hcju/isabelleps huggingface dataset:
inducts: fixes f :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b"   and x :: "'b"   and P :: "'b \<Rightarrow> bool" assumes "segmentp f a b x"   and "f a \<noteq> b \<Longrightarrow> P (f a)"   and "\<And>x. \<lbrakk>segmentp f a b x; P x; f x \<noteq> b\<rbrakk> \<Longrightarrow> P (f x)" shows "P x"
plus_1_eSuc_2: fixes q :: "enat" shows "q + 1 = eSuc q"
simps_2: fixes ns :: "C list" shows "funC (C (C [] # ns)) = (\<lambda>i. replicate i (C ns))"
rep_eq: shows "Rep_coreflexive top = (1::'a)"
perzeta_altdef1: fixes s :: "complex"   and q' :: "real" assumes "1 < Re s" shows "perzeta q' s = eval_fds (fds_perzeta q') s"
hyps: fixes child_node :: "(_) node_ptr"   and parent :: "(_) object_ptr" assumes "cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^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 child_node = child"   and "h \<turnstile> get_parent child_node \<rightarrow>\<^sub>r Some parent"   and "parent \<in> set c" shows "h \<turnstile> get_root_node parent \<rightarrow>\<^sub>r root_ptr"
Height_Bounded_Labeling_def: fixes c :: "nat \<times> nat \<Rightarrow> 'a"   and s :: "nat"   and t :: "nat"   and f :: "nat \<times> nat \<Rightarrow> 'a"   and l :: "nat \<Rightarrow> nat" shows "Height_Bounded_Labeling c s t f l \<equiv> Labeling c s t f l \<and> Height_Bounded_Labeling_axioms c l"
nat_cut_le_less_conv: fixes I :: "nat set"   and t :: "nat" shows "I \<down>\<le> t = I \<down>< Suc t"
intro: fixes S :: "'c set" assumes "open S" shows "submanifold_axioms S"
inject_5: fixes x61 :: "('a, 'b) MWLfCom"   and x62 :: "('a, 'b) MWLfCom list"   and y61 :: "('a, 'b) MWLfCom"   and y62 :: "('a, 'b) MWLfCom list" shows "(fork x61 x62 = fork y61 y62) = (x61 = y61 \<and> x62 = y62)"
i_aggregate_i_drop_mult: fixes n :: "nat"   and k :: "nat"   and f :: "nat \<Rightarrow> 'b"   and ag :: "'b list \<Rightarrow> 'b" shows "i_aggregate (f \<Up> (n * k)) k ag = i_aggregate f k ag \<Up> n"
subset: shows "{p. prime p \<and> p \<le> nat \<lfloor>x\<rfloor>} \<subseteq> {2, 3} \<union> ((\<lambda>k. 6 * k + 1) ` {0<..<nat \<lfloor>(x + 5) / 6\<rfloor>} \<union> (\<lambda>k. 6 * k + 5) ` {..<nat \<lfloor>(x + 1) / 6\<rfloor>})"
bet_neq21__neq: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Bet A B C"   and "B \<noteq> A" shows "A \<noteq> C"
len_gt_pref_is_pref: fixes l1 :: "'a list"   and l2 :: "'a list"   and l :: "'a list" assumes "length l1 < length l2"   and "prefix l1 l"   and "prefix l2 l" shows "prefix l1 l2"
vrt: shows "Id_on (vertices G\<^sub>1) `` vertices G\<^sub>1 \<subseteq> vertices G\<^sub>2"
il_sub_inf_right_isotone_var: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<sqinter> y \<squnion> x \<sqinter> z \<le> x \<sqinter> (y \<squnion> z)"
dim_le_card: fixes V :: "('a, 'b) vec set"   and W :: "('a, 'b) vec set" assumes "V \<subseteq> vec.span W"   and "finite W" shows "vec.dim V \<le> card W"
hd_last_sorted: fixes le :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and xs :: "'a list" assumes "linorder_list0.sortedP le xs"   and "1 < length xs" shows "le (hd xs) (last xs)"
supp_8: fixes x2 :: "pi"   and x1 :: "pi" shows "supp (x2 \<parallel> x1) = supp x2 \<union> supp x1"
