hcju/isabelleps huggingface dataset:
nn_integral_ge_point: fixes x :: "'c"   and A :: "'c set"   and p :: "'c \<Rightarrow> ennreal" assumes "x \<in> A" shows "p x \<le> integral\<^sup>N (count_space A) p"
evalDdb_subst: fixes e :: "db"   and s :: "db"   and x :: "nat"   and \<rho> :: "nat \<rightarrow> ValD" shows "evalDdb (e<s/x>)\<cdot>\<rho> = evalDdb e\<cdot>(\<Lambda> i. if x < i then \<rho>\<cdot>(i - 1) else if i = x then evalDdb s\<cdot>\<rho> else \<rho>\<cdot>i)"
simps_1: fixes x21 :: "'a"   and x22 :: "'a list"   and y21 :: "'a"   and y22 :: "'a list" shows "(x21 # x22 = y21 # y22) = (x21 = y21 \<and> x22 = y22)"
inj_on_image_eqpoll_self: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set" assumes "inj_on f A" shows "f ` A \<approx> A"
numeral_inc: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and x :: "num" assumes "class.numeral plus" shows "numeral.numeral one plus (Num.inc x) = plus (numeral.numeral one plus x) one"
mult_less_iff1: fixes z :: "'a"   and x :: "'a"   and y :: "'a" assumes "(0::'a) < z" shows "(x * z < y * z) = (x < y)"
IH1: assumes "Vagree \<nu> \<nu>' (FVT \<theta>\<^sub>1)"   and "Iagree I J {Inl x |x. x \<in> SIGT \<theta>\<^sub>1}" shows "dterm_sem I \<theta>\<^sub>1 \<nu> = dterm_sem J \<theta>\<^sub>1 \<nu>'"
is_nextElem_sublistI: fixes x :: "'a"   and y :: "'a"   and vs :: "'a list" assumes "is_sublist [x, y] vs" shows "is_nextElem vs x y"
right_total_def: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> bool" shows "right_total R = (\<forall>y. \<exists>x. R x y)"
quotient_of_Fract: fixes a :: "int"   and b :: "int" shows "quotient_of (Fract a b) = Rat.normalize (a, b)"
approximating_semantics_imp_fun: fixes \<gamma> :: "('b \<Rightarrow> 'c \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'c \<Rightarrow> bool)"   and p :: "'c"   and rs :: "'b rule list"   and s :: "state"   and t :: "state" assumes "\<gamma>,p\<turnstile> \<langle>rs, s\<rangle> \<Rightarrow>\<^sub>\<alpha> t" shows "approximating_bigstep_fun \<gamma> p rs s = t"
borel_measurable_euclidean_space: fixes f :: "'b \<Rightarrow> 'c"   and M :: "'b measure" shows "(f \<in> borel_measurable M) = (\<forall>i\<in>Basis. (\<lambda>x. f x \<bullet> i) \<in> borel_measurable M)"
smult_smult_assoc: fixes a :: "'b"   and b :: "'b"   and v :: "'b vec" shows "a \<cdot>\<^sub>v (b \<cdot>\<^sub>v v) = a * b \<cdot>\<^sub>v v"
properties_for_sort: fixes ys :: "'a list"   and xs :: "'a list" assumes "mset ys = mset xs"   and "sorted ys" shows "sort xs = ys"
uniform_limit2': fixes x :: "real"   and y :: "real" assumes "- 2 < x" shows "uniform_limit {z. Re z \<in> {x..y}} (\<lambda>n z. complex_interval_lebesgue_integral lborel (ereal (exp (- 1))) (ereal (1 - (1 / 2) ^ Suc n)) (f z)) (\<lambda>z. complex_interval_lebesgue_integral lborel (ereal (exp (- 1))) 1 (f z)) sequentially"
closedin_derived_set_of: fixes X :: "'c topology"   and S :: "'c set" assumes "Hausdorff_space X" shows "closedin X (X derived_set_of S)"
that: fixes g :: "'a \<Rightarrow> 'b" assumes "continuous_on S g"   and "g ` S \<subseteq> U"   and "\<And>x. x \<in> T \<Longrightarrow> g x = f x" shows "thesis"
\<delta>_def: fixes f :: "'a" shows "\<delta> f \<equiv> if arr f then \<d>[cod f] \<cdot> f else null"
assms_2: fixes x :: "nat" shows "\<phi> i x \<up>"
