hcju/isabelleps huggingface dataset:
ennreal_top_eq_mult_iff: fixes a :: "ennreal"   and b :: "ennreal" shows "(\<top> = a * b) = (a = \<top> \<and> b \<noteq> 0 \<or> b = \<top> \<and> a \<noteq> 0)"
isTable_subset: fixes qs :: "('a \<times> 'b) list"   and ps :: "('a \<times> 'b) list"   and E :: "'a \<Rightarrow> 'b"   and vs :: "'a list" assumes "set qs \<subseteq> set ps"   and "isTable E vs ps" shows "isTable E vs qs"
reduce_basis_4: fixes fs :: "int vec list" assumes "LLL_Impl.reduce_basis \<alpha> fs_init = fs" shows "length fs = m"
nu_roll: fixes g :: "'a \<Rightarrow> 'a"   and f :: "'a \<Rightarrow> 'a" assumes "isotone g"   and "has_greatest_fixpoint (f \<circ> g)"   and "has_greatest_fixpoint (g \<circ> f)" shows "\<nu> (g \<circ> f) = g (\<nu> (f \<circ> g))"
weight_swapLeaves: fixes t :: "'a tree"   and a :: "'a"   and b :: "'a"   and w\<^sub>a :: "nat"   and w\<^sub>b :: "nat" assumes "consistent t"   and "a \<noteq> b" shows "if a \<in> alphabet t then if b \<in> alphabet t then weight (swapLeaves t w\<^sub>a a w\<^sub>b b) + freq t a + freq t b = weight t + w\<^sub>a + w\<^sub>b else weight (swapLeaves t w\<^sub>a a w\<^sub>b b) + freq t a = weight t + w\<^sub>b else if b \<in> alphabet t then weight (swapLeaves t w\<^sub>a a w\<^sub>b b) + freq t b = weight t + w\<^sub>a else weight (swapLeaves t w\<^sub>a a w\<^sub>b b) = weight t"
smember_code: fixes x :: "'a"   and y :: "'a"   and s :: "'a stream" shows "Stream.smember x (y ## s) = (if x = y then True else Stream.smember x s)"
plossless_comp_converter: fixes \<I> :: "('i, 'j) \<I>"   and \<I>' :: "('k, 'l) \<I>"   and conv :: "('i, 'j, 'k, 'l) converter"   and \<I>'' :: "('m, 'n) \<I>"   and conv' :: "('k, 'l, 'm, 'n) converter" assumes "plossless_converter \<I> \<I>' conv"   and "plossless_converter \<I>' \<I>'' conv'"   and "\<I>, \<I>' \<turnstile>\<^sub>C conv \<surd>"   and "\<I>', \<I>'' \<turnstile>\<^sub>C conv' \<surd>" shows "plossless_converter \<I> \<I>'' (conv \<odot> conv')"
wcsr_rfns_def: fixes G :: "('e, 'f) Group_scheme"   and r :: "nat"   and f :: "nat \<Rightarrow> 'e set"   and s :: "nat" shows "wcsr_rfns G r f s = {h. tw_cmpser G (s * r) h \<and> (\<forall>i\<le>r. h (i * s) = f i)}"
cases: fixes a1 :: "jvm_state"   and a2 :: "jvm_state"   and P :: "jvm_prog" assumes "P \<turnstile> a1 -jvm\<rightarrow>\<^sub>1 a2" obtains \<sigma> :: "jvm_state" and \<sigma>' :: "jvm_state" where "a1 = \<sigma>" and "a2 = \<sigma>'" and "exec (P, \<sigma>) = \<lfloor>\<sigma>'\<rfloor>"
supp_Abs_2: fixes as :: "atom set"   and x :: "'a" shows "supp ([as]res. x) = supp x - as"
inject_7: fixes x8 :: "'a OCL_Types.type"   and y8 :: "'a OCL_Types.type" shows "(Sequence x8 = Sequence y8) = (x8 = y8)"
MtransC_mayDiverge: fixes tval :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and aval :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and c :: "('a, 'c) com"   and s :: "'b"   and c' :: "('a, 'c) com"   and s' :: "'b" assumes "PL.MtransC tval aval (c, s) (c', s')"   and "PL.mayDiverge tval aval c' s'" shows "PL.mayDiverge tval aval c s"
