hcju/isabelleps huggingface dataset:
run_properties_2: fixes \<chi> :: "'a ltl"   and \<phi> :: "'a ltl"   and i :: "nat"   and q :: "'a ltl_prop_equiv_quotient" assumes "\<chi> \<in> \<^bold>G \<phi>" shows "the (snd (run (\<A>.\<delta>\<^sub>\<A> \<Sigma>) (\<A>.\<iota>\<^sub>\<A> \<phi>) w i) \<chi>) q = semi_mojmir_def.state_rank \<Sigma> \<up>af\<^sub>G (Abs (theG \<chi>)) w q i"
unat_plus_if_size: fixes x :: "'b word"   and y :: "'b word" shows "unat (x + y) = (if unat x + unat y < 2 ^ size x then unat x + unat y else unat x + unat y - 2 ^ size x)"
InitObject\<^sub>1: fixes sh :: "sheap"   and C :: "char list"   and sfs :: "sfields"   and sh' :: "sheap"   and P :: "nat exp prog"   and C' :: "char list"   and Cs :: "char list list"   and e :: "nat exp"   and h :: "heap"   and l :: "val list"   and e' :: "nat exp"   and s' :: "heap \<times> val list \<times> sheap" assumes "sh C = \<lfloor>(sfs, Prepared)\<rfloor>"   and "C = Object"   and "sh' = sh(C \<mapsto> (sfs, Processing))"   and "P \<turnstile>\<^sub>1 \<langle>INIT C' (C # Cs,True) \<leftarrow> e,(h, l, sh')\<rangle> \<Rightarrow> \<langle>e',s'\<rangle>" shows "P \<turnstile>\<^sub>1 \<langle>INIT C' (C # Cs,False) \<leftarrow> e,(h, l, sh)\<rangle> \<Rightarrow> \<langle>e',s'\<rangle>"
assms_2: shows "set xs \<subseteq> A \<times> UNIV"
swapcols_mat: fixes A :: "'b mat"   and nr :: "nat"   and n :: "nat"   and k :: "nat"   and l :: "nat" assumes "A \<in> carrier_mat nr n"   and "k < n"   and "l < n" shows "swapcols k l A = A * swaprows_mat n k l"
poly_smult: fixes a :: "'a"   and p :: "'a poly"   and x :: "'a" shows "poly (smult a p) x = a * poly p x"
BinOp_\<tau>red1r_xt2: fixes uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and h :: "'heap"   and e2 :: "'addr expr1"   and xs :: "'addr val list"   and e2' :: "'addr expr1"   and xs' :: "'addr val list"   and v :: "'addr val"   and bop :: "bop" assumes "\<tau>red1gr uf P t h (e2, xs) (e2', xs')" shows "\<tau>red1gr uf P t h (Val v \<guillemotleft>bop\<guillemotright> e2, xs) (Val v \<guillemotleft>bop\<guillemotright> e2', xs')"
affine_scheme_is_scheme: fixes R :: "'d set"   and addition :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and multiplication :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and unit :: "'d"   and X :: "'e set"   and is_open :: "'e set \<Rightarrow> bool"   and \<O>\<^sub>X :: "'e set \<Rightarrow> 'f set"   and \<rho> :: "'e set \<Rightarrow> 'e set \<Rightarrow> 'f \<Rightarrow> 'f"   and b :: "'f"   and add_str :: "'e set \<Rightarrow> 'f \<Rightarrow> 'f \<Rightarrow> 'f"   and mult_str :: "'e set \<Rightarrow> 'f \<Rightarrow> 'f \<Rightarrow> 'f"   and zero_str :: "'e set \<Rightarrow> 'f"   and one_str :: "'e set \<Rightarrow> 'f"   and f :: "'e \<Rightarrow> 'd set"   and \<phi>\<^sub>f :: "'d set set \<Rightarrow> ('d set \<Rightarrow> ('d \<times> 'd) set) \<Rightarrow> 'f" assumes "affine_scheme R addition multiplication zero unit X is_open \<O>\<^sub>X \<rho> b add_str mult_str zero_str one_str f \<phi>\<^sub>f" shows "scheme R addition multiplication zero unit X is_open \<O>\<^sub>X \<rho> b add_str mult_str zero_str one_str"
