hcju/isabelleps huggingface dataset:
induct: fixes P1 :: "('c, 'd) Nested_Datatype.term \<Rightarrow> bool"   and P2 :: "('c, 'd) Nested_Datatype.term list \<Rightarrow> bool"   and "term" :: "('c, 'd) Nested_Datatype.term" assumes "\<And>x. P1 (Var x)"   and "\<And>x1 x2. P2 x2 \<Longrightarrow> P1 (App x1 x2)"   and "P2 []"   and "\<And>x1 x2. \<lbrakk>P1 x1; P2 x2\<rbrakk> \<Longrightarrow> P2 (x1 # x2)" shows "P1 term"
absolutely_integrable_norm: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "f absolutely_integrable_on S" shows "norm \<circ> f absolutely_integrable_on S"
bot: shows "(f \<longlongrightarrow> a) at_bot"
b: shows "b \<in> ideal_gen_by_prod \<aa> \<bb>"
finite_set_has_max: fixes s :: "nat set" assumes "finite s"   and "s \<noteq> {}" shows "\<exists>k\<in>s. \<forall>p\<in>s. p \<le> k"
minus_mult_left: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.ring plus zero minus uminus times" shows "uminus (times a b) = times (uminus a) b"
domain_make_vert: fixes n :: "nat" shows "domain_block (make_vert_block n) = int n"
mk: shows "odd m \<and> n - 1 = 2 ^ k * m"
add_lessD2: fixes i :: "nat"   and j :: "nat"   and k :: "nat" assumes "i + j < k" shows "j < k"
option_rel_Some2: fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'a option"   and y :: "'b" shows "rel_option A x (pure y) = (\<exists>x'. x = pure x' \<and> A x' y)"
abs_le_square_iff: fixes 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 times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and x :: "'b"   and y :: "'b" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less_eq (abs x) (abs y) = less_eq (power.power one times x 2) (power.power one times y 2)"
intra_proc_additional_edge: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and a :: "'a"   and a' :: "'a" assumes "CFG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main"   and "valid_edge a"   and "a' \<in> get_return_edges a" shows "\<exists>a''. valid_edge a'' \<and> sourcenode a'' = targetnode a \<and> targetnode a'' = sourcenode a' \<and> kind a'' = (\<lambda>cf. False)\<^sub>\<surd>"
simps_22: fixes x51 :: "bexp"   and x52 :: "com"   and x31 :: "com"   and x32 :: "com" shows "WHILE x51 DO x52 \<noteq> x31;; x32"
tab_simps_5: shows "cod \<sigma> = s \<star> s\<^sub>0"
disj_nb0: fixes p :: "fm"   and q :: "fm" assumes "bound0 p"   and "bound0 q" shows "bound0 (Parametric_Ferrante_Rackoff.disj p q)"
LVE_LVE: shows "LVE \<subseteq> \<Union> (funStar SS ` LVE)"
scc_root_is_Min_disc: fixes v :: "'v"   and scc :: "'v set" assumes "scc_root s v scc" shows "Min (\<delta> s ` (scc \<inter> dom (discovered s))) = \<delta> s v"
lx_def: shows "lx = x + (l - (x + y)) / 2"
map_sel_1: fixes a :: "'a list"   and f :: "'a \<Rightarrow> 'b" assumes "a \<noteq> []" shows "hd (map f a) = f (hd a)"
induct: fixes P :: "('c match_expr \<Rightarrow> 'c match_expr option) \<Rightarrow> 'c rule list \<Rightarrow> bool"   and a0 :: "'c match_expr \<Rightarrow> 'c match_expr option"   and a1 :: "'c rule list" assumes "\<And>uu_. P uu_ []"   and "\<And>f m a rs. \<lbrakk>f m = None \<Longrightarrow> P f rs; \<And>x2. f m = Some x2 \<Longrightarrow> P f rs\<rbrakk> \<Longrightarrow> P f (Rule m a # rs)" shows "P a0 a1"
