hcju/isabelleps huggingface dataset:
pSilDecVal_def: fixes v :: "bool"   and p :: "'p"   and c :: "('p, 'v, 's) configuration" shows "pSilDecVal v p c \<equiv> initReachable c \<and> (\<exists>c'. qReachable c (Proc - {p}) c' \<and> initReachable c' \<and> <\<bottom>, outM v> \<in># msgs c')"
Max_def: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "class.linorder less_eq less" shows "linorder.Max less_eq \<equiv> semilattice_set.F (ord.max less_eq)"
nonnegative_absolutely_integrable: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set" assumes "f integrable_on A"   and "\<And>x b. \<lbrakk>x \<in> A; b \<in> Basis\<rbrakk> \<Longrightarrow> 0 \<le> f x \<bullet> b" shows "f absolutely_integrable_on A"
non_sdominate_succ: fixes G :: "nat graph_rec"   and i :: "nat"   and j :: "nat"   and k :: "nat" assumes "cfg_doms G"   and "(i, j) \<in> g_E G"   and "k \<noteq> i"   and "cfg_doms.non_strict_dominate G k i" shows "cfg_doms.non_strict_dominate G k j"
Plus_FullL: fixes r :: "'a language" shows "Plus \<Sigma>\<^sup>* r = \<Sigma>\<^sup>*"
of_match_fields_safe_eq: fixes m :: "of_match_field set" assumes "all_prerequisites m" shows "OF_match_fields_safe m = OF_match_fields_unsafe m"
bind_gpv_def: shows "(\<bind>) \<equiv> \<lambda>uu uua. corec_gpv (\<lambda>(r, f). map_spmf (map_generat id id ((\<circ>) (\<lambda>uu. case uu of Inl uu \<Rightarrow> Inl (id uu) | Inr r \<Rightarrow> Inr (r, f)))) (the_gpv r \<bind> case_generat (\<lambda>x. map_spmf (map_generat id id ((\<circ>) Inl)) (the_gpv (f x))) (\<lambda>out c. return_spmf (IO out (\<lambda>input. Inr (c input)))))) (uu, uua)"
emeasure_PiM: fixes I :: "'i set"   and A :: "'i \<Rightarrow> 'a set" assumes "finite I"   and "\<And>i. i \<in> I \<Longrightarrow> A i \<in> M.events" shows "emeasure (Pi\<^sub>M I (\<lambda>_. borel)) (Pi\<^sub>E I A) = (\<Prod>i\<in>I. emeasure borel (A i))"
Max_image_cong: fixes S :: "'c set"   and T :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "S = T"   and "\<And>x. x \<in> T \<Longrightarrow> f x = g x" shows "Max (f ` S) = Max (g ` T)"
Ord_linear_le: fixes k :: "V"   and l :: "V" assumes "Ord k"   and "Ord l" obtains "k \<subseteq>\<^sub>\<circ> l"   | "l \<subseteq>\<^sub>\<circ> k"
is_singletonE: fixes A :: "'c set" assumes "is_singleton A" obtains x :: "'c" where "A = {x}"
not_single_rt_fresher: fixes rt1 :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option"   and ip :: "nat"   and rt2 :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option" assumes "\<not> the (rt1 ip) \<sqsubseteq> the (rt2 ip)" shows "\<not> rt1 \<sqsubseteq>\<^bsub>ip\<^esub> rt2"
psi_ubound_log_double_cases: fixes m :: "nat"   and c :: "real"   and c' :: "real"   and m' :: "nat" assumes "\<forall>n\<le>m. psi n \<le> c * ln 2 * real n"   and "1 + c / 2 + (4 * ln (real m + 1) + 3) / (ln 2 * (real m + 1)) \<le> c'"   and "m' = 2 * m"   and "c \<le> c'"   and "0 \<le> c"   and "1 \<le> m" shows "\<forall>n\<le>m'. psi n \<le> c' * ln 2 * real n"
