hcju/isabelleps huggingface dataset:
inj_on_Prod_primes: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and A :: "'b set set" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize"   and "\<And>P p. \<lbrakk>P \<in> A; p \<in> P\<rbrakk> \<Longrightarrow> normalization_semidom.prime zero times one normalize p"   and "\<And>P. P \<in> A \<Longrightarrow> finite P" shows "inj_on (comm_monoid_mult.prod times one (\<lambda>x. x)) A"
finite_subset_image: fixes B :: "'d set"   and f :: "'e \<Rightarrow> 'd"   and A :: "'e set" assumes "finite B"   and "B \<subseteq> f ` A" shows "\<exists>C\<subseteq>A. finite C \<and> B = f ` C"
isUnitClause_def: fixes uClause :: "Literal list"   and uLiteral :: "Literal"   and valuation :: "Literal list" shows "isUnitClause uClause uLiteral valuation \<equiv> uLiteral el uClause \<and> \<not> literalTrue uLiteral valuation \<and> \<not> literalFalse uLiteral valuation \<and> (\<forall>literal. literal el uClause \<and> literal \<noteq> uLiteral \<longrightarrow> literalFalse literal valuation)"
ssubst_sclose_commute: fixes s :: "char list"   and t' :: "sterm"   and p :: "char list"   and x :: "char list"   and n :: "nat"   and t :: "sterm" assumes "s \<notin> FV t'"   and "p \<notin> FV t'"   and "x \<noteq> s"   and "x \<noteq> p" shows "[x \<rightarrow> t'] {n \<leftarrow> [s,p]} t = {n \<leftarrow> [s,p]} [x \<rightarrow> t'] t"
sum_shift_lb_Suc0_0: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and f :: "nat \<Rightarrow> 'a"   and k :: "nat" assumes "class.comm_monoid_add plus zero"   and "f 0 = zero" shows "comm_monoid_add.sum plus zero f {Suc 0..k} = comm_monoid_add.sum plus zero f {0..k}"
intro: fixes split_fun :: "('a btnode ref option \<times> 'a) array \<times> nat \<Rightarrow> 'a \<Rightarrow> nat Heap" assumes "\<And>xs c a n p. sorted_less (separators xs) \<Longrightarrow> <is_pfa c xs (a, n)> split_fun (a, n) p <\<lambda>r. is_pfa c xs (a, n) * \<up> (r \<le> n \<and> (\<forall>j<r. snd (xs ! j) < p) \<and> (r < n \<longrightarrow> p \<le> snd (xs ! r)))>\<^sub>t" shows "imp_split_smeq split_fun"
iprev_le_mono: fixes a :: "nat"   and b :: "nat"   and I :: "nat set" assumes "a \<le> b"   and "a \<in> I"   and "b \<in> I" shows "iprev a I \<le> iprev b I"
a_get_tag_name_def: fixes element_ptr :: "(_) element_ptr" shows "a_get_tag_name element_ptr = get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t element_ptr tag_name"
canon_ang_plus_pi2: fixes \<alpha> :: "real" assumes "- 2 * pi < \<alpha>"   and "\<alpha> \<le> 0" shows "\<downharpoonright>\<alpha> + pi\<downharpoonleft> = \<alpha> + pi"
disc_iff_2: fixes xs :: "'b llist" shows "(\<not> lnull (lalternate xs)) = (\<not> lnull xs)"
hurwitz_zeta_def: fixes a :: "real"   and s :: "complex" shows "hurwitz_zeta a s = (if s = 1 then 0 else pre_zeta a s + complex_of_real a powr (1 - s) / (s - 1))"
homeomorphic_Borsukian: fixes S :: "'a set"   and T :: "'b set" assumes "Borsukian S"   and "S homeomorphic T" shows "Borsukian T"
