hcju/isabelleps huggingface dataset:
rPar1: fixes \<Psi>\<^sub>Q :: "'b"   and P' :: "('a, 'b, 'c) psi"   and A\<^sub>Q :: "name list" assumes "\<Psi> \<otimes> \<Psi>\<^sub>Q \<rhd> P \<longmapsto> \<alpha> \<prec> P'"   and "extractFrame Q = \<langle>A\<^sub>Q, \<Psi>\<^sub>Q\<rangle>"   and "distinct A\<^sub>Q"   and "A\<^sub>Q \<sharp>* \<Psi>"   and "A\<^sub>Q \<sharp>* P"   and "A\<^sub>Q \<sharp>* \<alpha>"   and "A\<^sub>Q \<sharp>* C" shows "Prop \<alpha> (P' \<parallel> Q)"
Abs_viewres_inverse_None: shows "Rep_viewres (Abs_viewres None) = None"
cobounded2: fixes a :: "'c DBMEntry"   and b :: "'c DBMEntry" shows "linordered_monoid.min a b \<preceq> b"
member_methd: fixes G :: "prog"   and sig :: "sig"   and m :: "qtname \<times> methd"   and C :: "qtname" assumes "G \<turnstile>Methd sig m member_of C"   and "ws_prog G" shows "methd G C sig = Some m"
unat_word_ariths_2: fixes a :: "'a word"   and b :: "'a word" shows "unat (a * b) = unat a * unat b mod 2 ^ LENGTH('a)"
all_post: fixes self :: "'\<AA> state \<times> '\<AA> state \<Rightarrow> '\<alpha>0" shows "\<forall>\<sigma> \<sigma>' \<sigma>''. ((\<sigma>, \<sigma>') \<Turnstile> PRE self) = ((\<sigma>, \<sigma>'') \<Turnstile> PRE self)"
star_rtc2: fixes x :: "'a" shows "(1' + x ; x \<le> x) = (x = x\<^sup>\<star>)"
assms_2: shows "(v, w) \<in> edges_of_walk xs"
psubst_append: fixes \<theta>\<^sub>1 :: "(name \<times> trm) list"   and \<theta>\<^sub>2 :: "(name \<times> trm) list"   and t :: "trm" assumes "supp (map fst (\<theta>\<^sub>1 @ \<theta>\<^sub>2)) \<sharp>* map snd (\<theta>\<^sub>1 @ \<theta>\<^sub>2)" shows "(\<theta>\<^sub>1 @ \<theta>\<^sub>2)\<lparr>t\<rparr> = \<theta>\<^sub>2\<lparr>\<theta>\<^sub>1\<lparr>t\<rparr>\<rparr>"
cInf_lower: fixes x :: "'c multiset"   and X :: "'c multiset set" assumes "x \<in> X"   and "subset_mset.bdd_below X" shows "\<Sqinter> X \<subseteq># x"
impl_of_inverse: fixes t :: "('c, 'd) mapping_rbt" shows "Mapping_RBT (impl_of t) = t"
intro: fixes m1_ops :: "('a, 'b, 'c, 'd) map_ops_scheme"   and m2_ops :: "('a, 'e, 'b, 'f) map_ops_scheme"   and s3_ops :: "('g, 'e, 'h) set_ops_scheme"   and m1_mvif :: "('a \<Rightarrow> 'b \<Rightarrow> 'b option) \<Rightarrow> 'c \<Rightarrow> 'c" assumes "StdMap m1_ops"   and "StdMap m2_ops"   and "StdSet s3_ops"   and "map_value_image_filter (map_op_\<alpha> m1_ops) (map_op_invar m1_ops) (map_op_\<alpha> m1_ops) (map_op_invar m1_ops) m1_mvif" shows "GraphByMap m1_ops m2_ops s3_ops m1_mvif"
euclidean_representation: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and norm :: "'a \<Rightarrow> real"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and sgn :: "'a \<Rightarrow> 'a"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and inner :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and Basis :: "'a set"   and x :: "'a" assumes "class.euclidean_space minus dist norm plus zero uminus scaleR sgn uniformity open inner Basis" shows "comm_monoid_add.sum plus zero (\<lambda>b. scaleR (inner x b) b) Basis = x"
