hcju/isabelleps huggingface dataset:
Par2: fixes \<Psi> :: "'b"   and \<Psi>\<^sub>P :: "'b"   and Q :: "('a, 'b, 'c) psi"   and \<alpha> :: "'a action"   and Q' :: "('a, 'b, 'c) psi"   and P :: "('a, 'b, 'c) psi"   and A\<^sub>P :: "name list" assumes "\<Psi> \<otimes> \<Psi>\<^sub>P \<rhd> Q \<longmapsto> \<alpha> \<prec> Q'"   and "extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>"   and "bn \<alpha> \<sharp>* P"   and "A\<^sub>P \<sharp>* \<Psi>"   and "A\<^sub>P \<sharp>* Q"   and "A\<^sub>P \<sharp>* \<alpha>" shows "\<Psi> \<rhd> P \<parallel> Q \<longmapsto> \<alpha> \<prec> P \<parallel> Q'"
sparse_row_mult_spvec_spmat: fixes a :: "(nat \<times> 'a) list"   and B :: "(nat \<times> (nat \<times> 'a) list) list"   and c :: "(nat \<times> 'a) list" assumes "sorted_spvec a"   and "sorted_spvec B" shows "sparse_row_vector (mult_spvec_spmat c a B) = sparse_row_vector c + sparse_row_vector a * sparse_row_matrix B"
pdevs_of_ivl_real: fixes a :: "real"   and b :: "real" shows "pdevs_of_ivl a b = singleton_pdevs ((b - a) / 2)"
add_cf_def: fixes S :: "('e, 'f) Ring_scheme"   and c :: "nat \<times> (nat \<Rightarrow> 'e)"   and d :: "nat \<times> (nat \<Rightarrow> 'e)" shows "add_cf S c d = (if fst c < fst d then (fst d, \<lambda>j. if j \<le> fst c then snd c j \<plusminus>\<^bsub>S\<^esub> snd d j else snd d j) else if fst c = fst d then (fst c, \<lambda>j. snd c j \<plusminus>\<^bsub>S\<^esub> snd d j) else (fst c, \<lambda>j. if j \<le> fst d then snd c j \<plusminus>\<^bsub>S\<^esub> snd d j else snd c j))"
freshChanEqChain_1: fixes xvec :: "name list"   and M :: "'a"   and N :: "'a" assumes "xvec \<sharp>* M"   and "xvec \<sharp>* N" shows "xvec \<sharp>* (M \<leftrightarrow> N)"
reduce_root_pCons: fixes a :: "'a"   and c :: "'a"   and p :: "'a poly" shows "reduce_root a (pCons c p) = pCons c (smult a (reduce_root a p))"
rel_boundary_of_convex_hull: fixes S :: "'b set" assumes "\<not> affine_dependent S" shows "convex hull S - rel_interior (convex hull S) = (\<Union>a\<in>S. convex hull (S - {a}))"
FinGroupRepresentation_def: fixes G :: "'a set"   and smult :: "('b, 'a) aezfun \<Rightarrow> 'c \<Rightarrow> 'c"   and V :: "'c set" shows "FinGroupRepresentation G smult V \<equiv> FGModule G smult V \<and> FinGroupRepresentation_axioms G smult V"
mprod_mem: fixes n :: "nat"   and e :: "nat \<Rightarrow> int"   and f :: "nat \<Rightarrow> 'a" assumes "\<forall>j\<le>n. e j \<in> Zset"   and "\<forall>j\<le>n. f j \<in> carrier K - {\<zero>}" shows "mprod_exp K e f n \<in> carrier K - {\<zero>}"
r_neg: fixes x :: "nat \<Rightarrow> int" assumes "x \<in> carrier Zp" shows "x \<oplus> \<ominus> x = \<zero>"
ltl_prop_equiv_lt_ltl_lang_equiv: shows "(\<sim>\<^sub>P) \<le> (\<sim>\<^sub>L)"
subst_nil_iff: fixes \<Gamma> :: "(Vars.var \<times> exp) list"   and x :: "Vars.var"   and z :: "Vars.var" shows "(\<Gamma>[x::h=z] = []) = (\<Gamma> = [])"
