hcju/isabelleps huggingface dataset:
compatible_iff_seq: fixes g :: "('obj, 'obj) concrete_category.arr"   and f :: "('obj, 'obj) concrete_category.arr" shows "(domains g \<inter> codomains f \<noteq> {}) = seq g f"
assms_1: shows "R = {(=), (<), (\<le>), (\<noteq>)}"
widen_subcls: fixes C :: "char list"   and D :: "char list"   and P :: "'a prog" assumes "P \<turnstile> C \<preceq>\<^sup>* D" shows "subtype P (Class C) (Class D)"
compl_pmdlI: fixes compl :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> nat \<times> 'g \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> 'f) list \<times> 'g" assumes "\<And>gs bs ps sps data. \<lbrakk>is_Groebner_basis (fst ` set gs); sps \<noteq> []; set sps \<subseteq> set ps; unique_idx (gs @ bs) data\<rbrakk> \<Longrightarrow> fst ` set (fst (compl gs bs (ps -- sps) sps data)) \<subseteq> pmdl (args_to_set (gs, bs, ps))" shows "compl_pmdl compl"
simps: fixes r :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and a1 :: "'g"   and a2 :: "'f" shows "r\<inverse>\<inverse> a1 a2 = (\<exists>a b. a1 = b \<and> a2 = a \<and> r a b)"
power2_minus: fixes a :: "'a" shows "(- a)\<^sup>2 = a\<^sup>2"
induct: fixes x1 :: "'a"   and x2 :: "'b set"   and x3 :: "'c multiset"   and fg :: "('a, 'c, 'd, 'b, 'e) flowgraph_rec_scheme"   and k :: "nat"   and P :: "'a \<Rightarrow> 'b set \<Rightarrow> 'c multiset \<Rightarrow> bool" assumes "(x1, x2, x3) \<in> S_cs fg k"   and "\<And>p. P (entry fg p) {} {#}"   and "\<And>u a v M P. \<lbrakk>(u, Base a, v) \<in> edges fg; (u, M, P) \<in> S_cs fg k; P u M P\<rbrakk> \<Longrightarrow> P v M P"   and "\<And>u q v M P Ms Ps P'. \<lbrakk>(u, Call q, v) \<in> edges fg; (u, M, P) \<in> S_cs fg k; P u M P; (return fg q, Ms, Ps) \<in> S_cs fg k; P (return fg q) Ms Ps; P' \<subseteq># P + Ps; size P' \<le> k\<rbrakk> \<Longrightarrow> P v (mon fg q \<union> M \<union> Ms) P'"   and "\<And>u q v M P P'. \<lbrakk>(u, Spawn q, v) \<in> edges fg; (u, M, P) \<in> S_cs fg k; P u M P; P' \<subseteq># {#q#} + P; size P' \<le> k\<rbrakk> \<Longrightarrow> P v M P'" shows "P x1 x2 x3"
OrdNotEqP_cong: fixes H :: "fm set"   and x :: "tm"   and x' :: "tm"   and y :: "tm"   and y' :: "tm" assumes "H \<turnstile> x EQ x'"   and "H \<turnstile> y EQ y'" shows "H \<turnstile> x NEQ y IFF x' NEQ y'"
simps_1: fixes n :: "nat" shows "mk_scheme (TVar n) = FVar n"
\<G>\<^sub>F\<^sub>G_Only_G: fixes \<phi> :: "'b ltl"   and w :: "nat \<Rightarrow> 'b set" shows "Only_G (\<G>\<^sub>F\<^sub>G \<phi> w)"
run_ask_option: fixes ask :: "('c \<Rightarrow> 'a) \<Rightarrow> 'a"   and f :: "'c \<Rightarrow> ('b, 'a) optionT" shows "run_option (ask_option ask f) = ask (\<lambda>r. run_option (f r))"
FOREACHoci_def: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and \<Phi> :: "'b set \<Rightarrow> 'a \<Rightarrow> bool"   and S :: "'b set"   and c :: "'a \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'a \<Rightarrow> 'a nres"   and \<sigma>0 :: "'a" shows "FOREACH\<^sub>O\<^sub>C\<^bsup>R,\<Phi>\<^esup> S c f \<sigma>0 \<equiv> ASSERT (finite S) \<bind> (\<lambda>_. SPEC (\<lambda>xs. distinct xs \<and> S = set xs \<and> sorted_wrt R xs) \<bind> (\<lambda>xs. WHILE\<^sub>T\<^bsup>\<lambda>(it, \<sigma>). \<exists>xs'. xs = xs' @ it \<and> \<Phi> (set it) \<sigma>\<^esup> (FOREACH_cond c) (FOREACH_body f) (xs, \<sigma>0) \<bind> (\<lambda>(uu_, \<sigma>). RETURN \<sigma>)))"
