hcju/isabelleps huggingface dataset:
coprime_crossproduct: fixes one :: "'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and a :: "'a"   and d :: "'a"   and b :: "'a"   and c :: "'a" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize"   and "algebraic_semidom.coprime times one a d"   and "algebraic_semidom.coprime times one b c" shows "(times (normalize a) (normalize c) = times (normalize b) (normalize d)) = (normalize a = normalize b \<and> normalize c = normalize d)"
padr_inj: fixes m :: "nat"   and n :: "nat" shows "inj_on (\<lambda>v. v @\<^sub>v 0\<^sub>v m) (carrier_vec n)"
IsoCompDef: fixes f :: "'b"   and g :: "'b" assumes "f \<approx>> g"   and "ciso f"   and "ciso g" shows "Cinv g \<approx>> Cinv f"
InfoAt_def: fixes c :: "('d, 'c, 'e) configuration_scheme"   and k :: "nat"   and p :: "'d"   and q :: "'d" shows "InfoAt c k p q = (if 0 \<le> k \<and> k < length (c_msg c p q) then c_msg c p q ! k else {#}\<^sub>z)"
gcd_mult_left_right_cancel: fixes b :: "'a"   and c :: "'a"   and a :: "'a" assumes "coprime b c" shows "gcd (a * c) b = gcd a b"
fctrs: shows "factorize_complex_main (reflect_poly q) = Some fctrs"
interpretation_comp_1: fixes \<theta> :: "'a \<Rightarrow> ('b, 'a) Term.term"   and \<sigma> :: "'a \<Rightarrow> ('b, 'a) Term.term" assumes "interpretation\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<theta>" shows "interpretation\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t (\<sigma> \<circ>\<^sub>s \<theta>)"
lexord_simps_3: fixes a :: "'g"   and x :: "'g list"   and b :: "'g"   and y :: "'g list"   and r :: "('g \<times> 'g) set" shows "((a # x, b # y) \<in> lexord r) = ((a, b) \<in> r \<or> a = b \<and> (x, y) \<in> lexord r)"
r_into_rtrancl: fixes p :: "'c \<times> 'c"   and r :: "('c \<times> 'c) set" assumes "p \<in> r" shows "p \<in> r\<^sup>*"
translation_Int: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and a :: "'a"   and s :: "'a set"   and t :: "'a set" assumes "class.ab_group_add plus zero minus uminus" shows "plus a ` (s \<inter> t) = plus a ` s \<inter> plus a ` t"
lborel_f_measurable_add_dim: fixes n :: "nat"   and x :: "real" shows "(\<lambda>f. f(n := x)) \<in> lborel_f n \<rightarrow>\<^sub>M lborel_f (Suc n)"
ballE: fixes A :: "'f set"   and P :: "'f \<Rightarrow> bool"   and x :: "'f" assumes "\<forall>x\<in>A. P x" obtains "P x"   | "x \<notin> A"
not_is_zero_one_double: shows "\<not> Code_Double.is_zero 1"
finsum_insert: fixes F :: "'c set"   and a :: "'c"   and f :: "'c \<Rightarrow> 'a vec" assumes "finite F"   and "a \<notin> F"   and "f \<in> F \<rightarrow> carrier_vec n"   and "f a \<in> carrier_vec n" shows "finsum V f (insert a F) = f a + finsum V f F"
ghd: fixes x :: "'v" shows "local.ground_heads (Var x) = UNIV"
finsum_reindex: fixes f :: "'d \<Rightarrow> 'a"   and h :: "'e \<Rightarrow> 'd"   and A :: "'e set" assumes "f \<in> h ` A \<rightarrow> carrier R"   and "inj_on h A" shows "finsum R f (h ` A) = (\<Oplus>x\<in>A. f (h x))"
