hcju/isabelleps huggingface dataset:
assms_4: shows "R \<in> carrier B \<rightarrow> carrier A"
char_eq_0_iff: fixes G :: "('c, 'd) monoid_scheme"   and \<chi> :: "'c \<Rightarrow> complex"   and a :: "'c" assumes "character G \<chi>" shows "(\<chi> a = 0) = (a \<notin> carrier G)"
lfp_eqI: fixes F :: "'c \<Rightarrow> 'c"   and x :: "'c" assumes "mono F"   and "F x = x"   and "\<And>z. F z = z \<Longrightarrow> x \<le> z" shows "lfp F = x"
intros_5: fixes a1 :: "aexp"   and s :: "char list \<Rightarrow> val"   and r1 :: "real"   and a2 :: "aexp"   and r2 :: "real" assumes "taval a1 s (Rv r1)"   and "taval a2 s (Rv r2)" shows "tbval (Less a1 a2) s (r1 < r2)"
measure_eqI_generator_eq: fixes E :: "'c set set"   and \<Omega> :: "'c set"   and M :: "'c measure"   and N :: "'c measure"   and A :: "nat \<Rightarrow> 'c set" assumes "Int_stable E"   and "E \<subseteq> Pow \<Omega>"   and "\<And>X. X \<in> E \<Longrightarrow> emeasure M X = emeasure N X"   and "sets M = sigma_sets \<Omega> E"   and "sets N = sigma_sets \<Omega> E"   and "range A \<subseteq> E"   and "\<Union> (range A) = \<Omega>"   and "\<And>i. emeasure M (A i) \<noteq> \<infinity>" shows "M = N"
singleton_subset_singleton_conv: fixes a :: "'a"   and b :: "'a" shows "({a} \<subseteq> {b}) = (a = b)"
ii: shows "((\<lambda>x. g ((v - u) * x + u)) has_vector_derivative (v - u) *\<^sub>R vector_derivative g (at ((v - u) * x + u))) (at x)"
preserves_hom: fixes f :: "'j"   and a :: "'j"   and b :: "'j" assumes "\<guillemotleft>f : a \<rightarrow>\<^sub>J b\<guillemotright>" shows "\<guillemotleft>\<chi> f : \<chi>.A.map a \<rightarrow>\<^sub>[\<^sub>A\<^sub>,\<^sub>B\<^sub>] D b\<guillemotright>"
Cons_2: assumes "\<forall>a\<in>set (zip ss ts). case a of (s, t) \<Rightarrow> s = t \<or> fst (f s t) \<or> fst (f t s)" shows "ss = ts \<or> fst (lex_ext_unbounded f ss ts) \<or> fst (lex_ext_unbounded f ts ss)"
pot_comp: fixes rep_nat_term :: "'a \<Rightarrow> (nat, nat) pp \<times> nat"   and splus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and is_scalar :: "'a itself \<Rightarrow> bool"   and lex_comp :: "'a \<Rightarrow> 'a \<Rightarrow> order"   and deg_comp :: "('a \<Rightarrow> 'a \<Rightarrow> order) \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> order"   and pot_comp :: "('a \<Rightarrow> 'a \<Rightarrow> order) \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> order"   and cmp :: "'a \<Rightarrow> 'a \<Rightarrow> order" assumes "class.nat_term_compare rep_nat_term splus less_eq less is_scalar lex_comp deg_comp pot_comp" shows "pot_comp cmp = (\<lambda>x y. case comparator_of (snd (rep_nat_term x)) (snd (rep_nat_term y)) of Eq \<Rightarrow> cmp x y | Lt \<Rightarrow> Lt | Gt \<Rightarrow> Gt)"
NIL_neq_CONS: fixes M :: "('b, unit) Old_Datatype.node set"   and N :: "('b, unit) Old_Datatype.node set"   and R :: "bool" assumes "NIL = CONS M N" shows "R"
