hcju/isabelleps huggingface dataset:
induct: fixes P :: "('a \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c) \<Rightarrow> ('a \<times> 'c) list \<Rightarrow> ('a \<times> 'c) list \<Rightarrow> bool"   and a0 :: "'a \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and a1 :: "('a \<times> 'c) list"   and a2 :: "('a \<times> 'c) list" assumes "\<And>f k v as k' v' bs. \<lbrakk>c k' k = Eq \<Longrightarrow> P f as bs; c k' k = Lt \<Longrightarrow> P f ((k, v) # as) bs; c k' k = Gt \<Longrightarrow> P f as ((k', v') # bs)\<rbrakk> \<Longrightarrow> P f ((k, v) # as) ((k', v') # bs)"   and "\<And>f uu_. P f [] uu_"   and "\<And>f uv_. P f uv_ []" shows "P a0 a1 a2"
FAccThrow\<^sub>1: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> nat exp) list) list"   and e :: "nat exp"   and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> val list"   and e' :: "nat exp"   and s\<^sub>1 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> val list"   and F :: "char list"   and D :: "char list" assumes "P \<turnstile>\<^sub>1 \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>throw e',s\<^sub>1\<rangle>" shows "P \<turnstile>\<^sub>1 \<langle>e\<bullet>F{D},s\<^sub>0\<rangle> \<Rightarrow> \<langle>throw e',s\<^sub>1\<rangle>"
equivalence_in_bicategory_axioms: shows "S.internal_equivalence (S.UP f) (S.UP g) (S.inv (S.cmp\<^sub>U\<^sub>P (g, f)) \<cdot>\<^sub>S S.UP \<eta> \<cdot>\<^sub>S UP.unit (src f)) (S.inv (UP.unit (trg f)) \<cdot>\<^sub>S S.UP \<epsilon> \<cdot>\<^sub>S S.cmp\<^sub>U\<^sub>P (f, g))"
drinks_vend_insufficient: fixes r :: "nat \<Rightarrow>f value option"   and x1 :: "int" assumes "r $ 2 = Some (Num x1)"   and "x1 < 100" shows "possible_steps drinks 1 r STR ''vend'' [] = {|(1, vend_fail)|}"
ifbd_sup_pres: shows "sup_pres bd\<^sup>-\<^sub>\<F>"
assms_2: shows "fst (cs c cid) \<noteq> fst (cs c' cid)"
normalize_pos: fixes m :: "'a common_primitive match_expr" assumes "normalized_nnf_match m"   and "\<not> has_disc_negated disc False m" shows "match_list \<gamma> (normalize_pos m) a p = matches \<gamma> m a p"
psi_ubound_log_double_cases': fixes m :: "nat"   and c :: "real"   and n :: "nat"   and m' :: "nat"   and c' :: "real" assumes "\<And>n. n \<le> m \<Longrightarrow> psi n \<le> c * ln 2 * real n"   and "n \<le> m'"   and "m' = 2 * m"   and "c \<le> c'"   and "0 \<le> c"   and "1 \<le> m"   and "1 + c / 2 + (4 * ln (real m + 1) + 3) / (ln 2 * (real m + 1)) \<le> c'" shows "psi n \<le> c' * ln 2 * real n"
path_cons: fixes e :: "('v \<times> 'outPort) \<times> 'v \<times> 'inPort"   and v' :: "'v"   and pth :: "(('v \<times> 'outPort) \<times> 'v \<times> 'inPort) list" assumes "e \<in> edges"   and "path (edge_end e) v' pth" shows "path (edge_begin e) v' (e # pth)"
Phi_diff_split_min: fixes t :: "'a tree"   and x :: "'a"   and t' :: "'a tree" assumes "Root_Balanced_Tree.split_min t = (x, t')"   and "t \<noteq> \<langle>\<rangle>" shows "\<Phi> t' - \<Phi> t \<le> 6 * e * real (height t)"
bind_assoc: fixes x :: "'c option"   and f :: "'c \<Rightarrow> 'b option"   and g :: "'b \<Rightarrow> 'a option" shows "x \<bind> f \<bind> g = x \<bind> (\<lambda>y. f y \<bind> g)"
