hcju/isabelleps huggingface dataset:
intro: fixes return :: "'b \<Rightarrow> 'c"   and bind :: "'c \<Rightarrow> ('b \<Rightarrow> 'c) \<Rightarrow> 'c"   and get :: "(('d \<Rightarrow> 'b option) \<Rightarrow> 'c) \<Rightarrow> 'c"   and put :: "('d \<Rightarrow> 'b option) \<Rightarrow> 'c \<Rightarrow> 'c" assumes "monad_state return bind get put" shows "memoization return bind get put"
dis_rews_1: shows "is_A\<cdot>\<bottom> = \<bottom>"
simps_1: fixes ys :: "'g list" shows "shuffles [] ys = {ys}"
succ_rel_swap_head: fixes ref :: "'b"   and list :: "'b list"   and n :: "'b option"   and oid :: "'b" assumes "distinct (ref # list)"   and "(ref, n) \<in> succ_rel (ref # list)" shows "succ_rel (oid # list) = {(oid, n)} \<union> succ_rel list"
lookup_map_entry_neq: fixes x :: "'c"   and y :: "'c"   and f :: "'d \<Rightarrow> 'd"   and m :: "('c, 'd) mapping" assumes "x \<noteq> y" shows "Mapping.lookup (Mapping.map_entry x f m) y = Mapping.lookup m y"
lunit_hcomp_2: fixes f :: "'b"   and g :: "'b" assumes "D.ide f"   and "D.ide g"   and "src\<^sub>D f = trg\<^sub>D g" shows "\<a>\<^sub>D\<^sup>-\<^sup>1[trg\<^sub>D f, f, g] \<cdot>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[f \<star>\<^sub>D g] = \<l>\<^sub>D\<^sup>-\<^sup>1[f] \<star>\<^sub>D g"
qe_pres_def: shows "qe_pres = Z.lift_dnf_qe (qe_pres\<^sub>1 \<circ> hd_coeffs1)"
split: fixes P :: "'a \<Rightarrow> bool"   and f1 :: "'a"   and f2 :: "'a"   and f3 :: "'a"   and finite_3 :: "finite_3" shows "P (case finite_3 of a\<^sub>1 \<Rightarrow> f1 | a\<^sub>2 \<Rightarrow> f2 | a\<^sub>3 \<Rightarrow> f3) = ((finite_3 = a\<^sub>1 \<longrightarrow> P f1) \<and> (finite_3 = a\<^sub>2 \<longrightarrow> P f2) \<and> (finite_3 = a\<^sub>3 \<longrightarrow> P f3))"
f_closed: fixes x :: "'a"   and y :: "'a" assumes "x \<in> D"   and "y \<in> D" shows "x \<cdot> y \<in> D"
f_2: shows "finprod G f ((\<lambda>a. generate G {a}) ` A) = \<one>"
Zconseq: fixes P :: "(char list \<Rightarrow> int) \<Rightarrow> enat"   and c :: "com"   and Q :: "(char list \<Rightarrow> int) \<Rightarrow> enat"   and k :: "nat"   and P' :: "(char list \<Rightarrow> int) \<Rightarrow> enat"   and Q' :: "(char list \<Rightarrow> int) \<Rightarrow> enat" assumes "\<turnstile>\<^sub>Z {P} c {Q}"   and "\<And>s. P s \<le> enat k * P' s"   and "\<And>s. enat k * Q' s \<le> Q s"   and "0 < k" shows "\<turnstile>\<^sub>Z {P'} c {Q'}"
bounded_clinear_add: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "bounded_clinear f"   and "bounded_clinear g" shows "bounded_clinear (\<lambda>x. f x + g x)"
snd_below_iff: fixes x :: "'c \<times> 'b"   and y :: "'b" shows "(snd x \<sqsubseteq> y) = (x \<sqsubseteq> (fst x, y))"
power_one_over: fixes divide :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and inverse :: "'e \<Rightarrow> 'e"   and one :: "'e"   and times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and uminus :: "'e \<Rightarrow> 'e"   and a :: "'e"   and n :: "nat" assumes "class.division_ring divide inverse one times plus zero minus uminus" shows "power.power one times (divide one a) n = divide one (power.power one times a n)"
