hcju/isabelleps huggingface dataset:
succ_chain_tableau_equiv': fixes l :: "('i, 'a) state list"   and i :: "nat"   and j :: "nat"   and v :: "nat \<Rightarrow> 'a" assumes "rel_chain l {(s, s'). \<triangle> (\<T> s) \<and> \<diamond> s \<and> \<Turnstile>\<^sub>n\<^sub>o\<^sub>l\<^sub>h\<^sub>s s \<and> \<nabla> s \<and> s \<succ>\<^sub>x s' \<and> \<B>\<^sub>i s' = \<B>\<^sub>i s \<and> \<U>\<^sub>c s' = \<U>\<^sub>c s}"   and "i \<le> j"   and "j < length l" shows "v \<Turnstile>\<^sub>t \<T> (l ! i) = v \<Turnstile>\<^sub>t \<T> (l ! j)"
bit_unset_bit_iff: fixes m :: "nat"   and a :: "'a"   and n :: "nat" shows "bit (unset_bit m a) n = (bit a n \<and> m \<noteq> n)"
stocks_of: fixes S :: "'c set"   and A :: "'c \<Rightarrow> nat \<Rightarrow> 'd \<Rightarrow> real" assumes "UNIV \<noteq> S" shows "stocks (discrete_market_of S A) = S"
bet__suma: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and P :: "'p" assumes "A \<noteq> B"   and "B \<noteq> C"   and "P \<noteq> B"   and "Bet A B C" shows "A B P P B C SumA A B C"
finsert_absorb: fixes a :: "'a"   and A :: "'a fset" assumes "a |\<in>| A" shows "finsert a A = A"
geq: shows "\<forall>v. dim_vec v = dim_col A \<longrightarrow> 0 \<le> inner_prod v (A *\<^sub>v v)"
simps_3: fixes A :: "('e, 'f) ioa"   and s :: "'e" shows "is_exec_frag_of A (s, []) = True"
acceptI: fixes r :: "nat \<Rightarrow> 'a node"   and w :: "nat \<Rightarrow> 'a set" assumes "gba.is_acc_run r"   and "\<And>i. local.gba.L (r i) (w i)" shows "local.gba.accept w"
v0_no_deadend: fixes G :: "('c, 'd) ParityGame_scheme"   and P :: "'c llist"   and v0 :: "'c"   and p :: "Player"   and \<sigma> :: "'c \<Rightarrow> 'c" assumes "vmc_path_no_deadend G P v0 p \<sigma>" shows "\<not> \<not> (\<exists>w\<in>V\<^bsub>G\<^esub>. v0 \<rightarrow>\<^bsub>G\<^esub> w)"
CFG_axioms: shows "CFG src trg (lift_valid_edge valid_edge sourcenode targetnode kind Entry Exit) NewEntry"
proots_count_of_root_set: fixes P :: "'b poly"   and R :: "'b set"   and S :: "'b set" assumes "P \<noteq> 0"   and "R \<subseteq> S"   and "\<And>x. x \<in> R \<Longrightarrow> poly P x = (0::'b)" shows "card R \<le> proots_count P S"
add: fixes b1 :: "('b, 'n) vec"   and b2 :: "('b, 'n) vec" shows "f (b1 + b2) = f b1 + f b2"
generate_subgroup_incl: fixes G :: "('b, 'c) monoid_scheme"   and K :: "'b set"   and H :: "'b set" assumes "Group.group G"   and "K \<subseteq> H"   and "subgroup H G" shows "generate G K \<subseteq> H"
zdiv_zminus1_eq_if: fixes b :: "int"   and a :: "int" assumes "b \<noteq> 0" shows "- a div b = (if a mod b = 0 then - (a div b) else - (a div b) - 1)"
NotActualImplD: fixes v :: "i"   and \<phi> :: "\<o>"   and \<psi> :: "\<o>" assumes "\<not> [\<^bold>\<A>(\<phi> \<^bold>\<rightarrow> \<psi>) in v]" shows "\<not> [\<^bold>\<A>\<phi> \<^bold>\<rightarrow> \<^bold>\<A>\<psi> in v]"
