hcju/isabelleps huggingface dataset:
pirreducible_pow_pdivides_iff: fixes K :: "'a list set"   and p :: "'a list list"   and q :: "'a list list"   and r :: "'a list list"   and n :: "nat" assumes "subfield K (poly_ring R)"   and "p \<in> carrier (K [X]\<^bsub>poly_ring R\<^esub>)"   and "q \<in> carrier (K [X]\<^bsub>poly_ring R\<^esub>)"   and "r \<in> carrier (K [X]\<^bsub>poly_ring R\<^esub>)"   and "pirreducible\<^bsub>poly_ring R\<^esub> K p"   and "\<not> p pdivides\<^bsub>poly_ring R\<^esub> q" shows "p [^]\<^bsub>K [X]\<^bsub>poly_ring R\<^esub>\<^esub> n pdivides\<^bsub>poly_ring R\<^esub> q \<otimes>\<^bsub>K [X]\<^bsub>poly_ring R\<^esub>\<^esub> r = p [^]\<^bsub>K [X]\<^bsub>poly_ring R\<^esub>\<^esub> n pdivides\<^bsub>poly_ring R\<^esub> r"
vars_of_eq_a: fixes e1 :: "('a \<Rightarrow> 'b) \<Rightarrow> 'b"   and e2 :: "('a \<Rightarrow> 'b) \<Rightarrow> 'b" shows "Bexp.vars (\<lambda>\<sigma>. e1 \<sigma> = e2 \<sigma>) \<subseteq> Aexp.vars e1 \<union> Aexp.vars e2"
preserves_comp_1: fixes A :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and B :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and F :: "'a \<Rightarrow> 'c"   and G :: "'a \<Rightarrow> 'c"   and \<tau> :: "'a \<Rightarrow> 'c"   and f' :: "'a"   and f :: "'a" assumes "natural_transformation A B F G \<tau>"   and "partial_magma.arr A (A f' f)" shows "\<tau> (A f' f) = B (G f') (\<tau> f)"
atLeast_Suc_lessThan_Suc_shift: fixes g :: "nat \<Rightarrow> 'b"   and m :: "nat"   and n :: "nat" shows "sum g {Suc m..<Suc n} = sum (g \<circ> Suc) {m..<n}"
nba_g_ipath_run: fixes A :: "('d, 'c) nba"   and r :: "nat \<Rightarrow> 'c" assumes "ipath (g_E (nba_g A)) r" obtains w :: "'d stream" where "run A (w ||| smap (r \<circ> Suc) nats) (r 0)"
cobounded2: fixes b :: "'t"   and a :: "'t" shows "b \<preceq>\<^sub>t ord_term_lin.max a b"
spmf_rel_map1: fixes R :: "'i \<Rightarrow> 'h \<Rightarrow> bool"   and f :: "'j \<Rightarrow> 'i"   and x :: "'j spmf" shows "rel_spmf R (map_spmf f x) = rel_spmf (\<lambda>x. R (f x)) x"
ltl_llist_of: fixes xs :: "'b list" shows "ltl (llist_of xs) = llist_of (tl xs)"
integral_fst: fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'd" assumes "integrable (count_space A \<Otimes>\<^sub>M count_space B') (\<lambda>(x, y). f x y)" shows "LINT x|count_space A. integral\<^sup>L (count_space B') (f x) = LINT (x, y)|(count_space A \<Otimes>\<^sub>M count_space B'). f x y"
det_identical_rows: fixes i :: "'c"   and i' :: "'c"   and A :: "('d, 'c) sq_matrix" assumes "i \<noteq> i'"   and "Square_Matrix.row A i = Square_Matrix.row A i'" shows "Square_Matrix.det A = (0::'d)"
BrnFT_elim: fixes ii :: "nat"   and cl :: "('test, 'atom, 'choice) cmd list"   and dl :: "('d, 'e, 'f) cmd list" assumes "ii \<in> BrnFT cl dl" obtains n :: "nat" and i :: "nat" where "n \<in> theFTOne cl dl" and "i < brn (cl ! n)" and "ii = brnL cl n + i"
scaleR: fixes r :: "real"   and b :: "'a" shows "g (r *\<^sub>R b) = r *\<^sub>R g b"