p: fixes c :: "'a" assumes "c \<in> set cs" shows "c \<in> carrier G \<and> irreducible G c"
assms_1: shows "\<Gamma> \<turnstile> e, eP, eV : \<tau>"
interior_of_eq_empty_alt: fixes X :: "'c topology"   and S :: "'c set" shows "(X interior_of S = {}) = (\<forall>T. openin X T \<and> T \<noteq> {} \<longrightarrow> T - S \<noteq> {})"
pos_mod_bound: fixes b :: "'b"   and a :: "'b" assumes "(0::'b) < b" shows "a mod b < b"
aezfun_setspan_proj_in_setspan: fixes A :: "'b set"   and x :: "('a, 'b) aezfun" shows "aezfun_setspan_proj A x \<in> aezfun_setspan A"
ord_isom_func: fixes D :: "('c, 'd) Order_scheme"   and E :: "('e, 'f) Order_scheme"   and f :: "'c \<Rightarrow> 'e" assumes "Order D"   and "Order E"   and "ord_isom D E f" shows "f \<in> carrier D \<rightarrow> carrier E"
module_hom_snd: fixes s1 :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and s2 :: "'a \<Rightarrow> 'c \<Rightarrow> 'c" assumes "module_prod s1 s2" shows "module_hom (module_prod.scale s1 s2) s2 snd"
weak_respection_of_barbs_and_closures_3: fixes Rel :: "('a \<times> 'a) set"   and CWB :: "('a, 'b) calculusWithBarbs" assumes "rel_weakly_respects_barbs Rel CWB" shows "rel_weakly_respects_barbs (Rel\<^sup>+) CWB"
permute_perm_def: fixes p :: "'b perm"   and q :: "'b perm" shows "permute_perm p q = p + q - p"
ht: fixes i :: "'a" assumes "i \<in> I" shows "homotopic_with (p i) (X i) (Y i) (f i) (g i)"
update_accessor_congruence_foldE: fixes upd :: "('a \<Rightarrow> 'a) \<Rightarrow> 'b \<Rightarrow> 'b"   and ac :: "'b \<Rightarrow> 'a"   and r :: "'b"   and r' :: "'b"   and v' :: "'a"   and f :: "'a \<Rightarrow> 'a"   and f' :: "'a \<Rightarrow> 'a" assumes "iso_tuple_update_accessor_cong_assist upd ac"   and "r = r'"   and "ac r' = v'"   and "\<And>v. v' = v \<Longrightarrow> f v = f' v" shows "upd f r = upd f' r'"
new_discover: fixes s :: "('v, 'es) state_scheme"   and v :: "'v"   and s' :: "('v, 'es) state_scheme" assumes "DFS_invar G param s"   and "I s"   and "cond s"   and "\<not> is_break param s"   and "v \<notin> dom (discovered s)"   and "discovered s' = discovered s(v \<mapsto> counter s)"   and "finished s' = finished s"   and "counter s' = Suc (counter s)"   and "stack s' = v # stack s"   and "back_edges s' = back_edges s"   and "cross_edges s' = cross_edges s"   and "tree_edges s \<subseteq> tree_edges s'"   and "state.more s' = state.more s" shows "on_new_root param v s' \<le>\<^sub>n SPEC (\<lambda>x. DFS_invar G param (s'\<lparr>state.more := x\<rparr>) \<longrightarrow> I (s'\<lparr>state.more := x\<rparr>))"
ftif_cons_botI: fixes P :: "cld list"   and ctx :: "ctx"   and cl :: "cl"   and f :: "char list"   and fd_list :: "fd list" assumes "find_type P ctx cl None" shows "ftype_in_fds P ctx ([fd_def cl f] @ fd_list) f ty_opt_bot_bot"
resultant_const_1: fixes a :: "'b"   and q :: "'b poly" shows "resultant [:a:] q = a ^ degree q"