Prefixes_conc: fixes B :: "'a list set"   and A :: "'a list set" assumes "B \<noteq> {}" shows "Prefixes (A @@ B) = Prefixes A \<union> A @@ Prefixes B"
inf_sup_aci_1: fixes inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b" assumes "class.lattice inf less_eq less sup" shows "inf x y = inf y x"
plus_tree_simps_2: fixes t :: "'b tree"   and t' :: "'b tree" shows "left (t + t') = left t + left t'"
is_orthogonal_closure_cspan: fixes X :: "'a set"   and Y :: "'a set"   and x :: "'a"   and y :: "'a" assumes "\<And>x y. \<lbrakk>x \<in> X; y \<in> Y\<rbrakk> \<Longrightarrow> is_orthogonal x y"   and "x \<in> closure (cspan X)"   and "y \<in> closure (cspan Y)" shows "is_orthogonal x y"
assms_1: shows "a \<in>\<^sub>\<circ> set A"
assms: shows "run_one_step d i (s, vs, ves, $Return) = (s', vs', res)"
indep_in_iff_subset_basis_in: fixes X :: "'a set" shows "indep_in \<E> X = (\<exists>B. basis_in \<E> B \<and> X \<subseteq> B)"
abs_eq: fixes xa :: "Vars.var set"   and x :: "Vars.var set" shows "xa G\<times> x = Abs_CoCall (xa \<times> x \<union> x \<times> xa)"
Game_ext: fixes l1 :: "game zet"   and r1 :: "game zet"   and l2 :: "game zet"   and r2 :: "game zet" shows "(Game l1 r1 = Game l2 r2) = (l1 = l2 \<and> r1 = r2)"
rsp: shows "eq_onp (\<lambda>f. \<forall>\<^sub>\<infinity>n. f n = (0::'b)) (\<lambda>_. 0::'b) (\<lambda>_. 0::'b)"
wt_app_eff_def: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and app :: "nat \<Rightarrow> 'b \<Rightarrow> bool"   and step :: "nat \<Rightarrow> 'b \<Rightarrow> (nat \<times> 'b) list"   and \<tau>s :: "'b list" shows "wt_app_eff r app step \<tau>s = (\<forall>p<length \<tau>s. app p (\<tau>s ! p) \<and> (\<forall>(q, \<tau>)\<in>set (step p (\<tau>s ! p)). \<tau> \<sqsubseteq>\<^bsub>r\<^esub> \<tau>s ! q))"
poly_roots_of_int_normalize: fixes p :: "int poly" shows "poly_roots (of_int_poly (normalize p)) = poly_roots (of_int_poly p)"
dom_is_obj: fixes x :: "'arr" assumes "x \<in> Arr" shows "D x \<in> Obj"
extend_def: fixes f :: "'a \<Rightarrow> real" shows "extend f = (SOME f'. f' \<in> diff_fun_space \<and> (\<forall>x\<in>closure B. f x = f' x))"
"1_2": assumes "\<not> Q x" shows "local.dom (h x)"
\<psi>_char_4: shows "\<exists>!\<psi>. \<guillemotleft>\<psi> : \<I>\<^sub>D \<rightarrow>\<^sub>D F \<I>\<^sub>C\<guillemotright> \<and> D.iso \<psi> \<and> \<psi> \<cdot>\<^sub>D \<iota>\<^sub>D = \<iota>\<^sub>1 \<cdot>\<^sub>D (\<psi> \<otimes>\<^sub>D \<psi>)"
M_reverse: fixes a :: "(real, 3) vec"   and b :: "(real, 3) vec" shows "a \<bullet> (M *v b) = b \<bullet> (M *v a)"
arr_dom_iff_arr: fixes f :: "'s" shows "arr (local.dom f) = arr f"
nat_add_distrib: fixes z :: "int"   and z' :: "int" assumes "0 \<le> z"   and "0 \<le> z'" shows "nat (z + z') = nat z + nat z'"
bound_indices: shows "f 0 = a \<and> f (card Q - 1) = c"