less_2p_is_upper_bits_unset: fixes p :: "'b word"   and n :: "nat" shows "(p < 2 ^ n) = (n < LENGTH('b) \<and> (\<forall>n'\<ge>n. n' < LENGTH('b) \<longrightarrow> \<not> p !! n'))"
Repr_Neg_axioms: shows "Repr_Neg var trm fmla Var FvarsT substT Fvars subst eql cnj imp all exi fls num prv bprv enc N"
split_int_gt_real': fixes a :: "int"   and b :: "real" shows "(0 < real_of_int a + b) = (0 < real_of_int a + real_of_int \<lfloor>b\<rfloor> \<or> real_of_int a + real_of_int \<lfloor>b\<rfloor> = 0 \<and> real_of_int \<lfloor>b\<rfloor> - b < 0)"
mono_neutral_left: fixes T :: "'c set"   and S :: "'c set"   and g :: "'c \<Rightarrow> 'a" assumes "finite T"   and "S \<subseteq> T"   and "\<forall>i\<in>T - S. g i = \<^bold>1" shows "F g S = F g T"
u_4: shows "(\<lambda>x. ereal (u x)) \<longlonglongrightarrow> b"
stopping_time_le_const: fixes \<Omega> :: "'c set"   and F :: "'d \<Rightarrow> 'c measure"   and T :: "'c \<Rightarrow> 'd"   and s :: "'d"   and t :: "'d" assumes "filtration \<Omega> F"   and "stopping_time F T"   and "s \<le> t" shows "Measurable.pred (F t) (\<lambda>\<omega>. T \<omega> \<le> s)"
map_upd_eqD1: fixes m :: "'a \<Rightarrow> 'b option"   and a :: "'a"   and x :: "'b"   and n :: "'a \<Rightarrow> 'b option"   and y :: "'b" assumes "m(a \<mapsto> x) = n(a \<mapsto> y)" shows "x = y"
meet_commute: fixes x :: "'a"   and y :: "'a" shows "x \<sqinter> y = y \<sqinter> x"
assign_local_def: fixes upd :: "('b list \<Rightarrow> 'b list) \<Rightarrow> 'a control_state_scheme \<Rightarrow> 'a control_state_scheme"   and rhs :: "'a control_state_scheme \<Rightarrow> 'b" shows "assign_local upd rhs = assign (\<lambda>\<sigma>. (upd \<circ> map_hd) (\<lambda>_. rhs \<sigma>) \<sigma>)"
harry_meas_step: fixes xs :: "bool list" assumes "True \<in> set xs" shows "harry_meas (harry_step xs) = harry_meas xs - 1"
agree_trans: fixes \<nu> :: "(real, 'd) vec \<times> (real, 'd) vec"   and \<mu> :: "(real, 'd) vec \<times> (real, 'd) vec"   and A :: "('d + 'd) set"   and \<omega> :: "(real, 'd) vec \<times> (real, 'd) vec"   and B :: "('d + 'd) set" assumes "Vagree \<nu> \<mu> A"   and "Vagree \<mu> \<omega> B" shows "Vagree \<nu> \<omega> (A \<inter> B)"
objI_src: fixes a :: "'a" assumes "B.arr a"   and "src\<^sub>B a = a" shows "B.obj a"
reindex_cong: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and z :: "'b"   and l :: "'c \<Rightarrow> 'd"   and B :: "'c set"   and A :: "'d set"   and g :: "'d \<Rightarrow> 'b"   and h :: "'c \<Rightarrow> 'b" assumes "abel_semigroup_set f z"   and "inj_on l B"   and "A = l ` B"   and "\<And>x. x \<in> B \<Longrightarrow> g (l x) = h x" shows "abel_semigroup_set.F f z g A = abel_semigroup_set.F f z h B"
axioms_1: fixes \<alpha> :: "V"   and \<CC> :: "V" assumes "digraph \<alpha> \<CC>" shows "\<Z> \<alpha>"
