hcju/isabelleps huggingface dataset:
eq_newpkt_eq: fixes newpkt :: "nat \<times> nat \<Rightarrow> 'd"   and eq_newpkt :: "'d \<Rightarrow> bool"   and d :: "nat"   and i :: "nat" assumes "class.msg newpkt eq_newpkt" shows "eq_newpkt (newpkt (d, i))"
ang_vec_def: fixes z1 :: "complex"   and z2 :: "complex" shows "\<angle> z1 z2 \<equiv> \<downharpoonright>arg z2 - arg z1\<downharpoonleft>"
rep_eq: fixes x :: "circline_mat"   and xa :: "circline_mat" shows "circline_eq_clmat x xa = circline_eq_cmat (Rep_circline_mat x) (Rep_circline_mat xa)"
intros_4: shows "CK_nf_pos (I \<circ> K)"
relcomppE: fixes r :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and s :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'a"   and c :: "'c" assumes "(r OO s) a c" obtains b :: "'b" where "r a b" and "s b c"
interior_rel_interior_gen: fixes S :: "'a set" shows "interior S = (if aff_dim S = int DIM('a) then rel_interior S else {})"
app_subst_Cons: fixes S :: "nat \<Rightarrow> typ"   and x :: "'b"   and l :: "'b list" shows "$ S (x # l) = $ S x # $ S l"
simps_2: fixes l :: "('d \<times> 'e) tree"   and a :: "'d"   and uu :: "'e"   and r :: "('d \<times> 'e) tree" shows "Tree2.set_tree \<langle>l, (a, uu), r\<rangle> = {a} \<union> Tree2.set_tree l \<union> Tree2.set_tree r"
dual_comp_neg: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and dual :: "'b \<Rightarrow> 'b"   and dual_star :: "'b \<Rightarrow> 'b"   and omega :: "'b \<Rightarrow> 'b"   and star :: "'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and x :: "'b"   and y :: "'b" assumes "class.mbt_algebra one times inf less_eq less sup dual dual_star omega star bot top" shows "sup (times (dual x) y) (times (mbt_algebra.neg_assert one times inf dual bot x) top) = times (dual x) y"
d: fixes \<sigma> :: "'Var \<Rightarrow> 'Val option" assumes "dom \<sigma> = differing_vars_lists mem\<^sub>1 mem\<^sub>2 mems k" shows "\<exists>c\<^sub>f' mem\<^sub>f'. (cms\<^sub>2 ! k, subst \<sigma> (snd (mems ! k))) \<leadsto> \<langle>c\<^sub>f', snd (cms\<^sub>1' ! k), mem\<^sub>f'\<rangle> \<and> (cms\<^sub>1' ! k, subst (g1 \<sigma>) mem\<^sub>1') \<approx> \<langle>c\<^sub>f', snd (cms\<^sub>1' ! k), mem\<^sub>f'\<rangle>"
lsum_list_def: fixes xs :: "'a llist" shows "lsum_list xs = (if lfinite xs then sum_list (list_of xs) else (0::'a))"
max_0_1_2: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "ord.max less_eq one zero = one"
ipurge_tr_ref_aux_failures: fixes P :: "'e process"   and I :: "('f \<times> 'f) set"   and D :: "'e \<Rightarrow> 'f"   and xs :: "'e list"   and X :: "'e set"   and U :: "'f set" assumes "secure P I D"   and "(xs, X) \<in> failures P" shows "(ipurge_tr_aux I D U xs, ipurge_ref_aux I D U xs X) \<in> failures P"
\<dagger>: shows "\<forall>\<^sub>F t in at_top. pi / 2 - (LBINT u=0..\<infinity>. exp (- (u * t)) * (u * sin t + cos t) / (1 + u\<^sup>2)) = Si t"