distinct_11: fixes x31 :: "'b match_expr"   and x32 :: "'b match_expr" shows "MatchAnd x31 x32 \<noteq> MatchAny"
Zero_Eq_HPairE: fixes x :: "tm"   and y :: "tm"   and H :: "fm set"   and A :: "fm" shows "insert (Zero EQ HPair x y) H \<turnstile> A"
powers_2: fixes a :: "'b"   and b :: "'b" shows "a + b = b + a"
induct: fixes P :: "'g list \<Rightarrow> bool"   and list :: "'g list" assumes "P []"   and "\<And>x1 x2. P x2 \<Longrightarrow> P (x1 # x2)" shows "P list"
one: shows "r|\<sigma>1| -s r \<down>m {#\<beta>#} = {#}"
select_convs_3: fixes tdir :: "'b"   and xdir :: "'b"   and ydir :: "'b"   and zdir :: "'b"   and more :: "'c" shows "ydir \<lparr>tdir = tdir, xdir = xdir, ydir = ydir, zdir = zdir, \<dots> = more\<rparr> = ydir"
isomorphic_src: fixes a :: "'c" assumes "C.obj a" shows "D.isomorphic (src\<^sub>D (F a)) (F a)"
set_mset_inter: fixes A :: "'c multiset"   and B :: "'c multiset" shows "set_mset (A \<inter># B) = set_mset A \<inter> set_mset B"
axiom_def: shows "Axioms.axiom \<equiv> \<lambda>\<phi>. \<forall>v. [\<phi> in v]"
bind_assoc: fixes f :: "('a, 'b, 'e) prog"   and g :: "'e \<Rightarrow> ('a, 'b, 'd) prog"   and h :: "'d \<Rightarrow> ('a, 'b, 'c) prog" shows "Heap_Error_Monad.bind (Heap_Error_Monad.bind f g) h = Heap_Error_Monad.bind f (\<lambda>x. Heap_Error_Monad.bind (g x) h)"
transpose_code: fixes A :: "(('b, 'a) vec, 'c) vec" shows "($) (Finite_Cartesian_Product.transpose A) = transpose_row A"
suc: fixes t :: "trm" assumes "t \<in> num" shows "suc t \<in> num"
G_def: shows "G = \<Union> (\<G>_F ` Q)"
path_def: fixes g :: "real \<Rightarrow> 'a" shows "path g = continuous_on {0..1} g"
a_consistentE: fixes Aexp :: "exp \<Rightarrow> Arity \<rightarrow> Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and Aheap :: "(Vars.var \<times> exp) list \<Rightarrow> exp \<Rightarrow> Arity \<rightarrow> Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and ae :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and a :: "Arity"   and as :: "Arity list"   and \<Gamma> :: "(Vars.var \<times> exp) list"   and e :: "exp"   and S :: "stack_elem list" assumes "ArityAnalysisLetSafe Aexp Aheap"   and "ArityAnalysisLetSafe.a_consistent Aexp (ae, a, as) (\<Gamma>, e, S)" obtains "edom ae \<subseteq> domA \<Gamma> \<union> upds S" and "Astack S \<sqsubseteq> a" and "(ArityAnalysis.AEstack Aexp as S \<squnion> (Aexp e\<cdot>a \<squnion> ArityAnalysis.ABinds Aexp \<Gamma>\<cdot>ae)) f|` (domA \<Gamma> \<union> upds S) \<sqsubseteq> ae" and "stack_consistent as S"
equivalence_of_categories_axioms: shows "equivalence_of_categories hom'.comp hom.comp (\<lambda>\<mu>. e\<^sub>1 \<star> \<mu> \<star> d\<^sub>0) (\<lambda>\<mu>'. d\<^sub>1 \<star> \<mu>' \<star> e\<^sub>0) \<phi> \<psi>"
assms_2: shows "f \<in> {f \<in> A \<rightarrow>\<^sub>E B. inj_on f A}"
conjugate_square_eq_0_2: fixes x :: "'c" shows "(conjugate x * x = (0::'c)) = (x = (0::'c))"