convergence: fixes j :: "nat" assumes "t j \<le> t0 + e1'" shows "dist (solution (t j)) (euler f x0 t j) \<le> sqrt (real DIM('a)) * (B + 1) / 2 * (exp (B' * e1' + 1) - 1) * max_stepsize j"
bind_option_def: fixes fail :: "'e"   and x :: "'f option"   and f :: "'f \<Rightarrow> 'e" shows "local.bind_option fail x f = (case x of None \<Rightarrow> fail | Some x' \<Rightarrow> f x')"
FtorCompIdDef: fixes X :: "'a"   and F :: "('a, 'd, 'b, 'e, 'c, 'f) Functor"   and G :: "('d, 'g, 'e, 'h, 'f, 'i) Functor" assumes "X \<in> obj\<^bsub>CatDom F\<^esub>"   and "PreFunctor (F ;;: G)"   and "F \<approx>>;;; G" shows "F ;;: G @@ X = G @@ F @@ X"
simple_closed_path_winding_number_pos: fixes \<gamma> :: "real \<Rightarrow> complex"   and z :: "complex" assumes "simple_path \<gamma>"   and "pathfinish \<gamma> = pathstart \<gamma>"   and "z \<notin> path_image \<gamma>"   and "0 < Re (winding_number \<gamma> z)" shows "winding_number \<gamma> z = 1"
cases: fixes r :: "'b arrow_of_spans_data" obtains Chn :: "'b" and Dom :: "'b span_data" and Cod :: "'b span_data" where "r = \<lparr>Chn = Chn, Dom = Dom, Cod = Cod\<rparr>"
nonstutseq_def: fixes s :: "nat \<Rightarrow> 'b" shows "nonstutseq s \<equiv> \<forall>i. s i = s (Suc i) \<longrightarrow> (\<forall>j>i. s i = s j)"
has_integral_stretch_cart: fixes f :: "(real, 'a) vec \<Rightarrow> 'b"   and i :: "'b"   and a :: "(real, 'a) vec"   and b :: "(real, 'a) vec"   and m :: "'a \<Rightarrow> real" assumes "(f has_integral i) (cbox a b)"   and "\<And>k. m k \<noteq> 0" shows "((\<lambda>x. f (\<chi>k. m k * x $ k)) has_integral i /\<^sub>R \<bar>prod m UNIV\<bar>) ((\<lambda>x. \<chi>k. x $ k / m k) ` cbox a b)"
assoc'_simps_1: fixes f :: "'a"   and g :: "'a"   and h :: "'a" assumes "B.ide f"   and "B.ide g"   and "B.ide h"   and "src\<^sub>B f = trg\<^sub>B g"   and "src\<^sub>B g = trg\<^sub>B h" shows "B.arr (B.\<a>' f g h)"
that: fixes M' :: "nat \<Rightarrow> nat \<Rightarrow> real DBMEntry" assumes "[M]\<^bsub>v,n\<^esub> = [M']\<^bsub>v,n\<^esub>"   and "[DBM_Normalization.norm M (\<lambda>x. real ((k \<circ> v') x)) n]\<^bsub>v,n\<^esub> = [DBM_Normalization.norm M' (\<lambda>x. real ((k \<circ> v') x)) n]\<^bsub>v,n\<^esub>"   and "\<forall>i\<le>n. M' i i = \<one>"   and "canonical M' n" shows "thesis"
a0i_inv1_iagreeI: fixes x :: "'b a0n_state" assumes "\<And>h d. \<lbrakk>set h \<subseteq> good; d \<notin> corrupted x\<rbrakk> \<Longrightarrow> signals x (Commit h d) \<le> signals x (Running h d)" shows "x \<in> a0i_inv1_iagree"
vmatch_dom: fixes pat :: "pat"   and v :: "value"   and env :: "(name, value) fmap" assumes "vmatch pat v = Some env" shows "fmdom env = patvars pat"
assms_3: shows "\<parallel>c\<parallel>\<^bsub>lnth t i\<^esub>"
group_congruence_axioms: shows "group_congruence G (\<cdot>) \<one> E"
le_iff_join: fixes x :: "int"   and y :: "int" assumes "x \<in> UNIV"   and "y \<in> UNIV" shows "(x \<le> y) = (x = x \<sqinter>\<^bsub>\<lparr>carrier = UNIV, eq = (=), le = (\<le>)\<rparr>\<^esub> y)"
