hcju/isabelleps huggingface dataset:
mset_set_Union: fixes A :: "'k set"   and B :: "'k set" assumes "finite A"   and "finite B"   and "A \<inter> B = {}" shows "mset_set (A \<union> B) = mset_set A + mset_set B"
id: shows "smult c p = smult c (primitive_part q * smult d (primitive_part r))"
\<sigma>_integer: fixes n :: "nat"   and fs :: "'a vec list"   and l :: "nat"   and j :: "nat"   and i :: "nat" assumes "gram_schmidt_fs_int n fs"   and "l \<le> j"   and "j \<le> i"   and "i < length fs" shows "gram_schmidt_fs_lin_indpt.\<sigma> n fs l i j \<in> \<int>"
maximal_prime: fixes mx :: "'a set" assumes "maximal_ideal R mx" shows "prime_ideal R mx"
Pow_top: fixes A :: "'b set" shows "A \<in> Pow A"
mono_sup: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "mono f"   and "mono g" shows "mono (f \<squnion> g)"
hyps: fixes Q :: "(char list \<Rightarrow> int) \<Rightarrow> enat" assumes "vc C Q" shows "\<turnstile>\<^sub>2 {pre C Q} strip C {Q}"
assms_2: fixes i :: "nat" assumes "Y 0 = false"   and "Y (Suc i) \<sqsubseteq> Y i" shows "P"
cyclic_on_singleI: fixes s :: "'b"   and S :: "'b set"   and f :: "'b \<Rightarrow> 'b" assumes "s \<in> S"   and "S = orbit f s" shows "cyclic_on f S"
equality: fixes r :: "('b, 'c) set_arrow_scheme"   and r' :: "('b, 'c) set_arrow_scheme" assumes "set_dom r = set_dom r'"   and "set_func r = set_func r'"   and "set_cod r = set_cod r'"   and "set_arrow.more r = set_arrow.more r'" shows "r = r'"
out_diff1: fixes A :: "'p"   and B :: "'p"   and C :: "'p" assumes "A Out B C" shows "B \<noteq> A"
R_G_cfg_on_valid: fixes cfg :: "('s \<times> ('c \<Rightarrow> real) set) cfg" assumes "cfg \<in> R_G_cfg_on_div s'" shows "cfg \<in> R_G.valid_cfg"
obtain_domain_permutation_for_two_bijections: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and B :: "'b set"   and f' :: "'a \<Rightarrow> 'b" assumes "bij_betw f A B"   and "bij_betw f' A B" obtains p :: "'a \<Rightarrow> 'a" where "p permutes A" and "\<forall>a\<in>A. f a = f' (p a)"
extended_simps_2: fixes x' :: "'b"   and y' :: "'b" shows "(\<lbrakk>x'\<rbrakk> < \<lbrakk>y'\<rbrakk>) = (x' < y')"
i: shows "i < dim_vec (Matrix.row (B * V) 0)"
uw\<theta>w'\<theta>'\<beta>: shows "ide u \<and> ide w \<and> ide w' \<and> \<guillemotleft>\<theta> : f \<star> w \<Rightarrow> u\<guillemotright> \<and> \<guillemotleft>\<theta>' : f \<star> w' \<Rightarrow> u\<guillemotright> \<and> \<guillemotleft>\<beta> : g \<star> w \<Rightarrow> g \<star> w'\<guillemotright> \<and> (r \<star> \<theta>) \<cdot> \<a>[r, f, w] \<cdot> (\<rho> \<star> w) = (r \<star> \<theta>') \<cdot> \<a>[r, f, w'] \<cdot> (\<rho> \<star> w') \<cdot> \<beta>"
1: shows "unlabel (dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t (transaction_strand T)) = T1 @ T2 @ T3 @ T4 @ T5"
intros_4: fixes \<pi> :: "nat \<Rightarrow> 'a"   and m :: "nat"   and prob :: "('a, 'b, 'c, 'd) ifc_problem"   and \<pi>' :: "nat \<Rightarrow> 'a"   and m' :: "nat"   and n :: "nat"   and v :: "'b" assumes "(\<pi>, m) \<in> IFC_def.scp prob"   and "(\<pi>', m') \<in> IFC_def.scp prob"   and "IFC_def.is_dcdi_via prob n \<pi> v m \<pi>' m'" shows "(\<pi>, n) \<in> IFC_def.scp prob"