vproductD_1: fixes f :: "V"   and I :: "V"   and A :: "V \<Rightarrow> V" assumes "f \<in>\<^sub>\<circ> vproduct I A" shows "vsv f"
locally_path_connected_space_def: fixes X :: "'c topology" shows "locally_path_connected_space X \<equiv> neighbourhood_base_of (path_connectedin X) X"
case_prod_beta': fixes f :: "'e \<Rightarrow> 'f \<Rightarrow> 'g" shows "(\<lambda>(x, y). f x y) = (\<lambda>x. f (fst x) (snd x))"
term_to_nterm'_def: fixes t :: "Term_Class.term" shows "term_to_nterm' t = frun_fresh (term_to_nterm [] t) (frees t)"
not_SN_onI: fixes f :: "nat \<Rightarrow> 'b"   and X :: "'b set"   and R :: "('b \<times> 'b) set" assumes "f 0 \<in> X"   and "\<forall>i. (f i, f (Suc i)) \<in> R" shows "\<not> SN_on R X"
ith: shows "(t # ts) ! i = (p, is, xs, sb, \<D>, \<O>, \<R>)"
desargues_config_not_col_9: fixes A :: "Points"   and B :: "Points"   and C :: "Points"   and A' :: "Points"   and B' :: "Points"   and C' :: "Points"   and M :: "Points"   and N :: "Points"   and P :: "Points"   and R :: "Points" assumes "desargues_config A B C A' B' C' M N P R" shows "\<not> col B A A'"
simps_1: fixes b :: "bexp"   and c :: "com"   and Q :: "(char list \<Rightarrow> int) \<Rightarrow> bool"   and s :: "char list \<Rightarrow> int" shows "wpw b c 0 Q s = (\<not> bval b s \<and> Q s)"
uniform_limitD: fixes f :: "'c \<Rightarrow> 'd \<Rightarrow> 'e"   and S :: "'d set"   and l :: "'d \<Rightarrow> 'e"   and F :: "'c filter"   and e :: "real" assumes "uniform_limit S f l F"   and "0 < e" shows "\<forall>\<^sub>F n in F. \<forall>x\<in>S. dist (f n x) (l x) < e"
two_composable_identity_arrows_of_spans_axioms: shows "two_composable_identity_arrows_of_spans (\<cdot>) prj0 prj1 f (src f)"
subst_mset_def: fixes \<sigma> :: "'e \<Rightarrow> ('c, 'd) Term.term"   and M :: "(('c, 'e) Term.term \<times> ('c, 'e) Term.term) multiset" shows "subst_mset \<sigma> M = {#(fst p \<cdot> \<sigma>, snd p \<cdot> \<sigma>). p \<in># M#}"
diff_mult_distrib2: fixes k :: "nat"   and m :: "nat"   and n :: "nat" shows "k * (m - n) = k * m - k * n"
deepL_inv: fixes n :: "nat"   and m :: "('b, 'c) FingerTreeStruc"   and sf :: "('b, 'c) Digit"   and pr :: "('b, 'c) Node list" assumes "is_leveln_ftree (Suc n) m \<and> is_measured_ftree m"   and "is_leveln_digit n sf \<and> is_measured_digit sf"   and "\<forall>x\<in>set pr. (is_measured_node x \<and> is_leveln_node n x) \<and> length pr \<le> 4" shows "is_leveln_ftree n (deepL pr m sf) \<and> is_measured_ftree (deepL pr m sf)"
complement_incomplete: shows "incomplete_design \<V> \<B>\<^sup>C (des.\<v> - \<k>)"
mergesort_by_rel_split_length: fixes xs1 :: "'b list"   and xs2 :: "'b list"   and xs :: "'b list" shows "length (fst (mergesort_by_rel_split (xs1, xs2) xs)) = length xs1 + length xs div 2 + length xs mod 2 \<and> length (snd (mergesort_by_rel_split (xs1, xs2) xs)) = length xs2 + length xs div 2"
vec_lambda_unique: fixes f :: "('b, 'a) Finite_Cartesian_Product.vec"   and g :: "'a \<Rightarrow> 'b" shows "(\<forall>i. f $h i = g i) = (vec_lambda g = f)"
