hcju/isabelleps huggingface dataset:
ck_unifier_thm: fixes t :: "'c trm"   and s :: "'c trm"   and \<sigma> :: "('c \<times> 'c trm) list"   and k :: "inferences" assumes "ck_unifier t s \<sigma> k" shows "t \<lhd> \<sigma> = s \<lhd> \<sigma>"
cfWritePreservesUnallocatedBlocksInvariant: fixes cfile1 :: "CFile"   and byteIndex :: "nat"   and "value" :: "byte"   and cfile2 :: "CFile" assumes "reachabilityInvariant cfile1"   and "cfWrite cfile1 byteIndex value = Some cfile2" shows "unallocatedBlocksInvariant cfile2"
substAbs_simp: fixes Y :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and X :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and xs :: "'varSort"   and ys :: "'varSort"   and x :: "'var"   and y :: "'var" assumes "good Y"   and "good X"   and "xs \<noteq> ys \<or> x \<noteq> y"   and "fresh xs x Y" shows "Abs xs x X $[Y / y]_ys = Abs xs x (X #[Y / y]_ys)"
path_append: fixes u :: "'v"   and ls1 :: "enat list"   and ls2 :: "enat list"   and v :: "'v" shows "path u (ls1 @ ls2) v = (\<exists>w. path u ls1 w \<and> path w ls2 v)"
SeqHRP_lemma: fixes m :: "name"   and x :: "tm"   and x' :: "tm"   and s :: "tm"   and k :: "tm"   and n :: "name"   and sm :: "name"   and sm' :: "name"   and sn :: "name"   and sn' :: "name" assumes "atom m \<sharp> (x, x', s, k, n, sm, sm', sn, sn')"   and "atom n \<sharp> (x, x', s, k, sm, sm', sn, sn')"   and "atom sm \<sharp> (x, x', s, k, sm', sn, sn')"   and "atom sm' \<sharp> (x, x', s, k, sn, sn')"   and "atom sn \<sharp> (x, x', s, k, sn')"   and "atom sn' \<sharp> (x, x', s, k)" shows "{SeqHRP x x' s k} \<turnstile> OrdP x AND WRP x x' OR SyntaxN.Ex m (SyntaxN.Ex n (SyntaxN.Ex sm (SyntaxN.Ex sm' (SyntaxN.Ex sn (SyntaxN.Ex sn' (Var m IN k AND Var n IN k AND SeqHRP (Var sm) (Var sm') s (Var m) AND SeqHRP (Var sn) (Var sn') s (Var n) AND x EQ HPair (Var sm) (Var sn) AND x' EQ Q_HPair (Var sm') (Var sn')))))))"
eq_add_iff1: fixes a :: "'a"   and e :: "'a"   and c :: "'a"   and b :: "'a"   and d :: "'a" shows "(a * e + c = b * e + d) = ((a - b) * e + c = d)"
Red_F_of_subset: fixes N :: "('a literal multiset \<times> label) set"   and N' :: "('a literal multiset \<times> label) set" assumes "N \<subseteq> N'" shows "FL.Red_F_\<G> N \<subseteq> FL.Red_F_\<G> N'"
var: fixes \<Gamma> :: "(name, value) fmap"   and name :: "name"   and val :: "value"   and rs :: "(name \<times> value) list" assumes "fmlookup \<Gamma> name = Some val" shows "rs, \<Gamma> \<turnstile>\<^sub>v Svar name \<down> val"
weak_llessI: fixes R :: "('d, 'e) gorder_scheme"   and x :: "'d"   and y :: "'d" assumes "x \<sqsubseteq>\<^bsub>R\<^esub> y"   and "x .\<noteq>\<^bsub>R\<^esub> y" shows "x \<sqsubset>\<^bsub>R\<^esub> y"
assms_1: fixes c_o :: "'c_o"   and c_a :: "'c_a"   and s :: "'s" assumes "(c_o, c_a, s) \<in> set_spmf init" shows "interaction_bounded_by' (Not \<circ> isl) (\<A> c_a) q"
UP\<^sub>0_simps_2: fixes a :: "'a" assumes "B.obj a" shows "arr (UP\<^sub>0 a)"
simps_1: fixes e :: "'v \<times> 'v" shows "h_minus_aux 0 e = 0"
