hcju/isabelleps huggingface dataset:
vimage_comp: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'c"   and x :: "'c set" shows "f -` g -` x = (g \<circ> f) -` x"
le_prod_Pair_conv: fixes a\<^sub>1 :: "'c"   and b\<^sub>1 :: "'d"   and r\<^sub>A :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and r\<^sub>B :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and a\<^sub>2 :: "'c"   and b\<^sub>2 :: "'d" shows "((a\<^sub>1, b\<^sub>1) <=(r\<^sub>A,r\<^sub>B) (a\<^sub>2, b\<^sub>2)) = (a\<^sub>1 \<sqsubseteq>\<^bsub>r\<^sub>A\<^esub> a\<^sub>2 \<and> b\<^sub>1 \<sqsubseteq>\<^bsub>r\<^sub>B\<^esub> b\<^sub>2)"
and_or_s: fixes w1 :: "word32"   and w2 :: "word32" assumes "get_S w1 = 0 \<and> w2 AND 128 = 0" shows "get_S (w1 AND 4279238655 OR w2) = 0"
interpretation_subst_exists': fixes X :: "'a set" shows "\<exists>\<theta>. subst_domain \<theta> = X \<and> ground (subst_range \<theta>)"
PSP: fixes F :: "'c program"   and A :: "'c set"   and A' :: "'c set"   and B :: "'c set"   and B' :: "'c set" assumes "F \<in> A \<longmapsto>w A'"   and "F \<in> B Co B'" shows "F \<in> A \<inter> B' \<longmapsto>w A' \<inter> B \<union> (B' - B)"
h7: shows "set (characterize_consistent_signs_at_roots p (qs1 @ qs2)) \<subseteq> set (get_signs_R (snd (combine_systems_R p (qs1, matrix1, subsets1, signs1) (qs2, matrix2, subsets2, signs2))))"
set_paint: fixes c :: "color"   and t :: "('b \<times> color) tree" shows "Tree2.set_tree (paint c t) = Tree2.set_tree t"
mset_le_addE: fixes xs :: "'a multiset"   and ys :: "'a multiset" assumes "xs \<subseteq># ys" obtains zs :: "'a multiset" where "ys = xs + zs"
Nml_Nmlize_1: fixes t :: "'a local.term" assumes "Arr t" shows "Nml \<^bold>\<lfloor>t\<^bold>\<rfloor>"
NSDeMoivre_ext: fixes a :: "real star"   and n :: "nat star" shows "hcis a pow n = hcis (hypreal_of_hypnat n * a)"
T_del_min_def: fixes ts :: "'b tree list" shows "T_del_min ts = T_get_min_rest ts + (case get_min_rest ts of (Node xa x ts\<^sub>1, ts\<^sub>2) \<Rightarrow> T_rev ts\<^sub>1 + T_merge (rev ts\<^sub>1) ts\<^sub>2) + 1"
InfCard_iff: fixes \<kappa> :: "V" shows "InfCard \<kappa> = (Card \<kappa> \<and> infinite (elts \<kappa>))"
inject_3: fixes x31 :: "nat"   and x32 :: "exp"   and y31 :: "nat"   and y32 :: "exp" shows "(ELam x31 x32 = ELam y31 y32) = (x31 = y31 \<and> x32 = y32)"
hom_prod: fixes f :: "'b \<Rightarrow> 'a"   and X :: "'b set" shows "[:prod f X:] = (\<Prod>x\<in>X. [:f x:])"
eval\<^sub>l_bij: fixes b_of_a :: "'c \<Rightarrow> 'd"   and E :: "char list \<Rightarrow> 'c"   and F :: "char list \<Rightarrow> 'c list \<Rightarrow> 'c"   and G :: "char list \<Rightarrow> 'c list \<Rightarrow> bool"   and l :: "fterm literal" assumes "bij b_of_a" shows "eval\<^sub>l (E_conv b_of_a E) (F_conv b_of_a F) (G_conv b_of_a G) l = eval\<^sub>l E F G l"
extreme_singleton: fixes x :: "'a"   and y :: "'a" assumes "x \<in> A" shows "extreme {x} (\<sqsubseteq>) y = (x = y)"
p_1: shows "poly (map_poly of_int p) x = (0::'a)"
hatom_of_nat_def: shows "hatom_of_nat \<equiv> inv nat_of_hatom"