diagonal_in_associates: fixes associates :: "'a set"   and residues :: "'a \<Rightarrow> 'a set"   and H :: "(('a, 'b) vec, 'b) vec"   and i :: "'b" assumes "Hermite associates residues H"   and "invertible H"   and "upper_triangular H" shows "H $ i $ i \<in> associates"
eq: fixes k :: "nat" assumes "0 < k" shows "Suc n choose k = n choose k + (n choose (k - 1))"
equal_neg_zero: fixes x :: "'a" assumes "x \<in> carrier R" shows "(x = \<ominus> x) = (x = \<zero>)"
leg0_in_hom_2: shows "\<guillemotleft>f : f \<Rightarrow>\<^sub>C f\<guillemotright>"
3: shows "A *\<^sub>v v = 0\<^sub>v n"
assms: shows "RModule FG (\<cdot>) U \<and> U \<subseteq> V"
plus_separate_comp_bot: fixes x :: "'a"   and y :: "'a" assumes "x \<sqinter> y = bot" shows "(x \<squnion> y) \<sqinter> top = x \<sqinter> top \<squnion> y \<sqinter> top \<squnion> y \<sqinter> top \<sqinter> (x \<sqinter> top)"
pow_eq_id: fixes x :: "'a"   and n :: "nat" assumes "x \<in> carrier G" shows "(x [^] n = \<one>) = (G.ord x dvd n)"
l_get_disconnected_document_def: fixes get_disconnected_document :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) document_ptr) prog"   and get_disconnected_document_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_get_disconnected_document get_disconnected_document get_disconnected_document_locs \<equiv> ((\<forall>node_ptr h h'. reads get_disconnected_document_locs (get_disconnected_document node_ptr) h h') \<and> (\<forall>h node_ptr. h \<turnstile> ok get_disconnected_document node_ptr \<longrightarrow> node_ptr |\<in>| node_ptr_kinds h)) \<and> (\<forall>node_ptr h. pure (get_disconnected_document node_ptr) h) \<and> (\<forall>h child_node disconnected_document. h \<turnstile> get_disconnected_document child_node \<rightarrow>\<^sub>r disconnected_document \<longrightarrow> disconnected_document |\<in>| document_ptr_kinds h)"
valid_Act_fresh: fixes \<alpha> :: "('act, 'effect) L_action"   and P :: "('state, 'effect) L_state"   and x :: "('a, 'pred, ('act, 'effect) L_action) Formula.formula" assumes "bn \<alpha> \<sharp>* P" shows "L_transform.valid P (Formula.Act \<alpha> x) = (\<exists>P'. P \<rightarrow>\<^sub>L \<langle>\<alpha>,P'\<rangle> \<and> L_transform.valid P' x)"
\<Gamma>_prefix_conv: fixes p :: "'a prefix"   and s :: "'a trace"   and s' :: "'a trace"   and i :: "nat" assumes "prefix_of p s"   and "prefix_of p s'"   and "i < plen p" shows "\<Gamma> s i = \<Gamma> s' i"
in_idem: fixes A :: "'c DBMEntry set"   and x :: "'c DBMEntry" assumes "finite A"   and "x \<in> A" shows "linordered_monoid.max x (linordered_monoid.Max A) = linordered_monoid.Max A"
erf_minus: fixes z :: "'a" shows "erf (- z) = - erf z"
