hcju/isabelleps huggingface dataset:
map_def: fixes A :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and B :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and g :: "'e"   and f :: "'d" assumes "constant_transformation A B g" shows "constant_transformation.map A B g f \<equiv> if partial_magma.arr A f then g else partial_magma.null B"
is_expansion_map_aux: fixes xs :: "('a \<times> real) msllist"   and g :: "real \<Rightarrow> real"   and b :: "real \<Rightarrow> real"   and basis :: "(real \<Rightarrow> real) list"   and f :: "'a \<Rightarrow> 'b"   and basis' :: "(real \<Rightarrow> real) list" assumes "is_expansion_aux xs g (b # basis)"   and "\<And>C. is_expansion C basis \<Longrightarrow> is_expansion (f C) basis'"   and "length basis' = expansion_level TYPE('b)"   and "\<And>C. eval (f C) = eval C" shows "is_expansion_aux (map_ms_aux f xs) g (b # basis')"
inf2D1: fixes A :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and B :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'c"   and y :: "'d" assumes "(A \<sqinter> B) x y" shows "A x y"
LEX_def: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and f :: "nat \<Rightarrow> 'a"   and g :: "nat \<Rightarrow> 'a" shows "LEX P f g = (\<exists>i. P (f i) (g i) \<and> (\<forall>j<i. f j = g j))"
inject: fixes x :: "('c \<times> ('c, 'd) rose_tree\<^sub>m list, 'd \<times> 'd rose_tree\<^sub>h list) blindable\<^sub>m"   and ya :: "('c \<times> ('c, 'd) rose_tree\<^sub>m list, 'd \<times> 'd rose_tree\<^sub>h list) blindable\<^sub>m" shows "(Tree\<^sub>m x = Tree\<^sub>m ya) = (x = ya)"
residue_add_closed': fixes k :: "nat"   and x :: "int"   and y :: "int" shows "x \<oplus>\<^bsub>residue_ring (p ^ k)\<^esub> y \<in> {0..<p ^ k}"
Compl_insert: fixes x :: "'b"   and A :: "'b set" shows "- (x \<triangleright> A) = - A - {x}"
prefix_length_prefix: fixes ps :: "'b list"   and xs :: "'b list"   and qs :: "'b list" assumes "prefix ps xs"   and "prefix qs xs"   and "length ps \<le> length qs" shows "prefix ps qs"
bisimilar_equivp: shows "equivp (\<sim>\<cdot>)"
pid_gc_state_eq: fixes pgc :: "('d, 'e, 'b) pid_global_config" shows "pid_global_config.state pgc = global_config.state (pidgc_\<alpha> pgc)"
homeomorphism_cong: fixes X :: "'c set"   and Y :: "'d set"   and f :: "'c \<Rightarrow> 'd"   and g :: "'d \<Rightarrow> 'c"   and X' :: "'c set"   and Y' :: "'d set"   and f' :: "'c \<Rightarrow> 'd"   and g' :: "'d \<Rightarrow> 'c" assumes "homeomorphism X Y f g"   and "X' = X"   and "Y' = Y"   and "\<And>x. x \<in> X \<Longrightarrow> f' x = f x"   and "\<And>y. y \<in> Y \<Longrightarrow> g' y = g y" shows "homeomorphism X' Y' f' g'"
0: shows "\<forall>i<length (annos C2). annos C1 ! i \<le> annos C2 ! i"
connectedin_subset_separated_union: fixes X :: "'c topology"   and C :: "'c set"   and S :: "'c set"   and T :: "'c set" assumes "connectedin X C"   and "separatedin X S T"   and "C \<subseteq> S \<union> T" shows "C \<subseteq> S \<or> C \<subseteq> T"
