hcju/isabelleps huggingface dataset:
composite_pseudonatural_transformation_def: fixes V\<^sub>C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and H\<^sub>C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<a>\<^sub>C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<i>\<^sub>C :: "'c \<Rightarrow> 'c"   and src\<^sub>C :: "'c \<Rightarrow> 'c"   and trg\<^sub>C :: "'c \<Rightarrow> 'c"   and V\<^sub>D :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and H\<^sub>D :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and \<a>\<^sub>D :: "'d \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and \<i>\<^sub>D :: "'d \<Rightarrow> 'd"   and src\<^sub>D :: "'d \<Rightarrow> 'd"   and trg\<^sub>D :: "'d \<Rightarrow> 'd"   and F :: "'c \<Rightarrow> 'd"   and \<Phi>\<^sub>F :: "'c \<times> 'c \<Rightarrow> 'd"   and G :: "'c \<Rightarrow> 'd"   and \<Phi>\<^sub>G :: "'c \<times> 'c \<Rightarrow> 'd"   and H :: "'c \<Rightarrow> 'd"   and \<Phi>\<^sub>H :: "'c \<times> 'c \<Rightarrow> 'd"   and \<sigma>\<^sub>0 :: "'c \<Rightarrow> 'd"   and \<sigma>\<^sub>1 :: "'c \<Rightarrow> 'd"   and \<rho>\<^sub>0 :: "'c \<Rightarrow> 'd"   and \<rho>\<^sub>1 :: "'c \<Rightarrow> 'd" shows "composite_pseudonatural_transformation V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D F \<Phi>\<^sub>F G \<Phi>\<^sub>G H \<Phi>\<^sub>H \<sigma>\<^sub>0 \<sigma>\<^sub>1 \<rho>\<^sub>0 \<rho>\<^sub>1 \<equiv> (bicategory V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C \<and> bicategory V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D \<and> pseudofunctor V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D F \<Phi>\<^sub>F) \<and> (pseudofunctor V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D G \<Phi>\<^sub>G \<and> pseudofunctor V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D H \<Phi>\<^sub>H) \<and> pseudonatural_transformation V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D F \<Phi>\<^sub>F G \<Phi>\<^sub>G \<sigma>\<^sub>0 \<sigma>\<^sub>1 \<and> pseudonatural_transformation V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D G \<Phi>\<^sub>G H \<Phi>\<^sub>H \<rho>\<^sub>0 \<rho>\<^sub>1"
oo: fixes U :: "'a set" assumes "closedin (top_of_set S) U" shows "closedin (top_of_set T) (f ` U)"
vfsequence_vcons_vproductD_1: fixes xs :: "V"   and x :: "V"   and n :: "V"   and A :: "V \<Rightarrow> V" assumes "xs #\<^sub>\<circ> x \<in>\<^sub>\<circ> vproduct n A"   and "n \<in>\<^sub>\<circ> \<omega>" shows "xs \<in>\<^sub>\<circ> vproduct (vcard xs) A"
wf_FV_LESS: fixes idx :: "'n"   and \<phi> :: "('a, 'k) aformula"   and v :: "nat"   and k :: "'k" assumes "local.wf idx \<phi>"   and "v \<in> FV \<phi> k" shows "LESS k v idx"
bot_def: shows "bot = \<Sqinter>\<^sub>f\<^sub>i\<^sub>n UNIV"
is_iso_def: fixes G :: "('d, 'e) monoid_scheme"   and H :: "('f, 'g) monoid_scheme" shows "G \<cong> H = (Group.iso G H \<noteq> {})"