divide_left: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'b"   and h :: "'a \<Rightarrow> 'b" assumes "\<forall>\<^sub>F x in F. f x \<noteq> (0::'b)"   and "\<forall>\<^sub>F x in F. g x \<noteq> (0::'b)"   and "g \<in> L F f" shows "(\<lambda>x. h x / f x) \<in> L F (\<lambda>x. h x / g x)"
simps_3: fixes x3 :: "'e"   and y3 :: "'e" shows "(Fault x3 = Fault y3) = (x3 = y3)"
Int_Un_eq_3: fixes S :: "'d set"   and T :: "'d set" shows "S \<union> S \<inter> T = S"
F_def: fixes i :: "nat" shows "F i = {x \<in> space M. i < f x}"
eval_finalId: fixes e :: "char list exp"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and s :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)" assumes "final e" shows "P \<turnstile> \<langle>e,s\<rangle> \<Rightarrow> \<langle>e,s\<rangle>"
surjective: fixes r :: "('d, 'e, 'f) automaton_scheme" shows "r = \<lparr>init = init r, trans = automaton.trans r, \<dots> = more r\<rparr>"
that: fixes q :: "'b list"   and r :: "'b list" assumes "cas u q y"   and "cas y r v"   and "p = q @ r"   and "awalk_verts u q = xs @ [y]"   and "awalk_verts y r = y # ys" shows "thesis"
Suc_mult_le_cancel1: fixes k :: "nat"   and m :: "nat"   and n :: "nat" shows "(Suc k * m \<le> Suc k * n) = (m \<le> n)"
blinding_of_on_axioms: shows "blinding_of_on UNIV h bo"
AE_2_3: shows "AE x in M. (\<lambda>n. - (u n x / real n)) \<longlonglongrightarrow> - subcocycle_lim u x"
sccs_verts_subsets: fixes S :: "'a set" assumes "S \<in> H.sccs_verts" shows "S \<subseteq> verts H"
dist_triangle: fixes dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and x :: "'b"   and z :: "'b"   and y :: "'b" assumes "class.metric_space dist uniformity open" shows "dist x z \<le> dist x y + dist y z"
withChannel'_def: fixes gl :: "bool"   and v :: "String.literal"   and idx :: "integer option"   and f :: "nat \<Rightarrow> channel \<Rightarrow> 'a"   and g :: "'b gState_scheme"   and p :: "pState" shows "withChannel' gl v idx f g p = (let error = \<lambda>_. abortv STR ''Variable is not a channel: '' v (\<lambda>_. f 0 InvChannel); abort = \<lambda>_. abortv STR ''Channel already closed / invalid: '' v (\<lambda>_. f 0 InvChannel) in withVar' gl v idx (\<lambda>i. let i = nat_of_integer i in if length (gState.channels g) \<le> i then error () else let c = gState.channels g ! i in case c of InvChannel \<Rightarrow> abort () | _ \<Rightarrow> f i c) g p)"
move_defensive_by_Even_def: fixes m :: "'a"   and p :: "'a list" shows "move_defensive_by_Even m p \<equiv> even (length p) \<longrightarrow> \<not> winning_position_Odd (p @ [m])"
smooth_compat_D1: fixes k :: "enat"   and c1 :: "('c, 'd) chart"   and c2 :: "('c, 'd) chart" assumes "k-smooth_compat c1 c2" shows "k-smooth_on (apply_chart c1 ` (domain c1 \<inter> domain c2)) (apply_chart c2 \<circ> inv_chart c1)"
is_natural_transformation: fixes A :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and F :: "'a \<Rightarrow> 'b"   and G :: "'a \<Rightarrow> 'b"   and H :: "'a \<Rightarrow> 'b"   and \<sigma> :: "'a \<Rightarrow> 'b"   and \<tau> :: "'a \<Rightarrow> 'b" assumes "vertical_composite A B F G H \<sigma> \<tau>" shows "natural_transformation A B F H (vertical_composite.map A B \<sigma> \<tau>)"
