hcju/isabelleps huggingface dataset:
bind_transfer: fixes R0 :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and R1 :: "'c \<Rightarrow> 'd \<Rightarrow> bool" shows "(crel_vs R0 ===> (R0 ===>\<^sub>T R1) ===>\<^sub>T R1) (\<lambda>v f. f v) (\<bind>)"
sum_integral_diff_series_monoseq: shows "monoseq sum_integral_diff_series"
inj_on_index: fixes xs :: "'c list" shows "inj_on (index xs) (set xs)"
path_img: shows "path_image \<gamma> \<subseteq> s"
cinner_simps_5: fixes r :: "complex"   and x :: "'a"   and y :: "'a" shows "r *\<^sub>C x \<bullet>\<^sub>C y = cnj r * (x \<bullet>\<^sub>C y)"
monoT: fixes t :: "('s \<Rightarrow> real) \<Rightarrow> 's \<Rightarrow> real"   and u :: "('s \<Rightarrow> real) \<Rightarrow> 's \<Rightarrow> real" assumes "le_trans t u"   and "\<And>P. sound P \<Longrightarrow> sound (t P)"   and "\<And>P. sound P \<Longrightarrow> sound (u P)" shows "le_trans (T t) (T u)"
power_eq_0_iff: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and a :: "'c"   and n :: "nat" assumes "class.semiring_1_no_zero_divisors one times plus zero" shows "(power.power one times a n = zero) = (a = zero \<and> 0 < n)"
a_is_weakly_dom_component_safe_def: fixes S\<^sub>a\<^sub>r\<^sub>g :: "(_) object_ptr set"   and S\<^sub>r\<^sub>e\<^sub>s\<^sub>u\<^sub>l\<^sub>t :: "(_) object_ptr set"   and h :: "(_) heap"   and h' :: "(_) heap" shows "a_is_weakly_dom_component_safe S\<^sub>a\<^sub>r\<^sub>g S\<^sub>r\<^sub>e\<^sub>s\<^sub>u\<^sub>l\<^sub>t h h' = (let removed_pointers = fset (object_ptr_kinds h) - fset (object_ptr_kinds h'); added_pointers = fset (object_ptr_kinds h') - fset (object_ptr_kinds h); arg_components = \<Union>ptr\<in>(\<Union>ptr\<in>S\<^sub>a\<^sub>r\<^sub>g. set |h \<turnstile> a_get_component ptr|\<^sub>r) \<inter> fset (object_ptr_kinds h). set |h \<turnstile> a_get_component ptr|\<^sub>r; arg_components' = \<Union>ptr\<in>(\<Union>ptr\<in>S\<^sub>a\<^sub>r\<^sub>g. set |h \<turnstile> a_get_component ptr|\<^sub>r) \<inter> fset (object_ptr_kinds h'). set |h' \<turnstile> a_get_component ptr|\<^sub>r in removed_pointers \<subseteq> arg_components \<and> S\<^sub>r\<^sub>e\<^sub>s\<^sub>u\<^sub>l\<^sub>t \<subseteq> arg_components' \<union> added_pointers \<and> (\<forall>outside_ptr\<in>fset (object_ptr_kinds h) \<inter> fset (object_ptr_kinds h') - (\<Union>ptr\<in>S\<^sub>a\<^sub>r\<^sub>g. set |h \<turnstile> a_get_component ptr|\<^sub>r). preserved (get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t outside_ptr id) h h'))"
stern_brocot_denominator_non_zero: fixes path :: "dir list" shows "case root (traverse_tree path stern_brocot_recurse) of (m, n) \<Rightarrow> 0 < m \<and> 0 < n"
