hcju/isabelleps huggingface dataset:
transfer_correctness: fixes Qi :: "'d set"   and \<delta> :: "('d, 'e) ta_rule set"   and P :: "'d set \<times> 'd set \<times> (('d, 'e) ta_rule \<Rightarrow> nat option) \<Rightarrow> bool" assumes "\<forall>s. s \<in> wa_invar (bre'_algo Qi \<delta>) \<and> s \<notin> wa_cond (bre'_algo Qi \<delta>) \<longrightarrow> P s" shows "\<forall>sc. sc \<in> wa_invar (brw_algo Qi \<delta>) \<and> sc \<notin> wa_cond (brw_algo Qi \<delta>) \<longrightarrow> P (brw_\<alpha> sc)"
sub2: shows "(\<Union>y\<in>{y. Inl (Inr y) \<in> SIGO ODE2}. FVT (\<sigma> y)) \<subseteq> (\<Union>y\<in>{y. Inl (Inr y) \<in> SIGO (OProd ODE1 ODE2)}. FVT (\<sigma> y))"
Map_dom: fixes f :: "('a2 \<Rightarrow> 'b, 'a2 \<Rightarrow> 'b) concrete_category.arr" assumes "A2_B.arr f" shows "A2_B.Map (A2_B.dom f) = A2_B.Dom f"
calculus_axioms: shows "calculus Bot_F Inf_F (\<lambda>N1 N2. \<G>_Fset N1 \<Turnstile>G \<G>_Fset N2) Red_I_\<G> empt_ord.Red_F_\<G>"
append_take_drop_id: fixes n :: "nat"   and xs :: "'d list" shows "take n xs @ drop n xs = xs"
snoc_1: fixes xs :: "'a list"   and zs :: "'a list" assumes "filter P xs = ys @ zs" shows "\<exists>us vs. xs = us @ vs \<and> filter P us = ys \<and> filter P vs = zs"
map_spmf_conv_bind_spmf: fixes f :: "'b \<Rightarrow> 'a"   and p :: "'b spmf" shows "map_spmf f p = bind_spmf p (\<lambda>x. return_spmf (f x))"
rep_eq: fixes x :: "('c, 'd) uexpr"   and xa :: "('c, 'd) uexpr" shows "(x \<le> xa) = (\<forall>A. \<lbrakk>x\<rbrakk>\<^sub>e A \<le> \<lbrakk>xa\<rbrakk>\<^sub>e A)"
init_fin_simps_5: fixes final :: "'a \<Rightarrow> bool"   and r :: "('d, 'b, 'a, 'c, 'e, 'f) semantics"   and t :: "'b"   and xm :: "(status \<times> 'a) \<times> 'c"   and ta :: "('d, 'b, status \<times> 'a, 'c, 'e, 'f action) thread_action"   and x' :: "'a"   and m' :: "'c" shows "multithreaded_base.init_fin final r t xm ta ((Running, x'), m') = ((\<exists>x m. xm = ((Running, x), m) \<and> (\<exists>taa. ta = convert_TA_initial (convert_obs_initial taa) \<and> r t (x, m) taa (x', m'))) \<or> xm = ((PreStart, x'), m') \<and> ta = \<lbrace>InitialThreadAction\<rbrace>)"
tendsto_infnorm: fixes f :: "'b \<Rightarrow> 'c"   and a :: "'c"   and F :: "'b filter" assumes "(f \<longlongrightarrow> a) F" shows "((\<lambda>x. infnorm (f x)) \<longlongrightarrow> infnorm a) F"
iall_triv: fixes I :: "nat set"   and P :: "bool" shows "(\<box> t I. P) = ((\<exists>t. t \<in> I) \<longrightarrow> P)"
f_def: fixes fs :: "'a list"   and i :: "nat" shows "??.Newton_Interpolation.f fs i = fs ! i"
lexordp_eq_simps_2: fixes less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'b list" shows "ord.lexordp_eq less xs [] = (xs = [])"
measure_lborel_Ioo: fixes l :: "real"   and u :: "real" assumes "l \<le> u" shows "content {l<..<u} = u - l"
less_eq_multiset_empty_left: fixes M :: "'b multiset" shows "{#} \<le> M"
