hcju/isabelleps huggingface dataset:
carrier_vs_is_self: fixes W :: "'a mod_ring poly set" shows "carrier (vector_space_poly.vs W) = W"
matrix_A_R_def: fixes sign_list :: "rat list list"   and subset_list :: "(nat list \<times> nat list) list" shows "matrix_A_R sign_list subset_list = mat_of_rows_list (length sign_list) (map (mtx_row_R sign_list) subset_list)"
assms_6: shows "((\<xi>, p'), a, \<xi>', q) \<in> automaton.trans A"
les_qe_inf_helper: fixes les :: "(real \<times> real \<times> real) list"   and q :: "real" assumes "\<forall>(d, e, f)\<in>set les. \<exists>y'>q. \<forall>x\<in>{q<..y'}. d * x\<^sup>2 + e * x + f < 0" shows "\<exists>y'>q. \<forall>(d, e, f)\<in>set les. \<forall>x\<in>{q<..y'}. d * x\<^sup>2 + e * x + f < 0"
f_const_has_max_value_1: assumes "const 0 \<or> const 1" shows "(cmod (\<Sum>k<2 ^ n. (- 1) ^ f k))\<^sup>2 = real (2 ^ (2 * n))"
select_convs_9: fixes counter :: "nat"   and discovered :: "'d \<Rightarrow> nat option"   and finished :: "'d \<Rightarrow> nat option"   and pending :: "('d \<times> 'd) set"   and stack :: "'d list"   and tree_edges :: "('d \<times> 'd) set"   and back_edges :: "('d \<times> 'd) set"   and cross_edges :: "('d \<times> 'd) set"   and more :: "'c" shows "state.more \<lparr>counter = counter, discovered = discovered, finished = finished, pending = pending, stack = stack, tree_edges = tree_edges, back_edges = back_edges, cross_edges = cross_edges, \<dots> = more\<rparr> = more"
set_bd_2: fixes x :: "('l, 'm, 'k, 'n) G" shows "|Gset2 x| \<le>o bd_G"
finprod_insert: fixes F :: "'c set"   and a :: "'c"   and f :: "'c \<Rightarrow> 'a list" assumes "finite F"   and "a \<notin> F"   and "f \<in> F \<rightarrow> carrier (K [X])"   and "f a \<in> carrier (K [X])" shows "finsum (K [X]) f (insert a F) = f a \<oplus>\<^bsub>K [X]\<^esub> finsum (K [X]) f F"
pprod_pp_of_term: shows "pprod.pp_of_term = fst"
Throw_star: fixes \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and s :: "'d"   and cfg\<^sub>2 :: "('d, 'e, 'f) com \<times> ('d, 'f) xstate" assumes "\<Gamma> \<turnstile> (Throw, Normal s) \<rightarrow>\<^sup>* cfg\<^sub>2" shows "cfg\<^sub>2 = (Throw, Normal s)"
low_eq_low_mds_eq: fixes mem\<^sub>1 :: "'Var \<Rightarrow> 'Val"   and mem\<^sub>2 :: "'Var \<Rightarrow> 'Val" shows "mem\<^sub>1 =\<^sup>l mem\<^sub>2 = mem\<^sub>1 =\<^bsub>\<lambda>m. {}\<^esub>\<^sup>l mem\<^sub>2"
rename_Increasing: fixes h :: "'c \<Rightarrow> 'd"   and F :: "'c program"   and func :: "'d \<Rightarrow> 'e" assumes "bij h" shows "(rename h F \<in> Increasing func) = (F \<in> Increasing (func \<circ> h))"
le_imp_diff_is_add: fixes a :: "'k multiset"   and b :: "'k multiset"   and c :: "'k multiset" assumes "a \<subseteq># b"   and "a \<subseteq># b" shows "(b - a = c) = (b = c + a)"
vector_choose_dist: fixes c :: "real"   and x :: "'a" assumes "0 \<le> c" obtains y :: "'a" where "dist x y = c"
Sleator_Tarjan: fixes n :: "nat" shows "T_mtf n \<le> (\<Sum>i<n. 2 * c_A i + p_A i - f_A i) - int n"
