hcju/isabelleps huggingface dataset:
trans_mult2_alt_local: fixes X :: "'a multiset"   and Y :: "'a multiset"   and Z :: "'a multiset"   and s :: "('a \<times> 'a) set"   and ns :: "('a \<times> 'a) set"   and b1 :: "bool"   and b2 :: "bool" assumes "\<And>x y z. \<lbrakk>x \<in># X; y \<in># Y; z \<in># Z; (x, y) \<in> s; (y, z) \<in> s\<rbrakk> \<Longrightarrow> (x, z) \<in> s"   and "\<And>x y z. \<lbrakk>x \<in># X; y \<in># Y; z \<in># Z; (x, y) \<in> ns; (y, z) \<in> s\<rbrakk> \<Longrightarrow> (x, z) \<in> s"   and "\<And>x y z. \<lbrakk>x \<in># X; y \<in># Y; z \<in># Z; (x, y) \<in> s; (y, z) \<in> ns\<rbrakk> \<Longrightarrow> (x, z) \<in> s"   and "\<And>x y z. \<lbrakk>x \<in># X; y \<in># Y; z \<in># Z; (x, y) \<in> ns; (y, z) \<in> ns\<rbrakk> \<Longrightarrow> (x, z) \<in> ns"   and "(X, Y) \<in> mult2_alt b1 ns s"   and "(Y, Z) \<in> mult2_alt b2 ns s" shows "(X, Z) \<in> mult2_alt (b1 \<and> b2) ns s"
\<delta>s: fixes i :: "'a" shows "(MIN i. SOME \<delta>. 0 < \<delta> \<and> (\<forall>x2. x2 \<noteq> x1 \<and> dist x2 x1 < \<delta> \<longrightarrow> dist (f i x2) (f i x1) < \<epsilon> / real CARD('a))) \<le> (SOME \<delta>. 0 < \<delta> \<and> (\<forall>x2. x2 \<noteq> x1 \<and> dist x2 x1 < \<delta> \<longrightarrow> dist (f i x2) (f i x1) < \<epsilon> / real CARD('a)))"
drop_upd_irrelevant: fixes m :: "nat"   and n :: "nat"   and l :: "'b list"   and x :: "'b" assumes "m < n" shows "drop n (l[m := x]) = drop n l"
inj_on_inv_image_inv_of_rat_Pow_rat: shows "inj_on ((`) (inv of_rat)) (Pow \<rat>)"
invariant_def: fixes f :: "'b \<Rightarrow> 'b"   and X :: "'b set"   and M :: "'b set" assumes "auto_ll_on_open f X" shows "auto_ll_on_open.invariant f X M = (\<forall>x\<in>M. auto_ll_on_open.trapped f X x M)"
\<sigma>_def: fixes i :: "nat" shows "\<sigma> i = (\<Sum>l\<leftarrow>[3..<Suc i]. (n (l - 2) + n (i - 1) + 1) * (n (l - 1) + n (i - 1) + 1))"
rev_path_distinct_classes: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and C :: "char list"   and Cs :: "char list list"   and wf_md :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list \<Rightarrow> char list \<Rightarrow> char list \<times> ty list \<times> ty \<times> char list list \<times> expr \<Rightarrow> bool" assumes "Subobjs P C (rev Cs)"   and "wf_prog wf_md P" shows "distinct Cs"
parcs_mk_symmetric: fixes G :: "('c, 'd) pre_digraph" shows "parcs (mk_symmetric G) = (\<Union>e\<in>arcs G. {(tail G e, head G e), (head G e, tail G e)})"
sign_def: fixes x :: "'a" shows "sign x \<equiv> if (0::'a) < x then 1 else if x = (0::'a) then 0 else - 1"
fls_mult_subdegree_ge: fixes f :: "'a fls"   and g :: "'a fls" assumes "f * g \<noteq> 0" shows "fls_subdegree f + fls_subdegree g \<le> fls_subdegree (f * g)"
Abs_nebset_inverse: fixes y :: "'d set" assumes "y \<in> {A. A \<noteq> {} \<and> |A| <o natLeq +c |UNIV|}" shows "set_nebset (Abs_nebset y) = y"