list_all2_all_nthI: fixes a :: "'d list"   and b :: "'e list"   and P :: "'d \<Rightarrow> 'e \<Rightarrow> bool" assumes "length a = length b"   and "\<And>n. n < length a \<Longrightarrow> P (a ! n) (b ! n)" shows "list_all2 P a b"
prefixToLevel_auxIncreaseAuxilaryCounter: fixes i :: "nat"   and k :: "nat"   and M :: "('b \<times> bool) list"   and l :: "nat" assumes "i \<le> k" shows "prefixToLevel_aux M l i = prefixToLevel_aux M (l + (k - i)) k"
induct: fixes P :: "(nat \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> nat \<Rightarrow> bool"   and a0 :: "nat \<Rightarrow> 'b"   and a1 :: "'b"   and a2 :: "nat" assumes "\<And>f i n. (\<And>x. \<lbrakk>n \<noteq> 0; n \<noteq> 1; x \<in> {d. d dvd n \<and> d < n}\<rbrakk> \<Longrightarrow> P f i x) \<Longrightarrow> P f i n" shows "P a0 a1 a2"
Sup_llist_imp_exists_index: fixes x :: "'b"   and Xs :: "'b set llist" assumes "x \<in> Sup_llist Xs" shows "\<exists>i. enat i < llength Xs \<and> x \<in> lnth Xs i"
2125: fixes x :: "'a"   and y :: "'a" shows "- - (x \<squnion> y) \<sqinter> y = y"
integral_restrict_Int: fixes S :: "'a set"   and T :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "S \<in> sets lebesgue"   and "T \<in> sets lebesgue" shows "LINT x|lebesgue_on T. (if x \<in> S then f x else (0::'b)) = integral\<^sup>L (lebesgue_on (S \<inter> T)) f"
descriptions: fixes x :: "\<nu>"   and \<phi> :: "\<nu> \<Rightarrow> \<o>" shows "[[x\<^sup>P \<^bold>= (\<^bold>\<iota>x. \<phi> x) \<^bold>\<equiv> (\<^bold>\<forall>z. \<^bold>\<A>\<phi> z \<^bold>\<equiv> z \<^bold>= x)]]"
nlistToTree_list: fixes nl :: "('b, 'c) Node list" shows "toList (nlistToTree nl) = nlistToList nl"
mult_nat_left_at_top: fixes c :: "nat" assumes "0 < c" shows "filterlim ((*) c) sequentially sequentially"
vars_and_\<C>_def: fixes X :: "'Var set" shows "vars_and_\<C> X \<equiv> X \<union> vars_\<C> X"
paths_stay_in_winning_region: fixes G :: "('c, 'd) ParityGame_scheme"   and P :: "'c llist"   and v0 :: "'c"   and p :: "Player"   and \<sigma> :: "'c \<Rightarrow> 'c"   and \<sigma>' :: "'c \<Rightarrow> 'c" assumes "vmc_path G P v0 p \<sigma>"   and "ParityGame.strategy G p \<sigma>'"   and "ParityGame.winning_strategy G p \<sigma>' v0"   and "\<And>v. v \<in> ParityGame.winning_region G p \<Longrightarrow> \<sigma>' v = \<sigma> v" shows "lset P \<subseteq> ParityGame.winning_region G p"
s_id_def: shows "1\<^sub>\<sigma> \<equiv> \<Union>a. {(a, {a})}"
components_are_iso: fixes a :: "'c" assumes "C.ide a" shows "C.iso (C.map a)"
approx_minus_iff: fixes x :: "'c star"   and y :: "'c star" shows "(x \<approx> y) = (x - y \<approx> 0)"
rep_eq: fixes x :: "'a vec" shows "dim_vec x = fst (Rep_vec x)"
assms_2: shows "I.qmpt_factor proj1 M2 T2"
atLeastPlusOneAtMost_greaterThanAtMost_int: fixes l :: "int"   and u :: "int" shows "{l + 1..u} = {l<..u}"
