hcju/isabelleps huggingface dataset:
RootEx_def: fixes F :: "('a, 'b, 'c) seqauto set"   and G :: "'a \<Rightarrow> ('a, 'b, 'c) seqauto set option" shows "RootEx F G = (\<exists>!A. A \<in> F \<and> A \<notin> \<Union> (ran G))"
e_le_ee: fixes i :: "real" assumes "i \<in> kk" shows "e \<le> ee i"
compact_closure: fixes S :: "'b set" shows "compact (closure S) = bounded S"
inf_of_singleton_closed: fixes x :: "'a" assumes "x \<in> carrier L" shows "\<Sqinter>{x} \<in> carrier L"
ta_tb_wf_2: shows "wf\<^sub>t\<^sub>r\<^sub>m tb"
integrable_subinterval_real: fixes f :: "real \<Rightarrow> 'c"   and a :: "real"   and b :: "real"   and c :: "real"   and d :: "real" assumes "f integrable_on {a..b}"   and "{c..d} \<subseteq> {a..b}" shows "f integrable_on {c..d}"
add_nonneg_eq_0_iff: fixes x :: "'a multiset"   and y :: "'a multiset" assumes "{#} \<subseteq># x"   and "{#} \<subseteq># y" shows "(x + y = {#}) = (x = {#} \<and> y = {#})"
known_ptrs_is_l_known_ptrs: shows "l_known_ptrs ShadowRootClass.known_ptr ShadowRootClass.known_ptrs"
dom\<sigma>: shows "differing_vars mem\<^sub>1 mem\<^sub>2 \<subseteq> dom \<sigma>"
InvIncomingInfoUpright_def: fixes c :: "('c, 'd, 'e) configuration_scheme" shows "InvIncomingInfoUpright c = (\<forall>k p q. upright (IncomingInfo c k p q))"
simps_8: fixes x :: "nat"   and a :: "floatarith"   and b :: "floatarith"   and vs :: "real list" shows "isDERIV x (Max a b) vs = False"
normalize_quot_eq_iff': fixes x :: "'b \<times> 'b"   and y :: "'b \<times> 'b" assumes "snd x \<noteq> (0::'b)"   and "snd y \<noteq> (0::'b)" shows "(normalize_quot x = normalize_quot y) = (fst x * snd y = snd x * fst y)"
bisimE_2: fixes P :: "pi"   and Q :: "pi" assumes "P \<sim>\<^sub>e Q" shows "Q \<sim>\<^sub>e P"
stop_cxt: fixes \<Gamma> :: "'Var \<Rightarrow> Sec option"   and c :: "('Var, 'AExp, 'BExp) Stmt"   and \<Gamma>' :: "'Var \<Rightarrow> Sec option" assumes "\<turnstile> \<Gamma> {c} \<Gamma>'"   and "c = Stop" shows "\<Gamma> \<sqsubseteq>\<^sub>c \<Gamma>'"
inv_integer_mod_group: fixes x :: "int"   and n :: "nat" assumes "x \<in> carrier (integer_mod_group n)" shows "inv\<^bsub>integer_mod_group n\<^esub> x = - x mod int n"
simps_1: fixes t' :: "transition"   and t :: "transition"   and p :: "nat" shows "stored_reused_aux_per_reg t' t 0 p = (if is_generalised_output_of t' t 0 p then Some (0, p) else None)"
proots_within_upper_squarefree: fixes p :: "complex poly" assumes "rsquarefree p" shows "card (proots_within p {x. 0 < Im x}) = (let pp = smult (inverse (lead_coeff p)) p; pI = map_poly Im pp; pR = map_poly Re pp; g = gcd pR pI in nat ((int (degree p) - changes_R_smods g (pderiv g) - changes_R_smods pR pI) div 2))"
lt_in_keys: fixes p :: "('x option \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" assumes "p \<noteq> 0" shows "pp_pm.extended_ord.lpp p \<in> keys p"
right_translations_of_group_axioms: shows "right_translations_of_group G (\<cdot>) \<one>"
