hcju/isabelleps huggingface dataset:
assms: shows "((\<sigma>, s), a, \<sigma>', s') \<in> oseqp_sos \<Gamma> i"
a: shows "phiDefNodes_aux g v (filter (\<lambda>n. (n, v) \<notin> Mapping.keys (phis'_aux g v (set ms) phis)) (\<alpha>n g)) m \<subseteq> phiDefNodes_aux g v (filter (\<lambda>n. (n, v) \<notin> Mapping.keys phis) (\<alpha>n g)) m"
e_length_update: fixes b :: "nat"   and k :: "nat"   and v :: "nat" shows "e_length (e_update b k v) = e_length b"
lead_coeff_of_int: fixes k :: "int" shows "Polynomial.lead_coeff (of_int k) = of_int k"
fpslsts_def: fixes A :: "'b set" shows "A\<^sup>\<clubsuit> \<equiv> A\<^sup>\<star> - {LNil}"
HPhase1or2Write_HInv4c_q: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and d :: "Disk"   and q :: "Proc" assumes "HPhase1or2Write s s' p d"   and "HInv4c s q"   and "p \<noteq> q" shows "HInv4c s' q"
run_state_return: fixes x :: "'c"   and s :: "'d" shows "run_state (return x) s = (x, s)"
cases: fixes a1 :: "'a list"   and a2 :: "sequent" assumes "(a1, a2) \<in> nprovRules" obtains c :: "sequent" and F :: "char list" and x :: "NominalSequents.var" and A :: "form" and ps :: "'a list" where "a1 = ps" and "a2 = c" and "NominalSequents.mset c = \<LM> F \<nabla> [x].A  \<RM>" and "ps \<noteq> []"
ncoplanar_perm_20: 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 D B A C"
sf: shows "summable (\<lambda>n. 1 / real (Suc n))"
fs_sq_norm_ge_1: fixes n :: "nat"   and fs :: "'a vec list"   and i :: "nat" assumes "gram_schmidt_fs_int n fs"   and "i < length fs" shows "(1::'a) \<le> \<parallel>fs ! i\<parallel>\<^sup>2"
ext: fixes x :: "V"   and y :: "V" assumes "elts x = elts y" shows "x = y"
one_over_poly_monom_add: fixes a :: "'a"   and p :: "nat \<Rightarrow> 'a"   and n :: "nat" assumes "a \<in> carrier R"   and "a \<noteq> \<zero>"   and "p \<in> carrier P"   and "deg R p < n" shows "one_over_poly (p \<oplus>\<^bsub>P\<^esub> monom P a n) = monom P a 0 \<oplus>\<^bsub>P\<^esub> monom P \<one> (n - deg R p) \<otimes>\<^bsub>P\<^esub> one_over_poly p"
funcset_image: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and B :: "'d set" assumes "f \<in> A \<rightarrow> B" shows "f ` A \<subseteq> B"
lfp_fixpoint: fixes f :: "'d \<Rightarrow> 'd" assumes "mono f" shows "f (\<nu> f) = \<nu> f"
sl: shows "sgn (poly (real_of_int_poly p) (real_of_1 (p, l, r) - d)) = sgn (poly (real_of_int_poly p) (real_of_rat l)) \<or> 0 = sgn (poly (real_of_int_poly p) (real_of_rat l))"
shows_prec_ratfps_def: fixes p :: "nat"   and x :: "'b ratfps" shows "shows_prec p x = showsp_ratfps p x"
that: fixes a :: "real"   and q :: "'a pmf" assumes "a \<in> {0<..<1}"   and "set_pmf q = set_pmf p - {y}"   and "p = mix_pmf a q (return_pmf y)" shows "thesis"
can_implies_arr: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b local.term set" assumes "free_monoidal_category C"   and "free_monoidal_category.can C f" shows "partial_magma.arr (free_monoidal_category.comp C) f"
