hcju/isabelleps huggingface dataset:
rel_mset_Plus: fixes R :: "'b \<Rightarrow> 'e \<Rightarrow> bool"   and a :: "'b"   and b :: "'e"   and M :: "'b multiset"   and N :: "'e multiset" assumes "R a b"   and "rel_mset R M N" shows "rel_mset R (add_mset a M) (add_mset b N)"
simps_1: fixes e :: "(name, hf) finfun" shows "\<lbrakk>Zero\<rbrakk>e = 0"
singleton_rev_conv: fixes x :: "'c"   and xs :: "'c list" shows "([x] = rev xs) = (xs = [x])"
minus: fixes f :: "'a \<Rightarrow> real" assumes "f \<in> R" shows "(\<lambda>x. - f x) \<in> R"
simps_1: fixes S :: "'b fm set"   and C :: "'b fm set set"   and f :: "nat \<Rightarrow> 'b fm" shows "extend S C f 0 = S"
simply_connected_imp_connected: fixes S :: "'a set" assumes "simply_connected S" shows "connected S"
mem_Collect_eq: fixes a :: "'f"   and P :: "'f \<Rightarrow> bool" shows "(a \<in> Collect P) = P a"
interval_lebesgue_integral_mult_left: fixes M :: "real measure"   and a :: "ereal"   and b :: "ereal"   and f :: "real \<Rightarrow> 'a"   and c :: "'a" shows "LINT x=a..b|M. (f x * c) = interval_lebesgue_integral M a b f * c"
prec_frame_expl: fixes P :: "'a \<Rightarrow> assn"   and F1 :: "assn"   and F2 :: "assn"   and h :: "heap \<times> nat set"   and R1 :: "assn"   and R2 :: "assn"   and x :: "'a"   and y :: "'a" assumes "\<forall>x y. h \<Turnstile> P x * F1 \<and>\<^sub>A P y * F2 \<longrightarrow> x = y"   and "h \<Turnstile> R1 \<and>\<^sub>A R2"   and "R1 \<Longrightarrow>\<^sub>A P x * F1"   and "R2 \<Longrightarrow>\<^sub>A P y * F2" shows "x = y"
partial_saturation_is_preserved: fixes S :: "'at Literal set set"   and E1 :: "'at"   and E2 :: "'at"   and R :: "'at Literal set set" assumes "partial_saturation S E1 S"   and "partial_saturation S E2 (S \<union> R)"   and "all_fulfill (\<lambda>x. \<not> tautology x) S"   and "all_fulfill (in_all_resolvents_upon S E2) R" shows "partial_saturation (S \<union> R) E1 (S \<union> R)"
params: shows "num_params (tm_poly tg) \<le> length I"
ltl_semantics_min_dnf: fixes w :: "nat \<Rightarrow> 'a set"   and \<phi> :: "'a ltln" shows "w \<Turnstile>\<^sub>n \<phi> = (\<exists>C\<in>min_dnf \<phi>. \<forall>\<psi>. \<psi> |\<in>| C \<longrightarrow> w \<Turnstile>\<^sub>n \<psi>)"
wf_prog_dyn_override_prop: fixes G :: "prog"   and new :: "qtname \<times> mdecl"   and old :: "qtname \<times> mdecl" assumes "G \<turnstile> new overrides old"   and "wf_prog G" shows "accmodi old \<le> accmodi new"
B: shows "path_connected {x. \<exists>i\<in>Basis. a \<bullet> i < x \<bullet> i}"
maxim: shows "\<forall>i<n. strictly_maximal_wrt (As ! i) (Cs ! i)"
case_1: fixes f1 :: "nat \<Rightarrow> 'b \<Rightarrow> 'a"   and f2 :: "nat \<Rightarrow> 'b \<Rightarrow> 'a"   and x11 :: "nat"   and x12 :: "'b" shows "(case Leq x11 x12 of Leq x xa \<Rightarrow> f1 x xa | Geq x xa \<Rightarrow> f2 x xa) = f1 x11 x12"
map\<^sub>0_def: fixes a :: "'a" shows "UP.map\<^sub>0 a \<equiv> src (UP a)"
Un_upper1: fixes A :: "'b set"   and B :: "'b set" shows "A \<subseteq> A \<union> B"
