hcju/isabelleps huggingface dataset:
real_sqrt_power: fixes x :: "real"   and k :: "nat" shows "sqrt (x ^ k) = sqrt x ^ k"
div_cancel_aux: fixes c :: "nat"   and a :: "nat"   and d :: "nat"   and b :: "nat" assumes "c dvd a" shows "(d + a * b) div c = d div c + a div c * b"
nn_integral_fst: fixes f :: "'b \<times> real \<Rightarrow> ennreal"   and M1 :: "'b measure" assumes "f \<in> borel_measurable (M1 \<Otimes>\<^sub>M M)" shows "\<integral>\<^sup>+ x. \<integral>\<^sup>+ y. f (x, y) \<partial>M \<partial>M1 = integral\<^sup>N (M1 \<Otimes>\<^sub>M M) f"
matchC_C_mono: fixes cval :: "'a \<Rightarrow> 'b \<Rightarrow> real"   and indis :: "('b \<times> 'b) set"   and aval :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and tval :: "'d \<Rightarrow> 'b \<Rightarrow> bool"   and theta :: "(('d, 'c, 'a) cmd \<times> ('d, 'c, 'a) cmd) set"   and c :: "('d, 'c, 'a) cmd"   and d :: "('d, 'c, 'a) cmd"   and theta' :: "(('d, 'c, 'a) cmd \<times> ('d, 'c, 'a) cmd) set" assumes "PL_Indis cval indis"   and "PL_Indis.matchC_C aval tval cval indis theta c d"   and "theta \<subseteq> theta'" shows "PL_Indis.matchC_C aval tval cval indis theta' c d"
induct: fixes x :: "int \<Rightarrow> int"   and P :: "(int \<Rightarrow> int) \<Rightarrow> bool" assumes "x \<in> CofGroups.Ex1"   and "P upOne"   and "\<And>f. \<lbrakk>f \<in> CofGroups.Ex1; P f\<rbrakk> \<Longrightarrow> P (upOne \<circ> f)"   and "\<And>f. \<lbrakk>f \<in> CofGroups.Ex1; P f\<rbrakk> \<Longrightarrow> P (inv upOne \<circ> f)" shows "P x"
floor_le_one: fixes x :: "'a" shows "(\<lfloor>x\<rfloor> \<le> 1) = (x < (2::'a))"
u_in_hom: shows "\<guillemotleft>(tab\<^sub>0 h \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0 : src ((tab\<^sub>0 h \<star> Tgh.p\<^sub>0) \<star> TfTgh.p\<^sub>0) \<rightarrow> src (f \<star> g \<star> h)\<guillemotright>"
set_gen: fixes B :: "nat"   and n :: "nat" shows "set (gen B n) = {xs. length xs = n \<and> (\<forall>i<n. xs ! i \<le> B)}"
\<beta>0: fixes i :: "nat" shows "\<beta> i \<noteq> 0"
Labels_WhileExit: fixes b :: "expr"   and c' :: "cmd" shows "labels (while (b) c') 1 Skip"
simps_4: fixes x :: "brick list"   and xs :: "wall"   and y :: "brick list"   and ys :: "wall" shows "x * xs \<otimes> y * ys = (x \<otimes> y) * (xs \<otimes> ys)"
sshiftr_eq_funpow_sshiftr1: fixes w :: "'b word"   and n :: "nat" shows "w >>> n = (sshiftr1 ^^ n) w"
eqelem_imp_iff: fixes x :: "'c"   and y :: "'c"   and A :: "'c set" assumes "x = y" shows "(x \<in> A) = (y \<in> A)"
finite_mono_remains_stable_implies_strict_prefix: fixes f :: "nat \<Rightarrow> 'a" assumes "finite (range f)"   and "mono f"   and "\<forall>n. f n = f (Suc n) \<longrightarrow> f (Suc n) = f (Suc (Suc n))" shows "\<exists>N. (\<forall>n\<le>N. \<forall>m\<le>N. m < n \<longrightarrow> f m < f n) \<and> (\<forall>n\<ge>N. f N = f n)"
hosts: shows "hosts (generate_valid_stateful_policy_IFSACS_2 G M edgesList) = nodes G"
assms_2: shows "\<bar>v\<bar> \<in> Basis"
E': shows "E = lappend (llist_of (lift_start_obs h.start_tid h.start_heap_obs)) E'"
