hcju/isabelleps huggingface dataset:
exec_step_ind_Getfield_Null: fixes stk :: "val list"   and F :: "char list"   and C :: "char list"   and P :: "jvm_prog"   and h :: "heap"   and loc :: "val list"   and C\<^sub>0 :: "char list"   and M\<^sub>0 :: "char list"   and pc :: "nat"   and ics :: "init_call_status"   and frs :: "frame list"   and sh :: "sheap" assumes "hd stk = Null" shows "exec_step_ind (StepI (Getfield F C)) P h stk loc C\<^sub>0 M\<^sub>0 pc ics frs sh (\<lfloor>addr_of_sys_xcpt NullPointer\<rfloor>, h, (stk, loc, C\<^sub>0, M\<^sub>0, pc, ics) # frs, sh)"
sound_to_valid: fixes \<phi> :: "('sf, 'sc, 'sz) formula" assumes "sound ([], [], [\<phi>])" shows "valid \<phi>"
assms_2: shows "\<I>,\<I>' \<turnstile>\<^sub>C conv \<sim> conv'"
pmat_extension_carrier: fixes m :: "'a mat" shows "ps2_P.pmat_extension m \<in> carrier_mat ps2_P.d0 ps2_P.d0"
assms: shows "\<guillemotleft>\<mu> : src e\<^sub>0 \<rightarrow> src e\<^sub>1\<guillemotright>"
elims: fixes x :: "nat"   and xa :: "nat"   and y :: "'a" assumes "geod x xa = y" obtains "x = 0" and "xa = 0" and "y = x0"   | m :: "nat" where "x = 0" and "xa = Suc m__" and "y = y0"   | n :: "nat" and m :: "nat" where "x = Suc nb__" and "xa = 2 * ma__" and "y = geod nb__ ma__"   | n :: "nat" and m :: "nat" where "x = Suc n" and "xa = Suc (2 * m)" and "y = middle (geod n m) (geod n (Suc m))"
char_eq_0_iff: fixes a :: "'a \<Rightarrow> complex" shows "(principal_char (Characters G) a = 0) = (a \<notin> carrier (Characters G))"
Complexs_0: shows "(0::'b) \<in> \<complex>"
map_try_spmf: fixes f :: "'b \<Rightarrow> 'a"   and p :: "'b spmf"   and q :: "'b spmf" shows "map_spmf f (TRY p ELSE q) = TRY map_spmf f p ELSE map_spmf f q"
RECT_eq_REC: fixes body :: "('c \<Rightarrow> 'b) \<Rightarrow> 'c \<Rightarrow> 'b"   and x :: "'c" assumes "REC\<^sub>T body x \<noteq> top" shows "REC\<^sub>T body x = REC body x"
that: fixes r :: "int"   and N :: "nat" assumes "r \<in> carrier (residue_ring (p ^ n))"   and "\<And>m. N < m \<Longrightarrow> s m n = r" shows "thesis"
mset_subset_eq_add_right: fixes B :: "'f multiset"   and A :: "'f multiset" shows "B \<subseteq># A + B"
lub_belowI: fixes S :: "'a set"   and x :: "'a" assumes "\<And>y. y \<in> S \<Longrightarrow> y \<sqsubseteq> x" shows "lub S \<sqsubseteq> x"
M_def: shows "M = (1::'a) / v $ i * (bnd + (1::'a))"
m_in: fixes i :: "nat" assumes "i \<in> N" shows "m i \<in> (\<Union>l. M (V l))"
times_divide_eq_right: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "class.division_ring divide inverse one times plus zero minus uminus" shows "times a (divide b c) = divide (times a b) c"
open_imp_subset: fixes U :: "'a set" assumes "is_open U" shows "U \<subseteq> X"