gState_inv_def: fixes prog :: "program"   and g :: "'a gState_scheme" shows "gState_inv prog g = (length (procs g) \<le> max_procs \<and> (\<forall>p\<in>set (procs g). pState_inv prog p \<and> cl_inv (g, p)) \<and> length (gState.channels g) \<le> max_channels \<and> set (gState.channels g) \<subseteq> Collect channel_inv \<and> lm.ball (gState.vars g) (\<lambda>(k, v). variable_inv v))"
igFreshBinp_def: fixes MOD :: "('b, 'c, 'd, 'e, 'f, 'g, 'h, 'i, 'j) model_scheme"   and ys :: "'d"   and y :: "'g"   and binp :: "'k \<Rightarrow> 'i option" shows "igFreshBinp MOD ys y binp \<equiv> liftAll (igFreshAbs MOD ys y) binp"
lens_equiv_def: fixes X :: "'c \<Longrightarrow> 'd"   and Y :: "'e \<Longrightarrow> 'd" shows "X \<approx>\<^sub>L Y = (X \<subseteq>\<^sub>L Y \<and> Y \<subseteq>\<^sub>L X)"
simps: fixes init_nsc :: "('a \<times> 'b ns_constraint) list \<Rightarrow> 'c"   and assert_nsc :: "'a \<Rightarrow> 'c \<Rightarrow> 'a list + 'c"   and check_nsc :: "'c \<Rightarrow> 'a list + 'c"   and solution_nsc :: "'c \<Rightarrow> (nat, 'b) mapping"   and checkpoint_nsc :: "'c \<Rightarrow> 'd"   and backtrack_nsc :: "'d \<Rightarrow> 'c \<Rightarrow> 'c"   and invariant_nsc :: "('a \<times> 'b ns_constraint) list \<Rightarrow> 'a set \<Rightarrow> 'c \<Rightarrow> bool"   and checked_nsc :: "('a \<times> 'b ns_constraint) list \<Rightarrow> 'a set \<Rightarrow> 'c \<Rightarrow> bool"   and to_ns :: "('a \<times> constraint) list \<Rightarrow> ('a \<times> 'b ns_constraint) list"   and from_ns :: "(nat, 'b) mapping \<Rightarrow> 'b ns_constraint list \<Rightarrow> (nat, rat) mapping"   and cs :: "('a \<times> constraint) list"   and J :: "'a set"   and ds :: "'b ns_constraint list"   and s :: "'c" assumes "Incremental_NS_Constraint_Ops_To_Ns_For_Incremental_Simplex init_nsc assert_nsc check_nsc solution_nsc checkpoint_nsc backtrack_nsc invariant_nsc checked_nsc to_ns from_ns" shows "Incremental_NS_Constraint_Ops_To_Ns_For_Incremental_Simplex.checked_cs checked_nsc to_ns cs J (ds, s) = (ds = map snd (to_ns cs) \<and> checked_nsc (to_ns cs) J s)"
derangements_in_image_strong: fixes d :: "nat \<Rightarrow> nat"   and A :: "nat set"   and x :: "nat" assumes "d \<in> derangements A"   and "x \<in> A" shows "d x \<in> A - {x}"
the_elem_eq: fixes x :: "'g" shows "the_elem {x} = x"
Posix_Times: fixes s1 :: "'b list"   and r1 :: "'b rexp"   and v1 :: "'b val"   and s2 :: "'b list"   and r2 :: "'b rexp"   and v2 :: "'b val" assumes "s1 \<in> r1 \<rightarrow> v1"   and "s2 \<in> r2 \<rightarrow> v2"   and "\<nexists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> s1 @ s\<^sub>3 \<in> lang r1 \<and> s\<^sub>4 \<in> lang r2" shows "(s1 @ s2) \<in> Times r1 r2 \<rightarrow> Seq v1 v2"
wf_mbuf2_def: fixes i :: "nat"   and ja :: "nat"   and jb :: "nat"   and P :: "nat \<Rightarrow> 'c option list set \<Rightarrow> bool"   and Q :: "nat \<Rightarrow> 'c option list set \<Rightarrow> bool"   and buf :: "'c option list set list \<times> 'c option list set list" shows "wf_mbuf2 i ja jb P Q buf = (i \<le> ja \<and> i \<le> jb \<and> (case buf of (xs, ys) \<Rightarrow> list_all2 P [i..<ja] xs \<and> list_all2 Q [i..<jb] ys))"