tau_mpj_bilin1: fixes M :: "('e, 'a, 'f) Module_scheme"   and N :: "('g, 'a, 'f) Module_scheme"   and P :: "('e \<times> 'g, 'a, 'f) Module_scheme"   and x1 :: "'e"   and x2 :: "'e"   and y :: "'g" assumes "R module M"   and "R module N"   and "FM\<^bsub>R\<^esub> P M N"   and "x1 \<in> carrier M"   and "x2 \<in> carrier M"   and "y \<in> carrier N" shows "mpj P (TR\<^bsub>R\<^esub> M N P) (x1 \<plusminus>\<^bsub>M\<^esub> x2, y) = mpj P (TR\<^bsub>R\<^esub> M N P) (x1, y) \<plusminus>\<^bsub>M \<^bsub>P\<^esub>\<Otimes>\<^bsub>R\<^esub> N\<^esub> mpj P (TR\<^bsub>R\<^esub> M N P) (x2, y)"
WellB: shows "wo_rel (Restr r B) \<and> Well_order (Restr r B)"
inj_on: shows "inj_on f ({..<q} - {0})"
r_rev_aux_def: shows "r_rev_aux \<equiv> Pr 1 r_hd (Cn 3 r_prod_encode [Cn 3 r_nth [recf.Id 3 2, Cn 3 S [recf.Id 3 0]], recf.Id 3 1])"
"1_1": shows "v\<^sub>2 \<le>\<^sub>F\<^sub>I w"
primepow_odd_altdef: fixes n :: "nat" shows "primepow_odd n = (primepow n \<and> odd (multiplicity (aprimedivisor n) n) \<and> 1 < multiplicity (aprimedivisor n) n)"
coeff_poly_lift: fixes p :: "'c poly"   and i :: "nat" shows "coeff (poly_lift p) i = [:coeff p i:]"
assoc: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "a \<^bold>* b \<^bold>* c = a \<^bold>* (b \<^bold>* c)"
C_index: fixes vs :: "'e list"   and xs :: "'e list"   and M2 :: "('e, 'f, 'g) FSM"   and M1 :: "('e, 'f, 'h) FSM"   and \<Omega> :: "('e, 'f) ATC set"   and V :: "'e list set"   and m :: "nat"   and i :: "nat" assumes "vs @ xs \<in> C M2 M1 \<Omega> V m i"   and "mcp (vs @ xs) V vs" shows "Suc (length xs) = i"
Var_apps_eq_Var_apps_conv: fixes m :: "name"   and rs :: "lam list"   and n :: "name"   and ss :: "lam list" shows "(Var m \<degree>\<degree> rs = Var n \<degree>\<degree> ss) = (m = n \<and> rs = ss)"
OrR2_in_SNa: fixes M :: "trm"   and a :: "coname"   and b :: "coname" assumes "SNa M" shows "SNa (OrR2 <a>.M b)"
minimal_exists: fixes A :: "'b set"   and P :: "'b set \<Rightarrow> bool" assumes "finite A"   and "P A" shows "\<exists>B\<subseteq>A. minimal B P"
lincomb_add: fixes b1 :: "'c set"   and b2 :: "'c set"   and x1 :: "'c"   and a1 :: "'c \<Rightarrow> 'a"   and x2 :: "'c"   and a2 :: "'c \<Rightarrow> 'a"   and x :: "'c" assumes "finite (b1 \<union> b2)"   and "b1 \<union> b2 \<subseteq> carrier V"   and "x1 = lincomb a1 b1"   and "a1 \<in> b1 \<rightarrow> carrier K"   and "x2 = lincomb a2 b2"   and "a2 \<in> b2 \<rightarrow> carrier K"   and "x = x1 \<oplus>\<^bsub>V\<^esub> x2" shows "lincomb (\<lambda>v. (if v \<in> b1 then a1 v else \<zero>\<^bsub>K\<^esub>) \<oplus>\<^bsub>K\<^esub> (if v \<in> b2 then a2 v else \<zero>\<^bsub>K\<^esub>)) (b1 \<union> b2) = x"
simps: fixes acc :: "'a list"   and xs :: "'a list" shows "permutations_of_list_impl_aux acc xs = (if xs = [] then [acc] else List.bind (remdups xs) (\<lambda>x. permutations_of_list_impl_aux (x # acc) (remove1 x xs)))"
Un_absorb: fixes A :: "'a set" shows "A \<union> A = A"
