hcju/isabelleps huggingface dataset:
Q_sub: shows "pp_of_term ` {v. v \<prec>\<^sub>t u \<and> d (pp_of_term v) \<le> dgrad_max d \<and> component_of_term v < length fs \<and> \<not> is_RB_in d rword G v} \<subseteq> dgrad_set d (dgrad_max d)"
fin_Pair: fixes x :: "'a" shows "finite (Pair x -` X)"
elims_2: fixes x :: "Core.term" assumes "beta_reducible x" obtains uu_ :: "typ" and uv_ :: "Core.term" and uw_ :: "Core.term" where "x = Abs uu_ uv_ $ uw_"   | ux_ :: "typ" and t :: "Core.term" where "x = Abs ux_ t" and "beta_reducible t"   | v :: "String.literal" and va :: "typ" and u :: "Core.term" where "x = Ct v__ va__ $ u__" and "beta_reducible (Ct v__ va__) \<or> beta_reducible u__"   | v :: "variable" and va :: "typ" and u :: "Core.term" where "x = Fv vb__ vaa__ $ ua__" and "beta_reducible (Fv vb__ vaa__) \<or> beta_reducible ua__"   | v :: "nat" and u :: "Core.term" where "x = Bv vc__ $ ub__" and "beta_reducible (Bv vc__) \<or> beta_reducible ub__"   | v :: "Core.term" and va :: "Core.term" and u :: "Core.term" where "x = v $ va $ u" and "beta_reducible (v $ va) \<or> beta_reducible u"
take_i_f_join_eq1: fixes n :: "nat"   and I :: "nat set"   and f :: "nat \<Rightarrow> 'b" assumes "n < card I" shows "f \<Join>\<^sub> I \<down> n = f \<Join>\<^sub> (I \<down>< (I \<rightarrow> n))"
r2f_f2r_galois: fixes f :: "'c \<Rightarrow> 'd set"   and R :: "('c \<times> 'd) set" shows "(\<R> f = R) = (\<F> R = f)"
lessThan_def: fixes u :: "'b" shows "{..<u} \<equiv> {x. x < u}"
aux0_0: fixes x :: "('a, 'b) Combinators list" assumes "singleCombinators x" shows "\<nexists>a b. a \<oplus> b \<in> set x"
step: fixes b :: "int"   and a :: "int" assumes "0 < b" shows "P (Fract a b)"
cmp_simps'_1: fixes \<mu> :: "'c"   and \<nu> :: "'c" assumes "C.arr \<mu>"   and "C.arr \<nu>"   and "src\<^sub>C \<mu> = trg\<^sub>C \<nu>" shows "D.arr (cmp (\<mu>, \<nu>))"
phiUsesI: fixes \<alpha>e :: "'a \<Rightarrow> ('b \<times> 'c \<times> 'b) set"   and \<alpha>n :: "'a \<Rightarrow> 'b list"   and invar :: "'a \<Rightarrow> bool"   and inEdges' :: "'a \<Rightarrow> 'b \<Rightarrow> ('b \<times> 'c) list"   and Entry :: "'a \<Rightarrow> 'b"   and defs :: "'a \<Rightarrow> 'b \<Rightarrow> 'd set"   and "uses" :: "'a \<Rightarrow> 'b \<Rightarrow> 'd set"   and phis :: "'a \<Rightarrow> 'b \<times> 'd \<Rightarrow> 'd list option"   and n' :: "'b"   and g :: "'a"   and v' :: "'d"   and vs :: "'d list"   and n :: "'b"   and v :: "'d" assumes "CFG_SSA \<alpha>e \<alpha>n invar inEdges' Entry defs uses phis"   and "n' \<in> set (\<alpha>n g)"   and "phis g (n', v') = Some vs"   and "(n, v) \<in> set (zip (graph_path_base.predecessors inEdges' g n') vs)" shows "v \<in> CFG_SSA_base.phiUses \<alpha>n inEdges' phis g n"
X_def: fixes M :: "'b measure"   and I :: "'c set"   and A :: "'c \<Rightarrow> 'b set"   and i :: "'c" assumes "prob_space_with_indicators M I A" shows "prob_space_with_indicators.X A i = rind (A i)"
