hcju/isabelleps huggingface dataset:
Module_axioms_def: fixes M :: "('e, 'f, 'g) Module_scheme"   and R :: "('f, 'h) Ring_scheme" shows "Module_axioms M R \<equiv> (Ring R \<and> (\<forall>a m. a \<in> carrier R \<longrightarrow> m \<in> carrier M \<longrightarrow> a \<cdot>\<^sub>s\<^bsub>M\<^esub> m \<in> carrier M) \<and> (\<forall>a b m. a \<in> carrier R \<longrightarrow> b \<in> carrier R \<longrightarrow> m \<in> carrier M \<longrightarrow> (a \<plusminus>\<^bsub>R\<^esub> b) \<cdot>\<^sub>s\<^bsub>M\<^esub> m = a \<cdot>\<^sub>s\<^bsub>M\<^esub> m \<plusminus>\<^bsub>M\<^esub> b \<cdot>\<^sub>s\<^bsub>M\<^esub> m)) \<and> (\<forall>a m n. a \<in> carrier R \<longrightarrow> m \<in> carrier M \<longrightarrow> n \<in> carrier M \<longrightarrow> a \<cdot>\<^sub>s\<^bsub>M\<^esub> (m \<plusminus>\<^bsub>M\<^esub> n) = a \<cdot>\<^sub>s\<^bsub>M\<^esub> m \<plusminus>\<^bsub>M\<^esub> a \<cdot>\<^sub>s\<^bsub>M\<^esub> n) \<and> (\<forall>a b m. a \<in> carrier R \<longrightarrow> b \<in> carrier R \<longrightarrow> m \<in> carrier M \<longrightarrow> (a \<cdot>\<^sub>r\<^bsub>R\<^esub> b) \<cdot>\<^sub>s\<^bsub>M\<^esub> m = a \<cdot>\<^sub>s\<^bsub>M\<^esub> (b \<cdot>\<^sub>s\<^bsub>M\<^esub> m)) \<and> (\<forall>m. m \<in> carrier M \<longrightarrow> 1\<^sub>r\<^bsub>R\<^esub> \<cdot>\<^sub>s\<^bsub>M\<^esub> m = m)"
sorted_sequences: fixes xs :: "'a list" shows "\<forall>x\<in>set (local.sequences xs). sorted (map key x)"
error_msg_ne_SUCCES_4: fixes \<phi> :: "char list \<Rightarrow> nat"   and i :: "'a"   and m :: "char list" assumes "poly_embed \<phi>" shows "\<not> is_cfound (error_msg i m)"
assms_1: shows "IO out' rpv \<in> set_spmf (the_gpv gpv)"
DFS_axioms: shows "DFS succs is_node invariant ins memb empt"
dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t\<^sub>p_inv_2: fixes l :: "'d strand_label"   and a :: "('e, 'f) stateful_strand_step"   and k :: "'d strand_label"   and b :: "('e, 'f) stateful_strand_step"   and t :: "('e, 'f) Term.term" assumes "dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t\<^sub>p (l, a) = (k, b)"   and "a = receive\<langle>t\<rangle>" shows "b = send\<langle>t\<rangle>"
simps: fixes B :: "ty"   and C :: "ty"   and X :: "ctrm set"   and Y :: "ctrm set" shows "ANDRIGHT (B AND C) X Y = {<c>:AndR <a>.M <b>.N c |c a b M N. fic (AndR <a>.M <b>.N c) c \<and> <a>:M \<in> X \<and> <b>:N \<in> Y}"
\<eta>_p_atm: shows "\<forall>i<Suc n. \<forall>A. A \<in> atms_of ((DA0' # CAs0') ! i) \<longrightarrow> A \<cdot>a (\<eta>0' # \<eta>s0') ! i = A \<cdot>a \<eta>"
reachNT_reach: fixes istate :: "'a"   and step :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<times> 'a"   and T :: "('a, 'b, 'c) trans \<Rightarrow> bool"   and s :: "'a" assumes "BD_Security.reachNT istate step T s" shows "IO_Automaton.reach istate step s"
