hcju/isabelleps huggingface dataset:
split: fixes P :: "'b \<Rightarrow> bool"   and f1 :: "bool \<Rightarrow> nat \<Rightarrow> nat set \<times> ((nat \<Rightarrow> nat option) \<Rightarrow> nat) \<Rightarrow> nat \<Rightarrow> nat set \<Rightarrow> nat set \<Rightarrow> nat set \<Rightarrow> nat set \<Rightarrow> 'b"   and f2 :: "bool \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'b"   and f3 :: "'c \<Rightarrow> 'c \<Rightarrow> instr list \<Rightarrow> 'b"   and f4 :: "nat set \<Rightarrow> nat set \<Rightarrow> nat set \<Rightarrow> nat set \<Rightarrow> 'b"   and memref :: "'c memref" shows "P (case memref of Write\<^sub>s\<^sub>b x xa xb xc xd xe xf xg \<Rightarrow> f1 x xa xb xc xd xe xf xg | Read\<^sub>s\<^sub>b x xa xb xc \<Rightarrow> f2 x xa xb xc | Prog\<^sub>s\<^sub>b x xa xb \<Rightarrow> f3 x xa xb | Ghost\<^sub>s\<^sub>b x xa xb xc \<Rightarrow> f4 x xa xb xc) = ((\<forall>x11 x12 x13 x14 x15 x16 x17 x18. memref = Write\<^sub>s\<^sub>b x11 x12 x13 x14 x15 x16 x17 x18 \<longrightarrow> P (f1 x11 x12 x13 x14 x15 x16 x17 x18)) \<and> (\<forall>x21 x22 x23 x24. memref = Read\<^sub>s\<^sub>b x21 x22 x23 x24 \<longrightarrow> P (f2 x21 x22 x23 x24)) \<and> (\<forall>x31 x32 x33. memref = Prog\<^sub>s\<^sub>b x31 x32 x33 \<longrightarrow> P (f3 x31 x32 x33)) \<and> (\<forall>x41 x42 x43 x44. memref = Ghost\<^sub>s\<^sub>b x41 x42 x43 x44 \<longrightarrow> P (f4 x41 x42 x43 x44)))"
in_indetsI: fixes x :: "'a"   and t :: "'a \<Rightarrow>\<^sub>0 nat"   and p :: "('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" assumes "x \<in> keys t"   and "t \<in> keys p" shows "x \<in> indets p"
hyps: fixes x2a :: "('b, 'a) Term.term" assumes "x2a \<in> set ts"   and "x \<in> vars_term x2a"   and "is_Fun x2a" shows "num_funs (\<sigma> x) < num_funs (x2a \<cdot> \<sigma>)"
eliminate_swap_global_lid: fixes l :: "'a"   and s :: "'b \<Rightarrow> (('a \<Rightarrow> ('b, 'a, 'c) val option) \<times> ('a \<Rightarrow> ('b, 'a, 'c) val option) \<times> ('b, 'a, 'c) expr) option"   and l' :: "'a" assumes "l \<notin> LID\<^sub>G s"   and "l' \<notin> LID\<^sub>G s" shows "\<R>\<^sub>G id (id(l := l', l' := l)) s = s"
borderD_pref: fixes x :: "'b list"   and w :: "'b list" assumes "x \<le>b w" shows "x \<le>p w"
D: shows "(Asset n, Crypt (SesK SK) (Num 0)) \<in> s"
wit_G_set2: fixes x :: "'n"   and l2 :: "'n" assumes "x \<in> set2_G (wit_G l2)" shows "x = l2"
chain_boundary_simplicial_subdivision: fixes p :: "nat"   and S :: "(nat \<Rightarrow> real) set"   and c :: "((nat \<Rightarrow> real) \<Rightarrow> nat \<Rightarrow> real) \<Rightarrow>\<^sub>0 int" assumes "simplicial_chain p S c" shows "chain_boundary p (simplicial_subdivision p c) = simplicial_subdivision (p - 1) (chain_boundary p c)"
assms: assumes "lt p = lt q" shows "lc p + lc q \<noteq> (0::'b)"
subset_of_opens: fixes U :: "'a set" assumes "U \<in> I" shows "is_open U"