simps_5: fixes b :: "expr"   and c :: "cmd" shows "#:while (b) c = #:c + 2"
insert_ops_remove1: fixes xs :: "('b \<times> 'b option) list"   and x :: "'b \<times> 'b option" assumes "insert_ops xs" shows "insert_ops (remove1 x xs)"
less_imp_le: fixes x :: "'x \<Rightarrow>\<^sub>0 nat"   and y :: "'x \<Rightarrow>\<^sub>0 nat" assumes "x \<prec> y" shows "x \<preceq> y"
prems_2: shows "eval (Pr n f g) (Suc y # xs) \<down>"
sinvar_no_offending: fixes G :: "'a graph"   and nP :: "'a \<Rightarrow> 'b" assumes "sinvar G nP" shows "set_offending_flows G nP = {}"
last_ConsL: fixes xs :: "'c list"   and x :: "'c" assumes "xs = []" shows "last (x # xs) = x"
sums_integral_count_space_nat: fixes f :: "nat \<Rightarrow> 'b" assumes "integrable (count_space UNIV) f" shows "f sums integral\<^sup>L (count_space UNIV) f"
rtrancl_refl: fixes a :: "'g"   and r :: "('g \<times> 'g) set" shows "(a, a) \<in> r\<^sup>*"
0: fixes xs :: "'a stream" assumes "\<phi>1 xs" shows "\<phi>2 xs"
prod_simps_2: fixes f0 :: "'a"   and f1 :: "'a" assumes "arr f0"   and "arr f1" shows "local.dom (f0 \<otimes> f1) = local.dom f0 \<otimes> local.dom f1"
sum_dcharacter_block: shows "sum (principal_dchar n) {..<n} = (if principal_dchar n = principal_dchar n then of_nat (totient n) else 0)"
SS_def2: fixes t1 :: "'trm"   and t2 :: "'trm" assumes "t1 \<in> trm"   and "t2 \<in> trm"   and "yy \<notin> FvarsT t1" shows "SS t1 t2 = subst (subst S t1 xx) t2 yy"
list_all2_append1: fixes P :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and xs :: "'b list"   and ys :: "'b list"   and zs :: "'c list" shows "list_all2 P (xs @ ys) zs = (\<exists>us vs. zs = us @ vs \<and> length us = length xs \<and> length vs = length ys \<and> list_all2 P xs us \<and> list_all2 P ys vs)"
inj_map_total: fixes \<pi> :: "'c \<Rightarrow> 'd" shows "inj_map (Some \<circ> \<pi>) = inj \<pi>"
non_null_if_pivot: fixes A :: "nat \<Rightarrow> nat \<Rightarrow> 'b"   and m :: "nat"   and n :: "nat"   and x :: "nat \<Rightarrow> 'b"   and q :: "nat" assumes "usolution A m n x"   and "q < m" shows "\<exists>p<m. A p q \<noteq> (0::'b)"
smc_SemiCAT_components_5: fixes \<alpha> :: "V" shows "smc_SemiCAT \<alpha>\<lparr>Comp\<rparr> = (\<lambda>\<GG>\<FF>\<in>\<^sub>\<circ>composable_arrs (dg_SemiCAT \<alpha>). \<GG>\<FF>\<lparr>[]\<^sub>\<circ>\<rparr> \<circ>\<^sub>D\<^sub>G\<^sub>H\<^sub>M \<GG>\<FF>\<lparr>1\<^sub>\<nat>\<rparr>)"
atLeastLessThan_def: fixes l :: "'b"   and u :: "'b" shows "{l..<u} \<equiv> {l..} \<inter> {..<u}"
GFP_closed: fixes f :: "'a \<Rightarrow> 'a" shows "GFP f \<in> carrier L"
rel_fun_const: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'c" shows "rel_fun R (=) (\<lambda>x. a) (\<lambda>y. a)"
simps_8: shows "subcomp sA8 = {sA81, sA82}"
not_Cons_self2: fixes x :: "'x"   and xs :: "'x list" shows "x # xs \<noteq> xs"