walk_first_edge': fixes G :: "('e, 'f) Graph_scheme"   and v :: "'e"   and xs :: "'e list" assumes "Graph G"   and "Graph.walk G (v # xs)"   and "xs \<noteq> []" shows "v \<rightarrow>\<^bsub>G\<^esub> hd xs"
wfactors_dividesI: fixes fs :: "'a list"   and a :: "'a"   and f :: "'a" assumes "wfactors (Ring_Divisibility.mult_of R) fs a"   and "set fs \<subseteq> carrier (Ring_Divisibility.mult_of R)"   and "a \<in> carrier (Ring_Divisibility.mult_of R)"   and "f \<in> set fs" shows "f divides\<^bsub>Ring_Divisibility.mult_of R\<^esub> a"
tl_Nil: fixes xs :: "'d list" shows "(tl xs = []) = (xs = [] \<or> (\<exists>x. xs = [x]))"
zero_mult_inf: shows "0\<^sub>h *\<^sub>h \<infinity>\<^sub>h = 1\<^sub>h"
TryRedsFail: fixes e\<^sub>1 :: "char list exp"   and s :: "heap \<times> sfields \<times> sheap"   and b :: "bool"   and a :: "nat"   and h :: "heap"   and l :: "sfields"   and sh :: "sheap"   and b' :: "bool"   and P :: "(char list list \<times> char list exp) prog"   and D :: "char list"   and fs :: "Objects.fields"   and C :: "char list"   and V :: "char list"   and e\<^sub>2 :: "char list exp" assumes "P \<turnstile> \<langle>e\<^sub>1,s,b\<rangle> \<rightarrow>* \<langle>Throw a,(h, l, sh),b'\<rangle>"   and "h a = \<lfloor>(D, fs)\<rfloor>"   and "(D, C) \<notin> (subcls1 P)\<^sup>*" shows "P \<turnstile> \<langle>try e\<^sub>1 catch(C V) e\<^sub>2,s,b\<rangle> \<rightarrow>* \<langle>Throw a,(h, l, sh),b'\<rangle>"
poly_eq_iff: fixes p :: "'a poly"   and q :: "'a poly" shows "(p = q) = (\<forall>n. poly.coeff p n = poly.coeff q n)"
a1: shows "atms_of E \<subseteq> \<Union> (atms_of ` set CAs) \<union> atms_of DA"
case_eq_if: fixes f :: "'b extended \<Rightarrow> 'b \<Rightarrow> 'b extended \<Rightarrow> 'c \<Rightarrow> 'a"   and insert_message :: "('b, 'c) insert_message" shows "(case insert_message of InsertMessage x xa xb xc \<Rightarrow> f x xa xb xc) = f (P insert_message) (I insert_message) (S insert_message) (\<Sigma> insert_message)"
bit_mask_iff: fixes m :: "nat"   and n :: "nat" shows "bit (mask m) n = (n < min LENGTH('b) m)"
vector_space_ow_axioms: shows "vector_space_ow U\<^sub>V\<^sub>S (+) (0::'b) (-) uminus (*s)"
connected_component_disjoint: fixes S :: "'b set"   and a :: "'b"   and b :: "'b" shows "(connected_component_set S a \<inter> connected_component_set S b = {}) = (a \<notin> connected_component_set S b)"
sup_lub: fixes A :: "'a set" assumes "A \<subseteq> carrier L" shows "is_lub L (\<Squnion>A) A"
normalize_lead_coeff: fixes p :: "'a list" assumes "length (local.normalize p) < length p" shows "lead_coeff p = \<zero>"
cases: fixes x :: "ereal \<times> ereal" obtains r :: "real" and p :: "real" where "x = (ereal r__, ereal p__)"   | a :: "ereal" where "x = (\<infinity>, a__)"   | a :: "ereal" where "x = (a, \<infinity>)"   | r :: "real" where "x = (ereal r, - \<infinity>)"   | p :: "real" where "x = (- \<infinity>, ereal p)"   | "x = (- \<infinity>, - \<infinity>)"
right_unique_def: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool" shows "right_unique R = (\<forall>x y z. R x y \<longrightarrow> R x z \<longrightarrow> y = z)"