det_zero_row_2: fixes j :: "'b"   and F :: "(('a, 'b) vec, 'b) vec" assumes "row j F = 0" shows "det F = (0::'a)"
assms_2: shows " t \<longrightarrow>\<^sub>\<beta>\<^sup>* t2"
lowlink_eq_disc_iff_scc_root: fixes v :: "'v" assumes "v \<in> dom (finished s) \<or> stack s \<noteq> [] \<and> v = hd (stack s) \<and> pending s `` {v} = {}" shows "(the_lowlink s v = the_discovered s v) = scc_root s v (scc_of E v)"
rat_inv_eq: fixes a :: "int"   and b :: "int"   and c :: "int"   and d :: "int" assumes "Fract a b = Fract c d"   and "0 < b"   and "0 < d"   and "coprime a b"   and "coprime c d" shows "a = c \<and> b = d"
assms_1: shows "\<Psi> \<rhd> P \<approx> Q"
MonotProblem_def: fixes wtFsym :: "'a \<Rightarrow> bool"   and wtPsym :: "'b \<Rightarrow> bool"   and arOf :: "'a \<Rightarrow> 'c list"   and resOf :: "'a \<Rightarrow> 'c"   and parOf :: "'b \<Rightarrow> 'c list"   and \<Phi> :: "('a, 'b) lit list set" shows "MonotProblem wtFsym wtPsym arOf resOf parOf \<Phi> \<equiv> M.Problem wtFsym wtPsym arOf resOf parOf \<Phi> \<and> MonotProblem_axioms wtFsym wtPsym arOf resOf parOf \<Phi>"
fmap_ex_cof2: fixes f' :: "'a \<Rightarrow> 'b option"   and f :: "'a \<Rightarrow> 'b option"   and P :: "'c \<Rightarrow> 'c \<Rightarrow> 'b option \<Rightarrow> 'b option \<Rightarrow> 'a \<Rightarrow> bool" assumes "dom f' = dom f"   and "\<forall>l\<in>dom f. \<exists>L. finite L \<and> (\<forall>s p. s \<notin> L \<and> p \<notin> L \<and> s \<noteq> p \<longrightarrow> P s p (f l) (f' l) l)" shows "\<exists>L. finite L \<and> (\<forall>l\<in>dom f. \<forall>s p. s \<notin> L \<and> p \<notin> L \<and> s \<noteq> p \<longrightarrow> P s p (f l) (f' l) l)"
prime_dvd_multD: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and p :: "'a"   and a :: "'a"   and b :: "'a" assumes "class.normalization_semidom divide plus minus zero times one unit_factor normalize"   and "normalization_semidom.prime zero times one normalize p"   and "dvd.dvd times p (times a b)" shows "dvd.dvd times p a \<or> dvd.dvd times p b"
f3: shows "\<forall>x1. (0 < x1) = (\<not> x1 \<le> 0)"
restrict_nonempty_product_eq: fixes ks :: "'c list"   and k :: "'c"   and ls :: "'c list"   and l :: "'c"   and ms :: "'c list"   and m :: "'c"   and f :: "'c \<times> 'c \<Rightarrow> 'd"   and g :: "'c \<times> 'c \<Rightarrow> 'd"   and h :: "'c \<times> 'c \<Rightarrow> 'd" assumes "\<not> List.member ks k"   and "\<not> List.member ls l"   and "\<not> List.member ms m" shows "((k # ks) \<langle>f\<rangle> (l # ls) \<odot> (l # ls) \<langle>g\<rangle> (m # ms) = (k # ks) \<langle>h\<rangle> (m # ms)) = ([k] \<langle>f\<rangle> [l] \<odot> [l] \<langle>g\<rangle> [m] \<oplus> [k] \<langle>f\<rangle> ls \<odot> ls \<langle>g\<rangle> [m] = [k] \<langle>h\<rangle> [m] \<and> [k] \<langle>f\<rangle> [l] \<odot> [l] \<langle>g\<rangle> ms \<oplus> [k] \<langle>f\<rangle> ls \<odot> ls \<langle>g\<rangle> ms = [k] \<langle>h\<rangle> ms \<and> ks \<langle>f\<rangle> [l] \<odot> [l] \<langle>g\<rangle> [m] \<oplus> ks \<langle>f\<rangle> ls \<odot> ls \<langle>g\<rangle> [m] = ks \<langle>h\<rangle> [m] \<and> ks \<langle>f\<rangle> [l] \<odot> [l] \<langle>g\<rangle> ms \<oplus> ks \<langle>f\<rangle> ls \<odot> ls \<langle>g\<rangle> ms = ks \<langle>h\<rangle> ms)"