prems_3: shows "\<alpha> \<le> ordertype A VWF"
trim_alt: fixes s :: "'state"   and rs :: "'rule stream" assumes "s \<in> S"   and "fair rs" shows "trim rs s = sdrop (minWait rs s) rs"
eq: shows "sin y / (cos y + 1) = tan y / (1 + sqrt (1 + (tan y)\<^sup>2))"
interleave_cons_2: fixes l :: "'b list"   and a :: "'b list"   and b :: "'b list"   and x :: "'b" assumes "l \<in> a \<otimes> b" shows "x # l \<in> a \<otimes> x # b"
assms_2: shows "\<forall>m x. (diff m has_real_derivative diff (Suc m) x) (at x)"
dvd_pderiv_iff: fixes p :: "'a poly" shows "(p dvd pderiv p) = (Polynomial.degree p = 0)"
fpair_inject: fixes a :: "V"   and b :: "V"   and a' :: "V"   and b' :: "V" assumes "[a, b]\<^sub>\<circ> = [a', b']\<^sub>\<circ>" obtains "a = a'" and "b = b'"
simps_1: fixes \<xi> :: "Preliminaries.var \<Rightarrow> univ"   and T1 :: "('fsym, 'tp) efsym trm"   and T2 :: "('fsym, 'tp) efsym trm" shows "satA \<xi> (Eq T1 T2) = (int \<xi> T1 = int \<xi> T2)"
id_fresh: fixes x :: "name"   and \<Gamma> :: "(name \<times> ty) list" assumes "x \<sharp> \<Gamma>" shows "x \<sharp> Lam_ml.id \<Gamma>"
Out_def: shows "FIG.Out = (\<lambda>(A, A'). \<exists>a. list_select FIG_alphabet.v A' = a \<and> a \<notin> idS A \<and> idS A' = idS A \<union> {list_select FIG_alphabet.v A'})"
supp_transform: fixes PropApp :: "'a \<Rightarrow> 'a"   and PropLam :: "'a \<Rightarrow> 'a"   and AnalLet :: "(Vars.var \<times> exp) list \<Rightarrow> exp \<Rightarrow> 'a \<Rightarrow> 'b"   and PropLetBody :: "'b \<Rightarrow> 'a"   and PropLetHeap :: "'b \<Rightarrow> Vars.var \<Rightarrow> 'a\<^sub>\<bottom>"   and PropIfScrut :: "'a \<Rightarrow> 'a"   and TransVar :: "'a \<Rightarrow> Vars.var \<Rightarrow> exp"   and TransApp :: "'a \<Rightarrow> exp \<Rightarrow> Vars.var \<Rightarrow> exp"   and TransLam :: "'a \<Rightarrow> Vars.var \<Rightarrow> exp \<Rightarrow> exp"   and TransLet :: "'b \<Rightarrow> (Vars.var \<times> exp) list \<Rightarrow> exp \<Rightarrow> exp"   and a :: "'a"   and e :: "exp" assumes "AbstractTransform PropApp PropLam AnalLet PropLetBody PropLetHeap PropIfScrut TransVar TransApp TransLam TransLet" shows "supp (AbstractTransform.transform PropApp PropLam AnalLet PropLetBody PropLetHeap PropIfScrut TransVar TransApp TransLam TransLet a e) \<subseteq> supp e"
intro: fixes \<C> :: "'b list set" assumes "\<And>xs ys. \<lbrakk>xs \<in> lists \<C>; ys \<in> lists \<C>; concat xs = concat ys\<rbrakk> \<Longrightarrow> xs = ys" shows "code \<C>"
S_transform_Tree\<^sub>\<alpha>_Not\<^sub>\<alpha>: fixes t\<^sub>\<alpha> :: "('a, 'c, 'b) Tree\<^sub>\<alpha>" shows "S_transform_Tree\<^sub>\<alpha> (Not\<^sub>\<alpha> t\<^sub>\<alpha>) = Not\<^sub>\<alpha> (S_transform_Tree\<^sub>\<alpha> t\<^sub>\<alpha>)"