Inf_to_Sup: shows "Inf = uminus \<circ> Sup \<circ> \<P> uminus"
assms_1: shows "Inl (x, s') \<in> set_spmf (inline1 callee gpv s)"
TID_sym: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and x :: "'a"   and y :: "'a" assumes "Turn_into_distance f" shows "turn_into_distance f x y = turn_into_distance f y x"
integrable_count_space: fixes X :: "'c set"   and f :: "'c \<Rightarrow> 'd" assumes "finite X" shows "integrable (count_space X) f"
cblinfun_apply_cblinfun_compose: fixes a :: "'e \<Rightarrow>\<^sub>C\<^sub>L 'c"   and b :: "'d \<Rightarrow>\<^sub>C\<^sub>L 'e"   and c :: "'d" shows "(a o\<^sub>C\<^sub>L b) *\<^sub>V c = a *\<^sub>V b *\<^sub>V c"
B_invariant: fixes A :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and B :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sim :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and PA :: "'c \<Rightarrow> bool"   and PB :: "'d \<Rightarrow> bool"   and a :: "'d"   and b :: "'d" assumes "Bisimulation_Invariant A B sim PA PB"   and "PB a"   and "B a b" shows "PB b"
FnCall_respects: fixes F :: "nat" shows "(\<lambda>Us. exprel `` {FNCALL F Us}) respects listrel exprel"
i_take_imp_iprefix: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'b" shows "f \<Down> n \<sqsubseteq> f"
proots_count_pos: fixes P :: "'a poly"   and S :: "'a set" assumes "0 < proots_count P S" shows "\<exists>x\<in>S. poly P x = (0::'a)"
add_assoc: fixes leq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and add :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and x :: "'c"   and y :: "'c"   and z :: "'c" assumes "class.plus_ord leq add" shows "add (add x y) z = add x (add y z)"
gbg_impl_rel_eext_internal_def: fixes Rm :: "('f \<times> 'h) set"   and Rv :: "('e \<times> 'g) set" shows "gbg_impl_rel_eext Rm Rv \<equiv> \<langle>Rm, \<langle>\<langle>Rv\<rangle>list_set_rel\<rangle>list_set_rel\<rangle>gen_gbg_impl_rel_eext"
quadratic_hensel_step_def: fixes C :: "int poly"   and q :: "int"   and S :: "int poly"   and T :: "int poly"   and D :: "int poly"   and H :: "int poly" shows "quadratic_hensel_step C q S T D H = hensel_step C q q S T D H S T D H"
mem_cbox_2D: fixes x\<^sub>0 :: "real"   and x :: "real"   and x\<^sub>1 :: "real"   and y\<^sub>0 :: "real"   and y :: "real"   and y\<^sub>1 :: "real" shows "(x\<^sub>0 \<le> x \<and> x \<le> x\<^sub>1 \<and> y\<^sub>0 \<le> y \<and> y \<le> y\<^sub>1) = ((x, y) \<in> cbox (x\<^sub>0, y\<^sub>0) (x\<^sub>1, y\<^sub>1))"
reduce_clauses_in_Q: fixes C :: "'a literal list"   and N :: "('a literal list \<times> nat) list"   and P :: "('a literal list \<times> nat) list"   and Q :: "('a literal list \<times> nat) list"   and Q' :: "('a literal list \<times> nat) list"   and n :: "nat" assumes "C \<in> fst ` set N"   and "\<forall>(E, k)\<in>set P. is_irreducible [C] E" shows "wstate_of_dstate (N, P, Q @ Q', n) \<leadsto>\<^sub>w\<^sup>* wstate_of_dstate (N, fst (reduce_all2 C Q') @ P, Q @ snd (reduce_all2 C Q'), n)"
