hcju/isabelleps huggingface dataset:
RSubmoduleI: fixes R :: "'a set"   and smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and M :: "'b set"   and N :: "'b set" assumes "RModule R smult M"   and "Group N \<and> N \<subseteq> M"   and "\<And>r n. \<lbrakk>r \<in> R; n \<in> N\<rbrakk> \<Longrightarrow> smult r n \<in> N" shows "RModule R smult N \<and> N \<subseteq> M"
scast_up_scast: fixes w :: "'b word" assumes "is_up SCAST('b \<rightarrow> 'c)" shows "SCAST('c \<rightarrow> 'd) (SCAST('b \<rightarrow> 'c) w) = SCAST('b \<rightarrow> 'd) w"
IH1_perm: assumes "Suc p \<le> hi"   and "hi < length xs'" shows "set (WB_Sort.sublist xs'' (Suc p) hi) = set (WB_Sort.sublist xs' (Suc p) hi)"
intro: fixes G :: "('e, 'f) ring_scheme" assumes "abelian_monoid G"   and "abelian_group_axioms G" shows "abelian_group G"
continuous_on_setdist: fixes T :: "'a set"   and S :: "'a set" shows "continuous_on T (\<lambda>y. setdist {y} S)"
runit_naturality: fixes f :: "'a" assumes "arr f" shows "runit (cod f) \<cdot> tensor f unity = f \<cdot> runit (local.dom f)"
filterlim_at_within_closure_implies_filterlim: fixes x :: "'b"   and s :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and l :: "'c filter" assumes "x \<in> closure s \<Longrightarrow> filterlim f l (at x within s)" shows "filterlim f l (at x within s)"
NDC_def: fixes d :: "'d"   and c :: "('exp, 'id) MWLsCom" shows "NDC d c \<equiv> \<forall>m m'. m =\<^bsub>d\<^esub> m' \<longrightarrow> \<lbrakk>c\<rbrakk>(m) =\<^bsub>d\<^esub> \<lbrakk>c\<rbrakk>(m')"
eventually_ln_at_top: fixes P :: "real \<Rightarrow> bool" shows "(\<forall>\<^sub>F x in at_top. P (ln x)) = eventually P at_top"
sl7: shows "f\<^sup>2 \<in> radical_sqrt"
Edges_def: fixes A :: "('a \<times> ('b, 'd) cconstraint \<times> ('b set \<times> 'a) pmf) set \<times> 'e" shows "Edges A \<equiv> \<Union> {edges t |t. t \<in> fst A}"
in_homE: fixes f :: "('a2 \<Rightarrow> 'b, 'a2 \<Rightarrow> 'b) concrete_category.arr"   and a :: "('a2 \<Rightarrow> 'b, 'a2 \<Rightarrow> 'b) concrete_category.arr"   and b :: "('a2 \<Rightarrow> 'b, 'a2 \<Rightarrow> 'b) concrete_category.arr" assumes "\<guillemotleft>f : a \<rightarrow>\<^sub>[\<^sub>A\<^sub>2\<^sub>,\<^sub>B\<^sub>] b\<guillemotright>" obtains "A2_B.arr f" and "A2_B.dom f = a" and "A2_B.cod f = b"
derived_is_normal: fixes H :: "'a set" assumes "H \<lhd> G" shows "derived G H \<lhd> G"
compatCh_def: fixes ch :: "'choice" shows "compatCh ch \<equiv> \<forall>s t. s \<approx> t \<longrightarrow> cval ch s = cval ch t"
approx_indep_ge: fixes x :: "'b pmf"   and y :: "'b pmf"   and \<R> :: "('b pmf \<times> 'b pmf) set"   and \<alpha> :: "real"   and outcomes :: "'b set" assumes "x \<approx>[\<R>] y"   and "\<alpha> \<in> {0..1}"   and "rational_preference (Neumann_Morgenstern_Utility_Theorem.\<P> outcomes) \<R>"   and "independent_vnm (Neumann_Morgenstern_Utility_Theorem.\<P> outcomes) \<R>" shows "\<forall>r\<in>Neumann_Morgenstern_Utility_Theorem.\<P> outcomes. mix_pmf \<alpha> y r \<succeq>[\<R>] mix_pmf \<alpha> x r"