LtKeys_parts_extractable: fixes I :: "msg set"   and K :: "msg set" assumes "I \<subseteq> valid" shows "parts (extractable K I) \<inter> range LtK = {}"
pick: fixes x :: "'a"   and X :: "'a set" assumes "x \<in> X" shows "pick X \<in> X"
m_cf_def: fixes S :: "('d, 'e) Ring_scheme"   and c :: "nat \<times> (nat \<Rightarrow> 'd)" shows "m_cf S c = (fst c, \<lambda>j. -\<^sub>a\<^bsub>S\<^esub> snd c j)"
split: fixes P :: "'a \<Rightarrow> bool"   and f1 :: "'a"   and f2 :: "'a"   and f3 :: "'b \<Rightarrow> 'a"   and vertex :: "'b vertex" shows "P (case vertex of SOURCE \<Rightarrow> f1 | vertex.SINK \<Rightarrow> f2 | \<langle>x\<rangle> \<Rightarrow> f3 x) = ((vertex = SOURCE \<longrightarrow> P f1) \<and> (vertex = vertex.SINK \<longrightarrow> P f2) \<and> (\<forall>x3. vertex = \<langle>x3\<rangle> \<longrightarrow> P (f3 x3)))"
prj_simps_13: shows "C.dom \<mu>_\<nu>_\<pi>\<rho>.Prj\<^sub>1 = H\<mu>H\<nu>H\<pi>\<rho>.chine"
kernel_is_ideal: fixes R :: "('c, 'd) ring_scheme"   and S :: "('e, 'f) ring_scheme"   and h :: "'c \<Rightarrow> 'e" assumes "ring_hom_ring R S h" shows "ideal (a_kernel R S h) R"
ct_list_eq_shared: fixes xs :: "ct list"   and as :: "t list"   and ys :: "ct list" assumes "ct_list_eq xs (to_ct_list as)"   and "ct_list_eq ys (to_ct_list as)" shows "ct_list_eq xs ys"
next_lins2_edge: fixes e :: "node fset \<times> command_gadget \<times> node fset"   and \<pi> :: "(node + node fset \<times> command_gadget \<times> node fset) list"   and S :: "node fset"   and V :: "node fset"   and \<Lambda> :: "node \<Rightarrow> assertion_gadget"   and E :: "(node fset \<times> command_gadget \<times> node fset) list"   and vs :: "node fset"   and c :: "command_gadget"   and ws :: "node fset" assumes "Inr e # \<pi> \<in> lins2 S (Graph V \<Lambda> E)"   and "vs |\<subseteq>| S"   and "e = (vs, c, ws)" shows "\<pi> \<in> lins2 (S |-| vs) (Graph (V |-| vs) \<Lambda> (removeAll e E))"
assms: shows "x \<in>\<^sub>\<circ> r \<restriction>\<^sup>l\<^sub>\<circ> A"
m_o_h: fixes m :: "'a \<Rightarrow> nat"   and h :: "nat"   and X :: "char list set"   and opt :: "'a st option" assumes "Measure1 m h"   and "finite X" shows "Measure1.m_o m h opt X \<le> h * card X + 1"
matchPres: fixes P :: "pi"   and Q :: "pi"   and a :: "name"   and b :: "name" assumes "P \<simeq> Q" shows "[a\<frown>b]P \<simeq> [a\<frown>b]Q"
restrict_space_singleton: fixes x :: "'c"   and M :: "'c measure" assumes "{x} \<in> sets M" shows "sets (restrict_space M {x}) = sets (count_space {x})"
fresh_star_def_2: fixes xs :: "'a set"   and c :: "'b" shows "xs \<sharp>* c \<equiv> \<forall>x\<in>xs. x \<sharp> c"
distinct_3: fixes x11 :: "agent"   and x12 :: "agent"   and x3 :: "agent" shows "sharK x11 x12 \<noteq> privK x3"
rb_aux_inv_is_RB_upt: fixes bs :: "('t \<Rightarrow>\<^sub>0 'b) list"   and ss :: "'t list"   and ps :: "(('t \<Rightarrow>\<^sub>0 'b) \<times> ('t \<Rightarrow>\<^sub>0 'b) + nat) list"   and u :: "'t" assumes "rb_aux_inv (bs, ss, ps)"   and "\<And>p. p \<in> set ps \<Longrightarrow> u \<preceq>\<^sub>t sig_of_pair p" shows "is_RB_upt dgrad rword (set bs) u"
