hcju/isabelleps huggingface dataset:
inject: fixes x1 :: "object atom formula"   and x2 :: "object ast_effect"   and y1 :: "object atom formula"   and y2 :: "object ast_effect" shows "(Ground_Action x1 x2 = Ground_Action y1 y2) = (x1 = y1 \<and> x2 = y2)"
length_power_polys: fixes mult_p :: "'b"   and u :: "'b"   and curr_p :: "'b"   and n :: "nat" shows "length (power_polys mult_p u curr_p n) = n"
between_h: assumes "\<exists>n<length (sorted_list_of_set {x. \<exists>q\<in>set qs. rpoly q x = 0}) - 1. sorted_list_of_set {x. \<exists>q\<in>set qs. rpoly q x = 0} ! n < x1 \<and> x1 < sorted_list_of_set {x. \<exists>q\<in>set qs. rpoly q x = 0} ! (n + 1)" shows "\<exists>w\<in>roots_of_coprime_r (cast_rat_list qs). sga = sign_vec qs w"
rep_eq: fixes x :: "'b mod_ring" shows "to_int_mod_ring x = Rep_mod_ring x"
bump_def: fixes phi :: "nat \<Rightarrow> nat"   and y :: "nat" shows "bump phi y = (case y of 0 \<Rightarrow> 0 | Suc n \<Rightarrow> Suc (phi n))"
delete_shadow_root_type_wf_preserved: fixes h :: "(_) heap"   and shadow_root_ptr :: "(_) shadow_root_ptr"   and h' :: "(_) heap" assumes "h \<turnstile> delete\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_M shadow_root_ptr \<rightarrow>\<^sub>h h'"   and "ShadowRootClass.type_wf h" shows "ShadowRootClass.type_wf h'"
update_convs_1: fixes Pol' :: "('d \<times> 'd) set \<Rightarrow> ('d \<times> 'd) set"   and Pol :: "('d \<times> 'd) set"   and Map :: "'c \<Rightarrow> 'd"   and Dom :: "'d"   and In :: "'c list"   and Out :: "'c list"   and more :: "'e" shows "Pol_update Pol' \<lparr>Pol = Pol, Map = Map, Dom = Dom, In = In, Out = Out, \<dots> = more\<rparr> = \<lparr>Pol = Pol' Pol, Map = Map, Dom = Dom, In = In, Out = Out, \<dots> = more\<rparr>"
compow_matrix_mul_diag_mat_eq: fixes n :: "nat"   and f :: "'e \<Rightarrow> 'd" shows "((**) (diag_mat f) ^^ n) (mat (1::'d)) = diag_mat (\<lambda>i. f i ^ n)"
surjective: fixes r :: "('a, 'b, 'c) Category_scheme" shows "r = \<lparr>Obj = obj\<^bsub>r\<^esub>, Mor = mor\<^bsub>r\<^esub>, Dom = Dom r, Cod = Cod r, Id = Category.Id r, Comp = (;;\<^bsub>r\<^esub>), \<dots> = Category.more r\<rparr>"
proj2_line_abs_rep: fixes l :: "proj2_line" shows "proj2_line_abs (proj2_line_rep l) = l"
unitK_simps: fixes v :: "'a ValK"   and c :: "'a ValK \<rightarrow> 'a" shows "unitK\<cdot>v\<cdot>c = c\<cdot>v"
ennreal_SUP_add_left: fixes I :: "'a set"   and f :: "'a \<Rightarrow> ennreal"   and c :: "ennreal" assumes "I \<noteq> {}" shows "(\<Squnion>i\<in>I. f i + c) = \<Squnion> (f ` I) + c"
presWls_fromIMor: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<times> 'c"   and MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) Recursion.model" assumes "ipresWls h (fromMOD MOD)" shows "presWls (fromIMor h) MOD"
word_clz_def: fixes w :: "'b word" shows "word_clz w \<equiv> length (takeWhile Not (to_bl w))"
D_1: shows "upper_asymptotic_density D \<le> 1 - l"