B_def: shows "B = (\<lambda>N. \<Union> (A ` {..<N}))"
Z: shows "\<Sqinter> {eNorm N1 f1 + eNorm N2 f2 |f1 f2. (0::'a) = f1 + f2} = 0"
add_cn: fixes P :: "pol"   and Q :: "pol" assumes "isnorm P"   and "isnorm Q" shows "isnorm (P \<langle>+\<rangle> Q)"
g_cong: fixes f :: "'d \<Rightarrow> 'e"   and merge :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and g :: "'d list \<Rightarrow> 'f"   and xs :: "'d list"   and ys :: "'d list" assumes "groupsort f merge g"   and "mset xs = mset ys" shows "g xs = g ys"
simps: fixes P1 :: "'e \<Rightarrow> bool"   and P2 :: "'f \<Rightarrow> bool"   and a :: "'e \<times> 'f" shows "pred_prod P1 P2 a = (\<exists>aa b. a = (aa, b) \<and> P1 aa \<and> P2 b)"
out_subseqs2_set_def: fixes tail :: "'a iarray"   and v :: "'b"   and js :: "nat list" shows "out_subseqs2_set tail v js \<equiv> {f (tail !! j) v |j. j \<in> set js}"
br_set_rel_alt: fixes s' :: "'a set"   and s :: "'b set"   and \<alpha> :: "'a \<Rightarrow> 'b"   and I :: "'a \<Rightarrow> bool" shows "((s', s) \<in> \<langle>br \<alpha> I\<rangle>set_rel) = (s = \<alpha> ` s' \<and> (\<forall>x\<in>s'. I x))"
power_eq_imp_eq_base: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and a :: "'a"   and n :: "nat"   and b :: "'a" assumes "class.linordered_semidom plus minus less_eq less zero times one"   and "power.power one times a n = power.power one times b n"   and "less_eq zero a"   and "less_eq zero b"   and "0 < n" shows "a = b"
l_neg: fixes x :: "nat \<Rightarrow> int" assumes "x \<in> carrier Zp" shows "\<ominus> x \<oplus> x = \<zero>"
id0: shows "0\<^sub>v (nr + (nc + nc)) = 0\<^sub>v nr @\<^sub>v 0\<^sub>v nc @\<^sub>v 0\<^sub>v nc"
induct: fixes x1 :: "('g, 'h, 'i, 'j) com list \<times> 'h option \<times> 'j"   and x2 :: "('g, 'i) seq_label"   and x3 :: "('g, 'h, 'i, 'j) com list \<times> 'h option \<times> 'j"   and P :: "('g, 'h, 'i, 'j) com list \<times> 'h option \<times> 'j \<Rightarrow> ('g, 'i) seq_label \<Rightarrow> ('g, 'h, 'i, 'j) com list \<times> 'h option \<times> 'j \<Rightarrow> bool" assumes "x1 \<rightarrow>\<^bsub>x2\<^esub> x3"   and "\<And>\<alpha> action s s' val \<beta> l cs uu_. \<lbrakk>\<alpha> = action s; s' \<in> val \<beta> s\<rbrakk> \<Longrightarrow> P (\<lbrace>l\<rbrace> Request action val # cs, uu_, s) \<guillemotleft>\<alpha>, \<beta>\<guillemotright> (cs, Some l, s')"   and "\<And>s' \<beta> action \<alpha> s l cs uv_. (s', \<beta>) \<in> action \<alpha> s \<Longrightarrow> P (\<lbrace>l\<rbrace> Response action # cs, uv_, s) \<guillemotright>\<alpha>, \<beta>\<guillemotleft> (cs, Some l, s')"   and "\<And>s' R s l cs uw_. s' \<in> R s \<Longrightarrow> P (\<lbrace>l\<rbrace> LocalOp