hcju/isabelleps huggingface dataset:
s_Case_inv: fixes \<pi> :: "Syntax.term list"   and e :: "Syntax.term"   and e\<^sub>1 :: "Syntax.term"   and e\<^sub>2 :: "Syntax.term"   and m :: "mode"   and \<pi>' :: "Syntax.term list"   and e' :: "Syntax.term" assumes "\<lless>\<pi>, Case e e\<^sub>1 e\<^sub>2\<ggreater> m\<rightarrow> \<lless>\<pi>', e'\<ggreater>" obtains e'a :: "Syntax.term" where "e' = Case e'a e\<^sub>1 e\<^sub>2" and "\<lless>\<pi>, e\<ggreater> m\<rightarrow> \<lless>\<pi>', e'a\<ggreater>"   | v :: "Syntax.term" where "e = Inj1 v__" and "\<pi>' = \<pi>" and "e' = App e\<^sub>1 v__" and "value v__"   | v :: "Syntax.term" where "e = Inj2 v" and "\<pi>' = \<pi>" and "e' = App e\<^sub>2 v" and "value v"
acc_lengths_shift: fixes acc :: "nat"   and as :: "'b list list" assumes "NO_MATCH (0::'a) acc" shows "acc_lengths acc as = map ((+) acc) (acc_lengths 0 as)"
mult_vec_distrib_right: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c"   and r :: "'t \<Rightarrow>\<^sub>0 'c" shows "(p + q) ** r = p ** r + q ** r"
bij_betw_msubset_of: fixes A :: "'c set"   and B :: "'d set" assumes "finite A" shows "bij_betw (msubset_of A) ((A \<rightarrow>\<^sub>E B) // domain_permutation A B) {M. set_mset M \<subseteq> B \<and> size M = card A}"
boundedE: fixes b :: "'a multiset"   and c :: "'a multiset"   and a :: "'a multiset" assumes "b \<union># c \<subseteq># a" obtains "b \<subseteq># a" and "c \<subseteq># a"
elims: fixes x :: "'b literal"   and y :: "'b set" assumes "vars_of_lit x = y" obtains e :: "'b equation" where "x = Pos e__" and "y = vars_of_eq e__"   | e :: "'b equation" where "x = Neg e" and "y = vars_of_eq e"
merge_prod: fixes rha :: "'c \<Rightarrow> 'd"   and boa :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and ma :: "'c \<Rightarrow> 'c \<Rightarrow> 'c option"   and rhb :: "'e \<Rightarrow> 'f"   and bob :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and mb :: "'e \<Rightarrow> 'e \<Rightarrow> 'e option" assumes "merge_on UNIV rha boa ma"   and "merge_on UNIV rhb bob mb" shows "merge_on UNIV (map_prod rha rhb) (rel_prod boa bob) (merge_prod ma mb)"
word_msb_alt: fixes w :: "'a word" shows "msb w = hd (to_bl w)"
simps_1: fixes x :: "'a" shows "lookup_pair [] x = (0::'d)"
trans: fixes b :: "'x"   and a :: "'x"   and c :: "'x" assumes "b \<le> a"   and "c \<le> b" shows "c \<le> a"
dist_fps_ge0: fixes a :: "'a fps"   and b :: "'a fps" shows "0 \<le> dist a b"
assms_3: shows "rkp' \<equiv> interpret_floatarith rkp vs"
lspasl_starl_der: fixes A :: "'a \<Rightarrow> bool"   and B :: "'a \<Rightarrow> bool"   and h0 :: "'a" assumes "(A \<and>* B) h0" shows "\<exists>h1 h2. (h1,h2\<triangleright>h0) \<and> A h1 \<and> B h2"
sum_goestozero_lemma: fixes a :: "nat \<Rightarrow> real"   and M :: "nat"   and N :: "nat"   and d :: "real" assumes "\<bar>\<Sum>i = M..N. a i / real i\<bar> < d"   and "\<And>n. a n + ln (real n) \<le> a (Suc n) + ln (real (Suc n))"   and "0 < M"   and "M < N" shows "a M \<le> d * real N / (real N - real M) + (real N - real M) / real M \<and> - a N \<le> d * real N / (real N - real M) + (real N - real M) / real M"