assms_2: fixes \<rho> :: "'a OCL_Types.type" assumes "\<tau> = Set \<rho>"   and "\<rho> < \<sigma>" shows "P"
reduced_GB_is_monic_set_Polys: fixes X :: "'x set"   and F :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set" assumes "finite X"   and "F \<subseteq> P[X]" shows "punit.is_monic_set (punit.reduced_GB F)"
arctan_upper_44_def: shows "arctan_upper_44 \<equiv> \<lambda>x. pi / 2 - (15159 + 147455 * x\<^sup>2 + 345345 * x ^ 4 + 225225 * x ^ 6) * x / (35 * (35 + 1260 * x\<^sup>2 + 6930 * x ^ 4 + 12012 * x ^ 6 + 6435 * x ^ 8))"
primrootI1: fixes u :: "'b list"   and r :: "'b list"   and k :: "nat" assumes "u = r \<^sup>@ Suc k"   and "primitive r" shows "\<rho> u = r"
R_def: fixes h :: "'g \<times> 'a"   and w :: "'h + 'b"   and e :: "'e" shows "\<Sigma>_OR.R h w e = init_OR h w \<bind> (\<lambda>(r, a). response_OR r w e \<bind> (\<lambda>z. return_spmf (a, e, z)))"
that: fixes x :: "('a \<times> 'c \<Rightarrow>\<^sub>L 'a) set" assumes "x \<in> xa" shows "poincare_mapsto P x S CXS PS"
qSwapAll_preserves_alphaAll: fixes X :: "('f, 'g, 'e, 'd, 'h) qTerm"   and A :: "('f, 'g, 'e, 'd, 'h) qAbs" assumes "FixVars TYPE('d) TYPE('e)" shows "(FixVars.qGood X \<longrightarrow> (\<forall>X' zs z1 z2. X #= X' \<longrightarrow> X #[[z1 \<and> z2]]_zs #= X' #[[z1 \<and> z2]]_zs)) \<and> (FixVars.qGoodAbs A \<longrightarrow> (\<forall>A' zs z1 z2. A $= A' \<longrightarrow> A $[[z1 \<and> z2]]_zs $= A' $[[z1 \<and> z2]]_zs))"
pow_eq_div2: fixes G :: "('c, 'd) monoid_scheme"   and x :: "'c"   and m :: "nat"   and n :: "nat" assumes "Group.group G"   and "x \<in> carrier G"   and "x [^]\<^bsub>G\<^esub> m = x [^]\<^bsub>G\<^esub> n" shows "x [^]\<^bsub>G\<^esub> (m - n) = \<one>\<^bsub>G\<^esub>"
rel_Grp: fixes A1 :: "'b set"   and f1 :: "'b \<Rightarrow> 'd"   and A2 :: "'c set"   and f2 :: "'c \<Rightarrow> 'e" shows "rel_sp\<^sub>\<mu> (BNF_Def.Grp A1 f1) (BNF_Def.Grp A2 f2) = BNF_Def.Grp {x. set1_sp\<^sub>\<mu> x \<subseteq> A1 \<and> set2_sp\<^sub>\<mu> x \<subseteq> A2} (map_sp\<^sub>\<mu> f1 f2)"
fromEnum_def: fixes x :: "'a" shows "fromEnum x \<equiv> the_index enum_class.enum x"
GodIsEssential: shows "\<lfloor>(\<lambda>w. \<forall>x. (\<forall>xa. \<P> xa w = (\<^bold>\<box>xa x) w) \<longrightarrow> (\<forall>xa. (\<^bold>\<box>xa x) w = (\<^bold>\<box>(\<lambda>v. \<forall>x. existsAt x v \<longrightarrow> (\<forall>xa. \<P> xa v = (\<^bold>\<box>xa x) v) \<longrightarrow> xa x v)) w))\<rfloor>"
rk_ext: fixes X :: "Points set" assumes "rk X \<le> 3" shows "\<exists>P. rk (X \<union> {P}) = rk X + 1"
onewpkt_completeTE: fixes \<sigma> :: "nat \<Rightarrow> 'd"   and s :: "'e net_state"   and i :: "nat"   and d :: "nat"   and di :: "nat"   and \<sigma>' :: "nat \<Rightarrow> 'd"   and s' :: "'e net_state"   and S :: "(((nat \<Rightarrow> 'd) \<times> 'e net_state) \<times> 'f node_action \<times> (nat \<Rightarrow> 'd) \<times> 'e net_state) set" assumes "((\<sigma>, s), i:newpkt(d, di), \<sigma>', s') \<in> ocnet_sos S" obtains K :: "nat set" where "((\<sigma>, s), {i}\<not>K:arrive(newpkt (d, di)), \<sigma>', s') \<in> S" and "\<forall>j. j \<notin> net_ips s \<longrightarrow> \<sigma>' j = \<sigma> j"