lmeasurable_iff_has_integral: fixes S :: "'c set" shows "(S \<in> lmeasurable) = (indicat_real S has_integral Sigma_Algebra.measure lebesgue S) UNIV"
IH: fixes vs :: "'a set"   and s :: "('a, 'b) fmap"   and b :: "('a, 'b) fmap \<times> ('a, 'b) fmap"   and s' :: "('a, 'b) fmap" assumes "vars_change as vs s = []"   and "sat_precond_as s as"   and "ListMem b as"   and "fmrestrict_set vs s = fmrestrict_set vs s'" shows "fmrestrict_set vs (snd b) \<subseteq>\<^sub>f fmrestrict_set vs s'"
sel_1: fixes x1 :: "'c"   and x2 :: "'d"   and x3 :: "nat"   and x4 :: "('c, 'd) SkewBinomialTree list" shows "val (Node x1 x2 x3 x4) = x1"
map_comp0: fixes g :: "'e \<Rightarrow> 'd"   and f :: "'c \<Rightarrow> 'e" shows "map_F (g \<circ> f) = map_F g \<circ> map_F f"
suminf_geometric: fixes c :: "'a" assumes "norm c < 1" shows "(\<Sum>n. c ^ n) = (1::'a) / ((1::'a) - c)"
lincomb_Nil: fixes R :: "'a set"   and smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and M :: "'b set"   and rs :: "'a list"   and ms :: "'b list" assumes "RModule R smult M"   and "rs = [] \<or> ms = []" shows "scalar_mult.lincomb smult rs ms = (0::'b)"
UNIV_uprod: shows "UNIV = range (\<lambda>x. Upair x x) \<union> (\<lambda>(x, y). Upair x y) ` (SIGMA x:UNIV. UNIV - {x})"
RedStaticCastFail: fixes C :: "char list"   and Cs :: "char list list"   and P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and E :: "char list \<Rightarrow> ty option"   and a :: "nat"   and s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)" assumes "C \<notin> set Cs"   and "(last Cs, C) \<notin> (subcls1 P)\<^sup>*" shows "P,E \<turnstile> \<langle>\<lparr>C\<rparr>ref (a, Cs),s\<rangle> \<rightarrow> \<langle>Throw (addr_of_sys_xcpt ClassCast, [ClassCast]),s\<rangle>"
list_subset_iff_projection_neutral: fixes l :: "'a list"   and X :: "'a set" shows "(set l \<subseteq> X) = (l \<upharpoonleft> X = l)"
fv_list\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s_append: fixes F :: "(('b, 'a) Term.term \<times> ('c, 'a) Term.term) list"   and G :: "(('b, 'a) Term.term \<times> ('c, 'a) Term.term) list" shows "fv_list\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s (F @ G) = fv_list\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s F @ fv_list\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s G"
greatest_ge_nonzero_row: fixes i :: "'b"   and k :: "nat"   and A :: "(('c, 'd) vec, 'b) vec" assumes "\<not> is_zero_row_upt_k i k A" shows "i \<le> (GREATEST m. \<not> is_zero_row_upt_k m k A)"
AbstFormP: fixes i :: "name"   and A :: "fm" shows "{} \<turnstile> AbstFormP \<guillemotleft>Var i\<guillemotright> Zero \<guillemotleft>A\<guillemotright> (quot_dbfm (trans_fm [i] A))"
Constrains_def: fixes A :: "'d set"   and B :: "'d set" shows "A Co B \<equiv> {F. F \<in> reachable F \<inter> A co B}"
hyps: shows "gen_check as = filter (suffs C as) (alls B as)"