span_legs_eq_4: shows "SPN_fgh.\<pi>.leg1 = THfgh.S\<^sub>1"
servTicket_authentic_Tgs: fixes B :: "agent"   and A :: "agent"   and servK :: "nat"   and Ts :: "nat"   and evs :: "event list" assumes "Crypt (shrK B) \<lbrace>Agent A, Agent B, Key servK, Number Ts\<rbrace> \<in> parts (knows Spy evs)"   and "B \<noteq> Tgs"   and "B \<notin> bad"   and "evs \<in> kerbIV" shows "\<exists>authK. Says Tgs A (Crypt authK \<lbrace>Key servK, Agent B, Number Ts, Crypt (shrK B) \<lbrace>Agent A, Agent B, Key servK, Number Ts\<rbrace>\<rbrace>) \<in> set evs"
scale: fixes c :: "complex"   and b :: "'a" shows "f M (c *\<^sub>C b) = c *\<^sub>C f M b"
newstat_lemma: fixes m :: "nat"   and q :: "nat"   and r :: "nat" shows "rec_exec rec_newstat [m, q, r] = newstat m q r"
boundedE: fixes A :: "'c Word.word set"   and x :: "'c Word.word" assumes "finite A"   and "A \<noteq> {}"   and "x \<le>s signed.Min A" obtains "\<And>a. a \<in> A \<Longrightarrow> x \<le>s a"
Pair_inject: fixes a :: "'x"   and b :: "'y"   and a' :: "'x"   and b' :: "'y" assumes "(a, b) = (a', b')" obtains "a = a'" and "b = b'"
adjointFO_def: fixes I :: "('sf, 'sc, 'sz) interp"   and \<sigma> :: "'a \<Rightarrow> ('sf, 'sz) trm"   and \<nu> :: "(real, 'sz) vec \<times> (real, 'sz) vec" shows "adjointFO I \<sigma> \<nu> = \<lparr>Functions = \<lambda>f. case f of Inl f' \<Rightarrow> Functions I f' | Inr f' \<Rightarrow> \<lambda>_. dterm_sem I (\<sigma> f') \<nu>, Predicates = Predicates I, Contexts = Contexts I, Programs = Programs I, ODEs = ODEs I, ODEBV = ODEBV I\<rparr>"
dependent_mono: fixes A :: "'b set"   and B :: "'b set" assumes "A \<subseteq> U\<^sub>M"   and "local.dependent B"   and "B \<subseteq> A" shows "local.dependent A"
inject_5: fixes x51 :: "sterm"   and x52 :: "Label"   and x53 :: "sterm"   and y51 :: "sterm"   and y52 :: "Label"   and y53 :: "sterm" shows "(Upd x51 x52 x53 = Upd y51 y52 y53) = (x51 = y51 \<and> x52 = y52 \<and> x53 = y53)"
neq_Nil_conv: fixes xs :: "'b list" shows "(xs \<noteq> \<epsilon>) = (\<exists>y ys. xs = y # ys)"
Mem_Eats_Iff: fixes H :: "fm set"   and x :: "tm"   and t :: "tm"   and u :: "tm" shows "H \<turnstile> x IN Eats t u IFF x IN t OR x EQ u"
ih: assumes "s \<noteq> []" shows "pderivs s (Star r) \<subseteq> Timess (pderivs_lang (PSuf s) r) (Star r)"
less_le_not_le: fixes x :: "'c"   and y :: "'c" shows "(x < y) = strict (\<le>) x y"
Dyn_Tab2_axioms: shows "Dyn_Tab2 a c"
fun_add_closed: fixes f :: "'a \<Rightarrow> 'a"   and g :: "'a \<Rightarrow> 'a" assumes "f \<in> carrier (function_ring (carrier R) R)"   and "g \<in> carrier (function_ring (carrier R) R)" shows "f \<oplus>\<^bsub>function_ring (carrier R) R\<^esub> g \<in> carrier (function_ring (carrier R) R)"
sorted_insort: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and xs :: "'c list" assumes "class.linorder less_eq less" shows "linorder.sorted less_eq (linorder.insort_key less_eq (\<lambda>x. x) x xs) = linorder.sorted less_eq xs"