assms_5: shows "e' mod order \<G> < e mod order \<G>"
ext_inject: fixes ok :: "bool"   and more :: "'c"   and ok' :: "bool"   and more' :: "'c" shows "(\<lparr>ok = ok, \<dots> = more\<rparr> = \<lparr>ok = ok', \<dots> = more'\<rparr>) = (ok = ok' \<and> more = more')"
det_row_mul: fixes k :: "'c"   and c :: "'b"   and a :: "'c \<Rightarrow> ('b, 'c) vec"   and b :: "'c \<Rightarrow> ('b, 'c) vec" shows "det (\<chi>i. if i = k then c *s a i else b i) = c * det (\<chi>i. if i = k then a i else b i)"
Ax: fixes k :: "'b"   and \<Gamma> :: "'b formula multiset"   and \<Delta> :: "'b formula multiset"   and n :: "nat" assumes "Atom k \<in># \<Gamma>"   and "Atom k \<in># \<Delta>" shows "\<Gamma> \<Rightarrow> \<Delta> \<down> Suc n"
substitution_code_def: fixes g :: "'g"   and "next" :: "'node \<times> 'val" shows "uninst_code.substitution_code phis g next = the (the_trivial (snd next) (the (Mapping.lookup (phis g) next)))"
DDH0_def: fixes \<G> :: "'b cyclic_group"   and \<A> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool spmf" shows "ddh.DDH0 \<G> \<A> = sample_uniform (order \<G>) \<bind> (\<lambda>s. sample_uniform (order \<G>) \<bind> (\<lambda>r. let h = \<^bold>g\<^bsub>\<G>\<^esub> [^]\<^bsub>\<G>\<^esub> s in \<A> h (\<^bold>g\<^bsub>\<G>\<^esub> [^]\<^bsub>\<G>\<^esub> r) (h [^]\<^bsub>\<G>\<^esub> r)))"
cong_sym: fixes divide :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and modulo :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and euclidean_size :: "'d \<Rightarrow> nat"   and division_segment :: "'d \<Rightarrow> 'd"   and b :: "'d"   and c :: "'d"   and a :: "'d" 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" shows "unique_euclidean_semiring.cong modulo c b a"
line_integral_exists_rec_join_cons: fixes F :: "real \<times> real \<Rightarrow> real \<times> real"   and basis :: "(real \<times> real) set"   and \<gamma> :: "real \<Rightarrow> real \<times> real"   and l :: "(int \<times> (real \<Rightarrow> real \<times> real)) list" assumes "line_integral_exists F basis (rec_join ((1, \<gamma>) # l))"   and "\<And>k' \<gamma>'. (k', \<gamma>') \<in> set ((1, \<gamma>) # l) \<Longrightarrow> valid_path \<gamma>'"   and "finite basis" shows "line_integral_exists F basis (\<gamma> +++ rec_join l)"
strong_cases: fixes a :: "lam"   and x :: "name" assumes "NF a" obtains ts :: "lam list" and x :: "name" where "a = Var x \<degree>\<degree> ts" and "listsp NF ts"   | t :: "lam" where "x__ \<sharp> a \<Longrightarrow> a = (Lam [x__].t) \<and> NF t"
A: shows "{\<cdot> p } \<le> S * {\<cdot> q } * \<top>"
fun_eq_iff: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" shows "(f = g) = (\<forall>x. f x = g x)"
not_eq_order_implies_strict: fixes a :: "'d multiset"   and b :: "'d multiset" assumes "a \<noteq> b"   and "a \<subseteq># b" shows "a \<subset># b"