Radj_repv3: fixes z :: "'sz"   and x :: "'sz"   and y :: "'sz"   and \<nu> :: "(real, 'sz) vec \<times> (real, 'sz) vec"   and r :: "real" assumes "z \<noteq> x"   and "z \<noteq> y" shows "Radj x y (repv \<nu> z r) = repv (Radj x y \<nu>) z r"
list_all2_tlI: fixes A :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and XS :: "'c list"   and y :: "'d list" assumes "list_all2 A XS y" shows "list_all2 A (tl XS) (tl y)"
that: fixes a :: "'a" assumes "atom a \<notin> X" shows "thesis"
offs_next_cons_neq: fixes index :: "('c \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> nat"   and key :: "'c \<Rightarrow> 'd"   and x :: "'c"   and ns :: "nat list"   and mi :: "'d"   and ma :: "'d"   and i :: "nat"   and xs :: "'c list"   and j :: "nat"   and ub :: "nat" assumes "index key x (length ns) mi ma = i"   and "offs_set_prev ns (x # xs) index key mi ma i \<noteq> {}"   and "Max (offs_set_prev ns (x # xs) index key mi ma i) = j" shows "offs_next (ns[i := Suc (ns ! i)]) ub xs index key mi ma j = (if 0 < offs_num (length ns) xs index key mi ma i then Suc (ns ! i) else offs_next ns ub (x # xs) index key mi ma i)"
DIAndCut34Elim2_def: shows "DIAndCut34Elim2 = (([[$\<alpha> vid1]](Pc pid3 && Pc pid4) \<rightarrow> Pc pid4) \<rightarrow> ([[$\<alpha> vid1]]Pc pid3 && Pc pid4) \<rightarrow> [[$\<alpha> vid1]]Pc pid4)"
rel_interior_injective_linear_image: fixes f :: "'a \<Rightarrow> 'b"   and S :: "'a set" assumes "bounded_linear f"   and "inj f" shows "rel_interior (f ` S) = f ` rel_interior S"
assms_2: shows "(f \<circ> s) \<longlonglongrightarrow> l"
dependent_eq_real: shows "module_on.dependent (*\<^sub>R) = ??.Real_Vector_Spaces.dependent"
monot_def: shows "monot \<equiv> (\<exists>intT intF intP. MModel intT intF intP) \<longrightarrow> (\<exists>intTI intFI intPI. IInfModel intTI intFI intPI)"
upper_plus: fixes A :: "'a interval"   and B :: "'a interval" shows "upper (A + B) = upper A + upper B"
Tinv_qmpt: assumes "I.invertible_qmpt" shows "qmpt M I.Tinv"
vrange_mono: fixes r :: "V"   and s :: "V" assumes "r \<subseteq>\<^sub>\<circ> s" shows "\<R>\<^sub>\<circ> r \<subseteq>\<^sub>\<circ> \<R>\<^sub>\<circ> s"
sorted_inorder_induct_last: fixes ts :: "('b btree \<times> 'b) list"   and t :: "'b btree" assumes "sorted_less (inorder (Node ts t))" shows "sorted_less (inorder t)"
map_poly_zero: fixes f :: "'a \<Rightarrow> 'b"   and p :: "'a poly" assumes "\<forall>c. f c = (0::'b) \<longrightarrow> c = (0::'a)" shows "(map_poly f p = 0) = (p = 0)"
is_ord_mkt_bal_l: fixes n :: "'b"   and l :: "'b tree"   and r :: "'b tree"   and h :: "nat" assumes "is_ord (MKT n l r h)" shows "is_ord (mkt_bal_l n l r)"
inj_on_image: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set set" assumes "inj_on f (\<Union> A)" shows "inj_on ((`) f) A"
commute: fixes a :: "'a list"   and b :: "'a list" shows "gcp a b = gcp b a"
simps_3: fixes sprFst :: "('a, 'b, 'c, 'd, 'e) BEState"   and sprLst :: "('a, 'b, 'c, 'd, 'e) BEState"   and sprCRel :: "(('a, 'b, 'c, 'd, 'e) BEState \<times> ('a, 'b, 'c, 'd, 'e) BEState) set"   and more :: "'f" shows "sprCRel \<lparr>sprFst = sprFst, sprLst = sprLst, sprCRel = sprCRel, \<dots> = more\<rparr> = sprCRel"