rcset: fixes x :: "'f cset" shows "rcset x \<in> {A. countable A}"
not_produces_imp_notin_production: fixes S :: "'b literal multiset \<Rightarrow> 'b literal multiset"   and N :: "'b literal multiset set"   and C :: "'b literal multiset"   and A :: "'b" assumes "ground_resolution_with_selection S"   and "ground_resolution_with_selection.production S N C \<noteq> {A}" shows "A \<notin> ground_resolution_with_selection.production S N C"
continuous_on_cases_local: fixes s :: "'c set"   and t :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd"   and P :: "'c \<Rightarrow> bool" assumes "closedin (top_of_set (s \<union> t)) s"   and "closedin (top_of_set (s \<union> t)) t"   and "continuous_on s f"   and "continuous_on t g"   and "\<And>x. x \<in> s \<and> \<not> P x \<or> x \<in> t \<and> P x \<Longrightarrow> f x = g x" shows "continuous_on (s \<union> t) (\<lambda>x. if P x then f x else g x)"
vertices_triangle: fixes f :: "face"   and a :: "nat" assumes "triangle f"   and "a \<in> \<V> f"   and "distinct (vertices f)" shows "\<V> f = {a, f \<bullet> a, f \<bullet> (f \<bullet> a)}"
ennsqrt_le: fixes x :: "ennreal"   and y :: "ennreal" shows "(ennsqrt x \<le> ennsqrt y) = (x \<le> y)"
alist_normalize_idem: fixes xs :: "('d, 'c) bintree option list" shows "elements (Binomial_Queue.normalize xs) = elements xs"
open_cover_of_open_subset_axioms_def: fixes is_open :: "'a set \<Rightarrow> bool"   and U :: "'a set" shows "open_cover_of_open_subset_axioms is_open U \<equiv> is_open U"
simps: fixes v :: "nat"   and c :: "nat" shows "local.fdr_alt v c = local.fdr_step v c \<bind> (\<lambda>(v', c'). if c' < n then return_spmf c' else local.fdr_alt (v' - n) (c' - n))"
map_id: fixes t :: "'a msllist" shows "mslmap id t = t"
pdata_DefaultPData: fixes D :: "'b dataspace" shows "(replicate (PartNum D) None, D) \<in> pdata"
functor_comp: fixes A :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and F :: "'a \<Rightarrow> 'b"   and C :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and G :: "'b \<Rightarrow> 'e" assumes "functor A B F"   and "functor B C G" shows "functor A C (G \<circ> F)"
rel_resource_Grp: fixes f :: "'g \<Rightarrow> 'e"   and g :: "'f \<Rightarrow> 'h" shows "rel_resource (BNF_Def.Grp UNIV f)\<inverse>\<inverse> (BNF_Def.Grp UNIV g) = BNF_Def.Grp UNIV (map_resource f g)"
farkas_minkowsky_weyl_theorem_2: fixes A :: "'a mat"   and nr :: "nat"   and Bnd :: "'a" assumes "A \<in> carrier_mat nr n" shows "\<exists>X\<subseteq>carrier_vec n. finite X \<and> polyhedral_cone A = cone X \<and> (A \<in> \<int>\<^sub>m \<inter> Bounded_mat Bnd \<longrightarrow> X \<subseteq> \<int>\<^sub>v \<inter> Bounded_vec (det_bound (n - 1) (max (1::'a) Bnd)))"
nth_mem_mset: fixes i :: "nat"   and ls :: "'a list" assumes "i < length ls" shows "ls ! i \<in># mset ls"
shows_prec_literal_def: fixes p :: "nat"   and s :: "String.literal" shows "shows_prec p s = shows_string (literal.explode s)"
