hcju/isabelleps huggingface dataset:
map\<^sub>1_in_hom_2: fixes f :: "'b" assumes "D.ide f" shows "\<guillemotleft>F.counit\<^sub>1 f : E.I\<^sub>C.map f \<star>\<^sub>D F.counit\<^sub>0 (src\<^sub>D f) \<Rightarrow>\<^sub>D F.counit\<^sub>0 (trg\<^sub>D f) \<star>\<^sub>D E.FG.map f\<guillemotright>"
memory_write_asi_low_equal_none: fixes s1 :: "'b sparc_state"   and s2 :: "'b sparc_state"   and addr :: "word32"   and bm :: "word4"   and data :: "word32" assumes "low_equal s1 s2"   and "memory_write_asi 10 addr bm data s1 = None" shows "memory_write_asi 10 addr bm data s2 = None"
simps: fixes M :: "('d, 'e, 'f) FSM" shows "observable M = (\<forall>t s1. succ M t s1 = {} \<or> (\<exists>s2. succ M t s1 = {s2}))"
HVZK_def: shows "\<Sigma>_OR.HVZK \<equiv> \<forall>e\<in>challenge_space. (\<forall>(h, w)\<in>Rel_OR. \<Sigma>_OR.R h w e = \<Sigma>_OR.S h e) \<and> (\<forall>h\<in>valid_pub_OR. \<forall>(a, z)\<in>set_spmf (S_OR h e). check_OR h a e z)"
count_eq_zero_iff: fixes M :: "'c multiset"   and x :: "'c" shows "(count M x = 0) = (x \<notin># M)"
wp_def: fixes \<pi> :: "char list \<Rightarrow> com option"   and c :: "com"   and Q :: "(char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> bool"   and s :: "char list \<Rightarrow> int \<Rightarrow> int" shows "wp \<pi> c Q s \<equiv> \<exists>t. \<pi>: (c, s) \<Rightarrow> t \<and> Q t"
Limit_omega_in_VsetI: fixes \<alpha> :: "V"   and a :: "nat" assumes "Limit \<alpha>" shows "a\<^sub>\<nat> \<in>\<^sub>\<circ> Vset \<alpha>"
run_raiseI: fixes s :: "String.literal"   and \<sigma> :: "heap option"   and r :: "'b" shows "run (raise s) \<sigma> None r"
symmetric_Q: fixes I :: "'b"   and a :: "'a"   and b :: "'a" assumes "Q I a b" shows "Q I b a"
config_rand_length: fixes A :: "('c list \<Rightarrow> 'd pmf) \<times> ('c list \<times> 'd \<Rightarrow> 'c \<Rightarrow> ((nat \<times> nat list) \<times> 'd) pmf)"   and init :: "'c list"   and qs :: "'c list" shows "\<forall>x\<in>set_pmf (Partial_Cost_Model.config'_rand A (fst A init \<bind> (\<lambda>is. return_pmf (init, is))) qs). length (fst x) = length init"
intro: fixes U\<^sub>1 :: "'a set"   and \<tau>\<^sub>1 :: "'a set \<Rightarrow> bool"   and U\<^sub>2 :: "'b set"   and \<tau>\<^sub>2 :: "'b set \<Rightarrow> bool" assumes "topological_space_ow U\<^sub>1 \<tau>\<^sub>1"   and "first_countable_topology_ow U\<^sub>2 \<tau>\<^sub>2" shows "ts_fct_ow U\<^sub>1 \<tau>\<^sub>1 U\<^sub>2 \<tau>\<^sub>2"
is_cindependent_set: fixes scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and scaleC :: "complex \<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 canonical_basis :: "'b list" assumes "class.basis_enum scaleR scaleC plus zero minus uminus canonical_basis" shows "\<not> cdependent (set canonical_basis)"
that: fixes E :: "('a \<times> 'a) set" assumes "E \<inter> E\<inverse> = {}"   and "G = E \<union> E\<inverse>" shows "thesis"