inline_try_gpv: fixes \<I> :: "('a, 'b) \<I>"   and gpv :: "('c, 'a, 'b) gpv"   and \<I>' :: "('d, 'e) \<I>"   and f :: "'c \<times> ('a, 'b, 'd, 'e) converter \<Rightarrow> 'f"   and x :: "'c"   and z :: "'f"   and s :: "('a, 'b, 'd, 'e) converter" assumes "\<I> \<turnstile>g gpv \<surd>"   and "pfinite_gpv \<I> gpv"   and "\<And>s. colossless_converter \<I> \<I>' s \<and> \<I>, \<I>' \<turnstile>\<^sub>C s \<surd> \<Longrightarrow> f (x, s) = z"   and "\<And>s x. \<lbrakk>x \<in> outs_\<I> \<I>; colossless_converter \<I> \<I>' s \<and> \<I>, \<I>' \<turnstile>\<^sub>C s \<surd>\<rbrakk> \<Longrightarrow> colossless_gpv \<I>' (run_converter s x)"   and "colossless_converter \<I> \<I>' s \<and> \<I>, \<I>' \<turnstile>\<^sub>C s \<surd>" shows "eq_\<I>_gpv (=) \<I>' (map_gpv f id (inline run_converter (TRY gpv ELSE Done x) s)) (TRY map_gpv f id (inline run_converter gpv s) ELSE Done z)"
TauPres: fixes P :: "pi"   and Q :: "pi" assumes "P \<equiv>\<^sub>e Q" shows "\<tau>.P \<equiv>\<^sub>e \<tau>.Q"
usubstappt_induct: fixes P :: "(char \<Rightarrow> trm option) \<times> (char \<Rightarrow> trm option) \<times> (char \<Rightarrow> fml option) \<times> (char \<Rightarrow> game option) \<Rightarrow> variable set \<Rightarrow> trm \<Rightarrow> bool"   and a0 :: "(char \<Rightarrow> trm option) \<times> (char \<Rightarrow> trm option) \<times> (char \<Rightarrow> fml option) \<times> (char \<Rightarrow> game option)"   and a1 :: "variable set"   and a2 :: "trm" assumes "\<And>\<sigma> U x. P \<sigma> U (Var x)"   and "\<And>\<sigma> U r. P \<sigma> U (Number r)"   and "\<And>\<sigma> U f. P \<sigma> U (Const f)"   and "\<And>\<sigma> U f \<theta>. \<lbrakk>P \<sigma> U \<theta>; \<And>x2 x2a. \<lbrakk>usubstappt \<sigma> U \<theta> = Aterm x2; SFuncs \<sigma> f = Aterm x2a; FVT x2a \<inter> U = {}\<rbrakk> \<Longrightarrow> P (dotsubstt x2) {} x2a\<rbrakk> \<Longrightarrow> P \<sigma> U (trm.Func f \<theta>)"   and "\<And>\<sigma> U \<theta> \<eta>. \<lbrakk>P \<sigma> U \<theta>; P \<sigma> U \<eta>\<rbrakk> \<Longrightarrow> P \<sigma> U (Plus \<theta> \<eta>)"   and "\<And>\<sigma> U \<theta> \<eta>. \<lbrakk>P \<sigma> U \<theta>; P \<sigma> U \<eta>\<rbrakk> \<Longrightarrow> P \<sigma> U (Times \<theta> \<eta>)"   and "\<And>\<sigma> U \<theta>. P \<sigma> Denotational_Semantics.allvars \<theta> \<Longrightarrow> P \<sigma> U (Differential \<theta>)" shows "P a0 a1 a2"
iso_tuple_update_accessor_cong_assist_def: fixes upd :: "('a \<Rightarrow> 'a) \<Rightarrow> 'b \<Rightarrow> 'b"   and ac :: "'b \<Rightarrow> 'a" shows "iso_tuple_update_accessor_cong_assist upd ac = ((\<forall>f v. upd (\<lambda>x. f (ac v)) v = upd f v) \<and> (\<forall>v. upd id v = v))"
vector_to_cblinfun_scaleC: fixes a :: "complex"   and \<psi> :: "'d" shows "vector_to_cblinfun (a *\<^sub>C \<psi>) = a *\<^sub>C vector_to_cblinfun \<psi>"
permutes_vimage: fixes f :: "'b \<Rightarrow> 'b"   and A :: "'b set" assumes "f permutes A" shows "f -` A = A"
