hcju/isabelleps huggingface dataset:
R_O_Id: fixes R :: "('a \<times> 'c) set" shows "R O Id = R"
phl_cons2: fixes w :: "'a"   and x :: "'a"   and z :: "'a"   and y :: "'a" assumes "w \<le> x"   and "z ; y \<le> y ; w" shows "z ; y \<le> y ; x"
simps_3: fixes x :: "'b"   and y :: "'b" shows "(extB.Val x \<le> extB.Val y) = (x \<le> y)"
power_mult: fixes a :: "'a"   and m :: "nat"   and n :: "nat" shows "a +^ (m * n) = (a +^ m) +^ n"
ladder_cut_prefix_n: fixes index :: "nat"   and L :: "(nat \<times> nat \<times> nat) list"   and d :: "nat" assumes "index < length L"   and "0 < length L" shows "ladder_n (ladder_cut_prefix d L) index = ladder_n L index"
comp_fun_commute_sample_var: fixes X :: "'v \<Rightarrow> int pmf" shows "comp_fun_commute (\<lambda>x m. bind (memo (sample_var X) x) (\<lambda>_. m))"
less_imp_le: fixes x :: "nat list"   and y :: "nat list" assumes "x <\<^sub>v y" shows "x \<le>\<^sub>v y"
below_Cinf: fixes r :: "C" shows "r \<sqsubseteq> C\<^sup>\<infinity>"
CompDiag_preserves_Diag_2: fixes t :: "'a local.term"   and u :: "'a local.term" assumes "Diag t"   and "Diag u"   and "Dom t = Cod u" shows "Dom (t \<^bold>\<lfloor>\<^bold>\<cdot>\<^bold>\<rfloor> u) = Dom u"
simps_2: fixes nil :: "'a"   and lnk :: "'b \<Rightarrow> 'a \<Rightarrow> 'a"   and lbl :: "'a \<Rightarrow> 'b"   and x :: "'a"   and y :: "'a"   and l :: "'a list" assumes "pointer nil" shows "pointer.stack nil lnk lbl x (y # l) = (x \<noteq> nil \<and> x = y \<and> x \<notin> set l \<and> pointer.stack nil lnk lbl (lnk (lbl x) x) l)"
add_uminus_conv_diff: fixes minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and uminus :: "'d \<Rightarrow> 'd"   and a :: "'d"   and b :: "'d" assumes "class.group_add minus plus zero uminus" shows "plus a (uminus b) = minus a b"
weakOutput: fixes \<Psi> :: "'b"   and M :: "'a"   and K :: "'a"   and Q :: "('a, 'b, 'c) psi"   and N :: "'a"   and P :: "('a, 'b, 'c) psi" assumes "\<Psi> \<turnstile> M \<leftrightarrow> K"   and "insertAssertion (extractFrame Q) \<Psi> \<hookrightarrow>\<^sub>F \<langle>\<epsilon>, \<Psi> \<otimes> \<one>\<rangle>" shows "\<Psi> : Q \<rhd> M\<langle>N\<rangle>.P \<Longrightarrow>K\<langle>N\<rangle> \<prec> P"
distinct_clearjunk_id: fixes al :: "('a \<times> 'b) list" assumes "distinct (map fst al)" shows "AList.clearjunk al = al"
det_permute_columns: fixes p :: "'a \<Rightarrow> 'a"   and A :: "(('b, 'a) vec, 'a) vec" assumes "p permutes UNIV" shows "det (\<chi>i j. A $ i $ p j) = of_int (sign p) * det A"
vector_uminus_component: fixes x :: "('d, 'e) vec"   and i :: "'e" shows "(- x) $ i = - x $ i"
d_min: fixes C :: "'a literal multiset" assumes "C \<in> N"   and "\<not> I_of N \<TTurnstile>e {C}" shows "D \<le> C"
exteqI: fixes f :: "recf"   and g :: "recf" assumes "arity f = arity g"   and "\<And>xs. length xs = arity f \<Longrightarrow> eval f xs = eval g xs" shows "f \<simeq> g"