f_pos: fixes x :: "'a" shows "0 \<le> f x"
prv_imp_scnj1: fixes \<phi> :: "'fmla" assumes "\<phi> \<in> fmla" shows "prv (imp \<phi> (scnj {\<phi>}))"
connectedin_path_image: fixes X :: "'a topology"   and g :: "real \<Rightarrow> 'a" assumes "pathin X g" shows "connectedin X (g ` {0..1})"
mopup_inv_steps: fixes n :: "nat"   and lm :: "nat list"   and s :: "nat"   and l :: "cell list"   and r :: "cell list"   and ires :: "cell list"   and stp :: "nat" assumes "n < length lm"   and "mopup_inv (s, l, r) lm n ires" shows "mopup_inv (steps0 (s, l, r) (mopup_a n @ shift mopup_b (2 * n)) stp) lm n ires"
irreducibility_hom_axioms: shows "irreducibility_hom (\<lambda>p. p \<circ>\<^sub>p x_y)"
simps_1: fixes le :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "linorder_list0.selsort le [] = []"
simps_3: fixes t :: "'a E.term"   and u :: "'a E.term" shows "E.Src (t \<^bold>\<star> u) = E.Src u"
take_bit_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 m :: "nat"   and n :: "nat"   and a :: "'a" assumes "class.semiring_bit_shifts plus minus zero times one divide modulo bit push_bit drop_bit take_bit" shows "take_bit m (drop_bit n a) = drop_bit n (take_bit (m + n) a)"
par_comp\<^sub>l\<^sub>s\<^sub>s\<^sub>t_def: fixes \<A> :: "('lbl strand_label \<times> ('fun, ('fun, 'atom) Term.term \<times> nat) stateful_strand_step) list"   and Secrets :: "('fun, ('fun, 'atom) Term.term \<times> nat) Term.term set" shows "par_comp\<^sub>l\<^sub>s\<^sub>s\<^sub>t \<A> Secrets \<equiv> (\<forall>l1 l2. l1 \<noteq> l2 \<longrightarrow> GSMP_disjoint (trms\<^sub>l\<^sub>s\<^sub>s\<^sub>t (proj l1 \<A>) \<union> pair ` setops\<^sub>s\<^sub>s\<^sub>t (proj_unl l1 \<A>)) (trms\<^sub>l\<^sub>s\<^sub>s\<^sub>t (proj l2 \<A>) \<union> pair ` setops\<^sub>s\<^sub>s\<^sub>t (proj_unl l2 \<A>)) Secrets) \<and> ground Secrets \<and> (\<forall>s\<in>Secrets. \<forall>s'\<in>subterms s. {} \<turnstile>\<^sub>c s' \<or> s' \<in> Secrets) \<and> (\<forall>(i, p)\<in>setops\<^sub>l\<^sub>s\<^sub>s\<^sub>t \<A>. \<forall>(j, q)\<in>setops\<^sub>l\<^sub>s\<^sub>s\<^sub>t \<A>. (\<exists>\<delta>. Unifier \<delta> (pair p) (pair q)) \<longrightarrow> i = j)"
g_def: shows "g \<equiv> \<lambda>r. poly pp \<circ> (\<lambda>x. complex_of_real (linepath (- r) (of_real r) x))"
permutes_in_funpow_image: fixes f :: "'c \<Rightarrow> 'c"   and S :: "'c set"   and x :: "'c"   and n :: "nat" assumes "f permutes S"   and "x \<in> S" shows "(f ^^ n) x \<in> S"
lg_lemma''': fixes y :: "nat"   and x :: "nat" assumes "\<not> Suc 0 < y" shows "rec_exec rec_lg [x, y] = lg x y"
card_number_partitions_with_only_parts_1_eq_0: fixes x :: "nat"   and n :: "nat" assumes "x < n" shows "card {N. (\<forall>n. n \<in># N \<longrightarrow> n = 1) \<and> number_partition n N \<and> size N \<le> x} = 0"