apply_adaptation\<^sub>A_2: fixes mem\<^sub>1\<^sub>C :: "'Var\<^sub>C \<Rightarrow> 'Val"   and A\<^sub>C :: "'Var\<^sub>C \<Rightarrow> ('Val \<times> 'Val) option" shows "mem\<^sub>A_of mem\<^sub>1\<^sub>C [\<parallel>\<^sub>2 A\<^sub>A_of A\<^sub>C] = mem\<^sub>A_of mem\<^sub>1\<^sub>C [\<parallel>\<^sub>2 A\<^sub>C]"
mkArr'_in_hom: fixes a :: "'s"   and b :: "'s"   and F :: "'s \<Rightarrow> 's" assumes "ide a"   and "ide b"   and "F \<in> hom unity a \<rightarrow> hom unity b" shows "\<guillemotleft>mkArr' a b F : a \<rightarrow> b\<guillemotright>"
assms: shows "sv \<in> Aexp.vars (adapt_aexp e s)"
Int: fixes a :: "'a set"   and M :: "'a measure"   and b :: "'a set" assumes "a \<in> fmeasurable M"   and "b \<in> fmeasurable M" shows "a \<inter> b \<in> fmeasurable M"
simps_1: fixes u :: "'b" shows "pawalk_verts u [] = [u]"
map_cong: fixes x :: "'a stream"   and ya :: "'a stream"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "x = ya"   and "\<And>z. z \<in> sset ya \<Longrightarrow> f z = g z" shows "smap f x = smap g ya"
cases: fixes A :: "('d, 'e, 'f, 'g) FSM_scheme"   and a1 :: "(('d \<times> 'e) \<times> 'f) list"   and a2 :: "'f" assumes "path A a1 a2" obtains p :: "'f" where "a1 = []" and "a2 = p__"   | a :: "('d \<times> 'e) \<times> 'f" and p :: "'f" and r :: "(('d \<times> 'e) \<times> 'f) list" where "a1 = a # r" and "a2 = p" and "snd a \<in> succ A (fst a) p" and "path A r (snd a)"
Catch_decomp_star: fixes \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and p\<^sub>1 :: "('d, 'e, 'f) com"   and p\<^sub>2 :: "('d, 'e, 'f) com"   and s :: "'d"   and p' :: "('d, 'e, 'f) com"   and s' :: "'d" assumes "\<Gamma> \<turnstile> (Catch p\<^sub>1 p\<^sub>2, Normal s) \<rightarrow>\<^sup>* (p', Normal s')"   and "final (p', Normal s')" shows "(\<exists>t. \<Gamma> \<turnstile> (p\<^sub>1, Normal s) \<rightarrow>\<^sup>* (Throw, Normal t) \<and> \<Gamma> \<turnstile> (p\<^sub>2, Normal t) \<rightarrow>\<^sup>* (p', Normal s')) \<or> \<Gamma> \<turnstile> (p\<^sub>1, Normal s) \<rightarrow>\<^sup>* (Skip, Normal s') \<and> p' = Skip"
trivialPhi_strict_dom: fixes v :: "'val"   and g :: "'g"   and v' :: "'val" assumes "v \<in> allVars g"   and "isTrivialPhi g v v'" shows "defNode g v' \<noteq> defNode g v \<and> def_dominates g v' v"
simps_2: fixes f :: "'b \<Rightarrow> 'a \<Rightarrow> 'a"   and x :: "'b"   and xs :: "'b list" shows "fold f (x # xs) = fold f xs \<circ> f x"
star_fun_def: fixes f :: "'b \<Rightarrow> 'b" shows "star_fun f = gfp (Omega_fun f id)"
sums_diff: fixes f :: "nat \<Rightarrow> 'c"   and a :: "'c"   and g :: "nat \<Rightarrow> 'c"   and b :: "'c" assumes "f sums a"   and "g sums b" shows "(\<lambda>n. f n - g n) sums (a - b)"
norm_axis: fixes i :: "'d"   and x :: "'c" shows "norm (axis i x) = norm x"
assms_3: fixes i :: "'a" assumes "i \<in> I" shows "Group.group (G i)"