br_run_sim: fixes \<alpha> :: "'b \<Rightarrow> 'a"   and I :: "'b \<Rightarrow> bool"   and r :: "nat \<Rightarrow> 'b" assumes "R\<inverse> = br \<alpha> I"   and "b.is_run r" shows "a.is_run (\<alpha> \<circ> r)"
lin_data_in': shows "linearize data_in' = linearize data_in @ [(nt, X)]"
lower_minus: fixes p :: "'a \<Rightarrow>\<^sub>0 'c"   and q :: "'a \<Rightarrow>\<^sub>0 'c"   and v :: "'a" shows "punit.lower (p - q) v = punit.lower p v - punit.lower q v"
Abs_ainj_all: fixes X :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and X' :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and xs :: "'varSort"   and x :: "'var"   and xs' :: "'varSort"   and x' :: "'var" assumes "good X"   and "good X'" shows "(Abs xs x X = Abs xs' x' X') = (xs = xs' \<and> (\<forall>y. (y = x \<or> fresh xs y X) \<and> (y = x' \<or> fresh xs y X') \<longrightarrow> X #[y \<and> x]_xs = X' #[y \<and> x']_xs))"
int_cong: fixes T :: "'fsym trm"   and \<xi>1 :: "Preliminaries.var \<Rightarrow> 'univ"   and \<xi>2 :: "Preliminaries.var \<Rightarrow> 'univ" assumes "\<And>x. x \<in> vars T \<Longrightarrow> \<xi>1 x = \<xi>2 x" shows "int \<xi>1 T = int \<xi>2 T"
assms: shows "locally_included R1 R2 (A + B) (C + D)"
twopow_le_central_binomial: fixes n :: "nat" shows "2 ^ n \<le> 2 * n choose n"
assms_4: shows "Prj\<^sub>0\<^sub>1 \<cdot> h = Prj\<^sub>0\<^sub>1 \<cdot> h'"
qring_ring: fixes R :: "('c, 'd) Ring_scheme"   and I :: "'c set" assumes "Ring R"   and "ideal R I" shows "Ring (R /\<^sub>r I)"
in_join_iff: fixes z :: "'a option list" shows "(z \<in> mmulti_join n A_pos A_neg (L_pos @ L_neg)) = (wf_tuple n (\<Union>A\<in>set A_pos. A) z \<and> list_all2 (\<lambda>A. (\<in>) (restrict A z)) A_pos (take (length A_pos) (L_pos @ L_neg)) \<and> list_all2 (\<lambda>A. (\<notin>) (restrict A z)) A_neg (drop (length A_pos) (L_pos @ L_neg)))"
sorted_distinct_set_unique: fixes less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and xs :: "'g list"   and ys :: "'g list" assumes "class.linorder less_eq less"   and "linorder.sorted less_eq xs"   and "distinct xs"   and "linorder.sorted less_eq ys"   and "distinct ys"   and "set xs = set ys" shows "xs = ys"
Abs_dlist_inject: fixes x :: "'c list"   and y :: "'c list" assumes "x \<in> {xs. equal_base.list_distinct ceq' xs \<or> ID CEQ('c) = None}"   and "y \<in> {xs. equal_base.list_distinct ceq' xs \<or> ID CEQ('c) = None}" shows "(Abs_dlist x = Abs_dlist y) = (x = y)"
transpose_mat: fixes n :: "'a" shows "Finite_Cartesian_Product.transpose (Finite_Cartesian_Product.mat n) = Finite_Cartesian_Product.mat n"
IH: fixes V :: "'a set" assumes "wf\<^sub>s\<^sub>t V (S @ S')" shows "wf\<^sub>s\<^sub>t (V \<union> wfvarsoccs\<^sub>s\<^sub>t S) S'"
Int: fixes a :: "'c set"   and M :: "'c measure"   and b :: "'c set" assumes "a \<in> fmeasurable M"   and "b \<in> fmeasurable M" shows "a \<inter> b \<in> fmeasurable M"
