hcju/isabelleps huggingface dataset:
assms_3: fixes \<xi> :: "'a" shows "U \<xi> \<xi>"
dom_eval: fixes f :: "'c local.term" assumes "Arr f" shows "D.dom \<lbrace>f\<rbrace> = \<lbrace>Dom f\<rbrace>"
inducts: fixes P :: "'a list \<Rightarrow> bool"   and list :: "'a list" assumes "P []"   and "\<And>x1 x2. P x2 \<Longrightarrow> P (x1 # x2)" shows "P list"
majs_def: shows "majs \<equiv> {S. N div 2 < card S}"
valid_app_fresh: fixes \<Delta> :: "(name \<times> ty) list"   and \<Gamma> :: "(name \<times> ty) list"   and x :: "name" assumes "valid (\<Delta> @ \<Gamma>)"   and "x \<in> supp \<Delta>" shows "x \<sharp> \<Gamma>"
exhaust: fixes y :: "atom" obtains x1 :: "atom_sort" and x2 :: "nat" where "y = Atom x1 x2"
update_triv: fixes al :: "('d \<times> 'c) list"   and k :: "'d"   and v :: "'c" assumes "map_of al k = Some v" shows "update k v al = al"
type_less_eq_x_Sequence: fixes \<tau> :: "'a OCL_Types.type"   and \<sigma> :: "'a OCL_Types.type" assumes "\<tau> \<le> Sequence \<sigma>" obtains \<rho> :: "'a OCL_Types.type" where "\<tau> = Sequence \<rho>" and "\<rho> \<le> \<sigma>"
ff: shows "set ((swap 0 ^^ ALG_P ss x y s1) s2) = {x, y}"
find_index_conv_takeWhile: fixes P :: "'b \<Rightarrow> bool"   and xs :: "'b list" shows "find_index P xs = length (takeWhile (Not \<circ> P) xs)"
bop: shows "binop bop v1 v2 = \<lfloor>Inr a\<rfloor>"
plain_floatarith_approx_not_None: fixes N :: "nat"   and fa :: "floatarith"   and XS :: "float interval option list"   and p :: "nat" assumes "plain_floatarith N fa"   and "N \<le> length XS"   and "\<And>i. i < N \<Longrightarrow> XS ! i \<noteq> None" shows "approx p fa XS \<noteq> None"
fls_nonzero_nth: fixes f :: "'a fls" shows "(f \<noteq> 0) = (\<exists>n. fls_nth f n \<noteq> (0::'a))"
std_eff_enabled: fixes name :: "char list"   and pres :: "(nat \<times> nat) list"   and effs :: "((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list"   and layer :: "nat"   and s :: "nat \<Rightarrow> nat option"   and problem :: "(char list \<times> nat option \<times> char list list) list \<times> nat list \<times> (nat \<times> nat) list \<times> (char list \<times> (nat \<times> nat) list \<times> ((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list \<times> nat) list" assumes "is_standard_operator (name, pres, effs, layer)"   and "s \<in> ast_problem.valid_states problem" shows "filter (ast_problem.eff_enabled problem s) effs = effs"
DF\<^sub>S\<^sub>K\<^sub>I\<^sub>P_unfold: fixes A :: "'b set" shows "DF\<^sub>S\<^sub>K\<^sub>I\<^sub>P A = ((\<sqinter>z\<in>A \<rightarrow>  DF\<^sub>S\<^sub>K\<^sub>I\<^sub>P A) |-| SKIP)"
divmod_numeral_simps_23: fixes n :: "num" shows "(num.One < num.Bit1 n) = True"
path_uncons: fixes E :: "('b \<times> 'b) set"   and u :: "'b"   and u' :: "'b"   and l :: "'b list"   and w :: "'b" assumes "path E u (u' # l) w" obtains v :: "'b" where "u' = u" and "(u, v) \<in> E" and "path E v l w"
Cons_nth_drop_Suc: fixes i :: "nat"   and xs :: "'c list" assumes "i < length xs" shows "xs ! i # drop (Suc i) xs = drop i xs"
