hcju/isabelleps huggingface dataset:
induct: fixes x :: "'g LDCFG_node \<times> ('h, 'i, 'j, 'k) edge_kind \<times> 'g LDCFG_node"   and get_return_edges :: "'l \<Rightarrow> 'l set"   and valid_edge :: "'l \<Rightarrow> bool"   and src :: "'l \<Rightarrow> 'g"   and trg :: "'l \<Rightarrow> 'g"   and knd :: "'l \<Rightarrow> ('h, 'i, 'j, 'k) edge_kind"   and e :: "'g LDCFG_node \<times> ('h, 'i, 'j, 'k) edge_kind \<times> 'g LDCFG_node"   and P :: "'g LDCFG_node \<times> ('h, 'i, 'j, 'k) edge_kind \<times> 'g LDCFG_node \<Rightarrow> bool" assumes "x \<in> lift_get_return_edges get_return_edges valid_edge src trg knd e"   and "\<And>a a' e'. \<lbrakk>e = (Node (src a), knd a, Node (trg a)); valid_edge a; a' \<in> get_return_edges a; e' = (Node (src a'), knd a', Node (trg a'))\<rbrakk> \<Longrightarrow> P e'" shows "P x"
ideduct_ik_eq: fixes M :: "('fun, 'var) Term.term set"   and M' :: "('fun, 'var) Term.term set"   and t :: "('fun, 'var) Term.term" assumes "\<forall>t\<in>M. M' \<turnstile> t" shows "(M' \<turnstile> t) = (M' \<union> M \<turnstile> t)"
normg: shows "1 \<le> \<parallel>local.g\<parallel>\<^sup>2"
square_free_multD_2: fixes f :: "'c poly"   and g :: "'c poly" assumes "square_free (f * g)" shows "square_free f"
pdg_vsubset_index_pdigraph_base: fixes J :: "V" assumes "J \<subseteq>\<^sub>\<circ> I" shows "pdigraph_base \<alpha> J \<AA>"
proper_part_irreflexivity: fixes x :: "i" assumes "MM3" shows "\<not> PP x x"
words_infI_construct: fixes w :: "'a stream"   and execute :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and enabled :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and p :: "'b" assumes "\<forall>v. v \<le>\<^sub>F\<^sub>I w \<longrightarrow> transition_system.path execute enabled v p" shows "transition_system.run execute enabled w p"
absorb_iff2: fixes a :: "'t"   and b :: "'t" shows "(a \<preceq>\<^sub>t b) = (ord_term_lin.max a b = b)"
hcomp_assoc: fixes \<mu> :: "'a"   and \<nu> :: "'a"   and \<tau> :: "'a" shows "(\<mu> \<star> \<nu>) \<star> \<tau> = \<mu> \<star> \<nu> \<star> \<tau>"
degree_map_poly_hom: fixes p :: "'a poly" shows "Polynomial.degree (map_poly (\<lambda>a. [:a:]) p) = Polynomial.degree p"
a': shows "( \<Gamma> \<Rightarrow>* Em, n) \<in> derivable R*"
Suc: shows "g\<cdot>(iterate i\<cdot>f\<cdot>\<bottom>) = iterate i\<cdot>h\<cdot>\<bottom>"
interv_is_compl_latt: fixes cl :: "'b potype"   and f :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "CLF cl f"   and "a \<in> pset cl"   and "b \<in> pset cl"   and "interval (order cl) a b \<noteq> {}" shows "\<lparr>pset = interval (order cl) a b, order = induced (interval (order cl) a b) (order cl)\<rparr> \<in> CompleteLattice"
valid_path_join: fixes g1 :: "real \<Rightarrow> 'a"   and g2 :: "real \<Rightarrow> 'a" assumes "valid_path g1"   and "valid_path g2"   and "pathfinish g1 = pathstart g2" shows "valid_path (g1 +++ g2)"