post_mult_proj_psi_lR: shows "post * proj_psi_l R = proj_psi_l R"
scale_eq_0_iff: fixes a :: "'d"   and x :: "('d, 'e) vec" shows "(a *s x = 0) = (a = (0::'d) \<or> x = 0)"
flow_solves_ode: fixes T :: "real set"   and f :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and X :: "'b set"   and t0 :: "real"   and x0 :: "'b" assumes "ll_on_open_it T f X"   and "t0 \<in> T"   and "x0 \<in> X" shows "(ll_on_open.flow T f X t0 x0 solves_ode f) (ll_on_open.existence_ivl T f X t0 x0) X"
snd_scaleC: fixes r :: "complex"   and A :: "'e \<times> 'd" shows "snd (r *\<^sub>C A) = r *\<^sub>C snd A"
fcompose_pCons: fixes a :: "'b"   and p :: "'b poly"   and q1 :: "'b poly"   and q2 :: "'b poly" shows "fcompose (pCons a p) q1 q2 = smult a (q2 ^ degree (pCons a p)) + q1 * fcompose p q1 q2"
0: fixes \<sigma> :: "'b \<Rightarrow> ('a, 'b) Term.term" assumes "subst_domain \<sigma> = set X"   and "ground (subst_range \<sigma>)" shows "\<sigma> \<circ>\<^sub>s (\<delta> \<circ>\<^sub>s \<theta>) = \<delta> \<circ>\<^sub>s (\<sigma> \<circ>\<^sub>s \<theta>)"
coboundedI1: fixes c :: "'e"   and a :: "'e"   and b :: "'e" assumes "c \<le> a" shows "c \<le> sup a b"
SubstArityAnalysis_axioms_def: fixes Aexp :: "exp \<Rightarrow> Arity \<rightarrow> Vars.var \<Rightarrow> Arity\<^sub>\<bottom>" shows "SubstArityAnalysis_axioms Aexp \<equiv> \<forall>x S y e a. x \<notin> S \<longrightarrow> y \<notin> S \<longrightarrow> Aexp e[x::=y]\<cdot>a f|` S = Aexp e\<cdot>a f|` S"
diff: shows "zs \<noteq> ys \<or> z \<noteq> y"
mult_isol_equiv_subdistl: shows "(\<forall>x y z. x \<le> y \<longrightarrow> \<lparr>z\<rparr>x\<lparr>z \<cdot> y\<rparr>) = (\<forall>x y z. \<lparr>z\<rparr>x\<lparr>z \<cdot> (x + y)\<rparr>)"
SUPSEQ_simps_2: shows "SUPSEQ {[]} = UNIV"
adjacent_sym: fixes v :: "'v"   and v' :: "'v" shows "adjacent v v' = adjacent v' v"
eval_subst_tm: fixes e :: "(name, hf) finfun"   and i :: "name"   and t :: "tm"   and u :: "tm" shows "\<lbrakk>subst i t u\<rbrakk>e = \<lbrakk>u\<rbrakk>(finfun_update e i \<lbrakk>t\<rbrakk>e)"
lessE: fixes a :: "'d"   and b :: "'d" assumes "a < b" obtains c :: "'d" where "b = a + c" and "c \<noteq> (0::'d)"
strong_induct: fixes x1 :: "(name \<times> ty) list"   and x2 :: "lam"   and x3 :: "ty"   and P :: "'a \<Rightarrow> (name \<times> ty) list \<Rightarrow> lam \<Rightarrow> ty \<Rightarrow> bool"   and z :: "'a" assumes "x1 \<turnstile> x2 : x3"   and "\<And>\<Gamma> x T y. \<lbrakk>valid \<Gamma>; (x, T) \<in> set \<Gamma>\<rbrakk> \<Longrightarrow> P y \<Gamma> (Var x) T"   and "\<And>\<Gamma> t\<^sub>1 T\<^sub>1 T\<^sub>2 t\<^sub>2 y. \<lbrakk>\<Gamma> \<turnstile> t\<^sub>1 : T\<^sub>1 \<rightarrow> T\<^sub>2; \<And>z. P z \<Gamma> t\<^sub>1 T\<^sub>1 \<rightarrow> T\<^sub>2; \<Gamma> \<turnstile> t\<^sub>2 : T\<^sub>1; \<And>z. P z \<Gamma> t\<^sub>2 T\<^sub>1\<rbrakk> \<Longrightarrow> P y \<Gamma> (App t\<^sub>1 t\<^sub>2) T\<^sub>2"   and "\<And>x \<Gamma> T\<^sub>1 t T\<^sub>2 y. \<lbrakk>x \<sharp> y; x \<sharp> \<Gamma>; (x, T\<^sub>1) # \<Gamma> \<turnstile> t : T\<^sub>2; \<And>z. P z ((x, T\<^sub>1) # \<Gamma>) t T\<^sub>2\<rbrakk> \<Longrightarrow> P y \<Gamma> Lam [x].t T\<^sub>1 \<rightarrow> T\<^sub>2" shows "P z x1 x2 x3"