Upper_is_closed: fixes L :: "('c, 'd) gorder_scheme"   and A :: "'c set" assumes "weak_partial_order L"   and "A \<subseteq> carrier L" shows "is_closed\<^bsub>L\<^esub> (Upper L A)"
singleton_ne_empty_zmset: fixes x :: "'a" shows "{#x#}\<^sub>z \<noteq> {#}\<^sub>z"
vsubset_vdoubleton_leftI: fixes a :: "V"   and A :: "V"   and b :: "V" assumes "a \<in>\<^sub>\<circ> A"   and "b \<in>\<^sub>\<circ> A" shows "set {a, b} \<subseteq>\<^sub>\<circ> A"
I: fixes ps :: "'a list"   and c :: "('b, 'c) sequent"   and M :: "'c"   and Ms :: "('b, 'c) form list" assumes "ps \<noteq> []"   and "ModalSequents.mset c = \<LM> Modal M Ms  \<RM>" shows "(ps, c) \<in> modRules2"
set_concat_lists: fixes xs :: "'c list list" shows "set (concat_lists xs) = {as. length as = length xs \<and> (\<forall>i<length xs. as ! i \<in> set (xs ! i))}"
algebraic_int_root: fixes y :: "'a"   and p :: "'a poly"   and x :: "'a" assumes "algebraic_int y"   and "poly p x = y"   and "\<forall>i. coeff p i \<in> \<int>"   and "monic p"   and "0 < degree p" shows "algebraic_int x"
supportE_preT: fixes C :: "acom"   and e :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> nat" shows "supportE (\<lambda>l. preT C (e l)) \<subseteq> supportE e"
logical_instr_result: fixes i :: "sparc_operation \<times> inst_operand list"   and s :: "'b sparc_state" assumes "fst i \<in> {logic_type ANDs, logic_type ANDcc, logic_type ANDN, logic_type ANDNcc, logic_type ORs, logic_type ORcc, logic_type ORN, logic_type XORs, logic_type XNOR}" shows "snd (logical_instr i s) = False"
reg_widen_Err: fixes P :: "'a prog"   and n :: "nat"   and LT :: "ty err list" shows "P \<turnstile> replicate n Err [\<le>\<^sub>\<top>] LT = (LT = replicate n Err)"
induct: fixes P :: "'b channel process \<Rightarrow> 'b channel process \<Rightarrow> 'b channel process \<Rightarrow> bool" assumes "adm (\<lambda>(COPY', SEND', REC'). P COPY' SEND' REC')"   and "P \<bottom> \<bottom> \<bottom>"   and "\<And>a aa b. P a aa b \<Longrightarrow> P (left`?`x \<rightarrow> right`!`x \<rightarrow> a) (left`?`x \<rightarrow> mid`!`x \<rightarrow> ack \<rightarrow> aa) (mid`?`x \<rightarrow> right`!`x \<rightarrow> ack \<rightarrow> b)" shows "P COPY' SEND' REC'"
columns_def: fixes A :: "(('b, 'd) vec, 'c) vec" shows "columns A = {column i A |i. i \<in> UNIV}"
closure_openin_Int_closure: fixes U :: "'a set"   and S :: "'a set"   and T :: "'a set" assumes "openin (top_of_set U) S"   and "T \<subseteq> U" shows "closure (S \<inter> closure T) = closure (S \<inter> T)"
invrules_3: shows "B\<^sub>1 \<notin> P\<^sub>1 \<union> P\<^sub>2 \<union> wrap B\<^sub>2"
add_nonneg_eq_0_iff: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'d"   and y :: "'d" assumes "class.ordered_comm_monoid_add plus zero less_eq less"   and "less_eq zero x"   and "less_eq zero y" shows "(plus x y = zero) = (x = zero \<and> y = zero)"