R # cs, uw_, s) \<tau> (cs, Some l, s')"   and "\<And>b s l c cs ux_. b s \<Longrightarrow> P (\<lbrace>l\<rbrace> IF b THEN c FI # cs, ux_, s) \<tau> (c # cs, Some l, s)"   and "\<And>b s l c cs uy_. \<not> b s \<Longrightarrow> P (\<lbrace>l\<rbrace> IF b THEN c FI # cs, uy_, s) \<tau> (cs, Some l, s)"   and "\<And>b s l c1 c2 cs uz_. b s \<Longrightarrow> P (\<lbrace>l\<rbrace> IF b THEN c1 ELSE c2 FI # cs, uz_, s) \<tau> (c1 # cs, Some l, s)"   and "\<And>b s l c1 c2 cs va_. \<not> b s \<Longrightarrow> P (\<lbrace>l\<rbrace> IF b THEN c1 ELSE c2 FI # cs, va_, s) \<tau> (c2 # cs, Some l, s)"   and "\<And>c cs s \<alpha> cs' s'. \<lbrakk>(c # LOOP DO c OD # cs, s) \<rightarrow>\<^bsub>\<alpha>\<^esub> (cs', s'); P (c # LOOP DO c OD # cs, s) \<alpha> (cs', s')\<rbrakk> \<Longrightarrow> P (LOOP DO c OD # cs, s) \<alpha> (cs', s')"   and "\<And>b s l c cs vb_. b s \<Longrightarrow> P (\<lbrace>l\<rbrace> WHILE b DO c OD # cs, vb_, s) \<tau> (c # \<lbrace>l\<rbrace> WHILE b DO c OD # cs, Some l, s)"   and "\<And>b s l c cs vc_. \<not> b s \<Longrightarrow> P (\<lbrace>l\<rbrace> WHILE b DO c OD # cs, vc_, s) \<tau> (cs, Some l, s)"   and "\<And>c1 c2 cs s \<alpha> cs' s'. \<lbrakk>(c1 # c2 # cs, s) \<rightarrow>\<^bsub>\<alpha>\<^esub> (cs', s'); P (c1 # c2 # cs, s) \<alpha> (cs', s')\<rbrakk> \<Longrightarrow> P (c1 ;; c2 # cs, s) \<alpha> (cs', s')"   and "\<And>c1 cs s \<alpha> cs' s' c2. \<lbrakk>(c1 # cs, s) \<rightarrow>\<^bsub>\<alpha>\<^esub> (cs', s'); P (c1 # cs, s) \<alpha> (cs', s')\<rbrakk> \<Longrightarrow> P (c1 \<oplus> c2 # cs, s) \<alpha> (cs', s')"   and "\<And>c2 cs s \<alpha> cs' s' c1. \<lbrakk>(c2 # cs, s) \<rightarrow>\<^bsub>\<alpha>\<^esub> (cs', s'); P (c2 # cs, s) \<alpha> (cs', s')\<rbrakk> \<Longrightarrow> P (c1 \<oplus> c2 # cs, s) \<alpha> (cs', s')" shows "P x1 x2 x3"
that: fixes as' :: "'edge list" assumes "n -as'\<rightarrow>\<^sub>\<iota>* n'"   and "set (sourcenodes as') \<subseteq> set (sourcenodes as)" shows "thesis"
Suc_mod_eq_add3_mod_numeral: fixes m :: "nat"   and v :: "num" shows "Suc (Suc (Suc m)) mod numeral v = (3 + m) mod numeral v"
show_r_zero: fixes x :: "'c"   and a :: "'c" assumes "x \<in> carrier V"   and "a \<in> carrier V" shows "(x \<oplus>\<^bsub>V\<^esub> a = x) = (a = \<zero>\<^bsub>V\<^esub>)"
Domain_empty_iff: fixes r :: "('d \<times> 'e) set" shows "(Domain r = {}) = (r = {})"
f_Exec_Stream_append_if: fixes trans_fun :: "'e \<Rightarrow> 'd \<Rightarrow> 'd"   and xs :: "'e list"   and ys :: "'e list"   and c :: "'d" shows "f_Exec_Comp_Stream trans_fun (xs @ ys) c = f_Exec_Comp_Stream trans_fun xs c @ f_Exec_Comp_Stream trans_fun ys (if xs = [] then c else last (f_Exec_Comp_Stream trans_fun xs c))"
trg_cod: fixes \<mu> :: "'a" assumes "C.arr \<mu>" shows "trg\<^sub>C (C.cod \<mu>) = trg\<^sub>C \<mu>"