conv_has_start_end_points: fixes x :: "'a" shows "has_start_end_points x = has_start_end_points (x\<^sup>T)"
hyps: fixes ws :: "'a list" assumes "ws \<in> set (subseqs vs)" shows "mset ws \<subseteq># mset vs"
new_element_child_nodes: fixes h :: "(_) heap"   and h' :: "(_) heap"   and new_element_ptr :: "(_) element_ptr" assumes "h \<turnstile> new_element \<rightarrow>\<^sub>h h'"   and "h \<turnstile> new_element \<rightarrow>\<^sub>r new_element_ptr" shows "h' \<turnstile> get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t new_element_ptr RElement.child_nodes \<rightarrow>\<^sub>r []"
bind_error_simps_1: fixes f :: "'c \<rightarrow> 'a\<cdot>'b error" shows "\<bottom> \<bind> f = \<bottom>"
distinct_2: fixes x21 :: "char list"   and x22 :: "expr" shows "x21:=x22 \<noteq> Skip"
exhaust: fixes y :: "meth_def" obtains x1 :: "meth_sig" and x2 :: "meth_body" where "y = meth_def_def x1 x2"
intro: fixes \<U> :: "'b set"   and \<A> :: "'b set multiset"   and \<V> :: "'b set"   and \<B> :: "'b set multiset" assumes "proper_sub_set_system \<U> \<A> \<V> \<B>"   and "incidence_system \<U> \<A>" shows "proper_sub_incidence_system \<U> \<A> \<V> \<B>"
rt_graph_vD_trans: fixes ip :: "nat"   and ip' :: "nat"   and \<sigma> :: "nat \<Rightarrow> state"   and dip :: "nat" assumes "(ip, ip') \<in> (rt_graph \<sigma> dip)\<^sup>+" shows "dip \<in> vD (rt (\<sigma> ip))"
SD_efficient_lottery_exists: fixes p :: "'b pmf" assumes "p \<in> lotteries_on alts'" obtains q :: "'b pmf" where "q \<in> lotteries_on alts'" and "Pareto (SD \<circ> R') p q" and "SD_efficient R' q"
\<phi>_\<psi>: fixes b :: "'d"   and a :: "'d"   and x :: "'s" assumes "D.ide b"   and "D.ide a"   and "x \<in> HomD.set (b, a)" shows "\<phi>D (b, a) (\<psi>D (b, a) x) = x"
buffer_defs_7: fixes ic :: "state \<Rightarrow> 'b"   and q :: "state \<Rightarrow> 'b list"   and oc :: "state \<Rightarrow> 'b" shows "BInv ic q oc \<equiv> List.last<$oc # $q> = $ic \<and> noreps<$oc # $q>"
Subgraph_Start_axioms: shows "Subgraph_Start E E'"
m_comm: fixes x :: "'a"   and y :: "'a" assumes "x \<in> UNIV"   and "y \<in> UNIV" shows "x * y = y * x"
complement_shunting: fixes z :: "'a"   and w :: "'a"   and x :: "'a"   and y :: "'a" assumes "complement z w" shows "(z \<sqinter> x \<le> y) = (x \<le> w \<squnion> y)"
fin: fixes s :: "'s1" assumes "wellformed1 s" shows "finite {fold delta1 w s |w. w \<in> lists (set alphabet1)}"
S_reachable: shows "S ({(v, u)} \<union> A) \<subseteq> nodes rg"
next_execs_consistent: shows "\<forall>s t execs. vpeq (current s) s t \<and> (\<forall>d\<in>involved (next_action s execs). vpeq d s t) \<and> current s = current t \<longrightarrow> fst (snd (control s (current s) (execs (current s)))) = fst (snd (control t (current s) (execs (current s))))"
butlast_Subobjs: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and C :: "char list"   and Cs :: "char list list"   and D :: "char list" assumes "Subobjs P C (Cs @ [D])"   and "Cs \<noteq> []" shows "Subobjs P C Cs"