ActualEquivI: fixes v :: "i"   and \<phi> :: "\<o>"   and \<psi> :: "\<o>" assumes "[\<^bold>\<A>\<phi> \<^bold>\<equiv> \<^bold>\<A>\<psi> in v]" shows "[\<^bold>\<A>(\<phi> \<^bold>\<equiv> \<psi>) in v]"
exhaust: fixes y :: "'b equation" obtains x1 :: "'b trm" and x2 :: "'b trm" where "y = Eq x1 x2"
assms_4: shows "B \<in>\<^sub>\<circ> Vset \<alpha>"
keys_ins: fixes k :: "'b"   and p :: "'c"   and t :: "('b \<times> 'c) tree" shows "keys (ins k p t) = insert k (keys t)"
redT_updWs_Some_otherD: fixes t' :: "'a"   and ws :: "'a \<Rightarrow> 'b wait_set_status option"   and was :: "('a, 'b) wait_set_action list"   and ws' :: "'a \<Rightarrow> 'b wait_set_status option"   and t :: "'a"   and w :: "'b wait_set_status" assumes "redT_updWs t' ws was ws'"   and "ws' t = \<lfloor>w\<rfloor>"   and "t \<noteq> t'" shows "case w of InWS w' \<Rightarrow> ws t = \<lfloor>InWS w'\<rfloor> | PostWS wake_up_status \<Rightarrow> ws t = \<lfloor>w\<rfloor> \<or> (\<exists>w'. ws t = \<lfloor>InWS w'\<rfloor>)"
add: fixes b1 :: "'vs_1"   and b2 :: "'vs_1" assumes "b1 \<in> U\<^sub>V\<^sub>S\<^sub>_\<^sub>1"   and "b2 \<in> U\<^sub>V\<^sub>S\<^sub>_\<^sub>1" shows "f (b1 +\<^sub>V\<^sub>S\<^sub>_\<^sub>1 b2) = f b1 +\<^sub>V\<^sub>S\<^sub>_\<^sub>2 f b2"
trans: fixes S :: "('a, 'b) eq_object_scheme"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "equivalence S"   and "x .=\<^bsub>S\<^esub> y"   and "y .=\<^bsub>S\<^esub> z"   and "x \<in> carrier S"   and "y \<in> carrier S"   and "z \<in> carrier S" shows "x .=\<^bsub>S\<^esub> z"
semiring_norm_11: fixes m :: "num" shows "m * num.One = m"
mono_SEQI2a: assumes "P \<sqsubseteq> Q" shows "\<forall>s. s \<notin> D (S `;` P) \<longrightarrow> Ra (S `;` Q) s \<subseteq> Ra (S `;` P) s"
discI_2: fixes literal :: "'b literal"   and x2 :: "'b" assumes "literal = Neg x2" shows "is_neg literal"
mult_sign_intros_2: fixes a :: "'c"   and b :: "'c" assumes "(0::'c) \<le> a"   and "b \<le> (0::'c)" shows "a * b \<le> (0::'c)"
RETURN_rel_conv2: fixes f :: "'c \<Rightarrow> 'd \<Rightarrow> 'e"   and g :: "'f \<Rightarrow> 'g \<Rightarrow> 'h"   and A :: "('c \<times> 'f) set"   and B :: "('d \<times> 'g) set"   and R :: "('e \<times> 'h) set" shows "((RETURN \<circ>\<circ> f, RETURN \<circ>\<circ> g) \<in> A \<rightarrow> B \<rightarrow> \<langle>R\<rangle>nres_rel) = ((f, g) \<in> A \<rightarrow> B \<rightarrow> R)"
permutes_vimage: fixes f :: "'c \<Rightarrow> 'c"   and A :: "'c set" assumes "f permutes A" shows "f -` A = A"
kE_satA: fixes at :: "('fsym, 'psym) atm"   and \<xi> :: "Preliminaries.var \<Rightarrow> univ" assumes "wtA at"   and "I.wtE (kE \<xi>)" shows "I.satA (kE \<xi>) at = F.satA \<xi> at"
HcompNml_Prim: fixes t :: "'b local.term"   and \<nu> :: "'b" assumes "\<not> is_Prim\<^sub>0 t" shows "\<^bold>\<langle>\<nu>\<^bold>\<rangle> \<^bold>\<lfloor>\<^bold>\<star>\<^bold>\<rfloor> t = \<^bold>\<langle>\<nu>\<^bold>\<rangle> \<^bold>\<star> t"
