hcju/isabelleps huggingface dataset:
branch_instr_sub1_def: fixes instr_name :: "sparc_operation"   and s :: "'b sparc_state" shows "branch_instr_sub1 instr_name s \<equiv> let n_val = get_icc_N (cpu_reg s PSR); z_val = get_icc_Z (cpu_reg s PSR); v_val = get_icc_V (cpu_reg s PSR) in Let (get_icc_C (cpu_reg s PSR)) (eval_icc instr_name n_val z_val v_val)"
H_eq_1: fixes s :: "'s" shows "recurrent s = (H s s = 1)"
normal_set_mult_subgroup: shows "subgroup (H <#> S) G"
proper_design_axioms: shows "proper_design \<V>\<^sup>+ \<B>\<^sup>+"
simps_4: fixes Vs :: "char list list"   and e\<^sub>1 :: "char list exp"   and bop :: "bop"   and e\<^sub>2 :: "char list exp" shows "compE\<^sub>1 Vs (e\<^sub>1 \<guillemotleft>bop\<guillemotright> e\<^sub>2) = compE\<^sub>1 Vs e\<^sub>1 \<guillemotleft>bop\<guillemotright> compE\<^sub>1 Vs e\<^sub>2"
assms: fixes x :: "'a" assumes "x \<in> set xs" shows "\<exists>y. f x y"
VLambda_eq_D2: fixes A :: "V"   and f :: "V \<Rightarrow> V"   and g :: "V \<Rightarrow> V"   and x :: "V" assumes "VLambda A f = VLambda A g"   and "x \<in> elts A" shows "f x = g x"
assms_3: shows "\<integral>\<^sup>+x\<in>A. ennreal (f x)\<partial>M < \<infinity>"
adds_term_pprod_def: shows "adds_term_pprod = pprod.adds_term"
pfinite_gpv_stop: fixes \<I> :: "('a, 'b) \<I>"   and gpv :: "('c, 'a, 'b) gpv" assumes "\<I> \<turnstile>g gpv \<surd>" shows "pfinite_gpv (stop_\<I> \<I>) (gpv_stop gpv) = pfinite_gpv \<I> gpv"
coprime_0_1: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and p :: "'a" assumes "class.comm_semiring_1 times one plus zero" shows "comm_monoid_mult.coprime times one p zero = dvd.dvd times p one"
SG: fixes \<phi> :: "('sf, 'sc, 'sz) formula list \<times> ('sf, 'sc, 'sz) formula list" assumes "List.member B \<phi>" shows "seq_sem I \<phi> = UNIV"
trans_sym: fixes r :: "'a"   and s :: "'a"   and t :: "'a" assumes "r = s"   and "t = s" shows "r = t"
continuous_on_empty: fixes f :: "'c \<Rightarrow> 'd" shows "continuous_on {} f"
assms: shows "\<alpha>\<^sub>0 (db\<^sub>s\<^sub>s\<^sub>t A I) n \<noteq> \<alpha>\<^sub>0 (db\<^sub>s\<^sub>s\<^sub>t (A @ B) I) n"
simps_2: fixes x :: "'b"   and y :: "'b"   and xs :: "'b list" shows "remove1 x (y # xs) = (if x = y then xs else y # remove1 x xs)"
prefs_wf: fixes i :: "'agent" assumes "i \<in> agents" shows "finite_total_preorder_on alts (R i)"
exhaust: fixes y :: "'b list" obtains "y = []"   | x21 :: "'b" and x22 :: "'b list" where "y = x21 # x22"
rep_eq: fixes x :: "'a"   and xa :: "'b"   and xb :: "'a \<Rightarrow>\<^sub>0 'b" shows "lookup (Poly_Mapping.update x xa xb) = (lookup xb)(x := xa)"
sys_block_sizes_obtain_bl: fixes x :: "int" assumes "x \<in> pbd.sys_block_sizes" shows "\<exists>bl\<in>#\<B> + mset_set {insert x g |g. g \<in> \<G>}. int (card bl) = x"
fih: fixes x :: "'a" assumes "x \<in> carrier A" shows "f (i x) = h x"