continuous_on_compose: fixes s :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and g :: "'c \<Rightarrow> 'd" assumes "continuous_on s f"   and "continuous_on (f ` s) g" shows "continuous_on s (g \<circ> f)"
continuous_map_of_snd: fixes X :: "'d topology"   and Y :: "'e topology"   and Z :: "'f topology"   and f :: "'e \<Rightarrow> 'f" shows "continuous_map (prod_topology X Y) Z (f \<circ> snd) = (topspace X = {} \<or> continuous_map Y Z f)"
map_tree_eq_Leaf_iff: fixes f :: "'c \<Rightarrow> 'b"   and t :: "'c tree" shows "(map_tree f t = Leaf) = (t = Leaf)"
subadditive_of_superadditive: fixes u :: "nat \<Rightarrow> real" assumes "superadditive u" shows "subadditive (\<lambda>n. - u n)"
cases: fixes x :: "(nat \<times> nat) set \<times> nat \<times> (nat list \<times> (nat \<times> nat) \<times> transition) fset \<times> (nat list \<Rightarrow> nat list \<Rightarrow> (nat list \<times> (nat \<times> nat) \<times> transition) fset \<Rightarrow> nat) \<times> (nat list \<Rightarrow> nat list \<Rightarrow> nat \<Rightarrow> (nat list \<times> (nat \<times> nat) \<times> transition) fset \<Rightarrow> (nat list \<times> (nat \<times> nat) \<times> transition) fset \<Rightarrow> (nat list \<times> (nat \<times> nat) \<times> transition) fset \<Rightarrow> (((nat \<times> nat) \<times> transition) fset \<Rightarrow> bool) \<Rightarrow> (nat list \<times> (nat \<times> nat) \<times> transition) fset option) \<times> (((nat \<times> nat) \<times> transition) fset \<Rightarrow> bool) \<times> ((nat list \<times> (nat \<times> nat) \<times> transition) fset \<Rightarrow> (nat \<times> (nat \<times> nat) \<times> (transition \<times> nat list) \<times> transition \<times> nat list) fset)" obtains failedMerges :: "(nat \<times> nat) set" and k :: "nat" and e :: "(nat list \<times> (nat \<times> nat) \<times> transition) fset" and r :: "nat list \<Rightarrow> nat list \<Rightarrow> (nat list \<times> (nat \<times> nat) \<times> transition) fset \<Rightarrow> nat" and m :: "nat list \<Rightarrow> nat list \<Rightarrow> nat \<Rightarrow> (nat list \<times> (nat \<times> nat) \<times> transition) fset \<Rightarrow> (nat list \<times> (nat \<times> nat) \<times> transition) fset \<Rightarrow> (nat list \<times> (nat \<times> nat) \<times> transition) fset \<Rightarrow> (((nat \<times> nat) \<times> transition) fset \<Rightarrow> bool) \<Rightarrow> (nat list \<times> (nat \<times> nat) \<times> transition) fset option" and check :: "((nat \<times> nat) \<times> transition) fset \<Rightarrow> bool" and np :: "(nat list \<times> (nat \<times> nat) \<times> transition) fset \<Rightarrow> (nat \<times> (nat \<times> nat) \<times> (transition \<times> nat list) \<times> transition \<times> nat list) fset" where "x = (failedMerges, k, e, r, m, check, np)"
inf_prefix_takeL: fixes xs :: "'b list"   and ys :: "'b list" shows "xs \<sqinter> ys = xs \<down> inf_prefix_length xs ys"
piD: shows "path_image J = path_image J1 \<union> {flow0 x t1<--<flow0 x t2}"