take_bit_add: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and division_segment :: "'a \<Rightarrow> 'a"   and bit :: "'a \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and drop_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and take_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and n :: "nat"   and a :: "'a"   and b :: "'a" assumes "class.unique_euclidean_semiring_with_bit_shifts divide plus minus zero times one modulo euclidean_size division_segment bit push_bit drop_bit take_bit" shows "take_bit n (plus (take_bit n a) (take_bit n b)) = take_bit n (plus a b)"
simps_2: fixes n :: "nat"   and x :: "'a" shows "x \<up> Suc n = x # x \<up> n"
munionM_commute: fixes x :: "'m"   and y :: "'m" shows "munionM x y = munionM y x"
elims_2: fixes x :: "arctic"   and xa :: "arctic" assumes "x \<le> xa" obtains xb :: "arctic" where "x = MinInfty" and "xa = xb__"   | y :: "int" and xb :: "int" where "x = Num_arc y" and "xa = Num_arc xb" and "y \<le> xb"
3: fixes u :: "real"   and A :: "real \<Rightarrow> bool" assumes "\<forall>x. u \<le> \<bar>x\<bar> \<longrightarrow> A x" shows "\<exists>N. \<forall>n\<le>N. A n"
affine_diffs_subspace: fixes S :: "'b set"   and a :: "'b" assumes "affine S"   and "a \<in> S" shows "subspace ((+) (- a) ` S)"
implies_mp_eq: fixes x :: "'a"   and y :: "'a" shows "x \<sqinter> (x \<leadsto> y) = x \<sqinter> y"
g: fixes i :: "nat" assumes "i \<in> N" shows "diff_fun k charts (g i)"
senior_monotonic_Suc: fixes \<Sigma> :: "'c set"   and \<delta> :: "'d \<Rightarrow> 'c \<Rightarrow> 'd"   and q\<^sub>0 :: "'d"   and w :: "nat \<Rightarrow> 'c"   and x :: "nat"   and n :: "nat" assumes "semi_mojmir \<Sigma> \<delta> q\<^sub>0 w"   and "x \<le> n" shows "semi_mojmir_def.senior \<delta> q\<^sub>0 w x (Suc n) \<le> semi_mojmir_def.senior \<delta> q\<^sub>0 w x n"
upd_cs_snoc_Return_Cons: fixes a :: "'edge"   and Q :: "('var \<Rightarrow> 'val option) \<times> 'ret \<Rightarrow> bool"   and p :: "'pname"   and f :: "('var \<Rightarrow> 'val option) \<Rightarrow> ('var \<Rightarrow> 'val option) \<Rightarrow> 'var \<Rightarrow> 'val option"   and cs :: "'edge list"   and as :: "'edge list"   and c' :: "'edge"   and cs' :: "'edge list" assumes "kind a = Q\<hookleftarrow>\<^bsub>p\<^esub>f"   and "upd_cs cs as = c' # cs'" shows "upd_cs cs (as @ [a]) = cs'"
swap_msgs_Recv_RecvMarker: fixes c :: "('p, 's, 'm) configuration"   and ev :: "('p, 's, 'm) event"   and d :: "('p, 's, 'm) configuration"   and ev' :: "('p, 's, 'm) event"   and e :: "('p, 's, 'm) configuration"   and d' :: "('p, 's, 'm) configuration"   and e' :: "('p, 's, 'm) configuration"   and i :: "nat" assumes "c \<turnstile> ev \<mapsto> d"   and "d \<turnstile> ev' \<mapsto> e"   and "isRecv ev"   and "isRecvMarker ev'"   and "c \<turnstile> ev' \<mapsto> d'"   and "d' \<turnstile> ev \<mapsto> e'"   and "occurs_on ev \<noteq> occurs_on ev'" shows "msgs e i = msgs e' i"
surj_f_inv_f: fixes f :: "'d \<Rightarrow> 'c"   and y :: "'c" assumes "surj f" shows "f (inv f y) = y"
measurable_fst: fixes M1 :: "'a measure"   and M2 :: "'b measure" shows "fst \<in> M1 \<Otimes>\<^sub>M M2 \<rightarrow>\<^sub>M M1"
