hcju/isabelleps huggingface dataset:
equivalence_of_monoidal_categories_def: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and T\<^sub>C :: "'b \<times> 'b \<Rightarrow> 'b"   and \<alpha>\<^sub>C :: "'b \<times> 'b \<times> 'b \<Rightarrow> 'b"   and \<iota>\<^sub>C :: "'b"   and D :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and T\<^sub>D :: "'d \<times> 'd \<Rightarrow> 'd"   and \<alpha>\<^sub>D :: "'d \<times> 'd \<times> 'd \<Rightarrow> 'd"   and \<iota>\<^sub>D :: "'d"   and F :: "'d \<Rightarrow> 'b"   and \<phi> :: "'d \<times> 'd \<Rightarrow> 'b"   and G :: "'b \<Rightarrow> 'd"   and \<eta> :: "'d \<Rightarrow> 'd"   and \<epsilon> :: "'b \<Rightarrow> 'b" shows "equivalence_of_monoidal_categories C T\<^sub>C \<alpha>\<^sub>C \<iota>\<^sub>C D T\<^sub>D \<alpha>\<^sub>D \<iota>\<^sub>D F \<phi> G \<eta> \<epsilon> \<equiv> (monoidal_category C T\<^sub>C \<alpha>\<^sub>C \<iota>\<^sub>C \<and> monoidal_category D T\<^sub>D \<alpha>\<^sub>D \<iota>\<^sub>D) \<and> equivalence_of_categories C D F G \<eta> \<epsilon> \<and> monoidal_functor D T\<^sub>D \<alpha>\<^sub>D \<iota>\<^sub>D C T\<^sub>C \<alpha>\<^sub>C \<iota>\<^sub>C F \<phi>"
zero_neq_one: fixes one :: "'e"   and zero :: "'e" assumes "class.zero_neq_one one zero" shows "zero \<noteq> one"
continuous_within_topologicalE: fixes x :: "'c"   and s :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and B :: "'d set" assumes "continuous (at x within s) f"   and "open B"   and "f x \<in> B" obtains A :: "'c set" where "open A" and "x \<in> A" and "\<And>y. \<lbrakk>y \<in> s; y \<in> A\<rbrakk> \<Longrightarrow> f y \<in> B"
incrIndexList_less: fixes ls :: "nat list"   and m :: "nat"   and nmax :: "nat"   and n :: "nat" assumes "incrIndexList ls m nmax"   and "Suc n < |ls|"   and "ls ! n \<noteq> ls ! Suc n" shows "ls ! n < ls ! Suc n"
annos: shows "P,E \<turnstile> es [\<leadsto>'] es'"
prems_3: shows "BSD \<V>i Tr\<^bsub>ESi\<^esub>"
square_free_separable_main: fixes f :: "'a poly" assumes "square_free f"   and "\<not> separable f" shows "\<exists>g k. f = g * k \<and> degree g \<noteq> 0 \<and> pderiv g = 0"
cases: fixes a :: "lam" assumes "NF a" obtains ts :: "lam list" and x :: "name" where "a = Var x__ \<degree>\<degree> ts" and "listsp NF ts"   | t :: "lam" and x :: "name" where "a = (Lam [x].t)" and "NF t"
ub_root: shows "\<forall>p\<in>set ps. \<forall>x. poly p x = 0 \<longrightarrow> x < ub"
inj_enat: fixes A :: "nat set" shows "inj_on enat A"
fst_vwb_lens: shows "vwb_lens fst\<^sub>L"
span_base: fixes scale :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and S :: "'b set" assumes "module scale"   and "a \<in> S" shows "a \<in> module.span scale S"
simps_6: fixes F :: "'b formula"   and G :: "'b formula" shows "nnf (\<^bold>\<not> (F \<^bold>\<rightarrow> G)) = nnf F \<^bold>\<and> nnf (\<^bold>\<not> G)"
zero_u_def: shows "0 = up\<cdot>(0::'a)"
sym_digraphI: assumes "symmetric (app_iso hom G)" shows "sym_digraph (app_iso hom G)"
