hcju/isabelleps huggingface dataset:
expand_aimpl_def: shows "expand_aimpl \<equiv> REC\<^sub>T (\<lambda>expand (n, ns). if new n = {} then if \<exists>n'\<in>ns. old n' = old n \<and> next n' = next n then RETURN (name n, upd_incoming n ns) else ASSERT (n \<notin> ns) \<bind> (\<lambda>_. ASSERT (name n \<notin> name ` ns) \<bind> (\<lambda>_. expand (\<lparr>name = expand_new_name (name n), incoming = {name n}, new = next n, old = {}, next = {}\<rparr>, insert n ns))) else SPEC (\<lambda>x. x \<in> new n) \<bind> (\<lambda>\<phi>. let n = n\<lparr>new := new n - {\<phi>}\<rparr> in case \<phi> of true\<^sub>r \<Rightarrow> expand (n\<lparr>old := {\<phi>} \<union> old n\<rparr>, ns) | false\<^sub>r \<Rightarrow> RETURN (name n, ns) | prop\<^sub>r(q) \<Rightarrow> if nprop\<^sub>r(q) \<in> old n then RETURN (name n, ns) else expand (n\<lparr>old := {\<phi>} \<union> old n\<rparr>, ns) | nprop\<^sub>r(q) \<Rightarrow> if prop\<^sub>r(q) \<in> old n then RETURN (name n, ns) else expand (n\<lparr>old := {\<phi>} \<union> old n\<rparr>, ns) | \<nu> and\<^sub>r \<mu> \<Rightarrow> expand (n\<lparr>new := insert \<nu> (insert \<mu> (new n)), old := {\<phi>} \<union> old n, next := next n\<rparr>, ns) | \<mu> or\<^sub>r \<nu> \<Rightarrow> expand (n\<lparr>new := insert \<mu> (new n), old := insert \<phi> (old n), next := {} \<union> next n\<rparr>, ns) \<bind> (\<lambda>(nm, nds). expand (n\<lparr>name := nm, new := {\<nu>} \<union> new n, old := {\<phi>} \<union> old n\<rparr>, nds)) | X\<^sub>r \<nu> \<Rightarrow> expand (n\<lparr>new := new n, old := {\<phi>} \<union> old n, next := insert \<nu> (next n)\<rparr>, ns) | \<mu> U\<^sub>r \<nu> \<Rightarrow> expand (n\<lparr>new := insert \<mu> (new n), old := insert \<phi> (old n), next := {\<phi>} \<union> next n\<rparr>, ns) \<bind> (\<lambda>(nm, nds). expand (n\<lparr>name := nm, new := {\<nu>} \<union> new n, old := {\<phi>} \<union> old n\<rparr>, nds)) | \<mu> R\<^sub>r \<nu> \<Rightarrow> expand (n\<lparr>new := insert \<nu> (new n), old := insert \<phi> (old n), next := {\<phi>} \<union> next n\<rparr>, ns) \<bind> (\<lambda>(nm, nds). expand (n\<lparr>name := nm, new := {\<mu>, \<nu>} \<union> new n, old := {\<phi>} \<union> old n\<rparr>, nds))))"
Fun_map: fixes g :: "'c"   and f :: "'c" assumes "C.arr g"   and "C.arr f" shows "S.Fun (Hom.map (g, f)) = restrict (\<phi> (C.dom g, C.cod f) \<circ> (\<lambda>h. f \<cdot> h \<cdot> g) \<circ> \<psi> (C.cod g, C.dom f)) (Hom.set (C.cod g, C.dom f))"
lesubstep_type_def: fixes A :: "(nat \<times> 'b) set"   and r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and B :: "(nat \<times> 'b) set" shows "A {\<sqsubseteq>\<^bsub>r\<^esub>} B \<equiv> \<forall>(p, \<tau>)\<in>A. \<exists>\<tau>'. (p, \<tau>') \<in> B \<and> \<tau> \<sqsubseteq>\<^bsub>r\<^esub> \<tau>'"