assms: shows "TE.wt (Fn (Tag \<sigma>) Tl)"
updates_2: fixes i :: "nat"   and j :: "nat" assumes "i < m"   and "j < i" shows "d\<mu> fs'' i j = (if i = i - 1 then d\<mu> fs i j else if i = i \<and> j \<noteq> i - 1 then d\<mu> fs (i - 1) j else if i < i \<and> j = i then (d fs (Suc i) * d\<mu> fs i (i - 1) - d\<mu> fs i (i - 1) * d\<mu> fs i j) div d fs i else if i < i \<and> j = i - 1 then (d\<mu> fs i (i - 1) * d\<mu> fs i j + d\<mu> fs i i * d fs (i - 1)) div d fs i else d\<mu> fs i j)"
homogeneousD_poly_deg: fixes p :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd"   and t :: "'c \<Rightarrow>\<^sub>0 nat" assumes "homogeneous p"   and "t \<in> keys p" shows "deg_pm t = poly_deg p"
conj_assoc: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(x \<^bold>\<sqinter> y) \<^bold>\<sqinter> z = x \<^bold>\<sqinter> y \<^bold>\<sqinter> z"
hull_mono: fixes s :: "'c set"   and t :: "'c set"   and S :: "'c set \<Rightarrow> bool" assumes "s \<subseteq> t" shows "S hull s \<subseteq> S hull t"
intro: fixes hom :: "'c \<Rightarrow> 'd" assumes "ring_isom hom" shows "comm_ring_isom hom"
is_reduced_GB_def: fixes B :: "('t \<Rightarrow>\<^sub>0 'd) set" shows "is_reduced_GB B \<equiv> is_Groebner_basis B \<and> is_auto_reduced B \<and> is_monic_set B \<and> 0 \<notin> B"
uw\<theta>: shows "ide \<chi>.chine \<and> \<guillemotleft>\<chi>.the_\<theta> : u\<^sub>0 \<star> \<chi>.chine \<Rightarrow> s\<^sub>0\<guillemotright>"
tm_inject_apps: fixes \<zeta> :: "('a, 'b) hd"   and ss :: "('a, 'b) tm list"   and \<xi> :: "('a, 'b) hd"   and ts :: "('a, 'b) tm list" shows "(apps (Hd \<zeta>) ss = apps (Hd \<xi>) ts) = (\<zeta> = \<xi> \<and> ss = ts)"
fps_divide_fps_X_minus1_sum_ring1: fixes a :: "'a fps" assumes "inverse (1::'a) = (1::'a)" shows "a / (1 - fps_X) = Abs_fps (\<lambda>n. sum (fps_nth a) {0..n})"
orthogonal_clauses_7: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and norm :: "'a \<Rightarrow> real"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and sgn :: "'a \<Rightarrow> 'a"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and inner :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and x :: "'a"   and a :: "'a"   and c :: "real" assumes "class.real_inner minus dist norm plus zero uminus scaleR sgn uniformity open inner"   and "real_inner.orthogonal inner x a" shows "real_inner.orthogonal inner (scaleR c x) a"
assms_1: shows "butlast (cs\<^bsup>\<pi>\<^esup> m) = butlast (cs\<^bsup>\<pi>\<^esup> n)"