hyps: fixes p :: "real \<times> real"   and x :: "(real \<times> real) \<times> real \<times> real"   and y :: "(real \<times> real) \<times> real \<times> real"   and z :: "(real \<times> real) \<times> real \<times> real" assumes "list_all (\<lambda>seg. 0 \<le> det3 (fst seg) (snd seg) p) (x # y # z # ws)"   and "convex_polygon (x # y # z # ws)"   and "ccw'.sortedP (fst x) (map snd (butlast (x # y # z # ws)))" shows "p \<in> convex hull set (map fst (x # y # z # ws))"
simps_4: fixes e :: "nat \<Rightarrow> 'a"   and f :: "'b \<Rightarrow> 'a list \<Rightarrow> 'a"   and g :: "'c \<Rightarrow> 'a list \<Rightarrow> bool"   and p :: "('b, 'c) form"   and q :: "('b, 'c) form" shows "eval e f g (And p q) = (eval e f g p \<and> eval e f g q)"
composites_are_arrows_1: shows "hseq f (g \<star> h)"
strT2_def: shows "strT2 = (\<lambda>(Kl, lab). case lab [] of Inl x \<Rightarrow> undefined x | Inr x \<Rightarrow> F2map id (\<lambda>k1. (BNF_Greatest_Fixpoint.Shift Kl (Inl k1), BNF_Greatest_Fixpoint.shift lab (Inl k1))) (\<lambda>k2. (BNF_Greatest_Fixpoint.Shift Kl (Inr k2), BNF_Greatest_Fixpoint.shift lab (Inr k2))) x)"
adom_succs: fixes i :: "nat"   and j :: "nat"   and k :: "nat" assumes "local.dom i j"   and "i \<noteq> j"   and "j \<in> succs k" shows "local.dom i k"
rep_eq: fixes x :: "'a scene" shows "Rep_scene (- x) = (\<lambda>xa y. Rep_scene x y xa)"
trans_5: shows "(R ===> mat_rel R ===> (=)) pivot_positions_gen pivot_positions_gen"
perm_exits_Vs: fixes i :: "name" assumes "i \<in> insert i__ V" shows "p' \<bullet> i \<notin> insert i__ V"
nonneg_incseq_Bseq_subseq_iff: fixes f :: "nat \<Rightarrow> real"   and g :: "nat \<Rightarrow> nat" assumes "\<And>x. 0 \<le> f x"   and "incseq f"   and "strict_mono g" shows "Bseq (\<lambda>x. f (g x)) = Bseq f"
ldistinct_lfilterD: fixes P :: "'c \<Rightarrow> bool"   and xs :: "'c llist"   and n :: "nat"   and m :: "nat"   and a :: "'c" assumes "ldistinct (lfilter P xs)"   and "enat n < llength xs"   and "enat m < llength xs"   and "P a"   and "lnth xs n = a"   and "lnth xs m = a" shows "m = n"
finite_deg_sect: fixes X :: "'b set"   and d :: "nat" assumes "finite X" shows "finite (deg_sect X d)"
int_mod_eq: fixes b :: "int"   and n :: "int"   and a :: "int" assumes "0 \<le> b"   and "b < n"   and "a mod n = b mod n" shows "a mod n = b"
1: shows "local.getGMState (hd sl) = do st u1 c1"
finite_UN_I: fixes A :: "'b set"   and B :: "'b \<Rightarrow> 'c set" assumes "finite A"   and "\<And>a. a \<in> A \<Longrightarrow> finite (B a)" shows "finite (\<Union> (B ` A))"