rComm1: fixes \<Psi> :: "'b"   and \<Psi>\<^sub>Q :: "'b"   and P :: "('a, 'b, 'c) psi"   and M :: "'a"   and N :: "'a"   and P' :: "('a, 'b, 'c) psi"   and A\<^sub>P :: "name list"   and \<Psi>\<^sub>P :: "'b"   and Q :: "('a, 'b, 'c) psi"   and K :: "'a"   and xvec :: "name list"   and Q' :: "('a, 'b, 'c) psi"   and A\<^sub>Q :: "name list"   and C :: "'d" assumes "\<Psi> \<otimes> \<Psi>\<^sub>Q \<rhd> P \<longmapsto> M\<lparr>N\<rparr> \<prec> P'"   and "extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>"   and "distinct A\<^sub>P"   and "\<And>C. Prop C (\<Psi> \<otimes> \<Psi>\<^sub>Q) P (M\<lparr>N\<rparr> \<prec> P') A\<^sub>P \<Psi>\<^sub>P"   and "\<Psi> \<otimes> \<Psi>\<^sub>P \<rhd> Q \<longmapsto> K\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> Q'"   and "extractFrame Q = \<langle>A\<^sub>Q, \<Psi>\<^sub>Q\<rangle>"   and "distinct A\<^sub>Q"   and "\<Psi> \<otimes> \<Psi>\<^sub>P \<otimes> \<Psi>\<^sub>Q \<turnstile> M \<leftrightarrow> K"   and "\<And>C. Prop C (\<Psi> \<otimes> \<Psi>\<^sub>P) Q (K\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> Q') A\<^sub>Q \<Psi>\<^sub>Q"   and "distinct xvec"   and "A\<^sub>P \<sharp>* \<Psi>"   and "A\<^sub>P \<sharp>* \<Psi>\<^sub>Q"   and "A\<^sub>P \<sharp>* P"   and "A\<^sub>P \<sharp>* M"   and "A\<^sub>P \<sharp>* N"   and "A\<^sub>P \<sharp>* P'"   and "A\<^sub>P \<sharp>* Q"   and "A\<^sub>P \<sharp>* Q'"   and "A\<^sub>P \<sharp>* A\<^sub>Q"   and "A\<^sub>P \<sharp>* xvec"   and "A\<^sub>Q \<sharp>* \<Psi>"   and "A\<^sub>Q \<sharp>* \<Psi>\<^sub>P"   and "A\<^sub>Q \<sharp>* P"   and "A\<^sub>Q \<sharp>* N"   and "A\<^sub>Q \<sharp>* P'"   and "A\<^sub>Q \<sharp>* Q"   and "A\<^sub>Q \<sharp>* K"   and "A\<^sub>Q \<sharp>* Q'"   and "A\<^sub>Q \<sharp>* xvec"   and "xvec \<sharp>* \<Psi>"   and "xvec \<sharp>* \<Psi>\<^sub>P"   and "xvec \<sharp>* \<Psi>\<^sub>Q"   and "xvec \<sharp>* P"   and "xvec \<sharp>* M"   and "xvec \<sharp>* Q"   and "xvec \<sharp>* K"   and "A\<^sub>P \<sharp>* C"   and "A\<^sub>Q \<sharp>* C"   and "xvec \<sharp>* C" shows "Prop C \<Psi> (P \<parallel> Q) (\<tau> \<prec> \<lparr>\<nu>*xvec\<rparr>P' \<parallel> Q') (A\<^sub>P @ A\<^sub>Q) (\<Psi>\<^sub>P \<otimes> \<Psi>\<^sub>Q)"
cod_pr0: fixes a :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and b :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "ide a"   and "ide b" shows "cod (pr0 a b) = b"
retraction_subset: fixes S :: "'b set"   and T :: "'b set"   and r :: "'b \<Rightarrow> 'b"   and s' :: "'b set" assumes "retraction S T r"   and "T \<subseteq> s'"   and "s' \<subseteq> S" shows "retraction s' T r"
N: shows "\<langle>l, x, r\<rangle> \<in> subtrees t"
T_Nil: fixes \<Gamma> :: "binding list" assumes "\<Gamma> \<turnstile>\<^sub>w\<^sub>f" shows "\<Gamma> \<turnstile> [] [:] []"
insert_fset_case: fixes S :: "'a fset"   and x :: "'a" assumes "x |\<notin>| S"   and "P S" shows "P (insert_fset x S)"