simps: fixes i :: "'a ifex"   and t :: "'a ifex"   and e :: "'a ifex" shows "ifex_ite_opt i t e = (case param_opt i t e of None \<Rightarrow> case lowest_tops [i, t, e] of None \<Rightarrow> case i of Trueif \<Rightarrow> t | Falseif \<Rightarrow> e | Some x \<Rightarrow> IFC x (ifex_ite_opt (restrict_top i x True) (restrict_top t x True) (restrict_top e x True)) (ifex_ite_opt (restrict_top i x False) (restrict_top t x False) (restrict_top e x False)) | Some b \<Rightarrow> b)"
refl_on_DomainD: fixes A :: "'b set"   and r :: "('b \<times> 'b) set" assumes "refl_on A r" shows "A = Domain r"
up: shows "a + liminf v \<le> liminf (\<lambda>n. u n + v n)"
stepReachable_def: fixes c1 :: "('p, 'v, 's) configuration"   and msg :: "('p, 'v) message"   and c2 :: "('p, 'v, 's) configuration" shows "stepReachable c1 msg c2 \<equiv> \<exists>c' c''. reachable c1 c' \<and> c' \<turnstile> msg \<mapsto> c'' \<and> reachable c'' c2"
less_le: fixes y :: "('d, 'e) Term.term"   and x :: "('d, 'e) Term.term" shows "y \<rhd> x = (y \<unrhd> x \<and> x \<noteq> y)"
rotate1_rl': fixes l :: "'b list"   and a :: "'b" shows "rotater1 (l @ [a]) = a # l"
assms: shows "Some a \<in> set (map (map_option priority) xs)"
weight_bind_spmf: fixes x :: "'c spmf"   and f :: "'c \<Rightarrow> 'a spmf" shows "weight_spmf (x \<bind> f) = integral\<^sup>L (measure_spmf x) (weight_spmf \<circ> f)"
remove_key_def: fixes k0 :: "'c"   and f :: "'c \<Rightarrow>\<^sub>0 'd" shows "remove_key k0 f = Abs_poly_mapping (\<lambda>k. lookup f k when k \<noteq> k0)"
bounded_countable_bipartite_web_of: shows "bounded_countable_bipartite_web (bipartite_web_of \<Gamma>)"
that: fixes K :: "'a set"   and g :: "'a \<Rightarrow> 'b" assumes "finite K"   and "K \<subseteq> L"   and "K \<subseteq> T"   and "disjnt K S"   and "continuous_on (T - K) g"   and "g ` (T - K) \<subseteq> sphere a r"   and "\<And>x. x \<in> S \<Longrightarrow> g x = f x" shows "thesis"
fold_mset_def: fixes f :: "'f \<Rightarrow> 'e \<Rightarrow> 'e"   and s :: "'e"   and M :: "'f multiset" shows "fold_mset f s M = Finite_Set.fold (\<lambda>x. f x ^^ count M x) s (set_mset M)"
not_outside_1: fixes x :: "'b"   and y :: "'b" assumes "map_relation f le x y" shows "x \<in> f -` carrier"
sum_linear: fixes x :: "'c \<times> 'c \<Rightarrow> 'd"   and y :: "'c \<times> 'c \<Rightarrow> 'd" shows "sum\<^sub>M x \<preceq> sum\<^sub>M y \<or> sum\<^sub>M y \<preceq> sum\<^sub>M x"
prems_1: shows "e\<langle>i:T\<rangle> \<tturnstile> bs @ [b] : Us"
Sup_bdd: shows "\<bar>Sup (ereal ` S)\<bar> \<noteq> \<infinity>"
F_Mprefix: fixes A :: "'b set"   and P :: "'b \<Rightarrow> 'b process" shows "F (Mprefix A P) = {(tr, ref). tr = [] \<and> ref \<inter> ev ` A = {}} \<union> {(tr, ref). tr \<noteq> [] \<and> hd tr \<in> ev ` A \<and> (\<exists>a. ev a = hd tr \<and> (tl tr, ref) \<in> F (P a))}"
exhaust: fixes y :: "'a interval" obtains x1 :: "'a" and x2 :: "'a" where "y = Interval x1 x2"
