hcju/isabelleps huggingface dataset:
fract_poly_dvdD: fixes p :: "'a poly"   and q :: "'a poly" assumes "map_poly to_fract p dvd map_poly to_fract q"   and "content p = (1::'a)" shows "p dvd q"
trancl_def: shows "trancl \<equiv> \<lambda>r. _Collect (xa, x) ((\<lambda>x xa. (x, xa) \<in> r)\<^sup>+\<^sup>+ xa x)"
j_2: shows "\<forall>n<m\<^sub>0. s ((\<lambda>x. if x = Suc n\<^sub>0 then Some 1 else Some 0) \<triangleright> n) \<down>= 0"
CauchyI: fixes X :: "nat \<Rightarrow> 'b" assumes "\<And>e. 0 < e \<Longrightarrow> \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. norm (X m - X n) < e" shows "Cauchy X"
simps_2: fixes h :: "'b hp" shows "merge_pairs [h] = Some h"
split_minD: fixes t :: "'c tree23"   and x :: "'c"   and t' :: "'c upD" assumes "split_min t = (x, t')"   and "complete t"   and "0 < height t" shows "x # inorder (treeD t') = inorder t"
inf_fun_def: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" shows "inf f g = (\<lambda>x. inf (f x) (g x))"
run_shift_elim: fixes r :: "'action list"   and s :: "'action stream"   and p :: "'state" assumes "run (r @- s) p" obtains "path r p" and "run s (target r p)"
wf\<^sub>c\<^sub>o\<^sub>n\<^sub>s\<^sub>t\<^sub>r_def: fixes S :: "('c, 'd) strand_step list"   and \<theta> :: "'d \<Rightarrow> ('c, 'd) Term.term" shows "wf\<^sub>c\<^sub>o\<^sub>n\<^sub>s\<^sub>t\<^sub>r S \<theta> \<equiv> wf\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<theta> \<and> wf\<^sub>s\<^sub>t {} S \<and> subst_domain \<theta> \<inter> vars\<^sub>s\<^sub>t S = {} \<and> range_vars \<theta> \<inter> bvars\<^sub>s\<^sub>t S = {} \<and> fv\<^sub>s\<^sub>t S \<inter> bvars\<^sub>s\<^sub>t S = {}"
is_FL_formula_eqvt: fixes F :: "'effect fs_set"   and x :: "('a, 'pred, 'act, 'effect) formula"   and p :: "perm" assumes "x \<in> \<A>[F]" shows "p \<bullet> x \<in> \<A>[p \<bullet> F]"
Cons_1: fixes Ks :: "'a list" assumes "set Ks \<subseteq> K"   and "combine Ks Us = \<zero>"   and "local.independent K Us" shows "set (take (length Us) Ks) \<subseteq> {\<zero>}"
u_servTicket_authentic: fixes B :: "agent"   and A :: "agent"   and servK :: "nat"   and Ts :: "nat"   and evs :: "event list" assumes "Crypt (shrK B) \<lbrace>Agent A, Agent B, Key servK, Number Ts\<rbrace> \<in> parts (knows Spy evs)"   and "B \<noteq> Tgs"   and "B \<notin> bad"   and "evs \<in> kerbIV_gets" shows "\<exists>Ta authK. Says Kas A (Crypt (shrK A) \<lbrace>Key authK, Agent Tgs, Number Ta, Crypt (shrK Tgs) \<lbrace>Agent A, Agent Tgs, Key authK, Number Ta\<rbrace>\<rbrace>) \<in> set evs \<and> Says Tgs A (Crypt authK \<lbrace>Key servK, Agent B, Number Ts, Crypt (shrK B) \<lbrace>Agent A, Agent B, Key servK, Number Ts\<rbrace>\<rbrace>) \<in> set evs \<and> servKlife + Ts \<le> authKlife + Ta"
fps_shift_fls_regpart_conv_fls_shift: fixes m :: "nat"   and f :: "'a fls" shows "fps_shift m (fls_regpart f) = fls_regpart (fls_shift (int m) f)"
vec_nth_inverse: fixes x :: "('a, 'b) vec" shows "vec_lambda (($) x) = x"