inner_ge_zero: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and norm :: "'b \<Rightarrow> real"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and sgn :: "'b \<Rightarrow> 'b"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and inner :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and x :: "'b" assumes "class.real_inner minus dist norm plus zero uminus scaleR sgn uniformity open inner" shows "0 \<le> inner x x"
finite_code_2: fixes rbt :: "'b set_rbt" shows "finite (RBT_set rbt) = (case ID ccompare of None \<Rightarrow> Code.abort STR ''finite RBT_set: ccompare = None'' (\<lambda>_. finite (RBT_set rbt)) | Some x \<Rightarrow> True)"
adjoint_pair_antipar_3: fixes f :: "'a"   and g :: "'a" assumes "B.adjoint_pair f g" shows "src f = trg g"
simps: fixes a1 :: "'node SDG_node"   and a2 :: "'node SDG_node list"   and a3 :: "'node SDG_node" shows "a1 is-a2\<rightarrow>\<^sub>d* a3 = ((\<exists>n. a1 = n \<and> a2 = [] \<and> a3 = n \<and> valid_SDG_node n) \<or> (\<exists>n ns n'' n'. a1 = n \<and> a2 = ns @ [n''] \<and> a3 = n' \<and> n is-ns\<rightarrow>\<^sub>d* n'' \<and> n'' s\<longrightarrow>\<^bsub>cd\<^esub> n') \<or> (\<exists>n ns n'' V n'. a1 = n \<and> a2 = ns @ [n''] \<and> a3 = n' \<and> n is-ns\<rightarrow>\<^sub>d* n'' \<and> n'' s-V\<rightarrow>\<^sub>d\<^sub>d n' \<and> n'' \<noteq> n') \<or> (\<exists>n ns n'' p n'. a1 = n \<and> a2 = ns @ [n''] \<and> a3 = n' \<and> n is-ns\<rightarrow>\<^sub>d* n'' \<and> n'' s-p\<rightarrow>\<^bsub>sum\<^esub> n'))"
ncloses_id: fixes \<Gamma> :: "(name \<times> ty) list"   and a :: "coname" shows "idn \<Gamma> a ncloses \<Gamma>"
is_num_normalize_8: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.neg_numeral minus plus zero uminus" shows "uminus (plus a b) = plus (uminus b) (uminus a)"
eq_fin_le_inf_concat_end: fixes w\<^sub>1 :: "'action list"   and w\<^sub>2 :: "'action list"   and w :: "'action stream" assumes "w\<^sub>1 =\<^sub>F w\<^sub>2" shows "w\<^sub>1 @- w \<preceq>\<^sub>I w\<^sub>2 @- w"
take_bit_of_0: fixes plus :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and minus :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and zero :: "'k"   and times :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and one :: "'k"   and divide :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and modulo :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and bit :: "'k \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'k \<Rightarrow> 'k"   and drop_bit :: "nat \<Rightarrow> 'k \<Rightarrow> 'k"   and take_bit :: "nat \<Rightarrow> 'k \<Rightarrow> 'k"   and n :: "nat" assumes "class.semiring_bit_shifts plus minus zero times one divide modulo bit push_bit drop_bit take_bit" shows "take_bit n zero = zero"
algebraic_nth_root: fixes n :: "nat"   and x :: "'b"   and y :: "'b" assumes "n \<noteq> 0"   and "algebraic x"   and "y ^ n = x" shows "algebraic y"
max_node_def: fixes l :: "('a \<times> 'b \<times> 'a) list" shows "max_node l \<equiv> Max (insert (0::'a) (fst ` set l \<union> (snd \<circ> snd) ` set l))"
