hcju/isabelleps huggingface dataset:
aff_dim_insert: fixes a :: "'a"   and S :: "'a set" shows "aff_dim (insert a S) = (if a \<in> affine hull S then aff_dim S else aff_dim S + 1)"
aHom_mem: fixes F :: "('e, 'f) aGroup_scheme"   and G :: "('g, 'h) aGroup_scheme"   and f :: "'e \<Rightarrow> 'g"   and a :: "'e" assumes "aGroup F"   and "aGroup G"   and "f \<in> aHom F G"   and "a \<in> carrier F" shows "f a \<in> carrier G"
hypnat_diff_0_eq_0: fixes n :: "nat star" shows "0 - n = 0"
absolutely_integrable_integrable_bound: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> real" assumes "\<And>x. x \<in> S \<Longrightarrow> norm (f x) \<le> g x"   and "f integrable_on S"   and "g integrable_on S" shows "f absolutely_integrable_on S"
mono_multpw: fixes ns :: "('a \<times> 'a) set"   and ns' :: "('a \<times> 'a) set" assumes "ns \<subseteq> ns'" shows "multpw ns \<subseteq> multpw ns'"
finite_measure_Diff: fixes A :: "'a set"   and B :: "'a set" assumes "A \<in> sets M"   and "B \<in> sets M"   and "B \<subseteq> A" shows "Sigma_Algebra.measure M (A - B) = Sigma_Algebra.measure M A - Sigma_Algebra.measure M B"
prems_5: shows "(mb # n_as) ! bj = (n, a)"
id: shows "p ^ l = p ^ local.ll * p ^ (l - local.ll)"
cone_def: fixes Vs :: "'a vec set" shows "next_dim.cone Vs = {x. \<exists>Ws. finite Ws \<and> Ws \<subseteq> Vs \<and> x \<in> next_dim.finite_cone Ws}"
loc_def: fixes w :: "'a el_step list" shows "loc w \<equiv> filter (\<lambda>e. case e of LOC a \<Rightarrow> True | ENV a \<Rightarrow> False) w"
objE: fixes a :: "'a" assumes "B.obj a" obtains "B.ide a" and "trg a = a" and "src a = a"
convergent: fixes prod :: "'d \<Rightarrow> 'e \<Rightarrow> 'f"   and X :: "nat \<Rightarrow> 'd"   and Y :: "nat \<Rightarrow> 'e" assumes "bounded_bilinear prod"   and "convergent X"   and "convergent Y" shows "convergent (\<lambda>n. prod (X n) (Y n))"
monom_altdef: fixes c :: "'a"   and n :: "nat" shows "Polynomial.monom c n = Polynomial.smult c ([:0::'a, 1::'a:] ^ n)"
truncate_down_mono: fixes x :: "real"   and y :: "real"   and p :: "nat" assumes "x \<le> y" shows "truncate_down p x \<le> truncate_down p y"
abs_eq: fixes xa :: "'a"   and x :: "'a" shows "Upair xa x = abs_uprod (xa, x)"
less_1: fixes xs :: "'a llist" assumes "the_enat (llength xs) < the_enat (llength xsa__)"   and "lfinite xs" shows "P xs"
add_nonneg_nonneg: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c"   and b :: "'c" assumes "class.ordered_comm_monoid_add plus zero less_eq less"   and "less_eq zero a"   and "less_eq zero b" shows "less_eq zero (plus a b)"
subformulas\<^sub>\<mu>_subfrmlsn: fixes \<phi> :: "'a ltln" shows "subformulas\<^sub>\<mu> \<phi> \<subseteq> subfrmlsn \<phi>"
IDE_mkarr_Ide: fixes a :: "'c local.term" assumes "Ide a" shows "IDE (mkarr a)"
iteration_slide: fixes x :: "'a"   and y :: "'a" shows "x \<cdot> (y \<cdot> x)\<^sup>\<infinity> = (x \<cdot> y)\<^sup>\<infinity> \<cdot> x"