base_simps_2: shows "hseq t\<^sub>0\<^sup>* u\<^sub>1"
equivpE: fixes R :: "'e \<Rightarrow> 'e \<Rightarrow> bool" assumes "equivp R" obtains "reflp R" and "symp R" and "transp R"
m_ivl_widen: fixes y :: "ivl"   and x :: "ivl" assumes "\<not> y \<sqsubseteq> x" shows "m_ivl (x \<nabla> y) < m_ivl x"
in_rel: fixes R :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and a :: "'d fset"   and b :: "'e fset" shows "rel_fset R a b = (\<exists>z. z \<in> {x. fset x \<subseteq> {(x, y). R x y}} \<and> fst |`| z = a \<and> snd |`| z = b)"
reduce_root_def: fixes a :: "'b"   and p :: "'b poly" shows "reduce_root a p = p \<circ>\<^sub>p [:0::'b, a:]"
N: fixes n :: "nat" assumes "N \<le> n" shows "harm n - ln (real n) < 13 / 22"
cases: fixes x :: "'a ext" obtains "x = Bot"   | uu_ :: "'a" where "x = Val uu_"   | "x = Top"
distinct_update: fixes al :: "('a \<times> 'b) list"   and k :: "'a"   and v :: "'b" assumes "distinct (list.map fst al)" shows "distinct (list.map fst (AList.update k v al))"
con_g1: shows "continuous_on ({0..1} - insert 1 ((*) 2 ` S)) (\<lambda>x. vector_derivative g1 (at x))"
simps_11: fixes \<phi> :: "'b ltln"   and \<psi> :: "'b ltln"   and \<nu> :: "'b set" shows "af_letter (\<phi> M\<^sub>n \<psi>) \<nu> = af_letter \<psi> \<nu> and\<^sub>n af_letter \<phi> \<nu> or\<^sub>n (\<phi> M\<^sub>n \<psi>)"
simps_10: fixes bs :: "real list"   and p :: "fm"   and q :: "fm" shows "Ifm bs (And p q) = (Ifm bs p \<and> Ifm bs q)"
min_enat2_conv_enat: fixes a :: "enat"   and b :: "nat" shows "min a (enat b) = enat (case a of enat a' \<Rightarrow> min a' b | \<infinity> \<Rightarrow> b)"
tree_map_def: fixes t :: "('c, 'd) tree" shows "tree_map t = map_of_alist (in_traverse_pairs t)"
Int_UN_distrib: fixes B :: "'g set"   and A :: "'h \<Rightarrow> 'g set"   and I :: "'h set" shows "B \<inter> \<Union> (A ` I) = (\<Union>i\<in>I. B \<inter> A i)"
is_empty_stack_impl_def: fixes s :: "('f, 'g) simple_state_scheme" shows "is_empty_stack_impl s \<equiv> ss_stack s = []"
intro: fixes set_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and type_wf :: "(_) heap \<Rightarrow> bool"   and set_disconnected_nodes :: "(_) document_ptr \<Rightarrow> (_) node_ptr list \<Rightarrow> ((_) heap, exception, unit) prog"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and known_ptrs :: "(_) heap \<Rightarrow> bool"   and get_parent :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr option) prog"   and get_parent_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" assumes "l_set_disconnected_nodes_get_child_nodes set_disconnected_nodes_locs get_child_nodes_locs"   and "l_set_disconnected_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf set_disconnected_nodes set_disconnected_nodes_locs"   and "l_get_parent\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr type_wf get_child_nodes get_child_nodes_locs known_ptrs get_parent get_parent_locs" shows "l_set_disconnected_nodes_get_parent\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr type_wf set_disconnected_nodes set_disconnected_nodes_locs get_child_nodes get_child_nodes_locs known_ptrs get_parent get_parent_locs"