boundOutputApp: fixes xvec :: "name list"   and yvec :: "name list"   and B :: "('d, 'e, 'f) boundOutput" shows "\<lparr>\<nu>*(xvec @ yvec)\<rparr>B = \<lparr>\<nu>*xvec\<rparr>\<lparr>\<nu>*yvec\<rparr>B"
theta_double_lemma: fixes n :: "nat" assumes "1200 \<le> n" shows "theta (n div 2) < theta n"
coboundedI1: fixes a :: "'b::semilattice_inf"   and c :: "'b::semilattice_inf"   and b :: "'b::semilattice_inf" assumes "(a::'b::semilattice_inf) \<le> (c::'b::semilattice_inf)" shows "(a::'b::semilattice_inf) \<sqinter> (b::'b::semilattice_inf) \<le> (c::'b::semilattice_inf)"
fixp_induct: fixes P :: "(real \<Rightarrow> nat spmf) \<Rightarrow> bool" assumes "spmf.admissible P"   and "P (\<lambda>geometric_spmf. return_pmf None)"   and "\<And>p. P p \<Longrightarrow> P (\<lambda>pa. bernoulli pa \<bind> (\<lambda>b. if b then return_spmf 0 else map_spmf ((+) 1) (p pa)))" shows "P geometric_spmf"
nf: shows "(\<lambda>x. norm (f x)) integrable_on S"
strongly_connected: shows "\<forall>p q. p \<noteq> q \<longrightarrow> has_channel\<^sup>+\<^sup>+ p q"
less_fract_def: fixes z :: "'b fract"   and w :: "'b fract" shows "(z < w) = (z \<le> w \<and> \<not> w \<le> z)"
compositionality_BSD: fixes ES1 :: "'b ES_rec"   and ES2 :: "'b ES_rec"   and \<V> :: "'b V_rec"   and \<V>1 :: "'b V_rec"   and \<V>2 :: "'b V_rec" assumes "Compositionality ES1 ES2 \<V> \<V>1 \<V>2"   and "BSD \<V>1 Tr\<^bsub>ES1\<^esub>"   and "BSD \<V>2 Tr\<^bsub>ES2\<^esub>" shows "BSD \<V> Tr\<^bsub>ES1 \<parallel> ES2\<^esub>"
expandExecutionReachable: fixes cfgLast :: "('p, 'v, 's) configuration"   and cfg :: "('p, 'v, 's) configuration"   and exec :: "('p, 'v, 's) configuration list"   and trace :: "('p, 'v) message list" assumes "reachable cfgLast cfg"   and "execution trans sends start exec trace"   and "cfgLast = last exec" shows "\<exists>exec' trace'. execution trans sends start exec' trace' \<and> (prefixList exec exec' \<and> prefixList trace trace' \<or> exec = exec' \<and> trace = trace') \<and> last exec' = cfg"
subcls_induct: fixes wf_md :: "'b prog \<Rightarrow> char list \<Rightarrow> 'b mdecl \<Rightarrow> bool"   and P :: "'b prog"   and Q :: "char list \<Rightarrow> bool"   and C :: "char list" assumes "wf_prog wf_md P"   and "\<And>C. \<forall>D. (C, D) \<in> (subcls1 P)\<^sup>+ \<longrightarrow> Q D \<Longrightarrow> Q C" shows "Q C"
assoc_in_hom: shows "\<guillemotleft>local.inv \<a>[b, b, f] : b \<star> b \<star> f \<Rightarrow>\<^sub>S (b \<star> b) \<star> f\<guillemotright>"
t_def: shows "t = min (Gromov_product_at e x y) (Gromov_product_at e y z)"
finite_fmdom': fixes m :: "('c, 'd) fmap" shows "finite (fmdom' m)"
hoare\<^sub>3'_def: fixes P :: "'d \<Rightarrow> bool"   and M :: "'d \<Rightarrow> ('e \<times> 'd) option" shows "\<lbrace>P\<rbrace> M\<dagger> \<equiv> \<forall>\<sigma>. P \<sigma> \<longrightarrow> (case M \<sigma> of None \<Rightarrow> True | Some a \<Rightarrow> False)"