coinduct: fixes X :: "'b \<Rightarrow> ('a, 'b, 'c) psi \<Rightarrow> ('a, 'b, 'c) psi \<Rightarrow> bool"   and x1 :: "'b"   and x2 :: "('a, 'b, 'c) psi"   and x3 :: "('a, 'b, 'c) psi" assumes "X x1 x2 x3"   and "\<And>x1 x2 x3. X x1 x2 x3 \<Longrightarrow> \<exists>P \<Psi> Q. x1 = \<Psi> \<and> x2 = P \<and> x3 = Q \<and> insertAssertion (extractFrame P) \<Psi> \<simeq>\<^sub>F insertAssertion (extractFrame Q) \<Psi> \<and> \<Psi> \<rhd> P \<leadsto>[{(xb, xa, x). X xb xa x \<or> xb \<rhd> xa \<sim> x}] Q \<and> (\<forall>\<Psi>'. X (\<Psi> \<otimes> \<Psi>') P Q \<or> \<Psi> \<otimes> \<Psi>' \<rhd> P \<sim> Q) \<and> (X \<Psi> Q P \<or> \<Psi> \<rhd> Q \<sim> P)" shows "x1 \<rhd> x2 \<sim> x3"
span_zero: fixes scale :: "'d \<Rightarrow> 'e \<Rightarrow> 'e"   and S :: "'e set" assumes "module scale" shows "(0::'e) \<in> module.span scale S"
assm: shows "0 < \<parallel>ext v\<parallel> \<and> |lan v| = 1 \<and> (\<forall>c. \<parallel>len v ts c\<parallel> = 0 \<or> restrict v (clm ts) c = bot \<and> restrict v (res ts) c = bot)"
liftI_0: fixes A :: "nat set" shows "0 \<in> lift A"
IH: shows "state (m * n) (A \<otimes>\<^bsup>n\<^esup>)"
hyps_3: shows "sh' sh'' = \<lfloor>(C, i_)\<rfloor>"
Cd_preferred: fixes x :: "'a"   and y :: "'a"   and d :: "'b"   and X :: "'a set" assumes "(x, y) \<in> Pd d"   and "x \<in> Cd d X"   and "y \<in> X" shows "y = x"
iso_points_map: fixes \<V> :: "'c set"   and \<B> :: "'c set multiset"   and \<V>' :: "'d set"   and \<B>' :: "'d set multiset"   and bij_map :: "'c \<Rightarrow> 'd" assumes "incidence_system_isomorphism \<V> \<B> \<V>' \<B>' bij_map" shows "bij_map ` \<V> = \<V>'"
bisimilar_simulation_step: fixes satisfies :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and transition :: "'a \<Rightarrow> ('c, 'a) residual \<Rightarrow> bool"   and P :: "'a"   and Q :: "'a"   and \<alpha> :: "'c"   and P' :: "'a" assumes "nominal_ts satisfies transition"   and "nominal_ts.bisimilar satisfies transition P Q"   and "bn \<alpha> \<sharp>* Q"   and "transition P \<langle>\<alpha>,P'\<rangle>" obtains Q' :: "'a" where "transition Q \<langle>\<alpha>,Q'\<rangle>" and "nominal_ts.bisimilar satisfies transition P' Q'"
surj_imp_imT_carrier: assumes "T ` carrier V = carrier W" shows "imT = carrier W"
implies_curry: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<leadsto> (y \<leadsto> z) = x \<sqinter> y \<leadsto> z"
C'def: shows "C' = (\<Gamma>C', \<Delta>C')"
M'_def: shows "M' \<equiv> \<lambda>cfg. distr (MDP.MC.T (t cfg)) (stream_space (count_space UNIV)) (smap absc)"
is_lasso_prpl_pre_conv: fixes G :: "('e, 'f) graph_rec_scheme"   and prpl :: "'e list \<times> 'e list" assumes "graph G" shows "graph.is_lasso_prpl_pre G prpl = (snd prpl \<noteq> [] \<and> graph.is_lasso_pre G (lasso_of_prpl prpl))"
degree_def: fixes x :: "'b pdevs" shows "degree x = (LEAST i. \<forall>j\<ge>i. pdevs_apply x j = (0::'b))"
