hcju/isabelleps huggingface dataset:
StaysEnabledStep: fixes cfg :: "('p, 'v, 's) configuration"   and msg :: "('p, 'v) message"   and n :: "nat" assumes "enabled cfg msg"   and "cfg = execConf n"   and "n < length exec" shows "enabled (execConf (n + 1)) msg \<or> n = length exec - 1 \<or> msg = execMsg n"
apex_def: shows "trg.apex \<equiv> C.dom trg.leg0"
linear_iff: fixes f :: "'e \<Rightarrow> 'f" shows "linear f = ((\<forall>x y. f (x + y) = f x + f y) \<and> (\<forall>c x. f (c *\<^sub>R x) = c *\<^sub>R f x))"
fw_fw_spec: fixes M :: "nat \<Rightarrow> nat \<Rightarrow> 'b"   and n :: "nat" shows "RETURN (FW M n) \<le> fw_spec n M"
approx_un_ne_None: fixes p :: "nat"   and f :: "float interval \<Rightarrow> float interval"   and r :: "(real \<times> real pdevs) \<times> real" shows "approx_un p (\<lambda>ivl. Some (f ivl)) (Some r) \<noteq> None"
assoc'_simps_1: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and prj0 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and prj1 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b arrow_of_spans_data"   and g :: "'b arrow_of_spans_data"   and h :: "'b arrow_of_spans_data" assumes "span_bicategory C prj0 prj1"   and "partial_magma.ide (span_vertical_category.vcomp C) f"   and "partial_magma.ide (span_vertical_category.vcomp C) g"   and "partial_magma.ide (span_vertical_category.vcomp C) h"   and "span_vertical_category.src C f = span_vertical_category.trg C g"   and "span_vertical_category.src C g = span_vertical_category.trg C h" shows "partial_magma.arr (span_vertical_category.vcomp C) \<lparr>Chn = three_composable_identity_arrows_of_spans.chine_assoc' C prj0 prj1 f g h, Dom = Cod (span_bicategory.hcomp C prj0 prj1 f (span_bicategory.hcomp C prj0 prj1 g h)), Cod = Dom (span_bicategory.hcomp C prj0 prj1 (span_bicategory.hcomp C prj0 prj1 f g) h)\<rparr>"
cont_nosp: fixes x :: "'a" assumes "norm x \<le> 1"   and "x \<in> affine hull S" shows "continuous (at x within cball (0::'a) 1 \<inter> affine hull S) (\<lambda>x. norm x *\<^sub>R surf (proj x))"
ltl_lzip: fixes xs :: "'c llist"   and ys :: "'d llist" assumes "\<not> lnull xs"   and "\<not> lnull ys" shows "ltl (lzip xs ys) = lzip (ltl xs) (ltl ys)"
w_def: shows "w = normalize_quot (fst x', snd y')"
dist_eqs_1: fixes a' :: "'r"   and slist' :: "[:'r:]" shows "[::] \<noteq> a' :# slist'"
ide_in_hom_2: fixes f :: "'b" assumes "D.ide f" shows "\<guillemotleft>f : f \<Rightarrow>\<^sub>D f\<guillemotright>"
antisym: fixes x :: "'c list"   and y :: "'c list" assumes "subseq x y"   and "subseq y x" shows "x = y"
alpha_preserves_qGood: fixes X :: "('f, 'g, 'e, 'd, 'h) qTerm"   and X' :: "('f, 'g, 'e, 'd, 'h) qTerm" assumes "FixVars TYPE('d) TYPE('e)"   and "X #= X'" shows "FixVars.qGood X = FixVars.qGood X'"
ne: shows "cs\<^bsup>\<pi>\<^esup> (k + n) \<noteq> cs\<^bsup>\<pi>'\<^esup> (k' + n')"
