH_k k \<in> carrier_mat d d"
hyps: fixes w :: "'a stream"   and x :: "'a stream" assumes "\<And>i. Suc i \<le> n \<Longrightarrow> w !! i = x !! i" shows "stake n w = stake n x"
fset_le_singleton_iff: fixes X :: "'a fset"   and x :: "'a" shows "(fset X \<subseteq> {x}) = (X = {||} \<or> X = {|x|})"
simps: fixes a :: "'a \<Rightarrow> real"   and X :: "'a set"   and I :: "'a \<Rightarrow> Regions.intv"   and r :: "('a \<times> 'a) set" shows "(a \<in> Regions.region X I r) = (\<exists>u X\<^sub>0. a = u \<and> (\<forall>x\<in>X. 0 \<le> u x) \<and> (\<forall>x\<in>X. Regions.intv_elem x u (I x)) \<and> X\<^sub>0 = {x \<in> X. \<exists>d. I x = Regions.intv.Intv d} \<and> (\<forall>x\<in>X\<^sub>0. \<forall>y\<in>X\<^sub>0. ((x, y) \<in> r) = (frac (u x) \<le> frac (u y))))"
cet_if: fixes \<Gamma> :: "nat \<Rightarrow> pdf_type"   and b :: "cexpr"   and e1 :: "cexpr"   and t :: "pdf_type"   and e2 :: "cexpr" assumes "\<Gamma> \<turnstile>\<^sub>c b : BOOL"   and "\<Gamma> \<turnstile>\<^sub>c e1 : t"   and "\<Gamma> \<turnstile>\<^sub>c e2 : t" shows "\<Gamma> \<turnstile>\<^sub>c IF\<^sub>c b THEN e1 ELSE e2 : t"
consec_intermediate1: fixes j :: "nat_int"   and k :: "nat_int"   and i :: "nat_int" shows "consec j k \<and> consec i (j \<squnion> k) \<longrightarrow> consec i j"
intros: fixes s :: "'b + 'c"   and x :: "'b" assumes "s = Inl x" shows "setlp s x"
trnl\<^sub>\<epsilon>_eq: fixes u :: "'a"   and v :: "'a"   and \<nu> :: "'a" assumes "ide u"   and "ide v"   and "src f = trg v"   and "src g = trg u"   and "\<nu> \<in> hom v (g \<star> u)" shows "trnl\<^sub>\<epsilon> u \<nu> = (\<epsilon> \<star> u) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, u] \<cdot> (f \<star> \<nu>)"
T_vrestr_composed_2: fixes M :: "'b measure"   and T :: "'b \<Rightarrow> 'b"   and A :: "'b set"   and m :: "nat" assumes "qmpt M T"   and "A \<in> sets M" shows "qmpt.vimage_restr M T (qmpt.vimage_restr M (T ^^ m) A) = qmpt.vimage_restr M (T ^^ (m + 1)) A"
abs_eq_iff: fixes x :: "'b \<times> 'c"   and y :: "'b \<times> 'c" shows "(abs_residual x = abs_residual y) = alpha_residual x y"
validE_1: fixes X :: "tyvrs"   and T :: "ty"   and \<Gamma> :: "binding list" assumes "\<turnstile> (TVarB X T # \<Gamma>) ok" obtains "\<turnstile> \<Gamma> ok" and "X \<sharp> ty_dom \<Gamma>" and "T closed_in \<Gamma>"
span: shows "span {x. i \<bullet> x = 0} = {x. i \<bullet> x = 0}"
simps: fixes \<Gamma> :: "('d, 'e) \<I>"   and a :: "('f, 'd, 'e) gpv" shows "\<Gamma> \<turnstile>g a \<surd> = (\<exists>gpv. a = GPV gpv \<and> (\<forall>x xa. IO x xa \<in> set_spmf gpv \<longrightarrow> x \<in> outs_\<I> \<Gamma> \<and> (\<forall>input\<in>responses_\<I> \<Gamma> x. \<Gamma> \<turnstile>g xa input \<surd>)))"
concat_bit_assoc_sym: fixes m :: "nat"   and n :: "nat"   and k :: "int"   and l :: "int"   and r :: "int" shows "concat_bit m (concat_bit n k l) r = concat_bit (min m n) k (concat_bit (m - n) l r)"
degree_le: fixes n :: "nat"   and p :: "'a Polynomial.poly" assumes "\<forall>i>n. poly.coeff p i = (0::'a)" shows "Polynomial.degree p \<le> n"
fixed_point_proof_def: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'b" shows "fixed_point_proof A f \<equiv> f ` A \<subseteq> A"
subset_Diff_insert: fixes A :: "'a set"   and B :: "'a set"   and x :: "'a"   and C :: "'a set" shows "(A \<subseteq> B - insert x C) = (A \<subseteq> B - C \<and> x \<notin> A)"
dependent_eq_real2: shows "module_on.dependent (*\<^sub>R) = dependent"
univ_poly_units': fixes R :: "('c, 'd) ring_scheme"   and K :: "'c set"   and p :: "'c list" assumes "domain R"   and "subfield K R" shows "(p \<in> Units (K [X]\<^bsub>R\<^esub>)) = (p \<in> carrier (K [X]\<^bsub>R\<^esub>) \<and> p \<noteq> [] \<and> degree p = 0)"
red_sp_imp_black_sp: fixes prb :: "('d, 'e, 'f) pre_RedBlack"   and rv1 :: "'d \<times> nat"   and res :: "('d \<times> nat) edge list"   and rv2 :: "'d \<times> nat" assumes "RedBlack prb"   and "SubRel.subpath (red prb) rv1 res rv2 (subs prb)" shows "Graph.subpath (black prb) (fst rv1) (ui_es res) (fst rv2)"
trnr\<^sub>\<eta>_def: fixes v :: "'a"   and \<mu> :: "'a" shows "trnr\<^sub>\<eta> v \<mu> \<equiv> (\<mu> \<star> f) \<cdot> \<a>\<^sup>-\<^sup>1[v, g, f] \<cdot> (v \<star> \<eta>) \<cdot> \<r>\<^sup>-\<^sup>1[v]"
intro: fixes \<G> :: "'b cyclic_group" assumes "list_cdh \<G>" shows "hashed_elgamal \<G>"