simps_2: fixes r :: "real"   and cs :: "real list"   and xs :: "real list" shows "I\<^sub>R (Eq r cs) xs = (r = \<langle>cs,xs\<rangle>)"
intros_6: fixes \<S> :: "s_context"   and cl :: "cl"   and tf :: "tf"   and \<C> :: "t_context" assumes "cl_typing \<S> cl tf" shows "\<S>\<bullet>\<C> \<turnstile> [Callcl cl] : tf"
s_antimono: fixes x :: "real"   and y :: "real" assumes "x \<le> y"   and "delayed_safe_distance.other.t_stop \<le> x" shows "delayed_safe_distance.other.s y \<le> delayed_safe_distance.other.s x"
lunit_tensor': fixes a :: "'a"   and b :: "'a" assumes "ide a"   and "ide b" shows "\<l>[a \<otimes> b] \<cdot> \<a>[\<I>, a, b] = \<l>[a] \<otimes> b"
bigo_plus_subset2: fixes A :: "('a \<Rightarrow> 'b) set"   and f :: "'a \<Rightarrow> 'b"   and B :: "('a \<Rightarrow> 'b) set" assumes "A \<subseteq> O(f)"   and "B \<subseteq> O(f)" shows "A + B \<subseteq> O(f)"
ardenL: fixes r :: "'a language"   and s :: "'a language"   and x :: "'a language" assumes "Plus r (Times s x) \<le> x" shows "Times (Star s) r \<le> x"
base_def: shows "base = real_of_rat (4 * \<alpha> / (4 + \<alpha>))"
hom_mult_eq_zero: fixes x :: "real poly"   and y :: "real poly" assumes "x * y = 0" shows "map_poly complex_of_real x * map_poly complex_of_real y = 0"
path_Append: fixes prog :: "cmd"   and n :: "w_node"   and as :: "(w_node \<times> (char list \<Rightarrow> val option) edge_kind \<times> w_node) list"   and n'' :: "w_node"   and as' :: "(w_node \<times> (char list \<Rightarrow> val option) edge_kind \<times> w_node) list"   and n' :: "w_node" assumes "While_CFG.path prog n as n''"   and "While_CFG.path prog n'' as' n'" shows "While_CFG.path prog n (as @ as') n'"
repeat_subst_fm: fixes A :: "fm"   and i :: "name"   and t :: "tm"   and u :: "tm" shows "(A(i::=t))(i::=u) = A(i::=subst i u t)"
addition_inc_add: fixes f :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and A :: "'e set"   and n :: "nat" shows "add_set f A n \<subseteq> addition_set f A"
le_finI: fixes w\<^sub>1 :: "'item list"   and v\<^sub>1 :: "'item list"   and w\<^sub>2 :: "'item list" assumes "w\<^sub>1 @ v\<^sub>1 =\<^sub>F w\<^sub>2" shows "w\<^sub>1 \<preceq>\<^sub>F w\<^sub>2"
assms_3: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qTerm set option" assumes "rho \<in> set rhoL" shows "goodEnv rho"
tmf_ivl_cs_length: fixes prec :: "nat"   and n :: "nat"   and A :: "float interval list"   and a :: "float list"   and f :: "floatarith"   and cs :: "float interval list" assumes "tmf_ivl_cs prec n A a f = Some cs" shows "length cs = n + 1"
Cons_1: fixes rv1 :: "'a \<times> nat" assumes "RedBlack prb"   and "SubRel.ces rv1 res rv2 (subs prb)" shows "Graph.ces (fst rv1) (ui_es res) (fst rv2)"
effect_refI: fixes r :: "'a ref"   and h' :: "heap"   and v :: "'a"   and h :: "heap" assumes "(r, h') = Ref.alloc v h" shows "effect (ref v) h h' r"
powser_continuous_suminf: fixes r :: "real"   and a :: "nat \<Rightarrow> 'a"   and \<xi> :: "'a" assumes "ereal r < conv_radius a" shows "continuous_on (cball \<xi> r) (\<lambda>x. \<Sum>i. a i * (x - \<xi>) ^ i)"
orderE: fixes a :: "'b::semilattice_inf"   and b :: "'b::semilattice_inf" assumes "(a::'b::semilattice_inf) \<le> (b::'b::semilattice_inf)" obtains "(a::'b::semilattice_inf) = a \<sqinter> (b::'b::semilattice_inf)"
