hcju/isabelleps huggingface dataset:
dvd_mod: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and k :: "'a"   and m :: "'a"   and n :: "'a" assumes "class.euclidean_semiring_cancel divide plus minus zero times one modulo euclidean_size"   and "dvd.dvd times k m"   and "dvd.dvd times k n" shows "dvd.dvd times k (modulo m n)"
simps_2: fixes x31 :: "'a local.term"   and x32 :: "'a local.term"   and y31 :: "'a local.term"   and y32 :: "'a local.term" shows "(x31 \<^bold>\<otimes> x32 = y31 \<^bold>\<otimes> y32) = (x31 = y31 \<and> x32 = y32)"
induct: fixes x1 :: "('c \<times> 'd) list"   and x2 :: "'d \<Rightarrow> 'c set"   and P :: "('c \<times> 'd) list \<Rightarrow> ('d \<Rightarrow> 'c set) \<Rightarrow> bool" assumes "crdt_ops x1 x2"   and "\<And>deps. P [] deps"   and "\<And>xs deps oid oper. \<lbrakk>crdt_ops xs deps; P xs deps; oid \<notin> set (map fst xs); \<forall>ref\<in>deps oper. ref \<in> set (map fst xs) \<and> ref < oid\<rbrakk> \<Longrightarrow> P (xs @ [(oid, oper)]) deps" shows "P x1 x2"
cqt_further_5: fixes v :: "i"   and \<phi> :: "'a \<Rightarrow> \<o>"   and \<psi> :: "'a \<Rightarrow> \<o>" shows "[(\<^bold>\<exists>\<alpha>. \<phi> \<alpha> \<^bold>& \<psi> \<alpha>) \<^bold>\<rightarrow> (\<^bold>\<exists>\<alpha>. \<phi> \<alpha>) \<^bold>& (\<^bold>\<exists>\<alpha>. \<psi> \<alpha>) in v]"
bit_iff_odd_drop_bit: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bit :: "'a \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and drop_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and take_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and n :: "nat" assumes "class.semiring_bit_shifts plus minus zero times one divide modulo bit push_bit drop_bit take_bit" shows "bit a n = (\<not> dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) (drop_bit n a))"
n_def: shows "n = nat \<lfloor>R / (E * real DIM('a))\<rfloor> + 1"
jump_const: fixes c :: "real"   and a :: "real" shows "jump (\<lambda>_. c) a = 0"
degree_find_eq: fixes var :: "nat"   and L :: "atom list"   and A :: "real mpoly list"   and L' :: "atom list" assumes "find_eq var L = (A, L')" shows "\<forall>p\<in>set A. MPoly_Type.degree p var = 1 \<or> MPoly_Type.degree p var = 2"
setbydecision_all_append_subset2: fixes rs1 :: "'a common_primitive rule list"   and rs2 :: "'a common_primitive rule list"   and iface :: "iface" assumes "simple_ruleset (rs1 @ rs2)" shows "setbydecision_all TYPE('b) iface rs1 FinalDeny \<union> (setbydecision_all TYPE('b) iface rs2 FinalDeny - setbydecision TYPE('b) iface rs1 FinalAllow) \<subseteq> setbydecision_all TYPE('b) iface (rs1 @ rs2) FinalDeny"
derivativeEqvtI: fixes P :: "pi"   and Q :: "pi"   and a :: "subject"   and x :: "name"   and Rel :: "(pi \<times> pi) set"   and perm :: "(name \<times> name) list" assumes "derivative P Q a x Rel"   and "eqvt Rel" shows "derivative (perm \<bullet> P) (perm \<bullet> Q) (perm \<bullet> a) (perm \<bullet> x) Rel"