mod_ring_mult64: shows "(mod_ring_rel64 ===> mod_ring_rel64 ===> mod_ring_rel64) (mult_p64 pp) (*)"
ids_def: shows "set ids = Collect is_valid_event_id \<and> sorted_wrt (to_ord happened_immediately_before) ids \<and> distinct ids"
line_extension_def: fixes K :: "'a set"   and a :: "'a"   and E :: "'a set" shows "R.line_extension K a E = K #> a <+> E"
coeff_monom: fixes a :: "int"   and m :: "nat"   and n :: "nat" assumes "a \<in> carrier INTEG" shows "up_ring.coeff (UP INTEG) (up_ring.monom (UP INTEG) a m) n = (if m = n then a else \<zero>\<^bsub>INTEG\<^esub>)"
encode12_inv1: fixes i :: "nat"   and j :: "nat" assumes "i < d1"   and "j < d2" shows "encode1 (encode12 (i, j)) = i"
poly_number_rootat_def: fixes p :: "rat poly" shows "poly_number_rootat p \<equiv> sgn (lead_coeff p)"
pelims: fixes x :: "'a literal"   and y :: "char list \<times> 'a list" assumes "get_atom x = y"   and "get_atom_dom x" obtains p :: "char list" and ts :: "'a list" where "x = Pos p__ ts__" and "y = (p__, ts__)" and "get_atom_dom (Pos p__ ts__)"   | p :: "char list" and ts :: "'a list" where "x = Neg p ts" and "y = (p, ts)" and "get_atom_dom (Neg p ts)"
ball_subset_cball: fixes x :: "'a"   and e :: "real" shows "ball x e \<subseteq> cball x e"
map_of_zip_is_None: fixes xs :: "'a list"   and ys :: "'b list"   and x :: "'a" assumes "length xs = length ys" shows "(map_of (zip xs ys) x = None) = (x \<notin> set xs)"
rebalance_middle_tree_order: fixes k :: "nat"   and sub :: "'a btree"   and ls :: "('a btree \<times> 'a) list"   and rs :: "('a btree \<times> 'a) list"   and t :: "'a btree"   and sep :: "'a" assumes "almost_order k sub"   and "\<forall>s\<in>set (subtrees (ls @ rs)). order k s"   and "order k t"   and "case rs of [] \<Rightarrow> True | (rsub, rsep) # list \<Rightarrow> height rsub = height t"   and "length (ls @ (sub, sep) # rs) \<le> 2 * k"   and "height sub = height t" shows "almost_order k (rebalance_middle_tree k ls sub sep rs t)"
convex_hull_finite: fixes S :: "'b set" assumes "finite S" shows "convex hull S = {y. \<exists>u. (\<forall>x\<in>S. 0 \<le> u x) \<and> sum u S = 1 \<and> (\<Sum>x\<in>S. u x *\<^sub>R x) = y}"
rep_eq: fixes x :: "double" shows "Code_Double.is_zero x = IEEE.is_zero (rep_double x)"
path_connected_eq_homotopic_points: fixes S :: "'b set" shows "path_connected S = (\<forall>a b. a \<in> S \<and> b \<in> S \<longrightarrow> homotopic_loops S (linepath a a) (linepath b b))"
Proc_CFG_WhileBody: fixes c' :: "cmd"   and n :: "label"   and et :: "p_edge_kind"   and n' :: "label"   and b :: "expr" assumes "c' \<turnstile> n -et\<rightarrow>\<^sub>p n'"   and "n \<noteq> Entry"   and "n' \<noteq> Exit" shows "while (b) c' \<turnstile> n \<oplus> 2 -et\<rightarrow>\<^sub>p n' \<oplus> 2"
A_def: shows "A = \<D>\<^sub>\<circ> f"
sse_invar_initial: fixes \<Sigma>i :: "'b set"   and R :: "('b \<times> 'b) set" shows "sse_initial \<Sigma>i \<in> sse_invar \<Sigma>i R"
index_vec: fixes i :: "nat"   and n :: "nat"   and f :: "nat \<Rightarrow> 'a" assumes "i < n" shows "Matrix.vec n f $ i = f i"
p_congruent_def: fixes u :: "p_point"   and v :: "p_point"   and u' :: "p_point"   and v' :: "p_point" shows "p_congruent u v u' v' = (p_dist u v = p_dist u' v')"
