hcju/isabelleps huggingface dataset:
B: fixes TRel :: "('procT \<times> 'procT) set" shows "(\<forall>(TP, TQ)\<in>TRel. Pred (TargetTerm TP, TargetTerm TQ)) = (\<forall>(P, Q)\<in>indRelRT TRel. Pred (P, Q))"
nprv_clear3_3: fixes var :: "'a set"   and trm :: "'b set"   and fmla :: "'c set"   and Var :: "'a \<Rightarrow> 'b"   and FvarsT :: "'b \<Rightarrow> 'a set"   and substT :: "'b \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b"   and Fvars :: "'c \<Rightarrow> 'a set"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'c"   and eql :: "'b \<Rightarrow> 'b \<Rightarrow> 'c"   and cnj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and imp :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and all :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and exi :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and fls :: "'c"   and dsj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and num :: "'b set"   and prv :: "'c \<Rightarrow> bool"   and \<phi>1 :: "'c"   and \<phi>2 :: "'c"   and \<psi> :: "'c"   and \<phi>3 :: "'c" assumes "Deduct_with_False_Disj var trm fmla Var FvarsT substT Fvars subst eql cnj imp all exi fls dsj num prv"   and "Deduct_with_False_Disj.nprv cnj imp fls prv {\<phi>1, \<phi>2} \<psi>"   and "\<phi>1 \<in> fmla"   and "\<phi>2 \<in> fmla"   and "\<phi>3 \<in> fmla"   and "\<psi> \<in> fmla" shows "Deduct_with_False_Disj.nprv cnj imp fls prv {\<phi>1, \<phi>2, \<phi>3} \<psi>"
rconjby_add: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "rconjby (x + y) z = rconjby y (rconjby x z)"
assms_1: shows "(prio', prio) \<in> A \<rightarrow> B"
linear: fixes x :: "'c word"   and y :: "'c word" shows "x \<le>s y \<or> y \<le>s x"
cmp_pol_coeff: fixes f :: "'a \<Rightarrow> 'g"   and B :: "('g, 'h) Ring_scheme"   and n :: "nat"   and c :: "nat \<Rightarrow> 'a" assumes "f \<in> rHom S B"   and "pol_coeff S (n, c)" shows "pol_coeff B (n, cmp f c)"
simps_5: fixes a :: "MIR.num"   and b :: "MIR.num" shows "numbound0 (Add a b) = (numbound0 a \<and> numbound0 b)"
inverse_eq_divide: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and a :: "'a" assumes "class.division_ring divide inverse one times plus zero minus uminus" shows "inverse a = divide one a"
\<phi>_simps_2: shows "src f\<^sub>0gh\<^sub>1.\<phi> = src f\<^sub>0gh\<^sub>1.p\<^sub>0"
fs: shows "fs = (if n = 0 then hs else (Polynomial.monom 1 1, n - 1) # hs)"
t: shows "finite t \<and> (\<lambda>x. f x \<bullet> j) C1_differentiable_on s - t"
get: shows "(get, (!)) \<in> arel \<rightarrow> nat_rel \<rightarrow> Id"
GB_subset: fixes G :: "('t \<Rightarrow>\<^sub>0 'c) set"   and G' :: "('t \<Rightarrow>\<^sub>0 'c) set" assumes "is_Groebner_basis G"   and "G \<subseteq> G'"   and "pmdl G' = pmdl G" shows "is_Groebner_basis G'"
eq_iff_diff_eq_0: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.group_add minus plus zero uminus" shows "(a = b) = (minus a b = zero)"
