hcju/isabelleps huggingface dataset:
step_thread_correct_1: fixes I :: "(('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set) set"   and \<sigma>_invar :: "'a \<Rightarrow> 't set \<Rightarrow> bool"   and \<sigma> :: "'a"   and s :: "('l, 't) locks \<times> ('m_t \<times> 'm) \<times> 'm_w \<times> 's_i"   and \<sigma>_\<alpha> :: "'a \<Rightarrow> 'b"   and update_state :: "('l, 't, 'x, 'm, 'w, 'o) thread_action \<Rightarrow> 'a"   and t :: "'t" assumes "\<alpha>.deterministic I"   and "\<sigma>_invar \<sigma> (dom (thr_\<alpha> (thr s)))"   and "state_invar s"   and "state_\<alpha> s \<in> I" shows "map_option (apsnd (apsnd \<sigma>_\<alpha>)) (step_thread update_state s t) = \<alpha>.step_thread (\<sigma>_\<alpha> \<circ> update_state) (state_\<alpha> s) t"
binary_functor_axioms: shows "binary_functor (\<cdot>\<^sup>o\<^sup>p) (\<cdot>) (\<cdot>\<^sub>S) local.map"
parts_cut_eq: fixes X :: "msg"   and H :: "msg set" assumes "X \<in> parts H" shows "parts (insert X H) = parts H"
n: fixes \<epsilon> :: "'a"   and n :: "'a" assumes "sqrt_approximation \<epsilon> n" shows "(0::'a) < n"
normal_min_inf: fixes \<phi> :: "atom fm" assumes "Z.normal \<phi>" shows "Z.normal (inf\<^sub>- \<phi>)"
new_character_data_get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t: fixes h :: "(_) heap"   and h' :: "(_) heap"   and new_character_data_ptr :: "(_) character_data_ptr"   and ptr :: "(_) object_ptr"   and getter :: "(_) RObject_ext \<Rightarrow> 'v" assumes "h \<turnstile> new_character_data \<rightarrow>\<^sub>h h'"   and "h \<turnstile> new_character_data \<rightarrow>\<^sub>r new_character_data_ptr"   and "ptr \<noteq> cast\<^sub>c\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>_\<^sub>d\<^sub>a\<^sub>t\<^sub>a\<^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 new_character_data_ptr" shows "preserved (get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t ptr getter) h h'"
simps: fixes as :: "nat"   and lm :: "nat list"   and s :: "nat"   and l :: "cell list"   and r :: "cell list"   and ires :: "cell list" shows "dec_right_move (as, lm) (s, l, r) ires = (\<exists>lm1 lm2 m ml mr rn. lm = lm1 @ [m] @ lm2 \<and> ml = Suc m \<and> mr = 0 \<and> (if lm1 = [] then l = Bk # Oc \<up> ml @ Bk # Bk # ires else l = Bk # Oc \<up> ml @ [Bk] @ <rev lm1> @ Bk # Bk # ires) \<and> (r = Bk # <lm2> @ Bk \<up> rn \<or> r = [] \<and> lm2 = []))"
visible_methods_exist: fixes P :: "'a prog"   and C :: "String.literal"   and Mm :: "String.literal \<Rightarrow> ((ty list \<times> ty \<times> 'a option) \<times> String.literal) option"   and M :: "String.literal"   and m :: "ty list \<times> ty \<times> 'a option"   and D :: "String.literal" assumes "P \<turnstile> C sees_methods Mm"   and "Mm M = \<lfloor>(m, D)\<rfloor>" shows "\<exists>D' fs ms. class P D = \<lfloor>(D', fs, ms)\<rfloor> \<and> map_of ms M = \<lfloor>m\<rfloor>"
distinct_51: fixes x3 :: "nat"   and x101 :: "nat"   and x102 :: "msg" shows "Nonce x3 \<noteq> Crypt x101 x102"
oarrivemsg_def: fixes P :: "(nat \<Rightarrow> 'd) \<Rightarrow> 'e \<Rightarrow> bool"   and \<sigma> :: "nat \<Rightarrow> 'd"   and a :: "'e node_action" shows "oarrivemsg P \<sigma> a \<equiv> case a of ii\<not>ni:arrive(m) \<Rightarrow> P \<sigma> m | _ \<Rightarrow> True"