ANR_closed_Un_local_aux: fixes U :: "'a set"   and S :: "'a set"   and T :: "'a set" assumes "closedin (top_of_set U) S"   and "closedin (top_of_set U) T"   and "ANR S"   and "ANR T"   and "ANR (S \<inter> T)" obtains V :: "'a set" where "openin (top_of_set U) V" and "S \<union> T retract_of V"
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" assumes "\<And>i. i < length T \<Longrightarrow> (T ! i, S ! i) \<in> timpl_closure' c"   and "length T = length S" shows "(Fun f T, Fun f S) \<in> timpl_closure' c"
diff_self: fixes V :: "'b set"   and x :: "'b" assumes "vectorspace V"   and "x \<in> V" shows "x - x = (0::'b)"
rtrner_def: fixes ch :: "state \<Rightarrow> PrIds \<Rightarrow> ('b, 'a) chan" shows "rtrner ch \<equiv> \<lambda>s p. (rbit (ch s p), res (ch s p))"
is_acc_type_is_type: fixes G :: "prog"   and P :: "pname"   and T :: "ty" assumes "is_acc_type G P T" shows "is_type G T"
f12_split_face: shows "f12 = fst (split_face f v u ws)"
imult_infinity_if: fixes n :: "enat" shows "\<infinity> * n = (if n = 0 then 0 else \<infinity>)"
M'_1: shows "\<forall>u. u \<turnstile>\<^bsub>v,n\<^esub> \<lambda>i' j'. if i' = v c1 \<and> j' = 0 then Le r else if i' = 0 \<and> j' = v c1 then Le (- r) else M i' j' \<longrightarrow> u \<turnstile>\<^bsub>v,n\<^esub> M"
exec_def: fixes G :: "nat graph_rec"   and n :: "nat"   and xs :: "nat list" assumes "cfg_doms G" shows "cfg_doms.exec G n xs = map (\<lambda>pc. (pc, transf n xs)) (rev (sorted_list_of_set (cfg_doms.succs G n)))"
assms_2: fixes i :: "nat" assumes "i \<le> degree p" shows "coeff p i = coeff q i"
sort_descending_set_inv: fixes k :: "'a \<Rightarrow> 'b"   and t :: "'a list" shows "set (sort_descending_key k t) = set t"
lower_asymptotic_density_one_finite_Intersection: fixes I :: "'b set"   and A :: "'b \<Rightarrow> nat set" assumes "finite I"   and "\<And>i. i \<in> I \<Longrightarrow> lower_asymptotic_density (A i) = 1" shows "lower_asymptotic_density (\<Inter> (A ` I)) = 1"
inv_Mp_rev: fixes f :: "int poly" assumes "\<And>n. 2 * \<bar>coeff f n\<bar> < m" shows "inv_Mp (Mp f) = f"
map_eq_append_conv: fixes f :: "'e \<Rightarrow> 'd"   and xs :: "'e list"   and ys :: "'d list"   and zs :: "'d list" shows "(map f xs = ys @ zs) = (\<exists>us vs. xs = us @ vs \<and> ys = map f us \<and> zs = map f vs)"
select_convs_1: fixes committed :: "nat set"   and justifying_exec :: "('b \<times> ('a, 'b) obs_event action) llist"   and justifying_ws :: "nat \<Rightarrow> nat"   and action_translation :: "nat \<Rightarrow> nat"   and more :: "'c" shows "action_translation \<lparr>committed = committed, justifying_exec = justifying_exec, justifying_ws = justifying_ws, action_translation = action_translation, \<dots> = more\<rparr> = action_translation"
m'_in: fixes x :: "'a"   and l :: "nat" assumes "x \<in> M (V l)" shows "m' x \<in> N"
sample_vars_insert: fixes A :: "'v set"   and X :: "'v \<Rightarrow> int pmf"   and x :: "'v"   and m :: "'m" assumes "finite A" shows "sample_vars X (insert x A) m = bind (memo (sample_var X) x) (\<lambda>_. sample_vars X A m)"