preserves_ide: fixes a :: "'a \<times> 'a" assumes "CC.ide a" shows "ide (T a)"
While_stepsnD: fixes b :: "bexp"   and c :: "com"   and s :: "char list \<Rightarrow> int"   and n :: "nat"   and c2 :: "com"   and t :: "char list \<Rightarrow> int" assumes "(WHILE b DO c, s) \<rightarrow>(n) (c2, t)" shows "c2 \<in> {WHILE b DO c, IF b THEN c;; WHILE b DO c ELSE SKIP, SKIP} \<or> (\<exists>c1. c2 = c1;; WHILE b DO c \<and> (\<exists>s1 s2. (c, s1) \<rightarrow>* (c1, s2)))"
deep: fixes rs :: "nat list" assumes "deep_model_correct_params rs" shows "3 \<le> length rs"
Var_right_vars_mset_less: fixes x :: "'c"   and t :: "('d, 'c) Term.term"   and M :: "(('d, 'c) Term.term \<times> ('d, 'c) Term.term) multiset" assumes "x \<notin> vars_term t" shows "vars_mset (subst_mset (subst x t) M) \<subset> vars_mset (add_mset (t, Var x) M)"
simps: fixes n :: "nat"   and c :: "'a list"   and b :: "bool list"   and uu :: "'a list" shows "\<phi> n (c, b, uu) = (\<Sum>(x, y)\<in>Inv c (s_A n). if b ! index init y then 2 else 1)"
Pj_im_subg: fixes G :: "('e, 'f) Group_scheme"   and H :: "'e set"   and K :: "'e set" assumes "Group G"   and "G \<guillemotright> H "   and "G \<triangleright> K"   and "K \<subseteq> H" shows "Pj G K ` H = carrier (\<natural>\<^bsub>G\<^esub>H / K)"
ta_upd_proj_simps_1: fixes ta :: "('a, 'b, 'c, 'd, 'e, 'f) thread_action"   and obs :: "'f" shows "\<lbrace>ta_update_obs ta obs\<rbrace>\<^bsub>l\<^esub> = \<lbrace>ta\<rbrace>\<^bsub>l\<^esub>"
fres_anti: fixes x :: "'b \<times> 'c"   and y :: "'b \<times> 'c"   and z :: "'b \<times> 'c" assumes "less_eq_prod x y" shows "less_eq_prod (prod.fres z y) (prod.fres z x)"
surjective: fixes r :: "('c, 'd) monoid_scheme" shows "r = \<lparr>carrier = carrier r, monoid.mult = (\<otimes>\<^bsub>r\<^esub>), one = \<one>\<^bsub>r\<^esub>, \<dots> = monoid.more r\<rparr>"
orthogonal_complement_UNIV: shows "orthogonal_complement UNIV = {0::'b}"
warp_on_def: fixes A :: "'b set"   and f :: "'b set \<Rightarrow> 'b set" shows "warp_on A f = (\<forall>(x, y)\<in>rsp_on A f. (y, x) \<notin> rwp_on A f)"
permute_Abs_lst: fixes p :: "perm"   and as :: "atom list"   and x :: "'a" shows "p \<bullet> ([as]lst. x) = [p \<bullet> as]lst. p \<bullet> x"
S_irrefl: fixes t :: "('f, 'b) Term.term" shows "\<not> S t t"
UP_ring_subring_add: fixes S :: "'a set"   and g :: "nat \<Rightarrow> 'a"   and f :: "nat \<Rightarrow> 'a" assumes "subring S R"   and "g \<in> carrier (UP (R\<lparr>carrier := S\<rparr>))"   and "f \<in> carrier (UP (R\<lparr>carrier := S\<rparr>))" shows "g \<oplus>\<^bsub>UP R\<^esub> f = g \<oplus>\<^bsub>UP (R\<lparr>carrier := S\<rparr>)\<^esub> f"
diff_eq2: fixes x :: "'a"   and y :: "'a" assumes "x \<in> E"   and "y \<in> E" shows "x + - y = x - y"