term_subst_eq: fixes t :: "('b, 'a) Term.term"   and \<sigma> :: "'a \<Rightarrow> ('b, 'c) Term.term"   and \<tau> :: "'a \<Rightarrow> ('b, 'c) Term.term" assumes "\<And>x. x \<in> fv t \<Longrightarrow> \<sigma> x = \<tau> x" shows "t \<cdot> \<sigma> = t \<cdot> \<tau>"
strict_coboundedI1: fixes a :: "'a"   and c :: "'a"   and b :: "'a" assumes "a < c" shows "min a b < c"
ltl_const_equiv_no_const_prop_equiv: fixes \<psi> :: "'a ltln"   and \<phi> :: "'a ltln" assumes "eval \<psi> = Maybe"   and "\<phi> \<sim>\<^sub>C \<psi>" shows "\<phi> \<sim>\<^sub>P \<psi>"
enn2ereal_e2ennreal: fixes x :: "ereal" assumes "0 \<le> x" shows "enn2ereal (e2ennreal x) = x"
edges_part: fixes u :: "'n"   and a :: "('p, 'ba) edgeAnnot"   and v :: "'n" assumes "(u, a, v) \<in> edges fg" shows "proc_of fg u = proc_of fg v"
bottom_in_unlr: fixes R :: "'a admS" shows "\<bottom> \<in> unlr R"
en_eq_5: fixes v :: "i"   and x :: "\<kappa>"   and F :: "\<Pi>\<^sub>1"   and y :: "\<kappa>"   and G :: "\<Pi>\<^sub>1" shows "[\<^bold>\<box>(\<lbrace>x,F\<rbrace> \<^bold>\<equiv> \<lbrace>y,G\<rbrace>) \<^bold>\<equiv> (\<^bold>\<box>\<lbrace>x,F\<rbrace> \<^bold>\<equiv> \<^bold>\<box>\<lbrace>y,G\<rbrace>) in v]"
onl_oinvariantD: fixes A :: "('e \<times> ('f, 'g, 'h, 'i) seqp, 'j) automaton"   and S :: "'e \<Rightarrow> 'e \<Rightarrow> 'j \<Rightarrow> bool"   and U :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and \<Gamma> :: "'h \<Rightarrow> ('f, 'g, 'h, 'i) seqp"   and P :: "'e \<times> 'i \<Rightarrow> bool"   and \<sigma> :: "'e"   and p :: "('f, 'g, 'h, 'i) seqp"   and l :: "'i" assumes "A \<Turnstile> (S, U \<rightarrow>) onl \<Gamma> P"   and "(\<sigma>, p) \<in> oreachable A S U"   and "l \<in> labels \<Gamma> p" shows "P (\<sigma>, l)"
diagonal_step_code_nth: fixes A :: "(('b, 'c) vec, 'a) vec"   and i :: "nat"   and j :: "nat"   and c :: "'b"   and v :: "'b" shows "($) (diagonal_step A i j c v) = diagonal_step_row A i j c v"
get_root_node_def: shows "get_root_node \<equiv> l_get_root_node\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_root_node get_parent"
distinct_5: shows "Lagrange \<noteq> Neville_Aitken"
s_top_mono: fixes G :: "('e, 'f) Group_scheme"   and H :: "'e set"   and K :: "'e set"   and H1 :: "'e set"   and K1 :: "'e set" assumes "Group G"   and "H \<subseteq> carrier G"   and "K \<subseteq> carrier G"   and "H1 \<subseteq> H"   and "K1 \<subseteq> K" shows "H1 \<diamondop>\<^bsub>G\<^esub> K1 \<subseteq> H \<diamondop>\<^bsub>G\<^esub> K"
abs_eq: shows "(\<squnion>) = (\<lambda>x y. Abs_atoms (Rep_atoms x \<union> Rep_atoms y))"
Unf_sound: fixes w :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltln" shows "w \<Turnstile>\<^sub>n Unf \<phi> = w \<Turnstile>\<^sub>n \<phi>"
induct: fixes P :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool"   and a0 :: "'a list"   and a1 :: "'a list" assumes "\<And>ys. P [] ys"   and "\<And>v va. P (v # va) []"   and "\<And>x xs y ys. \<lbrakk>x = y \<Longrightarrow> P xs ys; \<lbrakk>x \<noteq> y; x < y\<rbrakk> \<Longrightarrow> P xs (y # ys); \<lbrakk>x \<noteq> y; \<not> x < y\<rbrakk> \<Longrightarrow> P (x # xs) ys\<rbrakk> \<Longrightarrow> P (x # xs) (y # ys)" shows "P a0 a1"