exe_antisym_imp_antisym: fixes l :: "('b \<times> 'b) list" assumes "exe_antisym l" shows "antisym (set l)"
nths_range: fixes xs :: "'c list"   and A :: "nat set" shows "nths xs A = nths xs (A \<inter> {..<length xs})"
inf_def: fixes cs :: "com list"   and s :: "state" shows "PHoareTotal.inf cs s = (\<exists>f. f 0 = (cs, s) \<and> (\<forall>i. f i \<rightarrow> f (Suc i)))"
funpow_dist_prop: fixes y :: "'c"   and f :: "'c \<Rightarrow> 'c"   and x :: "'c" assumes "y \<in> orbit f x" shows "(f ^^ funpow_dist f x y) x = y"
gen_subst_commutes: fixes S :: "nat \<Rightarrow> typ"   and t :: "typ"   and A :: "type_scheme list" assumes "free_tv S \<inter> (free_tv t - free_tv A) = {}" shows "gen ($ S A) ($ S t) = $ S (gen A t)"
InfCard_is_Limit: fixes \<kappa> :: "V" assumes "InfCard \<kappa>" shows "Limit \<kappa>"
intensional_rews_5: fixes f :: "'c \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> 'b"   and x :: "'g \<Rightarrow> 'c"   and y :: "'g \<Rightarrow> 'd"   and z :: "'g \<Rightarrow> 'e"   and zz :: "'g \<Rightarrow> 'f"   and w :: "'g" shows "(f<x, y, z, zz>) w \<equiv> f (x w) (y w) (z w) (zz w)"
wt_subst_compose: fixes \<theta> :: "'var \<Rightarrow> ('fun, 'var) Term.term"   and \<delta> :: "'var \<Rightarrow> ('fun, 'var) Term.term" assumes "wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<theta>"   and "wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<delta>" shows "wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t (\<theta> \<circ>\<^sub>s \<delta>)"
is_lb_def: fixes \<delta> :: "'e \<Rightarrow> 'f \<times> 'g \<Rightarrow> 'e"   and s :: "'e"   and s1 :: "'e"   and s2 :: "'e" shows "pre_RDR.is_lb \<delta> s s1 s2 \<equiv> pre_RDR.less_eq \<delta> s s2 \<and> pre_RDR.less_eq \<delta> s s1"
snd_inv_letter: fixes x :: "accent \<times> 'a" shows "snd (inv_letter x) = snd x"
that: fixes g :: "'g" assumes "g \<in> G"   and "g \<noteq> (0::'g)" shows "thesis"
the_inv_leftinv: fixes f :: "'a \<Rightarrow> 'b" assumes "bij f" shows "the_inv f \<circ> f = id"
fresh_2: fixes a :: "name"   and x3 :: "name"   and x2 :: "name"   and x1 :: "pi" shows "a \<sharp> x3{x2}.x1 = (a \<sharp> x3 \<and> a \<sharp> x2 \<and> a \<sharp> x1)"
convex_Affine: fixes X :: "'a \<times> 'a pdevs" shows "convex (Affine X)"
isMinim_unique: fixes B :: "'a set"   and a :: "'a"   and a' :: "'a" assumes "isMinim B a"   and "isMinim B a'" shows "a = a'"
string_rel_def: shows "string_rel = {(x, y). y = literal.explode x}"
elims_3: fixes x :: "'b option"   and xa :: "'b option" assumes "\<not> x \<sqsubseteq> xa" obtains xb :: "'b" and y :: "'b" where "x = Some xb" and "xa = Some y" and "\<not> xb \<sqsubseteq> y"   | uu_ :: "'b" where "x = Some uu_" and "xa = None"
lemma2_6_7_a: fixes r :: "('a \<times> 'a) set"   and Q :: "'a multiset"   and N :: "'a multiset"   and M :: "'a multiset" assumes "trans r"   and "set_mset Q \<subseteq> r \<down>m N - r \<down>m M"   and "(M, N) \<in> mul_eq r" shows "(Q + M, N) \<in> mul_eq r"
domain_def: fixes U :: "nat set" shows "domain U \<equiv> {Det True, Det False} \<union> Indet ` U"