subcocycle_int_tendsto_avg_ereal: fixes u :: "nat \<Rightarrow> 'a \<Rightarrow> real" assumes "subcocycle u" shows "(\<lambda>x. ereal (LINT xa|M. u x xa / real x)) \<longlonglongrightarrow> subcocycle_avg_ereal u"
simps_4: fixes types :: "tf list"   and funcs :: "nat list"   and tab :: "nat option"   and mem :: "nat option"   and globs :: "nat list"   and more :: "'a" shows "inst.mem \<lparr>types = types, funcs = funcs, tab = tab, mem = mem, globs = globs, \<dots> = more\<rparr> = mem"
l12: shows "0 \<le> b\<^sup>2 - 4 * a * c \<and> ((- b + sqrt (b\<^sup>2 - 4 * a * c)) / (2 * a) = x \<or> (- b - sqrt (b\<^sup>2 - 4 * a * c)) / (2 * a) = x)"
subst_pp_zero: fixes f :: "'d \<Rightarrow> ('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c" shows "subst_pp f 0 = 1"
observable2_def: fixes ob :: "'a \<Rightarrow> 'b"   and P :: "'a set" shows "observable2 ob P \<equiv> \<exists>Q. P = ob -` Q"
comm_monoid_axioms: shows "comm_monoid (\<squnion>) bot"
uniform_real_def: shows "uniform_real = map_real_pair (\<lambda>l u. density (stock_measure REAL) (uniform_real_density (l, u) \<circ> extract_real)) (\<lambda>_. undefined)"
assms_3: shows "\<forall>j\<in>R. B j \<in> measurable_sets M"
assms_2: shows "\<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>rs, s\<rangle> \<Rightarrow> t'"
tp_L_eq: fixes k :: "nat" assumes "k < length \<alpha>s" shows "tp (L ! k) = \<omega> \<up> \<alpha>s ! k"
codomains_comp: fixes g :: "(bool, 'arr list) concrete_category.arr"   and f :: "(bool, 'arr list) concrete_category.arr" assumes "seq g f" shows "codomains (g \<cdot> f) = codomains g"
ordIso_symmetric: fixes r :: "('a \<times> 'a) set"   and r' :: "('b \<times> 'b) set" assumes "r =o r'" shows "r' =o r"
triangle_in_hom_2: shows "\<guillemotleft>(g \<star> \<epsilon>) \<bullet> \<a>[g, f, g] \<bullet> (\<eta> \<star> g) : trg g \<star> g \<Rightarrow> g \<star> src g\<guillemotright>"
bind_map_pmf: fixes f :: "'c \<Rightarrow> 'b"   and M :: "'c pmf"   and g :: "'b \<Rightarrow> 'a pmf" shows "map_pmf f M \<bind> g = M \<bind> (\<lambda>x. g (f x))"
G_parts_1: shows "verts G = GU \<union> GV"
sWlsVal_def: fixes SEM :: "('g, 'h, 'i, 'sort, 'j, 'k, 'l) semDom_scheme"   and val :: "'varSort \<Rightarrow> 'm \<Rightarrow> 'k" shows "sWlsVal SEM val \<equiv> \<forall>ys y. sWls SEM (asSort ys) (val ys y)"
lens_override_idem: fixes X :: "'d \<Longrightarrow> 'e"   and S :: "'e" assumes "vwb_lens X" shows "S \<triangleleft>\<^bsub>X\<^esub> S = S"
\<pi>\<^sub>\<A>_def: shows "\<pi>\<^sub>\<A> = (\<lambda>\<chi>. mojmir_def.smallest_accepting_rank \<Sigma> \<delta>\<^sub>M (q\<^sub>0\<^sub>M (theG \<chi>)) w {q. dom \<pi> \<up>\<Turnstile>\<^sub>P q}) |` \<G>"