lookup_default_update': fixes d :: "'b"   and k :: "'c"   and v :: "'b"   and m :: "('c, 'b) mapping"   and k' :: "'c" shows "lookup_default d (update k v m) k' = (if k = k' then v else lookup_default d m k')"
F_outside: fixes f :: "'v \<times> 'v \<Rightarrow> ennreal"   and e :: "'v \<times> 'v" assumes "f \<in> Field leq"   and "e \<notin> \<^bold>E" shows "f e = 0"
subsetD: fixes A :: "'c set"   and B :: "'c set"   and c :: "'c" assumes "A \<subseteq> B"   and "c \<in> A" shows "c \<in> B"
map_poly_eval_poly: fixes a :: "'a"   and p :: "'a poly poly" shows "map_poly (\<lambda>p. poly p a) p = eval_poly (\<lambda>aa. [:poly aa a:]) p [:0::'a, 1::'a:]"
fw_int_preservation: fixes n :: "nat"   and M :: "nat \<Rightarrow> nat \<Rightarrow> 'c DBMEntry"   and a :: "nat"   and b :: "nat"   and c :: "nat"   and i :: "nat"   and j :: "nat" assumes "dbm_int M n"   and "a \<le> n"   and "b \<le> n"   and "c \<le> n"   and "i \<le> n"   and "j \<le> n"   and "fw M n a b c i j \<noteq> \<infinity>" shows "get_const (fw M n a b c i j) \<in> \<int>"
interleave_ConsE: fixes x :: "'b"   and xs :: "'b list"   and ys :: "'b list"   and zs :: "'b list" assumes "x # xs \<in> ys \<otimes> zs" obtains xs :: "'b list" where "ys = x # xs" and "xsa__ \<in> xs \<otimes> zs"   | ysa :: "'b list" where "zs = x # ysa" and "xsa__ \<in> ys \<otimes> ysa"
R2_def: shows "\<R>\<^sup>2 \<equiv> {\<lambda>x y. eval r [x, y] |r. recfn 2 r \<and> Partial_Recursive.total r}"
prefix_length_less: fixes xs :: "'d list"   and ys :: "'d list" assumes "strict_prefix xs ys" shows "length xs < length ys"
group_hom_axioms: shows "group_hom (relative_homology_group (int n) (nsphere n) (equator n)) (homology_group (int n - 1) (subtopology (nsphere n) (equator n))) (hom_boundary (int n) (nsphere n) (equator n))"
cases: fixes x :: "atom" obtains uu_ :: "nat" where "x = Less 0 uu_"   | v :: "nat" where "x = Less (Suc v__) 0"   | i :: "nat" and j :: "nat" where "x = Less (Suc i__) (Suc j__)"   | "x = Eq 0 0"   | v :: "nat" where "x = Eq 0 (Suc va__)"   | v :: "nat" where "x = Eq (Suc v) 0"   | i :: "nat" and j :: "nat" where "x = Eq (Suc i) (Suc j)"
ratrel_def: shows "ratrel = (\<lambda>x y. snd x \<noteq> 0 \<and> snd y \<noteq> 0 \<and> fst x * snd y = fst y * snd x)"
pred_unfold: fixes n :: "nat" shows "pred n = n - 1"
dims_A'_1: fixes rs :: "nat list"   and y :: "nat"   and shared_weights :: "bool"   and ws :: "nat \<Rightarrow> real" assumes "deep_model_correct_params_y rs y" shows "dim_row (deep_model_correct_params_y.A' shared_weights rs y ws) = prod_list (nths (dims (deep_model_correct_params_y.A shared_weights rs y ws)) (Collect even))"
incident_pole_polar: fixes p :: "proj2"   and l :: "proj2_line" assumes "proj2_incident p l" shows "proj2_incident (pole l) (polar p)"
simps: fixes v :: "(real, 'a) vec \<times> (real, 'a) vec"   and x :: "'a"   and r :: "real" shows "repv v x r = (\<chi>y. if x = y then r else fst v $ y, snd v)"