term_ok_tvars_sort_ok: fixes \<Theta> :: "((String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)) \<times> Core.term set"   and t :: "Core.term"   and x :: "variable"   and S :: "String.literal set" assumes "wf_theory \<Theta>"   and "term_ok \<Theta> t"   and "(x, S) \<in> tvs t" shows "wf_sort (subclass (osig (sig \<Theta>))) S"
remdupsListDiff: fixes a :: "'a list"   and b :: "'a list" shows "remdups (list_diff a b) = list_diff (remdups a) (remdups b)"
mult_inv_character: fixes k :: "'a" shows "\<chi> k * inv_character \<chi> k = principal_char G k"
DERIV_add: fixes f :: "'a \<Rightarrow> 'a"   and D :: "'a"   and x :: "'a"   and s :: "'a set"   and g :: "'a \<Rightarrow> 'a"   and E :: "'a" assumes "(f has_field_derivative D) (at x within s)"   and "(g has_field_derivative E) (at x within s)" shows "((\<lambda>x. f x + g x) has_field_derivative D + E) (at x within s)"
keys_eq_empty: fixes c :: "'b \<Rightarrow>\<^sub>0 'c" shows "(keys c = {}) = (c = 0)"
field_differentiable_at_csqrt: fixes z :: "complex" assumes "z \<notin> \<real>\<^sub>\<le>\<^sub>0" shows "csqrt field_differentiable at z"
Field_oprod: fixes r :: "('c \<times> 'c) set"   and r' :: "('d \<times> 'd) set" shows "Field (r *o r') = Field r \<times> Field r'"
ordertype_inc_eq: fixes A :: "'a set"   and r :: "('a \<times> 'a) set"   and \<pi> :: "'a \<Rightarrow> 'b"   and s :: "('b \<times> 'b) set" assumes "small A"   and "\<And>x y. \<lbrakk>x \<in> A; y \<in> A; (x, y) \<in> r\<rbrakk> \<Longrightarrow> (\<pi> x, \<pi> y) \<in> s"   and "wf r"   and "total_on A r"   and "wf s" shows "ordertype (\<pi> ` A) s = ordertype A r"
inj_on_imageI2: fixes f' :: "'i \<Rightarrow> 'h"   and f :: "'g \<Rightarrow> 'i"   and A :: "'g set" assumes "inj_on (f' \<circ> f) A" shows "inj_on f A"
induct: fixes P :: "'b poly \<Rightarrow> bool"   and a0 :: "'b poly" assumes "\<And>n. P (Bound n)"   and "\<And>a. P a \<Longrightarrow> P (Neg a)"   and "\<And>a b. \<lbrakk>P a; P b\<rbrakk> \<Longrightarrow> P (Add a b)"   and "\<And>a b. \<lbrakk>P a; P b\<rbrakk> \<Longrightarrow> P (Sub a b)"   and "\<And>a b. \<lbrakk>P a; P b\<rbrakk> \<Longrightarrow> P (Mul a b)"   and "\<And>p n. P p \<Longrightarrow> P (Pw p n)"   and "\<And>c n p. \<lbrakk>P c; P p\<rbrakk> \<Longrightarrow> P (CN c n p)"   and "\<And>v. P (v)\<^sub>p" shows "P a0"
Gr_def: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'c" shows "Gr A f = {(a, f a) |a. a \<in> A}"
nsets_compose_image_funcset: fixes f :: "'a set \<Rightarrow> 'b"   and T :: "'a set"   and k :: "nat"   and D :: "'b set"   and g :: "'c \<Rightarrow> 'a"   and S :: "'c set" assumes "f \<in> [T]\<^bsup>k\<^esup> \<rightarrow> D"   and "g \<in> S \<rightarrow> T"   and "inj_on g S" shows "f \<circ> (`) g \<in> [S]\<^bsup>k\<^esup> \<rightarrow> D"
intros_6: shows "UnlimitedNatural \<sqsubset>\<^sub>B Integer"