permute_C_def: fixes p :: "perm"   and c :: "C" shows "p \<bullet> c = c"
leadsTo_Trans: fixes F :: "'b program"   and A :: "'b set"   and B :: "'b set"   and C :: "'b set" assumes "F \<in> A \<longmapsto> B"   and "F \<in> B \<longmapsto> C" shows "F \<in> A \<longmapsto> C"
vs_confI: fixes vs :: "'addr \<times> addr_loc \<Rightarrow> 'addr val set"   and P :: "'b prog"   and h :: "'heap" assumes "\<And>ad al v. v \<in> vs (ad, al) \<Longrightarrow> \<exists>T. h.addr_loc_type P h ad al T \<and> h.conf P h v T" shows "h.vs_conf P h vs"
lm089: fixes f :: "('d \<times> 'c) set"   and X :: "'d set" shows "Range f - f `` X \<subseteq> Range (Outside' X f)"
inject: fixes x1 :: "'e"   and y1 :: "'e" shows "(prot_atom.Atom x1 = prot_atom.Atom y1) = (x1 = y1)"
int_iffD1: fixes F :: "'a \<Rightarrow> bool"   and G :: "'a \<Rightarrow> bool" assumes "\<turnstile> F = G" shows "\<turnstile> F \<longrightarrow> G"
HBL1_rev_prv: fixes \<phi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "Fvars \<phi> = {}"   and "prv (PP \<langle>\<phi>\<rangle>)" shows "prv \<phi>"
less_trans: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c"   and z :: "'c" assumes "class.preorder less_eq less"   and "less x y"   and "less y z" shows "less x z"
RangeI: fixes a :: "'c"   and b :: "'d"   and r :: "('c \<times> 'd) set" assumes "(a, b) \<in> r" shows "b \<in> Range r"
hom_add: fixes hom :: "'c \<Rightarrow> 'd"   and x :: "'c"   and y :: "'c" assumes "plus_hom hom" shows "hom (x + y) = hom x + hom y"
subsumes_list_alt: fixes Ls :: "('b, 'c) Term.term literal list"   and Ks :: "('b, 'c) Term.term literal list"   and \<sigma> :: "'c \<Rightarrow> ('b, 'c) Term.term option" shows "subsumes_list Ls Ks \<sigma> = subsumes_list_modulo Ls Ks \<sigma>"
frontier_subset_compact: fixes S :: "'b set" assumes "compact S" shows "frontier S \<subseteq> S"
trans: fixes a :: "('a, 'b) Term.term"   and b :: "('a, 'b) Term.term"   and c :: "('a, 'b) Term.term" assumes "a \<sqsubseteq> b"   and "b \<sqsubseteq> c" shows "a \<sqsubseteq> c"
simps_1: fixes x1 :: "'b"   and y1 :: "'b" shows "(flat_complete_lattice.Value x1 = flat_complete_lattice.Value y1) = (x1 = y1)"
rspp_Actual_out_notin_slice: fixes x :: "nat"   and a :: "'edge"   and xs :: "'var list"   and S :: "'node SDG_node set"   and f :: "'var \<Rightarrow> 'val option"   and g :: "'var \<Rightarrow> 'val option" assumes "x < length (ParamDefs (targetnode a))"   and "valid_edge a"   and "length (ParamDefs (targetnode a)) = length xs"   and "Actual_out (targetnode a, x) \<notin> S" shows "rspp (targetnode a) S xs f g (ParamDefs (targetnode a) ! x) = f (ParamDefs (targetnode a) ! x)"
orderE: fixes a :: "'b \<times> 'c"   and b :: "'b \<times> 'c" assumes "less_eq_prod a b" obtains "a = inf_prod a b"
tup_to_sum_to_tup: fixes F :: "('g discr \<rightarrow> 'h) \<times> ('i discr \<rightarrow> 'h)" shows "sum_to_tup\<cdot>(tup_to_sum\<cdot>F) = F"