igSubstInp_def: fixes MOD :: "('f, 'g, 'h, 'i, 'j, 'k, 'e, 'l, 'm) model_scheme"   and ys :: "'h"   and Y :: "'e"   and y :: "'k"   and inp :: "'d \<Rightarrow> 'e option" shows "igSubstInp MOD ys Y y inp \<equiv> lift (igSubst MOD ys Y y) inp"
Ex1_def: shows "CofGroups.Ex1 \<equiv> {x. Ex1p x}"
dim_mono: fixes V :: "('a, 'b) vec set"   and W :: "('a, 'b) vec set" assumes "V \<subseteq> vec.span W" shows "vec.dim V \<le> vec.dim W"
proj2_abs_abs_mult: fixes v :: "(real, 3) vec"   and w :: "(real, 3) vec" assumes "proj2_abs v = proj2_abs w"   and "w \<noteq> 0" shows "\<exists>c. v = c *\<^sub>R w"
sorted_list_of_set_lessThan_Suc: fixes k :: "nat" shows "list_of {..<Suc k} = list_of {..<k} @ [k]"
UP_zero_closed: shows "\<zero>\<^bsub>UP R\<^esub> \<in> carrier (UP R)"
replay''_sound: fixes thy :: "((String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)) \<times> Core.term set"   and res :: "Core.term" assumes "wf_theory thy"   and "\<And>x. x \<in> set [] \<Longrightarrow> term_ok thy x"   and "\<And>x. x \<in> set [] \<Longrightarrow> typ_of x = Some propT"   and "finite (fst ` fv_Proof P)"   and "fst ` FV (set []) \<subseteq> fst ` fv_Proof P"   and "fst ` fv_Proof P \<subseteq> fst ` fv_Proof P"   and "fst ` set [] \<subseteq> fst ` fv_Proof P"   and "replay'' thy [] (fst ` fv_Proof P) [] P = Some res" shows "thy,set [] \<turnstile> res"
rel: fixes X :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set"   and Y :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set" assumes "good X"   and "(X, Y) \<in> rel" shows "good Y \<and> skel Y = skel X"
space_C: fixes x :: "nat \<Rightarrow> 'a"   and n :: "nat"   and m :: "nat" assumes "x \<in> space (Pi\<^sub>M {0..<n} M)" shows "space (C n m x) = space (Pi\<^sub>M {0..<n + m} M)"
ldm: fixes M :: "('a \<times> 'a) list list" assumes "M \<in> set (all_maps_list G)" shows "lists_digraph_map G M"
assms_6: shows "free_vars e2 \<subseteq> V \<union> V'"
refl: fixes A :: "('d \<times> ('e, real) cconstraint \<times> 'f \<times> 'e list \<times> 'd) set \<times> ('d \<Rightarrow> ('e, real) cconstraint)"   and l :: "'d"   and Z :: "nat \<Rightarrow> nat \<Rightarrow> real DBMEntry"   and k :: "nat \<Rightarrow> nat"   and v :: "'e \<Rightarrow> nat"   and n :: "nat" shows "A \<turnstile> \<langle>l, Z\<rangle> \<leadsto>\<^bsub>k,v,n\<^esub>* \<langle>l, Z\<rangle>"
projection_def: fixes l :: "'a list"   and E :: "'a set" shows "l \<upharpoonleft> E \<equiv> filter (\<lambda>x. x \<in> E) l"
ide_in_hom: fixes a :: "'a \<times> 'a" shows "VV.ide a = VV.in_hom a a a"
absorb_iff1: fixes b :: "'a"   and a :: "'a" shows "(b \<le> a) = (max a b = a)"
f_def: fixes x :: "bool" shows "f x = pmf_of_set {True, False}"
length_merge: fixes xs :: "'b list"   and ys :: "'b list" shows "length (merge xs ys) = length xs + length ys"
vD_Some: fixes dip :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option" assumes "dip \<in> vD rt" shows "\<exists>dsn dsk hops nhip. rt dip = Some (dsn, dsk, val, hops, nhip)"
finite_set_of_finite_maps: fixes A :: "'a set"   and B :: "'b set" assumes "finite A"   and "finite B" shows "finite {m. dom m = A \<and> ran m \<subseteq> B}"