wadjust_backto_standard_pos_B_Oc_Oc_via_O: fixes m :: "nat"   and rs :: "nat"   and c :: "cell list"   and list :: "cell list" assumes "wadjust_backto_standard_pos_O m rs (c, Oc # list)"   and "c \<noteq> []"   and "hd c = Oc" shows "wadjust_backto_standard_pos_O m rs (tl c, Oc # Oc # list)"
hyps_2: shows "Formula.hereditarily_fs (L_transform_Tree\<^sub>\<alpha> t\<^sub>\<alpha>)"
tl_drop: fixes n :: "nat"   and xs :: "'c list" shows "tl (drop n xs) = drop n (tl xs)"
lm040: fixes P :: "('c \<times> 'd) set"   and X :: "'c set"   and Y :: "'c set"   and Q :: "('c \<times> 'd) set"   and Z :: "'c set" shows "Outside' (X \<union> Y) P \<inter> Q || X = {} \<and> Outside' X P \<inter> Q || (X \<inter> Z) = {}"
C1: fixes \<Psi>' :: "'b"   and R :: "('a, 'b, 'c) psi"   and S :: "('a, 'b, 'c) psi"   and yvec :: "name list" assumes "(\<Psi>', R, S) \<in> Rel"   and "yvec \<sharp>* \<Psi>'" shows "(\<Psi>', \<lparr>\<nu>*yvec\<rparr>R, \<lparr>\<nu>*yvec\<rparr>S) \<in> Rel"
of_nat_less_iff: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and m :: "nat"   and n :: "nat" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "less (semiring_1.of_nat one plus zero m) (semiring_1.of_nat one plus zero n) = (m < n)"
simple_fw_iff_generalized_fw_drop: fixes fw :: "'b simple_rule list"   and p :: "('b, 'c) simple_packet_scheme" shows "(simple_fw fw p = Decision FinalDeny) = (\<exists>r. generalized_sfw (map simple_rule_dtor fw) p = Some (r, Drop))"
of_nat_1_eq_iff: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and n :: "nat" assumes "class.semiring_char_0 one times plus zero" shows "(one = semiring_1.of_nat one plus zero n) = (n = 1)"
refl_on_domain: fixes A :: "'b set"   and r :: "('b \<times> 'b) set"   and a :: "'b"   and b :: "'b" assumes "refl_on A r"   and "a \<succeq>[r] b" shows "a \<in> A \<and> b \<in> A"
Limit_VPow_in_VsetI: fixes \<alpha> :: "V"   and A :: "V" assumes "Limit \<alpha>"   and "A \<in>\<^sub>\<circ> Vset \<alpha>" shows "VPow A \<in>\<^sub>\<circ> Vset \<alpha>"
img: shows "\<forall>w\<in>space M. pseudo_proj_True 0 w = sconst True"
collapse: fixes option :: "'e option" assumes "option \<noteq> None" shows "Some (the option) = option"
SHOmsgVectors_def: fixes A :: "('e, 'g, 'f, 'h) CHOAlgorithm_scheme"   and r :: "nat"   and p :: "'e"   and cfg :: "'e \<Rightarrow> 'g"   and HOp :: "'e set"   and SHOp :: "'e set" shows "SHOmsgVectors A r p cfg HOp SHOp \<equiv> {\<mu>. (\<forall>q. (q \<in> HOp) = (\<mu> q \<noteq> None)) \<and> (\<forall>q. q \<in> SHOp \<inter> HOp \<longrightarrow> \<mu> q = Some (sendMsg A r q p (cfg q)))}"
induct: fixes P :: "(nat \<times> nat) list \<Rightarrow> bool"   and a0 :: "(nat \<times> nat) list" assumes "P []"   and "\<And>x y ps. \<lbrakk>\<And>xa. P ps; \<And>xa xb. xb = ExcessNotAtRecList ps xa \<Longrightarrow> P (deleteAround xa x ps)\<rbrakk> \<Longrightarrow> P ((x, y) # ps)" shows "P a0"