birkhoff_sum_cocycle: fixes f :: "'a \<Rightarrow> 'b"   and n :: "nat"   and m :: "nat"   and x :: "'a" shows "birkhoff_sum f (n + m) x = birkhoff_sum f n x + birkhoff_sum f m ((T ^^ n) x)"
norm_axis: fixes ia :: "'b" shows "norm (axis ia (1::'a)) = 1"
a_2: shows "f \<in> \<Theta>(\<lambda>x. real x powr p * (1 + integral {a..real x} (\<lambda>u. u powr p * ln u powr p' / u powr (p + 1))))"
silent_moves_into_\<tau>rtrancl3p: fixes s :: "'s2"   and s' :: "'s2" assumes "s -\<tau>2\<rightarrow>* s'" shows "s -\<tau>2-[]\<rightarrow>* s'"
monotone_eadd1: fixes y :: "enat" shows "monotone (\<le>) (\<le>) (\<lambda>x. x + y)"
map_sum_sel_1: fixes s :: "'a + 'b"   and f :: "'a \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'd" assumes "isl s" shows "projl (map_sum f g s) = f (projl s)"
H_assign_var: fixes P :: "(char list \<Rightarrow> 'a) \<Rightarrow> bool"   and Q :: "(char list \<Rightarrow> 'a) \<Rightarrow> bool"   and v :: "char list"   and e :: "(char list \<Rightarrow> 'a) \<Rightarrow> 'a" assumes "\<forall>s. P s \<longrightarrow> Q (s(v := e s))" shows "rel_aka.H \<lceil>P\<rceil> (v ::= e) \<lceil>Q\<rceil>"
list_of_dlist_Dlist: fixes xs :: "'a list" shows "list_of_dlist (Dlist xs) = remdups xs"
assms_4: shows "(G ===> (=)) invar1 invar2"
redundant_code: fixes g :: "'g" shows "uninst_code.ssa.redundant (u_g g s1_uses) (p_g g s1_phis) g = uninst_code.ssa.redundant_code (p_g g s1_phis) g"
bbind_case_distr_strict: fixes f :: "'b \<rightarrow> 'a"   and g :: "'c\<^sub>\<bottom>"   and h :: "'c \<rightarrow> 'b" assumes "f\<cdot>\<bottom> = \<bottom>" shows "f\<cdot>(g >>= h) = g >>= (\<Lambda> x. f\<cdot>(h\<cdot>x))"
strictD: fixes f :: "'g \<Rightarrow> 'h" assumes "strict f" shows "f bot = bot"
316: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "- (- x \<sqinter> y) \<sqinter> (x \<sqinter> z) = x \<sqinter> z"
rel_with_target_impl_transC_TRel_is_weak_reduction_correspondence_simulation: fixes Rel :: "(('procS, 'procT) Proc \<times> ('procS, 'procT) Proc) set"   and TRel :: "('procT \<times> 'procT) set" assumes "weak_reduction_correspondence_simulation Rel (STCal Source Target)"   and "\<forall>T1 T2. (T1, T2) \<in> TRel \<longrightarrow> (TargetTerm T1, TargetTerm T2) \<in> Rel"   and "\<forall>T1 T2. (TargetTerm T1, TargetTerm T2) \<in> Rel \<longrightarrow> (T1, T2) \<in> TRel\<^sup>+" shows "weak_reduction_correspondence_simulation (TRel\<^sup>+) Target"
simp_num_pair_def: shows "simp_num_pair \<equiv> \<lambda>(t, n). if n = 0 then (C 0, 0) else let t' = simpnum t; g = numgcd t' in if 1 < g then let g' = gcd n g in if g' = 1 then (t', n) else (reducecoeffh t' g', n div g') else (t', n)"
valid_line_aux: fixes n :: "nat" shows "{l. length l = n \<and> valid l} \<noteq> {}"
hyps_1: shows "P \<turnstile> \<langle>e1,s1,b1\<rangle> \<rightarrow> \<langle>e2,s2,b2\<rangle>"
g_def: fixes \<psi> :: "nat"   and x :: "'a" shows "g \<psi> x = f \<psi> x / (\<Sum>i\<in>{j \<in> N. x \<in> W j}. f i x)"