preservation_if: fixes x :: "'a list"   and l1 :: "'a list"   and xa :: "'a"   and l2 :: "'a list"   and \<sigma> :: "'a set" assumes "wsorted x"   and "set x = carrier"   and "x = l1 @ xa # l2"   and "I_minWeightBasis (\<sigma>, set (xa # l2))"   and "indep \<sigma>"   and "xa \<in> carrier"   and "indep (insert xa \<sigma>)" shows "I_minWeightBasis (insert xa \<sigma>, set l2)"
Rep_process: fixes x :: "'b process" shows "Rep_process x \<in> {p. is_process p}"
UnlimitedNaturalLiteralTE: fixes \<Gamma> :: "String.literal \<rightharpoonup>\<^sub>f 'a OCL_Types.type"   and c :: "unat"   and \<tau> :: "'a OCL_Types.type" assumes "\<Gamma> \<turnstile>\<^sub>E Literal (UnlimitedNaturalLiteral c) : \<tau>" obtains "\<tau> = UnlimitedNatural[1]"
replicate_length_filter: fixes x :: "'g"   and xs :: "'g list" shows "replicate (length (filter ((=) x) xs)) x = filter ((=) x) xs"
dens: shows "has_density M (stock_measure (PRODUCT INTEG INTEG)) \<delta>"
0: shows "S.arr (local.map (gf' \<odot> gf))"
noFault_c1: shows "\<Gamma>\<turnstile> \<langle>c1,Normal s\<rangle> \<Rightarrow>\<notin>Fault ` F"
set_coeffs_map_poly: fixes f :: "'a \<Rightarrow> 'b"   and p :: "'a poly" assumes "\<And>x. (f x = (0::'b)) = (x = (0::'a))" shows "set (coeffs (map_poly f p)) = f ` set (coeffs p)"
\<TT>\<yy>\<pp>\<ee>\<OO>\<ff>\<OO>\<pp>_\<AA>\<rr>\<ii>\<tt>\<yy>: fixes opubx :: "'opubx" shows "\<AA>\<rr>\<ii>\<tt>\<yy> (\<DD>\<ee>\<II>\<nn>\<ll> (\<BB>\<oo>\<xx> opubx)) = length (fst (\<TT>\<yy>\<pp>\<ee>\<OO>\<ff>\<OO>\<pp> opubx))"
Cons_1: fixes i :: "nat" assumes "x \<in> lvars t" shows "let idx = eq_idx_for_lvar_aux t x i in i \<le> idx \<and> idx < i + length t \<and> lhs (t ! (idx - i)) = x"
sums_mult_D: fixes c :: "'a"   and f :: "nat \<Rightarrow> 'a"   and a :: "'a" assumes "(\<lambda>n. c * f n) sums a"   and "c \<noteq> (0::'a)" shows "f sums (a / c)"
diff_Real: fixes X :: "nat \<Rightarrow> rat"   and Y :: "nat \<Rightarrow> rat" assumes "Real.cauchy X"   and "Real.cauchy Y" shows "Real.Real X - Real.Real Y = Real.Real (\<lambda>n. X n - Y n)"
translation_inverse_monoid_isomorphism: shows "monoid_isomorphism (\<lambda>a\<in>G. transformation.symmetric.inverse (a)\<^sub>R) G (\<cdot>) \<one> Translations (compose G) (identity G)"
set_of_lvivl_def: fixes ivl :: "real list \<times> real list" shows "set_of_lvivl ivl = set_of_ivl (map_prod eucl_of_list eucl_of_list ivl)"
red_lam_inv: fixes x :: "nat"   and e :: "exp"   and e' :: "exp"   and P :: "bool" assumes "ELam x e \<longrightarrow> e'" shows "P"
signed_take_bit_0: fixes a :: "'a" shows "signed_take_bit 0 a = - (a mod (2::'a))"
update_type: fixes t :: "('c \<times> 'd) bro"   and h :: "nat"   and x :: "'c"   and y :: "'d" assumes "t \<in> B h" shows "update x y t \<in> B h \<union> B (Suc h)"
u_def: shows "u = Tensor.mat_of_cols_list 2 (map (map complex_of_real) [[1 / sqrt 2, 1 / sqrt 2]])"
ord_2: fixes i :: "nat" shows "0 \<le> g i"