inv_into_into: fixes x :: "'a"   and f :: "'b \<Rightarrow> 'a"   and A :: "'b set" assumes "x \<in> f ` A" shows "inv_into A f x \<in> A"
nonrec_def: fixes \<Gamma> :: "(Vars.var \<times> exp) list" shows "nonrec \<Gamma> = (\<exists>x e. \<Gamma> = [(x, e)] \<and> x \<notin> fv e)"
join: fixes subst :: "('a, 'b, 'c) expr \<Rightarrow> 'c \<Rightarrow> ('a, 'b, 'c) expr \<Rightarrow> ('a, 'b, 'c) expr"   and s :: "'a \<Rightarrow> (('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('a, 'b, 'c) expr) option"   and r :: "'a"   and \<sigma> :: "'b \<Rightarrow> ('a, 'b, 'c) val option"   and \<tau> :: "'b \<Rightarrow> ('a, 'b, 'c) val option"   and \<E> :: "('a, 'b, 'c) cntxt"   and r' :: "'a"   and \<sigma>' :: "'b \<Rightarrow> ('a, 'b, 'c) val option"   and \<tau>' :: "'b \<Rightarrow> ('a, 'b, 'c) val option"   and v :: "('a, 'b, 'c) val" assumes "substitution subst"   and "s r = Some (\<sigma>, \<tau>, \<E> [Rjoin (VE (Rid r'))])"   and "s r' = Some (\<sigma>', \<tau>', VE v)" shows "substitution.revision_step subst r s ((s(r \<mapsto> (\<sigma>, \<tau>;;\<tau>', \<E> [VE (CV Unit)])))(r' := None))"
psubset_insert_iff: fixes A :: "'b set"   and x :: "'b"   and B :: "'b set" shows "(A \<subset> insert x B) = (if x \<in> B then A \<subset> B else if x \<in> A then A - {x} \<subset> B else A \<subseteq> B)"
s_chop: fixes i :: "nat"   and t :: "('a, 'b, 'c) event list"   and c :: "('a, 'b, 'c) configuration" assumes "i \<le> length t" shows "s c t i = s c (take i t) i"
subset_diff: fixes B :: "'c set"   and A :: "'c set"   and g :: "'c \<Rightarrow> 'd" assumes "B \<subseteq> A"   and "finite A" shows "prod g A = prod g (A - B) * prod g B"
simul: shows "\<forall>i\<le>k. if pcss i = [] then CSS (f i) = css \<and> CS (f i) = pcs i @ cs else CS (f i) = pcs i \<and> CSS (f i) = butlast (pcss i) @ [(fst (last (pcss i)) @ cs, snd (last (pcss i)) @ cs)] @ css"
hom_a_inv: fixes G :: "('e, 'f) ring_scheme"   and H :: "('g, 'h) ring_scheme"   and h :: "'e \<Rightarrow> 'g"   and x :: "'e" assumes "abelian_group_hom G H h"   and "x \<in> carrier G" shows "h (\<ominus>\<^bsub>G\<^esub> x) = \<ominus>\<^bsub>H\<^esub> h x"
in_hom_char: fixes f :: "'a"   and a :: "'a"   and b :: "'a" shows "L.in_hom f a b = (L.arr a \<and> L.arr b \<and> L.arr f \<and> \<guillemotleft>f : a \<rightarrow> b\<guillemotright>)"
nAct_enat_the_nat: fixes n :: "enat"   and c :: "'id"   and t :: "cnf llist" assumes "n \<noteq> \<infinity>" shows "enat (the_enat \<langle>c #\<^bsub>n\<^esub>t\<rangle>) = \<langle>c #\<^bsub>n\<^esub>t\<rangle>"
winningAllocationPossible: fixes b :: "'a \<times> 'b set \<Rightarrow> 'c"   and G :: "'b set"   and N :: "'a set" shows "winningAllocationRel N G (\<subseteq>) b (allAllocations N G)"
logic_actual_nec_2: fixes \<phi> :: "\<o>"   and \<psi> :: "\<o>" shows "[[\<^bold>\<A>(\<phi> \<^bold>\<rightarrow> \<psi>) \<^bold>\<equiv> (\<^bold>\<A>\<phi> \<^bold>\<rightarrow> \<^bold>\<A>\<psi>)]]"