mod_or_dist: fixes P :: "assn"   and Q :: "assn"   and h :: "heap \<times> nat set" shows "(h \<Turnstile> P \<or>\<^sub>A Q) = (h \<Turnstile> P \<or> h \<Turnstile> Q)"
wlconfD: fixes G :: "prog"   and s :: "st"   and vs :: "'a \<Rightarrow> val option"   and Ts :: "'a \<Rightarrow> ty option"   and n :: "'a"   and T :: "ty"   and v :: "val" assumes "G,s\<turnstile>vs[\<sim>\<Colon>\<preceq>]Ts"   and "Ts n = Some T"   and "vs n = Some v" shows "G,s\<turnstile>v\<Colon>\<preceq>T"
IH: fixes t :: "'a trm"   and s :: "'a trm" assumes "subterm t q s" shows "subterm (t \<lhd> \<sigma>) q (s \<lhd> \<sigma>)"
rat_is_int_def: fixes q :: "rat" shows "is_int q = (\<exists>n. q = rat_of_int n)"
simps_2: fixes t :: "nat"   and k :: "nat" shows "var_to_dimacs (State t k) = 1 + Suc (length ast\<delta>) * Suc h + t + k * Suc h"
simps: fixes p :: "rat poly"   and xs :: "rat list"   and fs :: "rat poly list" shows "factorize_root_free_main p xs fs = (case xs of [] \<Rightarrow> let l = lead_coeff p; q = smult (inverse l) p in (l, if q = 1 then fs else q # fs) | x # xs \<Rightarrow> if poly p x = 0 then factorize_root_free_main (p div [:- x, 1:]) (x # xs) ([:- x, 1:] # fs) else factorize_root_free_main p xs fs)"
hoare_step: fixes P :: "'a \<Rightarrow> 'c"   and D :: "'a \<times> 'b \<Rightarrow> 'd \<Rightarrow> 'c"   and Q :: "'b \<Rightarrow> 'd" shows "(\<forall>i j. \<Turnstile> P i{| D (i, j) |}Q j) = \<Turnstile> P{| step D |}Q"
axioms_1: fixes S :: "'c set"   and T :: "'d set" assumes "real_vector_space_pair_on S T" shows "real_vector_space_on S"
Z_def: shows "Z \<equiv> abs_var (PMF (return_pmf (True, undefined)))"
fv\<^sub>s\<^sub>s\<^sub>t_cons_subset: fixes A :: "('d, 'c) stateful_strand_step list"   and a :: "('d, 'c) stateful_strand_step" shows "fv\<^sub>s\<^sub>s\<^sub>t A \<subseteq> fv\<^sub>s\<^sub>s\<^sub>t (a # A)"
adjacent_Cons_r2: fixes w :: "nat \<times> nat"   and t :: "'c tree"   and x :: "'b tree"   and xs :: "'b tree list" assumes "adjacent (fst w, r t) (x # xs)"   and "0 < fst w"   and "fst w \<le> snd w"   and "r t \<le> snd w"   and "local.atomics (max (fst w) (Suc (r t))) (snd w) = []" shows "adjacent w (x # xs)"
assms_2: shows "(perm \<bullet> P, perm \<bullet> Q) \<in> Rel"
strictly_preserves_\<iota>: shows "G \<iota>\<^sub>C = \<iota>\<^sub>D"
n: shows "x \<in> {real (nth_prime n)..<real (nth_prime (Suc n))}"
ctor_rec: fixes s :: "('d, ('d G \<times> 'c) F) G0 \<Rightarrow> 'c"   and x :: "('d, 'd G F) G0" shows "ctor_rec_G s (ctor_G x) = s (map_pre_G id <id, ctor_rec_G s> x)"
inc_above_def: fixes b :: "nat"   and i :: "nat"   and x :: "nat" shows "inc_above b i x = (if x < b then x else x + i)"
Cauchy_vec_nth: fixes X :: "nat \<Rightarrow> ('a, 'b) vec"   and i :: "'b" assumes "Cauchy X" shows "Cauchy (\<lambda>n. X n $ i)"
eval_wp_DC: fixes a :: "bool \<Rightarrow> ('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real"   and b :: "bool \<Rightarrow> ('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real"   and P :: "'c \<Rightarrow> real" shows "wp (a \<Sqinter> b) P = (\<lambda>s. min (wp a P s) (wp b P s))"