mk_rtrancl_def: fixes init :: "'b" shows "mk_rtrancl init \<equiv> mk_rtrancl_main init {}"
simps: fixes backtrack_s :: "'c \<Rightarrow> 'd \<Rightarrow> 'b"   and c :: "'c"   and asi_tv_ui :: "'a"   and s :: "'d" shows "backtrack_nsc backtrack_s c (asi_tv_ui, s) = (asi_tv_ui, backtrack_s c s)"
f_xs: shows "perm_restrict f (set xs) = list_succ xs"
gfp_least: fixes f :: "'b \<times> 'c \<Rightarrow> 'b \<times> 'c"   and X :: "'b \<times> 'c" assumes "\<And>u. less_eq_prod u (f u) \<Longrightarrow> less_eq_prod u X" shows "less_eq_prod (prod.gfp f) X"
accessible_in_RefT_simp: fixes G :: "prog"   and r :: "ref_ty"   and pack :: "pname" shows "G \<turnstile> RefT r accessible_in pack = G \<turnstile> r accessible_in' pack"
FORWEAK_elementwise_rule: fixes d :: "'a nres"   and XX :: "'b set"   and Inf_spec :: "'b \<Rightarrow> 'a nres"   and Q :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and comb :: "'a \<Rightarrow> 'a \<Rightarrow> 'a nres" assumes "nofail d"   and "\<And>X. X \<in> XX \<Longrightarrow> Inf_spec X \<le> SPEC (Q X)"   and "\<And>a b X Y. Q X a \<Longrightarrow> comb a b \<le> SPEC (Q X)"   and "\<And>a b X Y. Q X b \<Longrightarrow> comb a b \<le> SPEC (Q X)" shows "FORWEAK XX d Inf_spec comb \<le> SPEC (\<lambda>i. \<forall>x\<in>XX. Q x i)"
ran_restrict_def: fixes p :: "'c \<Rightarrow> 'd decision option"   and S :: "'d decision set" shows "p \<triangleright> S \<equiv> \<lambda>x. if p x \<in> Some ` S then p x else \<bottom>"
comp_def: fixes f :: "'s \<Rightarrow> 'r"   and g :: "'q \<Rightarrow> 's" shows "f \<circ> g = (\<lambda>x. f (g x))"
lookup_update: fixes S :: "'b Abs_State.st"   and x :: "char list"   and y :: "'b" shows "lookup (update S x y) = (lookup S)(x := y)"
intros_1: fixes F :: "'b formula" assumes "(F::'b formula) \<in> AX0" shows "(F::'b formula) \<in> AX10"
ipresAbs_def: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'd"   and hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> 'e"   and MOD :: "('f, 'g, 'varSort, 'h, 'i, 'var, 'd, 'e, 'j) model_scheme" shows "ipresAbs h hA MOD \<equiv> \<forall>xs x s X. isInBar (xs, s) \<and> wls s X \<longrightarrow> hA (Abs xs x X) = igAbs MOD xs x (h X)"
permutation_orbit_step: fixes f :: "'e \<Rightarrow> 'e"   and s :: "'e" assumes "permutation f" shows "orbit f (f s) = orbit f s"
2: shows "coreflexive (x\<^sup>\<star> \<sqinter> x\<^sup>T * x)"
simps_9: fixes \<phi> :: "formula"   and I :: "\<I>"   and \<psi> :: "formula" shows "convert_multiway (Until \<phi> I \<psi>) = Until (convert_multiway \<phi>) I (convert_multiway \<psi>)"