UnD2: fixes c :: "'a"   and A :: "'a set"   and B :: "'a set" assumes "c \<in> A \<union> B"   and "c \<notin> B" shows "c \<in> A"
tp_const: fixes c :: "real" shows "trigpoly (\<lambda>x. c)"
part_def: fixes J :: "'a set"   and P :: "'a set set" shows "part J P \<equiv> \<Union> P = J \<and> (\<forall>J1 J2. J1 \<in> P \<and> J2 \<in> P \<and> J1 \<noteq> J2 \<longrightarrow> J1 \<inter> J2 = {})"
wf_tuple_def: fixes n :: "nat"   and V :: "nat set"   and x :: "'a option list" shows "wf_tuple n V x = (length x = n \<and> (\<forall>i<n. (x ! i = None) = (i \<notin> V)))"
In0_def: fixes M :: "('c, 'd) node set" shows "In0 M \<equiv> Scons (Numb 0) M"
strict_iff_order: fixes b :: "'a word"   and a :: "'a word" shows "(b <s a) = (b \<le>s a \<and> a \<noteq> b)"
distinct_5: fixes x31 :: "ty_Rep"   and x32 :: "ty_Rep" shows "TUnit_Rep \<noteq> Arrow_Rep x31 x32"
simps_8: fixes v :: "cmsg"   and vb :: "cmsg" shows "catomic (cEnc v vb) = False"
add_pow_def: fixes R :: "('c, 'd) ring_scheme"   and k :: "'e"   and a :: "'c" shows "[k] \<cdot>\<^bsub>R\<^esub> a = a [^]\<^bsub>add_monoid R\<^esub> k"
transfer: shows "pcr_ell2 (=) (\<lambda>_. 0) 0"
1: fixes x1 :: "'a"   and y1 :: "'a"   and x2 :: "'b"   and y2 :: "'b" assumes "(x1, y1) \<in> K1.acc"   and "(x2, y2) \<in> K2.acc"   and "x1 \<in> C1"   and "y1 \<in> C1"   and "x2 \<in> C2"   and "y2 \<in> C2" shows "((x1, x2), y1, y2) \<in> (SIGMA x:UNIV. set_pmf (Kp x))\<^sup>*"
cInf_lower2: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and x :: "'b"   and X :: "'b set"   and y :: "'b" assumes "class.conditionally_complete_lattice Inf Sup inf less_eq less sup"   and "x \<in> X"   and "less_eq x y"   and "preorder.bdd_below less_eq X" shows "less_eq (Inf X) y"
face_of_affine_trivial: fixes S :: "'b set"   and T :: "'b set" assumes "affine S"   and "T face_of S" shows "T = {} \<or> T = S"
sep_impl_def: fixes P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool" shows "P \<longrightarrow>* Q \<equiv> \<lambda>h. \<forall>h'. h ## h' \<and> P h' \<longrightarrow> Q (h + h')"
gcd_mute_fun_def: fixes x :: "'b"   and y :: "'b" shows "gcd_mute_fun x y = (let g = gcd x y in (x div g, y div g, g))"
polyG: fixes X :: "'a set" assumes "X \<in> \<G>" shows "polytope X"
cSUP_least: fixes A :: "'c set"   and f :: "'c \<Rightarrow> 'a"   and M :: "'a" assumes "A \<noteq> {}"   and "\<And>x. x \<in> A \<Longrightarrow> f x \<le> M" shows "\<Squnion> (f ` A) \<le> M"
simplicial_chain_chain_map: fixes q :: "nat"   and X :: "('c \<Rightarrow> real) set"   and f :: "(nat \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real"   and p :: "nat"   and c :: "((nat \<Rightarrow> real) \<Rightarrow> nat \<Rightarrow> real) \<Rightarrow>\<^sub>0 int" assumes "simplicial_simplex q X f"   and "simplicial_chain p (standard_simplex q) c" shows "simplicial_chain p X (chain_map p f c)"