push_down_oldest_token_token_run: fixes q :: "'b"   and n :: "nat"   and x :: "nat" assumes "oldest_token q n = Some x" shows "token_run x n = q"
closed_translation_subtract: fixes S :: "'c set"   and a :: "'c" assumes "closed S" shows "closed ((\<lambda>x. x - a) ` S)"
system_U_implies_vNM_utility: fixes outcomes :: "'a set"   and \<R> :: "('a pmf \<times> 'a pmf) set" assumes "finite outcomes"   and "outcomes \<noteq> {}"   and "rational_preference (\<P> outcomes) \<R>"   and "independent_vnm (\<P> outcomes) \<R>"   and "continuous_vnm (\<P> outcomes) \<R>" shows "\<exists>u. ordinal_utility (\<P> outcomes) \<R> (\<lambda>x. measure_pmf.expectation x u)"
props: fixes v :: "'var"   and t :: "('fun, 'var) Term.term" assumes "\<I> v = t" shows "\<Gamma> (Var v) = \<Gamma> t \<and> public_ground_wf_term t"
cblinfun_euclidean_eqI: fixes x :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'b"   and y :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'b" assumes "\<And>i. i \<in> CBasis \<Longrightarrow> cblinfun_apply x i = cblinfun_apply y i" shows "x = y"
lNactLe_nact: fixes n' :: "nat"   and n :: "nat"   and c :: "'id"   and t :: "nat \<Rightarrow> cnf" assumes "n' = n \<or> n' < n \<and> \<not> (\<exists>k\<ge>n'. latestAct_cond c t n k)" shows "\<langle>c \<Leftarrow> t\<rangle>\<^bsub>n\<^esub> \<le> n'"
index_def: fixes xs :: "'b list" shows "index xs = (\<lambda>a. find_index (\<lambda>x. x = a) xs)"
converse_rtranclp_induct2: fixes r :: "'d \<times> 'e \<Rightarrow> 'd \<times> 'e \<Rightarrow> bool"   and ax :: "'d"   and ay :: "'e"   and bx :: "'d"   and "by" :: "'e"   and P :: "'d \<Rightarrow> 'e \<Rightarrow> bool" assumes "r\<^sup>*\<^sup>* (ax, ay) (bx, by)"   and "P bx by"   and "\<And>a b aa ba. \<lbrakk>r (a, b) (aa, ba); r\<^sup>*\<^sup>* (aa, ba) (bx, by); P aa ba\<rbrakk> \<Longrightarrow> P a b" shows "P ax ay"
A1: shows "P \<R>\<mapsto><Rel\<inverse>,Cal> Q"
meet_assoc: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<in> carrier L"   and "y \<in> carrier L"   and "z \<in> carrier L" shows "x \<sqinter> y \<sqinter> z = x \<sqinter> (y \<sqinter> z)"
Union_upper: fixes B :: "'b set"   and A :: "'b set set" assumes "(B::'b set) \<in> (A::'b set set)" shows "(B::'b set) \<subseteq> \<Union> (A::'b set set)"
filter_comm: fixes P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool"   and xs :: "'a list" shows "filter (\<lambda>x. P x \<and> Q x) xs = filter (\<lambda>x. Q x \<and> P x) xs"
has_lock_has_locks_conv: fixes l :: "('d \<times> nat) option"   and t :: "'d" shows "has_lock l t = (\<exists>n. has_locks l t = Suc n)"