mult_aform'_float_2: fixes p :: "nat"   and x :: "(real \<times> real pdevs) \<times> real"   and y :: "(real \<times> real pdevs) \<times> real"   and a :: "real"   and b :: "real pdevs"   and ba :: "real" assumes "mult_aform' p x y = ((a, b), ba)" shows "ba \<in> float"
assoc: fixes a :: "'d DBMEntry"   and b :: "'d DBMEntry"   and c :: "'d DBMEntry" shows "dbm_add (dbm_add a b) c = dbm_add a (dbm_add b c)"
in_set_product_lists_length: fixes xs :: "'a list"   and xss :: "'a list list" assumes "xs \<in> set (product_lists xss)" shows "length xs = length xss"
refl: fixes lc :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)" shows "\<langle>fst (fst lc), snd (fst lc), snd lc\<rangle> \<in> loc_reach lc"
closure_interior: fixes S :: "'b set" shows "closure S = - interior (- S)"
moebius_comp_inv_right: fixes M :: "moebius" shows "moebius_comp M (moebius_inv M) = id_moebius"
const_onD: fixes f :: "'c \<Rightarrow> 'd"   and S :: "'c set"   and x :: "'d"   and y :: "'c" assumes "const_on f S x"   and "y \<in> S" shows "f y = x"
lens_plus_ub: fixes Y :: "'a \<Longrightarrow> 'b"   and X :: "'c \<Longrightarrow> 'b" assumes "wb_lens Y" shows "X \<subseteq>\<^sub>L X +\<^sub>L Y"
elims: fixes x :: "nat"   and xa :: "'v \<times> 'v"   and y :: "ennreal" assumes "h_plus x xa = y" obtains xb :: "'v" and ya :: "'v" where "x = 0" and "xa = (xb__, ya__)" and "y = (if xb__ = source \<Delta> then g (xb__, ya__) else 0)"   | i :: "nat" and xb :: "'v" and ya :: "'v" where "x = Suc i" and "xa = (xb, ya)" and "y = (if enum_v (Suc i) = xb \<and> d_OUT (h_plus i) xb < d_IN (h_plus i) xb then let total = d_IN (h_plus i) xb - d_OUT (h_plus i) xb; share = g (xb, ya) - h_plus i (xb, ya); shares = d_OUT g xb - d_OUT (h_plus i) xb in h_plus i (xb, ya) + share * total / shares else h_plus i (xb, ya))"
BuildSeq3_exI: fixes B :: "hf \<Rightarrow> hf \<Rightarrow> hf \<Rightarrow> bool"   and t :: "hf"   and t' :: "hf"   and t'' :: "hf"   and C :: "hf \<Rightarrow> hf \<Rightarrow> hf \<Rightarrow> hf \<Rightarrow> hf \<Rightarrow> hf \<Rightarrow> hf \<Rightarrow> hf \<Rightarrow> hf \<Rightarrow> bool" assumes "B t t' t''" shows "\<exists>s k. BuildSeq3 B C s k t t' t''"
reflection: shows "rel_reflects_binary_pred Rel (\<lambda>b. hasBarb b CWB)"
U\<^sub>f_mult_without_empty_summands_sum_odd: fixes i :: "nat"   and j :: "nat"   and A :: "complex Matrix.mat" assumes "i < dim_row U\<^sub>f"   and "j < dim_col A"   and "odd i"   and "dim_col U\<^sub>f = dim_row A" shows "(\<Sum>k = 0..<dim_row A. U\<^sub>f $$ (i, k) * A $$ (k, j)) = (\<Sum>k\<in>{i - 1, i}. U\<^sub>f $$ (i, k) * A $$ (k, j))"
rep_eq: shows "fpxs_nth 1 = (\<lambda>r. if r = 0 then 1::'b else (0::'b))"
WP1rd: fixes p :: "('a, 'b) Combinators list" assumes "wellformed_policy1_strong p" shows "wellformed_policy1_strong (remdups p)"
preserves_dom: fixes f :: "'b" assumes "D.arr f" shows "D.dom (src\<^sub>D f) = src\<^sub>D (D.dom f)"