one_closed: fixes G :: "('f, 'g) monoid_scheme" assumes "Group.monoid G" shows "\<one>\<^bsub>G\<^esub> \<in> carrier G"
p_red: shows "(red G)\<^sup>*\<^sup>* p r"
delete_aux_eq_Nil_conv: fixes k :: "'c"   and ts :: "('c \<times> 'd) list" shows "(AList.delete_aux k ts = []) = (ts = [] \<or> (\<exists>v. ts = [(k, v)]))"
cil_cons2: fixes w :: "'d list"   and wa :: "'d list"   and \<alpha> :: "'d \<Rightarrow> 'e set \<times> 'e set"   and wb :: "'d list"   and e :: "'d" assumes "w \<in> wa \<otimes>\<^bsub>\<alpha>\<^esub> wb"   and "fst (\<alpha> e) \<inter> mon_pl (map \<alpha> wa) = {}" shows "e # w \<in> wa \<otimes>\<^bsub>\<alpha>\<^esub> e # wb"
sym: fixes s :: "'c"   and t :: "'c" assumes "s = t" shows "t = s"
C_def: fixes A :: "real mat"   and n :: "nat"   and m :: "nat"   and lam :: "complex"   and n_as :: "(nat \<times> complex) list"   and off :: "nat"   and k :: "nat" assumes "spectral_radius_1_jnf_max A n m lam n_as" shows "spectral_radius_1_jnf_max.C m n_as off k = spectral_radius_1_jnf_max.c m / real (spectral_radius_1_jnf_max.K m n_as off k) ^ (m - 1)"
fps_X_power_nth: fixes k :: "nat"   and n :: "nat" shows "fps_nth (fps_X ^ k) n = (if n = k then 1::'a else (0::'a))"
assms_3: shows "natural_transformation (\<cdot>\<^sub>A\<^sub>1) (\<cdot>\<^sub>[\<^sub>A\<^sub>2,\<^sub>B\<^sub>]) F G \<tau>"
StaticDownDynCast: 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 E :: "char list \<Rightarrow> ty option"   and e :: "expr"   and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and a :: "nat"   and Cs :: "char list list"   and C :: "char list"   and Cs' :: "char list list"   and s\<^sub>1 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)" assumes "P,E \<turnstile> \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>ref (a, Cs @ [C] @ Cs'),s\<^sub>1\<rangle>" shows "P,E \<turnstile> \<langle>Cast C e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>ref (a, Cs @ [C]),s\<^sub>1\<rangle>"
lfilter_code_2: fixes P :: "'c \<Rightarrow> bool"   and x :: "'c"   and xs :: "'c llist" shows "lfilter P (LCons x xs) = (if P x then LCons x (lfilter P xs) else lfilter P xs)"
A\<^sub>A_sec: shows "\<forall>x. dma\<^sub>A mem\<^sub>A_of mem\<^sub>1\<^sub>C [\<parallel>\<^sub>1 A\<^sub>A_of A\<^sub>C] x = Low \<and> (x \<notin> mds\<^sub>A_of mds\<^sub>C AsmNoReadOrWrite \<or> x \<in> \<C>\<^sub>A) \<longrightarrow> mem\<^sub>A_of mem\<^sub>1\<^sub>C [\<parallel>\<^sub>1 A\<^sub>A_of A\<^sub>C] x = mem\<^sub>A_of mem\<^sub>2\<^sub>C [\<parallel>\<^sub>2 A\<^sub>A_of A\<^sub>C] x"
split: fixes P :: "'a \<Rightarrow> bool"   and f1 :: "'a"   and f2 :: "'a"   and bool :: "bool" shows "P (case bool of True \<Rightarrow> f1 | False \<Rightarrow> f2) = ((bool = True \<longrightarrow> P f1) \<and> (bool = False \<longrightarrow> P f2))"