subformulas\<^sub>\<mu>_list_set: fixes \<phi> :: "'a ltln" shows "set (subformulas\<^sub>\<mu>_list \<phi>) = subformulas\<^sub>\<mu> \<phi>"
lcoset_conv_set: fixes a :: "'a"   and b :: "'a"   and A :: "'a set" assumes "a \<in> b +o A" shows "- b + a \<in> A"
pt_open_on_def: fixes Q :: "('c \<Rightarrow> 'd) \<Rightarrow> bool"   and A :: "('c \<Rightarrow> 'd) set" shows "pt_open_on Q A = (\<forall>f\<in>A. Q f = (\<exists>n. \<forall>g\<in>A. (\<forall>i<n. g i = f i) \<longrightarrow> Q g))"
D_PUT: fixes x :: "Var"   and f :: "Field"   and e :: "Expr"   and G :: "((Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool) set" assumes "CONTEXT x = low"   and "GAMMA f = low"   and "Expr_low e" shows "(G, Put x f e, Sec (\<lambda>(s, t, \<beta>). \<exists>r l C F h. fst r x = RVal (Loc l) \<and> r \<equiv>\<^sub>\<beta> t \<and> lookup (snd r) l = Some (C, F) \<and> h = (l, C, (f, evalE e (fst r)) # F) # snd r \<and> s = (fst r, h))) \<in> Deriv"
While: fixes Q :: "(char list \<Rightarrow> int) \<Rightarrow> enat" shows "\<turnstile>\<^sub>2 {wp\<^sub>2 c Q} c {Q}"
continuous_on_Lambert_W': shows "continuous_on {- exp (- 1)..<0} Lambert_W'"
cdens_ctxt_invar_def: fixes vs :: "nat list"   and vs' :: "nat list"   and \<Gamma> :: "nat \<Rightarrow> pdf_type"   and \<delta> :: "cexpr" shows "cdens_ctxt_invar vs vs' \<Gamma> \<delta> \<equiv> distinct (vs @ vs') \<and> free_vars \<delta> \<subseteq> set (vs @ vs') \<and> \<Gamma> \<turnstile>\<^sub>c \<delta> : REAL \<and> nonneg_cexpr (set vs \<union> set vs') \<Gamma> \<delta> \<and> subprob_cexpr (set vs) (set vs') \<Gamma> \<delta>"
dimXA: shows "\<forall>n. X (n + k) \<in> carrier_mat m m \<and> A \<in> carrier_mat m m"
Node_in_tree_sigma: fixes X :: "('b \<times> 'b tree \<times> 'b tree) set"   and M :: "'b measure" assumes "X \<in> sets (M \<Otimes>\<^sub>M tree_sigma M \<Otimes>\<^sub>M tree_sigma M)" shows "{\<langle>l, v, r\<rangle> |l v r. (v, l, r) \<in> X} \<in> sets (tree_sigma M)"
Figure_5: fixes k :: "nat"   and i :: "nat"   and js :: "nat list"   and xs :: "nat list"   and ys :: "nat list" assumes "foldl1 f1 (map (h1 k i) js) = One"   and "js = xs @ ys" shows "\<not> (i \<in> set xs \<and> i \<in> set ys)"
le_iff_inf: fixes inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "class.semilattice_inf inf less_eq less" shows "less_eq x y = (inf x y = x)"
learn_total_wrt_goedel: fixes \<chi> :: "nat \<Rightarrow> nat \<Rightarrow> nat option"   and \<psi> :: "nat \<Rightarrow> nat \<Rightarrow> nat option"   and U :: "(nat \<Rightarrow> nat option) set"   and s :: "nat \<Rightarrow> nat option" assumes "goedel_numbering \<chi>"   and "learn_total \<psi> U s" shows "\<exists>t. learn_total \<chi> U t"