fold_induct: fixes P :: "'b \<Rightarrow> bool"   and s :: "'b"   and xs :: "'c list"   and f :: "'c \<Rightarrow> 'b \<Rightarrow> 'b" assumes "P s"   and "\<And>x s. \<lbrakk>x \<in> set xs; P s\<rbrakk> \<Longrightarrow> P (f x s)" shows "P (fold f xs s)"
flrestriction_fconst_on: fixes A :: "V"   and c :: "V"   and B :: "V" shows "fconst_on A c \<restriction>\<^sup>l\<^sub>\<bullet> B = fconst_on B c \<restriction>\<^sup>l\<^sub>\<bullet> A"
subseteq_guards_Cond: fixes c :: "('d, 'e, 'f) com"   and b :: "'d set"   and c1 :: "('d, 'e, 'f) com"   and c2 :: "('d, 'e, 'f) com" assumes "c \<subseteq>\<^sub>g Cond b c1 c2" shows "\<exists>c1' c2'. c = Cond b c1' c2' \<and> (c1' \<subseteq>\<^sub>g c1) \<and> (c2' \<subseteq>\<^sub>g c2)"
Follows_def: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" shows "f Fols g \<equiv> Increasing g \<inter> Increasing f \<inter> Always {s. f s \<le> g s} \<inter> (\<Inter>k. {s. k \<le> g s} \<longmapsto>w {s. k \<le> f s})"
psubset_card_mono: fixes B :: "'a set"   and A :: "'a set" assumes "finite B"   and "A \<subset> B" shows "card A < card B"
subset_cd: shows "(\<lambda>a. \<gamma> a \<bullet> i) ` {0..1} \<subseteq> {c..d}"
elts_VSigma: fixes a :: "V"   and b :: "V \<Rightarrow> V" shows "elts (VSigma a b) = (\<lambda>(x, y). \<langle>x, y\<rangle>) ` (SIGMA x:elts a. elts (b x))"
cs_prepend_rule: fixes l :: "'a list"   and p :: "'a node ref option"   and x :: "'a" shows "<cs_list l p> cs_prepend x p <cs_list (x # l)>"
B: shows "path_image \<gamma> \<subseteq> ball 0 B"
assms_2: fixes R :: "pi"   and S :: "pi" assumes "Prop R S" shows "Prop S R"
n_o_def: fixes opt :: "'av st option"   and X :: "char list set" shows "n\<^sub>o opt X = (case opt of None \<Rightarrow> 0 | Some S \<Rightarrow> n\<^sub>s S X + 1)"
mem_simps_4: fixes c :: "'b"   and A :: "'b set"   and B :: "'b set" shows "(c \<in> A \<inter> B) = (c \<in> A \<and> c \<in> B)"
assms_2: shows "A1 B1 C1 LtA A2 B2 C2"
usubstr_cons_undef2: fixes \<sigma> :: "(char \<Rightarrow> trm option) \<times> (char \<Rightarrow> trm option) \<times> (char \<Rightarrow> fml option) \<times> (char \<Rightarrow> game option)"   and A :: "fml"   and B :: "fml list"   and C :: "fml" assumes "usubstr \<sigma> (A # B, C) \<noteq> None" shows "usubstr \<sigma> ([A], C) \<noteq> None \<and> usubstr \<sigma> (B, C) \<noteq> None"
nn_transfer_operator_intTn_g: fixes f :: "'a \<Rightarrow> ennreal"   and g :: "'a \<Rightarrow> ennreal"   and n :: "nat" assumes "f \<in> borel_measurable M"   and "g \<in> borel_measurable M" shows "\<integral>\<^sup>+ x. f x * (nn_transfer_operator ^^ n) g x \<partial>M = \<integral>\<^sup>+ x. f ((T ^^ n) x) * g x \<partial>M"
gcd_l: shows "gcd m n * (n div gcd m n) = n"
fset_eq_empty_iff: fixes M :: "'a fset" shows "(M = {||}) = (\<forall>x. x |\<notin>| M)"
extra_wt_subseteq: fixes s :: "('s, 'v) tm"   and t :: "('s, 'v) tm"   and \<rho> :: "'v \<Rightarrow> ('s, 'v) tm" assumes "vars_mset s \<subseteq># vars_mset t" shows "local.kbo_std.extra_wt \<rho> s \<le> local.kbo_std.extra_wt \<rho> t"