Le_Lt_dbm_lt_D: fixes a :: "'b"   and b :: "'b" assumes "Le a \<prec> Le b" shows "a < b"
sup_of_singleton_closed: fixes x :: "'a" assumes "x \<in> carrier L" shows "\<Squnion>{x} \<in> carrier L"
upto_aux_def: fixes i :: "int"   and j :: "int"   and js :: "int list" shows "upto_aux i j js = [i..j] @ js"
strip_comb_app: fixes size :: "'b \<Rightarrow> nat"   and frees :: "'b \<Rightarrow> name fset"   and subst :: "'b \<Rightarrow> (name, 'b) fmap \<Rightarrow> 'b"   and "consts" :: "'b \<Rightarrow> name fset"   and app :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unapp :: "'b \<Rightarrow> ('b \<times> 'b) option"   and const :: "name \<Rightarrow> 'b"   and unconst :: "'b \<Rightarrow> name option"   and free :: "name \<Rightarrow> 'b"   and unfree :: "'b \<Rightarrow> name option"   and abs_pred :: "('b \<Rightarrow> bool) \<Rightarrow> 'b \<Rightarrow> bool"   and t :: "'b"   and u :: "'b" assumes "class.term size frees subst consts app unapp const unconst free unfree abs_pred" shows "fst (term.strip_comb unapp (app t u)) = fst (term.strip_comb unapp t)"
has: shows "P \<turnstile> C has F,NonStatic:TF in D"
Sup_pres_ladj_aux: fixes f :: "'c \<Rightarrow> 'd" assumes "Sup_pres f" shows "f \<stileturn> radj f"
cmp_rfn0: fixes r :: "nat"   and s :: "nat"   and f :: "nat \<Rightarrow> 'a set"   and g :: "nat \<Rightarrow> 'a set"   and i :: "nat"   and j :: "nat" assumes "0 < r"   and "0 < s"   and "compseries G r f"   and "compseries G s g"   and "i \<le> r - 1"   and "j \<le> s - 1" shows "G \<guillemotright> f (Suc i) \<diamondop> f i \<inter> g j "
induct: fixes P1 :: "Term.var \<Rightarrow> bool"   and P2 :: "expr \<Rightarrow> bool"   and P3 :: "stmt \<Rightarrow> bool"   and expr :: "expr" assumes "\<And>x. P1 (LVar x)"   and "\<And>x1 x2 x3 x4 x5. P2 x4 \<Longrightarrow> P1 ({x1,x2,x3}x4..x5)"   and "\<And>x1 x2. \<lbrakk>P2 x1; P2 x2\<rbrakk> \<Longrightarrow> P1 (x1.[x2])"   and "\<And>x1 x2. \<lbrakk>P3 x1; P1 x2\<rbrakk> \<Longrightarrow> P1 (InsInitV x1 x2)"   and "\<And>x. P2 (NewC x)"   and "\<And>x1 x2. P2 x2 \<Longrightarrow> P2 (New x1[x2])"   and "\<And>x1 x2. P2 x2 \<Longrightarrow> P2 (Cast x1 x2)"   and "\<And>x1 x2. P2 x1 \<Longrightarrow> P2 (x1 InstOf x2)"   and "\<And>x. P2 (Lit x)"   and "\<And>x1 x2. P2 x2 \<Longrightarrow> P2 (UnOp x1 x2)"   and "\<And>x1 x2 x3. \<lbrakk>P2 x2; P2 x3\<rbrakk> \<Longrightarrow> P2 (BinOp x1 x2 x3)"   and "P2 Super"   and "\<And>x. P1 x \<Longrightarrow> P2 (Acc x)"   and "\<And>x1 x2. \<lbrakk>P1 x1; P2 x2\<rbrakk> \<Longrightarrow> P2 (x1:=x2)"   and "\<And>x1 x2 x3. \<lbrakk>P2 x1; P2 x2; P2 x3\<rbrakk> \<Longrightarrow> P2 (x1 ? x2 : x3)"   and "\<And>x1 x2 x3 x4 x5 x6 x7. \<lbrakk>P2 x4; \<And>x7a. x7a \<in> set x7 \<Longrightarrow> P2 x7a\<rbrakk> \<Longrightarrow> P2 ({x1,x2,x3}x4\<cdot>x5( {x6}x7))"   and "\<And>x1 x2. P2 (Methd x1 x2)"   and "\<And>x1 x2. P3 x2 \<Longrightarrow> P2 (Body x1 x2)"   and "\<And>x1 