simps: fixes a1 :: "enat"   and a2 :: "enat" shows "Le_enatp a1 a2 = ((\<exists>n. a1 = 0 \<and> a2 = n) \<or> (\<exists>m n k. a1 = eSuc m \<and> a2 = n + k \<and> Le_enatp m n \<and> k \<noteq> 0))"
arrow_of_spans_axioms: shows "arrow_of_spans (\<cdot>) \<rho>"
types_preserved_e: fixes s :: "s"   and vs :: "v list"   and es :: "e list"   and i :: "nat"   and s' :: "s"   and vs' :: "v list"   and es' :: "e list"   and \<S> :: "s_context"   and ts :: "t list" assumes "\<lparr>s;vs;es\<rparr> \<leadsto>_ i \<lparr>s';vs';es'\<rparr>"   and "store_typing s \<S>"   and "\<S>\<bullet>None \<tturnstile>_ i vs;es : ts" shows "\<S>\<bullet>None \<tturnstile>_ i vs';es' : ts"
mono: fixes t :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and u :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" assumes "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 (T t) (T u)"
ord_divides': fixes a :: "nat"   and d :: "nat"   and n :: "nat" shows "[a ^ d = Suc 0] (mod n) = (Pocklington.ord n a dvd d)"
interchange: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and tensor :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unity :: "'b"   and lunit :: "'b \<Rightarrow> 'b"   and runit :: "'b \<Rightarrow> 'b"   and assoc :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and g :: "'b"   and f :: "'b"   and g' :: "'b"   and f' :: "'b" assumes "elementary_monoidal_category C tensor unity lunit runit assoc"   and "partial_magma.arr C (C g f)"   and "partial_magma.arr C (C g' f')" shows "C (tensor g g') (tensor f f') = tensor (C g f) (C g' f')"
choose_subspace_of_subspace: fixes n :: "nat"   and S :: "('a, 'b) vec set" assumes "n \<le> vec.dim S" obtains T :: "('a, 'b) vec set" where "vec.subspace T" and "T \<subseteq> vec.span S" and "vec.dim T = n"
ref_def: fixes l :: "Location" shows "ref l = (case l of objLoc cf a \<Rightarrow> objV (cls cf) a | staticLoc f \<Rightarrow> nullV | arrLenLoc T a \<Rightarrow> arrV T a | arrLoc T a i \<Rightarrow> arrV T a)"
subset_boolean_algebra_axioms: shows "class.subset_boolean_algebra (\<lambda>x. ! (d x)) (\<squnion>)"
pdevs_apply_set_slist: fixes x :: "nat \<times> 'd"   and xs :: "(nat, 'd) slist" assumes "x \<in> set_slist xs" shows "snd x = pdevs_apply (Pdevs xs) (fst x)"
dterm_sterm_dfree: fixes \<theta> :: "('a, 'b) trm"   and I :: "('a, 'c, 'b) interp"   and \<nu> :: "(real, 'b) vec"   and \<nu>' :: "(real, 'b) vec" assumes "dfree \<theta>" shows "sterm_sem I \<theta> \<nu> = dterm_sem I \<theta> (\<nu>, \<nu>')"
dec_def: fixes dec_alg :: "'b \<Rightarrow> 'c \<Rightarrow> 'a option" shows "cipher.dec dec_alg \<equiv> CNV (stateless_callee (\<lambda>_. Pause (Inr ()) (\<lambda>c'. case c' of Inl a \<Rightarrow> Done None | Inr None \<Rightarrow> Done None | Inr (Some c) \<Rightarrow> Pause (Inl ()) Done \<bind> (\<lambda>k. Done (dec_alg (projl k) c))))) ()"
lconcat_LNil: shows "lconcat LNil = LNil"