mult_scaleQ_left: fixes a :: "'a"   and x :: "'a['b, 'd]"   and y :: "'a['c, 'd]" shows "(a *\<^sub>Q x) \<^bold>\<cdot> y = a *\<^sub>Q x \<^bold>\<cdot> y"
dvd_if_abs_eq: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and l :: "'a"   and k :: "'a" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one"   and "abs l = abs k" shows "dvd.dvd times l k"
negneg_D: fixes H :: "fmla set"   and A :: "fmla" assumes "H \<turnstile> neg (neg A)" shows "H \<turnstile> A"
add_def: fixes R :: "('c, 'd) ring_scheme"   and a :: "'c"   and p\<^sub>1 :: "'c point"   and p\<^sub>2 :: "'c point" assumes "ell_field R" shows "ell_field.add R a p\<^sub>1 p\<^sub>2 = (case p\<^sub>1 of Infinity \<Rightarrow> p\<^sub>2 | Point x\<^sub>1 y\<^sub>1 \<Rightarrow> case p\<^sub>2 of Infinity \<Rightarrow> p\<^sub>1 | Point x\<^sub>2 y\<^sub>2 \<Rightarrow> if x\<^sub>1 = x\<^sub>2 then if y\<^sub>1 = \<ominus>\<^bsub>R\<^esub> y\<^sub>2 then Infinity else let l = (\<guillemotleft>3\<guillemotright>\<^bsub>R\<^esub> \<otimes>\<^bsub>R\<^esub> x\<^sub>1 [^]\<^bsub>R\<^esub> 2 \<oplus>\<^bsub>R\<^esub> a) \<oslash>\<^bsub>R\<^esub> (\<guillemotleft>2\<guillemotright>\<^bsub>R\<^esub> \<otimes>\<^bsub>R\<^esub> y\<^sub>1); x\<^sub>3 = l [^]\<^bsub>R\<^esub> 2 \<ominus>\<^bsub>R\<^esub> \<guillemotleft>2\<guillemotright>\<^bsub>R\<^esub> \<otimes>\<^bsub>R\<^esub> x\<^sub>1 in Point x\<^sub>3 (\<ominus>\<^bsub>R\<^esub> y\<^sub>1 \<ominus>\<^bsub>R\<^esub> l \<otimes>\<^bsub>R\<^esub> (x\<^sub>3 \<ominus>\<^bsub>R\<^esub> x\<^sub>1)) else let l = (y\<^sub>2 \<ominus>\<^bsub>R\<^esub> y\<^sub>1) \<oslash>\<^bsub>R\<^esub> (x\<^sub>2 \<ominus>\<^bsub>R\<^esub> x\<^sub>1); x\<^sub>3 = l [^]\<^bsub>R\<^esub> 2 \<ominus>\<^bsub>R\<^esub> x\<^sub>1 \<ominus>\<^bsub>R\<^esub> x\<^sub>2 in Point x\<^sub>3 (\<ominus>\<^bsub>R\<^esub> y\<^sub>1 \<ominus>\<^bsub>R\<^esub> l \<otimes>\<^bsub>R\<^esub> (x\<^sub>3 \<ominus>\<^bsub>R\<^esub> x\<^sub>1)))"
assms_1: shows "x @ y <\<^sub>v u @ v"
length_pos_if_in_set: fixes x :: "'b"   and xs :: "'b list" assumes "x \<in> set xs" shows "0 < length xs"
span_image_scale: fixes S :: "'b set"   and c :: "'b \<Rightarrow> complex" assumes "finite S"   and "\<And>x. x \<in> S \<Longrightarrow> c x \<noteq> 0" shows "cspan ((\<lambda>x. c x *\<^sub>C x) ` S) = cspan S"
L_deltas: fixes w :: "'a list"   and s :: "'s" shows "L (fold delta w s) = fold Deriv w (L s)"
contour_integrable_subpath: fixes f :: "complex \<Rightarrow> complex"   and g :: "real \<Rightarrow> complex"   and u :: "real"   and v :: "real" assumes "f contour_integrable_on g"   and "valid_path g"   and "u \<in> {0..1}"   and "v \<in> {0..1}" shows "f contour_integrable_on subpath u v g"
Var\<^sub>1: fixes ls :: "val list"   and i :: "nat"   and v :: "val"   and P :: "nat exp prog"   and h :: "heap"   and sh :: "sheap" assumes "ls ! i = v"   and "i < length ls" shows "P \<turnstile>\<^sub>1 \<langle>Var i,(h, ls, sh)\<rangle> \<Rightarrow> \<langle>Val v,(h, ls, sh)\<rangle>"