simps_1: fixes uu :: "int" shows "is_Num (Num uu) = True"
add_mult_distrib_mat_vec: fixes A :: "'a mat"   and nr :: "nat"   and nc :: "nat"   and B :: "'a mat"   and v :: "'a vec" assumes "A \<in> carrier_mat nr nc"   and "B \<in> carrier_mat nr nc"   and "v \<in> carrier_vec nc" shows "(A + B) *\<^sub>v v = A *\<^sub>v v + B *\<^sub>v v"
WCR_defs: fixes r :: "('b \<times> 'b) set"   and A :: "'b set" shows "WCR_on r A = (\<forall>a\<in>A. \<forall>b c. (a, b) \<in> r \<and> (a, c) \<in> r \<longrightarrow> (b, c) \<in> r\<^sup>\<down>)"
real_root_gt_zero: fixes n :: "nat"   and x :: "real" assumes "0 < n"   and "0 < x" shows "0 < root n x"
a_in: shows "a \<in> read_only (\<S> \<oplus>\<^bsub>W\<^esub> R)"
Sup_inner_def: fixes X :: "'b set"   and y :: "'b" shows "Sup_inner X y = SPEC (\<lambda>r. \<forall>x\<in>X. x \<bullet> y \<le> r)"
simps: fixes a1 :: "'a list"   and a2 :: "'a list"   and a3 :: "'a list" shows "append2 a1 a2 a3 = ((\<exists>xs. a1 = [] \<and> a2 = xs \<and> a3 = xs) \<or> (\<exists>xs ys zs x. a1 = x # xs \<and> a2 = ys \<and> a3 = x # zs \<and> append2 xs ys zs))"
ide_in_hom: fixes a :: "'a1 \<times> 'a2" shows "ide a = \<guillemotleft>a : a \<rightarrow> a\<guillemotright>"
sum_upper: fixes I :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and P :: "'b \<Rightarrow> bool" assumes "finite I"   and "\<And>i. i \<in> I \<Longrightarrow> (0::'c) \<le> f i" shows "(\<Sum>i | i \<in> I \<and> P i. f i) \<le> sum f I"
retract_ofE: fixes T :: "'b set"   and S :: "'b set" assumes "T retract_of S" obtains r :: "'b \<Rightarrow> 'b" where "T = r ` S" and "r ` S \<subseteq> S" and "continuous_on S r" and "\<And>x. x \<in> S \<Longrightarrow> r (r x) = r x"
assms: shows "is_trail v ((v1, w, v2) # ps) v'"
While: fixes P :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and x :: "char list"   and c :: "com"   and b :: "bexp" assumes "\<turnstile>\<^sub>t {\<lambda>l. P (l(x := Suc (l x)))} c {P}"   and "\<forall>l s. 0 < l x \<and> P l s \<longrightarrow> bval b s"   and "\<forall>l s. l x = 0 \<and> P l s \<longrightarrow> \<not> bval b s" shows "\<turnstile>\<^sub>t {\<lambda>l s. \<exists>n. P (l(x := n)) s} WHILE b DO c {\<lambda>l. P (l(x := 0))}"
f: shows "f = (\<lambda>stp. (steps0 (Suc 0, l, r) (findnth n) stp, n))"
divides_mult_zero: fixes a :: "'a" assumes "a \<in> carrier R"   and "a divides\<^bsub>Ring_Divisibility.mult_of R\<^esub> \<zero>" shows "a = \<zero>"
F_one: fixes g :: "'c \<Rightarrow> 'a"   and A :: "'c set" shows "F g A \<^bold>* \<^bold>1 = F g A"
brw_cond_def: fixes Qi :: "'a set" shows "brw_cond Qi \<equiv> {(Q, W, rcm). W \<noteq> {} \<and> Qi \<inter> dom Q = {}}"
zero_closed: fixes H :: "'a set"   and G :: "('a, 'b) ring_scheme" assumes "additive_subgroup H G" shows "\<zero>\<^bsub>G\<^esub> \<in> H"
ib_butlast: fixes w :: "'a list" assumes "2 \<le> length w" shows "length (intrinsic_border w) \<le> length (intrinsic_border (butlast w)) + 1"