exec_mark_guards_to_exec: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and f :: "'f"   and c :: "('e, 'd, 'f) com"   and s :: "('e, 'f) xstate"   and t :: "('e, 'f) xstate" assumes "\<Gamma>\<turnstile> \<langle>mark_guards f c,s\<rangle> \<Rightarrow> t" shows "\<exists>t'. \<Gamma>\<turnstile> \<langle>c,s\<rangle> \<Rightarrow> t' \<and> (isFault t \<longrightarrow> isFault t') \<and> (t' = Fault f \<longrightarrow> t' = t) \<and> (isFault t' \<longrightarrow> isFault t) \<and> (\<not> isFault t' \<longrightarrow> t' = t)"
ordinal_oSuc_minus2: fixes x :: "ordinal"   and y :: "ordinal" assumes "x < y" shows "oSuc x - y = x - y"
Cd_greatest: fixes Xd :: "'d \<Rightarrow> 'e"   and Xh :: "'d \<Rightarrow> 'f"   and Pd :: "'e \<Rightarrow> ('d \<times> 'd) set"   and Ch :: "'f \<Rightarrow> 'd set \<Rightarrow> 'd set"   and d :: "'e"   and X :: "'d set" assumes "Contracts Xd Xh Pd Ch" shows "Contracts.Cd Pd d X = greatest (Pd d) (X \<inter> Field (Pd d))"
rep_iso_class: fixes f :: "'a \<times> 'a" assumes "VV.ide f" shows "VV.isomorphic (VVV.iso_class_rep (VV.iso_class f)) f"
inf_bot_left: fixes inf :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sup :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and bot :: "'d"   and x :: "'d" assumes "class.bounded_lattice_bot inf less_eq less sup bot" shows "inf bot x = bot"
PsiInv_gamma2: shows "\<turnstile> gamma2 \<and> $PsiInv \<longrightarrow> PsiInv$"
spr_simTrans: fixes t :: "'s Trace"   and ec :: "'s odlist"   and a :: "'a" assumes "t \<in> SPR.jkbpC"   and "spr_simAbs ec = SPRsingle.sim_equiv_class a t" shows "spr_simAbs ` set (local.spr_simTrans a ec) = rel_ext (\<lambda>uu_. \<exists>t' s. uu_ = SPRsingle.sim_equiv_class a (t' \<leadsto> s) \<and> t' \<leadsto> s \<in> SPR.jkbpC \<and> spr_jview a t' = spr_jview a t)"
AsWhileSkip: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "'d \<Rightarrow> ('e, 'd, 'f) ann list option"   and Q :: "'e set"   and A :: "'e set"   and r :: "'e set"   and i :: "'e set"   and p :: "('e, 'd, 'f) ann"   and b :: "'e set"   and e :: "('e, 'd, 'f) com" shows "assertionsR \<Gamma> \<Theta> Q A (AnnWhile r i p) (While b e) Q"
same_carrier: fixes C :: "'a set"   and A :: "'a set"   and B :: "'a set"   and g :: "'a \<Rightarrow> 'b"   and h :: "'a \<Rightarrow> 'b" assumes "finite C"   and "A \<subseteq> C"   and "B \<subseteq> C"   and "\<And>a. a \<in> C - A \<Longrightarrow> g a = (0::'b)"   and "\<And>b. b \<in> C - B \<Longrightarrow> h b = (0::'b)" shows "(sum g A = sum h B) = (sum g C = sum h C)"
exp_add_not_zero_imp_left: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bit :: "'a \<Rightarrow> nat \<Rightarrow> bool"   and m :: "nat"   and n :: "nat" assumes "class.semiring_bits plus minus zero times one divide modulo bit"   and "power.power one times (numeral.numeral one plus (num.Bit0 num.One)) (m + n) \<noteq> zero" shows "power.power one times (numeral.numeral one plus (num.Bit0 num.One)) m \<noteq> zero"