B: shows "\<exists>u\<in>U. \<exists>v\<in>sources_aux I D (sources_aux I D V [x]) xs. (u, v) \<in> I"
dg_prod_components_1: fixes I :: "V"   and \<AA> :: "V \<Rightarrow> V" shows "dg_prod I \<AA>\<lparr>Obj\<rparr> = (\<Prod>\<^sub>\<circ>i\<in>\<^sub>\<circ>I. \<AA> i\<lparr>Obj\<rparr>)"
lhd_is: shows "lhd Ns = lnth Ns 0"
image_UN: fixes f :: "'c \<Rightarrow> 'b"   and B :: "'d \<Rightarrow> 'c set"   and A :: "'d set" shows "f ` \<Union> (B ` A) = (\<Union>x\<in>A. f ` B x)"
conjunctiveD: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<in> conjunctive.conjunctive (\<sqinter>) (\<sqinter>) (*)" shows "x * (y \<sqinter> z) = x * y \<sqinter> (x * z)"
con_comp_pol_def: fixes I :: "('e \<times> 'e) set" shows "con_comp_pol I \<equiv> {(Some d, Some e) |d e. (d, e) \<in> I} \<union> {(u, v). v = None}"
map_of_filter_apply: fixes xs :: "('a \<times> 'b) list"   and P :: "'a \<times> 'b \<Rightarrow> bool"   and k :: "'a" assumes "distinct (map fst xs)" shows "map_of (filter P xs) k = (case map_of xs k of None \<Rightarrow> None | Some v \<Rightarrow> if P (k, v) then Some v else None)"
join_def: fixes x :: "'c"   and y :: "'c" shows "x \<squnion> y = (THE sup. is_sup x y sup)"
servK_authentic_ter: fixes A :: "agent"   and authK :: "nat"   and Ta :: "nat"   and authTicket :: "msg"   and evs :: "event list"   and servK :: "nat"   and B :: "agent"   and Ts :: "nat"   and servTicket :: "msg" assumes "Says Kas A (Crypt (shrK A) \<lbrace>Key authK, Agent Tgs, Number Ta, authTicket\<rbrace>) \<in> set evs"   and "Crypt authK \<lbrace>Key servK, Agent B, Number Ts, servTicket\<rbrace> \<in> parts (knows Spy evs)"   and "Key authK \<notin> analz (knows Spy evs)"   and "evs \<in> kerbIV" shows "Says Tgs A (Crypt authK \<lbrace>Key servK, Agent B, Number Ts, servTicket\<rbrace>) \<in> set evs"
sel_2: fixes x2 :: "'c" shows "projr (Inr x2) = x2"
r_inv: fixes G :: "('c, 'd) monoid_scheme"   and x :: "'c" assumes "Group.group G"   and "x \<in> carrier G" shows "x \<otimes>\<^bsub>G\<^esub> inv\<^bsub>G\<^esub> x = \<one>\<^bsub>G\<^esub>"
eq_\<I>_attach_on: fixes \<I>' :: "('a, 'b) \<I>"   and res :: "('a, 'b) resource"   and A :: "'c set"   and conv :: "('c, 'd, 'a, 'b) converter"   and conv' :: "('c, 'd, 'a, 'b) converter" assumes "\<I>' \<turnstile>res res \<surd>"   and "\<I>_uniform A UNIV,\<I>' \<turnstile>\<^sub>C conv \<sim> conv'" shows "A \<turnstile>\<^sub>R attach conv res \<sim> attach conv' res"
that: fixes A :: "'a multiset" assumes "\<And>x. x \<in># A \<Longrightarrow> prime x"   and "normalize (\<Prod>\<^sub># A) = normalize x" shows "thesis"
put_put: fixes return :: "'d \<Rightarrow> 'e"   and bind :: "'e \<Rightarrow> ('d \<Rightarrow> 'e) \<Rightarrow> 'e"   and get :: "('f \<Rightarrow> 'e) \<Rightarrow> 'e"   and put :: "'f \<Rightarrow> 'e \<Rightarrow> 'e"   and s :: "'f"   and s' :: "'f"   and m :: "'e" assumes "monad_state return bind get put" shows "put s (put s' m) = put s' m"
intro: fixes hcond :: "'a hrel \<Rightarrow> 'a hrel" assumes "Monotonic hcond" shows "utp_theory_mono_axioms hcond"