\<Sigma>_protocol_def: fixes Rel :: "('a \<times> 'b) set"   and valid_pub :: "'a set"   and init :: "'a \<Rightarrow> 'b \<Rightarrow> ('c \<times> 'd) spmf"   and response :: "'c \<Rightarrow> 'b \<Rightarrow> 'e \<Rightarrow> 'f spmf"   and check :: "'a \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and S_raw :: "'a \<Rightarrow> 'e \<Rightarrow> ('d \<times> 'f) spmf"   and \<A>ss :: "'a \<Rightarrow> 'd \<times> 'e \<times> 'f \<Rightarrow> 'd \<times> 'e \<times> 'f \<Rightarrow> 'b spmf"   and challenge_space :: "'e set" assumes "\<Sigma>_protocols_base Rel valid_pub" shows "\<Sigma>_protocols_base.\<Sigma>_protocol init response check Rel S_raw \<A>ss challenge_space valid_pub \<equiv> \<Sigma>_protocols_base.completeness init response check Rel challenge_space \<and> \<Sigma>_protocols_base.special_soundness check Rel \<A>ss challenge_space valid_pub \<and> \<Sigma>_protocols_base.HVZK init response check Rel S_raw challenge_space valid_pub"
bdd_above_image_mono: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set" assumes "mono f"   and "bdd_above A" shows "bdd_above (f ` A)"
map_of_fv_subset: fixes x :: "'d"   and \<Gamma> :: "('d \<times> 'e) list" assumes "x \<in> domA \<Gamma>" shows "fv (the (map_of \<Gamma> x)) \<subseteq> fv \<Gamma>"
v11_vconverse_app_app: fixes a :: "V" assumes "a \<in>\<^sub>\<circ> \<AA>\<lparr>Arr\<rparr>" shows "(\<FF>\<lparr>ArrMap\<rparr>)\<inverse>\<^sub>\<circ>\<lparr>\<FF>\<lparr>ArrMap\<rparr>\<lparr>a\<rparr>\<rparr> = a"
Austin_Lemma: fixes \<D> :: "'a set set" assumes "finite \<D>"   and "\<And>D. D \<in> \<D> \<Longrightarrow> \<exists>k a b. D = cbox a b \<and> (\<forall>i\<in>Basis. b \<bullet> i - a \<bullet> i = k)" obtains \<C> :: "'a set set" where "\<C> \<subseteq> \<D>" and "disjoint \<C>" and "Sigma_Algebra.measure lebesgue (\<Union> \<D>) / 3 ^ DIM('a) \<le> Sigma_Algebra.measure lebesgue (\<Union> \<C>)"
singleton_conv: fixes a :: "'a" shows "{x. x = a} = {a}"
canonical_basis_length_ell2: shows "length canonical_basis = CARD('b)"
heap_is_wellformed_children_distinct: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and heap_is_wellformed :: "(_) heap \<Rightarrow> bool"   and parent_child_rel :: "(_) heap \<Rightarrow> ((_) object_ptr \<times> (_) object_ptr) set"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and h :: "(_) heap"   and ptr :: "(_) object_ptr"   and children :: "(_) node_ptr list" assumes "l_heap_is_wellformed type_wf known_ptr heap_is_wellformed parent_child_rel get_child_nodes get_disconnected_nodes"   and "heap_is_wellformed h"   and "h \<turnstile> get_child_nodes ptr \<rightarrow>\<^sub>r children" shows "distinct children"
scaleRat_one: fixes x :: "'a" shows "1 *R x = x"
fmmerge_commut: fixes xm :: "'b \<rightharpoonup>\<^sub>f 'a"   and f :: "'a \<Rightarrow> 'a \<Rightarrow> 'c"   and ym :: "'b \<rightharpoonup>\<^sub>f 'a" assumes "\<And>x y. x \<in> fmran' xm \<Longrightarrow> f x y = f y x" shows "fmmerge f xm ym = fmmerge f ym xm"
tensors_from_net_Conv_all1: fixes m :: "real mat convnet"   and i :: "nat"   and nr :: "nat" assumes "valid_net' m"   and "i < nr" shows "tensors_from_net (Conv (all1_matrix nr (output_size' m)) m) $ i = listsum (input_sizes m) (list_of_vec (tensors_from_net m))"