defensive_move_exists_for_Even: fixes p :: "'a list" assumes "position p" shows "winning_position_Odd p \<or> (\<exists>m. move_defensive_by_Even m p)"
cong_mult_mod_2: fixes a :: "'a"   and b :: "'a"   and n :: "'a"   and c :: "'a" shows "[a * (b mod n) = c] (mod n) = [a * b = c] (mod n)"
least_mod_type: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and one :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and Rep :: "'a \<Rightarrow> int"   and Abs :: "int \<Rightarrow> 'a"   and n :: "'a" assumes "class.mod_type times minus plus zero uminus one less_eq less Rep Abs" shows "less_eq zero n"
star_unfoldl_eq: fixes x :: "'a" shows "(1::'a) + x \<cdot> x\<^sup>\<star> = x\<^sup>\<star>"
copy_distrib: fixes k :: "nat"   and x :: "'a"   and y :: "'a" shows "k \<times> (x \<squnion> y) = k \<times> x \<squnion> k \<times> y"
sorted_wrt_linorder_index_less_iff: fixes A :: "'b set"   and R :: "('b \<times> 'b) set"   and xs :: "'b list"   and x :: "'b"   and y :: "'b" assumes "linorder_on A R"   and "set xs \<subseteq> A"   and "Linorder_Relations.sorted_wrt R xs"   and "x \<in> set xs"   and "y \<in> set xs" shows "(index xs x < index xs y) = ((y, x) \<notin> R)"
intro: fixes remove_shadow_root :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, unit) prog"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_shadow_root :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, (_) shadow_root_ptr option) prog"   and set_shadow_root :: "(_) element_ptr \<Rightarrow> (_) shadow_root_ptr option \<Rightarrow> ((_) heap, exception, unit) prog"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and remove_shadow_root_locs :: "(_) element_ptr \<Rightarrow> (_) shadow_root_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and set_shadow_root_locs :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, unit) prog set" assumes "remove_shadow_root = l_remove_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_remove_shadow_root get_child_nodes get_shadow_root set_shadow_root get_disconnected_nodes"   and "remove_shadow_root_locs = l_remove_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_remove_shadow_root_locs set_shadow_root_locs" shows "l_remove_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms get_child_nodes get_shadow_root set_shadow_root set_shadow_root_locs get_disconnected_nodes remove_shadow_root remove_shadow_root_locs"
not_is_zero_row_upt_suc: fixes i :: "'a"   and k :: "nat"   and A :: "(('b, 'c) vec, 'a) vec" assumes "\<not> is_zero_row_upt_k i (Suc k) A"   and "\<forall>i. A $ i $ mod_type_class.from_nat k = (0::'b)" shows "\<not> is_zero_row_upt_k i k A"
simps_7: fixes p :: "fm"   and q :: "fm" shows "qfree (Iff p q) = (qfree p \<and> qfree q)"
preserves_hom: fixes f :: "'a"   and a :: "'a"   and b :: "'a" assumes "\<tau>.A.in_hom f a b" shows "\<tau>.B.in_hom (\<tau> f) (F a) (G b)"