disjointPartitioning_helper: fixes As :: "'b set set"   and ss :: "'b set list" assumes "disjoint As" shows "disjoint (partitioning ss As)"
chine_composite_2: shows "H\<nu>H\<pi>\<rho>.chine = \<nu>.leg0 \<down>\<down> \<pi>.leg1 \<cdot> \<pi>\<rho>.prj\<^sub>1"
poly_altdef: fixes p :: "'a poly"   and x :: "'a" shows "poly p x = (\<Sum>i\<le>Polynomial.degree p. poly.coeff p i * x ^ i)"
orderE: fixes b :: "'c multiset"   and a :: "'c multiset" assumes "b \<subseteq># a" obtains "a = a \<union># b"
mplus_llist_simps_2: fixes ys :: "'a\<cdot>llist" shows "mplus\<cdot>LNil\<cdot>ys = ys"
is_class_def: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'a) list) list"   and C :: "char list" shows "is_class P C \<equiv> class P C \<noteq> None"
conj_one_left: fixes x :: "'a" shows "- (1::'a) AND x = x"
mset_set_eq_iff: fixes A :: "'c set"   and B :: "'c set" assumes "finite A"   and "finite B" shows "(mset_set A = mset_set B) = (A = B)"
surjective_pairing: fixes t :: "'a \<times> 'b" shows "t = (fst t, snd t)"
ID_def: fixes t :: "'a"   and t' :: "'a"   and T :: "'b itself" shows "ID t t' T \<equiv> t = t'"
footprint_ge: fixes a :: "recf"   and p :: "abc_inst list"   and arity :: "nat"   and fp :: "nat" assumes "rec_ci a = (p, arity, fp)" shows "arity < fp"
RPDs_Test_2: fixes \<phi> :: "nat" shows "RPDs (MTestPos \<phi>) \<subseteq> {MTestPos \<phi>}"
cont: fixes i :: "nat" shows "continuous_on (topspace (Euclidean_space m) \<inter> S) (\<lambda>x. f x i)"
homotopic_paths_loop_parts: fixes S :: "'a set"   and p :: "real \<Rightarrow> 'a"   and q :: "real \<Rightarrow> 'a"   and a :: "'a" assumes "homotopic_loops S (p +++ reversepath q) (linepath a a)"   and "path q" shows "homotopic_paths S p q"
compile_pr_correct': fixes x :: "nat"   and g :: "recf"   and xs :: "nat list"   and n :: "nat"   and f :: "recf"   and gft :: "nat"   and gar :: "nat"   and gap :: "abc_inst list"   and fft :: "nat"   and far :: "nat"   and fap :: "abc_inst list"   and anything :: "nat list" assumes "\<forall>y<x. terminate g (xs @ [y, rec_exec (Pr n f g) (xs @ [y])])"   and "\<forall>y<x. \<forall>anything. {\<lambda>nl. nl = xs @ y # rec_exec (Pr n f g) (xs @ [y]) # 0 \<up> (gft - gar) @ anything} gap {\<lambda>nl. nl = xs @ y # rec_exec (Pr n f g) (xs @ [y]) # rec_exec g (xs @ [y, rec_exec (Pr n f g) (xs @ [y])]) # 0 \<up> (gft - Suc gar) @ anything}"   and "terminate f xs"   and "\<And>anything. {\<lambda>nl. nl = xs @ 0 \<up> (fft - far) @ anything} fap {\<lambda>nl. nl = xs @ rec_exec f xs # 0 \<up> (fft - Suc far) @ anything}"   and "length xs = n"   and "rec_ci f = (fap, far, fft)"   and "rec_ci g = (gap, gar, gft)" shows "{\<lambda>nl. nl = xs @ x # 0 \<up> (max (n + 3) (max fft gft) - n) @ anything} mv_box n (max (n + 3) (max fft gft)) [+] (fap [+] (mv_box n (Suc n) [+] ([Dec (max (n + 3) (max fft gft)) (length gap + 7)] [+] (gap [+] [Inc n, Dec (Suc n) 3, Goto (Suc 0)]) @ [Dec (Suc (Suc n)) 0, Inc (Suc n), Goto (length gap + 4)]))) {\<lambda>nl. nl = xs @ x # rec_exec (Pr n f g) (xs @ [x]) # 0 \<up> (max (n + 3) (max fft gft) - Suc n) @ anything}"