cqt_basic_9: fixes v :: "i"   and \<phi> :: "'a \<Rightarrow> \<o>"   and \<psi> :: "'a \<Rightarrow> \<o>"   and \<chi> :: "'a \<Rightarrow> \<o>" shows "[(\<^bold>\<forall>\<alpha>. \<phi> \<alpha> \<^bold>\<rightarrow> \<psi> \<alpha>) \<^bold>& (\<^bold>\<forall>\<alpha>. \<psi> \<alpha> \<^bold>\<rightarrow> \<chi> \<alpha>) \<^bold>\<rightarrow> (\<^bold>\<forall>\<alpha>. \<phi> \<alpha> \<^bold>\<rightarrow> \<chi> \<alpha>) in v]"
summable: shows "summable (\<lambda>x. rate ^ x / fact x)"
well_founded_ordering_axioms: shows "well_founded_ordering UNIV (\<le>) (<)"
coreflexive_pseudo_complement: fixes x :: "'a"   and y :: "'a" assumes "coreflexive x" shows "(x \<sqinter> y = bot) = (x \<le> y ')"
ideal_generated_pair_UNIV: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and s :: "'b"   and a :: "'b"   and t :: "'b"   and b :: "'b"   and u :: "'b" assumes "class.comm_ring_1 times one plus zero minus uminus"   and "plus (times s a) (times t b) = u"   and "dvd.dvd times u one" shows "ring.ideal_generated plus zero uminus times {a, b} = UNIV"
netgmap_subnet_split2: fixes sr :: "'e \<Rightarrow> 'c \<times> 'd"   and s1 :: "'e net_state"   and s2 :: "'e net_state"   and n1 :: "net_tree"   and n2 :: "net_tree"   and \<sigma> :: "nat \<Rightarrow> 'c"   and \<zeta> :: "'d net_state" assumes "netgmap sr (SubnetS s1 s2) = netmask (net_tree_ips (n1 \<parallel> n2)) (\<sigma>, \<zeta>)"   and "net_ips s1 = net_tree_ips n1"   and "net_ips s2 = net_tree_ips n2" shows "netgmap sr s2 = netmask (net_tree_ips n2) (\<sigma>, snd (netgmap sr s2))"
pick_eqv: fixes a :: "'b" shows "pick \<lfloor>a\<rfloor> \<sim> a"
flow_initial_time_if: fixes t0 :: "real"   and x0 :: "'a" shows "flow t0 x0 t0 = (if t0 \<in> T \<and> x0 \<in> X then x0 else (0::'a))"
take_zip: fixes n :: "nat"   and xs :: "'a list"   and ys :: "'b list" shows "take n (zip xs ys) = zip (take n xs) (take n ys)"
conf_clinit_Invoke: fixes P :: "jvm_prog"   and sh :: "sheap"   and stk :: "val list"   and loc :: "val list"   and C :: "char list"   and M :: "char list"   and pc :: "nat"   and ics :: "init_call_status"   and frs :: "frame list"   and M' :: "char list"   and stk' :: "val list"   and loc' :: "val list"   and C' :: "char list"   and pc' :: "nat" assumes "conf_clinit P sh ((stk, loc, C, M, pc, ics) # frs)"   and "M' \<noteq> clinit" shows "conf_clinit P sh ((stk', loc', C', M', pc', No_ics) # (stk, loc, C, M, pc, No_ics) # frs)"
prv_cnj_imp_monoR2: fixes \<phi> :: "'fmla"   and \<chi> :: "'fmla"   and \<psi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "\<chi> \<in> fmla"   and "\<psi> \<in> fmla"   and "bprv (imp \<phi> (imp \<chi> \<psi>))" shows "bprv (imp (cnj \<phi> \<chi>) \<psi>)"
take_is_prefix: fixes n :: "nat"   and xs :: "'b list" shows "take n xs \<le>p xs"
distinct_27: fixes x41 :: "bexp"   and x42 :: "acom"   and x43 :: "acom"   and x61 :: "((char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool) \<times> ((char list \<Rightarrow> int) \<Rightarrow> char list \<Rightarrow> int) \<times> ((char list \<Rightarrow> int) \<Rightarrow> nat)"   and x62 :: "bexp"   and x63 :: "acom" shows "IF x41 THEN x42 ELSE x43 \<noteq> {x61} WHILE x62 DO x63"