dg_prod_2_Dom_vsv: shows "vsv ((\<AA> \<times>\<^sub>D\<^sub>G \<BB>)\<lparr>Dom\<rparr>)"
LIM_at_top_divide: fixes f :: "'a \<Rightarrow> real"   and a :: "real"   and F :: "'a filter"   and g :: "'a \<Rightarrow> real" assumes "(f \<longlongrightarrow> a) F"   and "0 < a"   and "(g \<longlongrightarrow> 0) F"   and "\<forall>\<^sub>F x in F. 0 < g x" shows "LIM x F. f x / g x :> at_top"
not_eq_zero': shows "vec_to_iarray A !! a \<noteq> (0::'a)"
mult_nat_less_add_less: fixes m :: "nat"   and n :: "nat"   and \<alpha> :: "V"   and \<gamma> :: "V"   and \<beta> :: "V" assumes "m < n"   and "\<alpha> < \<gamma>"   and "\<beta> < \<gamma>"   and "Ord \<alpha>"   and "Ord \<beta>"   and "Ord \<gamma>" shows "\<gamma> * ord_of_nat m + \<alpha> < \<gamma> * ord_of_nat n + \<beta>"
assms_2: fixes t :: "sterm"   and u' :: "sterm" assumes "s => t"   and "u => u'"   and "z = Call t l u'" shows "Q"
rel_prod_inject: fixes R1 :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and R2 :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and a :: "'c"   and b :: "'e"   and c :: "'d"   and d :: "'f" shows "rel_prod R1 R2 (a, b) (c, d) = (R1 a c \<and> R2 b d)"
push_simps: fixes x :: "'a"   and y :: "'b"   and z :: "'c" shows "push ((x, y), z) = (x, y, z)"
Cd_mono: fixes Xd :: "'d \<Rightarrow> 'e"   and Xh :: "'d \<Rightarrow> 'f"   and Pd :: "'e \<Rightarrow> ('d \<times> 'd) set"   and Ch :: "'f \<Rightarrow> 'd set \<Rightarrow> 'd set"   and x :: "'d"   and d :: "'e"   and X :: "'d set"   and Y :: "'d set" assumes "Contracts Xd Xh Pd Ch"   and "x \<in> Contracts.Cd Pd d X" shows "\<exists>y\<in>Contracts.Cd Pd d (X \<union> Y). (x, y) \<in> Pd d"
degree_aform_acc_err: fixes X :: "('a \<times> 'a pdevs) \<times> real"   and d :: "nat"   and p :: "nat"   and e :: "real" assumes "degree_aform (fst X) \<le> d" shows "degree_aform (fst (acc_err p X e)) \<le> d"
infdist_zero: fixes a :: "'c"   and A :: "'c set" assumes "a \<in> A" shows "infdist a A = 0"
frp_invar_def: fixes T1 :: "('a, 'c) tree_automaton_rec"   and T2 :: "('b, 'c) tree_automaton_rec" shows "frp_invar T1 T2 \<equiv> frp_invar_add (ta_rules T1) (ta_rules T2) \<inter> {s. frp_\<alpha> s \<in> dfs_invar (ta_initial T1 \<times> ta_initial T2) (f_succ (\<delta>_prod (ta_rules T1) (ta_rules T2)))}"
normalized_cancels_to: fixes l :: "(bool \<times> 'c) list" shows "cancels_to l (Cancelation.normalize l)"
assms_3: fixes x :: "'a" assumes "x \<in> A" shows "f x = g x"
igAbsCongS_def: fixes MOD :: "('c, 'd, 'varSort, 'sort, 'e, 'f, 'g, 'h, 'i) model_scheme" shows "igAbsCongS MOD \<equiv> \<forall>xs x x' y s X X'. isInBar (xs, s) \<and> igWls MOD s X \<and> igWls MOD s X' \<longrightarrow> igFresh MOD xs y X \<and> igFresh MOD xs y X' \<and> igSwap MOD xs y x X = igSwap MOD xs y x' X' \<longrightarrow> igAbs MOD xs x X = igAbs MOD xs x' X'"
eq_up_to_apply: fixes \<pi> :: "nat \<Rightarrow> 'n"   and k :: "nat"   and \<pi>' :: "nat \<Rightarrow> 'n"   and j :: "nat" assumes "\<pi> =\<^bsub>k\<^esub> \<pi>'"   and "j \<le> k" shows "\<pi> j = \<pi>' j"