lang\<^sub>M\<^sub>2\<^sub>L_def: fixes \<Sigma> :: "'b list"   and n :: "nat"   and \<phi> :: "'b formula" assumes "formula \<Sigma>" shows "formula.lang\<^sub>M\<^sub>2\<^sub>L \<Sigma> n \<phi> = {formula.enc (w, I) |w I. length I = n \<and> formula.wf_interp_for_formula \<Sigma> (w, I) \<phi> \<and> formula.satisfies (w, I) \<phi>}"
S3S4S5S6LeadstoS6: fixes rmhist :: "state \<Rightarrow> PrIds \<Rightarrow> histState"   and p :: "PrIds"   and sigma :: "behavior" assumes "sigma \<Turnstile> S3 rmhist p \<leadsto> S4 rmhist p \<or> S6 rmhist p"   and "sigma \<Turnstile> S4 rmhist p \<and> ires!p = #NotAResult \<leadsto> S4 rmhist p \<and> ires!p \<noteq> #NotAResult \<or> S5 rmhist p"   and "sigma \<Turnstile> S4 rmhist p \<and> ires!p \<noteq> #NotAResult \<leadsto> S5 rmhist p"   and "sigma \<Turnstile> S5 rmhist p \<leadsto> S6 rmhist p" shows "sigma \<Turnstile> S3 rmhist p \<or> S4 rmhist p \<or> S5 rmhist p \<or> S6 rmhist p \<leadsto> S6 rmhist p"
S4: shows "sf (c_sum (c_pair x y)) \<le> sf (x + y) + x"
trace_equivalent: fixes e1 :: "((nat \<times> nat) \<times> transition) fset"   and e2 :: "((nat \<times> nat) \<times> transition) fset" assumes "\<forall>t. accepts_trace e1 0 <> t = accepts_trace e2 0 <> t" shows "trace_equivalent e1 e2"
Red_fv: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and E :: "char list \<Rightarrow> ty option"   and e :: "expr"   and h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and l :: "char list \<Rightarrow> val option"   and e' :: "expr"   and h' :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and l' :: "char list \<Rightarrow> val option" assumes "wwf_prog P"   and "P,E \<turnstile> \<langle>e,(h, l)\<rangle> \<rightarrow> \<langle>e',(h', l')\<rangle>" shows "fv e' \<subseteq> fv e"
acyclic_5a_5b: fixes x :: "'a" shows "acyclic_5a x = acyclic_5b x"
pac_step_rel_assn_alt_def: fixes R1 :: "'a \<Rightarrow> 'b \<Rightarrow> assn"   and R2 :: "'c \<Rightarrow> 'd \<Rightarrow> assn"   and R3 :: "'e \<Rightarrow> 'f \<Rightarrow> assn"   and x :: "('c, 'e, 'a) pac_step"   and y :: "('d, 'f, 'b) pac_step" shows "pac_step_rel_assn R1 R2 R3 x y = (case (x, y) of (Add p1 p2 i r, Add p1' p2' i' r') \<Rightarrow> R1 p1 p1' * R1 p2 p2' * R1 i i' * R2 r r' | (Add p1 p2 i r, _) \<Rightarrow> false | (Mult p1 p2 i r, Mult p1' p2' i' r') \<Rightarrow> R1 p1 p1' * R2 p2 p2' * R1 i i' * R2 r r' | (Mult p1 p2 i r, _) \<Rightarrow> false | (Extension i x p1, Extension i' x' p1') \<Rightarrow> R1 i i' * R3 x x' * R2 p1 p1' | (Extension i x p1, _) \<Rightarrow> false | (Del p1, Del p1') \<Rightarrow> R1 p1 p1' | (Del p1, _) \<Rightarrow> false)"
reachable_BA: fixes y :: "'v"   and x :: "'v" assumes "reachable y"   and "(x, y) \<in> \<^bold>E"   and "\<not> reachable x" shows "g (x, y) = 0"
path_Append: fixes n :: "'b LDCFG_node"   and as :: "('b LDCFG_node \<times> ('c, 'd, 'e, 'f) edge_kind \<times> 'b LDCFG_node) list"   and n'' :: "'b LDCFG_node"   and as' :: "('b LDCFG_node \<times> ('c, 'd, 'e, 'f) edge_kind \<times> 'b LDCFG_node) list"   and n' :: "'b LDCFG_node" assumes "CFGExit.path n as n''"   and "CFGExit.path n'' as' n'" shows "CFGExit.path n (as @ as') n'"