"3_1": shows "x # ns' = ns\<^sub>3 @ x # ns\<^sub>2"
rcos_module_imp: fixes H :: "'d set"   and G :: "('d, 'e) monoid_scheme"   and x :: "'d"   and x' :: "'d" assumes "subgroup H G"   and "Group.group G"   and "x \<in> carrier G"   and "x' \<in> H #>\<^bsub>G\<^esub> x" shows "x' \<otimes>\<^bsub>G\<^esub> inv\<^bsub>G\<^esub> x \<in> H"
homologous_rel_singular_relcycle: fixes p :: "nat"   and X :: "'b topology"   and S :: "'b set"   and c1 :: "((nat \<Rightarrow> real) \<Rightarrow> 'b) \<Rightarrow>\<^sub>0 int"   and c2 :: "((nat \<Rightarrow> real) \<Rightarrow> 'b) \<Rightarrow>\<^sub>0 int" assumes "homologous_rel p X S c1 c2" shows "singular_relcycle p X S c1 = singular_relcycle p X S c2"
errMOD_igSwapAllIPresIGWlsAllSTR: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'a, 'b) model" assumes "igSwapAllIPresIGWlsAll MOD"   and "igWlsAllDisj MOD"   and "igWlsAbsIsInBar MOD" shows "igSwapAllIPresIGWlsAllSTR (errMOD MOD)"
rbt_comp_delete: fixes c :: "'c \<Rightarrow> 'c \<Rightarrow> order" assumes "comparator c" shows "rbt_comp_delete c = ord.rbt_delete (lt_of_comp c)"
keys_cmul: fixes c :: "int"   and a :: "'c \<Rightarrow>\<^sub>0 int" shows "Poly_Mapping.keys (frag_cmul c a) \<subseteq> Poly_Mapping.keys a"
strict_sorted_iff: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and l :: "'c list" assumes "class.linorder less_eq less" shows "linorder.strict_sorted less l = (linorder.sorted less_eq l \<and> distinct l)"
elims_3: fixes x :: "'b option"   and xa :: "'b option" assumes "\<not> x \<le> xa" obtains xb :: "'b" where "x = Some xb__" and "xa = None"   | xb :: "'b" and y :: "'b" where "x = Some xb" and "xa = Some y" and "\<not> xb \<le> y"
set_delete_index_distinct: fixes xs :: "'b list"   and n :: "nat" assumes "distinct xs"   and "n < length xs" shows "set (delete_index n xs) = set xs - {xs ! n}"
simps_1: fixes uu :: "'b \<Rightarrow> bool" shows "successive uu [] = True"
rel_cases: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'c llist"   and b :: "'d llist" assumes "lrel R a b" obtains "a = LNil" and "b = LNil"   | x1 :: "'c" and x2 :: "'c llist" and y1 :: "'d" and y2 :: "'d llist" where "a = LCons x1 x2" and "b = LCons y1 y2" and "R x1 y1" and "lrel R x2 y2"
dim_row_transfer_rule: fixes A :: "'e mat"   and A' :: "(('e, 'f) Finite_Cartesian_Product.vec, 'g) Finite_Cartesian_Product.vec" assumes "Mod_Type_Connect.HMA_M A A'" shows "dim_row A = CARD('g)"
of_real_mult: fixes x :: "real"   and y :: "real" shows "of_real (x * y) = of_real x * of_real y"
dom_phis'_code: fixes g :: "'g"   and "next" :: "'node \<times> 'val" shows "dom (Mapping.lookup (phis'_code g next)) = dom (Mapping.lookup (phis g)) - {v. snd v = snd next}"
assms_3: shows "mset (map snd xs) = mset (map snd ys)"
l2_inv2_def: shows "l2_inv2 \<equiv> {s. \<forall>R. in_progress (progress s R) xsk \<longrightarrow> (\<exists>X N. sklvl1.guessed_frame R xsk = Some (Exp X (NonceF (R $ N))))}"