add_arc_def: fixes G :: "('c, 'd) pre_digraph"   and a :: "'d" shows "pre_digraph.add_arc G a = \<lparr>verts = verts G \<union> {tail G a, head G a}, arcs = insert a (arcs G), tail = tail G, head = head G\<rparr>"
simps_3: fixes h :: "'a \<Rightarrow> 'b list \<Rightarrow> bool"   and a :: "'a"   and xs :: "'b list" shows "interpret h (Atom a) xs = h a xs"
cases: fixes a :: "'b list list" assumes "map_head_side a" obtains xs :: "'b list list" where "a = xs" and "\<And>x. x \<noteq> []"
interleaves_length: fixes P :: "'b \<Rightarrow> 'b list \<Rightarrow> bool"   and xs :: "'b list"   and ys :: "'b list"   and zs :: "'b list" assumes "xs \<simeq> {ys, zs, P}" shows "length xs = length ys + length zs"
sel_1: fixes x :: "quat" shows "quat.Re (- x) = - quat.Re x"
no_trailing_Nil: fixes P :: "'b \<Rightarrow> bool" shows "no_trailing P []"
rtrancl_listrel1_ConsI1: fixes xs :: "'g list"   and ys :: "'g list"   and r :: "('g \<times> 'g) set"   and x :: "'g" assumes "(xs, ys) \<in> (listrel1 r)\<^sup>*" shows "(x # xs, x # ys) \<in> (listrel1 r)\<^sup>*"
parent_child_rel_impl: shows "Shadow_DOM.parent_child_rel = l_heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_parent_child_rel Shadow_DOM.get_child_nodes"
ran_m_fmempty: shows "ran_m fmempty = {#}"
in_cenum: fixes cEnum :: "('b list \<times> (('b \<Rightarrow> bool) \<Rightarrow> bool) \<times> (('b \<Rightarrow> bool) \<Rightarrow> bool)) option"   and enum :: "'b list"   and rest :: "(('b \<Rightarrow> bool) \<Rightarrow> bool) \<times> (('b \<Rightarrow> bool) \<Rightarrow> bool)"   and f :: "'b" assumes "class.cenum cEnum"   and "ID cEnum = Some (enum, rest)" shows "f \<in> set enum"
assms_2: shows "\<S>\<bullet>\<C> \<turnstile> [$C ConstInt32 c, $Br_table is i] : ts _> ts'"
os_ts1324__os: fixes A :: "'p"   and X :: "'p"   and Y :: "'p"   and Z :: "'p" assumes "A X OS Y Z"   and "A Y TS X Z" shows "A Z OS X Y"
subspace_0: fixes S :: "('b, 'c) vec set" assumes "vec.subspace S" shows "0 \<in> S"
sigma_finite_countable: fixes M :: "'c measure" assumes "sigma_finite_measure M" shows "\<exists>A. countable A \<and> A \<subseteq> sets M \<and> \<Union> A = space M \<and> (\<forall>a\<in>A. emeasure M a \<noteq> \<infinity>)"
extf_irrefl: fixes ground_heads_var :: "'a \<Rightarrow> 'b set"   and gt_sym :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and arity_sym :: "'b \<Rightarrow> enat"   and arity_var :: "'a \<Rightarrow> enat"   and wt_sym :: "'b \<Rightarrow> 'c"   and \<epsilon> :: "nat"   and \<delta> :: "nat"   and extf :: "'b \<Rightarrow> (('b, 'a) tm \<Rightarrow> ('b, 'a) tm \<Rightarrow> bool) \<Rightarrow> ('b, 'a) tm list \<Rightarrow> ('b, 'a) tm list \<Rightarrow> bool"   and xs :: "('b, 'a) tm list"   and gt :: "('b, 'a) tm \<Rightarrow> ('b, 'a) tm \<Rightarrow> bool"   and f :: "'b" assumes "kbo_std_basis ground_heads_var gt_sym arity_sym arity_var wt_sym \<epsilon> \<delta> extf"   and "\<forall>x\<in>set xs. \<not> gt x x" shows "\<not> extf f gt xs xs"