fund_cont_desc: fixes x :: "\<nu>"   and \<phi> :: "\<nu> \<Rightarrow> \<o>" shows "[x\<^sup>P \<^bold>= (\<^bold>\<iota>x. \<phi> x) \<^bold>\<equiv> (\<^bold>\<forall>z. \<phi> z \<^bold>\<equiv> z \<^bold>= x) in dw]"
\<nu>\<upsilon>_def: shows "\<nu>\<upsilon> \<equiv> case_\<nu> \<omega>\<upsilon> (\<sigma>\<upsilon> \<circ> \<alpha>\<sigma>)"
ms_1: fixes i :: "'b"   and j :: "'c" assumes "I i j" shows "G i j \<in> Pow \<Omega>"
suminf_ennreal_2dimen: fixes g :: "nat \<Rightarrow> ennreal"   and f :: "nat \<times> nat \<Rightarrow> ennreal" assumes "\<And>m. g m = (\<Sum>n. f (m, n))" shows "(\<Sum>i. f (prod_decode i)) = suminf g"
wp_under_wlp_Abort: shows "wp_under_wlp Abort"
pdevs_apply_map_slist: fixes f :: "'e \<Rightarrow> 'd"   and xs :: "(nat, 'e) slist"   and i :: "nat" assumes "f (0::'e) = (0::'d)" shows "pdevs_apply (Pdevs (map_slist f xs)) i = f (pdevs_apply (Pdevs xs) i)"
induct: fixes P :: "'a list \<Rightarrow> bool"   and a0 :: "'a list" assumes "P []"   and "\<And>uu_. P [uu_]"   and "\<And>x1 x2 xs. \<lbrakk>x1 * x2 < (0::'a) \<Longrightarrow> P (x2 # xs); \<lbrakk>\<not> x1 * x2 < (0::'a); x2 = (0::'a)\<rbrakk> \<Longrightarrow> P (x1 # xs); \<lbrakk>\<not> x1 * x2 < (0::'a); x2 \<noteq> (0::'a)\<rbrakk> \<Longrightarrow> P (x2 # xs)\<rbrakk> \<Longrightarrow> P (x1 # x2 # xs)" shows "P a0"
Lv_def: fixes K :: "('c, 'd) Ring_scheme"   and v :: "'c \<Rightarrow> ant" shows "Lv K v = AMin {x \<in> v ` carrier K. 0 < x}"
naturality: fixes f :: "'c" assumes "C.arr f" shows "\<epsilon> (C.cod f) \<cdot>\<^sub>C (F \<circ> G) f = C.map f \<cdot>\<^sub>C \<epsilon> (C.dom f)"
simps_1: fixes g :: "nat \<Rightarrow> nat"   and h :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat"   and x :: "nat" shows "PrimRecOp g h 0 x = g x"
carrier_eq: shows "alts = {x. Pareto (permute_profile (permutation_of_list xs) R) x x}"
trcl_pair_unconsE: fixes s :: "'a"   and c :: "'b"   and e :: "'c"   and w :: "'c list"   and s' :: "'a"   and c' :: "'b"   and S :: "(('a \<times> 'b) \<times> 'c \<times> 'a \<times> 'b) set" assumes "((s, c), e # w, s', c') \<in> trcl S" obtains sh :: "'a" and ch :: "'b" where "((s, c), e, sh, ch) \<in> S" and "((sh, ch), w, s', c') \<in> trcl S"
Quotient_complex_homo_coords: shows "Quotient (eq_onp (\<lambda>v. v \<noteq> vec_zero)) Abs_complex_homo_coords Rep_complex_homo_coords cr_complex_homo_coords"
cball_translation: fixes a :: "'b"   and c :: "'b"   and r :: "real" shows "cball (a + c) r = (+) a ` cball c r"
distinct_4: fixes x31 :: "'b"   and x32 :: "('a, 'b) form list"   and x1 :: "nat" shows "Modal x31 x32 \<noteq> At x1"
sum_plus_zero: fixes f :: "'c \<times> 'c \<Rightarrow> 'd" shows "sum\<^sub>M f \<oplus>\<^sub>M mzero = sum\<^sub>M f"
vec_times_mat_eqD_2: fixes y :: "'a vec"   and A :: "'a mat"   and c :: "'a vec" assumes "[y\<^sub>v*A]=c" shows "dim_col A\<^sup>T = dim_vec y"
monofun_cfun_fun: fixes f :: "'d \<rightarrow> 'e"   and g :: "'d \<rightarrow> 'e"   and x :: "'d" assumes "f \<sqsubseteq> g" shows "f\<cdot>x \<sqsubseteq> g\<cdot>x"
extend_set_strict_mono: fixes A :: "'a set"   and B :: "'a set" shows "(extend_set h A \<subseteq> extend_set h B) = (A \<subseteq> B)"