iwlsFSb_def: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'a, 'b, 'c, 'd) model_scheme" shows "iwlsFSb MOD \<equiv> igWlsAllDisj MOD \<and> igWlsAbsIsInBar MOD \<and> igConsIPresIGWls MOD \<and> igSubstAllIPresIGWlsAll MOD \<and> igFreshCls MOD \<and> igSubstCls MOD \<and> igAbsRen MOD"
invert_side_of_triangle_2: fixes h :: "'c"   and f :: "'c"   and g :: "'c" assumes "C.arr h"   and "f \<cdot>\<^sub>C g = h"   and "C.iso g" shows "C.seq h (C.inv g) \<and> f = h \<cdot>\<^sub>C C.inv g"
emeasure_lim: fixes J :: "'i set"   and X :: "('i \<Rightarrow> 'a) set" assumes "finite J"   and "J \<subseteq> I"   and "X \<in> sets (Pi\<^sub>M J M)"   and "\<And>J X. \<lbrakk>\<And>i. J i \<subseteq> I; incseq J; \<And>i. finite (J i); \<And>i. X i \<in> sets (Pi\<^sub>M (J i) M); decseq (\<lambda>i. emb I (J i) (X i)); 0 < (INF i. emeasure (Pi\<^sub>M (J i) M) (X i))\<rbrakk> \<Longrightarrow> (\<Inter>i. emb I (J i) (X i)) \<noteq> {}" shows "emeasure local.lim (emb I J X) = emeasure (Pi\<^sub>M J M) X"
wtl_all: fixes "is" :: "'b list"   and c :: "'a list"   and s :: "'a"   and pc :: "nat" assumes "wtl is c 0 s \<noteq> \<top>"   and "pc < length is" shows "wtc c pc (wtl (take pc is) c 0 s) \<noteq> \<top>"
composition_series_prefix_closed: fixes i :: "nat" assumes "i \<le> length \<HH>"   and "0 < i" shows "composition_series (G\<lparr>carrier := \<HH> ! (i - 1)\<rparr>) (take i \<HH>)"
longestP: shows "longestImpWitnessPath c loc1 loc2 xs t"
f0: shows "(\<Sum>j\<in>{0, 1}. (cmod (jozsa_algo $$ (j, 0)))\<^sup>2) = 1"
UP_a_inv_closed: fixes R :: "('a, 'b) ring_scheme"   and p :: "nat \<Rightarrow> 'a" assumes "UP_ring R"   and "p \<in> carrier (UP R)" shows "\<ominus>\<^bsub>UP R\<^esub> p \<in> carrier (UP R)"
acyclic_6_irreflexive: fixes x :: "'a" assumes "acyclic_6 x" shows "irreflexive x"
sets_T: fixes s :: "'s" shows "sets (T s) = sets (stream_space (count_space UNIV))"
replicating_price_process: fixes pf :: "'b \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> real"   and der :: "'a \<Rightarrow> real"   and matur :: "nat" assumes "replicating_portfolio pf der matur"   and "support_adapt Mkt pf" shows "price_structure der matur (initial_value pf) (cls_val_process Mkt pf)"
simps_2: fixes f :: "'a"   and ss1 :: "('a, 'b) Term.term list"   and C :: "('a, 'b) ctxt"   and ss2 :: "('a, 'b) Term.term list"   and s :: "('a, 'b) Term.term" shows "(More f ss1 C ss2)\<langle>s\<rangle> = Fun f (ss1 @ C\<langle>s\<rangle> # ss2)"
assms: shows "locally_trans R (A + B) (C + D) (E + F)"
ctermsl_ex_trans: fixes p :: "('a, 'b, 'c, 'd) seqp"   and r :: "('a, 'b, 'c, 'd) seqp" assumes "\<exists>q\<in>ctermsl p. r \<in> ctermsl q" shows "r \<in> ctermsl p"
sublinear_wp_loop: fixes body :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and G :: "'a \<Rightarrow> bool" assumes "healthy (wp body)"   and "nearly_healthy (wlp body)"   and "sublinear (wp body)"   and "bd_cts (wp body)" shows "sublinear (wp (\<mu>x. body ;; x \<^bsub>\<guillemotleft> G \<guillemotright>\<^esub>\<oplus> Skip))"
