hcju/isabelleps huggingface dataset:
tauInduct: fixes P :: "pi"   and P' :: "pi"   and F :: "'a \<Rightarrow> pi \<Rightarrow> pi \<Rightarrow> bool"   and C :: "'a" assumes "P \<longmapsto>\<^sub>l \<tau> \<prec>\<^sub>l P'"   and "\<And>P C. F C (\<tau>.P) P"   and "\<And>P P' c C. \<lbrakk>P \<longmapsto>\<^sub>l \<tau> \<prec>\<^sub>l P'; \<And>C. F C P P'\<rbrakk> \<Longrightarrow> F C ([c\<frown>c]P) P'"   and "\<And>P P' c d C. \<lbrakk>P \<longmapsto>\<^sub>l \<tau> \<prec>\<^sub>l P'; \<And>C. F C P P'; c \<noteq> d\<rbrakk> \<Longrightarrow> F C ([c\<noteq>d]P) P'"   and "\<And>P P' Q C. \<lbrakk>P \<longmapsto>\<^sub>l \<tau> \<prec>\<^sub>l P'; \<And>C. F C P P'\<rbrakk> \<Longrightarrow> F C (P \<oplus> Q) P'"   and "\<And>Q Q' P C. \<lbrakk>Q \<longmapsto>\<^sub>l \<tau> \<prec>\<^sub>l Q'; \<And>C. F C Q Q'\<rbrakk> \<Longrightarrow> F C (P \<oplus> Q) Q'"   and "\<And>P P' Q C. \<lbrakk>P \<longmapsto>\<^sub>l \<tau> \<prec>\<^sub>l P'; \<And>C. F C P P'\<rbrakk> \<Longrightarrow> F C (P \<parallel> Q) (P' \<parallel> Q)"   and "\<And>Q Q' P C. \<lbrakk>Q \<longmapsto>\<^sub>l \<tau> \<prec>\<^sub>l Q'; \<And>C. F C Q Q'\<rbrakk> \<Longrightarrow> F C (P \<parallel> Q) (P \<parallel> Q')"   and "\<And>P a x P' Q b Q' C. \<lbrakk>P \<longmapsto>\<^sub>l a<x> \<prec>\<^sub>l P'; Q \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l Q'; x \<sharp> P; x \<sharp> Q; x \<sharp> C\<rbrakk> \<Longrightarrow> F C (P \<parallel> Q) (P'[x::=b] \<parallel> Q')"   and "\<And>P a b P' Q x Q' C. \<lbrakk>P \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l P'; Q \<longmapsto>\<^sub>l a<x> \<prec>\<^sub>l Q'; x \<sharp> P; x \<sharp> Q; x \<sharp> C\<rbrakk> \<Longrightarrow> F C (P \<parallel> Q) (P' \<parallel> Q'[x::=b])"   and "\<And>P a x P' Q y Q' C. \<lbrakk>P \<longmapsto>\<^sub>l a<x> \<prec>\<^sub>l P'; Q \<longmapsto>\<^sub>l a<\<nu>y> \<prec>\<^sub>l Q'; x \<sharp> P; x \<sharp> Q; x \<sharp> C; y \<sharp> P; y \<sharp> Q; y \<sharp> C; x \<noteq> y\<rbrakk> \<Longrightarrow> F C (P \<parallel> Q) (<\<nu>y>(P'[x::=y] \<parallel> Q'))"   and "\<And>P a y P' Q x Q' C. \<lbrakk>P \<longmapsto>\<^sub>l a<\<nu>y> \<prec>\<^sub>l P'; Q \<longmapsto>\<^sub>l a<x> \<prec>\<^sub>l Q'; x \<sharp> P; x \<sharp> Q; x \<sharp> C; y \<sharp> P; y \<sharp> Q; y \<sharp> C; x \<noteq> y\<rbrakk> \<Longrightarrow> F C (P \<parallel> Q) (<\<nu>y>(P' \<parallel> Q'[x::=y]))"   and "\<And>P P' x C. \<lbrakk>P \<longmapsto>\<^sub>l \<tau> \<prec>\<^sub>l P'; x \<sharp> C; \<And>C. F C P P'\<rbrakk> \<Longrightarrow> F C (<\<nu>x>P) (<\<nu>x>P')"   and "\<And>P P' C. \<lbrakk>P \<parallel> !P \<longmapsto>\<^sub>l \<tau> \<prec>\<^sub>l P'; \<And>C. F C (P \<parallel> !P) P'\<rbrakk> \<Longrightarrow> F C (!P) P'" shows "F C P P'"
preserves_arr: fixes f :: "'a \<times> 'a \<times> 'a" assumes "B.VVV.arr f" shows "B.arr (B.HoHV f)"