fid_on_def: fixes A :: "V" shows "fid_on A = set {[a, a]\<^sub>\<circ> |a. a \<in>\<^sub>\<circ> A}"
mat_kernel: fixes A :: "'b mat"   and nr :: "nat"   and nc :: "nat" assumes "A \<in> carrier_mat nr nc" shows "mat_kernel A = {v \<in> carrier_vec nc. A *\<^sub>v v = 0\<^sub>v nr}"
vsubset_vintersection_rightI: fixes A :: "V"   and B :: "V"   and C :: "V" assumes "A \<subseteq>\<^sub>\<circ> B"   and "A \<subseteq>\<^sub>\<circ> C" shows "A \<subseteq>\<^sub>\<circ> B \<inter>\<^sub>\<circ> C"
col_mat_of_cols_list: fixes j :: "nat"   and l :: "complex list list" assumes "j < length l" shows "Matrix.col (Tensor.mat_of_cols_list (length (l ! j)) l) j = vec_of_list (l ! j)"
bidirected_digraph_axioms: shows "bidirected_digraph G arev"
worst_degenerate_in_worst_overall: shows "\<exists>x\<in>local.degenerate_lotteries. \<forall>y\<in>local.\<P>. y \<succeq>[\<R>] x"
has_integral_spike_finite: fixes S :: "'a set"   and T :: "'a set"   and g :: "'a \<Rightarrow> 'b"   and f :: "'a \<Rightarrow> 'b"   and y :: "'b" assumes "finite S"   and "\<And>x. x \<in> T - S \<Longrightarrow> g x = f x"   and "(f has_integral y) T" shows "(g has_integral y) T"
bshiftr1_eq: fixes b :: "bool"   and w :: "'b word" shows "bshiftr1 b w = of_bl (b # butlast (to_bl w))"
simps_2: fixes A :: "'b mat"   and As :: "'b mat list" shows "diag_block_mat (A # As) = (let B = diag_block_mat As in four_block_mat A (0\<^sub>m (dim_row A) (dim_col B)) (0\<^sub>m (dim_row B) (dim_col A)) B)"
bind_returns_result_I: fixes h :: "'m"   and f :: "('m, 'n, 'o) prog"   and x :: "'o"   and h' :: "'m"   and g :: "'o \<Rightarrow> ('m, 'n, 'p) prog"   and y :: "'p" assumes "h \<turnstile> f \<rightarrow>\<^sub>r x"   and "h \<turnstile> f \<rightarrow>\<^sub>h h'"   and "h' \<turnstile> g x \<rightarrow>\<^sub>r y" shows "h \<turnstile> Heap_Error_Monad.bind f g \<rightarrow>\<^sub>r y"
classes_above_has_fields_dne: fixes C :: "char list"   and P :: "'b prog"   and P' :: "'b prog" assumes "classes_above P C \<inter> classes_changed P P' = {}" shows "(\<forall>FDTs. \<not> P \<turnstile> C has_fields FDTs) = (\<forall>FDTs. \<not> P' \<turnstile> C has_fields FDTs)"
powr_inj: fixes a :: "real"   and x :: "real"   and y :: "real" assumes "0 < a"   and "a \<noteq> 1" shows "(a powr x = a powr y) = (x = y)"
runiq_wrt_ex1: fixes R :: "('c \<times> 'd) set" shows "runiq R = (\<forall>a\<in>Domain R. \<exists>!b. (a, b) \<in> R)"
pair_return_pmf1: fixes x :: "'a"   and y :: "'b pmf" shows "pair_pmf (return_pmf x) y = map_pmf (Pair x) y"
AR_homeomorphic_AR: fixes T :: "'a set"   and S :: "'b set" assumes "AR T"   and "S homeomorphic T" shows "AR S"
assms_1: shows "\<forall>i xs. i \<le> n \<and> set xs \<subseteq> {0..n} \<longrightarrow> \<one> \<le> len M i i xs"
divide_def: fixes x :: "nat \<Rightarrow> int"   and y :: "nat \<Rightarrow> int" shows "local.divide x y = (if x = \<zero> then \<zero> else [p] \<cdot> \<one> [^] nat (ord_Zp x - ord_Zp y) \<otimes> ac_Zp x \<otimes> inv ac_Zp y)"