9: shows "sset (gtrace r u) \<subseteq> greachable A w (clean A w V) u"
jonsson2c: shows "(\<forall>x y. x \<lhd> y = x \<cdot> ((1::'a) \<lhd> y)) = (\<forall>x y. y \<rhd> x = x \<rhd> y \<rhd> (1::'a))"
intros_8: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and rs :: "'c rule list"   and X :: "final_decision" shows "\<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>rs, Decision X\<rangle> \<Rightarrow> Decision X"
lift_clw_rel_map: fixes A :: "('e \<times> 'f set) set"   and B :: "('g \<times> 'h set) set"   and fi :: "'e \<Rightarrow> 'g"   and f :: "'f set \<Rightarrow> 'h set" assumes "single_valued A"   and "single_valued B"   and "(fi, f) \<in> A \<rightarrow> B"   and "\<And>X. X \<subseteq> Range A \<Longrightarrow> f (\<Union> X) = \<Union> (f ` X)" shows "(map fi, f) \<in> clw_rel A \<rightarrow> clw_rel B"
homeomorphic_closedin_convex: fixes S :: "'a set" assumes "aff_dim S < int DIM('b)" obtains U :: "'b set" and T :: "'b set" where "convex U" and "U \<noteq> {}" and "closedin (top_of_set U) T" and "S homeomorphic T"
Sin: shows "((\<lambda>x. sin (eval C x)) expands_to Sin) basis"
almost_full_on_with_bot: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a set" assumes "almost_full_on P A" shows "almost_full_on (option_le P) A\<^sub>\<bottom>"
isVar_def: fixes s :: "'sort"   and X :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set" shows "isVar s X \<equiv> \<exists>xs x. s = asSort xs \<and> X = Var xs x"
inSet_not_MT: fixes a :: "'a"   and p :: "'a list" assumes "a \<in> set p" shows "p \<noteq> []"
eq_eqvt: fixes pi :: "(name \<times> name) list"   and x :: "'d"   and y :: "'d" shows "pi \<bullet> (x = y) = (pi \<bullet> x = pi \<bullet> y)"
f9: shows "s'' \<otimes> s \<otimes> (s'' \<otimes> s') = s'' \<otimes> s'' \<otimes> (s \<otimes> s')"
WTrtAAccNT: fixes P :: "'addr J_prog"   and h :: "'heap"   and E :: "String.literal \<Rightarrow> ty option"   and a :: "'addr expr"   and i :: "'addr expr"   and T :: "ty" assumes "P,E,h \<turnstile> a : NT"   and "P,E,h \<turnstile> i : Integer" shows "P,E,h \<turnstile> a\<lfloor>i\<rceil> : T"
quasi_measure_preserving_composition: fixes f :: "'d \<Rightarrow> 'e"   and M :: "'d measure"   and N :: "'e measure"   and g :: "'e \<Rightarrow> 'f"   and P :: "'f measure" assumes "f \<in> quasi_measure_preserving M N"   and "g \<in> quasi_measure_preserving N P" shows "(\<lambda>x. g (f x)) \<in> quasi_measure_preserving M P"
const_poly_dvd_const_poly_iff: fixes a :: "'b"   and b :: "'b" shows "([:a:] dvd [:b:]) = (a dvd b)"
g_image_rw: shows "g ` {0..<Suc n} = insert (g n) (g ` {0..<n})"
inv_fromD: fixes p :: "'a mat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> bool"   and A :: "'a mat"   and m :: "nat"   and i :: "nat"   and j :: "nat" assumes "local.inv_from p A m"   and "i < n"   and "j < n"   and "m < j" shows "p A i j"
monom_mult_zero_left: fixes t :: "'x \<Rightarrow>\<^sub>0 nat"   and p :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" shows "local.punit.monom_mult (0::'b) t p = 0"