orderE: fixes a :: "'g"   and b :: "'g" assumes "a \<le> b" obtains "a = a \<sqinter> b"
induct: fixes P :: "'a rexp \<Rightarrow> 'a rexp \<Rightarrow> bool"   and a0 :: "'a rexp"   and a1 :: "'a rexp" assumes "\<And>uu_. P Zero uu_"   and "P One Zero"   and "\<And>v. P (Atom v) Zero"   and "\<And>v va. P (Plus v va) Zero"   and "\<And>v va. P (Times v va) Zero"   and "\<And>v. P (Star v) Zero"   and "P One One"   and "\<And>v. P One (Atom v)"   and "\<And>v va. P One (Plus v va)"   and "\<And>v va. P One (Times v va)"   and "\<And>v. P One (Star v)"   and "\<And>v. P (Atom v) One"   and "\<And>v va. P (Plus v va) One"   and "\<And>v va. P (Times v va) One"   and "\<And>v. P (Star v) One"   and "\<And>a b. P (Atom a) (Atom b)"   and "\<And>uy_ v va. P (Atom uy_) (Plus v va)"   and "\<And>uy_ v va. P (Atom uy_) (Times v va)"   and "\<And>uy_ v. P (Atom uy_) (Star v)"   and "\<And>v vc vb_. P (Plus v vc) (Atom vb_)"   and "\<And>v vc vb_. P (Times v vc) (Atom vb_)"   and "\<And>v vb_. P (Star v) (Atom vb_)"   and "\<And>r s. P r s \<Longrightarrow> P (Star r) (Star s)"   and "\<And>vc_ v va. P (Star vc_) (Plus v va)"   and "\<And>vc_ v va. P (Star vc_) (Times v va)"   and "\<And>v va vf_. P (Plus v va) (Star vf_)"   and "\<And>v va vf_. P (Times v va) (Star vf_)"   and "\<And>r r' s s'. \<lbrakk>r = s \<Longrightarrow> P r' s'; r \<noteq> s \<Longrightarrow> P r s\<rbrakk> \<Longrightarrow> P (Plus r r') (Plus s s')"   and "\<And>vg_ vh_ v va. P (Plus vg_ vh_) (Times v va)"   and "\<And>v va vk_ vl_. P (Times v va) (Plus vk_ vl_)"   and "\<And>r r' s s'. \<lbrakk>r = s \<Longrightarrow> P r' s'; r \<noteq> s \<Longrightarrow> P r s\<rbrakk> \<Longrightarrow> P (Times r r') (Times s s')" shows "P a0 a1"
low_mds_eq\<^sub>C: shows "conc.low_mds_eq mds\<^sub>C mem\<^sub>1\<^sub>C mem\<^sub>2\<^sub>C"
\<mu>_V_less_inf: fixes v :: "'a" assumes "v \<in> verts G"   and "dist v \<noteq> \<infinity>" shows "\<mu> c s v \<noteq> \<infinity>"
Constrains_INT: fixes I :: "'b set"   and F :: "'c program"   and A :: "'b \<Rightarrow> 'c set"   and A' :: "'b \<Rightarrow> 'c set" assumes "\<And>i. i \<in> I \<Longrightarrow> F \<in> A i Co A' i" shows "F \<in> \<Inter> (A ` I) Co \<Inter> (A' ` I)"
sum_distrib_right: fixes f :: "'c \<Rightarrow> 'a"   and A :: "'c set"   and r :: "'a" shows "comm_monoid_add.sum (\<squnion>) bot f A \<sqinter> r = comm_monoid_add.sum (\<squnion>) bot (\<lambda>n. f n \<sqinter> r) A"
f_expand_append: fixes xs :: "'b message_af list"   and ys :: "'b message_af list"   and k :: "nat" shows "(xs @ ys) \<odot> k = xs \<odot> k @ ys \<odot> k"
fullchar: fixes xs :: "'c list"   and q :: "'c"   and i :: "nat"   and n :: "nat" assumes "index xs q < length xs" shows "(i < length xs) = (index xs q < i \<and> i < length xs \<or> index xs q = i \<or> index xs q - n \<le> i \<and> i < index xs q \<or> i < index xs q - n)"