InvId: fixes C :: "('a, 'b, 'c) Category_scheme"   and X :: "'a" assumes "Category C"   and "X \<in> obj\<^bsub>C\<^esub>" shows "cinv\<^bsub>C\<^esub> id\<^bsub>C\<^esub> X id\<^bsub>C\<^esub> X"
wbalanced_bal_tree: fixes n :: "nat"   and t :: "'a tree" assumes "n \<le> size t" shows "wbalanced (bal_tree n t)"
nf_ACI_nFAND: fixes \<phi>s :: "('a, 'b) aformula list" assumes "sorted \<phi>s"   and "distinct \<phi>s"   and "list_all nf_ACI \<phi>s"   and "list_all (\<lambda>x. \<not> is_FAnd x) \<phi>s" shows "nf_ACI (nFAND \<phi>s)"
simps_7: fixes i :: "poly"   and a :: "tm" shows "tmbound0 (tm.Mul i a) = tmbound0 a"
start_def: fixes dm :: "nat" shows "start dm = replicate dm (0, 1)"
swapped_preserves_good: fixes X :: "('c, 'd, 'varSort, 'var, 'e) qTerm set"   and Y :: "('c, 'd, 'varSort, 'var, 'e) qTerm set" assumes "good X"   and "(X, Y) \<in> swapped" shows "good Y"
strict_order_mult_1: fixes R :: "('a \<times> 'a) set" assumes "irrefl R"   and "trans R" shows "irrefl (mult R)"
h_vert_disj: shows "\<H> \<inter> two_chain_vertical_boundary two_chain = {}"
cindex_poly_cross: fixes a :: "real"   and b :: "real"   and p :: "real poly" assumes "a < b"   and "poly p a \<noteq> 0"   and "poly p b \<noteq> 0" shows "cindex_poly a b 1 p = cross p a b"
ksimplex_0: fixes p :: "nat"   and n :: "nat"   and base :: "nat \<Rightarrow> nat"   and upd :: "nat \<Rightarrow> nat"   and s :: "(nat \<Rightarrow> nat) set" assumes "kuhn_simplex p n base upd s"   and "n = 0" shows "s = {\<lambda>x. p}"
vec1I_def: fixes ze :: "'b"   and on :: "'b"   and n :: "nat"   and i :: "nat" shows "vec1I ze on n i \<equiv> replicate i ze @ on # replicate (n - 1 - i) ze"
open_inter: fixes U :: "'a set set"   and V :: "'a set set" assumes "is_zariski_open U"   and "is_zariski_open V" shows "is_zariski_open (U \<inter> V)"
CatExpNatTransCompDef: fixes A :: "('g, 'i, 'k) Category_scheme"   and B :: "('h, 'j, 'l) Category_scheme"   and f :: "('g, 'h, 'i, 'j, 'k, 'l) NatTrans"   and g :: "('g, 'h, 'i, 'j, 'k, 'l) NatTrans" assumes "f \<approx>>\<^bsub>CatExp A B\<^esub> g" shows "f \<approx>>\<bullet> g"
setdist_eq_0I: fixes x :: "'c"   and S :: "'c set"   and T :: "'c set" assumes "x \<in> S"   and "x \<in> T" shows "setdist S T = 0"
sortedP_right_of_last: fixes le :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and zs :: "'b list"   and y :: "'b" assumes "linorder_list0.sortedP le zs"   and "y \<in> set zs"   and "y \<noteq> last zs" shows "le y (last zs)"
assms_8: shows "pyf \<in> borel_measurable (F matur)"
induct: fixes x1 :: "'node SDG_node"   and x2 :: "'node SDG_node list"   and x3 :: "'node SDG_node"   and P :: "'node SDG_node \<Rightarrow> 'node SDG_node list \<Rightarrow> 'node SDG_node \<Rightarrow> bool" assumes "x1 cc-x2\<rightarrow>\<^sub>d* x3"   and "\<And>n. valid_SDG_node n \<Longrightarrow> P n [] n"   and "\<And>n ns n'' n'. \<lbrakk>n cc-ns\<rightarrow>\<^sub>d* n''; P n ns n''; n'' \<longrightarrow>\<^bsub>cd\<^esub> n'\<rbrakk> \<Longrightarrow> P n (ns @ [n'']) n'"   and "\<And>n ns n'' p n'. \<lbrakk>n cc-ns\<rightarrow>\<^sub>d* n''; P n ns n''; n'' -p\<rightarrow>\<^bsub>call\<^esub> n'\<rbrakk> \<Longrightarrow> P n (ns @ [n'']) n'" shows "P x1 x2 x3"