add_less_imp_less_left: fixes c :: "complex"   and a :: "complex"   and b :: "complex" assumes "c + a <\<^sub>\<complex> c + b" shows "a <\<^sub>\<complex> b"
OK_iff_ok: fixes I :: "'c set"   and F :: "'c \<Rightarrow> 'd program" shows "OK I F = (\<forall>i\<in>I. \<forall>j\<in>I - {i}. F i ok F j)"
arr_Set_ArrDom_in_Vset: shows "T\<lparr>ArrDom\<rparr> \<in>\<^sub>\<circ> Vset \<alpha>"
upper_triangular_adjugate: fixes A :: "(('a, 'b) vec, 'b) vec" assumes "upper_triangular A" shows "upper_triangular (adjugate A)"
induces_adjunction: shows "adjunction (\<cdot>\<^sub>C) (\<cdot>\<^sub>D) replete_setcat.comp \<phi>\<psi>.\<phi>C \<phi>\<psi>.\<phi>D F G \<phi> \<psi> \<phi>\<psi>.\<eta> \<phi>\<psi>.\<epsilon> \<phi>\<psi>.\<Phi> \<phi>\<psi>.\<Psi>"
fib_rec_even': fixes n :: "nat" shows "fib (2 * n) = (2 * fib (n - 1) + fib n) * fib n"
dep_in_subI: fixes X :: "'a set"   and \<E>' :: "'a set"   and \<E> :: "'a set" assumes "X \<subseteq> \<E>'"   and "\<not> X_insert.indep_in \<E>' X" shows "\<not> X_insert.indep_in \<E> X"
simps: fixes x :: "'a"   and y :: "'b Filters.filter"   and z :: "'a"   and w :: "'b Filters.filter" shows "pairs_sup (x, y) (z, w) = (x \<squnion> z, y \<sqinter> w)"
case_prodE2: fixes Q :: "'d \<Rightarrow> bool"   and P :: "'e \<Rightarrow> 'f \<Rightarrow> 'd"   and z :: "'e \<times> 'f" assumes "Q (case z of (a, b) \<Rightarrow> P a b)" obtains x :: "'e" and y :: "'f" where "z = (x, y)" and "Q (P x y)"
is_Write\<^sub>s\<^sub>b_simps_3: fixes p :: "'b"   and p' :: "'b"   and "is" :: "instr list" shows "is_Write\<^sub>s\<^sub>b (Prog\<^sub>s\<^sub>b p p' is) = False"
Inf_greatest: fixes A :: "('d \<times> 'e) set"   and z :: "'d \<times> 'e" assumes "\<And>x. x \<in> A \<Longrightarrow> less_eq_prod z x" shows "less_eq_prod z (Inf_prod A)"
LIMSEQ_le_const: fixes X :: "nat \<Rightarrow> 'a"   and x :: "'a"   and a :: "'a" assumes "X \<longlonglongrightarrow> x"   and "\<exists>N. \<forall>n\<ge>N. a \<le> X n" shows "a \<le> x"
ev_2: shows "\<forall>\<^sub>F x in at_right 0. g' x \<noteq> 0"
padic_add_closed: fixes p :: "int"   and x :: "nat \<Rightarrow> int"   and y :: "nat \<Rightarrow> int" assumes "prime p"   and "x \<in> carrier (padic_int p)"   and "y \<in> carrier (padic_int p)" shows "x \<oplus>\<^bsub>padic_int p\<^esub> y \<in> carrier (padic_int p)"
MATCH1_singleton_Var_2: fixes \<sigma> :: "'f \<Rightarrow> ('d, 'e) Term.term option"   and x :: "'f"   and t :: "('d, 'e) Term.term" assumes "\<sigma> x = Some t" shows "MATCH1 ({#(Var x, t)#}, \<sigma>) ({#}, \<sigma>(x \<mapsto> t))"
AbstrLevels_S12_A91: fixes i :: "AbstrLevelsID" assumes "sS12 \<in> AbstrLevel i" shows "sA91 \<notin> AbstrLevel i"
iterates_above_def: shows "iterates_above \<equiv> \<lambda>a. {x. iterates_abovep a x}"
S_reduced_imp_distinct_lconjseq': fixes ss :: "'w list" assumes "ss \<in> lists S"   and "\<not> distinct (lconjseq ss)" shows "\<not> S_reduced ss"