lconcat_lappend: fixes xss :: "'a llist llist"   and yss :: "'a llist llist" assumes "lfinite xss" shows "lconcat (lappend xss yss) = lappend (lconcat xss) (lconcat yss)"
dist_fs_triangle_infinite3: fixes a :: "complex"   and b :: "complex" shows "cmod (a - b) / (sqrt (1 + (cmod a)\<^sup>2) * sqrt (1 + (cmod b)\<^sup>2)) \<le> 1 / sqrt (1 + (cmod a)\<^sup>2) + 1 / sqrt (1 + (cmod b)\<^sup>2)"
span_insert_cong: fixes A :: "('t \<Rightarrow>\<^sub>0 'c) set"   and B :: "('t \<Rightarrow>\<^sub>0 'c) set"   and p :: "'t \<Rightarrow>\<^sub>0 'c" assumes "pmdl A = pmdl B" shows "pmdl (insert p A) = pmdl (insert p B)"
S_def: shows "S = (SIGMA i:{i. 0 < i \<and> real i \<le> log 2 x}. {p. prime p \<and> real p \<le> root i x})"
has_integral_altD: fixes f :: "'a \<Rightarrow> 'b"   and y :: "'b"   and i :: "'a set"   and e :: "real" assumes "(f has_integral y) i"   and "\<nexists>a b. i = cbox a b"   and "0 < e" obtains B :: "real" where "0 < B" and "\<forall>a b. ball (0::'a) B \<subseteq> cbox a b \<longrightarrow> (\<exists>z. ((\<lambda>x. if x \<in> i then f x else (0::'b)) has_integral z) (cbox a b) \<and> norm (z - y) < e)"
Units_closed: fixes x :: "'a list" assumes "x \<in> Units (poly_ring R)" shows "x \<in> carrier (poly_ring R)"
intro: fixes cmp :: "'b \<Rightarrow> 'b \<Rightarrow> comp" assumes "\<And>a. cmp a a = Equiv"   and "\<And>a b c. \<lbrakk>cmp a b = Equiv; cmp b c = Equiv\<rbrakk> \<Longrightarrow> cmp a c = Equiv"   and "\<And>a b c. \<lbrakk>cmp a b = Less; cmp b c = Less\<rbrakk> \<Longrightarrow> cmp a c = Less"   and "\<And>b a. (cmp b a = Greater) = (cmp a b = Less)" shows "comparator cmp"
lt_alt: fixes p :: "'t \<Rightarrow>\<^sub>0 'c" assumes "p \<noteq> 0" shows "lt p = ord_term_lin.Max (keys p)"
bfs_reachability_exists: fixes r :: "'a"   and s :: "'a" assumes "regular r \<and> regular s \<and> vector s" shows "\<exists>t. asymmetric t \<and> t \<le> r \<and> t\<^sup>T\<^sup>\<star> * s = r\<^sup>T\<^sup>\<star> * s"
distinct_6: fixes x41 :: "('a, 'b, 'c) Stmt"   and x42 :: "('a, 'b, 'c) Stmt"   and x11 :: "'a"   and x12 :: "'b" shows "x41 ;; x42 \<noteq> x11 \<leftarrow> x12"
diff_0_right: fixes minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and uminus :: "'d \<Rightarrow> 'd"   and a :: "'d" assumes "class.group_add minus plus zero uminus" shows "minus a zero = a"
induct: fixes x :: "db"   and P :: "db \<Rightarrow> bool" assumes "val x"   and "\<And>n. P (DBNum n)"   and "P DBff"   and "P DBtt"   and "\<And>e. P (DBAbsN e)"   and "\<And>e. P (DBAbsV e)" shows "P x"
nth_chain: shows "\<forall>j. enat (j + 1) < llength xs \<longrightarrow> R (lnth xs j) (lnth xs (j + 1))"
Int_closed_segment: fixes b :: "'a"   and a :: "'a"   and c :: "'a" assumes "b \<in> closed_segment a c \<or> \<not> collinear {a, b, c}" shows "closed_segment a b \<inter> closed_segment b c = {b}"