static_to_dynamic_accessible_from_static: fixes G :: "prog"   and accC :: "qtname"   and m :: "qtname \<times> memberdecl"   and statC :: "qtname" assumes "G \<turnstile> m of statC accessible_from accC"   and "is_static m"   and "wf_prog G" shows "G \<turnstile> m in declclass m dyn_accessible_from accC"
NotS: fixes v :: "i"   and \<phi> :: "\<o>" shows "[\<^bold>\<not>\<phi> in v] = (\<not> [\<phi> in v])"
permute_rep_abs_Tree\<^sub>p: fixes t :: "('a, 'b, 'c, 'd) Tree" obtains p :: "perm" where "rep_Tree\<^sub>p (abs_Tree\<^sub>p t) = p \<bullet> t"
assoc: fixes a :: "'b list"   and b :: "'b list"   and c :: "'b list" shows "append (append a b) c = append a (append b c)"
ext: shows "extendRule S r = (Ps,  \<Gamma> \<oplus> F \<nabla> [x].A \<Rightarrow>* \<Delta>)"
simps_1: fixes carrier :: "'a set"   and eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and le :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and more :: "'b" shows "le \<lparr>carrier = carrier, eq = eq, le = le, \<dots> = more\<rparr> = le"
explode_Elem: fixes x :: "ZF"   and X :: "ZF" shows "(x \<in> explode X) = Elem x X"
disc_22: fixes x41 :: "'a rexp"   and x42 :: "'a rexp" shows "is_Plus (Plus x41 x42)"
assms: fixes x :: "'a"   and y :: "'a" assumes "x \<in> X"   and "y \<in> Y" shows "is_orthogonal x y"
SUP_L_P_def: fixes X :: "'a \<Rightarrow> 'b \<Rightarrow> 'e"   and u :: "'c"   and i :: "'b" shows "SUP_L_P X u i = (SUP v\<in>{v. pair v i \<sqsubset> u}. X v i)"
ideal_n_prod_primeTr: fixes P :: "'a set"   and n :: "nat"   and J :: "nat \<Rightarrow> 'a set" assumes "prime_ideal R P" shows "(\<forall>k\<le>n. ideal R (J k)) \<longrightarrow> i\<Pi>\<^bsub>R,n\<^esub> J \<subseteq> P \<longrightarrow> (\<exists>i\<le>n. J i \<subseteq> P)"
LPDs_refl: fixes r :: "mregex" shows "r \<in> LPDs r"
rel_cong: fixes x :: "'a option"   and ya :: "'a option"   and y :: "'b option"   and xa :: "'b option"   and R :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and Ra :: "'a \<Rightarrow> 'b \<Rightarrow> bool" assumes "x = ya"   and "y = xa"   and "\<And>z yb. \<lbrakk>z \<in> set_option ya; yb \<in> set_option xa\<rbrakk> \<Longrightarrow> R z yb = Ra z yb" shows "rel_option R x y = rel_option Ra ya xa"
fds_abs_converges_integral: fixes f :: "'b fds"   and s :: "'b"   and c :: "'b" assumes "fds_abs_converges f s" shows "fds_abs_converges (fds_integral c f) s"
linorder_neqE_nat: fixes x :: "nat"   and y :: "nat" assumes "x \<noteq> y" obtains "x < y"   | "y < x"
R: shows "F, \<Gamma> \<Rightarrow> G, \<Delta>"
tendsto_eq_rhs: fixes "open" :: "'c set \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'c"   and x :: "'c"   and F :: "'d filter"   and y :: "'c" assumes "class.topological_space open"   and "filterlim f (topological_space.nhds open x) F"   and "x = y" shows "filterlim f (topological_space.nhds open y) F"
smallo_def: fixes F :: "'b filter"   and g :: "'b \<Rightarrow> 'c" shows "o[F](g) = {f. \<forall>c>0. \<forall>\<^sub>F x in F. norm (f x) \<le> c * norm (g x)}"