read_tmps_distinct: fixes 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" assumes "i < length ts\<^sub>s\<^sub>b"   and "ts\<^sub>s\<^sub>b ! i = (p, is, \<theta>, sb, \<D>, \<O>, \<R>)" shows "distinct_read_tmps sb"
sum_of_2_squares_nat_mult: fixes x :: "nat"   and y :: "nat" assumes "sum_of_2_squares_nat x"   and "sum_of_2_squares_nat y" shows "sum_of_2_squares_nat (x * y)"
strategyproofness: shows "pmf (sds R14) d + pmf (sds R14) a \<le> pmf (sds R18) d + pmf (sds R18) a"
mono_trans_def: fixes t :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" shows "mono_trans t \<equiv> \<forall>P Q. sound P \<and> sound Q \<and> P \<tturnstile> Q \<longrightarrow> t P \<tturnstile> t Q"
simps: fixes a :: "'c set" shows "finite a = (a = {} \<or> (\<exists>A aa. a = insert aa A \<and> finite A))"
preserves_cod: fixes f :: "'a \<times> 'a" assumes "VoV.arr f" shows "VoV.cod (VoV.map f) = VoV.map (VoV.cod f)"
set_bit_negative_int_iff: fixes n :: "nat"   and k :: "int" shows "(set_bit n k < 0) = (k < 0)"
S_O_S_1: shows "S O S \<subseteq> S"
simps_2: fixes \<xi> :: "Preliminaries.var \<Rightarrow> univ"   and a :: "('fsym, 'psym) atm" shows "I.satL \<xi> (Neg a) = (\<not> I.satA \<xi> a)"
is_real_arg2: fixes z :: "complex" assumes "is_real z" shows "arg z = 0 \<or> arg z = pi"
deg_f2: fixes \<sigma> :: "nat \<Rightarrow> nat" assumes "\<sigma> permutes {0..<m + n}"   and "\<sigma> \<noteq> id" shows "degree (f \<sigma>) < m * n"
keys_dehomogenizeE: fixes t :: "'c \<Rightarrow>\<^sub>0 nat"   and x :: "'c"   and p :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd" assumes "t \<in> keys (dehomogenize x p)" obtains s :: "'c \<Rightarrow>\<^sub>0 nat" where "s \<in> keys p" and "t = except s {x}"
has_right_adjoint_functor: shows "right_adjoint_functor (\<cdot>\<^sub>C) (\<cdot>\<^sub>D) F"
Min_elem_dest: fixes A :: "'c DBMEntry set"   and x :: "'c DBMEntry" assumes "finite A"   and "A \<noteq> {}"   and "x = linordered_monoid.Min A" shows "x \<in> A"
restrictA_append: fixes S :: "'a set"   and a :: "('a \<times> 'b) list"   and b :: "('a \<times> 'b) list" shows "restrictA S (a @ b) = restrictA S a @ restrictA S b"
tendsto_Re: fixes g :: "'a \<Rightarrow> complex"   and a :: "complex"   and F :: "'a filter" assumes "(g \<longlongrightarrow> a) F" shows "((\<lambda>x. complex.Re (g x)) \<longlongrightarrow> complex.Re a) F"
hc_def: fixes F :: "'a formula" shows "hc F \<equiv> AX10 \<turnstile>\<^sub>H F"
smooth_on_le: fixes k :: "enat"   and S :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and l :: "enat" assumes "k-smooth_on S f"   and "l \<le> k" shows "l-smooth_on S f"
Suc_le': fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and one :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and Rep :: "'b \<Rightarrow> int"   and Abs :: "int \<Rightarrow> 'b"   and a :: "'b" assumes "class.mod_type times minus plus zero uminus one less_eq less Rep Abs"   and "plus a one \<noteq> zero" shows "less a (plus a one)"