oracle_def: fixes len :: "nat"   and \<sigma> :: "'b \<Rightarrow> bool list option"   and x :: "'b" shows "oracle len \<sigma> x = (case \<sigma> x of None \<Rightarrow> spmf_of_set (nlists UNIV len) \<bind> (\<lambda>bs. return_spmf (bs, \<sigma>(x \<mapsto> bs))) | Some bs \<Rightarrow> return_spmf (bs, \<sigma>))"
matrix_vector_mult_diff_rdistrib: fixes A :: "(('b, 'd) vec, 'c) vec"   and B :: "(('b, 'd) vec, 'c) vec"   and x :: "('b, 'd) vec" shows "(A - B) *v x = A *v x - B *v x"
simp: shows "\<forall>(term, d)\<in>set (chop (to_list v p)). f v ^ (d - i) * f v ^ i = f v ^ d"
simps_1: fixes x :: "nat" shows "V (TVar x) = {e. val e}"
prems_1: shows "\<forall>X\<in>set (X # Xs). X \<subseteq> A"
list_e_eq_lel_1: fixes e :: "'a"   and l1 :: "'a list"   and e' :: "'a"   and l2 :: "'a list" shows "([e] = l1 @ e' # l2) = (l1 = [] \<and> e' = e \<and> l2 = [])"
valid_Act: fixes P :: "'state"   and \<alpha> :: "'act"   and x :: "('c, 'pred, 'act) formula" shows "P \<Turnstile> Act \<alpha> x = (\<exists>\<alpha>'' x' P'. Act \<alpha> x = Act \<alpha>'' x' \<and> P \<rightarrow> \<langle>\<alpha>'',P'\<rangle> \<and> P' \<Turnstile> x')"
set: fixes x :: "'a list" shows "set_array (Array x) = set x"
derivative_component_fun_component_at_within_gen: fixes s :: "real set"   and \<gamma> :: "real \<Rightarrow> 'a"   and i :: "'a" assumes "\<forall>x\<in>s. \<gamma> differentiable at x"   and "s \<subseteq> {0..1}" shows "\<forall>x\<in>s. vector_derivative \<gamma> (at x within {0..1}) \<bullet> i = vector_derivative (\<lambda>x. \<gamma> x \<bullet> i) (at x within {0..1})"
o_case_sum: fixes h :: "'h \<Rightarrow> 'g"   and f :: "'e \<Rightarrow> 'h"   and g :: "'f \<Rightarrow> 'h" shows "h \<circ> case_sum f g = case_sum (h \<circ> f) (h \<circ> g)"
\<rho>_in_terms_of_rep: shows "\<rho> = (trnr\<^sub>\<epsilon> (g \<star> f\<^sup>*) \<rho> \<star> f) \<cdot> trnr\<^sub>\<eta> g (g \<star> f\<^sup>*)"
admD: fixes P :: "'a \<Rightarrow> bool"   and Y :: "nat \<Rightarrow> 'a" assumes "adm P"   and "chain Y"   and "\<And>i. P (Y i)" shows "P (\<Squnion>i. Y i)"
that_2: fixes rs\<^sub>1 :: "'a rule list"   and m :: "'a match_expr"   and rs\<^sub>2 :: "'a rule list" assumes "rs = rs\<^sub>1 @ [Rule m Return] @ rs\<^sub>2"   and "process_ret rs = rs\<^sub>1 @ process_ret ([Rule m Return] @ rs\<^sub>2)" shows "thesis"
SUP_cong_simp: fixes A :: "'d set"   and B :: "'d set"   and C :: "'d \<Rightarrow> 'e"   and D :: "'d \<Rightarrow> 'e"   and Sup :: "'e set \<Rightarrow> 'e" assumes "A = B"   and "\<And>x. x \<in> B =simp=> C x = D x" shows "Sup (C ` A) = Sup (D ` B)"
cong_ex_bigtheta: fixes L :: "'b filter \<Rightarrow> ('b \<Rightarrow> 'c) \<Rightarrow> ('b \<Rightarrow> 'c) set"   and L' :: "'d filter \<Rightarrow> ('d \<Rightarrow> 'c) \<Rightarrow> ('d \<Rightarrow> 'c) set"   and Lr :: "'b filter \<Rightarrow> ('b \<Rightarrow> real) \<Rightarrow> ('b \<Rightarrow> real) set"   and f1 :: "'b \<Rightarrow> 'c"   and F :: "'b filter"   and f2 :: "'b \<Rightarrow> 'c"   and g1 :: "'b \<Rightarrow> 'c"   and g2 :: "'b \<Rightarrow> 'c" assumes "landau_symbol L L' Lr"   and "f1 \<in> \<Theta>[F](f2)"   and "g1 \<in> \<Theta>[F](g2)" shows "(f1 \<in> L F g1) = (f2 \<in> L F g2)"