sorted_wrt_rev: fixes P :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and xs :: "'c list" shows "sorted_wrt P (rev xs) = sorted_wrt (\<lambda>x y. P y x) xs"
polyhedron_affine_hull: fixes S :: "'b set" shows "polyhedron (affine hull S)"
W5b: fixes a :: "'a"   and b :: "'a" shows "(-r a l\<rightarrow> -r b) l\<rightarrow> b r\<rightarrow> a = (1::'a)"
map_run_refine: shows "(map_run_impl, (\<circ>)) \<in> (Id \<rightarrow> Id) \<rightarrow> ce_rel \<rightarrow> mce_rel"
card_UNIV_bool_def: shows "card_UNIV_class.card_UNIV = Phantom(bool) 2"
assms_3: fixes x1 :: "'a acom"   and x2 :: "'a acom" assumes "P x1"   and "P x2" shows "P (x1 \<triangle> x2)"
simps: fixes a1 :: "'b LDCFG_node"   and a2 :: "('b LDCFG_node \<times> 'c edge_kind \<times> 'b LDCFG_node) list"   and a3 :: "'b LDCFG_node" shows "CFGExit_wf.path a1 a2 a3 = ((\<exists>n. a1 = n \<and> a2 = [] \<and> a3 = n \<and> CFGExit_wf.valid_node n) \<or> (\<exists>n'' as n' a n. a1 = n \<and> a2 = a # as \<and> a3 = n' \<and> CFGExit_wf.path n'' as n' \<and> lift_valid_edge valid_edge sourcenode targetnode kind Entry Exit a \<and> src a = n \<and> trg a = n''))"
simps_14: fixes p :: "fm"   and q :: "fm" shows "bound0 (Imp p q) = (bound0 p \<and> bound0 q)"
uniform_alt_def_all: shows "\<forall>bl\<in>#\<B>\<^sup>+. int (card bl) = \<k>"
le_fract: fixes b :: "'b"   and d :: "'b"   and a :: "'b"   and c :: "'b" assumes "b \<noteq> (0::'b)"   and "d \<noteq> (0::'b)" shows "(Fract a b \<le> Fract c d) = (a * d * (b * d) \<le> c * b * (b * d))"
dropWhile_eq_Nil_conv: fixes P :: "'b \<Rightarrow> bool"   and xs :: "'b list" shows "(dropWhile P xs = []) = (\<forall>x\<in>set xs. P x)"
induct: fixes P :: "int \<Rightarrow> int tree \<Rightarrow> bool"   and a0 :: "int"   and a1 :: "int tree" assumes "\<And>i. P i \<langle>\<rangle>"   and "\<And>i l k r. \<lbrakk>P i l; P (k + 1) r\<rbrakk> \<Longrightarrow> P i \<langle>l, k, r\<rangle>" shows "P a0 a1"
getEnv_updEnv1: fixes ys :: "'varSort"   and xs :: "'varSort"   and y :: "'var"   and x :: "'var"   and rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qTerm set option"   and X :: "('a, 'b, 'varSort, 'var, 'c) qTerm set" assumes "ys \<noteq> xs \<or> y \<noteq> x" shows "(rho [x \<leftarrow> X]_xs) ys y = rho ys y"
bin_rep_aux_coeff: fixes m :: "nat"   and n :: "nat"   and i :: "nat" assumes "m < 2 ^ n"   and "i \<le> n"   and "0 \<le> m" shows "bin_rep_aux n m ! i = 0 \<or> bin_rep_aux n m ! i = 1"
nth_subdegree_mult: fixes f :: "'a fps"   and g :: "'a fps" shows "fps_nth (f * g) (subdegree f + subdegree g) = fps_nth f (subdegree f) * fps_nth g (subdegree g)"
summable_Cauchy: fixes f :: "nat \<Rightarrow> 'd" shows "summable f = (\<forall>e>0. \<exists>N. \<forall>m\<ge>N. \<forall>n. norm (sum f {m..<n}) < e)"
assms_4: shows "trg\<^sub>C f = trg\<^sub>C f'"
simps_3: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and E :: "char list \<Rightarrow> ty option"   and h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and v :: "expr"   and va :: "expr list" shows "(P,E,h \<turnstile> v # va [:]\<^bsub>NT\<^esub> []) = False"
