hcju/isabelleps huggingface dataset:
redundancy_criterion_standard_extension_saturated_upto_iff: fixes \<Gamma> :: "'b inference set"   and \<Gamma>' :: "'b inference set"   and Rf :: "'b literal multiset set \<Rightarrow> 'b literal multiset set"   and Ri :: "'b literal multiset set \<Rightarrow> 'b inference set"   and M :: "'b literal multiset set" assumes "\<Gamma> \<subseteq> \<Gamma>'"   and "redundancy_criterion \<Gamma> Rf Ri" shows "redundancy_criterion.saturated_upto \<Gamma> Rf Ri M = redundancy_criterion.saturated_upto \<Gamma>' Rf (\<lambda>N. Ri N \<union> (\<Gamma>' - \<Gamma>)) M"
path_connectedin: fixes X :: "'b topology"   and S :: "'b set" shows "path_connectedin X S = (S \<subseteq> topspace X \<and> (\<forall>x\<in>S. \<forall>y\<in>S. \<exists>g. pathin X g \<and> g ` {0..1} \<subseteq> S \<and> g 0 = x \<and> g 1 = y))"
multpw_single: fixes x :: "'a"   and y :: "'a"   and ns :: "('a \<times> 'a) set" assumes "(x, y) \<in> ns" shows "({#x#}, {#y#}) \<in> multpw ns"
has_field_mono: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'a) list) list"   and C :: "char list"   and F :: "char list"   and T :: "ty"   and D :: "char list"   and C' :: "char list" assumes "P \<turnstile> C has F:T in D"   and "P \<turnstile> C' \<preceq>\<^sup>* C" shows "P \<turnstile> C' has F:T in D"
Ana_nonempty_inv: fixes t :: "(('fun, 'atom, 'sets) prot_fun, 'a) Term.term" assumes "Ana t \<noteq> ([], [])" shows "\<exists>f T. t = Fun (Fu f) T \<and> arity\<^sub>f f = length T \<and> 0 < arity\<^sub>f f \<and> (\<exists>K M. Ana\<^sub>f f = (K, M) \<and> Ana t = (K \<cdot>\<^sub>l\<^sub>i\<^sub>s\<^sub>t (!) T, map ((!) T) M))"
\<epsilon>_map_simp: fixes x :: "'a" assumes "A.ide x" shows "transformation_by_components.map (\<cdot>\<^sub>A) (\<cdot>\<^sub>A) FG.map \<epsilon>o x = F (G x)"
unat_mod_ten: shows "unat (n mod 10) = unat n mod 10"
left_cancel: fixes a :: "'b"   and b :: "'b"   and c :: "'b" shows "(a + b = a + c) = (b = c)"
conversepI: fixes r :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'a"   and b :: "'b" assumes "r a b" shows "r\<inverse>\<inverse> b a"
pmf_embed_pmf: fixes f :: "'b \<Rightarrow> real"   and x :: "'b" assumes "\<And>x. 0 \<le> f x"   and "\<integral>\<^sup>+ x. ennreal (f x) \<partial>count_space UNIV = 1" shows "pmf (local.embed_pmf f) x = f x"
assms_2: shows "\<forall>t. subalgebra (F t) (G t)"
not_lfinite_llength: fixes xs :: "'a llist" assumes "\<not> lfinite xs" shows "llength xs = \<infinity>"
subst_eq: fixes k :: "nat"   and u :: "dB" shows "Var k[u/k] = u"
simps_1: fixes w :: "brick list" shows "kauff_mat (basic w) = blockmat w"
continuous_map_iff_continuous: fixes S :: "'a set"   and g :: "'a \<Rightarrow> 'b" shows "continuous_map (top_of_set S) euclidean g = continuous_on S g"
trg_hcomp': fixes \<nu> :: "'a"   and \<mu> :: "'a" assumes "\<nu> \<star> \<mu> \<noteq> null" shows "trg (\<nu> \<star> \<mu>) = trg \<nu>"