heap_WHILET_rule'': fixes R :: "('a \<times> 'a) set"   and P :: "assn"   and I :: "'a \<Rightarrow> assn"   and s :: "'a"   and bi :: "'a \<Rightarrow> bool Heap"   and b :: "'a \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'a Heap"   and Q :: "'a \<Rightarrow> assn" assumes "wf R"   and "P \<Longrightarrow>\<^sub>A I s"   and "\<And>s. <I s * true> bi s <\<lambda>r. I s * \<up> (r = b s)>\<^sub>t"   and "\<And>s. b s \<Longrightarrow> <I s * true> f s <\<lambda>r. I r * \<up> ((r, s) \<in> R)>\<^sub>t"   and "\<And>s. \<not> b s \<Longrightarrow> I s \<Longrightarrow>\<^sub>A Q s" shows "<P> heap_WHILET bi f s <Q>\<^sub>t"
vrange_iff_vdomain: fixes b :: "V"   and r :: "V" shows "(b \<in>\<^sub>\<circ> \<R>\<^sub>\<circ> r) = (\<exists>a\<in>elts (\<D>\<^sub>\<circ> r). \<langle>a, b\<rangle> \<in>\<^sub>\<circ> r)"
\<phi>_biuniversal_prop_2: fixes u :: "'a"   and v :: "'a"   and w :: "'a"   and w' :: "'a"   and \<theta> :: "'a"   and \<theta>' :: "'a"   and \<beta> :: "'a" assumes "ide u"   and "ide v"   and "ide w"   and "ide w'"   and "\<guillemotleft>\<theta> : p\<^sub>0 \<star> w \<Rightarrow> u\<guillemotright>"   and "\<guillemotleft>\<theta>' : p\<^sub>0 \<star> w' \<Rightarrow> u\<guillemotright>"   and "\<guillemotleft>\<beta> : p\<^sub>1 \<star> w \<Rightarrow> p\<^sub>1 \<star> w'\<guillemotright>"   and "(h \<star> \<theta>) \<cdot> \<a>[h, p\<^sub>0, w] \<cdot> (\<phi> \<star> w) \<cdot> \<a>\<^sup>-\<^sup>1[k, p\<^sub>1, w] = (h \<star> \<theta>') \<cdot> \<a>[h, p\<^sub>0, w'] \<cdot> (\<phi> \<star> w') \<cdot> \<a>\<^sup>-\<^sup>1[k, p\<^sub>1, w'] \<cdot> (k \<star> \<beta>)" shows "\<exists>!\<gamma>. \<guillemotleft>\<gamma> : w \<Rightarrow> w'\<guillemotright> \<and> \<theta> = \<theta>' \<cdot> (p\<^sub>0 \<star> \<gamma>) \<and> p\<^sub>1 \<star> \<gamma> = \<beta>"
r: shows "\<turnstile>\<^sub>! (r \<^bold>\<longleftrightarrow>\<^sub>! r')"
\<tau>move2LAssRed2: fixes P :: "'a prog"   and h :: "'heap"   and stk :: "'addr val list"   and V :: "nat"   and e :: "'addr expr1" shows "\<tau>move2 P h stk (V:=e) (Suc (length (compE2 e))) None"
Pdrop_maximal: fixes n :: "nat" shows "maximal_path (ldropn n P)"
simps_13: fixes n :: "nat"   and t :: "tm"   and p :: "fm" shows "subst n t (A p) = A (subst (Suc n) (incrtm0 t) p)"
integrable_measure: fixes X :: "'a \<Rightarrow> 's stream set"   and I :: "'a set"   and s :: "'s" assumes "disjoint_family_on X I"   and "countable I" shows "integrable (count_space I) (\<lambda>i. T.prob s (X i))"
homeomorphic_map_path_connectedness: fixes X :: "'b topology"   and Y :: "'c topology"   and f :: "'b \<Rightarrow> 'c"   and U :: "'b set" assumes "homeomorphic_map X Y f"   and "U \<subseteq> topspace X" shows "path_connectedin Y (f ` U) = path_connectedin X U"
\<delta>3m: shows "set (labels_conv \<delta>3) \<subseteq> r \<down>s {fst \<alpha>_step, fst \<beta>_step}"
assms_2: fixes n :: "'c" shows "x n \<in> s"
\<mu>_def: shows "\<mu> \<equiv> \<lambda>f. if \<F>\<^sub>SC.arr f then f else \<F>\<^sub>SC.null"
strict_trans1: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<preceq> b"   and "b \<prec> c" shows "a \<prec> c"
case_distrib: fixes h :: "'c \<Rightarrow> 'a"   and f :: "'d \<Rightarrow> 'e \<Rightarrow> 'c"   and prod :: "'d \<times> 'e" shows "h (case prod of (x, xa) \<Rightarrow> f x xa) = (case prod of (x1, x2) \<Rightarrow> h (f x1 x2))"