distinct_1: fixes x11 :: "('a, 'b) graph_node"   and x12 :: "'a"   and x13 :: "nat"   and x14 :: "'c"   and x21 :: "nat"   and x22 :: "'c" shows "I x11 x12 x13 x14 \<noteq> H x21 x22"
h6: shows "mat_of_rows_list 3 [[1, 1, 1], [0, 1, 0], [1, 0, - 1]] = M_mat_R [[1], [0], [- 1]] [([], []), ([0], []), ([], [0])]"
prob_algebra_def: fixes K :: "'d measure" shows "prob_algebra K = restrict_space (subprob_algebra K) {M. prob_space M}"
path2_rev_induct: fixes g :: "'g"   and n :: "'node"   and ns :: "'node list"   and m :: "'node"   and P :: "'node \<Rightarrow> 'node list \<Rightarrow> 'node \<Rightarrow> bool" assumes "g \<turnstile> n-ns\<rightarrow>m"   and "n \<in> set (\<alpha>n g) \<Longrightarrow> P n [n] n"   and "\<And>ns m' m. \<lbrakk>g \<turnstile> n-ns\<rightarrow>m'; P n ns m'; m' \<in> set (predecessors g m)\<rbrakk> \<Longrightarrow> P n (ns @ [m]) m" shows "P n ns m"
simps: fixes var :: "nat"   and a :: "real mpoly"   and b :: "real mpoly"   and c :: "real mpoly"   and d :: "real mpoly"   and F :: "atom fm" shows "quadratic_sub_fm var a b c d F = quadratic_sub_fm_helper var a b c d F 0"
3: shows "ladder_stepdown_\<alpha>_0 \<alpha> D L = Derive (ladder_\<alpha> \<alpha> D L 1) [D ! ladder_n L 0]"
simps: fixes a1 :: "('a, 'b) Proc"   and a2 :: "('a, 'b) Proc"   and Enc :: "'a \<Rightarrow> 'b"   and TRel :: "('b \<times> 'b) set" shows "((a1, a2) \<in> encoding.indRelLT Enc TRel) = ((\<exists>S. a1 = TargetTerm (Enc S) \<and> a2 = SourceTerm S) \<or> (\<exists>T1 T2. a1 = TargetTerm T1 \<and> a2 = TargetTerm T2 \<and> (T1, T2) \<in> TRel))"
s_mul_ext_def: fixes ns :: "('a \<times> 'a) set"   and s :: "('a \<times> 'a) set" shows "s_mul_ext ns s \<equiv> (mult2_alt_s (ns\<inverse>) (s\<inverse>))\<inverse>"
vec_to_iarray_def: fixes A :: "('c, 'd) vec" shows "vec_to_iarray A = IArray.of_fun (\<lambda>i. A $ mod_type_class.from_nat i) CARD('d)"
1: shows "nat (Max (sq_norm ` set fs_init)) \<in> (nat \<circ> sq_norm) ` set fs_init"
picard_lindeloef_constant: fixes c :: "'b"   and t\<^sub>0 :: "real" shows "picard_lindeloef (\<lambda>t s. c) UNIV UNIV t\<^sub>0"
reachable_def: shows "A.reachable \<equiv> A.step\<^sup>*\<^sup>* as\<^sub>0"
bad: shows "rel_fun (\<lambda>x y. bad1 x = bad2 y \<and> (\<not> bad2 y \<longrightarrow> A x = B y)) (=) bad1 bad2"
eval_simps_2: fixes b :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and c :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "ide b"   and "ide c" shows "local.dom (hfsetcat.eval b c) = HF'.prod (hfsetcat.exp b c) b"
eq_equiv_class: fixes r :: "('e \<times> 'e) set"   and a :: "'e"   and b :: "'e"   and A :: "'e set" assumes "r `` {a} = r `` {b}"   and "equiv A r"   and "b \<in> A" shows "(a, b) \<in> r"
length_dropWhile_le: fixes P :: "'c \<Rightarrow> bool"   and xs :: "'c list" shows "length (dropWhile P xs) \<le> length xs"
ext: fixes g :: "'a \<times> 'a \<times> 'a"   and f :: "'a \<times> 'a \<times> 'a" assumes "VVV.comp g f \<noteq> VVV.null" shows "VVV.seq g f"
err_def: fixes A :: "'b set" shows "err A = insert Err {OK x |x. x \<in> A}"