E_sup_const: fixes c :: "ennreal"   and s :: "'s" assumes "0 \<le> c" shows "E_sup s (\<lambda>_. c) = c"
in_gamma_update: fixes s :: "char list \<Rightarrow> int"   and S :: "char list \<Rightarrow> 'av"   and i :: "int"   and a :: "'av"   and x :: "char list" assumes "s \<in> \<gamma>\<^sub>f S"   and "i \<in> \<gamma> a" shows "s(x := i) \<in> \<gamma>\<^sub>f (S(x := a))"
simps_3: fixes t :: "Term_Class.term" shows "frees (\<Lambda> t) = frees t"
lubH_is_fixp: fixes cl :: "'b potype"   and f :: "'b \<Rightarrow> 'b"   and H :: "'b set" assumes "CLF cl f"   and "H = {x \<in> pset cl. (x, f x) \<in> order cl}" shows "lub H cl \<in> fix f (pset cl)"
sset_sconst: fixes x :: "'d" shows "sset (sconst x) = {x}"
that: fixes k :: "'u"   and v :: "'v" assumes "max s P = Some (k, v)"   and "(k, v) \<in> rel_of (\<alpha> s) P"   and "\<forall>(k', v')\<in>rel_of (\<alpha> s) P. k' \<le> k" shows "thesis"
del_edge_undirected_degree_plus: fixes g :: "('c, 'd) graph"   and v :: "'c"   and e :: "'d"   and v' :: "'c" assumes "finite (edges g)"   and "(v, e, v') \<in> edges g"   and "(v', e, v) \<in> edges g" shows "degree v (del_unEdge v e v' g) + 1 = degree v g"
simps_2: fixes disc :: "'c \<Rightarrow> bool"   and a :: "'c" shows "has_disc disc (Match a) = disc a"
update_cons_nil_env: fixes x :: "'c"   and opt :: "('a, 'b, 'c) env option"   and b :: "'b"   and es :: "'c \<Rightarrow> ('a, 'b, 'c) env option" shows "update [x] opt (Env b es) = Env b (es(x := opt))"
new_mm: shows "\<lfloor>real d * mahler_measure f\<rfloor> \<le> new_mm"
prems: shows "Vagree \<nu> \<omega> (\<Union>i\<in>{i. Inr i \<in> SIGT ($f x1a x2a)}. FVT (\<sigma> i))"
chain_sucD: fixes X :: "'a set" assumes "chain_on P X {x \<in> A. \<not> Q x}" shows "B.suc X \<subseteq> {x \<in> A. \<not> Q x} \<and> chain_on P (B.suc X) {x \<in> A. \<not> Q x}"
exec_If\<^sub>C_If\<^sub>S\<^sub>E: fixes \<sigma> :: "'a control_state_scheme"   and P :: "'a control_state_scheme \<Rightarrow> bool"   and B\<^sub>1 :: "'a control_state_scheme \<Rightarrow> ('b \<times> 'a control_state_scheme) option"   and B\<^sub>2 :: "'a control_state_scheme \<Rightarrow> ('b \<times> 'a control_state_scheme) option" assumes "\<not> exec_stop \<sigma>" shows "(_if_SECLEAN P B\<^sub>1 B\<^sub>2) \<sigma> = (_if_SE P B\<^sub>1 B\<^sub>2) \<sigma>"
mirror_point_self_inverse: fixes p :: "'a"   and x :: "'a" shows "mirror_point p (mirror_point p x) = x"
Rep_\<I>_inverse: fixes x :: "('d, 'e) \<I>" shows "Abs_\<I> (Rep_\<I> x) = x"
pure_B_conv: fixes g :: "('f \<Rightarrow> 'd) af"   and f :: "('e \<Rightarrow> 'f) af"   and x :: "'e af" shows "pure (\<lambda>g f x. g (f x)) \<diamondop> g \<diamondop> f \<diamondop> x = g \<diamondop> (f \<diamondop> x)"