list_product_finite: fixes A :: "'b list set"   and B :: "'b list set" assumes "finite A"   and "finite B" shows "finite (A \<otimes>\<^sub>l B)"
lunit_char_2: fixes f :: "'a" assumes "B.ide f" shows "R (B.lunit f) = (f \<star> \<i>[src f]) \<cdot> B.\<a>' (src f) (src f) f"
tl_def: fixes list :: "'b list" shows "tl list = (case list of [] \<Rightarrow> [] | x21 # x22 \<Rightarrow> x22)"
CH_def: fixes X :: "X3 set" shows "Theorem_9.CH X = (\<Union>h. CX3h h X)"
not_bet_out: fixes A :: "'p"   and B :: "'p"   and C :: "'p" assumes "Col A B C"   and "\<not> Bet A B C" shows "B Out A C"
extended_simps_2: fixes x' :: "'a"   and y' :: "'a" shows "(\<lbrakk>x'\<rbrakk> < \<lbrakk>y'\<rbrakk>) = (x' < y')"
ih2: assumes "wt t \<ge>\<^sub>p wt s2" shows "vars s2 \<subseteq> vars t"
log2_of_power_eq: fixes m :: "nat"   and n :: "nat" assumes "m = 2 ^ n" shows "real n = log 2 (real m)"
runWriterT_bindWT: fixes m :: "'h\<cdot>('g, 'f) writerT"   and k :: "'h \<rightarrow> 'e\<cdot>('g, 'f) writerT" shows "runWriterT\<cdot>(bindWT\<cdot>m\<cdot>k) = runWriterT\<cdot>m \<bind> (\<Lambda> (Writer\<cdot>w\<cdot>x). runWriterT\<cdot>(k\<cdot>x) \<bind> (\<Lambda> (Writer\<cdot>w'\<cdot>y). return\<cdot>(Writer\<cdot>(mappend\<cdot>w\<cdot>w')\<cdot>y)))"
assms_4: shows "q \<equiv> Poly (map (\<lambda>k. (\<Sum>i\<le>min N k. c (N - i) * f (k - i)) - g k) [0..<N + m])"
pivotandupdate_valuation_nonlhs: fixes eq_idx_for_lvar :: "(nat \<times> linear_poly) list \<Rightarrow> nat \<Rightarrow> nat"   and pivot_and_update :: "nat \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> ('b, 'a) state \<Rightarrow> ('b, 'a) state"   and s :: "('b, 'a) state"   and x\<^sub>i :: "nat"   and x\<^sub>j :: "nat"   and x :: "nat"   and c :: "'a" assumes "PivotAndUpdate eq_idx_for_lvar pivot_and_update"   and "\<triangle> (\<T> s)"   and "\<nabla> s"   and "x\<^sub>i \<in> lvars (\<T> s)"   and "x\<^sub>j \<in> rvars_eq (EqForLVar.eq_for_lvar eq_idx_for_lvar (\<T> s) x\<^sub>i)" shows "x \<notin> lvars (\<T> s) - {x\<^sub>i} \<union> {x\<^sub>j} \<longrightarrow> Mapping.lookup (\<V> (pivot_and_update x\<^sub>i x\<^sub>j c s)) x = (if x = x\<^sub>i then Some c else Mapping.lookup (\<V> s) x)"
euclidean_inner: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and norm :: "'a \<Rightarrow> real"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and sgn :: "'a \<Rightarrow> 'a"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and inner :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and Basis :: "'a set"   and x :: "'a"   and y :: "'a" assumes "class.euclidean_space minus dist norm plus zero uminus scaleR sgn uniformity open inner Basis" shows "inner x y = (\<Sum>b\<in>Basis. inner x b * inner y b)"