pfa_insert_grow_rule: fixes i :: "nat"   and n :: "nat"   and c :: "nat"   and l :: "'b list"   and a :: "'b array"   and x :: "'b" assumes "i \<le> n" shows "<is_pfa c l (a, n)> pfa_insert_grow (a, n) i x <\<lambda>r. case r of (a', n') \<Rightarrow> is_pfa (max c (n + 1)) (take i l @ x # drop i l) (a', n') * \<up> (n' = n + 1 \<and> n \<le> c)>\<^sub>t"
locks_ticket_simps_4: fixes k :: "nat"   and k' :: "nat" shows "locks_ticket (REL k) k' = (k' = k)"
isSnapshot_def: fixes event :: "('a, 'b, 'c) event" shows "isSnapshot event = (\<exists>x4. event = Snapshot x4)"
2: shows "{i. n cd\<^bsup>\<pi>\<^esup>\<rightarrow> i \<and> i < k} = {i. j cd\<^bsup>\<pi>\<^esup>\<rightarrow> i} \<union> {j}"
Impl1': fixes A :: "triple set"   and P :: "state \<Rightarrow> cname \<times> mname \<Rightarrow> assn"   and Q :: "state \<Rightarrow> cname \<times> mname \<Rightarrow> assn"   and Ms :: "(cname \<times> mname) set"   and Cm :: "cname \<times> mname"   and Z :: "state" assumes "\<forall>Z. A \<union> (\<Union>Z. (\<lambda>Cm. (P Z Cm, Impl Cm, Q Z Cm)) ` Ms) |\<turnstile> (\<lambda>Cm. (P Z Cm, body Cm, Q Z Cm)) ` Ms"   and "Cm \<in> Ms" shows "A \<turnstile> {P Z Cm} Impl Cm {Q Z Cm}"
bangCases: fixes P :: "pi"   and Rs :: "residual" assumes "!P \<longmapsto> Rs" obtains Pa :: "pi" where "!P = !Pa" and "Pa \<parallel> !Pa \<longmapsto> Rs"
AndR: fixes \<Gamma> :: "'b formula multiset"   and F :: "'b formula"   and \<Delta> :: "'b formula multiset"   and n :: "nat"   and G :: "'b formula" assumes "\<Gamma> \<Rightarrow> F, \<Delta> \<down> n"   and "\<Gamma> \<Rightarrow> G, \<Delta> \<down> n" shows "\<Gamma> \<Rightarrow> F \<^bold>\<and> G, \<Delta> \<down> Suc n"
freshness_lemma_unique: fixes h :: "'b \<Rightarrow> 'a" assumes "pt TYPE('a) TYPE('b)"   and "at TYPE('b)"   and "finite (supp h)"   and "\<exists>a. a \<sharp> (h, h a)" shows "\<exists>!fr. \<forall>a. a \<sharp> h \<longrightarrow> h a = fr"
trm_ord_vars: fixes t :: "'a trm"   and s :: "'a trm" assumes "(t, s) \<in> trm_ord" shows "vars_of t \<subseteq> vars_of s"
asIMOD_igOp: fixes SEM :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'b) semDom"   and delta :: "'opSym"   and inp :: "'index \<Rightarrow> (('varSort \<Rightarrow> 'var \<Rightarrow> 'b) \<Rightarrow> 'b) option"   and binp :: "'bindex \<Rightarrow> (('varSort \<Rightarrow> 'var \<Rightarrow> 'b) \<Rightarrow> ('varSort, 'b) sAbs) option" shows "igOp (asIMOD SEM) delta inp binp = (\<lambda>val. if sWlsVal SEM val then sOp SEM delta (lift (\<lambda>X. X val) inp) (lift (\<lambda>A. A val) binp) else undefined)"
sel_2: fixes x31 :: "'e"   and x32 :: "'f" shows "un_Yield1 (Yield x31 x32) = x31"
abs_eq: shows "\<epsilon> = abs_interface \<epsilon>\<^sub>c"
pre_subdivFace'_Some2: fixes g :: "graph"   and f :: "face"   and v' :: "nat"   and v :: "nat"   and u :: "nat"   and vol :: "nat option list" assumes "pre_subdivFace' g f v' v 0 (Some u # vol)" shows "pre_subdivFace' g f v' u 0 vol"