secure_implies_ref_event_consistency: fixes P :: "'a process"   and I :: "('b \<times> 'b) set"   and D :: "'a \<Rightarrow> 'b"   and xs :: "'a list"   and ys :: "'a list"   and x :: "'a" assumes "secure P I D"   and "(xs, ys) \<in> rel_ipurge P I D (D x)"   and "{x} \<in> refusals P xs" shows "{x} \<in> refusals P ys"
error_returns_heap: fixes h :: "'a"   and e :: "'b"   and h' :: "'a" shows "\<not> h \<turnstile> error e \<rightarrow>\<^sub>h h'"
iterated_singular_subdivision: fixes p :: "nat"   and X :: "'c topology"   and c :: "((nat \<Rightarrow> real) \<Rightarrow> 'c) \<Rightarrow>\<^sub>0 int"   and n :: "nat" assumes "singular_chain p X c" shows "(singular_subdivision p ^^ n) c = frag_extend (\<lambda>f. chain_map p f ((simplicial_subdivision p ^^ n) (frag_of (restrict id (standard_simplex p))))) c"
unadjoin_correct_2: fixes x :: "'a" assumes "x \<in> adjoin G H a" shows "snd (unadjoin x) < subgroup_indicator G H a"
select_convs_14: fixes getSATFlag :: "ExtendedBool"   and getF :: "Literal list list"   and getM :: "(Literal \<times> bool) list"   and getConflictFlag :: "bool"   and getConflictClause :: "nat"   and getQ :: "Literal list"   and getReason :: "Literal \<Rightarrow> nat option"   and getWatch1 :: "nat \<Rightarrow> Literal option"   and getWatch2 :: "nat \<Rightarrow> Literal option"   and getWatchList :: "Literal \<Rightarrow> nat list"   and getC :: "Literal list"   and getCl :: "Literal"   and getCll :: "Literal"   and getCn :: "nat"   and more :: "'a" shows "getCn \<lparr>getSATFlag = getSATFlag, getF = getF, getM = getM, getConflictFlag = getConflictFlag, getConflictClause = getConflictClause, getQ = getQ, getReason = getReason, getWatch1 = getWatch1, getWatch2 = getWatch2, getWatchList = getWatchList, getC = getC, getCl = getCl, getCll = getCll, getCn = getCn, \<dots> = more\<rparr> = getCn"
star_denest: fixes x :: "'a ppath set"   and y :: "'a ppath set" shows "pp_star (x \<union> y) = pp_star (pp_prod (pp_star x) (pp_star y))"
weak_ranking_index_unique: fixes xs :: "'b set list"   and i :: "nat"   and j :: "nat"   and x :: "'b" assumes "is_weak_ranking xs"   and "i < length xs"   and "j < length xs"   and "x \<in> xs ! i"   and "x \<in> xs ! j" shows "i = j"
AND_mod: fixes x :: "int"   and n :: "nat" shows "x AND 2 ^ n - 1 = x mod 2 ^ n"
distinct_rev_path: fixes ps :: "('a \<times> 'b \<times> 'a) list" shows "distinct (rev_path ps) = distinct ps"
dims2_def: shows "st.dims2 = nths dims0 (- vars1')"
add_right: fixes a :: "'d"   and b :: "'b"   and b' :: "'b" shows "g a ** (b + b') = g a ** b + g a ** b'"
hb_completion_shift1: fixes s :: "('l, 'thread_id) locks \<times> (('thread_id \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('thread_id \<Rightarrow> 'w wait_set_status option) \<times> 'thread_id set"   and E :: "('thread_id \<times> ('addr, 'thread_id) obs_event action) list"   and t :: "'thread_id"   and ta :: "('l, 'thread_id, 'x, 'm, 'w, ('addr, 'thread_id) obs_event action) thread_action"   and s' :: "('l, 'thread_id) locks \<times> (('thread_id \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('thread_id \<Rightarrow> 'w wait_set_status option) \<times> 'thread_id set" assumes "hb_completion s E"   and "s -t\<triangleright>ta\<rightarrow> s'"   and "non_speculative P (w_values P (\<lambda>_. {}) (map snd E)) (llist_of \<lbrace>ta\<rbrace>\<^bsub>o\<^esub>)" shows "hb_completion s' (E @ map (Pair t) \<lbrace>ta\<rbrace>\<^bsub>o\<^esub>)"