Subring_tOp_ring_tOp: fixes S :: "('a, 'e) Ring_scheme"   and a :: "'a"   and b :: "'a" assumes "Subring R S"   and "a \<in> carrier S"   and "b \<in> carrier S" shows "a \<cdot>\<^sub>r\<^bsub>S\<^esub> b = a \<cdot>\<^sub>r b"
insert3_insert2_d: fixes n :: "nat"   and dl :: "nat"   and t :: "'a tree"   and x :: "'a" assumes "bal_i (n + dl) (height t)" shows "insert3_d x (t, n, dl) = insert2_d x (t, n, dl)"
d: fixes p :: "int poly" shows "roots_of_2 p = concat (map roots_of_2_irr (factors_of_int_poly p))"
assms_4: shows "restrict (A \<inter> J) zJ \<in> restrict J ` Set.filter (isSameIntersection zI (A \<inter> I)) X"
fin_ex_inf_eq_limit: fixes A :: "'a set"   and w :: "nat \<Rightarrow> 'a" assumes "finite A" shows "(\<exists>\<^sub>\<infinity>i. w i \<in> A) = (limit w \<inter> A \<noteq> {})"
hom_graph_compl: fixes f :: "'a \<Rightarrow> 'b"   and S :: "'a set"   and x :: "'a formula"   and a :: "'b" assumes "hom_graph f S x a" shows "hom_graph f S (- x) (- a)"
t4_space: fixes "open" :: "'a set \<Rightarrow> bool"   and S :: "'a set"   and T :: "'a set" assumes "class.t4_space open"   and "topological_space.closed open S"   and "topological_space.closed open T"   and "S \<inter> T = {}" shows "\<exists>U V. open U \<and> open V \<and> S \<subseteq> U \<and> T \<subseteq> V \<and> U \<inter> V = {}"
foundation16': fixes \<tau> :: "'d state \<times> 'd state"   and X :: "'d state \<times> 'd state \<Rightarrow> 'e" shows "(\<tau> \<Turnstile> \<delta> X) = (X \<tau> \<noteq> invalid \<tau> \<and> X \<tau> \<noteq> null \<tau>)"
cmp_def: fixes x :: "'b"   and y :: "'b" shows "cmp x y = (if x < y then LT else if x = y then EQ else GT)"
p: shows "standard_simplex p \<subseteq> \<Union> \<C>"
pnormal_def: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and p :: "'b list" assumes "class.semiring_0 plus zero times" shows "semiring_0.pnormal zero p = (semiring_0.pnormalize zero p = p \<and> p \<noteq> [])"
upb1_def: shows "upb1 = loc_upb (c_snd (c_snd n)) x"
preserves_hhom: fixes \<mu> :: "'a" assumes "C.arr \<mu>" shows "\<guillemotleft>F \<mu> : F.map\<^sub>0 (src\<^sub>C \<mu>) \<rightarrow>\<^sub>D F.map\<^sub>0 (trg\<^sub>C \<mu>)\<guillemotright>"
one_sq_mtx_def: shows "1 \<equiv> id (to_mtx (mat 1))"
SD_preorder: fixes q :: "'a pmf"   and p :: "'a pmf" shows "SD le q p = (p \<in> lotteries_on carrier \<and> q \<in> lotteries_on carrier \<and> (\<forall>x\<in>carrier. measure_pmf.prob q (preferred_alts le x) \<le> measure_pmf.prob p (preferred_alts le x)))"
bdd_belowI2: fixes A :: "'a set"   and m :: "'b multiset"   and f :: "'a \<Rightarrow> 'b multiset" assumes "\<And>x. x \<in> A \<Longrightarrow> m \<subseteq># f x" shows "subset_mset.bdd_below (f ` A)"
override_on_apply_in: fixes a :: "'c"   and A :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "a \<in> A" shows "override_on f g A a = g a"
negligible_mono: fixes g :: "nat \<Rightarrow> real"   and f :: "nat \<Rightarrow> real" assumes "negligible g"   and "f \<in> O(g)" shows "negligible f"