ord_\<Gamma>_contradiction_Rf: fixes Sts :: "(('a literal multiset \<times> nat) multiset \<times> ('a literal multiset \<times> nat) multiset \<times> ('a literal multiset \<times> nat) multiset \<times> nat) llist"   and N :: "'a literal multiset set" assumes "full_chain (\<leadsto>\<^sub>w) Sts"   and "wrp.P_of_wstate (lhd Sts) = {}"   and "wrp.Q_of_wstate (lhd Sts) = {}"   and "{#} \<in> N" shows "standard_redundancy_criterion.Ri (ground_resolution_with_selection.ord_\<Gamma> (wrp.S_gQ Sts)) N = ground_resolution_with_selection.ord_\<Gamma> (wrp.S_gQ Sts)"
Keys_bad_insert_keys_of: fixes Ik :: "msg set"   and Bad :: "agent set"   and A :: "agent" assumes "Keys_bad Ik Bad" shows "Keys_bad (keys_of A \<union> Ik) (insert A Bad)"
irreflD: fixes x :: "'a" assumes "x \<bowtie> x" shows "x \<notin> A"
resl_trans: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(x \<leftarrow> y) \<cdot> (y \<leftarrow> z) \<le> x \<leftarrow> z"
pair_spmf_of_set: fixes A :: "'a set"   and B :: "'b set" shows "pair_spmf (spmf_of_set A) (spmf_of_set B) = spmf_of_set (A \<times> B)"
refl: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b" shows "equivpartp r x x"
rtranclp_eq_rtrancl_path: fixes r :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and x :: "'k"   and y :: "'k" shows "r\<^sup>*\<^sup>* x y = (\<exists>xs. rtrancl_path r x xs y)"
rel_sum_map1: fixes R1 :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and R2 :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and f1 :: "'i \<Rightarrow> 'e"   and f2 :: "'j \<Rightarrow> 'g"   and x :: "'i + 'j"   and y :: "'f + 'h" shows "rel_sum R1 R2 (map_sum f1 f2 x) y = rel_sum (\<lambda>x. R1 (f1 x)) (\<lambda>x. R2 (f2 x)) x y"
vsingleton_inject: fixes x :: "V"   and y :: "V" assumes "set {x} = set {y}" shows "x = y"
Union_Eats: fixes x :: "tm"   and y :: "tm"   and z :: "tm"   and a :: "tm" shows "{UnionP x y z} \<turnstile> UnionP (Eats x a) y (Eats z a)"
order_refl: fixes x :: "nat \<Rightarrow> nat" shows "x \<lesssim> x"
dom_dom: fixes f :: "'b" assumes "C'.arr f" shows "C'.dom (C'.dom f) = C'.dom f"
inject_2: fixes x21 :: "pat"   and x22 :: "pat"   and y21 :: "pat"   and y22 :: "pat" shows "(\<langle>\<langle>x21, x22\<rangle>\<rangle> = \<langle>\<langle>y21, y22\<rangle>\<rangle>) = (x21 = y21 \<and> x22 = y22)"
simps_3: fixes f1 :: "'a"   and f2 :: "'a" shows "(case \<zero> of \<zero> \<Rightarrow> f1 | \<one> \<Rightarrow> f2) = f1"
is_monomial_monomial_ordered: fixes p :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" assumes "is_monomial p" obtains c :: "'b" and v :: "'x \<Rightarrow>\<^sub>0 nat" where "c \<noteq> (0::'b)" and "lcf p = c" and "lpp p = v" and "p = monomial c v"
nat_to_sch_at_pos: fixes x :: "nat" assumes "0 < x" shows "nat_to_sch x = (let u = c_fst x mod 7; v = c_snd x; v1 = c_fst v; v2 = c_snd v; sch1 = nat_to_sch v1 in Let (nat_to_sch v2) (loc_f u sch1))"
assms_4: shows "set lss = valid_states (prob_proj PROB vs)"
prems: shows "\<exists>k\<le>Suc n. c k \<noteq> (0::'a)"
simps: fixes pat :: "[:'r:]" shows "matches\<cdot>pat = smap\<cdot>slength oo sfilter\<cdot>(endswith\<cdot>pat) oo sinits"
