hcju/isabelleps huggingface dataset:
restrict_nonempty_eq: fixes k :: "'c"   and ks :: "'c list"   and f :: "'c \<times> 'c \<Rightarrow> 'd"   and l :: "'c"   and ls :: "'c list"   and g :: "'c \<times> 'c \<Rightarrow> 'd" shows "((k # ks) \<langle>f\<rangle> (l # ls) = (k # ks) \<langle>g\<rangle> (l # ls)) = ([k] \<langle>f\<rangle> [l] = [k] \<langle>g\<rangle> [l] \<and> [k] \<langle>f\<rangle> ls = [k] \<langle>g\<rangle> ls \<and> ks \<langle>f\<rangle> [l] = ks \<langle>g\<rangle> [l] \<and> ks \<langle>f\<rangle> ls = ks \<langle>g\<rangle> ls)"
simps: fixes cms :: "(('Var, 'AExp, 'BExp) Stmt \<times> (Mode \<Rightarrow> 'Var set)) list"   and mem :: "'Var \<Rightarrow> 'Val" shows "sound_mode_use (cms, mem) = (list_all (\<lambda>cm. locally_sound_mode_use (cm, mem)) cms \<and> globally_sound_mode_use (cms, mem))"
CFG_SSA_wf_defs_10: fixes g :: "'g"   and v :: "'val" shows "trivial g v \<equiv> \<exists>v'\<in>allVars g. isTrivialPhi g v v'"
C1: fixes R :: "('a, 'b, 'c) psi" shows "(\<Psi>, R, R) \<in> Rel"
rel_state_elim: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and f :: "(heap, 'c) state"   and g :: "['d]"   and heap :: "heap" assumes "local.rel_state R f g"   and "Q heap" obtains heap' :: "heap" and v :: "'c" and v' :: "'d" where "run_state f heap = (v, heap')" and "execute g heap = Some (v', heap')" and "R v v'" and "Q heap'"
path_connected_space_topspace_empty: fixes X :: "'c topology" assumes "topspace X = {}" shows "path_connected_space X"
hcomp_in_hhom': fixes \<mu> :: "'a"   and \<nu> :: "'a"   and a :: "'a"   and c :: "'a" assumes "arr \<mu>"   and "arr \<nu>"   and "src \<mu> = a"   and "trg \<nu> = c"   and "src \<nu> = trg \<mu>" shows "\<guillemotleft>\<nu> \<star> \<mu> : a \<rightarrow> c\<guillemotright>"
compEs1_conv_map: fixes Vs :: "String.literal list"   and es :: "'a expr list" shows "compEs1 Vs es = map (compE1 Vs) es"
x_not_i: shows "x \<noteq> mod_type_class.to_nat i"
post_def: fixes C :: "'b acom" shows "post C = last (annos C)"
induct: fixes P :: "binding \<Rightarrow> bool"   and binding :: "binding" assumes "\<And>vrs ty. P (VarB vrs ty)"   and "\<And>tyvrs ty. P (TVarB tyvrs ty)" shows "P binding"
op_induct': fixes P :: "[:'r:] \<Rightarrow> bool"   and us :: "[:'r:]" assumes "\<And>us. (\<And>us'. lt\<cdot>(slength\<cdot>us')\<cdot>(slength\<cdot>us) = TT \<Longrightarrow> P us') \<Longrightarrow> P us" shows "P us"
crit_pair_cs_imp_crit_pair_cbelow_on: fixes d :: "'a \<Rightarrow> nat"   and F :: "('t \<Rightarrow>\<^sub>0 'c) set"   and m :: "nat"   and p :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c" assumes "dickson_grading d"   and "F \<subseteq> dgrad_p_set d m"   and "p \<in> dgrad_p_set d m"   and "q \<in> dgrad_p_set d m"   and "relation.cs (red F) (fst (crit_pair p q)) (snd (crit_pair p q))" shows "crit_pair_cbelow_on d m F p q"
fcomp_comp: fixes f :: "'a \<Rightarrow> 'c"   and g :: "'c \<Rightarrow> 'b" shows "fcomp f g = g \<circ> f"