valid_stmtI: fixes A :: "'b triple set"   and G :: "prog"   and c :: "stmt"   and P :: "'b assn"   and Q :: "'b assn" assumes "\<And>n s0 L accC C s1 Y Z. \<lbrakk>\<forall>t\<in>A. G\<Turnstile>n\<Colon>t; s0\<Colon>\<preceq>(G, L); normal s0 \<Longrightarrow> \<lparr>prg = G, cls = accC, lcl = L\<rparr>\<turnstile>c\<Colon>\<surd>; normal s0 \<Longrightarrow> \<lparr>prg = G, cls = accC, lcl = L\<rparr>\<turnstile> dom (locals (snd s0)) \<guillemotright>In1r c\<guillemotright> C; G\<turnstile>s0 \<midarrow>c\<midarrow>n\<rightarrow> s1; P Y s0 Z\<rbrakk> \<Longrightarrow> Q \<diamondsuit> s1 Z \<and> s1\<Colon>\<preceq>(G, L)" shows "G,A|\<Turnstile>\<Colon>{{P} .c. {Q}}"
adjacent_half_chamber_system_image_gf: fixes C :: "'a set"   and D :: "'a set" assumes "C \<in> g \<turnstile> folding_g.\<C>"   and "D \<in> f \<turnstile> folding_g.\<C>"   and "C \<sim> D" shows "g ` D = C"
Sup_V_def: fixes X :: "V set" shows "\<Squnion> X \<equiv> if small X \<Rightarrow> set (\<Union> (elts ` X)) | otherwise \<Rightarrow> 0"
split_asm: fixes P :: "'a \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'b list \<Rightarrow> 'a"   and ta_rule :: "('b, 'c) ta_rule" shows "P (case ta_rule of x \<rightarrow> xa xb \<Rightarrow> f x xa xb) = (\<nexists>x1 x2 x3. ta_rule = x1 \<rightarrow> x2 x3 \<and> \<not> P (f x1 x2 x3))"
s3_noaction_hlp: fixes ac :: "32 simple_match"   and p :: "(32, 'b) simple_packet_ext_scheme"   and x1 :: "16 word"   and ad :: "char list"   and ba :: "simple_action"   and ifs :: "char list list" assumes "simple_match_valid ac"   and "\<not> simple_matches ac p"   and "match_iface (oiface ac) (p_oiface p)" shows "OF_match_linear OF_match_fields_safe (map (\<lambda>x. split3 OFEntry (x1, x, case ba of Accept \<Rightarrow> [Forward ad] | Drop \<Rightarrow> [])) (simple_match_to_of_match ac ifs)) p = NoAction"
order_trans: fixes less_eq :: "'h \<Rightarrow> 'h \<Rightarrow> bool"   and less :: "'h \<Rightarrow> 'h \<Rightarrow> bool"   and x :: "'h"   and y :: "'h"   and z :: "'h" assumes "class.preorder less_eq less"   and "less_eq x y"   and "less_eq y z" shows "less_eq x z"
gcd_proj1_if_dvd: 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 "dvd.dvd times a b" shows "gcd a b = normalize a"
If_eq: fixes y :: "'a"   and A :: "'a set"   and g :: "'a \<Rightarrow> 'c \<Rightarrow> 'b"   and h1 :: "'a \<Rightarrow> 'c"   and h2 :: "'a \<Rightarrow> 'c" assumes "y \<in> A"   and "finite A" shows "(\<Prod>x\<in>A. g x (if x = y then h1 x else h2 x)) = g y (h1 y) * (\<Prod>x\<in>A - {y}. g x (h2 x))"
mems_agree: shows "list_all2 mem_agree (s.mem s) (s_mem \<S>)"
in_homE: fixes f :: "'a"   and a :: "'a"   and b :: "'a" assumes "Right_a.in_hom f a b" obtains "Right_a.arr f" and "Right_a.dom f = a" and "Right_a.cod f = b"