range_prefix_match_def: fixes pfx :: "'b prefix_match"   and rg :: "'b wordinterval" shows "range_prefix_match pfx rg \<equiv> let pfxrg = prefix_to_wordinterval pfx in (wordinterval_intersection rg pfxrg, wordinterval_setminus rg pfxrg)"
c4: shows "c4 = map (\<lambda>d. \<forall>[]\<langle>\<or>\<noteq>: [(pair (t, s), pair d)]\<rangle>\<^sub>s\<^sub>t) (unlabel (filter (\<lambda>d. d \<notin> set Di) (dbproj i D)))"
gcd_dvd_prod: fixes a :: "'a"   and b :: "'a"   and k :: "'a" shows "gcd a b dvd k * b"
ennreal_mult_eq_top_iff: fixes a :: "ennreal"   and b :: "ennreal" shows "(a * b = top) = (a = top \<and> b \<noteq> 0 \<or> b = top \<and> a \<noteq> 0)"
mult1E: fixes N :: "'k multiset"   and M :: "'k multiset"   and r :: "('k \<times> 'k) set" assumes "(N, M) \<in> mult1 r" obtains a :: "'k" and M0 :: "'k multiset" and K :: "'k multiset" where "M = add_mset a M0" and "N = M0 + K" and "\<And>b. b \<in># K \<Longrightarrow> (b, a) \<in> r"
inline1_unfold: fixes callee :: "'i \<Rightarrow> 'm \<Rightarrow> ('l \<times> 'i, 'j, 'k) gpv"   and gpv :: "('h, 'm, 'l) gpv"   and s :: "'i" shows "inline1 callee gpv s = the_gpv gpv \<bind> case_generat (\<lambda>x. return_spmf (Inl (x, s))) (\<lambda>out rpv. the_gpv (callee s out) \<bind> case_generat (\<lambda>(x, y). inline1 callee (rpv x) y) (\<lambda>out rpv'. return_spmf (Inr (out, rpv', rpv))))"
abs_ast_goal_def: shows "abs_ast_goal \<equiv> map_of astG"
sup1I2: fixes B :: "'a \<Rightarrow> bool"   and x :: "'a"   and A :: "'a \<Rightarrow> bool" assumes "B x" shows "sup A B x"
lemma_4_8_iii: fixes a :: "'a"   and b :: "'a" shows "(a l\<rightarrow> b) l\<rightarrow> b l\<rightarrow> a = b l\<rightarrow> a"
sp_optim: fixes n :: "nat"   and p1 :: "('dyn, 'c, 'fun, 'op, 'opinl, 'opubx, 'ubx1, 'ubx2) Ubx.instr list"   and \<Sigma>1 :: "Unboxed.type option list"   and p2 :: "('dyn, 'c, 'fun, 'op, 'opinl, 'opubx, 'ubx1, 'ubx2) Ubx.instr list"   and \<Sigma>2 :: "Unboxed.type option list"   and F :: "'fun \<Rightarrow> 'd fundef option" assumes "optim n p1 \<Sigma>1 = Ok (p2, \<Sigma>2)"   and "\<And>x. rel_option (\<lambda>ar fd. arity fd = ar) (get_arity x) (F x)" shows "Subx.sp F p2 \<Sigma>1 = Ok \<Sigma>2"
rtranclp_tranclp_tranclp: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "r\<^sup>*\<^sup>* x y"   and "r\<^sup>+\<^sup>+ y z" shows "r\<^sup>+\<^sup>+ x z"
c2: shows "(\<exists>p\<in>set qs. 0 < degree p) \<longrightarrow> poly_f qs \<noteq> 0"
PDG_path_Append: fixes n'' :: "'node"   and n' :: "'node"   and n :: "'node" assumes "n'' \<longrightarrow>\<^sub>d* n'"   and "n \<longrightarrow>\<^sub>d* n''" shows "n \<longrightarrow>\<^sub>d* n'"
proper_of_compact: fixes S :: "'b set" assumes "compact S" shows "proper S"
map'_try_gpv: fixes f :: "'g \<Rightarrow> 'd"   and g :: "'h \<Rightarrow> 'e"   and h :: "'f \<Rightarrow> 'i"   and gpv :: "('g, 'h, 'i) gpv"   and gpv' :: "('g, 'h, 'i) gpv" shows "map_gpv' f g h (TRY gpv ELSE gpv') = TRY map_gpv' f g h gpv ELSE map_gpv' f g h gpv'"
preserves_section_retraction: fixes e :: "'a"   and m :: "'a" assumes "Adj.C.ide (e \<cdot>\<^sub>C m)" shows "Adj.D.ide (G e \<cdot>\<^sub>D G m)"