filtrated_prob_space_axioms: shows "filtrated_prob_space M F"
subset: shows "\<forall>clause. clause el formula' \<longrightarrow> clause el formula"
minus_ms_def: fixes F :: "'a ms"   and G :: "'a ms" shows "F - G = F + - G"
dg_Rel_Dom_app: fixes \<alpha> :: "V"   and x :: "V" assumes "arr_Rel \<alpha> x" shows "dg_Rel \<alpha>\<lparr>Dom\<rparr>\<lparr>x\<rparr> = x\<lparr>ArrDom\<rparr>"
that: fixes c :: "'x" assumes "c \<sharp> x" shows "thesis"
If: fixes f :: "'f"   and A :: "'t"   and B :: "'t" assumes "f \<in> Sig iS : A \<rightarrow> B" shows "Fn\<lbrakk>f\<rbrakk> maps\<^bsub>iC\<^esub> Ty\<lbrakk>A\<rbrakk> to Ty\<lbrakk>B\<rbrakk>"
map_prod_surj_on: fixes f :: "'e \<Rightarrow> 'd"   and A :: "'e set"   and A' :: "'d set"   and g :: "'g \<Rightarrow> 'f"   and B :: "'g set"   and B' :: "'f set" assumes "f ` A = A'"   and "g ` B = B'" shows "map_prod f g ` (A \<times> B) = A' \<times> B'"
HEndPhase1_HInv4d_q: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and q :: "Proc" assumes "HEndPhase1 s s' p"   and "HInv4d s q"   and "p \<noteq> q" shows "HInv4d s' q"
empty_correct_1: shows "rs.\<alpha> (rs.empty ()) = {}"
ec': shows "simAbs ec' \<in> simAbs ` set X"
Always_weaken: fixes F :: "'b program"   and A :: "'b set"   and B :: "'b set" assumes "F \<in> Always A"   and "A \<subseteq> B" shows "F \<in> Always B"
one_reducibility_via_is_many: fixes A :: "nat set"   and B :: "nat set"   and f :: "nat \<Rightarrow> nat" assumes "one_reducible_to_via A B f" shows "many_reducible_to_via A B f"
numeral_nat_3: fixes n :: "num" shows "numeral (num.Bit1 n) = Suc (numeral (num.Bit0 n))"
simps_8: fixes I1 :: "WS1S_Nameful.fo \<Rightarrow> nat"   and I2 :: "WS1S_Nameful.so \<Rightarrow> nat fset"   and x :: "WS1S_Nameful.fo"   and \<phi> :: "WS1S_Nameful.ws1s" shows "satisfies I1 I2 (All1 x \<phi>) = (\<forall>n. satisfies (I1(x := n)) I2 \<phi>)"
Aki0: shows "A $ from_nat k $ from_nat i = (0::'a)"
mon_w_def: fixes fg :: "('b, 'c, 'd, 'a, 'e) flowgraph_rec_scheme"   and w :: "('c, 'f) label list" shows "mon_w fg w \<equiv> \<Union> {mon_e fg e |e. e \<in> set w}"
left_diff_distrib: fixes x :: "(real, 3) vec"   and y :: "(real, 3) vec"   and z :: "(real, 3) vec" shows "(x - y) \<times> z = x \<times> z - y \<times> z"
s_def: shows "s \<equiv> \<lambda>x. if findr x \<down>= e_length x then Some 0 else Some (e_take (Suc (the (findr x))) x)"
run_strict: shows "Fixrec.run\<cdot>\<bottom> = \<bottom>"
nn_integral_count_space_reindex: fixes f :: "'k \<Rightarrow> 'l"   and A :: "'k set"   and g :: "'l \<Rightarrow> ennreal" assumes "inj_on f A" shows "integral\<^sup>N (count_space (f ` A)) g = (\<Sum>\<^sup>+ x\<in>A. g (f x))"
assms: shows "u \<rightarrow>\<^bsub>G\<^esub> v"
res_opp_fold_A_\<C>_A_im_opp_fold_A_\<C>_A_im: shows "restrict1 opp_fold_A (\<Union> A) \<turnstile> ChamberComplex.chamber_system A = opp_fold_A \<turnstile> ChamberComplex.chamber_system A"