problem_plan_bound_works: fixes PROB :: "(('d, bool) fmap \<times> ('d, bool) fmap) set"   and s :: "('d, bool) fmap"   and as :: "(('d, bool) fmap \<times> ('d, bool) fmap) list" assumes "finite PROB"   and "s \<in> valid_states PROB"   and "as \<in> valid_plans PROB" shows "\<exists>as'. exec_plan s as = exec_plan s as' \<and> subseq as' as \<and> length as' \<le> problem_plan_bound PROB"
induct: fixes P :: "'a common_primitive match_expr \<Rightarrow> bool"   and a0 :: "'a common_primitive match_expr" assumes "P MatchAny"   and "\<And>uu_ uv_. P (Match (Dst (IpAddrRange uu_ uv_)))"   and "\<And>uw_. P (Match (Dst (IpAddr uw_)))"   and "\<And>ux_ uy_. P (Match (Dst (IpAddrNetmask ux_ uy_)))"   and "\<And>v. P (Match (Src v))"   and "\<And>v. P (Match (IIface v))"   and "\<And>v. P (Match (OIface v))"   and "\<And>v. P (Match (Prot v))"   and "\<And>v. P (Match (Src_Ports v))"   and "\<And>v. P (Match (Dst_Ports v))"   and "\<And>v. P (Match (MultiportPorts v))"   and "\<And>v. P (Match (L4_Flags v))"   and "\<And>v. P (Match (CT_State v))"   and "\<And>v. P (Match (Extra v))"   and "\<And>va_. P (MatchNot (Match (Dst va_)))"   and "\<And>v. P (MatchNot (Match (Src v)))"   and "\<And>v. P (MatchNot (Match (IIface v)))"   and "\<And>v. P (MatchNot (Match (OIface v)))"   and "\<And>v. P (MatchNot (Match (Prot v)))"   and "\<And>v. P (MatchNot (Match (Src_Ports v)))"   and "\<And>v. P (MatchNot (Match (Dst_Ports v)))"   and "\<And>v. P (MatchNot (Match (MultiportPorts v)))"   and "\<And>v. P (MatchNot (Match (L4_Flags v)))"   and "\<And>v. P (MatchNot (Match (CT_State v)))"   and "\<And>v. P (MatchNot (Match (Extra v)))"   and "\<And>m1 m2. \<lbrakk>P m1; P m2\<rbrakk> \<Longrightarrow> P (MatchAnd m1 m2)"   and "\<And>vc_ vd_. P (MatchNot (MatchAnd vc_ vd_))"   and "\<And>ve_. P (MatchNot (MatchNot ve_))"   and "P (MatchNot MatchAny)" shows "P a0"
simps_1: fixes thy :: "((String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)) \<times> Core.term set"   and uu :: "(variable \<times> typ) list"   and uv :: "variable set"   and Hs :: "Core.term list"   and t :: "Core.term"   and Tis :: "((variable \<times> String.literal set) \<times> typ) list" shows "replay' thy uu uv Hs (PAxm t Tis) = (if inst_ok thy Tis \<and> term_ok thy t then if t \<in> axioms thy then Some (forall_intro_vars (subst_typ' Tis t) []) else None else None)"