absorb1: fixes a :: "'a Word.word"   and b :: "'a Word.word" assumes "a \<le>s b" shows "signed.min a b = a"
diff_delta_ln_upper_5_def: shows "diff_delta_ln_upper_5 \<equiv> \<lambda>x. (x - 1) ^ 5 / ((3 * x\<^sup>2 + 6 * x + 1)\<^sup>2 * x)"
induct: fixes P :: "'a + 'b \<Rightarrow> bool"   and sum :: "'a + 'b" assumes "\<And>a. P (Inl a)"   and "\<And>b. P (Inr b)" shows "P sum"
SeqHR_hpair: fixes x1 :: "hf"   and x3 :: "hf"   and s1 :: "hf"   and k1 :: "hf"   and x2 :: "hf"   and x4 :: "hf"   and s2 :: "hf"   and k2 :: "hf" assumes "SeqHR x1 x3 s1 k1"   and "SeqHR x2 x4 s2 k2" shows "\<exists>s k. SeqHR \<langle>x1, x2\<rangle> (q_HPair x3 x4) s k"
merge_fun_def: fixes f :: "'c \<Rightarrow> 'b\<^sub>m"   and g :: "'c \<Rightarrow> 'b\<^sub>m" shows "local.merge_fun f g = (if \<forall>x. m (f x) (g x) \<noteq> None then Some (\<lambda>x. the (m (f x) (g x))) else None)"
of_real_of_nat_eq: fixes n :: "nat" shows "of_real (real n) = of_nat n"
cdijkstra_def: fixes g :: "'G"   and v0 :: "'V" shows "cdijkstra g v0 \<equiv> cdinit g v0 \<bind> (\<lambda>\<sigma>0. WHILE\<^sub>T (\<lambda>(wl, uu_). \<not> qw.isEmpty wl) (\<lambda>\<sigma>. cpop_min \<sigma> \<bind> (\<lambda>(v, wv, \<sigma>'). cupdate g v wv \<sigma>')) \<sigma>0 \<bind> (\<lambda>(uu_, res). RETURN res))"
int_floor_fl_def: shows "int_floor_fl \<equiv> map_fun real_of_float id floor"
Well_Scoped_Instantiation_axioms: shows "Well_Scoped_Instantiation freshenLC renameLCs lconsts closed subst subst_lconsts subst_renameLCs anyP inPorts outPorts nodeOf hyps vertices vidx inst edges local_vars"
diameter_le: fixes S :: "'b set"   and d :: "real" assumes "S \<noteq> {} \<or> 0 \<le> d"   and "\<And>x y. \<lbrakk>x \<in> S; y \<in> S\<rbrakk> \<Longrightarrow> norm (x - y) \<le> d" shows "diameter S \<le> d"
Field_oprod: fixes r :: "('d \<times> 'd) set"   and r' :: "('e \<times> 'e) set" shows "Field (r *o r') = Field r \<times> Field r'"
ptrm_alpha_equiv_reflp: shows "reflp (\<approx>)"
inter_distr2: fixes j :: "nat_int"   and k :: "nat_int"   and i :: "nat_int" shows "consec j k \<longrightarrow> (j \<squnion> k) \<sqinter> i = j \<sqinter> i \<squnion> k \<sqinter> i"
hide_rules_3: fixes c :: "'c \<Rightarrow> 'b"   and a :: "'c"   and B :: "'b set"   and P :: "'b process" assumes "c a \<in> B" shows "(c`!`a \<rightarrow> P) \ B = P \ B"
isomorphic_imp_GRep: fixes G :: "'a set"   and smult :: "('b, 'a) aezfun \<Rightarrow> 'c \<Rightarrow> 'c"   and V :: "'c set"   and smult' :: "('b, 'a) aezfun \<Rightarrow> 'd \<Rightarrow> 'd"   and W :: "'d set" assumes "FinGroupRepresentation G smult V"   and "\<exists>T. FGModuleIso G smult V smult' T W" shows "FinGroupRepresentation G smult' W"
arcsin_plus_arccos: fixes x :: "real" assumes "- 1 \<le> x"   and "x \<le> 1" shows "arcsin x + arccos x = pi / 2"