lemma4: fixes x :: "Syntax.var"   and \<tau>' :: "ty"   and \<Gamma> :: "(Syntax.var, ty) fmap"   and e :: "Syntax.term"   and eP :: "Syntax.term"   and eV :: "Syntax.term"   and \<tau> :: "ty"   and v :: "Syntax.term"   and vP :: "Syntax.term"   and vV :: "Syntax.term" assumes "\<Gamma>(x $$:= \<tau>') \<turnstile> e, eP, eV : \<tau>"   and "{$$} \<turnstile> v, vP, vV : \<tau>'"   and "value v"   and "value vP"   and "value vV" shows "\<Gamma> \<turnstile> e[v / x], eP[vP / x], eV[vV / x] : \<tau>"
rel_cong: fixes x :: "'d \<rightharpoonup>\<^sub>f 'e"   and ya :: "'d \<rightharpoonup>\<^sub>f 'e"   and y :: "'d \<rightharpoonup>\<^sub>f 'f"   and xa :: "'d \<rightharpoonup>\<^sub>f 'f"   and R :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and Ra :: "'e \<Rightarrow> 'f \<Rightarrow> bool" assumes "x = ya"   and "y = xa"   and "\<And>z yb. \<lbrakk>z \<in> fmran' ya; yb \<in> fmran' xa\<rbrakk> \<Longrightarrow> R z yb = Ra z yb" shows "fmrel R x y = fmrel Ra ya xa"
list_all2_induct: fixes P :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and xs :: "'c list"   and ys :: "'d list"   and R :: "'c list \<Rightarrow> 'd list \<Rightarrow> bool" assumes "list_all2 P xs ys"   and "R [] []"   and "\<And>x xs y ys. \<lbrakk>P x y; list_all2 P xs ys; R xs ys\<rbrakk> \<Longrightarrow> R (x # xs) (y # ys)" shows "R xs ys"
ring_iso_set_trans: fixes f :: "'a \<Rightarrow> 'b"   and R :: "('a, 'c) ring_scheme"   and S :: "('b, 'd) ring_scheme"   and g :: "'b \<Rightarrow> 'e"   and Q :: "('e, 'f) ring_scheme" assumes "f \<in> ring_iso R S"   and "g \<in> ring_iso S Q" shows "g \<circ> f \<in> ring_iso R Q"
path_connectedin_singleton: fixes X :: "'d topology"   and a :: "'d" shows "path_connectedin X {a} = (a \<in> topspace X)"
induct: fixes P :: "('d \<Rightarrow> 'e \<Rightarrow> 'f) \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> ('g \<times> 'd) list \<Rightarrow> ('g \<times> 'e) list \<Rightarrow> bool"   and a0 :: "'d \<Rightarrow> 'e \<Rightarrow> 'f"   and a1 :: "'d"   and a2 :: "'e"   and a3 :: "('g \<times> 'd) list"   and a4 :: "('g \<times> 'e) list" assumes "\<And>f z1 z2. P f z1 z2 [] []"   and "\<And>f z1 z2 i x xs. P f z1 z2 xs [] \<Longrightarrow> P f z1 z2 ((i, x) # xs) []"   and "\<And>f z1 z2 i y ys. P f z1 z2 [] ys \<Longrightarrow> P f z1 z2 [] ((i, y) # ys)"   and "\<And>f z1 z2 i x xs j y ys. \<lbrakk>i = j \<Longrightarrow> P f z1 z2 xs ys; \<lbrakk>i \<noteq> j; j < i\<rbrakk> \<Longrightarrow> P f z1 z2 xs ((j, y) # ys); \<lbrakk>i \<noteq> j; \<not> j < i\<rbrakk> \<Longrightarrow> P f z1 z2 ((i, x) # xs) ys\<rbrakk> \<Longrightarrow> P f z1 z2 ((i, x) # xs) ((j, y) # ys)" shows "P a0 a1 a2 a3 a4"
LOr2: fixes b :: "coname"   and a :: "coname"   and M :: "trm"   and N1 :: "trm"   and N2 :: "trm"   and y :: "name"   and x1 :: "name"   and x2 :: "name" assumes "b \<sharp> ([a].M, N1, N2, a)"   and "y \<sharp> ([x1].N1, [x2].N2, M, x1, x2)"   and "x1 \<sharp> (M, N2)"   and "x2 \<sharp> (M, N1)"   and "a \<sharp> (N1, N2)"   and "x1 \<noteq> x2" shows "Cut <b>.OrR2 <a>.M b y.OrL x1.N1 x2.N2 y \<longrightarrow>\<^sub>l Cut <a>.M x2.N2"