psi_hom: fixes h1 :: "'c"   and h2 :: "'c" assumes "h1 \<in> carrier H"   and "h2 \<in> carrier H" shows "\<psi> (h1 \<otimes>\<^bsub>H\<^esub> h2) = \<psi> h1 \<otimes> \<psi> h2"
slice_len: fixes "from" :: "nat"   and to :: "nat"   and xs :: "'a list" assumes "from \<le> to"   and "to \<le> length xs" shows "length (Misc.slice from to xs) = to - from"
0: shows "f \<ominus> g = [p] \<cdot> \<one> [^] k \<otimes> [p] \<cdot> \<one> [^] m \<otimes> ac_Zp (f \<ominus> g)"
subsumes_def: fixes C :: "('a, 'b) Term.term literal multiset"   and D :: "('a, 'b) Term.term literal multiset" shows "subsumes C D = (\<exists>\<sigma>. subst_cls C \<sigma> \<subseteq># D)"
varsClauseVarsSet: fixes clause :: "Literal list" shows "vars clause = vars (set clause)"
Amicable_Example_Pythagoras: shows "220 Amic 284"
fv_Alts: fixes e1 :: "exp"   and e2 :: "exp" shows "fv (Alts e1 e2) = fv e1 \<union> fv e2"
fin: shows "finite {f \<in> set ft. \<gamma> (ofe_fields f) p \<and> (\<forall>fo\<in>set ft. ofe_prio f < ofe_prio fo \<longrightarrow> \<not> \<gamma> (ofe_fields fo) p)}"
fimage_fconst_on_vempty: fixes A :: "V"   and B :: "V"   and c :: "V" assumes "A \<inter>\<^sub>\<circ> B = []\<^sub>\<circ>" shows "fconst_on A c `\<^sub>\<bullet> B = []\<^sub>\<circ>"
rational_proper_factor_def: fixes p :: "rat poly" shows "rational_proper_factor p = (if degree p \<le> 1 then None else if degree p = 2 then case rat_roots2 p of [] \<Rightarrow> None | x # xs \<Rightarrow> Some [:- x, 1:] else if degree p = 3 then case rational_root_test p of None \<Rightarrow> None | Some x \<Rightarrow> Some [:- x, 1:] else kronecker_factorization_rat p)"
I: assumes "X \<noteq> {}"   and "Y \<noteq> {}" shows "X \<inter> Y \<noteq> {}"
unitary11_sgn_det: fixes k :: "complex"   and a :: "complex"   and b :: "complex"   and M :: "complex \<times> complex \<times> complex \<times> complex"   and A :: "complex"   and B :: "complex"   and C :: "complex"   and D :: "complex" assumes "k \<noteq> 0"   and "mat_det (a, b, cnj b, cnj a) \<noteq> 0"   and "M = k *\<^sub>s\<^sub>m (a, b, cnj b, cnj a)"   and "M = (A, B, C, D)" shows "sgn (Re (mat_det (a, b, cnj b, cnj a))) = (if b = 0 then 1 else sgn (Re (A * D / (B * C)) - 1))"
linear_subspace_image: fixes f :: "'c \<Rightarrow> 'd"   and S :: "'c set" assumes "clinear f"   and "csubspace S" shows "csubspace (f ` S)"
qstar_inductl: fixes z :: "'b"   and x :: "'b"   and y :: "'b" assumes "z \<squnion> x \<cdot> y \<le> y" shows "qstar x \<cdot> z \<le> y"
numeral_eq_one_iff: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and n :: "num" assumes "class.semiring_char_0 one times plus zero" shows "(numeral.numeral one plus n = one) = (n = num.One)"
ide_self_inverse: fixes a :: "'s" assumes "ide a" shows "inverse_arrows a a"
\<I>_trivialI: fixes \<I> :: "('a, 'b) \<I>" assumes "\<And>x. x \<in> outs_\<I> \<I>" shows "\<I>_trivial \<I>"
permute_map_list: fixes p :: "perm"   and l :: "'a list" shows "p \<bullet> l = map (permute p) l"