cases: fixes a1 :: "Var list \<times> (int \<times> Instr) list \<times> int"   and a2 :: "int"   and a3 :: "Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list"   and a4 :: "nat"   and a5 :: "(Addr \<times> Class \<times> (Field \<times> Val) list) list"   and a6 :: "Val" assumes "(a1, a2, a3, a4, a5, a6) \<in> Exec" obtains M :: "Var list \<times> (int \<times> Instr) list \<times> int" and l :: "int" and v :: "Val" and os :: "Val list" and s :: "(Var \<times> Val) list" and h :: "(Addr \<times> Class \<times> (Field \<times> Val) list) list" where "a1 = Ma__" and "a2 = la__" and "a3 = (va__ # os, sa__, ha__)" and "a4 = 1" and "a5 = ha__" and "a6 = va__" and "get_ins Ma__ la__ = Some vreturn"   | M :: "Var list \<times> (int \<times> Instr) list \<times> int" and l :: "int" and s :: "Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list" and n :: "nat" and ll :: "int" and t :: "Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list" and m :: "nat" and h :: "(Addr \<times> Class \<times> (Field \<times> Val) list) list" and v :: "Val" and k :: "nat" where "a1 = M" and "a2 = l" and "a3 = s" and "a4 = k" and "a5 = h" and "a6 = v" and "(M, l, s, n, ll, t) \<in> Step" and "(M, ll, t, m, h, v) \<in> Exec" and "k = max n m + 1"
g1: shows "R_Chop(r,r3,Abs_real_int (left r4, right r2))"
wp_under_wlpD: fixes prog :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and P :: "'a \<Rightarrow> real" assumes "wp_under_wlp prog"   and "unitary P" shows "wp prog P \<tturnstile> wlp prog P"
iterate_add_Numeral1: fixes n :: "nat" shows "iterate_add n Numeral1 = of_nat n"
heads_le_length: fixes xs :: "bool list" shows "heads xs \<le> length xs"
toCard_inj: fixes A :: "'a set"   and r :: "('b \<times> 'b) set"   and x :: "'a"   and y :: "'a" assumes "|A| \<le>o r"   and "Card_order r"   and "x \<in> A"   and "y \<in> A" shows "(toCard A r x = toCard A r y) = (x = y)"
arg_of_real: fixes x :: "real" shows "arg (cor x) = (if x < 0 then pi else 0)"
ex_bij_betw_nat_finite: fixes M :: "'a set" assumes "finite M" shows "\<exists>h. bij_betw h {0..<card M} M"
cUN_cUN_flatten: fixes C :: "'e \<Rightarrow> 'd cset"   and B :: "'f \<Rightarrow> 'e cset"   and A :: "'f cset" shows "cUnion (cimage C (cUnion (cimage B A))) = cUnion (cimage (\<lambda>y. cUnion (cimage C (B y))) A)"
conj_one_right: fixes x :: "'c" shows "x AND - (1::'c) = x"
simps_4: fixes x :: "('c, 'd) Node"   and xs :: "('c, 'd) Node list"   and v :: "('c, 'd) Node" shows "app3 (Single x) xs (Single v) = nlcons x (lconsNlist xs (Single v))"
ran_map_add: fixes m1 :: "'a \<Rightarrow> 'b option"   and m2 :: "'a \<Rightarrow> 'b option" assumes "dom m1 \<inter> dom m2 = {}" shows "ran (m1 ++ m2) = ran m1 \<union> ran m2"
unat_arith_simps_2: fixes a :: "'a word"   and b :: "'a word" shows "(a < b) = (unat a < unat b)"
simps_1: shows "htuple 0 = \<langle>0, 0\<rangle>"
sum_list_map_filter_sum: fixes f :: "'b \<Rightarrow> 'a"   and g :: "'b \<Rightarrow> bool"   and xs :: "'b list" shows "sum_list (map f (filter g xs)) + sum_list (map f (filter (Not \<circ> g) xs)) = sum_list (map f xs)"