l2_inv5_def: shows "l2_inv5 \<equiv> {s. \<forall>Rb A B gnx Ni. hmac \<langle>Number 1, Exp Gen (NonceF (Rb $ ny)), gnx, Messages.Agent A, Messages.Agent B\<rangle> (hmac Ni (NonceF (Rb $ nr))) \<in> parts (extr (bad s) (ik s) (chan s)) \<longrightarrow> guessed_runs Rb = \<lparr>role = Resp, owner = B, partner = A\<rparr> \<longrightarrow> A \<notin> bad s \<and> B \<notin> bad s \<longrightarrow> (\<exists>Ra. guessed_runs Ra = \<lparr>role = Init, owner = A, partner = B\<rparr> \<and> in_progressS (progress s Ra) {xnx, xni, xnr, xgnx, xgny, xsk, xEnd} \<and> sklvl1.guessed_frame Ra xgnx = Some gnx \<and> sklvl1.guessed_frame Ra xni = Some Ni \<and> sklvl1.guessed_frame Ra xnr = Some (NonceF (Rb $ nr)) \<and> sklvl1.guessed_frame Ra xgny = Some (Exp Gen (NonceF (Rb $ ny))))}"
HPhase1or2ReadThen_maxBalInp: fixes s :: "state"   and s' :: "state"   and q :: "Proc"   and d :: "Disk"   and p :: "Proc"   and b :: "nat"   and v :: "InputsOrNi" assumes "HPhase1or2ReadThen s s' q d p"   and "maxBalInp s b v" shows "maxBalInp s' b v"
bind_assoc: fixes m :: "'d \<Rightarrow> ('g \<times> 'd) option"   and k :: "'g \<Rightarrow> 'd \<Rightarrow> ('f \<times> 'd) option"   and h :: "'f \<Rightarrow> 'd \<Rightarrow> ('e \<times> 'd) option" shows "bind\<^sub>S\<^sub>E (bind\<^sub>S\<^sub>E m k) h = _bind_SE x m (bind\<^sub>S\<^sub>E (k x) h)"
squash_real_of_rat: fixes x :: "rat" shows "squash (real_of_rat x) = squash x"
prod_rule: fixes p :: "'a \<times> 'b"   and S :: "'a \<Rightarrow> 'b \<Rightarrow> 'c nres"   and \<Phi> :: "'c \<Rightarrow> bool" assumes "\<And>a b. p = (a, b) \<Longrightarrow> S a b \<le> SPEC \<Phi>" shows "(case p of (x, xa) \<Rightarrow> S x xa) \<le> SPEC \<Phi>"
H_skip: fixes p :: "'a" shows "H p (1::'a) p"
deg_nonzero_represents: fixes p :: "int poly" assumes "degree p \<noteq> 0" shows "\<exists>x. p represents x"
simps: fixes succs :: "'e \<Rightarrow> 'e list"   and ins :: "'e \<Rightarrow> 'd \<Rightarrow> 'd"   and memb :: "'e \<Rightarrow> 'd \<Rightarrow> bool"   and S :: "'d"   and wl :: "'e list" shows "gen_dfs succs ins memb S wl = (case wl of [] \<Rightarrow> S | x # xs \<Rightarrow> if memb x S then gen_dfs succs ins memb S xs else gen_dfs succs ins memb (ins x S) (succs x @ xs))"
mult_botr: fixes x :: "'c \<times> 'd" shows "times_prod' x bot_prod = bot_prod"
rep_eq: fixes x :: "'a fls" shows "fls_prpart x = Abs_poly (\<lambda>n. if n = 0 then 0::'a else fls_nth x (- int n))"
local_lipschitz_congI: fixes s' :: "'d set"   and t' :: "'e set"   and g' :: "'d \<Rightarrow> 'e \<Rightarrow> 'f"   and s :: "'d set"   and t :: "'e set"   and g :: "'d \<Rightarrow> 'e \<Rightarrow> 'f" assumes "local_lipschitz s' t' g'"   and "s' = s"   and "t' = t"   and "\<And>x y. \<lbrakk>x \<in> s; y \<in> t\<rbrakk> \<Longrightarrow> g' x y = g x y" shows "local_lipschitz s t g"