less_fun_def: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" shows "(f < g) = strict (\<le>) f g"
vector_fraction_eq_iff: fixes u :: "real"   and v :: "real"   and a :: "'b"   and x :: "'b" shows "((u / v) *\<^sub>R a = x) = (if v = 0 then x = (0::'b) else u *\<^sub>R a = v *\<^sub>R x)"
no_matching_Goto_append1: fixes \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and rs1 :: "'c rule list"   and rs2 :: "'c rule list" assumes "no_matching_Goto \<gamma> p (rs1 @ rs2)" shows "no_matching_Goto \<gamma> p rs1"
"case": fixes f :: "'y \<Rightarrow> 'z \<Rightarrow> 'x"   and x1 :: "'y"   and x2 :: "'z" shows "(case (x1, x2) of (x, xa) \<Rightarrow> f x xa) = f x1 x2"
intro: assumes "\<And>y x. PP y x \<Longrightarrow> \<exists>z. P z x \<and> \<not> O z y" shows "MM_axioms"
abs_eq: shows "ounit_circle = abs_ocircline unit_circle_clmat"
and_one_eq: fixes a :: "'a" shows "a AND (1::'a) = a mod (2::'a)"
jumpF_eq: fixes y :: "real"   and S :: "real set" assumes "p \<noteq> 0" shows "jumpF (\<lambda>x. poly q x / poly p x) (at y within S) = jumpF g (at y within S)"
that: fixes V :: "'a set"   and W :: "'a set" assumes "openin (top_of_set U) V"   and "closedin (top_of_set U) W"   and "S \<subseteq> V"   and "V \<subseteq> W"   and "S retract_of W" shows "thesis"
Bia: shows "B $ i $ a \<noteq> (0::'a)"
size_14: fixes x7 :: "'b ptrm" shows "size (PSnd x7) = size x7 + Suc 0"
left_impl_ded: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and imp_l :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and one :: "'b"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "class.left_complemented_monoid_algebra times inf imp_l less_eq less one" shows "imp_l (times a b) c = imp_l a (imp_l b c)"
P1: shows "Bet A B D \<and> Cong B D B C"
max_less: fixes b :: "'a list"   and B :: "'a list set"   and b' :: "'a list" assumes "b \<in> B"   and "finite B"   and "length b' < length b" shows "length b' < length (MAX B)"
main_connect: assumes "length (map of_int_hom.vec_hom fs') \<le> n" shows "gram_schmidt n (map of_int_hom.vec_hom fs') = map gs2.gso [0..<length (map of_int_hom.vec_hom fs')]"
residue_mult_group_kernel_partition_3: fixes m :: "nat"   and n :: "nat" assumes "1 < m"   and "0 < n"   and "m dvd n" shows "card (kernel (residue_mult_group n) (residue_mult_group m) (\<lambda>k. k mod m)) = totient n div totient m"
is_valid_operator_sas_plus_then_4: fixes \<Psi> :: "('c, 'd) sas_plus_problem"   and op :: "('c, 'd) sas_plus_operator" assumes "is_valid_operator_sas_plus \<Psi> op" shows "\<forall>(v, a)\<in>set (effect_of op). \<R>\<^sub>+ \<Psi> v \<noteq> {} \<and> a \<in> \<R>\<^sub>+ \<Psi> v"
minus_cancel_vec: fixes v :: "'c vec"   and n :: "nat" assumes "v \<in> carrier_vec n" shows "v - v = 0\<^sub>v n"
bounded_clinear_scaleC_const: fixes g :: "'d \<Rightarrow> complex"   and x :: "'e" assumes "bounded_clinear g" shows "bounded_clinear (\<lambda>xa. g xa *\<^sub>C x)"