has_vderiv_on_subset: fixes f :: "real \<Rightarrow> 'b"   and f' :: "real \<Rightarrow> 'b"   and S :: "real set"   and T :: "real set" assumes "(f has_vderiv_on f') S"   and "T \<subseteq> S" shows "(f has_vderiv_on f') T"
PROTECT2_def: fixes x :: "'j"   and y :: "prop" shows "PROTECT2 x PROP y \<equiv> x"
insertE: fixes a :: "'a"   and b :: "'a"   and A :: "'a set" assumes "a \<in> b \<triangleright> A" obtains "a = b"   | "a \<in> A"
1: shows "(c1 ;; c2, s) \<rightarrow>*c (d', t')"
finite_dec_above_ne: fixes f :: "nat \<Rightarrow> 'b"   and c :: "'b"   and b :: "nat"   and i :: "nat" assumes "finite {x. f x \<noteq> c}" shows "finite {x. f (dec_above b i x) \<noteq> c}"
L2_set_mono: fixes K :: "'b set"   and f :: "'b \<Rightarrow> real"   and g :: "'b \<Rightarrow> real" assumes "\<And>i. i \<in> K \<Longrightarrow> f i \<le> g i"   and "\<And>i. i \<in> K \<Longrightarrow> 0 \<le> f i" shows "L2_set f K \<le> L2_set g K"
iso_map\<^sub>1_ide: fixes f :: "'b" assumes "D.ide f" shows "D.iso (F.counit\<^sub>1 f)"
nofail_simps_2: fixes X :: "'c set" shows "nofail (RES X) = True"
doctor_optimal_match_stable_on: fixes ds :: "'d set"   and Y :: "'x set" assumes "doctor_optimal_match ds Y" shows "stable_on ds Y"
not_single_rt_fresh_asI2: fixes rt2 :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option"   and dip :: "nat"   and rt1 :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option" assumes "\<not> the (rt2 dip) \<sqsubseteq> the (rt1 dip)" shows "\<not> rt1 \<approx>\<^bsub>dip\<^esub> rt2"
current_lock_used: fixes c :: "nat"   and n :: "nat"   and ts :: "nat \<Rightarrow> thread"   and t :: "nat"   and k :: "nat" assumes "A.reachable (c, n, ts)"   and "t < N"   and "ts t = WAIT k" obtains t' :: "nat" where "t' < N" and "has_ticket (ts t') c"
reduce_less_plus_consts: fixes x :: "nat"   and n1 :: "nat"   and y :: "nat"   and n2 :: "nat" assumes "x + n1 < y + n2" shows "x < y + (n2 - n1)"
Prop_2_step_L7: fixes g :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and xs :: "'b list" shows "map (\<lambda>k. foldl1 g (map ((!) xs) [0..<k + 1])) [0..<length xs] = map (\<lambda>k. foldl1 g (take (k + 1) xs)) [0..<length xs]"
prime_imp_not_one: fixes divide :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and unit_factor :: "'d \<Rightarrow> 'd"   and normalize :: "'d \<Rightarrow> 'd"   and x :: "'d" assumes "class.normalization_semidom divide plus minus zero times one unit_factor normalize"   and "ASSUMPTION (normalization_semidom.prime zero times one normalize x)" shows "x \<noteq> one"
relation_subset_crossp: shows "relation \<subseteq> \<P> outcomes \<times> \<P> outcomes"
swap_image: fixes i :: "'b"   and j :: "'b"   and f :: "'b \<Rightarrow> 'a"   and A :: "'b set" shows "Fun.swap i j f ` A = (if i \<in> A then if j \<in> A then f ` A else f ` (A - {i} \<union> {j}) else if j \<in> A then f ` (A - {j} \<union> {i}) else f ` A)"
intros_4: fixes K :: "key"   and X :: "msg"   and H :: "msg set" assumes "Crypt K X \<in> analz H"   and "Key (invKey K) \<in> analz H" shows "X \<in> analz H"