unique_eigen_vector_real: fixes lam :: "real"   and A :: "((real, 'a) Finite_Cartesian_Product.vec, 'a) Finite_Cartesian_Product.vec"   and v :: "(real, 'a) Finite_Cartesian_Product.vec"   and w :: "(real, 'a) Finite_Cartesian_Product.vec" assumes "order lam (charpoly A) = 1"   and "eigen_vector A v lam"   and "eigen_vector A w lam" shows "\<exists>a. v = a *s w"
real_vector_affinity_eq: fixes m :: "real"   and x :: "'c"   and c :: "'c"   and y :: "'c" assumes "m \<noteq> 0" shows "(m *\<^sub>R x + c = y) = (x = y /\<^sub>R m - c /\<^sub>R m)"
star_of_of_nat: fixes n :: "nat" shows "star_of (of_nat n) = of_nat n"
HSem_restr: fixes ESem :: "'f \<Rightarrow> ('d \<Rightarrow> 'e) \<rightarrow> 'e"   and h :: "('d \<times> 'f) list"   and \<rho> :: "'d \<Rightarrow> 'e" shows "has_ESem.HSem ESem h\<cdot>(\<rho> f|` (- domA h)) = has_ESem.HSem ESem h\<cdot>\<rho>"
GF_advice_\<nu>LTL_2: fixes \<phi> :: "'b ltln"   and X :: "'b ltln set" assumes "\<phi> \<in> \<nu>LTL" shows "\<phi>[X]\<^sub>\<nu> = \<phi>"
I02: fixes s :: "'d"   and z :: "'e nres" assumes "s \<in> S"   and "SPEC (I S) \<le> z" shows "f s \<le> z"
tr_compat: shows "NS\<^sup>* O S \<subseteq> S"
express_poly: fixes p :: "real mpoly"   and var :: "nat" assumes "MPoly_Type.degree p var = 1 \<or> MPoly_Type.degree p var = 2" shows "p = isolate_variable_sparse p var 2 * (Var var)\<^sup>2 + isolate_variable_sparse p var 1 * Var var + isolate_variable_sparse p var 0"
insert: fixes A :: "'b set"   and x :: "'b" assumes "finite A"   and "A \<noteq> {}" shows "Min (insert x A) = min x (Min A)"
minus_eq: fixes x :: "nat \<Rightarrow> int"   and y :: "nat \<Rightarrow> int" shows "x \<ominus> y = x \<oplus> \<ominus> y"
bdd_relator_def: fixes p :: "(((nat \<Rightarrow> bool) \<Rightarrow> bool) \<times> nat) set"   and s :: "bddi" shows "bdd_relator p s \<equiv> \<exists>\<^sub>Acs. is_bdd_impl cs s * \<up> (p \<subseteq> bddmi_rel cs \<and> bdd_sane cs) * true"
quotient_is_cring: fixes I :: "'a set"   and R :: "('a, 'b) ring_scheme" assumes "ideal I R"   and "cring R" shows "cring (R Quot I)"
from_nat_def: fixes i :: "nat" shows "from_nat i = univ_list ! i"
VA': shows "Vagree \<nu> \<omega> (\<Union>i\<in>SIGT \<theta>. case SFunctions \<sigma> i of None \<Rightarrow> {} | Some a \<Rightarrow> FVT a)"
fbrelation_flrestriction: fixes r :: "V"   and A :: "V" shows "fbrelation (r \<restriction>\<^sup>l\<^sub>\<bullet> A)"
distinct_1: fixes x1 :: "'a"   and x2 :: "dB" shows "Opaque x1 \<noteq> Pure x2"
varFIX: fixes b :: "BExpr"   and A :: "(Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool"   and \<Phi> :: "(Var \<Rightarrow> Val) \<times> (Var \<Rightarrow> Val) \<Rightarrow> bool"   and s :: "Var \<Rightarrow> Val"   and t :: "Var \<Rightarrow> Val" assumes "(b, A, \<Phi>, s, t) \<in> var" shows "PhiWhileP A b \<Phi> (s, t)"
simps_8: fixes x :: "'c" shows "norm_instr (Inca.instr.ICall x) = Std.instr.ICall x"
Int_insert_right_if1: fixes a :: "'a"   and A :: "'a set"   and B :: "'a set" assumes "a \<in> A" shows "A \<inter> insert a B = insert a (A \<inter> B)"