emeasure_interval_measure_Ioc: fixes a :: "real"   and b :: "real"   and F :: "real \<Rightarrow> real" assumes "a \<le> b"   and "\<And>x y. x \<le> y \<Longrightarrow> F x \<le> F y"   and "\<And>a. continuous (at_right a) F" shows "emeasure (interval_measure F) {a<..b} = ennreal (F b - F a)"
unfold_congs_3: fixes r :: "'b heap_scheme"   and r' :: "'b heap_scheme"   and v' :: "nat"   and f :: "nat \<Rightarrow> nat"   and f' :: "nat \<Rightarrow> nat" assumes "r = r'"   and "lim r' = v'"   and "\<And>v. v = v' \<Longrightarrow> f v = f' v" shows "lim_update f r = lim_update f' r'"
morgan: fixes Cn :: "'b set \<Rightarrow> 'b set"   and true_PL :: "'b"   and false_PL :: "'b"   and imp_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and not_PL :: "'b \<Rightarrow> 'b"   and conj_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and disj_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and equiv_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and p :: "'b"   and q :: "'b"   and A :: "'b set" assumes "Supraclassical_logic Cn true_PL false_PL imp_PL not_PL conj_PL disj_PL equiv_PL" shows "(not_PL (conj_PL p q) \<in> Cn A) = (disj_PL (not_PL p) (not_PL q) \<in> Cn A)"
assms_2: fixes a :: "'a"   and b :: "'a"   and w :: "'a stream" assumes "P (a ## b ## w)" shows "b \<le> a \<and> P (b ## w)"
inv_aug_addition: fixes f :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and z :: "'e"   and i :: "'e \<Rightarrow> 'e"   and A :: "'e set" assumes "commute_bpp f (aug_pm_set z i A)"   and "assoc_bpp (aug_pm_set z i A) f"   and "ipp_cond1 A i"   and "ipp_cond2 z A i f"   and "ipp_cond3 z i"   and "inv_ipp z i f A"   and "commute_bpp f (aug_pm_set z i A)"   and "zeroA z i f A z" shows "\<forall>a\<in>addition_set f (aug_pm_set z i A). \<^sub>i- a \<^sub>f+ a = z"
FW_sym: fixes p :: "nat"   and q :: "nat" shows "FW_word p q = FW_word q p"
scale_left_imp_eq: fixes a :: "'a"   and x :: "('a, 'b) vec"   and y :: "('a, 'b) vec" assumes "a \<noteq> (0::'a)"   and "a *s x = a *s y" shows "x = y"
echelon_form_echelon_form_of: fixes bezout :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a \<times> 'a \<times> 'a"   and A :: "(('a, 'b) vec, 'c) vec" assumes "is_bezout_ext bezout" shows "echelon_form (echelon_form_of A bezout)"
Smith_normal_form_diagonal_to_Smith: fixes bezout :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a \<times> 'a \<times> 'a"   and A :: "(('a, 'b) vec, 'c) vec" assumes "is_bezout_ext bezout"   and "isDiagonal A" shows "Smith_normal_form (diagonal_to_Smith A bezout)"
st_vec_inverse: fixes x :: "'a st_vec" shows "Abs_st_vec (st_vec x) = x"
viewL_aux_def: fixes t :: "('c, 'd) FingerTree" shows "viewL_aux t \<equiv> Abs_viewres (FingerTreeStruc.viewL (Rep_FingerTree t))"
mHom_0: fixes M :: "('e, 'f, 'g) Module_scheme"   and R :: "('f, 'h) Ring_scheme"   and N :: "('i, 'f, 'j) Module_scheme"   and f :: "'e \<Rightarrow> 'i" assumes "R module M"   and "R module N"   and "f \<in> mHom R M N" shows "f \<zero>\<^bsub>M\<^esub> = \<zero>\<^bsub>N\<^esub>"