M_to_Z: fixes M :: "('f, 'a, 'g) Module_scheme"   and f :: "'f \<Rightarrow> 'h"   and e :: "'h"   and g :: "'f \<Rightarrow> 'h" assumes "R module M"   and "f \<in> mHom R M (Zm R e)"   and "g \<in> mHom R M (Zm R e)" shows "f = g"
IH: assumes "wf\<^sub>s\<^sub>t V S"   and "P []"   and "\<And>S t. \<lbrakk>wf\<^sub>s\<^sub>t V S; P S\<rbrakk> \<Longrightarrow> P (S @ [send\<langle>t\<rangle>\<^sub>s\<^sub>t])"   and "\<And>S t. \<lbrakk>wf\<^sub>s\<^sub>t V S; P S; fv t \<subseteq> V \<union> wfvarsoccs\<^sub>s\<^sub>t S\<rbrakk> \<Longrightarrow> P (S @ [receive\<langle>t\<rangle>\<^sub>s\<^sub>t])"   and "\<And>S t' t. \<lbrakk>wf\<^sub>s\<^sub>t V S; P S; fv t' \<subseteq> V \<union> wfvarsoccs\<^sub>s\<^sub>t S\<rbrakk> \<Longrightarrow> P (S @ [\<langle>assign: t \<doteq> t'\<rangle>\<^sub>s\<^sub>t])"   and "\<And>S t t'. \<lbrakk>wf\<^sub>s\<^sub>t V S; P S\<rbrakk> \<Longrightarrow> P (S @ [\<langle>check: t \<doteq> t'\<rangle>\<^sub>s\<^sub>t])"   and "\<And>S X F. \<lbrakk>wf\<^sub>s\<^sub>t V S; P S\<rbrakk> \<Longrightarrow> P (S @ [\<forall>X\<langle>\<or>\<noteq>: F\<rangle>\<^sub>s\<^sub>t])" shows "P S"
fun_sum_commute_canc: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'d \<Rightarrow> 'b"   and A :: "'d set" assumes "\<And>x y. f (x + y) = f x + f y" shows "f (sum g A) = (\<Sum>a\<in>A. f (g a))"
mat_monoid: fixes nr :: "nat"   and nc :: "nat"   and b :: "'d" shows "Group.monoid (mat_monoid nr nc b)"
val_Zp_dist_infty': fixes a :: "nat \<Rightarrow> int"   and b :: "nat \<Rightarrow> int" assumes "a \<in> carrier Zp"   and "b \<in> carrier Zp"   and "a = b" shows "val_Zp_dist a b = \<infinity>"
path_delete: fixes p :: "bool list"   and ds :: "bool list"   and T :: "tree" assumes "path p (delete ds T)" shows "path p T"
x_alpha_tau: fixes x :: "'a" shows "\<nu> x + \<tau> x = x"
axioms_1: fixes x :: "'b \<Longrightarrow> 'c" assumes "ief_lens x" shows "weak_lens x"
triangle_left: shows "(\<epsilon> \<star>\<^sub>C f) \<cdot>\<^sub>C \<a>\<^sub>C\<^sup>-\<^sup>1[f, g, f] \<cdot>\<^sub>C (f \<star>\<^sub>C \<eta>) = \<l>\<^sub>C\<^sup>-\<^sup>1[f] \<cdot>\<^sub>C \<r>\<^sub>C[f]"
f_def: shows "f = (\<lambda>k. \<Sum>m = 1..x. unity_root n (- int m * k))"
bisim: shows "(\<forall>P Q P'. P \<lesssim>\<lbrakk>\<cdot>\<rbrakk>LT<TRel> Q \<and> P \<longmapsto>STCal Source Target P' \<longrightarrow> (\<exists>Q'. Q \<longmapsto>STCal Source Target Q' \<and> P' \<lesssim>\<lbrakk>\<cdot>\<rbrakk>LT<TRel> Q')) \<and> (\<forall>P Q Q'. P \<lesssim>\<lbrakk>\<cdot>\<rbrakk>LT<TRel> Q \<and> Q \<longmapsto>STCal Source Target Q' \<longrightarrow> (\<exists>P'. P \<longmapsto>STCal Source Target P' \<and> P' \<lesssim>\<lbrakk>\<cdot>\<rbrakk>LT<TRel> Q'))"
hyperplane_face_of_halfspace_ge: fixes a :: "'b"   and b :: "real" shows "{x. a \<bullet> x = b} face_of {x. b \<le> a \<bullet> x}"
simps: fixes E :: "(nat \<times> nat) set"   and a1 :: "nat list"   and a2 :: "nat" shows "path_entry E a1 a2 = (a1 = [] \<and> a2 = 0 \<or> (\<exists>u v l. a1 = u # l \<and> a2 = v \<and> (u, v) \<in> E \<and> path_entry E l u))"