rel_is_equivalence: shows "equivalence (Sigma I \<FF>) {(x, y). x \<sim> y}"
natural_transformation_axioms: shows "natural_transformation (\<cdot>\<^sub>J) (\<cdot>) mkCone_\<chi>.A.map local.map (mkCone (\<chi> J.AA) (\<chi> J.BB))"
lookup_prod_zero: fixes f :: "'d \<Rightarrow> 'c \<Rightarrow>\<^sub>0 'b"   and I :: "'d set" shows "lookup (prod f I) (0::'c) = (\<Prod>i\<in>I. lookup (f i) (0::'c))"
count_take: fixes n :: "nat"   and cs :: "'a list"   and x :: "'a" shows "count_list (take n cs) x \<le> count_list cs x"
hyps: shows "step_score xs e s = foldr (\<lambda>a. case a of (id1, id2) \<Rightarrow> \<lambda>acc. let score = s id1 id2 e in if score = 0 then 0 else score + acc) xs 0"
map_prod_ordIso: fixes r :: "('d \<times> 'd) set"   and f :: "'d \<Rightarrow> 'e" assumes "Well_order r"   and "inj_on f (Field r)" shows "map_prod f f ` r =o r"
simps_38: fixes x41 :: "('a, 'b, 'c) Stmt"   and x42 :: "('a, 'b, 'c) Stmt"   and x61 :: "'c"   and x62 :: "('a, 'b, 'c) Stmt" shows "x41 ;; x42 \<noteq> While x61 x62"
is_document_ptr_def: fixes ptr :: "(_) document_ptr" shows "is_document_ptr\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr = (case ptr of document_ptr.Ref x \<Rightarrow> True | document_ptr.Ext document_ptr \<Rightarrow> False)"
sub_branch_def: fixes f :: "'g \<Rightarrow> 'f"   and blocks :: "(('e, 'g) fm list \<times> 'g) list" shows "sub_branch f blocks \<equiv> map (sub_block f) blocks"
has_sgnx_derivative_at_left: fixes g :: "real \<Rightarrow> real"   and c :: "real"   and x :: "real" assumes "(g has_real_derivative c) (at x)"   and "g x = 0"   and "c \<noteq> 0" shows "(g has_sgnx - sgn c) (at_left x)"
complex_quadratic_equation_monic_roots: fixes \<xi> :: "complex"   and b :: "complex"   and c :: "complex" assumes "\<xi> = (- b + ccsqrt (b\<^sup>2 - 4 * c)) / 2 \<or> \<xi> = (- b - ccsqrt (b\<^sup>2 - 4 * c)) / 2" shows "\<xi>\<^sup>2 + b * \<xi> + c = 0"
sorted_wrt_linorder_unique: fixes A :: "'a set"   and R :: "('a \<times> 'a) set"   and xs :: "'a list"   and ys :: "'a list" assumes "linorder_on A R"   and "mset xs = mset ys"   and "Linorder_Relations.sorted_wrt R xs"   and "Linorder_Relations.sorted_wrt R ys" shows "xs = ys"
Zero_neq_One: shows "(0::'a) \<noteq> One"
chine_assoc_in_hom: shows "\<guillemotleft>\<mu>\<nu>\<pi>.chine_assoc : HH\<mu>\<nu>\<pi>.chine \<rightarrow>\<^sub>C H\<mu>H\<nu>\<pi>.chine\<guillemotright>"
bij_inv_project_act_eq: fixes h :: "'c \<Rightarrow> 'd" assumes "bij h" shows "inv (project_act (\<lambda>(x, u). inv h x)) = project_act (\<lambda>(x, u). h x)"
elims_3: fixes x :: "('a, 'b) rose_tree\<^sub>m" assumes "\<not> is_blinded x" obtains va :: "'a \<times> ('a, 'b) rose_tree\<^sub>m list" where "x = Tree\<^sub>m (Unblinded va)"
append_eq_mapE: fixes fl :: "'a list"   and fl' :: "'a list"   and f :: "'b \<Rightarrow> 'a"   and ls :: "'b list" assumes "fl @ fl' = map f ls" obtains l :: "'b list" and l' :: "'b list" where "ls = l @ l'" and "map f l = fl" and "map f l' = fl'"