leg_simps_4: shows "C.dom T.leg1 = T.apex"
update_convs_1: fixes cx' :: "int \<Rightarrow> int"   and cx :: "int"   and cy :: "int"   and more :: "'a" shows "cx_update cx' \<lparr>cx = cx, cy = cy, \<dots> = more\<rparr> = \<lparr>cx = cx' cx, cy = cy, \<dots> = more\<rparr>"
valid_path_def: fixes as :: "('b LDCFG_node \<times> ('c, 'd, 'e, 'f) edge_kind \<times> 'b LDCFG_node) list" shows "CFGExit.valid_path as \<equiv> CFGExit.valid_path_aux [] as"
V2_def: shows "V2 = filter (\<lambda>v. except g v \<noteq> 0) V"
prob_space_axioms: fixes s :: "'b" shows "prob_space (K2.T s)"
runit'_in_vhom: fixes f :: "'d"   and a :: "'d" assumes "D.ide f"   and "src\<^sub>D f = a" shows "\<guillemotleft>\<r>\<^sub>D\<^sup>-\<^sup>1[f] : f \<Rightarrow>\<^sub>D f \<star>\<^sub>D a\<guillemotright>"
simps_2: fixes wtFsym :: "'f \<Rightarrow> bool"   and wtPsym :: "'g \<Rightarrow> bool"   and at :: "('h, 'i) atm" assumes "Signature TYPE('e) wtFsym wtPsym" shows "fsymsL (Neg at) = fsymsA at"
sub: shows "B \<subseteq> hom_deg_set z A - {0}"
conj_mono_right: fixes c\<^sub>0 :: "'b::refinement_lattice"   and c\<^sub>1 :: "'b::refinement_lattice"   and d :: "'b::refinement_lattice" assumes "(c\<^sub>0::'b::refinement_lattice) \<sqsubseteq> (c\<^sub>1::'b::refinement_lattice)" shows "(d::'b::refinement_lattice) \<iinter> (c\<^sub>0::'b::refinement_lattice) \<sqsubseteq> d \<iinter> (c\<^sub>1::'b::refinement_lattice)"
primepow_factors_altdef: fixes x :: "'a" assumes "x \<noteq> (0::'a)" shows "primepow_factors x = {p ^ k |p k. p \<in># prime_factorization x \<and> k \<in> {0<..multiplicity p x}}"
permutes_image: fixes p :: "'b \<Rightarrow> 'b"   and S :: "'b set" assumes "p permutes S" shows "p ` S = S"
T3: shows "serial (\<lambda>w v. w r v \<and> (\<exists>x. existsAt x v \<and> (\<forall>xa. (\<P> xa \<^bold>\<rightarrow> xa x) v)))"
l2_secrecy_def: shows "l2_secrecy \<equiv> {s. dy_fake_msg (bad s) (ik s) (chan s) \<inter> secret s = {}}"
monad_fail_alt_def: fixes return :: "'a \<Rightarrow> 'b"   and bind :: "'b \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b"   and fail :: "'b"   and alt :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" shows "monad_fail_alt return bind fail alt \<equiv> monad_fail return bind fail \<and> monad_alt return bind alt \<and> monad_fail_alt_axioms fail alt"
convex_box_1: fixes a :: "'b"   and b :: "'b" shows "convex (cbox a b)"
fls_mult_distrib_subdegree_ge_0_1: fixes f :: "'a fls"   and g :: "'a fls"   and h :: "'a fls" assumes "0 \<le> fls_subdegree f"   and "0 \<le> fls_subdegree g"   and "0 \<le> fls_subdegree h" shows "(f + g) * h = f * h + g * h"
simps_3: fixes css :: "(name, (Term_Class.term \<times> sterm) list) fmap"   and name :: "name"   and \<Gamma> :: "(name, value) fmap" shows "vwelldefined' (Vrecabs css name \<Gamma>) = (pred_fmap id (fmmap vwelldefined' \<Gamma>) \<and> pred_fmap (\<lambda>cs. list_all (\<lambda>(pat, t). consts t |\<subseteq>| fmdom \<Gamma> |\<union>| (C |\<union>| fmdom css)) cs \<and> fdisjnt C (fmdom \<Gamma>)) css \<and> name |\<in>| fmdom css \<and> fdisjnt C (fmdom css))"