basis_exists: fixes V :: "('b, 'c) vec set" obtains B :: "('b, 'c) vec set" where "B \<subseteq> V" and "vec.independent B" and "V \<subseteq> vec.span B" and "card B = vec.dim V"
take_bit_int_eq_self: fixes k :: "int"   and n :: "nat" assumes "0 \<le> k"   and "k < 2 ^ n" shows "take_bit n k = k"
unfold_congs_2: fixes r :: "('b, 'c) state_scheme"   and r' :: "('b, 'c) state_scheme"   and v' :: "nat"   and f :: "nat \<Rightarrow> nat"   and f' :: "nat \<Rightarrow> nat" assumes "r = r'"   and "l r' = v'"   and "\<And>v. v = v' \<Longrightarrow> f v = f' v" shows "l_update f r = l_update f' r'"
until_frmlsr_correct: fixes \<phi> :: "'a ltlr" shows "until_frmlsr \<phi> = {(\<mu>, \<eta>). \<mu> U\<^sub>r \<eta> \<in> subfrmlsr \<phi>}"
aBy_def: fixes v :: "Value" shows "aBy v = (case v of byteV by \<Rightarrow> by)"
hom_connected_1: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<nu> :: "'b"   and \<mu> :: "'b" assumes "weak_composition V H" shows "(H \<nu> \<mu> \<noteq> partial_magma.null V) = (H (partial_magma.dom V \<nu>) \<mu> \<noteq> partial_magma.null V)"
generate: fixes I :: "'c set"   and R :: "('c, 'd) ring_scheme" assumes "principalideal I R" shows "\<exists>i\<in>carrier R. I = Idl\<^bsub>R\<^esub> {i}"
bij_moebius_pt: fixes M :: "moebius" shows "bij (moebius_pt M)"
not_HInfinite_one: shows "1 \<notin> HInfinite"
saturated_saturate: fixes N :: "'a literal multiset set" shows "saturated (saturate N)"
G_def: shows "G = \<Union> (range F) \<union> {b}"
nn_integral_has_integral_lebesgue': fixes \<Omega> :: "'b set"   and f :: "'b \<Rightarrow> real"   and I :: "real" assumes "\<And>x. x \<in> \<Omega> \<Longrightarrow> 0 \<le> f x"   and "(f has_integral I) \<Omega>" shows "\<integral>\<^sup>+x\<in>\<Omega>. ennreal (f x)\<partial>lborel = ennreal I"
tdghm_NTDom: shows "\<NN>'\<lparr>NTDom\<rparr> = \<FF>'"
hom_add: fixes x :: "'b poly"   and y :: "'b poly" shows "pCons (0::'b) (x + y) = pCons (0::'b) x + pCons (0::'b) y"
is_lub_set: fixes S :: "'b set set" shows "S <<| \<Union> S"
distrib: fixes a :: "'ag"   and b :: "'ag"   and c :: "'ag" assumes "a \<in> U"   and "b \<in> U"   and "c \<in> U" shows "(a +\<^sub>o\<^sub>w b) *\<^sub>o\<^sub>w c = a *\<^sub>o\<^sub>w c +\<^sub>o\<^sub>w b *\<^sub>o\<^sub>w c"
T_Sub_5: fixes S' :: "POPLmark.type" assumes "\<Gamma> \<turnstile> U <: S"   and "VarB S \<Colon> \<Gamma> \<turnstile> s : S'"   and "\<Gamma> \<turnstile> \<down>\<^sub>\<tau> 1 0 S' <: U'" shows "P"
conj_ac_1: fixes a :: "'b"   and b :: "'b"   and c :: "'b" shows "(a AND b) AND c = a AND b AND c"
prems_2: shows "local.prefixes (t + Suc d) j \<down>= b'"
checker_eq_checker2: fixes s\<^sub>e :: "real"   and v\<^sub>e :: "real"   and a\<^sub>e :: "real"   and s\<^sub>o :: "real"   and v\<^sub>o :: "real"   and a\<^sub>o :: "real" shows "checker s\<^sub>e v\<^sub>e a\<^sub>e s\<^sub>o v\<^sub>o a\<^sub>o = checker2 s\<^sub>e v\<^sub>e a\<^sub>e s\<^sub>o v\<^sub>o a\<^sub>o"
