hcju/isabelleps huggingface dataset:
reachable_app_iso_eq: fixes hom :: "('a, 'b, 'e, 'f) digraph_isomorphism"   and u :: "'e"   and v :: "'e" assumes "digraph_isomorphism hom"   and "u \<in> iso_verts hom ` verts G"   and "v \<in> iso_verts hom ` verts G" shows "(u \<rightarrow>\<^sup>*\<^bsub>app_iso hom G\<^esub> v) = (iso_verts (inv_iso hom) u \<rightarrow>\<^sup>* iso_verts (inv_iso hom) v)"
upset_set_prop_var: fixes X :: "'a set" shows "\<Up> X = \<Union> (\<up> ` X)"
exp_conj_sound: fixes P :: "'c \<Rightarrow> real"   and Q :: "'c \<Rightarrow> real" assumes "sound P"   and "sound Q" shows "sound (P && Q)"
le_multiset_empty_left: fixes M :: "'b multiset" assumes "M \<noteq> {#}" shows "{#} < M"
flowsto_unionE: fixes X0 :: "('a \<times> 'c \<Rightarrow>\<^sub>L 'a) set"   and T :: "real set"   and CX :: "('a \<times> 'c \<Rightarrow>\<^sub>L 'a) set"   and Y :: "('a \<times> 'c \<Rightarrow>\<^sub>L 'a) set"   and Z :: "('a \<times> 'c \<Rightarrow>\<^sub>L 'a) set" assumes "flowsto X0 T CX (Y \<union> Z)" obtains X1 :: "('a \<times> 'c \<Rightarrow>\<^sub>L 'a) set" and X2 :: "('a \<times> 'c \<Rightarrow>\<^sub>L 'a) set" where "X0 = X1 \<union> X2" and "flowsto X1 T CX Y" and "flowsto X2 T CX Z"
one_less_inverse: 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" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less zero a"   and "less a one" shows "less one (inverse a)"
filter_append: fixes P :: "'b \<Rightarrow> bool"   and xs :: "'b list"   and ys :: "'b list" shows "filter P (xs @ ys) = filter P xs @ filter P ys"
ci_nth: fixes ly :: "nat list"   and aprog :: "abc_inst list"   and as :: "nat"   and ins :: "abc_inst" assumes "ly = layout_of aprog"   and "abc_fetch as aprog = Some ins" shows "ci ly (start_of ly as) ins = tms_of aprog ! as"
G_def: shows "G = {a. (0::'b) \<le> a \<and> a \<le> u}"
basic_cqe_conv1_5: fixes c :: "'a" shows "(\<exists>x. poly [:c:] x = (0::'a)) = (c = (0::'a))"
map_eq_zero_iff: fixes f :: "'d \<Rightarrow> 'c"   and p :: "'b \<Rightarrow>\<^sub>0 'd" shows "(Poly_Mapping.map f p = 0) = (\<forall>k\<in>keys p. f (lookup p k) = (0::'c))"
pw_sup_inres: fixes a :: "'b nres"   and b :: "'b nres"   and x :: "'b" shows "inres (sup a b) x = (inres a x \<or> inres b x)"
repr_inj: fixes x :: "'a"   and y :: "'a" shows "(repr x = repr y) = (x = y)"
min_rvar_incdec_eq_def: fixes dir :: "('a, 'b) Direction"   and s :: "('a, 'b) state"   and eq :: "nat \<times> linear_poly" shows "min_rvar_incdec_eq dir s eq = (let rvars = Abstract_Linear_Poly.vars_list (rhs eq) in case min_satisfying (\<lambda>x. 0 < coeff (rhs eq) x \<and> \<lhd>\<^sub>u\<^sub>b (Direction.lt dir) (\<langle>\<V> s\<rangle> x) (Direction.UB dir s x) \<or> coeff (rhs eq) x < 0 \<and> \<rhd>\<^sub>l\<^sub>b (Direction.lt dir) (\<langle>\<V> s\<rangle> x) (Direction.LB dir s x)) rvars of None \<Rightarrow> Inl (unsat_indices dir s rvars eq) | Some x\<^sub>j \<Rightarrow> Inr x\<^sub>j)"