neutral_right: fixes f :: "'a \<Rightarrow> 'b" assumes "f \<in> (\<lambda>f. restrict f A) ` S - Pi\<^sub>E A B" shows "(\<Prod>xa\<in>A. pmf (p xa) (f xa)) = 0"
error_pure: fixes e :: "'y"   and h :: "'x" shows "pure (error e) h"
assms: shows "wf\<^sub>s\<^sub>t\<^sub>s' \<S> \<A>"
restrict_nonempty_right: fixes ks :: "'c list"   and f :: "'c \<times> 'c \<Rightarrow> 'd"   and l :: "'c"   and ls :: "'c list" shows "ks \<langle>f\<rangle> (l # ls) = ks \<langle>f\<rangle> [l] \<oplus> ks \<langle>f\<rangle> ls"
tts_AB_transfer: fixes f :: "'e \<Rightarrow> 'd"   and \<UU>A :: "'e set"   and \<UU>B :: "'d set"   and A :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and B :: "'d \<Rightarrow> 'g \<Rightarrow> bool" assumes "f ` \<UU>A \<subseteq> \<UU>B"   and "Domainp A = (\<lambda>x. x \<in> \<UU>A)"   and "bi_unique A"   and "right_total A"   and "Domainp B = (\<lambda>x. x \<in> \<UU>B)"   and "bi_unique B"   and "right_total B" shows "\<exists>rcdt. rel_fun A B f rcdt"
REAL_E: fixes v :: "val" assumes "val_type v = REAL" obtains b :: "real" where "v = RealVal b"
add_mset_lt_right_lt: fixes A :: "'a multiset"   and B :: "'a multiset"   and a :: "'a" assumes "A < B" shows "add_mset a A < add_mset a B"
iprev_nth_iMin_fix: fixes I :: "nat set"   and a :: "nat"   and b :: "nat" assumes "I \<noteq> {}"   and "I \<leftarrow> a = iMin I"   and "a \<le> b" shows "I \<leftarrow> b = iMin I"
rel_interior_inter_aux: fixes I :: "'a set set" shows "\<Inter> {rel_interior S |S. S \<in> I} \<subseteq> \<Inter> I"
gen_dfs_pre_imp_fin: fixes E :: "('a \<times> 'a) set"   and U :: "'a set"   and S :: "'a set"   and V :: "'a set"   and u0 :: "'a" assumes "gen_dfs_pre E U S V u0" shows "finite (E `` {u0})"
invariant_1D_4: fixes plr :: "int poly \<times> rat \<times> rat" assumes "invariant_1 plr" shows "\<exists>!x. root_cond plr x"
cil_length: fixes w :: "'a list"   and wa :: "'a list"   and \<alpha> :: "'a \<Rightarrow> 'b set \<times> 'b set"   and wb :: "'a list" assumes "w \<in> wa \<otimes>\<^bsub>\<alpha>\<^esub> wb" shows "length w = length wa + length wb"
div_0: 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 a :: "'c" assumes "class.semidom_divide divide plus minus zero times one" shows "divide zero a = zero"
bi_unique_rel_gpv: fixes A :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and B :: "'f \<Rightarrow> 'g \<Rightarrow> bool" assumes "bi_unique A"   and "bi_unique B" shows "bi_unique (rel_gpv A B)"
cases: fixes a1 :: "('s, 'v) tm"   and a2 :: "('s, 'v) tm" assumes "a1 >\<^sub>t a2" obtains t :: "('s, 'v) tm" and s :: "('s, 'v) tm" where "a1 = tb__" and "a2 = sb__" and "vars_mset sb__ \<subseteq># vars_mset tb__" and "wt sb__ < wt tb__"   | t :: "('s, 'v) tm" and s :: "('s, 'v) tm" where "a1 = tc__" and "a2 = sc__" and "vars_mset sc__ \<subseteq># vars_mset tc__" and "wt tc__ = wt sc__" and "head tc__ >\<^sub>h\<^sub>d head sc__"   | t :: "('s, 'v) tm" and s :: "('s, 'v) tm" where "a1 = t" and "a2 = s" and "vars_mset s \<subseteq># vars_mset t" and "wt t = wt s" and "head t = head s" and "\<forall>f\<in>local.ground_heads (head s). extf f (>\<^sub>t) (args t) (args s)"