out546_suma__conga: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p"   and E :: "'p"   and F :: "'p"   and G :: "'p"   and H :: "'p"   and I :: "'p" assumes "A B C D E F SumA G H I"   and "E Out D F" shows "A B C CongA G H I"
simps_8: fixes v :: "'a" shows "local.le_rexp One (Atom v) = True"
A: shows "index xs q - n < index (mtf2 n q xs) x"
\<section>: fixes x :: "'a" assumes "B \<le> i \<bullet> x" shows "x \<in> S"
content_pos_lt: fixes a :: "'a"   and b :: "'a" assumes "\<forall>i\<in>Basis. a \<bullet> i < b \<bullet> i" shows "0 < content (cbox a b)"
ideal_generated_pair_exists_UNIV: fixes a :: "'a"   and b :: "'a" shows "(ideal_generated {a, b} = ideal_generated {1::'a}) = (\<exists>p q. p * a + q * b = (1::'a))"
total_ES1_C1_inter_Upsilon1_inter_N2_inter_Delta2: shows "EventSystems.total ES1 (C\<^bsub>\<V>1\<^esub> \<inter> \<Upsilon>\<^bsub>\<Gamma>1\<^esub> \<inter> N\<^bsub>\<V>2\<^esub> \<inter> \<Delta>\<^bsub>\<Gamma>2\<^esub>)"
assume_def: fixes p :: "'a"   and q :: "'a" shows "[. p .] q \<equiv> - p \<squnion> q"
val_Zp_p_pow: fixes n :: "nat" shows "val_Zp ([p] \<cdot> \<one> [^] n) = eint (int n)"
zadd_zle_mono: fixes w' :: "int"   and w :: "int"   and z' :: "int"   and z :: "int" assumes "w' \<le> w"   and "z' \<le> z" shows "w' + z' \<le> w + z"
assms: shows "\<exists>x. [A*\<^sub>vx]\<le>b"
dirichlet_prod_prime_power: fixes p :: "nat"   and f :: "nat \<Rightarrow> 'a"   and g :: "nat \<Rightarrow> 'a"   and k :: "nat" assumes "prime p" shows "dirichlet_prod f g (p ^ k) = (\<Sum>i\<le>k. f (p ^ i) * g (p ^ (k - i)))"
inf_right_idem: fixes x :: "'h multiset"   and y :: "'h multiset" shows "x \<inter># y \<inter># y = x \<inter># y"
mtSequence_def: shows "Sequence{} \<equiv> \<lambda>\<tau>. Abs_Sequence\<^sub>b\<^sub>a\<^sub>s\<^sub>e \<lfloor>\<lfloor>[]\<rfloor>\<rfloor>"
suf_same_len: fixes u :: "'b list"   and v :: "'b list" assumes "u \<le>s v"   and "\<^bold>|u\<^bold>| = \<^bold>|v\<^bold>|" shows "u = v"
document_put_get: fixes h :: "(_) heap"   and document_ptr :: "(_) document_ptr"   and setter :: "('w \<Rightarrow> 'w) \<Rightarrow> ('n, 'l, 'm, 'u option) RDocument_scheme \<Rightarrow> ('n, 'l, 'm, 'u option) RDocument_scheme"   and v :: "'w"   and h' :: "(_) heap"   and getter :: "('n, 'l, 'm, 'u option) RDocument_scheme \<Rightarrow> 'w" assumes "h \<turnstile> put_M\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t document_ptr setter v \<rightarrow>\<^sub>h h'"   and "\<And>x. getter (setter (\<lambda>_. v) x) = v" shows "h' \<turnstile> get_M\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t document_ptr getter \<rightarrow>\<^sub>r v"
AttributeCallTE: fixes \<Gamma> :: "String.literal \<rightharpoonup>\<^sub>f 'a OCL_Types.type"   and src :: "'a expr"   and k :: "call_kind"   and "prop" :: "String.literal"   and \<tau> :: "'a OCL_Types.type" assumes "\<Gamma> \<turnstile>\<^sub>E AttributeCall src k prop : \<tau>" obtains \<C> :: "'a" and \<D> :: "'a" where "k = DotCall" and "\<Gamma> \<turnstile>\<^sub>E src : \<langle>\<C>\<rangle>\<^sub>\<T>[1]" and "attribute \<C> prop \<D> \<tau>"