r_both_def: shows "r_both \<equiv> Cn 4 r_ifz [Cn 4 r_result1 [recf.Id 4 0, recf.Id 4 1, recf.Id 4 3], Cn 4 r_ifz [Cn 4 r_result1 [recf.Id 4 0, recf.Id 4 2, recf.Id 4 3], Cn 4 r_prod_encode [r_constn 3 2, r_constn 3 0], Cn 4 r_prod_encode [r_constn 3 1, Cn 4 r_dec [Cn 4 r_result1 [recf.Id 4 0, recf.Id 4 2, recf.Id 4 3]]]], Cn 4 r_prod_encode [r_constn 3 0, Cn 4 r_dec [Cn 4 r_result1 [recf.Id 4 0, recf.Id 4 1, recf.Id 4 3]]]]"
empty_ran_restrict: fixes f :: "'c \<Rightarrow> 'b option"   and A :: "'c set" shows "(ran (f |` A) = {}) = (\<forall>x\<in>A. f x = None)"
echelon_form_of_upt_k_det_def: fixes bezout :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<times> 'b \<times> 'b \<times> 'b \<times> 'b"   and A' :: "'e \<times> (('b, 'c) vec, 'd) vec"   and k :: "nat" shows "echelon_form_of_upt_k_det bezout A' k = (let A = snd A'; f = foldl (echelon_form_of_column_k_det bezout) (1::'b, A, 0) [0..<Suc k] in (fst f, fst (snd f)))"
degree_diff_le: fixes p :: "'b poly"   and n :: "nat"   and q :: "'b poly" assumes "degree p \<le> n"   and "degree q \<le> n" shows "degree (p - q) \<le> n"
fcard_funion_finter: fixes A :: "'b fset"   and B :: "'b fset" shows "fcard A + fcard B = fcard (A |\<union>| B) + fcard (A |\<inter>| B)"
l1_step4_def: fixes Rb :: "fid_t"   and A :: "agent"   and B :: "agent"   and gnx :: "msg" shows "l1_step4 Rb A B gnx \<equiv> {(s, s'). guessed_runs Rb = \<lparr>role = Resp, owner = B, partner = A\<rparr> \<and> progress s Rb = Some {xny, xgnx, xgny, xsk} \<and> dhlvl1.guessed_frame Rb xgnx = Some gnx \<and> (can_signal s A B \<longrightarrow> (\<exists>Ra. guessed_runs Ra = \<lparr>role = Init, owner = A, partner = B\<rparr> \<and> in_progressS (progress s Ra) {xnx, xgnx, xgny, xsk, xEnd} \<and> dhlvl1.guessed_frame Ra xgnx = Some gnx \<and> dhlvl1.guessed_frame Ra xgny = Some (Exp Gen (NonceF (Rb $ ny))))) \<and> (Rb = test \<longrightarrow> Exp gnx (NonceF (Rb $ ny)) \<notin> synth (analz (ik s))) \<and> s' = s\<lparr>progress := progress s(Rb \<mapsto> {xny, xgnx, xgny, xsk, xEnd}), secret := {x. x = Exp gnx (NonceF (Rb $ ny)) \<and> Rb = test} \<union> secret s, signalsResp := if can_signal s A B then addSignal (signalsResp s) (Commit A B (Exp gnx (NonceF (Rb $ ny)))) else signalsResp s\<rparr>}"
req: shows "r = (Ps,  \<Empt> \<Rightarrow>* \<LM> A \<and>* B  \<RM>)"
prems_1: fixes S :: "'a set" assumes "Separation G v0 v1 S" shows "Suc n \<le> card S"
assms_1: shows "wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<theta>"
pos_alt: fixes x :: "'a"   and y :: "'a"   and e :: "'a" assumes "D x y"   and "e \<in> E"   and "x \<oplus> y = e" shows "x = e"
add_diff_assoc_eint: fixes z :: "eint"   and y :: "eint"   and x :: "eint" assumes "z \<le> y" shows "x + (y - z) = x + y - z"
fs_int_indpt_def: fixes n :: "nat"   and fs :: "int vec list" shows "fs_int_indpt n fs \<equiv> cof_vec_space.lin_indpt_list n (map of_int_hom.vec_hom fs)"
mult_vv_commute: fixes v1 :: "complex \<times> complex"   and v2 :: "complex \<times> complex" shows "v1 *\<^sub>v\<^sub>v v2 = v2 *\<^sub>v\<^sub>v v1"