conj_intro: fixes P :: "'b \<Rightarrow> bool"   and x :: "'b"   and Q :: "'b \<Rightarrow> bool" assumes "P x"   and "Q x" shows "(P \<and> Q) x"
hyps_3: assumes "P' \<noteq> P \<parallel> !P" shows "!P \<Longrightarrow>\<^sub>\<tau> P'"
assms_3: shows "\<forall>\<Psi>'. \<Psi> \<otimes> \<Psi>' \<rhd> P \<sim> Q"
ssubst_Pair_rhs: fixes r :: "'c"   and s :: "'d"   and R :: "('c \<times> 'd) set"   and s' :: "'d" assumes "(r, s) \<in> R"   and "s' = s" shows "(r, s') \<in> R"
lift_map_eq_iff: fixes i :: "nat"   and s :: "'c"   and f :: "nat \<Rightarrow> 'c"   and uu :: "'d"   and i' :: "nat"   and s' :: "'c"   and f' :: "nat \<Rightarrow> 'c"   and uu' :: "'d" shows "(lift_map i (s, f, uu) = lift_map i' (s', f', uu')) = (uu = uu' \<and> insert_map i s f = insert_map i' s' f')"
real_seq_to_hypreal_Infinitesimal: fixes X :: "nat \<Rightarrow> 'a"   and x :: "'a" assumes "\<forall>n. norm (X n - x) < inverse (real (Suc n))" shows "star_n X - star_of x \<in> Infinitesimal"
cnj_mult_vv: fixes v1 :: "complex \<times> complex"   and v2 :: "complex \<times> complex" shows "cnj (v1 *\<^sub>v\<^sub>v v2) = vec_cnj v1 *\<^sub>v\<^sub>v vec_cnj v2"
lens_plus_mono_right: fixes X :: "'d \<Longrightarrow> 'e"   and Z :: "'f \<Longrightarrow> 'e"   and Y :: "'g \<Longrightarrow> 'e" assumes "X \<bowtie> Z"   and "Y \<subseteq>\<^sub>L Z" shows "X +\<^sub>L Y \<subseteq>\<^sub>L X +\<^sub>L Z"
IH: shows "normalize (\<Prod>\<^sub># (image_mset normalize A)) = normalize (\<Prod>\<^sub># A)"
raw_match_term_restriction_on_tvs: fixes t :: "Core.term"   and u :: "Core.term"   and subs :: "variable \<times> String.literal set \<Rightarrow> typ option"   and subs' :: "variable \<times> String.literal set \<Rightarrow> typ option" assumes "raw_match_term t u subs = Some subs'" shows "raw_match_term t u (subs |` tvs t) = Some (subs' |` tvs t)"
test_eq4: fixes p :: "'a"   and q :: "'a"   and x :: "'a" assumes "test p"   and "test q" shows "(!p \<cdot> x \<cdot> !q = x \<cdot> !q) = (p \<cdot> x \<cdot> !q = (0::'a))"
Spq: shows "set (p @ q) \<subseteq> set (xvec @ yvec) \<times> set ((p @ q) \<bullet> (xvec @ yvec))"
WT_real: shows "\<I>_real \<oplus>\<^sub>\<I> \<I>_common \<turnstile>res real_resource \<surd>"
axioms_1: fixes R :: "('e, 'f) ring_scheme" assumes "cring R" shows "ring R"
ExecSeqError: fixes s\<^sub>1 :: "stmt"   and \<sigma> :: "char list \<Rightarrow> nat option"   and s\<^sub>2 :: "stmt" assumes "s\<^sub>1 \<rhd> \<sigma> \<leadsto> None" shows "Seq s\<^sub>1 s\<^sub>2 \<rhd> \<sigma> \<leadsto> None"
rel: shows "(xs, ys) \<in> s_mul_ext ns s"
complete_development_pairE: fixes A :: "'a trm"   and B :: "'a trm"   and X :: "'a trm" assumes "SimplyTyped.Pair A B >>> X" shows "\<exists>A' B'. A >>> A' \<and> B >>> B' \<and> X = SimplyTyped.Pair A' B'"
unitary_def: fixes U :: "'c \<Rightarrow>\<^sub>C\<^sub>L 'd" shows "unitary U = (U* o\<^sub>C\<^sub>L U = id_cblinfun \<and> U o\<^sub>C\<^sub>L U* = id_cblinfun)"
zero_fun_def: shows "0 = (\<lambda>x. 0::'c)"
subset_inj_on: fixes f :: "'g \<Rightarrow> 'h"   and B :: "'g set"   and A :: "'g set" assumes "inj_on f B"   and "A \<subseteq> B" shows "inj_on f A"