hom_one: shows "h \<one>\<^bsub>free_Abelian_group S\<^esub> = \<one>\<^bsub>free_Abelian_group T\<^esub>"
U_par_nc: shows "nc \<cdot> 1\<^sub>\<pi> \<parallel> U = U"
"case": fixes f :: "('b \<Rightarrow> 'c) \<Rightarrow> 'a"   and x :: "'b \<Rightarrow> 'c" shows "(case StateT x of StateT x \<Rightarrow> f x) = f x"
not_hd_in_tl: fixes x :: "'a"   and xs :: "'a list" assumes "x \<noteq> hd xs"   and "x \<in> set xs" shows "x \<in> set (tl xs)"
simps_1: fixes M :: "nat \<Rightarrow> nat \<Rightarrow> 'c DBMEntry"   and f :: "nat \<Rightarrow> nat \<Rightarrow> 'c DBMEntry"   and uu :: "nat" shows "DBM_upd M f 0 0 uu = DBM_update M 0 0 (f 0 0)"
prod_orA_def: fixes p1 :: "'d \<Rightarrow> 'f decision option"   and p2 :: "'e \<Rightarrow> 'g decision option" shows "p1 \<Otimes>\<^sub>\<or>\<^sub>A p2 = (\<lambda>(x, y). case p1 x of \<bottom> \<Rightarrow> \<bottom> | \<lfloor>allow d1\<rfloor> \<Rightarrow> case p2 y of \<bottom> \<Rightarrow> \<bottom> | \<lfloor>allow d2\<rfloor> \<Rightarrow> \<lfloor>allow (d1, d2)\<rfloor> | \<lfloor>deny d2\<rfloor> \<Rightarrow> \<lfloor>allow (d1, d2)\<rfloor> | \<lfloor>deny d1\<rfloor> \<Rightarrow> case p2 y of \<bottom> \<Rightarrow> \<bottom> | \<lfloor>allow d2\<rfloor> \<Rightarrow> \<lfloor>allow (d1, d2)\<rfloor> | \<lfloor>deny d2\<rfloor> \<Rightarrow> \<lfloor>deny (d1, d2)\<rfloor>)"
assms_3: shows "\<not> (are_all_operators_applicable_in s ops \<and> SAS_Plus_Semantics.are_all_operator_effects_consistent ops)"
preListpreSet: fixes xs :: "('a \<times> ((char list \<Rightarrow> int) \<Rightarrow> nat)) list"   and l :: "'a \<Rightarrow> nat"   and C :: "acom"   and s :: "char list \<Rightarrow> int" assumes "preSet (set xs) C l s" shows "preList xs C l s"
fresh_star_Pair: fixes as :: "atom set"   and x :: "'a"   and y :: "'b" shows "as \<sharp>* (x, y) = (as \<sharp>* x \<and> as \<sharp>* y)"
Lim_transform_eq: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and a :: "'b" assumes "((\<lambda>x. f x - g x) \<longlongrightarrow> (0::'b)) F" shows "(f \<longlongrightarrow> a) F = (g \<longlongrightarrow> a) F"
ipPartition_def: fixes A :: "'b set set"   and B :: "'b set set" shows "ipPartition A B \<equiv> \<forall>a\<in>A. \<forall>b\<in>B. a \<inter> b = {} \<or> b \<subseteq> a"
cindexE_combine: fixes f :: "real \<Rightarrow> real"   and a :: "real"   and c :: "real"   and b :: "real" assumes "finite_jumpFs f a c"   and "a \<le> b"   and "b \<le> c" shows "cindexE a c f = cindexE a b f + cindexE b c f"
eventually_nonzero_at_top: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter" assumes "filterlim f at_top F" shows "eventually_nonzero F f"
open_p_point_def: fixes U :: "p_point set" shows "open U = (\<forall>x\<in>U. \<forall>\<^sub>F (x', y) in uniformity. x' = x \<longrightarrow> y \<in> U)"
cod_comp: fixes g :: "'b"   and f :: "'b" assumes "B.seq g f" shows "B.cod (g \<cdot>\<^sub>B f) = B.cod g"