cball_min_Int: fixes a :: "'b"   and r :: "real"   and s :: "real" shows "cball a (min r s) = cball a r \<inter> cball a s"
scopeExtSum: fixes x :: "name"   and P :: "pi"   and Q :: "pi" assumes "x \<sharp> P" shows "<\<nu>x>(P \<oplus> Q) \<sim> P \<oplus> <\<nu>x>Q"
Pbij_inverse_extends_twice: fixes \<beta> :: "(Location \<times> Location) set" shows "Pbij_extends (Pbij_inverse (Pbij_inverse \<beta>)) \<beta>"
case_sum_o_inj_1: fixes f :: "'e \<Rightarrow> 'f"   and g :: "'g \<Rightarrow> 'f" shows "case_sum f g \<circ> Inl = f"
abs_inverse: fixes a :: "'b" shows "\<bar>inverse a\<bar> = inverse \<bar>a\<bar>"
closest_supchamberD_2: fixes X :: "'b set set"   and F :: "'b set"   and D :: "'b set" assumes "ChamberComplex X"   and "F \<in> X"   and "SimplicialComplex.maxsimp X D" shows "order.greater_eq (ChamberComplex.closest_supchamber X F D) F"
tendsto_of_int_ceiling: fixes f :: "'a \<Rightarrow> 'b"   and l :: "'b"   and F :: "'a filter" assumes "(f \<longlongrightarrow> l) F"   and "l \<notin> \<int>" shows "((\<lambda>x. of_int \<lceil>f x\<rceil>) \<longlongrightarrow> of_int \<lceil>l\<rceil>) F"
drlex_pm_strict_alt: fixes s :: "'a \<Rightarrow>\<^sub>0 'b"   and t :: "'a \<Rightarrow>\<^sub>0 'b" shows "drlex_pm_strict s t = (\<not> drlex_pm t s)"
ideal_eq_left_ideal: fixes I :: "'a set" shows "ideal I = left_ideal I"
pverts_mk_symmetric: fixes G :: "('c, 'd) pre_digraph" shows "pverts (mk_symmetric G) = verts G"
image2_eqI: fixes b :: "'d"   and f :: "'e \<Rightarrow> 'd"   and x :: "'e"   and c :: "'f"   and g :: "'e \<Rightarrow> 'f"   and A :: "'e set" assumes "b = f x"   and "c = g x"   and "x \<in> A" shows "(b, c) \<in> BNF_Greatest_Fixpoint.image2 A f g"
build_rqrm_def: fixes \<delta> :: "('a, 'b) ta_rule dlist" shows "build_rqrm \<delta> \<equiv> ls.iterate \<delta> (\<lambda>r res. foldl (\<lambda>res q. rqrm_add q r res) res (rhsq r)) (hm.empty ())"
member_of_subcls: fixes G :: "prog"   and old :: "qtname \<times> memberdecl"   and C :: "qtname"   and new :: "qtname \<times> memberdecl"   and D :: "qtname" assumes "G \<turnstile> old member_of C"   and "G \<turnstile> new member_of D"   and "memberid new = memberid old"   and "G\<turnstile>D\<preceq>\<^sub>C C"   and "G\<turnstile>declclass new\<prec>\<^sub>C declclass old"   and "ws_prog G" shows "G\<turnstile>D\<prec>\<^sub>C C"
diaR: fixes A :: "(C, BD) form" shows "([ \<Empt> \<Rightarrow>* \<LM> A  \<RM>],  \<Empt> \<Rightarrow>* \<LM> \<diamond> A  \<RM>) \<in> g3mod2"
mat_extension_def: fixes m :: "'a mat" shows "ps_P.mat_extension m = ps_P.tensor_mat m (1\<^sub>m ps_P.d2)"
scale_right_imp_eq: fixes x :: "('a, 'b) vec"   and a :: "'a"   and b :: "'a" assumes "x \<noteq> 0"   and "a *s x = b *s x" shows "a = b"
union_commute: fixes M :: "'a multiset"   and N :: "'a multiset" shows "M + N = N + M"