reduce_rulesI: fixes r :: "('a, 'b) ta_rule"   and \<delta> :: "('a, 'b) ta_rule set"   and P :: "'a set" assumes "r \<in> \<delta>"   and "rule_states r \<subseteq> P" shows "r \<in> reduce_rules \<delta> P"
section_retraction_of_iso_1: fixes f :: "'a \<times> 'a"   and g :: "'a \<times> 'a" assumes "CC.iso f"   and "CC.ide (CC.comp g f)" shows "CC.inverse_arrows f g"
nonzero_divide_mult_cancel_left: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inverse :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c" assumes "class.field divide inverse times one plus zero minus uminus"   and "a \<noteq> zero" shows "divide a (times a b) = divide one b"
g_phi_less: fixes x :: "real" assumes "x \<in> {- pi..pi}" shows "\<bar>g x - (\<Sum>k\<le>n. a k * sin (real k * x) + b k * cos (real k * x))\<bar> < e / 6"
kozen_induct_r: fixes y :: "'a"   and x :: "'a"   and z :: "'a" assumes "y \<odot> x + z \<le> y" shows "z \<odot> x\<^sup>\<star> \<le> y"
assms: shows "x * z \<le> z * y\<^sup>\<star> \<squnion> w"
of_hypnat_less_iff: fixes m :: "nat star"   and n :: "nat star" shows "(of_hypnat m < of_hypnat n) = (m < n)"
one_eq_prime_power_iff: fixes divide :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and unit_factor :: "'d \<Rightarrow> 'd"   and normalize :: "'d \<Rightarrow> 'd"   and p :: "'d"   and n :: "nat" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize"   and "normalization_semidom.prime zero times one normalize p" shows "(one = power.power one times p n) = (n = 0)"
subset_Q_neg: fixes n :: "nat"   and V :: "nat set"   and Q :: "(nat set \<times> 'b option list set) set"   and Qn :: "(nat set \<times> 'b option list set) set"   and QQn :: "(nat set \<times> 'b option list set) set" assumes "rwf_query n V Q Qn"   and "QQn \<subseteq> Qn" shows "rwf_query n V Q QQn"
induct: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> nat \<Rightarrow> bool"   and a0 :: "'b"   and a1 :: "'b"   and a2 :: "nat" assumes "\<And>x y n. (\<And>xa xb ya. \<lbrakk>\<not> n \<le> 1; xa = divmod_nat n 2; (xb, ya) = xa\<rbrakk> \<Longrightarrow> P x y xb) \<Longrightarrow> P x y n" shows "P a0 a1 a2"
scale_left_diff_distrib: fixes a :: "complex"   and b :: "complex"   and x :: "'d" shows "(a - b) *\<^sub>C x = a *\<^sub>C x - b *\<^sub>C x"
collapse: fixes tree :: "('c, 'd) tree" assumes "tree \<noteq> Tip" shows "Node (lsub tree) (key tree) (nval tree) (rsub tree) = tree"
ClockAutoDFS_def: shows "ClockAutoDFS \<equiv> \<lambda>agents jkbp envInit envAction envTrans envVal envObs a. alg_dfs acts_MapOps trans_MapOps (ClockView.clock_simObs envObs a) (ClockView.clock_simTrans agents jkbp envAction envTrans envVal envObs a) (ClockView.clock_simAction jkbp envVal envObs a) (map (ClockView.clock_simInit envInit envObs a \<circ> envObs a) envInit)"
r0: fixes x :: "'a" assumes "x \<in> S \<inter> T" shows "r0 x = x"