diff_union_swap2: fixes y :: "'c"   and M :: "'c multiset"   and x :: "'c" assumes "y \<in># M" shows "add_mset x M - {#y#} = add_mset x (M - {#y#})"
Collect_mono_iff: fixes P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool" shows "(Collect P \<subseteq> Collect Q) = (\<forall>x. P x \<longrightarrow> Q x)"
bit_equality: shows "((ip >> 24) && 255 << 24) + ((ip >> 16) && 255 << 16) + ((ip >> 8) && 255 << 8) + (ip && 255) = of_bl (take 8 (to_bl ip) @ take 8 (drop 8 (to_bl ip)) @ take 8 (drop 16 (to_bl ip)) @ drop 24 (to_bl ip))"
proposition3_4_step: fixes r :: "('a \<times> 'a) set"   and \<beta> :: "'a"   and \<alpha> :: "'a"   and \<sigma>1 :: "'a list"   and \<sigma>2 :: "'a list"   and \<sigma>3 :: "'a list" assumes "trans r"   and "irrefl r"   and "LD_1' r \<beta> \<alpha> \<sigma>1 \<sigma>2 \<sigma>3" shows "(r|\<sigma>1 @ \<sigma>2 @ \<sigma>3| -s r \<down>m {#\<beta>#}, r|[\<alpha>]|) \<in> mul_eq r"
select_convs_2: fixes access :: "acc_modi"   and more :: "'a" shows "decl.more \<lparr>access = access, \<dots> = more\<rparr> = more"
r_zero: fixes x :: "complex vec" assumes "x \<in> carrier_vec dA" shows "x + 0\<^sub>v dA = x"
extremum_uniqueI: fixes a :: "'d" assumes "top \<le> a" shows "a = top"
atLeastatMost_subset_iff: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b"   and c :: "'b"   and d :: "'b" assumes "class.preorder less_eq less" shows "(ord.atLeastAtMost less_eq a b \<subseteq> ord.atLeastAtMost less_eq c d) = (\<not> less_eq a b \<or> less_eq c a \<and> less_eq b d)"
induct: fixes P :: "nat \<Rightarrow> bool"   and a0 :: "nat" assumes "P 0"   and "P (Suc 0)"   and "\<And>va. P (Suc (Suc va) div 2) \<Longrightarrow> P (Suc (Suc va))" shows "P a0"
t1: shows "t1 \<in> existence_ivl t0 x0"
fund_circuit_inI: fixes \<E> :: "'a set"   and B :: "'a set"   and x :: "'a"   and C :: "'a set" assumes "\<E> \<subseteq> \<E>"   and "\<E>.basis_in \<E> B"   and "x \<in> \<E> - B"   and "\<E>.circuit_in \<E> C"   and "C \<subseteq> insert x B" shows "\<E>.fund_circuit_in \<E> x B = C"
convolution_symm_eq_sum_over_supp_left: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "f \<in> aezfun_set" shows "convolution f g = (\<lambda>x. \<Sum>y\<in>supp f. f y * g (- y + x))"
cbelow_on_second_in: fixes A :: "'b set"   and ord :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and z :: "'b"   and rel :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b" assumes "cbelow_on A ord z rel a b" shows "b \<in> A"
rec_dummyfac_def: shows "rec_dummyfac = Pr 1 (constn 1) (Cn 3 rec_mult [recf.id 3 2, Cn 3 s [recf.id 3 1]])"
sing_pow_lists: fixes a :: "'b"   and A :: "'b set"   and n :: "nat" assumes "a \<in> A" shows "[a] \<^sup>@ n \<in> lists A"
llength_def: shows "LList2.llength \<equiv> finlsts_rec 0 (\<lambda>a r. Suc)"