sigma_algebra_completion: fixes M :: "'b measure" shows "sigma_algebra (space M) {uu_. \<exists>S N N'. uu_ = S \<union> N \<and> S \<in> sets M \<and> N' \<in> null_sets M \<and> N \<subseteq> N'}"
of_nat_0_neq: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and n :: "nat" assumes "class.semiring_char_0 one times plus zero" shows "zero \<noteq> semiring_1.of_nat one plus zero (Suc n)"
create_optim_in_sols_2: fixes X :: "(nat, rat) mapping"   and x :: "rat vec"   and y :: "rat vec" assumes "create_optimal_solutions A b c = Inr X"   and "(x, y) = split_n_m_x (dim_vec c) (dim_vec b) X" shows "b \<bullet> y \<in> {b \<bullet> y |y. [y\<^sub>v*A]=c \<and> (\<forall>i<dim_vec y. 0 \<le> y $ i)}"
GrpE: fixes A :: "'n set"   and f :: "'n \<Rightarrow> 'o"   and x :: "'n"   and y :: "'o" assumes "Grp A f x y" obtains "f x = y" and "x \<in> A"
r_phi': fixes f :: "recf"   and x :: "nat" assumes "recfn 1 f" shows "eval r_phi [encode f, x] = eval f [x]"
ordered_semiring_record_simps_11: fixes carrier :: "'c set"   and mult :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and zero :: "'c"   and add :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and more :: "'d" shows "\<zero>\<^bsub>\<lparr>carrier = carrier, mult = mult, one = one, zero = zero, add = add, \<dots> = more\<rparr>\<^esub> = zero"
fmlookup_drop: fixes a :: "'d"   and m :: "'d \<rightharpoonup>\<^sub>f 'c"   and x :: "'d" shows "fmlookup (fmdrop a m) x = (if x \<noteq> a then fmlookup m x else None)"
le_sup_iff: fixes x :: "'a upred"   and y :: "'a upred"   and z :: "'a upred" shows "(z \<sqsubseteq> (x \<or> y)) = (z \<sqsubseteq> x \<and> z \<sqsubseteq> y)"
assms_1: shows "dip \<in> vD (rt2 nhip)"
card_sccs_verts: shows "card sccs_verts = card sccs"
block_closed: fixes a :: "'a"   and A :: "'a set" assumes "a \<in> A"   and "A \<in> left.transformation.orbit.Partition" shows "a \<in> G"
simps_2: fixes x :: "accent \<times> 'b"   and xs :: "(accent \<times> 'b) list" shows "local.lconv (x # xs) = local.lstep x O local.lconv xs"
comp_null_2: fixes f :: "'s" shows "f \<cdot> null = null"
sumAssocRight: fixes Rel :: "(pi \<times> pi) set"   and P :: "pi"   and Q :: "pi"   and R :: "pi" assumes "Id \<subseteq> Rel" shows "P \<oplus> Q \<oplus> R \<leadsto>[Rel] (P \<oplus> Q) \<oplus> R"
equal_val_Zp': fixes a :: "nat \<Rightarrow> int"   and b :: "nat \<Rightarrow> int"   and c :: "nat \<Rightarrow> int" assumes "a \<in> carrier Zp"   and "b \<in> carrier Zp"   and "c \<in> carrier Zp"   and "val_Zp a = val_Zp b"   and "val_Zp b < val_Zp c" shows "val_Zp (a \<oplus> c) = val_Zp b"
simps_1: fixes x :: "'b" shows "- Fin x = Fin (- x)"
assoc'_simps_3: fixes f :: "'b"   and g :: "'b"   and h :: "'b" assumes "D.ide f"   and "D.ide g"   and "D.ide h"   and "src\<^sub>D f = trg\<^sub>D g"   and "src\<^sub>D g = trg\<^sub>D h" shows "trg\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[f, g, h] = trg\<^sub>D f"
summable'_1: fixes A :: "nat set" shows "(\<lambda>n. f n / nat_power n s) abs_summable_on A"