simps: fixes size :: "'a \<Rightarrow> nat"   and frees :: "'a \<Rightarrow> name fset"   and subst :: "'a \<Rightarrow> (name, 'a) fmap \<Rightarrow> 'a"   and "consts" :: "'a \<Rightarrow> name fset"   and app :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and unapp :: "'a \<Rightarrow> ('a \<times> 'a) option"   and const :: "name \<Rightarrow> 'a"   and unconst :: "'a \<Rightarrow> name option"   and free :: "name \<Rightarrow> 'a"   and unfree :: "'a \<Rightarrow> name option"   and abs_pred :: "('a \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> bool"   and t\<^sub>1 :: "Term_Class.term"   and t\<^sub>2 :: "'a"   and u :: "'a" assumes "class.term size frees subst consts app unapp const unconst free unfree abs_pred" shows "term.rewrite_step subst unapp unconst (t\<^sub>1, t\<^sub>2) u = map_option (subst t\<^sub>2) (term.match unapp unconst t\<^sub>1 u)"
assms_2: shows "z \<noteq> 0\<^sub>h"
D''_def: shows "D'' = abstr g (\<lambda>i j. \<infinity>) v"
assms_1: shows "rhs \<equiv> expectation_gpv 0 \<I> (\<lambda>_. y)"
sig_red_regular_rtrancl_lc: fixes top_tail :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and F :: "('t \<Rightarrow>\<^sub>0 'b) set"   and p :: "'t \<Rightarrow>\<^sub>0 'b"   and q :: "'t \<Rightarrow>\<^sub>0 'b" assumes "(sig_red (\<prec>\<^sub>t) top_tail F)\<^sup>*\<^sup>* p q" shows "lc q = lc p"
len_help: shows "length (pivot_positions A) = card {i. i < dim_row A \<and> row A i \<noteq> 0\<^sub>v (dim_col A)}"
better_OrR2_substc: fixes d :: "coname"   and a :: "coname"   and M :: "trm"   and z :: "name"   and P :: "trm" assumes "d \<sharp> [a].M" shows "OrR2 <a>.M d{d:=z.P} = fresh_fun (\<lambda>a'. Cut <a'>.OrR2 <a>.M a' z.P)"
disconnected_nodes_h: shows "h \<turnstile> get_disconnected_nodes owner_document \<rightarrow>\<^sub>r disconnected_nodes_h"
transpose_matrix_vector: fixes A :: "(('b, 'c) Finite_Cartesian_Product.vec, 'd) Finite_Cartesian_Product.vec"   and x :: "('b, 'd) Finite_Cartesian_Product.vec" shows "Finite_Cartesian_Product.transpose A *v x = x v* A"
card2: shows "card X = 2 \<longrightarrow> (\<exists>f. TernaryOrdering.ordering f ord X)"
def_cont_fix_eq: fixes f :: "'a"   and F :: "'a \<Rightarrow> 'a" assumes "f \<equiv> fix_syn F"   and "cont F" shows "f = F f"
view_trees_for_def: fixes view :: "view" shows "local.view_trees_for view = map view_of_zipper\<^sub>m (filter (\<lambda>(uu_, t). participant \<in> view_recipients t) (zippers_view ([], view)))"
1: shows "c \<bullet> z \<in> {c \<bullet> x |x. x \<in> sat_primal A b}"
intros: fixes A :: "fm"   and i :: "name"   and x :: "tm" shows "A(i::=x) IMP SyntaxN.Ex i A \<in> special_axioms"
dgrad_p_set_le_plus: fixes d :: "'a \<Rightarrow> nat"   and p :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c" shows "dgrad_p_set_le d {p + q} {p, q}"