borel_measurable_eq: fixes f :: "'c \<Rightarrow> 'd"   and M :: "'c measure"   and g :: "'c \<Rightarrow> 'd" assumes "f \<in> borel_measurable M"   and "g \<in> borel_measurable M" shows "{w \<in> space M. f w = g w} \<in> sets M"
N_L: fixes n :: "nat" assumes "N_L \<le> n" shows "\<bar>\<Sum>i. L (i + n)\<bar> < r / 3"
fermat_liar_def: fixes a :: "nat"   and n :: "nat" shows "fermat_liar a n = [a ^ (n - 1) = 1] (mod n)"
is_unit_smult_iff: fixes c :: "'b"   and p :: "'b poly" shows "(smult c p dvd 1) = (c dvd (1::'b) \<and> p dvd 1)"
f_inv_into_f: fixes y :: "'b"   and f :: "'c \<Rightarrow> 'b"   and A :: "'c set" assumes "y \<in> f ` A" shows "S\<^sub>O inv_into A f y = y"
simps_5: fixes f1 :: "'b"   and f2 :: "'c \<Rightarrow> 'b"   and x2 :: "'c" shows "(case \<lfloor>x2\<rfloor> of None \<Rightarrow> f1 | \<lfloor>x\<rfloor> \<Rightarrow> f2 x) = f2 x2"
val_ord_Zp: fixes a :: "nat \<Rightarrow> int" assumes "a \<noteq> \<zero>" shows "val_Zp a = eint (ord_Zp a)"
element_ptr_kinds_M_def: shows "element_ptr_kinds_M \<equiv> l_ptr_kinds_M.a_ptr_kinds_M element_ptr_kinds"
fundamental_theorem_of_calculus: fixes a :: "real"   and b :: "real"   and f :: "real \<Rightarrow> 'c"   and f' :: "real \<Rightarrow> 'c" assumes "a \<le> b"   and "\<And>x. x \<in> {a..b} \<Longrightarrow> (f has_vector_derivative f' x) (at x within {a..b})" shows "(f' has_integral f b - f a) {a..b}"
less_eq_list_induct: fixes x1 :: "'a list"   and x2 :: "'a list"   and Pa :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool" assumes "x1 \<preceq> x2"   and "\<And>ys. Pa [] ys"   and "\<And>xs ys y. \<lbrakk>xs \<preceq> ys; Pa xs ys\<rbrakk> \<Longrightarrow> Pa xs (y # ys)"   and "\<And>x y xs ys. \<lbrakk>x = y; xs \<preceq> ys; Pa xs ys\<rbrakk> \<Longrightarrow> Pa (x # xs) (y # ys)" shows "Pa x1 x2"
prime_power_eq_one_iff: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and p :: "'b"   and n :: "nat" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize"   and "normalization_semidom.prime zero times one normalize p" shows "(power.power one times p n = one) = (n = 0)"
isDERIV_max_Var_floatarithI: fixes n :: "nat"   and f :: "floatarith"   and ys :: "real list"   and xs :: "real list" assumes "isDERIV n f ys"   and "\<And>i. i < max_Var_floatarith f \<Longrightarrow> xs ! i = ys ! i" shows "isDERIV n f xs"
conjugate_circline_cmat_def: shows "conjugate_circline_cmat = mat_cnj"
rb_aux_inv1_distinct_lt: fixes bs :: "('t \<Rightarrow>\<^sub>0 'b) list" assumes "rb_aux_inv1 bs" shows "distinct (map lt bs)"
OrdP_graph_aux_def: shows "OrdP_graph_aux \<equiv> lfp (\<lambda>p x1 x2. \<exists>OrdP_aux y x z. x1 = x \<and> x2 = All2 y x (Var y SUBS x AND All2 z (Var y) (Var z SUBS Var y)) \<and> atom y \<sharp> (x, z) \<and> atom z \<sharp> x)"