IH: fixes k :: "nat" assumes "k \<le> n" shows "ns k \<in> carrier Zp \<and> val_Zp (f' \<bullet> ns k) = val_Zp (f' \<bullet> a) \<and> eint 2 * val_Zp (f' \<bullet> a) + eint (2 ^ k) * t \<le> val_Zp (f \<bullet> ns k)"
step: fixes m :: "enat"   and n :: "enat" assumes "(m, n) \<in> X" shows "m = 0 \<and> n = 0 \<or> (\<exists>m' n'. m = eSuc m' \<and> n = eSuc n' \<and> ((m', n') \<in> X \<or> m' = n'))"
cC: fixes n :: "nat" shows "dist c ((f ^^ n) x) \<le> C"
iter_hcomp_cong: fixes vs :: "'a fset"   and \<phi> :: "'a \<Rightarrow> interface"   and \<phi>' :: "'a \<Rightarrow> interface" assumes "\<forall>v\<in>fset vs. \<phi> v = \<phi>' v" shows "iter_hcomp vs \<phi> = iter_hcomp vs \<phi>'"
simps_5: fixes f1 :: "'a"   and f2 :: "'b \<Rightarrow> 'b list \<Rightarrow> 'a"   and x21 :: "'b"   and x22 :: "'b list" shows "(case x21 # x22 of [] \<Rightarrow> f1 | x # xa \<Rightarrow> f2 x xa) = f2 x21 x22"
steps_Abrupt_prop: fixes \<Gamma> :: "'b \<Rightarrow> ('a, 'b, 'c) com option"   and c :: "('a, 'b, 'c) com"   and s :: "('a, 'c) xstate"   and c' :: "('a, 'b, 'c) com"   and s' :: "('a, 'c) xstate"   and t :: "'a" assumes "\<Gamma>\<turnstile> (c, s) \<rightarrow>\<^sup>* (c', s')"   and "s = Abrupt t" shows "s' = Abrupt t"
prems_1: fixes ea :: "exp" assumes "ea \<in> snd ` set \<Gamma>" shows "eqvt_at cCCexp_sumC ea"
split_conv: fixes f :: "'c \<Rightarrow> 'd \<Rightarrow> 'a"   and a :: "'c"   and b :: "'d" shows "(case (a, b) of (c, d) \<Rightarrow> f c d) = f a b"
Union_finite_multiset: fixes M :: "'a multiset" shows "finite (\<Union> {supp x |x. x \<in># M})"
SigmaE: fixes c :: "'b \<times> 'c"   and A :: "'b set"   and B :: "'b \<Rightarrow> 'c set" assumes "c \<in> Sigma A B" obtains x :: "'b" and y :: "'c" where "x \<in> A" and "y \<in> B x" and "c = (x, y)"
intros_24: fixes i :: "nat"   and \<C> :: "t_context"   and t :: "t" assumes "i < length (local \<C>)"   and "local \<C> ! i = t" shows "\<C> \<turnstile> [Get_local i] : [] _> [t]"
same_insert: fixes e1 :: "'a \<times> 'b \<times> bool"   and e2 :: "'a \<times> 'b \<times> bool"   and xs :: "('a \<times> ('a, 'b) operation) event list"   and i :: "nat"   and i1 :: "'a"   and n1 :: "'a option"   and i2 :: "'a"   and n2 :: "'a option" assumes "fst e1 = fst e2"   and "xs prefix of i"   and "(i1, Insert e1 n1) \<in> set (node_deliver_messages xs)"   and "(i2, Insert e2 n2) \<in> set (node_deliver_messages xs)" shows "Insert e1 n1 = Insert e2 n2"
insert: fixes A :: "'b fset"   and x :: "'b" assumes "A \<noteq> {||}" shows "fMin (finsert x A) = min x (fMin A)"
assms_1: shows "summable (\<lambda>n. norm (g n))"
sa: shows "Sig aS\<^bsub>T\<^esub> \<triangleright> Vx : A \<turnstile> e : B"
simps_2: fixes a :: "'b"   and h :: "'b tree" shows "U (Insert a) [h] = log 2 (real (size h + 1)) + 1"
strip_list_comb: fixes f :: "'a"   and ys :: "'a list" shows "strip_comb (list_comb f ys) = (fst (strip_comb f), snd (strip_comb f) @ ys)"