completeness: fixes p :: "('c, 'b) fm"   and i :: "'b" assumes "infinite UNIV"   and "\<forall>M g w. M, g, w \<Turnstile> p" shows "nominals p, 1 \<turnstile> [([\<^bold>\<not> p], i)]"
SPRAY_event: fixes x :: "'a" assumes "SPRAY x \<noteq> {}" shows "x \<in> \<E>"
dep_in_subI: fixes X :: "'a set"   and \<E>' :: "'a set"   and \<E> :: "'a set" assumes "X \<subseteq> \<E>'"   and "\<not> \<E>.indep_in \<E>' X" shows "\<not> \<E>.indep_in \<E> X"
mult1_def: fixes r :: "('c \<times> 'c) set" shows "mult1 r = {(N, M). \<exists>a M0 K. M = add_mset a M0 \<and> N = M0 + K \<and> (\<forall>b. b \<in># K \<longrightarrow> (b, a) \<in> r)}"
mem_equal_unchanged: fixes s1 :: "'b sparc_state"   and s2 :: "'b sparc_state"   and pa :: "word36"   and addr :: "word36"   and val :: "word8" assumes "mem_equal s1 s2 pa" shows "mem_equal (s1\<lparr>mem := (mem s1)(10 := mem s1 10(addr \<mapsto> val), 11 := (mem s1 11)(addr := None))\<rparr>) (s2\<lparr>mem := (mem s2)(10 := mem s2 10(addr \<mapsto> val), 11 := (mem s2 11)(addr := None))\<rparr>) pa"
expectation_return_pmf: fixes x :: "'c"   and f :: "'c \<Rightarrow> 'b" shows "measure_pmf.expectation (return_pmf x) f = f x"
d1: fixes x' :: "'a" assumes "x' \<in> S"   and "dist x' x < d1" shows "f x' \<subseteq> T \<and> f x' \<subseteq> (\<Union>a\<in>f x. \<Union>b\<in>ball (0::'b) e. {a + b})"
flow_of_current_def: fixes \<Delta> :: "('a, 'b) network_scheme"   and f :: "('a \<times> 'a) \<times> 'a \<times> 'a \<Rightarrow> ennreal"   and e :: "'a \<times> 'a" shows "flow_of_current \<Delta> f e = max (d_OUT f e) (d_IN f e)"
no_multi_arcs: fixes G :: "('c, 'd) pre_digraph"   and e1 :: "'d"   and e2 :: "'d" assumes "nomulti_digraph G"   and "e1 \<in> arcs G"   and "e2 \<in> arcs G"   and "arc_to_ends G e1 = arc_to_ends G e2" shows "e1 = e2"
Inter_faces_finite_altbound: fixes T :: "'b set set"   and S :: "'b set" assumes "\<And>c. c \<in> T \<Longrightarrow> c face_of S" shows "\<exists>F'. finite F' \<and> F' \<subseteq> T \<and> card F' \<le> DIM('b) + 2 \<and> \<Inter> F' = \<Inter> T"
same_suffix_nil: fixes ys :: "'b list"   and xs :: "'b list" shows "suffix (ys @ xs) xs = (ys = [])"
WT: shows "exception_\<I> (\<I>_real \<oplus>\<^sub>\<I> \<I>_common) \<turnstile>g \<A> \<surd>"
vdomain_atD: fixes a :: "V" assumes "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> vrat_inverse" shows "\<exists>b\<in>elts (\<R>\<^sub>\<circ> vrat_inverse). a\<inverse>\<^sub>\<rat> = b"
append_prefixD: fixes xs :: "'c list"   and ys :: "'c list"   and zs :: "'c list" assumes "xs @ ys \<le> zs" shows "xs \<le> zs"
digraphD_4: fixes \<alpha> :: "V"   and \<CC> :: "V" assumes "digraph \<alpha> \<CC>" shows "vsv (\<CC>\<lparr>Cod\<rparr>)"
uminus_floatarith_def: fixes f :: "floatarith" shows "- f = Minus f"
rat_floor_lemma: fixes a :: "int"   and b :: "int" shows "rat_of_int (a div b) \<le> Fract a b \<and> Fract a b < rat_of_int (a div b + 1)"
66281: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "- (x \<squnion> (- y \<squnion> z)) = - - y \<sqinter> - (x \<squnion> z)"