rep_eq: shows "lookup 0 = (\<lambda>k. 0::'d)"
select_convs_3: fixes Pol :: "('b \<times> 'b) set"   and Map :: "'c \<Rightarrow> 'b"   and Doms :: "'b set"   and List :: "'c list"   and ListOp :: "'c list option"   and Set :: "'c set"   and more :: "'d" shows "Doms \<lparr>Pol = Pol, Map = Map, Doms = Doms, List = List, ListOp = ListOp, Set = Set, \<dots> = more\<rparr> = Doms"
cases: fixes a1 :: "nat"   and a2 :: "nat" assumes "eq_pred a1 a2" obtains n :: "nat" where "a1 = n__" and "a2 = n__"   | n :: "nat" and m :: "nat" where "a1 = na__" and "a2 = s1 ma__" and "eq_pred na__ ma__"   | n :: "nat" and m :: "nat" where "a1 = n" and "a2 = s2 m" and "eq_pred n m"
gt_same: fixes s :: "('s, 'v) tm"   and t :: "('s, 'v) tm" assumes "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 t). extf f (>\<^sub>t) (args t) (args s)" shows "t >\<^sub>t s"
assms_2: fixes i :: "int" assumes "k < i"   and "\<And>j. \<lbrakk>k < j; j < i\<rbrakk> \<Longrightarrow> P j" shows "P i"
Diff_insert0: fixes x :: "'q"   and A :: "'q set"   and B :: "'q set" assumes "x \<notin> A" shows "A - insert x B = A - B"
min_top: fixes x :: "'d upred" shows "ord.min (\<le>) true x = x"
hom: fixes f :: "'a \<Rightarrow> 'c" assumes "continuous_on s f"   and "f ` s \<subseteq> U"   and "P f" shows "\<exists>c. homotopic_with_canon P s U f (\<lambda>x. c)"
assms_3: fixes xs :: "V"   and x :: "V" assumes "vfsequence xs"   and "P xs" shows "P (xs #\<^sub>\<circ> x)"
is_iso_dghm_op: shows "op_dghm \<FF> : op_dg \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>i\<^sub>s\<^sub>o\<^bsub>\<alpha>\<^esub> op_dg \<BB>"
STA_drop_block: fixes ps :: "('c, 'd) fm list"   and ps' :: "('c, 'd) fm list"   and a :: "'d"   and branch :: "(('c, 'd) fm list \<times> 'd) list"   and A :: "'d set"   and n :: "nat" assumes "set ps \<subseteq> set ps'"   and "(ps', a) \<in>. branch"   and "A, n \<turnstile> (ps, a) # branch" shows "A, Suc n \<turnstile> branch"
join_table: fixes n :: "nat"   and A :: "nat set"   and X :: "'b option list set"   and B :: "nat set"   and Y :: "'b option list set"   and b :: "bool"   and C :: "nat set" assumes "table n A X"   and "table n B Y"   and "\<not> b \<Longrightarrow> B \<subseteq> A"   and "A \<union> B = C" shows "table n C (join X b Y)"
simps: fixes A :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and C :: "'i \<Rightarrow> 'j \<Rightarrow> bool"   and R :: "'k \<Rightarrow> 'l \<Rightarrow> bool"   and a1 :: "('g, 'i, 'k) gpv"   and a2 :: "('h, 'j, 'l) gpv" shows "rel_gpv'' A C R a1 a2 = (\<exists>gpv gpv'. a1 = gpv \<and> a2 = gpv' \<and> rel_spmf (rel_generat A C (R ===> rel_gpv'' A C R)) (the_gpv gpv) (the_gpv gpv'))"
is_partial_magma_2: shows "partial_magma.null (\<star>) = null"
sum_idempotence: fixes x :: "i" shows "x \<oplus> x = x"
greaterThanLessThan_eq: fixes less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a" shows "ord.greaterThanLessThan less a b = ord.greaterThan less a \<inter> ord.lessThan less b"