vector_7_4: fixes x1 :: "'b"   and x2 :: "'b"   and x3 :: "'b"   and x4 :: "'b"   and x5 :: "'b"   and x6 :: "'b"   and x7 :: "'b" shows "vector [x1, x2, x3, x4, x5, x6, x7] $ 4 = x4"
reduce: fixes p :: "fm" shows "valid p = valid_in {1..card (props p)} p"
ht: fixes P :: "'s \<Rightarrow> real" assumes "sound P" shows "sound (t P)"
simps: fixes rv :: "rat vec" shows "vec_to_lpoly rv = list_to_lpoly (list_of_vec rv)"
is_applicable_then_subsumes: fixes s :: "nat \<Rightarrow> nat option"   and name :: "char list"   and pres :: "(nat \<times> nat) list"   and effs :: "((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list"   and layer :: "nat" assumes "s \<in> valid_states"   and "is_operator_applicable_in s (abs_ast_operator (name, pres, effs, layer))" shows "s \<in> subsuming_states (map_of pres)"
eq_iff: fixes x :: "'a multiset"   and y :: "'a multiset" shows "(x = y) = (x \<subseteq># y \<and> y \<subseteq># x)"
sums_unique2: fixes f :: "nat \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "f sums a"   and "f sums b" shows "a = b"
K'_map: fixes p :: "'a pfm"   and q :: "'a pfm"   and i :: "'a" assumes "\<turnstile>\<^sub>! (p \<^bold>\<longrightarrow>\<^sub>! q)" shows "\<turnstile>\<^sub>! (K\<^sub>! i p \<^bold>\<longrightarrow>\<^sub>! K\<^sub>! i q)"
mult_add_distrib_mat_vec: fixes A :: "'a mat"   and nr :: "nat"   and nc :: "nat"   and v\<^sub>1 :: "'a vec"   and v\<^sub>2 :: "'a vec" assumes "A \<in> carrier_mat nr nc"   and "v\<^sub>1 \<in> carrier_vec nc"   and "v\<^sub>2 \<in> carrier_vec nc" shows "A *\<^sub>v (v\<^sub>1 + v\<^sub>2) = A *\<^sub>v v\<^sub>1 + A *\<^sub>v v\<^sub>2"
abs_m2_less_half: shows "\<bar>real_of_int m2 * 2 powr (real_of_int e2 - real_of_int e1 + real k1 + 1)\<bar> < 1 / 2"
permutes_insert_lemma: fixes p :: "'b \<Rightarrow> 'b"   and a :: "'b"   and S :: "'b set" assumes "p permutes insert a S" shows "Fun.swap a (p a) id \<circ> p permutes S"
singular_chain_singular_subdivision: fixes p :: "nat"   and X :: "'c topology"   and c :: "((nat \<Rightarrow> real) \<Rightarrow> 'c) \<Rightarrow>\<^sub>0 int" assumes "singular_chain p X c" shows "singular_chain p X (singular_subdivision p c)"
monotone_def: fixes orda :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and ordb :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'd" shows "monotone orda ordb f = (\<forall>x y. orda x y \<longrightarrow> ordb (f x) (f y))"
zero_adds_pp: fixes pair_of_term :: "'a \<Rightarrow> 'b \<times> 'c"   and term_of_pair :: "'b \<times> 'c \<Rightarrow> 'a"   and v :: "'a" assumes "term_powerprod pair_of_term term_of_pair" shows "term_powerprod.adds_pp pair_of_term (0::'b) v"
R\<^sub>N_recurrence: fixes n :: "nat"   and u :: "nat"   and l :: "nat" assumes "0 < n"   and "p \<in> {0<..1}" shows "R\<^sub>N n u l = bernoulli_pmf p \<bind> (\<lambda>b. if b then map_pmf (\<lambda>n. n + l) (R\<^sub>N (n - 1) u l) else binomial_pmf (n - 1) (1 - p) \<bind> (\<lambda>m. map_pmf (\<lambda>n. n + u) (R\<^sub>N (m + 1) u l)))"