inj_on_Utn: shows "inj_on Utn (\<Union> (range J))"
1: shows "path ((\<lambda>z. (x1, h z)) +++ (\<lambda>z. (g z, y2)))"
cr_prod1_Grp: fixes c :: "'d"   and A :: "'b set"   and f :: "'b \<Rightarrow> 'c" shows "cr_prod1 c (BNF_Def.Grp A f) = BNF_Def.Grp A (\<lambda>b. (f b, c))"
prove_elts_imp_prove_All2: fixes t :: "tm"   and A :: "fm"   and i :: "name" assumes "ground t"   and "\<And>u. u \<in> elts t \<Longrightarrow> {} \<turnstile> A(i::=u)" shows "{} \<turnstile> All2 i t A"
three_in_long_chain: fixes f :: "nat \<Rightarrow> 'a"   and X :: "'a set" assumes "long_ch_by_ord f X"   and "finite X" obtains x :: "'a" and y :: "'a" and z :: "'a" where "x \<in> X" and "y \<in> X" and "z \<in> X" and "x \<noteq> y" and "x \<noteq> z" and "y \<noteq> z"
power_commutes: fixes a :: "'a"   and n :: "nat" shows "a ^ n \<cdot> a = a \<cdot> a ^ n"
linear_continuous_at: fixes f :: "'a \<Rightarrow> 'b"   and a :: "'a" assumes "bounded_linear f" shows "isCont f a"
eq_ivl_iff: fixes p1 :: "int extended \<times> int extended"   and p2 :: "int extended \<times> int extended" shows "eq_ivl p1 p2 = (is_empty_rep p1 \<and> is_empty_rep p2 \<or> p1 = p2)"
le_iff_sup: fixes sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "class.semilattice_sup sup less_eq less" shows "less_eq x y = (sup x y = y)"
strip_transl: fixes C :: "Nielson_VCG.acom" shows "Nielson_VCGi.strip (transl C) = Nielson_VCG.strip C"
select_sets_def: fixes M :: "'c measure measure" shows "select_sets M = (SOME N. sets M = sets (subprob_algebra N))"
swap_msgs_Send_Trans: fixes c :: "('a, 'b, 'c) configuration"   and ev :: "('a, 'b, 'c) event"   and d :: "('a, 'b, 'c) configuration"   and ev' :: "('a, 'b, 'c) event"   and e :: "('a, 'b, 'c) configuration"   and d' :: "('a, 'b, 'c) configuration"   and e' :: "('a, 'b, 'c) configuration"   and i :: "nat" assumes "c \<turnstile> ev \<mapsto> d"   and "d \<turnstile> ev' \<mapsto> e"   and "isSend ev"   and "isTrans ev'"   and "c \<turnstile> ev' \<mapsto> d'"   and "d' \<turnstile> ev \<mapsto> e'"   and "occurs_on ev \<noteq> occurs_on ev'" shows "msgs e i = msgs e' i"
set_of_map_def: fixes m :: "'e \<Rightarrow> 'f option" shows "set_of_map m = {(k, v) |k v. m k = Some v}"
defined: shows "A \<inter> (\<Union>x\<in>set_pmf p. set_pmf (f x)) \<noteq> {}"
simps: fixes \<Sigma> :: "'a set set"   and \<phi> :: "'a ltl" shows "F\<^sub>\<A> \<Sigma> \<phi> = {(M_fin \<pi> \<union> \<Union> {Acc_fin \<Sigma> \<pi> \<chi> |\<chi>. \<chi> \<in> dom \<pi>}, {Acc_inf \<pi> \<chi> |\<chi>. \<chi> \<in> dom \<pi>}) |\<pi>. dom \<pi> \<subseteq> \<^bold>G \<phi> \<and> (\<forall>\<chi>\<in>dom \<pi>. the (\<pi> \<chi>) < max_rank_of \<Sigma> \<chi>)}"