synth_analz_NI_I_K_synth_analz_NI_E: fixes K :: "msg set"   and Bad :: "agent set"   and NI :: "msg set"   and I :: "msg set" assumes "Keys_bad K Bad"   and "NI \<subseteq> payload"   and "K \<subseteq> range LtK"   and "I \<subseteq> valid" shows "synth (analz (NI \<union> I \<union> K \<union> Tags)) \<subseteq> synth (analz (extr Bad NI (local.abs I))) \<union> - payload"
alphaEnv_sym: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm option"   and rho' :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm option" assumes "rho &= rho'" shows "rho' &= rho"
dWHILEIT_def: shows "dWHILEIT \<equiv> iWHILEIT (\<bind>) dRETURN"
one_chain_j_integrals: shows "one_chain_line_integral F {j} one_chain_typeII = one_chain_line_integral F {j} one_chain_typeI \<and> (\<forall>(k, \<gamma>)\<in>one_chain_typeII. line_integral_exists F {j} \<gamma>) \<and> (\<forall>(k, \<gamma>)\<in>one_chain_typeI. line_integral_exists F {j} \<gamma>)"
oth_class_taut_2: fixes v :: "i"   and \<phi> :: "\<o>" shows "[\<phi> \<^bold>\<or> \<^bold>\<not>\<phi> in v]"
prveqlPT_nprv_instInp_instInpP: fixes \<phi> :: "'fmla"   and \<tau> :: "'fmla"   and t :: "'trm" assumes "\<phi> \<in> fmla"   and "Fvars \<phi> \<subseteq> {inp}"   and "\<tau> \<in> ptrm 0"   and "t \<in> trm"   and "FvarsT t = {}"   and "prveqlPT \<tau> t" shows "nprv {instInpP \<phi> 0 \<tau>} (instInp \<phi> t)"
image_subgroup: fixes G :: "'a set" assumes "subgroup G G (\<cdot>) \<one>" shows "subgroup ((\<lambda>a\<in>G. transformation.symmetric.inverse (a)\<^sub>R) ` G) Translations (compose G) (identity G)"
assms_3: shows "\<not> validate_ground_clause (int_clset S) (subst_cl C \<sigma>)"
maximal_independent_subset_extend: fixes scale :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and S :: "'b set"   and V :: "'b set" assumes "vector_space scale"   and "S \<subseteq> V"   and "\<not> module.dependent scale S" obtains B :: "'b set" where "S \<subseteq> B" and "B \<subseteq> V" and "\<not> module.dependent scale B" and "V \<subseteq> module.span scale B"
induct: fixes x1 :: "'a"   and x2 :: "'b set"   and x3 :: "'b set"   and x4 :: "'b \<Rightarrow> 'b set"   and fg :: "('a, 'c, 'd, 'b, 'e) flowgraph_rec_scheme"   and U :: "'a set"   and P :: "'a \<Rightarrow> 'b set \<Rightarrow> 'b set \<Rightarrow> ('b \<Rightarrow> 'b set) \<Rightarrow> bool" assumes "(x1, x2, x3, x4) \<in> RU_cs fg U"   and "\<And>u. u \<in> U \<Longrightarrow> P u {} {} (\<lambda>x. {})"   and "\<And>u p u' v M P Ml Me h. \<lbrakk>(u, Call p, u') \<in> edges fg; proc_of fg v = p; (v, M, P) \<in> S_cs fg 0; (v, Ml, Me, h) \<in> RU_cs fg U; P v Ml Me h; mon_n fg u \<inter> Me = {}\<rbrakk> \<Longrightarrow> P u (mon fg p \<union> M \<union> Ml) Me (ah_update h (mon fg p, M) (Ml \<union> Me))"   and "\<And>u p u' v M P q Ml Me h. \<lbrakk>(u, Call p, u') \<in> edges fg; proc_of fg v = p; (v, M, P) \<in> S_cs fg 1; q \<in># P; (entry fg q, Ml, Me, h) \<in> RU_cs fg U; P (entry fg q) Ml Me h; (mon_n fg u \<union> mon fg p) \<inter> (Ml \<union> Me) = {}\<rbrakk> \<Longrightarrow> P u (mon fg p \<union> M) (Ml \<union> Me) (ah_update h (mon fg p, M) (Ml \<union> Me))" shows "P x1 x2 x3 x4"