inject_4: fixes x5 :: "'d"   and y5 :: "'d" shows "(Std.instr.IOp x5 = Std.instr.IOp y5) = (x5 = y5)"
cases: fixes H :: "chmsg \<Rightarrow> bool"   and a :: "atom" assumes "atomsp H a" obtains c :: "secprop set" and A :: "agent" and B :: "agent" and M :: "atom list" and At :: "atom" where "a = Ata__" and "H (StatCh c__ A B (Msg M__))" and "Ata__ \<in> set M__"   | c :: "secprop set" and K :: "key" and M :: "atom list" and At :: "atom" where "a = At" and "H (DynCh c K (Msg M))" and "At \<in> set M"
elims_1: fixes x :: "('a \<times> ('b \<Rightarrow> 'c) \<times> 'd) list"   and xa :: "'a \<Rightarrow> 'c"   and xb :: "'b"   and y :: "bool" assumes "postList x xa xb = y" obtains l :: "'a \<Rightarrow> 'c" and s :: "'b" where "x = []" and "xa = l__" and "xb = s__" and "y"   | xc :: "'a" and e :: "'b \<Rightarrow> 'c" and v :: "'d" and xs :: "('a \<times> ('b \<Rightarrow> 'c) \<times> 'd) list" and l :: "'a \<Rightarrow> 'c" and s :: "'b" where "x = (xc, e, v) # xs" and "xa = l" and "xb = s" and "y = (l xc = e s \<and> postList xs l s)"
cases: fixes x :: "('b \<times> nat) tree \<times> 'b \<times> ('b \<times> nat) tree" obtains l :: "('b \<times> nat) tree" and a :: "'b" and r :: "('b \<times> nat) tree" where "x = (l, a, r)"
continuous_imp_measurable_on_sets_lebesgue: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "continuous_on S f"   and "S \<in> sets lebesgue" shows "f \<in> borel_measurable (lebesgue_on S)"
bisim_def: fixes Retr :: "('b \<times> 'b) set \<Rightarrow> ('b \<times> 'b) set"   and theta :: "('b \<times> 'b) set" shows "bisim Retr theta \<equiv> sym theta \<and> simul Retr theta"
prv_rawpsubst_sdsj: fixes F :: "'fmla set"   and txs :: "('trm \<times> 'var) list" assumes "F \<subseteq> fmla"   and "finite F"   and "snd ` set txs \<subseteq> var"   and "fst ` set txs \<subseteq> trm" shows "prv (eqv (rawpsubst (sdsj F) txs) (sdsj ((\<lambda>\<phi>. rawpsubst \<phi> txs) ` F)))"
ord_iff_2: 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"   and y :: "'a" assumes "class.nat minus plus zero equal less_eq less rep_nat abs_nat" shows "less x y = (rep_nat x < rep_nat y)"
cr_one_star: fixes t :: "lam"   and t2 :: "lam"   and t1 :: "lam" assumes " t \<longrightarrow>\<^sub>1\<^sup>* t2"   and " t \<longrightarrow>\<^sub>1\<^sup>* t1" shows "\<exists>t3.  t1 \<longrightarrow>\<^sub>1\<^sup>* t3 \<and>  t2 \<longrightarrow>\<^sub>1\<^sup>* t3"
res2: shows "\<alpha>_i' \<upharpoonleft> V\<^bsub>\<V>i\<^esub> = (\<gamma> @ \<delta>') \<upharpoonleft> V\<^bsub>\<V>i\<^esub>"
fifo_push_def: fixes f :: "nat \<times> nat \<Rightarrow> 'capacity"   and l :: "nat \<Rightarrow> nat"   and Q :: "nat list" shows "fifo_push f l Q \<equiv> \<lambda>(u, v). assert (push_precond f l (u, v)) \<bind> (\<lambda>_. assert (Labeling c s t f l) \<bind> (\<lambda>_. let Q = if v \<noteq> s \<and> v \<noteq> t \<and> excess f v = (0::'capacity) then Q @ [v] else Q in return (push_effect f (u, v), Q)))"