induct: fixes P :: "nat \<Rightarrow> bool list bdd \<Rightarrow> bool"   and a0 :: "nat"   and a1 :: "bool list bdd" assumes "\<And>i q. P i (Leaf q)"   and "\<And>l r. P 0 (Branch l r)"   and "\<And>i l r. \<lbrakk>P i l; P i r\<rbrakk> \<Longrightarrow> P (Suc i) (Branch l r)" shows "P a0 a1"
integral_def: fixes i :: "'c set"   and f :: "'c \<Rightarrow> 'b" shows "integral i f = (SOME y. (f has_integral y) i \<or> \<not> f integrable_on i \<and> y = (0::'b))"
ord_FO_\<Gamma>_infer_from_Collect_eq: fixes DD :: "'a literal multiset set"   and C :: "'a literal multiset" shows "{\<gamma> \<in> ord_FO_\<Gamma> S. infer_from (DD \<union> {C}) \<gamma> \<and> C \<in># prems_of \<gamma>} = {\<gamma> \<in> ord_FO_\<Gamma> S. \<exists>D\<in>DD \<union> {C}. prems_of \<gamma> = {#C, D#}}"
ge2_events: fixes Q :: "'a set"   and a :: "'a" assumes "Q \<in> \<P>"   and "a \<in> Q" shows "\<exists>b\<in>Q. b \<noteq> a"
init_AND_def: fixes X :: "'pub0 \<times> 'pub1"   and W :: "'witness0 \<times> 'witness1" shows "init_AND X W = (let (x0, x1) = X; (w0, w1) = W in init0 x0 w0 \<bind> (\<lambda>(r0, a0). init1 x1 w1 \<bind> (\<lambda>(r1, a1). return_spmf ((r0, r1), a0, a1))))"
mds\<^sub>s_A_of_mds\<^sub>0: shows "mds\<^sub>s = mds\<^sub>A_of mds\<^sub>0"
total2_def: fixes f :: "nat \<Rightarrow> nat \<Rightarrow> nat option" shows "total2 f \<equiv> \<forall>x y. f x y \<down>"
det_fold_map_aux: fixes l :: "('f \<times> 'g) list"   and R :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and c :: "'h \<Rightarrow> bool"   and f :: "'f \<times> 'g \<Rightarrow> 'h \<Rightarrow> 'h"   and \<sigma> :: "'h"   and result :: "('f \<Rightarrow> 'g option) \<Rightarrow> 'h"   and m :: "'f \<Rightarrow> 'g option" assumes "\<lbrakk>distinct (map fst l); sorted_wrt (key_rel R) l\<rbrakk> \<Longrightarrow> foldli l c f \<sigma> = result (map_of l)"   and "RETURN l \<le> it_to_sorted_list (key_rel R) (map_to_set m)" shows "foldli l c f \<sigma> = result m"
has_bochner_integral_reflect_real_lemma: fixes a :: "real"   and b :: "real"   and f :: "real \<Rightarrow> 'c"   and i :: "'c" assumes "has_bochner_integral (lebesgue_on {a..b}) f i" shows "has_bochner_integral (lebesgue_on {- b..- a}) (\<lambda>x. f (- x)) i"
hcomp_in_vhom: fixes \<mu> :: "'a arrow_of_spans_data"   and f :: "'a arrow_of_spans_data"   and g :: "'a arrow_of_spans_data"   and \<nu> :: "'a arrow_of_spans_data"   and h :: "'a arrow_of_spans_data"   and k :: "'a arrow_of_spans_data" assumes "\<guillemotleft>\<mu> : f \<Rightarrow> g\<guillemotright>"   and "\<guillemotleft>\<nu> : h \<Rightarrow> k\<guillemotright>"   and "src h = trg f" shows "\<guillemotleft>\<nu> \<star> \<mu> : h \<star> f \<Rightarrow> k \<star> g\<guillemotright>"
A: fixes v :: "nat \<Rightarrow> 'a" shows "(abs_Gromov_completion v = to_Gromov_completion x \<and> Gromov_completion_rel v v) = (v = (\<lambda>n. x))"