matroid_ax_1b: fixes X :: "Points set" shows "rk X \<le> card X"
mat_swaprows_def: fixes k :: "nat"   and l :: "nat"   and A :: "'c mat" shows "swaprows k l A = mat (dim_row A) (dim_col A) (\<lambda>(i, j). if k = i then A $$ (l, j) else if l = i then A $$ (k, j) else A $$ (i, j))"
returns_error_def: fixes h :: "'a"   and p :: "('a, 'b, 'c) prog"   and e :: "'b" shows "h \<turnstile> p \<rightarrow>\<^sub>e e = (case h \<turnstile> p of Inl e' \<Rightarrow> e = e' | Inr x \<Rightarrow> False)"
app: fixes P :: "'c \<Rightarrow> bool"   and t :: "'c"   and u :: "'c" assumes "simple_syntactic_or P" shows "P (app t u) = (P t \<or> P u)"
circ_decompose_4: fixes x :: "'a"   and y :: "'a" shows "(x\<^sup>\<circ> * y\<^sup>\<circ>)\<^sup>\<circ> = x\<^sup>\<circ> * (y\<^sup>\<circ> * x\<^sup>\<circ>)\<^sup>\<circ>"
between_not_r2: fixes vs :: "'a list"   and ram2 :: "'a"   and ram1 :: "'a" assumes "distinct vs" shows "ram2 \<notin> set (between vs ram1 ram2)"
intro: fixes K :: "'e set"   and R :: "('e, 'f) ring_scheme" assumes "subdomain K R"   and "subfield_axioms K R" shows "subfield K R"
of_bl_length_less: fixes x :: "bool list"   and k :: "nat" assumes "length x = k"   and "k < LENGTH('a)" shows "of_bl x < 2 ^ k"
simps_19: fixes f1 :: "(('b \<Rightarrow> 'c option) \<Rightarrow> 'b \<Rightarrow> 'c option) \<Rightarrow> 'a"   and f2 :: "(('b \<Rightarrow> 'c option) \<Rightarrow> bool) \<Rightarrow> 'a"   and f3 :: "(('b \<Rightarrow> 'c option) \<times> 'd \<Rightarrow> bool) \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> (('b \<Rightarrow> 'c option) \<Rightarrow> 'c option) list \<Rightarrow> 'a"   and f4 :: "(('b \<Rightarrow> 'c option) \<times> 'd \<Rightarrow> bool) \<Rightarrow> 'e \<Rightarrow> (('b \<Rightarrow> 'c option) \<Rightarrow> ('b \<Rightarrow> 'c option) \<Rightarrow> 'b \<Rightarrow> 'c option) \<Rightarrow> 'a"   and x31 :: "('b \<Rightarrow> 'c option) \<times> 'd \<Rightarrow> bool"   and x32 :: "'d"   and x33 :: "'e"   and x34 :: "(('b \<Rightarrow> 'c option) \<Rightarrow> 'c option) list" shows "(case x31:x32\<hookrightarrow>\<^bsub>x33\<^esub>x34 of \<Up>x \<Rightarrow> f1 x | (x)\<^sub>\<surd> \<Rightarrow> f2 x | x:xa\<hookrightarrow>\<^bsub>xb\<^esub>xc \<Rightarrow> f3 x xa xb xc | x\<hookleftarrow>\<^bsub>xa\<^esub>xb \<Rightarrow> f4 x xa xb) = f3 x31 x32 x33 x34"
fdomainI: fixes a :: "V"   and b :: "V"   and r :: "V" assumes "[a, b]\<^sub>\<circ> \<in>\<^sub>\<circ> r" shows "a \<in>\<^sub>\<circ> \<D>\<^sub>\<bullet> r"
vpeq_trans: fixes u :: "partition_id_t"   and s1 :: "state_t"   and s2 :: "state_t"   and s3 :: "state_t" assumes "vpeq u s1 s2"   and "vpeq u s2 s3" shows "vpeq u s1 s3"
coerce_writerT_abs: fixes x :: "'h\<cdot>'j writer\<cdot>'i" shows "COERCE('h\<cdot>('i, 'j) writerT, 'e\<cdot>('f, 'g) writerT)\<cdot>(writerT_abs\<cdot>x) = writerT_abs\<cdot>(COERCE('h\<cdot>'j writer\<cdot>'i, 'e\<cdot>'g writer\<cdot>'f)\<cdot>x)"