indets_prod_subset: fixes f :: "'d \<Rightarrow> ('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c"   and A :: "'d set" shows "indets (prod f A) \<subseteq> (\<Union>a\<in>A. indets (f a))"
1: shows "snd (snd z) \<in> (snd \<circ> snd) ` set_pmf (config'_rand BIT (fst BIT init \<bind> (\<lambda>is. return_pmf (init, is))) qs)"
assms_4: fixes a :: "'a"   and b :: "'a" assumes "x = a * b"   and "\<not> a dvd (1::'a)"   and "\<not> b dvd (1::'a)" shows "thesis"
system_needed_asses_include_action_needed_asses: fixes a :: "('a, 'b) fmap \<times> 'c"   and PROB :: "(('a, 'b) fmap \<times> 'c) set"   and s :: "('a, 'b) fmap" assumes "a \<in> PROB" shows "action_needed_asses a (system_needed_asses PROB s) = action_needed_asses a s"
size_fmap_def: fixes f :: "'d \<Rightarrow> nat"   and g :: "'e \<Rightarrow> nat"   and m :: "('d, 'e) fmap" shows "size_fmap f g m = size_fset (\<lambda>(a, b). f a + g b) (fset_of_fmap m)"
in_lub: fixes x :: "Vars.var"   and y :: "Vars.var"   and S :: "CoCalls set" shows "x--y\<in>lub S = (\<exists>G\<in>S. x--y\<in>G)"
assms: shows "\<exists>A' C' D' F'. B Out A' A \<and> B Out C' C \<and> E Out D' D \<and> E Out F' F \<and> A' B C' Cong3 D' E F'"
l: shows "lhs t \<noteq> trm_rep (lhs t) S \<longrightarrow> (trm_rep (lhs t) S, lhs t) \<in> trm_ord"
get_def: shows "get = State (\<lambda>s. (s, s))"
map_project_ENC: fixes X :: "nat set"   and n :: "nat"   and Z :: "('a \<times> bool list) list set" assumes "X \<subseteq> {0..<n + 1}"   and "Z \<subseteq> lists ((set \<circ> \<sigma> \<Sigma>) (n + 1))" shows "map \<pi> ` (Z \<inter> lang (n + 1) (ENC (n + 1) X) - {[]}) = (if 0 \<in> X then map \<pi> ` (Z \<inter> lang (n + 1) (ENC (n + 1) {0})) \<inter> lang n (ENC n ((\<lambda>x. x - 1) ` (X - {0}))) - {[]} else map \<pi> ` Z \<inter> lang n (ENC n ((\<lambda>x. x - 1) ` (X - {0}))) - {[]})"
g: fixes x :: "real" assumes "x \<in> {a..b}" shows "(g has_vector_derivative f x) (at x within {a..b})"
compact_imp_closed: fixes "open" :: "'b set \<Rightarrow> bool"   and s :: "'b set" assumes "class.t2_space open"   and "topological_space.compact open s" shows "topological_space.closed open s"
refl: fixes stop :: "'a"   and eval :: "((('a \<times> (Mode \<Rightarrow> 'b set)) \<times> ('b \<Rightarrow> 'c)) \<times> ('a \<times> (Mode \<Rightarrow> 'b set)) \<times> ('b \<Rightarrow> 'c)) set"   and some_val :: "'c"   and some_val' :: "'c"   and lc :: "('a \<times> (Mode \<Rightarrow> 'b set)) \<times> ('b \<Rightarrow> 'c)" assumes "sifum_security stop eval some_val some_val'" shows "((fst (fst lc), snd (fst lc)), snd lc) \<in> sifum_security.loc_reach eval lc"
assms: shows "get_S w1 = 0 \<and> w2 AND 128 = 0 \<and> w3 AND 128 = 0 \<and> w4 AND 128 = 0 \<and> w5 AND 128 = 0"
integrable: shows "integrable (count_space A \<Otimes>\<^sub>M count_space B') (\<lambda>(x, y). indicat_real (B x) y *\<^sub>R f (x, y))"