fdia_fbox_demorgan: fixes R :: "('e \<times> 'f) set"   and f :: "'f \<Rightarrow> 'd"   and x :: "'e" shows " |R\<rangle> f x = - |R] (\<lambda>y. - f y) x"
gcd_dvd2: fixes one :: "'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and gcd :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and lcm :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit_factor :: "'c \<Rightarrow> 'c"   and normalize :: "'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize" shows "dvd.dvd times (gcd a b) b"
simps: fixes A :: "'d atMost \<Rightarrow> 'e atMost \<Rightarrow> 'c"   and B :: "'d atMost \<Rightarrow> 'e atMost \<Rightarrow> 'c" shows "matrix_plus (Matrix A) (Matrix B) = Matrix (\<lambda>i j. A i j + B i j)"
mono_sync2a: fixes r :: "'c event list"   and P :: "'c process"   and A :: "'c set"   and S :: "'c process" assumes "r \<in> min_elems (D (P \<lbrakk>A\<rbrakk> S))" shows "\<exists>t u. r setinterleaves ((t, u), insert tick (ev ` A)) \<and> (t \<in> min_elems (D P) \<and> u \<in> T S \<or> t \<in> min_elems (D S) \<and> u \<in> T P \<and> u \<in> T P - (D P - min_elems (D P)))"
intros_3: fixes \<S> :: "s_context"   and \<C> :: "t_context"   and es :: "e list"   and t1s :: "t list"   and t2s :: "t list"   and ts :: "t list" assumes "\<S>\<bullet>\<C> \<turnstile> es : t1s _> t2s" shows "\<S>\<bullet>\<C> \<turnstile> es : ts @ t1s _> ts @ t2s"
EF_def: fixes f :: "'a set" shows "EF f \<equiv> lfp (\<lambda>Z. f \<union> EX' Z)"
e'': shows "P \<turnstile> \<langle>e'',s''\<rangle> \<Rightarrow> \<langle>throw e\<^sub>t,s'\<rangle>"
vfinite_vcard_omega: fixes A :: "V" assumes "vfinite A" shows "vcard A \<in>\<^sub>\<circ> \<omega>"
nat_diff_left_cancel_eq1: fixes k :: "nat"   and m :: "nat"   and n :: "nat" assumes "k - m = k - n"   and "m < k" shows "m = n"
add_match_split: fixes m :: "'c match_expr"   and rs1 :: "'c rule list"   and rs2 :: "'c rule list" shows "add_match m (rs1 @ rs2) = add_match m rs1 @ add_match m rs2"
cases: fixes a :: "trm" assumes "a \<in> value" obtains T :: "POPLmark.type" and t :: "trm" where "a = (\<lambda>:Tb__. tb__)"   | T :: "POPLmark.type" and t :: "trm" where "a = (\<lambda><:T. t)"
inj2_hom: fixes R :: "('g, 'h) ring_scheme"   and M1 :: "('g, 'i) module"   and M2 :: "('g, 'j) module" assumes "module R M1"   and "module R M2" shows "mod_hom R M2 (direct_sum M1 M2) (inj2 M1 M2)"
hypreal_of_real_le_add_Infininitesimal_cancel2: fixes u :: "real star"   and v :: "real star"   and x :: "real"   and y :: "real" assumes "u \<in> Infinitesimal"   and "v \<in> Infinitesimal"   and "hypreal_of_real x + u \<le> hypreal_of_real y + v" shows "x \<le> y"
Comm1: fixes P :: "pi"   and b :: "name"   and P'' :: "pi"   and a :: "name"   and x :: "name"   and P' :: "pi"   and Q :: "pi"   and Q' :: "pi" assumes "P \<Longrightarrow>\<^sub>lb in P''\<rightarrow>a<x> \<prec> P'"   and "Q \<Longrightarrow>\<^sub>l a[b] \<prec> Q'" shows "P \<parallel> Q \<Longrightarrow>\<^sub>l \<tau> \<prec> P' \<parallel> Q'"