poly_mapping_keys_eqI: fixes p :: "'c \<Rightarrow>\<^sub>0 'd"   and q :: "'c \<Rightarrow>\<^sub>0 'd" assumes "keys p = keys q"   and "\<And>t. t \<in> keys p \<Longrightarrow> lookup p t = lookup q t" shows "p = q"
factorization_lattice_diag_nonzero_RAT: fixes d :: "nat"   and u :: "int poly"   and k :: "int"   and i :: "nat" assumes "d = degree u"   and "d \<le> n"   and "u \<noteq> 0"   and "k \<noteq> 0"   and "i < n" shows "map of_int_hom.vec_hom (factorization_lattice u (n - d) k) ! i $ i \<noteq> 0"
FOREACHoci_refine_rcg: fixes \<alpha> :: "'c \<Rightarrow> 'd"   and S :: "'c set"   and S' :: "'d set"   and \<sigma>0 :: "'e"   and \<sigma>0' :: "'f"   and R :: "('e \<times> 'f) set"   and RR :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and RR' :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and \<Phi>' :: "'d set \<Rightarrow> 'f \<Rightarrow> bool"   and \<Phi> :: "'c set \<Rightarrow> 'e \<Rightarrow> bool"   and c :: "'e \<Rightarrow> bool"   and c' :: "'f \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'e \<Rightarrow> 'e nres"   and f' :: "'d \<Rightarrow> 'f \<Rightarrow> 'f nres" assumes "inj_on \<alpha> S"   and "S' = \<alpha> ` S"   and "(\<sigma>0, \<sigma>0') \<in> R"   and "\<And>x y. \<lbrakk>x \<in> S; y \<in> S; RR x y\<rbrakk> \<Longrightarrow> RR' (\<alpha> x) (\<alpha> y)"   and "\<And>it \<sigma> it' \<sigma>'. \<lbrakk>it' = \<alpha> ` it; it \<subseteq> S; it' \<subseteq> S'; \<Phi>' it' \<sigma>'; \<Phi> it \<sigma>; (\<sigma>, \<sigma>') \<in> R\<rbrakk> \<Longrightarrow> c \<sigma> = c' \<sigma>'"   and "\<And>it \<sigma> it' \<sigma>'. \<lbrakk>it' = \<alpha> ` it; it \<subseteq> S; it' \<subseteq> S'; \<Phi>' it' \<sigma>'; (\<sigma>, \<sigma>') \<in> R\<rbrakk> \<Longrightarrow> \<Phi> it \<sigma>"   and "\<And>x it \<sigma> x' it' \<sigma>'. \<lbrakk>\<forall>y\<in>it - {x}. RR x y; x' = \<alpha> x; x \<in> it; x' \<in> it'; it' = \<alpha> ` it; it \<subseteq> S; it' \<subseteq> S'; \<Phi> it \<sigma>; \<Phi>' it' \<sigma>'; c \<sigma>; c' \<sigma>'; (\<sigma>, \<sigma>') \<in> R\<rbrakk> \<Longrightarrow> f x \<sigma> \<le> \<Down> R (f' x' \<sigma>')" shows "FOREACH\<^sub>O\<^sub>C\<^bsup>RR,\<Phi>\<^esup> S c f \<sigma>0 \<le> \<Down> R (FOREACH\<^sub>O\<^sub>C\<^bsup>RR',\<Phi>'\<^esup> S' c' f' \<sigma>0')"
permute_pure: fixes permute :: "perm \<Rightarrow> 'e \<Rightarrow> 'e"   and p :: "perm"   and x :: "'e" assumes "class.pure permute" shows "permute p x = x"
weak_map_of_SomeI: fixes k :: "'c"   and x :: "'d"   and l :: "('c \<times> 'd) list" assumes "(k, x) \<in> set l" shows "\<exists>x. table_of l k = Some x"
good_item_simps_34: fixes inp :: "'a \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qTerm set option"   and binpa :: "'b \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qAbs set option"   and inp' :: "'a \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qTerm set option"   and binp' :: "'b \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qAbs set option"   and delta :: "'c"   and delta' :: "'c" assumes "goodInp inp"   and "goodBinp binpa"   and "goodInp inp'"   and "goodBinp binp'" shows "(Op delta inp binpa = Op delta' inp' binp') = (delta = delta' \<and> inp = inp' \<and> binpa = binp')"