assms_4: shows "prv (imp \<phi> (imp \<chi> \<psi>))"
assms_4: shows "eval_fds f s \<noteq> (0::'a)"
1: shows "is_subobj P (C, Cs @ [D, D'] @ rev Cs')"
rem_implicit_pres_ops_numVars: fixes prob :: "(char list \<times> nat option \<times> char list list) list \<times> nat list \<times> (nat \<times> nat) list \<times> (char list \<times> (nat \<times> nat) list \<times> ((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list \<times> nat) list" shows "ast_problem.numVars (rem_implicit_pres_ops prob) = ast_problem.numVars prob"
simple_distributedI: fixes X :: "'key \<Rightarrow> 'a"   and P :: "'a \<Rightarrow> real" assumes "simple_function (point_measure keys (\<lambda>x. ennreal (K x))) X"   and "\<And>x. 0 \<le> P x"   and "\<And>x. x \<in> X ` space (point_measure keys (\<lambda>x. ennreal (K x))) \<Longrightarrow> P x = K.prob (X -` {x} \<inter> space (point_measure keys (\<lambda>x. ennreal (K x))))" shows "simple_distributed (point_measure keys (\<lambda>x. ennreal (K x))) X P"
intros_2: fixes a :: "'a"   and \<alpha> :: "'b"   and b :: "'a"   and ars :: "('a \<times> 'b \<times> 'a) set"   and ss :: "(bool \<times> 'b \<times> 'a) list" assumes "(a, \<alpha>, b) \<in> ars"   and "(b, ss) \<in> conv ars" shows "(a, (True, \<alpha>, b) # ss) \<in> conv ars"
subst2: fixes x :: "name"   and M :: "'a"   and xvec :: "name list"   and Tvec :: "'a list" assumes "x \<sharp> M"   and "x \<sharp> xvec"   and "x \<sharp> Tvec" shows "x \<sharp> substTerm M xvec Tvec"
lm048: fixes b :: "'e \<times> 'c set \<Rightarrow> 'd"   and i :: "'e"   and a :: "('e \<times> 'c set) set"   and X :: "'e set" assumes "bidMonotonicity b i"   and "runiq a" shows "sum b ({i} \<times> Outside' X a `` {i}) \<le> sum b ({i} \<times> {\<Union> (a `` (X \<union> {i}))})"
ord_pres_mono: fixes f :: "'b \<Rightarrow> 'c" shows "mono f = ord_pres f"
from_list_correct: fixes empty :: "'l"   and get :: "'l \<Rightarrow> 'm \<Rightarrow> 'n option"   and add :: "'l \<Rightarrow> 'm \<Rightarrow> 'n \<Rightarrow> 'l"   and to_list :: "'l \<Rightarrow> ('m \<times> 'n) list"   and xs :: "('m \<times> 'n) list"   and k :: "'m" assumes "env empty get add to_list" shows "map_of xs k = get (env.from_list empty add xs) k"
AR_def: fixes S :: "'a set" shows "AR S \<equiv> \<forall>U S'. S homeomorphic S' \<and> closedin (top_of_set U) S' \<longrightarrow> S' retract_of U"
close_closed: fixes C :: "'b fm set set" shows "subset_closed (close C)"
S_Min_Le: shows "S_Min_Le = {u c' - d |c' d. 0 < v c' \<and> v c' \<le> n \<and> c \<noteq> c' \<and> M (v c') (v c) = Le d} \<union> {- d |d. M 0 (v c) = Le d}"
set_insort: fixes x :: "'c"   and xs :: "'c list" shows "set (insort x xs) = {x} \<union> set xs"