rep_eq: fixes x :: "'a" shows "(*\<^sub>V) (cblinfun.prod_left x) = (\<lambda>a. a *\<^sub>V x)"
PiE: fixes f :: "'d \<Rightarrow> 'e"   and A :: "'d set"   and B :: "'d \<Rightarrow> 'e set"   and x :: "'d" assumes "f \<in> Pi A B" obtains "f x \<in> B x"   | "x \<notin> A"
rel_fundefs_Some2: fixes Fstd_empty :: "'a"   and Fstd_get :: "'a \<Rightarrow> 'b \<Rightarrow> ('c, 'd, 'b, 'e) Std.instr fundef option"   and Fstd_add :: "'a \<Rightarrow> 'b \<Rightarrow> ('c, 'd, 'b, 'e) Std.instr fundef \<Rightarrow> 'a"   and Fstd_to_list :: "'a \<Rightarrow> ('b \<times> ('c, 'd, 'b, 'e) Std.instr fundef) list"   and Finca_empty :: "'f"   and Finca_get :: "'f \<Rightarrow> 'b \<Rightarrow> ('c, 'd, 'b, 'e, 'g) Inca.instr fundef option"   and Finca_add :: "'f \<Rightarrow> 'b \<Rightarrow> ('c, 'd, 'b, 'e, 'g) Inca.instr fundef \<Rightarrow> 'f"   and Finca_to_list :: "'f \<Rightarrow> ('b \<times> ('c, 'd, 'b, 'e, 'g) Inca.instr fundef) list"   and heap_empty :: "'h"   and heap_get :: "'h \<Rightarrow> 'd \<times> 'c \<Rightarrow> 'c option"   and heap_add :: "'h \<Rightarrow> 'd \<times> 'c \<Rightarrow> 'c \<Rightarrow> 'h"   and heap_to_list :: "'h \<Rightarrow> (('d \<times> 'c) \<times> 'c) list"   and is_true :: "'c \<Rightarrow> bool"   and is_false :: "'c \<Rightarrow> bool"   and \<OO>\<pp> :: "'e \<Rightarrow> 'c list \<Rightarrow> 'c"   and \<AA>\<rr>\<ii>\<tt>\<yy> :: "'e \<Rightarrow> nat"   and \<II>\<nn>\<ll>\<OO>\<pp> :: "'g \<Rightarrow> 'c list \<Rightarrow> 'c"   and \<II>\<nn>\<ll> :: "'e \<Rightarrow> 'c list \<Rightarrow> 'g option"   and \<II>\<ss>\<II>\<nn>\<ll> :: "'g \<Rightarrow> 'c list \<Rightarrow> bool"   and \<DD>\<ee>\<II>\<nn>\<ll> :: "'g \<Rightarrow> 'e"   and f :: "'i \<Rightarrow> ('j, 'k, 'l, 'e) Std.instr fundef option"   and g :: "'i \<Rightarrow> ('j, 'k, 'l, 'e, 'g) Inca.instr fundef option"   and x :: "'i"   and y :: "('j, 'k, 'l, 'e, 'g) Inca.instr fundef" assumes "std_inca_simulation Fstd_empty Fstd_get Fstd_add Fstd_to_list Finca_empty Finca_get Finca_add Finca_to_list heap_empty heap_get heap_add heap_to_list is_true is_false \<OO>\<pp> \<AA>\<rr>\<ii>\<tt>\<yy> \<II>\<nn>\<ll>\<OO>\<pp> \<II>\<nn>\<ll> \<II>\<ss>\<II>\<nn>\<ll> \<DD>\<ee>\<II>\<nn>\<ll>"   and "std_inca_simulation.rel_fundefs \<DD>\<ee>\<II>\<nn>\<ll> f g"   and "g x = Some y" shows "\<exists>z. f x = Some z \<and> rel_fundef (\<lambda>x y. x = std_inca_simulation.norm_instr \<DD>\<ee>\<II>\<nn>\<ll> y) z y"
\<zeta>_def: shows "\<zeta> \<equiv> \<lambda>s. \<Sum>n. (of_nat n + complex_of_real a) powr - s"
PrefixListHasTail: fixes l1 :: "'a list"   and l2 :: "'a list" assumes "prefixList l1 l2" shows "\<exists>l. l2 = l1 @ l \<and> l \<noteq> []"