all_chains_no_call_upd_orig: fixes c :: "char list"   and \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and rs :: "'c rule list"   and x :: "'c rule list"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and s :: "state"   and t :: "state" assumes "all_chains (no_call_to c) \<Gamma> rs" shows "\<Gamma>(c \<mapsto> x),\<gamma>,p\<turnstile> \<langle>rs, s\<rangle> \<Rightarrow> t = \<Gamma>,\<gamma>,p\<turnstile> \<langle>rs, s\<rangle> \<Rightarrow> t"
simps: fixes a1 :: "'a"   and a2 :: "'a" shows "((a1, a2) \<in> hull_rel) = (\<exists>p x. a1 = p \<bullet> x \<and> a2 = x)"
cases: fixes a1 :: "(Vars.var \<times> exp) list \<times> exp \<times> stack_elem list"   and a2 :: "(Vars.var \<times> exp) list \<times> exp \<times> stack_elem list" assumes "a1 \<Rightarrow> a2" obtains \<Gamma> :: "(Vars.var \<times> exp) list" and e :: "exp" and x :: "Vars.var" and S :: "stack_elem list" where "a1 = (\<Gamma>'''__, App ea__ x__, S__)" and "a2 = (\<Gamma>'''__, ea__, Arg x__ # S__)"   | \<Gamma> :: "(Vars.var \<times> exp) list" and y :: "Vars.var" and e :: "exp" and x :: "Vars.var" and S :: "stack_elem list" where "a1 = (\<Gamma>''''__, Lam [y]. eb__, Arg xa__ # Sa__)" and "a2 = (\<Gamma>''''__, eb__[y::=xa__], Sa__)"   | \<Gamma> :: "(Vars.var \<times> exp) list" and x :: "Vars.var" and e :: "exp" and S :: "stack_elem list" where "a1 = (\<Gamma>'''''__, Var xb__, Sb__)" and "a2 = (delete xb__ \<Gamma>'''''__, ec__, Upd xb__ # Sb__)" and "map_of \<Gamma>'''''__ xb__ = Some ec__"   | x :: "Vars.var" and \<Gamma> :: "(Vars.var \<times> exp) list" and e :: "exp" and S :: "stack_elem list" where "a1 = (\<Gamma>''''''__, ed__, Upd x # Sc__)" and "a2 = ((x, ed__) # \<Gamma>''''''__, ed__, Sc__)" and "x \<notin> domA \<Gamma>''''''__" and "isVal ed__"   | \<Delta> :: "(Vars.var \<times> exp) list" and \<Gamma> :: "(Vars.var \<times> exp) list" and S :: "stack_elem list" and e :: "exp" where "a1 = (\<Gamma>'''''''__, Let \<Delta> e, Sd__)" and "a2 = (\<Delta> @ \<Gamma>'''''''__, e, Sd__)" and "atom ` domA \<Delta> \<sharp>* \<Gamma>'''''''__" and "atom ` domA \<Delta> \<sharp>* Sd__"   | \<Gamma> :: "(Vars.var \<times> exp) list" and scrut :: "exp" and e1 :: "exp" and e2 :: "exp" and S :: "stack_elem list" where "a1 = (\<Gamma>''''''''__, scrut ? e1__ : e2__, Se__)" and "a2 = (\<Gamma>''''''''__, scrut, Alts e1__ e2__ # Se__)"   | \<Gamma> :: "(Vars.var \<times> exp) list" and b :: "bool" and e1 :: "exp" and e2 :: "exp" and S :: "stack_elem list" where "a1 = (\<Gamma>, Bool b, Alts e1 e2 # S)" and "a2 = (\<Gamma>, if b then e1 else e2, S)"
lessThan_Int_lessThan: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a" assumes "class.linorder less_eq less" shows "ord.greaterThan less a \<inter> ord.greaterThan less b = ord.greaterThan less (ord.max less_eq a b)"
The_on_SomeI': fixes U :: "'c set"   and P :: "'c \<Rightarrow> bool" assumes "\<exists>!x. x \<in> U \<and> P x" obtains x :: "'c" where "(THE x on U. P x) = Some x" and "P x"