gcd_impl_primitive_def: fixes G1 :: "'a poly"   and G2 :: "'a poly" shows "gcd_impl_primitive G1 G2 = normalize (primitive_part (fst (subresultant_prs dichotomous_Lazard G1 G2)))"
setops\<^sub>s\<^sub>s\<^sub>t_proj_subset_1: fixes n :: "'c"   and A :: "('c strand_label \<times> ('a, 'b) stateful_strand_step) list" shows "setops\<^sub>s\<^sub>s\<^sub>t (proj_unl n A) \<subseteq> setops\<^sub>s\<^sub>s\<^sub>t (unlabel A)"
refl_r: fixes x :: "'a" shows "x \<sqsubseteq>\<^bsub>r\<^esub> x"
finite_convex_hull_iff_convex_hull_list: fixes Vs :: "'a vec set"   and Vsl' :: "'a vec list" assumes "Vs \<subseteq> carrier_vec n"   and "Vs = set Vsl'" shows "convex_hull Vs = convex_hull_list Vsl'"
drop_replicate: fixes i :: "nat"   and k :: "nat"   and x :: "'b" shows "drop i (replicate k x) = replicate (k - i) x"
iterates_above_Sup_aux: fixes lub :: "'b set \<Rightarrow> 'b"   and leq :: "('b \<times> 'b) set"   and f :: "'b \<Rightarrow> 'b"   and M :: "'b set"   and M' :: "'b set" assumes "bourbaki_witt_fixpoint lub leq f"   and "M \<in> Chains leq"   and "M \<noteq> {}"   and "M' \<in> Chains leq"   and "M' \<noteq> {}"   and "\<And>x. x \<in> M \<Longrightarrow> x \<in> bourbaki_witt_fixpoint.iterates_above lub leq f (lub M') \<or> lub M' \<in> bourbaki_witt_fixpoint.iterates_above lub leq f x" shows "(lub M, lub M') \<in> leq \<or> lub M \<in> bourbaki_witt_fixpoint.iterates_above lub leq f (lub M')"
in_idem: fixes A :: "'b set"   and x :: "'b" assumes "finite A"   and "x \<in> A" shows "max x (Max A) = Max A"
fr_nStore_nReturn3: fixes n :: "nat"   and nSend :: "nat \<Rightarrow> nat \<Rightarrow> 'a Frame list"   and recv :: "nat \<Rightarrow> 'a Frame list"   and nReturn :: "nat \<Rightarrow> nat \<Rightarrow> 'a Frame list"   and nC :: "nat \<Rightarrow> Config"   and nStore :: "nat \<Rightarrow> nat \<Rightarrow> 'a Frame list"   and nGet :: "nat \<Rightarrow> nat \<Rightarrow> nat list"   and t :: "nat"   and k :: "nat" assumes "Cable n nSend recv"   and "\<forall>i<n. FlexRayController (nReturn i) recv (nC i) (nStore i) (nSend i) (nGet i)"   and "DisjointSchedules n nC"   and "IdenticCycleLength n nC"   and "t mod cycleLength (nC k) mem schedule (nC k)"   and "k < n" shows "\<forall>j. j < n \<and> j \<noteq> k \<longrightarrow> nStore j t = nReturn k t"
bound_indices: shows "f 0 = a \<and> f (card X - 1) = c"
step_14: shows "InvariantVarsM (getM state') F0 Vbl"
wf_Ext_foo: shows "wf_mdecl tprg Ext Ext_foo"
append_Cons: fixes x :: "'e"   and xs :: "'e list"   and ys :: "'e list" shows "(x # xs) @ ys = x # xs @ ys"
id_matrix_dim_2: fixes nr :: "nat"   and nc :: "nat" shows "dim_col (id_matrix nr nc) = nc"
map_append: fixes f :: "'b \<Rightarrow> 'a"   and xs :: "'b list"   and ys :: "'b list" shows "map f (xs @ ys) = map f xs @ map f ys"