tl_P_new_2: shows "hd (tl P_new) \<notin> second_vertices"
a1: shows "a $ 1 \<noteq> (0::'a)"
exec_meth_drop_xt_pc: fixes ci :: "('addr, 'heap) check_instr"   and P :: "(nat \<times> nat \<times> 'addr instr list \<times> (nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list) prog"   and ins :: "'addr instr list"   and ins' :: "'addr instr list"   and xt :: "(nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list"   and xt' :: "(nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list"   and t :: "'thread_id"   and h :: "'heap"   and stk :: "'addr val list"   and loc :: "'addr val list"   and pc :: "nat"   and xcp :: "'addr option"   and ta :: "('addr, 'thread_id, 'heap) jvm_thread_action"   and h' :: "'heap"   and stk' :: "'addr val list"   and loc' :: "'addr val list"   and pc' :: "nat"   and xcp' :: "'addr option"   and n' :: "nat" assumes "exec_meth ci P (ins @ ins') (xt @ shift (length ins) xt') t h (stk, loc, pc, xcp) ta h' (stk', loc', pc', xcp')"   and "length ins \<le> pc"   and "pcs xt \<subseteq> {..<length ins}"   and "jump_ok ins' 0 n'" shows "length ins \<le> pc'"
SUP_apply: fixes f :: "'c \<Rightarrow> 'b \<Rightarrow> 'a"   and A :: "'c set"   and x :: "'b" shows "(\<Squnion> (f ` A)) x = (\<Squnion>y\<in>A. f y x)"
assms_4: shows "InvariantWatchesEl (getF state) (getWatch1 state) (getWatch2 state)"
\<i>'_extends_\<i>_3: fixes a :: "'a" assumes "weak_unit a"   and "obj a" shows "\<i>' a = \<i>[a]"
in_quotient_imp_subset: fixes A :: "'a set"   and r :: "('a \<times> 'a) set"   and X :: "'a set" assumes "equiv A r"   and "X \<in> A // r" shows "X \<subseteq> A"
terminates_removeAll_raw: fixes g :: "'e \<Rightarrow> ('f, 'e) step"   and b :: "'f" assumes "terminates g" shows "terminates (removeAll_raw b g)"
ordinal_div_plus_mod: fixes y :: "ordinal"   and x :: "ordinal" shows "y * (x div y) + x mod y = x"
Singl: fixes trn :: "('state, 'act, 'out) trans" assumes "validTrans trn" shows "valid [trn]"
measurable_compose_Pair1: fixes x :: "'c"   and M1 :: "'c measure"   and g :: "'c \<times> 'd \<Rightarrow> 'e"   and M2 :: "'d measure"   and L :: "'e measure" assumes "x \<in> space M1"   and "g \<in> M1 \<Otimes>\<^sub>M M2 \<rightarrow>\<^sub>M L" shows "(\<lambda>y. g (x, y)) \<in> M2 \<rightarrow>\<^sub>M L"
cfg_onD_state: fixes K :: "'a \<Rightarrow> 'a pmf set"   and cfg :: "'a cfg"   and s :: "'a" assumes "Markov_Decision_Process K"   and "cfg \<in> Markov_Decision_Process.cfg_on K s" shows "state cfg = s"
presSubst_def: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'c"   and MOD :: "('index, 'bindex, 'varSort, 'd, 'opSym, 'var, 'c, 'e, 'f) Recursion.model_scheme" shows "presSubst h MOD \<equiv> \<forall>ys Y y s X. wls (asSort ys) Y \<and> wls s X \<longrightarrow> h (X #[Y / y]_ys) = gSubst MOD ys Y (h Y) y X (h X)"
rel_frontier_eq_empty: fixes S :: "'a set" shows "(rel_frontier S = {}) = affine S"