B_def: shows "B = {y. \<exists>x. y = (x, x)}"
simps_19: fixes f1 :: "'b \<Rightarrow> 'a"   and f2 :: "'b \<Rightarrow> 'a"   and f3 :: "'b \<Rightarrow> 'b \<Rightarrow> 'a"   and f4 :: "'b \<Rightarrow> 'c \<Rightarrow> 'a"   and x31 :: "'b"   and x32 :: "'b" shows "(case Less x31 x32 of Fo x \<Rightarrow> f1 x | Z x \<Rightarrow> f2 x | Less x xa \<Rightarrow> f3 x xa | In x xa \<Rightarrow> f4 x xa) = f3 x31 x32"
bot_unique: fixes a :: "'c list" shows "prefix a [] = (a = [])"
nth_prime_partition': fixes x :: "real" assumes "2 \<le> x" shows "\<exists>k. x \<in> {real (nth_prime k)..<real (nth_prime (Suc k))}"
hyps_2: fixes f :: "'a \<Rightarrow> 'c list \<Rightarrow> 'c" shows "\<exists>p\<in>set (Neg (A[App n []/0]) # G). \<not> eval e f g p"
induct: fixes P :: "name list \<Rightarrow> nat \<Rightarrow> name \<Rightarrow> bool"   and a0 :: "name list"   and a1 :: "nat"   and a2 :: "name" assumes "\<And>n x. P [] n x"   and "\<And>y ys n x. (x \<noteq> y \<Longrightarrow> P ys (Suc n) x) \<Longrightarrow> P (y # ys) n x" shows "P a0 a1 a2"
is_singletonI: fixes x :: "'b" shows "is_singleton {x}"
is_inf_greatest: fixes x :: "'a"   and y :: "'a"   and i :: "'a"   and z :: "'a" assumes "is_inf x y i"   and "z \<sqsubseteq> x"   and "z \<sqsubseteq> y" shows "z \<sqsubseteq> i"
not_empty_cyc_free: fixes n :: "nat"   and v :: "'d \<Rightarrow> nat"   and M :: "nat \<Rightarrow> nat \<Rightarrow> 'e DBMEntry" assumes "\<forall>k\<le>n. 0 < k \<longrightarrow> (\<exists>c. v c = k)"   and "[M]\<^bsub>v,n\<^esub> \<noteq> {}" shows "\<forall>i xs. i \<le> n \<and> set xs \<subseteq> {0..n} \<longrightarrow> \<one> \<le> len M i i xs"
rCase: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and M :: "'a"   and N :: "'a"   and P' :: "('a, 'b, 'c) psi"   and \<phi> :: "'c"   and Cs :: "('c \<times> ('a, 'b, 'c) psi) list"   and C :: "'d" assumes "\<Psi> \<rhd> P \<longmapsto> M\<lparr>N\<rparr> \<prec> P'"   and "\<And>C. Prop C \<Psi> P M N P'"   and "(\<phi>, P) mem Cs"   and "\<Psi> \<turnstile> \<phi>"   and "guarded P" shows "Prop C \<Psi> (Cases Cs) M N P'"
uminus_mat_def: fixes A :: "'a mat" shows "- A \<equiv> mat (dim_row A) (dim_col A) (\<lambda>ij. - A $$ ij)"
artanh_real_has_field_derivative: fixes x :: "real"   and A :: "real set" assumes "\<bar>x\<bar> < 1" shows "(artanh has_real_derivative 1 / (1 - x\<^sup>2)) (at x within A)"
add_assocD_var2: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "D x y \<and> D (x \<oplus> y) z" shows "D y z \<and> D x (y \<oplus> z)"
simps: fixes a1 :: "pat"   and a2 :: "trm"   and a3 :: "trm list" shows "(\<turnstile> a1 \<rhd> a2 \<Rightarrow> a3) = ((\<exists>T t. a1 = PVar T \<and> a2 = t \<and> a3 = [t]) \<or> (\<exists>fps fs ts. a1 = PRcd fps \<and> a2 = Rcd fs \<and> a3 = ts \<and> \<turnstile> fps [\<rhd>] fs \<Rightarrow> ts))"
not_arr_null: shows "\<not> C.VxVxV.arr C.VxVxV.null"
more_than_one_mset_mset_diff: fixes a :: "'b"   and M :: "'b multiset" assumes "a \<in># remove1_mset a M" shows "set_mset (remove1_mset a M) = set_mset M"