c1: shows "Partial_Cost_Model.config'_rand BIT (type0 init x y) [x] = type0 init x y"
hom_zero: shows "map_poly (pCons (0::'f)) 0 = 0"
real_power_diff: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and nat_power :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and norm :: "'a \<Rightarrow> real"   and scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and sgn :: "'a \<Rightarrow> 'a"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and inner :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and real_power :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and d :: "real"   and a :: "'a"   and b :: "'a" assumes "class.nat_power_normed_field one times plus zero nat_power divide inverse minus uminus dist norm scaleR sgn uniformity open inner real_power"   and "0 < d" shows "real_power d (minus a b) = divide (real_power d a) (real_power d b)"
trail_arc_balanceE: fixes G :: "('c, 'd) pre_digraph"   and u :: "'c"   and p :: "'d list"   and v :: "'c" assumes "wf_digraph G"   and "pre_digraph.trail G u p v" obtains "\<And>w. \<lbrakk>u = v \<or> w \<noteq> u \<and> w \<noteq> v; w \<in> verts G\<rbrakk> \<Longrightarrow> pre_digraph.arc_set_balance G w (set p) = 0" and "u \<noteq> v \<Longrightarrow> pre_digraph.arc_set_balance G u (set p) = - 1" and "u \<noteq> v \<Longrightarrow> pre_digraph.arc_set_balance G v (set p) = 1"
box_below: fixes below :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'d"   and b :: "'d"   and c :: "'d"   and d :: "'d" assumes "class.po below"   and "below a b"   and "below c a"   and "below b d" shows "below c d"
pderiv_iszero: fixes p :: "'a poly" assumes "pderiv p = 0" shows "\<exists>h. p = [:h:]"
singular_relboundary_def: fixes p :: "nat"   and X :: "'b topology"   and S :: "'b set" shows "singular_relboundary p X S \<equiv> \<lambda>c. \<exists>d. singular_chain (Suc p) X d \<and> (chain_boundary (Suc p) d, c) \<in> mod_subset p (subtopology X S)"
ovr_overshadow_left: fixes x :: "'s"   and y :: "'s"   and z :: "'s" shows "x \<triangleright> y \<triangleright> z = x \<triangleright> z"
dgrad_p_set_le_red_rtrancl: fixes d :: "'a \<Rightarrow> nat"   and F :: "('t \<Rightarrow>\<^sub>0 'c) set"   and p :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c" assumes "dickson_grading d"   and "(red F)\<^sup>*\<^sup>* p q" shows "dgrad_p_set_le d {q} (insert p F)"
Close1: fixes P :: "pi"   and a :: "name"   and x :: "name"   and P' :: "pi"   and Q :: "pi"   and y :: "name"   and Q' :: "pi" assumes "P \<longmapsto>\<^sub>l a<x> \<prec>\<^sub>l P'"   and "Q \<longmapsto>\<^sub>l a<\<nu>y> \<prec>\<^sub>l Q'"   and "y \<sharp> P" shows "P \<parallel> Q \<longmapsto>\<^sub>l \<tau> \<prec>\<^sub>l <\<nu>y>(P'[x::=y] \<parallel> Q')"
sortedByWeight_insortTree: fixes ts :: "'a tree list"   and t :: "'a tree" assumes "sortedByWeight ts"   and "height t = 0"   and "height\<^sub>F ts = 0" shows "sortedByWeight (insortTree t ts)"