less_imp_not_less: fixes x :: "'b list"   and y :: "'b list" assumes "x <p y" shows "(\<not> y <p x) = True"
iprev_nth_card_iMin': fixes I :: "nat set"   and n :: "nat" assumes "finite I"   and "I \<noteq> {}"   and "card I - Suc 0 \<le> n" shows "I \<leftarrow> n = iMin I"
list_slice2_1: fixes xs :: "'b list" shows "list_slice2 xs (Suc 0) = map (\<lambda>x. [x]) xs"
in_homI: fixes f :: "'c local.term set"   and a :: "'c local.term set"   and b :: "'c local.term set" assumes "\<F>\<^sub>SC.arr f"   and "\<F>\<^sub>SC.dom f = a"   and "\<F>\<^sub>SC.cod f = b" shows "\<guillemotleft>f : a \<rightarrow>\<^sub>S b\<guillemotright>"
GreatestI2: fixes P :: "'a \<Rightarrow> bool"   and a :: "'a"   and Q :: "'a \<Rightarrow> bool" assumes "P a"   and "\<And>x. P x \<Longrightarrow> Q x" shows "Q (Greatest P)"
interior_of_maximal: fixes T :: "'c set"   and S :: "'c set"   and X :: "'c topology" assumes "T \<subseteq> S"   and "openin X T" shows "T \<subseteq> X interior_of S"
simps: fixes a :: "'a" shows "the_In1 (In1 a) = a"
rm_vars_img: fixes V :: "'d set"   and s :: "'d \<Rightarrow> ('c, 'd) Term.term" shows "subst_range (rm_vars V s) = s ` subst_domain (rm_vars V s)"
eventually_filtercomap_nhds: fixes P :: "'c \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'd"   and x :: "'d" shows "eventually P (filtercomap f (nhds x)) = (\<exists>S. open S \<and> x \<in> S \<and> (\<forall>x. f x \<in> S \<longrightarrow> P x))"
intros_8: fixes \<Gamma> :: "'b formula multiset"   and F :: "'b formula"   and G :: "'b formula"   and \<Delta> :: "'b formula multiset"   and n :: "nat" assumes "\<Gamma> \<Rightarrow> F, G, \<Delta> \<down> n" shows "\<Gamma> \<Rightarrow> F \<^bold>\<or> G, \<Delta> \<down> Suc n"
d_restrict_iff: fixes x :: "'a"   and y :: "'a" shows "(x \<le> y) = (x \<le> ad (ad x) \<cdot> y)"
properfactor_cong_l: fixes x' :: "'a list"   and x :: "'a list"   and y :: "'a list" assumes "x' \<sim>\<^bsub>poly_ring R\<^esub> x"   and "properfactor (poly_ring R) x y"   and "x \<in> carrier (poly_ring R)"   and "x' \<in> carrier (poly_ring R)"   and "y \<in> carrier (poly_ring R)" shows "properfactor (poly_ring R) x' y"
encR: fixes S :: "'procS"   and TRel :: "('procT \<times> 'procT) set" shows "SourceTerm S \<R>\<lbrakk>\<cdot>\<rbrakk>RT<TRel> TargetTerm (\<lbrakk>S\<rbrakk>)"
up_set_idem: shows "\<Up> \<circ> \<Up> = \<Up>"
get_new_root_def: fixes s :: "('v, 'es) state_scheme" shows "get_new_root s \<equiv> SPEC (\<lambda>v. v \<in> V0 \<and> \<not> is_discovered v s)"
rbt_delete_in_tree: fixes t :: "('a, 'c) rbt"   and k :: "'a"   and v :: "'c"   and x :: "'a" assumes "is_rbt t" shows "((k, v) \<in> set (entries (rbt_delete x t))) = (x \<noteq> k \<and> (k, v) \<in> set (entries t))"
edges_are_in_V_2: fixes v :: "'a"   and w :: "'a" assumes "v \<rightarrow> w" shows "w \<in> V"