wpo_stable: fixes s :: "('f, 'v) Term.term"   and t :: "('f, 'v) Term.term"   and \<delta> :: "'v \<Rightarrow> ('f, 'v) Term.term" shows "(s \<succ> t \<longrightarrow> s \<cdot> \<delta> \<succ> t \<cdot> \<delta>) \<and> (s \<succeq> t \<longrightarrow> s \<cdot> \<delta> \<succeq> t \<cdot> \<delta>)"
monom_mult_lookup_not_zero_subset_keys: fixes p :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd"   and f :: "'b \<Rightarrow> ('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd" shows "{t. MPoly_Type_Class.punit.monom_mult (lookup p t) 0 (subst_pp f t) \<noteq> 0} \<subseteq> keys p"
not_Nil_listrel1: fixes xs :: "'g list"   and r :: "('g \<times> 'g) set" shows "([], xs) \<notin> listrel1 r"
\<pi>_pos: fixes x :: "real" assumes "2 \<le> x" shows "0 < primes_pi x"
closed_seqs_diff_closed: fixes s :: "nat \<Rightarrow> nat \<Rightarrow> int"   and a :: "nat \<Rightarrow> int"   and m :: "nat" assumes "s \<in> carrier Zp\<^bsup>\<omega>\<^esup>"   and "a \<in> carrier Zp" shows "s m \<ominus> a \<in> carrier Zp"
assms_3: shows "\<pi> n \<noteq> \<pi> 0"
compact_diff: fixes S :: "'a set"   and T :: "'a set" assumes "compact S"   and "open T" shows "compact (S - T)"
l_inv: fixes x :: "'a \<Rightarrow> 'b" assumes "x \<in> carrier (DirProds G I)" shows "inv\<^bsub>DirProds G I\<^esub> x \<otimes>\<^bsub>DirProds G I\<^esub> x = \<one>\<^bsub>DirProds G I\<^esub>"
sumSym: fixes P :: "pi"   and Q :: "pi" shows "P \<oplus> Q \<sim>\<^sub>e Q \<oplus> P"
simps_2: fixes v :: "'b" shows "Polynomial_Expression.degree (v)\<^sub>p = 0"
no_infinite_computation_impl_terminates: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'a, 'c) com option"   and c :: "('b, 'a, 'c) com"   and s :: "('b, 'c) xstate" assumes "\<not> \<Gamma>\<turnstile> (c, s) \<rightarrow> \<dots>(\<infinity>)" shows "\<Gamma>\<turnstile>c \<down> s"
span_listI: fixes qs :: "('a \<Rightarrow>\<^sub>0 'c) list"   and bs :: "('t \<Rightarrow>\<^sub>0 'c) list" shows "sum_list (map2 (\<odot>) qs bs) \<in> pmdl (set bs)"
pth_1: fixes x :: "'a" shows "x \<equiv> 1 *\<^sub>R x"
T3: fixes w :: "i"   and F :: "\<o>" shows "[\<lparr>F\<rparr> in w] = (\<exists>r. Some r = d\<^sub>0 F \<and> ex0 r w)"
CondFalse: fixes s :: "'a"   and b :: "'a set"   and \<Gamma> :: "'b \<Rightarrow> ('a, 'b, 'c) com option"   and c\<^sub>2 :: "('a, 'b, 'c) com"   and t :: "('a, 'c) xstate"   and c\<^sub>1 :: "('a, 'b, 'c) com" assumes "s \<notin> b"   and "\<Gamma>\<turnstile> \<langle>c\<^sub>2,Normal s\<rangle> \<Rightarrow> t" shows "\<Gamma>\<turnstile> \<langle>Cond b c\<^sub>1 c\<^sub>2,Normal s\<rangle> \<Rightarrow> t"
delta: fixes a :: "'c"   and g :: "'c \<Rightarrow> 'b" shows "(\<Prod>b. if b = a then g b else (1::'b)) = g a"