assms_2: shows "enat x = \<langle>c #\<^bsub>enat n'\<^esub>t\<rangle>"
subsequence_nth: fixes k :: "nat"   and j :: "nat"   and i :: "nat"   and w :: "nat \<Rightarrow> 'b" assumes "k < j - i" shows "w [i \<rightarrow> j] ! k = w (i + k)"
set_to_list_def: shows "set_to_list \<equiv> inv set"
coeff_not_in_monomials: fixes x :: "nat \<Rightarrow>\<^sub>0 nat"   and m :: "'a mpoly" assumes "x \<notin> monomials m" shows "MPoly_Type.coeff m x = (0::'a)"
c'_def: shows "c' = Inf ({a..b} \<inter> g -` {c..d})"
ground_hd_in_ground_heads: fixes s :: "('s, 'v) tm" assumes "ground s" shows "hd.sym (head s) \<in> local.ground_heads (head s)"
Pair_eqD2: fixes x :: "'a"   and y :: "'b"   and x' :: "'a"   and y' :: "'b" assumes "(x, y) = (x', y')" shows "y = y'"
subset_Zorn_nonempty: fixes \<A> :: "'c set set" assumes "\<A> \<noteq> {}"   and "\<And>\<C>. \<lbrakk>\<C> \<noteq> {}; subset.chain \<A> \<C>\<rbrakk> \<Longrightarrow> \<Union> \<C> \<in> \<A>" shows "\<exists>M\<in>\<A>. \<forall>X\<in>\<A>. M \<subseteq> X \<longrightarrow> X = M"
\<theta>_img: shows "subst_range \<theta> = insert t (subst_range \<sigma>)"
lcm_mult_gcd: fixes a :: "'a"   and b :: "'a" shows "lcm a b * gcd a b = normalize a * normalize b"
tmax_1: fixes t :: "real" assumes "t \<in> cball t__ et" shows "t \<le> tmax"
Abs_swap_fresh: fixes X :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and xs :: "'varSort"   and x' :: "'var"   and x :: "'var" assumes "good X"   and "fresh xs x' X" shows "Abs xs x X = Abs xs x' (X #[x' \<and> x]_xs)"
get_M\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t_ok: fixes h :: "(_) heap"   and ptr :: "(_) document_ptr"   and getter :: "('n, 'l, 'm, 'u option) RDocument_scheme \<Rightarrow> 'w" assumes "DocumentClass.type_wf h"   and "ptr |\<in>| document_ptr_kinds h" shows "h \<turnstile> ok get_M\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t ptr getter"
nrename_OrR1: fixes R :: "trm"   and x :: "name"   and y :: "name"   and c :: "coname"   and N :: "trm"   and d :: "coname" assumes "R[x\<turnstile>n>y] = OrR1 <c>.N d"   and "c \<sharp> (R, d)" shows "\<exists>N'. R = OrR1 <c>.N' d \<and> N'[x\<turnstile>n>y] = N"
hyps_1: fixes c2 :: "('a, 'b, 'c) com"   and c :: "('a, 'b, 'c) com"   and s :: "('a, 'c) xstate" assumes "(a1 \<inter>\<^sub>g c2) = Some c"   and "\<Gamma>\<turnstile>a1 \<down> s" shows "\<Gamma>\<turnstile>c \<down> s"
S: shows "openin (topology (arbitrary union_of (finite intersection_of P relative_to U))) S"
F_start: shows "F ds (XD_smallest ds X, dual (XH_largest ds X)) = (XD_smallest ds X, dual (XH_largest ds X))"
apply_comp_disjunctive: shows "Apply.disjunctive = Comp.disjunctive"
cINF_less_iff: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and A :: "'b set"   and f :: "'b \<Rightarrow> 'a"   and a :: "'a" assumes "class.conditionally_complete_linorder Inf Sup inf less_eq less sup"   and "A \<noteq> {}"   and "preorder.bdd_below less_eq (f ` A)" shows "less (Inf (f ` A)) a = (\<exists>x\<in>A. less (f x) a)"