FDERIV: fixes f :: "'d \<Rightarrow> 'a"   and f' :: "'d \<Rightarrow> 'a"   and x :: "'d"   and s :: "'d set"   and g :: "'d \<Rightarrow> 'b"   and g' :: "'d \<Rightarrow> 'b" assumes "(f has_derivative f') (at x within s)"   and "(g has_derivative g') (at x within s)" shows "((\<lambda>x. f x ** g x) has_derivative (\<lambda>h. f x ** g' h + f' h ** g x)) (at x within s)"
iso_group_mul_gen: fixes G :: "('b, 'c) monoid_scheme"   and A :: "'b set"   and B :: "'b set" assumes "Group.group G"   and "A \<lhd> G"   and "B \<lhd> G" shows "((\<lambda>(x, y). x \<otimes>\<^bsub>G\<^esub> y) \<in> Group.iso (subgroup_generated G A \<times>\<times> subgroup_generated G B) G) = (A \<inter> B \<subseteq> {\<one>\<^bsub>G\<^esub>} \<and> A <#>\<^bsub>G\<^esub> B = carrier G)"
HAStates_SA_mem: fixes SA :: "('d, 'e, 'f) seqauto"   and A :: "('d, 'e, 'f) hierauto"   and S :: "'d" assumes "SA \<in> SAs A"   and "S \<in> States SA" shows "S \<in> HAStates A"
llast'_def: shows "llast' \<equiv> wrap\<cdot>(fix\<cdot>llast_worker)"
clock_repRels_def: fixes jkbp :: "'b \<Rightarrow> ('b, 'c, 'd) GC list"   and agents :: "'b odlist"   and envObs :: "'b \<Rightarrow> 'e \<Rightarrow> 'f" assumes "FiniteLinorderEnvironment jkbp agents" shows "FiniteLinorderEnvironment.clock_repRels envObs \<equiv> \<lambda>a. rel_ext (\<lambda>(s, s'). envObs a s = envObs a s')"
Struct_def: fixes wtFsym :: "'a \<Rightarrow> bool"   and wtPsym :: "'b \<Rightarrow> bool"   and arOf :: "'a \<Rightarrow> 'c list"   and resOf :: "'a \<Rightarrow> 'c"   and intT :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and intF :: "'a \<Rightarrow> 'd list \<Rightarrow> 'd"   and intP :: "'b \<Rightarrow> 'd list \<Rightarrow> bool" shows "M.Struct wtFsym wtPsym arOf resOf intT intF intP \<equiv> Signature TYPE('c) wtFsym wtPsym \<and> M.Tstruct intT \<and> Struct_axioms wtFsym arOf resOf intT intF intP"
injec_mHom_image: fixes R :: "('e, 'f) Ring_scheme"   and N :: "('g, 'e, 'h) Module_scheme"   and M1 :: "('i, 'e, 'j) Module_scheme"   and M2 :: "('k, 'e, 'l) Module_scheme"   and x :: "'g \<Rightarrow> 'k"   and f :: "'i \<Rightarrow> 'k" assumes "Ring R"   and "R module N"   and "R module M1"   and "R module M2"   and "x \<in> mHom R N M2"   and "f \<in> mHom R M1 M2"   and "x ` carrier N \<subseteq> f ` carrier M1"   and "injec\<^bsub>M1,M2\<^esub> f" shows "(\<lambda>n\<in>carrier N. SOME m. m \<in> carrier M1 \<and> x n = f m) \<in> mHom R N M1 \<and> compos N f (\<lambda>n\<in>carrier N. SOME m. m \<in> carrier M1 \<and> x n = f m) = x"
sup_commute: fixes sup :: "'m \<Rightarrow> 'm \<Rightarrow> 'm"   and less_eq :: "'m \<Rightarrow> 'm \<Rightarrow> bool"   and less :: "'m \<Rightarrow> 'm \<Rightarrow> bool"   and x :: "'m"   and y :: "'m" assumes "class.semilattice_sup sup less_eq less" shows "sup x y = sup y x"
oLimit: fixes f :: "nat \<Rightarrow> ordinal" shows "F (oLimit f) = oLimit (\<lambda>n. F (f n))"