finsert_absorb: fixes a :: "'e"   and A :: "'e fset" assumes "a |\<in>| A" shows "finsert a A = A"
replicate_Suc: fixes n :: "nat"   and x :: "'a" shows "replicate (Suc n) x = x # replicate n x"
iT_infinite_2: fixes m :: "nat"   and r :: "nat" assumes "0 < m" shows "infinite [ r, mod m ]"
mcont_monoD: fixes luba :: "'a set \<Rightarrow> 'a"   and orda :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and lubb :: "'b set \<Rightarrow> 'b"   and ordb :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'b"   and x :: "'a"   and y :: "'a" assumes "mcont luba orda lubb ordb f"   and "orda x y" shows "ordb (f x) (f y)"
assms_2: shows "GatewayReq req dt a stop lose d ack i vc"
add_le_same_cancel2: fixes a :: "'a multiset"   and b :: "'a multiset" shows "(a + b \<subseteq># b) = (a \<subseteq># {#})"
induct: fixes P :: "'b list \<Rightarrow> bool"   and a0 :: "'b list" assumes "\<And>i j cs. P (j # cs) \<Longrightarrow> P (i # j # cs)"   and "P []"   and "\<And>v. P [v]" shows "P a0"
uminus_Sup: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and A :: "'a set" assumes "class.complete_boolean_algebra Inf Sup inf less_eq less sup bot top minus uminus" shows "uminus (Sup A) = Inf (uminus ` A)"
pp_increasing: fixes uminus :: "'c \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and x :: "'c" assumes "class.p_algebra uminus inf less_eq less sup bot top" shows "less_eq x (uminus (uminus x))"
assms_1: fixes i :: "'c" assumes "i \<in> Y" shows "f i \<in> carrier R"
closed_sequentially: fixes s :: "'c set"   and f :: "nat \<Rightarrow> 'c"   and l :: "'c" assumes "closed s"   and "\<forall>n. f n \<in> s"   and "f \<longlonglongrightarrow> l" shows "l \<in> s"
commute: fixes a :: "'g"   and b :: "'g" shows "a * b = b * a"
simps_2: fixes x :: "nat"   and t :: "tree"   and u :: "tree" shows "labels (Node x t u) = {x} \<union> labels t \<union> labels u"
not_AE_zero_E: fixes f :: "'c \<Rightarrow> real"   and M :: "'c measure" assumes "AE x in M. 0 \<le> f x"   and "\<not> (AE x in M. f x = 0)"   and "f \<in> borel_measurable M" shows "\<exists>A e. A \<in> sets M \<and> 0 < e \<and> 0 < emeasure M A \<and> (\<forall>x\<in>A. e \<le> f x)"
sum_list_simps_1: shows "sum_list [] = (0::'a)"
trans_nchotomy: fixes c :: "('test, 'atom) com"   and s :: "'state" shows "(\<exists>c' s'. (c, s) \<rightarrow>c (c', s')) \<or> (\<exists>s'. (c, s) \<rightarrow>t s')"
sup_absorb2: fixes sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "class.semilattice_sup (sup::'b \<Rightarrow> 'b \<Rightarrow> 'b) (less_eq::'b \<Rightarrow> 'b \<Rightarrow> bool) (less::'b \<Rightarrow> 'b \<Rightarrow> bool)"   and "(less_eq::'b \<Rightarrow> 'b \<Rightarrow> bool) (x::'b) (y::'b)" shows "(sup::'b \<Rightarrow> 'b \<Rightarrow> 'b) (x::'b) (y::'b) = y"
poly_gcd_iff: fixes p :: "'a poly"   and q :: "'a poly"   and x :: "'a" shows "(poly (gcd p q) x = (0::'a)) = (poly p x = (0::'a) \<and> poly q x = (0::'a))"