prems_3: shows "free_vars (Exp $$ e) \<subseteq> set vs \<union> set vs'"
shift2_inv_def: fixes qs :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list" shows "shift2_inv qs = (valid_decomp X qs \<and> standard_decomp k qs \<and> exact_decomp (Suc m) qs \<and> (\<forall>d0<d. card {q \<in> set qs. poly_deg (fst q) = d0 \<and> m < card (snd q)} \<le> 1))"
while_hoare_r: fixes p :: "'b upred"   and b :: "'b upred"   and S :: "'b hrel" assumes "\<lbrace>p \<and> b\<rbrace> S \<lbrace>p\<rbrace>\<^sub>u" shows "\<lbrace>p\<rbrace> while\<^sup>\<top> b do S od \<lbrace>\<not> b \<and> p\<rbrace>\<^sub>u"
widen_RefT2: fixes G :: "prog"   and S :: "ty"   and R :: "ref_ty" assumes "G\<turnstile>S\<preceq>RefT R" shows "\<exists>t. S = RefT t"
set_def: fixes r :: "'b ref"   and x :: "'b" shows "Ref.set r x = refs_update (\<lambda>h. h(TYPEREP('b) := (h TYPEREP('b))(addr_of_ref r := to_nat x)))"
lm09: fixes X :: "'c set" assumes "trivial X" shows "Pow X \<subseteq> {{}, X}"
second_def: fixes s :: "nat \<Rightarrow> 'a" shows "second s \<equiv> s 1"
parent_child_rel_impl: fixes known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and type_wf :: "(_) heap \<Rightarrow> bool"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and heap_is_wellformed :: "(_) heap \<Rightarrow> bool"   and parent_child_rel :: "(_) heap \<Rightarrow> ((_) object_ptr \<times> (_) object_ptr) set" assumes "l_heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr type_wf get_child_nodes get_child_nodes_locs get_disconnected_nodes get_disconnected_nodes_locs heap_is_wellformed parent_child_rel" shows "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 get_child_nodes"
insert_relcomp: fixes a :: "'b"   and b :: "'d"   and A :: "('b \<times> 'd) set"   and B :: "('d \<times> 'c) set" shows "insert (a, b) A O B = A O B \<union> {a} \<times> {c. (b, c) \<in> B}"
equality: fixes r :: "('k, 'l) web_scheme"   and r' :: "('k, 'l) web_scheme" assumes "edge r = edge r'"   and "weight r = weight r'"   and "A r = A r'"   and "B r = B r'"   and "web.more r = web.more r'" shows "r = r'"
foo_cases1: fixes x1a :: "'b list"   and x2a :: "'c list" shows "foo x1a x2a = (case x1a of [] \<Rightarrow> case x2a of [] \<Rightarrow> 3::'a | x2fa # x1fa \<Rightarrow> 1::'a | x2ea # x1ea \<Rightarrow> case x2a of [] \<Rightarrow> 0::'a | x4a # x3a \<Rightarrow> foo [] [])"
f_props_2: shows "\<forall>(k, \<gamma>)\<in>one_chain1. if k = 1 then chain_subdiv_path \<gamma> (f (k, \<gamma>)) else chain_subdiv_path (reversepath \<gamma>) (f (k, \<gamma>))"
r1_leq: shows "r'\<^sub>1 \<le> r1"
simplicial_chain_def: fixes p :: "nat"   and S :: "('a \<Rightarrow> real) set"   and c :: "((nat \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real) \<Rightarrow>\<^sub>0 'b" shows "simplicial_chain p S c \<equiv> Poly_Mapping.keys c \<subseteq> Collect (simplicial_simplex p S)"