intros_3: fixes dims :: "nat list"   and P :: "complex mat"   and Q :: "complex mat"   and R :: "complex mat"   and S1 :: "com"   and S2 :: "com" assumes "state_sig.is_quantum_predicate dims P"   and "state_sig.is_quantum_predicate dims Q"   and "state_sig.is_quantum_predicate dims R"   and "state_sig.hoare_partial dims P S1 Q"   and "state_sig.hoare_partial dims Q S2 R" shows "state_sig.hoare_partial dims P (S1;; S2) R"
assms: shows "affine_parallel (affine hull S) (affine hull T)"
mat_empty_column_length: fixes id :: "'b"   and f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and nr :: "nat"   and nc :: "nat"   and M :: "'c list list" assumes "Matrix_Tensor.mult id f"   and "mat nr nc M"   and "M = []" shows "nc = 0"
c_main: shows "c_main \<le> (\<Sum>jj = 0..<i. 2 * n + 2 + 4 * jj + 1)"
var_rename_inv_comp: fixes t :: "('a, 'b \<times> nat) Term.term"   and n :: "nat" shows "t \<cdot> var_rename n \<circ>\<^sub>s var_rename_inv n = t"
subcring_def: fixes H :: "'e set"   and R :: "('e, 'f) ring_scheme" shows "subcring H R \<equiv> subring H R \<and> subcring_axioms H R"
pos_poly_total: fixes p :: "'a poly" shows "p = 0 \<or> pos_poly p \<or> pos_poly (- p)"
simps_1: fixes t :: "nat" shows "valid_sops_stmt t Skip = True"
adjunction_in_bicategory_axioms: shows "adjunction_in_bicategory (\<cdot>) (\<star>) \<a> \<i> src trg f g \<eta> \<epsilon>"
distr_PiM_reindex: fixes K :: "'a set"   and M :: "'a \<Rightarrow> 'b measure"   and f :: "'c \<Rightarrow> 'a"   and I :: "'c set" assumes "\<And>i. i \<in> K \<Longrightarrow> prob_space (M i)"   and "inj_on f I"   and "f \<in> I \<rightarrow> K" shows "distr (Pi\<^sub>M K M) (Pi\<^sub>M I (\<lambda>i. M (f i))) (\<lambda>\<omega>. \<lambda>n\<in>I. \<omega> (f n)) = Pi\<^sub>M I (\<lambda>i. M (f i))"
enumdom_nonempty_domain_1: fixes i :: "nat"   and x\<^sub>0 :: "nat"   and y :: "nat" assumes "\<phi> i x\<^sub>0 \<down>" shows "eval r_enumdom [i, y] \<down>"
ind_add_str_def: fixes S :: "'c set"   and is_open :: "'c set \<Rightarrow> bool"   and \<FF> :: "'c set \<Rightarrow> 'd set"   and \<rho> :: "'c set \<Rightarrow> 'c set \<Rightarrow> 'd \<Rightarrow> 'd"   and b :: "'d"   and add_str :: "'c set \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and mult_str :: "'c set \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and zero_str :: "'c set \<Rightarrow> 'd"   and one_str :: "'c set \<Rightarrow> 'd"   and U :: "'c set"   and V :: "'c set" assumes "ind_sheaf S is_open \<FF> \<rho> b add_str mult_str zero_str one_str U" shows "ind_sheaf.ind_add_str add_str U V \<equiv> add_str (U \<inter> V)"
trans: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "r a b"   and "r b c" shows "r a c"
sum_list_simps_2: fixes x :: "'b"   and xs :: "'b list" shows "sum_list (x # xs) = x + sum_list xs"
eval_if0: fixes \<rho> :: "(nat \<times> bval) list"   and e1 :: "exp"   and e3 :: "exp"   and v3 :: "bval"   and e2 :: "exp" assumes "\<rho> \<turnstile> e1 \<Down> BNat 0"   and "\<rho> \<turnstile> e3 \<Down> v3" shows "\<rho> \<turnstile> EIf e1 e2 e3 \<Down> v3"