eval_implies_machine_star_ctx: fixes t :: "lam"   and t' :: "lam"   and Es :: "ctx list" assumes "t \<Down> t'" shows "<t,Es> \<mapsto>* <t',Es>"
simps: fixes x :: "'a"   and b :: "'a node ref option" shows "os_rem x b = (case b of None \<Rightarrow> return None | Some p \<Rightarrow> !p \<bind> (\<lambda>n. os_rem x (next n) \<bind> (\<lambda>q. if val n = x then return q else p := Node (val n) q \<bind> (\<lambda>_. return (Some p)))))"
proj_idem: fixes l :: "'a"   and A :: "('a strand_label \<times> 'b) list" shows "proj l (proj l A) = proj l A"
closed_insert: fixes "open" :: "'a set \<Rightarrow> bool"   and S :: "'a set"   and a :: "'a" assumes "class.t1_space open"   and "topological_space.closed open S" shows "topological_space.closed open (insert a S)"
Rat_interval_closure: fixes x :: "real"   and y :: "real" assumes "x < y" shows "closure ({x<..<y} \<inter> \<rat>) = {x..y}"
sig_red_singleD6: fixes sing_reg :: "'t \<Rightarrow> 't \<Rightarrow> bool"   and top_tail :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and p :: "'t \<Rightarrow>\<^sub>0 'b"   and q :: "'t \<Rightarrow>\<^sub>0 'b"   and f :: "'t \<Rightarrow>\<^sub>0 'b"   and t :: "'a" assumes "sig_red_single sing_reg top_tail p q f t" shows "sing_reg (t \<oplus> lt f) (lt p)"
simps_2: fixes x :: "char list"   and a :: "aexp"   and Q :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool" shows "vc (x ::= a) Q = True"
is_ring_morphism: fixes U :: "'a set" assumes "is_open U" shows "ring_homomorphism (fam_morphisms U) (\<FF> U) +\<^bsub>U\<^esub> \<cdot>\<^bsub>U\<^esub> \<zero>\<^bsub>U\<^esub> \<one>\<^bsub>U\<^esub> (\<FF>' U) +'\<^bsub>U\<^esub> \<cdot>'\<^bsub>U\<^esub> \<zero>'\<^bsub>U\<^esub> \<one>'\<^bsub>U\<^esub>"
inject_8: fixes x81 :: "formula"   and x82 :: "formula"   and y81 :: "formula"   and y82 :: "formula" shows "(And x81 x82 = And y81 y82) = (x81 = y81 \<and> x82 = y82)"
bf_restrict_def: fixes i :: "'a"   and val :: "bool"   and f :: "('a \<Rightarrow> bool) \<Rightarrow> bool" shows "bf_restrict i val f \<equiv> \<lambda>v. f (v(i := val))"
unstables_def: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and step :: "nat \<Rightarrow> 'b \<Rightarrow> (nat \<times> 'b) list"   and \<tau>s :: "'b list" shows "unstables r step \<tau>s = {p. p < length \<tau>s \<and> \<not> stable r step \<tau>s p}"
fps_compose_fps_X: fixes a :: "'b fps" shows "a oo fps_X = a"
phl_skip: fixes x :: "'a" shows "\<lparr>x\<rparr>1::'a\<lparr>(1::'a) \<cdot> x\<rparr>"
conf_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 h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and v :: "val"   and T :: "ty" shows "P,h \<turnstile> v :\<le> T \<equiv> \<exists>T'. typeof\<^bsub>h\<^esub> v = \<lfloor>T'\<rfloor> \<and> P \<turnstile> T' \<le> T"
triangle_right: shows "(g \<star> \<epsilon>) \<cdot> \<a>[g, f, g] \<cdot> (\<eta> \<star> g) = g"