subset_insert: fixes x :: "'a"   and A :: "'a set"   and B :: "'a set" assumes "x \<notin> A" shows "(A \<subseteq> insert x B) = (A \<subseteq> B)"
SD_pref_profileI: fixes i :: "'agent"   and p :: "'alt pmf"   and q :: "'alt pmf" assumes "i \<in> agents"   and "p \<in> lotteries"   and "q \<in> lotteries"   and "\<And>x. x \<in> alts \<Longrightarrow> lottery_prob q (preferred_alts (R i) x) \<le> lottery_prob p (preferred_alts (R i) x)" shows "SD (R i) q p"
reparam_path_image: fixes \<gamma>1 :: "real \<Rightarrow> 'a"   and \<gamma>2 :: "real \<Rightarrow> 'a" assumes "reparam \<gamma>1 \<gamma>2" shows "path_image \<gamma>1 = path_image \<gamma>2"
trans: fixes A :: "'c set"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c"   and z :: "'c" assumes "transitive A less_eq"   and "less_eq x y"   and "less_eq y z"   and "x \<in> A"   and "y \<in> A"   and "z \<in> A" shows "less_eq x z"
aux4: fixes M :: "trm"   and M' :: "trm" assumes "M = M'" shows "M \<longrightarrow>\<^sub>a* M'"
seq_refine_pred: fixes P :: "('a, 'b) urel"   and b :: "'a upred"   and s :: "'b upred"   and Q :: "('b, 'c) urel"   and c :: "'c upred" assumes "(\<lceil>b\<rceil>\<^sub>< \<Rightarrow> \<lceil>s\<rceil>\<^sub>>) \<sqsubseteq> P"   and "(\<lceil>s\<rceil>\<^sub>< \<Rightarrow> \<lceil>c\<rceil>\<^sub>>) \<sqsubseteq> Q" shows "(\<lceil>b\<rceil>\<^sub>< \<Rightarrow> \<lceil>c\<rceil>\<^sub>>) \<sqsubseteq> P ;; Q"
nat_pow_one: fixes n :: "nat" shows "\<one>\<^bsub>G\<lparr>carrier := H\<rparr>\<^esub> [^]\<^bsub>G\<lparr>carrier := H\<rparr>\<^esub> n = \<one>\<^bsub>G\<lparr>carrier := H\<rparr>\<^esub>"
mk_valid_prefix: fixes base :: "'b word"   and len :: "nat" shows "valid_prefix (PrefixMatch (base && ~~ mask (LENGTH('b) - len)) len)"
assms_2: shows "p\<^sub>1 \<in> ps\<^sub>R"
inpt_str_comp_aux_a: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> 'c \<Rightarrow> 'e \<Rightarrow> bool"   and r' :: "'b \<Rightarrow> 'b \<Rightarrow> 'e \<Rightarrow> 'd \<Rightarrow> bool"   and u :: "nat \<Rightarrow> 'a \<times> 'b"   and x :: "nat \<Rightarrow> 'c"   and n :: "nat" assumes "\<forall>n. (\<forall>i<n. inpt_st (\<lambda>(u, v) (u', v'). r u u' OO r' v v') (u i) (u (Suc i)) (x i)) \<longrightarrow> inpt_st r (fst (u n)) (fst (u (Suc n))) (x n) \<and> (\<forall>y. r (fst (u n)) (fst (u (Suc n))) (x n) y \<longrightarrow> inpt_st r' (snd (u n)) (snd (u (Suc n))) y)" shows "inpt_st r ((fst \<circ> u) n) ((fst \<circ> u) (Suc n)) (x n) \<and> (\<forall>y. r (fst (u n)) (fst (u (Suc n))) (x n) y \<longrightarrow> inpt_st r' (snd (u n)) (snd (u (Suc n))) y)"