wp_loop_step_mono: fixes body :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and t :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and u :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and G :: "'a \<Rightarrow> bool" assumes "healthy (wp body)"   and "le_trans t u"   and "\<And>P. sound P \<Longrightarrow> sound (t P)"   and "\<And>P. sound P \<Longrightarrow> sound (u P)" shows "le_trans (wp (body ;; Embed t \<^bsub>\<guillemotleft> G \<guillemotright>\<^esub>\<oplus> Skip)) (wp (body ;; Embed u \<^bsub>\<guillemotleft> G \<guillemotright>\<^esub>\<oplus> Skip))"
to_nat_on_from_nat_into: fixes n :: "nat"   and A :: "'a set" assumes "n \<in> to_nat_on A ` A" shows "to_nat_on A (from_nat_into A n) = n"
in_Sts_in_Sts_Suc: shows "\<forall>l\<ge>i. enat (Suc l) < llength Sts \<longrightarrow> D \<in> Q_of_state (lnth Sts l) \<longrightarrow> D \<in> Q_of_state (lnth Sts (Suc l))"
subst_typ_simulates_tsubstT_gen': fixes l :: "(variable \<times> String.literal set) list"   and T :: "typ"   and \<rho> :: "variable \<Rightarrow> String.literal set \<Rightarrow> typ" assumes "distinct l"   and "tvsT T \<subseteq> set l" shows "tsubstT T \<rho> = subst_typ (map (\<lambda>(x, y). ((x, y), \<rho> x y)) l) T"
particular_denial_def: fixes A :: "'a set"   and B :: "'a set" shows "A Z B \<equiv> \<exists>b\<in>B. b \<notin> A"
sym_P2R: fixes \<phi> :: "'c \<Rightarrow> 'c \<Rightarrow> bool" shows "sym (P2R \<phi>) = symP \<phi>"
decisive7_2: fixes x :: "'a"   and y :: "'a"   and z :: "'a"   and A :: "'a set"   and swf :: "('b \<Rightarrow> ('a \<times> 'a) set) \<Rightarrow> ('a \<times> 'a) set"   and Is :: "'b set"   and j :: "'b" assumes "hasw [x, y, z] A"   and "iia swf A Is"   and "SWF swf A Is universal_domain"   and "weak_pareto swf A Is universal_domain"   and "semidecisive swf A Is {j} x y" shows "decisive swf A Is {j} z x"
description_axiom: fixes \<psi> :: "i \<Rightarrow> bool"   and \<phi> :: "i \<Rightarrow> bool" assumes "atomic \<psi>" shows "\<psi> (\<^bold>\<tau>x. \<phi> x) \<equiv> \<^bold>\<exists>x. \<phi> x \<and> (\<^bold>\<forall>xa. \<phi> xa \<longrightarrow> x \<^bold>= xa) \<and> \<psi> x"
isSome_def: fixes opt :: "'c option" shows "isSome opt \<equiv> case opt of None \<Rightarrow> False | Some x \<Rightarrow> True"
ccProd_mono2: fixes S' :: "Vars.var set"   and S'' :: "Vars.var set"   and S :: "Vars.var set" assumes "S' \<subseteq> S''" shows "S G\<times> S' \<sqsubseteq> S G\<times> S''"
v11_veqpollI: fixes r :: "V"   and A :: "V"   and B :: "V" assumes "v11 r"   and "\<D>\<^sub>\<circ> r = A"   and "\<R>\<^sub>\<circ> r = B" shows "A \<approx>\<^sub>\<circ> B"
ipresIGAbs_termMOD: fixes Delta :: "('b, 'c, 'd, 'e, 'f) signature"   and h :: "('b, 'c, 'd, 'a, 'f) qTerm set \<Rightarrow> 'g"   and hA :: "('b, 'c, 'd, 'a, 'f) qAbs set \<Rightarrow> 'h"   and MOD :: "('i, 'j, 'd, 'k, 'l, 'a, 'g, 'h, 'm) model_scheme" assumes "FixSyn TYPE('a) Delta" shows "FixSyn.ipresIGAbs Delta h hA (FixSyn.termMOD Delta) MOD = FixSyn.ipresAbs Delta h hA MOD"