sim: shows "\<forall>P Q P'. P \<lesssim>\<lbrakk>\<cdot>\<rbrakk>RT<TRel> Q \<and> P \<longmapsto>STCal Source Target P' \<longrightarrow> (\<exists>Q'. Q \<longmapsto>STCal Source Target Q' \<and> P' \<lesssim>\<lbrakk>\<cdot>\<rbrakk>RT<TRel> Q')"
is_rflow_def: fixes N :: "nat"   and f :: "nat \<times> nat \<Rightarrow> int"   and cfi :: "int Heap.array" shows "is_rflow N f cfi \<equiv> \<exists>\<^sub>Acf. asmtx_assn N int_assn cf cfi * \<up> (RGraph c s t cf \<and> f = flow_of_cf cf)"
intros_6: fixes \<Gamma> :: "'b formula set"   and F :: "'b formula"   and G :: "'b formula" assumes "\<Gamma> \<turnstile> F \<^bold>\<and> G" shows "\<Gamma> \<turnstile> G"
disc_iff_2: fixes P :: "'c \<Rightarrow> bool"   and xs :: "'c llist" shows "(\<not> lnull (ltakeWhile P xs)) = (\<not> lnull xs \<and> P (lhd xs))"
induct: fixes P :: "nat \<Rightarrow> Core.term \<Rightarrow> bool"   and a0 :: "nat"   and a1 :: "Core.term" assumes "\<And>lev i. P lev (Bv i)"   and "\<And>lev T t. P (lev + 1) t \<Longrightarrow> P lev (Abs T t)"   and "\<And>lev t u. \<lbrakk>P lev t; P lev u\<rbrakk> \<Longrightarrow> P lev (t $ u)"   and "\<And>uu_ v va. P uu_ (Ct v va)"   and "\<And>uu_ v va. P uu_ (Fv v va)" shows "P a0 a1"
separation_t2: fixes "open" :: "'b set \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "class.t2_space open" shows "(x \<noteq> y) = (\<exists>U V. open U \<and> open V \<and> x \<in> U \<and> y \<in> V \<and> U \<inter> V = {})"
assms_1: shows "i < 2 ^ n \<or> j < 2 ^ n"
compose_actions_def: fixes fs :: "((predicate \<times> object list \<Rightarrow> bool) \<Rightarrow> (predicate \<times> object list \<Rightarrow> bool) option) list" shows "compose_actions fs \<equiv> fold compose_action fs Some"
word_diff_ls''_3: fixes y :: "'b word"   and x :: "'b word"   and w :: "'b word" assumes "y - x \<le> w + x - x"   and "x \<le> w + x" shows "y \<le> w + x"
greaterThanLessThan_eq_iff: fixes r :: "real"   and s :: "real"   and t :: "real"   and u :: "real" shows "({r<..<s} = {t<..<u}) = (s \<le> r \<and> u \<le> t \<or> r = t \<and> s = u)"
user_reg_low_equal: fixes s1 :: "'b sparc_state"   and s2 :: "'b sparc_state" assumes "low_equal s1 s2" shows "user_reg s1 = user_reg s2"
strand_sem_stateful_induct: fixes P :: "('fun, 'var) Term.term set \<Rightarrow> (('fun, 'var) Term.term \<times> ('fun, 'var) Term.term) set \<Rightarrow> ('fun, 'var) stateful_strand_step list \<Rightarrow> bool"   and a0 :: "('fun, 'var) Term.term set"   and a1 :: "(('fun, 'var) Term.term \<times> ('fun, 'var) Term.term) set"   and a2 :: "('fun, 'var) stateful_strand_step list" assumes "\<And>M D. P M D []"   and "\<And>M D t S. (\<And>x. P M D S) \<Longrightarrow> P M D (send\<langle>t\<rangle> # S)"   and "\<And>M D t S. (\<And>x. P (insert (t \<cdot> x) M) D S) \<Longrightarrow> P M D (receive\<langle>t\<rangle> # S)"   and "\<And>M D uu_ t t' S. (\<And>x. P M D S) \<Longrightarrow> P M D (\<langle>uu_: t \<doteq> t'\<rangle> # S)"   and "\<And>M D t s S. (\<And>x. P M (insert (t \<cdot> x, s \<cdot> x) D) S) \<Longrightarrow> P M D (insert\<langle>t,s\<rangle> # S)"   and "\<And>M D t s S. (\<And>x. P M (D - {(t \<cdot> x, s \<cdot> x)}) S) \<Longrightarrow> P M D (delete\<langle>t,s\<rangle> # S)"   and "\<And>M D uv_ t s S. (\<And>x. P M D S) \<Longrightarrow> P M D (\<langle>uv_: t \<in> s\<rangle> # S)"   and "\<And>M D X F F' S. (\<And>x. P M D S) \<Longrightarrow> P M D (\<forall>X\<langle>\<or>\<noteq>: F \<or>\<notin>: F'\<rangle> # S)" shows "P a0 a1 a2"
one_power2: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b" assumes "class.semiring_1 one times plus zero" shows "power.power one times one 2 = one"
simps_1: fixes uu :: "'a" shows "is_Inl (Inl uu) = True"
\<eta>o_def: fixes y :: "'d" shows "\<eta>o y = \<phi> y (F y)"
simps_1: fixes p :: "nat"   and a :: "nat"   and r :: "nat" shows "build_fpc p a r [] = [Triple p a r]"
subsumes_trans: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "subsumes a b"   and "subsumes b c" shows "subsumes a c"
split_asm: fixes P :: "'c \<Rightarrow> bool"   and f1 :: "'c"   and f2 :: "'d \<Rightarrow> 'd \<Rightarrow> 'c"   and point :: "'d point" shows "P (case point of Infinity \<Rightarrow> f1 | Point x xa \<Rightarrow> f2 x xa) = (\<not> (point = Infinity \<and> \<not> P f1 \<or> (\<exists>x21 x22. point = Point x21 x22 \<and> \<not> P (f2 x21 x22))))"
deg_h: shows "poly_deg h = poly_deg s + poly_deg h0"
eval_poly_poly: fixes p :: "int poly"   and x :: "int" shows "eval_poly of_int p (of_int x) = of_int (poly p x)"
WT_parallel_resource1_wiring: fixes \<I>1 :: "('i, 'l) \<I>"   and \<I>2 :: "('j, 'm) \<I>"   and \<I>3 :: "('k, 'n) \<I>" shows "\<I>1 \<oplus>\<^sub>\<I> (\<I>2 \<oplus>\<^sub>\<I> \<I>3), \<I>2 \<oplus>\<^sub>\<I> (\<I>1 \<oplus>\<^sub>\<I> \<I>3) \<turnstile>\<^sub>C parallel_resource1_wiring \<surd>"
wp1_aux1aa: fixes p :: "('a, 'b) Combinators list" assumes "wellformed_policy1_strong p" shows "DenyAll \<in> set p"
distrib_lattice_axioms_def: fixes inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" shows "class.distrib_lattice_axioms inf sup \<equiv> \<forall>x y z. sup x (inf y z) = inf (sup x y) (sup x z)"
transitivity_L: fixes A :: "'a set" shows "Cn (Cn A) \<subseteq> Cn A"
hom_0: fixes x :: "'b" assumes "[:x:] = 0" shows "x = (0::'b)"
comm_semiring_ow_axioms: shows "comm_semiring_ow U (+\<^sub>o\<^sub>w) (*\<^sub>o\<^sub>w)"
split_cong: fixes q :: "'a \<times> 'b"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and g :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and p :: "'a \<times> 'b" assumes "\<And>x y. (x, y) = q \<Longrightarrow> f x y = g x y"   and "p = q" shows "(case p of (x, xa) \<Rightarrow> f x xa) = (case q of (x, xa) \<Rightarrow> g x xa)"
case_prod_beta: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'a"   and p :: "'b \<times> 'c" shows "(case p of (x, xa) \<Rightarrow> f x xa) = f (fst p) (snd p)"