real_sqrt_mult_self: fixes a :: "real" shows "sqrt a * sqrt a = \<bar>a\<bar>"
sorted_inl: fixes X :: "'b \<times> (real \<times> real) pdevs" shows "ccw.sortedP 0 (ccw.selsort 0 (inl (snd X)))"
cofactor_def: fixes A :: "'b mat"   and i :: "nat"   and j :: "nat" shows "cofactor A i j = (- (1::'b)) ^ (i + j) * det (mat_delete A i j)"
discI: fixes option :: "'x option"   and x2 :: "'x" assumes "option = Some x2" shows "option \<noteq> None"
nrename_NotR: fixes R :: "trm"   and x :: "name"   and y :: "name"   and z :: "name"   and N :: "trm"   and c :: "coname" assumes "R[x\<turnstile>n>y] = NotR z.N c"   and "z \<sharp> (R, x, y)" shows "\<exists>N'. R = NotR z.N' c \<and> N'[x\<turnstile>n>y] = N"
finitePairSecondRange: fixes R :: "('a \<times> 'b set) set" shows "(\<forall>pair\<in>R. finite (snd pair)) = (\<forall>y\<in>Range R. finite y)"
runit_naturality: fixes \<mu> :: "'b" assumes "D.arr \<mu>" shows "\<mu> \<cdot>\<^sub>D \<r>\<^sub>D[D.dom \<mu>] = \<r>\<^sub>D[D.cod \<mu>] \<cdot>\<^sub>D (\<mu> \<star>\<^sub>D src\<^sub>D \<mu>)"
that: fixes h :: "'a"   and h' :: "'a" assumes "h \<in> H"   and "inv_into (carrier G) f (f a) = h \<otimes> a"   and "h' \<in> H"   and "inv_into (carrier G) f (f a) = a \<otimes> h'" shows "thesis"
lossless2: fixes cipher :: "bool list \<times> bool list"   and \<sigma> :: "'a" shows "lossless_gpv \<I>_full (\<A>2 cipher \<sigma>)"
P_H2: shows "P_H2 \<in> carrier_mat (dim_row A) (dim_row A)"
3: shows "card S = card ((*\<^sub>v) A ` S)"
sifum_security_init_axioms: shows "sifum_security_init dma\<^sub>A \<C>_vars\<^sub>A eval\<^sub>A"
eq_\<I>_gpv_Pause: fixes out :: "'a"   and \<I> :: "('a, 'b) \<I>"   and A :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and rpv :: "('c, 'a, 'b) rpv"   and rpv' :: "('d, 'a, 'b) rpv" assumes "out \<in> outs_\<I> \<I>"   and "\<And>input. input \<in> responses_\<I> \<I> out \<Longrightarrow> eq_\<I>_gpv A \<I> (rpv input) (rpv' input)" shows "eq_\<I>_gpv A \<I> (Generative_Probabilistic_Value.Pause out rpv) (Generative_Probabilistic_Value.Pause out rpv')"
add_incr: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and b :: "'a"   and a :: "'a" assumes "class.weight plus zero less_eq less"   and "less_eq zero b" shows "less_eq a (plus a b)"
InfinitesimalD: fixes x :: "'a star" assumes "x \<in> Infinitesimal" shows "\<forall>r\<in>\<real>. 0 < r \<longrightarrow> hnorm x < r"
map_option_case: fixes f :: "'d \<Rightarrow> 'c"   and y :: "'d option" shows "map_option f y = (case y of None \<Rightarrow> None | Some x \<Rightarrow> Some (f x))"
incomparable_def: fixes A :: "'b set" shows "incomparable A = (\<forall>x\<in>A. \<forall>y\<in>A. x \<noteq> y \<longrightarrow> \<not> x < y \<and> \<not> y < x)"
override_on_cont2: fixes m :: "'c \<Rightarrow> 'd"   and S :: "'c set" shows "cont (override_on_syn m S)"
AE_increasing_then_invariant: fixes f :: "'a \<Rightarrow> 'c" assumes "AE x in M. f x \<le> f (T x)"   and "f \<in> borel_measurable M" shows "AE x in M. f (T x) = f x"