elims: fixes x :: "nat"   and xa :: "nat"   and y :: "rat" assumes "fs.gs.\<mu> x xa = y" obtains i :: "nat" and j :: "nat" where "x = i" and "xa = j" and "y = (if j < i then map of_int_hom.vec_hom fs ! i \<bullet> fs.gs.gso j / \<parallel>fs.gs.gso j\<parallel>\<^sup>2 else if i = j then 1 else 0)"
basic: fixes t :: "'a set" assumes "t \<in> A" shows "t open"
beta_l_real: fixes l :: "nat" shows "beta_l l \<in> \<real>"
uf_1: fixes n :: "nat" shows "uf n \<in> {eNorm N1 f1 + eNorm N2 f2 |f1 f2. f = f1 + f2}"
cong_less_imp_eq_nat: fixes a :: "nat"   and m :: "nat"   and b :: "nat" assumes "0 \<le> a"   and "a < m"   and "0 \<le> b"   and "b < m"   and "[a = b] (mod m)" shows "a = b"
simps: fixes a :: "['b]" shows "distinct a = (a = [] \<or> (\<exists>x xs. a = x : xs \<and> distinct xs \<and> x \<notin> set xs))"
sup_bot_left: fixes x :: "'a" shows "\<top> \<sqinter> x = x"
le_spvec_iff_sparse_row_le: fixes a :: "(nat \<times> 'a) list"   and b :: "(nat \<times> 'a) list" assumes "sorted_spvec a"   and "sorted_spvec b" shows "le_spvec a b = (sparse_row_vector a \<le> sparse_row_vector b)"
is_mset_set: fixes X :: "'d set" shows "is_mset_set (mset_set X)"
widen_rules_4: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'a) list) list"   and h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and loc :: "val list"   and LT :: "ty err list"   and LT' :: "ty err list" assumes "P,h \<turnstile> loc [:\<le>\<^sub>\<top>] LT"   and "P \<turnstile> LT [\<le>\<^sub>\<top>] LT'" shows "P,h \<turnstile> loc [:\<le>\<^sub>\<top>] LT'"
ptensor_mat_mult: fixes m1 :: "'b mat"   and m2 :: "'b mat"   and m3 :: "'b mat"   and m4 :: "'b mat" assumes "m1 \<in> carrier_mat d1 d1"   and "m2 \<in> carrier_mat d1 d1"   and "m3 \<in> carrier_mat d2 d2"   and "m4 \<in> carrier_mat d2 d2" shows "ptensor_mat (m1 * m2) (m3 * m4) = ptensor_mat m1 m3 * ptensor_mat m2 m4"
g_reduction: shows "\<parallel>gs2.gso (i - 1)\<parallel>\<^sup>2 < reduction * \<parallel>fs.gs.gso (i - 1)\<parallel>\<^sup>2"
h5: shows "invertible_mat (mat_of_rows_list 3 [[1, 1, 1], [0, 1, 0], [1, 0, - 1]])"
prems_2: shows "(l # ts) ! i = (p, is, xs, sb, \<D>, \<O>, \<R>)"
naturality: fixes f :: "'a \<times> 'a" assumes "VV.arr f" shows "S.cmp\<^sub>U\<^sub>P (VV.cod f) \<cdot>\<^sub>S UP.H\<^sub>DoFF.map f = UP.FoH\<^sub>C.map f \<cdot>\<^sub>S S.cmp\<^sub>U\<^sub>P (VV.dom f)"
ac_ne: shows "cbox a (\<chi>i. if i = m then b $ m + b $ n else b $ i) \<noteq> {}"
pt_set_infinite_ineq: fixes pi :: "('b \<times> 'b) list"   and X :: "'a set" assumes "pt TYPE('a) TYPE('b)"   and "at TYPE('b)" shows "infinite (pi \<bullet> X) = infinite X"
ImpL_inv: fixes F :: "'b formula"   and G :: "'b formula"   and \<Gamma> :: "'b formula multiset"   and \<Delta> :: "'b formula multiset" assumes "F \<^bold>\<rightarrow> G, \<Gamma> \<Rightarrow> \<Delta>" shows "\<Gamma> \<Rightarrow> F, \<Delta> \<and> G, \<Gamma> \<Rightarrow> \<Delta>"