c_card_is_pr: shows "c_card \<in> PrimRec1"
endpoints_refine: fixes xi :: "nat \<times> int \<times> nat"   and x :: "'edge" assumes "\<alpha> xi = x" shows "endpoints xi \<le> \<Down> Id (a_endpoints x)"
mono_addLPD: fixes r :: "mregex" shows "mono (\<lambda>S. insert r S \<union> (insert r S \<bind> LPD))"
ss_adversary_def: fixes x' :: "'a \<times> 'a"   and c1 :: "('a \<times> 'a) \<times> nat \<times> nat"   and c2 :: "('a \<times> 'a) \<times> nat \<times> nat" shows "ss_adversary x' c1 c2 = (let ((a, a'), e, z) = c1; ((b, b'), e', z') = c2 in return_spmf (if e' mod order \<G> < e mod order \<G> then nat ((int z - int z') * Number_Theory_Aux.inverse ((e mod order \<G> - e' mod order \<G>) mod order \<G>) (order \<G>) mod int (order \<G>)) else nat ((int z' - int z) * Number_Theory_Aux.inverse ((e' mod order \<G> - e mod order \<G>) mod order \<G>) (order \<G>) mod int (order \<G>))))"
subs_wf_sub_rel: fixes prb :: "('d, 'e, 'f) pre_RedBlack" assumes "RedBlack prb" shows "wf_sub_rel (subs prb)"
case_1: fixes f1 :: "bool \<Rightarrow> nat \<Rightarrow> nat set \<times> ((nat \<Rightarrow> nat option) \<Rightarrow> nat) \<Rightarrow> nat \<Rightarrow> nat set \<Rightarrow> nat set \<Rightarrow> nat set \<Rightarrow> nat set \<Rightarrow> 'b"   and f2 :: "bool \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'b"   and f3 :: "'c \<Rightarrow> 'c \<Rightarrow> instr list \<Rightarrow> 'b"   and f4 :: "nat set \<Rightarrow> nat set \<Rightarrow> nat set \<Rightarrow> nat set \<Rightarrow> 'b"   and x11 :: "bool"   and x12 :: "nat"   and x13 :: "nat set \<times> ((nat \<Rightarrow> nat option) \<Rightarrow> nat)"   and x14 :: "nat"   and x15 :: "nat set"   and x16 :: "nat set"   and x17 :: "nat set"   and x18 :: "nat set" shows "(case Write\<^sub>s\<^sub>b x11 x12 x13 x14 x15 x16 x17 x18 of Write\<^sub>s\<^sub>b x xa xb xc xd xe xf xg \<Rightarrow> f1 x xa xb xc xd xe xf xg | Read\<^sub>s\<^sub>b x xa xb xc \<Rightarrow> f2 x xa xb xc | Prog\<^sub>s\<^sub>b x xa xb \<Rightarrow> f3 x xa xb | Ghost\<^sub>s\<^sub>b x xa xb xc \<Rightarrow> f4 x xa xb xc) = f1 x11 x12 x13 x14 x15 x16 x17 x18"
abel_semigroup_axioms: shows "abel_semigroup (\<squnion>)"
multiset_induct: fixes P :: "'c multiset \<Rightarrow> bool"   and M :: "'c multiset" assumes "P {#}"   and "\<And>x M. P M \<Longrightarrow> P (add_mset x M)" shows "P M"
simps_1: fixes t1 :: "'b trm"   and t2 :: "'b trm" shows "rhs (t1 \<cdot> t2) = t2"
image_comp: fixes f :: "'d \<Rightarrow> 'c"   and g :: "'f \<Rightarrow> 'd"   and r :: "'f set" shows "f ` g ` r = (f \<circ> g) ` r"
fun_type_inv_1: fixes t :: "('fun, ('fun, 'atom) Term.term \<times> nat) Term.term"   and f :: "'fun"   and T :: "('fun, 'atom) Term.term list" assumes "\<Gamma> t = Fun f T" shows "0 < arity f"
assms_2: shows "\<mu> = rot90 (z2 - z1)"