alt_env_def: fixes m1 :: "('c, 'm) envT"   and m2 :: "('c, 'm) envT" shows "local.alt_env m1 m2 = EnvT (\<lambda>r. alt (run_env m1 r) (run_env m2 r))"
evaln_Suc: fixes c :: "com"   and s :: "state"   and n :: "nat"   and s' :: "state" assumes "<c,s> -n-> s'" shows "<c,s> -Suc n-> s'"
intro: fixes less_eq :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and top :: "'k" assumes "\<And>a. less_eq a top" shows "ordering_top_axioms less_eq top"
typeof_NoSuchMethodError: fixes h :: "heap" assumes "preallocated h" shows "typeof\<^bsub>h\<^esub> (Addr (addr_of_sys_xcpt NoSuchMethodError)) = \<lfloor>Class NoSuchMethodError\<rfloor>"
divide_cancel_right: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inverse :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and a :: "'c"   and c :: "'c"   and b :: "'c" assumes "class.field divide inverse times one plus zero minus uminus" shows "(divide a c = divide b c) = (c = zero \<or> a = b)"
context_le_trans: fixes \<Gamma> :: "'Var \<Rightarrow> Sec option"   and \<Gamma>' :: "'Var \<Rightarrow> Sec option"   and \<Gamma>'' :: "'Var \<Rightarrow> Sec option" assumes "\<Gamma> \<sqsubseteq>\<^sub>c \<Gamma>'"   and "\<Gamma>' \<sqsubseteq>\<^sub>c \<Gamma>''" shows "\<Gamma> \<sqsubseteq>\<^sub>c \<Gamma>''"
denormal_imp_not_zero: fixes f :: "('e, 'f) IEEE.float" assumes "is_denormal f" shows "\<not> is_zero f"
image2p_in_rel: fixes f :: "'e \<Rightarrow> 'c"   and g :: "'f \<Rightarrow> 'd"   and R :: "('e \<times> 'f) set" shows "BNF_Greatest_Fixpoint.image2p f g (in_rel R) = in_rel (map_prod f g ` R)"
empty: fixes g :: "'h \<Rightarrow> 'g" shows "prod g {} = (1::'g)"
unitr_ex: fixes x :: "'a" shows "\<exists>e\<in>E. D x e \<and> x \<oplus> e = x"
Nat_atleast_newC: fixes s :: "state"   and aa :: "loc"   and v :: "val"   and n :: "nat"   and obj :: "obj" assumes "heap s aa = None"   and "s:v \<ge> n" shows "hupd(aa\<mapsto>obj) s:v \<ge> n"
subst_codomain_is_finite: fixes A :: "'b set"   and \<eta> :: "('b \<times> 'b trm) list" assumes "finite A" shows "finite (subst_codomain \<eta> A)"
vproduct_def: fixes I :: "V"   and A :: "V \<Rightarrow> V" shows "vproduct I A = ZFC_in_HOL.set {f. vsv f \<and> \<D>\<^sub>\<circ> f = I \<and> (\<forall>i\<in>elts I. f\<lparr>i\<rparr> \<in>\<^sub>\<circ> A i)}"
A5: shows "weak_reduction_bisimulation Rel (STCal Source Target)"
irreducibleD': fixes a :: "'b"   and b :: "'b" assumes "irreducible a"   and "b dvd a" shows "a dvd b \<or> b dvd (1::'b)"
gpv: shows "eq_\<I>_gpv A \<I> gpv1 gpv2"
IH: assumes "bvars\<^sub>s\<^sub>s\<^sub>t S \<inter> subst_domain \<theta> = {}" shows "setops\<^sub>s\<^sub>s\<^sub>t (S \<cdot>\<^sub>s\<^sub>s\<^sub>t \<theta>) = setops\<^sub>s\<^sub>s\<^sub>t S \<cdot>\<^sub>p\<^sub>s\<^sub>e\<^sub>t \<theta>"