farkas_coefficients_atoms_tableau_mono: fixes as :: "'a atom set"   and bs :: "'a atom set"   and t :: "(nat \<times> linear_poly) list"   and C :: "(rat \<times> 'a atom) list" assumes "as \<subseteq> bs"   and "farkas_coefficients_atoms_tableau as t C" shows "farkas_coefficients_atoms_tableau bs t C"
of_rat_def: 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" assumes "class.field_char_0 divide inverse times one plus zero minus uminus" shows "field_char_0.of_rat divide one plus zero minus \<equiv> map_fun Rep_Rat id (\<lambda>x. divide (ring_1.of_int one plus zero minus (fst x)) (ring_1.of_int one plus zero minus (snd x)))"
special_cosets_def: shows "\<P> \<equiv> \<Union>T\<in>Pow S. \<Union>w\<in>W. {w +o \<langle>T\<rangle>}"
coeff_poly_times_one_minus_x: fixes g :: "'a poly"   and n :: "nat" shows "coeff g n = sum (coeff (g * [:1::'a, - (1::'a):])) {..n}"
eq: fixes f :: "'b \<Rightarrow> V" shows "f ` (\<pi> ` A \<inter> {y. (y, \<pi> x) \<in> s}) = (f \<circ> \<pi>) ` (A \<inter> {y. (y, x) \<in> r})"
neq_left_right_of: fixes a :: "real \<times> real"   and b :: "real \<times> real"   and c :: "real \<times> real"   and d :: "real \<times> real" assumes "ccw' a b c"   and "ccw' a c d" shows "b \<noteq> d"
graph_level_finite: fixes A :: "('b, 'a) nba"   and w :: "'b stream"   and k :: "nat"   and l :: "nat" assumes "finite (nodes A)" shows "finite (level A w k l)"
eq_neutr_iff: fixes a :: "'b"   and b :: "'b" shows "(infi a b = topi) = (a = topi \<and> b = topi)"
mset_subset_of_empty: fixes A :: "'b multiset" shows "(A \<subset># {#}) = False"
degree_one_imp_pirreducible: fixes K :: "'a list set"   and p :: "'a list list" assumes "subfield K (poly_ring R)"   and "p \<in> carrier (K [X]\<^bsub>poly_ring R\<^esub>)"   and "degree p = 1" shows "pirreducible\<^bsub>poly_ring R\<^esub> K p"
simps_1: fixes a :: "atom"   and \<Gamma> :: "real list" shows "eval (fm.Atom a) \<Gamma> = aEval a \<Gamma>"
vprojection_def: fixes I :: "V"   and A :: "V \<Rightarrow> V"   and i :: "V" shows "vprojection I A i = (\<lambda>f\<in>\<^sub>\<circ>vproduct I A. f\<lparr>i\<rparr>)"
dom_allow_pfun: fixes f :: "'c \<Rightarrow> 'd option" shows "dom (A\<^sub>p f) = dom f"
construct_lhs_vector_R_def: fixes p :: "real poly"   and qs :: "real poly list"   and signs :: "rat list list" shows "construct_lhs_vector_R p qs signs \<equiv> vec_of_list (map (\<lambda>w. rat_of_int (int (length (filter (\<lambda>v. v = w) (map (consistent_sign_vec qs) (characterize_root_list_p p)))))) signs)"
add_mult_distrib1: fixes x :: "'a"   and y :: "'a"   and a :: "real" assumes "x \<in> E"   and "y \<in> E" shows "a \<cdot> (x + y) = a \<cdot> x + a \<cdot> y"
rtrancl_induct2: fixes ax :: "'a"   and ay :: "'b"   and bx :: "'a"   and "by" :: "'b"   and r :: "(('a \<times> 'b) \<times> 'a \<times> 'b) set"   and P :: "'a \<Rightarrow> 'b \<Rightarrow> bool" assumes "((ax, ay), bx, by) \<in> r\<^sup>*"   and "P ax ay"   and "\<And>a b aa ba. \<lbrakk>((ax, ay), a, b) \<in> r\<^sup>*; ((a, b), aa, ba) \<in> r; P a b\<rbrakk> \<Longrightarrow> P aa ba" shows "P bx by"
eq_iff_17: fixes hash :: "hash"   and hasha :: "hash" shows "(Hash hash = Hash hasha) = (hash = hasha)"
prim_iff_rotate: fixes w :: "'b list" assumes "w \<noteq> \<epsilon>" shows "primitive w = (\<forall>n. 0 < n \<and> n < \<^bold>|w\<^bold>| \<longrightarrow> rotate n w \<noteq> w)"