liftAll_def: fixes phi :: "'f \<Rightarrow> bool"   and inp :: "'g \<Rightarrow> 'f option" shows "liftAll phi inp \<equiv> \<forall>i v. inp i = Some v \<longrightarrow> phi v"
merge_heap_is_heap: fixes l :: "'c Tree"   and r :: "'c Tree" assumes "is_heap l"   and "is_heap r" shows "is_heap (merge l r)"
reachable_tot_eq: fixes F :: "'b program" shows "reachable (totalize F) = reachable F"
rbl_mul_simps_2: fixes ys :: "bool list" shows "rbl_mul [] ys = []"
SystemClasses_def: shows "SystemClasses \<equiv> [ObjectC, NullPointerC, ClassCastC, OutOfMemoryC, NoClassDefFoundC, IncompatibleClassChangeC, NoSuchFieldC, NoSuchMethodC]"
nontriv_solution_def: shows "nontriv_solution = (\<lambda>(aa, b). (aa, b) \<noteq> (1::'a, 0::'a) \<and> aa\<^sup>2 = (1::'a) + of_nat (a\<^sup>2 - Suc 0) * b\<^sup>2)"
nfv_Exists: fixes \<phi> :: "'c MFOTL.formula" shows "MFOTL.nfv (formula.Exists \<phi>) = MFOTL.nfv \<phi> - 1"
lselect_llength_le: fixes w :: "'b llist"   and s :: "nat set" shows "llength (lselect s w) \<le> esize s"
simps_30: fixes x6 :: "ty" shows "x6\<lfloor>\<rceil> \<noteq> NT"
lhs_def: shows "lhs \<equiv> \<lambda>plus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 zero\<^sub>V\<^sub>S\<^sub>_\<^sub>1 minus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 scale\<^sub>1 basis\<^sub>1. finite_dimensional_vector_space_pair_ow (Collect (Domainp B\<^sub>1)) plus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 zero\<^sub>V\<^sub>S\<^sub>_\<^sub>1 minus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 scale\<^sub>1 basis\<^sub>1 (Collect (Domainp B\<^sub>2))"
WFS0: fixes S :: "(ZF, ZF) Signature" shows "WellFormedToSet S 0 = {Vx}"
Union_set_lists: fixes A :: "'a set" shows "\<Union> (set ` lists A) = A"
parPres: fixes P :: "pi"   and Q :: "pi"   and R :: "pi" assumes "P \<simeq> Q" shows "P \<parallel> R \<simeq> Q \<parallel> R"
sprod_l_distr: fixes a :: "'b"   and b :: "'b"   and m :: "'a" assumes "a \<in> carrier R"   and "b \<in> carrier R"   and "m \<in> carrier M" shows "(a \<plusminus>\<^bsub>R\<^esub> b) \<cdot>\<^sub>s m = a \<cdot>\<^sub>s m \<plusminus> b \<cdot>\<^sub>s m"
qFresh_preserves_alpha: fixes X :: "('a, 'b, 'varSort, 'var, 'c) qTerm"   and X' :: "('a, 'b, 'varSort, 'var, 'c) qTerm"   and zs :: "'varSort"   and z :: "'var" assumes "qGood X \<or> qGood X'"   and "X #= X'" shows "qFresh zs z X = qFresh zs z X'"
locally_compact_homeomorphism_projection_closed: fixes S :: "'c set" assumes "locally compact S" obtains T :: "('c \<times> 'd) set" and f :: "'c \<Rightarrow> 'c \<times> 'd" where "closed T" and "homeomorphism S T f fst"
ev_induct_strong: fixes \<phi> :: "'d stream \<Rightarrow> bool"   and x :: "'d stream"   and P :: "'d stream \<Rightarrow> bool" assumes "ev \<phi> x"   and "\<And>xs. \<phi> xs \<Longrightarrow> P xs"   and "\<And>xs. \<lbrakk>ev \<phi> (stl xs); \<not> \<phi> xs; P (stl xs)\<rbrakk> \<Longrightarrow> P xs" shows "P x"