Suc_numeral: fixes n :: "Num.num" shows "Suc (numeral n) = numeral (n + num.One)"
Nat_casedist: fixes x :: "nat discr Box" obtains "x = \<bottom>"   | "x = 0"   | n :: "nat discr Box" where "x = n + 1"
assms: shows "\<forall>PROB as s. finite PROB \<and> P PROB \<and> s \<in> valid_states PROB \<and> as \<in> valid_plans PROB \<longrightarrow> (\<exists>as'. exec_plan s as = exec_plan s as' \<and> subseq as' as \<and> length as' < f PROB s)"
arr_cod_iff_arr: fixes f :: "'s" shows "arr (cod f) = arr f"
f12_def: shows "f12 = fst (split_face f' v a ws)"
assms_1: shows "local_ring_morphism f A addA multA zeroA oneA B addB multB zeroB oneB"
wf: shows "\<forall>y. (y, x) \<in> measure FOL_Fitting.size_form \<longrightarrow> (y \<in> H \<longrightarrow> closed 0 y \<longrightarrow> eval e HApp (\<lambda>a ts. Pred a (terms_of_hterms ts) \<in> H) y) \<and> (Neg y \<in> H \<longrightarrow> closed 0 y \<longrightarrow> eval e HApp (\<lambda>a ts. Pred a (terms_of_hterms ts) \<in> H) (Neg y))"
D: shows "\<forall>A\<in>fst (snd SK). \<exists>i C. (Asset i, \<lbrace>Num 2, PubKey A\<rbrace>) \<in> s \<and> (Asset i, Token i (Auth_PriK i) A C SK) \<in> s"
swap_apply_3: fixes c :: "'a"   and a :: "'a"   and b :: "'a"   and f :: "'a \<Rightarrow> 'b" assumes "c \<noteq> a"   and "c \<noteq> b" shows "Fun.swap a b f c = f c"
map\<^sub>0_in_hom_2: fixes a :: "'a" assumes "C.obj a" shows "\<guillemotleft>\<sigma>.map\<^sub>0 a : \<sigma>.map\<^sub>0 a \<Rightarrow>\<^sub>D \<sigma>.map\<^sub>0 a\<guillemotright>"
prems: shows "\<forall>y. y \<in> set (a # xs) \<longrightarrow> (0::'a) < y"
refine: fixes n :: "nat"   and Xi :: "real list \<times> real list"   and X :: "'b set"   and sctni :: "real list sctn"   and sctn :: "'b sctn" assumes "DIM_precond TYPE('b) n"   and "(Xi, X) \<in> lvivl_rel"   and "(sctni, sctn) \<in> \<langle>lv_rel\<rangle>sctn_rel" shows "(nres_of (split_ivl_at_halfspace_impl n sctni Xi), split_ivl_at_halfspace sctn X) \<in> \<langle>lvivl_rel \<times>\<^sub>r lvivl_rel\<rangle>nres_rel"
mono_neutral_cong_left: fixes T :: "'a set"   and S :: "'a set"   and h :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "finite T"   and "S \<subseteq> T"   and "\<forall>i\<in>T - S. h i = (0::'b)"   and "\<And>x. x \<in> S \<Longrightarrow> g x = h x" shows "sum g S = sum h T"
is_parallel_solution_for_problem_def: fixes \<Pi> :: "('b, 'c) strips_problem_scheme"   and \<pi> :: "'b strips_operator list list" shows "is_parallel_solution_for_problem \<Pi> \<pi> \<equiv> \<Pi>\<^sub>G \<subseteq>\<^sub>m execute_parallel_plan (\<Pi>\<^sub>I) \<pi> \<and> list_all (list_all (\<lambda>op. ListMem op (\<Pi>\<^sub>\<O>))) \<pi>"
fold_rec: fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and A :: "'a set"   and x :: "'a"   and z :: "'b" assumes "comp_fun_commute f"   and "finite A"   and "x \<in> A" shows "Finite_Set.fold f z A = f x (Finite_Set.fold f z (A - {x}))"