wordinterval_empty_set_eq: fixes r :: "'a wordinterval" shows "wordinterval_empty r = (wordinterval_to_set r = {})"
nth_shiftl: fixes w :: "'b word"   and m :: "nat"   and n :: "nat" shows "(w << m) !! n = (n < LENGTH('b) \<and> m \<le> n \<and> w !! (n - m))"
minus_mult_vec_commute: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c" shows "- p ** q = p ** - q"
lang_dual_subset_lists: fixes \<Sigma> :: "'a list"   and n :: "nat"   and r :: "'a atom rexp_dual" assumes "wf_dual \<Sigma> n r" shows "lang_dual \<Sigma> n r \<subseteq> lists ((set \<circ> \<sigma> \<Sigma>) n)"
ee_factorsI: fixes a :: "'a"   and b :: "'a"   and as :: "'a list"   and bs :: "'a list" assumes "a \<sim> b"   and "factors G as a"   and "a \<notin> Units G"   and "factors G bs b"   and "b \<notin> Units G"   and "set as \<subseteq> carrier G"   and "set bs \<subseteq> carrier G" shows "essentially_equal G as bs"
fmdom'_drop: fixes a :: "'a"   and m :: "('a, 'b) fmap" shows "fmdom' (fmdrop a m) = fmdom' m - {a}"
continuous_map_prod_snd: fixes i :: "'a"   and I :: "'a set"   and X :: "'b topology"   and Y :: "'c topology" assumes "i \<in> I" shows "continuous_map (prod_topology X (product_topology (\<lambda>i. Y) I)) Y (\<lambda>x. snd x i)"
dvd_imp_degree_le: fixes p :: "'a poly"   and q :: "'a poly" assumes "p dvd q"   and "q \<noteq> 0" shows "Polynomial.degree p \<le> Polynomial.degree q"
update_def: fixes s :: "Store"   and l :: "Location"   and v :: "Value" shows "s\<langle>l := v\<rangle> \<equiv> if alive (ref l) s \<and> alive v s \<and> \<tau> v \<le> ltype l then Abs_Store (Rep_Store s\<lparr>vals := (vals (Rep_Store s))(l := v)\<rparr>) else s"
select_convs_2: fixes mstate_i :: "nat"   and mstate_m :: "('a, 'b) mformula"   and mstate_n :: "nat"   and more :: "'c" shows "mstate_m \<lparr>mstate_i = mstate_i, mstate_m = mstate_m, mstate_n = mstate_n, \<dots> = more\<rparr> = mstate_m"
chine_composite_1: shows "Chn ((local.dom \<mu> \<star> local.dom \<nu>) \<star> local.dom \<pi>) = doms.\<nu>.leg0 \<cdot> doms.\<mu>\<nu>.prj\<^sub>0 \<down>\<down> doms.\<pi>.leg1"
star_of_minus: fixes x :: "'c" shows "star_of (- x) = - star_of x"
prems_2: shows "g \<turnstile> last (ns @ [n])-ms\<rightarrow>m"
boxR: fixes A :: "(C, BD) form" shows "g3mod1p [ \<Empt> \<Rightarrow>* \<LM> A  \<RM>] ( \<Empt> \<Rightarrow>* \<LM> \<box> A  \<RM>)"
collinear_affine_hull: fixes S :: "'a set" shows "collinear S = (\<exists>u v. S \<subseteq> affine hull {u, v})"
monoid_add_ow: shows "class.monoid_add = monoid_add_ow UNIV"
sorted_list_of_set_eq_Nil_iff: fixes A :: "'a word set" assumes "finite A" shows "(signed.sorted_list_of_set A = []) = (A = {})"
update_valuate_rhs: fixes e :: "nat \<times> linear_poly"   and s :: "('c, 'a) state"   and x :: "nat"   and c :: "'a" assumes "e \<in> set (\<T> s)"   and "\<triangle> (\<T> s)" shows "rhs e \<lbrace> \<langle>\<V> (update x c s)\<rangle> \<rbrace> = rhs e \<lbrace> \<langle>\<V> s\<rangle>(x := c) \<rbrace>"