Joints_swap_Cons_append: fixes xs :: "'b list"   and XS :: "('b \<times> 'b pdevs) list"   and x :: "'b"   and ys :: "'b list"   and X :: "'b \<times> 'b pdevs"   and YS :: "('b \<times> 'b pdevs) list" assumes "length xs = length XS" shows "(x # ys @ xs \<in> Joints (X # YS @ XS)) = (ys @ x # xs \<in> Joints (YS @ X # XS))"
elims: fixes x :: "'b option"   and y :: "'b\<^sub>\<bottom>" assumes "option2up x = y" obtains "x = None" and "y = \<bottom>"   | a :: "'b" where "x = Some a" and "y = up\<cdot>a"
P1_advantages_eq: fixes n :: "nat"   and x :: "'grp \<times> 'grp"   and y :: "bool"   and z :: "'d"   and \<A> :: "('grp \<times> 'grp \<Rightarrow> 'grp \<Rightarrow> 'grp \<Rightarrow> 'grp \<Rightarrow> 'grp \<Rightarrow> 'grp \<Rightarrow> 'd \<Rightarrow> (('grp \<times> 'grp \<times> 'grp) \<times> 'e) spmf) \<times> ('grp \<Rightarrow> 'grp \<Rightarrow> 'grp \<Rightarrow> 'grp \<Rightarrow> 'grp \<Rightarrow> 'grp \<times> 'grp \<Rightarrow> 'e \<Rightarrow> ((('grp \<times> 'grp) \<times> 'grp \<times> 'grp) \<times> 'e) spmf) \<times> ('e \<Rightarrow> 'f spmf)"   and D :: "'f \<times> 'grp \<Rightarrow> bool spmf" shows "mal_def.adv_P1 n x y z (P1_S1 n, P1_S2) \<A> D = P1_adv_real_ideal_model n D x y \<A> z"
rep_eq: fixes x :: "float" shows "real_of_float (- x) = - real_of_float x"
frac_eq_0_iff: fixes x :: "'d" shows "(frac x = (0::'d)) = (x \<in> \<int>)"
fPow_top: fixes A :: "'b fset" shows "A |\<in>| fPow A"
rel_fundefs_Some1: fixes f :: "'c \<Rightarrow> ('d, 'e, 'f, 'op) Std.instr fundef option"   and g :: "'c \<Rightarrow> ('d, 'e, 'f, 'op, 'opinl) Inca.instr fundef option"   and x :: "'c"   and y :: "('d, 'e, 'f, 'op) Std.instr fundef" assumes "rel_fundefs f g"   and "f x = Some y" shows "\<exists>z. g x = Some z \<and> rel_fundef (\<lambda>x y. x = norm_instr y) y z"
d'_3: shows "\<forall>t\<in>S_Max_Le. d' \<le> t"
consistent_sign_vectors_def: fixes qs :: "rat poly list"   and S :: "real set" shows "consistent_sign_vectors qs S = sign_vec qs ` S"
fin_crename: fixes M :: "trm"   and a :: "coname"   and b :: "coname"   and x :: "name" assumes "fin (M[a\<turnstile>c>b]) x" shows "fin M x"
inf_fun_def: fixes f :: "'d \<Rightarrow> 'e"   and g :: "'d \<Rightarrow> 'e" shows "f \<sqinter> g = (\<lambda>x. f x \<sqinter> g x)"
gt_same: fixes s :: "('s, 'v) tm"   and t :: "('s, 'v) tm" assumes "vars_mset s \<subseteq># vars_mset t"   and "wt t = wt s"   and "head t = head s"   and "\<forall>f\<in>local.ground_heads (head s). extf f (>\<^sub>t) (args t) (args s)" shows "t >\<^sub>t s"
finite_Image: fixes R :: "('a \<times> 'b) set"   and A :: "'a set" assumes "finite R" shows "finite (R `` A)"
mangoldt_odd_def: fixes d :: "nat" shows "mangoldt_odd d = (if primepow_odd d then ln (real (aprimedivisor d)) else 0)"
the_\<nu>_props_1: shows "\<guillemotleft>TTfgh_TfTgh.the_\<nu> : (tab\<^sub>1 f \<star> f\<^sub>0g\<^sub>1.p\<^sub>1) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1 \<rightarrow> (tab\<^sub>1 f \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1) \<star> TTfgh_TfTgh.chine\<guillemotright>"