set_insert_tries: fixes key :: "'c \<Rightarrow> 'd list"   and xs :: "'c list"   and t :: "('d, 'c list) trie" shows "set_tries (inserts_tries key xs t) = set xs \<union> set_tries t"
assms_1: shows "q = (1 + r - d) / (u - d)"
mat_x_leq_vec_sol: fixes A :: "rat mat"   and b :: "rat vec"   and x :: "(nat, rat) mapping"   and i :: "nat" assumes "\<langle>x\<rangle> \<Turnstile>\<^sub>c\<^sub>s set (mat_x_leq_vec A b)"   and "i < dim_vec b" shows "matrix_to_lpolies A ! i \<lbrace> \<langle>x\<rangle> \<rbrace> \<le> b $ i"
bounded_clinear_CBlinfun_apply: fixes f :: "'a \<Rightarrow> 'b" assumes "bounded_clinear f" shows "cblinfun_apply (CBlinfun f) = f"
comp_tfr\<^sub>s\<^sub>t_def: fixes arity :: "'a \<Rightarrow> nat"   and Ana :: "('a, 'b \<times> nat) Term.term \<Rightarrow> ('a, 'b \<times> nat) Term.term list \<times> 'c"   and \<Gamma> :: "('a, 'b \<times> nat) Term.term \<Rightarrow> ('d, 'e) Term.term"   and M :: "('a, 'b \<times> nat) Term.term list"   and S :: "('a, 'b \<times> nat) strand_step list" shows "comp_tfr\<^sub>s\<^sub>t arity Ana \<Gamma> M S \<equiv> list_all (comp_tfr\<^sub>s\<^sub>t\<^sub>p \<Gamma>) S \<and> list_all (wf\<^sub>t\<^sub>r\<^sub>m' arity) (trms_list\<^sub>s\<^sub>t S) \<and> has_all_wt_instances_of \<Gamma> (trms\<^sub>s\<^sub>t S) (set M) \<and> comp_tfr\<^sub>s\<^sub>e\<^sub>t arity Ana \<Gamma> M"
assms_2: shows "\<forall>s. s \<le>ns w \<longrightarrow> s \<noteq> w \<longrightarrow> w <lex s"
step_elim_cases_9: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'a, 'c) com option"   and c :: "'b \<Rightarrow> ('b, 'a, 'c) com"   and s :: "('b, 'c) xstate"   and u :: "('b, 'a, 'c) com \<times> ('b, 'c) xstate" assumes "\<Gamma>\<turnstile> (DynCom c, s) \<rightarrow> u" obtains sa :: "'b" where "s = Normal sa" and "u = (c sa, Normal sa)"   | f :: "'c" where "s = Fault f__" and "u = (Skip, Fault f__)"   | "s = Stuck" and "u = (Skip, Stuck)"   | f :: "'b" where "s = Abrupt f" and "u = (Skip, Abrupt f)"
AGMC_SC_axioms: shows "AGMC_SC Cn (\<div>) \<top> \<bottom> (.\<longrightarrow>.) .\<not> (.\<and>.) (.\<or>.) (.\<longleftrightarrow>.)"
primeidealI: fixes I :: "'c set"   and R :: "('c, 'd) ring_scheme" assumes "ideal I R"   and "cring R"   and "carrier R \<noteq> I"   and "\<And>a b. \<lbrakk>a \<in> carrier R; b \<in> carrier R; a \<otimes>\<^bsub>R\<^esub> b \<in> I\<rbrakk> \<Longrightarrow> a \<in> I \<or> b \<in> I" shows "primeideal I R"
dual_linorder: shows "class.linorder (\<lambda>x y. y \<le>lex x) (\<lambda>x y. y <lex x)"
False: shows "\<not> (fst instr \<in> {sreg_type RDPSR, sreg_type RDWIM, sreg_type RDTBR} \<or> fst instr = sreg_type RDASR \<and> privileged_ASR (get_operand_w5 (snd instr ! 0)))"
simps_1: fixes x11 :: "nat"   and x12 :: "nat"   and y11 :: "nat"   and y12 :: "nat" shows "(Less x11 x12 = Less y11 y12) = (x11 = y11 \<and> x12 = y12)"
typing_inv_auto_1: fixes \<Gamma> :: "(name \<times> ty) list"   and x :: "name"   and t :: "trm"   and T :: "ty" assumes " \<Gamma> \<turnstile> Lam [x].t : T " obtains xa :: "name" and T\<^sub>1 :: "ty" and t :: "trm" and T\<^sub>2 :: "ty" where "[x].ta__ = [xa].t" and "T = T\<^sub>1\<rightarrow>T\<^sub>2" and "xa \<sharp> \<Gamma>" and " (xa, T\<^sub>1) # \<Gamma> \<turnstile> t : T\<^sub>2 "