abs_eq: fixes x :: "agent" shows "Messages.Agent x = Ab (cAgent x)"
map_dds_id: fixes S :: "('b, 'c) dds" shows "map_dds id id S = S"
w_lt_n: shows "\<forall>x\<in>set w. x < n"
mod_hom_def: fixes R :: "('g, 'h) ring_scheme"   and M :: "('g, 'i, 'j) module_scheme"   and N :: "('g, 'k, 'l) module_scheme"   and f :: "'i \<Rightarrow> 'k" shows "mod_hom R M N f \<equiv> module R M \<and> module R N \<and> mod_hom_axioms R M N f"
even_numeral: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and n :: "num" assumes "class.semiring_parity plus minus zero times one divide modulo" shows "dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) (numeral.numeral one plus (num.Bit0 n))"
fls_inverse_deriv_divring: fixes a :: "'a fls" shows "fls_deriv (inverse a) = - inverse a * fls_deriv a * inverse a"
strong_induct: fixes P :: "'a \<Rightarrow> lam \<Rightarrow> bool"   and z :: "'a"   and lam :: "lam" assumes "\<And>name z. P z (Var name)"   and "\<And>lam1 lam2 z. \<lbrakk>\<And>z. P z lam1; \<And>z. P z lam2\<rbrakk> \<Longrightarrow> P z (App lam1 lam2)"   and "\<And>name lam z. \<lbrakk>name \<sharp> z; \<And>z. P z lam\<rbrakk> \<Longrightarrow> P z Lam [name].lam" shows "P z lam"
simps_2: fixes p :: "fm"   and q :: "fm" shows "\<zeta> (Or p q) = lcm (\<zeta> p) (\<zeta> q)"
diag_mat_def: fixes A :: "'b mat" shows "diag_mat A = map (\<lambda>i. A $$ (i, i)) [0..<dim_row A]"
ord_zero_strong_imp_rel_triv: fixes A :: "'a set" assumes "A \<subseteq> carrier G"   and "\<forall>a\<in>A. ord a = 0"   and "\<forall>f\<in>\<Pi>\<^sub>E a\<in>A. generate G {a}. finprod G f A = \<one> \<longrightarrow> (\<forall>a\<in>A. f a = \<one>)" shows "relations A = {\<lambda>_\<in>A. 0}"
norm_lt_1_iff_in_hyp2: fixes v :: "(real, 2) vec" shows "(norm v < 1) = (proj2_pt v \<in> hyp2)"
minarc_regular: fixes x :: "'a" shows "regular (minarc x)"
im_jointfunTr2: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'c"   and m :: "nat"   and g :: "nat \<Rightarrow> 'c" shows "jointfun n f m g ` nset (Suc n) (Suc (n + m)) = g ` {j. j \<le> m}"
C: fixes y :: "'a" shows "monotone option.le_fun option_ord (C y)"
carr_span: shows "local.span (gso ` {0..<i}) \<subseteq> carrier_vec n"
set_less_eq_aux_empty: fixes A :: "'a set" shows "(A \<sqsubseteq>' {}) = (A = {})"
intros_9: fixes \<Gamma> :: "'b formula multiset"   and F :: "'b formula"   and \<Delta> :: "'b formula multiset"   and G :: "'b formula" assumes "\<Gamma> \<Rightarrow> F, \<Delta>"   and "G, \<Gamma> \<Rightarrow> \<Delta>" shows "F \<^bold>\<rightarrow> G, \<Gamma> \<Rightarrow> \<Delta>"
face_ofD: fixes T :: "'b set"   and S :: "'b set"   and x :: "'b"   and a :: "'b"   and b :: "'b" assumes "T face_of S"   and "x \<in> open_segment a b"   and "a \<in> S"   and "b \<in> S"   and "x \<in> T" shows "a \<in> T \<and> b \<in> T"