mult_mono: fixes plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and a :: "'e"   and b :: "'e"   and c :: "'e"   and d :: "'e" assumes "class.ordered_semiring plus zero less_eq less times"   and "less_eq a b"   and "less_eq c d"   and "less_eq zero b"   and "less_eq zero c" shows "less_eq (times a c) (times b d)"
dividend_less_times_div: fixes n :: "nat"   and m :: "nat" assumes "0 < n" shows "m < n + n * (m div n)"
nc_scomp_closed_alt: fixes x :: "'a"   and y :: "'a" shows "(x \<sqinter> nc) \<cdot> (y \<sqinter> nc) \<sqinter> nc = (x \<sqinter> nc) \<cdot> (y \<sqinter> nc)"
is_measured_ftree_def: shows "is_measured_ftree \<equiv> rec_FingerTreeStruc True is_measured_node (\<lambda>a l m r ma. (is_measured_digit l \<and> ma \<and> is_measured_digit r) \<and> a = gmd l + gmft m + gmd r)"
prefix_trie_append: fixes ps :: "bool list"   and qs :: "bool list"   and t :: "trie" shows "prefix_trie (ps @ qs) t = prefix_trie ps (prefix_trie qs t)"
seqI: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and f :: "'a"   and g :: "'a" assumes "category C"   and "partial_magma.arr C f"   and "partial_magma.arr C g"   and "partial_magma.dom C g = partial_magma.cod C f" shows "partial_magma.arr C (C g f)"
c1_def: fixes x :: "'e" shows "c1 x = (1::'d)"
boundOutputFreshSimps_3: fixes xvec :: "name list"   and B :: "('d, 'e, 'f) boundOutput" shows "xvec \<sharp>* (\<lparr>\<nu>*xvec\<rparr>B)"
is_prefix_f: fixes a :: "'a"   and as :: "'a list"   and vs :: "'a list" shows "is_prefix (a # as) (a # vs) = is_prefix as vs"
The_on_None: fixes U :: "'c set"   and P :: "'c \<Rightarrow> bool" assumes "\<nexists>!x. x \<in> U \<and> P x" shows "(THE x on U. P x) = None"
landau_symbol_at_top_imp_at_right: fixes f :: "real \<Rightarrow> 'f"   and a :: "real"   and g :: "real \<Rightarrow> 'f" assumes "(\<lambda>x. f (a + inverse x)) \<in> L' at_top (\<lambda>x. g (a + inverse x))" shows "f \<in> L (at_right a) g"
set_iterator_abs_rule_insert_P: assumes "set_iterator_abs \<alpha> invar it S0"   and "I {} \<sigma>0"   and "\<And>S \<sigma> x. \<lbrakk>True; invar x; \<alpha> x \<in> S0 - S; I S \<sigma>; S \<subseteq> S0\<rbrakk> \<Longrightarrow> I (insert (\<alpha> x) S) (f x \<sigma>)"   and "\<And>\<sigma>. I S0 \<sigma> \<Longrightarrow> P \<sigma>"   and "\<And>\<sigma> S. \<lbrakk>S \<subseteq> S0; S \<noteq> S0; \<not> True; I S \<sigma>\<rbrakk> \<Longrightarrow> P \<sigma>" shows "P (it (\<lambda>_. True) f \<sigma>0)"
weak_gt_mono: fixes xys :: "('a \<times> 'a) list" assumes "\<forall>x y. (x, y) \<in> set xys \<longrightarrow> weak_gt x y" shows "\<exists>gt. SN_strict_mono_ordered_semiring_1 default gt mono \<and> (\<forall>x y. (x, y) \<in> set xys \<longrightarrow> gt x y)"
tauActionI: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi" shows "\<exists>P'. \<Psi> \<rhd> \<tau>.P \<longmapsto> \<tau> \<prec> P' \<and> \<Psi> \<rhd> P \<sim> P'"