orthogonal_def: fixes x :: "'b"   and y :: "'b" shows "orthogonal x y = (x \<bullet> y = 0)"
identity_functor_axioms: shows "identity_functor (\<cdot>\<^sub>J)"
f_def: fixes y :: "'b" shows "f y = (LEAST n. nth_item n = Rep y)"
sqrt_rat_ceiling_code: fixes x :: "rat" shows "sqrt_rat_ceiling x = - sqrt_rat_floor (- x)"
discI_1: fixes tllist :: "('a, 'b) tllist"   and x1 :: "'b" assumes "tllist = TNil x1" shows "is_TNil tllist"
Sigma_Suc: fixes w :: "nat"   and f :: "nat list \<Rightarrow> nat"   and xs :: "nat list" assumes "\<forall>k<Suc w. f (xs @ [k]) = Suc 0" shows "UF.Sigma f (xs @ [w]) = Suc w"
l1_step2_refines_a0i_running_skip: fixes A :: "agent"   and B :: "agent"   and KE :: "msg"   and Rb :: "fid_t" shows "{R01ia} (a0i_running A B \<langle>KE, NonceF (Rb $ sk)\<rangle>)\<^sup>=, l1_step2 Rb A B KE {> R01ia}"
tranclp_trans_induct: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b"   and P :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "r\<^sup>+\<^sup>+ x y"   and "\<And>x y. r x y \<Longrightarrow> P x y"   and "\<And>x y z. \<lbrakk>r\<^sup>+\<^sup>+ x y; P x y; r\<^sup>+\<^sup>+ y z; P y z\<rbrakk> \<Longrightarrow> P x z" shows "P x y"
can_extend_partial_order: fixes r :: "('a \<times> 'a) set"   and x :: "'a"   and y :: "'a" assumes "strict_partial_order r"   and "(x, y) \<notin> r"   and "(y, x) \<notin> r"   and "x \<noteq> y" shows "strict_ext r ((r \<union> {(x, y)})\<^sup>+)"
simps_1: fixes x :: "three" shows "f1 x Zero = x"
coeff_le_sq_norm: fixes f :: "'d poly"   and i :: "nat" shows "\<bar>coeff f i\<bar>\<^sup>2 \<le> \<parallel>f\<parallel>\<^sup>2"
simps_1: fixes I :: "('e, 'f, 'd) interp"   and t1 :: "('e, 'd) trm"   and t2 :: "('e, 'd) trm" shows "fml_sem I (Geq t1 t2) = {v. dterm_sem I t2 v \<le> dterm_sem I t1 v}"
elims_3: fixes x :: "'fsym trm" assumes "\<not> Ik.wt x" obtains f :: "'fsym" and Tl :: "'fsym trm list" where "x = Fn f Tl" and "\<not> (wtFsym f \<and> list_all Ik.wt Tl \<and> arOf f = map Ik.tpOf Tl)"
cka_1: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and zero :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and star :: "'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b" assumes "class.commutative_kleene_algebra plus times one zero less_eq less star" shows "less_eq (star (times x y)) (times (star x) (star y))"
ltln_strong_weak_stable_words_2: fixes w :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltln"   and \<psi> :: "'b ltln" shows "w \<Turnstile>\<^sub>n \<phi> M\<^sub>n \<psi> = w \<Turnstile>\<^sub>n \<phi> R\<^sub>n (\<psi> and\<^sub>n F\<^sub>n \<phi>[\<F>\<G> \<phi> w]\<^sub>\<Sigma>\<^sub>1)"
eval_Pr_diverg_Suc: fixes n :: "nat"   and f :: "recf"   and g :: "recf"   and xs :: "nat list"   and x :: "nat" assumes "recfn (Suc n) (Pr n f g)"   and "n = length xs"   and "eval (Pr n f g) (x # xs) \<up>" shows "eval (Pr n f g) (Suc x # xs) \<up>"