that: fixes i :: "'a" assumes "i \<in> Basis" shows "\<bar>x \<bullet> i\<bar> \<le> \<bar>y \<bullet> i\<bar>"
foldli_transform: shows "\<exists>l0. distinct l0 \<and> S = set l0 \<and> sorted_wrt R l0 \<and> it = foldli l0"
invariant_imp_Always: fixes F :: "'b program"   and A :: "'b set" assumes "F \<in> invariant A" shows "F \<in> Always A"
fundamental_theorem_algebra_factorized: fixes p :: "complex poly" shows "\<exists>as. Polynomial.smult (Polynomial.lead_coeff p) (\<Prod>a\<leftarrow>as. [:- a, 1:]) = p \<and> length as = Polynomial.degree p"
S_respects_labels: fixes B :: "'c set"   and \<phi> :: "'a \<Rightarrow> 'c"   and s :: "'a permutation"   and v :: "'a" assumes "label_wrt B \<phi>"   and "s \<in> (\<Union>(f, g)\<in>fundfoldpairs. {Abs_induced_automorph f g})"   and "v \<in> \<Union> X" shows "\<phi> (s \<rightarrow> v) = \<phi> v"
SeqWRP_SUCC_lemma: fixes y' :: "name"   and s :: "tm"   and k :: "tm"   and y :: "tm" assumes "atom y' \<sharp> (s, k, y)" shows "{SeqWRP s (SUCC k) y} \<turnstile> SyntaxN.Ex y' (SeqWRP s k (Var y') AND y EQ Q_Succ (Var y'))"
rep_eq: fixes x :: "'c fset" shows "sorted_list_of_fset x = sorted_list_of_set (fset x)"
Y_meas: fixes n :: "nat" shows "Y n \<in> sets M"
rep_eq: fixes x :: "'b fset" shows "fthe_elem' x = the_elem' (fset x)"
f1: shows "lookup (tensors_from_net (Conv (id_matrix nr (output_size' m)) m) $ i) is = evaluate_net (Conv (id_matrix nr (output_size' m)) m) (base_input (Conv (id_matrix nr (output_size' m)) m) is) $ i"
pow_order_eq_1: fixes G :: "('a, 'b) monoid_scheme"   and a :: "'a" assumes "Group.group G"   and "a \<in> carrier G" shows "a [^]\<^bsub>G\<^esub> order G = \<one>\<^bsub>G\<^esub>"
cases: fixes a1 :: "('a, 'b, 'x, 'c, 'd, 'e) thread_action"   and a2 :: "('a, 'b, 'f, 'c, 'd, 'e) thread_action" assumes "actions_subset a1 a2" obtains ta' :: "('a, 'b, 'x, 'c, 'd, 'e) thread_action" and ta :: "('a, 'b, 'f, 'c, 'd, 'e) thread_action" where "a1 = ta'" and "a2 = ta" and "collect_locks' \<lbrace>ta'\<rbrace>\<^bsub>l\<^esub> \<subseteq> collect_locks \<lbrace>ta\<rbrace>\<^bsub>l\<^esub>" and "collect_cond_actions \<lbrace>ta'\<rbrace>\<^bsub>c\<^esub> \<subseteq> collect_cond_actions \<lbrace>ta\<rbrace>\<^bsub>c\<^esub>" and "collect_interrupts \<lbrace>ta'\<rbrace>\<^bsub>i\<^esub> \<subseteq> collect_interrupts \<lbrace>ta\<rbrace>\<^bsub>i\<^esub>"
distinct_23: fixes x21 :: "POPLmarkRecord.type"   and x22 :: "trm"   and x71 :: "trm"   and x72 :: "char list" shows "(\<lambda>:x21. x22) \<noteq> x71..x72"
inject_2: fixes x2 :: "vrs"   and x3 :: "trm"   and x1 :: "ty"   and y2 :: "vrs"   and y3 :: "trm"   and y1 :: "ty" shows "((\<lambda>x2:x1. x3) = (\<lambda>y2:y1. y3)) = ([x2].x3 = [y2].y3 \<and> x1 = y1)"
identity_\<Pi>\<^sub>2_def: shows "(\<^bold>=) \<equiv> (\<^bold>=\<^sub>2)"
C12: fixes x :: "'b"   and y :: "'b" shows "(x \<cdot> y)\<^sup>\<star> = (1::'b) + x \<cdot> (y \<cdot> x)\<^sup>\<star> \<cdot> y"