bigoI_tendsto: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c"   and c :: "'c"   and F :: "'b filter" assumes "((\<lambda>x. f x / g x) \<longlongrightarrow> c) F"   and "\<forall>\<^sub>F x in F. g x \<noteq> (0::'c)" shows "f \<in> O[F](g)"
MGU_is_Unifier: fixes t :: "('a, 'b) Term.term"   and \<sigma> :: "'b \<Rightarrow> ('a, 'b) Term.term"   and u :: "('a, 'b) Term.term" assumes "MGU \<sigma> t u" shows "Unifier \<sigma> t u"
monoid_mult_hom_def: fixes hom :: "'c \<Rightarrow> 'd" shows "monoid_mult_hom hom \<equiv> one_hom hom \<and> semigroup_mult_hom hom"
iso_bij_betwn_block_sets: shows "bij_betw ((`) \<pi>) (set_mset \<B>) (set_mset \<B>')"
degree_one: fixes x :: "nat" shows "MPoly_Type.degree (Var x) x = 1"
simps_3: fixes hs1 :: "'b tree"   and uy :: "'b"   and hs2 :: "'b tree"   and uz :: "'b"   and v :: "'b tree"   and va :: "'b"   and vb :: "'b tree" shows "ub_pass\<^sub>1 \<langle>hs1, uy, \<langle>hs2, uz, \<langle>v, va, vb\<rangle>\<rangle>\<rangle> = 2 * log 2 (real (size hs1 + size hs2 + size \<langle>v, va, vb\<rangle> + 2)) - 2 * log 2 (real (size \<langle>v, va, vb\<rangle>)) - 2 + ub_pass\<^sub>1 \<langle>v, va, vb\<rangle>"
smult_neg: fixes R :: "'a set"   and smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and M :: "'b set"   and r :: "'a"   and m :: "'b" assumes "RModule R smult M"   and "r \<in> R"   and "m \<in> M" shows "smult r (- m) = - smult r m"
npowf_exp_add: fixes x :: "'a"   and i :: "int"   and j :: "int" assumes "x \<in> carrier K"   and "x \<noteq> \<zero>" shows "x\<^bsub>K\<^esub>\<^bsup>(i + j)\<^esup> = x\<^bsub>K\<^esub>\<^bsup>i\<^esup> \<cdot>\<^sub>r x\<^bsub>K\<^esub>\<^bsup>j\<^esup>"
ket_plus_eval: fixes i :: "nat" assumes "i < 2" shows "ket_plus $ i = 1 / csqrt 2"
Cons_prefix_Cons: fixes x :: "'b"   and xs :: "'b list"   and y :: "'b"   and ys :: "'b list" shows "(x # xs \<le> y # ys) = (x = y \<and> xs \<le> ys)"
rep_eq: shows "semilattice_set_apply sup_sls = sup"
le: fixes x :: "'n"   and k :: "nat" assumes "x \<in> S" shows "norm (f k x) \<le> h x"
max_word_not_less: fixes x :: "'b word" shows "\<not> - 1 < x"
unit_def: fixes a :: "'b" shows "FoH.unit a \<equiv> THE \<psi>. \<guillemotleft>\<psi> : FoH.map\<^sub>0 a \<Rightarrow>\<^sub>D FoH.map a\<guillemotright> \<and> D.iso \<psi> \<and> \<psi> \<cdot>\<^sub>D \<i>\<^sub>D[FoH.map\<^sub>0 a] = (FoH.map \<i>\<^sub>B[a] \<cdot>\<^sub>D FoH.cmp (a, a)) \<cdot>\<^sub>D (\<psi> \<star>\<^sub>D \<psi>)"
bdd_above_plus: fixes f :: "'c \<Rightarrow> real"   and S :: "'c set"   and g :: "'c \<Rightarrow> real" assumes "bdd_above (f ` S)"   and "bdd_above (g ` S)" shows "bdd_above ((\<lambda>x. f x + g x) ` S)"