digit_altdef: fixes x :: "nat"   and i :: "nat" shows "digit x i = x div base ^ i mod base"
solve_lhr_def: fixes cs :: "complex list"   and fs :: "complex list" shows "solve_lhr cs fs = (if cs = [] \<or> length fs < length cs - 1 then None else let m = length fs + 1 - length cs; p = lhr_fps_numerator m cs ((!) fs); q = lr_fps_denominator' cs in map_option (solve_factored_ratfps' p) (factorize_complex_main q))"
continuous_on_neq_split: fixes s :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and y :: "'b" assumes "\<forall>x\<in>s. f x \<noteq> y"   and "continuous_on s f"   and "connected s" shows "(\<forall>x\<in>s. y < f x) \<or> (\<forall>x\<in>s. f x < y)"
sgn_1_neg: 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 sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and a :: "'b" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "(sgn a = uminus one) = less a zero"
Pd_linear': fixes d :: "'b" shows "Linear_order (override_on Pd Pds' ds' d)"
distrib: fixes g :: "'b \<Rightarrow> 'a"   and h :: "'b \<Rightarrow> 'a"   and A :: "'b set" shows "(\<Sum>x\<in>A. g x + h x) = sum g A + sum h A"
orthonormal_coeff_def: fixes S :: "'a set"   and w :: "nat \<Rightarrow> 'a \<Rightarrow> real"   and f :: "'a \<Rightarrow> real"   and n :: "nat" shows "orthonormal_coeff S w f n = l2product S (w n) f"
f_outside: fixes e :: "'a \<times> 'a" assumes "e \<notin> \<^bold>E" shows "f e = 0"
dist_fs_infinite2: fixes z1 :: "complex" shows "dist_fs \<infinity>\<^sub>h (of_complex z1) = 2 / sqrt (1 + (cmod z1)\<^sup>2)"
sat_push_edge_action_bound': fixes f :: "nat \<times> nat \<Rightarrow> 'capacity"   and l :: "nat \<Rightarrow> nat"   and p :: "pr_operation' list"   and f' :: "nat \<times> nat \<Rightarrow> 'capacity"   and l' :: "nat \<Rightarrow> nat"   and e :: "nat \<times> nat" assumes "((f, l), p, f', l') \<in> trcl pr_algo_lts'" shows "length (filter ((=) (SAT_PUSH' e)) p) \<le> 2 * card V"
RPHIL\<^sub>n\<^sub>o\<^sub>r\<^sub>m_rec: fixes i :: "nat" shows "RPHIL\<^sub>n\<^sub>o\<^sub>r\<^sub>m i = (\<lambda>s. Mprefix (Tr\<^sub>r\<^sub>p i s) (\<lambda>e. RPHIL\<^sub>n\<^sub>o\<^sub>r\<^sub>m i (Up\<^sub>r\<^sub>p i s e)))"
rtrancl_subset: fixes R :: "('a \<times> 'a) set"   and S :: "('a \<times> 'a) set" assumes "R \<subseteq> S"   and "S \<subseteq> R\<^sup>*" shows "S\<^sup>* = R\<^sup>*"
inj_on_def: fixes f :: "'k \<Rightarrow> 'l"   and A :: "'k set" shows "inj_on f A = (\<forall>x\<in>A. \<forall>y\<in>A. f x = f y \<longrightarrow> x = y)"
sup_cancel_left1: fixes x :: "'a"   and a :: "'a"   and b :: "'a" shows "x + a + (- x + b) = 1"
distinct_Ex1: fixes xs :: "'c list"   and x :: "'c" assumes "distinct xs"   and "x \<in> set xs" shows "\<exists>!i. i < length xs \<and> xs ! i = x"