elims_3: fixes x :: "nat \<times> nat" assumes "\<not> invar x" obtains n :: "nat" and l :: "nat" where "x = (n, l)" and "\<not> (l0 \<le> l \<and> (int l0 \<le> \<lfloor>real l / c\<rfloor> \<longrightarrow> f1 * real l \<le> real n) \<and> real n \<le> f2 * real l)"
finvimage_frange_vsubset: fixes r :: "V"   and B :: "V" assumes "\<R>\<^sub>\<bullet> r \<subseteq>\<^sub>\<circ> B" shows "r -`\<^sub>\<bullet> B = \<D>\<^sub>\<bullet> r"
trans_le_add1_hmset: fixes i :: "hmultiset"   and j :: "hmultiset"   and m :: "hmultiset" assumes "i \<le> j" shows "i \<le> j + m"
prefix_length_prefix: fixes ps :: "'a list"   and xs :: "'a list"   and qs :: "'a list" assumes "prefix ps xs"   and "prefix qs xs"   and "length ps \<le> length qs" shows "prefix ps qs"
assms_2: shows "\<psi> \<in> subfrmlsn \<phi>"
inv_comp_left_1: fixes g :: "'d"   and f :: "'d" assumes "D.iso (g \<cdot>\<^sub>D f)"   and "D.iso g" shows "D.inv (g \<cdot>\<^sub>D f) = D.inv f \<cdot>\<^sub>D D.inv g"
leq_antisym: fixes x :: "'a"   and y :: "'a" assumes "leq x y"   and "leq y x" shows "x = y"
remove_insert_weights: fixes s :: "bool"   and m :: "(nat \<times> nat) convnet"   and w :: "nat \<Rightarrow> real" shows "remove_weights (insert_weights s m w) = m"
assms: shows "\<forall>i j. i \<noteq> j \<longrightarrow> f (i, j) = bot"
eval_pmod_var_2: fixes L :: "('c multiset \<Rightarrow> ('a list \<times> nat) multiset \<Rightarrow> 'a) ring"   and i :: "'c"   and q :: "('c multiset \<Rightarrow> ('a list \<times> nat) multiset \<Rightarrow> 'a) list" assumes "field L"   and "\<And>P. P \<in> carrier L \<Longrightarrow> M.carrier_coeff P"   and "\<And>P. P \<in> carrier L \<Longrightarrow> M.index_free P i"   and "\<zero>\<^bsub>L\<^esub> = M.indexed_const \<zero>\<^bsub>M\<^esub>"   and "M.indexed_const \<in> ring_hom M L"   and "q \<in> carrier (poly_ring L)"   and "1 < degree q" shows "M.indexed_var i \<in> carrier (image_ring (M.eval_pmod L i q) (poly_ring L))"
simps: fixes uu :: "unit"   and k :: "nat"   and "is" :: "int list"   and i :: "int"   and d :: "int" shows "decr0 uu k (presb.Eq is i d) = presb.Eq (take k is @ drop (Suc k) is) i d"
vector_minus_component: fixes x :: "('b, 'c) vec"   and y :: "('b, 'c) vec"   and i :: "'c" shows "(x - y) $ i = x $ i - y $ i"
gcd_coprime_exists: fixes one :: "'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize"   and "gcd a b \<noteq> zero" shows "\<exists>a' b'. a = times a' (gcd a b) \<and> b = times b' (gcd a b) \<and> algebraic_semidom.coprime times one a' b'"
Diff_eq_empty_iff: fixes A :: "'d set"   and B :: "'d set" shows "(A - B = {}) = (A \<subseteq> B)"
porder_linorder_tranclpE: fixes A :: "'b set"   and r :: "('b \<times> 'b) set"   and B :: "'b set"   and s :: "('b \<times> 'b) set"   and x :: "'b"   and y :: "'b" assumes "partial_order_on A r"   and "linear_order_on B s"   and "order_consistent r s"   and "B \<subseteq> A"   and "(x, y) \<in> (r \<union> s)\<^sup>+" obtains "(x, y) \<in> r"   | u :: "'b" and v :: "'b" where "(x, u) \<in> r" and "(u, v) \<in> s" and "(v, y) \<in> r"