simps_2: fixes x :: "'b"   and xs :: "['b]" shows "tails\<cdot>(x : xs) = (x : xs) : tails\<cdot>xs"
fls_base_factor_subdegree: fixes f :: "'a fls" shows "fls_subdegree (fls_base_factor f) = 0"
rep_eq: fixes x :: "'a rexp" shows "Rep_reg_lan (r_lang x) = lang x"
concat_eq_append_conv: fixes xss :: "'g list list"   and ys :: "'g list"   and zs :: "'g list" shows "(concat xss = ys @ zs) = (if xss = [] then ys = [] \<and> zs = [] else \<exists>xss1 xs xs' xss2. xss = xss1 @ (xs @ xs') # xss2 \<and> ys = concat xss1 @ xs \<and> zs = xs' @ concat xss2)"
simps_5: fixes x3 :: "nat \<times> nat" shows "SAT_PUSH' x3 \<noteq> RELABEL'"
cons_rule: fixes P :: "assn"   and P' :: "assn"   and Q :: "'a \<Rightarrow> assn"   and Q' :: "'a \<Rightarrow> assn"   and c :: "'a Heap" assumes "P \<Longrightarrow>\<^sub>A P'"   and "\<And>x. Q x \<Longrightarrow>\<^sub>A Q' x"   and "<P'> c <Q>" shows "<P> c <Q'>"
nAct_llength_proj: fixes n :: "nat"   and c :: "'id"   and t :: "nat \<Rightarrow> cnf" assumes "\<exists>i\<ge>n. \<parallel>c\<parallel>\<^bsub>t i\<^esub>" shows "eSuc \<langle>c #\<^bsub>enat n\<^esub>inf_llist t\<rangle> \<le> llength (\<pi>\<^bsub>c\<^esub>inf_llist t)"
simps_2: fixes x :: "('b, 'c) Combinators" shows "list2FWpolicy [x] = x"
degree_i_def: fixes pp :: "'a list" shows "degree_i pp \<equiv> length pp - 1"
ball_imageD: fixes f :: "'e \<Rightarrow> 'd"   and A :: "'e set"   and P :: "'d \<Rightarrow> bool" assumes "\<forall>x\<in>f ` A. P x" shows "\<forall>x\<in>A. P (f x)"
rel_FGcontra_def: fixes L1 :: "'s \<Rightarrow> 'af \<Rightarrow> bool"   and Co1 :: "'t \<Rightarrow> 'ag \<Rightarrow> bool"   and Co2 :: "'ab \<Rightarrow> 'ao \<Rightarrow> bool"   and Co3 :: "'u \<Rightarrow> 'ah \<Rightarrow> bool"   and Co4 :: "'v \<Rightarrow> 'ai \<Rightarrow> bool"   and Co5 :: "'w \<Rightarrow> 'aj \<Rightarrow> bool"   and Contra1 :: "'x \<Rightarrow> 'ak \<Rightarrow> bool"   and Contra2 :: "'y \<Rightarrow> 'al \<Rightarrow> bool"   and Contra3 :: "'z \<Rightarrow> 'am \<Rightarrow> bool"   and Contra4 :: "'aa \<Rightarrow> 'an \<Rightarrow> bool"   and Contra5 :: "'ad \<Rightarrow> 'ap \<Rightarrow> bool" shows "rel_FGcontra L1 Co1 Co2 Co3 Co4 Co5 Contra1 Contra2 Contra3 Contra4 Contra5 = rel_F L1 Co1 Co3 Co1 Co4 Co5 (rel_G Contra1 Contra2 Contra3 Contra4 Co1 Co2) Contra1 Contra5"
LHS_def: shows "LHS \<equiv> fls_prpart (fls_shift m f)"
prems_2: shows "(\<forall>X<:T\<^sub>2. T\<^sub>1) closed_in \<Gamma>"
M: shows "det (subresultant'_mat J k f g) = det (mat (df + dg) (df + dg) (\<lambda>(i, j). if i = df + dg - 1 then if j < dg then coeff_int f (int k - int (dg - 1) + int j) else coeff_int g (int k - int (df - 1) + int (j - dg)) else M i j))"
tranclp_rtranclp_absorb: fixes R :: "'d \<Rightarrow> 'd \<Rightarrow> bool" shows "R\<^sup>+\<^sup>+\<^sup>*\<^sup>* = R\<^sup>*\<^sup>*"
Lxy_notin: fixes xs :: "'b list"   and S :: "'b set" assumes "set xs \<inter> S = {}" shows "Lxy xs S = []"