simps: fixes a :: "'d" shows "the_gpv (Generative_Probabilistic_Value.Done a) = return_spmf (Pure a)"
frontier_of_disjoint_eq: fixes S :: "'c set"   and X :: "'c topology" assumes "S \<subseteq> topspace X" shows "(X frontier_of S \<inter> S = {}) = openin X S"
det_tree_automaton_axioms: shows "det_tree_automaton TA A"
VDMCall: fixes A :: "(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"   and 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" assumes "({A} \<union> G) \<rhd> body : A" shows "G \<rhd> Call : A"
induct: fixes x1 :: "(ty \<times> nat) list \<times> nat"   and x2 :: "exp"   and x3 :: "ty"   and P :: "(ty \<times> nat) list \<times> nat \<Rightarrow> exp \<Rightarrow> ty \<Rightarrow> bool" assumes "x1 \<turnstile> x2 : x3"   and "\<And>\<Gamma> n. P \<Gamma> (ENat n) TNat"   and "\<And>\<Gamma> n \<tau>. lookup \<Gamma> n = Some \<tau> \<Longrightarrow> P \<Gamma> (EVar n) \<tau>"   and "\<And>\<Gamma> e \<sigma> \<tau> e'. \<lbrakk>\<Gamma> \<turnstile> e : \<sigma> \<rightarrow> \<tau>; P \<Gamma> e (\<sigma> \<rightarrow> \<tau>); \<Gamma> \<turnstile> e' : \<sigma>; P \<Gamma> e' \<sigma>\<rbrakk> \<Longrightarrow> P \<Gamma> (EApp e e') \<tau>"   and "\<And>\<sigma> \<Gamma> e \<tau>. \<lbrakk>push_ty \<sigma> \<Gamma> \<turnstile> e : \<tau>; P (push_ty \<sigma> \<Gamma>) e \<tau>\<rbrakk> \<Longrightarrow> P \<Gamma> (ELam \<sigma> e) (\<sigma> \<rightarrow> \<tau>)"   and "\<And>\<sigma> \<tau> \<Gamma> e. \<lbrakk>push_ty (\<sigma> \<rightarrow> \<tau>) \<Gamma> \<turnstile> e : \<sigma> \<rightarrow> \<tau>; P (push_ty (\<sigma> \<rightarrow> \<tau>) \<Gamma>) e (\<sigma> \<rightarrow> \<tau>)\<rbrakk> \<Longrightarrow> P \<Gamma> (EFix (\<sigma> \<rightarrow> \<tau>) e) (\<sigma> \<rightarrow> \<tau>)"   and "\<And>\<Gamma> e \<tau>. \<lbrakk>push_tyvar \<Gamma> \<turnstile> e : \<tau>; P (push_tyvar \<Gamma>) e \<tau>\<rbrakk> \<Longrightarrow> P \<Gamma> (EAbs e) (Forall \<tau>)"   and "\<And>\<Gamma> e \<tau> \<sigma>. \<lbrakk>\<Gamma> \<turnstile> e : Forall \<tau>; P \<Gamma> e (Forall \<tau>)\<rbrakk> \<Longrightarrow> P \<Gamma> (EInst e \<sigma>) (subst 0 \<sigma> \<tau>)" shows "P x1 x2 x3"
indoc_def: fixes a :: "'item"   and u :: "'item list" shows "indoc a u \<equiv> \<exists>u\<^sub>1 u\<^sub>2. u = u\<^sub>1 @ [a] @ u\<^sub>2 \<and> a \<notin> set u\<^sub>1 \<and> Ind {a} (set u\<^sub>1)"
simps_1: fixes f1 :: "'c \<Rightarrow> 'a"   and f2 :: "'d \<Rightarrow> 'b"   and a :: "'c" shows "map_sum f1 f2 (Inl a) = Inl (f1 a)"
assms_2: shows "f (0::'a) = (0::'b)"
det_mult: fixes A :: "'a mat"   and n :: "nat"   and B :: "'a mat" assumes "A \<in> carrier_mat n n"   and "B \<in> carrier_mat n n" shows "det (A * B) = det A * det B"
real_cond_exp_pos: fixes f :: "'a \<Rightarrow> real" assumes "AE x in M. 0 \<le> f x"   and "f \<in> borel_measurable M" shows "AE x in M. 0 \<le> real_cond_exp M Invariants f x"
comm_monoid_axioms: shows "comm_monoid (\<^bold>*) \<^bold>1"
is_functor: shows "functor (\<cdot>\<^sub>J) (\<cdot>) Cod_f.map"
fresh_3: fixes a :: "vrs" shows "a \<sharp> Top = True"
idiff_infinity: fixes n :: "enat" shows "\<infinity> - n = \<infinity>"
Bfun_def: fixes f :: "'c \<Rightarrow> 'd"   and F :: "'c filter" shows "Bfun f F = (\<exists>K>0. \<forall>\<^sub>F x in F. norm (f x) \<le> K)"