L9_Controller: fixes s :: "nat \<Rightarrow> nat list"   and l :: "nat \<Rightarrow> bit"   and z :: "nat \<Rightarrow> bit list"   and i :: "nat"   and r :: "nat" assumes "Controller_L s (fin_inf_append [Zero] l) l z"   and "fin_make_untimed (inf_truncate z i) ! (length (fin_make_untimed (inf_truncate z i)) - Suc 0) = Zero"   and "last (fin_make_untimed (inf_truncate z i)) = l i"   and "hd (s (Suc i)) = hd (s i) - r"   and "fin_make_untimed (inf_truncate z i) \<noteq> []"   and "r \<le> 10" shows "200 \<le> hd (s (Suc i))"
Abs_formula_eqvt: fixes t\<^sub>\<alpha> :: "('a, 'b, 'c, 'd) Tree\<^sub>\<alpha>"   and p :: "perm" assumes "hereditarily_fs t\<^sub>\<alpha>" shows "p \<bullet> Abs_formula t\<^sub>\<alpha> = Abs_formula (p \<bullet> t\<^sub>\<alpha>)"
simps_2: fixes s :: "'c"   and p :: "'d \<times> 'c"   and ps :: "('d \<times> 'c) list" shows "last_state (s, Cons p ps) = snd p"
atLeastLessThan_iff: fixes i :: "'b"   and l :: "'b"   and u :: "'b" shows "(i \<in> {l..<u}) = (l \<le> i \<and> i < u)"
adiv_def: fixes x :: "ant"   and y :: "ant" shows "x adiv y = ant (tna x div tna y)"
prems_1: fixes x :: "'a \<times> 'c" assumes "x \<in> set (a # xs)" shows "g (f (fst x)) = fst x"
semiring_normalization_rules_24: fixes a :: "'a"   and c :: "'a" shows "a + c = c + a"
dyn_equiv_stat: fixes Bot :: "'a set"   and Inf :: "'a inference set"   and entails :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool"   and Red_I :: "'a set \<Rightarrow> 'a inference set"   and Red_F :: "'a set \<Rightarrow> 'a set" assumes "calculus Bot Inf entails Red_I Red_F" shows "dynamically_complete_calculus Bot Inf entails Red_I Red_F = statically_complete_calculus Bot Inf entails Red_I Red_F"
S10: shows "2 ^ (x - y) + u div 2 ^ y = u div 2 ^ y + 2 ^ (x - y - 1) * 2"
tp_assoc: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<in> carrier R"   and "b \<in> carrier R"   and "c \<in> carrier R" shows "a \<cdot>\<^sub>r b \<cdot>\<^sub>r c = a \<cdot>\<^sub>r (b \<cdot>\<^sub>r c)"
assms_3: shows "(\<lambda>x. (x, a)) ` S \<subseteq> U"
closest_supchamber_def: fixes X :: "'b set set"   and F :: "'b set"   and D :: "'b set" assumes "ChamberComplex X" shows "ChamberComplex.closest_supchamber X F D = (ARG_MIN (\<lambda>C. ChamberComplex.chamber_distance X C D) C. SimplicialComplex.maxsimp X C \<and> order.greater_eq C F)"
awalk_last_in_verts: fixes u :: "'a"   and p :: "('a \<times> 'a) list"   and v :: "'a" assumes "S.awalk u p v" shows "v \<in> pverts (subdivide G e w)"
sets_M: shows "sets M = sets (stock_measure (PRODUCT REAL REAL))"
A: shows "from_hma\<^sub>m A \<in> carrier_mat CARD('n) CARD('n)"
sigma_sets_subset': fixes a :: "'d set set"   and M :: "'d measure"   and \<Omega>' :: "'d set" assumes "a \<subseteq> sets M"   and "\<Omega>' \<in> sets M" shows "sigma_sets \<Omega>' a \<subseteq> sets M"