l_coset_consistent: fixes H :: "'c set"   and G :: "('c, 'd) monoid_scheme"   and h :: "'c"   and I :: "'c set" shows "h <#\<^bsub>G\<lparr>carrier := H\<rparr>\<^esub> I = h <#\<^bsub>G\<^esub> I"
\<omega>_sum_replicate: fixes m :: "nat"   and a :: "V" shows "\<omega>_sum (replicate m a) = \<omega> \<up> a * ord_of_nat m"
vseq_implies_hpar_1: fixes \<nu> :: "'a arrow_of_spans_data"   and \<mu> :: "'a arrow_of_spans_data" assumes "seq \<nu> \<mu>" shows "src \<nu> = src \<mu>"
length21: shows "length \<GG> - 2 + 1 = length \<GG> - 1"
pullback_commutes': fixes f :: "'a"   and g :: "'a" assumes "cospan f g" shows "f \<cdot> \<p>\<^sub>1[f, g] = g \<cdot> \<p>\<^sub>0[f, g]"
star_1r: fixes x :: "'a" shows "x\<^sup>\<star> ; x \<le> x\<^sup>\<star>"
lfilter_empty_conv: fixes P :: "'c \<Rightarrow> bool"   and xs :: "'c llist" shows "(lfilter P xs = LNil) = (\<forall>x\<in>lset xs. \<not> P x)"
invff: fixes x :: "'b" assumes "x \<in> Field s" shows "the_inv_into (Field s) f (f x) = x"
simps_3: fixes x :: "'a"   and b :: "'a"   and A :: "'a tree"   and B :: "'a tree"   and CD :: "'a tree" assumes "x < b" shows "splay x \<langle>\<langle>A, x, B\<rangle>, b, CD\<rangle> = \<langle>A, x, \<langle>B, b, CD\<rangle>\<rangle>"
map_insertl: fixes f :: "'b \<Rightarrow> 'a"   and i :: "nat"   and x :: "'b"   and xs :: "'b list" shows "map f (insertl i x xs) = insertl i (f x) (map f xs)"
atMost_def: fixes less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and u :: "'g" shows "ord.atMost less_eq u \<equiv> {x. less_eq x u}"
bounded_linear_blinfun_matrix: fixes j :: "'d"   and i :: "'e" shows "bounded_linear (\<lambda>x. blinfun_apply x j \<bullet> i)"
split_float_interval_bounds_3: fixes X :: "float interval" shows "upper (fst (split_float_interval X)) = mid X"
add_diff_assoc: fixes a :: "'k multiset"   and b :: "'k multiset"   and c :: "'k multiset" assumes "a \<subseteq># b" shows "c + (b - a) = c + b - a"
polyn_Hom_def: fixes R :: "('e, 'g) Ring_scheme"   and S :: "('e, 'h) Ring_scheme"   and X :: "'e"   and A :: "('f, 'i) Ring_scheme"   and B :: "('f, 'j) Ring_scheme"   and Y :: "'f" shows "pHom R S X, A B Y = {f \<in> rHom R A. f ` carrier S \<subseteq> carrier B \<and> f X = Y}"
green_typeI_typeII_chain_axioms: shows "green_typeI_typeII_chain s i j F two_chain_typeI two_chain_typeII"
stable_pair_onI: fixes XD_XH :: "'a set \<times> 'a set"   and ds :: "'b set" assumes "fst XD_XH = - Rf CH (snd XD_XH)"   and "snd XD_XH = - Rf (CD_on ds) (fst XD_XH)" shows "stable_pair_on ds XD_XH"
oa_facts_1: fixes v :: "i"   and x :: "\<nu>" shows "[\<lparr>O!,x\<^sup>P\<rparr> \<^bold>\<rightarrow> \<^bold>\<box>\<lparr>O!,x\<^sup>P\<rparr> in v]"
matrix_inv_right: fixes M :: "(('c, 'd) vec, 'e) vec" assumes "invertible M" shows "M ** matrix_inv M = mat (1::'c)"