ranked: fixes q :: "'Q"   and f :: "'L"   and qs :: "'Q list" assumes "q \<rightarrow> f qs \<in> \<delta>" shows "A f = Some (length qs)"
simps_3: fixes f1 :: "'b"   and f2 :: "'b" shows "(case color.R of color.R \<Rightarrow> f1 | color.B \<Rightarrow> f2) = f1"
capacity_finite: fixes e :: "'v \<times> 'v" shows "capacity \<Delta> e \<noteq> \<top>"
mono_neutral_right: fixes T :: "'c set"   and S :: "'c set"   and g :: "'c \<Rightarrow> 'a" assumes "finite T"   and "S \<subseteq> T"   and "\<forall>i\<in>T - S. g i = \<^bold>1" shows "F g T = F g S"
transaction_projection_for_def: fixes t :: "transaction" shows "local.transaction_projection_for t = (let tvs = local.transaction_views_for t in if tvs = [] then blind_transaction t else the (transaction.Merge (set tvs)))"
rel_compp: fixes R1 :: "'a \<Rightarrow> 'e \<Rightarrow> bool"   and S1 :: "'e \<Rightarrow> 'c \<Rightarrow> bool"   and R2 :: "'b \<Rightarrow> 'f \<Rightarrow> bool"   and S2 :: "'f \<Rightarrow> 'd \<Rightarrow> bool" shows "rel_prod (R1 OO S1) (R2 OO S2) = rel_prod R1 R2 OO rel_prod S1 S2"
sublist_Cons_right: fixes xs :: "'b list"   and y :: "'b"   and ys :: "'b list" shows "xs \<le>f y # ys = (xs \<le>p y # ys \<or> xs \<le>f ys)"
antisympD: fixes r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c"   and b :: "'c" assumes "antisymp r"   and "r a b"   and "r b a" shows "a = b"
non_atomic_Aenc: fixes x :: "msg"   and y :: "msg" shows "composed (Aenc x y)"
antecedent: shows "P x y \<and> \<not> O x (y \<otimes> z)"
upd_invs_Some_eq: fixes ts :: "'g \<Rightarrow> ('h \<times> 'i \<Rightarrow>f nat) option"   and tas :: "('g, 'j, 'k) new_thread_action list"   and t :: "'g"   and x :: "'h \<times> 'i \<Rightarrow>f nat"   and I :: "'g \<Rightarrow> 'l option"   and Q :: "'l \<Rightarrow> 'g \<Rightarrow> 'j \<Rightarrow> 'k \<Rightarrow> bool" assumes "thread_oks ts tas"   and "ts t = \<lfloor>x\<rfloor>" shows "upd_invs I Q tas t = I t"
spaceN_iff: fixes x :: "'a"   and N :: "'a quasinorm" shows "(x \<in> space\<^sub>N N) = (eNorm N x < \<infinity>)"
acomplete_def: fixes t :: "'a tree" shows "acomplete t = (height t - min_height t \<le> 1)"
triangle_F: shows "Adj'.\<epsilon>FoF\<eta>.map = F"
dot_cross_self_1: fixes x :: "(real, 3) vec"   and y :: "(real, 3) vec" shows "x \<bullet> x \<times> y = 0"
subsums_trans: fixes c1 :: "('a, 'b) conf"   and c2 :: "('a, 'b) conf"   and c3 :: "('a, 'b) conf" assumes "c1 \<sqsubseteq> c2"   and "c2 \<sqsubseteq> c3" shows "c1 \<sqsubseteq> c3"
fb1: fixes a :: "real"   and b :: "real" assumes "0 \<le> a"   and "a \<le> 1"   and "0 \<le> b"   and "b \<le> 1" shows "(1 - a) * f b + a * b \<le> 1"
configuration_distinct: fixes q :: "'b"   and q' :: "'b"   and n :: "nat" assumes "q \<noteq> q'" shows "configuration q n \<inter> configuration q' n = {}"
le_utransD: fixes t :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and u :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and P :: "'a \<Rightarrow> real" assumes "le_utrans t u"   and "unitary P" shows "t P \<tturnstile> u P"