norm_bound_Basis_le: fixes b :: "'c"   and x :: "'c"   and e :: "real" assumes "b \<in> Basis"   and "norm x \<le> e" shows "\<bar>inner x b\<bar> \<le> e"
map_filter_map_filter: fixes f :: "'b \<Rightarrow> 'a"   and P :: "'b \<Rightarrow> bool"   and xs :: "'b list" shows "map f (filter P xs) = List.map_filter (\<lambda>x. if P x then Some (f x) else None) xs"
pmf_neq_exists_less: fixes M :: "'a pmf"   and N :: "'a pmf" assumes "M \<noteq> N" shows "\<exists>x. pmf M x < pmf N x"
filterlim_at_split: fixes f :: "'c \<Rightarrow> 'd"   and F :: "'d filter"   and x :: "'c" shows "filterlim f F (at x) = (filterlim f F (at_left x) \<and> filterlim f F (at_right x))"
region_not_empty: fixes X :: "'b set"   and k :: "'b \<Rightarrow> nat"   and I :: "'b \<Rightarrow> intv"   and r :: "('b \<times> 'b) set" assumes "finite X"   and "valid_region X k I r" shows "\<exists>u. u \<in> region X I r"
dom_char: fixes f :: "'c local.term set \<times> 'c local.term set" shows "CC.dom f = (if arr (fst f) \<and> arr (snd f) then (local.dom (fst f), local.dom (snd f)) else (null, null))"
simps_2: fixes xs :: "'a list"   and ys :: "'a list"   and yss :: "'a list list" shows "vec_mat_Tensor xs (ys # yss) = vec_vec_Tensor xs ys # vec_mat_Tensor xs yss"
otherE: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and I :: "'b set"   and \<sigma> :: "'b \<Rightarrow> 'a"   and \<sigma>' :: "'b \<Rightarrow> 'a"   and R :: "('b \<Rightarrow> 'a) \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> bool" assumes "other P I \<sigma> \<sigma>'"   and "\<lbrakk>\<forall>i\<in>I. \<sigma>' i = \<sigma> i; \<forall>j. j \<notin> I \<longrightarrow> P (\<sigma> j) (\<sigma>' j)\<rbrakk> \<Longrightarrow> R \<sigma> \<sigma>'" shows "R \<sigma> \<sigma>'"
Sup_upper: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and 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 bot :: "'b"   and top :: "'b"   and x :: "'b"   and A :: "'b set" assumes "class.complete_lattice (Inf::'b set \<Rightarrow> 'b) (Sup::'b set \<Rightarrow> 'b) (inf::'b \<Rightarrow> 'b \<Rightarrow> 'b) (less_eq::'b \<Rightarrow> 'b \<Rightarrow> bool) (less::'b \<Rightarrow> 'b \<Rightarrow> bool) (sup::'b \<Rightarrow> 'b \<Rightarrow> 'b) (bot::'b) (top::'b)"   and "(x::'b) \<in> (A::'b set)" shows "(less_eq::'b \<Rightarrow> 'b \<Rightarrow> bool) (x::'b) ((Sup::'b set \<Rightarrow> 'b) (A::'b set))"
h2: shows "\<forall>x\<in>X. \<forall>y\<in>X. \<exists>z\<in>X. x \<le> z \<and> y \<le> z"