rearrange: fixes j :: "nat"   and p :: "nat"   and k :: "nat" assumes "j * (p * j) = k * k" shows "k * k = p * (j * j)"
Rep_cases: fixes Rep :: "'c \<Rightarrow> 'd"   and Abs :: "'d \<Rightarrow> 'c"   and A :: "'d set"   and y :: "'d" assumes "type_definition Rep Abs A"   and "y \<in> A" obtains x :: "'c" where "y = Rep x"
pseudonatural_transformation_axioms: shows "pseudonatural_transformation (\<cdot>\<^sub>D) (\<star>\<^sub>D) \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D (\<cdot>\<^sub>D) (\<star>\<^sub>D) \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D I\<^sub>D.map I\<^sub>D.cmp E.GF.map E.GF.cmp \<eta>\<^sub>0 \<eta>\<^sub>1"
S_def: fixes G :: "real \<Rightarrow> real"   and x :: "real"   and y :: "real" assumes "G 1 = 1"   and "\<And>x. 0 < x \<Longrightarrow> G (x + 1) = x * G x"   and "\<And>x. 0 < x \<Longrightarrow> 0 < G x"   and "convex_on {0<..} (ln \<circ> G)" shows "??.Gamma_Function.S G x y = (ln (G y) - ln (G x)) / (y - x)"
eq1: shows "W P (Suc k) = adjoint M0 * P * M0 + adjoint M1 * WS (W P k) * M1"
coeffs_Poly: fixes as :: "'c list" shows "coeffs (Poly as) = strip_while ((=) (0::'c)) as"
induced_subgraph_altdef: fixes H :: "('a, 'b) pre_digraph" shows "induced_subgraph H (G \<restriction> {v. u \<rightarrow>\<^sup>* v}) = (subgraph H (G \<restriction> {v. u \<rightarrow>\<^sup>* v}) \<and> (\<forall>H'. subgraph H' (G \<restriction> {v. u \<rightarrow>\<^sup>* v}) \<longrightarrow> verts H' \<noteq> verts H \<or> arcs H' \<subseteq> arcs H))"
mult_1: fixes a :: "'d" shows "(1::'d) * a = a"
simps_1: fixes i :: "nat"   and s :: "'a FOL_Fitting.term"   and k :: "nat" shows "Var i[s/k] = (if k < i then Var (i - 1) else if i = k then s else Var i)"
simps_3: fixes n :: "nat"   and f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and y :: "'b"   and x :: "'b" assumes "n \<noteq> 0"   and "even n" shows "efficient_funpow f y x n = efficient_funpow f y (f x x) (n div 2)"
vinsert_insert: fixes a :: "V"   and A :: "V" shows "vinsert a A = set (insert a (elts A))"
rep_eq: fixes x :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and xa :: "('c, 'd) RBT.rbt" shows "fold1_keys x xa = rbt_fold1_keys x (rbt.impl_of xa)"
assms_1: fixes I :: "'a set" assumes "I \<in> S" shows "subcring I R"
simps_14: fixes prec :: "nat"   and x :: "nat"   and a :: "floatarith"   and vs :: "float interval option list" shows "isDERIV_approx prec x (Ln a) vs = (isDERIV_approx prec x a vs \<and> (case approx prec a vs of None \<Rightarrow> False | Some ivl \<Rightarrow> 0 < lower ivl))"
periodic: fixes a :: "nat" shows "\<Phi> (a + d) = \<Phi> a"
card: shows "CARD('a bit1) = card {0..<1 + 2 * int CARD('a)}"
induct: fixes P :: "('a \<Rightarrow> 'b) \<Rightarrow> 'b list \<Rightarrow> 'a list \<Rightarrow> bool"   and a0 :: "'a \<Rightarrow> 'b"   and a1 :: "'b list"   and a2 :: "'a list" assumes "\<And>f b. P f b []"   and "\<And>f b x xs. P f b xs \<Longrightarrow> P f b (x # xs)" shows "P a0 a1 a2"