fun_belowI: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "\<And>x. f x \<sqsubseteq> g x" shows "f \<sqsubseteq> g"
eq_Eq_conv: fixes x :: "'a"   and y :: "'a" shows "(comp x y = Eq) = (x = y)"
mult_col_div_row_def: fixes a :: "'a"   and k :: "nat"   and A :: "'a mat" shows "mult_col_div_row a k A = multrow k (inverse a) (multcol k a A)"
M_perp_unique: fixes a :: "proj2"   and b :: "proj2"   and l :: "proj2_line"   and m :: "proj2_line"   and n :: "proj2_line" assumes "a \<in> hyp2"   and "b \<in> hyp2"   and "proj2_incident a l"   and "proj2_incident b m"   and "proj2_incident b n"   and "M_perp l m"   and "M_perp l n" shows "m = n"
PLam_insert: fixes i :: "nat"   and I :: "nat set"   and F :: "nat \<Rightarrow> ('c \<times> (nat \<Rightarrow> 'c) \<times> 'd) program" shows "PLam (insert i I) F = lift i (F i) \<squnion> PLam I F"
intros_7: fixes A :: "nat set"   and L :: "nat set"   and R :: "nat set"   and W :: "nat set"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and x :: "unit"   and m :: "nat \<Rightarrow> nat"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<R> :: "unit"   and \<S> :: "nat \<Rightarrow> bool option" shows "(Ghost A L R W # is, \<theta>, x, m, \<D>, \<O>, \<R>, \<S>) \<rightarrow>\<^sub>v (is, \<theta>, x, m, \<D>, \<O> \<union> A - R, \<S> \<oplus>\<^bsub>W\<^esub> R \<ominus>\<^bsub>A\<^esub> L)"
term_variants_pred_refl: fixes P :: "'c \<Rightarrow> 'c list"   and t :: "('c, 'd) Term.term" shows "term_variants_pred P t t"
M_up: fixes m :: "'m"   and a :: "'a"   and b :: "'b list" assumes "M m" shows "M (up a b m)"
Guarantee: fixes P :: "'d set"   and g :: "'d set"   and R :: "'d set"   and \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and \<Theta> :: "('d set \<times> 'e \<times> 'd set \<times> 'd set) set"   and F :: "'f set"   and c :: "('d, 'e, 'f) com"   and Q :: "'d set"   and A :: "'d set"   and f :: "'f" assumes "P \<subseteq> {s. s \<in> g \<longrightarrow> s \<in> R}"   and "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>R c Q,A"   and "f \<in> F" shows "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P Guard f g c Q,A"
Lambert_W_ex1: fixes x :: "real" assumes "- exp (- 1) \<le> x" shows "\<exists>!w. - 1 \<le> w \<and> w * exp w = x"
assoc_closed: fixes f :: "'a"   and g :: "'a"   and h :: "'a" assumes "(B.arr f \<and> src\<^sub>B f = a \<and> trg\<^sub>B f = a) \<and> B.ide f"   and "(B.arr g \<and> src\<^sub>B g = a \<and> trg\<^sub>B g = a) \<and> B.ide g"   and "(B.arr h \<and> src\<^sub>B h = a \<and> trg\<^sub>B h = a) \<and> B.ide h"   and "src\<^sub>B f = trg\<^sub>B g"   and "src\<^sub>B g = trg\<^sub>B h" shows "B.arr \<a>\<^sub>B[f, g, h] \<and> src\<^sub>B \<a>\<^sub>B[f, g, h] = a \<and> trg\<^sub>B \<a>\<^sub>B[f, g, h] = a"
R43_ad_2: shows "pmf (sds R43) d = 1 / 2"
card_eq_dim_span_imp_same_span: fixes W :: "'a vec set"   and V :: "'a vec set" assumes "W \<subseteq> carrier_vec n"   and "V \<subseteq> local.span W"   and "lin_indpt V"   and "card V = dim_span W" shows "local.span V = local.span W"