N_def: fixes i :: "nat" shows "local.N i = local.Xij 0 i"
1: shows "1 = \<one>\<^bsub>units_of (residue_ring (p ^ m))\<^esub>"
Inf_nat_def: fixes X :: "nat set" shows "\<Sqinter> X = least X"
star_def: fixes A :: "'a list set" shows "star A = \<Union> (range ((^^) A))"
initial_state_def: fixes m :: "nat"   and fs_init :: "int vec list" shows "initial_state m fs_init = (let dmus = d\<mu>_impl fs_init; ds = IArray.of_fun (\<lambda>i. if i = 0 then 1 else let i1 = i - 1 in dmus !! i1 !! i1) (Suc m); dmus' = IArray.of_fun (\<lambda>i. let row_i = dmus !! i in IArray.of_fun ((!!) row_i) i) m in (([], fs_init), dmus', ds))"
gen_iapath_triv_path: fixes u :: "'a"   and p :: "('a \<times> 'a) list"   and v :: "'a" assumes "gen_iapath (pverts G) u p v" shows "p = [(u, v)]"
P_eq_2: fixes i :: "nat" shows "space (P (J i)) = space (Pi\<^sub>M (J i) (\<lambda>i. borel))"
power_commutes: fixes a :: "'a"   and n :: "nat" shows "power.power (1::'a) (\<odot>) a n \<odot> a = a \<odot> power.power (1::'a) (\<odot>) a n"
ide_MkIde: fixes A :: "'o" assumes "A \<in> Obj" shows "ide (MkIde A)"
H: shows "ptrm_infer_type \<Gamma> (PSnd P) = Some \<tau>"
select_convs_1: fixes carrier :: "'a set"   and more :: "'b" shows "carrier \<lparr>carrier = carrier, \<dots> = more\<rparr> = carrier"
exhaust: fixes y :: "char" obtains x1 :: "bool" and x2 :: "bool" and x3 :: "bool" and x4 :: "bool" and x5 :: "bool" and x6 :: "bool" and x7 :: "bool" and x8 :: "bool" where "y = Char x1 x2 x3 x4 x5 x6 x7 x8"
disjoint_RF: fixes j' :: "nat"   and i' :: "nat"   and j :: "nat"   and i :: "nat" assumes "\<not> disjnt (RF j' i') (RF j i)"   and "j' \<le> k"   and "j \<le> k"   and "i' < m"   and "i < m" shows "i' = i \<and> j' = j"
ex_iter: fixes R :: "'a \<Rightarrow> bool"   and g :: "'a \<Rightarrow> 'a"   and a :: "'a" shows "(\<exists>n. R (iter g a n)) = (if R a then True else \<exists>n. R (iter g (g a) n))"
meld'_def: fixes z :: "('c, 'd) bintree option"   and xs :: "('c, 'd) bintree option list"   and ys :: "('c, 'd) bintree option list" shows "meld' z xs ys = add z (meld xs ys)"
that: fixes r :: "real" assumes "x = of_real r"   and "r \<le> 0" shows "thesis"
allAllocationsIntersection: fixes G :: "'d set"   and N :: "'c set" shows "allAllocations N G = allAllocations'' N G"
prefix_imp_sublist: fixes xs :: "'b list"   and ys :: "'b list" assumes "xs \<le>p ys" shows "xs \<le>f ys"
null_set_Int2: fixes B :: "'b set"   and M :: "'b measure"   and A :: "'b set" assumes "B \<in> null_sets M"   and "A \<in> sets M" shows "B \<inter> A \<in> null_sets M"
r_hom: fixes U :: "'a set" assumes "U \<in> I" shows "ring_homomorphism (\<psi> U) (\<FF> U) +\<^bsub>U\<^esub> \<cdot>\<^bsub>U\<^esub> \<zero>\<^bsub>U\<^esub> \<one>\<^bsub>U\<^esub> A add mult zero one"
\<phi>_\<psi>: fixes b :: "'d"   and a :: "'d"   and x :: "(('c + 'd) set, 'c + 'd \<Rightarrow> 'c + 'd) concrete_category.arr" assumes "D.ide b"   and "D.ide a"   and "x \<in> HomD.set (b, a)" shows "inD (HomD.\<psi> (b, a) x) = x"
