hcju/isabelleps huggingface dataset:
abs_eq: fixes x :: "'a rexp" shows "r_lang x = Abs_reg_lan (lang x)"
Units_l_inv: fixes x :: "nat" assumes "x \<in> Units R" shows "inv\<^bsub>R\<^esub> x \<otimes>\<^bsub>R\<^esub> x = \<one>\<^bsub>R\<^esub>"
3: shows "\<G>\<F> \<phi> w = \<G>\<F> \<phi> (suffix i w)"
Cong3_def: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and A' :: "'p"   and B' :: "'p"   and C' :: "'p" shows "A B C Cong3 A' B' C' \<equiv> Cong A B A' B' \<and> Cong A C A' C' \<and> Cong B C B' C'"
divide_divide_eq_left': fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inverse :: "'c \<Rightarrow> 'c"   and one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c"   and c :: "'c" assumes "class.division_ring divide inverse one times plus zero minus uminus" shows "divide (divide a b) c = divide a (times c b)"
has_integral_nonneg: fixes f :: "'a \<Rightarrow> real"   and i :: "real"   and S :: "'a set" assumes "(f has_integral i) S"   and "\<And>x. x \<in> S \<Longrightarrow> 0 \<le> f x" shows "0 \<le> i"
alts_eq: fixes agents :: "'a set"   and alts :: "'b set"   and scf :: "('a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'b set"   and A1 :: "'a"   and A2 :: "'a"   and A3 :: "'a"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "fb_impossibility_3_3 agents alts scf A1 A2 A3 a b c" shows "alts = {a, b, c}"
no_pending_refine: fixes s :: "('a, 'e) simple_state_scheme"   and s' :: "('a, 'e) simple_state_scheme"   and u :: "'a" assumes "(s, s') \<in> rel"   and "\<not> is_empty_stack_impl s'"   and "hd (ss_stack s) = (u, {})" shows "hd (ss_stack s') = (u, {})"
n_def: shows "n = length (filter ((=) a) xs) - 1"
comp_null_1: fixes f :: "nat" shows "J.comp J.null f = J.null"
map_simp_ide: fixes a :: "'a" assumes "A.ide a" shows "\<sigma>\<sigma>'.map a = B (\<sigma>' a) (\<sigma> a)"
bij_betw_disjoint_Un: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and C :: "'b set"   and g :: "'a \<Rightarrow> 'b"   and B :: "'a set"   and D :: "'b set" assumes "bij_betw f A C"   and "bij_betw g B D"   and "A \<inter> B = {}"   and "C \<inter> D = {}" shows "bij_betw (\<lambda>x. if x \<in> A then f x else g x) (A \<union> B) (C \<union> D)"
cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r_inject: fixes x :: "(_) element_ptr"   and y :: "(_) element_ptr" shows "(cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r x = cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r y) = (x = y)"
t1_space_subtopology: fixes X :: "'c topology"   and U :: "'c set" assumes "t1_space X" shows "t1_space (subtopology X U)"
nLt_le_conv: fixes comp :: "'a \<Rightarrow> 'a \<Rightarrow> order"   and x :: "'a"   and y :: "'a" assumes "comparator comp" shows "(comp x y \<noteq> Lt) = le_of_comp comp y x"
assms_1: shows "f \<in> borel_measurable (nat_filtration n)"