fresh_swap_exchange1: fixes X :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and xs :: "'varSort"   and x2 :: "'var"   and x1 :: "'var" assumes "good X" shows "fresh xs x2 (X #[x1 \<and> x2]_xs) = fresh xs x1 X"
C1: fixes R :: "ccs"   and S :: "ccs" assumes "(R, S) \<in> Rel" shows "R \<leadsto>\<^sup>^<Rel> S"
extensible_from_chainI: fixes S :: "nat \<Rightarrow> ('e, 'f) labeled_graph"   and C :: "('e, 'g) labeled_graph"   and f :: "('f \<times> 'g) set" assumes "chain S"   and "graph_homomorphism (S 0) C f"   and "\<And>f i. graph_homomorphism (S i) C f \<Longrightarrow> \<exists>h. graph_homomorphism (S (Suc i)) C h \<and> agree_on (S i) f h" shows "extensible (S 0, chain_sup S) C f"
r: shows "(\<Sum>i. ereal (f i)) = ereal r"
hyps_3: fixes s :: "char list \<Rightarrow> int option"   and n :: "nat" assumes "P' (s, n)" shows "P (s, n)"
rel_fun_Sup_distr: fixes G :: "('c \<times> 'd \<Rightarrow> 'e) set"   and f :: "'c \<times> 'd \<Rightarrow> 'e" shows "ps_prod.rel_partial_semigroup.times_rel_fun (\<Squnion>G) f = \<Squnion>{ps_prod.rel_partial_semigroup.times_rel_fun g f |g. g \<in> G}"
assms: fixes x :: "Preliminaries.var" assumes "x \<in> varsC c" shows "\<pi>1 x = \<pi>2 x"
g_pos: fixes x :: "'a" assumes "x \<in> space M" shows "0 < g x"
zero_canonical: shows "\<downharpoonright>0\<downharpoonleft> = 0"
diff_conv_add_uminus: fixes minus :: "'q \<Rightarrow> 'q \<Rightarrow> 'q"   and plus :: "'q \<Rightarrow> 'q \<Rightarrow> 'q"   and zero :: "'q"   and uminus :: "'q \<Rightarrow> 'q"   and a :: "'q"   and b :: "'q" assumes "class.group_add minus plus zero uminus" shows "minus a b = plus a (uminus b)"
distrib_sup_le: fixes x :: "'c"   and y :: "'c"   and z :: "'c" shows "sup x (inf y z) \<le> inf (sup x y) (sup x z)"
finite_opset: fixes opset :: "('c \<times> 'd) set"   and deps :: "'d \<Rightarrow> 'c set" assumes "opset opset deps" shows "finite opset"
quasi_isometry_on_compose: fixes lambda :: "real"   and C :: "real"   and X :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and mu :: "real"   and D :: "real"   and Y :: "'c set"   and g :: "'c \<Rightarrow> 'd" assumes "lambda C -quasi_isometry_on X f"   and "mu D -quasi_isometry_on Y g"   and "f ` X \<subseteq> Y" shows "(lambda * mu) (C * mu + D) -quasi_isometry_on X (g \<circ> f)"
order_gt_0: fixes G :: "('c, 'd) monoid_scheme" assumes "finite_group G" shows "0 < order G"
reflFreshAll_def: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'c"   and hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> 'd"   and MOD :: "('index, 'bindex, 'varSort, 'e, 'opSym, 'var, 'c, 'd, 'f) Recursion.model_scheme" shows "reflFreshAll h hA MOD \<equiv> reflFresh h MOD \<and> reflFreshAbs hA MOD"
Supi_def: fixes X :: "'a set" shows "Supi X = \<Sqinter> {y. \<forall>x\<in>X. x \<le> y}"
join_False_alt: fixes X :: "'b option list set"   and Y :: "'b option list set" shows "join X False Y = X - join X True Y"