eucl_less_le_not_le: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and norm :: "'b \<Rightarrow> real"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and sgn :: "'b \<Rightarrow> 'b"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and inner :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and Basis :: "'b set"   and abs :: "'b \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "class.ordered_euclidean_space Inf Sup minus dist norm plus zero uminus scaleR sgn uniformity open inner Basis abs inf sup less_eq less" shows "less x y = (less_eq x y \<and> \<not> less_eq y x)"
shuffle_singleton: fixes x :: "'a" shows "shuffle [x] = return_pmf [x]"
singular_chain_subtopology: fixes p :: "nat"   and X :: "'c topology"   and S :: "'c set"   and c :: "((nat \<Rightarrow> real) \<Rightarrow> 'c) \<Rightarrow>\<^sub>0 int" shows "singular_chain p (subtopology X S) c = (singular_chain p X c \<and> (\<forall>f\<in>Poly_Mapping.keys c. f ` standard_simplex p \<subseteq> S))"
assms_4: fixes \<sigma> :: "'a \<Rightarrow> 'b"   and \<zeta> :: "'a \<Rightarrow> 'b"   and \<sigma>' :: "'a \<Rightarrow> 'b"   and \<zeta>' :: "'a \<Rightarrow> 'b"   and s :: "'c"   and a :: "'d"   and s' :: "'c" assumes "\<forall>j\<in>J. \<sigma> j = \<zeta> j"   and "\<forall>j\<in>J. \<sigma>' j = \<zeta>' j"   and "P ((\<sigma>, s), a, \<sigma>', s')" shows "P ((\<zeta>, s), a, \<zeta>', s')"
steps_imp_not_SN: fixes t :: "'c \<Rightarrow> 'd"   and f :: "'c \<Rightarrow> 'c"   and R :: "('d \<times> 'd) set" assumes "\<And>x. (t x, t (f x)) \<in> R" shows "\<not> SN R"
simps_3: fixes x41 :: "stmt"   and x42 :: "stmt"   and y41 :: "stmt"   and y42 :: "stmt" shows "(Seq x41 x42 = Seq y41 y42) = (x41 = y41 \<and> x42 = y42)"
v0_V: shows "v0 \<in> V\<^bsub>H_x\<^esub>"
frac_le: fixes y :: "'a"   and x :: "'a"   and w :: "'a"   and z :: "'a" assumes "(0::'a) \<le> y"   and "x \<le> y"   and "(0::'a) < w"   and "w \<le> z" shows "x / z \<le> y / w"
a_redu_ImpL_elim: fixes a :: "coname"   and M :: "trm"   and y :: "name"   and N :: "trm"   and z :: "name"   and R :: "trm" assumes "ImpL <a>.M y.N z \<longrightarrow>\<^sub>a R" shows "(\<exists>M'. R = ImpL <a>.M' y.N z \<and> M \<longrightarrow>\<^sub>a M') \<or> (\<exists>N'. R = ImpL <a>.M y.N' z \<and> N \<longrightarrow>\<^sub>a N')"
isLastAssertedCharacterization_1: fixes literal :: "Literal"   and literalList :: "Literal list"   and valuation :: "Literal list" assumes "isLastAssertedLiteral literal (oppositeLiteralList literalList) valuation" shows "opposite literal el literalList"
comp_Rel_components_2: fixes S :: "V"   and T :: "V" shows "(S \<circ>\<^sub>R\<^sub>e\<^sub>l T)\<lparr>ArrDom\<rparr> = T\<lparr>ArrDom\<rparr>"