assms_2: shows "f ` cbox (0::'a) One \<subseteq> cbox (0::'a) One"
zero_le_numeral: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and n :: "num" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "less_eq zero (numeral.numeral one plus n)"
lower_closed: fixes x :: "'a" assumes "x \<in> carrier \<X>" shows "\<pi>\<^sup>* x \<in> carrier \<Y>"
finprod_one_eqI: fixes A :: "'c set"   and f :: "'c \<Rightarrow> 'a" assumes "\<And>x. x \<in> A \<Longrightarrow> f x = \<zero>" shows "finsum R f A = \<zero>"
Suc_div_unat_helper: fixes sz :: "nat"   and us :: "nat" assumes "sz < LENGTH('a)"   and "us \<le> sz" shows "2 ^ (sz - us) = Suc (unat ((2 ^ sz - 1) div 2 ^ us))"
lexordp_simps_3: fixes less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and xs :: "'a list"   and y :: "'a"   and ys :: "'a list" shows "ord.lexordp less (x # xs) (y # ys) = (less x y \<or> \<not> less y x \<and> ord.lexordp less xs ys)"
convex_box_1: fixes a :: "'c"   and b :: "'c" shows "convex (cbox a b)"
Cons_3: shows "SubRel.subpath (red prb) rv1 (re # res) rv2 (subs prb)"
distinct_9: fixes x2 :: "nat"   and x4 :: "nat" shows "WAIT x2 \<noteq> REL x4"
Red_I_to_Inf: fixes N :: "('f \<times> 'l) set" shows "Red_I N \<subseteq> Inf_FL"
col_permutation_3: fixes A :: "'p"   and B :: "'p"   and C :: "'p" assumes "Col A B C" shows "Col C B A"
xz2: shows "- det2 a11 a12 a21 a22 * x2 + det2 a12 a13 a22 a23 * z2 = (0::'a)"
Gl: fixes X :: "real \<Rightarrow> 'a" assumes " D X = (\<lambda>\<tau>. f \<tau> (X \<tau>)) on U (X t\<^sub>0)" shows "\<forall>\<tau>\<in>U (X t\<^sub>0). \<tau> < t\<^sub>0 \<longrightarrow> \<mu>' (X \<tau>) \<le> \<nu>' (X \<tau>)"
hyps_3: assumes "f \<in> A \<rightarrow> carrier G" shows "finprod G f A \<in> carrier G"
ps_chains_def: fixes G :: "diagram" shows "ps_chains G \<equiv> mk_ps_chain \<lbrace> initial_ps G \<rbrace> ` lins G"
C_eq_s_ext: fixes p :: "((int \<times> int) set set, int) Combinators list"   and a :: "int \<times> (int \<times> int) \<times> (int \<times> int) \<times> DummyContent" assumes "p \<noteq> []" shows "C (list2FWpolicy (separate p)) a = C (list2FWpolicy p) a"
simps_3: fixes v :: "bool" shows "fromA (C v) = None"
as_needed_asses_submap_exec_v: fixes v :: "'a"   and f :: "('a, 'b) fmap"   and g :: "('a, 'b) fmap" assumes "v \<in> fmdom' f"   and "f \<subseteq>\<^sub>f g" shows "v \<in> fmdom' g"
ind: assumes "steps0 (Suc 0, Bk \<up> l, <lm>) tp stp = (0, Bk \<up> m, Oc \<up> rs @ Bk \<up> n)" shows "\<exists>stp. nonstop (code tp) (bl2wc (<lm>)) stp = 0 \<and> (\<forall>stp'. nonstop (code tp) (bl2wc (<lm>)) stp' = 0 \<longrightarrow> stp \<le> stp')"
Domainp_applyI: fixes A :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and B :: "'i \<Rightarrow> 'j \<Rightarrow> bool"   and f :: "'g \<Rightarrow> 'i"   and g :: "'h \<Rightarrow> 'j"   and x :: "'g"   and y :: "'h" assumes "(A ===> B) f g"   and "A x y" shows "Domainp B (f x)"