of_real_Re: fixes z :: "complex" assumes "z \<in> \<real>" shows "cor (Re z) = z"
pn_props_def: fixes ps :: "'b ltlr set" shows "pn_props ps \<equiv> FOREACH\<^bsup>\<lambda>it (P, N). P = {p. prop\<^sub>r(p) \<in> ps - it} \<and> N = {p. nprop\<^sub>r(p) \<in> ps - it}\<^esup> ps (\<lambda>p (P, N). case p of prop\<^sub>r(p) \<Rightarrow> RETURN (insert p P, N) | nprop\<^sub>r(p) \<Rightarrow> RETURN (P, insert p N) | _ \<Rightarrow> RETURN (P, N)) ({}, {})"
set_orbit_list': fixes f :: "'b \<Rightarrow> 'b"   and x :: "'b" assumes "permutation f" shows "set (orbit_list f x) = orbit f x"
S1: shows "nat_to_list u = [c_snd (u - 1)]"
init_is_s_t_0: fixes channel :: "nat \<Rightarrow> ('d \<times> 'd) option"   and trans :: "'d \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> bool"   and send :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and recv :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and init :: "('d, 'e, 'f) configuration"   and final :: "('d, 'e, 'f) configuration"   and t :: "('d, 'e, 'f) event list" assumes "computation channel trans send recv init final"   and "distributed_system.trace channel trans send recv init t final" shows "init = distributed_system.s channel trans send recv init t 0"
word_less_minus_mono_left: fixes y :: "'b word"   and z :: "'b word"   and x :: "'b word" assumes "y < z"   and "x \<le> y" shows "y - x < z - x"
OBTAIN_def: fixes P :: "'c \<Rightarrow> bool" shows "OBTAIN P \<equiv> ASSERT (\<exists>x. P x) \<bind> (\<lambda>_. SPEC P)"
Dirichlet_L_principal: fixes s :: "complex" shows "Dirichlet_L n (principal_dchar n) s = (\<Prod>x\<in>{p. prime p \<and> p dvd n}. 1 - 1 / of_nat x powr s) * zeta s"
lt_trans: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "lt\<cdot>x\<cdot>y = TT"   and "lt\<cdot>y\<cdot>z = TT" shows "lt\<cdot>x\<cdot>z = TT"
decseq_convergent: fixes X :: "nat \<Rightarrow> real"   and B :: "real" assumes "decseq X"   and "\<forall>i. B \<le> X i" obtains L :: "real" where "X \<longlonglongrightarrow> L" and "\<forall>i. L \<le> X i"
that: fixes p :: "(nat \<times> nat) list" assumes "cf.isSimplePath u p s" shows "thesis"
lconf_upd: 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 h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and l :: "'a \<Rightarrow> val option"   and E :: "'a \<Rightarrow> ty option"   and v :: "val"   and T :: "ty"   and V :: "'a" assumes "P,h \<turnstile> l (:\<le>)\<^sub>w E"   and "P,h \<turnstile> v :\<le> T"   and "E V = \<lfloor>T\<rfloor>" shows "P,h \<turnstile> l(V \<mapsto> v) (:\<le>)\<^sub>w E"
rfr_base: fixes r :: "real"   and w :: "'b" shows "disc_rfr_proc r 0 w = 1"
plus_except: fixes p :: "'a \<Rightarrow>\<^sub>0 'b"   and t :: "'a" shows "p = monomial (lookup p t) t + except p {t}"
dist_les_2: fixes discr :: "bool discr"   and cfun' :: "(C \<rightarrow> CValue) \<rightarrow> C \<rightarrow> CValue" shows "CB\<cdot>discr \<notsqsubseteq> CFn\<cdot>cfun'"