orthogonal_to_span: fixes a :: "'a"   and S :: "'a set"   and x :: "'a" assumes "a \<in> span S"   and "\<And>y. y \<in> S \<Longrightarrow> orthogonal x y" shows "orthogonal x a"
emeasure_CI': fixes J :: "nat set"   and X :: "(nat \<Rightarrow> 'a) set" assumes "finite J"   and "X \<in> sets (Pi\<^sub>M J (\<lambda>_. M))" shows "emeasure (CI J) X = emeasure (C 0 (up_to J) (\<lambda>_. undefined)) (PF.emb {0..<up_to J} J X)"
valid_Not: fixes P :: "'state"   and x :: "('b, 'pred, 'act) formula" shows "P \<Turnstile> Formula.Not x = (\<not> P \<Turnstile> x)"
power_mult: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and m :: "nat"   and n :: "nat" assumes "class.monoid_mult one times" shows "power.power one times a (m * n) = power.power one times (power.power one times a m) n"
impl_of: fixes x :: "('c, 'd) mapping_rbt" shows "impl_of x \<in> {t. ord.is_rbt cless t \<or> ID ccompare = None}"
line_through_K2_intersect_S_again: fixes p :: "proj2"   and l :: "proj2_line"   and q :: "proj2" assumes "p \<in> hyp2"   and "proj2_incident p l" shows "\<exists>r. r \<noteq> q \<and> r \<in> S \<and> proj2_incident r l"
App: fixes ts :: "lam list"   and x :: "name" assumes "listsp NF ts" shows "NF (Var x \<degree>\<degree> ts)"
l_get_parent_get_host_wf_is_get_parent_get_host_wf: shows "l_get_parent_get_host_wf Shadow_DOM.heap_is_wellformed Shadow_DOM.get_parent get_shadow_root get_host Shadow_DOM.get_child_nodes"
ipresIGWls_termMOD: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'h"   and MOD :: "('i, 'j, 'k, 'sort, 'l, 'm, 'h, 'n, 'o) model_scheme" shows "ipresIGWls h termMOD MOD = ipresWls h MOD"
norm_of_complex: fixes c :: "complex" shows "norm (of_complex c) = cmod c"
scaleR2_rel_br: fixes a :: "'a \<Rightarrow> ('b \<times> 'c) set"   and I :: "'a \<Rightarrow> bool" shows "\<langle>br a I\<rangle>scaleR2_rel = br (\<lambda>((x, xa), y). scaleR2 x xa (a y)) (\<lambda>((l, u), y). I y \<and> ereal -` {l..u} \<noteq> {})"
sums_Re_zeta_of_nat_offset: fixes n :: "nat"   and r :: "nat" assumes "1 < n" shows "(\<lambda>k. 1 / real ((r + k + 1) ^ n)) sums (Re (zeta (of_nat n)) - (\<Sum>k = 1..r. 1 / real (k ^ n)))"
inf_bot_right: fixes inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and x :: "'c" assumes "class.bounded_lattice_bot inf less_eq less sup bot" shows "inf x bot = bot"
distinct_1: fixes x1 :: "nat" shows "Label x1 \<noteq> Entry"
simps: shows "reverse = foldl\<cdot>(flip\<cdot>Cons)\<cdot>[]"
tranclp_into_tranclp2: fixes r :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and a :: "'e"   and b :: "'e"   and c :: "'e" assumes "r a b"   and "r\<^sup>+\<^sup>+ b c" shows "r\<^sup>+\<^sup>+ a c"
RF_essential: fixes \<Gamma> :: "('k, 'l) web_scheme"   and S :: "'k set" shows "RF\<^bsub>\<Gamma>\<^esub> (\<E>\<^bsub>\<Gamma>\<^esub> S) = RF\<^bsub>\<Gamma>\<^esub> S"