Esigmanotzero: fixes n :: "nat" assumes "1 \<le> n" shows "1 \<le> Esigma n"
redT_updW_not_Suspend_Some: fixes t :: "'b"   and ws :: "'b \<Rightarrow> 'c wait_set_status option"   and wa :: "('b, 'c) wait_set_action"   and ws' :: "'b \<Rightarrow> 'c wait_set_status option"   and w' :: "'c wait_set_status"   and w :: "'c wait_set_status" assumes "redT_updW t ws wa ws'"   and "ws' t = \<lfloor>w'\<rfloor>"   and "ws t = \<lfloor>w\<rfloor>"   and "\<And>w. wa \<noteq> Suspend w" shows "w' = w \<or> (\<exists>w'' w'''. w = InWS w'' \<and> w' = PostWS w''')"
mod_exhaust_less_4: fixes m :: "nat" shows "m mod 4 = 0 \<or> m mod 4 = 1 \<or> m mod 4 = 2 \<or> m mod 4 = 3"
ord_resolve_rename_max_side_prems: fixes CAs :: "'a literal multiset list"   and DA :: "'a literal multiset"   and AAs :: "'a multiset list"   and As :: "'a list"   and \<sigma> :: "'s"   and E :: "'a literal multiset" assumes "ord_resolve_rename CAs DA AAs As \<sigma> E" shows "length CAs \<le> size DA"
induct: fixes \<phi>s :: "Formula.formula list"   and x1 :: "mregex \<Rightarrow> mregex"   and x2 :: "Formula.formula Regex.regex \<Rightarrow> Formula.formula Regex.regex"   and P :: "(mregex \<Rightarrow> mregex) \<Rightarrow> (Formula.formula Regex.regex \<Rightarrow> Formula.formula Regex.regex) \<Rightarrow> bool" assumes "ok_rctxt \<phi>s x1 x2"   and "P id id"   and "\<And>\<kappa> \<kappa>' mr. \<lbrakk>ok_rctxt \<phi>s \<kappa> \<kappa>'; P \<kappa> \<kappa>'\<rbrakk> \<Longrightarrow> P (\<lambda>t. \<kappa> (MTimes mr t)) (\<lambda>t. \<kappa>' (regex.Times (from_mregex mr \<phi>s) t))" shows "P x1 x2"
take_bit_eq_mod: fixes plus :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and minus :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and zero :: "'k"   and times :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and one :: "'k"   and divide :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and modulo :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and bit :: "'k \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'k \<Rightarrow> 'k"   and drop_bit :: "nat \<Rightarrow> 'k \<Rightarrow> 'k"   and take_bit :: "nat \<Rightarrow> 'k \<Rightarrow> 'k"   and n :: "nat"   and a :: "'k" assumes "class.semiring_bit_shifts plus minus zero times one divide modulo bit push_bit drop_bit take_bit" shows "take_bit n a = modulo a (power.power one times (numeral.numeral one plus (num.Bit0 num.One)) n)"
assms_2: shows "rqrm_prop (ls.\<alpha> \<delta>) rqrm"
assms_2: shows "x0 \<in> carrier (\<G> n)"
sel_1: fixes e :: "((nat \<times> nat) \<times> transition) fset"   and s :: "nat option"   and d :: "nat \<Rightarrow>f value option"   and p :: "value option list"   and i :: "(String.literal \<times> value list) stream" shows "shd (make_full_observation e s d p i) = \<lparr>statename = s, datastate = d, action = shd i, output = p\<rparr>"
abc_comp_frist_steps_halt_eq: fixes lm :: "nat list"   and A :: "abc_inst list"   and n :: "nat"   and lm' :: "nat list"   and B :: "abc_inst list" assumes "abc_steps_l (0, lm) A n = (length A, lm')" shows "\<exists>n'. abc_steps_l (0, lm) (A [+] B) n' = (length A, lm')"