\<mu>_cong: fixes j :: "nat"   and i :: "nat"   and f1 :: "'a vec list"   and f2 :: "'a vec list"   and n :: "nat" assumes "\<And>k. \<lbrakk>j < i; k \<le> j\<rbrakk> \<Longrightarrow> f1 ! k = f2 ! k"   and "j < i \<Longrightarrow> f1 ! i = f2 ! i" shows "gram_schmidt_fs.\<mu> n f1 i j = gram_schmidt_fs.\<mu> n f2 i j"
obtain_shortest_path: fixes u :: "nat"   and v :: "nat" assumes "connected u v" obtains p :: "(nat \<times> nat) list" where "isShortestPath u p v"
div_lt_nat: fixes j :: "nat"   and x :: "nat"   and y :: "nat" assumes "j < x * y" shows "j div x < y"
valuate_update_x_lemma: fixes f :: "nat \<Rightarrow> rat"   and x :: "nat"   and v1 :: "nat \<Rightarrow> 'a"   and v2 :: "nat \<Rightarrow> 'a" assumes "\<forall>y. f y \<noteq> 0 \<longrightarrow> y \<noteq> x \<longrightarrow> v1 y = v2 y"   and "finite {v. f v \<noteq> 0}" shows "(\<Sum>x\<in>{v. f v \<noteq> 0}. f x *R v1 x) + f x *R (v2 x - v1 x) = (\<Sum>x\<in>{v. f v \<noteq> 0}. f x *R v2 x)"
prog: shows "{X. G \<in> stable X} \<in> progress_set F UNIV B"
semiring_norm_12: fixes n :: "num" shows "num.One * n = n"
orth_at_distincts: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and X :: "'a"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and U :: "'a"   and V :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.OrthAt Bet Cong X A B C U V" shows "A \<noteq> B \<and> B \<noteq> C \<and> A \<noteq> C \<and> U \<noteq> V"
poincare_between_poincare_line_uvz: fixes u :: "complex_homo"   and v :: "complex_homo"   and z :: "complex_homo" assumes "u \<noteq> v"   and "u \<in> unit_disc"   and "v \<in> unit_disc"   and "z \<in> unit_disc"   and "poincare_between u v z" shows "z \<in> circline_set (poincare_line u v)"
Min_snd_alist_of: fixes q :: "('c, 'd) pq" assumes "\<not> is_empty q" shows "Min (snd ` set (alist_of q)) = snd (hd (alist_of q))"
RedCASFail: fixes h :: "'heap"   and a :: "'addr"   and D :: "String.literal"   and F :: "String.literal"   and v'' :: "'addr val"   and v :: "'addr val"   and extTA :: "('addr, 'thread_id, 'heap) external_thread_action \<Rightarrow> ('addr, 'thread_id, 'b, 'heap) Jinja_thread_action"   and P :: "'addr J_prog"   and t :: "'thread_id"   and v' :: "'addr val"   and l :: "String.literal \<Rightarrow> 'addr val option" assumes "heap_read h a (CField D F) v''"   and "v \<noteq> v''" shows "extTA,P,t \<turnstile> \<langle>addr a\<bullet>compareAndSwap(D\<bullet>F, Val v, Val v'),(h, l)\<rangle> -\<lbrace>ReadMem a (CField D F) v''\<rbrace>\<rightarrow> \<langle>false,(h, l)\<rangle>"
set_mult_closed: fixes H :: "'a set"   and K :: "'a set" assumes "H \<subseteq> carrier R"   and "K \<subseteq> carrier R" shows "H <#>\<^bsub>add_monoid R\<^esub> K \<subseteq> carrier R"
divide_fract_def: fixes q :: "'b fract"   and r :: "'b fract" shows "q / r = q * inverse r"
trancl_Image_unfold_right: fixes E :: "('a \<times> 'a) set"   and S :: "'a set" shows "E\<^sup>+ `` S = E `` E\<^sup>* `` S"
assms: fixes x :: "'a" assumes "x \<in> vars_term t" shows "\<sigma> x = \<tau> x"