fact_0: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a" assumes "class.semiring_char_0 one times plus zero" shows "semiring_char_0.fact one plus zero 0 = one"
suminf: fixes f :: "'d \<Rightarrow> 'e"   and X :: "nat \<Rightarrow> 'd" assumes "bounded_linear f"   and "summable X" shows "f (\<Sum>n. X n) = (\<Sum>n. f (X n))"
S6EnvUnch: fixes p :: "PrIds"   and rmhist :: "state \<Rightarrow> PrIds \<Rightarrow> histState" shows "\<turnstile> [ENext p]_(e p) \<and> $S6 rmhist p \<longrightarrow> unchanged e p"
set_6: fixes x61 :: "'b formula"   and x62 :: "'b formula" shows "atoms (x61 \<^bold>\<rightarrow> x62) = atoms x61 \<union> atoms x62"
add_minus_eq: fixes x :: "'a"   and y :: "'a" assumes "x \<in> E"   and "y \<in> E"   and "x - y = (0::'a)" shows "x = y"
uadmit_fml_adjoint: fixes \<sigma> :: "('sf, 'sc, 'sz) subst"   and \<phi> :: "('sf, 'sc, 'sz) formula"   and U :: "('sz + 'sz) set"   and \<nu> :: "(real, 'sz) vec \<times> (real, 'sz) vec"   and \<omega> :: "(real, 'sz) vec \<times> (real, 'sz) vec"   and I :: "('sf, 'sc, 'sz) interp" assumes "FUadmit \<sigma> \<phi> U"   and "Vagree \<nu> \<omega> (- U)"   and "fsafe \<phi>"   and "ssafe \<sigma>"   and "is_interp I" shows "fml_sem (local.adjoint I \<sigma> \<nu>) \<phi> = fml_sem (local.adjoint I \<sigma> \<omega>) \<phi>"
return_bind: fixes x :: "'a option"   and f :: "'a option \<Rightarrow> 'm" shows "bind (return x) f = f x"
IF_3: shows "v \<notin> ifex_var_set (IF v1 t e)"
sup_le: fixes a :: "'a"   and b :: "'a" shows "a \<le> a \<squnion> b"
simps_3: fixes v :: "ty"   and va :: "ty list" shows "conformable (v # va) [] = False"
list_all2_elem_filter: fixes P :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and xs :: "'b list"   and us :: "'c list"   and x :: "'b" assumes "list_all2 P xs us"   and "x \<in> set xs" shows "1 \<le> length (filter (P x) us)"
upd_def: fixes i :: "nat"   and x :: "'b"   and a :: "'b array" shows "Array.upd i x a = Heap_Monad.guard (\<lambda>h. i < Array.length h a) (\<lambda>h. (a, Array.update a i x h))"
object_ptr_kinds_M_eq3_h': shows "object_ptr_kinds h3 = object_ptr_kinds h'"
S: fixes x :: "'a" assumes "x \<in> S" shows "B \<le> f x \<bullet> rot n"
fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s_Cons: fixes s :: "('d, 'c) Term.term"   and t :: "('e, 'c) Term.term"   and F :: "(('d, 'c) Term.term \<times> ('e, 'c) Term.term) list" shows "fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s ((s, t) # F) = fv s \<union> fv t \<union> fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s F"
prems: shows "h \<turnstile> get_root_node_si c \<rightarrow>\<^sub>r cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r child"
hyps: fixes fs :: "int vec list" assumes "LLL_bound_invariant False True i fs"   and "\<mu>_small_row i fs j"   and "\<mu>_bound_row_inner fs i j"   and "basis_reduction_add_rows_loop i fs j = fs'"   and "i < m"   and "j \<le> i" shows "LLL_bound_invariant True False i fs'"