max_pos_imp_Interp: fixes C :: "'a literal multiset"   and A :: "'a" assumes "C \<in> N"   and "C \<noteq> {#}"   and "Max_mset C = Pos A"   and "S C = {#}" shows "Interp C \<Turnstile> C"
not_empty_cyc_free: fixes n :: "nat"   and v :: "'c \<Rightarrow> nat"   and M :: "nat \<Rightarrow> nat \<Rightarrow> 'd DBMEntry" assumes "\<forall>k\<le>n. 0 < k \<longrightarrow> (\<exists>c. v c = k)"   and "[M]\<^bsub>v,n\<^esub> \<noteq> {}" shows "\<forall>i xs. i \<le> n \<and> set xs \<subseteq> {0..n} \<longrightarrow> \<one> \<le> len M i i xs"
P1_adv_real_ideal_model_def: fixes D :: "'e \<times> 'a \<Rightarrow> bool spmf"   and M :: "'a \<times> 'a"   and \<sigma> :: "bool"   and \<A> :: "('a \<times> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'f \<Rightarrow> (('a \<times> 'a \<times> 'a) \<times> 'g) spmf) \<times> ('a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<Rightarrow> 'g \<Rightarrow> ((('a \<times> 'a) \<times> 'a \<times> 'a) \<times> 'g) spmf) \<times> ('g \<Rightarrow> 'e spmf)"   and z :: "'f" shows "P1_adv_real_ideal_model D M \<sigma> \<A> z = \<bar>spmf (P1_real_model M \<sigma> z \<A> \<bind> D) True - spmf (P1_ideal_model M \<sigma> z \<A> \<bind> D) True\<bar>"
sheaf_spec_extensional: fixes R :: "'c set"   and addition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and multiplication :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and unit :: "'c"   and a :: "'c set \<Rightarrow> ('c \<times> 'c) set"   and U :: "'c set set" assumes "comm_ring R addition multiplication zero unit"   and "a \<in> comm_ring.sheaf_spec R addition multiplication zero unit U" shows "a \<in> extensional U"
not_Some_eq: fixes x :: "'l option" shows "(\<forall>y. x \<noteq> Some y) = (x = None)"
dvd_poly_i_def: fixes ops :: "'b arith_ops_record"   and xs :: "'b list"   and ys :: "'b list" shows "dvd_poly_i ops xs ys = (\<exists>zs. is_poly ops zs \<and> ys = times_poly_i ops xs zs)"
assms_9: shows "l \<circ> h \<circ> g \<le>\<le> g \<circ> h \<circ> l"
inducts: fixes M :: "'b measure"   and f :: "'b \<Rightarrow> 'c" assumes "integrable M f" obtains x :: "'c" where "has_bochner_integral M f x"
fBex_cong: fixes A :: "'a fset"   and B :: "'a fset"   and P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool" assumes "A = B"   and "\<And>x. x |\<in>| B \<Longrightarrow> P x = Q x" shows "fBex A P = fBex B Q"
program_typedef_6: fixes F :: "'b program" shows "Acts F \<equiv> case Rep_Program F of (init, acts, allowed) \<Rightarrow> acts"
D_COMP_H: fixes G :: "((Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool) set"   and c :: "OBJ"   and d :: "OBJ" assumes "(G, c, HighSec) \<in> Deriv"   and "(G, d, HighSec) \<in> Deriv" shows "(G, Comp c d, HighSec) \<in> Deriv"
in_homE: fixes f :: "'a \<times> 'a"   and a :: "'a \<times> 'a"   and b :: "'a \<times> 'a" assumes "B.VV.in_hom f a b" obtains "B.VV.arr f" and "B.VV.dom f = a" and "B.VV.cod f = b"