pcompose_0': fixes p :: "'c poly" shows "p \<circ>\<^sub>p 0 = [:poly.coeff p 0:]"
bind_assoc_pmf: fixes A :: "'j pmf"   and B :: "'j \<Rightarrow> 'i pmf"   and C :: "'i \<Rightarrow> 'h pmf" shows "A \<bind> B \<bind> C = A \<bind> (\<lambda>x. B x \<bind> C)"
freshBoundOutputTransition: fixes P :: "pi"   and a :: "name"   and x :: "name"   and P' :: "pi"   and c :: "name" assumes "P \<Longrightarrow>\<^sub>l a<\<nu>x> \<prec> P'"   and "c \<sharp> P"   and "c \<noteq> x" shows "c \<sharp> P'"
prime_pow_divides_iff: fixes p :: "'a list"   and a :: "'a list"   and b :: "'a list"   and n :: "nat" assumes "p \<in> carrier (Ring_Divisibility.mult_of (K [X]))"   and "a \<in> carrier (Ring_Divisibility.mult_of (K [X]))"   and "b \<in> carrier (Ring_Divisibility.mult_of (K [X]))"   and "prime (Ring_Divisibility.mult_of (K [X])) p"   and "\<not> p divides\<^bsub>Ring_Divisibility.mult_of (K [X])\<^esub> a" shows "p [^]\<^bsub>Ring_Divisibility.mult_of (K [X])\<^esub> n divides\<^bsub>Ring_Divisibility.mult_of (K [X])\<^esub> a \<otimes>\<^bsub>K [X]\<^esub> b = p [^]\<^bsub>Ring_Divisibility.mult_of (K [X])\<^esub> n divides\<^bsub>Ring_Divisibility.mult_of (K [X])\<^esub> b"
check1: shows "\<not> (w1 = NEG_INF \<and> w2 = NEG_INF)"
g: shows "LIM n sequentially. g s0 (cruel n) :> at_top"
empty_Diff: fixes A :: "'b set" shows "{}\<setminus>A = {}"
lower_triangular_def: fixes A :: "'b mat" shows "lower_triangular A = (\<forall>i j. i < j \<and> i < dim_row A \<and> j < dim_col A \<longrightarrow> A $$ (i, j) = (0::'b))"
ground_var_denotts: fixes ts :: "fterm list"   and E :: "char list \<Rightarrow> 'a"   and F :: "char list \<Rightarrow> 'a list \<Rightarrow> 'a"   and E' :: "char list \<Rightarrow> 'a" assumes "ground\<^sub>t\<^sub>s ts" shows "eval\<^sub>t\<^sub>s E F ts = eval\<^sub>t\<^sub>s E' F ts"
length_fresh_rename_ns: fixes G :: "variable set"   and n :: "nat"   and B :: "Core.term"   and insts :: "((variable \<times> typ) \<times> Core.term) list" assumes "finite G" shows "length (fresh_rename_idns n B insts G) = n"
assms_2: shows "v * - v\<^sup>T \<sqinter> g = bot"
Ints_minus: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and a :: "'b" assumes "class.ring_1 one times plus zero minus uminus"   and "a \<in> ring_1.Ints one plus zero minus" shows "uminus a \<in> ring_1.Ints one plus zero minus"
preserves_comp_def: fixes G :: "('o, 'a, 'a set, 'a set_arrow) functor" shows "preserves_comp G \<equiv> \<forall>f\<in>Ar. \<forall>g\<in>Ar. Cod f = Dom g \<longrightarrow> G \<^bsub>\<a>\<^esub> (g \<bullet> f) = G \<^bsub>\<a>\<^esub> g \<bullet>\<^bsub>BB\<^esub> G \<^bsub>\<a>\<^esub> f"
RU_init: fixes u :: "'a"   and U :: "'a set"   and fg :: "('a, 'c, 'd, 'b, 'e) flowgraph_rec_scheme" assumes "u \<in> U" shows "(u, {}, {}, \<lambda>x. {}) \<in> RU_cs fg U"
quot_ideal_imp_ring_ideal: fixes I :: "'a set"   and J :: "'a set set" assumes "ideal I R"   and "ideal J (R Quot I)" shows "ideal (\<Union> J) R"