add_update: fixes i :: "nat"   and ns :: "nat list"   and m :: "nat" assumes "i < length ns" shows "foldl (+) m (ns[i := Suc (ns ! i)]) = Suc (foldl (+) m ns)"
coboundedI2: fixes c :: "'a Word.word"   and b :: "'a Word.word"   and a :: "'a Word.word" assumes "c \<le>s b" shows "c \<le>s signed.max a b"
LIMSEQ_minus_fract_mult: fixes r :: "real" shows "(\<lambda>n. r * (1 - 1 / real (Suc (Suc n)))) \<longlonglongrightarrow> r"
a: shows "graph G \<and> vertices G \<noteq> {} \<and> ident_rel S_Idt G \<and> getRel S_Bot G = {} \<and> getRel S_Top G = {(x, y). x \<in> vertices G \<and> y \<in> vertices G} \<and> (\<forall>(l, v)\<in>(\<lambda>c. (S_Const c, Inl c)) ` C. getRel l G = {(v, v)})"
get_ancestors_never_empty: fixes h :: "(_) heap"   and child :: "(_) object_ptr"   and ancestors :: "(_) object_ptr list" assumes "heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M h"   and "h \<turnstile> Shadow_DOM.get_ancestors child \<rightarrow>\<^sub>r ancestors" shows "ancestors \<noteq> []"
strong_induct: fixes x1 :: "lam"   and x2 :: "lam"   and P :: "'a \<Rightarrow> lam \<Rightarrow> lam \<Rightarrow> bool"   and z :: "'a" assumes " x1 \<longrightarrow>\<^sub>d x2"   and "\<And>x y. P y (Var x) (Var x)"   and "\<And>t s x y. \<lbrakk>x \<sharp> y;  t \<longrightarrow>\<^sub>d s; \<And>z. P z t s\<rbrakk> \<Longrightarrow> P y (Lam [x].t) (Lam [x].s)"   and "\<And>t1 t2 s1 s2 y. \<lbrakk>\<nexists>y t'. t1 = Lam [y].t';  t1 \<longrightarrow>\<^sub>d t2; \<And>z. P z t1 t2;  s1 \<longrightarrow>\<^sub>d s2; \<And>z. P z s1 s2\<rbrakk> \<Longrightarrow> P y (App t1 s1) (App t2 s2)"   and "\<And>x s1 s2 t1 t2 y. \<lbrakk>x \<sharp> y; x \<sharp> (s1, s2);  t1 \<longrightarrow>\<^sub>d t2; \<And>z. P z t1 t2;  s1 \<longrightarrow>\<^sub>d s2; \<And>z. P z s1 s2\<rbrakk> \<Longrightarrow> P y (App (Lam [x].t1) s1) (t2[x::=s2])" shows "P z x1 x2"
Var_apps_neq_Abs_apps: fixes n :: "nat"   and ts :: "dB list"   and r :: "dB"   and ss :: "dB list" shows "Var n \<degree>\<degree> ts \<noteq> Abs r \<degree>\<degree> ss"
oreachable_other: fixes s :: "'d \<times> 'e"   and A :: "('d \<times> 'e, 'f) automaton"   and S :: "'d \<Rightarrow> 'd \<Rightarrow> 'f \<Rightarrow> bool"   and U :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and \<sigma>' :: "'d" assumes "s \<in> oreachable A S U"   and "U (fst s) \<sigma>'" shows "(\<sigma>', snd s) \<in> oreachable A S U"
prec_F_grounding: fixes q :: "'a literal multiset set"   and C2 :: "'a literal multiset"   and C1 :: "'a literal multiset" assumes "q \<in> UNIV"   and "C2 \<prec>\<cdot> C1" shows "\<G>_F C1 \<subseteq> \<G>_F C2"
diff_fun_space_extensionalD: fixes g :: "'a \<Rightarrow> real" assumes "g \<in> sub.diff_fun_space" shows "extensional0 sub.carrier g"
merge_pres: fixes ss :: "(nat \<times> 'a) list"   and x :: "'a"   and c :: "'a list"   and pc :: "nat" assumes "snd ` set ss \<subseteq> A"   and "x \<in> A" shows "merge c pc ss x \<in> A"