id: fixes B :: "'a mat"   and nc :: "nat" assumes "B \<in> carrier_mat nr nc" shows "eliminate_entries_rec B i is = P * B"
simps_2: fixes ps :: "'b list"   and r :: "'b"   and rs :: "'b list" shows "remdups_rev_code ps (r # rs) = (if r \<in> set ps then remdups_rev_code ps rs else r # remdups_rev_code (r # ps) rs)"
add_mset_add_mset_same_iff: fixes a :: "'f"   and A :: "'f multiset"   and B :: "'f multiset" shows "(add_mset a A = add_mset a B) = (A = B)"
induct: fixes x :: "'a"   and \<Gamma> :: "('b, 'c) \<I>"   and xa :: "('a, 'b, 'c) gpv"   and P :: "('a, 'b, 'c) gpv \<Rightarrow> bool" assumes "x \<in> results_gpv \<Gamma> xa"   and "\<And>gpv. Pure x \<in> set_spmf (the_gpv gpv) \<Longrightarrow> P gpv"   and "\<And>out c gpv input. \<lbrakk>IO out c \<in> set_spmf (the_gpv gpv); input \<in> responses_\<I> \<Gamma> out; x \<in> results_gpv \<Gamma> (c input); P (c input)\<rbrakk> \<Longrightarrow> P gpv" shows "P xa"
Abs_aezfun_inverse: fixes y :: "'a \<Rightarrow> 'b" assumes "y \<in> aezfun_set" shows "aezfun (Abs_aezfun y) = y"
IH: shows "\<exists>\<Gamma>''\<subseteq>F \<triangleright> \<Gamma>. finite \<Gamma>'' \<and> (\<Gamma>'' \<turnstile> \<bottom>)"
m: shows "mat_kernel (1\<^sub>m 0) = {0\<^sub>v 0}"
bounded: shows "\<exists>K. \<forall>a b. norm (g a ** b) \<le> norm a * norm b * K"
singleton_group_def: fixes a :: "'e" shows "singleton_group a = \<lparr>carrier = {a}, mult = \<lambda>x y. a, one = a\<rparr>"
alpha_other_17: fixes vb :: "('e, 'f, 'g) Tree"   and v :: "'g"   and va :: "('e, 'f, 'g) Tree" shows "(tNot vb =\<^sub>\<alpha> tAct v va) = False"
pseudominimal_comp: fixes P :: "'a set"   and x :: "'a"   and y :: "'a" assumes "pseudominimal_in P x"   and "pseudominimal_in P y"   and "y \<succeq> x" shows "x = y"
cancel_semigroup_pair_def: fixes f\<^sub>a :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and f\<^sub>b :: "'d \<Rightarrow> 'd \<Rightarrow> 'd" shows "cancel_semigroup_pair f\<^sub>a f\<^sub>b \<equiv> cancel_semigroup f\<^sub>a \<and> cancel_semigroup f\<^sub>b"
assms_2: shows "N \<subseteq>\<^sub>N N2"
analzp_analz_eq: fixes H :: "msg set" shows "analzp (\<lambda>x. x \<in> H) = (\<lambda>x. x \<in> analz H)"
And_append_S: fixes S :: "'b ltl set"   and xs :: "'b ltl list"   and ys :: "'b ltl list" shows "S \<Turnstile>\<^sub>P And (xs @ ys) = S \<Turnstile>\<^sub>P And xs and And ys"
ag_pOp_add_r: fixes A :: "('c, 'd) aGroup_scheme"   and a :: "'c"   and b :: "'c"   and c :: "'c" assumes "aGroup A"   and "a \<in> carrier A"   and "b \<in> carrier A"   and "c \<in> carrier A"   and "a = b" shows "a \<plusminus>\<^bsub>A\<^esub> c = b \<plusminus>\<^bsub>A\<^esub> c"
lossless_response: fixes r :: "'rand"   and w :: "'witness"   and e :: "'challenge" shows "lossless_spmf (response r w e)"