Approximation1_5Tr7: fixes a :: "'a"   and vv :: "nat \<Rightarrow> 'a \<Rightarrow> ant"   and x :: "'a"   and n :: "nat" assumes "a \<in> carrier K"   and "vv 0 a = 1"   and "x \<in> carrier K" shows "vals_nonequiv K (Suc n) vv \<and> Ostrowski_elem K (Suc n) vv x \<longrightarrow> (\<exists>l. \<forall>m>l. \<forall>j\<in>nset (Suc 0) (Suc n). vv j ((1\<^sub>r \<plusminus> -\<^sub>a x)^\<^bsup>K m\<^esup> \<plusminus> a \<cdot>\<^sub>r x^\<^bsup>K m\<^esup>) = 0)"
VSigma_iff: fixes a :: "V"   and b :: "V"   and A :: "V"   and B :: "V \<Rightarrow> V" shows "(\<langle>a, b\<rangle> \<in>\<^sub>\<circ> VSigma A B) = (a \<in>\<^sub>\<circ> A \<and> b \<in>\<^sub>\<circ> B a)"
eSubstBinp_simp1: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) model"   and ys :: "'varSort"   and Y :: "'c"   and delta :: "'opSym"   and binp :: "'bindex \<Rightarrow> 'd option"   and y :: "'var" assumes "igWls MOD (asSort ys) Y"   and "igWlsBinp MOD delta binp" shows "eSubstBinp MOD ys (OK Y) y (OKI binp) = OKI (igSubstBinp MOD ys Y y binp)"
c6: fixes x :: "'a" shows "x \<cdot> 1\<^sub>\<pi> \<le> 1\<^sub>\<pi>"
le_add_same_cancel1: fixes a :: "'c multiset"   and b :: "'c multiset" shows "(a \<subseteq># a + b) = ({#} \<subseteq># b)"
IteratorTE: fixes \<Gamma> :: "String.literal \<rightharpoonup>\<^sub>f 'a OCL_Types.type"   and src :: "'a expr"   and its :: "String.literal list"   and body :: "'a OCL_Types.type option \<times> 'a expr"   and ys :: "'a OCL_Types.type \<times> 'a OCL_Types.type \<times> 'a OCL_Types.type" assumes "\<Gamma> \<turnstile>\<^sub>I (src, its, body) : ys" obtains \<tau> :: "'a OCL_Types.type" and \<sigma> :: "'a OCL_Types.type" and its_ty :: "'a OCL_Types.type" and bodya :: "'a expr" and \<rho> :: "'a OCL_Types.type" where "body = (Some its_ty, bodya)" and "ys = (\<tau>, \<sigma>, \<rho>)" and "\<Gamma> \<turnstile>\<^sub>E src : \<tau>" and "element_type \<tau> \<sigma>" and "\<sigma> \<le> its_ty" and "\<Gamma> ++\<^sub>f fmap_of_list (map (\<lambda>it. (it, its_ty)) its) \<turnstile>\<^sub>E bodya : \<rho>"
concat_eq_append_conv: fixes xss :: "'e list list"   and ys :: "'e list"   and zs :: "'e 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)"
absorb_iff2: fixes a :: "'a"   and b :: "'a" shows "order.greater_eq b a = (sup a b = b)"
ord_eq_1: fixes G :: "('c, 'd) monoid_scheme"   and x :: "'c" assumes "Group.group G"   and "x \<in> carrier G" shows "(group.ord G x = 1) = (x = \<one>\<^bsub>G\<^esub>)"
Borsukian_eq_simply_connected: fixes S :: "complex set" assumes "open S" shows "Borsukian S = (\<forall>C\<in>components S. simply_connected C)"