x2. \<lbrakk>P3 x1; P2 x2\<rbrakk> \<Longrightarrow> P2 (InsInitE x1 x2)"   and "\<And>x1 x2. P2 x2 \<Longrightarrow> P2 (Callee x1 x2)"   and "P3 Skip"   and "\<And>x. P2 x \<Longrightarrow> P3 (Expr x)"   and "\<And>x1 x2. P3 x2 \<Longrightarrow> P3 (x1\<bullet> x2)"   and "\<And>x1 x2. \<lbrakk>P3 x1; P3 x2\<rbrakk> \<Longrightarrow> P3 (x1;; x2)"   and "\<And>x1 x2 x3. \<lbrakk>P2 x1; P3 x2; P3 x3\<rbrakk> \<Longrightarrow> P3 (If(x1) x2 Else x3)"   and "\<And>x1 x2 x3. \<lbrakk>P2 x2; P3 x3\<rbrakk> \<Longrightarrow> P3 (x1\<bullet> While(x2) x3)"   and "\<And>x. P3 (Jmp x)"   and "\<And>x. P2 x \<Longrightarrow> P3 (Throw x)"   and "\<And>x1 x2 x3 x4. \<lbrakk>P3 x1; P3 x4\<rbrakk> \<Longrightarrow> P3 (Try x1 Catch(x2 x3) x4)"   and "\<And>x1 x2. \<lbrakk>P3 x1; P3 x2\<rbrakk> \<Longrightarrow> P3 (x1 Finally x2)"   and "\<And>x1 x2. P3 x2 \<Longrightarrow> P3 (FinA x1 x2)"   and "\<And>x. P3 (Init x)" shows "P2 expr"
source_sink: shows "source \<Delta> \<noteq> sink \<Delta>"
sx0: shows "sx \<cdot> (s1 \<cdot> r2 - s2 \<cdot> r1) = \<zero>"
incident_own_polar_in_S: fixes p :: "proj2" shows "proj2_incident p (polar p) = (p \<in> S)"
finite_proj_image': fixes x :: "'e"   and P :: "'e \<Rightarrow>\<^sub>F 'f"   and S :: "'e set" assumes "x \<notin> domain P" shows "finite ((P)\<^sub>F ` S)"
h: shows "\<forall>x. x \<in> P \<longrightarrow> derivable_ecl x S \<and> ((\<forall>x\<in>S. finite (cl_ecl x)) \<longrightarrow> finite (cl_ecl x))"
IfHigh: fixes G :: "((Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool) set"   and c :: "OBJ"   and d :: "OBJ"   and b :: "BExpr" assumes "G \<rhd> c : HighSec"   and "G \<rhd> d : HighSec" shows "G \<rhd> Iff b c d : HighSec"
Cons_acc_listrel1I: fixes x :: "'a"   and r :: "('a \<times> 'a) set"   and xs :: "'a list" assumes "x \<in> Wellfounded.acc r"   and "xs \<in> Wellfounded.acc (listrel1 r)" shows "x # xs \<in> Wellfounded.acc (listrel1 r)"
widen_antisym: fixes m :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list \<Rightarrow> char list \<Rightarrow> char list \<times> ty list \<times> ty \<times> 'b \<Rightarrow> bool"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list"   and T :: "ty"   and U :: "ty" assumes "wf_prog m P"   and "subtype P T U"   and "subtype P U T" shows "T = U"
IH: fixes x :: "'a prefix_match option"   and y :: "'a wordinterval"   and x2 :: "'a prefix_match" assumes "\<not> wordinterval_empty rs"   and "(x, y) = wordinterval_CIDR_split1 rs"   and "x = Some x2"   and "a \<in> set (wordinterval_CIDR_split_prefixmatch y)"   and "b \<in> set (wordinterval_CIDR_split_prefixmatch y)"   and "a \<noteq> b" shows "prefix_to_wordset a \<inter> prefix_to_wordset b = {}"
length_filter_less: fixes x :: "'b"   and xs :: "'b list"   and P :: "'b \<Rightarrow> bool" assumes "x \<in> set xs"   and "\<not> P x" shows "length (filter P xs) < length xs"
set_integrable_def: fixes M :: "'a measure"   and A :: "'a set"   and f :: "'a \<Rightarrow> 'd" shows "set_integrable M A f \<equiv> integrable M (\<lambda>x. indicat_real A x *\<^sub>R f x)"