simps: fixes G :: "prog"   and a1 :: "ty"   and a2 :: "ty" shows "G\<turnstile>a1\<succ>a2 = ((\<exists>C D. a1 = Class D \<and> a2 = Class C \<and> G\<turnstile>C\<preceq>\<^sub>C D) \<or> (\<exists>C I. a1 = Class C \<and> a2 = Iface I \<and> \<not> G\<turnstile>C\<leadsto>I) \<or> (\<exists>T. a1 = Class Object \<and> a2 = T.[]) \<or> (\<exists>I C. a1 = Iface I \<and> a2 = Class C) \<or> (\<exists>I J mrt. a1 = Iface I \<and> a2 = Iface J \<and> (imethds G I hidings imethds G J entails \<lambda>(md, mh) (md', mh'). G\<turnstile>mrt mh\<preceq>mrt mh') \<and> (I, J) \<notin> (subint1 G)\<^sup>*) \<or> (\<exists>S T. a1 = RefT S.[] \<and> a2 = RefT T.[] \<and> G\<turnstile>RefT S\<succ>RefT T))"
drlex_pp_alt: fixes s :: "('a, 'b) pp"   and t :: "('a, 'b) pp" shows "drlex_pp s t = (deg_pp s < deg_pp t \<or> deg_pp s = deg_pp t \<and> lex_pp t s)"
subseqs_subformulas\<^sub>\<nu>_restricted_list: fixes Y :: "'b ltln set"   and \<phi> :: "'b ltln" shows "(Y \<subseteq> subformulas\<^sub>\<nu> \<phi> \<inter> restricted_subformulas \<phi>) = (\<exists>ys. Y = set ys \<and> ys \<in> set (subseqs (filter (\<lambda>x. x \<in> restricted_subformulas \<phi>) (subformulas\<^sub>\<nu>_list \<phi>))))"
matrix_left_invertible_ker: fixes A :: "(('b, 'd) vec, 'c) vec" shows "(\<exists>B. B ** A = mat (1::'b)) = (\<forall>x. A *v x = 0 \<longrightarrow> x = 0)"
ResPres: fixes P :: "pi"   and Q :: "pi"   and x :: "name" assumes "P \<equiv>\<^sub>e Q" shows "<\<nu>x>P \<equiv>\<^sub>e <\<nu>x>Q"
iTILL_add: fixes i :: "nat"   and k :: "nat" shows "[\<dots>i] \<oplus> k = [k\<dots>,i]"
Longest_common_prefix_prefix: fixes xs :: "'a list"   and L :: "'a list set" assumes "xs \<in> L" shows "prefix (Longest_common_prefix L) xs"
iT_0_3: fixes n :: "nat" shows "[n\<dots>,0] = {n}"
vcons_eqD_2: fixes xs\<^sub>1 :: "V"   and x\<^sub>1 :: "V"   and xs\<^sub>2 :: "V"   and x\<^sub>2 :: "V" assumes "xs\<^sub>1 #\<^sub>\<circ> x\<^sub>1 = xs\<^sub>2 #\<^sub>\<circ> x\<^sub>2"   and "vfsequence xs\<^sub>1"   and "vfsequence xs\<^sub>2" shows "x\<^sub>1 = x\<^sub>2"
is_inf_def: fixes x :: "int"   and y :: "int"   and i :: "int" shows "int.is_inf x y i = (i \<le> x \<and> i \<le> y \<and> (\<forall>z. z \<le> x \<and> z \<le> y \<longrightarrow> z \<le> i))"
finite_incidence_system_axioms_def: fixes point_set :: "'b set" shows "finite_incidence_system_axioms point_set \<equiv> finite point_set"
locally_faithful_iff_faithful: shows "F.locally_faithful = faithful_functor (\<cdot>\<^sub>C) (\<cdot>\<^sub>D) F"
assms_1: shows "M \<longrightarrow>\<^sub>l M'"
doesnt_read_def: fixes c :: "'Com"   and x :: "'Var" shows "doesnt_read c x = (\<forall>mds mem c' mds' mem'. \<langle>c, mds, mem\<rangle> \<leadsto> \<langle>c', mds', mem'\<rangle> \<longrightarrow> (\<forall>v. \<langle>c, mds, mem(x := v)\<rangle> \<leadsto> \<langle>c', mds', mem'(x := v)\<rangle>) \<or> (\<forall>v. \<langle>c, mds, mem(x := v)\<rangle> \<leadsto> \<langle>c', mds', mem'\<rangle>))"
nonneg_Reals_add_I: fixes a :: "'a"   and b :: "'a" assumes "a \<in> \<real>\<^sub>\<ge>\<^sub>0"   and "b \<in> \<real>\<^sub>\<ge>\<^sub>0" shows "a + b \<in> \<real>\<^sub>\<ge>\<^sub>0"