cases: fixes r :: "('a, 'b) TermEquivClT" obtains TDomain :: "'a" and TExprSet :: "('a, 'b) Expression set" and TCodomain :: "'a" where "r = \<lparr>TDomain = TDomain, TExprSet = TExprSet, TCodomain = TCodomain\<rparr>"
S_automorphism: fixes s :: "'a permutation" assumes "s \<in> (\<Union>(f, g)\<in>fundfoldpairs. {Abs_induced_automorph f g})" shows "ChamberComplexAutomorphism X ((\<rightarrow>) s)"
reachablep_reachable_eq: fixes A :: "('e, 'f, 'd, 'g) FSM_scheme"   and p :: "'d" shows "transition_system.reachablep (\<lambda>a p. snd a) (\<lambda>a p. snd a \<in> succ A (fst a) p) p = (\<lambda>x. x \<in> reachable A p)"
collapse_5: fixes event :: "('a, 'b, 'c) event" assumes "isRecvMarker event" shows "RecvMarker (getId event) (occurs_on event) (partner event) = event"
addition2: fixes f :: "nat \<Rightarrow> 'a"   and n :: "nat"   and h :: "nat \<Rightarrow> nat" assumes "f \<in> {j. j \<le> Suc n} \<rightarrow> carrier A"   and "h \<in> {j. j \<le> Suc n} \<rightarrow> {j. j \<le> Suc n}"   and "inj_on h {j. j \<le> Suc n}" shows "\<Sigma>\<^sub>e A cmp f h Suc n = \<Sigma>\<^sub>e A f Suc n"
Basis_list_impl_def: fixes n :: "nat" shows "Basis_list_impl n = (let zs = replicate n (0::'b) in map (\<lambda>i. zs[i := 1::'b]) [0..<n])"
has_integral_reflect: fixes f :: "'a \<Rightarrow> 'b"   and i :: "'b"   and b :: "'a"   and a :: "'a" shows "((\<lambda>x. f (- x)) has_integral i) (cbox (- b) (- a)) = (f has_integral i) (cbox a b)"
conj_com_Cn: fixes p :: "'a"   and q :: "'a" shows "Cn {p .\<and>. q} = Cn {q .\<and>. p}"
reachable_step: fixes s :: "'a"   and A :: "('a, 'b) automaton"   and I :: "'b \<Rightarrow> bool"   and a :: "'b"   and s' :: "'a" assumes "s \<in> reachable A I"   and "(s, a, s') \<in> automaton.trans A"   and "I a" shows "s' \<in> reachable A I"
eq_UNIV_I: fixes Z :: "'b set" assumes "rel_set A {a. Domainp A a} Z" shows "Z = UNIV"
ComplI: fixes c :: "'f"   and A :: "'f set" assumes "c \<in> A \<Longrightarrow> False" shows "c \<in> - A"
mk_regular1_def: fixes P :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and F :: "'b \<Rightarrow> 'a \<Rightarrow> 'a"   and I :: "'a set" shows "mk_regular1 P F I = I \<union> {F q x |q x. x \<in> I \<and> P q x}"
po_eq_conv: fixes below :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "class.po below" shows "(x = y) = (below x y \<and> below y x)"
hyps: fixes A :: "(('b, 'c) vec, 'a) vec" assumes "n \<le> mod_type_class.to_nat i"   and "mod_type_class.to_nat a < n" shows "Hermite_reduce_above A n i j res $ a $ b = row_add A a i ((res (A $ i $ j) (A $ a $ j) - A $ a $ j) div A $ i $ j) $ a $ b"
cases: fixes x :: "(int \<times> int) \<times> int \<times> int" obtains p\<^sub>0 :: "int \<times> int" and p\<^sub>1 :: "int \<times> int" where "x = (p\<^sub>0, p\<^sub>1)"
invertible_matrix_mult_left_rank: fixes P :: "(('a, 'b) vec, 'b) vec"   and A :: "(('a, 'c) vec, 'b) vec" assumes "invertible P" shows "rank (P ** A) = rank A"