sim_rangeI: fixes M' :: "('c, 'd, 'b) KripkeStructure"   and f :: "'e \<Rightarrow> 'b"   and M :: "('c, 'd, 'e) KripkeStructure" assumes "worlds M' = f ` worlds M"   and "\<And>a. relations M' a \<subseteq> worlds M' \<times> worlds M'" shows "sim_range M M' f"
eq_equiv_class_iff: fixes A :: "'e set"   and r :: "('e \<times> 'e) set"   and x :: "'e"   and y :: "'e" assumes "equiv A r"   and "x \<in> A"   and "y \<in> A" shows "(r `` {x} = r `` {y}) = ((x, y) \<in> r)"
simps_19: fixes f1 :: "'b \<Rightarrow> 'a"   and f2 :: "'b \<Rightarrow> 'a"   and f3 :: "'c \<Rightarrow> 'a"   and f4 :: "'a" shows "(case Stuck of Normal x \<Rightarrow> f1 x | Abrupt x \<Rightarrow> f2 x | Fault x \<Rightarrow> f3 x | Stuck \<Rightarrow> f4) = f4"
pos_recurrent_def: fixes x :: "'s \<times> 's option" shows "KN.pos_recurrent x = (KN.recurrent x \<and> KN.U' x x \<noteq> \<infinity>)"
surjective_section_eq_homeomorphic_map: fixes X :: "'a topology"   and Y :: "'b topology"   and f :: "'a \<Rightarrow> 'b" shows "(section_map X Y f \<and> f ` topspace X = topspace Y) = homeomorphic_map X Y f"
big_step_t3_same: fixes C :: "com"   and ps :: "char list \<Rightarrow> int option"   and m :: "nat"   and ps' :: "char list \<Rightarrow> int option" assumes "(C, ps) \<Rightarrow>\<^sub>A m \<Down> ps'" shows "ps = ps' on UNIV - lvars C"
assms_1: shows "n -as\<rightarrow>* (_Exit_)"
simps_20: fixes v :: "'b rexp"   and vc :: "'b rexp"   and vb :: "'b" shows "le_rexp (Plus v vc) (Atom vb) = False"
is_generator_set: fixes scaleR :: "real \<Rightarrow> 'd \<Rightarrow> 'd"   and scaleC :: "complex \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and canonical_basis :: "'d list" assumes "class.basis_enum scaleR scaleC plus zero minus uminus canonical_basis" shows "cspan (set canonical_basis) = UNIV"
N\<^sub>r_def: fixes l :: "real" shows "N\<^sub>r l = Num (float_of l)"
inverse_less_iff: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs" shows "less (inverse a) (inverse b) = ((less zero (times a b) \<longrightarrow> less b a) \<and> (less_eq (times a b) zero \<longrightarrow> less a b))"
initialize_def: fixes body :: "'c_o \<Rightarrow> 'c_a \<Rightarrow> 's \<Rightarrow> nat \<Rightarrow> bool spmf" shows "initialize body = init \<bind> (\<lambda>(c_o, c_a, s). spmf_of_set {..<q} \<bind> body c_o c_a s)"
diagonal_to_Smith_PQ_exists_internalize_sort: fixes A :: "(('b, 'c) Finite_Cartesian_Product.vec, 'd) Finite_Cartesian_Product.vec" assumes "class.bezout_ring (*) (1::'b) (+) (0::'b) (-) uminus"   and "isDiagonal A" shows "\<exists>P Q. invertible P \<and> invertible Q \<and> Smith_normal_form (P ** A ** Q)"
\<omega>_simps_2: shows "src \<rho>' = src w'"
locks_a_conv: fixes ls :: "'a lock_actions"   and ntsjswss :: "('b, 'c, 'd) new_thread_action list \<times> 'b conditional_action list \<times> ('b, 'e) wait_set_action list \<times> 'b interrupt_action list \<times> 'f list" shows "\<lbrace>(ls, ntsjswss)\<rbrace>\<^bsub>l\<^esub> = ls"