effect_nthI: fixes i :: "nat"   and h :: "heap"   and a :: "'a array"   and h' :: "heap"   and r :: "'a" assumes "i < Array.length h a"   and "h' = h"   and "r = Array.get h a ! i" shows "effect (Array.nth a i) h h' r"
cis_plus_2pi: fixes x :: "real" shows "cis (x + 2 * pi) = cis x"
abs_eq: fixes xa :: "int"   and x :: "int" shows "(integer_of_int xa \<le> integer_of_int x) = (xa \<le> x)"
nfin2: shows "infinite {x. a * x\<^sup>2 + b * x + c = 0}"
real_lebesgue_integral_def: fixes M :: "'d measure"   and f :: "'d \<Rightarrow> real" assumes "integrable M f" shows "integral\<^sup>L M f = enn2real (\<integral>\<^sup>+ x. ennreal (f x) \<partial>M) - enn2real (\<integral>\<^sup>+ x. ennreal (- f x) \<partial>M)"
ev: shows "\<forall>\<^sub>F x in F. g x \<in> s"
bhv_dn_context: fixes uid :: "'nid"   and t :: "nat \<Rightarrow> cnf"   and n :: "nat" assumes "\<not> honest uid"   and "\<parallel>uid\<parallel>\<^bsub>t n\<^esub>"   and "\<exists>n'. latestAct_cond uid t n n'" shows "\<exists>nid'. \<parallel>nid'\<parallel>\<^bsub>t \<langle>uid \<leftarrow> t\<rangle>\<^bsub>n\<^esub>\<^esub> \<and> (mining (\<sigma>\<^bsub>uid\<^esub>t n) \<and> (\<exists>b. prefix (bc (\<sigma>\<^bsub>uid\<^esub>t n)) (bc (\<sigma>\<^bsub>nid'\<^esub>t \<langle>uid \<leftarrow> t\<rangle>\<^bsub>n\<^esub>) @ [b])) \<or> \<not> mining (\<sigma>\<^bsub>uid\<^esub>t n) \<and> prefix (bc (\<sigma>\<^bsub>uid\<^esub>t n)) (bc (\<sigma>\<^bsub>nid'\<^esub>t \<langle>uid \<leftarrow> t\<rangle>\<^bsub>n\<^esub>)))"
arr_cod: fixes f :: "'a1 \<times> 'a2" assumes "A1xA2.arr f" shows "A1xA2.arr (A1xA2.cod f)"
sg_i_m_closed: fixes H :: "'a set"   and a :: "'a"   and b :: "'a" assumes "G \<guillemotright> H "   and "a \<in> H"   and "b \<in> H" shows "\<rho> a \<cdot> b \<in> H"
essential_classD2: fixes C :: "'s set"   and x :: "'s"   and y :: "'s" assumes "essential_class C"   and "x \<in> C"   and "(x, y) \<in> acc" shows "y \<in> C"
standard_lifting_axioms: shows "standard_lifting Inf_FL Bot_G (Inf_G_q q) (entails_q q) (Red_I_q q) (Red_F_q q) q_lifting.Bot_FL q_lifting.\<G>_F_L q_lifting.\<G>_I_L"
powser_ms_aux': fixes cs :: "real msllist"   and f :: "real \<Rightarrow> real"   and basis :: "(real \<Rightarrow> real) list"   and G :: "('a \<times> real) msllist"   and g :: "real \<Rightarrow> real" assumes "convergent_powser' cs f"   and "basis_wf basis"   and "is_expansion_aux G g basis"   and "ms_exp_gt 0 (ms_aux_hd_exp G)" shows "is_expansion_aux (powser_ms_aux cs G) (f \<circ> g) basis"
sp_generalize: fixes F :: "'o \<Rightarrow> 'p fundef option"   and xs :: "('c, 'q, 'o, 'e, 'f, 'g, 'h, 'i) instr list"   and \<Sigma> :: "Unboxed.type option list"   and \<Sigma>' :: "Unboxed.type option list" assumes "sp F xs \<Sigma> = Ok \<Sigma>'" shows "sp F (map generalize_instr xs) (map Map.empty \<Sigma>) = Ok (map Map.empty \<Sigma>')"