PartialOrder_def: shows "PartialOrder = {P. refl_on (pset P) (order P) \<and> antisym (order P) \<and> trans (order P)}"
test_de_morgan: fixes x :: "'a"   and y :: "'a" shows "!(t x \<cdot> t y) = !x + !y"
Liminf_ereal_mult_right: fixes F :: "'b filter"   and c :: "real"   and f :: "'b \<Rightarrow> ereal" assumes "F \<noteq> bot"   and "0 \<le> c" shows "Liminf F (\<lambda>n. f n * ereal c) = Liminf F f * ereal c"
preserves_dom_def: fixes G :: "('a, 'b, 'a, 'b) functor" shows "preserves_dom G \<equiv> \<forall>f\<in>Ar. G \<^bsub>\<o>\<^esub> (Dom f) = Dom\<^bsub>BB\<^esub> G \<^bsub>\<a>\<^esub> f"
sorted_wrt_gen: fixes B :: "nat"   and n :: "nat" shows "sorted_wrt (<\<^sub>r\<^sub>l\<^sub>e\<^sub>x) (gen B n)"
changes_0: fixes xs :: "'b list" shows "changes ((0::'b) # xs) = changes xs"
False: shows "{a. en a q} \<noteq> {a. ren a q}"
im_sheaf_def: fixes S :: "'d set"   and is_open :: "'d set \<Rightarrow> bool"   and \<FF> :: "'d set \<Rightarrow> 'e set"   and \<rho> :: "'d set \<Rightarrow> 'd set \<Rightarrow> 'e \<Rightarrow> 'e"   and b :: "'e"   and add_str :: "'d set \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and mult_str :: "'d set \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero_str :: "'d set \<Rightarrow> 'e"   and one_str :: "'d set \<Rightarrow> 'e"   and S' :: "'f set"   and is_open' :: "'f set \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'f" shows "im_sheaf S is_open \<FF> \<rho> b add_str mult_str zero_str one_str S' is_open' f \<equiv> sheaf_of_rings S is_open \<FF> \<rho> b add_str mult_str zero_str one_str \<and> continuous_map S is_open S' is_open' f"
one_power2: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c" assumes "class.semiring_1 one times plus zero" shows "power.power one times one 2 = one"
f1: shows "\<exists>x. x \<in> (r |\<^bsub>rel\<^esub> s)"
env_delete_def: fixes x :: "'c"   and m :: "'c \<Rightarrow> 'd" shows "env_delete x m = m(x := \<bottom>)"
non_empty_cycle_free: fixes M :: "nat \<Rightarrow> nat \<Rightarrow> 'a DBMEntry" assumes "[M]\<^bsub>v,n\<^esub> \<noteq> {}" shows "cycle_free M n"
prob_space_K_lim: fixes x :: "real \<times> 'a" shows "prob_space (K.lim_stream x)"
d: shows "tensor_P (1\<^sub>m N) (mat_incr K) * Q2 * adjoint (tensor_P (1\<^sub>m N) (mat_incr K)) \<in> carrier_mat d d"
npeSum2_sub_mulx: fixes x :: "'a"   and y :: "'a"   and n :: "nat" assumes "x \<in> carrier R"   and "y \<in> carrier R" shows "x \<cdot>\<^sub>r \<Sigma>\<^sub>e R (\<lambda>i.  (n choose i) \<times>\<^bsub>R\<^esub> (x^\<^bsup>R (n - i)\<^esup> \<cdot>\<^sub>r y^\<^bsup>R i\<^esup>)) n = \<Sigma>\<^sub>e R (\<lambda>i.  (n choose Suc i) \<times>\<^bsub>R\<^esub> (x^\<^bsup>R (Suc n - Suc i)\<^esup> \<cdot>\<^sub>r y^\<^bsup>R Suc i\<^esup>)) n \<plusminus>  (Suc n choose 0) \<times>\<^bsub>R\<^esub> (x^\<^bsup>R (Suc n - 0)\<^esup> \<cdot>\<^sub>r y^\<^bsup>R 0\<^esup>)"