cases: fixes a :: "('a, 'b) trm" assumes "dfree a" obtains i :: "'b" where "a = Var i__"   | r :: "real" where "a = Const r"   | args :: "'b \<Rightarrow> ('a, 'b) trm" and i :: "'a" where "a = $f i args" and "\<And>i. dfree (args i)"   | \<theta>\<^sub>1 :: "('a, 'b) trm" and \<theta>\<^sub>2 :: "('a, 'b) trm" where "a = Plus \<theta>\<^sub>1__ \<theta>\<^sub>2__" and "dfree \<theta>\<^sub>1__" and "dfree \<theta>\<^sub>2__"   | \<theta>\<^sub>1 :: "('a, 'b) trm" and \<theta>\<^sub>2 :: "('a, 'b) trm" where "a = Times \<theta>\<^sub>1 \<theta>\<^sub>2" and "dfree \<theta>\<^sub>1" and "dfree \<theta>\<^sub>2"
ncoplanar_perm_2: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "\<not> Tarski_neutral_dimensionless.Coplanar Bet A B C D" shows "\<not> Tarski_neutral_dimensionless.Coplanar Bet A C B D"
case_prodI2': fixes p :: "'a \<times> 'b"   and c :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c" assumes "\<And>a b. (a, b) = p \<Longrightarrow> c a b x" shows "(case p of (a, b) \<Rightarrow> c a b) x"
filter_IFS_no_violations_def: fixes G :: "'b graph"   and M :: "'b SecurityInvariant_configured list"   and Es :: "('b \<times> 'b) list" shows "filter_IFS_no_violations G M Es = filter_IFS_no_violations_accu G M [] Es"
cfg_of_cfg_onI: fixes D :: "'s pmf"   and s :: "'s"   and c :: "'s \<Rightarrow> 's cfg" assumes "D \<in> K s"   and "\<And>t. t \<in> set_pmf D \<Longrightarrow> c t \<in> cfg_on t" shows "Cfg s D c \<in> cfg_on s"
conseq: fixes P :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool"   and c :: "com"   and Q :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool"   and P' :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool"   and Q' :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool" assumes "\<turnstile>\<^sub>3\<^sub>' {P} c { Q}"   and "\<And>s n. P' (s, n) \<Longrightarrow> P (s, 1 * n)"   and "\<And>s n. Q (s, n) \<Longrightarrow> Q' (s, n div 1)"   and "0 < 1" shows "\<turnstile>\<^sub>3\<^sub>' {P'} c { Q'}"
simps_1: fixes bq :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree list" shows "SkewBinomialHeapStruc.meldUniq [] bq = bq"
weakly_bisimilar_weak_simulation_step: fixes P :: "'state"   and Q :: "'state"   and \<alpha> :: "('act, 'pred) S_action"   and P' :: "'state" assumes "P \<approx>\<cdot>\<^sub>S Q"   and "bn \<alpha> \<sharp>* Q"   and "P \<Rightarrow>\<langle>\<alpha>\<rangle>\<^sub>S P'" obtains Q' :: "'state" where "Q \<Rightarrow>\<langle>\<alpha>\<rangle>\<^sub>S Q'" and "P' \<approx>\<cdot>\<^sub>S Q'"
fst_of_sp_is_vert: fixes g :: "('d, 'e) rgraph_scheme"   and v1 :: "'d"   and es :: "'d edge list"   and v2 :: "'d" assumes "Graph.subpath g v1 es v2" shows "v1 \<in> Graph.vertices g"
shift1_invD_2: fixes qs :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list"   and B :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) set" assumes "shift1_inv (qs, B)" shows "shift2_inv qs"
closure_of_closedin: fixes X :: "'b topology"   and S :: "'b set" assumes "closedin X S" shows "X closure_of S = S"
Me: shows "(\<lambda>\<sigma>. expr_sem \<sigma> e) \<in> state_measure (V \<union> V') \<Gamma> \<rightarrow>\<^sub>M subprob_algebra (stock_measure t)"