int_shiftr_ge_0: fixes i :: "int"   and n :: "nat" shows "(0 \<le> i >> n) = (0 \<le> i)"
divisor_def: fixes n :: "nat"   and m :: "nat" shows "n divisor m \<equiv> 1 \<le> n \<and> n \<le> m \<and> n dvd m"
log_divide_eq: fixes A :: "real"   and B :: "real" shows "log b (A / B) = (if 0 < A * B then log b \<bar>A\<bar> - log b \<bar>B\<bar> else log b 0)"
Mid_cases: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Midpoint Bet Cong A B C \<or> Tarski_neutral_dimensionless.Midpoint Bet Cong A C B" shows "Tarski_neutral_dimensionless.Midpoint Bet Cong A B C"
assms_2: shows "tfr\<^sub>s\<^sub>s\<^sub>t (unlabel A)"
D: shows "\<forall>x. \<bar>pbernpoly (2 * N + 1) x\<bar> \<le> D"
in_own_span: fixes S :: "'c set" assumes "S \<subseteq> carrier V" shows "S \<subseteq> local.module.span S"
induct: fixes P :: "'b tree \<Rightarrow> 'b tree \<Rightarrow> bool"   and a0 :: "'b tree"   and a1 :: "'b tree" assumes "P \<langle>\<rangle> \<langle>\<rangle>"   and "\<And>l2 a2 r2. P r2 \<langle>\<rangle> \<Longrightarrow> P \<langle>\<rangle> \<langle>l2, a2, r2\<rangle>"   and "\<And>l1 a1 r1. P r1 \<langle>\<rangle> \<Longrightarrow> P \<langle>l1, a1, r1\<rangle> \<langle>\<rangle>"   and "\<And>l1 a1 r1 l2 a2 r2. \<lbrakk>a1 \<le> a2 \<Longrightarrow> P \<langle>l2, a2, r2\<rangle> r1; \<not> a1 \<le> a2 \<Longrightarrow> P \<langle>l1, a1, r1\<rangle> r2\<rbrakk> \<Longrightarrow> P \<langle>l1, a1, r1\<rangle> \<langle>l2, a2, r2\<rangle>" shows "P a0 a1"
subst_def: fixes xs :: "'varSort"   and X :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and x :: "'var" shows "subst xs X x \<equiv> psubst (idEnv [x \<leftarrow> X]_xs)"
equiv'_rotate_1: fixes a :: "'a"   and b :: "'b" assumes "A_B.equiv' a b" shows "B_A.equiv' b a"
rep_eq: fixes x :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'b"   and xa :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'b" shows "(*\<^sub>V) (x - xa) = (\<lambda>xb. x *\<^sub>V xb - xa *\<^sub>V xb)"
tendsto_uminus_ereal: fixes f :: "'b \<Rightarrow> ereal"   and x :: "ereal"   and F :: "'b filter" assumes "(f \<longlongrightarrow> x) F" shows "((\<lambda>x. - f x) \<longlongrightarrow> - x) F"
simps_7: fixes v :: "'b" shows "(Fin v \<le> -\<infinity>) = False"
resOutput: fixes x :: "name"   and a :: "name"   and b :: "name"   and P :: "pi" assumes "x \<noteq> a"   and "x \<noteq> b" shows "<\<nu>x>a{b}.P \<sim> a{b}.(<\<nu>x>P)"
distinct_1: fixes x1 :: "'b"   and x2 :: "'b match_expr" shows "Match x1 \<noteq> MatchNot x2"
distinct_83: fixes x5 :: "load_store_type"   and x9 :: "ctrl_type" shows "load_store_type x5 \<noteq> ctrl_type x9"
divide_divide_eq_left': fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "a / b / c = a / (c * b)"
subadditive_imp_eventually_subadditive: fixes u :: "nat \<Rightarrow> real" assumes "subadditive u" shows "eventually_subadditive u 0"