bigtheta_trans1: fixes f :: "'b \<Rightarrow> 'c"   and F :: "'b filter"   and g :: "'b \<Rightarrow> 'c"   and h :: "'b \<Rightarrow> 'c" assumes "f \<in> O[F](g)"   and "g \<in> \<Theta>[F](h)" shows "f \<in> O[F](h)"
lb_cos_def: fixes prec :: "nat"   and x :: "float" shows "lb_cos prec x = (let horner = \<lambda>x. lb_sin_cos_aux prec (get_even (prec div 4 + 1)) 1 1 (x * x); half = \<lambda>x. if x < 0 then - 1 else float_plus_down prec (Float 1 1 * x * x) (- 1) in if x < Float 1 (- 1) then horner x else if x < 1 then half (horner (x * Float 1 (- 1))) else half (half (horner (x * Float 1 (- 2)))))"
weak_sharing_consis_tl: fixes t :: "'b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'b memref list \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)"   and ts :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'b memref list \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list" assumes "weak_sharing_consis (t # ts)" shows "weak_sharing_consis ts"
add_less_cancel_left: fixes c :: "'a multiset"   and a :: "'a multiset"   and b :: "'a multiset" shows "(c + a \<subset># c + b) = (a \<subset># b)"
\<Sigma>_protocols_base_def: fixes Rel :: "('a \<times> 'b) set"   and valid_pub :: "'a set" shows "\<Sigma>_protocols_base Rel valid_pub \<equiv> Domain Rel \<subseteq> valid_pub"
dist_les_1: fixes a' :: "'a"   and slist' :: "[:'a:]" shows "[::] \<notsqsubseteq> a' :# slist'"
card_bind: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'b set" assumes "finite S"   and "\<forall>X\<in>S. finite (f X)"   and "disjoint_family_on f S" shows "card (S \<bind> f) = (\<Sum>x\<in>S. card (f x))"
is_bisimulation_def: fixes R :: "'state \<Rightarrow> 'state \<Rightarrow> bool" shows "is_bisimulation R \<equiv> symp R \<and> (\<forall>P Q. R P Q \<longrightarrow> (\<forall>\<phi>. P \<turnstile> \<phi> \<longrightarrow> Q \<turnstile> \<phi>)) \<and> (\<forall>P Q. R P Q \<longrightarrow> (\<forall>\<alpha> P'. bn \<alpha> \<sharp>* Q \<longrightarrow> P \<rightarrow> \<langle>\<alpha>,P'\<rangle> \<longrightarrow> (\<exists>Q'. Q \<rightarrow> \<langle>\<alpha>,Q'\<rangle> \<and> R P' Q')))"
Pbefore_in_def: fixes x :: "'c"   and y :: "'c"   and A :: "('c list \<Rightarrow> 'd pmf) \<times> ('c list \<times> 'd \<Rightarrow> 'c \<Rightarrow> ((nat \<times> nat list) \<times> 'd) pmf)"   and qs :: "'c list"   and init :: "'c list" shows "Pbefore_in x y A qs init = map_pmf (\<lambda>p. x < y in fst p) (Partial_Cost_Model.config'_rand A (fst A init \<bind> (\<lambda>is. return_pmf (init, is))) qs)"
no_step_final': fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and cfg :: "('e, 'd, 'f) com \<times> ('e, 'f) xstate"   and cfg' :: "('e, 'd, 'f) com \<times> ('e, 'f) xstate"   and P :: "bool" assumes "\<Gamma>\<turnstile> cfg \<rightarrow> cfg'"   and "final cfg" shows "P"
continuous_openin_preimage: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and T :: "'b set"   and U :: "'b set" assumes "continuous_on S f"   and "f ` S \<subseteq> T"   and "openin (top_of_set T) U" shows "openin (top_of_set S) (S \<inter> f -` U)"