IH: assumes "\<lbrakk>M; map (\<lambda>d. \<forall>X\<langle>\<or>\<noteq>: [(pair (t, s), pair (snd d))]\<rangle>\<^sub>s\<^sub>t) (filter (\<lambda>d. d \<notin> set Di) (dbproj i D))\<rbrakk>\<^sub>d \<I>"   and "\<forall>a\<in>{(i, t, s)} \<union> set D \<union> set Di. case a of (j, p) \<Rightarrow> \<forall>a\<in>{(i, t, s)} \<union> set D \<union> set Di. case a of (k, q) \<Rightarrow> (\<exists>\<delta>. Unifier \<delta> (pair p) (pair q)) \<longrightarrow> j = k" shows "\<lbrakk>M; map (\<lambda>d. \<forall>X\<langle>\<or>\<noteq>: [(pair (t, s), pair (snd d))]\<rangle>\<^sub>s\<^sub>t) (filter (\<lambda>d. d \<notin> set Di) D)\<rbrakk>\<^sub>d \<I>"
labels_sterms_labels: fixes \<Gamma> :: "'e \<Rightarrow> ('f, 'g, 'e, 'h) seqp"   and p' :: "('f, 'g, 'e, 'h) seqp"   and p :: "('f, 'g, 'e, 'h) seqp"   and l :: "'h" assumes "wellformed \<Gamma>"   and "p' \<in> sterms \<Gamma> p"   and "l \<in> labels \<Gamma> p'" shows "l \<in> labels \<Gamma> p"
skl1_step2_def: fixes Rb :: "fid_t"   and A :: "agent"   and B :: "agent"   and Ni :: "msg"   and gnx :: "msg" shows "skl1_step2 Rb A B Ni gnx \<equiv> {(s, s'). guessed_runs Rb = \<lparr>role = Resp, owner = B, partner = A\<rparr> \<and> Rb \<notin> dom (progress s) \<and> sklvl1.guessed_frame Rb xgnx = Some gnx \<and> sklvl1.guessed_frame Rb xni = Some Ni \<and> sklvl1.guessed_frame Rb xsk = Some (Exp gnx (NonceF (Rb $ ny))) \<and> s' = s\<lparr>progress := progress s(Rb \<mapsto> {xny, xni, xnr, xgny, xgnx, xsk}), signalsInit := if can_signal s A B then addSignal (signalsInit s) (Running A B \<langle>Ni, NonceF (Rb $ nr), Exp gnx (NonceF (Rb $ ny))\<rangle>) else signalsInit s, signalsInit2 := if can_signal s A B then addSignal (signalsInit2 s) (Running A B (Exp gnx (NonceF (Rb $ ny)))) else signalsInit2 s\<rparr>}"
hyps: shows "snd (foldl (echelon_form_of_column_k_det bezout) (n, A, 0) [0..<k]) = foldl (echelon_form_of_column_k bezout) (A, 0) [0..<k]"
locally_irrefl_def: fixes R :: "('b \<times> 'b) set"   and A :: "'b multiset" shows "locally_irrefl R A \<equiv> \<forall>t. t \<in># A \<longrightarrow> (t, t) \<notin> R"
cond_alt: shows "cond = (\<lambda>s. (V0 \<subseteq> dom (discovered s) \<longrightarrow> stack s \<noteq> []) \<and> \<not> is_break param s)"
ideal_ring_multiple1: fixes R :: "('d, 'e) Ring_scheme"   and I :: "'d set"   and x :: "'d"   and r :: "'d" assumes "Ring R"   and "ideal R I"   and "x \<in> I"   and "r \<in> carrier R" shows "x \<cdot>\<^sub>r\<^bsub>R\<^esub> r \<in> I"
ffb_eq: fixes F :: "'a \<Rightarrow> 'b set"   and S :: "'b set" shows "fb\<^sub>\<F> F S = {s. \<forall>s'. s2p (F s) s' \<longrightarrow> s2p S s'}"
assoc_naturality\<^sub>A\<^sub>W\<^sub>C: fixes \<tau> :: "'a"   and \<mu> :: "'a"   and \<nu> :: "'a" assumes "\<tau> \<star> \<mu> \<noteq> null"   and "\<mu> \<star> \<nu> \<noteq> null" shows "\<a>[cod \<tau>, cod \<mu>, cod \<nu>] \<cdot> ((\<tau> \<star> \<mu>) \<star> \<nu>) = (\<tau> \<star> \<mu> \<star> \<nu>) \<cdot> \<a>[local.dom \<tau>, local.dom \<mu>, local.dom \<nu>]"
not_dbm_le_le_impl: fixes a :: "'b"   and b :: "'b" assumes "\<not> Le a \<prec> Le b" shows "b \<le> a"