jAction_simAbs_cong: fixes t :: "'s Trace"   and ec :: "'rep"   and a :: "'a"   and ec' :: "'rep" assumes "t \<in> jkbpC"   and "simAbs ec = sim_equiv_class a t"   and "simAbs ec = simAbs ec'" shows "set (simAction a ec) = set (simAction a ec')"
bres_canc1: fixes x :: "'b \<times> 'c"   and y :: "'b \<times> 'c" shows "less_eq_prod (times_prod' x (prod.bres x y)) y"
sets_stream_space_sstart: fixes S :: "'b set" assumes "countable S" shows "sets (stream_space (count_space S)) = sets (sigma (streams S) (sstart S ` lists S \<union> {{}}))"
deducible_clause_preserve_redundancy: fixes C :: "'at Literal set"   and S :: "'at Literal set set"   and R :: "'at Literal set \<Rightarrow> 'at Literal set \<Rightarrow> 'at Literal set \<Rightarrow> bool" assumes "redundant C S" shows "redundant C (add_all_deducible_clauses R S)"
simps_4: fixes uw :: "'a" shows "(extC.Val uw \<le> extC.Top) = True"
is_mgu_def: fixes subst_atm :: "'b \<Rightarrow> 'c \<Rightarrow> 'b"   and comp_subst :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<sigma> :: "'c"   and AAA :: "'b set set" shows "substitution_ops.is_mgu subst_atm comp_subst \<sigma> AAA = (substitution_ops.is_unifiers subst_atm \<sigma> AAA \<and> (\<forall>\<tau>. substitution_ops.is_unifiers subst_atm \<tau> AAA \<longrightarrow> (\<exists>\<gamma>. \<tau> = comp_subst \<sigma> \<gamma>)))"
ldistinct_ltlI: fixes xs :: "'c llist" assumes "ldistinct xs" shows "ldistinct (ltl xs)"
i: shows "on_triple f `` E \<subseteq> edges (S i)"
cases: fixes a :: "'a"   and A :: "'a set"   and r :: "('a \<times> 'a) set" assumes "a \<in> directed_towards A r" obtains x :: "'a" where "a = xa__" and "xa__ \<in> A"   | x :: "'a" and y :: "'a" where "a = x" and "y \<in> directed_towards A r" and "(x, y) \<in> r"
gcd_add1: fixes m :: "'a"   and n :: "'a" shows "gcd (m + n) n = gcd m n"
typeof_unboxed_inversion_1: fixes x :: "('a, 'b, 'c) unboxed" assumes "typeof x = None" shows "\<exists>d. x = OpDyn d"
add_left: fixes a :: "'b \<Rightarrow>\<^sub>C\<^sub>L 'a"   and a' :: "'b \<Rightarrow>\<^sub>C\<^sub>L 'a"   and b :: "'b" shows "cblinfun_apply (a + a') b = cblinfun_apply a b + cblinfun_apply a' b"
rtrancl_reflcl_absorb: fixes R :: "('a \<times> 'a) set" shows "(R\<^sup>*)\<^sup>= = R\<^sup>*"
xt: shows "x \<notin> FvarsT t \<and> snd ` set txs \<inter> FvarsT t = {}"
coboundedI1: fixes a :: "'b multiset"   and c :: "'b multiset"   and b :: "'b multiset" assumes "a \<subseteq># c" shows "a \<inter># b \<subseteq># c"
vector_scaleR_component: fixes r :: "real"   and x :: "('a, 'b) vec"   and i :: "'b" shows "(r *\<^sub>R x) $ i = r *\<^sub>R x $ i"
fls_inverse_eq_0_iff: fixes f :: "'a fls" shows "(inverse f = 0) = (fls_nth f (fls_subdegree f) = (0::'a))"
assms_2: shows "\<forall>i. i \<notin> {0..<n} \<longrightarrow> f i = i"
ordinal_minusI: fixes x :: "ordinal"   and y :: "ordinal"   and z :: "ordinal" assumes "x = y + z" shows "x - y = z"
f_3: shows "\<forall>A\<in>M. f A \<noteq> \<infinity>"
is_Hcomp_def: fixes "term" :: "'b local.term" shows "is_Hcomp term = (\<exists>x31 x32. term = x31 \<^bold>\<star> x32)"