simps: fixes s :: "('a, 'b) Term.term"   and t :: "('a, 'b) Term.term" shows "kbo s t = (if vars_term_ms (KBO.SCF t) \<subseteq># vars_term_ms (KBO.SCF s) \<and> weight t \<le> weight s then if weight t < weight s then (True, True) else case s of Var y \<Rightarrow> (False, case t of Var x \<Rightarrow> x = y | Fun g ts \<Rightarrow> ts = [] \<and> least weights g) | Fun f ss \<Rightarrow> case t of Var x \<Rightarrow> (True, True) | Fun g ts \<Rightarrow> if pr_strict weights (f, length ss) (g, length ts) then (True, True) else if (pr_strict weights)\<^sup>=\<^sup>= (f, length ss) (g, length ts) then lex_ext_unbounded kbo ss ts else (False, False) else (False, False))"
degree_X: assumes "\<one> \<noteq> \<zero>" shows "deg Zp (X_poly Zp) = 1"
Crypt_parts_imp_used: fixes K :: "nat"   and X :: "msg"   and evs :: "event list" assumes "Crypt K X \<in> parts (knows Spy evs)"   and "K \<in> symKeys"   and "evs \<in> set_mr" shows "Key K \<in> used evs"
ffieldI2: fixes a :: "V"   and b :: "V"   and r :: "V" assumes "[a, b]\<^sub>\<circ> \<in>\<^sub>\<circ> r" shows "a \<in>\<^sub>\<circ> \<F>\<^sub>\<bullet> r"
insert_elt_def: fixes a :: "'b array" shows "insert_elt a = (\<lambda>j. (a.(j)) \<bind> (\<lambda>key. ref j \<bind> (\<lambda>i. shiftr a key i \<bind> (\<lambda>_. !i \<bind> (\<lambda>i'. a.(i') \<leftarrow> key)))))"
ex_certain_iff_singleton_support: fixes p :: "'b pmf" shows "(\<exists>x. pmf p x = 1) = (card (set_pmf p) = 1)"
wt_Seq: fixes c1 :: "('test, 'atom, 'choice) cmd"   and c2 :: "('test, 'atom, 'choice) cmd"   and s :: "'state" shows "wt (c1 ;; c2) s = wt c1 s"
chkvar_def: fixes t :: "('s, 'v) tm"   and s :: "('s, 'v) tm" shows "chkvar t s = (vars_hd (head s) \<subseteq> vars t)"
b_e_type_comp2_unlift: fixes \<S> :: "s_context"   and \<C> :: "t_context"   and e1 :: "b_e"   and e2 :: "b_e"   and t1s :: "t list"   and t2s :: "t list" assumes "\<S>\<bullet>\<C> \<turnstile> [$e1, $e2] : t1s _> t2s" shows "\<exists>ts'. \<C> \<turnstile> [e1] : t1s _> ts' \<and> \<C> \<turnstile> [e2] : ts' _> t2s"
Polygamma_real_mono: fixes x :: "real"   and y :: "real"   and n :: "nat" assumes "0 < x"   and "x \<le> y"   and "even n" shows "Polygamma n x \<le> Polygamma n y"
reaches1_steps_iff: fixes x :: "'a"   and y :: "'a" shows "(x \<rightarrow>\<^sup>+ y) = (\<exists>xs. steps (x # xs @ [y]))"
\<sigma>'_def: shows "\<sigma>' = \<sigma>\<^sup>2"
preserves_seq: fixes g :: "'a"   and f :: "'a" assumes "B.seq g f" shows "B.seq (src\<^sub>B g) (src\<^sub>B f)"
mkarr_Runit: fixes t :: "'c local.term" assumes "Arr t" shows "\<F>C.mkarr \<^bold>\<r>\<^bold>[t\<^bold>] = \<F>C.\<rho> (\<F>C.mkarr t)"
mat_of_cblinfun_plus: fixes F :: "'b \<Rightarrow>\<^sub>C\<^sub>L 'c"   and G :: "'b \<Rightarrow>\<^sub>C\<^sub>L 'c" shows "mat_of_cblinfun (F + G) = mat_of_cblinfun F + mat_of_cblinfun G"