GodIsEssential: shows "\<lfloor>(\<lambda>w. \<forall>x. (\<forall>xa. (\<P> xa \<^bold>\<rightarrow> xa x) w) \<longrightarrow> (\<forall>xa. (\<P> xa \<^bold>\<rightarrow> xa x) w) \<and> (\<forall>xa. (xa x \<^bold>\<rightarrow> \<^bold>\<box>(\<lambda>v. \<forall>x. existsAt x v \<longrightarrow> (\<forall>xa. (\<P> xa \<^bold>\<rightarrow> xa x) v) \<longrightarrow> xa x v)) w))\<rfloor>"
Eps_Opt_eq_Some: fixes P :: "'b \<Rightarrow> bool"   and x :: "'b" assumes "\<And>x'. \<lbrakk>P x; P x'\<rbrakk> \<Longrightarrow> x' = x" shows "(Eps_Opt P = Some x) = P x"
iD_2: shows "interpret_floatarith fa (list_of_eucl x @ params) \<notin> \<int>"
termMOD_igAbsRen: shows "igAbsRen termMOD"
valid_foldl_eval_step_mmuaux': fixes args :: "Monitor.args"   and cur :: "nat"   and dt :: "nat"   and aux :: "nat \<times> nat queue \<times> nat \<times> bool list \<times> bool list \<times> ('a option list, nat) mapping \<times> (nat, ('a option list, nat + nat) mapping) mapping \<times> 'a option list set list \<times> nat"   and auxlist :: "(nat \<times> 'a option list set \<times> 'a option list set) list"   and tss :: "nat list"   and tss' :: "nat list" assumes "valid_mmuaux' args cur dt aux auxlist"   and "lin_ts_mmuaux aux = tss @ tss'"   and "\<And>ts. ts \<in> set (take (length tss) (lin_ts_mmuaux aux)) \<Longrightarrow> enat ts + right (args_ivl args) < enat dt" shows "valid_mmuaux' args cur dt (foldl (\<lambda>aux _. eval_step_mmuaux aux) aux tss) auxlist \<and> lin_ts_mmuaux (foldl (\<lambda>aux _. eval_step_mmuaux aux) aux tss) = tss'"
cPar1B: fixes P :: "pi"   and a :: "name"   and x :: "name"   and P' :: "pi"   and Q :: "pi"   and C :: "'a" assumes "P \<longmapsto> a<x> \<prec> P'"   and "x \<sharp> P"   and "x \<sharp> Q"   and "x \<noteq> a"   and "\<And>C. F C P a x P'" shows "F C (P \<parallel> Q) a x (P' \<parallel> Q)"
homeomorphic_arc_image_interval: fixes g :: "real \<Rightarrow> 'b"   and a :: "real"   and b :: "real" assumes "arc g"   and "a < b" shows "path_image g homeomorphic {a..b}"
antisymmetric_axioms: shows "antisymmetric X (\<sqsubseteq>)"
tree_edge_impl_parenthesis: fixes G :: "('c, 'd) graph_rec_scheme"   and param :: "('c, ('c, 'e) state_scheme, 'e) gen_parameterization"   and s :: "('c, 'e) state_scheme"   and v :: "'c"   and w :: "'c" assumes "DFS_invar G param s"   and "(v, w) \<in> tree_edges s"   and "v \<in> dom (finished s)" shows "w \<in> dom (finished s) \<and> the_discovered s v < the_discovered s w \<and> the_finished s w < the_finished s v"
assms_1: shows "\<forall>x\<in>X. f x = g x"
boundOutputChainAlpha'': fixes p :: "(name \<times> name) list"   and xvec :: "name list"   and M :: "'d"   and P :: "('d, 'e, 'f) psi"   and yvec :: "name list" assumes "(p \<bullet> xvec) \<sharp>* M"   and "(p \<bullet> xvec) \<sharp>* P"   and "set p \<subseteq> set xvec \<times> set (p \<bullet> xvec)"   and "set xvec \<subseteq> set yvec" shows "\<lparr>\<nu>*yvec\<rparr>M \<prec>' P = \<lparr>\<nu>*p \<bullet> yvec\<rparr>(p \<bullet> M) \<prec>' (p \<bullet> P)"