lconv_append: fixes xs :: "(accent \<times> 'b) list"   and ys :: "(accent \<times> 'b) list" shows "local.lconv (xs @ ys) = local.lconv xs O local.lconv ys"
notin_range_Inner: fixes x :: "'a vertex" shows "(x \<notin> range Inner) = (x = SOURCE \<or> x = vertex.SINK)"
IH: fixes gpv :: "('c, 'a, 'b) gpv"   and gpv' :: "('d, 'a, 'b) gpv"   and s :: "'s" assumes "eq_\<I>_gpv A \<I> gpv gpv'"   and "I s" shows "rel_spmf (rel_prod A (eq_onp I)) (exec_gpv' gpv s) (exec_gpv'' gpv' s)"
rep_eq: fixes x :: "'b cl_op_im"   and xa :: "'b cl_op_im" shows "Rep_cl_op_im (x \<cdot> xa) = cl_op (Rep_cl_op_im x \<cdot> Rep_cl_op_im xa)"
IH: fixes i :: "nat"   and "is" :: "nat list" assumes "i < hd (dims A)"   and "is \<lhd> dims (subtensor A i)" shows "lookup (\<alpha> \<cdot> subtensor A i) is = \<alpha> * lookup (subtensor A i) is"
finite_intersection_of_relative_to: fixes P :: "'c set \<Rightarrow> bool"   and U :: "'c set" shows "finite intersection_of P relative_to U = finite intersection_of (P relative_to U) relative_to U"
preT_linear: fixes C :: "acom"   and k :: "nat"   and e :: "(char list \<Rightarrow> int) \<Rightarrow> nat" shows "preT C (\<lambda>s. k * e s) = (\<lambda>s. k * preT C e s)"
ts'_U: shows "((t', n'), s', m') \<in> set (alluopairs U)"
disc_20: fixes x51 :: "val"   and x52 :: "val" shows "is_PairVal <|x51, x52|>"
finite_nextl: fixes Q :: "hf set"   and u :: "'a list" shows "finite (nextl Q u)"
reachable_ensures: fixes F :: "'d program"   and A :: "'d set"   and B :: "'d set" assumes "F \<in> A ensures B" shows "F \<in> reachable F \<inter> A ensures B"
retraction_prop_fix: fixes f :: "'b \<Rightarrow> 'b" assumes "f \<circ> f = f" shows "range f = Fix f"
inner_Basis: fixes u :: "'a"   and v :: "'a" assumes "u \<in> Basis"   and "v \<in> Basis" shows "inner u v = (if u = v then 1 else 0)"
gen2_nth0: fixes A :: "nat"   and m :: "nat"   and B :: "nat"   and n :: "nat" shows "gen2 A B m n ! 0 = (zeroes m, zeroes n)"
IH_1: assumes "Vagree \<nu> \<nu>' (Inl ` FVO ODE1)"   and "Iagree I J ({Inl x |x. Inl x \<in> SIGO ODE1} \<union> {Inr (Inr x) |x. Inr x \<in> SIGO ODE1})" shows "ODE_sem I ODE1 (fst \<nu>) = ODE_sem J ODE1 (fst \<nu>')"
continuous_map_from_subtopology: fixes X :: "'a topology"   and X' :: "'b topology"   and f :: "'a \<Rightarrow> 'b"   and S :: "'a set" assumes "continuous_map X X' f" shows "continuous_map (subtopology X S) X' f"
subprob_not_empty: fixes M :: "'a measure" assumes "subprob_space M" shows "space M \<noteq> {}"
Ag_ind_aHom: fixes A :: "('c, 'd) aGroup_scheme"   and f :: "'c \<Rightarrow> 'e"   and D :: "'e set" assumes "aGroup A"   and "f \<in> carrier A \<rightarrow> D"   and "bij_to f (carrier A) D" shows "Agii A f \<in> aHom A (Ag_ind A f)"
Domain_iff: fixes a :: "'g"   and r :: "('g \<times> 'h) set" shows "(a \<in> Domain r) = (\<exists>y. (a, y) \<in> r)"