error_returns_heap: fixes h :: "'l"   and e :: "'m"   and h' :: "'l" shows "\<not> h \<turnstile> error e \<rightarrow>\<^sub>h h'"
result_eq: shows "(rel, aux') = update_matchP n I mr mrs rels (\<tau> \<sigma> ne) aux"
insert_3: shows "finite {f. (\<forall>x. x \<notin> R \<longrightarrow> f x = c) \<and> (\<forall>x. x \<in> R \<longrightarrow> f x \<in> A)}"
sorted_distinct_set_unique: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'b list"   and ys :: "'b list" assumes "class.linorder less_eq less"   and "linorder.sorted less_eq xs"   and "distinct xs"   and "linorder.sorted less_eq ys"   and "distinct ys"   and "ran\<^sub>l xs = ran\<^sub>l ys" shows "xs = ys"
elims_2: fixes x :: "'b trm \<Rightarrow> 'b trm \<Rightarrow> bool"   and xa :: "'b equation" assumes "validate_ground_eq x xa" obtains I :: "'b trm \<Rightarrow> 'b trm \<Rightarrow> bool" and t :: "'b trm" and s :: "'b trm" where "x = I" and "xa = Eq t s" and "I t s"
rec_accum_ex: fixes rf :: "recf"   and xs :: "nat list"   and x :: "nat" assumes "primerec rf (Suc (length xs))" shows "(rec_exec (rec_accum rf) (xs @ [x]) = 0) = (\<exists>t\<le>x. rec_exec rf (xs @ [t]) = 0)"
lset_ltake_prefix: fixes n :: "enat"   and m :: "enat"   and xs :: "'c llist" assumes "n \<le> m" shows "lset (ltake n xs) \<subseteq> lset (ltake m xs)"
sum_mset_dvd: fixes M :: "'a multiset"   and k :: "'b"   and f :: "'a \<Rightarrow> 'b" assumes "\<forall>m\<in>#M. k dvd f m" shows "k dvd \<Sum>\<^sub># (image_mset f M)"
ordinal_times_left_cancel_le: fixes w :: "ordinal"   and x :: "ordinal"   and y :: "ordinal" assumes "0 < w" shows "(w * x \<le> w * y) = (x \<le> y)"
simps_3: fixes f :: "'d"   and args :: "'c \<Rightarrow> ('d, 'c) trm" shows "FVT ($f f args) = (\<Union>i. FVT (args i))"
simps: fixes a1 :: "aexp"   and a2 :: "char list \<Rightarrow> val"   and a3 :: "val" shows "taval a1 a2 a3 = ((\<exists>i s. a1 = Ic i \<and> a2 = s \<and> a3 = Iv i) \<or> (\<exists>r s. a1 = Rc r \<and> a2 = s \<and> a3 = Rv r) \<or> (\<exists>x s. a1 = V x \<and> a2 = s \<and> a3 = s x) \<or> (\<exists>a1a s i1 a2a i2. a1 = Plus a1a a2a \<and> a2 = s \<and> a3 = Iv (i1 + i2) \<and> taval a1a s (Iv i1) \<and> taval a2a s (Iv i2)) \<or> (\<exists>a1a s r1 a2a r2. a1 = Plus a1a a2a \<and> a2 = s \<and> a3 = Rv (r1 + r2) \<and> taval a1a s (Rv r1) \<and> taval a2a s (Rv r2)))"
rel_game_def: fixes \<A> :: "'a \<Rightarrow> nat spmf" shows "Schnorr_\<Sigma>_commit.rel_game \<A> = TRY G \<bind> (\<lambda>(h, w). \<A> h \<bind> (\<lambda>w'. return_spmf ((h, w') \<in> R_DL))) ELSE return_spmf False"