dualization_unl_1: fixes F :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool"   and w :: "nat \<Rightarrow> 'a" shows "(\<not> (\<box>F) w) = (\<diamond>\<not> F) w"
a_inv_closed: fixes x :: "'a" assumes "x \<in> carrier G" shows "h (\<ominus> x) \<in> carrier H"
finite_code: fixes A :: "'e set" assumes "class.finite TYPE('e)" shows "finite A = True"
thunksE: fixes x :: "Vars.var"   and \<Gamma> :: "(Vars.var \<times> exp) list"   and e :: "exp" assumes "x \<in> thunks \<Gamma>"   and "map_of \<Gamma> x = Some e" shows "\<not> isVal e"
p_eq: fixes p :: "((int \<times> int) set set, int) Combinators list" assumes "p \<noteq> []" shows "list2policy (map C (rev p)) = C (list2FWpolicy p)"
Is': assumes "LI dir s' (lhs (eq_for_lvar (\<T> s') x\<^sub>i)) \<in> indices_state s'" shows "set I \<subseteq> indices_state s'"
iMODb_imp_iMOD: fixes x :: "nat"   and r :: "nat"   and m :: "nat"   and c :: "nat" assumes "x \<in> [ r, mod m, c ]" shows "x \<in> [ r, mod m ]"
roots_def: shows "roots = {x. poly p x = 0} - {x}"
minus_in_Bounded_vec: fixes x :: "'b vec"   and b :: "'b" shows "(- x \<in> Bounded_vec b) = (x \<in> Bounded_vec b)"
assms_5: shows "X #[y // x]_xs = X' #[y // x']_xs"
integrable_spike: fixes f :: "'b \<Rightarrow> 'c"   and T :: "'b set"   and S :: "'b set"   and g :: "'b \<Rightarrow> 'c" assumes "f integrable_on T"   and "negligible S"   and "\<And>x. x \<in> T - S \<Longrightarrow> g x = f x" shows "g integrable_on T"
mem_existence_ivl_iv_defined_2: fixes t :: "real"   and x0 :: "'a" assumes "t \<in> existence_ivl t0 x0" shows "x0 \<in> X"
Gromov_boundary_abs_converging: fixes x :: "'a Gromov_completion"   and u :: "nat \<Rightarrow> 'a" assumes "x \<in> Gromov_boundary"   and "abs_Gromov_completion u = x"   and "Gromov_completion_rel u u" shows "Gromov_converging_at_boundary u"
simps_1: fixes v :: "'b"   and s :: "('b \<times> 'b trm) list" shows "Var v \<lhd> s = assoc v (Var v) s"
rep_eq: fixes x :: "('a \<Rightarrow>\<^sub>0 'c) \<Rightarrow> ('a \<Rightarrow>\<^sub>0 'c) \<Rightarrow> bool"   and xa :: "'a \<Rightarrow>\<^sub>0 'c"   and xb :: "'a \<Rightarrow>\<^sub>0 'c" shows "dord_pm x xa xb = dord_fun (map_fun Abs_poly_mapping id \<circ> x \<circ> Abs_poly_mapping) (lookup xa) (lookup xb)"
id_stake_snth_sdrop: fixes s :: "'c stream"   and i :: "nat" shows "s = stake i s @- s !! i ## sdrop (Suc i) s"
mem_val_alt_10_mem_equal_0: fixes s1 :: "'b sparc_state"   and s2 :: "'b sparc_state"   and pa :: "word36" assumes "mem_equal s1 s2 pa" shows "mem_val_alt 10 (pa AND 68719476732) s1 = mem_val_alt 10 (pa AND 68719476732) s2"
equation_5: fixes x1 :: "com \<times> (char list \<Rightarrow> int)"   and x2 :: "nat"   and x3 :: "char list \<Rightarrow> int" shows "x1 \<Rightarrow> x2 \<Down> x3 = Predicate.holds (big_step_t_Pii_i_i x1 x2 x3)"