assoc_iff_assoc_mult: fixes a :: "'a"   and b :: "'a" assumes "a \<in> carrier R"   and "b \<in> carrier R" shows "a \<sim> b = a \<sim>\<^bsub>Ring_Divisibility.mult_of R\<^esub> b"
assms: shows "\<Gamma>,High \<turnstile> while (b) c'"
IH_1: fixes l :: "float poly"   and r :: "float poly" assumes "(l, r) = split_by_prec prec c" shows "Ipoly args (map_poly real_of_float c) = Ipoly args (map_poly real_of_float l) + Ipoly args (map_poly real_of_float r) \<and> num_params l \<le> num_params c \<and> num_params r \<le> num_params c"
div_distrib: fixes i :: "nat"   and n :: "nat"   and a :: "nat"   and b :: "nat" assumes "i < n" shows "(a * base ^ n + b) div base ^ i mod base = b div base ^ i mod base"
right_mult_zero_mat': fixes A :: "'a mat"   and n :: "nat"   and nc :: "nat" assumes "dim_col A = n" shows "A * 0\<^sub>m n nc = 0\<^sub>m (dim_row A) nc"
add: fixes b1 :: "('a, 'm) vec"   and b2 :: "('a, 'm) vec" shows "f (b1 + b2) = f b1 + f b2"
that: fixes A :: "'a multiset"   and B :: "'a multiset" shows "f (A + B) = bind (f A) (\<lambda>x. bind (f B) (\<lambda>y. return (x + y)))"
invariant_after_cswitch: shows "\<forall>s n. invariant s \<longrightarrow> invariant (cswitch n s)"
moebius_rotation_inv_rotation: fixes \<phi> :: "real"   and z :: "complex_homo" shows "moebius_pt (moebius_rotation \<phi>) (moebius_pt (moebius_rotation (- \<phi>)) z) = z"
simps_2: fixes s :: "('s, 'v) tm"   and t :: "('s, 'v) tm" shows "local.kbo_std.wt (App s t) = local.kbo_std.wt s - 0 + local.kbo_std.wt t"
shift1_nz: fixes p :: "'b poly" shows "shift1 p \<noteq> 0\<^sub>p"
rejectD: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and r :: "'c rule list"   and s :: "state"   and t :: "state"   and m :: "'c match_expr" assumes "\<Gamma>,\<gamma>,p\<turnstile> \<langle>r, s\<rangle> \<Rightarrow> t"   and "r = [Rule m Reject]"   and "matches \<gamma> m p"   and "s = Undecided" shows "t = Decision FinalDeny"
Fract_conv_to_fract: fixes a :: "'a"   and b :: "'a" shows "Fraction_Field.Fract a b = to_fract a / to_fract b"
terminal_chain_decision: fixes rs :: "'c rule list"   and \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and t :: "state" assumes "terminal_chain rs"   and "\<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>rs, Undecided\<rangle> \<Rightarrow> t" shows "\<exists>X. t = Decision X"
ind_one_str_def: fixes V :: "'a set" shows "ind_one_str V \<equiv> \<one>\<^bsub>U \<inter> V\<^esub>"
l_i_unique: fixes a :: "'a"   and b :: "'a" assumes "a \<in> carrier G"   and "b \<in> carrier G"   and "b \<cdot> a = \<one>" shows "\<rho> a = b"
RTranClp_RTranCl_eq: fixes succs :: "'b \<Rightarrow> 'b list" shows "RTranClp succs = (\<lambda>x. in_RTranCl x succs)"
A: shows "AE x in S \<Otimes>\<^sub>M T. Px (fst x) = 0 \<longrightarrow> Pxy x = 0"