lsu_sup_closed_var: fixes x :: "'b"   and y :: "'b" shows "\<nu>\<^sup>\<natural> (\<nu>\<^sup>\<natural> x \<squnion> \<nu>\<^sup>\<natural> y) = \<nu>\<^sup>\<natural> x \<squnion> \<nu>\<^sup>\<natural> y"
prems: shows "((\<lambda>xs. \<not> P xs) until alw P) \<omega>"
in_idem: fixes a :: "'a"   and A :: "'a set" assumes "a \<in> A" shows "a \<^bold>* F A = F A"
ty_vrs_fresh: fixes x :: "vrs"   and T :: "ty" shows "x \<sharp> T"
is_tdghm_axioms: shows "\<NN> : \<FF> \<mapsto>\<^sub>D\<^sub>G\<^sub>H\<^sub>M \<GG> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<BB>"
sinl_strict: shows "sinl\<cdot>\<bottom> = \<bottom>"
log2_mono: fixes x :: "real"   and y :: "real" assumes "0 < x"   and "x \<le> y" shows "log 2 x \<le> log 2 y"
compatible_sym: fixes x :: "'a"   and y :: "'a" assumes "compatible x y" shows "compatible y x"
axioms_1: fixes step1 :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and step2 :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and final1 :: "'d \<Rightarrow> bool"   and final2 :: "'e \<Rightarrow> bool"   and load1 :: "'f \<Rightarrow> 'd \<Rightarrow> bool"   and load2 :: "'g \<Rightarrow> 'e \<Rightarrow> bool"   and order :: "'h \<Rightarrow> 'h \<Rightarrow> bool"   and match :: "'h \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> bool"   and compile :: "'f \<Rightarrow> 'g option" assumes "compiler step1 step2 final1 final2 load1 load2 order match compile" shows "language step1 final1"
keys_shift_map_keys_subset: fixes t :: "'a"   and c :: "'c"   and xs :: "('t, 'c) oalist_ntm" shows "keys (MP_oalist (shift_map_keys t ((*) c) xs)) \<subseteq> (\<oplus>) t ` keys (MP_oalist xs)"
assoc_in_homs_5: shows "\<guillemotleft>\<lparr>Chn = \<mu>\<nu>_\<pi>_\<rho>.chine_assoc, Dom = Dom (((\<mu> \<star> \<nu>) \<star> \<pi>) \<star> \<rho>), Cod = Cod ((\<mu> \<star> \<nu>) \<star> \<pi> \<star> \<rho>)\<rparr> : ((\<mu> \<star> \<nu>) \<star> \<pi>) \<star> \<rho> \<Rightarrow> (\<mu> \<star> \<nu>) \<star> \<pi> \<star> \<rho>\<guillemotright>"
scaleR_minus_right: fixes a :: "real"   and x :: "'b" shows "a *\<^sub>R - x = - (a *\<^sub>R x)"
Card_order_Un_infinite: fixes r :: "('b \<times> 'b) set"   and p :: "('b \<times> 'b) set" assumes "infinite (Field r)"   and "Card_order r"   and "p \<le>o r" shows "|Field r \<union> Field p| =o r \<and> |Field p \<union> Field r| =o r"
LeafI: fixes a :: "'b" shows "Leaf a \<in> sexp"
coincide_on_def: fixes \<sigma> :: "('b \<times> 'b trm) list"   and \<eta> :: "('b \<times> 'b trm) list"   and V :: "'b set" shows "coincide_on \<sigma> \<eta> V = (\<forall>x\<in>V. Var x \<lhd> \<sigma> = Var x \<lhd> \<eta>)"
Liouville_weak_inverse: fixes f :: "complex \<Rightarrow> complex" assumes "f holomorphic_on UNIV"   and "\<And>B. \<forall>\<^sub>F x in at_infinity. B \<le> cmod (f x)" obtains z :: "complex" where "f z = 0"