closed_imp_fip_compact: fixes S :: "'b set"   and \<F> :: "'b set set" assumes "closed S"   and "\<And>T. T \<in> \<F> \<Longrightarrow> compact T"   and "\<And>\<F>'. \<lbrakk>finite \<F>'; \<F>' \<subseteq> \<F>\<rbrakk> \<Longrightarrow> S \<inter> \<Inter> \<F>' \<noteq> {}" shows "S \<inter> \<Inter> \<F> \<noteq> {}"
jonsson1b: shows "(\<forall>x y. x \<rhd> y = (x \<rhd> (1::'a)) \<cdot> y) = (\<forall>x y. x \<cdot> y = x \<rhd> (1::'a) \<rhd> y)"
simps_1: fixes uu :: "'b option" shows "None + uu = None"
loop_stepC: fixes P\<^sub>1 :: "'a set set"   and P\<^sub>2 :: "'a set set"   and B\<^sub>1 :: "'a set"   and B\<^sub>2 :: "'a set"   and V :: "'a set"   and u :: "'a" assumes "inv\<^sub>3 P\<^sub>1 P\<^sub>2 B\<^sub>1 B\<^sub>2 V"   and "B\<^sub>1 \<noteq> {}"   and "u \<in> V \<inter> S"   and "W B\<^sub>1 + w u \<le> real c" shows "inv\<^sub>3 P\<^sub>1 P\<^sub>2 (B\<^sub>1 \<union> {u}) B\<^sub>2 (V - {u})"
nat_diff_distrib: fixes z' :: "int"   and z :: "int" assumes "0 \<le> z'"   and "z' \<le> z" shows "nat (z - z') = nat z - nat z'"
lm059: fixes a :: "('a \<times> 'b set) set"   and G :: "'b set"   and N :: "'a set"   and bids :: "'a \<times> 'b set \<Rightarrow> 'c" assumes "a \<in> allAllocations N G" shows "sum (summedBidVector bids N G) a = sum snd (summedBid bids ` (N \<times> (Pow G - {{}}) \<inter> a))"
R\<^sub>1_def: shows "rs.R\<^sub>1 = \<lbrakk>\<lbrakk>tab\<^sub>1 r\<rbrakk>\<rbrakk>"
simps_11: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and x2 :: "'a"   and y2 :: "'b" shows "rel_option R (Some x2) (Some y2) = R x2 y2"
plus_def: fixes A :: "'b tensor"   and B :: "'b tensor" shows "A + B = (if dims A = dims B then plus_base A B else undefined)"
reachable_mono: fixes H :: "('a, 'b) pre_digraph"   and u :: "'a"   and v :: "'a" assumes "u \<rightarrow>\<^sup>*\<^bsub>H\<^esub> v"   and "subgraph H (G \<restriction> {w. ua__ \<rightarrow>\<^sup>* w})" shows "u \<rightarrow>\<^sup>*\<^bsub>G \<restriction> {w. ua__ \<rightarrow>\<^sup>* w}\<^esub> v"
card_eq_1_iff: fixes A :: "'c set" shows "(card A = 1) = (\<exists>x. A = {x})"
TCC: shows "T --` C \<Delta> C \<in> null_sets M"
c2_2: shows "B2 \<in> carrier_mat n1 nc2"
conv_galois_2: fixes y :: "'a"   and x :: "'a"   and z :: "'a" shows "(y ; x \<le> z) = (y \<le> - (- z ; x\<^sup>\<smile>))"
powsum_split: fixes x :: "'a"   and m :: "nat"   and i :: "nat"   and n :: "nat" shows "x\<^bsub>m\<^esub>\<^bsup>i + Suc n\<^esup> = x\<^bsub>m\<^esub>\<^bsup>i\<^esup> + x\<^bsub>m + Suc i\<^esub>\<^bsup>n\<^esup>"
d2: fixes x :: "real"   and y :: "real" assumes "g x = g y"   and "v \<le> x"   and "x \<le> u"   and "v \<le> y"   and "y \<le> u" shows "x = y"
nb\<^sub>2: shows "\<forall>v. (\<Psi>\<^sub>I\<^sub>+) v \<noteq> None \<longrightarrow> the ((\<Psi>\<^sub>I\<^sub>+) v) \<in> set (the (range_of \<Psi> v))"
eq2: shows "indets (monomial c t + p) = keys t \<union> indets p"
m_def: shows "Mapping.lookup a2_map (tp - len) = Some m"