neg_less_eq_nonneg: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c" assumes "class.linordered_ab_group_add plus zero minus uminus less_eq less" shows "less_eq (uminus a) a = less_eq zero a"
i1: shows "ideal_generated {a, b, c} = ideal_generated {d}"
pair_pmf_cong: fixes p :: "'a pmf"   and p' :: "'a pmf"   and q :: "'b pmf"   and q' :: "'b pmf" assumes "p = p'"   and "q = q'" shows "pair_pmf p q = pair_pmf p' q'"
assms_6: shows "strengthenings prb = (\<lambda>rv \<sigma>. True)"
ndet_id: fixes P :: "'b process" shows "(P |-| P) = P"
inc_item_def: fixes x :: "item"   and k :: "nat" shows "inc_item x k = Item (item_rule x) (item_dot x + 1) (item_origin x) k"
distinct_26: fixes x81 :: "pat"   and x82 :: "trm"   and x83 :: "trm"   and x21 :: "POPLmarkRecord.type"   and x22 :: "trm" shows "(LET x81 = x82 IN x83) \<noteq> (\<lambda>:x21. x22)"
awlast_append: fixes u :: "'a"   and p :: "'b list"   and q :: "'b list" shows "awlast u (p @ q) = awlast (awlast u p) q"
C_def: shows "C = (\<Sum>p. ln (real (p + 1)) * (1 / real (p * (p - 1))))"
body: shows "P,E(this \<mapsto> Class (last Ds), pns [\<mapsto>] Ts) \<turnstile> \<langle>body,(h\<^sub>2, l\<^sub>2')\<rangle> \<rightarrow>* \<langle>ef,(h\<^sub>3, l\<^sub>3)\<rangle>"
O_mono2: fixes R :: "('a \<times> 'b) set"   and R' :: "('a \<times> 'b) set"   and T :: "('b \<times> 'c) set" assumes "R \<subseteq> R'" shows "R O T \<subseteq> R' O T"
monom_mult_sum_right: fixes c :: "'d"   and t :: "'a"   and f :: "'e \<Rightarrow> 't \<Rightarrow>\<^sub>0 'd"   and P :: "'e set" shows "monom_mult c t (sum f P) = (\<Sum>p\<in>P. monom_mult c t (f p))"
exhaust: fixes y :: "type_scheme" obtains x1 :: "nat" where "y = FVar x1"   | x2 :: "nat" where "y = BVar x2"   | x31 :: "type_scheme" and x32 :: "type_scheme" where "y = x31 =-> x32"
and_neg_mask_plus_mask_mono: fixes p :: "'b word"   and n :: "nat" shows "p \<le> (p AND NOT (mask n)) + mask n"
sup_commute: fixes x :: "'d multiset"   and y :: "'d multiset" shows "x \<union># y = y \<union># x"
eqvt: fixes x1 :: "(Syntax.var, ty) fmap"   and x2 :: "Syntax.term"   and x3 :: "Syntax.term"   and x4 :: "Syntax.term"   and x5 :: "ty"   and p :: "perm" assumes "x1 \<turnstile> x2, x3, x4 : x5" shows "(p \<bullet> x1) \<turnstile> p \<bullet> x2, p \<bullet> x3, p \<bullet> x4 : (p \<bullet> x5)"
suffix_imp_sublist: fixes xs :: "'b list"   and ys :: "'b list" assumes "xs \<le>s ys" shows "xs \<le>f ys"
aprimedivisor_not_Suc_0_nat: fixes d :: "nat" assumes "Suc 0 < d" shows "aprimedivisor d \<noteq> Suc 0"
skip_im_Tr0: fixes x :: "nat"   and n :: "nat" assumes "x \<in> {i. i \<le> n}" shows "skip 0 x = Suc x"
factors_int_poly_represents: fixes p :: "int poly"   and x :: "'a" assumes "p represents x" shows "\<exists>q\<in>set (factors_of_int_poly p). q represents x \<and> irreducible q \<and> 0 < lead_coeff q \<and> degree q \<le> degree p"