assms: fixes A :: "'Q set" assumes "A \<in> F" shows "finite A"
succ_4: shows "ord_rec_Seq T G s (succ k) y"
deformation_retract_relative_homology_group_isomorphism_id: fixes X :: "'a topology"   and Y :: "'a topology"   and r :: "'a \<Rightarrow> 'a"   and V :: "'a set"   and U :: "'a set"   and p :: "int" assumes "retraction_maps X Y r id"   and "V \<subseteq> U"   and "r ` U \<subseteq> V"   and "homotopic_with (\<lambda>h. h ` U \<subseteq> U) X X r id" shows "hom_induced p Y V X U id \<in> Group.iso (relative_homology_group p Y V) (relative_homology_group p X U)"
gt'_wf: shows "wfP (\<lambda>x y. gt' y x)"
mono_neutral_cong_right: fixes T :: "'d set"   and S :: "'d set"   and g :: "'d \<Rightarrow> 'e"   and h :: "'d \<Rightarrow> 'e" assumes "finite T"   and "S \<subseteq> T"   and "\<forall>i\<in>T - S. g i = (0::'e)"   and "\<And>x. x \<in> S \<Longrightarrow> g x = h x" shows "sum g T = sum h S"
split_half_ts: shows "split_half ts = (ls, (sub, sep) # rs)"
\<epsilon>_leg_simps_2: shows "\<epsilon>.cod.leg1 = f.dtrg"
grI: fixes K :: "('b, 'c) pre_digraph" assumes "K\<^bsub>3,3\<^esub> K \<or> K\<^bsub>5\<^esub> K" shows "graph K"
antisym: fixes b :: "'d multiset"   and a :: "'d multiset" assumes "b \<subseteq># a"   and "a \<subseteq># b" shows "a = b"
take_hd_drop: fixes n :: "nat"   and xs :: "'g list" assumes "n < length xs" shows "take n xs @ [hd (drop n xs)] = take (Suc n) xs"
internal_iface_name_wildcard_longest_def: fixes i1 :: "char list"   and i2 :: "char list" shows "internal_iface_name_wildcard_longest i1 i2 = (if take (min (length i1 - 1) (length i2 - 1)) i1 = take (min (length i1 - 1) (length i2 - 1)) i2 then Some (if length i1 \<le> length i2 then i2 else i1) else None)"
all_sample_points_prop: fixes qs :: "real poly list"   and S :: "real list" assumes "check_all_const_deg qs = False"   and "S = characterize_root_list_p (pderiv (prod_list_var qs) * prod_list_var qs * [:real_of_int (- crb (prod_list_var qs)), 1:] * [:real_of_int (crb (prod_list_var qs)), 1:])" shows "consistent_sign_vectors_R qs UNIV = consistent_sign_vectors_R qs (set S)"
fst_sym_preproc: fixes gs :: "('t \<Rightarrow>\<^sub>0 'd) list"   and fs :: "('t \<Rightarrow>\<^sub>0 'd) list" shows "fst (sym_preproc gs fs) = Keys_to_list (snd (sym_preproc gs fs))"
subsets_3: shows "{m. coeff (p + p') m \<noteq> (0::'a)} \<subseteq> S"
reads_writes_separate_backwards: fixes SR :: "('x \<Rightarrow> 'x \<Rightarrow> bool) set"   and getter :: "('x, 'y, 'z) prog"   and h :: "'x"   and h' :: "'x"   and SW :: "('x, 'aa, 'ab) prog set"   and setter :: "('x, 'aa, 'ac) prog"   and x :: "'z" assumes "reads SR getter h h'"   and "writes SW setter h h'"   and "h \<turnstile> setter \<rightarrow>\<^sub>h h'"   and "h' \<turnstile> getter \<rightarrow>\<^sub>r x"   and "\<And>h h'. \<forall>w\<in>SW. h \<turnstile> w \<rightarrow>\<^sub>h h' \<longrightarrow> (\<forall>r\<in>SR. r h h')" shows "h \<turnstile> getter \<rightarrow>\<^sub>r x"