vle2: fixes x :: "V"   and y :: "V" assumes "x \<unlhd> y" shows "x \<sqsubseteq> y"
dvd_div_eq_iff: 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 c :: "'a"   and a :: "'a"   and b :: "'a" assumes "class.semidom_divide divide plus minus zero times one"   and "dvd.dvd times c a"   and "dvd.dvd times c b" shows "(divide a c = divide b c) = (a = b)"
flow_attainability_axioms_def: fixes \<Delta> :: "('a, 'b) network_scheme" shows "flow_attainability_axioms \<Delta> \<equiv> (\<forall>x. x \<noteq> sink \<Delta> \<longrightarrow> d_IN (capacity \<Delta>) x \<noteq> \<top> \<or> d_OUT (capacity \<Delta>) x \<noteq> \<top>) \<and> (\<forall>x. \<not> edge \<Delta> x x) \<and> (\<forall>x. \<not> edge \<Delta> x (source \<Delta>))"
degree_smult_le: fixes a :: "'a"   and p :: "'a poly" shows "degree (Polynomial.smult a p) \<le> degree p"
finite_incidence_system_axioms: shows "finite_incidence_system bl \<B>\<^sup>D"
rcons_def: fixes t :: "('c, 'd) FingerTreeStruc"   and a :: "'c \<times> 'd" shows "t \<rhd> a = nrcons t (Tip (fst a) (snd a))"
put_put: fixes x :: "'d \<Longrightarrow> 'e"   and \<sigma> :: "'e"   and v :: "'d"   and u :: "'d" assumes "mwb_lens x" shows "put\<^bsub>x\<^esub> (put\<^bsub>x\<^esub> \<sigma> v) u = put\<^bsub>x\<^esub> \<sigma> u"
\<nu>_simps_3: shows "src \<nu> = src (src r)"
distinct_update: fixes al :: "('c \<times> 'd) list"   and k :: "'c"   and v :: "'d" assumes "distinct (map fst al)" shows "distinct (map fst (AList.update k v al))"
is_SDG_path_valid_SDG_node_1: fixes n :: "'node SDG_node"   and ns :: "'node SDG_node list"   and n' :: "'node SDG_node" assumes "n is-ns\<rightarrow>\<^sub>d* n'" shows "valid_SDG_node n"
uvabc: shows "a\<^sup>2 = u\<^sup>2 - v\<^sup>2 \<and> b\<^sup>2 = 2 * u * v \<and> \<bar>c\<bar> = u\<^sup>2 + v\<^sup>2 \<and> coprime u v"
elims_3: fixes x :: "'d"   and xa :: "'e"   and xb :: "'d set list"   and xc :: "('d, 'e) ta_rule" assumes "\<not> r_matchc x xa xb xc" obtains q :: "'d" and l :: "'e" and Qs :: "'d set list" and qr :: "'d" and lr :: "'e" and qsr :: "'d list" where "x = q" and "xa = l" and "xb = Qs" and "xc = qr \<rightarrow> lr qsr" and "\<not> (q = qr \<and> l = lr \<and> r_match Qs qsr)"
that: fixes P'' :: "ccs"   and P''' :: "ccs" assumes "P \<Longrightarrow>\<^sub>\<tau> P''"   and "P'' \<longmapsto>\<alpha> \<prec> P'''"   and "P''' \<Longrightarrow>\<^sub>\<tau> P'" shows "thesis"
simps_1: fixes bs :: "('a \<times> ('b, 'a) Term.term) list" shows "unify [] bs = Some bs"
exhaust: fixes y :: "jump" obtains x1 :: "label" where "y = Break x1"   | x2 :: "label" where "y = Cont x2"   | "y = Ret"
comm_semiring_1_hom_def: fixes hom :: "'a \<Rightarrow> 'b" shows "comm_semiring_1_hom hom \<equiv> comm_monoid_mult_hom hom \<and> zero_hom hom"