finite_vars_term: fixes t :: "('b, 'a) Term.term" shows "finite (vars_term t)"
coeff_swap_poly: fixes b :: "nat"   and i :: "nat"   and mp :: "'b mpoly"   and x :: "nat \<Rightarrow>\<^sub>0 nat" shows "MPoly_Type.coeff (swap_poly b i mp) x = MPoly_Type.coeff mp (swap0 b i x)"
hypreal_setle_less_trans: fixes S :: "real star set"   and x :: "real star"   and y :: "real star" assumes "S *<= x"   and "x < y" shows "S *<= y"
reduce_simple_not_value: fixes es :: "e list"   and es' :: "e list"   and vs :: "v list" assumes "\<lparr>es\<rparr> \<leadsto> \<lparr>es'\<rparr>" shows "es \<noteq> $$* vs"
a_host_shadow_root_rel_shadow_root: fixes h :: "(_) heap"   and host :: "(_) element_ptr"   and shadow_root :: "(_) shadow_root_ptr" assumes "h \<turnstile> get_shadow_root host \<rightarrow>\<^sub>r Some shadow_root" shows "(cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^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 host, cast\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^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 shadow_root) \<in> host_shadow_root_rel h"
variety_ofI: fixes F :: "(('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd) set"   and a :: "'c \<Rightarrow> 'd" assumes "\<And>f. f \<in> F \<Longrightarrow> poly_eval a f = (0::'d)" shows "a \<in> \<V> F"
ring_associated_iff: fixes a :: "'a"   and b :: "'a" assumes "a \<in> carrier R"   and "b \<in> carrier R" shows "a \<sim> b = (\<exists>u\<in>Units R. a = u \<otimes> b)"
zero_less_power_eq_numeral: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and a :: "'a"   and w :: "num" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less zero (power.power one times a (numeral w)) = (numeral w = 0 \<or> even (numeral w) \<and> a \<noteq> zero \<or> odd (numeral w) \<and> less zero a)"
rel_FGf_neg_distr_imp: fixes Co1 :: "'p \<Rightarrow> 'q \<Rightarrow> bool"   and Co1' :: "'q \<Rightarrow> 'r \<Rightarrow> bool"   and Co2 :: "'s \<Rightarrow> 't \<Rightarrow> bool"   and Co2' :: "'t \<Rightarrow> 'u \<Rightarrow> bool"   and Contra1 :: "'w \<Rightarrow> 'x \<Rightarrow> bool"   and Contra1' :: "'x \<Rightarrow> 'y \<Rightarrow> bool"   and Contra2 :: "'z \<Rightarrow> 'aa \<Rightarrow> bool"   and Contra2' :: "'aa \<Rightarrow> 'ab \<Rightarrow> bool"   and tytok_F :: "('ad \<times> 'ae \<times> 'af \<times> 'ag \<times> 'ah \<times> 'ai \<times> 'aj \<times> 'aj \<times> 'aj \<times> ('ak, 'aj, 'al, 'v, 'am, 'ac, 'an) G) itself"   and tytok_FGf :: "('ad \<times> 'ae \<times> 'af \<times> 'ag \<times> 'ah \<times> 'ai \<times> 'ak \<times> 'aj \<times> 'al \<times> 'v \<times> 'am \<times> 'ac \<times> 'an) itself" assumes "rel_F_neg_distr_cond Co1 Co1' Co2 Co2' (=) (=) Contra1 Contra1' Contra2 Contra2' (=) (=) tytok_F" shows "rel_FGf_neg_distr_cond Co1 Co1' Co2 Co2' Contra1 Contra1' Contra2 Contra2' tytok_FGf"
simulation_silent2_aux: fixes s1 :: "'s1"   and s2 :: "'s2"   and s2' :: "'s2"   and \<mu>2 :: "'s2 \<Rightarrow> 's2 \<Rightarrow> bool" assumes "s1 \<approx> s2"   and "s2 -\<tau>2\<rightarrow> s2'" shows "s1 \<approx> s2' \<and> \<mu>2\<^sup>+\<^sup>+ s2' s2 \<or> (\<exists>s1'. s1 -\<tau>1\<rightarrow>+ s1' \<and> s1' \<approx> s2')"