\<pi>: fixes x :: "'a"   and y :: "'a" assumes "x \<in> A"   and "y \<in> A"   and "(x, y) \<in> r" shows "\<pi> x < \<pi> y"
assms_1: shows "circuit_in \<E> C\<^sub>1"
iIN_iFROM_append_union_Suc: fixes n :: "nat"   and d :: "nat" shows "[n\<dots>,d] \<union> [Suc (n + d)\<dots>] = [n\<dots>]"
runit_char_4: fixes f :: "'a" assumes "ide f" shows "\<exists>!\<mu>. \<guillemotleft>\<mu> : f \<star> src f \<Rightarrow> f\<guillemotright> \<and> \<mu> \<star> src f = (f \<star> \<i>' (src f)) \<cdot> \<a>[f, src f, src f]"
ctrm_monom: fixes R :: "('c, 'd) ring_scheme"   and a :: "'c"   and k :: "nat" assumes "UP_cring R"   and "a \<in> carrier R" shows "monom (UP R) (monom (UP R) a (Suc k) 0) 0 = \<zero>\<^bsub>UP R\<^esub>"
XD_XH_gfp_F_1: 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 ds :: "'e set"   and XD_XH :: "'d set \<times> 'd set" assumes "ContractsWithSubstitutes Xd Xh Pd Ch"   and "Contracts.stable_pair_on Pd Ch ds XD_XH" shows "fst XD_XH \<subseteq> fst (ContractsWithSubstitutes.gfp_F Pd Ch ds)"
graph_pred_it_def: fixes \<alpha> :: "'a \<Rightarrow> ('b, 'c) graph"   and invar :: "'a \<Rightarrow> bool"   and pred_list_it :: "'a \<Rightarrow> 'b \<Rightarrow> (('b \<times> 'c) list \<Rightarrow> bool) \<Rightarrow> ('b \<times> 'c \<Rightarrow> ('b \<times> 'c) list \<Rightarrow> ('b \<times> 'c) list) \<Rightarrow> ('b \<times> 'c) list \<Rightarrow> ('b \<times> 'c) list" shows "graph_pred_it \<alpha> invar pred_list_it \<equiv> graph \<alpha> invar \<and> graph_pred_it_axioms \<alpha> invar pred_list_it"
simps_5: fixes x :: "('a, 'b) Node"   and xs :: "('a, 'b) Node list"   and v :: "'b"   and va :: "('a, 'b) Digit"   and vb :: "('a, 'b) FingerTreeStruc"   and vc :: "('a, 'b) Digit" shows "FingerTreeStruc.app3 (Single x) xs (Deep v va vb vc) = FingerTreeStruc.nlcons x (FingerTreeStruc.lconsNlist xs (Deep v va vb vc))"
OS_def: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and P :: "'a"   and Q :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong" shows "Tarski_neutral_dimensionless.OS Bet A B P Q \<equiv> \<exists>R. Tarski_neutral_dimensionless.TS Bet A B P R \<and> Tarski_neutral_dimensionless.TS Bet A B Q R"
fv_def: fixes e :: "'b" shows "fv e = {v. atom v \<in> supp e}"
normal_def: fixes H :: "'e set"   and G :: "('e, 'f) monoid_scheme" shows "H \<lhd> G \<equiv> subgroup H G \<and> Group.group G \<and> normal_axioms H G"
l_set_disconnected_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and set_disconnected_nodes :: "(_) document_ptr \<Rightarrow> (_) node_ptr list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, unit) prog set" shows "l_set_disconnected_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf set_disconnected_nodes set_disconnected_nodes_locs \<equiv> type_wf = DocumentClass.type_wf \<and> set_disconnected_nodes = l_set_disconnected_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_set_disconnected_nodes \<and> set_disconnected_nodes_locs = l_set_disconnected_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_set_disconnected_nodes_locs"