measure_nonneg': fixes M :: "'b measure"   and A :: "'b set" shows "\<not> Sigma_Algebra.measure M A < 0"
zf_2: shows "z = (interval_bij (a, b) (- 1, 1) \<circ> f) zf"
mult_scalar_monomial: fixes t :: "'x \<Rightarrow>\<^sub>0 nat"   and c :: "'b"   and p :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" shows "local.punit.mult_scalar (monomial c t) p = local.punit.monom_mult c t p"
rel_cases: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'b ref"   and b :: "'c ref" assumes "rel_ref R a b" obtains "a = Null" and "b = Null"   | x :: "'b" and y :: "'c" where "a = Ref x" and "b = Ref y" and "R x y"
that: fixes a :: "V"   and b :: "V" assumes "f : a \<mapsto>\<^bsub>\<CC>\<^esub> b"   and "a \<in>\<^sub>\<circ> \<CC>\<lparr>Obj\<rparr>"   and "b \<in>\<^sub>\<circ> \<CC>\<lparr>Obj\<rparr>" shows "thesis"
ap_spec_fst_subset: fixes ap :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> nat \<times> 'g \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and gs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and 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"   and hs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and data :: "nat \<times> 'g" assumes "ap_spec ap" shows "fst ` set (ap gs bs ps hs data) \<subseteq> fst ` set ps \<union> set hs"
a_assoc: fixes x :: "'a vec"   and y :: "'a vec"   and z :: "'a vec" assumes "x \<in> carrier_vec n"   and "y \<in> carrier_vec n"   and "z \<in> carrier_vec n" shows "x + y + z = x + (y + z)"
assms: shows "z * x \<le> y\<^sup>+ * z \<squnion> w"
find_crossing_edge_rtrancl: fixes u :: "'a"   and v :: "'a"   and g :: "'a ugraph"   and P :: "'a \<Rightarrow> bool" assumes "(u, v) \<in> (edges g)\<^sup>*"   and "\<not> P u"   and "P v" obtains u' :: "'a" and v' :: "'a" where "(u', v') \<in> edges g" and "\<not> P u'" and "P v'"
newton_seq_fact6: fixes l :: "nat" shows "f' \<bullet> ns l \<in> nonzero Zp"
merge_list: fixes h :: "'c \<Rightarrow> 'd"   and bo :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and m :: "'c \<Rightarrow> 'c \<Rightarrow> 'c option" assumes "merge_on UNIV h bo m" shows "merge_on UNIV (map h) (list_all2 bo) (merge_list m)"
take_bit_take_bit: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bit :: "'c \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and drop_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and take_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and m :: "nat"   and n :: "nat"   and a :: "'c" assumes "class.semiring_bit_shifts plus minus zero times one divide modulo bit push_bit drop_bit take_bit" shows "take_bit m (take_bit n a) = take_bit (min m n) a"
suffix_suffix: fixes m :: "nat"   and k :: "nat"   and x :: "nat \<Rightarrow> 'b" shows "suffix m (suffix k x) = suffix (k + m) x"
vcpowerD: fixes f :: "V"   and A :: "V"   and n :: "V" assumes "f \<in>\<^sub>\<circ> A ^\<^sub>\<times> n" shows "f \<in>\<^sub>\<circ> (\<Prod>\<^sub>\<circ>i\<in>\<^sub>\<circ>n. A)"
singleton_conv2: fixes a :: "'a" shows "{x. a = x} = {a}"
add_stalk_def: shows "stfx.add_stalk \<equiv> stfx.add_rel"
lt: shows "lt b = term_of_pair (0::'a, component_of_term u)"