top_conj_2: fixes P :: "bool"   and x :: "'a" shows "(P \<and> \<top> x) = P"
le_iff_sup: fixes x :: "'b"   and y :: "'b" shows "(x \<le> y) = (x \<sqinter> y = y)"
simps: fixes uu :: "int poly"   and uv :: "rat"   and r :: "rat" shows "rai_ub (uu, uv, r) = r"
finite_vote_set: fixes r :: "nat"   and v_hist :: "nat \<Rightarrow> 'c \<Rightarrow> 'd option"   and S :: "'c set" assumes "\<forall>r'\<ge>r. v_hist r' = Map.empty"   and "finite S" shows "finite (vote_set v_hist S)"
cases: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and a1 :: "ty"   and a2 :: "ty" assumes "P \<turnstile> a1 \<le> a2" obtains T :: "ty" where "a1 = T" and "a2 = T"   | C :: "char list" and D :: "char list" where "a1 = Class Ca__" and "a2 = Class D" and "P \<turnstile> Path Ca__ to D unique"   | C :: "char list" where "a1 = NT" and "a2 = Class C"
has_next_def: fixes g :: "('d, 'c) generator" shows "list.has_next g = fst (generator g)"
leD: fixes y :: "'c"   and x :: "'c" assumes "y \<le> x" shows "\<not> x < y"
simps: fixes ord :: "'c \<Rightarrow> 'c \<Rightarrow> bool" shows "finite_chains ord = (\<forall>x. Complete_Partial_Order.chain ord x \<longrightarrow> finite x)"
assms_1: shows "0\<^sub>h \<in> circline_set H"
Start_nsys_xcpts1_7: shows "Start \<noteq> NoSuchMethodError"
eq_dfa_error: fixes n :: "nat"   and ks :: "int list"   and l :: "int"   and bss :: "bool list list" shows "\<not> dfa_accepting (eq_dfa n ks l) (dfa_steps (eq_dfa n ks l) (length (snd (dioph_dfs n ks l))) bss)"
continuous_on_closed_Union: fixes I :: "'b set"   and U :: "'b \<Rightarrow> 'c set"   and f :: "'c \<Rightarrow> 'd" assumes "finite I"   and "\<And>i. i \<in> I \<Longrightarrow> closed (U i)"   and "\<And>i. i \<in> I \<Longrightarrow> continuous_on (U i) f" shows "continuous_on (\<Union> (U ` I)) f"
all_nat_split_at: fixes k :: "'g"   and P :: "'g \<Rightarrow> bool" assumes "\<forall>i<k. P i"   and "P k"   and "\<forall>i>k. P i" shows "\<forall>i. P i"
chamberD_simplex: fixes S :: "'b set"   and x :: "'b set set" assumes "CoxeterComplex S"   and "SimplicialComplex.maxsimp (CoxeterComplex.TheComplex S) x" shows "x \<in> CoxeterComplex.TheComplex S"
on_discover_refine: fixes u :: "'a"   and v :: "'a"   and s0 :: "('a, 'c) state_scheme"   and s :: "('a, 'c) state_scheme"   and ui :: "'a"   and vi :: "'a"   and si :: "('a, 'd) simple_state_scheme"   and si' :: "('a, 'd) simple_state_scheme"   and s' :: "('a, 'c) state_scheme" assumes "a.a.gen_dfs.pre_discover u v s0 s"   and "(ui, u) \<in> Id"   and "(vi, v) \<in> Id"   and "(si, s) \<in> \<langle>ES\<rangle>simple_state_rel"   and "(si', s') \<in> \<langle>ES\<rangle>simple_state_rel"   and "nf_inres (gbs_discover a.a.gbs u v s) s'" shows "on_discover parami ui vi si' \<le> \<Down> ES (on_discover param u v s')"