create_opt_sol_sat_leq_mat: fixes b :: "rat vec"   and A :: "rat mat"   and c :: "rat vec"   and X :: "(nat, rat) mapping"   and x :: "rat vec"   and y :: "rat vec" assumes "dim_vec b = dim_row A"   and "create_optimal_solutions A b c = Inr X"   and "(x, y) = split_i_j_x (dim_col A) (dim_vec b) X" shows "[A*\<^sub>vx]\<le>b"
axioms_1: fixes R :: "('c, 'd) ring_scheme"   and S :: "('e, 'f) ring_scheme"   and h :: "'c \<Rightarrow> 'e" assumes "ring_hom_cring R S h" shows "cring R"
Abs_pq_inverse: fixes y :: "('c \<times> 'd) list" assumes "y \<in> {xs. distinct (map fst xs) \<and> sorted (map snd xs)}" shows "alist_of (Abs_pq y) = y"
sig_red_red: fixes sing_reg :: "'t \<Rightarrow> 't \<Rightarrow> bool"   and top_tail :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and F :: "('t \<Rightarrow>\<^sub>0 'b) set"   and p :: "'t \<Rightarrow>\<^sub>0 'b"   and q :: "'t \<Rightarrow>\<^sub>0 'b" assumes "sig_red sing_reg top_tail F p q" shows "punit.red (rep_list ` F) (rep_list p) (rep_list q)"
incl_incl_of_1: fixes a :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and b :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "incl_in a b" shows "incl (incl_of a b)"
in_Sup_wP_def: shows "in_Sup_wP = (\<lambda>i. (C, i) \<in> Sup_llist (lmap (set_mset \<circ> wP_of_wstate) (ldrop (enat t_C) Sts)))"
IntD2: fixes c :: "'k"   and A :: "'k set"   and B :: "'k set" assumes "c \<in> A \<inter> B" shows "c \<in> B"
iF_prems: shows "set (prems_of \<iota>F) \<subseteq> fst ` NL"
affine_hull_span_gen: fixes a :: "'a"   and s :: "'a set" assumes "a \<in> affine hull s" shows "affine hull s = (+) a ` span ((+) (- a) ` s)"
subgroup_generated_restrict: fixes G :: "('b, 'c) monoid_scheme"   and S :: "'b set" shows "subgroup_generated G (carrier G \<inter> S) = subgroup_generated G S"
F_power: fixes n :: "nat" shows "F ^ n = fps_X ^ n * fps_exp (real n)"
in_Field_r: fixes \<P> :: "'a set set" shows "(\<P> \<in> Field_r) = ((\<forall>C\<in>\<P>. - C \<notin> \<P>) \<and> (\<forall>A. finite A \<longrightarrow> A \<in> \<P>))"
filter_id_conv: fixes P :: "'c \<Rightarrow> bool"   and xs :: "'c list" shows "(filter P xs = xs) = (\<forall>x\<in>set xs. P x)"
monom_pow: fixes a :: "'a"   and n :: "nat"   and m :: "nat" assumes "a \<in> carrier R" shows "monom P a n [^]\<^bsub>P\<^esub> m = monom P (a [^] m) (n * m)"
CLS_REP: fixes F :: "('a, 'a set) arr" assumes "arr F" shows "\<lbrakk>\<lbrakk>REP F\<rbrakk>\<rbrakk> = F"
Nodes_levellist: fixes pret :: "dag"   and levellista :: "ref list list"   and var :: "ref \<Rightarrow> nat"   and nb :: "nat"   and p :: "ref" assumes "wf_ll pret levellista var"   and "nb < length levellista"   and "p \<in> Nodes nb levellista" shows "p \<notin> set (levellista ! nb)"
b: shows "integral {0..1} (\<lambda>x. D x * (F (\<gamma>2 (\<phi> x)) \<bullet> b * (vector_derivative \<gamma>2 (at (\<phi> x) within {0..1}) \<bullet> b))) = integral {\<phi> 0..\<phi> 1} (\<lambda>x. F (\<gamma>2 x) \<bullet> b * (vector_derivative \<gamma>2 (at x within {0..1}) \<bullet> b))"