state_ext: fixes m :: "('a, 'b) state"   and n :: "('a, 'b) state" assumes "\<And>s. run_state m s = run_state n s" shows "m = n"
assms_1: fixes i :: "'a" assumes "i \<in> I" shows "X i \<in> sets M"
simps_1: fixes u :: "'v"   and v :: "'v" shows "path u [] v = (u = v)"
p_and_not_p: fixes P :: "'a upred" shows "(P \<and> \<not> P) = false"
aHom_add: fixes F :: "('c, 'd) aGroup_scheme"   and G :: "('e, 'f) aGroup_scheme"   and f :: "'c \<Rightarrow> 'e"   and a :: "'c"   and b :: "'c" assumes "aGroup F"   and "aGroup G"   and "f \<in> aHom F G"   and "a \<in> carrier F"   and "b \<in> carrier F" shows "f (a \<plusminus>\<^bsub>F\<^esub> b) = f a \<plusminus>\<^bsub>G\<^esub> f b"
inf_import_p: fixes x :: "'a"   and y :: "'a" shows "x \<sqinter> - (x \<sqinter> y) = x \<sqinter> - y"
C1_differentiable_on_mult: fixes f :: "real \<Rightarrow> 'a"   and S :: "real set"   and g :: "real \<Rightarrow> 'a" assumes "f C1_differentiable_on S"   and "g C1_differentiable_on S" shows "(\<lambda>x. f x * g x) C1_differentiable_on S"
simps_9: fixes \<sigma> :: "(char list \<times> 'c list) trace"   and v :: "'c list"   and i :: "nat"   and \<phi> :: "'c MFOTL.formula"   and I :: "\<I>"   and \<psi> :: "'c MFOTL.formula" shows "MFOTL.sat \<sigma> v i (formula.Until \<phi> I \<psi>) = (\<exists>j\<ge>i. mem (\<tau> \<sigma> j - \<tau> \<sigma> i) I \<and> MFOTL.sat \<sigma> v j \<psi> \<and> (\<forall>k\<in>{i..<j}. MFOTL.sat \<sigma> v k \<phi>))"
Frec: fixes n :: "nat"   and x :: "'a" shows "F (n + 1) x - F n (T x) = max (- F n (T x)) (f x)"
sel_1: fixes x1 :: "nat"   and x2 :: "'b"   and x3 :: "'b tree list" shows "rank (Node x1 x2 x3) = x1"
not_None_eq: fixes x :: "'k option" shows "(x \<noteq> None) = (\<exists>y. x = Some y)"
lm072: fixes f :: "('c \<times> 'd) set"   and x :: "'c"   and A :: "'d set" assumes "f \<in> injectionsUniverse"   and "x \<notin> Domain f"   and "A \<inter> Range f = {}" shows "{f \<union> {(x, y)} |y. y \<in> A} \<subseteq> injectionsUniverse"
add_rel: fixes a :: "('a set \<times> 'b) set"   and b :: "('a set \<times> 'b) set" assumes "a \<in> carrier_direct_lim"   and "b \<in> carrier_direct_lim" shows "add_rel a b \<in> carrier_direct_lim"
cases: fixes x :: "fterm" obtains p :: "char list" and ts :: "fterm list" where "x = Fun p ts"   | v :: "char list" where "x = \<epsilon> v"
integrable_sin_cx: fixes c :: "real" shows "integrable (lebesgue_on {- pi..pi}) (\<lambda>x. sin (x * c))"
agree_on_empty: fixes f :: "('c \<times> 'd) set"   and g :: "('c \<times> 'd) set" shows "agree_on (LG {} {}) f g"
msgs_unchanged_for_other_is: fixes c :: "('p, 's, 'm) configuration"   and ev :: "('p, 's, 'm) event"   and c' :: "('p, 's, 'm) configuration"   and i :: "nat"   and i' :: "nat" assumes "c \<turnstile> ev \<mapsto> c'"   and "regular_event ev"   and "getId ev = i"   and "i' \<noteq> i" shows "msgs c i' = msgs c' i'"
all_info1: shows "set (characterize_consistent_signs_at_roots_copr p qs1) \<subseteq> set signs1"