cod_src_eq_dom_src: shows "HHfgh.cod.src = HHfgh.dsrc"
round_robin_reschedule_Cons_1: fixes final :: "'c \<Rightarrow> bool"   and r :: "('b, 'a, 'c, 'd, 'e, 'f) semantics"   and convert_RA :: "'b \<Rightarrow>f nat \<Rightarrow> 'f list"   and t0 :: "'a"   and queue :: "'a list"   and n0 :: "nat"   and s :: "('b, 'a) locks \<times> (('a \<Rightarrow> ('c \<times> 'b \<Rightarrow>f nat) option) \<times> 'd) \<times> ('a \<Rightarrow> 'e wait_set_status option) \<times> 'a set" shows "multithreaded_base.round_robin_reschedule final r convert_RA t0 (t0 # queue) n0 s = None"
set_vwalk_arcs_snoc: fixes p :: "'a list"   and a :: "'a" assumes "p \<noteq> []" shows "set (vwalk_arcs (p @ [a])) = insert (last p, a) (set (vwalk_arcs p))"
cases: fixes a1 :: "lam"   and a2 :: "lam" assumes " a1 \<guillemotright> a2" obtains t :: "lam" and s :: "lam" where "a1 = App t s" and "a2 = t"
cleanup_simps: fixes f :: "'a \<times> 'a \<Rightarrow> ennreal"   and a :: "'a"   and b :: "'a" shows "cleanup f (a, b) = (if f (b, a) < f (a, b) then f (a, b) - f (b, a) else 0)"
cases: fixes x :: "('a \<times> 'b) list" obtains "x = []"   | p :: "'a \<times> 'b" and ps :: "('a \<times> 'b) list" where "x = p # ps"
continuous_on_closed: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'b" shows "continuous_on S f = (\<forall>T. closedin (top_of_set (f ` S)) T \<longrightarrow> closedin (top_of_set S) (S \<inter> f -` T))"
ballI2: fixes A :: "('c \<times> 'd) set"   and P :: "'c \<Rightarrow> 'd \<Rightarrow> bool" assumes "\<And>x y. (x, y) \<in> A \<Longrightarrow> P x y" shows "\<forall>(x, y)\<in>A. P x y"
antisym_imp_exe_antisym: fixes l :: "('a \<times> 'a) list" assumes "antisym (set l)" shows "exe_antisym l"
gcd_add_dvd_left2: fixes a :: "'a"   and c :: "'a"   and b :: "'a" assumes "a dvd c" shows "gcd (b + c) a = gcd b a"
freeultrafilter_FreeUltrafilterNat: shows "freeultrafilter \<U>"
sum_list_nonneg: fixes xs :: "'a list" assumes "\<And>x. x \<in> set xs \<Longrightarrow> (0::'a) \<le> x" shows "(0::'a) \<le> sum_list xs"
s'_\<alpha>_s: shows "s' \<alpha>\<Longrightarrow>\<^bsub>SES\<^esub> s"
smC_def: fixes a :: "'p"   and b :: "'p"   and c :: "'p"   and d :: "'p" shows "a b \<congruent>\<^sub>s\<^sub>m c d \<equiv> dist a b = dist c d"
Rf_mono: fixes \<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 N :: "'b literal multiset set"   and N' :: "'b literal multiset set" assumes "redundancy_criterion \<Gamma> Rf Ri"   and "N \<subseteq> N'" shows "Rf N \<subseteq> Rf N'"
bilateral_substitutes_onI: fixes A :: "'x set"   and f :: "'x set \<Rightarrow> 'x set" assumes "\<And>B a b. \<lbrakk>B \<subseteq> A; a \<in> A; b \<in> A; Xd a \<notin> Xd ` B \<and> Xd b \<notin> Xd ` B \<and> b \<notin> f (B \<union> {b})\<rbrakk> \<Longrightarrow> b \<notin> f (B \<union> {a, b})" shows "bilateral_substitutes_on A f"