filter_min_append_subset: fixes rel :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and xs :: "'c list"   and ys :: "'c list" shows "set (filter_min_append rel xs ys) \<subseteq> set xs \<union> set ys"
streamValue4: fixes D :: "nat"   and str :: "nat \<Rightarrow> 'a"   and t :: "nat"   and y :: "nat"   and x :: "'a" assumes "\<forall>j\<le>D. str (t + j + Suc y) = x"   and "str (t + y) = x" shows "\<forall>j\<le>D. str (t + j + y) = x"
transient_imp_leadsTo: fixes F :: "'c program"   and A :: "'c set" assumes "F \<in> transient A" shows "F \<in> A \<longmapsto> - A"
plus_intercept_stop_def: fixes left :: "'n \<Rightarrow> 'o \<Rightarrow> ('q option \<times> 'n, 's, 't) gpv"   and right :: "'n \<Rightarrow> 'p \<Rightarrow> ('r option \<times> 'n, 's, 't) gpv" shows "plus_intercept_stop left right \<equiv> \<lambda>uu uua. (case uua of Inl x \<Rightarrow> \<lambda>s. map_gpv (apfst (map_option Inl)) id (left s x) | Inr x \<Rightarrow> \<lambda>s. map_gpv (apfst (map_option Inr)) id (right s x)) uu"
sesK_image_eq: fixes x :: "fresh_t"   and AA :: "fresh_t set" shows "(sesK x \<in> sesK ` AA) = (x \<in> AA)"
genby_lcoset_el_reduce0: fixes a :: "'b"   and S :: "'b set" assumes "(0::'b) \<in> a +o \<langle>S\<rangle>" shows "a +o \<langle>S\<rangle> = \<langle>S\<rangle>"
minus_expand: fixes x :: "'a"   and y :: "'a" shows "x - y = eucl_of_list [x $\<^sub>e 0 - y $\<^sub>e 0, x $\<^sub>e 1 - y $\<^sub>e 1]"
\<I>\<^sub>\<tau>_4: shows "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (subst_range \<I>\<^sub>\<tau>)"
assms_5: shows "eint (int n) < val_Zp_dist a c"
arbitrage_process_def: fixes Mkt :: "('a, 'c) discrete_market"   and p :: "'c \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> real" shows "arbitrage_process Mkt p = (\<exists>m. self_financing Mkt p \<and> trading_strategy p \<and> (\<forall>w\<in>space M. val_process Mkt p 0 w = 0) \<and> (AE w in M. 0 \<le> cls_val_process Mkt p m w) \<and> 0 < prob {w \<in> space M. 0 < cls_val_process Mkt p m w})"
abs_eq: fixes xa :: "int extended \<times> int extended"   and x :: "int extended \<times> int extended" shows "abs_ivl xa \<sqinter> abs_ivl x = abs_ivl (inf_rep xa x)"
intro: fixes f :: "'a \<Rightarrow> 'b"   and period :: "'a" assumes "\<And>x. f (x + period) = f x" shows "periodic_fun_simple f period"
fset_mset_mset_fset: fixes N :: "'c fset" shows "fset_mset (mset_fset N) = N"
quasi_inverse_simps_4: fixes f :: "'b" assumes "D.equivalence_map f" shows "trg\<^sub>D f\<^sup>~\<^sup>D = src\<^sub>D f"
HFun_Sigma_Mem_imp_HPair: fixes H :: "fm set"   and r :: "tm"   and a :: "tm"   and x :: "name"   and y :: "name" assumes "H \<turnstile> HFun_Sigma r"   and "H \<turnstile> a IN r"   and "atom x \<sharp> (y, a, r)"   and "atom y \<sharp> (a, r)" shows "H \<turnstile> SyntaxN.Ex x (SyntaxN.Ex y (a EQ HPair (Var x) (Var y)))"
steps_z_alt: fixes A :: "('d \<times> ('e, 'f) cconstraint \<times> 'g \<times> 'e list \<times> 'd) set \<times> ('d \<Rightarrow> ('e, 'f) cconstraint)"   and l :: "'d"   and Z :: "('e \<Rightarrow> 'f) set"   and l' :: "'d"   and Z' :: "('e \<Rightarrow> 'f) set"   and l'' :: "'d"   and Z'' :: "('e \<Rightarrow> 'f) set" assumes "A \<turnstile> \<langle>l, Z\<rangle> \<leadsto>* \<langle>l', Z'\<rangle>"   and "A \<turnstile> \<langle>l', Z'\<rangle> \<leadsto> \<langle>l'', Z''\<rangle>" shows "A \<turnstile> \<langle>l, Z\<rangle> \<leadsto>* \<langle>l'', Z''\<rangle>"