rel_F_conversep: fixes L1 :: "'z \<Rightarrow> 'p \<Rightarrow> bool"   and L2 :: "'aa \<Rightarrow> 'q \<Rightarrow> bool"   and L3 :: "'ab \<Rightarrow> 'r \<Rightarrow> bool"   and Co1 :: "'ac \<Rightarrow> 's \<Rightarrow> bool"   and Co2 :: "'ad \<Rightarrow> 't \<Rightarrow> bool"   and Co3 :: "'ae \<Rightarrow> 'u \<Rightarrow> bool"   and Contra1 :: "'af \<Rightarrow> 'v \<Rightarrow> bool"   and Contra2 :: "'ag \<Rightarrow> 'w \<Rightarrow> bool"   and Contra3 :: "'ah \<Rightarrow> 'x \<Rightarrow> bool" shows "rel_F L1\<inverse>\<inverse> L2\<inverse>\<inverse> L3\<inverse>\<inverse> Co1\<inverse>\<inverse> Co2\<inverse>\<inverse> Co3\<inverse>\<inverse> Contra1\<inverse>\<inverse> Contra2\<inverse>\<inverse> Contra3\<inverse>\<inverse> = (rel_F L1 L2 L3 Co1 Co2 Co3 Contra1 Contra2 Contra3)\<inverse>\<inverse>"
assms_7: shows "y \<cdot> (0::'a) = (0::'a)"
bot_less1: fixes X :: "'b process" shows "\<bottom> \<le> X"
Fract_normalize: fixes x :: "int"   and y :: "int" shows "Fract (fst (Rat.normalize (x, y))) (snd (Rat.normalize (x, y))) = Fract x y"
reduced_GB_ideal_Polys: fixes X :: "'x set"   and F :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set" assumes "finite X"   and "F \<subseteq> P[X]" shows "ideal (i.punit.reduced_GB F) = ideal F"
not_H: fixes t :: "'s" assumes "(s, t) \<in> acc_on (- H)" shows "t \<notin> H"
simps_1: fixes a :: "'b" shows "|Elem a| = 0"
A: shows "ranked_tree_automaton (ta_prod TA1 TA2) A"
add_diff_assoc2: fixes k :: "nat"   and j :: "nat"   and i :: "nat" assumes "k \<le> j" shows "j - k + i = j + i - k"
assms_3: fixes x :: "'a" assumes "x \<in> X" shows "dist (f x) (g x) \<le> D"
induct: fixes x1 :: "numeric_unop"   and x2 :: "'a OCL_Types.type"   and x3 :: "'b OCL_Types.type"   and P :: "numeric_unop \<Rightarrow> 'a OCL_Types.type \<Rightarrow> 'b OCL_Types.type \<Rightarrow> bool" assumes "numeric_unop_type x1 x2 x3"   and "\<And>\<tau>. \<tau> = Real[1] \<Longrightarrow> P UMinusOp \<tau> Real[1]"   and "\<And>\<tau>. \<tau> = UnlimitedNatural[1]\<midarrow>Integer[1] \<Longrightarrow> P UMinusOp \<tau> Integer[1]"   and "\<And>\<tau>. \<tau> = Real[1] \<Longrightarrow> P AbsOp \<tau> Real[1]"   and "\<And>\<tau>. \<tau> = UnlimitedNatural[1]\<midarrow>Integer[1] \<Longrightarrow> P AbsOp \<tau> Integer[1]"   and "\<And>\<tau>. \<tau> = UnlimitedNatural[1]\<midarrow>Real[1] \<Longrightarrow> P FloorOp \<tau> Integer[1]"   and "\<And>\<tau>. \<tau> = UnlimitedNatural[1]\<midarrow>Real[1] \<Longrightarrow> P RoundOp \<tau> Integer[1]"   and "\<And>\<tau>. \<tau> = UnlimitedNatural[1] \<Longrightarrow> P numeric_unop.ToIntegerOp \<tau> Integer[1]" shows "P x1 x2 x3"
dec_X': fixes i :: "nat"   and n :: "nat"   and j :: "nat" assumes "J i \<subseteq> {0..<n}"   and "J j \<subseteq> {0..<n}"   and "i \<le> j" shows "PF.emb {0..<n} (J j) (X' j) \<subseteq> PF.emb {0..<n} (J i) (X' i)"
le: fixes x :: "'a" assumes "x \<in> Y" shows "x \<le> f x"
intro: fixes comp1 :: "'a \<Rightarrow> 'a \<Rightarrow> order"   and comp2 :: "'a \<Rightarrow> 'a \<Rightarrow> order" assumes "comparator comp1"   and "comparator comp2" shows "comparator2 comp1 comp2"
OO_eq: fixes R :: "'b \<Rightarrow> 'd \<Rightarrow> bool" shows "R OO (=) = R"
index_add_mat_1: fixes i :: "nat"   and B :: "'m mat"   and j :: "nat"   and A :: "'m mat" assumes "i < dim_row B"   and "j < dim_col B" shows "(A + B) $$ (i, j) = A $$ (i, j) + B $$ (i, j)"
floor_correct: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and 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 floor :: "'b \<Rightarrow> int"   and x :: "'b" assumes "class.floor_ceiling divide inverse times one plus zero minus uminus less_eq less sgn abs floor" shows "less_eq (ring_1.of_int one plus zero minus (floor x)) x \<and> less x (ring_1.of_int one plus zero minus (floor x + 1))"