back_inj: fixes II1 :: "nat set"   and cl :: "('test, 'atom, 'choice) cmd list"   and n :: "nat"   and II2 :: "nat set" assumes "II1 \<subseteq> {brnL cl n..<brnL cl n + brn (cl ! n)}"   and "II2 \<subseteq> {brnL cl n..<brnL cl n + brn (cl ! n)}" shows "(back cl n II1 = back cl n II2) = (II1 = II2)"
prod_zero_def: fixes I :: "'d set"   and A :: "'d \<Rightarrow> ('e, 'f) aGroup_scheme" shows "prod_zero I A = (\<lambda>x\<in>I. \<zero>\<^bsub>A x\<^esub>)"
L'_def: shows "L' = lpp ` punit.reduced_GB L"
intros_3: fixes w1 :: "'b list"   and G :: "'b list set"   and w2 :: "'b list" assumes "w1 \<in> \<langle>G\<rangle>\<^sub>F"   and "w2 \<in> \<langle>G\<rangle>\<^sub>F" shows "w1 \<cdot> w2 \<in> \<langle>G\<rangle>\<^sub>F"
set_lin_tss': shows "\<forall>t\<in>set (linearize tss'). t \<le> nt \<and> enat nt \<le> enat t + right I"
assms_2: fixes x :: "'a"   and y :: "'a" shows "h (x * y) = h x * h y"
extended_Gromov_product_at_commute: fixes e :: "'a"   and x :: "'a Gromov_completion"   and y :: "'a Gromov_completion" shows "extended_Gromov_product_at e x y = extended_Gromov_product_at e y x"
P: shows "four_block_mat P1 (0\<^sub>m n m) (0\<^sub>m m n) P2 \<in> carrier_mat (n + m) (n + m)"
exec_redex_Fault: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'a, 'c) com option"   and c :: "('b, 'a, 'c) com"   and s :: "('b, 'c) xstate"   and f :: "'c" assumes "\<Gamma>\<turnstile> \<langle>redex c,s\<rangle> \<Rightarrow> Fault f" shows "\<Gamma>\<turnstile> \<langle>c,s\<rangle> \<Rightarrow> Fault f"
func_le_restrict: fixes f :: "'a \<Rightarrow> 'b option"   and g :: "'a \<Rightarrow> 'b option"   and X :: "'a set" assumes "f \<preceq> g"   and "X \<subseteq> dom f" shows "f |` X \<preceq> g"
associated_refl: fixes a :: "'a" assumes "a \<in> carrier (Ring_Divisibility.mult_of R)" shows "a \<sim>\<^bsub>Ring_Divisibility.mult_of R\<^esub> a"
A_4: shows "\<forall>i. f (A i) \<noteq> \<infinity>"
step_3: shows "subst_domain (compose ss) \<inter> range_vars (compose ss) = {}"
fds_converges_iff: fixes f :: "'b fds"   and s :: "'b" shows "fds_converges f s = (\<lambda>n. fds_nth f n / nat_power n s) sums eval_fds f s"
atLeastAtMost_singleton: fixes a :: "'a word" shows "signed.atLeastAtMost a a = {a}"
map_sum_o_inj_2: fixes f :: "'d \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'c" shows "map_sum f g \<circ> Inr = Inr \<circ> g"
size_gen: fixes xa :: "'c \<Rightarrow> nat"   and x :: "'d \<Rightarrow> nat"   and x1 :: "char list option"   and x2 :: "char list option"   and x3 :: "'c"   and x4 :: "'d" shows "Tagged_Prod_Sum.prod.size_prod xa x (prod.Prod x1 x2 x3 x4) = xa x3 + x x4 + Suc 0"
SeqStTermP_Var_diff: fixes s :: "name"   and k :: "name"   and v :: "tm"   and w :: "tm"   and i :: "tm" assumes "atom s \<sharp> (k, v, w, i)"   and "atom k \<sharp> (v, w, i)" shows "{VarP v, VarP w, Neg (v EQ w)} \<turnstile> SyntaxN.Ex s (SyntaxN.Ex k (SeqStTermP v i w w (Var s) (Var k)))"