surjective: fixes r :: "('a, 'b) strips_problem_scheme" shows "r = \<lparr>variables_of = r\<^sub>\<V>, operators_of = r\<^sub>\<O>, initial_of = r\<^sub>I, goal_of = r\<^sub>G, \<dots> = strips_problem.more r\<rparr>"
load_tmps_le: fixes i :: "nat"   and s :: "stmt"   and t :: "nat"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and sb :: "(stmt \<times> nat) memref list"   and \<D> :: "bool"   and \<O> :: "nat set \<times> (nat \<Rightarrow> bool option)" assumes "i < length ts"   and "ts ! i = ((s, t), is, \<theta>, sb, \<D>, \<O>)" shows "\<forall>i\<in>load_tmps is. i < t"
splus_assoc: fixes pair_of_term :: "'a \<Rightarrow> 'b \<times> 'c"   and term_of_pair :: "'b \<times> 'c \<Rightarrow> 'a"   and s :: "'b"   and t :: "'b"   and v :: "'a" assumes "term_powerprod pair_of_term term_of_pair" shows "term_powerprod.splus pair_of_term term_of_pair (s + t) v = term_powerprod.splus pair_of_term term_of_pair s (term_powerprod.splus pair_of_term term_of_pair t v)"
Re_winding_number_half_upper: fixes \<gamma> :: "real \<Rightarrow> complex"   and z :: "complex" assumes "\<forall>p\<in>path_image \<gamma>. Im z \<le> Im p"   and "valid_path \<gamma>"   and "z \<notin> path_image \<gamma>" shows "Re (winding_number \<gamma> z) = (Im (Ln (\<i> * z - \<i> * pathfinish \<gamma>)) - Im (Ln (\<i> * z - \<i> * pathstart \<gamma>))) / (2 * pi)"
Int_UNIV: fixes A :: "'c set"   and B :: "'c set" shows "(A \<inter> B = UNIV) = (A = UNIV \<and> B = UNIV)"
freq_swapLeaves: fixes t :: "'a tree"   and a :: "'a"   and b :: "'a"   and w\<^sub>a :: "nat"   and w\<^sub>b :: "nat" assumes "consistent t"   and "a \<noteq> b" shows "freq (swapLeaves t w\<^sub>a a w\<^sub>b b) = (\<lambda>c. if c = a then if b \<in> alphabet t then w\<^sub>a else 0 else if c = b then if a \<in> alphabet t then w\<^sub>b else 0 else freq t c)"
vec_of_poly_n_def: fixes p :: "'b poly"   and n :: "nat" shows "vec_of_poly_n p n = vec n (\<lambda>i. if i < n - degree p - 1 then 0::'b else coeff p (n - i - 1))"
max2_equals2: fixes a :: "'a Bonk_Schramm_extension_unfolded"   and b :: "'a Bonk_Schramm_extension_unfolded" assumes "a \<in> Field Bonk_Schramm_extension_unfolded_wo"   and "b \<in> Field Bonk_Schramm_extension_unfolded_wo" shows "(wo.max2 a b = b) = ((a, b) \<in> Bonk_Schramm_extension_unfolded_wo)"
mono_Seq: fixes P :: "('c, 'd) action" shows "mono ((`;`) P)"
Proc_CFG_IEdge_intra_kind: fixes prog :: "cmd"   and n :: "label"   and et :: "(char list, val, char list \<times> label, char list) edge_kind"   and n' :: "label" assumes "prog \<turnstile> n -IEdge et\<rightarrow>\<^sub>p n'" shows "intra_kind et"
affine_hull_eq_empty: fixes S :: "'b set" shows "(affine hull S = {}) = (S = {})"
monoE: fixes g :: "'s"   and f :: "'s"   and f' :: "'s" assumes "local.mono g"   and "seq g f"   and "seq g f'"   and "g \<cdot> f = g \<cdot> f'" shows "f' = f"