cr_wn': fixes n :: "nat"   and \<pi> :: "nat \<Rightarrow> 'n"   and m :: "nat"   and m' :: "nat" assumes "n cd\<^bsup>\<pi>\<^esup>\<rightarrow> m"   and "\<not> m' cd\<^bsup>\<pi>\<^esup>\<rightarrow> m"   and "m < m'" shows "n < m'"
assms: shows "natural_transformation (\<cdot>\<^sup>o\<^sup>p) (\<cdot>\<^sub>S) (Y a) F \<tau>"
hyps: assumes "distinct ms"   and "[l] \<langle>f\<rangle> [l] \<odot> [l] \<langle>g\<rangle> ms \<preceq> [l] \<langle>g\<rangle> ms" shows "[l] \<langle>star \<circ> f\<rangle> [l] \<odot> [l] \<langle>g\<rangle> ms \<preceq> [l] \<langle>g\<rangle> ms"
s0_\<beta>x_s: shows "s0\<^bsub>SES\<^esub> (\<beta> @ [x])\<Longrightarrow>\<^bsub>SES\<^esub> s"
fst_gb_schema_dummy_pmdl_init: fixes bs :: "(('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) list"   and data :: "nat \<times> 'd"   and rc :: "nat"   and D :: "((('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) \<times> ('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) set" assumes "struct_spec sel ap ab compl"   and "compl_pmdl compl"   and "is_Groebner_basis (fst ` set gs)"   and "unique_idx (gs @ ab gs [] bs data) data"   and "rem_comps_spec (gs @ ab gs [] bs data) (rc, data)" shows "pmdl (fst ` set (fst (gb_schema_dummy (rc, data) D (ab gs [] bs data) (ap gs [] [] bs data)))) = pmdl (fst ` set (gs @ bs))"
Lab: fixes E :: "env"   and dt :: "loc \<Rightarrow> ty option"   and c :: "stmt"   and l :: "jump" assumes "E,dt\<Turnstile>c\<Colon>\<surd>" shows "E,dt\<Turnstile>l\<bullet> c\<Colon>\<surd>"
remove: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and z :: "'b"   and A :: "'c set"   and x :: "'c"   and g :: "'c \<Rightarrow> 'b" assumes "abel_semigroup_set f z"   and "finite A"   and "x \<in> A" shows "abel_semigroup_set.F f z g A = f (g x) (abel_semigroup_set.F f z g (A - {x}))"
induct: fixes x :: "('e, 'f, 'g) Tree\<^sub>\<alpha>"   and P :: "('e, 'f, 'g) Tree\<^sub>\<alpha> \<Rightarrow> bool" assumes "hereditarily_fs x"   and "\<And>tset\<^sub>\<alpha>. \<lbrakk>finite (supp tset\<^sub>\<alpha>); \<And>t\<^sub>\<alpha>. t\<^sub>\<alpha> \<in> set_bset tset\<^sub>\<alpha> \<Longrightarrow> hereditarily_fs t\<^sub>\<alpha>; \<And>t\<^sub>\<alpha>. t\<^sub>\<alpha> \<in> set_bset tset\<^sub>\<alpha> \<Longrightarrow> P t\<^sub>\<alpha>\<rbrakk> \<Longrightarrow> P (Conj\<^sub>\<alpha> tset\<^sub>\<alpha>)"   and "\<And>t\<^sub>\<alpha>. \<lbrakk>hereditarily_fs t\<^sub>\<alpha>; P t\<^sub>\<alpha>\<rbrakk> \<Longrightarrow> P (Not\<^sub>\<alpha> t\<^sub>\<alpha>)"   and "\<And>\<phi>. P (Pred\<^sub>\<alpha> \<phi>)"   and "\<And>t\<^sub>\<alpha> \<alpha>. \<lbrakk>hereditarily_fs t\<^sub>\<alpha>; P t\<^sub>\<alpha>\<rbrakk> \<Longrightarrow> P (Act\<^sub>\<alpha> \<alpha> t\<^sub>\<alpha>)" shows "P x"
permute_minus_cancel_1: fixes permute :: "perm \<Rightarrow> 'e \<Rightarrow> 'e"   and p :: "perm"   and x :: "'e" assumes "class.pt permute" shows "permute p (permute (- p) x) = x"