mgu_subst_range_vars: fixes s :: "('b, 'a) Term.term"   and t :: "('b, 'a) Term.term"   and \<sigma> :: "'a \<Rightarrow> ('b, 'a) Term.term" assumes "mgu s t = Some \<sigma>" shows "range_vars \<sigma> \<subseteq> fv s \<union> fv t"
open_ereal_generated: shows "open = generate_topology (range lessThan \<union> range greaterThan)"
digit_to_char_simps_3: shows "digit_to_char 2 = CHR ''2''"
double_coreflexive_complement: fixes x :: "'a" shows "x ' ' = - x '"
powr_eq_0_iff: fixes w :: "'b"   and z :: "'b" shows "(w powr z = (0::'b)) = (w = (0::'b))"
card_le_dim_spanning: fixes B :: "'c set"   and V :: "'c set" assumes "B \<subseteq> V"   and "V \<subseteq> vs2.span B"   and "finite B"   and "card B \<le> vs2.dim V" shows "vs2.independent B"
entails_\<G>_def: fixes N1 :: "'f set"   and N2 :: "'f set" shows "(N1 \<Turnstile>\<inter>\<G> N2) = (\<forall>q\<in>Q. entails_q q (no_labels.\<G>_Fset_q q N1) (no_labels.\<G>_Fset_q q N2))"
OF_priority_match_ana_def: fixes \<gamma> :: "'d set \<Rightarrow> 'f \<Rightarrow> bool"   and flow_entries :: "('d, 'e) flow_entry_match list"   and packet :: "'f" shows "OF_priority_match_ana \<gamma> flow_entries packet \<equiv> let m = filter (\<lambda>f. \<gamma> (ofe_fields f) packet) flow_entries; m' = filter (\<lambda>f. \<forall>fo\<in>set m. ofe_prio fo \<le> ofe_prio f) m in case m' of [] \<Rightarrow> NoAction | [s] \<Rightarrow> Action s | s # aa # lista \<Rightarrow> Undefined"
norm_def: fixes M :: "nat \<Rightarrow> nat \<Rightarrow> 'c DBMEntry"   and k :: "nat \<Rightarrow> 'c"   and n :: "nat" shows "norm M k n \<equiv> \<lambda>i j. let ub = if 0 < i then k i else (0::'c); lb = if 0 < j then - k j else (0::'c) in if i \<le> n \<and> j \<le> n then norm_lower (norm_upper (M i j) ub) lb else M i j"
f1: shows "{X -` A \<inter> space M |A. A \<in> sets N} \<subseteq> Pow (space M)"
ffb_set: fixes f :: "'d \<Rightarrow> 'c set" shows "fb\<^sub>\<F> f = (\<lambda>Y. {x. f x \<subseteq> Y})"
iMODb_def: fixes r :: "nat"   and m :: "nat"   and c :: "nat" shows "[ r, mod m, c ] \<equiv> {x. x mod m = r mod m \<and> r \<le> x \<and> x \<le> r + m * c}"
special_subgroup_special_coset: fixes T :: "'w set" assumes "T \<in> Pow S" shows "\<langle>T\<rangle> \<in> \<P>"
dagger_trans_eq: fixes x :: "'a" shows "x\<^sup>\<star> \<cdot> x\<^sup>\<star> = x\<^sup>\<star>"
span_base: fixes Sa :: "'a set"   and a :: "'a" assumes "Sa \<subseteq> S"   and "a \<in> Sa" shows "a \<in> span Sa"
iprefix_eq_iprefix_take: fixes xs :: "'b list"   and f :: "nat \<Rightarrow> 'b" shows "(xs \<sqsubseteq> f) = (f \<Down> length xs = xs)"
has_fields_rec: fixes P :: "'a prog"   and C :: "String.literal"   and D :: "String.literal"   and fs :: "fdecl list"   and ms :: "'a option mdecl list"   and FDTs :: "((String.literal \<times> String.literal) \<times> ty \<times> fmod) list"   and FDTs' :: "((String.literal \<times> String.literal) \<times> ty \<times> fmod) list" assumes "class P C = \<lfloor>(D, fs, ms)\<rfloor>"   and "C \<noteq> Object"   and "P \<turnstile> D has_fields FDTs"   and "FDTs' = map (\<lambda>(F, Tm). ((F, C), Tm)) fs @ FDTs" shows "P \<turnstile> C has_fields FDTs'"