mrw_values_no_write_unchanged: fixes obs :: "('a, 'b) obs_event action list"   and adal :: "'a \<times> addr_loc"   and P :: "'c prog"   and vs :: "'a \<times> addr_loc \<Rightarrow> ('a val \<times> bool) option" assumes "\<And>w. \<lbrakk>w \<in> set obs; is_write_action w; adal \<in> action_loc_aux P w\<rbrakk> \<Longrightarrow> case vs adal of None \<Rightarrow> False | \<lfloor>(v, b)\<rfloor> \<Rightarrow> b \<and> is_new_action w" shows "mrw_values P vs obs adal = vs adal"
intersects_x_axis_iff: fixes H :: "circline" assumes "is_poincare_line H" shows "(\<exists>x\<in>unit_disc. x \<in> circline_set H \<inter> circline_set x_axis) = intersects_x_axis H"
mult_less_cancel_right: fixes 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 abs :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and c :: "'b"   and b :: "'b" assumes "class.linordered_ring_strict plus zero minus uminus less_eq less abs times" shows "less (times a c) (times b c) = ((less_eq zero c \<longrightarrow> less a b) \<and> (less_eq c zero \<longrightarrow> less b a))"
set_asList: fixes A :: "'a set" assumes "finite A" shows "set (asList A) = A"
wtC_def: shows "M.wtC \<equiv> list_all M.wtL"
cut_greater_Min_greater: fixes I :: "'b set"   and t :: "'b" assumes "I \<down>> t \<noteq> {}" shows "t < iMin (I \<down>> t)"
zero_iffD: fixes f :: "'a \<Rightarrow> 'b" assumes "\<And>x. (f x = (0::'b)) = (x = (0::'a))" shows "f (0::'a) = (0::'b)"
rb_spp_aux_Nil: fixes fs :: "('a \<Rightarrow>\<^sub>0 'c) list"   and rword_strict :: "'t \<times> ('a \<Rightarrow>\<^sub>0 'c) \<Rightarrow> 't \<times> ('a \<Rightarrow>\<^sub>0 'c) \<Rightarrow> bool"   and bs :: "('t \<times> ('a \<Rightarrow>\<^sub>0 'c)) list"   and ss :: "'t list"   and z :: "nat" shows "rb_spp_aux fs rword_strict ((bs, ss, []), z) = ((bs, ss, []), z)"
insert_Id_Acts: fixes F :: "'b program" shows "insert Id (Acts F) = Acts F"
join_time_def: fixes s :: "nat \<Rightarrow> 'a list"   and n :: "nat"   and t :: "nat" shows "join_time s n t \<equiv> case n of 0 \<Rightarrow> [] | Suc i \<Rightarrow> join_ti s (n * t) i"
limit_cone_def: fixes J :: "'ag \<Rightarrow> 'ag \<Rightarrow> 'ag"   and C :: "'ah \<Rightarrow> 'ah \<Rightarrow> 'ah"   and D :: "'ag \<Rightarrow> 'ah"   and a :: "'ah"   and \<chi> :: "'ag \<Rightarrow> 'ah" shows "limit_cone J C D a \<chi> \<equiv> (category C \<and> category J) \<and> diagram J C D \<and> Limit.cone J C D a \<chi> \<and> limit_cone_axioms J C D a \<chi>"
atLeast0_atMost_Suc: fixes g :: "nat \<Rightarrow> 'b"   and n :: "nat" shows "sum g {0..Suc n} = sum g {0..n} + g (Suc n)"
igFreshClsSTR_imp_igFreshCls: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'e, 'f, 'g, 'h) model_scheme" assumes "igFreshClsSTR MOD" shows "igFreshCls MOD"
assms: shows "coincide_on \<sigma> \<eta> (vars_of_cl C)"
distinct_87: fixes pi1 :: "pi"   and pi2 :: "pi"   and pi' :: "pi" shows "pi1 \<parallel> pi2 \<noteq> !pi'"
image_inv_f_f: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set" assumes "inj f" shows "inv f ` f ` A = A"