swap_cs_Snapshot_Recv: fixes c :: "('a, 'b, 'c) configuration"   and ev :: "('a, 'b, 'c) event"   and d :: "('a, 'b, 'c) configuration"   and ev' :: "('a, 'b, 'c) event"   and e :: "('a, 'b, 'c) configuration"   and d' :: "('a, 'b, 'c) configuration"   and e' :: "('a, 'b, 'c) configuration"   and i :: "nat" assumes "c \<turnstile> ev \<mapsto> d"   and "d \<turnstile> ev' \<mapsto> e"   and "isSnapshot ev"   and "isRecv ev'"   and "c \<turnstile> ev' \<mapsto> d'"   and "d' \<turnstile> ev \<mapsto> e'"   and "occurs_on ev \<noteq> occurs_on ev'" shows "cs e i = cs e' i"
below_ord_zero: fixes x :: "nat \<Rightarrow> int"   and n :: "nat" assumes "x \<in> carrier Zp"   and "x (Suc n) \<noteq> 0" shows "ord_Zp x \<le> int n"
eq_rat_1: fixes b :: "int"   and d :: "int"   and a :: "int"   and c :: "int" assumes "b \<noteq> 0"   and "d \<noteq> 0" shows "(Fract a b = Fract c d) = (a * d = c * b)"
that: fixes k :: "'a"   and Ks :: "'a list" assumes "k \<in> K - {\<zero>}"   and "set Ks \<subseteq> K"   and "length Ks = length Us"   and "combine (k # Ks) (a # Us) = \<zero>" shows "thesis"
sets_stream_space_cong: fixes M :: "'c measure"   and N :: "'c measure" assumes "sets M = sets N" shows "sets (stream_space M) = sets (stream_space N)"
type: shows "P,E,h \<turnstile> es [:]\<^bsub>NT\<^esub> T'' # Ts''"
assms_4: shows "ks \<notin> specKeysSecrets Q"
fmdom'_notD: fixes x :: "'a"   and m :: "('a, 'b) fmap" assumes "x \<notin> fmdom' m" shows "fmlookup m x = None"
inject_1: fixes x1 :: "'b"   and y1 :: "'b" shows "(Atom x1 = Atom y1) = (x1 = y1)"
a_redu_subst2: fixes M :: "trm"   and M' :: "trm"   and c :: "coname"   and y :: "name"   and P :: "trm" assumes "M \<longrightarrow>\<^sub>a M'" shows "M{c:=y.P} \<longrightarrow>\<^sub>a* M'{c:=y.P}"
Act_def: shows "Act \<equiv> (id ---> Rep_formula ---> Abs_formula) Act\<^sub>\<alpha>"
bezout_matrix_works1: fixes bezout :: "'d \<Rightarrow> 'd \<Rightarrow> 'd \<times> 'd \<times> 'd \<times> 'd \<times> 'd"   and a :: "'e"   and b :: "'e"   and A :: "(('d, 'f) vec, 'e) vec"   and j :: "'f" assumes "is_bezout_ext bezout"   and "a \<noteq> b" shows "(bezout_matrix A a b j bezout ** A) $ a $ j = snd (snd (snd (snd (bezout (A $ a $ j) (A $ b $ j)))))"
gpath_def: fixes A :: "('d, 'c) nba"   and w :: "'d stream"   and V :: "(nat \<times> 'c) set" shows "gpath A w V \<equiv> transition_system.path (\<lambda>x _. x) (\<lambda>u (k, p). w !! k \<in> alphabet A \<and> u \<in> {Suc k} \<times> transition A (w !! k) p \<inter> V)"
subs: shows "subspace {x + y |x y. x \<in> S \<and> a \<bullet> y = 0}"
flrestriction_fdomain: fixes r :: "V" assumes "fbrelation r" shows "r \<restriction>\<^sup>l\<^sub>\<bullet> \<D>\<^sub>\<bullet> r = r"
Rules_def: shows "Rules = transl_rules T \<union> standard_rules C L"
enum_fun_def: shows "enum = map (\<lambda>ys. the \<circ> map_of (zip enum ys)) (List.n_lists (length enum) enum)"
onorm_eq_0: fixes f :: "'b \<Rightarrow> 'c" assumes "bounded_linear f" shows "(onorm f = 0) = (\<forall>x. f x = (0::'c))"
simps_2: fixes \<phi> :: "'fmla"   and \<phi>s :: "'fmla list" shows "lcnj (\<phi> # \<phi>s) = cnj \<phi> (lcnj \<phi>s)"
assms_7: shows "common_reparam_exists one_chain1 one_chain2"