assms_3: shows "homotopic_with (\<lambda>h. True) X X (g \<circ> f) id"
det_code_def: fixes A :: "'a mat" shows "local.det_code A = (if dim_row A = dim_col A then case local.triangulize A of (m, A') \<Rightarrow> prod_list (diag_mat A') div m else (0::'a))"
lunit_simps_3: fixes f :: "'a" assumes "B.ide f" shows "src (B.lunit f) = src f"
surjective: fixes r :: "('b, 'c) nfa_scheme" shows "r = \<lparr>nfa.states = nfa.states r, init = nfa.init r, final = nfa.final r, nxt = nfa.nxt r, eps = eps r, \<dots> = nfa.more r\<rparr>"
pre: shows "wf_matchF_aux \<sigma> V n R I r aux ne (length (fst st) - 1)"
simps_11: fixes op :: "'opubx" shows "norm_instr (IOpUbx op) = Inca.instr.IOpInl (\<BB>\<oo>\<xx> op)"
enum_all_H2_def: fixes P :: "H2 \<Rightarrow> bool" shows "enum_class.enum_all P = (P H1 \<and> P H2)"
less_add_same_cancel1: fixes a :: "'k multiset"   and b :: "'k multiset" shows "(a \<subset># a + b) = ({#} \<subset># b)"
sub_def: fixes k :: "num"   and l :: "num" shows "sub k l = (if l \<le> k then Some (numeral k - numeral l) else None)"
closed_complex_atLeast: fixes a :: "complex" shows "closed {a..}"
offending_notevalD: fixes F :: "('c \<times> 'c) set"   and sinvar :: "'c graph \<Rightarrow> ('c \<Rightarrow> 'd) \<Rightarrow> bool"   and G :: "'c graph"   and nP :: "'c \<Rightarrow> 'd" assumes "F \<in> SecurityInvariant_withOffendingFlows.set_offending_flows sinvar G nP" shows "\<not> sinvar G nP"
span_finite: fixes scale :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and S :: "'b set" assumes "module scale"   and "finite S" shows "module.span scale S = range (\<lambda>u. \<Sum>v\<in>S. scale (u v) v)"
split: fixes P :: "'b \<Rightarrow> bool"   and f1 :: "'b"   and f2 :: "'c \<Rightarrow> 'b"   and option :: "'c option" shows "P (case option of None \<Rightarrow> f1 | \<lfloor>x\<rfloor> \<Rightarrow> f2 x) = ((option = None \<longrightarrow> P f1) \<and> (\<forall>x2. option = \<lfloor>x2\<rfloor> \<longrightarrow> P (f2 x2)))"
hs_dfs_invar_def: fixes \<Sigma>i :: "('a, unit) HashMap.hashmap"   and post :: "'a \<Rightarrow> 'a dlist" shows "hs_dfs_invar \<Sigma>i post \<equiv> {s. hs_dfs_\<alpha> s \<in> dfs_invar (hs.\<alpha> \<Sigma>i) (hs_R post)}"
NSCauchy_NSconvergent_iff: fixes X :: "nat \<Rightarrow> 'a" shows "NSCauchy X = NSconvergent X"
gen_lossless_bind_gpvI: fixes co :: "bool"   and \<I> :: "('d, 'e) \<I>"   and gpv :: "('f, 'd, 'e) gpv"   and f :: "'f \<Rightarrow> ('g, 'd, 'e) gpv" assumes "gen_lossless_gpv co \<I> gpv"   and "\<And>x. x \<in> results_gpv \<I> gpv \<Longrightarrow> gen_lossless_gpv co \<I> (f x)" shows "gen_lossless_gpv co \<I> (gpv \<bind> f)"
abelian_group_def: fixes G :: "('b, 'c) ring_scheme" shows "abelian_group G \<equiv> abelian_monoid G \<and> abelian_group_axioms G"
simps_2: fixes a :: "'b wordinterval"   and b :: "'b wordinterval" shows "wordinterval_card (RangeUnion a b) = wordinterval_card a + wordinterval_card b"