COMB_OPT2': fixes x :: "nat"   and y :: "nat"   and \<sigma> :: "nat list" assumes "x \<noteq> y"   and "set \<sigma> \<subseteq> {x, y}" shows "T\<^sub>p_on_rand' (COMB []) (fst (COMB []) [x, y] \<bind> (\<lambda>is. return_pmf ([x, y], is))) \<sigma> \<le> 16 / 10 * real (T\<^sub>p_opt [x, y] \<sigma>) + 16 / 10"
simps: fixes G :: "'c set"   and N :: "'d set" shows "possible_allocations_rel G N = \<Union> {injections Y N |Y. Y \<in> all_partitions G}"
\<omega>_simps_4: shows "cod \<r>\<^sup>-\<^sup>1[r] = r \<star> src r"
c_fst_le_arg: fixes u :: "nat" shows "c_fst u \<le> u"
BSI_implies_SI_for_modified_view: fixes \<V>' :: "'e V_rec" assumes "BSI \<V>' Tr\<^bsub>(induceES SES)\<^esub>"   and "\<V>' = \<lparr>V = V\<^bsub>\<V>'\<^esub> \<union> N\<^bsub>\<V>'\<^esub>, N = {}, C = C\<^bsub>\<V>'\<^esub>\<rparr>" shows "SI \<V>' Tr\<^bsub>(induceES SES)\<^esub>"
impE: fixes cid :: "'nid"   and t :: "nat \<Rightarrow> cnf"   and t' :: "nat \<Rightarrow> 'ND"   and n :: "nat"   and \<gamma> :: "(nat \<Rightarrow> 'ND) \<Rightarrow> nat \<Rightarrow> bool"   and \<gamma>' :: "(nat \<Rightarrow> 'ND) \<Rightarrow> nat \<Rightarrow> bool" assumes "eval cid t t' n (\<gamma> \<longrightarrow>\<^sup>b \<gamma>')" shows "eval cid t t' n \<gamma> \<longrightarrow> eval cid t t' n \<gamma>'"
affine_closed: fixes S :: "'a set" assumes "affine S" shows "closed S"
DAimplieMR: fixes p :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators list"   and x :: "int \<times> (int \<times> int \<times> protocol) \<times> (int \<times> int \<times> protocol) \<times> DummyContent" assumes "DenyAll \<in> set p" shows "applied_rule_rev Cp x p \<noteq> \<bottom>"
relabel_preserve_other: fixes u :: "nat"   and v :: "nat" assumes "u \<noteq> v" shows "relabel_effect f (relabel_effect f l u) u v = relabel_effect f l u v"
interleave_ccFromList: fixes xs :: "Vars.var list"   and ys :: "Vars.var list"   and zs :: "Vars.var list" assumes "xs \<in> ys \<otimes> zs" shows "ccFromList xs = ccFromList ys \<squnion> ccFromList zs \<squnion> set ys G\<times> set zs"
SbisL_update: fixes cl :: "('test, 'atom, 'choice) cmd list"   and dl :: "('test, 'atom, 'choice) cmd list"   and c' :: "('test, 'atom, 'choice) cmd"   and d' :: "('test, 'atom, 'choice) cmd"   and n :: "nat" assumes "SbisL cl dl"   and "c' \<approx>s d'" shows "SbisL (cl[n := c']) (dl[n := d'])"
trans: fixes r :: "'k"   and s :: "'k"   and t :: "'k" assumes "r = s"   and "s = t" shows "r = t"
exhaust: fixes y :: "('a, 'b) L_action" obtains x1 :: "'a" where "y = L_action.Act x1"   | x2 :: "'b" where "y = Eff x2"
TermEquivClGen_def: fixes T :: "('a, 'b) Axioms"   and A :: "'a"   and e :: "('a, 'b) Expression"   and B :: "'a" shows "TermEquivClGen T A e B \<equiv> {e'. Vx : A \<turnstile> e'\<equiv>e : B \<in> Axioms.Theory T}"
lfp_exp_def: fixes t :: "('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real" shows "lfp_exp t = Inf_exp {P. sound P \<and> t P \<tturnstile> P}"