nn_integral_density: fixes f :: "'a \<Rightarrow> ennreal"   and M :: "'a measure"   and g :: "'a \<Rightarrow> ennreal" assumes "f \<in> borel_measurable M"   and "g \<in> borel_measurable M" shows "integral\<^sup>N (density M f) g = \<integral>\<^sup>+ x. f x * g x \<partial>M"
HPhase1or2Write_def: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and d :: "Disk" shows "HPhase1or2Write s s' p d = (Phase1or2Write s s' p d \<and> HNextPart s s')"
term_brackets_ConsI: fixes term_bracket :: "nat \<Rightarrow> dB \<Rightarrow> dB option"   and vs :: "nat list"   and t :: "dB"   and t' :: "dB"   and v :: "nat"   and t'' :: "dB" assumes "bracket_abstraction term_bracket"   and "bracket_abstraction.term_brackets term_bracket vs t = Some t'"   and "term_bracket v t' = Some t''" shows "bracket_abstraction.term_brackets term_bracket (v # vs) t = Some t''"
in_production_imp_produces: fixes A :: "'b"   and N :: "'b literal multiset set"   and C :: "'b literal multiset" assumes "A \<in> production N C" shows "production N C = {A}"
weakening_more_automated: fixes \<Gamma> :: "binding list"   and S :: "ty"   and T :: "ty"   and \<Delta> :: "binding list" assumes "\<Gamma>\<turnstile>S<:T"   and "\<turnstile> \<Delta> ok"   and "\<Delta> extends \<Gamma>" shows "\<Delta>\<turnstile>S<:T"
alw_def: shows "alw \<equiv> \<lambda>\<phi>. gfp (\<lambda>p x. \<exists>xs. x = xs \<and> \<phi> xs \<and> p (stl xs))"
Abs_check_instr_inverse: fixes y :: "'a instr \<Rightarrow> (nat \<times> nat \<times> 'a instr list \<times> (nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list) prog \<Rightarrow> 'b \<Rightarrow> 'a val list \<Rightarrow> 'a val list \<Rightarrow> String.literal \<Rightarrow> String.literal \<Rightarrow> nat \<Rightarrow> 'a frame list \<Rightarrow> bool" assumes "y \<in> Collect wf_ci" shows "ci_app (Abs_check_instr y) = y"
set_walks_def: shows "set_walks = {p. awalk u p v}"
simps: fixes i :: "char list" shows "internal_iface_name_to_set i = (if \<not> iface_name_is_wildcard i then {i} else {butlast i @ cs |cs. True})"
subst_no_occs: fixes v :: "'b"   and t :: "'b trm"   and s :: "'b trm" assumes "\<not> Var v \<prec> t"   and "Var v \<noteq> t" shows "t \<lhd> [(v, s)] = t"
tensor_mat_adjoint: fixes m1 :: "'b mat"   and m2 :: "'b mat" assumes "m1 \<in> carrier_mat ps_P.d1 ps_P.d1"   and "m2 \<in> carrier_mat ps_P.d2 ps_P.d2" shows "adjoint (ps_P.tensor_mat m1 m2) = ps_P.tensor_mat (adjoint m1) (adjoint m2)"
consistent_def: fixes ls :: "lit set" shows "consistent ls = (\<nexists>x. Pos x \<in> ls \<and> Neg x \<in> ls)"
finfun_All_const: fixes b :: "bool" shows "finfun_All (finfun_const b) = b"
assms_2: shows "\<I>_ideal \<equiv> \<lambda>\<eta>. \<I>_uniform UNIV {None, Some \<eta>}"
map_comp0: fixes g1 :: "'i \<Rightarrow> 'e"   and f1 :: "'c \<Rightarrow> 'i"   and g2 :: "'j \<Rightarrow> 'h"   and f2 :: "'g \<Rightarrow> 'j"   and g3 :: "'k \<Rightarrow> 'f"   and f3 :: "'d \<Rightarrow> 'k" shows "map_pre_coG\<^sub>M (g1 \<circ> f1) (g2 \<circ> f2) (g3 \<circ> f3) = map_pre_coG\<^sub>M g1 g2 g3 \<circ> map_pre_coG\<^sub>M f1 f2 f3"