limsup_le_liminf_real: fixes X :: "nat \<Rightarrow> real"   and L :: "real" assumes "limsup (\<lambda>x. ereal (X x)) \<le> ereal L"   and "ereal L \<le> liminf (\<lambda>x. ereal (X x))" shows "X \<longlonglongrightarrow> L"
lm072: fixes a :: "('a \<times> 'b set) set"   and G :: "'b set"   and N :: "'a set" assumes "a \<in> allAllocations N G" shows "Domain (pseudoAllocation a) \<subseteq> N \<and> Range (pseudoAllocation a) = finestpart G"
iIN_iMOD_subset_conv: fixes d' :: "nat"   and n' :: "nat"   and r :: "nat"   and m :: "nat" assumes "0 < d'" shows "([n'\<dots>,d'] \<subseteq> [ r, mod m ]) = (r \<le> n' \<and> m = Suc 0)"
disc_6: fixes x3 :: "'a" shows "is_wrong (Goes_wrong x3)"
QTrans: shows "Q \<Longrightarrow>\<^sub>lb in Q''\<rightarrow>a<x> \<prec> Q'"
locally_trans_multpw: fixes ns :: "('b \<times> 'b) set"   and S :: "'b multiset"   and T :: "'b multiset"   and U :: "'b multiset" assumes "locally_trans ns S T U"   and "(S, T) \<in> multpw ns"   and "(T, U) \<in> multpw ns" shows "(S, U) \<in> multpw ns"
primerec_rec_mult_2: shows "primerec rec_mult (Suc (Suc 0))"
Tau: shows "\<T> = generate_valid_stateful_policy_IFSACS_2 G M edgesList"
add_le_imp_le_left: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and c :: "'a"   and a :: "'a"   and b :: "'a" assumes "class.ordered_ab_semigroup_add_imp_le plus minus less_eq less"   and "less_eq (plus c a) (plus c b)" shows "less_eq a b"
B\<^sub>2_at_least_two_objects: 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 "u \<in> V \<inter> S"   and "real c < W B\<^sub>2 + w u" shows "2 \<le> card B\<^sub>2"
compute_rapprox_rat: fixes prec :: "nat"   and x :: "int"   and y :: "int" shows "rapprox_rat prec x y = - lapprox_rat prec (- x) y"
singleTauChain: fixes P :: "pi"   and P' :: "pi" assumes "P \<longmapsto> \<tau> \<prec> P'" shows "P \<Longrightarrow>\<^sub>\<tau> P'"
Dube_is_GB_cofactor_bound_pp: fixes fs :: "(('x, nat) pp \<Rightarrow>\<^sub>0 'b) list" shows "pp_pm.is_GB_cofactor_bound (Poly_Mapping.map_key PP ` set fs) (Dube (Suc (length (Indets_pp fs))) (max_list (map poly_deg_pp fs)))"
distinct_list: fixes A :: "'a set" assumes "finite A" shows "distinct (list A)"
not_in_first_image: fixes x :: "'c"   and s :: "('c \<times> 'd) set"   and p :: "'d" assumes "x \<notin> fst ` s" shows "(x, p) \<notin> s"
finite_enumerate_mono: fixes m :: "nat"   and n :: "nat"   and S :: "'a set" assumes "m < n"   and "finite S"   and "n < card S" shows "enum S m < enum S n"
end_point_iff2: fixes x :: "'a" assumes "path x" shows "is_point (end_points x) = has_end_points x"
sumlist_in_span: fixes W :: "rat vec set"   and xs :: "rat vec list" assumes "W \<subseteq> Rn"   and "\<And>x. x \<in> set xs \<Longrightarrow> x \<in> gs.span W" shows "gs.M.sumlist xs \<in> gs.span W"
Collect_mem_eq: fixes A :: "'b set" shows "{x::'b. x \<in> (A::'b set)} = A"
shortest_path_pos_cost_pred_axioms: shows "shortest_path_pos_cost_pred G dist c s num pred"