ih3: shows "pi \<bullet> (\<parallel>(B)\<parallel>) = \<parallel>(B)\<parallel>"
meld_invar: fixes bq1 :: "('c, 'd) SkewBinomialTree list"   and bq2 :: "('c, 'd) SkewBinomialTree list" assumes "invar bq1"   and "invar bq2" shows "invar (meld bq1 bq2)"
AbruptProp: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'a, 'c) com option"   and c :: "('b, 'a, 'c) com"   and s :: "'b" shows "\<Gamma>\<turnstile> \<langle>c,Abrupt s\<rangle> \<Rightarrow> Abrupt s"
fresh_abs_funI2: fixes x :: "'a"   and a :: "'b" assumes "pt TYPE('a) TYPE('b)"   and "at TYPE('b)"   and "finite (supp x)" shows "a \<sharp> [a].x"
Rep_assn_cases: fixes y :: "heap \<times> nat set \<Rightarrow> bool" assumes "y \<in> Collect proper" obtains x :: "assn" where "y = Rep_assn x"
arr_Par_ArrCod_in_Vset: fixes \<alpha> :: "V"   and T :: "V" assumes "arr_Par \<alpha> T" shows "T\<lparr>ArrCod\<rparr> \<in>\<^sub>\<circ> Vset \<alpha>"
inv_max_hi_split_min: fixes t :: "('b ivl \<times> 'b) tree" assumes "t \<noteq> \<langle>\<rangle>"   and "inv_max_hi t" shows "inv_max_hi (snd (split_min t))"
semimetric_def: fixes dist :: "'b \<Rightarrow> 'b \<Rightarrow> real" shows "semimetric dist \<equiv> (\<forall>x y. 0 \<le> dist x y) \<and> (\<forall>x y. (dist x y = 0) = (x = y)) \<and> (\<forall>x y. dist x y = dist y x)"
idiom_ext_rule_def: fixes extra_rule :: "'a itrm \<Rightarrow> 'a itrm \<Rightarrow> bool" shows "special_idiom.idiom_ext_rule extra_rule = sup idiom_rule extra_rule"
fourpow_gt_central_binomial: fixes n :: "nat" assumes "0 < n" shows "2 * n choose n < 4 ^ n"
1: shows "y \<otimes> local.divide x y = [p] \<cdot> \<one> [^] nat (ord_Zp y) \<otimes> ([p] \<cdot> \<one> [^] nat (ord_Zp x - ord_Zp y) \<otimes> ac_Zp x \<otimes> ac_Zp y \<otimes> inv ac_Zp y)"
qAFreshAll_simps_1: fixes xs :: "'d"   and x :: "'e"   and ys :: "'d"   and y :: "'e" shows "qAFresh xs x (qVar ys y) = (xs \<noteq> ys \<or> x \<noteq> y)"
measure_def: fixes M :: "'a measure"   and A :: "'a set" shows "Sigma_Algebra.measure M A = enn2real (emeasure M A)"
gauss_i_times_i: shows "\<i>\<^sub>\<int> * \<i>\<^sub>\<int> = - 1"
l_one: fixes G :: "('b, 'c) monoid_scheme"   and x :: "'b" assumes "Group.monoid G"   and "x \<in> carrier G" shows "\<one>\<^bsub>G\<^esub> \<otimes>\<^bsub>G\<^esub> x = x"
intro: fixes R :: "('e, 'f) Ring_scheme" assumes "\<And>a b. \<lbrakk>a \<in> carrier R; b \<in> carrier R; a \<cdot>\<^sub>r\<^bsub>R\<^esub> b = \<zero>\<^bsub>R\<^esub>\<rbrakk> \<Longrightarrow> a = \<zero>\<^bsub>R\<^esub> \<or> b = \<zero>\<^bsub>R\<^esub>" shows "Idomain_axioms R"
Range_def: shows "Range \<equiv> \<lambda>r. {x. Rangep (\<lambda>x xa. (x, xa) \<in> r) x}"
f_shrink_last_one: fixes k :: "nat"   and xs :: "'b list" assumes "0 < k"   and "length xs = k" shows "xs \<div>\<^sub>l k = [last xs]"
proj_k_dim: fixes k :: "nat" shows "proj_k k \<in> carrier_mat K K"