weakSimE: fixes P :: "ccs"   and Rel :: "(ccs \<times> ccs) set"   and Q :: "ccs"   and \<alpha> :: "act"   and Q' :: "ccs" assumes "P \<leadsto>\<^sup>^<Rel> Q"   and "Q \<longmapsto>\<alpha> \<prec> Q'" obtains P' :: "ccs" where "P \<Longrightarrow>\<^sup>^\<alpha> \<prec> P'" and "(P', Q') \<in> Rel"
accmethd_rT_is_type: fixes G :: "prog"   and S :: "qtname"   and C :: "qtname"   and sig :: "sig"   and m :: "qtname \<times> methd"   and y :: "class" assumes "wf_prog G"   and "accmethd G S C sig = Some m"   and "class G C = Some y" shows "is_type G (resTy m)"
invertible_right_inverse: fixes u :: "'a" assumes "additive.invertible u"   and "u \<in> R" shows "u - u = \<zero>"
integral_neg: fixes S :: "'b set"   and f :: "'b \<Rightarrow> 'a" shows "integral S (\<lambda>x. - f x) = - integral S f"
member_filter: fixes x :: "'a"   and P :: "'a \<Rightarrow> bool"   and A :: "'a set" shows "(x \<in> Set.filter P A) = (x \<in> A \<and> P x)"
d: shows " |x\<^sup>\<star>\<rangle> z \<le>  |x\<^sup>\<star>\<rangle> y"
primrootE: fixes x :: "'b list"   and r :: "'b list" assumes "x \<in>\<^sub>p r *" obtains k :: "nat" where "k \<noteq> 0" and "r \<^sup>@ k = x"
derived_points_index: fixes ps :: "'a set" assumes "ps \<subseteq> bl"   and "card ps = 2" shows "int (\<B>\<^sup>D index ps) = \<Lambda> - 1"
upd_col_transpose: fixes M :: "('c, 'd) sq_matrix"   and i :: "'d"   and p :: "('c, 'd) vec" shows "upd_col (Square_Matrix.transpose M) i p = Square_Matrix.transpose (upd_row M i p)"
linear_construct: fixes B :: "('a, 'b) vec set"   and f :: "('a, 'b) vec \<Rightarrow> ('a, 'c) vec" assumes "vec.independent B" shows "Vector_Spaces.linear (*s) (*s) (vec.construct B f)"
coeffs_smult: fixes a :: "'a"   and p :: "'a poly" shows "coeffs (Polynomial.smult a p) = (if a = (0::'a) then [] else map ((*) a) (coeffs p))"
the_gpv_parametric': fixes A :: "'m \<Rightarrow> 'p \<Rightarrow> bool"   and C :: "'n \<Rightarrow> 'q \<Rightarrow> bool"   and R :: "'o \<Rightarrow> 'r \<Rightarrow> bool" shows "(rel_gpv'' A C R ===> rel_spmf (rel_generat A C (R ===> rel_gpv'' A C R))) the_gpv the_gpv"
A_ma_k: shows "A $ ma $ mod_type_class.from_nat k \<noteq> (0::'a)"
is_processT8_S: fixes s :: "'a event list"   and P :: "'a process"   and X :: "'a event set" assumes "s \<in> D P" shows "(s, X) \<in> F P"
at_hchop_dist1: fixes c :: "cars"   and \<phi> :: "traffic \<Rightarrow> view \<Rightarrow> bool"   and \<psi> :: "traffic \<Rightarrow> view \<Rightarrow> bool" shows "\<forall>ts v. (\<forall>v'. v = c > v' \<longrightarrow> (\<exists>v u. v'=v\<parallel>u \<and> \<phi> ts v \<and> \<psi> ts u)) \<longrightarrow> (\<exists>va u. v=va\<parallel>u \<and> (\<forall>v'. va = c > v' \<longrightarrow> \<phi> ts v') \<and> (\<forall>v'. u = c > v' \<longrightarrow> \<psi> ts v'))"
vsv_vinsert: fixes a :: "V"   and A :: "V"   and b :: "V" assumes "a \<notin>\<^sub>\<circ> \<D>\<^sub>\<circ> (r \<restriction>\<^sup>l\<^sub>\<circ> A)" shows "vsv (vinsert \<langle>a, b\<rangle> (r \<restriction>\<^sup>l\<^sub>\<circ> A))"