a_in_ar_coset: fixes R :: "('c, 'd) Ring_scheme"   and I :: "'c set"   and a :: "'c" assumes "Ring R"   and "ideal R I"   and "a \<in> carrier R" shows "a \<in> a \<uplus>\<^bsub>R\<^esub> I"
minimum_elem_unique: fixes X :: "'a set"   and a1 :: "'a"   and a2 :: "'a" assumes "X \<subseteq> carrier D"   and "minimum_elem D X a1"   and "minimum_elem D X a2" shows "a1 = a2"
OUTfromChCorrect_data19: shows "OUTfromChCorrect data19"
nth_rotate_conv_nth1_conv_nth: fixes m :: "nat"   and xs :: "'b list" assumes "m < length xs" shows "rotate1 xs ! m = xs ! (Suc m mod length xs)"
divide_by_norm_row_def: fixes A :: "(('a, 'b) vec, 'c) vec"   and a :: "'c" shows "divide_by_norm_row A a = (\<chi>b. ((1 / norm (column b A)) *\<^sub>R column b A) $ a)"
max_int_eqvt: fixes pi :: "('a \<times> 'a) list"   and x :: "int"   and y :: "int" shows "pi \<bullet> max x y = max (pi \<bullet> x) (pi \<bullet> y)"
intro: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and S :: "'b span_data"   and T :: "'b span_data" assumes "partial_magma.cod C (Leg0 S) = partial_magma.cod C (Leg1 T)" shows "composite_span_axioms C S T"
valuesum_choice: fixes M :: "('c, 'd) mapping"   and x :: "'c"   and y :: "'c" assumes "finite (Mapping.keys M)"   and "x \<in> Mapping.keys M"   and "y \<in> Mapping.keys M" shows "the (Mapping.lookup M x) + valuesum (Mapping.delete x M) = the (Mapping.lookup M y) + valuesum (Mapping.delete y M)"
map_of_fv_subset: fixes x :: "'a"   and \<Gamma> :: "('a \<times> 'b) list" assumes "x \<in> domA \<Gamma>" shows "fv (the (map_of \<Gamma> x)) \<subseteq> fv \<Gamma>"
intro: fixes G :: "'a graph" assumes "fst ` edges G \<subseteq> nodes G"   and "snd ` edges G \<subseteq> nodes G"   and "finite (edges G)"   and "finite (nodes G)" shows "wf_graph G"
subset_Collect_conv: fixes S :: "'g set"   and P :: "'g \<Rightarrow> bool" shows "(S \<subseteq> Collect P) = (\<forall>x\<in>S. P x)"
leq_antisym: fixes leq :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and lub :: "'f set \<Rightarrow> 'f"   and x :: "'f"   and y :: "'f" assumes "partial_function_definitions leq lub"   and "leq x y"   and "leq y x" shows "x = y"
tiling_domino_finite: fixes t :: "(nat \<times> nat) set" assumes "t \<in> tiling domino" shows "finite t"
iterAbsSTR_def: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'd, 'c) model" shows "iterAbsSTR MOD \<equiv> Equiv_Relation2.univ (qInitAbs MOD)"
set_2: fixes x2 :: "'c" shows "RID\<^sub>V (Var x2) = {}"
inject_3: fixes x3 :: "'f"   and y3 :: "'f" shows "(Fault x3 = Fault y3) = (x3 = y3)"
exec_gpv_lift_resumption: fixes callee :: "'e \<Rightarrow> 'f \<Rightarrow> ('g \<times> 'e) spmf"   and r :: "('d, 'f, 'g) Resumption.resumption"   and s :: "'e" shows "exec_gpv callee (lift_resumption r) s = execp_resumption callee r s"
pred_intros_logic_1: fixes M :: "'c measure" shows "Measurable.pred M (\<lambda>x. x \<in> space M)"