prems_4: fixes k :: "nat"   and h :: "heap"   and h' :: "heap"   and r :: "'a" assumes "i \<le> k"   and "k < j"   and "I k h"   and "effect (f k) h h' r" shows "I (Suc k) h'"
remdups_adj_singleton: fixes xs :: "'a list"   and x :: "'a" assumes "remdups_adj xs = [x]" shows "xs = x \<up> length xs"
le_divide_eq_numeral1_1: fixes a :: "'a"   and b :: "'a"   and w :: "num" shows "(a \<le> b / numeral w) = (a * numeral w \<le> b)"
Rel_def: shows "Rel = indRelSTEQ SRel TRel - (_Collect (P, Q) (\<exists>S. \<lbrakk>S\<rbrakk> \<in>T P \<and> S \<in>S Q) \<union> _Collect (P, Q) (\<exists>S1 S2. S1 \<in>S P \<and> S2 \<in>S Q \<and> (S1, S2) \<notin> SRel) \<union> _Collect (P, Q) (\<exists>T1 T2. T1 \<in>T P \<and> T2 \<in>T Q \<and> (T1, T2) \<notin> TRel))"
th_rec: fixes g :: "nat \<Rightarrow> nat"   and \<alpha> :: "nat \<Rightarrow> nat \<Rightarrow> nat"   and h :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat"   and f :: "nat \<Rightarrow> nat \<Rightarrow> nat" assumes "g \<in> PrimRec1"   and "\<alpha> \<in> PrimRec2"   and "h \<in> PrimRec3"   and "\<forall>x y. \<alpha> y x \<le> y"   and "\<forall>x. f 0 x = g x"   and "\<forall>x y. f (Suc y) x = h y (f (\<alpha> y x) x) x" shows "f \<in> PrimRec2"
igFreshIGOpSTR_def: fixes MOD :: "('e, 'f, 'g, 'h, 'i, 'j, 'k, 'l, 'm) model_scheme" shows "igFreshIGOpSTR MOD \<equiv> \<forall>ys y delta inp binp. igFreshInp MOD ys y inp \<and> igFreshBinp MOD ys y binp \<longrightarrow> igFresh MOD ys y (igOp MOD delta inp binp)"
assms_11: shows "InvariantClCurrentLevel (getCl state) (getM state)"
deformation_retract_imp_homotopy_equivalent_space: fixes X :: "'a topology"   and r :: "'a \<Rightarrow> 'a"   and Y :: "'a topology" assumes "homotopic_with (\<lambda>x. True) X X r id"   and "retraction_maps X Y r id" shows "X homotopy_equivalent_space Y"
distinct_1: fixes x1 :: "ct list"   and x2 :: "t list" shows "TopType x1 \<noteq> Type x2"
tl_drop: fixes n :: "nat"   and xs :: "'g list" shows "tl (drop n xs) = drop n (tl xs)"
prems_1: shows "are_all_operators_applicable s (op # \<pi>\<^sub>1)"
eq_im_imp_diff_in_Ker: fixes G :: "'a set"   and smult :: "('b, 'a) aezfun \<Rightarrow> 'c \<Rightarrow> 'c"   and V :: "'c set"   and smult' :: "('b, 'a) aezfun \<Rightarrow> 'd \<Rightarrow> 'd"   and T :: "'c \<Rightarrow> 'd"   and g :: "'c"   and h :: "'c" assumes "FGModuleHom G smult V smult' T"   and "g \<in> V"   and "h \<in> V"   and "T g = T h" shows "g - h \<in> ker T \<inter> V"
simps_4: fixes f1 :: "'a"   and f2 :: "'a" shows "(case Bot of Top \<Rightarrow> f1 | Bot \<Rightarrow> f2) = f2"
has_field_derivative_ln_Gamma_real: fixes x :: "real" assumes "0 < x" shows "(ln_Gamma has_real_derivative Digamma x) (at x)"
simps_1: shows "size_wbt \<langle>\<rangle> = 0"
JVM_heap_conf_base'_axioms: shows "JVM_heap_conf_base' addr2thread_id thread_id2addr allocate typeof_addr heap_write P"
sub_distrib_pconj_def: fixes prog :: "bool \<Rightarrow> ('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real" shows "sub_distrib_pconj prog \<equiv> \<forall>P Q. unitary P \<longrightarrow> unitary Q \<longrightarrow> wlp prog P && wp prog Q \<tturnstile> wp prog (P && Q)"