Run: fixes 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" assumes "(M, l, s, n, ll, t) \<in> Step"   and "(M, ll, t, m, h, v) \<in> Exec"   and "k = max n m + 1" shows "(M, l, s, k, h, v) \<in> Exec"
apath_Cons_iff: fixes u :: "'a"   and e :: "'a \<times> 'a"   and es :: "('a \<times> 'a) list"   and w :: "'a" shows "apath u (e # es) w = (e \<in> parcs G \<and> fst e = u \<and> apath (snd e) es w \<and> fst e \<notin> set (pawalk_verts (snd e) es))"
Gets_imp_Says: fixes B :: "agent"   and X :: "msg"   and evs :: "event list" assumes "Gets B X \<in> set evs"   and "evs \<in> ns_public" shows "\<exists>A. Says A B X \<in> set evs"
min_var_value_def: shows "min_var_value = - (2 ^ 31)"
soundD_memv: fixes SG :: "(('sf, 'sc, 'sz) formula list \<times> ('sf, 'sc, 'sz) formula list) list"   and C :: "('sf, 'sc, 'sz) formula list \<times> ('sf, 'sc, 'sz) formula list"   and I :: "('sf, 'sc, 'sz) interp"   and \<nu> :: "(real, 'sz) vec \<times> (real, 'sz) vec" assumes "sound (SG, C)"   and "is_interp I"   and "\<And>\<phi> \<nu>. List.member SG \<phi> \<Longrightarrow> \<nu> \<in> seq_sem I \<phi>" shows "\<nu> \<in> seq_sem I C"
star_trans_eq: fixes x :: "'a" shows "x\<^sup>\<star> ; x\<^sup>\<star> = x\<^sup>\<star>"
resCasesB: fixes y :: "name"   and P :: "pi"   and a :: "subject"   and x :: "name"   and yP' :: "pi"   and F :: "subject \<Rightarrow> pi \<Rightarrow> bool" assumes "<\<nu>y>P \<longmapsto> a\<guillemotleft>x\<guillemotright> \<prec> yP'"   and "x \<noteq> y"   and "x \<sharp> P"   and "\<And>b P'. \<lbrakk>P \<longmapsto> b[y] \<prec> P'; b \<noteq> y; a = BoundOutputS b\<rbrakk> \<Longrightarrow> F (BoundOutputS b) ([(x, y)] \<bullet> P')"   and "\<And>P'. \<lbrakk>P \<longmapsto> a\<guillemotleft>x\<guillemotright> \<prec> P'; y \<sharp> a\<rbrakk> \<Longrightarrow> F a (<\<nu>y>P')" shows "F a yP'"
tab_agree_def: fixes \<S> :: "s_context"   and tcl :: "cl option" shows "tab_agree \<S> tcl = (case tcl of None \<Rightarrow> True | Some cl \<Rightarrow> \<exists>tf. cl_typing \<S> cl tf)"
Translations_transformation_monoid: shows "transformation_monoid (translation ` M) M"
assms_1: fixes n :: "nat" assumes "n \<le> 2 ^ k" shows "psi n \<le> c * ln 2 * real n"
eqprop: shows "\<forall>x. \<forall>(a, b, c)\<in>set a. a * x\<^sup>2 + b * x + c = 0"
squarefree_powerD: fixes m :: "nat"   and n :: "'b" assumes "0 < m"   and "squarefree (n ^ m)" shows "squarefree n"
sint_less: fixes w :: "'b word" shows "sint w < 2 ^ (LENGTH('b) - Suc 0)"
horner_eval_def: fixes f :: "nat \<Rightarrow> 'b"   and x :: "'b"   and n :: "nat" shows "horner_eval f x n = horner_eval' f x (0::'b) n"
callee: fixes out :: "'a"   and s :: "'s" assumes "out \<in> outs_\<I> \<I>"   and "I s" shows "lossless_spmf (callee s out)"
hom_top: fixes f :: "'b \<Rightarrow> 'a" shows "hom f \<top> = \<top>"