shift_var_set_Suc: fixes x :: "nat"   and V :: "nat set" shows "(Suc x \<in> shift_var_set V) = (x \<in> V)"
equalityD2: fixes A :: "'b set"   and B :: "'b set" assumes "A = B" shows "B \<subseteq> A"
A_def: shows "A \<equiv> connected_component_set (- S) a"
cr_spmf_gpv_def: fixes p :: "'e spmf"   and gpv :: "('e, 'f, 'g) gpv" shows "cr_spmf_gpv p gpv = (gpv = lift_spmf p)"
associated_refl: fixes a :: "'a" assumes "a \<in> carrier R" shows "a \<sim>\<^bsub>add_monoid R\<^esub> a"
simps_1: fixes c :: "int" shows "numbound0 (C c) = True"
normF_def: fixes F :: "'a set"   and x :: "'a \<Rightarrow> complex" shows "normF F x = L2_set (cmod \<circ> x) F"
elims: fixes x :: "arctic"   and xa :: "arctic"   and y :: "arctic" assumes "x * xa = y" obtains ya :: "arctic" where "x = MinInfty" and "xa = ya__" and "y = MinInfty"   | v :: "int" where "x = Num_arc v" and "xa = MinInfty" and "y = MinInfty"   | xb :: "int" and ya :: "int" where "x = Num_arc xb" and "xa = Num_arc ya" and "y = Num_arc (xb + ya)"
pareto_losers_lift_pref_profile: fixes agents :: "'a set"   and alts :: "'b set"   and R :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and agents' :: "'a set"   and alts' :: "'b set" assumes "pref_profile_wf agents alts R"   and "agents \<subseteq> agents'"   and "alts \<subseteq> alts'"   and "alts \<noteq> {}"   and "agents \<noteq> {}"   and "finite alts'" shows "pareto_losers (lift_pref_profile agents alts agents' alts' R) = pareto_losers R \<union> (alts' - alts)"
simps_2: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sl :: "'a list"   and x :: "'a"   and xs :: "'a list" shows "quicksort_by_rel R sl (x # xs) = (let (xs_s, xs_b) = partition_rev (\<lambda>y. R y x) ([], []) xs in quicksort_by_rel R (x # quicksort_by_rel R sl xs_b) xs_s)"
isCont_cblinfun_apply: fixes \<psi> :: "'a"   and A :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'b" shows "isCont ((*\<^sub>V) A) \<psi>"
le_of_nat_term_order: fixes to :: "'a nat_term_order"   and x :: "'a"   and y :: "'a" shows "le_of_nat_term_order to x y = (nat_term_compare to x y \<noteq> Gt)"
cet_cast_real_int: fixes \<Gamma> :: "nat \<Rightarrow> pdf_type"   and e :: "cexpr" assumes "\<Gamma> \<turnstile>\<^sub>c e : INTEG" shows "\<Gamma> \<turnstile>\<^sub>c \<langle>e\<rangle>\<^sub>c : REAL"
nat_pow_eone: fixes x :: "nat \<Rightarrow> int" assumes "x \<in> carrier Zp" shows "[1] \<cdot> x = x"
hd_append2: fixes xs :: "'c list"   and ys :: "'c list" assumes "xs \<noteq> []" shows "lead_coeff (xs @ ys) = lead_coeff xs"
projection_split_arbitrary_element: fixes \<tau> :: "'a list"   and X :: "'a set"   and \<alpha> :: "'a list"   and x :: "'a"   and \<beta> :: "'a list" assumes "\<tau> \<upharpoonleft> X = (\<alpha> @ [x] @ \<beta>) \<upharpoonleft> X"   and "x \<in> X" shows "\<exists>\<alpha>' \<beta>'. \<tau> = \<alpha>' @ [x] @ \<beta>' \<and> \<alpha>' \<upharpoonleft> X = \<alpha> \<upharpoonleft> X \<and> \<beta>' \<upharpoonleft> X = \<beta> \<upharpoonleft> X"