product_criterion: fixes ord :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and ord_strict :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and d :: "'e \<Rightarrow> nat"   and F :: "('e \<Rightarrow>\<^sub>0 'f) set"   and m :: "nat"   and p :: "'e \<Rightarrow>\<^sub>0 'f"   and q :: "'e \<Rightarrow>\<^sub>0 'f" assumes "gd_powerprod ord ord_strict"   and "dickson_grading d"   and "F \<subseteq> gd_term.dgrad_p_set to_pair_unit d m"   and "p \<in> F"   and "q \<in> F"   and "p \<noteq> 0"   and "q \<noteq> 0"   and "gcs (ordered_term.lt fst ord p) (ordered_term.lt fst ord q) = (0::'e)" shows "gd_term.crit_pair_cbelow_on to_pair_unit fst ord ord_strict d m F p q"
lfinite_lmapI: fixes xs :: "'c llist"   and f :: "'c \<Rightarrow> 'd" assumes "lfinite xs" shows "lfinite (lmap f xs)"
simps_5: fixes v :: "val" shows "init_exp_of (Val v) = None"
subseq_of_lengthD_2: fixes n :: "nat"   and xs :: "'c list"   and ys :: "'c list" assumes "subseq_of_length n xs ys" shows "length ys = n"
hom_gauss_jordan_single: fixes A :: "'a mat" shows "gauss_jordan_single (mat\<^sub>h A) = mat\<^sub>h (gauss_jordan_single A)"
assms_2: fixes x :: "real" assumes "a < x"   and "x < b" shows "\<exists>y. (f has_real_derivative y) (at x) \<and> y < 0"
psubst_dterm: fixes I :: "('sf, 'sc, 'sz) interp"   and \<theta> :: "('sf, 'sz) trm"   and \<sigma> :: "'b \<Rightarrow> ('sf, 'sc, 'sz) formula" assumes "is_interp I"   and "dsafe \<theta>" shows "dterm_sem I \<theta> = dterm_sem (PFadjoint I \<sigma>) \<theta>"
orthogonal_self: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and norm :: "'b \<Rightarrow> real"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and sgn :: "'b \<Rightarrow> 'b"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and inner :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and x :: "'b" assumes "class.real_inner minus dist norm plus zero uminus scaleR sgn uniformity open inner" shows "real_inner.orthogonal inner x x = (x = zero)"
kth_obs_unique: fixes prob :: "('a, 'b, 'c, 'd) ifc_problem"   and \<pi> :: "nat \<Rightarrow> 'a"   and k :: "nat"   and i :: "nat"   and j :: "nat" assumes "IFC prob"   and "IFC_def.is_kth_obs prob \<pi> k i"   and "IFC_def.is_kth_obs prob \<pi> k j" shows "i = j"
Basic: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and f :: "'e \<Rightarrow> 'e"   and cs :: "('e, 'd, 'f) com list"   and css :: "(('e, 'd, 'f) com list \<times> ('e, 'd, 'f) com list) list"   and s :: "'e" shows "\<Gamma>\<turnstile> (Basic f # cs, css, Normal s) \<rightarrow> (cs, css, Normal (f s))"
Sup_set_def: fixes A :: "'b set set" shows "\<Union> A = {x. Sup ((\<in>) x ` A)}"