zero_hom_axioms: shows "zero_hom (map_poly hom)"
p_def: shows "p \<equiv> poly_real_alg_1 rai"
hom_0_iff: fixes x :: "int" shows "(of_int x = (0::'b)) = (x = 0)"
insertE: fixes a :: "'b"   and b :: "'b"   and A :: "'b set" assumes "a \<in> insert b A" obtains "a = b"   | "a \<in> A"
reachable_def: fixes y :: "'f" shows "reachable y = (\<lambda>s. \<exists>x. x \<in> mut_roots s \<union> mut_ghost_honorary_root s \<union> tso_store_refs s \<and> (x reaches y) s)"
env_restr_def: fixes S :: "'d set"   and m :: "'d \<Rightarrow> 'e" shows "m f|` S = (\<lambda>x. if x \<in> S then m x else \<bottom>)"
rep_eq: fixes x :: "integer"   and xa :: "integer" shows "int_of_integer (x * xa) = int_of_integer x * int_of_integer xa"
ur: shows "\<exists>!x. root_cond (cf_pos_poly (poly_mult_rat r1 p2), l, r) x"
cases: fixes a1 :: "lam"   and a2 :: "lam" assumes " a1 \<longrightarrow>\<^sub>1 a2" obtains x :: "name" where "a1 = Var x__" and "a2 = Var x__"   | t1 :: "lam" and t2 :: "lam" and s1 :: "lam" and s2 :: "lam" where "a1 = App t1a__ s1__" and "a2 = App t2a__ s2__" and " t1a__ \<longrightarrow>\<^sub>1 t2a__" and " s1__ \<longrightarrow>\<^sub>1 s2__"   | t1 :: "lam" and t2 :: "lam" and x :: "name" where "a1 = Lam [xa__].t1b__" and "a2 = Lam [xa__].t2b__" and " t1b__ \<longrightarrow>\<^sub>1 t2b__"   | x :: "name" and s1 :: "lam" and s2 :: "lam" and t1 :: "lam" and t2 :: "lam" where "a1 = App (Lam [x].t1) s1" and "a2 = t2[x::=s2]" and "x \<sharp> (s1, s2)" and " t1 \<longrightarrow>\<^sub>1 t2" and " s1 \<longrightarrow>\<^sub>1 s2"
pos: shows "list_all3 (\<lambda>A Qi X. qtable n A P Qi X \<and> wf_set n A) A_pos Q_pos L_pos"
is_rewritableD_is_canon_rewriter: fixes bs :: "('t \<Rightarrow>\<^sub>0 'b) list"   and b :: "'t \<Rightarrow>\<^sub>0 'b"   and u :: "'t" assumes "rb_aux_inv1 bs"   and "is_rewritable bs b u" shows "\<not> is_canon_rewriter rword (set bs) u b"
assms_1: shows "Uvariation (F 0) \<omega> (U \<union> {RVar x, DVar x})"
Ide_in_Hom: fixes t :: "'a local.term" assumes "Ide t" shows "t \<in> Hom t t"
cis_2pi_int: fixes x :: "int" shows "cis (2 * pi * real_of_int x) = 1"
emp_UNlift01_UNpart1: fixes cl :: "('test, 'atom, 'choice) cmd list"   and dl :: "('test, 'atom, 'choice) cmd list"   and P :: "nat \<Rightarrow> nat set set"   and F :: "nat \<Rightarrow> nat set \<Rightarrow> nat set" assumes "length cl = length dl"   and "\<And>n. n < length cl \<Longrightarrow> part {..<brn (cl ! n)} (P n) \<and> {} \<notin> P n"   and "\<And>n. n < length dl \<Longrightarrow> {} \<notin> F n ` P n" shows "{} \<notin> UNlift01 cl dl P F ` UNpart1 cl dl P"
quad_part_1_leq: fixes var :: "nat"   and L :: "real list"   and p :: "real mpoly"   and deg :: "nat"   and D :: "real"   and a :: "real mpoly"   and A :: "real"   and b :: "real mpoly"   and B :: "real"   and d :: "real mpoly"   and C :: "real" assumes "var < length L"   and "MPoly_Type.degree p var = deg"   and "D \<noteq> 0"   and "\<forall>x. insertion (nth_default 0 (L[var := x])) a = A"   and "\<forall>x. insertion (nth_default 0 (L[var := x])) b = B"   and "\<forall>x. insertion (nth_default 0 (L[var := x])) d = D" shows "aEval (Leq p) (L[var := (A + B * C) / D]) = aEval (Leq (quadratic_part_1 var a b d (Leq p))) (L[var := C])"