p': shows "\<forall>y\<in>S - {p}. norm (v - p) < norm (v - y) \<and> v - p \<in> orthogonal_complement S"
IH: fixes S :: "(char list \<Rightarrow> int) set"   and S' :: "'av Abs_State.st option"   and c' :: "'av Abs_State.st option acom" assumes "S \<subseteq> \<gamma>\<^sub>o S'"   and "c1 \<le> \<gamma>\<^sub>c c'" shows "step S c1 \<le> \<gamma>\<^sub>c (step' S' c')"
enum_0_eq_Inf_finite: fixes N :: "nat set" assumes "finite N"   and "N \<noteq> {}" shows "enum N 0 = \<Sqinter> N"
at_rev_pi: fixes pi :: "('a \<times> 'a) list"   and c :: "'a" assumes "at TYPE('a)" shows "rev pi \<bullet> pi \<bullet> c = c"
convexI: fixes s :: "'c set" assumes "\<And>x y u v. \<lbrakk>x \<in> s; y \<in> s; 0 \<le> u; 0 \<le> v; u + v = 1\<rbrakk> \<Longrightarrow> u *\<^sub>R x + v *\<^sub>R y \<in> s" shows "convex s"
domain_def: fixes index :: "nat \<times> nat" shows "local.domain index \<equiv> {..<fst index}"
squarefree_part_pos: fixes n :: "nat" shows "0 < \<Prod>(squarefree_part n)"
assms_2: shows "{\<lambda>tp. tp = ([], <args>)} p \<up>"
right_one_impl: fixes a :: "'a" shows "(1::'a) l\<rightarrow> a = a"
B: assumes "c |\<notin>| A"   and "c |\<in>| B" shows "thesis"
FAccReds: fixes e :: "char list exp"   and s :: "heap \<times> sfields \<times> sheap"   and b :: "bool"   and e' :: "char list exp"   and s' :: "heap \<times> sfields \<times> sheap"   and b' :: "bool"   and P :: "(char list list \<times> char list exp) prog"   and F :: "char list"   and D :: "char list" assumes "P \<turnstile> \<langle>e,s,b\<rangle> \<rightarrow>* \<langle>e',s',b'\<rangle>" shows "P \<turnstile> \<langle>e\<bullet>F{D},s,b\<rangle> \<rightarrow>* \<langle>e'\<bullet>F{D},s',b'\<rangle>"
separated_by_not_empty: fixes w :: "'b set set"   and x :: "'b"   and y :: "'b" assumes "separated_by w x y" shows "w \<noteq> {}"
section_def: fixes f :: "'s" shows "section f \<equiv> \<exists>g. ide (g \<cdot> f)"
cases: fixes a1 :: "'thread_id"   and a2 :: "(status \<times> 'x) \<times> 'heap"   and a3 :: "('addr, 'thread_id, status \<times> 'x, 'heap, 'addr, ('addr, 'thread_id) obs_event action) thread_action"   and a4 :: "(status \<times> 'x) \<times> 'heap" assumes "a1 \<turnstile> a2 -a3\<rightarrow>i a4" obtains t :: "'thread_id" and x :: "'x" and m :: "'heap" and ta :: "('addr, 'thread_id, 'x, 'heap) Jinja_thread_action" and x' :: "'x" and m' :: "'heap" where "a1 = tb__" and "a2 = ((Running, x__), m__)" and "a3 = convert_TA_initial (convert_obs_initial ta)" and "a4 = ((Running, x'), m')" and "tb__ \<turnstile> \<langle>x__, m__\<rangle> -ta\<rightarrow> \<langle>x', m'\<rangle>"   | t :: "'thread_id" and x :: "'x" and m :: "'heap" where "a1 = tc__" and "a2 = ((PreStart, xa__), ma__)" and "a3 = \<lbrace>InitialThreadAction\<rbrace>" and "a4 = ((Running, xa__), ma__)"   | x :: "'x" and t :: "'thread_id" and m :: "'heap" where "a1 = t" and "a2 = ((Running, x), m)" and "a3 = \<lbrace>ThreadFinishAction\<rbrace>" and "a4 = ((Finished, x), m)" and "final x"