more_arith_simps_5: fixes a :: "'a" shows "(1::'a) * a = a"
exponential_orderI: fixes M :: "real"   and f :: "real \<Rightarrow> 'a"   and c :: "real" assumes "0 < M"   and "\<forall>\<^sub>F t in at_top. norm (f t) \<le> M * exp (c * t)" shows "exponential_order M c f"
assms_7: fixes k :: "'a" shows "f k (0::'b) (0::'c) = (0::'d)"
cases: fixes a1 :: "'e set set"   and a2 :: "(('f, 'g, 'e) prot_fun, 'h) Term.term"   and a3 :: "(('f, 'g, 'e) prot_fun, 'h) Term.term" assumes "timpls_transformable_to_pred a1 a2 a3" obtains A :: "'e set set" and x :: "'h" where "a1 = Aa__" and "a2 = Var x" and "a3 = Var x"   | f :: "('f, 'g, 'e) prot_fun" and T :: "(('f, 'g, 'e) prot_fun, 'h) Term.term list" and S :: "(('f, 'g, 'e) prot_fun, 'h) Term.term list" and A :: "'e set set" where "a1 = Ab__" and "a2 = Fun f T" and "a3 = Fun f S" and "\<not> is_Abs f" and "length T = length S" and "\<And>i. i < length T \<Longrightarrow> timpls_transformable_to_pred Ab__ (T ! i) (S ! i)"   | b :: "'e set" and A :: "'e set set" and a :: "'e set" where "a1 = A" and "a2 = Fun (Abs a) []" and "a3 = Fun (Abs b) []" and "b \<in> A"
\<Psi>_def: shows "\<Psi> \<equiv> \<lambda>x r. \<forall>d>0. \<exists>h>0. h < d \<and> r \<le> norm (i h x - f x)"
simps: fixes pth :: "(('vertex \<times> ('form, 'var) out_port) \<times> 'vertex \<times> ('form, 'var) antecedent) list" shows "hyps_along pth = ffUnion (extra_assms |`| snd |`| fset_from_list pth) |\<union>| global_assms TYPE('var)"
epi_def: fixes G :: "('c, 'e) monoid_scheme"   and H :: "('d, 'f) monoid_scheme" shows "epi G H = {f \<in> hom G H. f ` carrier G = carrier H}"
eucl_of_list_def: fixes xs :: "real list" shows "eucl_of_list xs = sum_list (map2 (*\<^sub>R) xs Basis_list)"
case_eq_if: fixes f1 :: "'a"   and f2 :: "('b, 'c) rbt \<Rightarrow> color \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> ('b, 'c) rbt \<Rightarrow> 'a"   and rbt :: "('b, 'c) rbt" shows "(case rbt of Leaf \<Rightarrow> f1 | Node x xa xb xc xd \<Rightarrow> f2 x xa xb xc xd) = (if rbt = Leaf then f1 else f2 (lsub rbt) (cl rbt) (key rbt) (val rbt) (rsub rbt))"
prime_natD: fixes p :: "nat"   and a :: "nat" assumes "prime p"   and "a dvd p" shows "a = 1 \<or> a = p"
finite_successors: fixes prog :: "cmd"   and n :: "w_node" shows "finite {n'. \<exists>a'. valid_edge prog a' \<and> sourcenode a' = n \<and> targetnode a' = n'}"
term_distinct_6: fixes name\<^sub>2 :: "name"   and name\<^sub>1 :: "name" shows "const name\<^sub>2 \<noteq> free name\<^sub>1"
simps: fixes m :: "nat"   and rs :: "nat"   and l :: "cell list"   and r :: "cell list" shows "wadjust_stop m rs (l, r) = (\<exists>rn. l = [Bk] \<and> r = Oc \<up> Suc m @ Bk # Oc \<up> Suc (Suc rs) @ Bk \<up> rn)"
opt_mru_vote_mru_of_set: fixes s :: "'a v_state_scheme"   and Q :: "process set" assumes "s \<in> Vinv1"   and "s \<in> SV_inv3" shows "opt_mru_vote (process_mru (votes s)) Q = mru_of_set (votes s) Q"
when_def: fixes zero :: "'d"   and a :: "'d"   and P :: "bool" shows "zero.when zero a P = (if P then a else zero)"