nsg_subset_elem: fixes H :: "'a set"   and h :: "'a" assumes "G \<triangleright> H"   and "h \<in> H" shows "h \<in> carrier G"
upper_triangularI: fixes A :: "'a mat" assumes "\<And>i j. \<lbrakk>j < i; i < dim_row A\<rbrakk> \<Longrightarrow> A $$ (i, j) = (0::'a)" shows "upper_triangular A"
swap_preserves_good: fixes X :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and xs :: "'varSort"   and x :: "'var"   and y :: "'var" assumes "good X" shows "good (X #[x \<and> y]_xs)"
simps_1: shows "ltree \<langle>\<rangle> = True"
order_lesseq_imp: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "class.order less_eq less" shows "(\<forall>z. less_eq x z \<longrightarrow> less_eq y z) = less_eq y x"
fst_apsnd: fixes f :: "'c \<Rightarrow> 'b"   and x :: "'a \<times> 'c" shows "fst (apsnd f x) = fst x"
language_state_inputs: fixes M :: "('d, 'e, 'f) FSM"   and io :: "('d \<times> 'e) list"   and q :: "'f" assumes "well_formed M"   and "io \<in> LS M q" shows "set (map fst io) \<subseteq> inputs M"
equal_div_mod: fixes j :: "nat"   and a :: "nat"   and i :: "nat" assumes "j div a = i div a"   and "j mod a = i mod a" shows "j = i"
Suc: shows "1 + int n gchoose n = 1 + int n"
is_auto_reduced_def: fixes B :: "('t \<Rightarrow>\<^sub>0 'c) set" shows "is_auto_reduced B \<equiv> \<forall>b\<in>B. \<not> is_red (B - {b}) b"
closed: shows "\<forall>s. set_pmf (\<K> s) \<subseteq> \<S>"
fac_body'_fac_body: shows "fac_body' = unwrapB oo fac_body oo wrapB"
diff_bound: assumes "4 \<le> 4" shows "sum_upto f 4 - \<MM> 4 \<in> {0..C}"
inducedHom: fixes F :: "('e, 'f) Group_scheme"   and G :: "('g, 'h) Group_scheme"   and f :: "'e \<Rightarrow> 'g" assumes "Group F"   and "Group G"   and "f \<in> gHom F G" shows "f\<dieresis>\<^bsub>F,G\<^esub> \<in> gHom (F / gker\<^bsub>F,G\<^esub> f) G"
fmap: fixes C_info :: "(name, dt_def) fmap"   and rs :: "(name \<times> pterm) fset" assumes "prules C_info rs" shows "is_fmap rs"
base: fixes P :: "('d, 'e, 'f) psi" shows "\<lparr>\<nu>*[]\<rparr>P = P"
cartesian_product: fixes A :: "'c set"   and B :: "'d set"   and g :: "'d \<Rightarrow> real"   and f :: "'c \<Rightarrow> real" assumes "finite A"   and "finite B" shows "sum g B * sum f A = (\<Sum>ab\<in>A \<times> B. f (fst ab) * g (snd ab))"
assms_6: fixes g1 :: "'t \<Rightarrow>\<^sub>0 'b"   and g2 :: "'t \<Rightarrow>\<^sub>0 'b" assumes "g1 \<in> G"   and "g2 \<in> G"   and "is_regular_spair g1 g2"   and "lt (spair g1 g2) \<prec>\<^sub>t u" shows "is_RB_in d rword G (lt (spair g1 g2))"
HPhase1or2Write_HInv4c_p: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and d :: "Disk" assumes "HPhase1or2Write s s' p d"   and "HInv4c s p"   and "Inv2c s" shows "HInv4c s' p"
rep_eq: shows "Rep_scene \<top>\<^sub>S = (\<lambda>x y. y)"