AbstTermP_Eats: fixes v :: "tm"   and i :: "tm"   and t1 :: "tm"   and u1 :: "tm"   and t2 :: "tm"   and u2 :: "tm" shows "{AbstTermP v i t1 u1, AbstTermP v i t2 u2} \<turnstile> AbstTermP v i (Q_Eats t1 t2) (Q_Eats u1 u2)"
hom_int_pow: fixes G :: "('a, 'b) monoid_scheme"   and H :: "('c, 'd) monoid_scheme"   and h :: "'a \<Rightarrow> 'c"   and x :: "'a"   and n :: "int" assumes "group_hom G H h"   and "x \<in> carrier G" shows "h (x [^]\<^bsub>G\<^esub> n) = h x [^]\<^bsub>H\<^esub> n"
assms: fixes x :: "'a nres" assumes "x \<in> set xs" shows "x \<le> SPEC P"
chop: shows "v'=v1'\<parallel>v2' \<and> \<phi> ts v1' \<and> \<psi> ts v2'"
lm097: fixes y :: "'b"   and f :: "('c \<times> 'b) set"   and x :: "'c" assumes "y \<in> f `` {x}"   and "runiq f" shows "f ,, x = y"
memberPsep: fixes x :: "('b, 'c) Combinators"   and p :: "('b, 'c) Combinators list" shows "memberP x (separate p) = memberP x p"
One_foldl1: fixes xs :: "three list" assumes "One \<in> set xs" shows "foldl1 f1 xs \<noteq> Zero"
test_bit_set_gen: fixes w :: "'b word"   and n :: "nat"   and x :: "bool"   and m :: "nat" shows "set_bit_class.set_bit w n x !! m = (if m = n then n < size w \<and> x else w !! m)"
simps_2: fixes l :: "nat"   and i :: "nat"   and j :: "nat" shows "\<sigma> (Suc l) i j = (d (Suc l) * \<sigma> l i j + \<mu>' i l * \<mu>' j l) / d l"
fv_subst_subset: fixes e :: "exp"   and y :: "Vars.var"   and x :: "Vars.var" shows "fv e[y::=x] \<subseteq> fv e - {y} \<union> {x}"
mono_SucI2: fixes X :: "nat \<Rightarrow> 'b" assumes "\<forall>n. X (Suc n) \<le> X n" shows "monoseq X"
set_filter_syzygy_basis: fixes n :: "nat"   and bs :: "('t \<Rightarrow>\<^sub>0 'c) list" shows "set (filter_syzygy_basis n bs) = {b \<in> set bs. proj_poly_syz n b = 0}"
order_refl: fixes U :: "'a set"   and le :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and ls :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a" assumes "preorder_ow U le ls"   and "x \<in> U" shows "le x x"
prob_T: fixes P :: "'s stream \<Rightarrow> bool"   and s :: "'s" assumes "Measurable.pred (stream_space (count_space UNIV)) P" shows "T.prob s {\<omega> \<in> space (T s). P \<omega>} = measure_pmf.expectation (K s) (\<lambda>t. T.prob t {\<omega> \<in> space (T t). P (t ## \<omega>)})"
funion_commute: fixes A :: "'a fset"   and B :: "'a fset" shows "A |\<union>| B = B |\<union>| A"
n_ivl_def: fixes iv :: "ivl" shows "n_ivl iv = 3 - m_ivl iv"
AnyIteratorTE: fixes \<Gamma> :: "String.literal \<rightharpoonup>\<^sub>f 'a OCL_Types.type"   and src :: "'a expr"   and k :: "call_kind"   and its :: "String.literal list"   and its_ty :: "'a OCL_Types.type option"   and body :: "'a expr"   and \<tau> :: "'a OCL_Types.type" assumes "\<Gamma> \<turnstile>\<^sub>E AnyIteratorCall src k its its_ty body : \<tau>" obtains \<tau>' :: "'a OCL_Types.type" and \<rho> :: "'a OCL_Types.type" where "k = ArrowCall" and "\<Gamma> \<turnstile>\<^sub>I (src, its, its_ty, body) : (\<tau>', \<tau>, \<rho>)" and "length its \<le> Suc 0" and "\<rho> \<le> Boolean[?]"