is_orthonormal: fixes scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and scaleC :: "complex \<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 dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and norm :: "'b \<Rightarrow> real"   and sgn :: "'b \<Rightarrow> 'b"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and cinner :: "'b \<Rightarrow> 'b \<Rightarrow> complex"   and canonical_basis :: "'b list" assumes "class.onb_enum scaleR scaleC plus zero minus uminus dist norm sgn uniformity open cinner canonical_basis" shows "is_ortho_set (set canonical_basis)"
bind_lzero: fixes f :: "'e \<Rightarrow> 'd option" shows "None \<bind> f = None"
part_equiv_refl_1: fixes R :: "('a \<times> 'a) set"   and x :: "'a"   and y :: "'a" assumes "part_equiv R"   and "(x, y) \<in> R" shows "(x, x) \<in> R"
unif_rtrancl_UNIF_conv: fixes E :: "(('c, 'd) Term.term \<times> ('c, 'd) Term.term) multiset"   and E' :: "(('c, 'd) Term.term \<times> ('c, 'd) Term.term) multiset" shows "((E, E') \<in> unif\<^sup>*) = (\<exists>ss. UNIF ss E E')"
Smith_normal_form_mat_def: fixes A :: "'b mat" shows "Smith_normal_form_mat A = ((\<forall>a. a + 1 < min (dim_row A) (dim_col A) \<longrightarrow> A $$ (a, a) dvd A $$ (a + 1, a + 1)) \<and> isDiagonal_mat A)"
mono_gamma_c: fixes ca :: "(char list \<Rightarrow> 'av) option acom"   and ca' :: "(char list \<Rightarrow> 'av) option acom" assumes "ca \<sqsubseteq> ca'" shows "\<gamma>\<^sub>c ca \<le> \<gamma>\<^sub>c ca'"
skel_def: fixes X :: "('a, 'b, 'c, 'd, 'e) qTerm set" shows "skel X \<equiv> qSkel (pick X)"
single_eq_single: fixes a :: "'a"   and b :: "'a" shows "({#a#} = {#b#}) = (a = b)"
accI_U: fixes x :: "'s"   and y :: "'s" assumes "0 < U x y" shows "(x, y) \<in> acc"
P_i_1: shows "P_i \<in> paths_with_new"
Ostrowski_baseTr0: fixes n :: "nat"   and vv :: "nat \<Rightarrow> 'a \<Rightarrow> ant"   and l :: "nat" assumes "vals_nonequiv K (Suc n) vv"   and "l \<le> Suc n" shows "0 < vv l (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) l) \<and> (\<forall>m\<in>{h. h \<le> Suc n} - {l}. 0 < vv m ((\<Omega>\<^bsub>K vv Suc n\<^esub>) l))"
wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s_code: fixes M :: "('fun, 'b) Term.term set" shows "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s M = wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s' arity M"
seq_mono_left: fixes seq :: "'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice"   and nil :: "'b::refinement_lattice"   and c\<^sub>0 :: "'b::refinement_lattice"   and c\<^sub>1 :: "'b::refinement_lattice"   and d :: "'b::refinement_lattice" assumes "seq_distrib_right (seq::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice) (nil::'b::refinement_lattice)"   and "(c\<^sub>0::'b::refinement_lattice) \<sqsubseteq> (c\<^sub>1::'b::refinement_lattice)" shows "(seq::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice) (c\<^sub>0::'b::refinement_lattice) (d::'b::refinement_lattice) \<sqsubseteq> seq (c\<^sub>1::'b::refinement_lattice) d"
opposite_dir_4: fixes a :: "'a"   and b :: "'a option" shows "\<unrhd>\<^sub>u\<^sub>b (\<lambda>x y. y < x) a b = \<unlhd>\<^sub>l\<^sub>b (<) a b"