vec_setE: fixes a :: "'b"   and v :: "'b vec" assumes "a \<in> set\<^sub>v v" obtains i :: "nat" where "v $ i = a" and "i < dim_vec v"
tfr\<^sub>s\<^sub>t_if_comp_tfr\<^sub>s\<^sub>t: fixes M :: "('fun, ('fun, 'atom) Term.term \<times> nat) Term.term list"   and S :: "('fun, ('fun, 'atom) Term.term \<times> nat) strand_step list" assumes "comp_tfr\<^sub>s\<^sub>t arity Ana \<Gamma> M S" shows "tfr\<^sub>s\<^sub>t S"
closed_orbitE': fixes x :: "'a" assumes "closed_orbit x" obtains T :: "real" where "0 < T" and "\<And>t x. flow0 x (t + T * real x) = flow0 x t"
orderI: fixes a :: "'b \<times> 'c"   and b :: "'b \<times> 'c" assumes "a = sup_prod a b" shows "less_eq_prod b a"
wf_mstate_minit: fixes \<phi> :: "Formula.formula"   and R :: "event_data list set" assumes "safe_formula \<phi>" shows "wf_mstate \<phi> pnil R (minit \<phi>)"
swaprows_mat: fixes A :: "'c mat"   and n :: "nat"   and nc :: "nat"   and k :: "nat"   and l :: "nat" assumes "A \<in> carrier_mat n nc"   and "k < n"   and "l < n" shows "swaprows k l A = swaprows_mat n k l * A"
Rep_Exp_inverse: fixes x :: "exp" shows "Abs_Exp (Rep_Exp x) = x"
F_ss: fixes G :: "('a, 'b) gb_graph_rec_scheme" assumes "gb_graph G" shows "gbg_F G \<subseteq> Pow (g_V G)"
rep_eq: fixes x :: "String.literal" shows "literal.explode (upChar_literal x) = upChar (literal.explode x)"
MGFnD: fixes G :: "prog"   and A :: "state triple set"   and n :: "nat"   and t :: "Term.term"   and P :: "state \<Rightarrow> bool" assumes "G,A\<turnstile>{=:n} t\<succ> {G\<rightarrow>}" shows "G,A\<turnstile>{(\<lambda>Y' s' s. s' = s \<and> P s) \<and>. G\<turnstile>init\<le>n} t\<succ> {(\<lambda>Y' s' s. G\<turnstile>s \<midarrow>t\<succ>\<rightarrow> (Y', s') \<and> P s) \<and>. G\<turnstile>init\<le>n}"
restrict_count_space_subset: fixes A :: "'b set"   and B :: "'b set" assumes "A \<subseteq> B" shows "restrict_space (count_space B) A = count_space A"
inject: fixes x21 :: "'d"   and x22 :: "'d list"   and y21 :: "'d"   and y22 :: "'d list" shows "(x21 # x22 = y21 # y22) = (x21 = y21 \<and> x22 = y22)"
mult_strict_mono: fixes a :: "'b"   and b :: "'b"   and c :: "'b"   and d :: "'b" assumes "a < b"   and "c < d"   and "(0::'b) < b"   and "(0::'b) \<le> c" shows "a * c < b * d"
finer_ex: fixes I0 :: "'a set"   and P :: "'a set set"   and Q :: "'a set set"   and I :: "'a set" assumes "part I0 P"   and "part I0 Q"   and "finer P Q"   and "I \<in> P" shows "\<exists>II. II \<in> Q \<and> I \<subseteq> II"
w\<^sub>g_is_map: shows "is_left_adjoint w\<^sub>g"
lgc_invar_infinity: fixes TNs :: "('f inference set \<times> ('f \<times> 'l) set) llist" assumes "\<not> lnull TNs"   and "\<forall>i. enat i < llength TNs \<longrightarrow> lgc_invar TNs (enat i)" shows "lgc_invar TNs \<infinity>"
scnj: fixes F :: "'fmla set" assumes "F \<subseteq> fmla"   and "finite F" shows "scnj F \<in> fmla"