three\<^sub>3_def: shows "\<^bold>3 = snd (snd threes)"
trancl_reflcl: fixes r :: "('d \<times> 'd) set" shows "(r\<^sup>=)\<^sup>+ = r\<^sup>*"
sqnf_def: fixes rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option"   and dip :: "nat" shows "sqnf rt dip \<equiv> case rt dip of None \<Rightarrow> unk | Some r \<Rightarrow> \<pi>\<^sub>3 r"
functor_axioms: shows "functor (\<cdot>\<acute>) (\<cdot>) \<Psi>"
verit_la_disequality: fixes a :: "'f"   and b :: "'f" shows "a = b \<or> \<not> a \<le> b \<or> \<not> b \<le> a"
simple_restrict_some: fixes i :: "nat"   and X :: "'b option list"   and A :: "nat set" assumes "i < length X"   and "i \<in> A" shows "restrict A X ! i = X ! i"
univ_poly_a_inv_length: fixes K :: "'a set"   and p :: "'a list" assumes "subring K R"   and "p \<in> carrier (K [X])" shows "length (\<ominus>\<^bsub>K [X]\<^esub> p) = length p"
alphaBinp_imp_asBinp_equal: fixes qbinp :: "'e \<Rightarrow> ('f, 'g, 'varSort, 'var, 'h) qAbs option"   and qbinp' :: "'e \<Rightarrow> ('f, 'g, 'varSort, 'var, 'h) qAbs option" assumes "qGoodBinp qbinp"   and "(qbinp, qbinp') \<in> alphaBinp" shows "asBinp qbinp = asBinp qbinp'"
ArrayCpy': fixes s' :: "char list \<Rightarrow> int \<Rightarrow> int"   and s :: "char list \<Rightarrow> int \<Rightarrow> int"   and x :: "char list"   and y :: "char list"   and \<pi> :: "char list \<Rightarrow> com option" assumes "s' = s(x := s y)" shows "\<pi>: (x[] ::= y, s) \<Rightarrow> s'"
list_emb_length: fixes P :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and xs :: "'c list"   and ys :: "'c list" assumes "list_emb P xs ys" shows "length xs \<le> length ys"
axioms_1: fixes return :: "'a \<Rightarrow> 'b"   and bind :: "'b \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b"   and ask :: "('c \<Rightarrow> 'b) \<Rightarrow> 'b"   and get :: "('d \<Rightarrow> 'b) \<Rightarrow> 'b"   and put :: "'d \<Rightarrow> 'b \<Rightarrow> 'b" assumes "monad_reader_state return bind ask get put" shows "monad_reader return bind ask"
solve_def: fixes F0 :: "Literal list list" shows "solve F0 = getSATFlag (solve_loop (initialize F0 initialState) (vars F0))"
null_eqI: fixes n :: "'a" assumes "\<And>f. L.comp n f = n \<and> L.comp f n = n" shows "n = L.null"
dim_eq_size_basis: fixes vs :: "'b list" assumes "set vs \<subseteq> V \<and> V = RSpan vs \<and> R_lin_independent vs" shows "length vs = dim_R V"
assms_2: assumes "\<gamma> t n"   and "\<gamma>' t n" shows "\<gamma>'' t n"
Special_axP_graph_aux_def: shows "Special_axP_graph_aux \<equiv> lfp (\<lambda>p x1 x2. \<exists>Special_axP_aux v p sx y ax x. x1 = p \<and> x2 = SyntaxN.Ex v (SyntaxN.Ex x (SyntaxN.Ex ax (SyntaxN.Ex y (SyntaxN.Ex sx (FormP (Var x) AND VarP (Var v) AND TermP (Var y) AND AbstFormP (Var v) Zero (Var x) (Var ax) AND SubstFormP (Var v) (Var y) (Var x) (Var sx) AND p EQ Q_Imp (Var sx) (Q_Ex (Var ax))))))) \<and> atom v \<sharp> (p, sx, y, ax, x) \<and> atom x \<sharp> (p, sx, y, ax) \<and> atom ax \<sharp> (p, sx, y) \<and> atom y \<sharp> (p, sx) \<and> atom sx \<sharp> p)"