dmu_ii: shows "dmu_ij_state state i (i - 1) = d\<mu> fs i (i - 1)"
lookup_cache: fixes n :: "'node"   and g :: "'g"   and f :: "'node \<Rightarrow> 'b" assumes "n \<in> set (\<alpha>n g)" shows "Mapping.lookup (cache g f) n = Some (f n)"
order_iff_strict: fixes a :: "'c DBMEntry"   and b :: "'c DBMEntry" shows "(a \<preceq> b) = (a \<prec> b \<or> a = b)"
prems_2: shows "\<forall>c. c \<in> set (clause # Wl') \<longrightarrow> 0 \<le> c \<and> c < length (getF state)"
connectedI_const: fixes S :: "'b set" assumes "\<And>P. continuous_on S P \<Longrightarrow> \<exists>c. \<forall>s\<in>S. P s = c" shows "connected S"
n_rw2: shows "nrows_iarray (IArray (map (\<lambda>x. IArray (map (\<lambda>i. A $ mod_type_class.from_nat x $ mod_type_class.from_nat i) [0..<CARD('b)])) [0..<CARD('c)])) = CARD('c)"
list_eq_iff_nth_eq: fixes xs :: "'g list"   and ys :: "'g list" shows "(xs = ys) = (length xs = length ys \<and> (\<forall>i<length xs. xs ! i = ys ! i))"
assms: shows "zs \<in> set_of_appr (XS @ YS)"
simps_5: fixes channel :: "nat \<Rightarrow> ('d \<times> 'd) option"   and trans :: "'d \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> bool"   and send :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and recv :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and c :: "('d, 'e, 'f) configuration"   and i :: "nat"   and p :: "'d"   and q :: "'d"   and s :: "'e"   and s' :: "'e"   and msg :: "'f"   and c' :: "('d, 'e, 'f) configuration" assumes "distributed_system channel" shows "distributed_system.next channel trans send recv c (Recv i p q s s' msg) c' = (distributed_system.can_occur channel trans send recv (Recv i p q s s' msg) c \<and> (states c p = s \<and> states c' p = s') \<and> (\<forall>r. r \<noteq> p \<longrightarrow> states c' r = states c r) \<and> msgs c i = Msg msg # msgs c' i \<and> (\<forall>i'. i \<noteq> i' \<longrightarrow> msgs c' i' = msgs c i') \<and> (\<forall>r. ps c r = ps c' r) \<and> (\<forall>i'. i' \<noteq> i \<longrightarrow> cs c i' = cs c' i') \<and> (if snd (cs c i) = Recording then cs c' i = (fst (cs c i) @ [msg], Recording) else cs c i = cs c' i))"
homotopy_equivalent_space_sym: fixes X :: "'b topology"   and Y :: "'c topology" shows "(X homotopy_equivalent_space Y) = (Y homotopy_equivalent_space X)"
split_sel: fixes P :: "'d \<Rightarrow> bool"   and f :: "('e \<Rightarrow> 'f) \<Rightarrow> 'd"   and stateT :: "('e, 'f) stateT" shows "P (case stateT of StateT x \<Rightarrow> f x) = (stateT = StateT (run_state stateT) \<longrightarrow> P (f (run_state stateT)))"
abs_eq: fixes xa :: "mregex list"   and x :: "mregex \<Rightarrow> 'b option list set" shows "mrtabulate xa x = Mapping (map_of (List.map_filter (\<lambda>k. let fk = x k in if fk = empty_table then None else Some (k, fk)) xa))"
dvd_smult_cancel: fixes p :: "'b poly"   and a :: "'b"   and q :: "'b poly" assumes "p dvd smult a q"   and "a \<noteq> (0::'b)" shows "p dvd q"