X_single: fixes x :: "real" assumes "x \<notin> J" shows "X = {x}"
is_le_rel_distinct: fixes rel :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "is_le_rel rel"   and "rel x y"   and "x \<noteq> y" shows "x < y"
sorted_spvec_step: fixes a :: "nat \<times> 'a"   and as :: "(nat \<times> 'a) list" shows "sorted_spvec (a # as) = (case as of [] \<Rightarrow> True | b # bs \<Rightarrow> fst a < fst b \<and> sorted_spvec as)"
assms: shows "split_n_m_x n m x = (a, b)"
divisors_int_def: fixes x :: "int" shows "divisors_int x \<equiv> let xs = divisors_int_pos x in xs @ map uminus xs"
power2_eq_imp_eq: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and x :: "'b"   and y :: "'b" assumes "class.linordered_semidom plus minus less_eq less zero times one"   and "power.power one times x 2 = power.power one times y 2"   and "less_eq zero x"   and "less_eq zero y" shows "x = y"
subgroup_conjugation_is_bij: fixes g :: "'a" assumes "g \<in> carrier G" shows "bij_betw (\<lambda>H\<in>{H. subgroup H G}. g <# H #> inv g) {H. subgroup H G} {H. subgroup H G}"
mono_sinvar: fixes sinvar :: "'c graph \<Rightarrow> ('c \<Rightarrow> 'd) \<Rightarrow> bool"   and N :: "'c set"   and E :: "('c \<times> 'c) set"   and E' :: "('c \<times> 'c) set"   and nP :: "'c \<Rightarrow> 'd" assumes "SecurityInvariant_preliminaries sinvar"   and "wf_graph \<lparr>nodes = N, edges = E\<rparr>"   and "E' \<subseteq> E"   and "sinvar \<lparr>nodes = N, edges = E\<rparr> nP" shows "sinvar \<lparr>nodes = N, edges = E'\<rparr> nP"
Int_commute: fixes A :: "'a set"   and B :: "'a set" shows "A \<inter> B = B \<inter> A"
card_of_well_order_on: fixes A :: "'a set" shows "well_order_on A |A|"
execn_merge_guards_to_execn: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "('e, 'd, 'f) com"   and s :: "('e, 'f) xstate"   and n :: "nat"   and t :: "('e, 'f) xstate" assumes "\<Gamma>\<turnstile> \<langle>merge_guards c,s\<rangle> =n\<Rightarrow> t" shows "\<Gamma>\<turnstile> \<langle>c,s\<rangle> =n\<Rightarrow> t"
distinct_5: fixes x21 :: "'a"   and x22 :: "'a convnet"   and x31 :: "'a convnet"   and x32 :: "'a convnet" shows "Conv x21 x22 \<noteq> Pool x31 x32"
simps_4: fixes c\<^sub>1 :: "('a, 'b, 'c) com"   and c\<^sub>2 :: "('a, 'b, 'c) com" shows "redexes (Seq c\<^sub>1 c\<^sub>2) = {Seq c\<^sub>1 c\<^sub>2} \<union> redexes c\<^sub>1"
a_r_coset_def': fixes G :: "('e, 'f) ring_scheme"   and H :: "'e set"   and a :: "'e" shows "H +>\<^bsub>G\<^esub> a \<equiv> \<Union>h\<in>H. {h \<oplus>\<^bsub>G\<^esub> a}"
cf_pos_poly_main: fixes f :: "int poly" shows "smult (sgn (lead_coeff f) * content f) (cf_pos_poly f) = f"
language_simulation_coinduction: fixes R :: "'a language \<Rightarrow> 'a language \<Rightarrow> bool"   and L :: "'a language"   and K :: "'a language" assumes "R L K"   and "\<And>L K. R L K \<Longrightarrow> \<oo> L \<le> \<oo> K \<and> (\<forall>x. R (\<dd> L x) (\<dd> K x))" shows "L \<le> K"