bin_rl_simp: fixes w :: "int" shows "(w div 2) BIT odd w = w"
full_functor_axioms: shows "full_functor (\<cdot>\<^sub>S) (\<cdot>\<^sub>S) R"
diff_set_Ex_iff: fixes X :: "'b multiset"   and M :: "'b multiset"   and N :: "'b multiset"   and Y :: "'b multiset" shows "(X \<noteq> {#} \<and> X \<subseteq># M \<and> N = M - X + Y) = (X \<noteq> {#} \<and> (\<exists>Z. M = Z + X \<and> N = Z + Y))"
strict_trans2: fixes b :: "'b"   and a :: "'b"   and c :: "'b" assumes "b < a"   and "c \<le> b" shows "c < a"
plus_length2: fixes ls :: "'c list"   and k :: "nat" shows "rotate (|ls| + k) ls = rotate k ls"
cnvalid_to_cvalid: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "('e set \<times> 'd \<times> 'e set \<times> 'e set) set"   and F :: "'f set"   and P :: "'e set"   and c :: "('e, 'd, 'f) com"   and Q :: "'e set"   and A :: "'e set" assumes "\<forall>n. \<Gamma>,\<Theta>\<Turnstile>n:\<^bsub>/F\<^esub> P c Q,A" shows "\<Gamma>,\<Theta>\<Turnstile>\<^bsub>/F\<^esub> P c Q,A"
cp_OclIf: fixes C :: "'b state \<times> 'b state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>"   and B\<^sub>1 :: "'b state \<times> 'b state \<Rightarrow> 'a"   and B\<^sub>2 :: "'b state \<times> 'b state \<Rightarrow> 'a"   and \<tau> :: "'b state \<times> 'b state" shows "(if C then B\<^sub>1 else B\<^sub>2 endif) \<tau> = (if (\<lambda>_. C \<tau>) then (\<lambda>_. B\<^sub>1 \<tau>) else (\<lambda>_. B\<^sub>2 \<tau>) endif) \<tau>"
diff_add_assoc: fixes a :: "'k multiset"   and b :: "'k multiset"   and c :: "'k multiset" assumes "a \<subseteq># b" shows "c + b - a = c + (b - a)"
ind_mult_str_def: fixes V :: "'a set" shows "ind_mult_str V \<equiv> \<cdot>\<^bsub>U \<inter> V\<^esub>"
subset_antisym: fixes A :: "'b set"   and B :: "'b set" assumes "A \<subseteq> B"   and "B \<subseteq> A" shows "A = B"
convex_on_closure: fixes C :: "'a set"   and f :: "'a \<Rightarrow> real" assumes "convex C"   and "convex_on C f"   and "continuous_on (closure C) f" shows "convex_on (closure C) f"
element_of_subseqs_then_subset: fixes l :: "'b list"   and l' :: "'b list" assumes "l \<in> set (subseqs l')" shows "set l \<subseteq> set l'"
ih2: fixes t :: "trm" assumes "e\<^sub>2 \<Down> t" shows "e\<^sub>2' = t"
hyps_2: fixes V :: "char list" assumes "V \<notin> fv e" shows "l' V = l V"
axb: shows "Derives1 a i r x \<and> Derivation x E b"
all_I: fixes H :: "fmla set"   and A :: "fmla"   and i :: "name" assumes "H \<turnstile> A"   and "\<forall>C\<in>H. atom i \<sharp> C" shows "H \<turnstile> all i A"
float_plus_down_le: fixes x :: "float"   and y :: "float"   and z :: "real"   and prec :: "nat" assumes "real_of_float x + real_of_float y \<le> z" shows "real_of_float (float_plus_down prec x y) \<le> z"
ldfix: shows "LeftDerivationFix \<alpha> i D j \<gamma>"
ide_def: fixes a :: "'a1" shows "C1.ide a \<equiv> a \<cdot>\<^sub>1 a \<noteq> C1.null \<and> (\<forall>f. (f \<cdot>\<^sub>1 a \<noteq> C1.null \<longrightarrow> f \<cdot>\<^sub>1 a = f) \<and> (a \<cdot>\<^sub>1 f \<noteq> C1.null \<longrightarrow> a \<cdot>\<^sub>1 f = f))"
nn_integral_suminf: fixes f :: "nat \<Rightarrow> 'a \<Rightarrow> ennreal"   and M :: "'a measure" assumes "\<And>i. f i \<in> borel_measurable M" shows "\<integral>\<^sup>+ x. (\<Sum>i. f i x) \<partial>M = (\<Sum>i. integral\<^sup>N M (f i))"