mset_set_set_mset_msubset: fixes A :: "'b multiset" shows "mset_set (set_mset A) \<subseteq># A"
is_abelian_group_hom: shows "abelian_group_hom R S h"
unowned_shared_def': fixes \<S> :: "nat \<Rightarrow> bool option"   and ts :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'c \<times> 'd \<times> nat set \<times> 'e) list" shows "unowned_shared \<S> ts = (\<forall>a. (\<forall>i<length ts. a \<notin> owned (ts ! i)) \<longrightarrow> a \<in> dom \<S>)"
zero_le: fixes x :: "'c multiset" shows "{#} \<subseteq># x"
\<a>'_def: fixes \<mu> :: "'a arrow_of_spans_data"   and \<nu> :: "'a arrow_of_spans_data"   and \<tau> :: "'a arrow_of_spans_data" shows "\<a>\<^sup>-\<^sup>1[\<mu>, \<nu>, \<tau>] \<equiv> \<alpha>' (\<mu>, \<nu>, \<tau>)"
s_hyp: shows "S \<subseteq> set (cols A\<^sup>H)"
simps: fixes a :: "fm" shows "(a \<in> special_axioms) = (\<exists>A i x. a = A(i::=x) IMP SyntaxN.Ex i A)"
l2_inv7_trans: shows "{l2_inv7 \<inter> l2_inv1 \<inter> l2_inv4 \<inter> l2_inv5 \<inter> l2_inv6 \<inter> l2_inv6'} TS.trans l2 {> l2_inv7}"
is_digit_def: fixes n :: "nat" shows "is_digit n = (n < 10)"
rep_eq: fixes x :: "'a \<Longrightarrow> 'b"   and xa :: "('c, 'b) uexpr" shows "(x \<natural> xa) = (\<forall>b b'. \<lbrakk>xa\<rbrakk>\<^sub>e (b' \<triangleleft>\<^bsub>x\<^esub> b) = \<lbrakk>xa\<rbrakk>\<^sub>e b)"
othrow_def: fixes e :: "'c" shows "othrow e = K (Some (Inl e))"
f4: shows "4 \<le> rk {A', B', P, R}"
le0: shows "adjoint M0 * P * M0 \<le>\<^sub>L adjoint M0 * M0"
map_fst_half_segments_aux_eq: fixes X :: "(real \<times> real) \<times> (real \<times> real) pdevs" assumes "[] \<noteq> half_segments_of_aform X" shows "map fst (half_segments_of_aform X) = fst (hd (half_segments_of_aform X)) # butlast (map snd (half_segments_of_aform X))"
inj_eq: fixes f :: "'d \<Rightarrow> 'e"   and x :: "'d"   and y :: "'d" assumes "inj f" shows "(f x = f y) = (x = y)"
connectedin_absolute: fixes X :: "'b topology"   and S :: "'b set" shows "connectedin (subtopology X S) S = connectedin X S"
asubg_mOp_closed: fixes A :: "('e, 'f) aGroup_scheme"   and H :: "'e set"   and x :: "'e" assumes "aGroup A"   and "A +> H"   and "x \<in> H" shows "-\<^sub>a\<^bsub>A\<^esub> x \<in> H"
list_rel_comp_Union_rel: fixes R :: "('c \<times> 'd) set"   and S :: "('d \<times> 'e set) set" assumes "single_valued R" shows "\<langle>R\<rangle>list_rel O \<langle>\<langle>Id\<rangle>list_wset_rel, S\<rangle>Union_rel = \<langle>\<langle>Id\<rangle>list_wset_rel, R O S\<rangle>Union_rel"
integrable_sum': fixes t :: "'b set"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'd"   and s :: "'c set" assumes "finite t"   and "\<forall>a\<in>t. f a integrable_on s" shows "(\<lambda>x. \<Sum>a\<in>t. f a x) integrable_on s"