i_Exec_Stream_Cons_rev: fixes trans_fun :: "'e \<Rightarrow> 'd \<Rightarrow> 'd"   and input :: "nat \<Rightarrow> 'e"   and c :: "'d" shows "[trans_fun (input 0) c] \<frown> i_Exec_Comp_Stream trans_fun (input \<Up> Suc 0) (trans_fun (input 0) c) = i_Exec_Comp_Stream trans_fun input c"
prv_eql_imp_trans_rev: fixes t1 :: "'trm"   and t2 :: "'trm"   and t3 :: "'trm" assumes "t1 \<in> trm"   and "t2 \<in> trm"   and "t3 \<in> trm" shows "prv (imp (eql t1 t2) (imp (eql t1 t3) (eql t2 t3)))"
of_int_power_less_of_int_cancel_iff: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and x :: "int"   and b :: "int"   and w :: "nat" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less (ring_1.of_int one plus zero minus x) (power.power one times (ring_1.of_int one plus zero minus b) w) = (x < b ^ w)"
int_in_Units_lucas_lehmer_ring_mod: fixes p :: "nat"   and x :: "nat" assumes "prime p"   and "0 < x"   and "x < p" shows "(x, 0) \<in> Units (lucas_lehmer_ring_mod p)"
R_consistent: shows "pcp {\<Gamma> |\<Gamma>. \<not> ((\<forall>\<gamma>\<in>\<Gamma>. is_cnf \<gamma>) \<longrightarrow> \<Union> (cnf ` \<Gamma>) \<turnstile> \<box>)}"
simps_2: fixes nd :: "('b, 'a) Node" shows "FingerTreeStruc.gmft (Single nd) = FingerTreeStruc.gmn nd"
hd_sort: fixes xs :: "'a list" assumes "xs \<noteq> []" shows "hd (sort xs) = Min (set xs)"
continuous_map_sum: fixes I :: "'c set"   and X :: "'d topology"   and f :: "'d \<Rightarrow> 'c \<Rightarrow> 'e" assumes "finite I"   and "\<And>i. i \<in> I \<Longrightarrow> continuous_map X euclidean (\<lambda>x. f x i)" shows "continuous_map X euclidean (\<lambda>x. sum (f x) I)"
distinct_13: fixes x41 :: "'b rexp"   and x42 :: "'b rexp" shows "One \<noteq> Plus x41 x42"
minus_diff_commute: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and b :: "'d"   and a :: "'d" assumes "class.ab_group_add plus zero minus uminus" shows "minus (uminus b) a = minus (uminus a) b"
inject: fixes x1 :: "'b match_expr"   and x2 :: "action"   and y1 :: "'b match_expr"   and y2 :: "action" shows "(Rule x1 x2 = Rule y1 y2) = (x1 = y1 \<and> x2 = y2)"
is_ground_cls_union: fixes subst_atm :: "'b \<Rightarrow> 'c \<Rightarrow> 'b"   and id_subst :: "'c"   and comp_subst :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and renamings_apart :: "'b literal multiset list \<Rightarrow> 'c list"   and atm_of_atms :: "'b list \<Rightarrow> 'b"   and C :: "'b literal multiset"   and D :: "'b literal multiset" assumes "substitution subst_atm id_subst comp_subst renamings_apart atm_of_atms" shows "substitution_ops.is_ground_cls subst_atm (C + D) = (substitution_ops.is_ground_cls subst_atm C \<and> substitution_ops.is_ground_cls subst_atm D)"