hyps: fixes child_node :: "(_) node_ptr"   and parent :: "(_) object_ptr" assumes "cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r child_node = child"   and "h \<turnstile> get_parent child_node \<rightarrow>\<^sub>r Some parent"   and "parent \<in> set c" shows "h \<turnstile> get_component parent \<rightarrow>\<^sub>r c"
nn_set_integral_set_mono: fixes A :: "'b set"   and B :: "'b set"   and M :: "'b measure"   and f :: "'b \<Rightarrow> ennreal" assumes "A \<subseteq> B" shows "set_nn_integral M A f \<le> set_nn_integral M B f"
comb_sat: shows "1 < length qs \<longrightarrow> p \<noteq> 0 \<and> 0 < length qs \<and> (\<forall>q. List.member qs q \<longrightarrow> coprime p q) \<longrightarrow> satisfies_properties p qs (get_subsets (snd (combine_systems p (take (length qs div 2) qs, calculate_data p (take (length qs div 2) qs)) (drop (length qs div 2) qs, calculate_data p (drop (length qs div 2) qs))))) (get_signs (snd (combine_systems p (take (length qs div 2) qs, calculate_data p (take (length qs div 2) qs)) (drop (length qs div 2) qs, calculate_data p (drop (length qs div 2) qs))))) (get_matrix (snd (combine_systems p (take (length qs div 2) qs, calculate_data p (take (length qs div 2) qs)) (drop (length qs div 2) qs, calculate_data p (drop (length qs div 2) qs)))))"
assms: fixes L :: "('a, 'b) Term.term literal"   and K :: "('a, 'c) Term.term literal"   and \<sigma> :: "'b \<Rightarrow> ('a, 'c) Term.term option"   and \<tau> :: "'b \<Rightarrow> ('a, 'c) Term.term option" assumes "L \<in> set Ls"   and "match_term_list [(atm_of L, atm_of K)] \<sigma> = Some \<tau>"   and "is_pos L = is_pos K" shows "P K"
assms_1: fixes x :: "'a"   and y :: "'a" assumes "x \<in> A"   and "y \<in> B" shows "is_orthogonal x y"
irrefl: fixes a :: "('a, 'b) Term.term" shows "\<not> a \<sqsubset> a"
invertibleI: fixes M :: "'c set"   and composition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit :: "'c"   and u :: "'c"   and v :: "'c" assumes "Group_Theory.monoid M composition unit"   and "composition u v = unit"   and "composition v u = unit"   and "u \<in> M"   and "v \<in> M" shows "monoid.invertible M composition unit u"
preserves_arr: fixes f :: "'a" assumes "arr f" shows "arr (L f)"
poly_shift_degree_zero: fixes p :: "nat \<Rightarrow> 'a" assumes "p \<in> carrier P"   and "deg R p = 0" shows "poly_shift p = \<zero>\<^bsub>P\<^esub>"
simps_2: fixes n :: "nat"   and x :: "'c" shows "replicate (Suc n) x = x # replicate n x"
eff_enabled'_def: fixes s :: "nat \<Rightarrow> nat option" shows "eff_enabled' s \<equiv> \<lambda>(pres, uu_, uu_, uu_). match_pres pres s"
ab_semigroup_add_hom_axioms: shows "ab_semigroup_add_hom of_real"
Red1New: fixes h' :: "'heap"   and a :: "'addr"   and h :: "'heap"   and C :: "String.literal"   and uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and l :: "'addr val list" assumes "(h', a) \<in> allocate h (Class_type C)" shows "uf,P,t \<turnstile>1 \<langle>new C,(h, l)\<rangle> -\<lbrace>NewHeapElem a (Class_type C)\<rbrace>\<rightarrow> \<langle>addr a,(h', l)\<rangle>"