le_supI2: fixes x :: "'a"   and b :: "'a"   and a :: "'a" assumes "x \<le> b" shows "x \<le> sup a b"
log_le_cancel_iff: fixes a :: "real"   and x :: "real"   and y :: "real" assumes "1 < a"   and "0 < x"   and "0 < y" shows "(Transcendental.log a x \<le> Transcendental.log a y) = (x \<le> y)"
bind_option_simps_1: fixes fail :: "'d"   and f :: "'e \<Rightarrow> 'd" shows "monad.bind_option fail None f = fail"
hom_uminus: fixes x :: "rat" shows "of_rat (- x) = - of_rat x"
image_Collect: fixes f :: "'c \<Rightarrow> 'b"   and P :: "'c \<Rightarrow> bool" shows "f ` {x. P x} = {f x |x. P x}"
loop_orientation_split: fixes y :: "'a"   and x :: "'a" shows "loop_orientation x y = split (x \<sqinter> (1::'a)) y x"
nths_Cons: fixes x :: "'m"   and l :: "'m list"   and A :: "nat set" shows "nths (x # l) A = (if 0 \<in> A then [x] else []) @ nths l {j. Suc j \<in> A}"
flip_altsublist_chain_length: fixes x :: "'w list"   and xs :: "'w list list"   and y :: "'w list" assumes "flip_altsublist_chain (x # xs @ [y])" shows "length y = length x"
g3: shows "monom_mult c s q = monom_mult c s p - monom_mult (lookup (monom_mult c s p) ((s + t) \<oplus> lt f) / lc f) (s + t) f"
ABRAW: fixes n :: "nat" shows "\<not> P (cbox (A (Suc n)) (B (Suc n))) \<and> (\<forall>i\<in>Basis. A n \<bullet> i \<le> A (Suc n) \<bullet> i \<and> A (Suc n) \<bullet> i \<le> B (Suc n) \<bullet> i \<and> B (Suc n) \<bullet> i \<le> B n \<bullet> i \<and> 2 * (B (Suc n) \<bullet> i - A (Suc n) \<bullet> i) \<le> B n \<bullet> i - A n \<bullet> i)"
fmrel_to_trancl: fixes r :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and xm :: "'e \<rightharpoonup>\<^sub>f 'd"   and ym :: "'e \<rightharpoonup>\<^sub>f 'd" assumes "reflp r"   and "fmrel r\<^sup>+\<^sup>+ xm ym" shows "(fmrel r)\<^sup>+\<^sup>+ xm ym"
qconv_plus: fixes c :: "'f \<Rightarrow>\<^sub>U 'e"   and x :: "'c['d, 'f]"   and y :: "'c['d, 'f]" shows "qconv c (x + y) = qconv c x + qconv c y"
s_le_nc: shows "1\<^sub>\<sigma> \<le> nc"
reads_consistent_flush_all_until_volatile_write: fixes \<S> :: "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"   and i :: "nat"   and p :: "'b"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and sb :: "'b memref list"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option"   and pending_write :: "bool"   and m :: "nat \<Rightarrow> nat" assumes "valid_ownership_and_sharing \<S> ts"   and "i < length ts"   and "ts ! i = (p, is, \<theta>, sb, \<D>, \<O>, \<R>)"   and "reads_consistent pending_write \<O> m sb" shows "reads_consistent True (acquired True (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb) \<O>) (flush_all_until_volatile_write ts m) (dropWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb)"
emeasure_embed_measure: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'d set"   and M :: "'c measure" assumes "inj f"   and "A \<in> sets (embed_measure M f)" shows "emeasure (embed_measure M f) A = emeasure M (f -` A \<inter> space M)"