mat_of_rows_list_dim_3: fixes n :: "nat"   and vs :: "'a list list" shows "dim_col (mat_of_rows_list n vs) = n"
assms_4: shows "test_tools M2 M1 FAIL PM V \<Omega>"
parallel_HSem_ind: fixes P :: "('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> bool"   and \<rho>\<^sub>1 :: "'a \<Rightarrow> 'b"   and \<Gamma>\<^sub>1 :: "('a \<times> 'c) list"   and ESem :: "'c \<Rightarrow> ('a \<Rightarrow> 'b) \<rightarrow> 'b"   and \<rho>\<^sub>2 :: "'a \<Rightarrow> 'b"   and \<Gamma>\<^sub>2 :: "('a \<times> 'c) list" assumes "adm (\<lambda>\<rho>'. P (fst \<rho>') (snd \<rho>'))"   and "P \<bottom> \<bottom>"   and "\<And>y z. P y z \<Longrightarrow> P (\<rho>\<^sub>1 ++\<^bsub>domA \<Gamma>\<^sub>1\<^esub> evalHeap \<Gamma>\<^sub>1 (\<lambda>e. ESem e\<cdot>y)) (\<rho>\<^sub>2 ++\<^bsub>domA \<Gamma>\<^sub>2\<^esub> evalHeap \<Gamma>\<^sub>2 (\<lambda>e. ESem e\<cdot>z))" shows "P (has_ESem.HSem ESem \<Gamma>\<^sub>1\<cdot>\<rho>\<^sub>1) (has_ESem.HSem ESem \<Gamma>\<^sub>2\<cdot>\<rho>\<^sub>2)"
parCompose: fixes P :: "pi"   and Rel :: "(pi \<times> pi) set"   and Q :: "pi"   and R :: "pi"   and Rel' :: "(pi \<times> pi) set"   and T :: "pi"   and Rel'' :: "(pi \<times> pi) set" assumes "P \<leadsto>\<^sup>^<Rel> Q"   and "R \<leadsto>\<^sup>^<Rel'> T"   and "(P, Q) \<in> Rel"   and "(R, T) \<in> Rel'"   and "\<And>P Q R T. \<lbrakk>(P, Q) \<in> Rel; (R, T) \<in> Rel'\<rbrakk> \<Longrightarrow> (P \<parallel> R, Q \<parallel> T) \<in> Rel''"   and "\<And>P Q a. (P, Q) \<in> Rel'' \<Longrightarrow> (<\<nu>a>P, <\<nu>a>Q) \<in> Rel''"   and "eqvt Rel"   and "eqvt Rel'"   and "eqvt Rel''" shows "P \<parallel> R \<leadsto>\<^sup>^<Rel''> Q \<parallel> T"
uniqElementsTrailImpliesUniqElementsPrefix: fixes p :: "('b \<times> bool) list"   and M :: "('b \<times> bool) list" assumes "isPrefix p M"   and "uniq (elements M)" shows "uniq (elements p)"
real_code_3: fixes x :: "'b list" shows "x \<noteq> [] \<longrightarrow> enc x = enc [hd x] @ enc (tl x)"
padic_mult_assoc0: fixes p :: "int"   and x :: "nat \<Rightarrow> int"   and y :: "nat \<Rightarrow> int"   and z :: "nat \<Rightarrow> int" assumes "prime p" shows "padic_mult p (padic_mult p x y) z = padic_mult p x (padic_mult p y z)"
that: fixes I :: "'a multiset"   and K :: "'a multiset"   and J :: "'a multiset" assumes "M = I + K"   and "N = I + J"   and "J \<noteq> {#}"   and "\<forall>k\<in>#K. \<exists>j\<in>#J. (k, j) \<in> r" shows "thesis"
succ_nodes: fixes q2 :: "'d"   and A :: "('e, 'f, 'd) FSM"   and w :: "'e \<times> 'f"   and q1 :: "'d" assumes "q2 \<in> succ A w q1"   and "q1 \<in> nodes A" shows "q2 \<in> nodes A"
mod_0_imp_div_mult_right_eq: fixes n :: "nat"   and m :: "nat"   and k :: "nat" assumes "n mod m = 0" shows "n * k div m = k * (n div m)"