is_pascal_under_alternate_vertices: fixes A :: "Points"   and B :: "Points"   and C :: "Points"   and A' :: "Points"   and B' :: "Points"   and C' :: "Points" assumes "pascal_prop"   and "is_pascal A B C A' B' C'" shows "is_pascal A B' C A' B C'"
assms_2: assumes "P a \<and> P' a'" shows "W"
negate_eq2a: fixes V :: "'b set"   and x :: "'b" assumes "vectorspace V"   and "x \<in> V" shows "- 1 \<cdot> x = - x"
cond_distr_var: fixes p :: "'a"   and q :: "'a"   and r :: "'a"   and x :: "'a"   and y :: "'a" assumes "test p"   and "test q"   and "test r" shows "(q \<cdot> p + r \<cdot> !p) \<cdot> (p \<cdot> x + !p \<cdot> y) = q \<cdot> p \<cdot> x + r \<cdot> !p \<cdot> y"
\<Omega>_def: shows "\<Omega> = set_pmf N \<union> (\<Union>i\<in>set_pmf N. set_pmf (K i))"
induct: fixes x :: "'e"   and G :: "('e, 'f) monoid_scheme"   and gens :: "'e set"   and P :: "'e \<Rightarrow> bool" assumes "x \<in> \<langle>gens\<rangle>\<^bsub>G\<^esub>"   and "P \<one>\<^bsub>G\<^esub>"   and "\<And>x. x \<in> gens \<Longrightarrow> P x"   and "\<And>x. \<lbrakk>x \<in> \<langle>gens\<rangle>\<^bsub>G\<^esub>; P x\<rbrakk> \<Longrightarrow> P (inv\<^bsub>G\<^esub> x)"   and "\<And>x y. \<lbrakk>x \<in> \<langle>gens\<rangle>\<^bsub>G\<^esub>; P x; y \<in> \<langle>gens\<rangle>\<^bsub>G\<^esub>; P y\<rbrakk> \<Longrightarrow> P (x \<otimes>\<^bsub>G\<^esub> y)" shows "P x"
andalso_weaken_left: fixes P :: "tr"   and Q :: "tr" assumes "P = TT \<Longrightarrow> Q = TT"   and "P = FF \<Longrightarrow> Q \<noteq> \<bottom>"   and "P = \<bottom> \<Longrightarrow> Q \<noteq> FF" shows "P = (Q andalso P)"
vsv_value: fixes a :: "V" assumes "a \<in>\<^sub>\<circ> \<BB>\<lparr>Obj\<rparr>" obtains b :: "V" where "\<GG>\<lparr>ObjMap\<rparr>\<lparr>a\<rparr> = b" and "b \<in>\<^sub>\<circ> \<R>\<^sub>\<circ> (\<GG>\<lparr>ObjMap\<rparr>)"
not_False_in_image_Ball: fixes P :: "'n \<Rightarrow> bool"   and A :: "'n set" shows "(False \<notin> P ` A) = Ball A P"
lnorm_def: fixes M :: "'a measure"   and p :: "real"   and f :: "'a \<Rightarrow> real" shows "lnorm M p f \<equiv> (LINT x|M. \<bar>f x\<bar> powr p) powr (1 / p)"
rel_comp_def: fixes r1 :: "'d \<Rightarrow> 'b \<Rightarrow> 'e \<Rightarrow> bool"   and r2 :: "'f \<Rightarrow> 'e \<Rightarrow> 'c \<Rightarrow> bool"   and i :: "'d \<times> 'f" shows "rel_comp r1 r2 i \<equiv> r1 (fst i) OO r2 (snd i)"
unit_eqI: fixes a :: "'b"   and \<mu> :: "'d" assumes "B.obj a"   and "\<guillemotleft>\<mu> : HoG.map\<^sub>0 a \<Rightarrow>\<^sub>D HoG.map a\<guillemotright>"   and "D.iso \<mu>"   and "\<mu> \<cdot>\<^sub>D \<i>\<^sub>D[HoG.map\<^sub>0 a] = (HoG.map \<i>\<^sub>B[a] \<cdot>\<^sub>D HoG.cmp (a, a)) \<cdot>\<^sub>D (\<mu> \<star>\<^sub>D \<mu>)" shows "\<mu> = HoG.unit a"