inv_begin_steps: fixes n :: "nat"   and cf :: "nat \<times> cell list \<times> cell list"   and stp :: "nat" assumes "inv_begin n cf"   and "0 < n" shows "inv_begin n (steps0 cf tcopy_begin stp)"
i_Exec_Stream_Acc_Output_take: fixes k :: "nat"   and n :: "nat"   and output_fun :: "'e \<Rightarrow> 'd message_af"   and trans_fun :: "'f message_af \<Rightarrow> 'e \<Rightarrow> 'e"   and input :: "nat \<Rightarrow> 'f message_af"   and c :: "'e" assumes "0 < k" shows "i_Exec_Comp_Stream_Acc_Output k output_fun trans_fun input c \<Down> n = f_Exec_Comp_Stream_Acc_Output k output_fun trans_fun (input \<Down> n) c"
split_fst: fixes R :: "'b \<Rightarrow> bool"   and p :: "'b \<times> 'c" shows "R (fst p) = (\<forall>x y. p = (x, y) \<longrightarrow> R x)"
Gromov_completion_converge_to_boundary: fixes x :: "'a Gromov_completion"   and u :: "nat \<Rightarrow> 'a" assumes "x \<in> Gromov_boundary" shows "(\<lambda>n. to_Gromov_completion (u n)) \<longlonglongrightarrow> x = (Gromov_completion_rel u u \<and> abs_Gromov_completion u = x)"
simps_28: fixes f1 :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> k \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> bool \<Rightarrow> 'a"   and f2 :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'a"   and f3 :: "(nat \<Rightarrow> nat option) \<Rightarrow> nat \<Rightarrow> 'a"   and f4 :: "nat \<Rightarrow> nat \<Rightarrow> 'a"   and f5 :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'a"   and x31 :: "nat \<Rightarrow> nat option"   and x32 :: "nat" shows "(case Rerr x31 x32 of Rreq x xa xb xc xd xe xf xg xh \<Rightarrow> f1 x xa xb xc xd xe xf xg xh | Rrep x xa xb xc xd \<Rightarrow> f2 x xa xb xc xd | Rerr x xa \<Rightarrow> f3 x xa | Newpkt x xa \<Rightarrow> f4 x xa | Pkt x xa xb \<Rightarrow> f5 x xa xb) = f3 x31 x32"
simps: fixes a1 :: "'b prog"   and a2 :: "ty"   and a3 :: "bop"   and a4 :: "ty"   and a5 :: "ty" shows "(a1 \<turnstile> a2\<guillemotleft>a3\<guillemotright>a4 :: a5) = ((\<exists>P T1 T2. a1 = P \<and> a2 = T1 \<and> a3 = Eq \<and> a4 = T2 \<and> a5 = Boolean \<and> (P \<turnstile> T1 \<le> T2 \<or> P \<turnstile> T2 \<le> T1)) \<or> (\<exists>P T1 T2. a1 = P \<and> a2 = T1 \<and> a3 = NotEq \<and> a4 = T2 \<and> a5 = Boolean \<and> (P \<turnstile> T1 \<le> T2 \<or> P \<turnstile> T2 \<le> T1)) \<or> (\<exists>P. a1 = P \<and> a2 = Integer \<and> a3 = LessThan \<and> a4 = Integer \<and> a5 = Boolean) \<or> (\<exists>P. a1 = P \<and> a2 = Integer \<and> a3 = LessOrEqual \<and> a4 = Integer \<and> a5 = Boolean) \<or> (\<exists>P. a1 = P \<and> a2 = Integer \<and> a3 = GreaterThan \<and> a4 = Integer \<and> a5 = Boolean) \<or> (\<exists>P. a1 = P \<and> a2 = Integer \<and> a3 = GreaterOrEqual \<and> a4 = Integer \<and> a5 = Boolean) \<or> (\<exists>P. a1 = P \<and> a2 = Integer \<and> a3 = Add \<and> a4 = Integer \<and> a5 = Integer) \<or> (\<exists>P. a1 = P \<and> a2 = Integer \<and> a3 = Subtract \<and> a4 = Integer \<and> a5 = Integer) \<or> (\<exists>P. a1 = P \<and> a2 = Integer \<and> a3 = Mult \<and> a4 = Integer \<and> a5 = Integer) \<or> (\<exists>P. a1 = P \<and> a2 = Integer \<and> a3 = Div \<and> a4 = Integer \<and> a5 = Integer) \<or> (\<exists>P. a1 = P \<and> a2 = Integer \<and> a3 = Mod \<and> a4 = Integer \<and> a5 = Integer) \<or> (\<exists>P. a1 = P \<and> a2 = Boolean \<and> a3 = BinAnd \<and> a4 = Boolean \<and> a5 = Boolean) \<or> (\<exists>P. a1 = P \<and> a2 = Integer \<and> a3 = BinAnd \<and> a4 = Integer \<and> a5 = Integer) \<or> (\<exists>P. a1 = P \<and> a2 = Boolean \<and> a3 = BinOr \<and> a4 = Boolean \<and> a5 = Boolean) \<or> (\<exists>P. a1 = P \<and> a2 = Integer \<and> a3 = BinOr \<and> a4 = Integer \<and> a5 = Integer) \<or> (\<exists>P. a1 = P \<and> a2 = Boolean \<and> a3 = BinXor \<and> a4 = Boolean \<and> a5 = Boolean) \<or> (\<exists>P. a1 = P \<and> a2 = Integer \<and> a3 = BinXor \<and> a4 = Integer \<and> a5 = Integer) \<or> (\<exists>P. a1 = P \<and> a2 = Integer \<and> a3 = ShiftLeft \<and> a4 = Integer \<and> a5 = Integer) \<or> (\<exists>P. a1 = P \<and> a2 = Integer \<and> a3 = ShiftRightZeros \<and> a4 = Integer \<and> a5 = Integer) \<or> (\<exists>P. a1 = P \<and> a2 = Integer \<and> a3 = ShiftRightSigned \<and> a4 = Integer \<and> a5 = Integer))"
UnionI: fixes X :: "'c set"   and C :: "'c set set"   and A :: "'c" assumes "X \<in> C"   and "A \<in> X" shows "A \<in> \<Union> C"
mp_PL: fixes Cn :: "'b set \<Rightarrow> 'b set"   and true_PL :: "'b"   and false_PL :: "'b"   and imp_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and not_PL :: "'b \<Rightarrow> 'b"   and conj_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and disj_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and equiv_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and p :: "'b"   and q :: "'b"   and A :: "'b set" assumes "Supraclassical_logic Cn true_PL false_PL imp_PL not_PL conj_PL disj_PL equiv_PL"   and "imp_PL p q \<in> Cn A"   and "p \<in> Cn A" shows "q \<in> Cn A"
min_dist_minD: fixes c :: "nat \<times> nat \<Rightarrow> 'b"   and v :: "nat"   and d :: "nat"   and v' :: "nat" assumes "Graph.dist c v d v'" shows "Graph.min_dist c v v' \<le> d"
lappend_llist_of_llist_of: fixes xs :: "'a list"   and ys :: "'a list" shows "llist_of xs $ llist_of ys = llist_of (xs @ ys)"
smultfunsetI: fixes G :: "'g set"   and FH :: "('f, 'g) aezfun set"   and f :: "('f, 'g) aezfun \<Rightarrow> 'v" assumes "\<forall>a. \<forall>g\<in>G. \<forall>x\<in>FH. f (a \<delta>\<delta> g * x) = (a \<delta>\<delta> g) \<cdot> f x" shows "f \<in> smultfunset G FH"
ennreal_numeral: fixes n :: "num" shows "ennreal (numeral n) = numeral n"
rotate1_is_Nil_conv: fixes xs :: "'d list" shows "(rotate1 xs = []) = (xs = [])"
\<delta>_3: shows "subst_range \<delta> \<subseteq> (\<lambda>c. Fun c []) ` ({f. public f} \<inter> \<C>) - (T \<union> subst_range \<sigma>)"
ZHMSet_plus: fixes xa :: "hmultiset zmultiset"   and x :: "hmultiset zmultiset" shows "ZHMSet (xa + x) = ZHMSet xa + ZHMSet x"