v11_VLambda: fixes f :: "V \<Rightarrow> V"   and A :: "V" assumes "inj_on f (elts A)" shows "v11 (VLambda A f)"
right_neutral: fixes a :: "'a \<Rightarrow> 'a set" shows "a \<circ>\<^sub>K \<eta> = a"
bounded_linear_imp_differentiable: fixes f :: "'a \<Rightarrow> 'b"   and net :: "'a filter" assumes "bounded_linear f" shows "f differentiable net"
prob_spaceI: fixes M :: "'b measure" assumes "emeasure M (space M) = 1" shows "prob_space M"
normalize_positive_src_ports_normalized_n_primitive: fixes m :: "'b common_primitive match_expr" assumes "normalized_nnf_match m"   and "\<not> has_disc_negated is_Src_Ports False m" shows "\<forall>m'\<in>set (normalize_positive_src_ports m). normalized_src_ports m'"
vert_P: fixes x :: "'b" assumes "x \<in> vertices G" shows "(x, Eps (P x)) \<in> getRel S_Idt G"
gamma_differentiable: shows "\<forall>x\<in>{0..1}. \<gamma> differentiable at x"
simps_1: fixes A :: "'a"   and uu :: "'a"   and uv :: "'a set" shows "Maps.Dom (Maps.MkArr A uu uv) = A"
map_sum_bot_mono: fixes B :: "'a list"   and C :: "'a \<Rightarrow> ('b \<Rightarrow> 'c +\<^sub>\<bottom> 'd) \<Rightarrow> 'c +\<^sub>\<bottom> 'e" assumes "\<And>y. y \<in> set B \<Longrightarrow> monotone sum_bot.le_fun sum_bot_ord (C y)" shows "monotone sum_bot.le_fun sum_bot_ord (\<lambda>f. map_sum_bot (\<lambda>y. C y f) B)"
update_cons_cons_env: fixes x :: "'h"   and y :: "'h"   and ys :: "'h list"   and opt :: "('f, 'g, 'h) env option"   and b :: "'g"   and es :: "'h \<Rightarrow> ('f, 'g, 'h) env option" shows "update (x # y # ys) opt (Env b es) = Env b (es(x := case es x of None \<Rightarrow> None | Some e \<Rightarrow> Some (update (y # ys) opt e)))"
integral_bounded_linear': fixes T :: "'d \<Rightarrow> 'e"   and T' :: "'e \<Rightarrow> 'd"   and M :: "'f measure"   and f :: "'f \<Rightarrow> 'd" assumes "bounded_linear T"   and "bounded_linear T'"   and "\<not> (\<forall>x. T x = (0::'e)) \<Longrightarrow> \<forall>x. T' (T x) = x" shows "LINT x|M. T (f x) = T (integral\<^sup>L M f)"
upper_triangular_factorization_lattice: fixes d :: "nat"   and u :: "'a poly"   and k :: "'a" assumes "d \<le> n"   and "d = degree u" shows "upper_triangular (mat_of_rows n (factorization_lattice u (n - d) k))"
matricize_add: fixes A :: "'b tensor"   and B :: "'b tensor"   and I :: "nat set" assumes "dims A = dims B" shows "matricize I A + matricize I B = matricize I (A + B)"
caprasse_def: shows "caprasse = R\<^sub>e 31801 / R\<^sub>e (10 ^ 4) + - Var 0 * Power (Var 2) 3 + R\<^sub>e 4 * Var 1 * Power (Var 2) 2 * Var 3 + R\<^sub>e 4 * Var 0 * Var 2 * Power (Var 3) 2 + R\<^sub>e 2 * Var 1 * Power (Var 3) 3 + R\<^sub>e 4 * Var 0 * Var 2 + R\<^sub>e 4 * Power (Var 2) 2 - R\<^sub>e 10 * Var 1 * Var 3 + - R\<^sub>e 10 * Power (Var 3) 2 + R\<^sub>e 2"
expectation_gpv2_def: shows "expectation_gpv2 \<equiv> expectation_gpv 0 \<I>'"
u_simps_2: shows "hseq (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) f\<^sub>0gh\<^sub>1.p\<^sub>0"