simps: fixes prec :: "nat"   and x :: "float" shows "lb_ln prec x = (if x \<le> 0 then None else if x < 1 then Some (- the (ub_ln prec (float_divr prec 1 x))) else let horner = \<lambda>x. float_round_down prec (x * lb_ln_horner prec (get_even prec) 1 x) in if x \<le> Float 3 (- 1) then Some (horner (x - 1)) else if x < Float 1 1 then Some (float_round_down prec (horner (Float 1 (- 1)) + horner (max (x * lapprox_rat prec 2 3 - 1) 0))) else let l = bitlen (mantissa x) - 1 in Some (float_plus_down prec (float_round_down prec (lb_ln2 prec * Float (exponent x + l) 0)) (horner (Float (mantissa x) (- l) - 1))))"
ref_validity_def: fixes A :: "'a attree" shows "\<turnstile>\<^sub>V A \<equiv> \<exists>A'. A \<sqsubseteq>\<^sub>V A'"
properties_for_sort: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and ys :: "'a list"   and xs :: "'a list" assumes "class.linorder less_eq less"   and "mset ys = mset xs"   and "linorder.sorted less_eq ys" shows "linorder.sort_key less_eq (\<lambda>x. x) xs = ys"
finite_bounded_log: fixes b :: "real"   and w :: "complex" shows "finite {z. cmod z \<le> b \<and> exp z = w}"
trans: fixes \<sigma> :: "nat \<Rightarrow> 'g"   and i :: "nat"   and s :: "'s"   and \<sigma>' :: "nat \<Rightarrow> 'g"   and s' :: "'s"   and a :: "'m seq_action" assumes "\<sigma> i = fst (sr s)"   and "\<sigma>' i = fst (sr s')"   and "(s, a, s') \<in> automaton.trans (np i)" shows "((\<sigma>, snd (sr s)), a, \<sigma>', snd (sr s')) \<in> automaton.trans (onp i)"
BinOp1Reds: fixes e :: "char list exp"   and s :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and e' :: "char list exp"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and bop :: "bop"   and e\<^sub>2 :: "char list exp" assumes "P \<turnstile> \<langle>e,s\<rangle> \<rightarrow>* \<langle>e',s'\<rangle>" shows "P \<turnstile> \<langle>e \<guillemotleft>bop\<guillemotright> e\<^sub>2,s\<rangle> \<rightarrow>* \<langle>e' \<guillemotleft>bop\<guillemotright> e\<^sub>2,s'\<rangle>"
for_all_def: fixes f :: "'a \<Rightarrow> bool"   and xs :: "'a list" shows "for_all f xs = (\<forall>x\<in>set xs. f x)"
gbm_ops_impl: shows "StdGraph hlg_gbm.gbm_ops"
input2state_exists: fixes x1 :: "nat"   and a :: "value" shows "\<exists>i. input2state i $ x1 = Some a"
utest_intro: fixes hcond :: "'a hrel \<Rightarrow> 'a hrel"   and utp_unit :: "'a hrel"   and P :: "'a hrel" assumes "utp_theory_unital hcond utp_unit"   and "utp_unit \<sqsubseteq> P" shows "utp_theory_units.utp_test utp_unit P"
negl_def: fixes a :: "'a" shows "negl a = a l\<rightarrow> (0::'a)"
supp_some: fixes x :: "'e" shows "supp (Some x) = supp x"