simps_2: fixes e :: "'c match_expr" shows "andfold_MatchExp [e] = e"
alpha_preserves_qGood2: fixes X :: "('d, 'e, 'varSort, 'var, 'f) qTerm"   and X' :: "('d, 'e, 'varSort, 'var, 'f) qTerm" assumes "X #= X'"   and "qGood X'" shows "qGood X"
inv_inter: fixes join :: "('b \<times> 'c) tree \<Rightarrow> 'b \<Rightarrow> ('b \<times> 'c) tree \<Rightarrow> ('b \<times> 'c) tree"   and inv :: "('b \<times> 'c) tree \<Rightarrow> bool"   and t1 :: "('b \<times> 'c) tree"   and t2 :: "('b \<times> 'c) tree" assumes "Set2_Join join inv"   and "inv t1"   and "inv t2" shows "inv (Set2_Join.inter join t1 t2)"
res_lc3: fixes x :: "'a"   and y :: "'a" shows "(x \<cdot> y \<leftarrow> y) \<cdot> y \<leftarrow> y = x \<cdot> y \<leftarrow> y"
1: assumes "distinct ms"   and "ms \<langle>g\<rangle> [l] \<odot> [l] \<langle>f\<rangle> [l] \<preceq> ms \<langle>g\<rangle> [l]" shows "ms \<langle>g\<rangle> [l] \<odot> [l] \<langle>star \<circ> f\<rangle> [l] \<preceq> ms \<langle>g\<rangle> [l]"
that: fixes x :: "'a" assumes "x \<in> set xs" shows "coprime a x"
l3_inv3_trans: shows "{l3_inv3} TS.trans l3 {> l3_inv3}"
that: fixes x' :: "'b" assumes "x = Some x'"   and "f x' = Some a" shows "thesis"
LatticeReflexivity: fixes F :: "'a \<Rightarrow> bool" shows "\<turnstile> F \<leadsto> F"
subcls1D: fixes C :: "char list"   and D :: "char list"   and P :: "'b prog" assumes "P \<turnstile> C \<prec>\<^sup>1 D" shows "C \<noteq> Object \<and> (\<exists>fs ms. class P C = \<lfloor>(D, fs, ms)\<rfloor>)"
dg_prod_components_3: fixes I :: "V"   and \<AA> :: "V \<Rightarrow> V" shows "dg_prod I \<AA>\<lparr>Dom\<rparr> = (\<lambda>f\<in>\<^sub>\<circ>\<Prod>\<^sub>\<circ>i\<in>\<^sub>\<circ>I. \<AA> i\<lparr>Arr\<rparr>. \<lambda>i\<in>\<^sub>\<circ>I. \<AA> i\<lparr>Dom\<rparr>\<lparr>f\<lparr>i\<rparr>\<rparr>)"
irreflclp_def: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b" shows "r\<^sup>\<noteq>\<^sup>\<noteq> a b = (r a b \<and> a \<noteq> b)"
assms_8: shows "r\<^sup>T \<le> v\<^sup>T"
q_ge_0: fixes p :: "nat"   and q :: "nat" assumes "QR p q" shows "0 < int q"
zero_rel_def: shows "zero_rel = \<zero> / \<one>"
sum_dcharacter_totatives: shows "sum (principal_dchar n) (totatives n) = (if principal_dchar n = principal_dchar n then of_nat (totient n) else 0)"
lower_asymptotic_density_eq_upper: fixes A :: "nat set"   and l :: "real" assumes "lower_asymptotic_density A = l"   and "upper_asymptotic_density A = l" shows "(\<lambda>n. real (card (A \<inter> {..<n})) / real n) \<longlonglongrightarrow> l"
add_increasing: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "class.ordered_comm_monoid_add plus zero less_eq less"   and "less_eq zero a"   and "less_eq b c" shows "less_eq b (plus a c)"
cp_in_scp: fixes \<sigma> :: "'var \<Rightarrow> 'val"   and k :: "nat"   and \<sigma>' :: "'var \<Rightarrow> 'val"   and k' :: "nat" assumes "((\<sigma>, k), \<sigma>', k') \<in> cp" shows "(path \<sigma>, k) \<in> scp \<and> (path \<sigma>', k') \<in> scp"