typing_implies_RED: fixes \<Gamma> :: "(name \<times> ty) list"   and t :: "trm"   and \<tau> :: "ty" assumes "\<Gamma> \<turnstile> t : \<tau>" shows "t \<in> RED \<tau>"
pmc_tautology: fixes K :: "'a set"   and A :: "'a set"   and \<phi> :: "'a" assumes "K = Cn A"   and "\<tturnstile> \<phi>" shows "K \<div>\<^bsub>\<gamma>\<^sub>A\<^sub>G\<^sub>M\<^esub> \<phi> = K"
Semi: fixes eval :: "exp \<Rightarrow> (loc \<Rightarrow> nat) \<Rightarrow> nat \<Rightarrow> (loc \<Rightarrow> nat) \<Rightarrow> bool"   and c0 :: "com"   and s :: "loc \<Rightarrow> nat"   and s2 :: "loc \<Rightarrow> nat"   and c1 :: "com"   and s1 :: "loc \<Rightarrow> nat" assumes "execp eval c0 s s2"   and "execp eval c1 s2 s1" shows "execp eval (c0;;c1) s s1"
leI: fixes x :: "'b"   and y :: "'b" assumes "\<not> x < y" shows "y \<le> x"
enum_in: fixes i :: "nat" assumes "i \<le> n" shows "enum i \<in> s"
lookup_empty': fixes ks :: "'d list" shows "lookup_trie (Trie None []) ks = None"
Q1: shows "\<forall>B'. A A' OS B B' \<and> Per B' A' A \<longrightarrow> Per B' A' C''"
is_ext_def: fixes y' :: "'c"   and g :: "'d"   and f :: "'c" shows "y\<eta>.is_ext y' g f \<equiv> \<guillemotleft>f : Fo y \<rightarrow>\<^sub>C y'\<guillemotright> \<and> g = G f \<cdot>\<^sub>D \<eta>o y"
float_numeral: fixes x :: "num" shows "real_of_float (numeral x) = numeral x"
multi_self_add_other_not_self_zmset: fixes M :: "'a zmultiset"   and x :: "'a" shows "(M = add_zmset x M) = False"
error_msg_not_equal_dom_def: fixes p :: "'a"   and q :: "'b"   and pq :: "'c"   and r :: "'d" shows "error_msg_not_equal_dom p q pq r = show p @ '' + '' @ show q @ '' = '' @ show pq @ '' not equal'' @ show r"
derivation_shift_empty: fixes \<NN> :: "symbol set"   and \<TT> :: "symbol set"   and \<RR> :: "(symbol \<times> symbol list) set"   and \<SS> :: "symbol"   and left :: "nat"   and right :: "nat" assumes "CFG \<NN> \<TT> \<RR> \<SS>" shows "derivation_shift [] left right = []"
styping_preservation: fixes c :: "com"   and s :: "char list \<Rightarrow> val"   and c' :: "com"   and s' :: "char list \<Rightarrow> val"   and \<Gamma> :: "char list \<Rightarrow> ty" assumes "(c, s) \<rightarrow> (c', s')"   and "\<Gamma> \<turnstile> c"   and "\<Gamma> \<turnstile> s" shows "\<Gamma> \<turnstile> s'"
psi_lR_is_beta: shows "psi_l R = \<beta>"
ih: shows "card (IDirProds G S I) = (\<Prod>i\<in>I. card (S i))"
simps_5: fixes v :: "val"   and e :: "'a exp" shows "init_switch (Val v) e = Val v"
return_def: fixes a :: "'b" shows "State_Monad.return a = State (Pair a)"
A: shows "y r\<rightarrow> x l\<rightarrow> z = (1::'a)"
IA2: shows "Iagree I J {Inl x |x. x \<in> SIGT \<theta>\<^sub>2}"
maybe_and_valid: fixes x :: "trilean"   and y :: "trilean" assumes "x \<and>? y \<noteq> invalid" shows "x \<noteq> invalid \<and> y \<noteq> invalid"
assms_2: shows "mod_type_class.to_nat a \<noteq> mod_type_class.to_nat b"