vars_scaleRat: fixes c :: "rat"   and p :: "linear_poly" assumes "c \<noteq> 0" shows "vars (c *R p) = vars p"
contour_integrable_neg: assumes "(\<lambda>x. - f x) contour_integrable_on g" shows "(\<lambda>x. - (- f x)) contour_integrable_on g"
poly2_by_map: fixes p :: "'a poly poly"   and x :: "'a" shows "poly2 p x = poly (map_poly (\<lambda>c. poly c x) p)"
dghm_const_def: fixes \<CC> :: "V"   and \<DD> :: "V"   and a :: "V"   and f :: "V" shows "dghm_const \<CC> \<DD> a f = [vconst_on (\<CC>\<lparr>Obj\<rparr>) a, vconst_on (\<CC>\<lparr>Arr\<rparr>) f, \<CC>, \<DD>]\<^sub>\<circ>"
domA_restrictA: fixes S :: "'a set"   and \<Gamma> :: "('a \<times> 'b) list" shows "domA (restrictA S \<Gamma>) = domA \<Gamma> \<inter> S"
times_matrix_def: shows "times_matrix = (\<lambda>((a, c), b, d) ((a', c'), b', d'). ((a * a' + b * c', c * a' + d * c'), a * b' + b * d', c * b' + d * d'))"
landau_product_preprocess_28: shows "Num.BitM num.One = num.One"
msubstpos_nb: fixes p :: "fm"   and t :: "tm"   and c :: "poly" assumes "islin p"   and "tmbound0 t" shows "bound0 (msubstpos p c t)"
constghom_def: fixes F :: "('c, 'e) Group_scheme"   and G :: "('d, 'f) Group_scheme" shows "1\<^bsub>F,G\<^esub> = (\<lambda>x\<in>carrier F. \<one>\<^bsub>G\<^esub>)"
simps_2: fixes s :: "nat \<Rightarrow> nat \<Rightarrow> int"   and a :: "nat \<Rightarrow> int"   and n :: "nat" shows "convergent_subseq_fun s a (Suc n) = (SOME k. convergent_subseq_fun s a n < k \<and> s k (Suc n) = a (Suc n))"
bit_and_iff: 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 push_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and drop_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and take_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and "and" :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and or :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and xor :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and mask :: "nat \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a"   and n :: "nat" assumes "class.semiring_bit_operations plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask" shows "bit (and a b) n = (bit a n \<and> bit b n)"
in_set_wall_crossings_decomp: fixes H :: "'a set set set"   and Cs :: "'a set list" assumes "H \<in> set (wall_crossings Cs)" shows "\<exists>As A B Bs. Cs = As @ [A, B] @ Bs \<and> H = THE_default {} (\<lambda>H. H \<in> (\<Union>(f, g)\<in>foldpairs. {{f \<turnstile> \<C>, g \<turnstile> \<C>}}) \<and> separated_by H A B)"
val_nonzero_z: fixes v :: "'a \<Rightarrow> ant"   and x :: "'a" assumes "valuation K v"   and "x \<in> carrier K"   and "x \<noteq> \<zero>" shows "\<exists>z. v x = ant z"
Rats_eq_int_div_nat: shows "\<rat> = {real_of_int i / real n |i n. n \<noteq> 0}"
lunit_def: fixes f :: "'a" shows "\<l>[f] \<equiv> THE \<mu>. \<guillemotleft>\<mu> : L f \<Rightarrow>\<^sub>S f\<guillemotright> \<and> L \<mu> = (\<iota> \<star> f) \<cdot>\<^sub>S local.inv \<a>[b, b, f]"
gen_lucas_lehmer_sequence'_Suc': fixes m :: "int"   and a :: "int"   and n :: "nat" shows "gen_lucas_lehmer_sequence' m a (Suc n) = ((gen_lucas_lehmer_sequence' m a n)\<^sup>2 - 2) mod m"