knows_Cons: fixes A :: "agent"   and ev :: "event"   and evs :: "event list" shows "knows A (ev # evs) = (if A = Spy then case ev of Says A' B X \<Rightarrow> insert X (knows Spy evs) | Gets A' X \<Rightarrow> knows Spy evs | Notes A' X \<Rightarrow> if A' \<in> bad then insert X (knows Spy evs) else knows Spy evs else case ev of Says A' B X \<Rightarrow> if A' = A then insert X (knows A evs) else knows A evs | Gets A' X \<Rightarrow> if A' = A then insert X (knows A evs) else knows A evs | Notes A' X \<Rightarrow> if A' = A then insert X (knows A evs) else knows A evs)"
min_int_poly_of_int: fixes n :: "int" shows "min_int_poly (of_int n) = [:- of_int n, 1:]"
thetaParWT_def: shows "thetaParWT \<equiv> {(c1 | c2, d1 | d2) |c1 c2 d1 d2. c1 \<approx>wT d1 \<and> c2 \<approx>wT d2}"
distinct_set_conv_list: fixes xs :: "'a list"   and g :: "'a \<Rightarrow> 'b" assumes "distinct xs" shows "sum g (list.set xs) = sum_list (map g xs)"
cc_le_Dube_aux: fixes j :: "nat" assumes "2 < card X"   and "0 < j"   and "j + 1 \<le> card X" shows "aa j + bb j \<le> Dube_aux (card X) (poly_deg f) j"
inner_sum_right: fixes x :: "'a"   and f :: "'b \<Rightarrow> 'a"   and A :: "'b set" shows "x \<bullet> sum f A = (\<Sum>y\<in>A. x \<bullet> f y)"
relImage_Gr: fixes R :: "('d \<times> 'd) set"   and A :: "'d set"   and f :: "'d \<Rightarrow> 'e" assumes "R \<subseteq> A \<times> A" shows "BNF_Greatest_Fixpoint.relImage R f = (BNF_Def.Gr A f)\<inverse> O R O BNF_Def.Gr A f"
cycle_remove_edge_1: fixes x :: "'a"   and s :: "'a"   and e :: "'a" assumes "cycle x"   and "point s"   and "point e"   and "e ; s\<^sup>T \<le> x" shows "path (x \<cdot> - (e ; s\<^sup>T))"
run_one_step_basic_unop_i_result: fixes d :: "nat"   and i :: "nat"   and s :: "s"   and vs :: "v list"   and ves :: "v list"   and x241 :: "t"   and x242 :: "unop_i"   and s' :: "s"   and vs' :: "v list"   and res :: "res_step" assumes "run_one_step d i (s, vs, ves, $Unop_i x241 x242) = (s', vs', res)" shows "(\<exists>r. res = RSNormal r) \<or> (\<exists>e. res = RSCrash e)"
two_realpow_ge_one: fixes n :: "nat" shows "1 \<le> 2 ^ n"
prolong_betw: fixes Q :: "'a set"   and a :: "'a"   and b :: "'a" assumes "Q \<in> \<P>"   and "a \<in> Q"   and "b \<in> Q"   and "a \<noteq> b" shows "\<exists>c\<in>\<E>. [[a b c]]"
CastRed: fixes e :: "char list exp"   and s :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and e' :: "char list exp"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and C :: "char list" assumes "P \<turnstile> \<langle>e,s\<rangle> \<rightarrow> \<langle>e',s'\<rangle>" shows "P \<turnstile> \<langle>Cast C e,s\<rangle> \<rightarrow> \<langle>Cast C e',s'\<rangle>"
abs_eq: fixes xa :: "nat"   and x :: "nat" shows "(natural_of_nat xa \<le> natural_of_nat x) = (xa \<le> x)"
sum_in_X: shows "(\<Sum>i\<in>U. i * f i) \<in> X"