t_mult_closure: fixes x :: "'a"   and y :: "'a" assumes "t x = x"   and "t y = y" shows "t (x \<cdot> y) = x \<cdot> y"
second_countable_topology_ow_axioms_def: fixes U :: "'c set"   and \<tau> :: "'c set \<Rightarrow> bool" shows "second_countable_topology_ow_axioms U \<tau> \<equiv> \<exists>B\<subseteq>Pow U. countable B \<and> (\<forall>S\<subseteq>U. \<tau> S = (in_topology_generated_by B on U : \<guillemotleft>open\<guillemotright> S))"
4: shows "EVAL (Dom (fst fg) \<^bold>\<star> Dom (snd fg)) = Map (fst fg) \<star>\<^sub>B Map (snd fg)"
\<phi>_simps_4: shows "local.dom f\<^sub>0g\<^sub>1.\<phi> = tab\<^sub>0 f \<star> f\<^sub>0g\<^sub>1.p\<^sub>1"
axioms_1: fixes \<eta> :: "('d, 'e, 'f, 'g, 'h, 'i) NatTrans" assumes "NatTrans \<eta>" shows "NatTransP \<eta>"
deg_leqI: fixes p :: "nat \<Rightarrow> 'a"   and k :: "nat" assumes "p \<in> carrier P"   and "\<And>n. k < n \<Longrightarrow> p n = \<zero>" shows "deg R p \<le> k"
dom_distrib: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and domain_op :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "class.domain_near_semiring plus times domain_op less_eq less" shows "plus (domain_op x) (times (domain_op y) (domain_op z)) = times (plus (domain_op x) (domain_op y)) (plus (domain_op x) (domain_op z))"
\<tau>moves2Hd: fixes P :: "'a prog"   and h :: "'heap"   and stk :: "'addr val list"   and e :: "'addr expr1"   and pc :: "nat"   and xcp :: "'addr option"   and es :: "'addr expr1 list" assumes "\<tau>move2 P h stk e pc xcp" shows "\<tau>moves2 P h stk (e # es) pc xcp"
homeomorphic_imp_homotopy_equivalent_space: fixes X :: "'a topology"   and Y :: "'b topology" assumes "X homeomorphic_space Y" shows "X homotopy_equivalent_space Y"
Units_eq: shows "Units (G\<lparr>carrier := generate G {a}\<rparr>) = carrier (G\<lparr>carrier := generate G {a}\<rparr>)"
dependent_def: fixes P :: "'b set" shows "local.dependent P = (\<exists>a\<in>P. a \<in> local.span (P - {a}))"
all_valid_fold: fixes G :: "'a gexp list"   and s :: "'a \<Rightarrow> value option" assumes "\<forall>g\<in>set G. gval g s \<noteq> invalid" shows "gval (fold gAnd G (Bc True)) s \<noteq> invalid"
countable_insert_eq: fixes x :: "'c"   and A :: "'c set" shows "countable (insert x A) = countable A"
prob_proj_comm: fixes PROB :: "(('a, 'b) fmap \<times> ('a, 'c) fmap) set"   and vs :: "'a set"   and vs' :: "'a set" shows "prob_proj (prob_proj PROB vs) vs' = prob_proj (prob_proj PROB vs') vs"
zmult_zless_mono_neg: fixes i :: "int"   and j :: "int"   and k :: "int" assumes "i < j"   and "k < 0" shows "j * k < i * k"
f_4: fixes x :: "'a" shows "(SUP i. f i x) = g x"
deleteMin_def: fixes q :: "('a, 'b) SkewBinomialHeap" shows "SkewBinomialHeap_loc.deleteMin q \<equiv> if q = SkewBinomialHeap_loc.empty then SkewBinomialHeap_loc.empty else Abs_SkewBinomialHeap (BsSkewBinomialHeapStruc.bs_deleteMin (Rep_SkewBinomialHeap q))"
subseqs_powset: fixes xs :: "'c list" shows "set ` set (subseqs xs) = Pow (set xs)"