evalF_arg2_cong: fixes x :: "formula"   and y :: "formula"   and M :: "model"   and p :: "vbl \<Rightarrow> object" assumes "x = y" shows "evalF M p x = evalF M p y"
inject_2: fixes x2 :: "'b"   and y2 :: "'b" shows "(\<^bold>\<langle>x2\<^bold>\<rangle> = \<^bold>\<langle>y2\<^bold>\<rangle>) = (x2 = y2)"
wf_eq_minimal2: fixes r :: "('a \<times> 'a) set" shows "wf r = (\<forall>A. A \<subseteq> Field r \<and> A \<noteq> {} \<longrightarrow> (\<exists>a\<in>A. \<forall>a'\<in>A. (a', a) \<notin> r))"
right_idem: fixes a :: "'a Word.word"   and b :: "'a Word.word" shows "signed.min (signed.min a b) b = signed.min a b"
dickson_grading_adds_imp_le: fixes d :: "'b \<Rightarrow> nat"   and s :: "'b"   and t :: "'b" assumes "dickson_grading d"   and "s adds t" shows "d s \<le> d t"
strictify_def: shows "strictify = (\<Lambda> f x. seq\<cdot>x\<cdot>(f\<cdot>x))"
add_less_same_cancel1: fixes b :: "'b multiset"   and a :: "'b multiset" shows "(b + a \<subset># b) = (a \<subset># {#})"
intro: fixes subs :: "(('c \<times> nat) \<times> 'c \<times> nat) set" assumes "\<And>sub. sub \<in> subs \<Longrightarrow> fst (subsumee sub) = fst (subsumer sub)"   and "\<forall>v\<in>subsumees subs. \<exists>!v'. (v, v') \<in> subs"   and "subsumers subs \<inter> subsumees subs = {}" shows "wf_sub_rel subs"
euclidean_size_greater_0_iff: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and b :: "'a" assumes "class.euclidean_semiring divide plus minus zero times one modulo euclidean_size" shows "(0 < euclidean_size b) = (b \<noteq> zero)"
tensor_mat_mult: fixes m1 :: "'b mat"   and m2 :: "'b mat"   and m3 :: "'b mat"   and m4 :: "'b mat" assumes "m1 \<in> carrier_mat d1 d1"   and "m2 \<in> carrier_mat d1 d1"   and "m3 \<in> carrier_mat d2 d2"   and "m4 \<in> carrier_mat d2 d2" shows "tensor_mat (m1 * m2) (m3 * m4) = tensor_mat m1 m3 * tensor_mat m2 m4"
bool_assn_proper_4: fixes P :: "heap \<times> nat set \<Rightarrow> bool"   and Q :: "heap \<times> nat set \<Rightarrow> bool" assumes "proper P"   and "proper Q" shows "proper (\<lambda>h. P h \<and> Q h)"
ex_leq_Suc: fixes j :: "nat"   and P :: "nat \<Rightarrow> bool" shows "(\<exists>i\<le>Suc j. P i) = (P 0 \<or> (\<exists>i\<le>j. P (Suc i)))"
finite_has_maximal2: fixes A :: "'b set"   and a :: "'b" assumes "finite A"   and "a \<in> A" shows "\<exists>m\<in>A. a \<le> m \<and> (\<forall>b\<in>A. m \<le> b \<longrightarrow> m = b)"
bij_betw_subset_of: fixes A :: "'b set"   and B :: "'c set" assumes "finite A"   and "finite B" shows "bij_betw (subset_of A) ({f \<in> A \<rightarrow>\<^sub>E B. inj_on f A} // domain_permutation A B) {X. X \<subseteq> B \<and> card X = card A}"
comp_in_homE: fixes g :: "'a"   and f :: "'a"   and a :: "'a"   and c :: "'a" assumes "\<guillemotleft>g \<cdot> f : a \<rightarrow> c\<guillemotright>" obtains b :: "'a" where "\<guillemotleft>f : a \<rightarrow> b\<guillemotright>" and "\<guillemotleft>g : b \<rightarrow> c\<guillemotright>"