divides_degree: fixes p :: "'a poly"   and q :: "'a poly" assumes "p dvd q" shows "degree p \<le> degree q \<or> q = 0"
xvecFreshK: shows "(p \<bullet> xvec) \<sharp>* K"
that: fixes e' :: "nat \<Rightarrow> real" assumes "pdevs_val e (pdevs_of_list (map snd (list_of_pdevs X))) = pdevs_val e' X"   and "e' \<in> UNIV \<rightarrow> I" shows "thesis"
tailrec_impl: shows "a.tailrec_impl \<le> a.c.gen_dfs"
P2_sec_OT_14_split: fixes m0 :: "bool"   and m1 :: "bool"   and m2 :: "bool"   and m3 :: "bool"   and c0 :: "bool"   and c1 :: "bool" shows "R2_14 (m0, m1, m2, m3) (c0, c1) = funct_OT_14 (m0, m1, m2, m3) (c0, c1) \<bind> (\<lambda>(out1, out2). S2_14 (c0, c1) out2)"
f_Exec_Comp_Stream_Acc_Output_def: fixes k :: "nat"   and output_fun :: "'e \<Rightarrow> 'd message_af"   and trans_fun :: "'f message_af \<Rightarrow> 'e \<Rightarrow> 'e"   and xs :: "'f message_af list"   and c :: "'e" shows "f_Exec_Comp_Stream_Acc_Output k output_fun trans_fun xs c \<equiv> map output_fun (f_Exec_Comp_Stream trans_fun (xs \<odot> k) c) \<div> k"
contours_in_ve_upds: fixes vs :: "(nat \<times> char list) list"   and ds :: "d list"   and ve :: "(nat \<times> char list) \<times> contour \<Rightarrow> d option"   and Q :: "contour \<Rightarrow> bool"   and b'' :: "contour" assumes "length vs = length ds"   and "\<forall>b'\<in>contours_in_ve ve. Q b'"   and "\<forall>d'\<in>set ds. \<forall>b'\<in>contours_in_d d'. Q b'" shows "\<forall>b'\<in>contours_in_ve (ve(map (\<lambda>v. (v, b'')) vs [\<mapsto>] ds)). Q b'"
game_0_def: fixes \<A> :: "(bool, '\<alpha>, '\<gamma>) gpv" shows "local.prf.game_0 \<A> = key_gen \<bind> (\<lambda>key. exec_gpv (local.prf.prf_oracle key) \<A> () \<bind> (\<lambda>(b, uu_). return_spmf b))"
unique1: 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 C :: "char list"   and Cs :: "char list list"   and C' :: "char list"   and Cs' :: "char list list"   and wf_md :: "(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 \<Rightarrow> char list \<Rightarrow> char list \<times> ty list \<times> ty \<times> char list list \<times> expr \<Rightarrow> bool" assumes "Subobjs P C (Cs @ C' # Cs')"   and "wf_prog wf_md P" shows "C' \<notin> set Cs'"
list_all2_mono: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'a list"   and ys :: "'b list"   and Q :: "'a \<Rightarrow> 'b \<Rightarrow> bool" assumes "list_all2 P xs ys"   and "\<And>xs ys. P xs ys \<Longrightarrow> Q xs ys" shows "list_all2 Q xs ys"
cases: fixes x :: "'edge list \<times> 'edge list" obtains cs :: "'edge list" where "x = (cs__, [])"   | cs :: "'edge list" and as :: "'edge list" and a :: "'edge" where "x = (cs, as @ [a])"
append_exec_def: fixes e :: "'b \<times> 'c list"   and e' :: "'d \<times> 'c list" shows "append_exec e e' \<equiv> (fst e, append (snd e') (snd e))"
eitherMarkedOrNotMarkedElement: fixes a :: "'b \<times> bool" shows "a = (element a, True) \<or> a = (element a, False)"