is_full: fixes a :: "'a"   and a' :: "'a"   and g :: "'a" assumes "ide a"   and "ide a'"   and "\<guillemotleft>g : R a' \<rightarrow> R a\<guillemotright>" shows "\<exists>f. \<guillemotleft>f : a' \<rightarrow> a\<guillemotright> \<and> R f = g"
subs_NL: shows "set (zip (prems_of \<iota>F) Ll) \<subseteq> NL"
dp_consistency_empty_axioms: shows "dp_consistency_empty lookup update P empty"
vertical_chop_singleton: fixes v :: "view"   and u :: "view"   and w :: "view" shows "v=u--w \<and> |lan v| = 1 \<longrightarrow> |lan u| = 0 \<or> |lan w| = 0"
AVar_lemma: fixes G :: "prog"   and x1 :: "abopt"   and s1 :: "st"   and e2 :: "expr"   and i :: "val"   and x2 :: "abopt"   and s2 :: "st"   and v :: "val"   and f :: "val \<Rightarrow> state \<Rightarrow> state"   and s2' :: "st"   and a :: "val"   and Ta :: "ty"   and L :: "lenv" assumes "wf_prog G"   and "G\<turnstile>(x1, s1) \<midarrow>e2-\<succ>i\<rightarrow> (x2, s2)"   and "((v, f), Norm s2') = avar G i a (x2, s2)"   and "x1 = None \<longrightarrow> G,s1\<turnstile>a\<Colon>\<preceq>Ta.[]"   and "(x2, s2)\<Colon>\<preceq>(G, L)"   and "s1\<le>|s2" shows "G,s2'\<turnstile>v\<Colon>\<preceq>Ta \<and> s2'\<le>|f\<preceq>Ta\<Colon>\<preceq>(G, L)"
butlast_conv_take: fixes xs :: "'d list" shows "butlast xs = take (length xs - 1) xs"
simps_3: fixes v :: "'b" shows "(-\<infinity> \<le> Fin v) = True"
simp_lifted_pair: fixes pf :: "('c, 'd) lifted_pair" shows "\<forall>f. Rep_lifted_pair pf f \<in> triple.pairs (Rep_phi f)"
monad_discard_def: fixes return :: "'a \<Rightarrow> 'b"   and bind :: "'b \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b" shows "monad_discard return bind \<equiv> monad return bind \<and> monad_discard_axioms bind"
supd_assn_def2: fixes f :: "state \<Rightarrow> state"   and P :: "'b assn"   and Y :: "vals"   and s :: "state" shows "(f .; P) Y s = P Y (f s)"
prod_rel_def: fixes R1 :: "('c \<times> 'e) set"   and R2 :: "('d \<times> 'f) set" shows "R1 \<times>\<^sub>r R2 \<equiv> {((a, b), a', b'). (a, a') \<in> R1 \<and> (b, b') \<in> R2}"
permutes_induct: fixes S :: "'d set"   and P :: "('d \<Rightarrow> 'd) \<Rightarrow> bool"   and p :: "'d \<Rightarrow> 'd" assumes "finite S"   and "P id"   and "\<And>a b p. \<lbrakk>a \<in> S; b \<in> S; P p; P p; permutation p\<rbrakk> \<Longrightarrow> P (Fun.swap a b id \<circ> p)"   and "p permutes S" shows "P p"
parallel_commute: fixes a :: "'c list"   and b :: "'c list" shows "(a \<parallel> b) = (b \<parallel> a)"
intro: fixes s1 :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and s2 :: "'c \<Rightarrow> 'e \<Rightarrow> 'e"   and B1 :: "'d set"   and B2 :: "'e set" assumes "vector_space_prod s1 s2"   and "finite_dimensional_vector_space_pair s1 B1 s2 B2" shows "finite_dimensional_vector_space_prod s1 s2 B1 B2"
\<sigma>_1: shows "D \<cdot> \<sigma> = C"
less_eq_enat_def: fixes m :: "enat"   and n :: "enat" shows "(m \<le> n) = (case n of enat n1 \<Rightarrow> case m of enat m1 \<Rightarrow> m1 \<le> n1 | \<infinity> \<Rightarrow> False | \<infinity> \<Rightarrow> True)"