exist_matrix'_implies_equivalent: fixes X :: "(('a, 'b) vec, 'b) vec"   and Y :: "(('a, 'c) vec, 'c) vec"   and f :: "('a, 'b) vec \<Rightarrow> ('a, 'c) vec"   and A :: "(('a, 'b) vec, 'c) vec"   and X' :: "(('a, 'b) vec, 'b) vec"   and Y' :: "(('a, 'c) vec, 'c) vec"   and B :: "(('a, 'b) vec, 'c) vec" assumes "matrix' X Y f = A"   and "matrix' X' Y' f = B"   and "is_basis (set_of_vector X)"   and "is_basis (set_of_vector Y)"   and "is_basis (set_of_vector X')"   and "is_basis (set_of_vector Y')"   and "linear (*s) (*s) f" shows "equivalent_matrices A B"
in_shared_sb_share_all_until_volatile_write: fixes \<S> :: "nat \<Rightarrow> bool option"   and i :: "nat"   and p :: "'a"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and sb :: "'a memref list"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option"   and a :: "nat" assumes "sharing_consis \<S> ts\<^sub>s\<^sub>b"   and "i < length ts\<^sub>s\<^sub>b"   and "ts\<^sub>s\<^sub>b ! i = (p, is, \<theta>, sb, \<D>, \<O>, \<R>)"   and "a \<in> \<O> \<union> all_acquired sb"   and "a \<in> dom (share sb \<S>)" shows "a \<in> dom (share (dropWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb) (share_all_until_volatile_write ts\<^sub>s\<^sub>b \<S>))"
Sup_union_distrib: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a"   and A :: "'a set"   and B :: "'a set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "Sup (A \<union> B) = sup (Sup A) (Sup B)"
prj1_in_hom': fixes f :: "'a"   and g :: "'a" assumes "cospan f g" shows "\<guillemotleft>\<p>\<^sub>1[f, g] : local.dom \<p>\<^sub>0[f, g] \<rightarrow> local.dom f\<guillemotright>"
split: fixes P :: "'b \<Rightarrow> bool"   and f1 :: "'b"   and f2 :: "'b"   and f3 :: "'c \<Rightarrow> 'b"   and dres :: "'c dres" shows "P (case dres of dSUCCEEDi \<Rightarrow> f1 | dFAILi \<Rightarrow> f2 | dRETURN x \<Rightarrow> f3 x) = ((dres = dSUCCEEDi \<longrightarrow> P f1) \<and> (dres = dFAILi \<longrightarrow> P f2) \<and> (\<forall>x3. dres = dRETURN x3 \<longrightarrow> P (f3 x3)))"
bind_eq_Some_conv: fixes f :: "'b option"   and g :: "'b \<Rightarrow> 'a option"   and x :: "'a" shows "(Option.bind f g = Some x) = (\<exists>y. f = Some y \<and> g y = Some x)"
dom_make_map: fixes ks :: "'c fset"   and v :: "'d" shows "dom (make_map ks v) = fset ks"
set_plus_imp_minus: fixes a :: "'b"   and b :: "'b"   and C :: "'b set" assumes "a \<in> b +o C" shows "a - b \<in> C"
sunflower_imp_union_intersect_unique: fixes S :: "'b set set"   and x :: "'b" assumes "sunflower S"   and "x \<in> \<Union> S - \<Inter> S" shows "\<exists>!A. A \<in> S \<and> x \<in> A"
fref_to_Down_curry: fixes f :: "'a \<Rightarrow> 'b nres"   and g :: "'c \<Rightarrow> 'd nres"   and P :: "'c \<Rightarrow> bool"   and A :: "('a \<times> 'c) set"   and B :: "('b \<times> 'd) set"   and x' :: "'c"   and x :: "'a" assumes "(f, g) \<in> [P]\<^sub>f A \<rightarrow> \<langle>B\<rangle>nres_rel"   and "P x'"   and "(x, x') \<in> A" shows "f x \<le> \<Down> B (g x')"