assms_3: shows "\<forall>x\<in>set sf. (is_measured_node x \<and> is_leveln_node n x) \<and> length sf \<le> 4"
set_homE_3: fixes f :: "'b set_arrow"   and U :: "'b set"   and A :: "'b set"   and B :: "'b set" assumes "f \<in> Hom\<^bsub>set_cat U\<^esub> A B" shows "set_dom f = A"
PREFER_D: fixes P :: "'c \<Rightarrow> bool"   and x :: "'c" assumes "PREFER P x" shows "P x"
less_linear: fixes x :: "'a Word.word"   and y :: "'a Word.word" shows "x <s y \<or> x = y \<or> y <s x"
Eq_subst_tm_Iff: fixes t :: "tm"   and u :: "tm"   and i :: "name"   and tm :: "tm" shows "{t EQ u} \<turnstile> subst i t tm EQ subst i u tm"
replicate_mset_minus_replicate_mset_same: fixes m :: "nat"   and x :: "'b"   and n :: "nat" shows "replicate_mset m x - replicate_mset n x = replicate_mset (m - n) x"
i_tj_stack_incr_disc: shows "is_invar (\<lambda>s. \<forall>k<length (tj_stack s). \<forall>j<k. \<delta> s (tj_stack s ! k) < \<delta> s (tj_stack s ! j))"
pointermap_getmki_def: fixes a :: "'b"   and m :: "'b pointermap_impl" shows "pointermap_getmki a m \<equiv> hm_lookup a (getentryi m) \<bind> (\<lambda>lo. case lo of None \<Rightarrow> return (snd (entriesi m)) \<bind> (\<lambda>p. arl_append (entriesi m) a \<bind> (\<lambda>ent. hm_update a p (getentryi m) \<bind> (\<lambda>lut. return \<lparr>entriesi = ent, getentryi = lut\<rparr> \<bind> (\<lambda>u. return (p, u))))) | Some l \<Rightarrow> return (l, m))"
stationary_distribution_exists: shows "\<exists>x. stationary_distribution (pmf_of_st_vec x)"
expansion_level_ms_def: fixes uu :: "'a ms itself" shows "expansion_level uu = Suc (expansion_level TYPE('a))"
p_subid_par_eq_meet_var: fixes x :: "'a"   and y :: "'a" shows "x \<cdot> 1\<^sub>\<pi> \<parallel> (y \<cdot> 1\<^sub>\<pi>) = x \<cdot> 1\<^sub>\<pi> \<sqinter> y \<cdot> 1\<^sub>\<pi>"
Resolution_unnecessary: fixes T :: "'a literal set set"   and S :: "'a literal set set"   and D :: "'a literal set" assumes "\<forall>C\<in>T. S \<turnstile> C" shows "(T \<union> S \<turnstile> D) = (S \<turnstile> D)"
induct: fixes x1 :: "'var"   and x2 :: "'node SDG_node"   and P :: "'var \<Rightarrow> 'node SDG_node \<Rightarrow> bool" assumes "x1 \<in> Use\<^bsub>SDG\<^esub> x2"   and "\<And>m V n. \<lbrakk>valid_node m; V \<in> Use m; n = CFG_node m\<rbrakk> \<Longrightarrow> P V n"   and "\<And>n m x V. \<lbrakk>valid_SDG_node n; n = Actual_in (m, x); V \<in> ParamUses m ! x\<rbrakk> \<Longrightarrow> P V n"   and "\<And>n m x p ins outs V. \<lbrakk>valid_SDG_node n; n = Formal_out (m, x); get_proc m = p; (p, ins, outs) \<in> set procs; V = outs ! x\<rbrakk> \<Longrightarrow> P V n" shows "P x1 x2"
valid_reqs1: fixes m :: "'b SecurityInvariant_configured"   and M :: "'b SecurityInvariant_configured list" assumes "valid_reqs (m # M)" shows "configured_SecurityInvariant m"