dirichlet_prod_inverse: fixes f :: "nat \<Rightarrow> 'b"   and i :: "'b" assumes "f 1 * i = (1::'b)" shows "dirichlet_prod f (dirichlet_inverse f i) = (\<lambda>n. if n = 1 then 1::'b else (0::'b))"
ide_in_hom_1: fixes f :: "'b" assumes "D.ide f" shows "\<guillemotleft>f : src\<^sub>D f \<rightarrow>\<^sub>D trg\<^sub>D f\<guillemotright>"
induct: fixes P :: "(('a \<aproc> \<times> 'a \<aproc> set list \<times> ((nat \<times> char list) \<times> 'a \<Rightarrow> 'a \<aproc> set) \<times> 'a + call \<times> (nat \<Rightarrow> 'a option) \<times> ((nat \<times> char list) \<times> 'a \<Rightarrow> 'a \<aproc> set) \<times> 'a) discr \<rightarrow> ('a \<aproc> \<times> 'a \<aproc> set list \<times> ((nat \<times> char list) \<times> 'a \<Rightarrow> 'a \<aproc> set) \<times> 'a + call \<times> (nat \<Rightarrow> 'a option) \<times> ((nat \<times> char list) \<times> 'a \<Rightarrow> 'a \<aproc> set) \<times> 'a) discr set) \<Rightarrow> bool" assumes "adm P"   and "P \<bottom>"   and "\<And>x. P x \<Longrightarrow> P (\<Lambda> x. case undiscr x of Inl (PC (Lambda lab vs c, \<beta>), as, ve, b) \<Rightarrow> if length vs = length as then let \<beta>' = \<beta>(lab \<mapsto> b); ve' = ve \<union>. \<Union>.map2 (\<lambda>v. smap_singleton (v, b)) vs as in {Discr (Inr (c, \<beta>', ve', b))} else \<bottom> | Inl (PP (Plus c), [], bb) \<Rightarrow> \<bottom> | Inl (PP (Plus c), [x], bb) \<Rightarrow> \<bottom> | Inl (PP (Plus c), [x, xa], bb) \<Rightarrow> \<bottom> | Inl (PP (Plus c), [x, xa, cnts], ve, b) \<Rightarrow> let b' = \<anb> b c; \<beta> = [c \<mapsto> b] in \<Union>cnt\<in>cnts. {Discr (Inl (cnt, [{}], ve, b'))} | Inl (PP (Plus c), x # xa # cnts # ac # lista, bb) \<Rightarrow> \<bottom> | Inl (PP (prim.If ct cf), [], bb) \<Rightarrow> \<bottom> | Inl (PP (prim.If ct cf), [x], bb) \<Rightarrow> \<bottom> | Inl (PP (prim.If ct cf), [x, cntts], bb) \<Rightarrow> \<bottom> | Inl (PP (prim.If ct cf), [x, cntts, cntfs], ve, b) \<Rightarrow> (let b' = \<anb> b ct; \<beta> = [ct \<mapsto> b] in \<Union>cnt\<in>cntts. {Discr (Inl (cnt, [], ve, b'))}) \<union> (let b' = \<anb> b cf; \<beta> = [cf \<mapsto> b] in \<Union>cnt\<in>cntfs. {Discr (Inl (cnt, [], ve, b'))}) | Inl (PP (prim.If ct cf), x # cntts # cntfs # ac # lista, bb) \<Rightarrow> \<bottom> | Inl (AStop, [], bb) \<Rightarrow> \<bottom> | Inl (AStop, [x], xa, xb) \<Rightarrow> {} | Inl (AStop, x # a # lista, bb) \<Rightarrow> \<bottom> | Inr (App lab f vs, \<beta>, ve, b) \<Rightarrow> let fs = \<aA> f \<beta> ve; as = map (\<lambda>v. \<aA> v \<beta> ve) vs; b' = \<anb> b lab in \<Union>f'\<in>fs. {Discr (Inl (f', as, ve, b'))} | Inr (call.Let lab ls c', \<beta>, ve, b) \<Rightarrow> let b' = \<anb> b lab; \<beta>' = \<beta>(lab \<mapsto> b'); ve' = ve \<union>. \<Union>.map (\<lambda>(v, l). { (v, b') := \<aA> (L l) \<beta>' ve}.) ls in {Discr (Inr (c', \<beta>', ve', b'))})" shows "P abs_R"