map_rev_map_nth_eq: fixes xs :: "nat list"   and as :: "'b list"   and f :: "'b \<Rightarrow> 'c" assumes "\<forall>x\<in>set xs. x < length as" shows "map f (rev (map ((!) as) xs)) = rev (map ((!) (map f as)) xs)"
alt2: fixes a :: "alt" shows "\<exists>b. b \<noteq> a"
order_eq_0_iff: fixes p :: "'a poly"   and x :: "'a" assumes "p \<noteq> 0" shows "(Polynomial.order x p = 0) = (poly p x \<noteq> (0::'a))"
assms_2: fixes a :: "'a"   and X' :: "'a set" assumes "a \<in> X"   and "X' = X - {a}" shows "f a \<le> SPEC (\<lambda>x. WHILE (\<lambda>(uu_, X). X \<noteq> {}) (\<lambda>(b, X). ASSERT (X \<noteq> {}) \<bind> (\<lambda>_. op_set_npick_remove X \<bind> (\<lambda>(a, X). f a \<bind> (\<lambda>b'. c b b' \<bind> (\<lambda>b. RETURN (b, X)))))) (x, X') \<le> SPEC (\<lambda>(b, uu_). RETURN b \<le> SPEC P))"
context_assms_3: shows "range w \<subseteq> \<Sigma>"
groupsort_axioms: shows "groupsort fst merge_primfun eval_primfuns"
permute_Abs_set: fixes p :: "perm"   and as :: "atom set"   and x :: "'b" shows "p \<bullet> ([as]set. x) = [p \<bullet> as]set. p \<bullet> x"
Nxt_def: fixes ic :: "state \<Rightarrow> 'b"   and q :: "state \<Rightarrow> 'b list"   and oc :: "state \<Rightarrow> 'b" shows "Nxt ic q oc \<equiv> Enq ic q oc \<or> Deq ic q oc"
even_push_bit_iff: fixes n :: "nat"   and a :: "'a" shows "even (push_bit n a) = (n \<noteq> 0 \<or> even a)"
assms_3: shows "pB \<in> io_targets B qB w"
mkarr_rep: fixes f :: "'c local.term set" assumes "arr f" shows "mkarr (rep f) = f"
shiftl_power: fixes x :: "nat"   and y :: "'b word" shows "(shiftl1 ^^ x) y = 2 ^ x * y"
simps_8: fixes v :: "char list"   and va :: "aexp"   and vb :: "'a"   and vc :: "bexp"   and vd :: "'a acom"   and ve :: "'a acom"   and vf :: "'a" shows "(v ::= va {vb} \<sqsubseteq> IF vc THEN vd ELSE ve {vf}) = False"
all_D: fixes H :: "fmla set"   and i :: "name"   and A :: "fmla"   and x :: "trm" assumes "H \<turnstile> all i A" shows "H \<turnstile> A(i::=x)"
simps_5: fixes \<phi> :: "'a ltln"   and \<xi> :: "'a ltln" shows "suspendable (\<phi> U\<^sub>n \<xi>) = (\<phi> = true\<^sub>n \<and> pure_universal \<xi> \<or> suspendable \<xi>)"
wt_elim_cases_22: fixes E :: "env"   and dt :: "loc \<Rightarrow> ty option"   and c1 :: "stmt"   and c2 :: "stmt"   and x :: "tys" assumes "E,dt\<Turnstile>In1r (c1;; c2)\<Colon>x" obtains "x = Inl (PrimT Void)" and "E,dt\<Turnstile>c1\<Colon>\<surd>" and "E,dt\<Turnstile>c2\<Colon>\<surd>"
rep_eq: fixes x :: "real_alg_3" shows "real_of_3 x = real_of_2 (rep_real_alg_3 x)"
xs_def: shows "xs = sorted_wrt_list_of_set R A"
obj_class_Arr: fixes T :: "ty"   and k :: "int"   and vs :: "vn \<Rightarrow> val option" shows "obj_class \<lparr>tag = Arr T k, values = vs\<rparr> = Object"
inf_greatest: fixes A :: "'a upred set"   and z :: "'a upred" assumes "A \<subseteq> carrier \<P>"   and "z \<in> carrier \<P>"   and "\<And>x. x \<in> A \<Longrightarrow> le \<P> z x" shows "le \<P> z (Lattice.inf \<P> A)"