add_diff_cancel_left': fixes plus :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and minus :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and a :: "'k"   and b :: "'k" assumes "class.cancel_ab_semigroup_add plus minus" shows "minus (plus a b) a = b"
carrier_vs_is_self: fixes N :: "'c set" shows "carrier (md N) = N"
mset_set_set_mset_subseteq: fixes A :: "'b multiset" shows "mset_set (set_mset A) \<subseteq># A"
f4: shows "List.gen_length 0 i2 \<le> List.gen_length 0 i1"
dims2_def: shows "st2.dims2 = nths dims (vars1 - {k})"
Berlekamp_subspace_fin_dim: shows "Berlekamp_subspace.fin_dim"
Base_foo_no_stat_override: fixes sig :: "sig"   and old :: "qtname \<times> methd"   and P :: "bool" assumes "tprg \<turnstile> qmdecl sig (Base, snd Base_foo) overrides\<^sub>S qmdecl sig old" shows "P"
vimage_const: fixes c :: "'c"   and A :: "'c set" shows "(\<lambda>x. c) -` A = (if c \<in> A then UNIV else {})"
len_def: fixes P :: "nat fset" shows "len P = (if P = {||} then 0 else Suc (fMax P))"
perm_8: fixes pi :: "(name \<times> name) list"   and x1 :: "('d, 'e, 'f) psi" shows "pi \<bullet> !x1 = !(pi \<bullet> x1)"
simps_4: fixes w :: "nat"   and t\<^sub>1 :: "'a tree"   and v :: "nat"   and va :: "'a tree"   and vb :: "'a tree"   and a :: "'a" shows "mergeSibling (Node w t\<^sub>1 (Node v va vb)) a = Node w (mergeSibling t\<^sub>1 a) (mergeSibling (Node v va vb) a)"
simps_7: fixes disc :: "'b common_primitive negation_type \<Rightarrow> bool"   and m1 :: "'b common_primitive match_expr"   and m2 :: "'b common_primitive match_expr" shows "abstract_primitive disc (MatchAnd m1 m2) = MatchAnd (abstract_primitive disc m1) (abstract_primitive disc m2)"
elims_3: fixes x :: "'a \<times> nat" assumes "\<not> is_valid_state_id x" obtains i :: "'a" and j :: "nat" where "x = (i, j)" and "\<not> j \<le> length (events i)"
sup_continuous_add: fixes f :: "'a \<Rightarrow> ereal"   and g :: "'a \<Rightarrow> ereal" assumes "\<And>x. 0 \<le> f x"   and "\<And>x. 0 \<le> g x"   and "sup_continuous f"   and "sup_continuous g" shows "sup_continuous (\<lambda>x. f x + g x)"
insertAt_nth: fixes i :: "nat"   and ls :: "'a list"   and n :: "'a" assumes "i \<le> length ls" shows "insertAt i n ls ! i = n"
iso_in_homE: fixes f :: "('c \<Rightarrow> 's, 'c \<Rightarrow> 's) concrete_category.arr"   and a :: "('c \<Rightarrow> 's, 'c \<Rightarrow> 's) concrete_category.arr"   and b :: "('c \<Rightarrow> 's, 'c \<Rightarrow> 's) concrete_category.arr" assumes "Cop_S.iso_in_hom f a b" obtains "\<guillemotleft>f : a \<rightarrow>\<^sub>[\<^sub>C\<^sub>o\<^sub>p\<^sub>,\<^sub>S\<^sub>] b\<guillemotright>" and "Cop_S.iso f"
mliftM2_def: fixes f :: "'d \<rightarrow> 'e \<rightarrow> 'f" shows "mliftM2 f \<equiv> \<Lambda> x y. x >>=\<^sub>M (\<Lambda> x'. y >>=\<^sub>M (\<Lambda> y'. Just\<cdot>(f\<cdot>x'\<cdot>y')))"