put_is_create: fixes x :: "'d \<Longrightarrow> 'e"   and \<sigma> :: "'e"   and v :: "'d" assumes "pbij_lens x" shows "put\<^bsub>x\<^esub> \<sigma> v = create\<^bsub>x\<^esub> v"
exec: shows "exec_moves_d P t (e # es) h (stk @ [v], loc, length (compE2 e) + pc, xcp) ta h' (stk', loc', pc', xcp')"
set_of_map_def: fixes m :: "'c \<Rightarrow> 'd option" shows "set_of_map m = {(k, v) |k v. m k = Some v}"
a: shows "A \<union> B1 = A \<union> B2"
assms_3: fixes i :: "nat" shows "f i \<in> S"
assms_5: fixes x :: "'a"   and ws :: "'a list"   and s :: "'b"   and r :: "'b" assumes "I x"   and "\<forall>x\<in>set ws. I x"   and "P ws (f x s) r" shows "P (x # ws) s r"
euler_C_def: shows "euler_C TYPE('a) = sqrt (real DIM('a)) * (B' * (B + 1) / 2)"
neg_le_divide_eq: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and c :: "'b"   and a :: "'b"   and b :: "'b" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less c zero" shows "less_eq a (divide b c) = less_eq b (times a c)"
l7_9: fixes A :: "'p"   and P :: "'p"   and X :: "'p"   and Q :: "'p" assumes "A Midpoint P X"   and "A Midpoint Q X" shows "P = Q"
edges_are_in_V_1: fixes v :: "'c"   and w :: "'c" assumes "v \<rightarrow>\<^bsub>T\<^esub> w" shows "v \<in> V\<^bsub>T\<^esub>"
merge_single: fixes x :: "'a"   and f :: "'a \<Rightarrow> 'm" shows "merge (csingle x) f = f x"
irreducibleD: fixes C :: "('s \<times> 's option) set"   and a :: "'s \<times> 's option"   and b :: "'s \<times> 's option" assumes "C \<in> UNIV // KN.communicating"   and "a \<in> C"   and "b \<in> C" shows "(a, b) \<in> KN.communicating"
ds_ord_refl: fixes x :: "'a" shows "domain_semigroup.ds_ord d (\<cdot>) x x"
mset_eq_imp_distinct_iff: fixes xs :: "'b list"   and ys :: "'b list" assumes "mset xs = mset ys" shows "distinct xs = distinct ys"
rcs_eq1: fixes H :: "'a set"   and a :: "'a"   and x :: "'a" assumes "G \<guillemotright> H "   and "a \<in> carrier G"   and "x \<in> H \<bullet> a" shows "H \<bullet> a = H \<bullet> x"
Vfrom_mono2: fixes a :: "V"   and i :: "V"   and j :: "V" shows "Vfrom a i \<le> Vfrom a (i \<squnion> j)"
fBex_mono: fixes P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool"   and S :: "'a fset" assumes "P \<le> Q" shows "fBex S P \<le> fBex S Q"
Node_def: shows "Node \<equiv> \<lambda>x1 x2 x3. ctor_tree (BNF_Composition.id_bnf (x1, x2, x3))"
q_def: shows "q = \<lfloor>t / T\<rfloor>"
expr_typing_pairE: fixes \<Gamma> :: "nat \<Rightarrow> pdf_type"   and e1 :: "expr"   and e2 :: "expr"   and t :: "pdf_type" assumes "\<Gamma> \<turnstile> <e1 ,  e2> : t" obtains t1 :: "pdf_type" and t2 :: "pdf_type" where "t = PRODUCT t1 t2" and "\<Gamma> \<turnstile> e1 : t1" and "\<Gamma> \<turnstile> e2 : t2"
right_adjoint_simps_1: fixes f :: "'a" assumes "C.is_left_adjoint f" shows "C.ide f\<^sup>*\<^sup>C"