is_volatile_Read\<^sub>s\<^sub>b_simps_2: fixes volatile :: "bool"   and a :: "nat"   and sop :: "nat set \<times> ((nat \<Rightarrow> nat option) \<Rightarrow> nat)"   and v :: "nat"   and A :: "nat set"   and L :: "nat set"   and R :: "nat set"   and W :: "nat set" shows "is_volatile_Read\<^sub>s\<^sub>b (Write\<^sub>s\<^sub>b volatile a sop v A L R W) = False"
notfinal: shows "abc_notfinal (abc_steps_l (0, lm) A n) A"
preserves_dom: fixes f :: "'a" assumes "B.arr f" shows "B.dom (trg\<^sub>B f) = trg\<^sub>B (B.dom f)"
Zfun_mult_left: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and a :: "'b" assumes "Zfun f F" shows "Zfun (\<lambda>x. f x * a) F"
assms_2: shows "preprm_compose \<pi> P =p preprm_compose \<sigma> S"
prime_elem_not_zeroI: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and p :: "'b" assumes "class.comm_semiring_1 times one plus zero"   and "comm_semiring_1.prime_elem times one zero p" shows "p \<noteq> zero"
Rep_perm_add: fixes p1 :: "perm"   and p2 :: "perm" shows "Rep_perm (p1 + p2) = Rep_perm p1 \<circ> Rep_perm p2"
IH: fixes A :: "('a \<times> ('a, 'b) list_op) list" assumes "list_ops A"   and "list_ops ops"   and "set A \<subseteq> set ops" shows "set (insertions A) \<subseteq> set (insertions ops)"
IH: fixes S :: "'node SDG_node set" assumes "n'' \<in> set ns"   and "nx \<in> S" shows "n'' \<in> HRB_slice S"
new_keys_not_used: fixes K :: "nat"   and evs :: "event list" assumes "Key K \<notin> used evs"   and "K \<in> symKeys"   and "evs \<in> kerbIV_gets" shows "K \<notin> keysFor (parts (knows Spy evs))"
Q: shows "Q \<equiv> P\<^sub>n\<^sub>o\<^sub>r\<^sub>m\<lbrakk>Tr\<^sub>2,Up\<^sub>2\<rbrakk>"
subspace_def: fixes U :: "'b set"   and V :: "'b set" shows "Subspace.subspace U V \<equiv> (U \<noteq> {} \<and> U \<subseteq> V) \<and> (\<forall>x y. x \<in> U \<longrightarrow> y \<in> U \<longrightarrow> x + y \<in> U) \<and> (\<forall>x a. x \<in> U \<longrightarrow> a \<cdot> x \<in> U)"
f_im: fixes R :: "('h, 'i) ring_scheme"   and M :: "('h, 'j, 'k) module_scheme"   and N :: "('h, 'l, 'm) module_scheme"   and f :: "'j \<Rightarrow> 'l"   and v :: "'j" assumes "mod_hom R M N f"   and "v \<in> carrier M" shows "f v \<in> carrier N"
prv_eqv_sym: fixes \<phi> :: "'fmla"   and \<chi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "\<chi> \<in> fmla"   and "prv (eqv \<phi> \<chi>)" shows "prv (eqv \<chi> \<phi>)"
decisive_onI: fixes A :: "'a set"   and f :: "'a set \<Rightarrow> 'a set" assumes "\<And>B. \<lbrakk>B \<subseteq> A; B \<noteq> {}\<rbrakk> \<Longrightarrow> f B \<noteq> {}" shows "decisive_on A f"
root_int_pos: fixes x :: "int"   and p :: "nat"   and y :: "int"   and ys :: "int list" assumes "0 \<le> x"   and "root_int p x = y # ys" shows "0 \<le> y"
dom_eq_empty_conv: fixes f :: "'a \<Rightarrow> 'b option" shows "(dom f = {}) = (f = \<epsilon>)"
simps_2: fixes i :: "iface" shows "compress_pos_interfaces [i] = Some i"
bounded_linear_imp_has_derivative: fixes f :: "'a \<Rightarrow> 'b"   and net :: "'a filter" assumes "bounded_linear f" shows "(f has_derivative f) net"