list_emb_Cons_iff2: fixes P :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c"   and xs :: "'c list"   and ys :: "'c list" assumes "\<not> P x y" shows "list_emb P (x # xs) (y # ys) = list_emb P (x # xs) ys"
rangeE: fixes b :: "'e"   and f :: "'f \<Rightarrow> 'e" assumes "b \<in> range f" obtains x :: "'f" where "b = f x"
successively_conv_sorted_wrt: fixes P :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and xs :: "'d list" assumes "transp P" shows "successively P xs = sorted_wrt P xs"
that_1: assumes "f x"   and "f (\<pi> \<bullet> x)" shows "thesis"
ReZ_of_nat: fixes n :: "nat" shows "ReZ (of_nat n) = int n"
fold_expand: shows "foldl (echelon_form_of_column_k_det bezout) (1::'a, A, 0) [0..<Suc k] = (fst (foldl (echelon_form_of_column_k_det bezout) (1::'a, A, 0) [0..<Suc k]), fst (snd (foldl (echelon_form_of_column_k_det bezout) (1::'a, A, 0) [0..<Suc k])), snd (snd (foldl (echelon_form_of_column_k_det bezout) (1::'a, A, 0) [0..<Suc k])))"
chop_assoc1: fixes r :: "real_int"   and r1 :: "real_int"   and r2 :: "real_int"   and r3 :: "real_int"   and r4 :: "real_int" shows "R_Chop(r,r1,r2) \<and> R_Chop(r2,r3,r4) \<longrightarrow> R_Chop(r,Abs_real_int (left r1, right r3),r4) \<and> R_Chop(Abs_real_int (left r1, right r3),r1,r3)"
coeff_poly_of_vec: fixes v :: "'b vec"   and i :: "nat" shows "coeff (poly_of_vec v) i = (if i < dim_vec v then v $ (dim_vec v - Suc i) else (0::'b))"
strip_step': fixes S :: "(char list \<Rightarrow> 'av) option"   and c :: "(char list \<Rightarrow> 'av) option acom" shows "strip (step' S c) = strip c"
INFM_iff_infinite: fixes P :: "'c \<Rightarrow> bool" shows "(\<exists>\<^sub>\<infinity>x. P x) = infinite {x. P x}"
simps_1: fixes P :: "'k \<Rightarrow> bool" shows "filter P [] = []"
simps: fixes I :: "'a set"   and v :: "nat \<Rightarrow> 'b"   and as :: "('a \<times> 'b atom) set" shows "(I, v) \<Turnstile>\<^sub>i\<^sub>a\<^sub>e\<^sub>s as = v \<Turnstile>\<^sub>a\<^sub>e\<^sub>s (snd ` (as \<inter> I \<times> UNIV))"
cond_spmf_try1: fixes q :: "'h spmf"   and A :: "'h set"   and p :: "'h spmf" assumes "set_spmf q \<inter> A = {}" shows "cond_spmf (TRY p ELSE q) A = cond_spmf p A"
extended_simps_9: fixes x' :: "'b" shows "\<not> \<lbrakk>x'\<rbrakk> \<le> \<turnstile>"
split_minNoneN0: fixes t :: "'b bro"   and h :: "nat" assumes "t \<in> B h"   and "split_min t = None" shows "t = N0"
igAbsIPresIGWls_def: fixes MOD :: "('c, 'd, 'varSort, 'sort, 'e, 'f, 'g, 'h, 'i) Iteration.model_scheme" shows "igAbsIPresIGWls MOD \<equiv> \<forall>xs s x X. isInBar (xs, s) \<and> igWls MOD s X \<longrightarrow> igWlsAbs MOD (xs, s) (igAbs MOD xs x X)"
constr_rep_1: fixes x2 :: "name"   and x1 :: "name" shows "Rep_freeRes (x2[x1]) = OutputR_Rep x2 x1"
AI_wn_def: shows "AI_wn = pfp_wn (local.step' \<top>)"
C3: shows "is_qfN (c\<^sup>2 + 3 * d\<^sup>2) 3"
valid_append: fixes as :: "'b list"   and t :: "'b tree"   and as' :: "'b list" assumes "valid as t" shows "valid (as @ as') t"