sets_P: fixes x :: "nat \<Rightarrow> 'a"   and n :: "nat" assumes "x \<in> space (Pi\<^sub>M {0..<n} M)" shows "sets (P n x) = sets (M n)"
push_down_rank_senior: fixes \<Sigma> :: "'c set"   and \<delta> :: "'d \<Rightarrow> 'c \<Rightarrow> 'd"   and q\<^sub>0 :: "'d"   and w :: "nat \<Rightarrow> 'c"   and x :: "nat"   and n :: "nat"   and y :: "nat"   and i :: "nat" assumes "semi_mojmir \<Sigma> \<delta> q\<^sub>0 w"   and "semi_mojmir_def.rank \<Sigma> \<delta> q\<^sub>0 w x n = semi_mojmir_def.rank \<Sigma> \<delta> q\<^sub>0 w y n"   and "semi_mojmir_def.rank \<Sigma> \<delta> q\<^sub>0 w x n = Some i" shows "semi_mojmir_def.senior \<delta> q\<^sub>0 w x n = semi_mojmir_def.senior \<delta> q\<^sub>0 w y n"
rref_condition3_equiv1: fixes A :: "(('b, 'c) vec, 'd) vec"   and i :: "nat"   and j :: "nat" assumes "reduced_row_echelon_form A"   and "i < j"   and "j < nrows A"   and "\<not> is_zero_row (mod_type_class.from_nat i) A"   and "\<not> is_zero_row (mod_type_class.from_nat j) A" shows "(LEAST n. A $ mod_type_class.from_nat i $ n \<noteq> (0::'b)) < (LEAST n. A $ mod_type_class.from_nat j $ n \<noteq> (0::'b))"
one_class: shows "\<one>\<^bsub>cring_class_ops\<^esub> = (1::'b)"
geometric_pmf_prob_greaterThan: fixes p :: "real"   and n :: "nat" assumes "p \<in> {0<..1}" shows "measure_pmf.prob (geometric_pmf p) {n<..} = (1 - p) ^ (n + 1)"
cut_less_cut_greater_ident: fixes t2 :: "'a"   and t1 :: "'a"   and I :: "'a set" assumes "t2 \<le> t1"   and "I \<inter> {t1..t2} = {}" shows "I \<down>< t1 \<union> I \<down>> t2 = I"
col_mat_of_col: fixes v :: "'b vec" shows "col (mat_of_col v) 0 = v"
chamber_adj: fixes C :: "'a set"   and D :: "'a set" assumes "chamber C"   and "D \<in> X"   and "C \<sim> D" shows "chamber D"
DataSpace_def: fixes L :: "'b set list" shows "DataSpace L = (distinct L \<and> (\<forall>D1\<in>set L. \<forall>D2\<in>set L. D1 \<noteq> D2 \<longrightarrow> D1 \<inter> D2 = {}) \<and> \<Union> (set L) = UNIV)"
proj_atomize_poly: fixes k :: "'k"   and p :: "'k \<Rightarrow>\<^sub>0 'a \<Rightarrow>\<^sub>0 'c" shows "proj_poly k (atomize_poly p) = lookup p k"
landau_function_family_def: fixes F :: "'a filter"   and H :: "('a \<Rightarrow> real) set" shows "landau_function_family F H \<equiv> (F \<noteq> bot \<and> (\<forall>h. h \<in> H \<longrightarrow> (\<forall>\<^sub>F x in F. 0 < h x))) \<and> (\<forall>h1 h2. h1 \<in> H \<longrightarrow> h2 \<in> H \<longrightarrow> h1 \<in> o[F](h2) \<or> h2 \<in> o[F](h1) \<or> h1 \<in> \<Theta>[F](h2)) \<and> (\<forall>h1 h2. h1 \<in> H \<longrightarrow> h2 \<in> H \<longrightarrow> (\<lambda>x. h1 x * h2 x) \<in> H) \<and> (\<forall>h. h \<in> H \<longrightarrow> (\<lambda>x. inverse (h x)) \<in> H)"