nat_pow_def: fixes G :: "('c, 'd) monoid_scheme"   and a :: "'c"   and n :: "nat" shows "a [^]\<^bsub>G\<^esub> n = rec_nat \<one>\<^bsub>G\<^esub> (\<lambda>u b. b \<otimes>\<^bsub>G\<^esub> a) n"
arrI: fixes f :: "'a" assumes "right a f" shows "S.arr f"
minus_add_cancel: fixes minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and uminus :: "'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c" assumes "class.group_add minus plus zero uminus" shows "plus (uminus a) (plus a b) = b"
Ri_subset_\<Gamma>: fixes N :: "'a literal multiset set" shows "Ri N \<subseteq> \<Gamma>"
in_Sup_multiset_iff: fixes A :: "'b multiset set"   and x :: "'b" assumes "subset_mset.bdd_above A" shows "(x \<in># Sup A) = (\<exists>X\<in>A. x \<in># X)"
F: shows "map (off P) (drop k (take (Suc (Min {k' \<in> {k..<length cfs}. \<not> (size P \<le> fst (cfs ! k') \<and> fst (cfs ! k') - size P < size (P' x))})) cfs)) ! (length (map (off P) (drop k (take (Suc (Min {k' \<in> {k..<length cfs}. \<not> (size P \<le> fst (cfs ! k') \<and> fst (cfs ! k') - size P < size (P' x))})) cfs))) - 1) = off P (cfs ! Min {k' \<in> {k..<length cfs}. \<not> (size P \<le> fst (cfs ! k') \<and> fst (cfs ! k') - size P < size (P' x))})"
prime_product_factor: fixes n :: "nat"   and sq :: "nat"   and p :: "nat" assumes "prime_product_factor n = (sq, p)" shows "n = sq * sq * p \<and> prime_product p"
HMA_diagonal_to_Smith_i_PQ: fixes i :: "nat"   and xs :: "nat list" assumes "i < min CARD('b) CARD('c)"   and "\<forall>j\<in>set xs. j < min CARD('b) CARD('c)" shows "((=) ===> HMA_M3 ===> HMA_M3) (diagonal_to_Smith_i_PQ_JNF xs i) (diagonal_to_Smith_i_PQ xs i)"
assms: fixes i' :: "nat"   and j' :: "nat" assumes "i' < n"   and "j' < j" shows "A $$ (i', j') = B $$ (i', j')"
NRO_validity_good: fixes NRO :: "msg"   and A :: "agent"   and B :: "agent"   and L :: "nat"   and C :: "msg"   and evs :: "event list" assumes "NRO = Crypt (priEK A) \<lbrace>Number f_nro, Agent B, Nonce L, C\<rbrace>"   and "NRO \<in> parts (knows Spy evs)"   and "A \<notin> bad"   and "evs \<in> zg" shows "Says A B \<lbrace>Number f_nro, Agent B, Nonce L, C, NRO\<rbrace> \<in> set evs"
boundedE: fixes A :: "'a set"   and x :: "'a" assumes "finite A"   and "A \<noteq> {}"   and "x \<le> Min A" obtains "\<And>a. a \<in> A \<Longrightarrow> x \<le> a"
\<eta>_leg_simps_1: shows "\<eta>.dom.leg0 = f.dsrc"
cblinfun_eq_mat_of_cblinfunI: fixes a :: "'b \<Rightarrow>\<^sub>C\<^sub>L 'c"   and b :: "'b \<Rightarrow>\<^sub>C\<^sub>L 'c" assumes "mat_of_cblinfun a = mat_of_cblinfun b" shows "a = b"
arity_zero_shape: fixes rs :: "(Term_Class.term list \<times> 'a) fset" assumes "Rewriting_Nterm.arity_compatibles rs"   and "arity rs = 0"   and "is_fmap rs"   and "rs \<noteq> {||}" obtains t :: "'a" where "rs = {|([], t)|}"
swap_nilpotent: fixes a :: "'a"   and b :: "'a"   and f :: "'a \<Rightarrow> 'b" shows "Fun.swap a b (Fun.swap a b f) = f"
iso_refl: fixes G :: "('e, 'f) monoid_scheme" shows "G \<cong> G"