from_P_to_Q: fixes Gs :: "'a literal multiset set llist"   and C :: "'a literal multiset"   and D :: "'a literal multiset"   and i :: "nat" assumes "chain (\<leadsto>) Sts"   and "fair_state_seq Sts"   and "Gs = lmap (\<lambda>St. grounding_of_clss (N_of_state St \<union> P_of_state St \<union> Q_of_state St)) Sts"   and "C \<in> Liminf_llist Gs - sr.Rf (Liminf_llist Gs)"   and "D \<in> P_of_state (lnth Sts i)"   and "enat i < llength Sts"   and "subsumes D C"   and "\<forall>E\<in>{E \<in> N_of_state (Sup_state Sts) \<union> P_of_state (Sup_state Sts) \<union> Q_of_state (Sup_state Sts). subsumes E C}. \<not> strictly_subsumes E D" shows "\<exists>l. D \<in> Q_of_state (lnth Sts l) \<and> enat l < llength Sts"
face_of_refl: fixes S :: "'a set" assumes "convex S" shows "S face_of S"
wls_freshAbs_simp: fixes s :: "'sort"   and X :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set"   and ys :: "'varSort"   and y :: "'var"   and xs :: "'varSort"   and x :: "'var" assumes "wls s X" shows "freshAbs ys y (Abs xs x X) = (ys = xs \<and> y = x \<or> fresh ys y X)"
authlife_def: shows "authlife \<equiv> SOME authlife. Suc 0 \<le> authlife"
fimage_inter_dom_2: fixes m :: "'f \<rightharpoonup>\<^sub>f 'e"   and A :: "'f fset" shows "fmimage m (A |\<inter>| fmdom m) = fmimage m A"
tail_arev: fixes a :: "'b \<times> 'b" assumes "a \<in> arcs (with_proj (list_digraph G_list))" shows "tail (with_proj (list_digraph G_list)) (edge_rev (to_map' (snd G_list) xss) a) = head (with_proj (list_digraph G_list)) a"
H: shows "\<not> (\<exists>epsilon>0. \<forall>U\<in>sets M. emeasure M U < ennreal epsilon \<longrightarrow> \<bar>set_lebesgue_integral M U h\<bar> < delta)"
igSwapCls_def: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'a, 'b, 'c, 'd) model_scheme" shows "igSwapCls MOD \<equiv> igSwapIGVar MOD \<and> igSwapIGAbs MOD \<and> igSwapIGOp MOD"
AX1_away: fixes \<Gamma> :: "'b formula set" shows "AX1 \<union> \<Gamma> = AX0 \<union> (\<Gamma> \<union> AX1)"
prems_2: shows "are_all_operators_non_interfering (op # op' # \<pi>)"
A_def: fixes f :: "nat \<Rightarrow> 'o"   and i :: "nat" shows "A f i = T (f i) (state f i) + \<Phi> (state f (i + 1)) - \<Phi> (state f i)"
fixp_unfold: fixes f :: "'a \<times> 'b \<Rightarrow> 'a \<times> 'b" assumes "monotone (rel_prod orda ordb) (rel_prod orda ordb) f" shows "ab.fixp f = f (ab.fixp f)"
sees_method_compP: fixes P :: "'a prog"   and C :: "char list"   and M :: "char list"   and b :: "staticb"   and Ts :: "ty list"   and T :: "ty"   and m :: "'a"   and D :: "char list"   and f :: "staticb \<Rightarrow> 'a \<Rightarrow> 'b" assumes "P \<turnstile> C sees M, b :  Ts\<rightarrow>T = m in D" shows "compP f P \<turnstile> C sees M, b :  Ts\<rightarrow>T = f b m in D"