chain_reparam_chain_def: fixes one_chain1 :: "('a \<times> (real \<Rightarrow> 'b)) set"   and one_chain2 :: "('a \<times> (real \<Rightarrow> 'b)) set" shows "chain_reparam_chain one_chain1 one_chain2 \<equiv> \<exists>f. bij f \<and> f ` one_chain1 = one_chain2 \<and> (\<forall>(k, \<gamma>)\<in>one_chain1. if k = fst (f (k, \<gamma>)) then reparam \<gamma> (snd (f (k, \<gamma>))) else reparam \<gamma> (reversepath (snd (f (k, \<gamma>)))))"
simps_7: fixes f1 :: "'d"   and f2 :: "nat \<Rightarrow> 'd \<Rightarrow> 'd"   and nat :: "nat" shows "rec_nat f1 f2 (Suc nat) = f2 nat (rec_nat f1 f2 nat)"
simps_1: fixes \<pi> :: "'b prm" shows "\<pi> \<bullet> PUnit = PUnit"
intros_3: fixes t :: "('a, 'b, 'c, 'd) Tree"   and f :: "'d"   and \<alpha> :: "'c" shows "(t, tAct f \<alpha> t) \<in> Tree_wf"
conuc_comp_closed: fixes f :: "'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b" assumes "conucleus f" shows "f (f x \<cdot> f y) = f x \<cdot> f y"
ht_hash_dom_map_of_disj_drop: fixes l :: "('a \<times> 'b) list list"   and i :: "nat" assumes "ht_hash l"   and "i < length l" shows "dom (map_of (l ! i)) \<inter> dom (map_of (concat (drop (Suc i) l))) = {}"
simps_1: fixes ys :: "'d list" shows "[] @ ys = ys"
le_fininf_range: fixes ind :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and w\<^sub>1 :: "'a list"   and w\<^sub>2 :: "'a stream" assumes "traces ind"   and "traces.le_fininf ind w\<^sub>1 w\<^sub>2" shows "set w\<^sub>1 \<subseteq> sset w\<^sub>2"
prod_algebraE_all: fixes A :: "('b \<Rightarrow> 'c) set"   and I :: "'b set"   and M :: "'b \<Rightarrow> 'c measure" assumes "A \<in> prod_algebra I M" obtains E :: "'b \<Rightarrow> 'c set" where "A = Pi\<^sub>E I E" and "E \<in> (\<Pi> i\<in>I. sets (M i))"
simps_1: fixes x11 :: "'c"   and x12 :: "'d"   and y11 :: "'c"   and y12 :: "'d" shows "(Add x11 x12 = Add y11 y12) = (x11 = y11 \<and> x12 = y12)"
UP_def: fixes x :: "'e" shows "UP x \<equiv> MkIde {x}"
zero_def: fixes x :: "'a"   and R :: "('a \<Rightarrow> 'b) set" shows "zero x R \<equiv> \<forall>u\<in>R. u x = (0::'b)"
coG_dtor_o_unfold: fixes s :: "'e \<Rightarrow> ('f, 'e coF) coG0" shows "dtor_coG \<circ> dtor_unfold_coG s = map_pre_coG id (dtor_unfold_coG s) \<circ> s"
count_mset_neg: fixes M :: "'a zmultiset"   and x :: "'a" shows "count (mset_neg M) x = nat (- zcount M x)"
empty_sb: fixes ts :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'b memref list \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list"   and \<S> :: "nat \<Rightarrow> bool option"   and i :: "nat"   and p :: "'b"   and "is" :: "instr list"   and xs :: "nat \<Rightarrow> nat option"   and sb :: "'b memref list"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option" assumes "initial\<^sub>s\<^sub>b ts \<S>"   and "i < length ts"   and "ts ! i = (p, is, xs, sb, \<D>, \<O>, \<R>)" shows "sb = []"