reachableI: fixes step :: "'b \<Rightarrow> 'd \<Rightarrow> 'b \<Rightarrow> bool"   and s :: "'b"   and p :: "'d list"   and s' :: "'b" assumes "LTS.path step s p s'" shows "LTS.reachable step s s'"
cases: fixes x :: "primfun" obtains n :: "nat" where "x = LnChain n"
q0_def: shows "q0 = monomial (lookup r' (t \<oplus> lt g) / lc g) t"
equivalence_is_bisimulation: shows "is_bisimulation (=\<cdot>)"
wary_AppE: fixes arity_sym :: "'a \<Rightarrow> enat"   and arity_var :: "'b \<Rightarrow> enat"   and s :: "('a, 'b) tm"   and t :: "('a, 'b) tm" assumes "arity.wary arity_sym arity_var (App s t)" obtains "arity.wary arity_sym arity_var s" and "arity.wary arity_sym arity_var t" and "enat (num_args s) < arity.arity_hd arity_sym arity_var (head s)"
eexp_add_mult: fixes a :: "ereal"   and b :: "ereal" assumes "\<not> (a = \<infinity> \<and> b = - \<infinity> \<or> a = - \<infinity> \<and> b = \<infinity>)" shows "eexp (a + b) = eexp a * eexp b"
ind: assumes "local.dom d u" shows "d \<in> set l \<or> d = u"
join_connection: fixes x :: "'a"   and y :: "'a" shows "(x \<sqsubseteq> y) = (x \<squnion> y = y)"
split_asm: fixes P :: "'a \<Rightarrow> bool"   and f1 :: "'b \<Rightarrow> 'a"   and f2 :: "'c \<Rightarrow> 'a"   and sum :: "'b + 'c" shows "P (case sum of Inl x \<Rightarrow> f1 x | Inr x \<Rightarrow> f2 x) = (\<not> ((\<exists>x1. sum = Inl x1 \<and> \<not> P (f1 x1)) \<or> (\<exists>x2. sum = Inr x2 \<and> \<not> P (f2 x2))))"
prefix_Cons: fixes xs :: "'b list"   and y :: "'b"   and ys :: "'b list" shows "prefix xs (y # ys) = (xs = [] \<or> (\<exists>zs. xs = y # zs \<and> prefix zs ys))"
rbd_rbb_demorgan: fixes R :: "('c \<times> 'd) set" shows "\<partial> \<circ> bd\<^sub>\<R> R = bb\<^sub>\<R> R \<circ> \<partial>"
assms_3: fixes f :: "'d"   and pred :: "'b" shows "P (Pred\<^sub>\<alpha> f pred)"
2: shows "h' \<turnstile> filter_M P xs \<rightarrow>\<^sub>r ys'"
S_def: shows "S = range (\<lambda>n. replicate n CHR ''A'')"
vmatch_env: fixes pat :: "pat"   and v :: "value"   and env :: "(name, value) fmap" assumes "vmatch pat v = Some env"   and "vwellformed v" shows "wellformed_venv env"
inclusion: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and Arr :: "'b \<Rightarrow> bool"   and f :: "'b" assumes "subcategory C Arr"   and "Arr f" shows "partial_magma.arr C f"
WT_gpv_inline1_1: fixes out :: "'call'"   and rpv :: "('ret \<times> 's, 'call', 'ret') rpv"   and rpv' :: "('b, 'call, 'ret) rpv"   and gpv :: "('b, 'call, 'ret) gpv"   and s :: "'s" assumes "Inr (out, rpv, rpv') \<in> set_spmf (inline1 callee gpv s)"   and "\<I> \<turnstile>g gpv \<surd>"   and "I s" shows "out \<in> outs_\<I> \<I>'"
noread_nowrite: fixes 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"   and x :: "'Var" assumes "\<langle>c, mds, mem\<rangle> \<leadsto> \<langle>c', mds', mem'\<rangle>"   and "\<And>v. \<langle>c, mds, mem(x := v)\<rangle> \<leadsto> \<langle>c', mds', mem'(x := v)\<rangle>" shows "mem x = mem' x"