converge_2: shows "cs\<^bsup>\<pi>\<^esup> m = cs\<^bsup>\<pi>'\<^esup> m'"
strict_monoidal_extension_to_free_monoidal_category_axioms: shows "strict_monoidal_extension_to_free_monoidal_category C D T\<^sub>D \<alpha>\<^sub>D \<iota>\<^sub>D V FoD.map"
\<Sigma>_protocol: shows "local.\<Sigma>_protocols_base.\<Sigma>_protocol"
that: fixes C :: "real"   and D :: "real" assumes "\<forall>\<^sub>F x in at_top. norm (EM_remainder' n f a x) \<le> C + D * g x" shows "thesis"
l_one: fixes x :: "'a list" assumes "x \<in> carrier (poly_ring R)" shows "\<one>\<^bsub>poly_ring R\<^esub> \<otimes>\<^bsub>poly_ring R\<^esub> x = x"
coeff_img_restrict: fixes p :: "'a list" shows "local.coeff p ` {..<length p} = set p"
globE: fixes \<gamma> :: "(nat \<Rightarrow> cnf) \<Rightarrow> nat \<Rightarrow> bool"   and t :: "nat \<Rightarrow> cnf"   and n :: "nat"   and n' :: "nat" assumes "(\<box>\<^sub>c\<gamma>) t n"   and "n \<le> n'" shows "\<gamma> t n'"
submonoid_pr_ideal: fixes R :: "'b set"   and I :: "'b set"   and addition :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and multiplication :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and unit :: "'b" assumes "pr_ideal R I addition multiplication zero unit" shows "submonoid (R\<setminus>I) R multiplication unit"
mult_cancel_left2: fixes 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"   and c :: "'c"   and a :: "'c" assumes "class.ring_1_no_zero_divisors one times plus zero minus uminus" shows "(times c a = c) = (c = zero \<or> a = one)"
jordan_block_carrier: fixes n :: "nat"   and k :: "'b" shows "jordan_block n k \<in> carrier_mat n n"
y: shows "map_of (zip (rev [0..<length B]) (rev B)) i = Some y"
connected_segment: fixes x :: "'a"   and y :: "'a" shows "connected (closed_segment x y)"
w_simps_3: shows "src \<chi>.chine = src s\<^sub>0"
open_bool_def: shows "open = generate_topology (range lessThan \<union> range greaterThan)"
emeasure_embed_measure_image: fixes f :: "'a \<Rightarrow> 'b"   and X :: "'a set"   and M :: "'a measure" assumes "inj f"   and "X \<in> sets M" shows "emeasure (embed_measure M f) (f ` X) = emeasure M X"
both_empty1: fixes t :: "'a ttree" shows "TTree.empty \<otimes>\<otimes> t = t"
bres_def: fixes x :: "'c \<times> 'd"   and z :: "'c \<times> 'd" shows "prod.bres x z = Sup_prod {y. less_eq_prod (times_prod' x y) z}"
neq_if_length_neq: fixes xs :: "'c list"   and ys :: "'c list" assumes "|xs| \<noteq> |ys|" shows "xs = ys \<equiv> False"
Gp_smult_sg_nsg: fixes G :: "('e, 'f) Group_scheme"   and H :: "'e set"   and N :: "'e set" assumes "Group G"   and "G \<guillemotright> H "   and "G \<triangleright> N" shows "Group (\<natural>\<^bsub>G\<^esub>H \<diamondop>\<^bsub>G\<^esub> N)"
access_arr_len_new_alloc: fixes s :: "Store"   and T :: "Arraytype"   and l :: "nat" shows "s\<langle>new_array T l\<rangle>@@arr_len (new s (new_array T l)) = intgV (int l)"