path_unique: fixes \<E> :: "'b set"   and \<P> :: "'b set set"   and ab :: "'b set"   and a :: "'b"   and b :: "'b"   and ab' :: "'b set" assumes "MinkowskiPrimitive \<E> \<P>"   and "ab \<in> \<P> \<and> a \<in> ab \<and> b \<in> ab \<and> a \<noteq> b"   and "ab' \<in> \<P> \<and> a \<in> ab' \<and> b \<in> ab' \<and> a \<noteq> b" shows "ab = ab'"
induct: fixes x1 :: "'b list"   and x2 :: "'b list"   and P :: "'b list \<Rightarrow> 'b list \<Rightarrow> bool" assumes "prefixList x1 x2"   and "\<And>x xs. P [] (x # xs)"   and "\<And>xa xb x. \<lbrakk>prefixList xa xb; P xa xb\<rbrakk> \<Longrightarrow> P (x # xa) (x # xb)" shows "P x1 x2"
ord_transl: fixes x :: "'a"   and y :: "'a" shows "(!x \<le> !y) = (!x \<sqsubseteq> !y)"
equalOn_Un: fixes A :: "'c set"   and B :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" shows "equalOn (A \<union> B) f g = (equalOn A f g \<and> equalOn B f g)"
bnd: fixes x :: "'b" shows "\<integral>\<^sup>+ y. f y x \<partial>M \<noteq> \<infinity>"
k_prop: shows "\<forall>x>k. a * x\<^sup>2 + b * x + c < 0"
hfref_cons: fixes f :: "'b \<Rightarrow> 'd Heap"   and g :: "'e \<Rightarrow> 'f nres"   and P :: "'e \<Rightarrow> bool"   and R :: "('e \<Rightarrow> 'b \<Rightarrow> assn) \<times> ('e \<Rightarrow> 'b \<Rightarrow> assn)"   and S :: "'f \<Rightarrow> 'd \<Rightarrow> assn"   and P' :: "'e \<Rightarrow> bool"   and R' :: "('e \<Rightarrow> 'b \<Rightarrow> assn) \<times> ('e \<Rightarrow> 'b \<Rightarrow> assn)"   and S' :: "'f \<Rightarrow> 'd \<Rightarrow> assn" assumes "(f, g) \<in> [P]\<^sub>a R \<rightarrow> S"   and "\<And>x. P' x \<Longrightarrow> P x"   and "\<And>x y. fst R' x y \<Longrightarrow>\<^sub>t fst R x y"   and "\<And>x y. snd R x y \<Longrightarrow>\<^sub>t snd R' x y"   and "\<And>x y. S x y \<Longrightarrow>\<^sub>t S' x y" shows "(f, g) \<in> [P']\<^sub>a R' \<rightarrow> S'"
errMOD_igAbsRenSTR: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) model" assumes "igVarIPresIGWls MOD"   and "igSubstIPresIGWls MOD"   and "igWlsDisj MOD"   and "igAbsRen MOD" shows "igAbsRenSTR (errMOD MOD)"
equiv'_rotate_2: fixes b :: "'b"   and a :: "'a" assumes "B_A.equiv' b a" shows "A_B.equiv' a b"
set_def: fixes al :: "('c, 'd) assoc_list" shows "Assoc_List.set al = set (assoc_list.impl_of al)"
vsv_value: fixes r :: "V"   and a :: "V" assumes "vsv r"   and "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> r" obtains b :: "V" where "r\<lparr>a\<rparr> = b" and "b \<in>\<^sub>\<circ> \<R>\<^sub>\<circ> r"
congruences_def: shows "congruences = {R. equivp R \<and> (\<forall>a b c d. R a b \<and> R c d \<longrightarrow> R (a * c) (b * d) \<and> R (a l\<rightarrow> c) (b l\<rightarrow> d) \<and> R (a r\<rightarrow> c) (b r\<rightarrow> d))}"
simps_8: fixes e :: "expr"   and F :: "char list"   and Cs :: "char list list" shows "fv (e\<bullet>F{Cs}) = fv e"