1: shows "H * e \<le> H * e * (top * H * e)\<^sup>\<star>"
Dup_head_oob: fixes p :: "('b, 'c) fm"   and i :: "'c"   and ps :: "('b, 'c) fm list"   and a :: "'c"   and branch :: "(('b, 'c) fm list \<times> 'c) list"   and xs :: "(nat \<times> nat) set" assumes "Dup p i ((ps, a) # branch) xs" shows "(length branch, length ps) \<notin> xs"
tl_aux_def: shows "tl_aux = drop (length done + 1) auxlist"
Quotient_alt_def: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and Abs :: "'a \<Rightarrow> 'b"   and Rep :: "'b \<Rightarrow> 'a"   and T :: "'a \<Rightarrow> 'b \<Rightarrow> bool" shows "Quotient R Abs Rep T = ((\<forall>a b. T a b \<longrightarrow> Abs a = b) \<and> (\<forall>b. T (Rep b) b) \<and> (\<forall>x y. R x y = (T x (Abs x) \<and> T y (Abs y) \<and> Abs x = Abs y)))"
factors_equal: fixes a :: "'a vec"   and b :: "'a vec"   and c :: "'a vec"   and d :: "'a vec" assumes "a = b"   and "c = d" shows "a + c = b + d"
A20: shows "A2 = 0\<^sub>m n (Suc m)"
monotone_max_enat2: fixes x :: "enat" shows "monotone (\<le>) (\<le>) (max x)"
simps: fixes n :: "integer"   and f :: "integer \<Rightarrow> 'b" shows "IArray.tabulate (n, f) = IArray (map (f \<circ> integer_of_nat) [0..<nat_of_integer n])"
G_def: fixes s :: "'s"   and t :: "'s" shows "G s t = \<integral>\<^sup>+ x. ennreal_of_enat (scount (HLD {t}) (s ## x)) \<partial>T s"
id_i_take_nth_i_drop: fixes f :: "nat \<Rightarrow> 'd"   and n :: "nat" shows "f = f \<Down> n \<frown> ([f n] \<frown> f) \<Up> Suc n"
to_list_p: fixes p :: "real mpoly"   and v :: "nat" shows "p = (\<Sum>(term, i)\<leftarrow>to_list v p. term * Var v ^ i)"
is_Lunit'_def: fixes "term" :: "'b local.term" shows "is_Lunit' term = (\<exists>x6. term = \<^bold>\<l>\<^sup>-\<^sup>1\<^bold>[x6\<^bold>])"
fmfilter_add_distrib: fixes P :: "'c \<Rightarrow> bool"   and m :: "('c, 'd) fmap"   and n :: "('c, 'd) fmap" shows "fmfilter P (m ++\<^sub>f n) = fmfilter P m ++\<^sub>f fmfilter P n"
x_stays_before_y_if_y_not_moved_to_front: fixes q :: "'c"   and xs :: "'c list"   and x :: "'c"   and y :: "'c"   and n :: "nat" assumes "q \<in> set xs"   and "distinct xs"   and "x \<in> set xs"   and "y \<in> set xs"   and "y \<noteq> q"   and "x < y in xs" shows "x < y in mtf2 n q xs"
T_spaceM_stable_2: fixes x :: "'a"   and n :: "nat" assumes "x \<in> space M" shows "(T ^^ n) x \<in> space M"
Collect_cong: fixes P :: "'a::type \<Rightarrow> bool"   and Q :: "'a::type \<Rightarrow> bool" assumes "\<And>x::'a::type. (P::'a::type \<Rightarrow> bool) x = (Q::'a::type \<Rightarrow> bool) x" shows "{x::'a::type. (P::'a::type \<Rightarrow> bool) x} = {x::'a::type. (Q::'a::type \<Rightarrow> bool) x}"
prems_4: shows "\<forall>n\<in>set (tl ms). var g v \<notin> var g ` allDefs g n"
set_remove1_eq: fixes xs :: "'b list"   and x :: "'b" assumes "distinct xs" shows "set (remove1 x xs) = set xs - {x}"
uvw_arcs1_2: shows "(w, v) \<in> parcs G"
le_less: fixes x :: "'a"   and y :: "'a" shows "cless_eq x y = (cless x y \<or> x = y)"