incrIndexList_1lem: fixes ls :: "nat list"   and m :: "nat"   and nmax :: "nat" assumes "incrIndexList ls m nmax" shows "Suc 0 < m"
loc_upb_6_z: fixes n :: "nat"   and x :: "nat" assumes "c_fst n mod 7 = 6"   and "c_fst x = 0" shows "loc_upb n x = c_pair (c_pair n x) (loc_upb (c_fst (c_snd n)) (c_snd x)) + 1"
symKeys_neq_imp_neq: fixes K :: "nat"   and K' :: "nat" assumes "(K \<in> symKeys) \<noteq> (K' \<in> symKeys)" shows "K \<noteq> K'"
L1: fixes \<tau> :: "real" assumes "\<tau> \<in> cball t \<epsilon>\<^sub>1 \<inter> T" shows "L1-lipschitz_on (cball s \<epsilon>\<^sub>1 \<inter> S) (f1 \<tau>)"
cert: shows "cert_ok c (length is) \<top> \<bottom> A"
IH: fixes v :: "'var" assumes "v \<in> wfrestrictedvars\<^sub>s\<^sub>t S"   and "\<lbrakk>{}; S\<rbrakk>\<^sub>c \<I>" shows "intruder_synth (ik\<^sub>s\<^sub>t S \<cdot>\<^sub>s\<^sub>e\<^sub>t \<I>) (\<I> v)"
simps_1: fixes e :: "((nat \<times> nat) \<times> transition) fset"   and s :: "nat"   and r :: "nat \<Rightarrow>f value option" shows "recognises_prim e s r [] = True"
ncf7: fixes y :: "real" assumes "0 \<le> y"   and "y \<le> x" shows "0 < numer_cf7 (- y)"
eucl_of_list_list_of_eucl: fixes x :: "'a" shows "eucl_of_list (list_of_eucl x) = x"
intro: fixes A :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and B :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and F :: "'b \<Rightarrow> 'c" assumes "faithful_functor A B F"   and "full_functor A B F" shows "fully_faithful_functor A B F"
adapt_stoch_proc_def: fixes F :: "'e \<Rightarrow> 'f measure"   and X :: "'e \<Rightarrow> 'f \<Rightarrow> 'g"   and N :: "'g measure" shows "adapt_stoch_proc F X N = (\<forall>t. X t \<in> F t \<rightarrow>\<^sub>M N)"
vector_minus_component: fixes x :: "('d, 'e) Finite_Cartesian_Product.vec"   and y :: "('d, 'e) Finite_Cartesian_Product.vec"   and i :: "'e" shows "(x - y) $h i = x $h i - y $h i"
noread_exists_change_respecting: fixes X :: "'Var set"   and c :: "'Com"   and mds :: "Mode \<Rightarrow> 'Var set"   and mem :: "'Var \<Rightarrow> 'Val"   and c' :: "'Com"   and mds' :: "Mode \<Rightarrow> 'Var set"   and mem' :: "'Var \<Rightarrow> 'Val" assumes "finite X"   and "\<langle>c, mds, mem\<rangle> \<leadsto> \<langle>c', mds', mem'\<rangle>"   and "\<forall>x\<in>X. doesnt_read_or_modify c x" shows "change_respecting \<langle>c, mds, mem\<rangle> \<langle>c', mds', mem'\<rangle> X"
le_neq_trans: fixes less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and a :: "'e"   and b :: "'e" assumes "class.order less_eq less"   and "less_eq a b"   and "a \<noteq> b" shows "less a b"
bot_Boolean_def: shows "UML_Types.bot_class.bot = Map.empty"
detTA_def: shows "detTA \<equiv> \<lparr>ta_initial = {s. s \<subseteq> Q \<and> s \<inter> Qi \<noteq> {}}, ta_rules = \<delta>ss\<rparr>"