gauge_integral_Fubini_universe_y_1: fixes f :: "'b \<times> 'c \<Rightarrow> 'd" assumes "integrable lborel f"   and "(\<lambda>x. integral UNIV (\<lambda>y. f (y, x))) \<in> borel_measurable lborel" shows "integral UNIV f = integral UNIV (\<lambda>x. integral UNIV (\<lambda>y. f (y, x)))"
distinct_2: fixes x21 :: "'a"   and x22 :: "'a convnet"   and x1 :: "nat" shows "Conv x21 x22 \<noteq> Input x1"
subst_cls_lists_Cons: fixes subst_atm :: "'b \<Rightarrow> 'c \<Rightarrow> 'b"   and id_subst :: "'c"   and comp_subst :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and renamings_apart :: "'b literal multiset list \<Rightarrow> 'c list"   and atm_of_atms :: "'b list \<Rightarrow> 'b"   and C :: "'b literal multiset"   and Cs :: "'b literal multiset list"   and \<sigma> :: "'c"   and \<sigma>s :: "'c list" assumes "substitution subst_atm id_subst comp_subst renamings_apart atm_of_atms" shows "substitution_ops.subst_cls_lists subst_atm (C # Cs) (\<sigma> # \<sigma>s) = substitution_ops.subst_cls subst_atm C \<sigma> # substitution_ops.subst_cls_lists subst_atm Cs \<sigma>s"
assms_1: shows "class_add P (C, cdec) \<turnstile> Object sees_methods Mm"
f_range_onI: fixes A :: "'d set"   and f :: "'d set \<Rightarrow> 'd set" assumes "\<And>B. B \<subseteq> A \<Longrightarrow> f B \<subseteq> B" shows "f_range_on A f"
push_forward_compose: fixes X :: "('c \<Rightarrow> real) \<Rightarrow> real"   and M1 :: "('c, 'd) chart set"   and k :: "enat"   and p :: "'c"   and M2 :: "('e, 'f) chart set"   and f :: "'c \<Rightarrow> 'e"   and M3 :: "('g, 'h) chart set"   and g :: "'e \<Rightarrow> 'g" assumes "X \<in> c_manifold.tangent_space M1 k p"   and "p \<in> manifold.carrier M1"   and "diff k M1 M2 f"   and "diff k M2 M3 g" shows "diff.push_forward k M2 M3 g (diff.push_forward k M1 M2 f X) = diff.push_forward k M1 M3 (g \<circ> f) X"
basis_det_nonzero: fixes n :: "nat"   and G :: "'a vec list" assumes "vectorspace.basis class_ring (module_vec TYPE('a) n) (set G)"   and "length G = n" shows "det (mat_of_rows n G) \<noteq> (0::'a)"
lcos_self: fixes G :: "('a, 'b) monoid_scheme"   and x :: "'a"   and H :: "'a set" assumes "Group.group G"   and "x \<in> carrier G"   and "subgroup H G" shows "x \<in> x <#\<^bsub>G\<^esub> H"
simps_1: fixes uu :: "'b"   and p :: "'a" shows "labelsOut uu (Reg p) = p"
sfilter_Stream: fixes P :: "'a \<Rightarrow> bool"   and x :: "'a"   and s :: "'a stream" shows "sfilter P (x ## s) = (if P x then x ## sfilter P s else sfilter P s)"
nomatch: fixes \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and m :: "'c match_expr"   and p :: "'d"   and \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and a :: "action" assumes "\<not> Semantics.matches \<gamma> m p" shows "\<Gamma>,\<gamma>,p\<turnstile> \<langle>[Rule m a], Undecided\<rangle> \<Rightarrow> Undecided"
closest_supchamber_def: fixes F :: "'b set"   and D :: "'b set" shows "codomain.closest_supchamber F D = (ARG_MIN (\<lambda>C. codomain.chamber_distance C D) C. codomain.chamber C \<and> order.greater_eq C F)"