nat_to_string_char_to_digit: fixes c :: "char" assumes "is_digit_char c" shows "nat_to_string (char_to_digit c) = [c]"
liftPoly_add: fixes w :: "nat"   and z :: "nat"   and a :: "'a mpoly"   and b :: "'a mpoly" shows "liftPoly w z (a + b) = liftPoly w z a + liftPoly w z b"
R2: fixes x :: "'b"   and xs :: "'b list"   and ys :: "'b list"   and z :: "'b"   and zs :: "'b list"   and X :: "'b list \<times> 'b list \<times> 'b list" assumes "(x # xs, ys, z # zs) \<in> l_sort_set X"   and "x \<le> z" shows "(xs, [], ys @ x # z # zs) \<in> l_sort_set X"
cos_total: fixes y :: "real" assumes "- 1 \<le> y"   and "y \<le> 1" shows "\<exists>!x. 0 \<le> x \<and> x \<le> pi \<and> cos x = y"
problem_plan_bound_def: fixes PROB :: "(('a, 'b) fmap \<times> ('a, 'b) fmap) set" shows "problem_plan_bound PROB \<equiv> Sup (MPLS PROB)"
assms: shows "\<forall>\<Psi>. \<Psi> \<rhd> P \<doteq> Q"
product_def: fixes A :: "'d set"   and B :: "'e set" shows "Product_Type.product A B = A \<times> B"
when_commute: fixes zero :: "'b"   and a :: "'b"   and Q :: "bool"   and P :: "bool" shows "zero.when zero (zero.when zero a Q) P = zero.when zero (zero.when zero a P) Q"
join_self_below_4: fixes y :: "'a"   and x :: "'a" shows "(y \<squnion> x = x) = (y \<sqsubseteq> x)"
set_pmf_of_set: fixes S :: "'c set" assumes "S \<noteq> {}"   and "finite S" shows "set_pmf (local.pmf_of_set S) = S"
vpair_nonzero: fixes x :: "V"   and y :: "V" shows "\<langle>x, y\<rangle> \<noteq> []\<^sub>\<circ>"
keys_cmul: fixes c :: "int"   and a :: "'b \<Rightarrow>\<^sub>0 int" shows "Poly_Mapping.keys (frag_cmul c a) \<subseteq> Poly_Mapping.keys a"
is_acc_run_def: fixes G :: "('a, 'b) gb_graph_rec_scheme"   and r :: "nat \<Rightarrow> 'a" assumes "gb_graph G" shows "gb_graph.is_acc_run G r \<equiv> graph_defs.is_run G r \<and> gb_graph.is_acc G r"
op_set_isEmpty_def: fixes s :: "'e set" shows "op_set_isEmpty s \<equiv> s = {}"
add_get_eq: fixes xa :: "char list"   and e :: "'b environment"   and U :: "'b"   and T :: "'b" assumes "xa \<notin> env_dom e"   and "ok e"   and "the e\<lparr>xa:U\<rparr>!xa = T" shows "U = T"
vs1: fixes v :: "'a mod_ring poly"   and u :: "'a mod_ring poly"   and i :: "nat" assumes "v \<in> set vs1"   and "u \<in> set us1 \<union> set divs"   and "i < CARD('a)" shows "gcd u (v - [:of_nat i:]) \<in> {1, u}"
flow0_cong: fixes Y :: "'a set"   and g :: "'a \<Rightarrow> 'a"   and t :: "real"   and x0 :: "'a" assumes "Y = Y"   and "\<And>x t. x \<in> Y \<Longrightarrow> g x = g x"   and "t \<in> Y.existence_ivl0 x0" shows "Y.flow0 x0 t = auto_ll_on_open.flow0 g Y x0 t"
powsum_shift: fixes x :: "'a"   and k :: "nat"   and m :: "nat"   and n :: "nat" shows "x\<^bsup>k\<^esup> \<cdot> x\<^bsub>m\<^esub>\<^bsup>n\<^esup> = x\<^bsub>k + m\<^esub>\<^bsup>n\<^esup>"
ext: shows "extendRule S r = (Ps,  \<Gamma> \<Rightarrow>* \<Delta> \<oplus> F \<nabla> [x].A)"