l_insert_before_wf_axioms_def: fixes heap_is_wellformed :: "(_) heap \<Rightarrow> bool"   and type_wf :: "(_) heap \<Rightarrow> bool"   and known_ptrs :: "(_) heap \<Rightarrow> bool"   and insert_before :: "(_) object_ptr \<Rightarrow> (_) node_ptr \<Rightarrow> (_) node_ptr option \<Rightarrow> ((_) heap, exception, unit) prog"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog" shows "l_insert_before_wf_axioms heap_is_wellformed type_wf known_ptrs insert_before get_child_nodes \<equiv> \<forall>h ptr ptr' node child h' children. heap_is_wellformed h \<longrightarrow> type_wf h \<longrightarrow> known_ptrs h \<longrightarrow> ptr \<noteq> ptr' \<longrightarrow> h \<turnstile> insert_before ptr node child \<rightarrow>\<^sub>h h' \<longrightarrow> h \<turnstile> get_child_nodes ptr' \<rightarrow>\<^sub>r node # children \<longrightarrow> h' \<turnstile> get_child_nodes ptr' \<rightarrow>\<^sub>r children"
is_write_seenD: fixes P :: "'a prog"   and E :: "('b \<times> ('c, 'b) obs_event action) llist"   and ws :: "nat \<Rightarrow> nat"   and a :: "nat"   and ad :: "'c"   and al :: "addr_loc"   and v :: "'c val" assumes "is_write_seen P E ws"   and "a \<in> read_actions E"   and "action_obs E a = NormalAction (ReadMem ad al v)" shows "ws a \<in> write_actions E \<and> (ad, al) \<in> action_loc P E (ws a) \<and> value_written P E (ws a) (ad, al) = v \<and> \<not> P,E \<turnstile> a \<le>hb ws a \<and> (is_volatile P al \<longrightarrow> \<not> P,E \<turnstile> a \<le>so ws a) \<and> (\<forall>a'\<in>write_actions E. (ad, al) \<in> action_loc P E a' \<and> (P,E \<turnstile> ws a \<le>hb a' \<and> P,E \<turnstile> a' \<le>hb a \<or> is_volatile P al \<and> P,E \<turnstile> ws a \<le>so a' \<and> P,E \<turnstile> a' \<le>so a) \<longrightarrow> a' = ws a)"
map_ide: fixes b :: "'d"   and a :: "'d" assumes "D.ide b"   and "D.ide a" shows "HomD.map (b, a) = S.mkIde (HomD.set (b, a))"
hnorm_mult_less: fixes x :: "'a star"   and r :: "real star"   and y :: "'a star"   and s :: "real star" assumes "hnorm x < r"   and "hnorm y < s" shows "hnorm (x * y) < r * s"
component_of_term_def: fixes v :: "'t" shows "component_of_term v = snd (pair_of_term v)"
AxEuclidean: fixes x :: "'a" assumes "(0::'a) \<le> x" shows "\<exists>r. sqrtTest x r"
invh_ins: fixes t :: "('b \<times> color) tree"   and x :: "'b" assumes "invh t" shows "invh (ins x t) \<and> bheight (ins x t) = bheight t"
take_bit_incr_eq: fixes n :: "nat"   and k :: "int" assumes "take_bit n k \<noteq> 2 ^ n - 1" shows "take_bit n (k + 1) = 1 + take_bit n k"
assms: shows "f \<in>\<^sub>\<circ> dg_prod I \<AA>\<lparr>Arr\<rparr>"
map_is_Nil_conv: fixes f :: "'b \<Rightarrow> 'a"   and xs :: "'b list" shows "(map f xs = []) = (xs = [])"
mk_minsky_PrimRec_2: fixes g :: "nat \<Rightarrow> nat \<Rightarrow> nat" assumes "g \<in> PrimRec2" shows "mk_minsky2 g"
accessible_on_imp_af: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set"   and x :: "'b" assumes "accessible_on P A x" shows "af A (\<lambda>u v. \<not> P v u \<or> \<not> P u x)"
simps_8: fixes p :: "('a, 'b) form" shows "FOL_Fitting.size_form (Forall p) = FOL_Fitting.size_form p + 1"
simps_2: fixes x2 :: "'c error_result"   and y2 :: "'c error_result" shows "(Rerr x2 = Rerr y2) = (x2 = y2)"