smallD: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'b"   and c :: "real" assumes "f \<in> o[F](g)"   and "0 < c" shows "\<forall>\<^sub>F x in F. norm (f x) \<le> c * norm (g x)"
pow_subgroup_indicator: fixes H :: "'a set"   and a :: "'a" assumes "subgroup H (G\<lparr>carrier := adjoin G H a\<rparr>)"   and "a \<in> carrier (G\<lparr>carrier := adjoin G H a\<rparr>)" shows "a [^]\<^bsub>G\<lparr>carrier := adjoin G H a\<rparr>\<^esub> subgroup_indicator (G\<lparr>carrier := adjoin G H a\<rparr>) H a \<in> H"
tl_drop: fixes n :: "nat"   and xs :: "'b list" shows "tl (drop n xs) = drop n (tl xs)"
TER_eq: fixes f :: "'v \<times> 'v \<Rightarrow> ennreal" shows "TER\<^bsub>\<Gamma>\<^esub> f = TER\<^bsub>\<Gamma>'\<^esub> f"
simps_9: fixes f1 :: "'b"   and f2 :: "'b"   and f3 :: "'b" shows "(case GT of LT \<Rightarrow> f1 | EQ \<Rightarrow> f2 | GT \<Rightarrow> f3) = f3"
bisimTransitiveCoinduct: fixes P :: "pi"   and Q :: "pi"   and X :: "(pi \<times> pi) set" assumes "(P, Q) \<in> X"   and "eqvt X"   and "\<And>R S. (R, S) \<in> X \<Longrightarrow> R \<leadsto>[(bisim O (X \<union> bisim) O bisim)] S"   and "\<And>R S. (R, S) \<in> X \<Longrightarrow> (S, R) \<in> bisim O (X \<union> bisim) O bisim" shows "P \<sim> Q"
subgroup_subset_complementary: fixes H :: "'a set"   and J :: "'a set"   and I :: "'a set" assumes "subgroup H G"   and "subgroup J G"   and "subgroup I G"   and "I \<subseteq> J"   and "complementary H J" shows "complementary H I"
unit_in_hom_1: shows "\<guillemotleft>\<eta>\<^sub>0 : src e\<^sub>0 \<rightarrow> src e\<^sub>0\<guillemotright>"
p2r_def: fixes P :: "'b \<Rightarrow> bool" shows "\<lceil>P\<rceil> = {(s, s) |s. P s}"
ts__coplanar: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "A B TS C D" shows "Coplanar A B C D"
empty_face_of: fixes S :: "'b set" shows "{} face_of S"
zero_divisor_iff_eq_0: fixes a :: "'b" shows "zero_divisor a = (a = (0::'b))"
in_theFTOne_theNFTBoth_2: fixes n1 :: "nat"   and cl :: "('d, 'e, 'f) cmd list"   and dl :: "('g, 'h, 'i) cmd list"   and n2 :: "nat" assumes "n1 \<in> theFTOne cl dl"   and "n2 \<in> theNFTBoth cl dl" shows "n2 \<noteq> n1"
nat_pow_pow: fixes G :: "('c, 'd) monoid_scheme"   and x :: "'c"   and n :: "nat"   and m :: "nat" assumes "Group.monoid G"   and "x \<in> carrier G" shows "(x [^]\<^bsub>G\<^esub> n) [^]\<^bsub>G\<^esub> m = x [^]\<^bsub>G\<^esub> (n * m)"
enat_the_enat_le: fixes x :: "enat" shows "enat (the_enat x) \<le> x"
cos_expr: shows "b = kernel G H f #>\<^bsub>G\<^esub> a \<and> a \<in> carrier G"
list_ex_intersection: fixes ys :: "'a list"   and xs :: "'a list" shows "list_ex (\<lambda>v. list_ex ((=) v) ys) xs = (set xs \<inter> set ys \<noteq> {})"
delete_ran_image: fixes j :: "nat"   and n :: "nat"   and f :: "nat \<Rightarrow> nat" assumes "j \<in> {0..<Suc n}"   and "f ` {0..<n} = {0..<Suc n} - {j}" shows "delete_ran f j ` {0..<n} = {0..<n}"
abs_eq: fixes x :: "(('c, 'd) vec, 'd) vec" shows "from_vec x = of_fun (\<lambda>i. ($) (x $ i))"