fixp_induct: fixes P :: "(nat \<Rightarrow> nat \<Rightarrow> nat spmf) \<Rightarrow> bool" assumes "spmf.admissible (\<lambda>fast_dice_roll. P (curry fast_dice_roll))"   and "P (\<lambda>fast_dice_roll v. return_pmf None)"   and "\<And>c. P c \<Longrightarrow> P (\<lambda>a b. if n \<le> a then if b < n then return_spmf b else c (a - n) (b - n) else coin_spmf \<bind> (\<lambda>ba. c (2 * a) (2 * b + (if ba then 1 else 0))))" shows "P local.fast_dice_roll"
iT_Min_1: fixes n :: "nat"   and d :: "nat" shows "iMin [n\<dots>,d] = n"
runit'_simps_1: fixes f :: "'c" assumes "C.ide f" shows "C.arr \<r>\<^sub>C\<^sup>-\<^sup>1[f]"
less_eq_sort_dir_def: shows "(\<le>) \<equiv> \<lambda>x y. rec_sort_dir (case_sort_dir False True) (case_sort_dir False False) x y \<or> x = y"
safe_formula_induct: fixes \<phi> :: "'c MFOTL.formula"   and P :: "'c MFOTL.formula \<Rightarrow> bool" assumes "safe_formula \<phi>"   and "\<And>t1 t2. trm.is_Const t1 \<Longrightarrow> P (formula.Eq t1 t2)"   and "\<And>t1 t2. trm.is_Const t2 \<Longrightarrow> P (formula.Eq t1 t2)"   and "\<And>x y. P (formula.Neg (formula.Eq (trm.Const x) (trm.Const y)))"   and "\<And>x y. x = y \<Longrightarrow> P (formula.Neg (formula.Eq (trm.Var x) (trm.Var y)))"   and "\<And>e ts. P (formula.Pred e ts)"   and "\<And>\<phi> \<psi>. \<lbrakk>\<not> (safe_formula (formula.Neg \<psi>) \<and> fv \<psi> \<subseteq> fv \<phi>); P \<phi>; P \<psi>\<rbrakk> \<Longrightarrow> P (MFOTL.And \<phi> \<psi>)"   and "\<And>\<phi> \<psi>. \<lbrakk>safe_formula \<psi>; fv \<psi> \<subseteq> fv \<phi>; P \<phi>; P \<psi>\<rbrakk> \<Longrightarrow> P (MFOTL.And_Not \<phi> \<psi>)"   and "\<And>\<phi> \<psi>. \<lbrakk>fv \<psi> = fv \<phi>; P \<phi>; P \<psi>\<rbrakk> \<Longrightarrow> P (formula.Or \<phi> \<psi>)"   and "\<And>\<phi>. P \<phi> \<Longrightarrow> P (formula.Exists \<phi>)"   and "\<And>I \<phi>. P \<phi> \<Longrightarrow> P (formula.Prev I \<phi>)"   and "\<And>I \<phi>. P \<phi> \<Longrightarrow> P (formula.Next I \<phi>)"   and "\<And>\<phi> I \<psi>. \<lbrakk>fv \<phi> \<subseteq> fv \<psi>; safe_formula \<phi>; P \<phi>; P \<psi>\<rbrakk> \<Longrightarrow> P (formula.Since \<phi> I \<psi>)"   and "\<And>\<phi> I \<psi>. \<lbrakk>fv (formula.Neg \<phi>) \<subseteq> fv \<psi>; \<not> safe_formula (formula.Neg \<phi>); P \<phi>; P \<psi>\<rbrakk> \<Longrightarrow> P (formula.Since (formula.Neg \<phi>) I \<psi>)"   and "\<And>\<phi> I \<psi>. \<lbrakk>fv \<phi> \<subseteq> fv \<psi>; safe_formula \<phi>; P \<phi>; P \<psi>\<rbrakk> \<Longrightarrow> P (formula.Until \<phi> I \<psi>)"   and "\<And>\<phi> I \<psi>. \<lbrakk>fv (formula.Neg \<phi>) \<subseteq> fv \<psi>; \<not> safe_formula (formula.Neg \<phi>); P \<phi>; P \<psi>\<rbrakk> \<Longrightarrow> P (formula.Until (formula.Neg \<phi>) I \<psi>)" shows "P \<phi>"
m_assoc: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<in> carrier R"   and "y \<in> carrier R"   and "z \<in> carrier R" shows "x \<oplus> y \<oplus> z = x \<oplus> (y \<oplus> z)"