unit_vecs_last: fixes n :: "nat" shows "unit_vecs n = unit_vecs_last n n"
benchmark_def: fixes \<alpha> :: "'a \<Rightarrow> natural set"   and invar :: "'a \<Rightarrow> bool"   and empty :: "unit \<Rightarrow> 'a"   and memb :: "natural \<Rightarrow> 'a \<Rightarrow> bool"   and ins :: "natural \<Rightarrow> 'a \<Rightarrow> 'a"   and rem :: "natural \<Rightarrow> 'a \<Rightarrow> 'a"   and size :: "'a \<Rightarrow> nat"   and filter :: "(natural \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> 'a" shows "benchmark \<alpha> invar empty memb ins rem size filter \<equiv> (set_empty \<alpha> invar empty \<and> set_memb \<alpha> invar memb \<and> set_ins \<alpha> invar ins) \<and> set_delete \<alpha> invar rem \<and> set_size \<alpha> invar size \<and> set_filter \<alpha> invar \<alpha> invar filter"
dist_pos_lt: fixes dist :: "'d \<Rightarrow> 'd \<Rightarrow> real"   and uniformity :: "('d \<times> 'd) filter"   and "open" :: "'d set \<Rightarrow> bool"   and x :: "'d"   and y :: "'d" assumes "class.metric_space dist uniformity open"   and "x \<noteq> y" shows "0 < dist x y"
one_poly_i_def: fixes ops :: "'a arith_ops_record" shows "one_poly_i ops = [arith_ops_record.one ops]"
goodInp_imp_qGoodInp_pickInp: fixes inp :: "'c \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm set option" assumes "goodInp inp" shows "qGoodInp (pickInp inp)"
dom_simp: fixes f :: "('a, 'a set) Maps.arr arrow_of_spans_data \<times> ('a, 'a set) Maps.arr arrow_of_spans_data" assumes "Span.VV.arr f" shows "Span.VV.dom f = Span.VxV.dom f"
assms_11: shows "callee_invariant_on oracle2 I2 \<I>"
finite_grab_iff: fixes N :: "nat set"   and n :: "nat" shows "finite (snd (grab N n)) = finite N"
abs_dvd_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 m :: "'b"   and k :: "'b" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "dvd.dvd times (abs m) k = dvd.dvd times m k"
list_all2_update_cong: fixes P :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and xs :: "'g list"   and ys :: "'h list"   and x :: "'g"   and y :: "'h"   and i :: "nat" assumes "list_all2 P xs ys"   and "P x y" shows "list_all2 P (xs[i := x]) (ys[i := y])"
hyps_1: shows "vmc_path_no_deadend G P v0 p \<sigma>"
rep_number_alt_def_all: shows "\<forall>x\<in>\<V>. \<B> rep x = \<r>"
map_vec_def: fixes f :: "'b \<Rightarrow> 'a"   and v :: "'b Matrix.vec" shows "map_vec f v \<equiv> Matrix.vec (dim_vec v) (\<lambda>i. f (v $ i))"
tid_simps: fixes C :: "CTypeId"   and a :: "nat" shows "tid (objV C a) = C"
signs_at_def: fixes qs :: "real poly list"   and x :: "real" shows "signs_at qs x \<equiv> map (squash \<circ> (\<lambda>q. poly q x)) qs"
diff_left_mono: 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 b :: "'a"   and a :: "'a"   and c :: "'a" assumes "class.ordered_ab_group_add plus zero minus uminus less_eq less"   and "less_eq b a" shows "less_eq (minus c a) (minus c b)"