id_is_local_ring_morphism: fixes A :: "'c set"   and add :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and mult :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and one :: "'c" assumes "local_ring A add mult zero one" shows "local_ring_morphism (identity A) A add mult zero one A add mult zero one"
eq: fixes x :: "'e"   and i :: "nat" assumes "x \<in> space N" shows "simple_bochner_integral M (\<lambda>y. s i (x, y)) = (\<Sum>z\<in>s i ` (space N \<times> space M). Sigma_Algebra.measure M {y \<in> space M. s i (x, y) = z} *\<^sub>R z)"
emb_step_under_args_emb_step: fixes p :: "dir list"   and t :: "('a, 'b) tm"   and d :: "dir" assumes "\<not> list_all (\<lambda>x. x = dir.Left) p"   and "position_of t (p @ [d])" obtains i :: "nat" where "i < num_args t" and "args t ! i \<rightarrow>\<^sub>e\<^sub>m\<^sub>b args (emb_step_at p d t) ! i" and "\<And>j. \<lbrakk>j < num_args t; i \<noteq> j\<rbrakk> \<Longrightarrow> args t ! j = args (emb_step_at p d t) ! j"
induced_subgraphI: fixes H :: "('c, 'd) pre_digraph"   and G :: "('c, 'd) pre_digraph" assumes "subgraph H G"   and "arcs H = {e \<in> arcs G. tail G e \<in> verts H \<and> head G e \<in> verts H}" shows "induced_subgraph H G"
roofed_circI: fixes x :: "'k"   and \<Gamma> :: "('k, 'l) web_scheme"   and T :: "'k set" assumes "x \<in> RF\<^bsub>\<Gamma>\<^esub> T"   and "x \<in> T \<Longrightarrow> \<not> essential \<Gamma> (B \<Gamma>) T x" shows "x \<in> RF\<^sup>\<circ>\<^bsub>\<Gamma>\<^esub> T"
inc_above_dec_above: fixes x :: "nat"   and b :: "nat"   and i :: "nat" assumes "x < b \<or> b + i \<le> x" shows "inc_above b i (dec_above b i x) = x"
trancl_insert2: fixes a :: "'a"   and b :: "'a"   and r :: "('a \<times> 'a) set" shows "(insert (a, b) r)\<^sup>+ = r\<^sup>+ \<union> {(x, y). ((x, a) \<in> r\<^sup>+ \<or> x = a) \<and> ((b, y) \<in> r\<^sup>+ \<or> y = b)}"
measurable_cong_sets: fixes M :: "'a measure"   and M' :: "'a measure"   and N :: "'b measure"   and N' :: "'b measure" assumes "sets M = sets M'"   and "sets N = sets N'" shows "M \<rightarrow>\<^sub>M N = M' \<rightarrow>\<^sub>M N'"
ln_approx_bounds: fixes x :: "real"   and n :: "nat" assumes "1 < x" shows "ln x \<in> {\<Sum>k<n. 2 * ((x - 1) / (x + 1)) ^ (2 * k + 1) / real (2 * k + 1)..(\<Sum>k<n. 2 * ((x - 1) / (x + 1)) ^ (2 * k + 1) / real (2 * k + 1)) + 2 * (((x - 1) / (x + 1)) ^ (2 * n + 1) / (1 - ((x - 1) / (x + 1))\<^sup>2) / real (2 * n + 1))}"
less_irrefl: fixes x :: "'a" shows "\<not> x \<prec> x"
vec_of_basis_enum_uminus: fixes b2 :: "'b" shows "vec_of_basis_enum (- b2) = - vec_of_basis_enum b2"
no_call_to_def: fixes c :: "char list"   and rs :: "'c rule list" shows "no_call_to c rs \<equiv> \<forall>r\<in>set rs. case get_action r of Call c' \<Rightarrow> c \<noteq> c' | _ \<Rightarrow> True"
fds_deriv_1: shows "fds_deriv 1 = 0"
img_mapsto: fixes a :: "'s" assumes "S.ide a" shows "S.some_img \<in> S.hom S.unity a \<rightarrow> S.Univ"
before_fun1: fixes f :: "'c \<Rightarrow> 'b"   and x :: "'a \<Rightarrow> 'c" shows "\<turnstile> $f<x> = f<$x>"