divide_cancel_right: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and a :: "'a"   and c :: "'a"   and b :: "'a" assumes "class.field divide inverse times one plus zero minus uminus" shows "(divide a c = divide b c) = (c = zero \<or> a = b)"
s_cf_pol_coeff: fixes p :: "'a" assumes "p \<in> carrier R" shows "pol_coeff S (s_cf R S X p)"
strong_induct: fixes x1 :: "Syntax.term list"   and x2 :: "Syntax.term"   and x3 :: "mode"   and x4 :: "nat"   and x5 :: "Syntax.term list"   and x6 :: "Syntax.term"   and P :: "'a \<Rightarrow> Syntax.term list \<Rightarrow> Syntax.term \<Rightarrow> mode \<Rightarrow> nat \<Rightarrow> Syntax.term list \<Rightarrow> Syntax.term \<Rightarrow> bool"   and c :: "'a" assumes "\<lless>x1, x2\<ggreater> x3\<rightarrow>x4 \<lless>x5, x6\<ggreater>"   and "\<And>\<pi> e uu_ c. P c \<pi> e uu_ 0 \<pi> e"   and "\<And>\<pi>\<^sub>1 e\<^sub>1 m i \<pi>\<^sub>2 e\<^sub>2 \<pi>\<^sub>3 e\<^sub>3 c. \<lbrakk>\<lless>\<pi>\<^sub>1, e\<^sub>1\<ggreater> m\<rightarrow>i \<lless>\<pi>\<^sub>2, e\<^sub>2\<ggreater>; \<And>c. P c \<pi>\<^sub>1 e\<^sub>1 m i \<pi>\<^sub>2 e\<^sub>2; \<lless>\<pi>\<^sub>2, e\<^sub>2\<ggreater> m\<rightarrow> \<lless>\<pi>\<^sub>3, e\<^sub>3\<ggreater>\<rbrakk> \<Longrightarrow> P c \<pi>\<^sub>1 e\<^sub>1 m (i + 1) \<pi>\<^sub>3 e\<^sub>3" shows "P c x1 x2 x3 x4 x5 x6"
abs_eq: fixes xa :: "('c, 'd) phi \<Rightarrow> 'c \<times> 'd Filters.filter"   and x :: "('c, 'd) phi \<Rightarrow> 'c \<times> 'd Filters.filter" assumes "eq_onp (\<lambda>pf. \<forall>f. pf f \<in> triple.pairs (Rep_phi f)) xa xa"   and "eq_onp (\<lambda>pf. \<forall>f. pf f \<in> triple.pairs (Rep_phi f)) x x" shows "Abs_lifted_pair xa \<sqinter> Abs_lifted_pair x = Abs_lifted_pair (\<lambda>f. triple.pairs_inf (xa f) (x f))"
extend_2: shows "[a] @- v\<^sub>2 =\<^sub>I l @- u"
all_envirD: fixes p :: "'d \<Rightarrow> bool"   and x :: "'d" assumes "all_envir p" shows "p x"
ev_stl_alt_def: fixes \<phi> :: "'b stream \<Rightarrow> bool"   and w :: "'b stream" shows "ev \<phi> (stl w) = (\<exists>u v. w = u @- v \<and> u \<noteq> [] \<and> \<phi> v)"
enum_obtain_index_finite: fixes x :: "nat"   and N :: "nat set" assumes "x \<in> N"   and "finite N" obtains i :: "nat" where "i < card N" and "x = enum N i"
F_entails_\<G>_iff: fixes N1 :: "'a literal multiset set"   and N2 :: "'a literal multiset set" shows "(N1 \<TTurnstile>\<G>e N2) = (\<Union> (\<G>_F ` N1) \<TTurnstile>e \<Union> (\<G>_F ` N2))"
simps: fixes xs :: "nat set"   and ps :: "('b, 'c) fm list"   and a :: "'c" shows "omit_block xs (ps, a) = (omit xs ps, a)"
valid_head_and_tail_valid_plan: fixes h :: "'a"   and PROB :: "'a set"   and as :: "'a list" assumes "h \<in> PROB"   and "as \<in> valid_plans PROB" shows "h # as \<in> valid_plans PROB"
comp_proj_stoch_proc_preimage: fixes X :: "nat \<Rightarrow> 'b \<Rightarrow> 'c"   and n :: "nat"   and x :: "'b" shows "proj_stoch_proc X n -` {proj_stoch_proc X n x} = (\<Inter>i\<in>{m. m \<le> n}. X i -` {X i x})"
weakly_bisimilar_is_weak_bisimulation: shows "is_weak_bisimulation (\<approx>\<cdot>)"