tdghm_dghm_comp_NTMap_app: fixes x :: "V"   and \<NN> :: "V" assumes "x \<in>\<^sub>\<circ> \<AA>\<lparr>Obj\<rparr>" shows "(\<NN> \<circ>\<^sub>T\<^sub>D\<^sub>G\<^sub>H\<^sub>M\<^sub>-\<^sub>D\<^sub>G\<^sub>H\<^sub>M \<FF>)\<lparr>NTMap\<rparr>\<lparr>x\<rparr> = \<NN>\<lparr>NTMap\<rparr>\<lparr>\<FF>\<lparr>ObjMap\<rparr>\<lparr>x\<rparr>\<rparr>"
fps_of_ratfps_1: shows "fps_of_ratfps 1 = 1"
g_def: shows "g = (\<lambda>y. SOME x. x \<in> X \<and> dist (f x) y \<le> C)"
WriteNonVolatile: fixes a :: "nat"   and \<O> :: "nat set"   and \<S> :: "nat \<Rightarrow> bool option"   and \<O>s :: "nat set list"   and i :: "nat"   and \<R>s :: "(nat \<Rightarrow> bool option) list"   and D :: "nat set"   and f :: "(nat \<Rightarrow> nat option) \<Rightarrow> nat"   and A :: "nat set"   and L :: "nat set"   and R :: "nat set"   and W :: "nat set"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and m :: "nat \<Rightarrow> nat"   and \<D> :: "bool" assumes "a \<in> \<O>"   and "a \<notin> dom \<S>"   and "\<forall>j<length \<O>s. i \<noteq> j \<longrightarrow> a \<notin> dom (\<R>s ! j)" shows "\<O>s,\<R>s,i\<turnstile> (Write False a (D, f) A L R W # is, \<theta>, m, \<D>, \<O>, \<S>) \<surd>"
countable_Collect_finite_subset: fixes T :: "'b set" assumes "countable T" shows "countable {A. finite A \<and> A \<subseteq> T}"
intro: fixes h :: "'d \<Rightarrow> 'e"   and G :: "('d, 'f) monoid_scheme"   and H :: "('e, 'g) monoid_scheme" assumes "h \<in> hom G H" shows "group_hom_axioms G H h"
blinding_hash_eq: fixes x :: "'a\<^sub>m"   and y :: "'a\<^sub>m" assumes "bo x y" shows "h x = h y"
between: fixes a :: "'a"   and b :: "'a"   and x :: "'a" shows "between (a, b) x = (dist a b = dist a x + dist x b)"
phase_rel_def: shows "phase_rel = {(in_sync, hp, ph, ph, []) |in_sync hp ph. handshake_phase_rel hp in_sync ph} \<union> {True} \<times> {(hp_IdleInit, ph_Init, ph_Idle, [mw_Phase ph_Init]), (hp_InitMark, ph_Mark, ph_Init, [mw_Phase ph_Mark]), (hp_IdleMarkSweep, ph_Sweep, ph_Mark, [mw_Phase ph_Sweep]), (hp_IdleMarkSweep, ph_Idle, ph_Mark, [mw_Phase ph_Sweep, mw_Phase ph_Idle]), (hp_IdleMarkSweep, ph_Idle, ph_Sweep, [mw_Phase ph_Idle])}"
Diff_Int2: fixes A :: "'c set"   and C :: "'c set"   and B :: "'c set" shows "A \<inter> C - B \<inter> C = A \<inter> C - B"
\<b>_zero: fixes ps :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list" assumes "ps \<noteq> []" shows "Suc (Max (poly_deg ` fst ` set ps)) \<le> \<b> ps 0"
B: fixes i :: "'n"   and j :: "'n" shows "0 \<le> B $h i $h j"
rep_inverse: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and equal :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and rep_nat :: "'a \<Rightarrow> nat"   and abs_nat :: "nat \<Rightarrow> 'a"   and x :: "'a" assumes "class.nat minus plus zero equal less_eq less rep_nat abs_nat" shows "abs_nat (rep_nat x) = x"
finite_cyclic_on: fixes f :: "'b \<Rightarrow> 'b"   and S :: "'b set" assumes "cyclic_on f S" shows "finite S"