sorted_hyp: shows "\<forall>q<length (sorted_list_of_set {x. \<exists>q\<in>set qs. rpoly q x = 0}). q + 1 < length (sorted_list_of_set {x. \<exists>q\<in>set qs. rpoly q x = 0}) \<longrightarrow> sorted_list_of_set {x. \<exists>q\<in>set qs. rpoly q x = 0} ! q < sorted_list_of_set {x. \<exists>q\<in>set qs. rpoly q x = 0} ! (q + 1)"
simps: fixes x :: "ordinal" shows "to_cnf2 x = (if x = 0 then [] else (oLog \<omega> x, inv ordinal_of_nat (x div \<omega> ** oLog \<omega> x)) # to_cnf2 (x mod \<omega> ** oLog \<omega> x))"
vcomp_eq: fixes \<nu> :: "'a arrow_of_spans_data"   and \<mu> :: "'a arrow_of_spans_data" assumes "V.seq \<nu> \<mu>" shows "\<nu> \<bullet> \<mu> = \<lparr>Chn = Chn \<nu> \<cdot> Chn \<mu>, Dom = Dom \<mu>, Cod = Cod \<nu>\<rparr>"
exec_less: shows "\<forall>i<r. 0 < rec_exec f (xs @ [i])"
ldropn_Suc_LCons: fixes n :: "nat"   and x :: "'g"   and xs :: "'g llist" shows "ldropn (Suc n) (LCons x xs) = ldropn n xs"
outpt: shows "outpt s' = (outpt s)(p := NotAnInput)"
constrains_INT: fixes I :: "'b set"   and F :: "'c program"   and A :: "'b \<Rightarrow> 'c set"   and A' :: "'b \<Rightarrow> 'c set" assumes "\<And>i. i \<in> I \<Longrightarrow> F \<in> A i co A' i" shows "F \<in> \<Inter> (A ` I) co \<Inter> (A' ` I)"
closure_of_restrict: fixes X :: "'a topology"   and S :: "'a set" shows "X closure_of S = X closure_of (topspace X \<inter> S)"
reduce_ivl_def: fixes X :: "('c \<times> 'd) set"   and b :: "'d" shows "reduce_ivl X b = ivl_rep X \<bind> (\<lambda>(i, s). CHECK (\<lambda>_. ST ''reduce_ivl strange basis'') (b \<in> set Basis_list) \<bind> (\<lambda>_. CHECK (\<lambda>_. ST ''reduce_ivl strange ivl'') (i \<le> s) \<bind> (\<lambda>_. let (i0, i1) = split_lv_rel i; (s0, s1) = split_lv_rel s; ivl2 = op_atLeastAtMost_ivl i1 s1 in project_set_ivl ivl2 b 0 \<bind> (\<lambda>P. ivl_rep P \<bind> (\<lambda>(iP, sP). if iP \<le> (0::'d) \<and> (0::'d) \<le> sP then if 0 < i1 \<bullet> b then let s = (i1 \<bullet> b) *\<^sub>R b in Let (op_atLeastAtMost_ivl (Pair_lv_rel i0 (iP + s)) (Pair_lv_rel s0 (sP + s))) (scaleRe_ivl_spec 1 \<infinity>) else if s1 \<bullet> b < 0 then let s = (s1 \<bullet> b) *\<^sub>R b in Let (op_atLeastAtMost_ivl (Pair_lv_rel i0 (iP + s)) (Pair_lv_rel s0 (sP + s))) (scaleRe_ivl_spec 1 \<infinity>) else scaleRe_ivl_spec 1 1 X else scaleRe_ivl_spec 1 1 X)))))"
1: shows "prv (imp (cnj (eqv \<phi> \<chi>) (eqv \<chi> \<psi>)) (imp \<phi> \<psi>))"
addpreRT_welldefined: fixes i :: "nat" shows "paodv i \<TTurnstile> onl \<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V (\<lambda>(\<xi>, l). (l \<in> {PRreq-:18..PRreq-:20} \<longrightarrow> dip \<xi> \<in> kD (rt \<xi>)) \<and> (l = PRreq-:19 \<longrightarrow> oip \<xi> \<in> kD (rt \<xi>)) \<and> (l = PRrep-:5 \<longrightarrow> dip \<xi> \<in> kD (rt \<xi>)) \<and> (l = PRrep-:6 \<longrightarrow> the (nhop (rt \<xi>) (dip \<xi>)) \<in> kD (rt \<xi>)))"
\<alpha>_def: shows "\<alpha> = \<Squnion> (value_flow \<Psi> ` {g. flow \<Psi> g})"
abs_eq: fixes x :: "int" shows "Uint8 (integer_of_int x) = Abs_uint8 (word_of_int x)"
pick_alphaAbs: fixes A :: "('d, 'e, 'varSort, 'var, 'f) qAbs set" assumes "goodAbs A" shows "pick A $= pick A"