negligible_times_poly: fixes f :: "nat \<Rightarrow> real"   and g :: "nat \<Rightarrow> real"   and n :: "real" assumes "negligible f"   and "g \<in> O(\<lambda>x. real x powr n)" shows "negligible (\<lambda>x. f x * g x)"
succs: shows "r1 \<in> succ A w p1 \<and> r2 \<in> succ B w p2"
matroid_def: fixes carrier :: "'b set"   and indep :: "'b set \<Rightarrow> bool" shows "matroid carrier indep \<equiv> indep_system carrier indep \<and> matroid_axioms indep"
not_coprime_iff_common_factor: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and p :: "'b"   and q :: "'b" assumes "class.comm_monoid_mult times one" shows "(\<not> comm_monoid_mult.coprime times one p q) = (\<exists>r. dvd.dvd times r p \<and> dvd.dvd times r q \<and> \<not> dvd.dvd times r one)"
weakSimE2: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi"   and Rel :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set"   and R :: "('a, 'b, 'c) psi"   and \<alpha> :: "'a action"   and Q' :: "('a, 'b, 'c) psi"   and \<Psi>' :: "'b" assumes "(\<Psi>, P, Q) \<in> Rel"   and "\<And>\<Psi>' R S. (\<Psi>', R, S) \<in> Rel \<Longrightarrow> \<Psi>' \<rhd> R \<leadsto><Rel> S"   and "\<Psi> : R \<rhd> Q \<Longrightarrow>\<alpha> \<prec> Q'"   and "bn \<alpha> \<sharp>* \<Psi>"   and "bn \<alpha> \<sharp>* P"   and "\<alpha> \<noteq> \<tau>" obtains P'' :: "('a, 'b, 'c) psi" and P' :: "('a, 'b, 'c) psi" where "\<Psi> : R \<rhd> P \<Longrightarrow>\<alpha> \<prec> P''" and "\<Psi> \<otimes> \<Psi>' \<rhd> P'' \<Longrightarrow>\<^sup>^\<^sub>\<tau> P'" and "(\<Psi> \<otimes> \<Psi>', P', Q') \<in> Rel"
is_span: shows "C.span H\<mu>H\<nu>\<pi>.leg0 H\<mu>H\<nu>\<pi>.leg1"
dbproj_Cons_2: fixes i :: "'a"   and j :: "'a"   and d :: "'b"   and D :: "('a \<times> 'b) list" assumes "i \<noteq> j" shows "dbproj j ((i, d) # D) = dbproj j D"
fls_base_factor_to_fps_right_inverse_nonzero: fixes y :: "'a"   and f :: "'a fls" assumes "y \<noteq> (0::'a)" shows "fls_base_factor_to_fps (fls_right_inverse f y) = fps_right_inverse (fls_base_factor_to_fps f) y"
distinct_43: fixes x4 :: "nat" shows "\<not>unicast x4 \<noteq> \<tau>\<^sub>s"
HInfinite_mult: fixes x :: "'a star"   and y :: "'a star" assumes "x \<in> HInfinite"   and "y \<in> HInfinite" shows "x * y \<in> HInfinite"
funion_assoc: fixes A :: "'a fset"   and B :: "'a fset"   and C :: "'a fset" shows "A |\<union>| B |\<union>| C = A |\<union>| (B |\<union>| C)"
mult_num_simps_1: fixes m :: "num" shows "m * num.One = m"
simps_11: fixes e\<^sub>1 :: "'a exp"   and e\<^sub>2 :: "'a exp"   and A :: "'a set option" shows "\<D> (e\<^sub>1;; e\<^sub>2) A = (\<D> e\<^sub>1 A \<and> \<D> e\<^sub>2 (A \<squnion> \<A> e\<^sub>1))"
zip_same: fixes a :: "'g"   and b :: "'g"   and xs :: "'g list" shows "((a, b) \<in> set (zip xs xs)) = (a \<in> set xs \<and> a = b)"
argy: shows "arg y = arg (croot n x) + 2 * real_of_int i * pi / real n"
cases: fixes x :: "('b, 'a) Term.term" obtains xa :: "'a" where "x = Var xa"   | f :: "'b" and T :: "('b, 'a) Term.term list" where "x = Fun f T"
hom_zero: shows "poly_x_mult_y 0 = 0"