collect_clki_def: fixes I :: "'e \<Rightarrow> ('c, 'd) cconstraint" shows "collect_clki I = \<Union> {collect_clock_pairs (I x) |x. True}"
ind: shows "\<forall>xc. {\<lambda>nl. nl = xs @ x # 0 \<up> (fft - far) @ xc} fap {\<lambda>nl. nl = xs @ x # rec_exec f (xs @ [x]) # 0 \<up> (fft - Suc far) @ xc}"
diff_self: fixes a :: "'a" shows "a - a = (0::'a)"
helper_defined_ruleset_calledby: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and rs :: "'c rule list"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and s :: "state" assumes "wf (called_by_chain \<Gamma>)"   and "\<forall>rsg\<in>ran \<Gamma> \<union> {rs}. wf_chain \<Gamma> rsg"   and "\<forall>rsg\<in>ran \<Gamma> \<union> {rs}. \<forall>r\<in>set rsg. (\<nexists>chain. get_action r = Goto chain) \<and> get_action r \<noteq> Unknown"   and "\<forall>r\<in>set rs. get_action r \<noteq> Return" shows "\<exists>t. \<Gamma>,\<gamma>,p\<turnstile> \<langle>rs, s\<rangle> \<Rightarrow> t"
lunit_simps_1: fixes f :: "'a" assumes "C.ide f" shows "C.arr \<l>\<^sub>C[f]"
lF_simps_5: fixes f :: "'a" assumes "ide f" shows "S.cod (UP.lF f) = S.UP f"
\<alpha>_ide_simp: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "ide a"   and "ide b"   and "ide c" shows "\<alpha> (a, b, c) = \<a>[a, b, c]"
that: fixes a :: "nat \<Rightarrow> real" assumes "homotopic_with (\<lambda>x. True) (nsphere p) (nsphere p) f (\<lambda>x. a)" shows "thesis"
sum_monom_mod: fixes b :: "nat"   and f :: "'b poly"   and g :: "nat \<Rightarrow> 'b" assumes "b < degree f" shows "(\<Sum>i\<le>b. monom (g i) i) mod f = (\<Sum>i\<le>b. monom (g i) i)"
assms_1: shows "y \<notin> subst_domain \<sigma>"
plus_lens_distr: fixes Z :: "'d \<Longrightarrow> 'e"   and X :: "'f \<Longrightarrow> 'd"   and Y :: "'g \<Longrightarrow> 'd" assumes "mwb_lens Z" shows "(X +\<^sub>L Y) ;\<^sub>L Z = X ;\<^sub>L Z +\<^sub>L Y ;\<^sub>L Z"
bounded_bilinear_blinfun_apply: shows "bounded_bilinear blinfun_apply"
right_unique_vimage2p: fixes P :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and g :: "'e \<Rightarrow> 'd"   and f :: "'f \<Rightarrow> 'c" assumes "right_unique P"   and "inj g" shows "right_unique (BNF_Def.vimage2p f g P)"
i_set0_indp_i_set0_ind_eq: shows "i_set0_indp = (\<lambda>x. x \<in> i_set0_ind)"
frac_non_empty: fixes a :: "'a"   and b :: "'a" assumes "(a, b) \<in> R \<times> (R\<setminus>\<pp>)" shows "qr.valid_frac (qr.frac a b)"
Diag_TensorE_1: fixes t :: "'a local.term"   and u :: "'a local.term" assumes "Diag (t \<^bold>\<otimes> u)" shows "\<^bold>\<langle>un_Prim t\<^bold>\<rangle> = t"
simps_5: fixes a :: "nat"   and b :: "nat"   and A :: "atom fm"   and B :: "atom fm" shows "swap_fm a b (Or A B) = Or (swap_fm a b A) (swap_fm a b B)"
that: fixes B :: "'b set" assumes "B \<subseteq> V"   and "independent B"   and "V \<subseteq> span B" shows "thesis"
lcosets_part_G: fixes H :: "'a set" assumes "subgroup H G" shows "\<Union> (lcosets H) = carrier G"
SumId: fixes P :: "pi" shows "P \<oplus> \<zero> \<equiv>\<^sub>s P"