fresh_8: fixes a :: "name"   and x2 :: "pi"   and x1 :: "pi" shows "a \<sharp> x2 \<parallel> x1 = (a \<sharp> x2 \<and> a \<sharp> x1)"
mult_mat_vec_smult_vec_assoc: fixes A :: "'a mat"   and n :: "nat"   and m :: "nat"   and w :: "'a vec"   and a :: "'a" assumes "A \<in> carrier_mat n m"   and "w \<in> carrier_vec m" shows "A *\<^sub>v (a \<cdot>\<^sub>v w) = a \<cdot>\<^sub>v (A *\<^sub>v w)"
collapse: fixes prod :: "'b \<times> 'c" shows "(hamlet prod, time prod) = prod"
less_imp_neq: fixes less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'d"   and y :: "'d" assumes "class.order less_eq less"   and "less x y" shows "x \<noteq> y"
arr_Rel_ArrDom_in_Vset: fixes \<alpha> :: "V"   and T :: "V" assumes "arr_Rel \<alpha> T" shows "T\<lparr>ArrDom\<rparr> \<in>\<^sub>\<circ> Vset \<alpha>"
prems_2: fixes a :: "'c"   and b :: "'d" assumes "a \<in> insert a' A"   and "b \<in> B" shows "g a b \<in> carrier G"
ngba_g_run_ipath: fixes A :: "('c, 'd) ngba"   and w :: "'c stream"   and r :: "'d stream"   and p :: "'d" assumes "run A (w ||| r) p" shows "ipath (g_E (ngba_g A)) ((!!) (p ## r))"
gState_progress_rel_channels_update_step: fixes prog :: "program"   and g :: "'a gState_scheme"   and g' :: "'a gState_scheme"   and c :: "channel"   and i :: "nat" assumes "gState_inv prog g"   and "(g, g') \<in> gState_progress_rel prog"   and "channel_inv c"   and "i < length (gState.channels g')" shows "(g, gState.channels_update (\<lambda>cs. cs[i := c]) g') \<in> gState_progress_rel prog"
finite_total_preorder_on_map: fixes carrier :: "'b set"   and le :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'b" assumes "finite_total_preorder_on carrier le"   and "finite (f -` carrier)" shows "finite_total_preorder_on (f -` carrier) (map_relation f le)"
stream_pure: fixes x :: "'a" shows "stream (pure x) = pure x"
C1: fixes x :: "'b" assumes "x \<in> C" shows "norm x = 1"
cases: fixes x :: "'b sort_key extended" obtains xa :: "'b sort_key" where "x = \<lbrakk>xa\<rbrakk>"   | "x = \<turnstile>"   | "x = \<stileturn>"
sublist_length_le: fixes xs :: "'d list"   and ys :: "'d list" assumes "sublist xs ys" shows "length xs \<le> length ys"
ennreal_Sup_countable_SUP: fixes A :: "ennreal set" assumes "A \<noteq> {}" shows "\<exists>f. incseq f \<and> range f \<subseteq> A \<and> Sup A = Sup (range f)"
discrete_evolution_def: fixes incr :: "real \<Rightarrow> real \<Rightarrow> 'e \<Rightarrow> 'e"   and t1 :: "real"   and t0 :: "real"   and x :: "'e" shows "discrete_evolution incr t1 t0 x = x + (t1 - t0) *\<^sub>R incr (t1 - t0) t0 x"
partial_state2_axioms: shows "partial_state2 {0..<Suc m} {Suc m}"
target: fixes T1 :: "'procT"   and T2 :: "'procT"   and TRel :: "('procT \<times> 'procT) set" assumes "(T1, T2) \<in> TRel" shows "TargetTerm T1 \<R>\<lbrakk>\<cdot>\<rbrakk>T<TRel> TargetTerm T2"
singlPart_def: fixes I :: "'c set" shows "singlPart I \<equiv> {{i} |i. i \<in> I}"
equal_sat_def: fixes A :: "'a sat"   and B :: "'a sat" shows "equal_class.equal A B = (nat_of A = nat_of B)"