assms: shows "\<forall>i\<in>set I. i < length T"
distrib_right: fixes a :: "'c \<times> 'c \<Rightarrow> 'd"   and b :: "'c \<times> 'c \<Rightarrow> 'd"   and c :: "'c \<times> 'c \<Rightarrow> 'd" shows "(a \<oplus> b) \<odot> c = a \<odot> c \<oplus> b \<odot> c"
sep_conjI: fixes P :: "'a \<Rightarrow> bool"   and p :: "'a"   and Q :: "'a \<Rightarrow> bool"   and q :: "'a"   and s :: "'a" assumes "P p"   and "Q q"   and "p \<oplus> q = Some s" shows "(P ** Q) s"
ord_less_eq_trans: fixes less :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and a :: "'k"   and b :: "'k"   and c :: "'k" assumes "less a b"   and "b = c" shows "less a c"
cr_float_def: shows "cr_float \<equiv> \<lambda>x y. x = real_of_float y"
refine: fixes succi :: "'b \<Rightarrow> 'b list"   and E :: "('b \<times> 'b) set"   and src :: "'b"   and tgt :: "'b" assumes "(succi, E) \<in> \<langle>Id\<rangle>succg_rel" shows "RETURN (dfs_impl succi src tgt) \<le> \<Down> bool_rel (dfs E src tgt)"
cases: fixes a1 :: "'a tree"   and a2 :: "'a stream" assumes "ipath a1 a2" obtains t :: "'a tree" and steps :: "'a stream" and t' :: "'a tree" where "a1 = t" and "a2 = steps" and "tree.root t = shd steps" and "t' |\<in>| cont t" and "ipath t' (stl steps)"
sp_rewrite_prog: fixes f :: "nat \<Rightarrow> 'i \<Rightarrow> Unboxed.type option list \<Rightarrow> ('c, ('d, 'e, 'f, 'op, 'opinl, 'opubx, 'g, 'h) Ubx.instr \<times> Unboxed.type option list) result"   and n :: "nat"   and p1 :: "'i list"   and \<Sigma>1 :: "Unboxed.type option list"   and p2 :: "('d, 'e, 'f, 'op, 'opinl, 'opubx, 'g, 'h) Ubx.instr list"   and \<Sigma>2 :: "Unboxed.type option list"   and F :: "'f \<Rightarrow> 'j fundef option" assumes "rewrite_prog f n p1 \<Sigma>1 = Ok (p2, \<Sigma>2)"   and "\<And>x \<Sigma>1 n x' \<Sigma>2. f n x \<Sigma>1 = Ok (x', \<Sigma>2) \<Longrightarrow> Subx.sp_instr F x' \<Sigma>1 = Ok \<Sigma>2" shows "Subx.sp F p2 \<Sigma>1 = Ok \<Sigma>2"
le_Vset_rank: fixes a :: "hf" shows "a \<le> Vset (rank a)"
sym: fixes S :: "('a, 'b) eq_object_scheme"   and x :: "'a"   and y :: "'a" assumes "equivalence S"   and "x .=\<^bsub>S\<^esub> y"   and "x \<in> carrier S"   and "y \<in> carrier S" shows "y .=\<^bsub>S\<^esub> x"
quadratic_sub_leq: fixes var :: "nat"   and L :: "real list"   and Dv :: "real"   and Cv :: "real"   and c :: "real mpoly"   and a :: "real mpoly"   and Av :: "real"   and b :: "real mpoly"   and Bv :: "real"   and d :: "real mpoly"   and p :: "real mpoly" assumes "var < length L"   and "Dv \<noteq> 0"   and "0 \<le> Cv"   and "var \<notin> vars c"   and "\<forall>x. insertion (nth_default 0 (L[var := x])) a = Av"   and "\<forall>x. insertion (nth_default 0 (L[var := x])) b = Bv"   and "\<forall>x. insertion (nth_default 0 (L[var := x])) c = Cv"   and "\<forall>x. insertion (nth_default 0 (L[var := x])) d = Dv" shows "aEval (Leq p) (L[var := (Av + Bv * sqrt Cv) / Dv]) = eval (quadratic_sub var a b c d (Leq p)) (L[var := sqrt Cv])"