HEndPhase1_HInv5: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and q :: "Proc" assumes "HEndPhase1 s s' p"   and "HInv5 s"   and "Inv1 s"   and "Inv2a s"   and "Inv2a s'"   and "Inv2b s"   and "Inv2c s"   and "HInv3 s"   and "HInv4 s" shows "HInv5_inner s' q"
poly_gt_def: fixes p :: "('b monom \<times> 'a) list"   and q :: "('b monom \<times> 'a) list" shows "p >p q = (\<forall>\<alpha>. pos_assign \<alpha> \<longrightarrow> eval_poly \<alpha> p \<succ> eval_poly \<alpha> q)"
gen_new_keys_not_analzd: fixes K :: "nat"   and evs :: "event list"   and KK :: "nat set" assumes "Key K \<notin> used evs"   and "K \<in> symKeys"   and "evs \<in> set_pur" shows "K \<notin> keysFor (analz (Key ` KK \<union> knows Spy evs))"
intros_4: fixes n :: "nat"   and P :: "nat \<Rightarrow> complex mat"   and Q :: "complex mat"   and S :: "com list"   and M :: "nat \<Rightarrow> complex mat" assumes "\<And>k. k < n \<Longrightarrow> is_quantum_predicate (P k)"   and "is_quantum_predicate Q"   and "\<And>k. k < n \<Longrightarrow> \<turnstile>\<^sub>p {P k} S ! k {Q}" shows "\<turnstile>\<^sub>p {matrix_sum d (\<lambda>k. adjoint (M k) * P k * M k) n} Measure n M S {Q}"
Red1FAss: fixes h :: "'heap"   and a :: "'addr"   and D :: "String.literal"   and F :: "String.literal"   and v :: "'addr val"   and h' :: "'heap"   and uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and l :: "'addr val list" assumes "heap_write h a (CField D F) v h'" shows "uf,P,t \<turnstile>1 \<langle>addr a\<bullet>F{D} := Val v,(h, l)\<rangle> -\<lbrace>WriteMem a (CField D F) v\<rbrace>\<rightarrow> \<langle>unit,(h', l)\<rangle>"
monoseq_def: fixes X :: "nat \<Rightarrow> 'b" shows "monoseq X = ((\<forall>m n. m \<le> n \<longrightarrow> X m \<le> X n) \<or> (\<forall>m n. m \<le> n \<longrightarrow> X n \<le> X m))"
filt_equiv_borel_AE_eq_iff: fixes F :: "'b \<Rightarrow> 'c measure"   and M :: "'c measure"   and N :: "'c measure"   and f :: "'c \<Rightarrow> real"   and t :: "'b"   and g :: "'c \<Rightarrow> real" assumes "filt_equiv F M N"   and "f \<in> borel_measurable (F t)"   and "g \<in> borel_measurable (F t)"   and "prob_space N"   and "prob_space M" shows "AEeq M f g = AEeq N f g"
mpt_factor_is_qmpt_factor: fixes proj :: "(int \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> real) \<Rightarrow> 'b"   and M2 :: "'b measure"   and T2 :: "'b \<Rightarrow> 'b" assumes "I.mpt_factor proj M2 T2" shows "I.qmpt_factor proj M2 T2"
disj_zero_right: fixes x :: "'a" shows "x \<^bold>\<squnion> \<zero> = x"
C: shows "0 < offs_num (length ns) (x # xs) index key mi ma j"
program_equalityE: fixes F :: "'b program"   and G :: "'b program" assumes "F = G" obtains "Init F = Init G" and "Acts F = Acts G" and "AllowedActs F = AllowedActs G"
L2_set_cong: fixes A :: "'c set"   and B :: "'c set"   and f :: "'c \<Rightarrow> real"   and g :: "'c \<Rightarrow> real" assumes "A = B"   and "\<And>x. x \<in> B \<Longrightarrow> f x = g x" shows "L2_set f A = L2_set g B"