balanced_alt_def_all: shows "\<forall>ps\<subseteq>\<V>. int (card ps) = 2 \<longrightarrow> int (\<B> index ps) = \<Lambda>"
distinct_153: shows "GreaterOrEqual \<noteq> BinOr"
rGamma_series_def: fixes z :: "'a"   and n :: "nat" shows "rGamma_series z n = pochhammer z (n + 1) / (fact n * exp (z * of_real (ln (real n))))"
h2: shows "\<forall>qa. List.member [q] qa \<longrightarrow> coprime p qa"
COPY'_def: shows "COPY' \<equiv> fst (fix_syn (\<lambda>(COPY', SEND', REC'). (left`?`x \<rightarrow> right`!`x \<rightarrow> COPY', left`?`x \<rightarrow> mid`!`x \<rightarrow> ack \<rightarrow> SEND', mid`?`x \<rightarrow> right`!`x \<rightarrow> ack \<rightarrow> REC')))"
c_def: shows "(c\<^sub>0, c\<^sub>1) = (if dist (case p\<^sub>0\<^sub>L of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) (case p\<^sub>1\<^sub>L of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) < dist (case p\<^sub>0\<^sub>R of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) (case p\<^sub>1\<^sub>R of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) then (p\<^sub>0\<^sub>L, p\<^sub>1\<^sub>L) else (p\<^sub>0\<^sub>R, p\<^sub>1\<^sub>R))"
relevant_atoms_same_semantics: fixes F :: "'b formula"   and \<A>\<^sub>1 :: "'b \<Rightarrow> bool"   and \<A>\<^sub>2 :: "'b \<Rightarrow> bool" assumes "\<forall>k\<in>atoms F. \<A>\<^sub>1 k = \<A>\<^sub>2 k" shows "\<A>\<^sub>1 \<Turnstile> F = \<A>\<^sub>2 \<Turnstile> F"
spec3_assoc: fixes a :: "'b"   and b :: "'b"   and p\<^sub>1 :: "'b point"   and p\<^sub>2 :: "'b point"   and p\<^sub>3 :: "'b point" assumes "on_curve a b p\<^sub>1"   and "on_curve a b p\<^sub>2"   and "on_curve a b p\<^sub>3"   and "is_generic p\<^sub>1 p\<^sub>2"   and "is_tangent p\<^sub>2 p\<^sub>3"   and "is_generic (ell_field_class.add a p\<^sub>1 p\<^sub>2) p\<^sub>3"   and "is_tangent p\<^sub>1 (ell_field_class.add a p\<^sub>2 p\<^sub>3)" shows "ell_field_class.add a p\<^sub>1 (ell_field_class.add a p\<^sub>2 p\<^sub>3) = ell_field_class.add a (ell_field_class.add a p\<^sub>1 p\<^sub>2) p\<^sub>3"
SF_imp_WF: fixes A :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool"   and v :: "'a \<Rightarrow> 'b" shows "\<turnstile> SF(A)_v \<longrightarrow> WF(A)_v"
incl_in_def: fixes S :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<AA> :: "('c \<times> 'c) set"   and a :: "'b"   and b :: "'b" assumes "set_category S \<AA>" shows "set_category.incl_in S \<AA> a b = (partial_magma.ide S a \<and> partial_magma.ide S b \<and> set_category_data.set S (set_category.some_img S \<AA>) a \<subseteq> set_category_data.set S (set_category.some_img S \<AA>) b)"
subsumes_def: fixes t :: "('a, 'b) Term.term"   and s :: "('a, 'b) Term.term" shows "(t <\<cdot> s) = strict (\<le>\<cdot>) t s"
final_size: shows "size (\<Sum>p\<in>#mset_set (\<V> - {x}). {#p#} \<times># filter_mset ((\<in>) p) (filter_mset ((\<in>) x) \<B>)) = size (mset_set (\<V> - {x})) * nat \<Lambda>"