map_ticket_addsimps_3: fixes f :: "nat \<Rightarrow> nat"   and t :: "thread"   and tk :: "nat" shows "(map_ticket f t = HOLD tk) = (\<exists>tk'. tk = f tk' \<and> t = HOLD tk')"
inj_idom_hom_map_poly: fixes hom :: "'b \<Rightarrow> 'c" assumes "inj_idom_hom hom" shows "inj_idom_hom (map_poly hom)"
HEndPhase1_HInv4d_p: fixes s :: "state"   and s' :: "state"   and p :: "Proc" assumes "HEndPhase1 s s' p"   and "HInv4d s p"   and "Inv2b s"   and "HInv4c s p" shows "HInv4d s' p"
length_map: fixes f :: "'c \<Rightarrow> 'b"   and xs :: "'c list" shows "length (map f xs) = length xs"
order_iff: fixes a :: "'t"   and b :: "'t" shows "(a \<preceq>\<^sub>t b) = (a = ord_term_lin.min a b)"
that: fixes f :: "('a \<times> 'c \<Rightarrow>\<^sub>L 'a) set \<Rightarrow> ('a \<times> 'c \<Rightarrow>\<^sub>L 'a) set" assumes "\<And>G. G \<in> Gs \<Longrightarrow> flowsto (f G) T CX (G \<union> trap)"   and "X = \<Union> (f ` Gs)" shows "thesis"
AE_I2: fixes M :: "'c measure"   and P :: "'c \<Rightarrow> bool" assumes "\<And>x. x \<in> space M \<Longrightarrow> P x" shows "almost_everywhere M P"
cytronMinimal_def: fixes g :: "'g" shows "cytronMinimal g \<equiv> \<forall>v\<in>allVars g. phi g v \<noteq> None \<longrightarrow> necessaryPhi' g v"
remdups_adj_rev: fixes xs :: "'a list" shows "remdups_adj (rev xs) = rev (remdups_adj xs)"
in_new_spairsE: fixes q :: "('t \<Rightarrow>\<^sub>0 'b) \<times> ('t \<Rightarrow>\<^sub>0 'b) + nat"   and bs :: "('t \<Rightarrow>\<^sub>0 'b) list"   and p :: "'t \<Rightarrow>\<^sub>0 'b" assumes "q \<in> set (new_spairs bs p)" obtains b :: "'t \<Rightarrow>\<^sub>0 'b" where "b \<in> set bs" and "is_regular_spair p b" and "q = Inl (p, b)"
SUP_constant: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and c :: "'b"   and A :: "'d set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "Sup ((\<lambda>y. c) ` A) = (if A = {} then bot else c)"
assms: shows "\<forall>\<sigma>. csmall P \<sigma> = {} \<or> (\<exists>o'. csmall P \<sigma> = {o'})"
simps_4: fixes a :: "('b, 'c) Combinators"   and v :: "'b"   and va :: "'b"   and vb :: "'c" shows "member a (AllowPortFromTo v va vb) = (a = AllowPortFromTo v va vb)"
zero_neq_neg_numeral: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and n :: "num" assumes "class.ring_char_0 one times plus zero minus uminus" shows "zero \<noteq> uminus (numeral.numeral one plus n)"
cod_char: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" assumes "span_vertical_category C" shows "partial_magma.cod (span_vertical_category.vcomp C) = (\<lambda>\<mu>. if partial_magma.arr (span_vertical_category.vcomp C) \<mu> then \<lparr>Chn = span_in_category.apex C (Cod \<mu>), Dom = Cod \<mu>, Cod = Cod \<mu>\<rparr> else partial_magma.null (span_vertical_category.vcomp C))"
simps_1: shows "subtrees \<langle>\<rangle> = {\<langle>\<rangle>}"
certificate: fixes lp :: "linear_poly" shows "LinearPoly (linear_poly_map lp) = lp"