trcl_single: fixes c :: "'a"   and a :: "'b"   and c' :: "'a"   and t :: "('a \<times> 'b \<times> 'a) set" shows "((c, [a], c') \<in> trcl t) = ((c, a, c') \<in> t)"
hd_drop_conv_nth: fixes n :: "nat"   and xs :: "'c list" assumes "n < length xs" shows "hd (drop n xs) = xs ! n"
ID_ccompare: fixes ccompare :: "('b \<Rightarrow> 'b \<Rightarrow> order) option"   and c :: "'b \<Rightarrow> 'b \<Rightarrow> order" assumes "class.ccompare ccompare"   and "ID ccompare = Some c" shows "class.linorder (le_of_comp c) (lt_of_comp c)"
right_diff_distrib': fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "a * (b - c) = a * b - a * c"
simps: fixes \<Gamma> :: "('d, 'e) \<I>"   and x :: "'f"   and a :: "('f, 'd, 'e) gpv" shows "resultsp_gpv \<Gamma> x a = ((\<exists>gpv. a = gpv \<and> Pure x \<in> set_spmf (the_gpv gpv)) \<or> (\<exists>out c gpv input. a = gpv \<and> IO out c \<in> set_spmf (the_gpv gpv) \<and> input \<in> responses_\<I> \<Gamma> out \<and> resultsp_gpv \<Gamma> x (c input)))"
assms: shows "(ic, p, q) \<notin> set (apply_ncrit crit data gs bs hs (xs @ (ic, q_in_bs, p, q) # ys))"
UP_subring_taylor_closed: fixes S :: "'a set"   and g :: "nat \<Rightarrow> 'a"   and a :: "'a" assumes "subring S R"   and "g \<in> carrier (UP R)"   and "\<And>n. g n \<in> S"   and "a \<in> S" shows "taylor_expansion R a g \<in> carrier (UP (R\<lparr>carrier := S\<rparr>))"
less_multiset\<^sub>D\<^sub>M_imp_mult: fixes M :: "'a multiset"   and N :: "'a multiset" assumes "less_multiset\<^sub>D\<^sub>M M N" shows "(M, N) \<in> mult {(x, y). x < y}"
interval_cart_1: fixes a :: "(real, 'a) vec"   and b :: "(real, 'a) vec" shows "box a b = {x. \<forall>i. a $ i < x $ i \<and> x $ i < b $ i}"
inverts_components: fixes a :: "'d" assumes "D.ide a" shows "C.inverse_arrows (\<tau> a) (\<tau>'.map a)"
tr\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s_has_pair_lists: fixes G :: "(('fun, 'var) Term.term \<times> ('fun, 'var) Term.term) list"   and F :: "(('fun, 'var) Term.term \<times> ('fun, 'var) Term.term) list"   and D :: "(('fun, 'var) Term.term \<times> ('fun, 'var) Term.term) list"   and g :: "('fun, 'var) Term.term \<times> ('fun, 'var) Term.term" assumes "G \<in> set (tr\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s F D)"   and "g \<in> set G" shows "\<exists>f\<in>set F. \<exists>d\<in>set D. g = (pair f, pair d)"
get_parent_reads_pointers: fixes ptr :: "(_) object_ptr" shows "preserved (get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t ptr RObject.nothing) \<in> get_parent_locs"
independent_explicit: fixes B :: "('c, 'd) vec set" shows "vec.independent B = (finite B \<and> (\<forall>c. (\<Sum>v\<in>B. c v *s v) = 0 \<longrightarrow> (\<forall>v\<in>B. c v = (0::'c))))"
assms_1: shows "f \<circ> Inl = g \<circ> Inl"