simps_3: fixes a :: "'a"   and b :: "'a" shows "Some a + Some b = Some (a + b)"
process_prop_1_def: fixes P :: "('a list \<times> 'a set) set \<times> 'a list set" shows "process_prop_1 P \<equiv> ([], {}) \<in> fst P"
divide_mod_ring_def: fixes x :: "'a mod_ring"   and y :: "'a mod_ring" shows "x div y = x * (if y = 0 then 0 else y ^ nat (int (CARD('a) - 2)))"
intros_4: fixes n :: "nat"   and l :: "'b tree234"   and m :: "'b tree234"   and m' :: "'b tree234"   and r :: "'b tree234"   and p :: "'b"   and q :: "'b"   and q' :: "'b" assumes "full n l"   and "full n m"   and "full n m'"   and "full n r" shows "full (Suc n) \<langle>l, p, m, q, m', q', r\<rangle>"
exhaust_disc: fixes nat :: "nat" obtains "nat = 0"   | "nat \<noteq> 0"
Efix: fixes \<tau> :: "ty"   and e :: "exp"   and \<rho> :: "val list" shows "E (EFix \<tau> e) \<rho> = Collect (iterate (E e) \<rho>)"
assms: shows "\<parallel>c\<parallel>\<^bsub>t i\<^esub>"
simps_6: fixes v :: "'a rexp" shows "local.le_rexp (Star v) Zero = False"
max_list_mem: fixes xs :: "nat list" assumes "xs \<noteq> []" shows "max_list xs \<in> set xs"
nonequiv_segment: fixes D :: "('c, 'd) Order_scheme"   and a :: "'c" assumes "Worder D"   and "a \<in> carrier D" shows "\<not> ord_equiv D (Iod D (segment D a))"
double_complement: fixes A :: "'b set" shows "- (- A) = A"
may_lock_upd_lock_conv: fixes l :: "('c \<times> nat) option"   and t :: "'c"   and L :: "lock_action" assumes "lock_action_ok l t L" shows "may_lock (upd_lock l t L) t = may_lock l t"
that: fixes q :: "'a \<Rightarrow> 'a" assumes "\<And>b. q b \<in> range h"   and "p = (\<Sum>b\<in>B. q b * b)" shows "thesis"
Rep_matrix_inverse: fixes x :: "'b matrix" shows "Abs_matrix (Rep_matrix x) = x"
VSigmaI: fixes a :: "V"   and A :: "V"   and b :: "V"   and B :: "V \<Rightarrow> V" assumes "a \<in>\<^sub>\<circ> A"   and "b \<in>\<^sub>\<circ> B a" shows "\<langle>a, b\<rangle> \<in>\<^sub>\<circ> VSigma A B"
enat_of_option_simps_4: fixes x :: "nat option" shows "(\<infinity> = enat_of_option x) = (x = None)"
vsv: fixes a :: "V"   and b :: "V"   and c :: "V" assumes "\<langle>a, b\<rangle> \<in>\<^sub>\<circ> xs #\<^sub>\<circ> x"   and "\<langle>a, c\<rangle> \<in>\<^sub>\<circ> xs #\<^sub>\<circ> x" shows "b = c"
vertices_conv_Union_edges2: fixes f :: "face" assumes "distinct (vertices f)" shows "\<V> f = (\<Union>\<^bsub>(a, b)\<in>\<E> f\<^esub> {b})"
splits_at_def: fixes \<delta> :: "symbol list"   and i :: "nat"   and \<alpha> :: "symbol list"   and N :: "symbol"   and \<beta> :: "symbol list" shows "splits_at \<delta> i \<alpha> N \<beta> = (i < length \<delta> \<and> \<alpha> = take i \<delta> \<and> N = \<delta> ! i \<and> \<beta> = drop (Suc i) \<delta>)"
7: shows "eint 3 \<le> val_Zp (f \<bullet> \<one>)"
map_closed: fixes a :: "'a set \<times> 'b" assumes "a \<in> Sigma I \<FF>" shows "rel.Class a \<in> rel.Partition"