connectedin_discrete_topology: fixes U :: "'c set"   and S :: "'c set" shows "connectedin (discrete_topology U) S = (S \<subseteq> U \<and> (\<exists>a. S \<subseteq> {a}))"
Rep_inject: fixes Rep :: "'e \<Rightarrow> 'f"   and Abs :: "'f \<Rightarrow> 'e"   and A :: "'f set"   and x :: "'e"   and y :: "'e" assumes "type_definition Rep Abs A" shows "(Rep x = Rep y) = (x = y)"
valid: shows "\<forall>Z. \<Gamma>\<Turnstile>\<^bsub>/F\<^esub> {s. s = Z \<and> s \<in> P} c {t. Z \<in> P \<longrightarrow> t \<in> Q},{t. Z \<in> P \<longrightarrow> t \<in> A}"
group_homomorphism_def: fixes \<eta> :: "'d \<Rightarrow> 'e"   and G :: "'d set"   and composition :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and unit :: "'d"   and G' :: "'e set"   and composition' :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and unit' :: "'e" shows "group_homomorphism \<eta> G composition unit G' composition' unit' \<equiv> monoid_homomorphism \<eta> G composition unit G' composition' unit' \<and> Group_Theory.group G composition unit \<and> Group_Theory.group G' composition' unit'"
trace_def: fixes A :: "'c mat" shows "trace A = (\<Sum>i = 0..<dim_row A. A $$ (i, i))"
finsum_index: fixes i :: "nat"   and f :: "'b vec \<Rightarrow> 'a vec"   and X :: "'b vec set" assumes "i < n"   and "f \<in> X \<rightarrow> carrier_vec n"   and "X \<subseteq> carrier_vec n" shows "finsum V f X $ i = (\<Sum>x\<in>X. f x $ i)"
P2: shows "A \<noteq> B \<and> C \<noteq> D \<and> Col X0 A B \<and> Col X0 C D \<and> (Col U A B \<and> Col V C D \<longrightarrow> Per U X0 V)"
xin: shows "x \<in> {l (- dr)<--<l dr}"
ha: shows "typeof_addr h a = \<lfloor>Class_type D\<rfloor>"
p_subid_lb2: fixes x :: "'a"   and y :: "'a" shows "x \<cdot> (0::'a) \<parallel> (y \<cdot> (0::'a)) \<le> y \<cdot> (0::'a)"
bitset_def: fixes n :: "nat" shows "bitset n = {i. odd (n div 2 ^ i)}"
intro: fixes block_collection :: "'b set multiset" assumes "\<And>bl. bl \<in># block_collection \<Longrightarrow> count block_collection bl = 1" shows "simple_incidence_system_axioms block_collection"
member_insert: fixes x :: "'c"   and xs :: "'c set_dlist" assumes "ID CEQ('c) \<noteq> None" shows "member (DList_Set.insert x xs) = (member xs)(x := True)"
plus_vec_def: shows "(+) \<equiv> \<lambda>x y. \<chi>i. x $ i + y $ i"
drop_bit_eq_div: fixes 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 divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bit :: "'a \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and drop_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and take_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and n :: "nat"   and a :: "'a" assumes "class.semiring_bit_shifts plus minus zero times one divide modulo bit push_bit drop_bit take_bit" shows "drop_bit n a = divide a (power.power one times (numeral.numeral one plus (num.Bit0 num.One)) n)"