int_nat_p_pow_minus: fixes p :: "int"   and k :: "nat" assumes "padic_integers p" shows "int (nat (p ^ k - 1)) = p ^ k - 1"
concurrentI: fixes s1 :: "'a"   and s2 :: "'a" assumes "\<not> s1 \<prec> s2"   and "\<not> s2 \<prec> s1" shows "s1 \<parallel> s2"
assms_4: shows "\<forall>l. Discl \<subseteq> M l"
case_prod_unfold: shows "case_prod = (\<lambda>c p. c (fst p) (snd p))"
intros: fixes F :: "'a"   and main :: "'fun"   and fd :: "('b, 'c, 'fun, 'd, 'e) Inca.instr fundef"   and H :: "'l" assumes "Finca_get F main = Some fd"   and "arity fd = 0" shows "Sinca.load (Prog F H main) (State F H [Frame main 0 []])"
restrict_rel_prod2: fixes R :: "'d \<Rightarrow> 'f \<Rightarrow> bool"   and S :: "'e \<Rightarrow> 'g \<Rightarrow> bool"   and I1 :: "'e \<Rightarrow> bool"   and I2 :: "'g \<Rightarrow> bool" shows "rel_prod R (S \<upharpoonleft> I1 \<otimes> I2) = rel_prod R S \<upharpoonleft> pred_prod (\<lambda>_. True) I1 \<otimes> pred_prod (\<lambda>_. True) I2"
get_shadow_root_pure: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and get_shadow_root :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, (_) shadow_root_ptr option) prog"   and get_shadow_root_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and element_ptr :: "(_) element_ptr"   and h :: "(_) heap" assumes "l_get_shadow_root type_wf get_shadow_root get_shadow_root_locs" shows "pure (get_shadow_root element_ptr) h"
mds\<^sub>A_of_def: fixes mds\<^sub>C :: "Mode \<Rightarrow> var\<^sub>C set" shows "mds\<^sub>A_of mds\<^sub>C \<equiv> \<lambda>m. inv Eg2_var\<^sub>C_of_Eg1 ` (range Eg2_var\<^sub>C_of_Eg1 \<inter> mds\<^sub>C m)"
zero_neq_neg_numeral: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and n :: "num" assumes "class.ring_char_0 one times plus zero minus uminus" shows "zero \<noteq> uminus (numeral.numeral one plus n)"
gcd_dvd2: fixes a :: "'b"   and b :: "'b" shows "gcd a b dvd b"
ivl_disj_un_12: fixes l :: "'c"   and u :: "'c" assumes "l < u" shows "{l<..<u} \<union> {u..} = {l<..}"
fresh_permute_iff: fixes p :: "perm"   and a :: "atom"   and x :: "'e" shows "p \<bullet> a \<sharp> p \<bullet> x = a \<sharp> x"
deriv_bodies: shows "\<forall>p\<in>Procs. \<forall>Z. \<Gamma>,\<Theta>'\<turnstile>\<^bsub>/F \<^esub>(P p Z) the (\<Gamma> p) (Q p Z),(A p Z)"
that: fixes v :: "'a vec" assumes "v \<in> carrier_vec (dim_row A)"   and "x = v \<^sub>v* A" shows "thesis"
wf_mopup: fixes n :: "nat" shows "tm_wf0 (mopup n)"
isomorphic_to_identity_is_equivalence: fixes \<phi> :: "'a \<Rightarrow> 'a" assumes "natural_isomorphism (\<cdot>) (\<cdot>) (\<lambda>f. T (\<I>, f)) local.map \<phi>" shows "equivalence_functor (\<cdot>) (\<cdot>) (\<lambda>f. T (\<I>, f))"
wfI_min: fixes R :: "('a \<times> 'a) set" assumes "\<And>x Q. x \<in> Q \<Longrightarrow> \<exists>z\<in>Q. \<forall>y. (y, z) \<in> R \<longrightarrow> y \<notin> Q" shows "wf R"