conseq: fixes P' :: "'b set"   and P :: "'b set"   and c :: "'b com"   and Q :: "'b set"   and Q' :: "'b set" assumes "P' \<subseteq> P"   and "\<turnstile> P c Q"   and "Q \<subseteq> Q'" shows "\<turnstile> P' c Q'"
assms: shows "A.ide (e \<cdot>\<^sub>A m)"
degV: fixes a :: "'a mod_ring poly"   and b :: "'a mod_ring poly"   and u :: "'b mod_ring poly"   and v :: "'c mod_ring poly"   and w :: "'c mod_ring poly"   and q_ty :: "'c itself" assumes "Knuth_ex_4_6_2_22_main a b u v w" shows "degree (#v + Polynomial.smult (of_nat (Knuth_ex_4_6_2_22_base.q q_ty)) (#Knuth_ex_4_6_2_22_main.v' q_ty b u v w)) = degree v"
set_of_def: fixes P :: "'b \<Rightarrow> 'c \<Rightarrow> bool" shows "set_of P = {(x, y). P x y}"
has_vderiv_on_sq_mtx_linear: fixes t\<^sub>0 :: "real"   and A :: "'b sq_mtx"   and s :: "(real, 'b) vec"   and t :: "real" shows " D (\<lambda>t. exp ((t - t\<^sub>0) *\<^sub>R A) *\<^sub>V s) = (\<lambda>t. A *\<^sub>V (exp ((t - t\<^sub>0) *\<^sub>R A) *\<^sub>V s)) on {t\<^sub>0--t}"
bilateral_substitutes_on_def: fixes A :: "'a set"   and f :: "'a set \<Rightarrow> 'a set" shows "Bossy''.bilateral_substitutes_on A f = (\<not> (\<exists>B\<subseteq>A. \<exists>a b. {a, b} \<subseteq> A \<and> Xd a \<notin> Xd ` B \<and> Xd b \<notin> Xd ` B \<and> b \<notin> f (B \<union> {b}) \<and> b \<in> f (B \<union> {a, b})))"
has_laplace_spike: fixes g :: "real \<Rightarrow> complex"   and L :: "complex"   and s :: "complex"   and T :: "real set"   and f :: "real \<Rightarrow> complex" assumes "(g has_laplace L) s"   and "negligible T"   and "\<And>t. \<lbrakk>t \<notin> T; 0 \<le> t\<rbrakk> \<Longrightarrow> f t = g t" shows "(f has_laplace L) s"
prems: shows "subst_domain \<theta> \<inter> bvars\<^sub>s\<^sub>t (a # A) = {}"
fac_triv': fixes p :: "'b list"   and x :: "'b list"   and q :: "'b list" assumes "p \<cdot> x \<cdot> q = x" shows "q = \<epsilon>"
eq_0_on_span: fixes b :: "'a set"   and x :: "'a" assumes "\<And>x. x \<in> b \<Longrightarrow> f x = (0::'b)"   and "x \<in> module.span (*\<^sub>R) b" shows "f x = (0::'b)"
eq_height_0: fixes t :: "'b tree" shows "(height t = 0) = (t = \<langle>\<rangle>)"
eng_add: fixes h1 :: "'c"   and R :: "('c, 'd) ring_scheme"   and H :: "'c set"   and h2 :: "'c" assumes "h1 \<in> generate_field R H"   and "h2 \<in> generate_field R H" shows "h1 \<oplus>\<^bsub>R\<^esub> h2 \<in> generate_field R H"
rtrancl3p_step: fixes r :: "'g \<Rightarrow> 'h \<Rightarrow> 'g \<Rightarrow> bool"   and a :: "'g"   and bs :: "'h list"   and a' :: "'g"   and b :: "'h"   and a'' :: "'g" assumes "rtrancl3p r a bs a'"   and "r a' b a''" shows "rtrancl3p r a (bs @ [b]) a''"
inconsist_for_V01: fixes f :: "nat \<Rightarrow> nat option"   and k :: "nat"   and e :: "nat" assumes "f \<in> V\<^sub>0\<^sub>1"   and "k = amalg01 f" shows "inconsist k e \<down>= (if \<exists>x<e_length e. \<phi> k x \<down>\<noteq> e_nth e x then LEAST x. x < e_length e \<and> \<phi> k x \<down>\<noteq> e_nth e x else e_length e)"