Fvars_fm_simps_2: fixes x :: "tm"   and y :: "tm" shows "Fvars (x EQ y) = Fvars x \<union> Fvars y"
least: shows "P \<turnstile> C has least M = (Ts, T, pns, body) via Cs'"
arr_char: fixes f :: "'a \<times> 'a \<times> 'a" shows "H.VVV.arr f = (V.arr (fst f) \<and> H.VV.arr (snd f) \<and> local.dom (fst f) = cod (fst (snd f)))"
GSMP_def: fixes arity :: "'a \<Rightarrow> nat"   and public :: "'a \<Rightarrow> bool"   and Ana :: "('a, 'b) Term.term \<Rightarrow> ('a, 'b) Term.term list \<times> ('a, 'b) Term.term list"   and \<Gamma> :: "('a, 'b) Term.term \<Rightarrow> ('a, 'c) Term.term"   and label_witness1 :: "'d"   and label_witness2 :: "'d"   and P :: "('a, 'b) Term.term set" assumes "labeled_typed_model arity public Ana \<Gamma> label_witness1 label_witness2" shows "labeled_typed_model.GSMP arity Ana \<Gamma> P \<equiv> {t \<in> typed_model.SMP arity Ana \<Gamma> P. fv t = {}}"
bijection_def: fixes f :: "'c \<Rightarrow> 'c" shows "bijection f \<equiv> bij f"
Collect_neg_eq: fixes P :: "'b \<Rightarrow> bool" shows "{x. \<not> P x} = - {x. P x}"
XD_smallest_def: fixes ds :: "'b set"   and X :: "'a set" shows "Singleton_for_d.XD_smallest ds X = - (Singleton_for_d.XH_largest ds X - X)"
id: fixes i :: "nat" shows "Axp [] ( \<LM> At i  \<RM> \<Rightarrow>* \<LM> At i  \<RM>)"
qe_pres\<^sub>1_def: fixes as :: "atom list" shows "qe_pres\<^sub>1 as = (let ds = filter is_dvd as; d = zlcms (map divisor ds); ls = lbounds as in if ls = [] then Disj [0..d - 1] (\<lambda>n. list_conj (map (Atom \<circ> asubst n []) ds)) else Disj ls (\<lambda>(li, lks). Disj [0..d - 1] (\<lambda>n. list_conj (map (Atom \<circ> asubst (li + n) (- lks)) as))))"
Sel_lower_bound: fixes A :: "(symbol \<times> character list) set"   and B :: "(symbol \<times> character list) set" assumes "A \<subseteq> B" shows "A \<subseteq> Sel A B"
irrefl: fixes a :: "'t" shows "\<not> a \<prec>\<^sub>t a"
emp_shift: fixes cl :: "('test, 'atom, 'choice) cmd list"   and n :: "nat"   and I :: "nat set" shows "(local.shift cl n I = {}) = (I = {})"
assms: fixes e :: "real" assumes "0 < e" shows "\<exists>B. \<forall>x. B \<le> norm x \<longrightarrow> dist (f x) l \<le> e"
assms: shows "z\<^sup>2 \<noteq> 1"
weak_join_assoc: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<in> carrier L"   and "y \<in> carrier L"   and "z \<in> carrier L" shows "x \<squnion> y \<squnion> z .= x \<squnion> (y \<squnion> z)"
mset_subset_eq_mono_add_right_cancel: fixes A :: "'k multiset"   and C :: "'k multiset"   and B :: "'k multiset" shows "(A + C \<subseteq># B + C) = (A \<subseteq># B)"
empty_valution_is_False: fixes v :: "sat_plan_variable" shows "\<A>\<^sub>0 v = False"
indspace_fspanning_set: fixes vs :: "'v list"   and hs :: "'g list" assumes "set vs \<subseteq> V"   and "V = R_scalar_mult.RSpan UNIV (\<sharp>\<cdot>) vs"   and "Supgroup.is_rcoset_replist G hs" shows "indV = R_scalar_mult.RSpan UNIV (\<currency>\<currency>) (concat (negHorbit_list hs induced_vector vs))"