right_zero_vec: fixes v :: "'a vec"   and n :: "nat" assumes "v \<in> carrier_vec n" shows "v + 0\<^sub>v n = v"
6: shows "p \<bullet> \<alpha>1 = \<alpha>2"
is_relation_order_red: fixes d :: "'a \<Rightarrow> nat"   and F :: "('t \<Rightarrow>\<^sub>0 'c) set"   and m :: "nat" assumes "dickson_grading d" shows "relation_order (red F) (\<prec>\<^sub>p) (dgrad_p_set d m)"
continuous_on_Re: fixes s :: "'a set"   and g :: "'a \<Rightarrow> complex" assumes "continuous_on s g" shows "continuous_on s (\<lambda>x. Re (g x))"
bin_lcp_pref_fst_snd: shows "\<alpha> \<le>p u\<^sub>0 \<cdot> u\<^sub>1"
injective_into_1d_eq_homeomorphism: fixes S :: "'b set"   and f :: "'b \<Rightarrow> real" assumes "continuous_on S f"   and "path_connected S" shows "inj_on f S = (\<exists>g. homeomorphism S (f ` S) f g)"
assms_2: shows "\<exists>k. hamlet (Rep_run r k c)"
field_differentiable_divide: fixes f :: "'a \<Rightarrow> 'a"   and a :: "'a"   and S :: "'a set"   and g :: "'a \<Rightarrow> 'a" assumes "f field_differentiable at a within S"   and "g field_differentiable at a within S"   and "g a \<noteq> (0::'a)" shows "(\<lambda>z. f z / g z) field_differentiable at a within S"
distinct_1: fixes nat' :: "nat" shows "0 \<noteq> Suc nat'"
higher_def: fixes p :: "'a \<Rightarrow>\<^sub>0 'c"   and t :: "'a" shows "punit.higher p t = except p {s. s \<preceq> t}"
assms_2: fixes i :: "'agent" assumes "i \<in> agents" shows "SD (R i) p q"
a6_def: shows "a6 = is_a_proper_intersec R A C' A' C"
Qpres: fixes \<sigma> :: "'state"   and \<sigma>' :: "'state"   and P :: "'prog" assumes "Q \<sigma>"   and "\<sigma>' \<in> small P \<sigma>" shows "Q \<sigma>'"
pmult_Nil: fixes l2 :: "'a list" shows "[] *** l2 = []"
doesnt_modify_def: fixes c :: "(var\<^sub>C, aexp\<^sub>C, bexp\<^sub>C) Stmt"   and x :: "var\<^sub>C" shows "conc.doesnt_modify c x = (\<forall>mds mem c' mds' mem'. eval_abv\<^sub>C \<langle>c, mds, mem\<rangle>\<^sub>C \<langle>c', mds', mem'\<rangle>\<^sub>C \<longrightarrow> mem x = mem' x \<and> dma\<^sub>C mem x = dma\<^sub>C mem' x)"
alphaAbs_qAbs_iff_ex_distinct_qFresh: fixes X :: "('d, 'e, 'varSort, 'var, 'f) qTerm"   and xs :: "'varSort"   and x :: "'var"   and xs' :: "'varSort"   and x' :: "'var"   and X' :: "('d, 'e, 'varSort, 'var, 'f) qTerm" assumes "qGood X" shows "(qAbs xs x X $= qAbs xs' x' X') = (xs = xs' \<and> (\<exists>y. y \<notin> {x, x'} \<and> qFresh xs y X \<and> qFresh xs y X' \<and> X #[[y \<and> x]]_xs #= X' #[[y \<and> x']]_xs))"
fresh_pgwt_same_type: fixes S :: "'fun set"   and t :: "('fun, 'var) Term.term" assumes "finite S"   and "wf\<^sub>t\<^sub>r\<^sub>m t" shows "\<Gamma> (fresh_pgwt S (\<Gamma> t)) = \<Gamma> t"
pathFailsP: fixes subs :: "'b \<Rightarrow> 'b set"   and P :: "(nat \<times> 'b) set \<Rightarrow> bool"   and gamma :: "'b"   and n :: "nat" assumes "inherited subs P"   and "fans subs"   and "\<not> P (tree subs gamma)" shows "\<not> P (tree subs (path subs gamma P n))"