assms_4: shows "revision_step r (s\<^sub>1(r' \<mapsto> y)) (s\<^sub>1(r' \<mapsto> y, r \<mapsto> x))"
unfold1: fixes s1 :: "'d \<Rightarrow> ('c, 'd, 'e) F1"   and s2 :: "'e \<Rightarrow> ('c, 'd, 'e) F2"   and z :: "'d" shows "dtor1 (unfold1 s1 s2 z) = F1map id (unfold1 s1 s2) (unfold2 s1 s2) (s1 z)"
mem_frontier_diff': fixes t :: "'t"   and N :: "'t zmultiset"   and M :: "'t zmultiset" assumes "t \<in>\<^sub>A frontier N - frontier M" shows "zcount (frontier_changes M N) t = - 1"
simps_2: fixes a :: "symbol list"   and d :: "nat \<times> symbol \<times> symbol list"   and D :: "(nat \<times> symbol \<times> symbol list) list"   and b :: "symbol list" shows "Derivation a (d # D) b = (\<exists>x. Derives1 a (fst d) (snd d) x \<and> Derivation x D b)"
poly_cutoff_1: fixes n :: "nat" shows "poly_cutoff n 1 = (if n = 0 then 0 else 1)"
count_poly_roots: fixes p :: "'b poly"   and x :: "'b" assumes "p \<noteq> 0" shows "count (poly_roots p) x = order x p"
valid_path_join: fixes g1 :: "real \<Rightarrow> 'b"   and g2 :: "real \<Rightarrow> 'b" assumes "valid_path g1"   and "valid_path g2"   and "pathfinish g1 = pathstart g2" shows "valid_path (g1 +++ g2)"
bs: shows "bs \<equiv> THE cs. length cs = length vs \<and> a \<cdot> v = cs \<bullet>\<cdot> vs"
irreducible_prime: fixes p :: "'a" assumes "irreducible G p"   and "p \<in> carrier G" shows "prime G p"
span_insert: fixes a :: "'a"   and S :: "'a set" shows "More_Modules.ideal (insert a S) = {x. \<exists>k. x - k * a \<in> More_Modules.ideal S}"
simps: fixes a1 :: "'loc"   and a2 :: "'loc"   and a3 :: "('loc \<times> 'sum \<times> 'loc) list" shows "path a1 a2 a3 = ((\<exists>l1 l2. a1 = l1 \<and> a2 = l2 \<and> a3 = [] \<and> l1 = l2) \<or> (\<exists>l1 l2 xs lbl l3. a1 = l1 \<and> a2 = l3 \<and> a3 = xs @ [(l2, lbl, l3)] \<and> path l1 l2 xs \<and> lbl \<in>\<^sub>A summary l2 l3))"
con_comp_secure_aux: fixes P :: "'e process"   and I :: "('f \<times> 'f) set"   and D :: "'e \<Rightarrow> 'f"   and Y :: "'g set"   and p :: "'e \<Rightarrow> 'g"   and ys :: "'g list"   and q :: "'h \<Rightarrow> 'g"   and U :: "'f option set"   and xs :: "'g list"   and E :: "'h \<Rightarrow> 'f" assumes "secure P I D"   and "Y \<subseteq> range p"   and "set ys \<subseteq> range p \<union> range q"   and "U \<subseteq> range Some"   and "(map (inv p) (filter (\<lambda>x. x \<in> range p) (xs @ ys)), inv p ` Y) \<in> failures P" shows "(map (inv p) (filter (\<lambda>x. x \<in> range p) xs) @ map (inv p) (ipurge_tr_aux_foldr (con_comp_pol I) (con_comp_map D E p q) (\<lambda>x. x \<in> range p) U ys), inv p ` ipurge_ref_aux_foldr (con_comp_pol I) (con_comp_map D E p q) (\<lambda>x. x \<in> range p) U ys Y) \<in> failures P"
ring_iso_set_sym: fixes R :: "('e, 'f) ring_scheme"   and h :: "'e \<Rightarrow> 'g"   and S :: "('g, 'h) ring_scheme" assumes "ring R"   and "h \<in> ring_iso R S" shows "inv_into (carrier R) h \<in> ring_iso S R"