rem_condless_valid_3: fixes s :: "('c, 'd) fmap"   and as :: "(('c, 'd) fmap \<times> ('c, 'd) fmap) list" shows "length (rem_condless_act s [] as) \<le> length as"
ih_13: shows "InvariantNoDecisionsWhenConflict (getF state') (getM state') (currentLevel (getM state'))"
set_replicate_Suc: fixes n :: "nat"   and x :: "'a" shows "set (replicate (Suc n) x) = {x}"
assms_7: shows "\<not> poincare_collinear {x, y, z}"
Union_of_def: fixes f :: "'b \<Rightarrow> 'a set"   and A :: "'b set" shows "Union_of f A \<equiv> \<Union> (f ` A)"
conservation_const: shows "\<forall>v\<in>V - {s, t}. sum f (incoming v) = sum f (outgoing v)"
simps_1: fixes ys :: "'c list" shows "[] \<cdot> ys = ys"
vdomain_vconverse: fixes r :: "V" shows "\<D>\<^sub>\<circ> (r\<inverse>\<^sub>\<circ>) = \<R>\<^sub>\<circ> r"
rell_T_alt_def: fixes L1 :: "'n \<Rightarrow> 't \<Rightarrow> bool" shows "rell_T L1 = rel_T L1 (=) (=) (=) (=)"
bipartite_web_of_sel_6: fixes \<Gamma> :: "('a, 'b) web_scheme" shows "A (bipartite_web_of \<Gamma>) = Inl ` (\<^bold>V\<^bsub>\<Gamma>\<^esub> - B \<Gamma>)"
simps: fixes f :: "nat \<Rightarrow> real" shows "polynomial f = (\<exists>n. f \<in> O(\<lambda>x. real x powr n))"
mm_equiv_intro: fixes \<R> :: "((('Com \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)) \<times> ('Com \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)) set"   and lc\<^sub>1 :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)"   and lc\<^sub>2 :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)" assumes "strong_low_bisim_mm \<R>"   and "(lc\<^sub>1, lc\<^sub>2) \<in> \<R>" shows "mm_equivp lc\<^sub>1 lc\<^sub>2"
a1: shows "s' = snd (fst (store_sub2 instr curr_win rd asi address s)) \<and> get_S (cpu_reg_val PSR s) = 0"
iT_Max_3: fixes r :: "nat"   and m :: "nat"   and c :: "nat" shows "Max [ r, mod m, c ] = r + m * c"
pack_OF_entries_def: fixes ifs :: "char list list"   and ard :: "(16 word \<times> 32 simple_match \<times> char list \<times> simple_action) list" shows "pack_OF_entries ifs ard \<equiv> map (split3 OFEntry) (lr_of_tran_s3 ifs ard)"
fun_left_comm: fixes y :: "'m"   and x :: "'m"   and z :: "'m" shows "funionM y (funionM x z) = funionM x (funionM y z)"
A_def: shows "A \<equiv> {x. poly p x = 0 \<and> a < x \<and> x < b}"
sumlist_vec_index: fixes xs :: "'a vec list"   and i :: "nat" assumes "\<And>x. x \<in> set xs \<Longrightarrow> x \<in> carrier_vec n"   and "i < n" shows "M.sumlist xs $ i = (\<Sum>x\<leftarrow>xs. x $ i)"
vdisjnt_vunion_right: fixes C :: "V"   and A :: "V"   and B :: "V" shows "(C \<inter>\<^sub>\<circ> (A \<union>\<^sub>\<circ> B) = []\<^sub>\<circ>) = (C \<inter>\<^sub>\<circ> A = []\<^sub>\<circ> \<and> C \<inter>\<^sub>\<circ> B = []\<^sub>\<circ>)"
intros_1: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b" assumes "r a b" shows "r\<^sup>+\<^sup>+ a b"
map_def: fixes f :: "'a" shows "\<Delta>o\<delta>_\<pi>o\<Delta>.map f = (if arr f then CC.comp ((\<pi>.map \<circ> Diag) (cod f)) ((Diag \<circ> dup) f) else CC.null)"