inf_absorb1: fixes inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "class.semilattice_inf inf less_eq less"   and "less_eq x y" shows "inf x y = x"
fps_compose_deriv: fixes b :: "'a fps"   and a :: "'a fps" assumes "fps_nth b 0 = (0::'a)" shows "fps_deriv (a oo b) = (fps_deriv a oo b) * fps_deriv b"
mset_get_min: fixes q :: "'b tree" assumes "braun q \<and> heap q"   and "mset_tree q \<noteq> {#}" shows "get_min q = Min_mset (mset_tree q)"
left_commute: fixes b :: "'d multiset"   and a :: "'d multiset"   and c :: "'d multiset" shows "b \<inter># (a \<inter># c) = a \<inter># (b \<inter># c)"
wrong_not_in_T: fixes \<tau> :: "ty"   and \<eta> :: "val set list" shows "Wrong \<notin> T \<tau> \<eta>"
antisym: assumes "i \<le> j"   and "j \<le> i" shows "i = j"
pfinite_INF_le_expectation_gpv: fixes \<I> :: "('a, 'b) \<I>"   and gpv :: "('c, 'a, 'b) gpv"   and f :: "'c \<Rightarrow> ennreal"   and fail :: "ennreal" assumes "pfinite_gpv \<I> gpv" shows "min (\<Sqinter> (f ` results_gpv \<I> gpv)) fail \<le> expectation_gpv fail \<I> f gpv"
simps: fixes P1 :: "'a \<Rightarrow> bool"   and P2 :: "'b \<Rightarrow> bool"   and a :: "'a \<times> 'b" shows "pred_prod P1 P2 a = (\<exists>aa b. a = (aa, b) \<and> P1 aa \<and> P2 b)"
mult_strict_mono': fixes a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" assumes "a < b"   and "c < d"   and "(0::'a) \<le> a"   and "(0::'a) \<le> c" shows "a * c < b * d"
components_are_iso: fixes a :: "'a" assumes "ide a" shows "local.iso (src a)"
strict_trans1: fixes a :: "nat"   and b :: "nat"   and c :: "nat" assumes "a dvd b"   and "b dvd c \<and> b \<noteq> c" shows "a dvd c \<and> a \<noteq> c"
determ_additiveD: fixes t :: "('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real" assumes "determ t" shows "Transformers.additive t"
ab: fixes i :: "'a" assumes "i \<in> Basis" shows "a \<bullet> i < b \<bullet> i"
fst_normalize_polynomial_subsetD: fixes a :: "char list list"   and b :: "int"   and p :: "(char list list \<times> int) list" assumes "(a, b) \<in> set (merge_coeffs p)" shows "a \<in> mononoms p"
infinite_computation_extract_head_Seq: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'a, 'c) com option"   and f :: "nat \<Rightarrow> ('b, 'a, 'c) com \<times> ('b, 'c) xstate"   and c\<^sub>1 :: "('b, 'a, 'c) com"   and c\<^sub>2 :: "('b, 'a, 'c) com"   and s :: "('b, 'c) xstate"   and k :: "nat" assumes "\<forall>i. \<Gamma>\<turnstile> f i \<rightarrow> f (i + 1)"   and "f 0 = (Seq c\<^sub>1 c\<^sub>2, s)"   and "\<forall>i<k. \<not> final (head (f i))" shows "\<forall>i<k. (\<exists>c' s'. f (i + 1) = (Seq c' c\<^sub>2, s')) \<and> \<Gamma>\<turnstile> head (f i) \<rightarrow> head (f (i + 1))"
swapInp_def: fixes xs :: "'varSort"   and x :: "'var"   and y :: "'var"   and inp :: "'d \<Rightarrow> ('e, 'f, 'varSort, 'var, 'g) qTerm set option" shows "inp %[x \<and> y]_xs \<equiv> lift (swap xs x y) inp"
SUPSEQ_subset: fixes A :: "'b list set" shows "A \<subseteq> SUPSEQ A"