bprules_1: shows "disjoint (P\<^sub>1 \<union> wrap B\<^sub>1 \<union> P\<^sub>2 \<union> wrap B\<^sub>2 \<union> {{v} |v. v \<in> V})"
fset_of_list_map: fixes f :: "'c \<Rightarrow> 'b"   and xs :: "'c list" shows "fset_of_list (map f xs) = f |`| fset_of_list xs"
less_irrefl: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a" assumes "class.preorder less_eq less" shows "\<not> less x x"
"2_4": shows "\<sigma>\<^bsup>k\<^esup> \<restriction> local.reads (\<pi> k) \<noteq> \<sigma>'\<^bsup>k'\<^esup> \<restriction> local.reads (\<pi> k)"
map_add_def: fixes m1 :: "'d \<Rightarrow> 'e option"   and m2 :: "'d \<Rightarrow> 'e option" shows "m1 ++ m2 = (\<lambda>x. case m2 x of None \<Rightarrow> m1 x | Some y \<Rightarrow> Some y)"
1: shows "is_rendered_commutative_by (\<chi> J.AA) (\<chi> J.BB)"
simps: fixes a1 :: "('a, 'b) Proc"   and a2 :: "('a, 'b) Proc"   and Enc :: "'a \<Rightarrow> 'b"   and TRel :: "('b \<times> 'b) set" shows "((a1, a2) \<in> encoding.indRelRTPO Enc TRel) = ((\<exists>S. a1 = SourceTerm S \<and> a2 = TargetTerm (Enc S)) \<or> (\<exists>S. a1 = SourceTerm S \<and> a2 = SourceTerm S) \<or> (\<exists>T1 T2. a1 = TargetTerm T1 \<and> a2 = TargetTerm T2 \<and> (T1, T2) \<in> TRel) \<or> (\<exists>P Q R. a1 = P \<and> a2 = R \<and> (P, Q) \<in> encoding.indRelRTPO Enc TRel \<and> (Q, R) \<in> encoding.indRelRTPO Enc TRel))"
Inf_uexpr_def: shows "Inf \<equiv> map_fun ((`) Rep_uexpr) mk\<^sub>e (\<lambda>PS A. \<Squnion>P\<in>PS. P A)"
C: fixes x :: "hf"   and y :: "hf"   and z :: "hf" assumes "C x y z" shows "C' x y z"
other: fixes j :: "'a" assumes "j \<notin> I" shows "P (\<sigma> j) (\<sigma>' j)"
less_finite_3_def: fixes x :: "Enum.finite_3"   and y :: "Enum.finite_3" shows "(x < y) = (case x of a\<^sub>1 \<Rightarrow> y \<noteq> a\<^sub>1 | a\<^sub>2 \<Rightarrow> y = a\<^sub>3 | a\<^sub>3 \<Rightarrow> False)"
lsu_coclop: shows "coclop \<nu>\<^sup>\<natural>"
qtable_join: fixes n :: "nat"   and A :: "nat set"   and P :: "'a option list \<Rightarrow> bool"   and Q1 :: "'a option list \<Rightarrow> bool"   and X :: "'a option list set"   and B :: "nat set"   and Q2 :: "'a option list \<Rightarrow> bool"   and Y :: "'a option list set"   and b :: "bool"   and C :: "nat set"   and Q :: "'a option list \<Rightarrow> bool" assumes "qtable n A P Q1 X"   and "qtable n B P Q2 Y"   and "\<not> b \<Longrightarrow> B \<subseteq> A"   and "C = A \<union> B"   and "\<And>x. \<lbrakk>wf_tuple n C x; P x\<rbrakk> \<Longrightarrow> P (restrict A x) \<and> P (restrict B x)"   and "\<And>x. \<lbrakk>b; wf_tuple n C x; P x\<rbrakk> \<Longrightarrow> Q x = (Q1 (restrict A x) \<and> Q2 (restrict B x))"   and "\<And>x. \<lbrakk>\<not> b; wf_tuple n C x; P x\<rbrakk> \<Longrightarrow> Q x = (Q1 (restrict A x) \<and> \<not> Q2 (restrict B x))" shows "qtable n C P Q (join X b Y)"
x2: shows "x\<^sub>2 ^ 3 = - (a * x\<^sub>2 + b)"