snd_conv: fixes x1 :: "'ah"   and x2 :: "'ag" shows "snd (x1, x2) = x2"
insert_Id_AllowedActs: fixes F :: "'b program" shows "insert Id (AllowedActs F) = AllowedActs F"
typing_alt_simps_39: fixes \<Gamma> :: "String.literal \<rightharpoonup>\<^sub>f 'a OCL_Types.type"   and a :: "'a expr"   and b :: "'a expr"   and \<tau> :: "'a OCL_Types.type" shows "(\<Gamma> \<turnstile>\<^sub>P CollectionRange a b : \<tau>) = (\<tau> = Integer[1] \<and> (\<exists>\<tau>. \<Gamma> \<turnstile>\<^sub>E a : \<tau> \<and> (\<exists>\<sigma>. \<Gamma> \<turnstile>\<^sub>E b : \<sigma> \<and> UnlimitedNatural[1] \<le> \<tau> \<and> \<tau> \<le> Integer[1] \<and> \<sigma> = UnlimitedNatural[1]\<midarrow>Integer[1])))"
euler_mascheroni_sequence_decreasing: assumes "0 < 1"   and "1 \<le> n" shows "harm n - ln (real n) \<le> harm 1 - ln (real 1)"
iso: fixes i :: "'a" assumes "i \<in> I" shows "G i \<cong> H i"
tendsto_iff: fixes dist :: "'d \<Rightarrow> 'd \<Rightarrow> real"   and uniformity :: "('d \<times> 'd) filter"   and "open" :: "'d set \<Rightarrow> bool"   and f :: "'e \<Rightarrow> 'd"   and l :: "'d"   and F :: "'e filter" assumes "class.metric_space dist uniformity open" shows "filterlim f (topological_space.nhds open l) F = (\<forall>e>0. \<forall>\<^sub>F x in F. dist (f x) l < e)"
cases: fixes x :: "'fsym trm" obtains xa :: "Preliminaries.var" where "x = Var xa"   | v :: "'fsym" and va :: "'fsym trm list" where "x = Fn v va"
rev_shiftr: fixes w :: "'a word"   and n :: "nat" shows "word_reverse w >> n = word_reverse (w << n)"
floor_frac_add_preservation': fixes d :: "real"   and a :: "real" assumes "0 \<le> d"   and "d < 1 - frac a" shows "\<lfloor>a\<rfloor> = \<lfloor>a + d\<rfloor>"
machine_implies_cbv_star_ctx: fixes e :: "lam"   and Es :: "ctx list"   and e' :: "lam"   and Es' :: "ctx list" assumes "<e,Es> \<mapsto> <e',Es'>" shows " Es\<down>\<lbrakk>e\<rbrakk> \<longrightarrow>cbv* Es'\<down>\<lbrakk>e'\<rbrakk>"
yeah: shows "(\<Sum>(x, y)\<in>{(x, y). x \<in> set init \<and> y \<in> set init \<and> x < y}. if Suc s < length (swaps ss init) then if swaps ss init ! s = x \<and> swaps ss init ! Suc s = y \<or> swaps ss init ! s = y \<and> swaps ss init ! Suc s = x then 1 else 0 else 0) = 1"
keysFor_insert_Crypt: fixes K :: "key"   and X :: "msg"   and H :: "msg set" shows "keysFor (insert (Crypt K X) H) = insert (invKey K) (keysFor H)"
fSup: shows "feasible (Sup_trans (fst ` {x. P (fst x) (snd x) \<and> feasible (fst x) \<and> (\<forall>Q. unitary Q \<longrightarrow> unitary (snd x Q)) \<and> le_trans (fst x) (lfp_trans (\<lambda>a. wp (body ;; Embed a \<^bsub>\<guillemotleft> G \<guillemotright>\<^esub>\<oplus> Skip))) \<and> le_utrans (gfp_trans (\<lambda>a. wlp (body ;; Embed a \<^bsub>\<guillemotleft> G \<guillemotright>\<^esub>\<oplus> Skip))) (snd x)}))"
assms: shows "\<langle>r, s\<rangle> \<in>\<^sub>\<circ> vrange A"
rp_defs_5: shows "R \<equiv> R3 \<circ> R2 \<circ> R1"