le_supI2: fixes sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and b :: "'a"   and a :: "'a" assumes "class.semilattice_sup sup less_eq less"   and "less_eq x b" shows "less_eq x (sup a b)"
ex_crit: fixes t :: "'t" assumes "t \<in> T" shows "\<exists>f. crit t f"
le_supI: fixes sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c"   and x :: "'c"   and b :: "'c" assumes "class.semilattice_sup sup less_eq less"   and "less_eq a x"   and "less_eq b x" shows "less_eq (sup a b) x"
pentagon: fixes f :: "'a"   and g :: "'a"   and h :: "'a"   and k :: "'a" assumes "ide f"   and "ide g"   and "ide h"   and "ide k"   and "src f = trg g"   and "src g = trg h"   and "src h = trg k" shows "(f \<star> \<a>[g, h, k]) \<cdot> \<a>[f, g \<star> h, k] \<cdot> (\<a>[f, g, h] \<star> k) = \<a>[f, g, h \<star> k] \<cdot> \<a>[f \<star> g, h, k]"
unit_simps_1: fixes a :: "'b" assumes "B.obj a" shows "D.arr (HoF.unit a)"
not_cong_3421: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "\<not> Cong A B C D" shows "\<not> Cong C D B A"
simps: fixes atom_ordering :: "('a \<times> 'a) set"   and I :: "'a set"   and C :: "'a Literal set" assumes "propositional_atoms atom_ordering" shows "propositional_atoms.validate_clause I C = (\<exists>L. L \<in> C \<and> propositional_atoms.validate_literal I L)"
simps_4: fixes n1 :: "nat"   and n2 :: "nat" shows "(R n1 < R n2) = (n1 < n2)"
chine_assoc_inverse: shows "C.inverse_arrows fgh.chine_assoc fgh.chine_assoc'"
out_var_prod_lens: fixes X :: "'d \<Longrightarrow> 'e"   and x :: "'f \<Longrightarrow> 'h"   and Y :: "'h \<Longrightarrow> 'g" assumes "wb_lens X" shows "$x\<acute> ;\<^sub>L X \<times> Y = $Y:x\<acute>"
proots_unbounded_line: fixes st :: "complex"   and tt :: "complex"   and p :: "complex poly" assumes "st \<noteq> tt"   and "p \<noteq> 0" shows "proots_count p (unbounded_line st tt) = (let pc = p \<circ>\<^sub>p [:st, tt - st:]; pR = map_poly Re pc; pI = map_poly Im pc; g = gcd pR pI in nat (changes_R_smods_ext g (pderiv g)))"
thetaSSeqD_Sretr: shows "thetaSSeqD \<subseteq> Sretr (thetaSSeqD \<union> Sbis)"
cod_comp: fixes g :: "'b"   and f :: "'b" assumes "D.seq g f" shows "D.cod (g \<cdot>\<^sub>D f) = D.cod g"
Int_empty_right: fixes A :: "'b set" shows "bot_homomorphism ((\<inter>) A)"
inf_compl_bot: fixes x :: "'b upred" shows "(x \<and> \<not> x) = false"
possible_steps_0_not_select: fixes a :: "String.literal"   and b :: "value list" assumes "a \<noteq> STR ''select''" shows "possible_steps drinks 0 <> a b = {||}"
naturality: fixes f :: "'d" assumes "D.arr f" shows "\<eta> (D.cod f) \<cdot>\<^sub>D D.map f = GF.map f \<cdot>\<^sub>D \<eta> (D.dom f)"
subC: shows "\<forall>M m D. Mm2 M = \<lfloor>(m, D)\<rfloor> \<longrightarrow> P \<turnstile> D \<preceq>\<^sup>* C"
simps_5: fixes p :: "fm"   and q :: "fm" shows "qfree (Or p q) = (qfree p \<and> qfree q)"
raw_interp_inverse: fixes x :: "('a, 'b, 'c) good_interp" shows "good_interp (raw_interp x) = x"
lprefix_antisym: fixes xs :: "'a llist"   and ys :: "'a llist" assumes "lprefix xs ys"   and "lprefix ys xs" shows "xs = ys"