set: fixes x1 :: "'a"   and x2 :: "'a stream" shows "sset (x1 ## x2) = insert x1 (sset x2)"
fbox_g_ode_subset: fixes f :: "'b \<Rightarrow> 'b"   and T :: "real set"   and S :: "'b set"   and \<phi> :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and U :: "'b \<Rightarrow> real set"   and G :: "'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool" assumes "local_flow f T S \<phi>"   and "\<And>s. s \<in> S \<Longrightarrow> 0 \<in> U s \<and> is_interval (U s) \<and> U s \<subseteq> T" shows "|x\<acute>=\<lambda>t. f & G on U S @ 0] Q = (\<lambda>s. s \<in> S \<longrightarrow> (\<forall>t\<in>U s. (\<forall>\<tau>\<in>down (U s) t. G (\<phi> \<tau> s)) \<longrightarrow> Q (\<phi> t s)))"
imonad1: fixes x :: "'b"   and f :: "'b \<Rightarrow> 'a" shows "bind (return x) f = f x"
ring_simprules_27: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier R"   and "y \<in> carrier R" shows "x \<otimes> \<ominus> y = \<ominus> (x \<otimes> y)"
lang_eq_ext: fixes r :: "'b rexp"   and s :: "'b rexp" shows "(lang r = lang s) = (\<forall>w\<in>lists (atoms r \<union> atoms s). (w \<in> lang r) = (w \<in> lang s))"
liminf_SUP_INF: fixes f :: "nat \<Rightarrow> 'c" shows "liminf f = (SUP n. Inf (f ` {n..}))"
Bseq_monoseq_convergent: fixes X :: "nat \<Rightarrow> real" assumes "Bseq X"   and "monoseq X" shows "convergent X"
inj_imp_dim_ker0: assumes "inj_on T (carrier V)" shows "Ker.dim = 0"
assms: shows "\<Psi> \<rhd> P \<leadsto><Rel> Q"
arccos_eq_iff: fixes x :: "real"   and y :: "real" assumes "\<bar>x\<bar> \<le> 1 \<and> \<bar>y\<bar> \<le> 1" shows "(arccos x = arccos y) = (x = y)"
setops\<^sub>l\<^sub>s\<^sub>s\<^sub>t\<^sub>p_subst: fixes a :: "'f strand_label \<times> ('e, 'd) stateful_strand_step"   and \<theta> :: "'d \<Rightarrow> ('e, 'd) Term.term" assumes "set (bvars\<^sub>s\<^sub>s\<^sub>t\<^sub>p (snd a)) \<inter> subst_domain \<theta> = {}" shows "setops\<^sub>l\<^sub>s\<^sub>s\<^sub>t\<^sub>p (a \<cdot>\<^sub>l\<^sub>s\<^sub>s\<^sub>t\<^sub>p \<theta>) = (\<lambda>p. (fst a, snd p \<cdot>\<^sub>p \<theta>)) ` setops\<^sub>l\<^sub>s\<^sub>s\<^sub>t\<^sub>p a"
integrable_add: fixes f :: "'a \<Rightarrow> 'b"   and S :: "'a set"   and g :: "'a \<Rightarrow> 'b" assumes "f integrable_on S"   and "g integrable_on S" shows "(\<lambda>x. f x + g x) integrable_on S"
abs_eq: fixes xa :: "(11, 52) IEEE.float"   and x :: "(11, 52) IEEE.float" shows "fcompare_double (abs_double xa) (abs_double x) = fcompare xa x"
scale_cancel_left: fixes a :: "rat"   and x :: "'a"   and y :: "'a" shows "(a *R x = a *R y) = (x = y \<or> a = 0)"
assms_1: shows "fv \<Gamma> \<subseteq> S"
inv: shows "\<forall>x\<in>space M. T' (T x) = x"
g1: shows "\<guillemotleft>ECC.prod f1 (local.dom f1) \<cdot> ECC.dup (local.dom f1) : local.dom f1 \<rightarrow> ECC.prod (cod f1) (local.dom f1)\<guillemotright>"