Suc: shows "local.eval (local.monom b n) a = b \<otimes> a [^] n"
uint_sint: fixes w :: "'c word" shows "uint w = take_bit LENGTH('c) (sint w)"
upd_elim: fixes c :: "(var\<^sub>C, aexp\<^sub>C, bexp\<^sub>C) Stmt"   and upd :: "var\<^sub>C ModeUpd"   and mds :: "Mode \<Rightarrow> var\<^sub>C set"   and mem :: "var\<^sub>C \<Rightarrow> nat"   and c' :: "(var\<^sub>C, aexp\<^sub>C, bexp\<^sub>C) Stmt"   and mds' :: "Mode \<Rightarrow> var\<^sub>C set"   and mem' :: "var\<^sub>C \<Rightarrow> nat" assumes "eval_abv\<^sub>C \<langle>c@[upd], mds, mem\<rangle>\<^sub>C \<langle>c', mds', mem'\<rangle>\<^sub>C" shows "eval_abv\<^sub>C \<langle>c, C.update_modes upd mds, mem\<rangle>\<^sub>C \<langle>c', mds', mem'\<rangle>\<^sub>C"
encode_problem_parallel_sound: fixes \<Pi> :: "'b strips_problem"   and \<A> :: "sat_plan_variable \<Rightarrow> bool"   and t :: "nat" assumes "is_valid_problem_strips \<Pi>"   and "\<A> \<Turnstile> \<Phi> \<Pi> t" shows "is_parallel_solution_for_problem \<Pi> (\<Phi>\<inverse> \<Pi> \<A> t)"
addsE: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and b :: "'c"   and a :: "'c" assumes "plus.adds plus b a" obtains k :: "'c" where "a = plus b k"
isomorphic_implies_hpar_4: fixes f :: "'b"   and f' :: "'b" assumes "D.isomorphic f f'" shows "trg\<^sub>D f = trg\<^sub>D f'"
intros_1: fixes f :: "'a \<Rightarrow> 'a"   and x :: "'a" shows "f x \<in> orbit f x"
XH_largest_def: fixes ds :: "'b set"   and X :: "'a set" shows "XH_largest ds X = {y. Xd y \<in> ds \<and> y \<in> Field (Pd (Xd y)) \<and> (\<forall>x\<in>dX X (Xd y). (x, y) \<in> Pd (Xd y))}"
meval_sched: shows "conc.meval_sched_abv gc\<^sub>C sched gc\<^sub>C'"
permutes_inverses_1: fixes p :: "'b \<Rightarrow> 'b"   and S :: "'b set"   and x :: "'b" assumes "p permutes S" shows "p (inv p x) = x"
vfinite_0_left: fixes b :: "V" shows "[]\<^sub>\<circ> \<times>\<^sub>\<bullet> b = []\<^sub>\<circ>"
assms_1: shows "crdt_ops (xs @ [(i1, r1)] @ [(i2, r2)]) deps"
nop: fixes \<C> :: "t_context" shows "\<C> \<turnstile> [Nop] : [] _> []"
valType: shows "\<forall>q. val[q,c] \<subseteq> {True, False}"
hypreal_hnorm_def: fixes r :: "real star" shows "hnorm r = \<bar>r\<bar>"
someinit_def: fixes i :: "nat" shows "opq.someinit i \<equiv> SOME x. x \<in> (fst \<circ> (\<lambda>(p, q). (fst (id p), snd (id p), q))) ` init (paodv i \<langle>\<langle> qmsg)"
enforce_map_option: fixes P :: "'c \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'c"   and x :: "'d option" shows "enforce_option P (map_option f x) = map_option f (enforce_option (P \<circ> f) x)"
hld_smap: fixes x :: "'a set"   and f :: "'b \<Rightarrow> 'a"   and s :: "'b stream" shows "HLD x (smap f s) = holds (\<lambda>y. f y \<in> x) s"