connected_contains_Icc: fixes A :: "'c set"   and a :: "'c"   and b :: "'c" assumes "connected A"   and "a \<in> A"   and "b \<in> A" shows "{a..b} \<subseteq> A"
convex_hull_caratheodory_aff_dim: fixes p :: "'b set" shows "convex hull p = {y. \<exists>S u. finite S \<and> S \<subseteq> p \<and> int (card S) \<le> aff_dim p + 1 \<and> (\<forall>x\<in>S. 0 \<le> u x) \<and> sum u S = 1 \<and> (\<Sum>v\<in>S. u v *\<^sub>R v) = y}"
reach_equiv_beh_states: fixes T :: "('a, 'b) TS_scheme" shows "reach T = \<Union> (set ` beh T)"
mark_loop_mo_locs_def: shows "mark_loop_mo_locs = prefixed ''mark_loop_mo''"
recurse_def: fixes f :: "'c \<Rightarrow> ('d, 'e) state" shows "recurse f = un_B \<circ> traverse_rev (B \<circ> f)"
hyps: fixes a :: "nat"   and b :: "nat" assumes "a + b < aa__ + ba__" shows "P a b"
Total_Id_Field: fixes r :: "('b \<times> 'b) set" assumes "Total r"   and "\<not> r \<subseteq> Id" shows "Field r = Field (r - Id)"
extremum_unique: fixes a :: "'g" shows "(a \<le> bot) = (a = bot)"
CD_on_def: fixes ds :: "'b set"   and X :: "'a set" shows "Bossy'.CD_on ds X = (\<Union>d\<in>ds. Bossy'.Cd d X)"
fun_lub_def: fixes L :: "'f set \<Rightarrow> 'e"   and A :: "('d \<Rightarrow> 'f) set" shows "fun_lub L A = (\<lambda>x. L {y. \<exists>f\<in>A. y = f x})"
prems_3: shows "P \<turnstile> \<langle>blocks (p # ps', Ts, vs, e),(h, l, sh)\<rangle> \<Rightarrow> \<langle>e',(h', l', sh')\<rangle>"
model_def: fixes e :: "nat \<Rightarrow> 'a"   and f :: "'b \<Rightarrow> 'a list \<Rightarrow> 'a"   and g :: "'c \<Rightarrow> 'a list \<Rightarrow> bool"   and ps :: "('b, 'c) form list"   and p :: "('b, 'c) form" shows "(e,f,g,ps \<Turnstile> p) = (list_all (eval e f g) ps \<longrightarrow> eval e f g p)"
simps: fixes a :: "trm \<Rightarrow> (char list \<times> trm) list" shows "(a \<in> rctxt) = ((\<exists>E l fs. a = (\<lambda>t. (l, E t) \<Colon> fs) \<and> E \<in> ctxt) \<or> (\<exists>v E l. a = (\<lambda>t. (l, v) \<Colon> E t) \<and> v \<in> value \<and> E \<in> rctxt))"
inter_Gp_nsg: fixes G :: "('e, 'f) Group_scheme"   and N :: "'e set"   and H :: "'e set" assumes "Group G"   and "G \<triangleright> N"   and "G \<guillemotright> H " shows "\<natural>\<^bsub>G\<^esub>H \<triangleright> H \<inter> N"
eq_split: fixes P :: "'a upred"   and Q :: "'a upred" assumes "`P \<Rightarrow> Q`"   and "`Q \<Rightarrow> P`" shows "P = Q"
nxtAct_active: fixes c :: "'id"   and t :: "nat \<Rightarrow> cnf"   and i :: "nat" assumes "\<parallel>c\<parallel>\<^bsub>t i\<^esub>" shows "\<langle>c \<rightarrow> t\<rangle>\<^bsub>i\<^esub> = i"
assms: fixes x :: "'a"   and e :: "real" assumes "x \<in> B"   and "0 < e" shows "\<exists>y\<in>B. norm (y - x) \<le> e \<and> y \<succeq>[P] x \<and> (x, y) \<notin> P"