abs_of_diff_is_pr: shows "abs_of_diff \<in> PrimRec2"
map_pmf_idI: fixes p :: "'a pmf"   and f :: "'a \<Rightarrow> 'a" assumes "\<And>x. x \<in> set_pmf p \<Longrightarrow> f x = x" shows "map_pmf f p = p"
locally_sound_mode_use_def2: fixes dma :: "('c \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> Sec"   and \<C>_vars :: "'c \<Rightarrow> 'c set"   and eval :: "((('e \<times> (Mode \<Rightarrow> 'c set)) \<times> ('c \<Rightarrow> 'd)) \<times> ('e \<times> (Mode \<Rightarrow> 'c set)) \<times> ('c \<Rightarrow> 'd)) set"   and lc :: "('e \<times> (Mode \<Rightarrow> 'c set)) \<times> ('c \<Rightarrow> 'd)" assumes "sifum_security_init dma \<C>_vars eval" shows "sifum_security_init.locally_sound_mode_use dma \<C>_vars eval lc \<equiv> \<forall>lc'\<in>sifum_security_init.loc_reach dma eval lc. sifum_security_init.respects_own_guarantees dma \<C>_vars eval (fst lc')"
Abs_real_int_inject: fixes x :: "real \<times> real"   and y :: "real \<times> real" assumes "x \<in> {r. fst r \<le> snd r}"   and "y \<in> {r. fst r \<le> snd r}" shows "(Abs_real_int x = Abs_real_int y) = (x = y)"
strict_order_iff: fixes b :: "'c"   and a :: "'c" shows "(b < a) = (a = sup a b \<and> a \<noteq> b)"
Block_closed: fixes a :: "'a" assumes "a \<in> S" shows "Block a \<in> P"
simps_1: fixes t :: "'b" shows "is_some (Some t) = True"
Recv_1: shows "\<exists>p q u u' m. t ! j = Recv cid q p u u' m"
get_root_node_locs_def: shows "get_root_node_locs = get_ancestors_locs"
ModelIkMcalc_axioms: shows "ModelIkMcalc wtFsym wtPsym arOf resOf parOf \<Phi> infTp Ik_intT Ik_intF Ik_intP"
assms_1: shows "P\<^sub>1 \<sqsubseteq> P\<^sub>2"
s: shows "supp \<beta> \<subseteq> atom ` Vs"
axioms: fixes A :: "V"   and f :: "V" assumes "pbinop A f" shows "pnop A (2\<^sub>\<nat>) f"
assms: shows "f \<in> space\<^sub>N (\<LL> p M)"
Units_l_cancel: fixes x :: "'a mod_ring poly"   and y :: "'a mod_ring poly"   and z :: "'a mod_ring poly" assumes "x \<in> Units R"   and "y \<in> carrier R"   and "z \<in> carrier R" shows "(x \<otimes>\<^bsub>R\<^esub> y = x \<otimes>\<^bsub>R\<^esub> z) = (y = z)"
PRECOND_OPT_tag_def: fixes P :: "'g" shows "PRECOND_OPT_tag P \<equiv> P"
assms_2: shows "fmdom' s2 \<subseteq> vs2"
runit'_simps_1: fixes f :: "'a" assumes "C.ide f" shows "C.arr \<r>\<^sub>C\<^sup>-\<^sup>1[f]"
repv_def_correct: fixes \<omega> :: "variable \<Rightarrow> real"   and x :: "variable"   and r :: "real" shows "repv \<omega> x r = (\<lambda>y. if x = y then r else \<omega> y)"
that: fixes s2 :: "'b" assumes "bisim12 s1 s2"   and "bisim23 s2 s3" shows "thesis"
keysFor_def: fixes H :: "msg set" shows "keysFor H = invKey ` {K. \<exists>X. Crypt K X \<in> H}"
abs_summable_on_Sigma_iff: fixes A :: "'c set"   and B :: "'c \<Rightarrow> 'd set"   and f :: "'c \<times> 'd \<Rightarrow> 'e" assumes "countable A"   and "\<And>x. x \<in> A \<Longrightarrow> countable (B x)" shows "(f abs_summable_on Sigma A B) = ((\<forall>x\<in>A. (\<lambda>y. f (x, y)) abs_summable_on B x) \<and> (\<lambda>x. \<Sum>\<^sub>ay\<in>B x. norm (f (x, y))) abs_summable_on A)"