valof_skip1: fixes e' :: "exp"   and s :: "loc \<Rightarrow> nat"   and v :: "nat"   and s' :: "loc \<Rightarrow> nat"   and e :: "exp" assumes "(e', s) -|-> (v, s')"   and "e' = VALOF SKIP RESULTIS e" shows "(e, s) -|-> (v, s')"
derf: fixes x :: "'a"   and n :: "nat" assumes "x \<in> S" shows "(f n has_derivative f' n x) (at x within S)"
subdegree_inverse: fixes f :: "'a fps" shows "subdegree (inverse f) = 0"
accept\<^sub>G\<^sub>R_def: fixes R :: "('d \<Rightarrow> 'e \<Rightarrow> 'd) \<times> 'd \<times> (('d \<times> 'e \<times> 'd) set \<times> ('d \<times> 'e \<times> 'd) set set) set"   and w :: "nat \<Rightarrow> 'e" shows "accept\<^sub>G\<^sub>R R w \<equiv> \<exists>(Fin, Inf)\<in>snd (snd R). accepting_pair\<^sub>G\<^sub>R (fst R) (fst (snd R)) (Fin, Inf) w"
domintros_3: fixes ks :: "'a list"   and v :: "(real, 'a) vec"   and vb :: "(real, 'a) vec"   and vc :: "(real, 'a) vec list" assumes "\<And>aa ba xa ya ab ac bb xc ad bc. \<lbrakk>(aa, ba) = widest_spread ks (v # vb # vc); (xa, ya) = widest_spread ks (v # vb # vc); (ab, ac, bb) = partition_by_median xa (v # vb # vc); (xc, ad, bc) = partition_by_median xa (v # vb # vc)\<rbrakk> \<Longrightarrow> build_dom (ks, xc)"   and "\<And>aa ba xa ya ab ac bb xc ad bc. \<lbrakk>(aa, ba) = widest_spread ks (v # vb # vc); (xa, ya) = widest_spread ks (v # vb # vc); (ab, ac, bb) = partition_by_median xa (v # vb # vc); (xc, ad, bc) = partition_by_median xa (v # vb # vc)\<rbrakk> \<Longrightarrow> build_dom (ks, bc)" shows "build_dom (ks, v # vb # vc)"
finprod_0': fixes f :: "nat \<Rightarrow> 'a"   and n :: "nat" assumes "f \<in> {..n} \<rightarrow> carrier G" shows "f 0 \<otimes> finprod G f {Suc 0..n} = finprod G f {..n}"
Step1_4_1: fixes p :: "PrIds"   and rmhist :: "state \<Rightarrow> PrIds \<Rightarrow> histState" shows "\<turnstile> ENext p \<and> $S1 rmhist p \<and> S2 rmhist p$ \<and> unchanged (c p, r p, m p) \<longrightarrow> unchanged (rtrner memCh!p, resbar rmhist!p)"
SUP_ereal_minus_right: fixes I :: "'a set"   and c :: "ereal"   and f :: "'a \<Rightarrow> ereal" assumes "I \<noteq> {}"   and "c \<noteq> - \<infinity>" shows "(SUP i\<in>I. c - f i) = c - Inf (f ` I)"
bind_returns_heap_E2: fixes h :: "'a"   and f :: "('a, 'b, 'd) prog"   and g :: "'d \<Rightarrow> ('a, 'b, 'c) prog"   and h' :: "'a" assumes "h \<turnstile> f \<bind> g \<rightarrow>\<^sub>h h'"   and "pure f h" obtains x :: "'d" where "h \<turnstile> f \<rightarrow>\<^sub>r x" and "h \<turnstile> g x \<rightarrow>\<^sub>h h'"
bounded_subset_box_symmetric: fixes S :: "'a set" assumes "bounded S" obtains a :: "'a" where "S \<subseteq> box (- a) a"
comp_eq_dest_lhs: fixes a :: "'h \<Rightarrow> 'g"   and b :: "'f \<Rightarrow> 'h"   and c :: "'f \<Rightarrow> 'g"   and v :: "'f" assumes "a \<circ> b = c" shows "a (b v) = c v"
det_upperdiagonal: fixes A :: "('b, 'a) sq_matrix" assumes "\<And>i j. j < i \<Longrightarrow> to_fun A i j = (0::'b)" shows "Square_Matrix.det A = (\<Prod>i\<in>UNIV. to_fun A i i)"