cases: fixes x :: "exp0" obtains uu_ :: "char list" where "x = Var (Short uu_)"   | oper :: "op0" and es :: "exp0 list" where "x = App oper es__"   | uv_ :: "char list" and es :: "exp0 list" where "x = Con (Some (Short uv_)) es"   | uw_ :: "char list" and e :: "exp0" where "x = Fun uw_ e__"   | e :: "exp0" and cs :: "(pat \<times> exp0) list" where "x = Mat e cs"   | v :: "exp0" where "x = Raise v__"   | v :: "exp0" and va :: "(pat \<times> exp0) list" where "x = Handle va__ vaa__"   | v :: "lit" where "x = Lit vb__"   | va :: "exp0 list" where "x = Con None vab__"   | vc :: "char list" and vd :: "(char list, char list) id0" and va :: "exp0 list" where "x = Con (Some (Long vc vd)) vac__"   | va :: "char list" and vb :: "(char list, char list) id0" where "x = Var (Long vad__ vba__)"   | v :: "lop" and va :: "exp0" and vb :: "exp0" where "x = Log ve__ vae__ vbb__"   | v :: "exp0" and va :: "exp0" and vb :: "exp0" where "x = exp0.If vf__ vaf__ vbc__"   | v :: "char list option" and va :: "exp0" and vb :: "exp0" where "x = exp0.Let vg__ vag__ vb"   | v :: "(char list \<times> char list \<times> exp0) list" and va :: "exp0" where "x = Letrec vh__ vah__"   | v :: "exp0" and va :: "t" where "x = Tannot vi__ vai__"   | v :: "exp0" and va :: "locn \<times> locn" where "x = Lannot v va"
gen_iapath_co_path: fixes e :: "'a \<times> 'a"   and w :: "'a"   and V :: "'a set"   and u :: "'a"   and p :: "('a \<times> 'a) list"   and v :: "'a" assumes "e \<in> parcs H"   and "w \<notin> pverts H"   and "V \<subseteq> pverts H"   and "pre_digraph.gen_iapath (with_proj (subdivide H e w)) V u p v" shows "H.gen_iapath V u (co_path e w p) v"
check_same_set_def: fixes xs :: "'b list"   and ys :: "'b list" shows "check_same_set xs ys = check_subseteq xs ys \<bind> (\<lambda>_. check_subseteq ys xs)"
assms: shows "\<R>\<^sub>\<circ> r \<subseteq>\<^sub>\<circ> B"
mult_le_cancel_left2_hmset: fixes y :: "hmultiset"   and x :: "hmultiset" assumes "y \<le> 1" shows "x * y \<le> x"
HEndPhase2_HInv4b_q: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and q :: "Proc" assumes "HEndPhase2 s s' p"   and "p \<noteq> q"   and "HInv4b s q" shows "HInv4b s' q"
distinct_list_of_fset: fixes s :: "'b fset" shows "distinct (list_of_fset s)"
intros_4: fixes K :: "nat"   and X :: "msg"   and H :: "msg set" assumes "Crypt K X \<in> analz H"   and "Key (invKey K) \<in> analz H" shows "X \<in> analz H"
int: fixes x :: "'a set"   and y :: "'a set" assumes "x open"   and "P x"   and "y open"   and "P y" shows "P (x \<inter> y)"
inject: fixes x3 :: "'a"   and y3 :: "'a" shows "(\<langle>x3\<rangle> = \<langle>y3\<rangle>) = (x3 = y3)"
le_iff_add: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a" assumes "class.canonically_ordered_monoid_add plus zero less_eq less" shows "less_eq a b = (\<exists>c. b = plus a c)"
simps: fixes a :: "'a"   and b :: "'c"   and c :: "'b"   and d :: "'d" shows "mat_transpose (a, b, c, d) = (a, c, b, d)"
in_multiset_in_set: fixes x :: "'k"   and xs :: "'k list" shows "(x \<in># mset xs) = (x \<in> set xs)"
simps_1: fixes uu :: "'l \<Rightarrow> 'm fundef option"   and \<Sigma> :: "Unboxed.type option list" shows "Subx.sp uu [] \<Sigma> = Ok \<Sigma>"
char_poly_matrix_four_block_mat: fixes A1 :: "'a mat"   and n :: "nat"   and A2 :: "'a mat"   and m :: "nat"   and A3 :: "'a mat"   and A4 :: "'a mat" assumes "A1 \<in> carrier_mat n n"   and "A2 \<in> carrier_mat n m"   and "A3 \<in> carrier_mat m n"   and "A4 \<in> carrier_mat m m" shows "char_poly_matrix (four_block_mat A1 A2 A3 A4) = four_block_mat (char_poly_matrix A1) (map_mat (\<lambda>x. [:- x:]) A2) (map_mat (\<lambda>x. [:- x:]) A3) (char_poly_matrix A4)"