assms: shows "dim_vec (u @\<^sub>v w) = dim_vec x"
is_heap_def: fixes xs :: "('b \<times> 'c) list" shows "is_heap xs = (\<forall>i j. eq_pred i j \<longrightarrow> j < length xs \<longrightarrow> snd (xs ! i) \<le> snd (xs ! j))"
span_breakdown_eq: fixes x :: "'b"   and a :: "'b"   and S :: "'b set" shows "(x \<in> span (insert a S)) = (\<exists>k. x - k *s a \<in> span S)"
flowD_capacity: fixes \<Delta> :: "('k, 'l) network_scheme"   and f :: "'k \<times> 'k \<Rightarrow> ennreal"   and e :: "'k \<times> 'k" assumes "flow \<Delta> f" shows "f e \<le> capacity \<Delta> e"
derivation: shows "LeftDerivation \<beta> (derivation_shift F 0 (Suc j)) b"
ent_disj_star_mono: fixes A :: "assn"   and C :: "assn"   and E :: "assn"   and B :: "assn"   and D :: "assn"   and F :: "assn" assumes "A \<or>\<^sub>A C \<Longrightarrow>\<^sub>A E"   and "B \<or>\<^sub>A D \<Longrightarrow>\<^sub>A F" shows "A * B \<or>\<^sub>A C * D \<Longrightarrow>\<^sub>A E * F"
sorted_spvec_minus_spmat: fixes A :: "(nat \<times> (nat \<times> 'a) list) list" assumes "sorted_spvec A" shows "sorted_spvec (minus_spmat A)"
U_LIMCONS_not_learn_cons: shows "\<not> learn_cons \<phi> U\<^bsub>LIM-CONS\<^esub> s"
all_dupE: fixes P :: "'a \<Rightarrow> bool"   and x :: "'a" assumes "\<forall>x. P x" obtains "P x" and "\<forall>x. P x"
iMOD_iMODb_inext_nth: fixes a :: "nat"   and c :: "nat"   and r :: "nat"   and m :: "nat" assumes "a \<le> c" shows "[ r, mod m, c ] \<rightarrow> a = [ r, mod m ] \<rightarrow> a"
matching_disjointness: fixes V :: "'b set"   and E :: "('b \<times> 'b) set"   and M :: "('b \<times> 'b) set"   and e1 :: "'b \<times> 'b"   and e2 :: "'b \<times> 'b" assumes "matching V E M"   and "e1 \<in> M"   and "e2 \<in> M"   and "e1 \<noteq> e2" shows "edge_as_set e1 \<inter> edge_as_set e2 = {}"
emp_UNlift_UNpart: fixes cl :: "('test, 'atom, 'choice) cmd list"   and dl :: "('test, 'atom, 'choice) cmd list"   and P :: "nat \<Rightarrow> nat set set"   and F :: "nat \<Rightarrow> nat set \<Rightarrow> nat set" assumes "length cl = length dl"   and "\<And>n. n < length cl \<Longrightarrow> part {..<brn (cl ! n)} (P n) \<and> {} \<notin> P n"   and "\<And>n. n < length dl \<Longrightarrow> {} \<notin> F n ` P n" shows "{} \<notin> UNlift cl dl P F ` UNpart cl P"
assms: shows "FI_RESULT m (p * up) (q * uq) f"
coActionSimps_1: fixes a :: "act" shows "coAction (coAction a) = a"
that: fixes i :: "nat" assumes "\<phi> i = f" shows "thesis"
fin_A: shows "finite {x. f x \<noteq> (0::'b)}"
assms: shows "(c', ctxts) \<in> decomposeLS s"
i_def: shows "i \<equiv> contour_integral (circlepath z e) f / c"
closure_rational_coordinates: shows "closure (\<Union>f\<in>Basis \<rightarrow> \<rat>. {\<Sum>i\<in>Basis. f i *\<^sub>R i}) = UNIV"