intro: fixes D :: "('c, 'd) Order_scheme" assumes "rel D \<subseteq> carrier D \<times> carrier D"   and "\<And>a. a \<in> carrier D \<Longrightarrow> (a, a) \<in> rel D"   and "\<And>a b. \<lbrakk>a \<in> carrier D; b \<in> carrier D; (a, b) \<in> rel D; (b, a) \<in> rel D\<rbrakk> \<Longrightarrow> a = b"   and "\<And>a b c. \<lbrakk>a \<in> carrier D; b \<in> carrier D; c \<in> carrier D; (a, b) \<in> rel D; (b, c) \<in> rel D\<rbrakk> \<Longrightarrow> (a, c) \<in> rel D" shows "Order D"
E: obtains u :: "'a \<Rightarrow> real" and e :: "real" where "s (flow0 x (u x)) = 0" and "u x = return_time {x \<in> S. s x = 0} x" and "\<And>y. y \<in> cball x e \<Longrightarrow> s (flow0 y (u y)) = 0" and "continuous_on (cball x e) u" and "(\<lambda>t. (t, u t)) ` cball x e \<subseteq> Sigma X existence_ivl0" and "0 < e" and "(u has_derivative blinfun_apply (- blinfun_scaleR_left (inverse (blinfun_apply (Ds (poincare_map {x \<in> S. s x = 0} x)) (f (poincare_map {x \<in> S. s x = 0} x)))) o\<^sub>L (Ds (poincare_map {x \<in> S. s x = 0} x) o\<^sub>L flowderiv x (return_time {x \<in> S. s x = 0} x)) o\<^sub>L embed1_blinfun)) (at x)"
pr_simps_2: fixes a :: "'a"   and b :: "'a" assumes "ide a"   and "ide b" shows "local.dom \<pp>\<^sub>0[a, b] = a \<otimes> b"
eq_alt_conversep2: shows "(=) = (BNF_Def.Grp UNIV id)\<inverse>\<inverse>\<inverse>\<inverse>"
wellorders_totally_ordered: fixes r :: "('c \<times> 'c) set"   and r' :: "('d \<times> 'd) set" assumes "Well_order r"   and "Well_order r'" shows "(\<exists>f. embed r r' f) \<or> (\<exists>f'. embed r' r f')"
l6_6: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and P :: "'a"   and A :: "'a"   and B :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Out Bet P A B" shows "Tarski_neutral_dimensionless.Out Bet P B A"
var_def: fixes i :: "'a" shows "var i = Abs_formula {A. i \<in> A}"
Setcompr_eq_image: fixes f :: "'c \<Rightarrow> 'b"   and A :: "'c set" shows "rel_ext (\<lambda>uu_. \<exists>x. uu_ = f x \<and> x \<in> A) = f ` A"
eq_fract_2: fixes a :: "'b" shows "Fract a (0::'b) = Fract (0::'b) (1::'b)"
reindex_nontrivial: fixes A :: "'a set"   and h :: "'a \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'c" assumes "finite A"   and "\<And>x y. \<lbrakk>x \<in> A; y \<in> A; x \<noteq> y; h x = h y\<rbrakk> \<Longrightarrow> g (h x) = (0::'c)" shows "sum g (h ` A) = sum (g \<circ> h) A"
bin_rest_lt0: fixes i :: "int" shows "(i div 2 < 0) = (i < 0)"
GreatestI: fixes P :: "'b \<Rightarrow> bool"   and k :: "'b" assumes "P k" shows "P (GREATEST x. P x)"
IH1: assumes "\<exists>pn. p1 \<in> subterms (\<Gamma> pn)"   and "((\<xi>, p1), a, \<xi>', p') \<in> seqp_sos \<Gamma>" shows "\<exists>pn. p' \<in> subterms (\<Gamma> pn)"
O2C_C2O: fixes n :: "C" shows "O2C (C2O n) = n"