integrate_insert_commute_insert_def: fixes M :: "('a, 'b) message set"   and a :: "'a extended \<Rightarrow> 'a sort_key extended"   and s :: "('a, 'b option) insert_message list"   and m :: "('a, 'b) insert_message" shows "integrate_insert_commute_insert M a s m \<equiv> integrate_insert_commute M a s \<and> integrate_insert_commute_insert_axioms M a m"
map\<^sub>0_simps_1: fixes a :: "'a" assumes "obj a" shows "Span.obj (SPN.map\<^sub>0 a)"
mat0_mult_right: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and nr :: "nat"   and nc :: "nat"   and m :: "'c list list"   and ncc :: "nat" assumes "class.semiring_0 plus zero times"   and "mat nr nc m" shows "mat_multI zero plus times nr m (mat0I zero nc ncc) = mat0I zero nr ncc"
InitPhi_def: shows "InitPhi \<equiv> x = #0 \<and> y = #0"
simps_16: fixes V :: "'a"   and T :: "ty"   and vo :: "'c val option"   and e :: "('a, 'b, 'c) exp" shows "fv {V:T=vo; e} = fv e - {V}"
mmerge_def: fixes return :: "'b multiset \<Rightarrow> 'a"   and bind :: "'a \<Rightarrow> ('b multiset \<Rightarrow> 'a) \<Rightarrow> 'a"   and A :: "'b multiset"   and f :: "'b \<Rightarrow> 'a" shows "mmerge return bind A f = fold_mset (\<lambda>m1 m2. bind m1 (\<lambda>A. bind m2 (\<lambda>B. return (A + B)))) (return {#}) (image_mset f A)"
prems_3: fixes i :: "nat" shows "i < Suc k \<longrightarrow> xs ! i = ys ! i"
c1: assumes "q = a" shows "q dvd prod_list_var (a # qs)"
weights_nat_def: shows "weights = \<lparr>w = Suc \<circ> prod_encode, w0 = 1, pr_strict = \<lambda>(f, n) (g, m). g < f \<or> f = g \<and> m < n, least = \<lambda>n. n = 0, scf = \<lambda>_ _. 1\<rparr>"
Bij_imp_extensional: fixes f :: "'c \<Rightarrow> 'c"   and S :: "'c set" assumes "f \<in> Bij S" shows "f \<in> extensional S"
prv_dsjR_fls: fixes \<phi> :: "'fmla" assumes "\<phi> \<in> fmla" shows "prv (eqv (dsj fls \<phi>) \<phi>)"
mem\<^sub>A_of_def: fixes mem\<^sub>C :: "'Var\<^sub>C \<Rightarrow> 'Val" shows "mem\<^sub>A_of mem\<^sub>C \<equiv> \<lambda>x\<^sub>A. mem\<^sub>C (var\<^sub>C_of x\<^sub>A)"
domain_def: fixes R :: "('e, 'f) ring_scheme" shows "domain R \<equiv> cring R \<and> domain_axioms R"
mpt_axioms: shows "mpt natural_extension_measure natural_extension_map"
dg_prod_2_ArrI: fixes \<alpha> :: "V"   and \<AA> :: "V"   and \<BB> :: "V"   and g :: "V"   and f :: "V" assumes "digraph \<alpha> \<AA>"   and "digraph \<alpha> \<BB>"   and "g \<in>\<^sub>\<circ> \<AA>\<lparr>Arr\<rparr>"   and "f \<in>\<^sub>\<circ> \<BB>\<lparr>Arr\<rparr>" shows "[g, f]\<^sub>\<circ> \<in>\<^sub>\<circ> (\<AA> \<times>\<^sub>D\<^sub>G \<BB>)\<lparr>Arr\<rparr>"
meet_le: fixes z :: "'a"   and x :: "'a"   and y :: "'a" assumes "z \<sqsubseteq> x"   and "z \<sqsubseteq> y"   and "x \<in> carrier L"   and "y \<in> carrier L"   and "z \<in> carrier L" shows "z \<sqsubseteq> x \<sqinter> y"
simps_3: fixes \<sigma> :: "'tp"   and al :: "univ list" shows "TE_intF (Wit \<sigma>) al = pickT \<sigma>"
one_less_exp_iff: fixes x :: "real" shows "(1 < exp x) = (0 < x)"
simps: fixes a :: "'a tree" shows "tfinite a = (\<exists>t. a = t \<and> (\<forall>x. x |\<in>| cont t \<longrightarrow> tfinite x))"
dy_fake_chan_def: fixes b :: "agent set"   and i :: "msg set"   and c :: "chan set" shows "dy_fake_chan b i c = fake b (dy_fake_msg b i c) c"