ESem_fresh_cong_below: fixes e :: "exp"   and \<rho> :: "Vars.var \<Rightarrow> C \<rightarrow> CValue"   and \<rho>' :: "Vars.var \<Rightarrow> C \<rightarrow> CValue" assumes "\<rho> f|` fv e \<sqsubseteq> \<rho>' f|` fv e" shows "\<N>\<lbrakk> e \<rbrakk>\<^bsub>\<rho>\<^esub> \<sqsubseteq> \<N>\<lbrakk> e \<rbrakk>\<^bsub>\<rho>'\<^esub>"
le0: shows "Re (\<i> * z + csqrt (1 - z\<^sup>2)) \<le> 0"
mem_case_prodI: fixes z :: "'c"   and c :: "'d \<Rightarrow> 'e \<Rightarrow> 'c set"   and a :: "'d"   and b :: "'e" assumes "z \<in> c a b" shows "z \<in> (case (a, b) of (d, e) \<Rightarrow> c d e)"
s'_pos: fixes x :: "real" assumes "0 \<le> x"   and "x < other.t_stop"   and "0 \<le> v\<^sub>o"   and "a\<^sub>o \<le> 0" shows "0 < other.s' x"
scale_zero_right: fixes a :: "'a" shows "s2 a (0::'c) = (0::'c)"
context_le_refl: fixes \<Gamma> :: "'Var \<Rightarrow> Sec option" shows "\<Gamma> \<sqsubseteq>\<^sub>c \<Gamma>"
inverse_mod_coprime: fixes x :: "int" assumes "prime m"   and "comm_monoid_mult_class.coprime x m" shows "M (inverse_mod x m * x) = 1"
A: shows "(yps, Y) \<in> futures (c_process step out s\<^sub>0) xps"
parts_spies_evs_revD2: fixes evs :: "event list" shows "parts (knows Spy (rev evs)) \<subseteq> parts (knows Spy evs)"
span_sum: fixes A :: "'d set"   and f :: "'d \<Rightarrow> 'b"   and S :: "'b set" assumes "\<And>x. x \<in> A \<Longrightarrow> f x \<in> m1.span S" shows "sum f A \<in> m1.span S"
integral_normal_moment_nz_1: fixes \<mu> :: "real" shows "LBINT x. normal_density \<mu> \<sigma> x * x = \<mu>"
fps_nth_deriv_neg: fixes n :: "nat"   and f :: "'a fps" shows "fps_nth_deriv n (- f) = - fps_nth_deriv n f"
gf: shows "\<forall>z\<in>h ` S. f z \<in> ball 0 1 \<and> g (f z) = z"
Cons_7: shows "getWatch2 stateA = getWatch2 stateB"
take_bit_int_less_exp: fixes n :: "nat"   and k :: "int" shows "take_bit n k < 2 ^ n"
gbinomial_minus': fixes a :: "'a"   and b :: "nat" shows "a + of_nat b gchoose b = (- (1::'a)) ^ b * (- (a + (1::'a)) gchoose b)"
remove_all_rev_subs: fixes x :: "'b"   and xs :: "'b list" shows "set (remove_all_rev x xs) \<subseteq> set xs"
ListWeakI: fixes R :: "(formula list \<times> formula list set) set"   and x :: "formula"   and Gamma :: "formula list" assumes "Perms \<subseteq> R"   and "Contrs \<subseteq> R"   and "x # Gamma @ Gamma \<in> deductions R" shows "x # Gamma \<in> deductions R"
simps_4: fixes v :: "poscheckvariant"   and va :: "('a, 'b) Term.term"   and vb :: "('a, 'b) Term.term"   and S :: "('a, 'b) strand_step list" shows "ik\<^sub>s\<^sub>t (\<langle>v: va \<doteq> vb\<rangle>\<^sub>s\<^sub>t # S) = ik\<^sub>s\<^sub>t S"
intro: fixes M :: "'b measure" assumes "finite_measure M"   and "finite_borel_measure'_axioms M" shows "finite_borel_measure' M"
mset_x_eq_poly_roots_Q: shows "image_mset f (mset_set {..<3}) = poly_roots Q"
e2ennreal_def: fixes x :: "ereal" shows "e2ennreal x = e2ennreal' (max 0 x)"
mset_un_single_un_cases: fixes a :: "'c"   and A :: "'c multiset"   and B :: "'c multiset"   and C :: "'c multiset" assumes "add_mset a A = B + C" obtains "a \<in># B" and "A = B - {#a#} + C"   | "a \<in># C" and "A = B + (C - {#a#})"