assms_4: shows "h \<turnstile> remove_child ptr' child \<rightarrow>\<^sub>h h'"
obj_simps_1: fixes a :: "'a" assumes "S.obj a" shows "S.arr a"
ZMod_mod: fixes m :: "int"   and a :: "int" shows "ZMod m a = ZMod m (a mod m)"
valid_cfg_def: shows "R_G.valid_cfg = \<Union> (R_G.cfg_on ` \<S>)"
in_minus_Collect: fixes a :: "'b"   and A :: "'b set"   and b :: "'b"   and B :: "'b set" assumes "a \<in> A"   and "b \<in> B" shows "a - b \<in> {x - y |x y. x \<in> A \<and> y \<in> B}"
CD_on_Xd: fixes x :: "'a"   and ds :: "'b set"   and A :: "'a set" assumes "x \<in> Singleton_for_d.CD_on ds A" shows "Xd x \<in> ds"
bLeq: shows "sint (SCAST(32 \<rightarrow> 64) w2) \<le> 2 ^ 31"
init_sim: fixes R :: "('h \<times> 'i) set"   and A :: "('h, 'j) graph_rec_scheme"   and B :: "('i, 'k) graph_rec_scheme"   and a0 :: "'h" assumes "simulation R A B"   and "a0 \<in> g_V0 A" shows "\<exists>b0. b0 \<in> g_V0 B \<and> (a0, b0) \<in> R"
cases: fixes x :: "32 Word.word \<times> 32 Word.word" obtains w1 :: "32 Word.word" and w2 :: "32 Word.word" where "x = (w1, w2)"
CFG_SSA_Transformed_code_def: fixes \<alpha>e :: "'a \<Rightarrow> ('b \<times> 'c \<times> 'b) set"   and \<alpha>n :: "'a \<Rightarrow> 'b list"   and invar :: "'a \<Rightarrow> bool"   and inEdges' :: "'a \<Rightarrow> 'b \<Rightarrow> ('b \<times> 'c) list"   and Entry :: "'a \<Rightarrow> 'b"   and oldDefs :: "'a \<Rightarrow> 'b \<Rightarrow> 'd set"   and oldUses :: "'a \<Rightarrow> 'b \<Rightarrow> 'd set"   and defs :: "'a \<Rightarrow> 'b \<Rightarrow> 'e set"   and "uses" :: "'a \<Rightarrow> ('b, 'e set) mapping"   and phis :: "'a \<Rightarrow> ('b \<times> 'e, 'e list) mapping"   and var :: "'a \<Rightarrow> 'e \<Rightarrow> 'd" shows "CFG_SSA_Transformed_code \<alpha>e \<alpha>n invar inEdges' Entry oldDefs oldUses defs uses phis var \<equiv> CFG_SSA_wf_code \<alpha>e \<alpha>n invar inEdges' Entry defs uses phis \<and> CFG_SSA_Transformed \<alpha>e \<alpha>n invar inEdges' Entry oldDefs oldUses defs (usesOf \<circ> uses) (\<lambda>g. Mapping.lookup (phis g)) var \<and> CFG_SSA_Transformed_code_axioms \<alpha>n uses"
nprv_allE0: fixes x :: "'var"   and \<phi> :: "'fmla"   and F :: "'fmla set"   and t :: "'trm"   and \<psi> :: "'fmla" assumes "all x \<phi> \<in> F"   and "nprv (insert (subst \<phi> t x) F) \<psi>"   and "F \<subseteq> fmla"   and "finite F"   and "\<phi> \<in> fmla"   and "t \<in> trm"   and "x \<in> var"   and "\<psi> \<in> fmla" shows "nprv F \<psi>"
nsI: fixes u :: "'b list"   and v :: "'b list" assumes "u \<noteq> \<epsilon>"   and "u \<le>s v" shows "u \<le>ns v"
part_disjoint_from_complement: fixes x :: "i"   and y :: "i" assumes "\<exists>v. \<not> O v x"   and "P y x" shows "\<not> O y (\<midarrow> x)"
assms_2: fixes t :: "real" assumes "0 < t"   and "t < 1" shows "Re (p1 t - z1) = Re (p2 t - z2)"
real_sqrt_ge_one: fixes x :: "real" assumes "1 \<le> x" shows "1 \<le> sqrt x"