factorize_int_last_nz_poly_def: fixes alg :: "int_poly_factorization_algorithm"   and f :: "int poly" shows "factorize_int_last_nz_poly alg f = (let df = degree f in if df = 0 then (coeff f 0, []) else if df = 1 then (content f, [(primitive_part f, 0)]) else internal_int_poly_factorization alg f)"
intros_3: fixes t1 :: "lam"   and t2 :: "lam"   and t3 :: "lam" assumes " t1 \<longrightarrow>\<^sub>1\<^sup>* t2"   and " t2 \<longrightarrow>\<^sub>1\<^sup>* t3" shows " t1 \<longrightarrow>\<^sub>1\<^sup>* t3"
ran: shows "p ` {0..<Suc n} = {0..<Suc n}"
reachable_adjI: fixes u :: "'a"   and v :: "'a" assumes "u \<rightarrow>\<^bsub>G\<^esub> v" shows "u \<rightarrow>\<^sup>*\<^bsub>G\<^esub> v"
Collect_empty_eq_bot: fixes P :: "'c \<Rightarrow> bool" shows "(Collect P = {}) = (P = \<bottom>)"
vors_3: shows "vs2 = between (vertices f) ram2 ram1"
ext_inject: fixes xc :: "int"   and yc :: "int"   and more :: "'a"   and xc' :: "int"   and yc' :: "int"   and more' :: "'a" shows "(\<lparr>xc = xc, yc = yc, \<dots> = more\<rparr> = \<lparr>xc = xc', yc = yc', \<dots> = more'\<rparr>) = (xc = xc' \<and> yc = yc' \<and> more = more')"
leading_coeff_neq_0: fixes p :: "'a poly" assumes "p \<noteq> 0" shows "lead_coeff p \<noteq> (0::'a)"
set_mset_union: fixes M :: "'b multiset"   and N :: "'b multiset" shows "set_mset (M + N) = set_mset M \<union> set_mset N"
rep_eq: fixes x :: "'a" shows "cblinfun_apply (cblinfun_mult_right x) = (*) x"
is_span: shows "C.span \<epsilon>.dom.leg0 \<epsilon>.dom.leg1"
less_max_iff_disj: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and z :: "'a"   and x :: "'a"   and y :: "'a" assumes "class.linorder less_eq less" shows "less z (ord.max less_eq x y) = (less z x \<or> less z y)"
rep_eq: fixes x :: "'d multiset"   and xa :: "'d multiset" shows "count (x + xa) = (\<lambda>a. count x a + count xa a)"
inject: fixes a :: "'h"   and b :: "'i"   and a' :: "'h"   and b' :: "'i" shows "((a, b) = (a', b')) = (a = a' \<and> b = b')"
p_def: fixes R :: "'d \<Rightarrow> 'd \<Rightarrow> real"   and s :: "'d"   and s' :: "'d"   and t :: "real" assumes "transition_rates R" shows "transition_rates.p R s s' t = \<P>(\<omega> in discrete_Markov_process.lim_stream (borel \<Otimes>\<^sub>M count_space UNIV) (transition_rates.K R) (0, s). transition_rates.trace_in {s'} t s \<omega>)"
in_space_T: fixes R :: "'d \<Rightarrow> 'd \<Rightarrow> real"   and x :: "(real \<times> 'd) stream" assumes "transition_rates R" shows "x \<in> space (stream_space (borel \<Otimes>\<^sub>M count_space UNIV))"
simps: fixes a :: "'b" shows "the_In1 (In1 a) = a"
W_AUB: shows "induced_map A --` W = induced_map A --` WA \<union> induced_map A --` WB"
S4a_successors: fixes rmhist :: "state \<Rightarrow> PrIds \<Rightarrow> histState"   and p :: "PrIds" shows "\<turnstile> $(S4 rmhist p \<and> ires!p = #NotAResult) \<and> ImpNext p \<and> [HNext rmhist p]_(c p, r p, m p, rmhist!p) \<and> (\<forall>l. $MemInv mm l) \<longrightarrow> (S4 rmhist p \<and> ires!p = #NotAResult)$ \<or> (S4 rmhist p \<and> ires!p \<noteq> #NotAResult \<or> S5 rmhist p)$"