interleave_assoc1: fixes a :: "'b list"   and xs :: "'b list"   and ys :: "'b list"   and b :: "'b list"   and zs :: "'b list" assumes "a \<in> xs \<otimes> ys"   and "b \<in> a \<otimes> zs" shows "\<exists>c. c \<in> ys \<otimes> zs \<and> b \<in> xs \<otimes> c"
elims_2: fixes x :: "('b, 'c) Combinators list" assumes "NetsCollected x" obtains xa :: "('b, 'c) Combinators" and xs :: "('b, 'c) Combinators list" where "x = xa # xs" and "(first_bothNet xa \<noteq> firstList xs \<longrightarrow> (\<forall>a\<in>set xs. first_bothNet xa \<noteq> first_bothNet a)) \<and> NetsCollected xs"   | "x = []"
assoc: fixes a :: "'e"   and b :: "'e"   and c :: "'e" shows "a * b * c = a * (b * c)"
distinct_insort: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'c"   and x :: "'d"   and xs :: "'d list" assumes "class.linorder less_eq less" shows "distinct (linorder.insort_key less_eq f x xs) = (x \<notin> set xs \<and> distinct xs)"
value_lemma: fixes r :: "nat" shows "rec_exec rec_valu [r] = valu r"
lideal: fixes I :: "'c set"   and R :: "'c set"   and addition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and multiplication :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and unit :: "'c"   and r :: "'c"   and a :: "'c" assumes "lideal I R addition multiplication zero unit"   and "r \<in> R"   and "a \<in> I" shows "multiplication r a \<in> I"
2: fixes x :: "V"   and y :: "V" assumes "x \<in> elts \<alpha>"   and "y \<in> elts \<alpha>"   and "(u + x, u + y) \<in> VWF" shows "x < y"
bb_disjnt: fixes q :: "nat"   and r :: "nat"   and i :: "nat"   and j :: "nat"   and k :: "nat"   and l :: "nat" assumes "q < r"   and "i < j"   and "j \<le> k"   and "r \<le> l"   and "j < r" shows "disjnt (b k (j, i)) (b l (r, q))"
mono_Span_append_1: fixes K :: "'a set"   and Us :: "'a list"   and Vs :: "'a list" assumes "subfield K R"   and "set Us \<subseteq> carrier R"   and "set Vs \<subseteq> carrier R" shows "R.Span K Us \<subseteq> R.Span K (Us @ Vs)"
\<gamma>_2: shows "pathstart \<gamma> = pathfinish h"
TH: fixes n :: "nat"   and t :: "'s"   and m :: "nat"   and s :: "'s" assumes "prob {\<omega> \<in> space M. X n \<omega> = t} \<noteq> 0"   and "prob {\<omega> \<in> space M. X m \<omega> = t} \<noteq> 0" shows "cond_prob M (\<lambda>\<omega>. X (Suc n) \<omega> = s) (\<lambda>\<omega>. X n \<omega> = t) = cond_prob M (\<lambda>\<omega>. X (Suc m) \<omega> = s) (\<lambda>\<omega>. X m \<omega> = t)"
pell_cnj_def: fixes D :: "nat" assumes "pell D" shows "pell_cnj = (\<lambda>(a, b). (a, - b))"
ring_1_ow_def: fixes U :: "'c set"   and one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c" shows "ring_1_ow U one times plus zero minus uminus \<equiv> ring_ow U plus zero minus uminus times \<and> zero_neq_one_ow U one zero \<and> monoid_mult_ow U one times"
map_tailrec_rev: fixes f :: "'b \<Rightarrow> 'a"   and as :: "'b list"   and bs :: "'a list" shows "map_tailrec_rev f as bs = rev (map f as) @ bs"
list_of_set_set_empty: shows "list_of_set_set {} = {[]}"