minus_set_def: fixes A :: "'e set"   and B :: "'e set" shows "A - B = Collect ((\<lambda>x. x \<in> A) - (\<lambda>x. x \<in> B))"
elims: fixes x :: "primfun \<times> 'a"   and xa :: "'a"   and y :: "primfun \<times> 'a" assumes "powr_primfun x xa = y" obtains xb :: "primfun" and a :: "'a" and e :: "'a" where "x = (xb, a)" and "xa = e" and "y = (xb, e * a)"
starfun_star_n: fixes f :: "'d \<Rightarrow> 'c"   and X :: "nat \<Rightarrow> 'd" shows "(*f* f) (star_n X) = star_n (\<lambda>n. f (X n))"
intros_11: fixes fop :: "binop_f"   and c1 :: "f64"   and c2 :: "f64"   and c :: "f64" assumes "app_binop_f fop c1 c2 = Some c" shows "\<lparr>[$C ConstFloat64 c1, $C ConstFloat64 c2, $Binop_f T_f64 fop]\<rparr> \<leadsto> \<lparr>[$C ConstFloat64 c]\<rparr>"
dp_aux1: fixes z :: "'a"   and w :: "'a"   and x :: "'a"   and y :: "'a" assumes "is_p_fun z"   and "is_total w"   and "x\<^sup>\<smile> ; z = 1" shows "(w ; x\<^sup>\<smile> \<cdot> y ; z\<^sup>\<smile>) ; z = y"
continuous_within_compose: fixes x :: "'a"   and s :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'c" assumes "continuous (at x within s) f"   and "continuous (at (f x) within f ` s) g" shows "continuous (at x within s) (g \<circ> f)"
S_diff_T_diff: fixes x :: "real" shows "S x - S (x / 2) \<le> T x - 2 * T (x / 2)"
is_offending_flows_min_set_def: fixes f :: "('a \<times> 'a) set"   and G :: "'a graph"   and nP :: "'a \<Rightarrow> 'b" shows "is_offending_flows_min_set f G nP \<equiv> is_offending_flows f G nP \<and> (\<forall>(e1, e2)\<in>f. \<not> sinvar (add_edge e1 e2 (delete_edges G f)) nP)"
less_cons: fixes s :: "'c list"   and t :: "'c list"   and a :: "'c" assumes "s < t" shows "a # s < a # t"
tdghm_op_tdghm_op_tdghm: shows "op_tdghm (op_tdghm \<NN>) = \<NN>"
finite_sub_comp_imp_card_eq_mult: fixes H :: "'a set"   and J :: "'a set" assumes "subgroup H G"   and "subgroup J G"   and "finite H"   and "finite J"   and "complementary H J" shows "card (H <#> J) = card J * card H"
suState: shows "Partial_Cost_Model.config' (rTS h0) ([x, y], rev a @ h) b = ([x, y], rev b @ rev a @ h)"
f4_red_aux_not_zero: fixes bs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and ps :: "((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list" shows "0 \<notin> set (f4_red_aux bs ps)"
Abs_freeletter_add: fixes s :: "'b"   and t :: "'b" shows "s [+] t = Abs_freelist [s, t]"
cases: fixes t :: "nat"   and a1 :: "nat \<times> nat \<times> (nat \<Rightarrow> thread)"   and a2 :: "nat \<times> nat \<times> (nat \<Rightarrow> thread)" assumes "alstep t a1 a2" obtains c :: "nat" and n :: "nat" and ts :: "nat \<Rightarrow> thread" and c' :: "nat" and n' :: "nat" and s' :: "thread" where "a1 = (c, n, ts)" and "a2 = (c', n', ts(t := s'))" and "t < N" and "astep_sng (c, n, ts t) (c', n', s')"
ofilter_embed: fixes r :: "('c \<times> 'c) set"   and A :: "'c set" assumes "Well_order r" shows "ofilter r A = (A \<subseteq> Field r \<and> embed (Restr r A) r id)"
graph_map_dom: fixes R :: "('a \<times> 'b) set" shows "dom (graph_map R) = fst ` R"
simps_1: fixes c :: "nat \<Rightarrow> ('a, 'b, 'c) com"   and p :: "'b" shows "seq c p 0 = Call p"