DERIV_gf_U: fixes z :: "real"   and x :: "'s"   and y :: "'s" assumes "0 < z"   and "z < 1" shows "(gf_U x y has_real_derivative gf_U' x y z) (at z)"
divide_i: fixes x :: "complex" shows "x / \<i> = - \<i> * x"
finite_SMP_representation_def: fixes arity :: "'a \<Rightarrow> nat"   and Ana :: "('a, 'b) Term.term \<Rightarrow> ('a, 'b) Term.term list \<times> 'c"   and \<Gamma> :: "('a, 'b) Term.term \<Rightarrow> ('d, 'e) Term.term"   and M :: "('a, 'b) Term.term list" shows "finite_SMP_representation arity Ana \<Gamma> M \<equiv> list_all (wf\<^sub>t\<^sub>r\<^sub>m' arity) M \<and> has_all_wt_instances_of \<Gamma> (subterms\<^sub>s\<^sub>e\<^sub>t (set M)) (set M) \<and> has_all_wt_instances_of \<Gamma> (\<Union> ((set \<circ> fst \<circ> Ana) ` set M)) (set M) \<and> is_TComp_var_instance_closed \<Gamma> M"
subset_cons: fixes z :: "'a list"   and z' :: "'a list"   and p :: "'a" assumes "set z \<subseteq> set z'" shows "set (p # z) \<subseteq> set (p # z')"
outputSwapFrameSubject: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and M :: "'a"   and xvec :: "name list"   and N :: "'a"   and P' :: "('a, 'b, 'c) psi"   and x :: "name"   and y :: "name" assumes "\<Psi> \<rhd> P \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> P'"   and "xvec \<sharp>* M"   and "x \<sharp> P"   and "y \<sharp> P" shows "[(x, y)] \<bullet> \<Psi> \<rhd> P \<longmapsto> ([(x, y)] \<bullet> M)\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> P'"
ord0_leqI: fixes x :: "ord0"   and y :: "ord0" assumes "\<forall>a. (a, x) \<in> ord0_prec\<^sup>+ \<longrightarrow> (a, y) \<in> ord0_leq O ord0_prec\<^sup>+" shows "(x, y) \<in> ord0_leq"
FAddSA_RootEx_def: fixes S :: "'d"   and G :: "'d \<Rightarrow> ('d, 'e, 'f) seqauto set option"   and SA :: "('d, 'e, 'f) seqauto"   and F :: "('d, 'e, 'f) seqauto set" assumes "S \<in> dom G"   and "States SA \<inter> dom G = {}" shows "RootEx F (G [f+] (S, SA)) = (\<exists>!A. A \<in> F \<and> A \<notin> insert SA (\<Union> (ran G)))"
asIMOD_igSubstIGVar2: fixes SEM :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'b) semDom" shows "igSubstIGVar2 (asIMOD SEM)"
mismatchTrans: fixes a :: "name"   and P :: "pi"   and Rs :: "residual" assumes "[a\<noteq>a]P \<longmapsto> Rs" shows "False"
equivalence_cancel_right: fixes e :: "'c"   and \<mu> :: "'c"   and \<mu>' :: "'c" assumes "C.equivalence_map e"   and "C.par \<mu> \<mu>'"   and "src\<^sub>C \<mu> = trg\<^sub>C e"   and "\<mu> \<star>\<^sub>C e = \<mu>' \<star>\<^sub>C e" shows "\<mu> = \<mu>'"
monom_mult_zero_right: fixes c :: "'d"   and t :: "'c" shows "punit.monom_mult c t 0 = 0"