density_unique_finite_measure: fixes f :: "'a \<Rightarrow> ennreal"   and f' :: "'a \<Rightarrow> ennreal" assumes "f \<in> borel_measurable (density M h)"   and "f' \<in> borel_measurable (density M h)"   and "AE x in density M h. 0 \<le> f x"   and "AE x in density M h. 0 \<le> f' x"   and "\<And>A. A \<in> sets (density M h) \<Longrightarrow> \<integral>\<^sup>+ x. f x * indicator A x \<partial>density M h = \<integral>\<^sup>+ x. f' x * indicator A x \<partial>density M h" shows "AE x in density M h. f x = f' x"
kill_short_induct: fixes G :: "nat set \<times> nat set set"   and k :: "nat"   and P :: "nat set \<times> nat set set \<Rightarrow> nat \<Rightarrow> bool" assumes "finite (uverts G)"   and "\<And>G. short_cycles G k = {} \<Longrightarrow> P G k"   and "\<And>G. \<lbrakk>finite (short_cycles G k); short_cycles G k \<noteq> {}; P (G -- choose_v G k) k\<rbrakk> \<Longrightarrow> P G k" shows "P G k"
assms_2: shows "\<forall>i\<le>n. s i \<le> t i"
Max_gr_iff: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a set"   and x :: "'a" assumes "class.linorder less_eq less"   and "finite A"   and "A \<noteq> {}" shows "less x (linorder.Max less_eq A) = (\<exists>a\<in>A. less x a)"
ih: fixes ta :: "('s, 'v) tm"   and sa :: "('s, 'v) tm" assumes "hsize ta + hsize sa < hsize t + hsize s"   and "t' >\<^sub>t ta" shows "App sa t' >\<^sub>t App sa ta"
nrename_nfresh: fixes x :: "name"   and y :: "name"   and M :: "trm" assumes "x \<sharp> y" shows "x \<sharp> M[x\<turnstile>n>y]"
Complex_mult_i: fixes a :: "real"   and b :: "real" shows "Complex a b * \<i> = Complex (- b) a"
responses_\<I>_uniform: fixes A :: "'i set"   and B :: "'h set"   and x :: "'i" shows "responses_\<I> (\<I>_uniform A B) x = (if x \<in> A then B else {})"
minimalize_offending_overapprox_boundnP: fixes m :: "'c graph \<Rightarrow> ('c \<Rightarrow> 'd) \<Rightarrow> bool"   and nP :: "'c \<Rightarrow> 'd"   and fs :: "('c \<times> 'c) list"   and keeps :: "('c \<times> 'c) list"   and G :: "'c graph" shows "TopoS_withOffendingFlows.minimalize_offending_overapprox (\<lambda>G. m G nP) fs keeps G = SecurityInvariant_withOffendingFlows.minimalize_offending_overapprox m fs keeps G nP"
skelBinp_def2: fixes binp :: "'a \<Rightarrow> ('b, 'c, 'varSort, 'var, 'd) qAbs set option" assumes "goodBinp binp" shows "skelBinp binp = lift skelAbs binp"
ins: shows "ins k p l = \<langle>l2, (k2, p2), r2\<rangle>"
scaleC: fixes r :: "complex"   and b :: "'c" shows "g (r *\<^sub>C b) = r *\<^sub>C g b"
hyperpow_le_le: fixes r :: "real star"   and n :: "nat star"   and N :: "nat star" assumes "0 \<le> r"   and "r \<le> 1"   and "n \<le> N" shows "r pow N \<le> r pow n"
arrE: fixes f :: "'a \<times> 'a" assumes "H.VxV.arr f" obtains "V.arr (fst f) \<and> V.arr (snd f)"
bounds_1: fixes x :: "real" assumes "a \<le> x"   and "x \<le> b" shows "g a \<le> g x"
circ_left_induct_mult_L: fixes x :: "'a"   and y :: "'a" assumes "L \<le> x"   and "x * y \<le> x" shows "x * y\<^sup>\<circ> \<le> x"
"case": fixes f :: "proj2 \<Rightarrow> 'a"   and x :: "proj2" shows "(case P2L x of P2L x \<Rightarrow> f x) = f x"
rq_sefl: fixes u :: "'b list" shows "u\<^sup><\<inverse>u  = \<epsilon>"
nreachable_def: fixes G :: "('c, 'd) pre_digraph"   and u :: "'c"   and n :: "nat"   and v :: "'c" shows "u \<rightarrow>\<^bsup>n\<^esup>\<^bsub>G\<^esub> v \<equiv> ntrancl_onp (verts G) (arcs_ends G) n u v"
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>T\<^sub>R\<^esub> \<phi> = K"