aux_onlyif: fixes xs :: "'b com option list"   and s :: "'b"   and ys :: "('b com option list \<times> 'b) list" assumes "(xs, s) # ys \<in> par_cptn" shows "\<exists>clist. length clist = length xs \<and> (xs, s) # ys \<propto> map (\<lambda>i. (fst i, s) # snd i) (zip xs clist) \<and> (\<forall>i<length xs. (xs ! i, s) # clist ! i \<in> cptn)"
HARoot_SAs: fixes HA :: "('a, 'b, 'c) hierauto" shows "HARoot HA \<in> SAs HA"
DIAndConcl_def: shows "DIAndConcl = ([], [(Pc pid1 && Pc pid2) \<rightarrow> ([[$\<alpha> vid1]]Pc pid3 && Pc pid4) \<rightarrow> [[$\<alpha> vid1]]Pc pid1 && Pc pid2])"
n_distrib_left: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "!x \<cdot> (!y + !z) = !x \<cdot> !y + !x \<cdot> !z"
of_nat_zhmset: fixes n :: "nat" shows "of_nat n = zhmset_of (of_nat n)"
data_dependence_def: fixes n :: "'node SDG_node"   and V :: "'var"   and n' :: "'node SDG_node" shows "n influences V in n' \<equiv> \<exists>as. V \<in> Def\<^bsub>SDG\<^esub> n \<and> V \<in> Use\<^bsub>SDG\<^esub> n' \<and> parent_node n -as\<rightarrow>\<^sub>\<iota>* parent_node n' \<and> (\<forall>n''. valid_SDG_node n'' \<and> parent_node n'' \<in> set (sourcenodes (tl as)) \<longrightarrow> V \<notin> Def\<^bsub>SDG\<^esub> n'')"
hyps_5: fixes b :: "binding list" assumes "\<turnstile> b ok"   and "b extends (TVarB X T\<^sub>1 # \<Gamma>)" shows "b\<turnstile>S\<^sub>2<:T\<^sub>2"
nesting_lemma': fixes P :: "'b set \<Rightarrow> 'b set \<Rightarrow> bool"   and bdy :: "'c \<Rightarrow> 'd set"   and mgf_call :: "'c \<Rightarrow> 'b"   and mgf :: "'d \<Rightarrow> 'b"   and U :: "'c set"   and uA :: "'b set"   and A :: "'b set"   and n :: "nat"   and t :: "'d" assumes "\<And>A ts. ts \<subseteq> A \<Longrightarrow> P A ts"   and "\<And>A pn. \<forall>b\<in>bdy pn. P (insert (mgf_call pn) A) {mgf b} \<Longrightarrow> P A {mgf_call pn}"   and "\<And>A t. \<forall>pn\<in>U. P A {mgf_call pn} \<Longrightarrow> P A {mgf t}"   and "finite U"   and "uA = mgf_call ` U"   and "A \<subseteq> uA"   and "n \<le> card uA"   and "card A = card uA - n" shows "P A {mgf t}"
le_floor_iff: fixes z :: "int"   and x :: "'b" shows "(z \<le> \<lfloor>x\<rfloor>) = (of_int z \<le> x)"
mult_divide_eq_ennreal: fixes b :: "ennreal"   and a :: "ennreal" assumes "b \<noteq> 0"   and "b \<noteq> \<top>" shows "a * b / b = a"
fds_zeta_times_moebius_mu: shows "fds_zeta * fds moebius_mu = 1"
assms_5: shows "igSwapAllIPresIGWlsAll MOD"
length_subseqs: fixes xs :: "'c list" shows "length (subseqs xs) = 2 ^ length xs"
order_refl: fixes x :: "'b multiset" shows "x \<subseteq># x"
fusion_spec_local_a: fixes a :: "'d"   and init :: "'d set"   and p :: "'d \<times> 'f \<Rightarrow> bool"   and r :: "'d \<times> 'f \<Rightarrow> 'e \<Rightarrow> bool"   and r' :: "'f \<Rightarrow> 'e \<Rightarrow> bool" assumes "a \<in> init" shows "([: x \<leadsto> (u, y) . u \<in> init \<and> x = y :] \<circ> {. p .} \<circ> [: r :]) \<parallel> [: r' :] = [: x \<leadsto> (u, y) . u \<in> init \<and> x = y :] \<circ> {. p .} \<circ> [: (u, x) \<leadsto> y . r (u, x) y \<and> r' x y :]"