simps: fixes a :: "('a, 'b, 'c, 'd) seqp"   and \<Gamma> :: "'c \<Rightarrow> ('a, 'b, 'c, 'd) seqp" shows "(a \<in> cterms \<Gamma>) = ((\<exists>p pn. a = p \<and> p \<in> sterms \<Gamma> (\<Gamma> pn)) \<or> (\<exists>pp p. a = p \<and> pp \<in> cterms \<Gamma> \<and> p \<in> dterms \<Gamma> pp))"
Ord_linear: fixes k :: "V"   and l :: "V" assumes "Ord k"   and "Ord l" shows "k \<in> elts l \<or> k = l \<or> l \<in> elts k"
list_hd_lastD: fixes xs :: "'a list" assumes "1 < length xs" obtains x :: "'a" and y :: "'a" and ys :: "'a list" where "xs = x # ys @ [y]"
has_complex_derivative_uniform_limit: fixes z :: "complex"   and r :: "real"   and f :: "'a \<Rightarrow> complex \<Rightarrow> complex"   and f' :: "'a \<Rightarrow> complex \<Rightarrow> complex"   and F :: "'a filter"   and g :: "complex \<Rightarrow> complex" assumes "\<forall>\<^sub>F n in F. continuous_on (cball z r) (f n) \<and> (\<forall>w\<in>ball z r. (f n has_field_derivative f' n w) (at w))"   and "uniform_limit (cball z r) f g F"   and "F \<noteq> bot"   and "0 < r" obtains g' :: "complex \<Rightarrow> complex" where "continuous_on (cball z r) g" and "\<And>w. w \<in> ball z r \<Longrightarrow> (g has_field_derivative g' w) (at w) \<and> ((\<lambda>n. f' n w) \<longlongrightarrow> g' w) F"
core_nhds: fixes U :: "'a llist set"   and x :: "'a llist" assumes "U \<subseteq> carrier"   and "x \<in> core U" shows "U \<in> nhds x"
exhaust_sel: fixes iface :: "iface" obtains "iface = Iface (iface_sel iface)"
Int_interval: fixes a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" shows "cbox a b \<inter> cbox c d = cbox (\<Sum>i\<in>Basis. max (a \<bullet> i) (c \<bullet> i) *\<^sub>R i) (\<Sum>i\<in>Basis. min (b \<bullet> i) (d \<bullet> i) *\<^sub>R i)"
S4: shows "c_unfold (length (x # ls)) (c_fold (x # ls)) = c_unfold (Suc (length ls)) (c_fold (x # ls))"
l_delete_shadow_root_get_child_nodes_def: fixes get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_delete_shadow_root_get_child_nodes get_child_nodes_locs \<equiv> \<forall>ptr' shadow_root_ptr h h' r. ptr' \<noteq> cast\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r shadow_root_ptr \<longrightarrow> h \<turnstile> delete\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_M shadow_root_ptr \<rightarrow>\<^sub>h h' \<longrightarrow> r \<in> get_child_nodes_locs ptr' \<longrightarrow> r h h'"
take_induct: fixes P :: "'e\<cdot>'f resT \<Rightarrow> bool"   and x :: "'e\<cdot>'f resT" assumes "adm P"   and "\<And>n. P (resT_take n\<cdot>x)" shows "P x"
IH: shows "Array.length h a = Array.length h' a \<and> sorted (take j (Array.get h' a)) \<and> mset (Array.get h a) = mset (Array.get h' a)"
assms: fixes i :: "nat" assumes "i \<in> s" shows "i < k"
wls_subst_Var_simp2: fixes xs :: "'varSort"   and Y :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set"   and x :: "'var" assumes "wls (asSort xs) Y" shows "Var xs x #[Y / x]_xs = Y"