\<C>_eq: shows "\<forall>x\<in>\<C>. mem\<^sub>1 x = mem\<^sub>2 x"
hdiff_iff: fixes u :: "hf"   and x :: "hf"   and y :: "hf" shows "(u \<^bold>\<in> x - y) = (u \<^bold>\<in> x \<and> u \<^bold>\<notin> y)"
obj_src: fixes \<mu> :: "'c" assumes "C.arr \<mu>" shows "C.obj (src\<^sub>C \<mu>)"
induced_modulus_dvd: fixes d :: "nat" assumes "induced_modulus d" shows "d dvd n"
\<alpha>_gen_remove: fixes As :: "'s \<times> natural \<times> natural"   and bound :: "natural"   and n :: "nat" assumes "invar (fst As)" shows "apfst \<alpha> (local.gen_remove bound n As) = Benchmark_Comparison.gen_remove bound n (apfst \<alpha> As)"
m_ivl_height: fixes ivl :: "ivl" shows "m_ivl ivl \<le> 2"
vec_coeff_append2: fixes i :: "nat"   and a :: "rat vec"   and b :: "rat vec" assumes "i \<in> {dim_vec a..<dim_vec (a @\<^sub>v b)}" shows "Abstract_Linear_Poly.coeff (vec_to_lpoly (a @\<^sub>v b)) i = b $ (i - dim_vec a)"
new_mm_def: shows "new_mm = (if k = 0 then mmm else min mm mmm)"
graph_homomorphism_def: fixes G\<^sub>1 :: "('b, 'c) labeled_graph"   and G\<^sub>2 :: "('b, 'd) labeled_graph"   and f :: "('c \<times> 'd) set" shows "graph_homomorphism G\<^sub>1 G\<^sub>2 f = (vertices G\<^sub>1 = Domain f \<and> graph G\<^sub>1 \<and> graph G\<^sub>2 \<and> f `` vertices G\<^sub>1 \<subseteq> vertices G\<^sub>2 \<and> univalent f \<and> edge_preserving f (edges G\<^sub>1) (edges G\<^sub>2))"
back_subst: fixes P :: "'c \<Rightarrow> bool"   and a :: "'c"   and b :: "'c" assumes "P a"   and "a = b" shows "P b"
simps_1: fixes l :: "'c tree"   and v :: "'c"   and r :: "'c tree" shows "left \<langle>l, v, r\<rangle> = l"
simps_1: fixes x :: "'a" shows "p_power x 0 = 1\<^sub>\<sigma>"
word_sle_def: shows "(\<le>s) \<equiv> map_fun Word.rep (map_fun Word.rep id) (\<lambda>k l. signed_take_bit (LENGTH('a) - Suc 0) k \<le> signed_take_bit (LENGTH('a) - Suc 0) l)"
Setcompr_eq_image: fixes f :: "'d \<Rightarrow> 'c"   and A :: "'d set" shows "{f x |x. s2p A x} = \<P> f A"
u: shows "u \<equiv> getFr [x, y] [t] [\<phi>]"
dlist_eq_iff: fixes dxs :: "'b dlist"   and dys :: "'b dlist" shows "(dxs = dys) = (list_of_dlist dxs = list_of_dlist dys)"
match_ex_table_not_pcs_None: fixes pc :: "nat"   and xt :: "(nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list"   and P :: "'a prog"   and C :: "String.literal" assumes "pc \<notin> pcs xt" shows "match_ex_table P C pc xt = None"
max_bot2: fixes x :: "'a" shows "max x (0::'a) = x"
exhaust: fixes y :: "'a stream" obtains x1 :: "'a" and x2 :: "'a stream" where "y = x1 ## x2"
hcomp_in_vhomE: fixes V :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and H :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and src :: "'c \<Rightarrow> 'c"   and trg :: "'c \<Rightarrow> 'c"   and \<nu> :: "'c"   and \<mu> :: "'c"   and f :: "'c"   and g :: "'c" assumes "horizontal_composition V H src trg"   and "partial_magma.in_hom V (H \<nu> \<mu>) f g" obtains "partial_magma.arr V \<mu>" and "partial_magma.arr V \<nu>" and "src \<nu> = trg \<mu>" and "src \<mu> = src f" and "src \<mu> = src g" and "trg \<nu> = trg f" and "trg \<nu> = trg g"