diamond_top_right_edge_def: shows "diamond_top_right_edge = (- 1, \<lambda>x. (diamond_x (1 / 2 * x + 1 / 2), - diamond_x (1 / 2 * x + 1 / 2) + d / 2))"
eq: shows "(\<lambda>x. (x, f x)) ` S = S \<times> T \<inter> (\<lambda>z. (f \<circ> fst) z - snd z) -` {0::'b}"
ord_ge_1: fixes G :: "('c, 'd) monoid_scheme"   and a :: "'c" assumes "Group.group G"   and "finite (carrier G)"   and "a \<in> carrier G" shows "1 \<le> group.ord G a"
bigE: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'b" assumes "f \<in> O[F](g)" obtains c :: "real" where "0 < c" and "\<forall>\<^sub>F x in F. norm (f x) \<le> c * norm (g x)"
exponential_def: fixes l :: "real" shows "exponential l = density lborel (\<lambda>x. ennreal (exponential_density l x))"
simps: fixes p :: "real poly"   and qs1 :: "real poly list"   and m1 :: "rat mat"   and sub1 :: "nat list list"   and sgn1 :: "rat list list"   and qs2 :: "real poly list"   and m2 :: "rat mat"   and sub2 :: "nat list list"   and sgn2 :: "rat list list" shows "combine_systems p (qs1, m1, sub1, sgn1) (qs2, m2, sub2, sgn2) = smash_systems p qs1 qs2 sub1 sub2 sgn1 sgn2 m1 m2"
sel: fixes x :: "'b" shows "the_tag_const (\<tau>\<^sub>c\<^sub>s\<^sub>t x) = x"
only_one_append: fixes C' :: "'a"   and Cs :: "'a list"   and Cs' :: "'a list"   and Ds :: "'a list"   and Ds' :: "'a list" assumes "C' \<notin> set Cs"   and "C' \<notin> set Cs'"   and "Ds @ C' # Ds' = Cs @ C' # Cs'" shows "Cs = Ds \<and> Cs' = Ds'"
poly_gcd_pderiv_iff: fixes p :: "'a poly"   and x :: "'a" shows "(poly (p div gcd p (pderiv p)) x = (0::'a)) = (poly p x = (0::'a))"
rexp_of'_def: fixes n :: "nat"   and \<phi> :: "'a formula" shows "rexp_of' n \<phi> = rexp.Inter (rexp_of_alt n \<phi>) (ENC n (FOV \<phi>))"
snd_foldl_ef_det_eq: fixes bezout :: "'e \<Rightarrow> 'e \<Rightarrow> 'e \<times> 'e \<times> 'e \<times> 'e \<times> 'e"   and n :: "'e"   and A :: "(('e, 'f) vec, 'g) vec"   and k :: "nat" shows "snd (foldl (echelon_form_of_column_k_det bezout) (n, A, 0) [0..<k]) = foldl (echelon_form_of_column_k bezout) (A, 0) [0..<k]"
Adm_def: fixes \<V> :: "'a V_rec"   and \<rho> :: "'a V_rec \<Rightarrow> 'a set"   and Tr :: "'a list set"   and \<beta> :: "'a list"   and e :: "'a" shows "Adm \<V> \<rho> Tr \<beta> e \<equiv> \<exists>\<gamma>. \<gamma> @ [e] \<in> Tr \<and> \<gamma> \<upharpoonleft> \<rho> \<V> = \<beta> \<upharpoonleft> \<rho> \<V>"
inject_4: fixes x51 :: "'b formula"   and x52 :: "'b formula"   and y51 :: "'b formula"   and y52 :: "'b formula" shows "(x51 \<^bold>\<or> x52 = y51 \<^bold>\<or> y52) = (x51 = y51 \<and> x52 = y52)"
prems: shows "x \<in> vars\<^sub>s\<^sub>s\<^sub>t (s # S \<cdot>\<^sub>s\<^sub>s\<^sub>t \<theta>)"
assms_1: shows "par_comp_prot_stateful \<P> Sec"
hd_set: fixes x :: "'b"   and a :: "'b"   and M :: "'b list" assumes "x \<in> set (a # M)" shows "x = a \<or> x \<in> set M"