strong_get_put: fixes x :: "'d \<Longrightarrow> 'e"   and \<sigma> :: "'e"   and \<rho> :: "'e" assumes "bij_lens x" shows "put\<^bsub>x\<^esub> \<sigma> (get\<^bsub>x\<^esub> \<rho>) = \<rho>"
\<tau>move1TryThrow: fixes P :: "'b prog"   and h :: "'heap"   and a :: "'addr"   and C :: "String.literal"   and V :: "'c"   and e :: "('c, 'd, 'addr) exp" shows "\<tau>move1 P h (try Throw a catch(C V) e)"
not_fin_crename_aux: fixes M :: "trm"   and y :: "name"   and a :: "coname"   and b :: "coname" assumes "fin M y" shows "fin (M[a\<turnstile>c>b]) y"
atan2_bounded: fixes y :: "real"   and x :: "real" shows "- pi \<le> atan2 y x \<and> atan2 y x < pi"
not_normalized_fracts_0_denom: fixes a :: "'a" shows "(a, 0::'a) \<notin> normalized_fracts"
ACIEQ_le_ACIcl: fixes r :: "'d rexp"   and s :: "'d rexp" assumes "r \<approx> s" shows "r ~~ s"
reduction_signs_is_get_signs_R: fixes p :: "real poly"   and qs :: "real poly list"   and signs :: "rat list list"   and subsets :: "(nat list \<times> nat list) list"   and m :: "rat mat" shows "reduction_signs_R p qs signs subsets m = get_signs_R (reduce_system_R p (qs, m, subsets, signs))"
rv_mult: fixes A :: "((complex, 'b) Finite_Cartesian_Product.vec, 'b) Finite_Cartesian_Product.vec"   and v :: "(real, 'b) Finite_Cartesian_Product.vec" assumes "real_non_neg_mat A" shows "A *v (\<chi>i. complex_of_real (v $h i)) = (\<chi>i. complex_of_real ((B A *v v) $h i))"
trans_MapOps_lookup_def: shows "trans_MapOps_lookup \<equiv> \<lambda>m k. Option.bind (trie_odlist_lookup m (fst (fst k))) (\<lambda>m. Option.bind (trie_odlist_lookup m (snd (fst k))) (\<lambda>m. Mapping.lookup m (snd k)))"
Gramian_determinant_div: fixes l :: "nat" assumes "l < length fs" shows "Gramian_determinant fs (Suc l) / Gramian_determinant fs l = \<parallel>gso l\<parallel>\<^sup>2"
one_blinop_def: shows "1 \<equiv> Abs_blinop 1\<^sub>L"
coprime_absorb_left: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and 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 x :: "'c"   and y :: "'c" assumes "class.algebraic_semidom divide plus minus zero times one"   and "dvd.dvd times x y" shows "algebraic_semidom.coprime times one x y = dvd.dvd times x one"
FDOM_state_succ: fixes a :: "('c, 'd) fmap \<times> ('c, 'd) fmap"   and s :: "('c, 'd) fmap" assumes "fmdom' (snd a) \<subseteq> fmdom' s" shows "fmdom' (state_succ s a) = fmdom' s"
index_infinite_set: fixes x :: "'c llist"   and m :: "nat"   and y :: "'c"   and n :: "nat" assumes "\<not> lfinite x"   and "x $ m = y"   and "\<And>i. x $ i = y \<Longrightarrow> \<exists>m>i. x $ m = y" shows "y \<in> lset (ldropn n x)"
type_less_eq_x_Boolean: fixes \<tau> :: "'a basic_type" assumes "\<tau> \<le> Boolean" obtains "\<tau> = OclVoid"   | "\<tau> = Boolean"
lossless_weight_spmfD: fixes p :: "'a spmf" assumes "lossless_spmf p" shows "weight_spmf p = 1"