eq_iff: fixes x :: "'b poly"   and y :: "'b poly" shows "(pCons (0::'b) x = pCons (0::'b) y) = (x = y)"
that: fixes h :: "complex \<Rightarrow> complex" assumes "h holomorphic_on ball 0 r"   and "\<And>z. cmod z < r \<Longrightarrow> f z = z * h z"   and "deriv f 0 = h 0" shows "thesis"
prod_vwb_lens: fixes X :: "'b \<Longrightarrow> 'c"   and Y :: "'d \<Longrightarrow> 'e" assumes "vwb_lens X"   and "vwb_lens Y" shows "vwb_lens (X \<times>\<^sub>L Y)"
reflpE: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a" assumes "reflp r" obtains "r x x"
preserves_ide: fixes a :: "'c local.term set" assumes "\<F>C.ide a" shows "ide (\<F>C.D a)"
comp_fun_idem: shows "comp_fun_idem (\<^bold>*)"
map\<^sub>0_simps_1: fixes a :: "'b" assumes "B.obj a" shows "C.obj (\<tau>.F.map\<^sub>0 a)"
poly_pfd_simple_def: fixes x :: "'b poly"   and cs :: "('b \<times> nat) list" shows "poly_pfd_simple x cs = (if cs = [] then (x, []) else let zs = map (\<lambda>i. let (c, n) = cs ! i in normalise_decomp_poly (decompose x (map (\<lambda>(c, n). [:1::'b, - c:] ^ Suc n) cs) ! i) [:1::'b, - c:] (n + 1)) [0..<length cs] in (sum_list (map fst zs), map (map (\<lambda>p. coeff p 0) \<circ> snd) zs))"
tendsto_const_iff: fixes "open" :: "'b set \<Rightarrow> bool"   and F :: "'c filter"   and a :: "'b"   and b :: "'b" assumes "class.t2_space open"   and "F \<noteq> bot" shows "(LIM x F. a :> topological_space.nhds open b) = (a = b)"
valid_path_statelist': fixes as :: "(('c, 'd) fmap \<times> ('c, 'd) fmap) list"   and Pi :: "(('c, 'd) fmap \<times> ('c, 'd) fmap) set"   and s :: "('c, 'd) fmap" assumes "as \<in> valid_plans Pi"   and "s \<in> valid_states Pi" shows "valid_path Pi (statelist' s as)"
simps_1: fixes E_ES :: "'b set"   and I_ES :: "'b set"   and O_ES :: "'b set"   and Tr_ES :: "'b list set"   and more :: "'c" shows "E_ES \<lparr>E_ES = E_ES, I_ES = I_ES, O_ES = O_ES, Tr_ES = Tr_ES, \<dots> = more\<rparr> = E_ES"
ChamberComplexRetraction_axioms: shows "ChamberComplexRetraction X g"
elims_3: fixes x :: "'a ext0"   and xa :: "'a ext0" assumes "\<not> x < xa" obtains x :: "'a ext0" and y :: "'a ext0" where "xa__ = x" and "xa = y" and "\<not> (x \<le> y \<and> \<not> y \<le> x)"
in_language_One: fixes xs :: "'b list" assumes "in_language One xs" shows "xs = []"
concat_parts': fixes a :: "nat"   and f :: "nat \<Rightarrow> 'b list"   and d :: "nat"   and i :: "nat" assumes "\<And>i. i < a \<Longrightarrow> length (f i) = d"   and "i < a" shows "fixed_length_sublist (concat (map f [0..<a])) d i = f i"
PQdepth: shows "depth P + depth Q \<le> Suc n"
mustT_Seq_L: fixes c1 :: "('test, 'atom) com"   and c2 :: "('test, 'atom) com"   and s :: "'state" assumes "mustT (c1 ;; c2) s" shows "mustT c1 s"
Plus_eq_empty_conv: fixes A :: "'h set"   and B :: "'i set" shows "(A <+> B = {}) = (A = {} \<and> B = {})"
complex_cnj_one_iff: fixes z :: "complex" shows "(cnj z = 1) = (z = 1)"
Sup_le_iff: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and A :: "'b set"   and b :: "'b" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "less_eq (Sup A) b = (\<forall>a\<in>A. less_eq a b)"
sccs_more_cong: fixes s :: "('c, 'd) tarjan_state_scheme"   and s' :: "('c, 'd) tarjan_state_scheme" assumes "state.more s = state.more s'" shows "sccs s = sccs s'"