weakly_sequential_def: fixes P :: "'a option process" shows "weakly_sequential P \<equiv> \<forall>xs\<in>traces P. None \<notin> set (butlast xs)"
divisors_nat_copy: shows "divisors_nat_copy = divisors_nat"
prems_2: fixes i :: "'a" assumes "i \<in> insert j I" shows "g i \<in> ideal B"
seq_char: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and Arr :: "'b \<Rightarrow> bool"   and g :: "'b"   and f :: "'b" assumes "subcategory C Arr" shows "partial_magma.arr (subcategory.comp C Arr) (subcategory.comp C Arr g f) = (partial_magma.arr (subcategory.comp C Arr) f \<and> partial_magma.arr (subcategory.comp C Arr) g \<and> partial_magma.arr C (C g f))"
simps_7: fixes x31 :: "'b"   and x32 :: "('a, 'b) form list"   and x1 :: "nat" shows "Modal x31 x32 \<noteq> At x1"
set_map_snd: assumes "distinct (map fst \<Gamma>)" shows "set (map snd \<Gamma>) = ran (map_of \<Gamma>)"
NetsEq_if_sameP_DD: fixes p :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators list"   and u :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators"   and v :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators"   and a :: "(int \<times> int \<times> protocol) set set"   and b :: "(int \<times> int \<times> protocol) set set"   and c :: "(int \<times> int \<times> protocol) set set"   and d :: "(int \<times> int \<times> protocol) set set"   and x :: "int \<times> (int \<times> int \<times> protocol) \<times> (int \<times> int \<times> protocol) \<times> DummyContent" assumes "allNetsDistinct p"   and "u \<in> set p"   and "v \<in> set p"   and "u = DenyAllFromTo a b"   and "v = DenyAllFromTo c d"   and "x \<in> dom (Cp u)"   and "x \<in> dom (Cp v)" shows "a = c \<and> b = d"
h_3: shows "(flow0 x0 h, Dflow x0 h o\<^sub>L d0) \<in> C"
abs_of_nonpos: fixes abs :: "'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b" assumes "class.ordered_ab_group_add_abs abs plus zero minus uminus less_eq less"   and "less_eq a zero" shows "abs a = uminus a"
valid_tuple_def: fixes tuple_since :: "('a option list, nat) mapping" shows "valid_tuple tuple_since = (\<lambda>(t, as). case Mapping.lookup tuple_since as of None \<Rightarrow> False | Some t' \<Rightarrow> t' \<le> t)"
q_def: shows "q \<equiv> \<lambda>a. if a \<in> FreeGroup S then BinOpSetGroup.Abs_G_perm (FreeGroup S // Restr (lcoset_rel \<langle>\<Union>g\<in>FreeGroup S. lconjby g ` P'\<rangle>) (FreeGroup S)) (+) (Restr (lcoset_rel \<langle>\<Union>g\<in>FreeGroup S. lconjby g ` P'\<rangle>) (FreeGroup S) `` {a}) else 0"
inject_3: fixes x41 :: "expr"   and x42 :: "cmd"   and x43 :: "cmd"   and y41 :: "expr"   and y42 :: "cmd"   and y43 :: "cmd" shows "(if (x41) x42 else x43 = if (y41) y42 else y43) = (x41 = y41 \<and> x42 = y42 \<and> x43 = y43)"
has_fps_expansion_inverse: fixes f :: "'a \<Rightarrow> 'a"   and F :: "'a fps" assumes "f has_fps_expansion F"   and "fps_nth F 0 \<noteq> (0::'a)" shows "(\<lambda>x. inverse (f x)) has_fps_expansion inverse F"
vdomain_vrange_is_vempty: shows "(\<D>\<^sub>\<circ> r = []\<^sub>\<circ>) = (\<R>\<^sub>\<circ> r = []\<^sub>\<circ>)"
rev_singleton_conv: fixes xs :: "'c list"   and x :: "'c" shows "(rev xs = [x]) = (xs = [x])"