foldl_LTLOr_prop_entailment: fixes S :: "'b ltl set"   and i :: "'b ltl"   and xs :: "'b ltl list" shows "S \<Turnstile>\<^sub>P foldl LTLOr i xs = (S \<Turnstile>\<^sub>P i \<or> (\<exists>y\<in>set xs. S \<Turnstile>\<^sub>P y))"
P4: shows "Per A' B' C' \<and> B C A LtA A' B' C'"
polymul_0_2: fixes p :: "poly"   and n0 :: "nat" assumes "isnpolyh p n0" shows "0\<^sub>p *\<^sub>p p = 0\<^sub>p"
weak_group_morphism_is_hom: fixes f :: "'a \<Rightarrow> 'd"   and H :: "'a set" assumes "weak_group_morphism f H G" shows "f \<in> hom G (image_group f G)"
C_AA: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "fixed_point_proof A f" shows "fixed_point_proof.C A less_eq f \<in> fixed_point_proof.AA A less_eq f"
empty_selected_part_def: fixes atom_ordering :: "('a \<times> 'a) set"   and Sel :: "'b set"   and S :: "'b Literal set set" assumes "propositional_atoms atom_ordering" shows "empty_selected_part Sel S = {C \<in> S. selected_part Sel C = {}}"
1: shows "S.arr (\<mu> \<star> \<nu>)"
valofn_ge_2p: shows "2 ^ IEEE.exponent x / 2 ^ bias TYPE(('a, 'b) IEEE.float) \<le> valofn x"
weakTransitionE: fixes \<Psi> :: "'b"   and Q :: "('a, 'b, 'c) psi"   and P :: "('a, 'b, 'c) psi"   and \<alpha> :: "'a action"   and P' :: "('a, 'b, 'c) psi" assumes "\<Psi> : Q \<rhd> P \<Longrightarrow>\<alpha> \<prec> P'" obtains P'' :: "('a, 'b, 'c) psi" where "\<Psi> \<rhd> P \<Longrightarrow>\<^sup>^\<^sub>\<tau> P''" and "insertAssertion (extractFrame Q) \<Psi> \<hookrightarrow>\<^sub>F insertAssertion (extractFrame P'') \<Psi>" and "\<Psi> \<rhd> P'' \<longmapsto> \<alpha> \<prec> P'"
set_intros_1: fixes x1 :: "'d"   and x2 :: "'d stream" shows "x1 \<in> sset (x1 ## x2)"
assms_2: shows "integrable (restr_to_subalg M F) f"
leg1_in_hom_2: shows "\<guillemotleft>r\<^sub>1 : r\<^sub>1 \<Rightarrow> r\<^sub>1\<guillemotright>"
right_closed: fixes a :: "'a"   and b :: "'a" assumes "(a, b) \<in> Ring_Congruence" shows "b \<in> R"
nn_integral_pair_pmf': fixes A :: "'b pmf"   and B :: "'c pmf"   and f :: "'b \<times> 'c \<Rightarrow> ennreal" shows "integral\<^sup>N (measure_pmf (pair_pmf A B)) f = \<integral>\<^sup>+ a. \<integral>\<^sup>+ b. f (a, b) \<partial>measure_pmf B \<partial>measure_pmf A"
simps_2: fixes t :: "('a, 'b) Term.term"   and f :: "'a"   and T :: "('a, 'b) Term.term list" shows "subterm_of t (Fun f T) = (t = Fun f T \<or> list_ex (subterm_of t) T)"
oalist_inv_list_of_oalist_ntm: fixes xs :: "('a, 'b) oalist_ntm" shows "ko_ntm.oalist_inv (list_of_oalist_ntm xs)"
power: fixes f :: "'a \<Rightarrow> real"   and n :: "nat" assumes "f \<in> R" shows "(\<lambda>x. f x ^ n) \<in> R"
IH: fixes xs :: "'a list" shows "bst_of_list (xs @ ys) = fold Tree_Set.insert ys (bst_of_list xs)"
asymptotics3: fixes x :: "real" assumes "x\<^sub>0 \<le> x" shows "(1 + ln x powr (- e / 2)) / 2 \<le> 1"
ex: shows "\<exists>n. A $ n $ j \<noteq> (0::'a) \<and> i \<le> n"