lin_depI: fixes A :: "'a mat"   and nc :: "nat"   and v :: "'a vec" assumes "A \<in> carrier_mat n nc"   and "v \<in> carrier_vec nc"   and "v \<noteq> 0\<^sub>v nc"   and "A *\<^sub>v v = 0\<^sub>v n"   and "distinct (cols A)" shows "lin_dep (set (cols A))"
commit_def: fixes x :: "'pub_input"   and e :: "'challenge" shows "commit x e = S x e \<bind> (\<lambda>(a, e, z). return_spmf (a, z))"
iMOD_iprev: fixes t :: "nat"   and r :: "nat"   and m :: "nat" assumes "t \<in> [ r, mod m ]"   and "r < t" shows "iprev t [ r, mod m ] = t - m"
inf_le1: fixes x :: "'c multiset"   and y :: "'c multiset" shows "x \<inter># y \<subseteq># x"
that_2: assumes "LENGTH('a) = 1"   and "a = 1"   and "sint 1 = - 1" shows "P"
transpose_mat_of_rows: fixes n :: "nat"   and vs :: "'a vec list" shows "(mat_of_rows n vs)\<^sup>T = mat_of_cols n vs"
assms_3: shows "(r', s') \<in> R \<times> S"
two_comparisons_into_case_order_9: fixes comp :: "'a \<Rightarrow> 'a \<Rightarrow> order"   and x :: "'a"   and y :: "'a"   and P :: "'b"   and R :: "'b"   and Q :: "'b" assumes "comparator comp" shows "(if x = y then P else if lt_of_comp comp y x then R else Q) = (case comp x y of Eq \<Rightarrow> P | Lt \<Rightarrow> Q | Gt \<Rightarrow> R)"
b3: shows "classical_operator (inv_map (Some \<circ> \<pi>)) o\<^sub>C\<^sub>L classical_operator (Some \<circ> \<pi>) = classical_operator (inv_map (Some \<circ> \<pi>) \<circ>\<^sub>m (Some \<circ> \<pi>))"
appNone: fixes i :: "instr"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list"   and mxs :: "nat"   and T\<^sub>r :: "ty"   and pc :: "nat"   and mpc :: "nat"   and et :: "(nat \<times> nat \<times> char list \<times> nat \<times> nat) list" shows "app i P mxs T\<^sub>r pc mpc et None = True"
alphaAbs_all_equal_or_qAFresh_def: fixes xs :: "'f"   and x :: "'g"   and X :: "('h, 'i, 'f, 'g, 'j) qTerm"   and xs' :: "'f"   and x' :: "'g"   and X' :: "('h, 'i, 'f, 'g, 'j) qTerm" shows "alphaAbs_all_equal_or_qAFresh xs x X xs' x' X' \<equiv> xs = xs' \<and> (\<forall>y. (y = x \<or> qAFresh xs y X) \<and> (y = x' \<or> qAFresh xs y X') \<longrightarrow> X #[[y \<and> x]]_xs #= X' #[[y \<and> x']]_xs)"
f'g: fixes x :: "'a" assumes "x \<in> C - A" shows "f' x = g x"
3: shows "(\<lambda>x. \<Sum>b\<in>basis. F (\<gamma> x) \<bullet> b * (vector_derivative \<gamma> (at x within {0..1}) \<bullet> b)) = (\<lambda>x. (\<Sum>b\<in>basis1. F (\<gamma> x) \<bullet> b * (vector_derivative \<gamma> (at x within {0..1}) \<bullet> b)) + (\<Sum>b\<in>basis2. F (\<gamma> x) \<bullet> b * (vector_derivative \<gamma> (at x within {0..1}) \<bullet> b)))"
assms_2: shows "Case_Labeling.CTXT IC ((''then'', IC, []) # CT') OC1 (ovalidNF L l Q)"
pheap_link: fixes t :: "'b tree" assumes "t \<noteq> \<langle>\<rangle>"   and "pheap t" shows "pheap (link t)"
ncoplanar_perm_23: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "\<not> Coplanar A B C D" shows "\<not> Coplanar D C B A"
conic_sgn_not_1_z_non_zero: fixes p :: "proj2" assumes "conic_sgn p \<noteq> 1" shows "z_non_zero p"