diff_add_eq: fixes plus :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and zero :: "'g"   and minus :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and uminus :: "'g \<Rightarrow> 'g"   and a :: "'g"   and b :: "'g"   and c :: "'g" assumes "class.ab_group_add plus zero minus uminus" shows "plus (minus a b) c = minus (plus a c) b"
distinct_19: fixes x4 :: "int"   and x5 :: "nat \<times> char list list" shows "Intg x4 \<noteq> Ref x5"
mult_minus_right: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c" assumes "class.ring plus zero minus uminus times" shows "times a (uminus b) = uminus (times a b)"
ub_sum: fixes f :: "'c \<Rightarrow> 'a"   and i :: "'c" shows "f i \<le> comm_monoid_add.sum (\<squnion>) bot f {k. True}"
MPAIR_imp_eqv_right: fixes X :: "freemsg"   and Y :: "freemsg"   and X' :: "freemsg"   and Y' :: "freemsg" assumes "MPAIR X Y \<sim> MPAIR X' Y'" shows "Y \<sim> Y'"
independent_and_consistent_def: fixes A :: "(('d, 'e) vec, 'f) vec"   and b :: "('d, 'f) vec" shows "independent_and_consistent A b = (consistent A b \<and> vec.dim (solution_set A 0) = 0)"
swapAbs_def: fixes xs :: "'varSort"   and x :: "'var"   and y :: "'var"   and A :: "('d, 'e, 'varSort, 'var, 'f) qAbs set" shows "A $[x \<and> y]_xs = asAbs (pick A $[[x \<and> y]]_xs)"
Aexp_edom': fixes e :: "exp"   and a :: "Arity" shows "edom (Aexp_syn' a e) \<subseteq> fv e"
in_trancl_closure_iff_in_trancl_fun: fixes a :: "'b"   and b :: "'b"   and TI :: "('b \<times> 'b) list" shows "((a, b) \<in> (set TI)\<^sup>+) = in_trancl TI a b"
discount_factor_borel_measurable: fixes r :: "real"   and n :: "nat"   and M :: "'c measure" shows "discount_factor r n \<in> borel_measurable M"
assoc_add_mat: fixes A :: "'c mat"   and nr :: "nat"   and nc :: "nat"   and B :: "'c mat"   and C :: "'c mat" assumes "A \<in> carrier_mat nr nc"   and "B \<in> carrier_mat nr nc"   and "C \<in> carrier_mat nr nc" shows "A + B + C = A + (B + C)"
assms: shows "X \<in> space M \<rightarrow> space N"
test_kat_2: fixes p :: "'a"   and q :: "'a"   and x :: "'a" assumes "test p"   and "test q" shows "(p \<cdot> x = p \<cdot> x \<cdot> q) = (p \<cdot> x \<le> x \<cdot> q)"
Collect_cong: fixes P :: "'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool" assumes "\<And>x. P x = Q x" shows "rel_ext P = rel_ext Q"
hseqI: fixes \<mu> :: "('a, 'a set) Maps.arr arrow_of_spans_data"   and a :: "('a, 'a set) Maps.arr arrow_of_spans_data"   and b :: "('a, 'a set) Maps.arr arrow_of_spans_data"   and \<nu> :: "('a, 'a set) Maps.arr arrow_of_spans_data"   and c :: "('a, 'a set) Maps.arr arrow_of_spans_data" assumes "Span.in_hhom \<mu> a b"   and "Span.in_hhom \<nu> b c" shows "Span.hseq \<nu> \<mu>"
thetaIfZOT_def: shows "thetaIfZOT \<equiv> {(if tst then c1 else c2, if tst then d1 else d2) |tst c1 c2 d1 d2. compatTst tst \<and> c1 \<approx>01T d1 \<and> c2 \<approx>01T d2}"
simps_2: fixes P :: "('a \<times> 'a) set"   and Pw :: "('a \<times> 'a) set"   and R :: "('a \<times> 'a) set"   and Rw :: "('a \<times> 'a) set" shows "SN_rel_ext_step P Pw R Rw top_ns = Pw"