nextl_init_state: fixes M :: "'b dfa"   and xs :: "'b list" assumes "dfa M" shows "dfa.nextl M (dfa.init M) xs \<in> dfa.states M"
items_le_def: fixes \<NN> :: "symbol set"   and \<TT> :: "symbol set"   and \<RR> :: "(symbol \<times> symbol list) set"   and \<SS> :: "symbol"   and Lex :: "symbol \<Rightarrow> character list \<Rightarrow> nat \<Rightarrow> nat set"   and Sel :: "(symbol \<times> character list) set \<Rightarrow> (symbol \<times> character list) set \<Rightarrow> (symbol \<times> character list) set"   and k :: "nat"   and I :: "item set" assumes "LocalLexing \<NN> \<TT> \<RR> \<SS> Lex Sel" shows "items_le k I = {x \<in> I. item_end x \<le> k}"
subdegree_greaterI: fixes f :: "'a fps"   and n :: "nat" assumes "f \<noteq> 0"   and "\<And>i. i \<le> n \<Longrightarrow> fps_nth f i = (0::'a)" shows "n < subdegree f"
class_of_0_eq: fixes U :: "'a set"   and U' :: "'a set" assumes "U \<in> {U. is_open\<^sub>X U \<and> x \<in> U}"   and "U' \<in> {U. is_open\<^sub>X U \<and> x \<in> U}" shows "stx.class_of U (zero_str\<^sub>X U) = stx.class_of U' (zero_str\<^sub>X U')"
MaxI: fixes k :: "'a"   and A :: "'a set" assumes "k \<in> A"   and "finite A" shows "Max A \<in> A"
per_double_cong: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and C' :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Per Bet Cong A B C"   and "Tarski_neutral_dimensionless.Midpoint Bet Cong B C C'" shows "Cong A C A C'"
length_apath: fixes u :: "'a"   and p :: "'b list"   and v :: "'a" assumes "apath u p v" shows "length p \<le> card (verts G)"
root_abs_power: fixes n :: "nat"   and y :: "real" assumes "0 < n" shows "\<bar>root n (y ^ n)\<bar> = \<bar>y\<bar>"
INF_absorb: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and k :: "'c"   and I :: "'c set"   and A :: "'c \<Rightarrow> 'b" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "k \<in> I" shows "inf (A k) (Inf (A ` I)) = Inf (A ` I)"
not_before_in: fixes x :: "'c"   and xs :: "'c list"   and y :: "'c" assumes "x \<in> set xs"   and "y \<in> set xs" shows "(\<not> x < y in xs) = (y < x in xs \<or> x = y)"
lm082: fixes X :: "'a set"   and Y :: "'a set"   and P :: "('a \<times> 'b) set" assumes "X \<inter> Y = {}" shows "Outside' Y P `` X = P `` X"
unity_root_sum_nonzero_iff: fixes k :: "nat"   and r :: "int" assumes "1 \<le> k"   and "r \<in> {- int k<..<int k}" shows "(unity_root_sum k r \<noteq> 0) = (r = 0)"
poly_f_is: shows "poly_f qs = pderiv (prod_list_var qs) * prod_list_var qs * [:real_of_int (- crb (prod_list_var qs)), 1:] * [:real_of_int (crb (prod_list_var qs)), 1:]"
restrict_chart_carrier: fixes x :: "('a, 'b) chart" assumes "x \<in> charts" shows "restrict_chart carrier x = x"
abelian_group_def: fixes G :: "('c, 'd) ring_scheme" shows "abelian_group G \<equiv> abelian_monoid G \<and> abelian_group_axioms G"
chain_not_lnull: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and xs :: "'a llist" assumes "chain R xs" shows "\<not> lnull xs"