gen_lucas_lehmer_sequence''_correct: fixes a :: "int"   and a' :: "int"   and p :: "nat"   and n :: "nat" assumes "[a = a'] (mod 2 ^ p - 1)" shows "[gen_lucas_lehmer_sequence'' p a n = gen_lucas_lehmer_sequence a' n] (mod 2 ^ p - 1)"
parallel_decomposition: fixes V :: "('exp, 'id) MWLfCom list"   and V' :: "('exp, 'id) MWLfCom list"   and d :: "'d" assumes "V \<approx>\<^bsub>d\<^esub> V'" shows "\<forall>i<length V. [V ! i] \<approx>\<^bsub>d\<^esub> [V' ! i]"
prems_1: fixes P''' :: "('a, 'b, 'c) psi" assumes "\<Psi> \<rhd> P'' \<Longrightarrow>\<^sup>^\<^sub>\<tau> P'''"   and "\<Psi> \<rhd> P'''__ \<sim> P'''" shows "thesis"
a_star_NotR: fixes M :: "trm"   and M' :: "trm"   and x :: "name"   and a :: "coname" assumes "M \<longrightarrow>\<^sub>a* M'" shows "NotR x.M a \<longrightarrow>\<^sub>a* NotR x.M' a"
succ_def: shows "succ \<equiv> \<Lambda> (ValN\<cdot>n). ValN\<cdot>(n + 1)"
intro: 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" assumes "Contracts Xd Xh Pd Ch"   and "ContractsWithIRC_axioms Ch" shows "ContractsWithIRC Xd Xh Pd Ch"
simps_2: fixes n :: "nat"   and f :: "'g \<Rightarrow> 'g" shows "f ^^ Suc n = (f \<circ>\<circ>\<circ> compow) n f"
WTrtCond: fixes P :: "(char list list \<times> char list exp) prog"   and h :: "heap"   and sh :: "sheap"   and E :: "char list \<Rightarrow> ty option"   and e :: "char list exp"   and e\<^sub>1 :: "char list exp"   and T\<^sub>1 :: "ty"   and e\<^sub>2 :: "char list exp"   and T\<^sub>2 :: "ty"   and T :: "ty" assumes "P,E,h,sh \<turnstile> e : Boolean"   and "P,E,h,sh \<turnstile> e\<^sub>1 : T\<^sub>1"   and "P,E,h,sh \<turnstile> e\<^sub>2 : T\<^sub>2"   and "P \<turnstile> T\<^sub>1 \<le> T\<^sub>2 \<or> P \<turnstile> T\<^sub>2 \<le> T\<^sub>1"   and "P \<turnstile> T\<^sub>1 \<le> T\<^sub>2 \<longrightarrow> T = T\<^sub>2"   and "P \<turnstile> T\<^sub>2 \<le> T\<^sub>1 \<longrightarrow> T = T\<^sub>1" shows "P,E,h,sh \<turnstile> if (e) e\<^sub>1 else e\<^sub>2 : T"
countable_Un_Int_1: fixes I :: "'a set"   and N :: "'a \<Rightarrow> 'b set"   and M :: "'b measure" assumes "\<And>i. i \<in> I \<Longrightarrow> N i \<in> sets M" shows "\<Union> (N ` I) \<in> sets M"
iT_Plus_neg_iWeakUntil_conv: fixes I :: "nat set"   and k :: "nat"   and P :: "nat \<Rightarrow> bool"   and Q :: "nat \<Rightarrow> bool" shows "(P t1. t1 \<W> t2 I \<oplus>- k. Q t2) = (P (t1 - k). t1 \<W> t2 I \<down>\<ge> k. Q (t2 - k))"
squarefree_nat_def: shows "squarefree_nat = squarefree"
nodes_sim: fixes R :: "('a \<times> 'b) set"   and A :: "('a, 'c) graph_rec_scheme"   and B :: "('b, 'd) graph_rec_scheme"   and a :: "'a"   and b :: "'b" assumes "simulation R A B"   and "a \<in> g_V A"   and "(a, b) \<in> R" shows "b \<in> g_V B"
prefix_snoc: fixes xs :: "'a list"   and ys :: "'a list"   and y :: "'a" shows "prefix xs (ys @ [y]) = (xs = ys @ [y] \<or> prefix xs ys)"
bounded_def: fixes Q :: "'a set" shows "bounded Q \<equiv> \<exists>Q\<^sub>b. is_bound Q\<^sub>b Q"