In1_def: fixes M :: "('b, 'c) Old_Datatype.node set" shows "In1 M \<equiv> Scons (Numb 1) M"
rbt_lookup: shows "rbt_lookup t1 = rbt_lookup t2"
pochhammer_rec': fixes z :: "'a"   and n :: "nat" shows "pochhammer z (Suc n) = (z + of_nat n) * pochhammer z n"
set_integrable_norm: fixes M :: "'c measure"   and k :: "'c set"   and f :: "'c \<Rightarrow> 'd" assumes "set_integrable M k f" shows "set_integrable M k (\<lambda>x. norm (f x))"
usubstappt_func2: fixes \<sigma> :: "(char \<Rightarrow> trm option) \<times> (char \<Rightarrow> trm option) \<times> (char \<Rightarrow> fml option) \<times> (char \<Rightarrow> game option)"   and f :: "char"   and r :: "trm"   and U :: "variable set"   and \<theta> :: "trm" assumes "SFuncs \<sigma> f = Aterm r"   and "FVT r \<inter> U \<noteq> {}" shows "usubstappt \<sigma> U (trm.Func f \<theta>) = undeft"
bind_put: fixes s :: "'s"   and m :: "'m"   and f :: "'a option \<Rightarrow> 'm" shows "bind (put s m) f = put s (bind m f)"
abs_eq: fixes x :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and xb :: "'a"   and xa :: "'a" assumes "eq_onp overrider x x" shows "xb \<oplus>\<^sub>S xa on Abs_scene x = x xb xa"
g: fixes x :: "'a" shows "g x = (if leq x bound then bot else f x)"
T_opt_def: fixes step :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'a"   and t :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> nat"   and s :: "'a"   and rs :: "'b list" shows "On_Off.T_opt step t s rs = \<Sqinter> {On_Off.T step t s rs as |as. length as = length rs}"
iffs: fixes carrier :: "'c set"   and more :: "'d"   and carrier' :: "'c set"   and more' :: "'d" shows "(\<lparr>carrier = carrier, \<dots> = more\<rparr> = \<lparr>carrier = carrier', \<dots> = more'\<rparr>) = (carrier = carrier' \<and> more = more')"
le_list_filter: fixes xs :: "'a list"   and ys :: "'a list"   and P :: "'a \<Rightarrow> bool" assumes "xs \<preceq> ys" shows "filter P xs \<preceq> filter P ys"
S2: assumes "\<And>n' x'. ((n', x'), fst p1, snd p1) \<in> lex_p \<Longrightarrow> c_assoc_have_key (pr_gr (loc_upb n' x')) (c_pair n' x') = 0" shows "c_assoc_have_key (pr_gr (loc_upb (fst p1) (snd p1))) (c_pair (fst p1) (snd p1)) = 0"
comp_null_2: fixes f :: "'c \<times> 'c" shows "C.VV.comp f C.VV.null = C.VV.null"
sees_field_def: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list"   and C :: "char list"   and F :: "char list"   and T :: "ty"   and D :: "char list" shows "P \<turnstile> C sees F:T in D \<equiv> \<exists>FDTs. P \<turnstile> C has_fields FDTs \<and> map_of (map (\<lambda>((F, D), T). (F, D, T)) FDTs) F = \<lfloor>(D, T)\<rfloor>"
mask_numeral: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bit :: "'b \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and drop_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and take_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and "and" :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and or :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and xor :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and mask :: "nat \<Rightarrow> 'b"   and n :: "num" assumes "class.semiring_bit_operations plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask" shows "mask (numeral n) = plus one (times (numeral.numeral one plus (num.Bit0 num.One)) (mask (pred_numeral n)))"