neq_Nil_conv2: fixes xs :: "'b SList.list" shows "(xs \<noteq> []) = (\<exists>y ys. xs = y # ys)"
silent_moves_intra_path_obs: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and Exit :: "'b"   and Def :: "'b \<Rightarrow> 'c set"   and Use :: "'b \<Rightarrow> 'c set"   and ParamDefs :: "'b \<Rightarrow> 'c list"   and ParamUses :: "'b \<Rightarrow> 'c set list"   and m' :: "'b"   and m :: "'b"   and S :: "'b SDG_node set"   and s :: "(('c \<Rightarrow> 'd option) \<times> 'e) list"   and msx' :: "'b list" assumes "SDG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main Exit Def Use ParamDefs ParamUses"   and "m' \<in> CFG.obs_intra sourcenode targetnode kind valid_edge m (SDG.SDG_to_CFG_set (SDG.HRB_slice sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses S))"   and "length s = length (m # msx')"   and "\<forall>m\<in>set msx'. CFG.return_node targetnode valid_edge get_return_edges m" obtains as' :: "'a list" where "SDG.silent_moves sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses S (SDG.slice_kind sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses S) (m # msx') s as' (m' # msx') s"
prems_5: shows "Marker \<in> set (msgs (S t i) cid)"
axioms_1: fixes A :: "V"   and n :: "V"   and f :: "V" assumes "nop_v11 A n f" shows "v11 f"
cong: fixes prf_fun :: "'a \<Rightarrow> bool list \<Rightarrow> bool list"   and prf_dlen :: "nat"   and upf_fun :: "'b \<Rightarrow> bool list \<Rightarrow> 'c" shows "simple_cipher.decrypt prf_fun prf_dlen upf_fun \<equiv> simple_cipher.decrypt prf_fun prf_dlen upf_fun"
guard_intros_2: fixes b :: "bool" assumes "\<not> b" shows "guard b = None"
ide_dom: fixes f :: "'s" assumes "S.arr f" shows "S.ide (S.dom f)"
landau_symbol_axioms: shows "landau_symbol smallo smallo smallo"
norm_hyp2_rep_lt_1: fixes p :: "hyp2" shows "norm (hyp2_rep p) < 1"
no_cs_change_if_no_event: fixes c :: "('a, 'b, 'c) configuration"   and ev :: "('a, 'b, 'c) event"   and c' :: "('a, 'b, 'c) configuration"   and i :: "nat" assumes "c \<turnstile> ev \<mapsto> c'"   and "isRecv ev \<longrightarrow> getId ev \<noteq> i"   and "regular_event ev" shows "cs c i = cs c' i"
ctr_1: fixes A :: "'c llist set" assumes "\<forall>x\<in>A. lnull x" shows "lSup A = LNil"
WHILE_bounded: shows "m0 \<bind> WHILE cond step = filter_nb cond (Sup {bounded_WHILE n cond step m0 |n. True})"
closedin_Union_locally_finite_closure: fixes X :: "'c topology"   and \<A> :: "'c set set" assumes "locally_finite_in X \<A>" shows "closedin X (\<Union> ((closure_of) X ` \<A>))"
ge: fixes Y :: "'a multiset" assumes "Y \<in> A" shows "Y \<subseteq># X"
distinct_1: fixes x1 :: "'b" shows "Atom x1 \<noteq> \<bottom>"
filter_set: fixes P :: "'c \<Rightarrow> bool"   and xs :: "'c list" shows "Set.filter P (set xs) = set (filter P xs)"
eqvt_lambda: fixes p :: "perm"   and f :: "'b \<Rightarrow> 'c" shows "p \<bullet> f \<equiv> \<lambda>x. p \<bullet> f (unpermute p x)"
collapse_2: fixes sum :: "'c + 'd" assumes "\<not> isl sum" shows "Inr (projr sum) = sum"