induct: fixes x :: "msg"   and H :: "msg set"   and P :: "msg \<Rightarrow> bool" assumes "x \<in> analz H"   and "\<And>X. X \<in> H \<Longrightarrow> P X"   and "\<And>X Y. \<lbrakk>\<lbrace>X, Y\<rbrace> \<in> analz H; P \<lbrace>X, Y\<rbrace>\<rbrakk> \<Longrightarrow> P X"   and "\<And>X Y. \<lbrakk>\<lbrace>X, Y\<rbrace> \<in> analz H; P \<lbrace>X, Y\<rbrace>\<rbrakk> \<Longrightarrow> P Y"   and "\<And>K X. \<lbrakk>Crypt K X \<in> analz H; P (Crypt K X); Key (invK K) \<in> analz H; P (Key (invK K))\<rbrakk> \<Longrightarrow> P X"   and "\<And>x y. \<lbrakk>Auth_Data x y \<in> analz H; P (Auth_Data x y); Pri_AgrK y \<in> analz H; P (Pri_AgrK y)\<rbrakk> \<Longrightarrow> P (Pri_AgrK x)"   and "\<And>x y. \<lbrakk>Auth_Data x y \<in> analz H; P (Auth_Data x y); Pri_AgrK x \<in> analz H; P (Pri_AgrK x)\<rbrakk> \<Longrightarrow> P (Pri_AgrK y)" shows "P x"
atLeastLessThan_upto: fixes i :: "int"   and j :: "int" shows "{i..<j} = set [i..j - 1]"
UN_simps_3: fixes A :: "'a set"   and B :: "'b \<Rightarrow> 'a set"   and C :: "'b set" shows "(\<Union>x\<in>C. A \<union> B x) = (if C = {} then {} else A \<union> \<Union> (B ` C))"
assms_4: shows "prv (eqv \<chi>1 \<chi>2)"
chain_sym_obtain: fixes f :: "nat \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a"   and c :: "'a"   and X :: "'a set" assumes "[f[a .. b ..  c]X]" obtains g :: "nat \<Rightarrow> 'a" where "[g[c .. b ..  a]X]" and "g = (\<lambda>n. f (card X - 1 - n))"
map_replicate_trivial: fixes x :: "'a"   and i :: "nat" shows "map (\<lambda>i. x) [0..<i] = replicate i x"
red_pres_fv: fixes e :: "exp"   and e' :: "exp" assumes "e \<longrightarrow> e'"   and "FV e = {}" shows "FV e' = {}"
Hpr_Suc: fixes R :: "('e, 'f) Ring_scheme"   and S :: "('e, 'g) Ring_scheme"   and X :: "'e"   and t :: "'e"   and R' :: "('e set, 'h) Ring_scheme"   and Y :: "'e set"   and f :: "'e"   and g :: "'e"   and h :: "'e"   and m :: "nat" shows "Hpr\<^bsub> R S X t R' Y f g h\<^esub> Suc m = Hen\<^bsub> R S X t R' Y f\<^esub> Suc m Hpr\<^bsub> R S X t R' Y f g h\<^esub> m"
G_perm_comp: fixes g :: "'a"   and h :: "'a" assumes "g \<in> G"   and "h \<in> G" shows "G_perm g \<circ> G_perm h = G_perm (binop g h)"
map_of_remap_SomeD: fixes t :: "(('c \<times> 'a) \<times> 'b) list"   and k :: "'c"   and k' :: "'a"   and x :: "'b" assumes "map_of (map (\<lambda>((k, k'), x). (k, k', x)) t) k = \<lfloor>(k', x)\<rfloor>" shows "map_of t (k, k') = \<lfloor>x\<rfloor>"
cl_inv_channels_update: fixes g :: "'a gState_scheme"   and p :: "pState"   and i :: "nat"   and c :: "channel" assumes "cl_inv (g, p)" shows "cl_inv (gState.channels_update (\<lambda>cs. cs[i := c]) g, p)"
split: fixes P :: "'c \<Rightarrow> bool"   and f1 :: "'c"   and f2 :: "enat \<Rightarrow> 'c"   and enat :: "enat" shows "P (case enat of 0 \<Rightarrow> f1 | eSuc x \<Rightarrow> f2 x) = ((enat = 0 \<longrightarrow> P f1) \<and> (\<forall>x2. enat = eSuc x2 \<longrightarrow> P (f2 x2)))"