DisjointPaths_extend: fixes P :: "'a list" assumes "v0 \<leadsto>P\<leadsto> v1"   and "\<And>xs v. \<lbrakk>xs \<in> paths; xs \<noteq> P; v \<in> set xs; v \<in> set P\<rbrakk> \<Longrightarrow> v = v0 \<or> v = v1" shows "DisjointPaths G v0 v1 (insert P paths)"
fag_single_free: fixes f :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and z :: "'e"   and i :: "'e \<Rightarrow> 'e"   and a :: "'e"   and n :: "int"   and m :: "int" assumes "fags_cond f z i a"   and "n \<noteq> m" shows "n\<Odot>a\<^bsub>f,i,z\<^esub> \<noteq> m\<Odot>a\<^bsub>f,i,z\<^esub>"
ex_post_efficient': fixes agents :: "'a set"   and alts :: "'b set"   and sds :: "('a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'b pmf"   and R :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "ex_post_efficient_sds agents alts sds"   and "pref_profile_wf agents alts R"   and "x \<prec>[Pareto R] y" shows "pmf (sds R) x = 0"
H2_4: fixes x :: "'a" shows "f1 x + C x = g2 x + u2 x - u2 (T x)"
inner_prod_distrib_right: fixes u :: "'c vec"   and n :: "nat"   and v :: "'c vec"   and w :: "'c vec" assumes "u \<in> carrier_vec n"   and "v \<in> carrier_vec n"   and "w \<in> carrier_vec n" shows "inner_prod u (v + w) = inner_prod u v + inner_prod u w"
check_eqv_sound: fixes \<Sigma> :: "'a list"   and n :: "nat"   and r :: "'a atom rexp"   and s :: "'a atom rexp" assumes "fast.check_eqvRE \<Sigma> n r s"   and "WS1S_Equivalence_Checking.wf \<Sigma> n r"   and "WS1S_Equivalence_Checking.wf \<Sigma> n s" shows "lang \<Sigma> n r = lang \<Sigma> n s"
that: fixes d :: "real" assumes "0 < d"   and "\<And>f x x'. \<lbrakk>f \<in> \<F>; x \<in> S; x' \<in> S; dist x' x < d\<rbrakk> \<Longrightarrow> dist (f x') (f x) < e" shows "thesis"
assms_4: shows "R\<^sup>+\<^sup>+ (f xm) (f ym)"
sndI: fixes x :: "'a \<times> 'b"   and y :: "'a"   and z :: "'b" assumes "x = (y, z)" shows "snd x = z"
row_carrier_vec: fixes i :: "nat"   and nr :: "nat"   and A :: "'b mat"   and nc :: "nat" assumes "i < nr"   and "A \<in> carrier_mat nr nc" shows "Matrix.row A i \<in> carrier_vec nc"
prems_2: shows "ifex_sat (IF v1 t e) = Some a"
surj_map_sum: fixes f :: "'i \<Rightarrow> 'h"   and g :: "'k \<Rightarrow> 'j" assumes "surj f"   and "surj g" shows "surj (map_sum f g)"
brec_construct_witness_def: fixes Qm :: "('b, 'a tree) HashMap.hashmap"   and r :: "('b, 'a) ta_rule" shows "brec_construct_witness Qm r \<equiv> NODE (rhsl r) (map (\<lambda>q. the (hm.lookup q Qm)) (rhsq r))"
map_poly_1: fixes f :: "'b \<Rightarrow> 'a" shows "map_poly f 1 = [:f (1::'b):]"
SUP_const: fixes A :: "'c set"   and f :: "'d \<times> 'e" assumes "A \<noteq> {}" shows "Sup_prod ((\<lambda>i. f) ` A) = f"
assms_3: fixes y :: "'a"   and s :: "'a stream" assumes "y \<in> sset (stl s)"   and "P y (stl s)" shows "P y s"
sum_telescope: fixes f :: "nat \<Rightarrow> 'a"   and i :: "nat" shows "(\<Sum>i\<le>i. f i - f (Suc i)) = f 0 - f (Suc i)"
SigmaI: fixes a :: "'g"   and A :: "'g set"   and b :: "'h"   and B :: "'g \<Rightarrow> 'h set" assumes "a \<in> A"   and "b \<in> B a" shows "(a, b) \<in> Sigma A B"