tauActTauChain: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and P' :: "('a, 'b, 'c) psi" assumes "\<Psi> \<rhd> P \<longmapsto> \<tau> \<prec> P'" shows "\<Psi> \<rhd> P \<Longrightarrow>\<^sup>^\<^sub>\<tau> P'"
assms_3: fixes x :: "'d"   and h :: "'a" assumes "x \<in> set xs" shows "pure (g x) h"
simps: fixes a :: "'c \<Rightarrow> real" shows "real_polynomial_function a = ((\<exists>f. a = f \<and> bounded_linear f) \<or> (\<exists>c. a = (\<lambda>x. c)) \<or> (\<exists>f g. a = (\<lambda>x. f x + g x) \<and> real_polynomial_function f \<and> real_polynomial_function g) \<or> (\<exists>f g. a = (\<lambda>x. f x * g x) \<and> real_polynomial_function f \<and> real_polynomial_function g))"
edc_var: fixes x :: "nat"   and vs :: "nat list"   and vs' :: "nat list"   and \<Gamma> :: "nat \<Rightarrow> pdf_type"   and \<delta> :: "cexpr" assumes "x \<in> set vs" shows "(vs, vs', \<Gamma>, \<delta>) \<turnstile>\<^sub>c Var x \<Rightarrow> marg_dens_cexpr \<Gamma> vs x \<delta>"
map: fixes f :: "'c \<Rightarrow> 'b"   and x1 :: "'c form multiset"   and x2 :: "'c form multiset" shows "map_sequent f ( x1 \<Rightarrow>* x2) = ( image_mset (map_form f) x1 \<Rightarrow>* image_mset (map_form f) x2)"
prim_impl_correct_2: fixes g :: "(('a \<times> ('a \<times> nat) list) \<times> color) tree"   and r :: "'a"   and \<pi>i :: "(('a \<times> 'a) \<times> color) tree" assumes "G.invar g"   and "prim_impl g r = \<pi>i" shows "is_MST (G.\<alpha>w g) (component_of (G.\<alpha>g g) r) (P.\<alpha>_MST r \<pi>i)"
plus_cont: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "f \<in> bcontfun"   and "g \<in> bcontfun" shows "(\<lambda>x. f x + g x) \<in> bcontfun"
simps_5: fixes test :: "'e \<Rightarrow> 'd \<Rightarrow> bool"   and i :: "'e"   and r :: "'d regex" shows "rpd test i (Star r) = TimesL (Star r) (rpd test i r)"
iszero_def: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and z :: "'a" assumes "class.ring_1 one times plus zero minus uminus" shows "ring_1.iszero zero z = (z = zero)"
GrpI: fixes f :: "'e \<Rightarrow> 'd"   and x :: "'e"   and y :: "'d"   and A :: "'e set" assumes "f x = y"   and "x \<in> A" shows "BNF_Def.Grp A f x y"
AB_ABB: shows "A \<union>\<^sub>\<circ> B = B \<union>\<^sub>\<circ> (A -\<^sub>\<circ> B)"
unwinding_2pi: fixes z :: "complex" shows "complex_of_real (2 * pi) * \<i> * of_int (unwinding z) = z - Ln (exp z)"
R_def: shows "R \<equiv> Id \<inter> {(uu_, n, ns). \<forall>n'\<in>ns. name n' < n}"
swaps_eq_nth_take_drop: fixes i :: "nat"   and xs :: "'b list" assumes "i < length xs" shows "swaps [0..<i] xs = xs ! i # take i xs @ drop (Suc i) xs"
assms_3: fixes s :: "'a net_state"   and t :: "'a net_state"   and a :: "'b node_action"   and s' :: "'a net_state"   and t' :: "'a net_state" assumes "SubnetS s t \<in> reachable (closed (pnet np (p1 \<parallel> p2))) I"   and "P s t"   and "(SubnetS s t, a, SubnetS s' t') \<in> automaton.trans (closed (pnet np (p1 \<parallel> p2)))"   and "I a" shows "P s' t'"