create_create_commute: fixes i1 :: "'d"   and e1 :: "'c"   and i2 :: "'d"   and e2 :: "'c" shows "\<langle>Create i1 e1\<rangle> \<rhd> \<langle>Create i2 e2\<rangle> = \<langle>Create i2 e2\<rangle> \<rhd> \<langle>Create i1 e1\<rangle>"
pick_in_set: fixes n :: "nat"   and S :: "nat set" assumes "n < card S \<or> infinite S" shows "pick S n \<in> S"
strategyproofness_2: shows "(pmf (sds R10) d < pmf (sds R5) d \<or> pmf (sds R10) d + (pmf (sds R10) b + pmf (sds R10) a) < pmf (sds R5) d + (pmf (sds R5) b + pmf (sds R5) a)) \<or> pmf (sds R10) d = pmf (sds R5) d \<and> pmf (sds R10) d + (pmf (sds R10) b + pmf (sds R10) a) = pmf (sds R5) d + (pmf (sds R5) b + pmf (sds R5) a)"
in_\<P>_charslength: fixes p :: "(symbol \<times> character list) list"   and k :: "nat"   and u :: "nat" assumes "p \<in> \<P> k u" shows "\<exists>v. p \<in> \<P> (charslength p) v"
semiring_hom_def: fixes hom :: "'c \<Rightarrow> 'd" shows "semiring_hom hom \<equiv> comm_monoid_add_hom hom \<and> monoid_mult_hom hom"
st_zero_le: fixes x :: "real star" assumes "0 \<le> x"   and "x \<in> HFinite" shows "0 \<le> st x"
ordLeq_total: fixes r :: "('a \<times> 'a) set"   and r' :: "('b \<times> 'b) set" assumes "Well_order r"   and "Well_order r'" shows "r \<le>o r' \<or> r' \<le>o r"
symmatrix_def: fixes M :: "(('a, 'b) vec, 'b) vec" shows "symmatrix M \<equiv> Finite_Cartesian_Product.transpose M = M"
mk_disjoint_insert: fixes a :: "'d"   and A :: "'d set" assumes "a \<in> A" shows "\<exists>B. A = insert a B \<and> a \<notin> B"
E5: shows "(\<Sum>(x, y)\<in>{(x, y). x \<in> set init \<and> y \<in> set init \<and> x < y}. (\<Sum>i | i < length qs \<and> qs ! i = y. ALG x qs i (swaps (snd (Strat ! i)) (steps' init qs Strat i), ())) + (\<Sum>i | i < length qs \<and> qs ! i = x. ALG y qs i (swaps (snd (Strat ! i)) (steps' init qs Strat i), ()))) = (\<Sum>(x, y)\<in>{(x, y). x \<in> set init \<and> y \<in> set init \<and> x < y}. \<Sum>i | i < length qs \<and> (qs ! i = y \<or> qs ! i = x). ALG y qs i (swaps (snd (Strat ! i)) (steps' init qs Strat i), ()) + ALG x qs i (swaps (snd (Strat ! i)) (steps' init qs Strat i), ()))"
bits_one_mod_two_eq_one: shows "(1::'a) mod (2::'a) = (1::'a)"
err_semilat_sup: fixes A :: "'b set"   and r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b err"   and n :: "nat" assumes "err_semilat (A, r, f)" shows "err_semilat (list n A, Listn.le r, Listn.sup f)"
f: shows "f \<in> L \<rightarrow>\<^sub>M N"
crypts_def: shows "crypts \<equiv> \<lambda>H. {x. cryptsp (\<lambda>x. x \<in> H) x}"
degree_of_X_plus: fixes R :: "('c, 'd) ring_scheme"   and a :: "'c" assumes "UP_cring R"   and "a \<in> carrier R"   and "\<one>\<^bsub>R\<^esub> \<noteq> \<zero>\<^bsub>R\<^esub>" shows "deg R (X_poly_plus R a) = 1"
elims: fixes x :: "bool"   and y :: "String.literal + unit" assumes "assert x = y" obtains "\<not> x" and "y = Inl STR ''Assertion failed.''"   | "x" and "y = Inr ()"