UniqueSucStates_SAStates_StepConf: fixes TS :: "('d \<times> (('d, 'e) expr \<times> ('f data \<Rightarrow> bool) \<times> 'e set \<times> 'f pupdate) \<times> 'd) set"   and ST :: "('d, 'e, 'f) status"   and S :: "'d"   and A :: "('d, 'e, 'f) seqauto"   and T :: "'d"   and U :: "'d" assumes "TS \<in> HPT ST"   and "S \<in> StepConf (HA ST) (Conf ST) TS"   and "A \<in> SAs (HA ST)"   and "A \<in> the (CompFun (HA ST) S)"   and "T \<in> States A"   and "U \<in> States A"   and "T \<in> StepConf (HA ST) (Conf ST) TS"   and "T \<noteq> U" shows "U \<notin> StepConf (HA ST) (Conf ST) TS"
R'_simps_5: fixes n :: "nat"   and x :: "nat"   and y :: "nat" assumes "Suc n < x \<or> Suc n < y" shows "(x, y) \<notin> R' n"
ordering_INTER_0: fixes A :: "nat \<Rightarrow> ordinal set" assumes "\<forall>n. normal_set (A n)"   and "\<forall>n. A (Suc n) \<subseteq> A n" shows "OrdinalVeblen.ordering (\<Inter> (range A)) 0 = oLimit (\<lambda>n. OrdinalVeblen.ordering (A n) 0)"
invertible_interchange_rows: fixes a :: "'e"   and b :: "'e" shows "invertible (interchange_rows (mat (1::'d)) a b)"
has_integral_AE: fixes \<Omega> :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c"   and x :: "'c" assumes "AE x\<in>\<Omega> in lborel. f x = g x" shows "(f has_integral x) \<Omega> = (g has_integral x) \<Omega>"
chambercomplex_opp_half_apartment: shows "ChamberComplex folding.Y"
Src_Inv: fixes t :: "'c local.term" assumes "Can t" shows "Src (Inv t) = Src t"
C_case_bot: shows "C_case\<cdot>\<bottom> = \<bottom>"
ntsem: fixes \<sigma> :: "'tp"   and b :: "univ" assumes "eintT \<sigma> b" shows "ntsem \<sigma> (eintF (Tag \<sigma>) [b]) = eintF (Tag \<sigma>) [b]"
godel_code'_get_nth: fixes i :: "nat"   and nl :: "nat list" assumes "i < length nl" shows "Max {u. Pi (Suc i) ^ u dvd godel_code' nl (Suc 0)} = nl ! i"
f_simps_4: fixes n :: "nat"   and y :: "'v" shows "f n (vertex.SINK, \<langle>y\<rangle>) = (if y \<in> A \<Gamma> then d_OUT (h 0) y - d_OUT (h n) y else 0)"
offs_pred_def: fixes ns :: "nat list"   and ub :: "nat"   and xs :: "'c list"   and index :: "('c \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> nat"   and key :: "'c \<Rightarrow> 'd"   and mi :: "'d"   and ma :: "'d" shows "offs_pred ns ub xs index key mi ma \<equiv> \<forall>i<length ns. offs_num (length ns) xs index key mi ma i \<le> offs_next ns ub xs index key mi ma i - ns ! i"
simps_2: fixes id :: "'a"   and f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and xs :: "'a list"   and ys :: "'a list"   and yss :: "'a list list" assumes "Matrix_Tensor.mult id f" shows "mult.vec_mat_Tensor f xs (ys # yss) = mult.vec_vec_Tensor f xs ys # mult.vec_mat_Tensor f xs yss"
inv_end_step: fixes x :: "nat"   and cf :: "nat \<times> cell list \<times> cell list" assumes "0 < x"   and "inv_end x cf" shows "inv_end x (step0 cf tcopy_end)"
simps: fixes x :: "'b"   and xl :: "'b bintree"   and xr :: "'b bintree" shows "t_left (Branch x xl xr) = xl"