assms_2: shows "(v, a) \<in> dom (\<phi>\<^sub>S \<Psi> s)"
threshold: shows "\<bar>valof a * valof b\<bar> < threshold TYPE(('e, 'f) IEEE.float)"
MkArr_Map: fixes f :: "('a2 \<Rightarrow> 'b, 'a2 \<Rightarrow> 'b) concrete_category.arr" assumes "A2_B.arr f" shows "A2_B.MkArr (A2_B.Dom f) (A2_B.Cod f) (A2_B.Map f) = f"
eq'': shows "(\<Sum>b\<in>rcosets\<^bsub>G\<^esub> kernel G H f. of_nat (totient k div totient d) * (\<Phi> (the_elem (f ` b)) * unity_root d (int (the_elem (f ` b))))) = (\<Sum>h\<in>carrier H. of_nat (totient k div totient d) * (\<Phi> h * unity_root d (int h)))"
comm_neutral: fixes a :: "'k" shows "a + (0::'k) = a"
less_ant_def: fixes z :: "ant"   and w :: "ant" shows "z < w \<equiv> z \<le> w \<and> z \<noteq> w"
def_coinduct: fixes A :: "'d"   and f :: "'d \<Rightarrow> 'd"   and X :: "'d" assumes "A \<equiv> gfp f"   and "mono f"   and "X \<le> f (X \<squnion> A)" shows "X \<le> A"
comm_add_vecset: fixes A :: "'b vec set"   and n :: "nat"   and B :: "'b vec set" assumes "A \<subseteq> carrier_vec n"   and "B \<subseteq> carrier_vec n" shows "A + B = B + A"
monad_alt_envT': assumes "monad_alt Monad_Overloading.return Monad_Overloading.bind alt" shows "monad_alt Monad_Overloading.return Monad_Overloading.bind alt"
prems_3: shows "(c, b) \<in> set (zip xs (y # ys))"
int_cancel_factors: fixes r :: "int"   and n :: "int" assumes "1 < r" shows "n = 0 \<or> (\<exists>k i. n = k * r ^ i \<and> \<not> r dvd k)"
sig_poly_rel_vec_of: shows "spp_rel sp (vec_of sp)"
f: shows "abc_steps_l (0, args @ 0 # 0 \<up> (c - Suc (length args)) @ anything) (a @ B) n = (s, nl)"
valid: fixes u :: "nat"   and v :: "nat" assumes "(u, v) \<in> cf.E" shows "relabel_effect f l u u \<le> relabel_effect f l u v + 1"
nn_integral_sfirst_finite': fixes s :: "'s"   and H :: "'s set" assumes "s \<notin> H"   and "finite (acc_on (- H) `` {s})"   and "almost_everywhere (T s) (ev (HLD H))" shows "\<integral>\<^sup>+ x. ennreal_of_enat (sfirst (HLD H) x) \<partial>T s \<noteq> \<infinity>"
compute_single: fixes k :: "'a"   and v :: "'b" shows "monomial v k = sparse\<^sub>0 [(k, v)]"
HaddP_graph_aux_def: shows "HaddP_graph_aux \<equiv> lfp (\<lambda>p x1 x2. \<exists>HaddP_aux s x y z. x1 = (x, y, z) \<and> x2 = SyntaxN.Ex s (SeqHaddP (Var s) x y z) \<and> atom s \<sharp> (x, y, z))"
simps_3: fixes \<sigma> :: "'a basic_type" shows "OclVoid \<squnion> \<sigma> = (if OclVoid \<le> \<sigma> then \<sigma> else if \<sigma> \<le> OclVoid then OclVoid else OclAny)"
b_rel_subtyping_merge: fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> assn"   and x :: "'a"   and y :: "'b"   and A' :: "'a \<Rightarrow> 'b \<Rightarrow> assn"   and Am :: "'a \<Rightarrow> 'b \<Rightarrow> assn"   and P :: "'a \<Rightarrow> bool"   and P' :: "'a \<Rightarrow> bool" assumes "hn_ctxt A x y \<or>\<^sub>A hn_ctxt A' x y \<Longrightarrow>\<^sub>t hn_ctxt Am x y" shows "hn_ctxt (b_assn A P) x y \<or>\<^sub>A hn_ctxt (b_assn A' P') x y \<Longrightarrow>\<^sub>t hn_ctxt (b_assn Am (\<lambda>x. P x \<or> P' x)) x y"