nsqn\<^sub>r_addpreRT_inv: fixes dip :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option"   and npre :: "nat set"   and dip' :: "nat" assumes "dip \<in> kD rt" shows "nsqn\<^sub>r (the (the (addpreRT rt dip npre) dip')) = nsqn\<^sub>r (the (rt dip'))"
that: fixes ys :: "'b list" assumes "(xs, ys) \<in> \<langle>R\<rangle>list_rel" shows "thesis"
Max_ge: fixes A :: "'a Word.word set"   and x :: "'a Word.word" assumes "finite A"   and "x \<in> A" shows "x \<le>s signed.Max A"
a2: shows " M2 \<longrightarrow>\<^sub>1\<^sup>* M3"
bound_insert: fixes x :: "'c"   and X :: "'c set"   and r :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and b :: "'d" shows "bound (insert x X) r b = (r x b \<and> bound X r b)"
fls_lr_inverse_unique_ring1_1: fixes f :: "'a fls"   and g :: "'a fls" assumes "f * g = 1"   and "fls_nth g (fls_subdegree g) * fls_nth f (fls_subdegree f) = (1::'a)" shows "fls_left_inverse g (fls_nth f (fls_subdegree f)) = f"
CSHOnextConfig_def: fixes A :: "('a, 'b, 'c, 'd) CHOAlgorithm_scheme"   and r :: "nat"   and cfg :: "'a \<Rightarrow> 'b"   and HO :: "'a \<Rightarrow> 'a set"   and SHO :: "'a \<Rightarrow> 'a set"   and coord :: "'a \<Rightarrow> 'a"   and cfg' :: "'a \<Rightarrow> 'b" shows "CSHOnextConfig A r cfg HO SHO coord cfg' \<equiv> \<forall>p. \<exists>\<mu>\<in>SHOmsgVectors A r p cfg (HO p) (SHO p). CnextState A r p (cfg p) \<mu> (coord p) (cfg' p)"
set_iterate: fixes m :: "nat"   and n :: "nat"   and f :: "'b \<Rightarrow> 'b"   and x :: "'b" shows "set (iterate m n f x) = (\<lambda>k. (f ^^ k) x) ` {m..<n}"
mono2mono_ltake2: fixes orda :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and t :: "'c \<Rightarrow> 'd llist"   and n :: "enat" assumes "monotone orda (\<sqsubseteq>) t" shows "monotone orda (\<sqsubseteq>) (\<lambda>x. ltake n (t x))"
natToVertexList_length: fixes es :: "nat list"   and f :: "face"   and v :: "nat" assumes "incrIndexList es |es| |vertices f|" shows "|natToVertexList v f es| = |es|"
FDOM_pre_subset_prob_dom_pair: fixes a :: "('a, 'b) fmap \<times> ('a, 'c) fmap"   and PROB :: "(('a, 'b) fmap \<times> ('a, 'c) fmap) set" assumes "a \<in> PROB" shows "fmdom' (fst a) \<subseteq> prob_dom PROB"
min_int_poly_of_rat: fixes r :: "rat" shows "min_int_poly (of_rat r) = poly_rat r"
Liminf_least: fixes F :: "'b filter"   and f :: "'b \<Rightarrow> 'd"   and x :: "'d" assumes "\<And>P. eventually P F \<Longrightarrow> Inf (f ` Collect P) \<le> x" shows "Liminf F f \<le> x"
intros: fixes q :: "'a"   and l :: "'b"   and qs :: "'a list"   and \<delta> :: "('a, 'b) ta_rule set" assumes "q \<rightarrow> l qs \<in> \<delta>"   and "\<And>x. x \<in> set qs \<Longrightarrow> x \<in> b_accessible \<delta>" shows "q \<in> b_accessible \<delta>"
LLL_measure_approx: fixes fs :: "int vec list"   and i :: "nat" assumes "LLL_invariant_weak fs"   and "g_bound fs"   and "4 / 3 < \<alpha>"   and "m \<noteq> 0" shows "real (LLL_measure i fs) \<le> real m + real (2 * m * m) * log base (real N)"