M: shows "subresultant'_mat J k f g = mat (df + dg) (df + dg) (\<lambda>(i, j). if i = df + dg - 1 then N j else M i j)"
linorder_char: shows "class.linorder (\<lambda>c d. of_char c \<le> of_char d) (\<lambda>c d. of_char c < of_char d)"
HOM_carrier: fixes M :: "('f, 'a, 'h) Module_scheme"   and N :: "('g, 'a, 'i) Module_scheme" shows "carrier (HOM\<^bsub>R\<^esub> M N) = mHom R M N"
hyps_2: fixes S :: "'node SDG_node set" assumes "n'' \<in> HRB_slice {n}"   and "nx \<in> S" shows "n'' \<in> HRB_slice S"
assms_1: shows "s \<in> {s. fmdom' s = insert v X}"
nth_list_update_neq: fixes i :: "nat"   and j :: "nat"   and xs :: "'e list"   and x :: "'e" assumes "i \<noteq> j" shows "xs[i := x] ! j = xs ! j"
if_distrib: fixes f :: "'c \<Rightarrow> 'b"   and c :: "bool"   and x :: "'c"   and y :: "'c" shows "f (if c then x else y) = (if c then f x else f y)"
as_good_as_sameIff: fixes carrier :: "'b set"   and relation :: "('b \<times> 'b) set"   and z :: "'b"   and y :: "'b" assumes "rational_preference carrier relation" shows "(z \<in> as_good_as y carrier relation) = z \<approx>[relation] y"
Inl_def: shows "Sum_Type.Inl = Abs_sum \<circ> Inl_Rep"
two: shows "x ; y \<cdot> - (x ; z) \<le> x ; y"
E_def: fixes k :: "nat" shows "E k = {x \<in> space M. \<bar>f x\<bar> \<le> real k}"
countable_lfp_apply: fixes F :: "('c \<Rightarrow> 'd set) \<Rightarrow> 'c \<Rightarrow> 'd set"   and x :: "'c" assumes "\<And>Y x. (\<And>x. countable (Y x)) \<Longrightarrow> countable (F Y x)"   and "sup_continuous F" shows "countable (lfp F x)"
prime_factorization_def: shows "prime_factorization \<equiv> map_fun id Abs_multiset (\<lambda>x p. if prime p then multiplicity p x else 0)"
goodInp_imp_qGoodInp_pickInp: fixes inp :: "'a \<Rightarrow> ('b, 'c, 'varSort, 'var, 'd) qTerm set option" assumes "goodInp inp" shows "qGoodInp (pickInp inp)"
crel_vs_def: fixes R :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and v :: "'e"   and s :: "(heap, 'f) state" shows "state_dp_consistency.crel_vs R v s \<equiv> \<forall>M. state_dp_consistency.cmem M \<and> P M \<longrightarrow> (case run_state s M of (v', M') \<Rightarrow> R v v' \<and> state_dp_consistency.cmem M' \<and> P M')"
g: fixes j :: "nat"   and r :: "nat"   and e :: "nat" shows "eval g [j, r, e] \<down>= (if e_nth e j \<noteq> 0 then j else if j = r then Suc j else r)"
f_Exec_Stream_nth_calc_Previous: fixes n :: "nat"   and xs :: "'d list"   and trans_fun :: "'d \<Rightarrow> 'e \<Rightarrow> 'e"   and c :: "'e" assumes "n < length xs" shows "f_Exec_Comp_Stream trans_fun xs c ! n = trans_fun (xs ! n) ((f_Exec_Comp_Stream trans_fun xs c)\<^bsup>\<leftarrow>' c\<^esup> n)"
neq_funs_neq_images: fixes s :: "'d set"   and f1 :: "'d \<Rightarrow> 'e"   and f2 :: "'d \<Rightarrow> 'e" assumes "\<forall>x. x \<in> s \<longrightarrow> (\<forall>y. y \<in> s \<longrightarrow> f1 x \<noteq> f2 y)"   and "\<exists>x. x \<in> s" shows "f1 ` s \<noteq> f2 ` s"
assms_1: shows "f \<in> borel_measurable Invariants"