label_wrt_def: fixes B :: "'b set"   and f :: "'a \<Rightarrow> 'b" shows "label_wrt B f \<equiv> \<forall>C\<in>\<C>. bij_betw f C B"
elims: fixes x :: "'b list"   and y :: "'b list list" assumes "permutations x = y" obtains "x = []" and "y = [[]]"   | v :: "'b" and va :: "'b list" where "x = v # va" and "y = concat (map (\<lambda>y. map ((#) y) (permutations (remove1 y (v # va)))) (v # va))"
finite_measure_mono: fixes A :: "real set"   and B :: "real set"   and k :: "nat" assumes "A \<subseteq> B"   and "B \<in> \<mu>.events k" shows "\<mu>.prob k A \<le> \<mu>.prob k B"
Sup_less_def: fixes Inf :: "'c set \<Rightarrow> 'c"   and Sup :: "'c set \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and x :: "'d \<Rightarrow> 'c"   and w :: "'d" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "complete_lattice.Sup_less Sup x w = Sup {y. \<exists>v<w. y = x v}"
prefix_of_appendD: fixes history :: "nat \<Rightarrow> 'c list"   and xs :: "'c list"   and ys :: "'c list"   and i :: "nat" assumes "node_histories history"   and "node_histories.prefix_of_node_history history (xs @ ys) i" shows "node_histories.prefix_of_node_history history xs i"
inherited_field_access: fixes G :: "prog"   and accC :: "qtname"   and membr :: "qtname \<times> memberdecl"   and statC :: "qtname"   and dynC :: "qtname" assumes "G \<turnstile> membr of statC accessible_from accC"   and "is_field membr"   and "G\<turnstile>dynC\<preceq>\<^sub>C statC" shows "G \<turnstile> membr in dynC dyn_accessible_from accC"
exhaust: fixes y :: "('b, 'c) rbt" obtains "y = rbt.Empty"   | x21 :: "color" and x22 :: "('b, 'c) rbt" and x23 :: "'b" and x24 :: "'c" and x25 :: "('b, 'c) rbt" where "y = Branch x21 x22 x23 x24 x25"
fls_subdegree_leI: fixes f :: "'b fls"   and n :: "int" assumes "f $$ n \<noteq> (0::'b)" shows "fls_subdegree f \<le> n"
wf\<^sub>c\<^sub>o\<^sub>n\<^sub>s\<^sub>t\<^sub>r_def: fixes S :: "('a, 'b) strand_step list"   and \<theta> :: "'b \<Rightarrow> ('a, 'b) Term.term" shows "wf\<^sub>c\<^sub>o\<^sub>n\<^sub>s\<^sub>t\<^sub>r S \<theta> \<equiv> wf\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<theta> \<and> wf\<^sub>s\<^sub>t {} S \<and> subst_domain \<theta> \<inter> vars\<^sub>s\<^sub>t S = {} \<and> range_vars \<theta> \<inter> bvars\<^sub>s\<^sub>t S = {} \<and> fv\<^sub>s\<^sub>t S \<inter> bvars\<^sub>s\<^sub>t S = {}"
fields_def: fixes P :: "'a prog"   and C :: "String.literal" shows "TypeRel.fields P C \<equiv> THE FDTs. P \<turnstile> C has_fields FDTs"
negligible_len: shows "negligible (\<lambda>\<eta>. 1 / 2 ^ len \<eta>)"
Prefix0_is_action: fixes c :: "'d \<Rightarrow> 'c" shows "R (true \<turnstile> do c \<and> (\<lambda>(A, A'). alpha_rp.more A' = alpha_rp.more A)) \<in> {p. is_CSP_process p}"