case_NegE: fixes P :: "Formula.formula \<Rightarrow> bool"   and \<phi> :: "Formula.formula" assumes "case \<phi> of formula.Neg \<phi>' \<Rightarrow> P \<phi>' | _ \<Rightarrow> False" obtains \<phi>' :: "Formula.formula" where "\<phi> = formula.Neg \<phi>'" and "P \<phi>'"
extensional_funcset_extend_domain_inj_onI: fixes x :: "'c"   and S :: "'c set"   and T :: "'d set" assumes "x \<notin> S" shows "inj_on (\<lambda>(y, g). g(x := y)) {(y, g). y \<in> T \<and> g \<in> S \<rightarrow>\<^sub>E T - {y} \<and> inj_on g S}"
has_real_derivative_powr: fixes z :: "real"   and r :: "real" assumes "0 < z" shows "((\<lambda>z. z powr r) has_real_derivative r * z powr (r - 1)) (at z)"
cases: fixes a :: "msg"   and H :: "msg set" assumes "a \<in> synth H" obtains X :: "msg" where "a = X__" and "X__ \<in> H"   | X :: "agent" where "a = Agent Xa__"   | n :: "nat" where "a = Number n"   | X :: "msg" where "a = Hash Xb__" and "Xb__ \<in> synth H"   | X :: "msg" and Y :: "msg" where "a = \<lbrace>Xc__, Y\<rbrace>" and "Xc__ \<in> synth H" and "Y \<in> synth H"   | X :: "msg" and K :: "nat" where "a = Crypt K X" and "X \<in> synth H" and "Key K \<in> H"   | x :: "rat" and y :: "rat" where "a = Auth_Data x y" and "Pri_AgrK x \<in> H" and "Pri_AgrK y \<in> H"
language_state_def: fixes M :: "('d, 'e, 'f) FSM"   and q :: "'f" shows "LS M q \<equiv> {map fst r |r. path M r q}"
weak_transition_tau_iff: fixes satisfies :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and transition :: "'a \<Rightarrow> ('c, 'a) residual \<Rightarrow> bool"   and \<tau> :: "'c"   and P :: "'a"   and P' :: "'a" assumes "weak_nominal_ts satisfies transition \<tau>" shows "weak_nominal_ts.weak_transition transition \<tau> P \<tau> P' = weak_nominal_ts.tau_transition transition \<tau> P P'"
map_graph_functional: fixes f :: "'c \<Rightarrow> 'd option" shows "functional (map_graph f)"
walk_drop: fixes xs :: "'a list"   and n :: "nat" assumes "walk xs" shows "walk (drop n xs)"
prod_def: shows "prod \<equiv> comm_monoid_set.F (*) (1::'a)"
p_eq_sum_p_f: fixes x :: "'s"   and y :: "'s"   and n :: "nat" shows "p x y n = (\<Sum>i\<le>n. p y y (n - i) * f x y i)"
HUN_E: fixes b :: "hf"   and A :: "hf"   and B :: "hf \<Rightarrow> hf" assumes "b \<^bold>\<in> \<Squnion>RepFun A B" obtains x :: "hf" where "x \<^bold>\<in> A" and "b \<^bold>\<in> B x"
idb_2: shows "i_b = 0 \<or> A $$ (i_b - 1, i_b) \<noteq> (1::'b)"
coprime_0_left_iff: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and a :: "'b" assumes "class.algebraic_semidom divide plus minus zero times one" shows "algebraic_semidom.coprime times one zero a = dvd.dvd times a one"
simps_3: fixes P :: "('a \<times> 'a) set"   and Pw :: "('a \<times> 'a) set"   and R :: "('a \<times> 'a) set"   and Rw :: "('a \<times> 'a) set" shows "SN_rel_ext_step P Pw R Rw normal_s = R"