simps_2: fixes p :: "Player"   and \<sigma> :: "'a \<Rightarrow> 'a"   and \<sigma>' :: "'a \<Rightarrow> 'a"   and v0 :: "'a" shows "lhd (greedy_conforming_path p \<sigma> \<sigma>' v0) = v0"
chamber_card: fixes X :: "'b set set"   and C :: "'b set"   and D :: "'b set" assumes "ChamberComplex X"   and "SimplicialComplex.maxsimp X C"   and "SimplicialComplex.maxsimp X D" shows "card C = card D"
simActBoundCases: fixes Rel :: "(pi \<times> pi) set"   and a :: "subject"   and P :: "pi"   and x :: "name"   and Q' :: "pi" assumes "eqvt Rel"   and "\<And>b. a = InputS b \<Longrightarrow> \<exists>P''. \<forall>u. \<exists>P'. P \<Longrightarrow>\<^sub>lu in P''\<rightarrow>b<x> \<prec> P' \<and> (P', Q'[x::=u]) \<in> Rel"   and "\<And>b. a = BoundOutputS b \<Longrightarrow> \<exists>P'. P \<Longrightarrow>\<^sub>l\<^sup>^b<\<nu>x> \<prec> P' \<and> (P', Q') \<in> Rel" shows "weakSimAct P (a\<guillemotleft>x\<guillemotright> \<prec> Q') P Rel"
suminf_pos_iff: fixes f :: "nat \<Rightarrow> 'a" assumes "summable f"   and "\<And>n. (0::'a) \<le> f n" shows "((0::'a) < suminf f) = (\<exists>i. (0::'a) < f i)"
pivot_tableau_normalized: fixes eq_idx_for_lvar :: "(nat \<times> linear_poly) list \<Rightarrow> nat \<Rightarrow> nat"   and pivot :: "nat \<Rightarrow> nat \<Rightarrow> ('a, 'b) state \<Rightarrow> ('a, 'b) state"   and s :: "('a, 'b) state"   and x\<^sub>i :: "nat"   and x\<^sub>j :: "nat" assumes "Pivot eq_idx_for_lvar pivot"   and "\<triangle> (\<T> 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 "\<triangle> (\<T> (pivot x\<^sub>i x\<^sub>j s))"
funStar_def: fixes f :: "'b \<Rightarrow> 'b set" shows "funStar f = (\<lambda>x. {y. (x, y) \<in> {(x, y). y \<in> f x}\<^sup>*})"
splitqsallg: fixes Strat :: "(nat \<times> nat list) list"   and a :: "'a"   and x :: "'a"   and y :: "'a"   and rest1 :: "'a list" assumes "Strat \<noteq> []"   and "a \<in> {x, y}" shows "t\<^sub>p [x, y] a (hd Strat) + (let L = step [x, y] a (hd Strat) in T\<^sub>p L rest1 (tl Strat)) = T\<^sub>p [x, y] (a # rest1) Strat"
bounded_by_imp_bounded: fixes b :: "real"   and P :: "'c \<Rightarrow> real" assumes "bounded_by b P" shows "Expectations.bounded P"
learn_bc_closed_subseteq: fixes \<psi> :: "nat \<Rightarrow> nat \<Rightarrow> nat option"   and U :: "(nat \<Rightarrow> nat option) set"   and s :: "nat \<Rightarrow> nat option"   and V :: "(nat \<Rightarrow> nat option) set" assumes "learn_bc \<psi> U s"   and "V \<subseteq> U" shows "learn_bc \<psi> V s"
prj_simps_3: shows "C.arr gfg.Prj\<^sub>0"
big_unclocked: fixes env :: "v sem_env"   and s :: "'b state"   and e :: "exp0"   and s' :: "'b state"   and r :: "(v, v) result"   and count1 :: "nat"   and count2 :: "nat" assumes "evaluate False env s e (s', r) \<Longrightarrow> r \<noteq> Rerr (Rabort Rtimeout_error)"   and "evaluate False env s e (s', r) \<Longrightarrow> clock s = clock s'"   and "evaluate False env (update_clock (\<lambda>_. count1) s) e (update_clock (\<lambda>_. count1) s', r)" shows "evaluate False env (update_clock (\<lambda>_. count2) s) e (update_clock (\<lambda>_. count2) s', r)"
assms: shows "run_one_step d i (s, vs, ves, $Set_global x18) = (s', vs', res)"
basis_left_null_space_iarrays_eq: fixes A :: "(('b, 'c) vec, 'd) vec" shows "basis_left_null_space_iarrays (matrix_to_iarray A) = basis_null_space_iarrays (transpose_iarray (matrix_to_iarray A))"