elims_3: fixes x :: "rat \<times> rat"   and xa :: "rat \<times> rat" assumes "\<not> sub_interval x xa" obtains l :: "rat" and r :: "rat" and l' :: "rat" and r' :: "rat" where "x = (l, r)" and "xa = (l', r')" and "\<not> (l' \<le> l \<and> r \<le> r')"
distinct_mset_size_2: fixes a :: "'b"   and b :: "'b" shows "distinct_mset {#a, b#} = (a \<noteq> b)"
lunit_def: fixes f :: "'a" shows "Left.lunit f \<equiv> THE \<mu>. Left.in_hom \<mu> (Left.L f) f \<and> Left.L \<mu> = Left.comp (\<i>[trg f] \<star> f) (local.inv \<a>[trg f, trg f, f])"
shift_map_keys_punit_def: shows "shift_map_keys_punit \<equiv> gd_nat_term.shift_map_keys to_pair_unit fst"
suffix_comparable_def: fixes u :: "'b list"   and v :: "'b list" shows "u \<bowtie>\<^sub>s v \<equiv> rev u \<bowtie> rev v"
bind_commute_spmf: fixes p :: "'d spmf"   and q :: "'e spmf"   and f :: "'d \<Rightarrow> 'e \<Rightarrow> 'b spmf" shows "p \<bind> (\<lambda>x. q \<bind> f x) = q \<bind> (\<lambda>y. p \<bind> (\<lambda>x. f x y))"
bounded_clinear_def: fixes f :: "'a \<Rightarrow> 'b" shows "bounded_clinear f \<equiv> clinear f \<and> bounded_clinear_axioms f"
finite_comm_group_adjoin_axioms: shows "finite_comm_group_adjoin G H a"
arev_eq: fixes G :: "('a, 'b) pre_digraph"   and arev :: "'b \<Rightarrow> 'b"   and a :: "'b" assumes "bidirected_digraph G arev"   and "a \<notin> arcs G" shows "arev a = a"
fset_rev_mp: fixes c :: "'a"   and A :: "'a fset"   and B :: "'a fset" assumes "c |\<in>| A"   and "A |\<subseteq>| B" shows "c |\<in>| B"
signed_take_bit_Suc_bit0: fixes n :: "nat"   and k :: "num" shows "signed_take_bit (Suc n) (numeral (num.Bit0 k)) = signed_take_bit n (numeral k) * 2"
nub_body_def: shows "nub_body \<equiv> \<mu> nub_body. \<Lambda> f v. Fixrec.run\<cdot>(match_lnil\<cdot>v\<cdot>(Fixrec.succeed\<cdot>lnil) +++ match_lcons\<cdot>v\<cdot>(\<Lambda> x xs. Fixrec.succeed\<cdot>(x :@ f\<cdot>(lfilter\<cdot>(neg oo Abs_cfun ((=\<^sub>B) x))\<cdot>xs))))"
qSwap_preserves_alpha: fixes X :: "('a, 'b, 'varSort, 'var, 'c) qTerm"   and X' :: "('a, 'b, 'varSort, 'var, 'c) qTerm"   and zs :: "'varSort"   and z1 :: "'var"   and z2 :: "'var" assumes "qGood X \<or> qGood X'"   and "X #= X'" shows "X #[[z1 \<and> z2]]_zs #= X' #[[z1 \<and> z2]]_zs"
p0: shows "p \<noteq> 0 \<or> x \<noteq> 1"
map\<^sub>0_simps_3: fixes a :: "'c" assumes "C.obj a" shows "trg\<^sub>D (\<rho>\<^sub>0 a) = H.map\<^sub>0 a"
plossless_gpv_mono: fixes \<I> :: "('a, 'b) \<I>"   and gpv :: "('c, 'a, 'b) gpv"   and \<I>' :: "('a, 'b) \<I>" assumes "plossless_gpv \<I> gpv"   and "\<I> \<le> \<I>'"   and "\<I> \<turnstile>g gpv \<surd>" shows "plossless_gpv \<I>' gpv"
plus_eint_simps_1: fixes m :: "int"   and n :: "int" shows "eint m + eint n = eint (m + n)"
ivl_disj_un_2: fixes u :: "'a" shows "{..<u} \<union> {u} = {..u}"
swb: fixes ka :: "nat"   and na :: "nat"   and Aa :: "'a mat" assumes "ka + na < dim_row Aa" shows "swb Aa ka na = mat (dim_row Aa) (dim_col Aa) (\<lambda>(i, j). let r = if i < ka \<or> ka + na < i then i else if i = ka + na then ka else Suc i in Aa $$ (r, j))"