canonical_proj_ext_is_hom: fixes n :: "nat"   and I :: "nat \<Rightarrow> 'a set" assumes "\<And>i. i \<le> n \<Longrightarrow> ideal (I i) R" shows "canonical_proj_ext I n \<in> ring_hom R (RDirProd_list (map (\<lambda>i. R Quot I i) [0..<Suc n]))"
arrI: fixes f :: "'a" assumes "\<guillemotleft>f : src (src f) \<rightarrow> src e\<guillemotright>" shows "hom.arr f"
Cons_2: shows "(name, rhs) \<in> set (r # rs)"
ub_def: shows "ub = (\<lambda>x. sqrt x / (ln 2)\<^sup>2 + 4 * x / (ln x)\<^sup>2)"
simps_1: fixes r :: "'b rexp" shows "pderivs [] r = {r}"
sum_upto_moebius_times_floor_linear: fixes x :: "real" shows "sum_upto (\<lambda>n. moebius_mu n * \<lfloor>x / real n\<rfloor>) x = (if 1 \<le> x then 1 else 0)"
Var: fixes x :: "'b"   and y :: "'c" shows "emv (Var x) (Var y)"
sem: shows "(\<nu>, \<omega>) \<in> prog_sem I a**"
sin_plus_pi: fixes z :: "'b" shows "sin (z + of_real pi) = - sin z"
exists_desc_box_1: fixes v :: "i"   and \<phi> :: "\<nu> \<Rightarrow> \<o>" shows "[(\<^bold>\<exists>y. y\<^sup>P \<^bold>= (\<^bold>\<iota>x. \<phi> x)) \<^bold>\<rightarrow> (\<^bold>\<exists>y. \<^bold>\<box>y\<^sup>P \<^bold>= (\<^bold>\<iota>x. \<phi> x)) in v]"
B': shows "BIT_inv (Partial_Cost_Model.config'_rand BIT Da qs) (last qs) [x, y]"
permute_list_ident: fixes xs :: "'b list" shows "permute_list (\<lambda>x. x) xs = xs"
ctxt_sub_def: fixes C :: "db"   and e :: "db" shows "C<e> \<equiv> closing_subst C (\<lambda>_. e) 0"
bind_returns_heap_I: fixes h :: "'a"   and f :: "('a, 'b, 'c) prog"   and x :: "'c"   and h' :: "'a"   and g :: "'c \<Rightarrow> ('a, 'b, 'd) prog"   and h'' :: "'a" assumes "h \<turnstile> f \<rightarrow>\<^sub>r x"   and "h \<turnstile> f \<rightarrow>\<^sub>h h'"   and "h' \<turnstile> g x \<rightarrow>\<^sub>h h''" shows "h \<turnstile> f \<bind> g \<rightarrow>\<^sub>h h''"
insert_absorb2: fixes x :: "'a"   and A :: "'a set" shows "insert x (insert x A) = insert x A"
m1a_def: shows "m1a \<equiv> \<lparr>init = m1a_init, trans = m1a_trans, obs = id\<rparr>"
select_convs_4: fixes NonceS :: "rat option"   and IntMapK :: "rat option"   and ExtMapK :: "rat option"   and IntAgrK :: "rat option"   and ExtAgrK :: "rat option"   and more :: "'a" shows "IntAgrK \<lparr>NonceS = NonceS, IntMapK = IntMapK, ExtMapK = ExtMapK, IntAgrK = IntAgrK, ExtAgrK = ExtAgrK, \<dots> = more\<rparr> = IntAgrK"
CFG_Use_SDG_Use: fixes m :: "'node"   and V :: "'var"   and n :: "'node SDG_node" assumes "valid_node m"   and "V \<in> Use m"   and "n = CFG_node m" shows "V \<in> Use\<^bsub>SDG\<^esub> n"
assms: fixes i :: "'i"   and j :: "'i" assumes "i \<in> I"   and "j \<in> I"   and "i \<noteq> j" shows "indep (A i) (A j)"
bounded_clinear_axioms_def: fixes f :: "'d \<Rightarrow> 'e" shows "bounded_clinear_axioms f \<equiv> \<exists>K. \<forall>x. norm (f x) \<le> norm x * K"
index_map_vec_2: fixes f :: "'c \<Rightarrow> 'b"   and v :: "'c vec" shows "dim_vec (map_vec f v) = dim_vec v"
coherent_Runit'_Ide: fixes a :: "'c local.term" assumes "Ide a" shows "coherent \<^bold>\<r>\<^sup>-\<^sup>1\<^bold>[a\<^bold>]"