simple_closed_path_wn2: fixes d :: "real"   and e :: "real"   and p :: "real \<Rightarrow> complex"   and a :: "complex" assumes "0 < d"   and "0 < e"   and "simple_path (p +++ linepath (a - complex_of_real d) (a + complex_of_real e))"   and "pathstart p = a + complex_of_real e"   and "pathfinish p = a - complex_of_real d" obtains z :: "complex" where "z \<in> inside (path_image (p +++ linepath (a - complex_of_real d) (a + complex_of_real e)))" and "cmod (winding_number (p +++ linepath (a - complex_of_real d) (a + complex_of_real e)) z) = 1"
intros_6: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "('e set \<times> 'd \<times> 'e set \<times> 'e set) set"   and F :: "'f set"   and P :: "'e set"   and b :: "'e set"   and c :: "('e, 'd, 'f) com"   and A :: "'e set" assumes "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>(P \<inter> b) c P,A" shows "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P While b c (P \<inter> - b),A"
apply_operations_def: fixes es :: "('a \<times> operation) event list" shows "apply_operations es \<equiv> hb.apply_operations (node_deliver_messages es) 0"
aux_comp_sum_divide_cmod: fixes g :: "nat \<Rightarrow> int"   and a :: "real"   and n :: "nat" shows "(cmod (complex_of_real (\<Sum>k<n. real_of_int (g k) / a)))\<^sup>2 = (cmod (of_int (sum g {..<n})) / a)\<^sup>2"
abs_eq: fixes x :: "'b pmf" shows "mapping_of_pmf x = mapping.Mapping (\<lambda>xa. if pmf x xa = 0 then None else Some (pmf x xa))"
normalizability_E: fixes a :: "'d"   and b :: "'d"   and A :: "('d \<times> 'd) set" assumes "(a, b) \<in> A\<^sup>!" obtains "(a, b) \<in> A\<^sup>*" and "b \<in> NF A"
rwof: fixes G :: "('a, 'b) graph_rec_scheme"   and param :: "('a, ('a, 'c) state_scheme, 'c) gen_parameterization"   and s :: "('a, 'c) state_scheme" assumes "DFS_invar G param s" shows "rwof (param_DFS_defs.init param) (param_DFS_defs.cond G param) (param_DFS_defs.step G param) s"
outs'_rpv_def: fixes rpv :: "('f, 'd, 'e) rpv" shows "outs'_rpv rpv = range rpv \<bind> outs'_gpv"
that: fixes r :: "'a \<Rightarrow> 'a" assumes "homotopic_with_canon (\<lambda>x. True) (T - {a}) (T - {a}) id r"   and "retraction (T - {a}) (rel_frontier S) r" shows "thesis"
ch2ch_monofun: fixes f :: "'a \<Rightarrow> 'b"   and Y :: "nat \<Rightarrow> 'a" assumes "monofun f"   and "chain Y" shows "chain (\<lambda>i. f (Y i))"
dom_char: fixes f :: "'c local.term set" shows "\<F>\<^sub>SC.dom f = (if \<F>\<^sub>SC.arr f then local.dom f else null)"
mk_tree_Suc: fixes n :: "nat" shows "mk_tree (Suc n) = \<triangle> \<^bold>l: mk_tree n \<^bold>r: mk_tree n"
invI_2: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier R"   and "y \<in> carrier R"   and "x \<otimes> y = \<one>" shows "x = inv y"
continuous_on_add: fixes s :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "continuous_on s f"   and "continuous_on s g" shows "continuous_on s (\<lambda>x. f x + g x)"
plus_adds_mono: fixes a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" assumes "a adds b"   and "c adds d" shows "a + c adds b + d"
that: fixes P'' :: "('a, 'b, 'c) psi" assumes "\<Psi> \<rhd> \<tau>.P \<Longrightarrow>\<^sup>^\<^sub>\<tau> P''"   and "\<Psi> \<rhd> P' \<sim> P''" shows "thesis"