cases: fixes a :: "('sequent, 'rule, 'link) ctree" assumes "cwf a" obtains l :: "'link" where "a = Link l" and "cwf (pointsTo l)"   | r :: "'rule" and s :: "'sequent" and cts :: "('sequent, 'rule, 'link) ctree fset" where "a = cNode (s, r) cts" and "r \<in> R" and "eff r s (seqOf |`| cts)" and "\<And>ct'. ct' |\<in>| cts \<Longrightarrow> cwf ct'"
match_rews_5: fixes k :: "'b \<rightarrow> [:'b:] \<rightarrow> 'a match" shows "match_scons\<cdot>[::]\<cdot>k = Fixrec.fail"
a_star_CutR: fixes N :: "trm"   and N' :: "trm"   and a :: "coname"   and M :: "trm"   and x :: "name" assumes "N \<longrightarrow>\<^sub>a* N'" shows "Cut <a>.M x.N \<longrightarrow>\<^sub>a* Cut <a>.M x.N'"
tj_stack_more_cong: fixes G :: "('c, 'd) graph_rec_scheme"   and s :: "('e, 'f) tarjan_state_scheme"   and s' :: "('e, 'f) tarjan_state_scheme" assumes "Tarjan G"   and "state.more s = state.more s'" shows "tj_stack s = tj_stack s'"
bounded_subset: fixes T :: "'c set"   and S :: "'c set" assumes "bounded T"   and "S \<subseteq> T" shows "bounded S"
inv\<^sub>1_def: fixes T :: "nat \<Rightarrow> nat"   and A :: "nat \<Rightarrow> nat set"   and j :: "nat" shows "inv\<^sub>1 T A j = (lb T A j \<and> j \<le> n \<and> (\<forall>T' A'. lb T' A' j \<longrightarrow> makespan T \<le> 2 * makespan T'))"
rat_precision_def: fixes prec :: "nat"   and x :: "int"   and y :: "int" shows "rat_precision prec x y = (let d = bitlen x - bitlen y in int prec - d + (if Float \<bar>x\<bar> 0 < Float \<bar>y\<bar> d then 1 else 0))"
add_0: fixes a :: "'c" shows "(0::'c) + a = a"
les_qe_backward: fixes les :: "(real \<times> real \<times> real) list" assumes "\<exists>x. \<forall>(a, b, c)\<in>set les. a * x\<^sup>2 + b * x + c < 0" shows "(\<forall>(a, b, c)\<in>set les. \<exists>x. \<forall>y<x. a * y\<^sup>2 + b * y + c < 0) \<or> (\<exists>(a', b', c')\<in>set les. a' = 0 \<and> b' \<noteq> 0 \<and> (\<forall>(d, e, f)\<in>set les. \<exists>y'>- (c' / b'). \<forall>x\<in>{- (c' / b')<..y'}. d * x\<^sup>2 + e * x + f < 0) \<or> a' \<noteq> 0 \<and> 4 * a' * c' \<le> b'\<^sup>2 \<and> ((\<forall>(d, e, f)\<in>set les. \<exists>y'>(sqrt (b'\<^sup>2 - 4 * a' * c') - b') / (2 * a'). \<forall>x\<in>{(sqrt (b'\<^sup>2 - 4 * a' * c') - b') / (2 * a')<..y'}. d * x\<^sup>2 + e * x + f < 0) \<or> (\<forall>(d, e, f)\<in>set les. \<exists>y'>(- b' - sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'). \<forall>x\<in>{(- b' - sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}. d * x\<^sup>2 + e * x + f < 0)))"
NewFail\<^sub>1: fixes h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> nat exp) list) list"   and C :: "char list"   and l :: "val list" assumes "new_Addr h = None" shows "P \<turnstile>\<^sub>1 \<langle>new C,(h, l)\<rangle> \<Rightarrow> \<langle>THROW OutOfMemory,(h, l)\<rangle>"
child_odd: fixes d :: "nat"   and p :: "(nat \<times> int) list"   and dir :: "dir" assumes "d < length p" shows "odd (ix (child p dir d) d)"
map_ident: fixes t :: "'b list" shows "map (\<lambda>x. x) t = t"
Abs_matrix_inverse: fixes y :: "nat \<Rightarrow> nat \<Rightarrow> 'b" assumes "y \<in> matrix" shows "Rep_matrix (Abs_matrix y) = y"
reciprocal_poly_def: fixes p :: "nat"   and P :: "'a poly" shows "reciprocal_poly p P = Poly (rev (coeffs P @ replicate (p - degree P) (0::'a)))"