farkas_assert_all_state: fixes t :: "(nat \<times> linear_poly) list"   and as :: "('d \<times> 'e atom) list" assumes "\<U> (AssertAllState''.assert_all_state init_state assert_bound_code check_code t as)"   and "\<triangle> t" shows "\<exists>C. farkas_coefficients_atoms_tableau (snd ` set as) t C"
axioms_3: fixes A :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and F :: "'a \<Rightarrow> 'b"   and G :: "'a \<Rightarrow> 'b"   and \<tau> :: "'a \<Rightarrow> 'b" assumes "natural_transformation A B F G \<tau>" shows "functor A B F"
not_lfinite_lprefix_conv_eq: fixes xs :: "'b llist"   and ys :: "'b llist" assumes "\<not> lfinite xs" shows "lprefix xs ys = (xs = ys)"
cY: shows "continuous_on Y (return_time {x \<in> S. s x = 0})"
sorted_on_linord_of_list_subseteq_r: fixes r :: "('a \<times> 'a) set"   and xs :: "'a list" assumes "Linear_order r"   and "sorted_on r xs"   and "distinct xs" shows "linord_of_list (rev xs) \<subseteq> r"
fmap_ball_all3: fixes f' :: "'a \<Rightarrow> 'b option"   and f :: "'a \<Rightarrow> 'b option"   and Px :: "'c \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> bool"   and P :: "'c \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'b option \<Rightarrow> 'b option \<Rightarrow> bool" assumes "dom f' = dom f"   and "\<forall>l\<in>dom f. \<forall>x y z. Px x y z \<longrightarrow> P x y z (f l) (f' l)" shows "\<forall>x y z. Px x y z \<longrightarrow> (\<forall>l\<in>dom f. P x y z (f l) (f' l))"
Un_assoc: fixes A :: "'e set"   and B :: "'e set"   and C :: "'e set" shows "A \<union> B \<union> C = A \<union> (B \<union> C)"
spair_alt_spp: fixes p :: "'t \<Rightarrow>\<^sub>0 'b"   and q :: "'t \<Rightarrow>\<^sub>0 'b" assumes "is_regular_spair p q" shows "spp_of (spair p q) = spair_spp (spp_of p) (spp_of q)"
sum_constant: fixes one :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and y :: "'d"   and A :: "'e set" assumes "class.semiring_1 one times plus zero" shows "comm_monoid_add.sum plus zero (\<lambda>x. y) A = times (semiring_1.of_nat one plus zero (card A)) y"
par_left_comm: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "A B Par C D" shows "B A Par C D"
eventually_nonneg_def: fixes F :: "'b filter"   and f :: "'b \<Rightarrow> 'c" shows "eventually_nonneg F f = (\<forall>\<^sub>F x in F. (0::'c) \<le> f x)"
f3: shows "f -` {b. b = (0::'b) \<or> b \<in> {}} = {a. f a = (0::'b)}"
safe_isFDERIV_append: fixes x :: "'c"   and xs :: "real list" assumes "x \<in> Csafe" shows "isFDERIV DIM('c) [0..<DIM('c)] ode_e (list_of_eucl x @ xs)"
assms_2: shows "incid A (line B' C') \<or> incid C' (line A B)"
homeomorphic_map_def: fixes X :: "'c topology"   and Y :: "'d topology"   and f :: "'c \<Rightarrow> 'd" shows "homeomorphic_map X Y f \<equiv> quotient_map X Y f \<and> inj_on f (topspace X)"
elims: fixes x :: "'a mat"   and xa :: "'a vec"   and y :: "'a mat" assumes "mat_append_col x xa = y" obtains A :: "'a mat" and b :: "'a vec" where "x = A" and "xa = b" and "y = mat_of_cols (dim_row A) (cols A @ [b])"
vfsequence_vcons_ex: fixes xs :: "V" assumes "vfsequence xs"   and "xs \<noteq> []\<^sub>\<circ>" obtains xs' :: "V" and x :: "V" where "xs = xs' #\<^sub>\<circ> x" and "vfsequence xs'"