hiding_tickFree: fixes s :: "'a event list"   and A :: "'a set" shows "tickFree s = tickFree (trace_hide s (ev ` A))"
simps_3: fixes sys :: "'b \<Rightarrow> ('a, 'c) process_term"   and a :: "'a"   and T :: "('a, 'b) process_term" shows "PREF sys (ACT a T) = a"
cong_trans: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and euclidean_size :: "'c \<Rightarrow> nat"   and division_segment :: "'c \<Rightarrow> 'c"   and b :: "'c"   and c :: "'c"   and a :: "'c"   and d :: "'c" assumes "class.unique_euclidean_semiring divide plus minus zero times one modulo euclidean_size division_segment"   and "unique_euclidean_semiring.cong modulo b c a"   and "unique_euclidean_semiring.cong modulo c d a" shows "unique_euclidean_semiring.cong modulo b d a"
distinct_conv_nth: fixes xs :: "'a list" shows "distinct xs = (\<forall>i<|xs|. \<forall>j<|xs|. i \<noteq> j \<longrightarrow> xs ! i \<noteq> xs ! j)"
set_conv_nth: fixes xs :: "'e list" shows "set xs = {xs ! i |i. i < length xs}"
trans: fixes x :: "'alt pmf"   and y :: "'alt pmf"   and z :: "'alt pmf" assumes "Pareto (SD \<circ> R) x y"   and "Pareto (SD \<circ> R) y z" shows "Pareto (SD \<circ> R) x z"
less_degree_pdevs_of_list_imp_less_length: fixes i :: "nat"   and xs :: "'b list" assumes "i < degree (pdevs_of_list xs)" shows "i < length xs"
known_ptrs_def: shows "DocumentClass.known_ptrs \<equiv> l_known_ptrs\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t.a_known_ptrs DocumentClass.known_ptr"
sel_2: fixes x21 :: "('c, 'd) tree"   and x22 :: "'c"   and x23 :: "'d"   and x24 :: "('c, 'd) tree" shows "key (Node x21 x22 x23 x24) = x22"
fr2: shows "supp q \<sharp>* ([as \<inter> supp x]set. x)"
cmp_extend_def: fixes D :: "'c set"   and cmp :: "'c \<Rightarrow> 'c \<Rightarrow> comp_res" shows "cmp_extend D cmp \<equiv> cmp_combine D cmp UNIV univ_cmp"
intro: fixes hom :: "'c \<Rightarrow> 'd" assumes "comm_monoid_add_hom hom" shows "map_poly_comm_monoid_add_hom hom"
ctor_rec: fixes s :: "('d, 'd F \<times> 'c) F0 \<Rightarrow> 'c"   and x :: "('d, 'd F) F0" shows "ctor_rec_F s (ctor_F x) = s (map_pre_F id <id, ctor_rec_F s> x)"
Nml_Trg: fixes t :: "'a local.term" assumes "Nml t" shows "Nml (Trg t)"
total_one: fixes x :: "'a" assumes "is_total x" shows "x ; 1 = 1"
elims: fixes x :: "nat"   and y :: "nat" assumes "pred x = y" obtains "x = 0" and "y = 0"   | n :: "nat" where "x = Suc n" and "y = n"
bilinear_func: fixes f :: "'g \<times> 'h \<Rightarrow> 'i"   and M :: "('g, 'a, 'j) Module_scheme"   and N :: "('h, 'a, 'j) Module_scheme"   and Z :: "('i, 'a, 'j) Module_scheme" assumes "bilinear_map f R M N Z" shows "f \<in> M \<times>\<^sub>c N \<rightarrow> carrier Z"
Ostrowski_base_nonzero: fixes n :: "nat"   and vv :: "nat \<Rightarrow> 'a \<Rightarrow> ant"   and j :: "nat" assumes "vals_nonequiv K (Suc n) vv"   and "j \<le> Suc n" shows "(\<Omega>\<^bsub>K vv Suc n\<^esub>) j \<noteq> \<zero>"