distinct_43: fixes psiCase :: "('d, 'e, 'f) psiCase"   and psi' :: "('d, 'e, 'f) psi" shows "Case psiCase \<noteq> !psi'"
add_inv_exists_mat: fixes A :: "'b mat"   and nr :: "nat"   and nc :: "nat" assumes "A \<in> carrier_mat nr nc" shows "\<exists>B\<in>carrier_mat nr nc. B + A = 0\<^sub>m nr nc \<and> A + B = 0\<^sub>m nr nc"
card_UNIV_set: shows "CARD('a set) = (if CARD('a) = 0 then 0 else 2 ^ CARD('a))"
one_poly_eq_simps_2: shows "([:1::'b:] = 1) = True"
lens_comp_def: fixes Y :: "'d \<Longrightarrow> 'f"   and X :: "'f \<Longrightarrow> 'e" shows "Y ;\<^sub>L X = \<lparr>lens_get = get\<^bsub>Y\<^esub> \<circ> get\<^bsub>X\<^esub>, lens_put = \<lambda>\<sigma> v. put\<^bsub>X\<^esub> \<sigma> (put\<^bsub>Y\<^esub> (get\<^bsub>X\<^esub> \<sigma>) v)\<rparr>"
hcomp_can: fixes t :: "'a E.term"   and u :: "'a E.term"   and v :: "'a E.term"   and w :: "'a E.term" assumes "E.Ide t"   and "E.Ide u"   and "E.Ide v"   and "E.Ide w"   and "\<^bold>\<lfloor>t\<^bold>\<rfloor> = \<^bold>\<lfloor>u\<^bold>\<rfloor>"   and "\<^bold>\<lfloor>v\<^bold>\<rfloor> = \<^bold>\<lfloor>w\<^bold>\<rfloor>"   and "E.Src t = E.Trg v"   and "E.Src u = E.Trg w" shows "B.can u t \<star>\<^sub>B B.can w v = B.can (u \<^bold>\<star> w) (t \<^bold>\<star> v)"
eq_imp_le: fixes m :: "nat"   and n :: "nat" assumes "(m::nat) = (n::nat)" shows "(m::nat) \<le> (n::nat)"
card_partition_on_size1_eq_0: fixes A :: "'a set"   and k :: "nat" assumes "finite A"   and "k < card A" shows "card {P. partition_on A P \<and> card P \<le> k \<and> (\<forall>X\<in>P. card X = 1)} = 0"
simps_5: fixes insts :: "((variable \<times> typ) \<times> Core.term) list"   and t :: "Core.term"   and u :: "Core.term" shows "subst_term insts (t $ u) = subst_term insts t $ subst_term insts u"
d1_a': fixes x :: "'a" shows "d x \<cdot> x = x"
adjugate_def: fixes A :: "(('b, 'c) vec, 'c) vec" shows "adjugate A = Finite_Cartesian_Product.transpose (cofactorM A)"
dom_char: fixes f :: "'a \<times> 'a" shows "B.VV.dom f = (if B.VV.arr f then B.VxV.dom f else B.VxV.null)"
id: fixes f :: "nat \<Rightarrow> 'a" shows "sum f {0..<Suc n} = f n + sum f {0..<n}"
simps_6: fixes p :: "('a, 'b) form"   and q :: "('a, 'b) form" shows "FOL_Fitting.size_form (Impl p q) = FOL_Fitting.size_form p + FOL_Fitting.size_form q + 1"
tna_def: fixes z :: "ant" shows "tna z = (if Rep_Ainteg z \<noteq> (0, 1) \<and> Rep_Ainteg z \<noteq> (0, - 1) then fst (Rep_Ainteg z) else undefined)"
subprob_space_axioms: fixes p :: "'a spmf" shows "subprob_space (measure_spmf p)"
arr_cod_iff_arr: fixes f :: "'d" shows "Dop.arr (Dop.cod f) = Dop.arr f"
antisym: fixes x :: "'a"   and y :: "'a" assumes "x \<sqsubseteq> y"   and "y \<sqsubseteq> x"   and "x \<in> P"   and "y \<in> P" shows "x = y"
exists_bezout_ext: fixes 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" assumes "class.bezout_ring times one plus zero minus uminus" shows "\<exists>bezout_ext. bezout_ring.is_bezout_ext times plus uminus bezout_ext"