bind_pmf_if': fixes C :: "bool pmf"   and A :: "'b pmf"   and B :: "'b pmf"   and D :: "'b \<Rightarrow> 'a pmf" shows "C \<bind> (\<lambda>c. (if c then A else B) \<bind> D) = C \<bind> (\<lambda>c. if c then A \<bind> D else B \<bind> D)"
BCh'_simps_4: 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 d1h1 :: "'d"   and d1h2 :: "'d"   and d1h3 :: "'d"   and d2h1 :: "'d"   and d2h2 :: "'d"   and d2h3 :: "'d"   and d3h1 :: "'d"   and d3h2 :: "'d"   and d3h3 :: "'d"   and ds :: "'e set"   and h :: "'f"   and X :: "'d set" assumes "ContractsWithBossyConstants Xd Xh Pd Ch d1h1 d1h2 d1h3 d2h1 d2h2 d2h3 d3h1 d3h2 d3h3 ds"   and "h \<notin> {Xh d1h1, Xh d1h2, Xh d1h3}" shows "ContractsWithBossyConstants.BCh' Xh d1h1 d1h2 d1h3 d2h1 d2h2 d2h3 d3h1 d3h2 d3h3 h X = {}"
AR_imp_absolute_retract: fixes S :: "'c set"   and S' :: "'d set"   and U :: "'d set" assumes "AR S"   and "S homeomorphic S'"   and "closedin (top_of_set U) S'" shows "S' retract_of U"
ide_in_hom: fixes a :: "'b" shows "B.ide a = \<guillemotleft>a : a \<rightarrow>\<^sub>B a\<guillemotright>"
bit_eq_iff: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bit :: "'c \<Rightarrow> nat \<Rightarrow> bool"   and a :: "'c"   and b :: "'c" assumes "class.semiring_bits plus minus zero times one divide modulo bit" shows "(a = b) = (\<forall>n. bit a n = bit b n)"
AA4: fixes A :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and v :: "'b \<Rightarrow> 'd"   and w :: "'b \<Rightarrow> 'c" shows "\<turnstile> \<diamond>\<langle>\<langle>A\<rangle>_v\<rangle>_w \<longrightarrow> \<diamond>\<langle>\<langle>A\<rangle>_w\<rangle>_v"
trace_adjoint_positive: fixes A :: "complex mat" shows "0 \<le> trace (A * adjoint A)"
invertible_right_cancel: fixes M :: "'c set"   and composition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit :: "'c"   and x :: "'c"   and y :: "'c"   and z :: "'c" assumes "Group_Theory.monoid M composition unit"   and "monoid.invertible M composition unit x"   and "x \<in> M"   and "y \<in> M"   and "z \<in> M" shows "(composition y x = composition z x) = (y = z)"
wset_conv: fixes ls :: "'b"   and ts :: "'c"   and m :: "'d"   and ws :: "'a"   and "is" :: "'e" shows "wset (ls, (ts, m), ws, is) = ws"
pair_wf_digraph_axioms: shows "pair_wf_digraph sG"
set_ConsD: fixes y :: "'a"   and x :: "'a"   and xs :: "'a list" assumes "y mem (x # xs)" shows "y = x \<or> y mem xs"
simps: fixes V :: "nat set"   and Q_pos :: "(nat set \<times> 'a option list set) set"   and Q_neg :: "(nat set \<times> 'a option list set) set" shows "genericJoin V Q_pos Q_neg = (if card V \<le> 1 then (\<Inter>(uu_, x)\<in>Q_pos. x) - (\<Union>(uu_, x)\<in>Q_neg. x) else let (I, J) = getIJ Q_pos Q_neg V; Q_I_pos = projectQuery I (filterQuery I Q_pos); Q_I_neg = filterQueryNeg I Q_neg; R_I = genericJoin I Q_I_pos Q_I_neg; Q_J_neg = Q_neg - Q_I_neg; Q_J_pos = filterQuery J Q_pos; X = {(t, genericJoin J (newQuery J Q_J_pos (I, t)) (newQuery J Q_J_neg (I, t))) |t. t \<in> R_I} in \<Union>(t, x)\<in>X. {merge xx t |xx. xx \<in> x})"