weakening_almost_automatic: fixes \<Gamma>1 :: "(name \<times> ty) list"   and t :: "llam"   and T :: "ty"   and \<Gamma>2 :: "(name \<times> ty) list" assumes " \<Gamma>1 \<turnstile> t : T "   and "\<Gamma>1 \<subseteq> \<Gamma>2"   and "valid \<Gamma>2" shows " \<Gamma>2 \<turnstile> t : T "
C: shows "C \<in> sets borel \<and> negligible N \<and> C \<union> N = U"
s_Inj2: fixes \<pi> :: "Syntax.term list"   and e :: "Syntax.term"   and m :: "mode"   and \<pi>' :: "Syntax.term list"   and e' :: "Syntax.term" assumes "\<lless>\<pi>, e\<ggreater> m\<rightarrow> \<lless>\<pi>', e'\<ggreater>" shows "\<lless>\<pi>, Inj2 e\<ggreater> m\<rightarrow> \<lless>\<pi>', Inj2 e'\<ggreater>"
trace_callee_resource_of_oracle2: shows "trace_callee run_resource (map_spmf (resource_of_oracle callee) (return_spmf s)) xs x = trace_callee callee (return_spmf s) xs x"
in_set_butlastD: fixes x :: "'e"   and xs :: "'e list" assumes "x \<in> set (butlast xs)" shows "x \<in> set xs"
bind_returns_heap_E2: fixes h :: "'aq"   and f :: "('aq, 'ar, 'at) prog"   and g :: "'at \<Rightarrow> ('aq, 'ar, 'as) prog"   and h' :: "'aq" assumes "h \<turnstile> Heap_Error_Monad.bind f g \<rightarrow>\<^sub>h h'"   and "pure f h" obtains x :: "'at" where "h \<turnstile> f \<rightarrow>\<^sub>r x" and "h \<turnstile> g x \<rightarrow>\<^sub>h h'"
poly_map_poly_eval_poly: fixes p :: "rat poly" shows "rpoly p = eval_poly of_rat p"
simps_1: fixes r1 :: "'b rexp"   and r2 :: "'b rexp"   and s :: "'b rexp" shows "ACI_nPlus (Plus r1 r2) s = ACI_nPlus r1 (ACI_nPlus r2 s)"
transfer_eq: fixes x :: "'b star"   and X :: "nat \<Rightarrow> 'b"   and y :: "'b star"   and Y :: "nat \<Rightarrow> 'b" assumes "x \<equiv> star_n X"   and "y \<equiv> star_n Y" shows "x = y \<equiv> \<forall>\<^sub>F n in \<U>. X n = Y n"
rel_cases: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'a literal"   and b :: "'b literal" assumes "rel_literal R a b" obtains x :: "'a" and y :: "'b" where "a = Pos x" and "b = Pos y" and "R x y"   | xa :: "'a" and ya :: "'b" where "a = Neg xa" and "b = Neg ya" and "R xa ya"
l_is_min: shows "heap l = Min_mset (arr_mset heap l r)"
sAsA': fixes acts :: "(nat \<times> nat list) list"   and qs :: "'c list"   and init :: "'c list"   and n :: "nat" assumes "BIT_Off acts qs init"   and "n < length (BIT_Off.paid_A' acts)" shows "BIT_Off.s_A' acts qs init n = BIT_Off.s_A acts qs init n"
is_functor: shows "functor (\<cdot>) (\<cdot>) C.map"
h1: shows "set (map (undo_factorize_polys (snd (factorize_polys qs))) (find_sgas (map real_of_rat_poly (fst (factorize_polys qs))))) = consistent_sign_vectors qs UNIV"
basis_finmap_def: shows "basis_finmap = {Pi' I S |I S. finite I \<and> (\<forall>i\<in>I. S i \<in> basis_proj)}"
norm_insert: fixes x :: "'c \<times> 'd \<times> 'c"   and xs :: "('c \<times> 'd \<times> 'c) list" shows "norm_insert x xs = (x \<in> set xs, List.insert x xs)"
is_unit_Lcm_fin_iff: fixes A :: "'a set" shows "is_unit (Lcm\<^sub>f\<^sub>i\<^sub>n A) = (Lcm\<^sub>f\<^sub>i\<^sub>n A = (1::'a))"
preserves_cod: fixes f :: "'a" assumes "F.A.arr f" shows "F.B.cod (F f) = F (F.A.cod f)"