continuous_def: fixes F :: "'a filter"   and f :: "'a \<Rightarrow> 'c" shows "continuous F f = (f \<longlongrightarrow> f (Lim F (\<lambda>x. x))) F"
assigned_nodes_def: shows "assigned_nodes \<equiv> l_assigned_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_assigned_nodes Shadow_DOM.get_child_nodes get_tag_name Shadow_DOM.get_root_node get_host find_slot"
sublist_def: fixes xs :: "'b list"   and ys :: "'b list" shows "xs \<le>f ys = (\<exists>ps ss. ys = ps \<cdot> xs \<cdot> ss)"
sys_block_sizes_def: shows "sys1.sys_block_sizes \<equiv> {int (card bl) |bl. bl \<in># \<B>}"
add_diff_inverse: fixes a :: "'b"   and b :: "'b" assumes "\<not> a < b" shows "b + (a - b) = a"
simps_1: fixes uu :: "'b match_expr \<Rightarrow> 'b match_expr option" shows "optimize_matches_option uu [] = []"
removelE: fixes s :: "'a list \<times> 'a list" assumes "fifo.invar s"   and "fifo.\<alpha> s \<noteq> []" obtains s' :: "'a list \<times> 'a list" where "fifo.removel s = (hd (fifo.\<alpha> s), s')" and "fifo.invar s'" and "fifo.\<alpha> s' = tl (fifo.\<alpha> s)"
bag_bound_V_empty: fixes t :: "'c" assumes "V = {}"   and "t \<in> V\<^bsub>T\<^esub>" shows "card (bag t) = 0"
assms_4: fixes x :: "Vars.var"   and c :: "'a"   and e :: "exp"   and exp :: "exp" assumes "{atom x} \<sharp>* c"   and "x \<notin> fv e"   and "\<And>c. P c e"   and "\<And>c. P c exp" shows "P c (let x be e in exp )"
strict: fixes z :: "real" assumes "0 < z"   and "z < 1" shows "gf_U x x z < 1"
fst_diag_id: fixes z :: "'a" shows "(fst \<circ> (\<lambda>x. (x, x))) z = id z"
f_3: shows "inj_on f {x. \<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x \<bullet> i = 0}"
assms_1: shows "s \<rightarrow>\<^bsub>T\<^esub> t"
simps: fixes S :: "'b trm set"   and \<sigma> :: "('b \<times> 'b trm) list" shows "subst_set S \<sigma> = {t \<lhd> \<sigma> |t. t \<in> S}"
simps: fixes less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a1 :: "'a list"   and a2 :: "'a list" shows "ord.lexordp_eq less a1 a2 = ((\<exists>ys. a1 = [] \<and> a2 = ys) \<or> (\<exists>x y xs ys. a1 = x # xs \<and> a2 = y # ys \<and> less x y) \<or> (\<exists>x y xs ys. a1 = x # xs \<and> a2 = y # ys \<and> \<not> less x y \<and> \<not> less y x \<and> ord.lexordp_eq less xs ys))"
is_Groebner_basis_def: fixes F :: "('a \<Rightarrow>\<^sub>0 'c) set" shows "punit.is_Groebner_basis F \<equiv> relation.is_ChurchRosser (punit.red F)"
diff_less_eq: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'d"   and b :: "'d"   and c :: "'d" assumes "class.ordered_ab_group_add plus zero minus uminus less_eq less" shows "less (minus a b) c = less a (plus c b)"
zero_middle: fixes p :: "nat"   and k :: "nat"   and g :: "nat \<Rightarrow> 'c"   and h :: "nat \<Rightarrow> 'c" assumes "1 \<le> p"   and "k \<le> p" shows "(\<Sum>j\<le>p. if j < k then g j else if j = k then 0::'c else h (j - Suc 0)) = (\<Sum>j\<le>p - Suc 0. if j < k then g j else h j)"