divide_nonpos_neg: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inverse :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sgn :: "'c \<Rightarrow> 'c"   and abs :: "'c \<Rightarrow> 'c"   and x :: "'c"   and y :: "'c" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less_eq x zero"   and "less y zero" shows "less_eq zero (divide x y)"
Upsilon2_subsetof_E2: shows "\<Upsilon>\<^bsub>\<Gamma>2\<^esub> \<subseteq> E\<^bsub>ES2\<^esub>"
oppOrdering_simps_2: shows "oppOrdering\<cdot>EQ = EQ"
fv_set_at_base: fixes l :: "'a list" shows "fv l = set l"
add_pos_pos: fixes a :: "'a multiset"   and b :: "'a multiset" assumes "{#} \<subset># a"   and "{#} \<subset># b" shows "{#} \<subset># a + b"
t_fusion_leftneutral: fixes x :: "'a \<times> ('b \<times> 'a) list" shows "t_fusion (first x, []) x = x"
quasi_isometry_onD_1: fixes lambda :: "real"   and C :: "real"   and X :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and x :: "'c"   and y :: "'c" assumes "lambda C -quasi_isometry_on X f"   and "x \<in> X"   and "y \<in> X" shows "dist (f x) (f y) \<le> lambda * dist x y + C"
Act\<^sub>\<alpha>: fixes t\<^sub>\<alpha> :: "('b, 'c, 'd) Tree\<^sub>\<alpha>"   and \<alpha> :: "'d" assumes "hereditarily_fs t\<^sub>\<alpha>" shows "hereditarily_fs (Act\<^sub>\<alpha> \<alpha> t\<^sub>\<alpha>)"
nn_integral_measure_pmf_support: fixes A :: "'b set"   and f :: "'b \<Rightarrow> ennreal"   and M :: "'b pmf" assumes "finite A"   and "\<And>x. x \<in> A \<Longrightarrow> 0 \<le> f x"   and "\<And>x. \<lbrakk>x \<in> set_pmf M; x \<notin> A\<rbrakk> \<Longrightarrow> f x = 0" shows "integral\<^sup>N (measure_pmf M) f = (\<Sum>x\<in>A. f x * ennreal (pmf M x))"
awalkE: fixes u :: "'a"   and p :: "('a \<times> 'a) list"   and v :: "'a" assumes "awalk u p v" obtains "set (pawalk_verts u p) \<subseteq> pverts G" and "set p \<subseteq> parcs G" and "pcas u p v" and "hd (pawalk_verts u p) = u" and "last (pawalk_verts u p) = v"
mdqdb: shows "bound0 (Cooper.disj (evaldjf (\<lambda>j. simpfm (subst0 (C j) (simpfm (minusinf (fst (unit p)))))) [1..snd (snd (unit p))]) (evaldjf (\<lambda>(b, j). simpfm (subst0 (Add b (C j)) (fst (unit p)))) (concat (map (\<lambda>b. map (Pair b) [1..snd (snd (unit p))]) (fst (snd (unit p)))))))"
stutter_extend_en_def: fixes en :: "'c \<Rightarrow> 'd set" shows "stutter_extend_en en \<equiv> \<lambda>s. let as = en s in if as = {} then {None} else Some ` as"
rep_eq: fixes x :: "'a"   and xa :: "'a set['b]" shows "set_bset (binsert x xa) = insert x (set_bset xa)"
Goal: fixes P'' :: "('a, 'b, 'c) psi" assumes "\<Psi> : Q \<rhd> \<tau>.P \<Longrightarrow>\<alpha> \<prec> P''"   and "\<Psi> \<rhd> P' \<sim> P''" shows "thesis"
\<alpha>_ide_simp: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "ide a"   and "ide b"   and "ide c" shows "assoc a b c = \<a>[a, b, c]"