elims: fixes x :: "('b \<Rightarrow> 'a) option"   and xa :: "'b option"   and y :: "'a option" assumes "Applicative.ap x xa = y" obtains f :: "'b \<Rightarrow> 'a" and xb :: "'b" where "x = pure f" and "xa = pure xb" and "y = pure (f xb)"   | uv_ :: "'b option" where "x = None" and "xa = uv_" and "y = None"   | uu_ :: "('b \<Rightarrow> 'a) option" where "x = uu_" and "xa = None" and "y = None"
list_eq_iff_nth_eq: fixes xs :: "'c list"   and ys :: "'c list" shows "(xs = ys) = (length xs = length ys \<and> (\<forall>i<length xs. xs ! i = ys ! i))"
is_functor: shows "functor (\<cdot>\<^sub>A\<^sub>2\<^sub>x\<^sub>A\<^sub>1) (\<cdot>\<^sub>A\<^sub>1\<^sub>x\<^sub>A\<^sub>2) S.map"
iso_blocks_map_inj: fixes x :: "'a set"   and y :: "'a set" assumes "x \<in># \<B>"   and "y \<in># \<B>"   and "\<pi> ` x = \<pi> ` y" shows "x = y"
leadsTo_Basis': fixes F :: "'b program"   and A :: "'b set"   and B :: "'b set" assumes "F \<in> A co A \<union> B"   and "F \<in> transient A" shows "F \<in> A \<longmapsto> B"
arbitrary_union_of_idempot: fixes P :: "'b set \<Rightarrow> bool" shows "arbitrary union_of arbitrary union_of P = arbitrary union_of P"
quat_eq_iff: fixes x :: "quat"   and y :: "quat" shows "(x = y) = (quat.Re x = quat.Re y \<and> Im1 x = Im1 y \<and> Im2 x = Im2 y \<and> Im3 x = Im3 y)"
assms_1: shows "evalRecvArgs rargs xs g p = (g', p')"
Inf_measure_def: fixes A :: "'a measure set" shows "Inf A = Sup {x. \<forall>a\<in>A. x \<le> a}"
connected_space_iff_components_eq: fixes X :: "'c topology" shows "connected_space X = (\<forall>C\<in>connected_components_of X. \<forall>C'\<in>connected_components_of X. C = C')"
lowest_vertex_eq_mirror_last: fixes X :: "(real \<times> real) \<times> (real \<times> real) pdevs" assumes "half_segments_of_aform X \<noteq> []" shows "lowest_vertex (fst X, nlex_pdevs (snd X)) = mirror_point (fst X) (snd (last (half_segments_of_aform X)))"
positively_invariant_pos_quad: shows "g.positively_invariant pos_quad"
span_mono: fixes A :: "'c set"   and B :: "'c set" assumes "A \<subseteq> B" shows "span A \<subseteq> span B"
right_neutral: fixes a :: "'a \<Rightarrow> ('b, 'a) Term.term" shows "a \<circ>\<^sub>s Var = a"
simps: 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 :: "char list \<Rightarrow> ty option"   and a2 :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and a3 :: "expr list"   and a4 :: "ty list" shows "(P,a1,a2 \<turnstile> a3 [:'] a4) = ((\<exists>E h. a1 = E \<and> a2 = h \<and> a3 = [] \<and> a4 = []) \<or> (\<exists>E h e T es Ts. a1 = E \<and> a2 = h \<and> a3 = e # es \<and> a4 = T # Ts \<and> P,E,h \<turnstile> e :' T \<and> P,E,h \<turnstile> es [:'] Ts))"
inf_sup_aci_6: fixes inf :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and sup :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and x :: "'g"   and y :: "'g"   and z :: "'g" assumes "class.lattice inf less_eq less sup" shows "sup (sup x y) z = sup x (sup y z)"
euclidean_dist_l2: fixes x :: "'a"   and y :: "'a" shows "dist x y = L2_set (\<lambda>i. dist (x \<bullet> i) (y \<bullet> i)) Basis"