cindexE_eq_cindex_polyE: fixes a :: "real"   and b :: "real"   and q :: "real poly"   and p :: "real poly" assumes "a < b" shows "cindexE a b (\<lambda>x. poly q x / poly p x) = cindex_polyE a b q p"
inverse_arrows_sym: fixes C :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and f :: "'d"   and g :: "'d" assumes "category C" shows "category.inverse_arrows C f g = category.inverse_arrows C g f"
cone_f_X_sub: shows "cone (f, X) \<subseteq> P[X]"
pgallery_def: fixes X :: "'b set set"   and xs :: "'b set list" assumes "ChamberComplex X" shows "SimplicialComplex.pmaxsimpchain X xs \<equiv> (\<forall>x\<in>set xs. SimplicialComplex.maxsimp X x) \<and> padjacentchain xs"
affine_imp_polyhedron: fixes S :: "'a set" assumes "affine S" shows "polyhedron S"
ide_iso_in_hom: fixes a :: "'d" assumes "D.ide a" shows "\<guillemotleft>a : a \<cong>\<^sub>D a\<guillemotright>"
Int_space_eq2: fixes x :: "'d set"   and M :: "'d measure" assumes "x \<in> sets M" shows "x \<inter> space M = x"
gcd_lcm_distrib: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and Gcd :: "'a set \<Rightarrow> 'a"   and Lcm :: "'a set \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "class.factorial_semiring_gcd divide plus minus zero times one unit_factor normalize gcd lcm Gcd Lcm" shows "gcd x (lcm y z) = lcm (gcd x y) (gcd x z)"
TS_subset: fixes i :: "nat"   and j :: "nat"   and M2 :: "('d, 'e, 'f) FSM"   and M1 :: "('d, 'e, 'g) FSM"   and \<Omega> :: "('d, 'e) ATC set"   and V :: "'d list set"   and m :: "nat" assumes "i \<le> j" shows "TS M2 M1 \<Omega> V m i \<subseteq> TS M2 M1 \<Omega> V m j"
leq_add_left: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "class.add_linorder_min plus minus zero less_eq less" shows "less_eq x (plus y x)"
ms2: shows "matrix_seq d (matrix_sum d (\<lambda>n. denote_while_n (x1 0) (x1 1) (denote x2) n \<rho>\<^sub>2))"
sum_defs_3: fixes b :: "hf" shows "HF.Inr b \<equiv> \<langle>1, b\<rangle>"
SAT_mono: fixes f :: "'k \<times> 'k \<Rightarrow> ennreal"   and g :: "'k \<times> 'k \<Rightarrow> ennreal"   and \<Gamma> :: "('k, 'l) web_scheme" assumes "\<And>e. f e \<le> g e" shows "SAT \<Gamma> f \<subseteq> SAT \<Gamma> g"
admissible_disj: fixes Sup :: "'b set \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and P :: "'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool" assumes "class.ccpo Sup less_eq less"   and "ccpo.admissible Sup less_eq P"   and "ccpo.admissible Sup less_eq Q" shows "ccpo.admissible Sup less_eq (\<lambda>x. P x \<or> Q x)"
prod_norm: fixes f :: "'e \<Rightarrow> 'f"   and A :: "'e set" shows "(\<Prod>x\<in>A. norm (f x)) = norm (prod f A)"
comp_simp: fixes g :: "'a1 \<times> 'a2"   and f :: "'a1 \<times> 'a2" assumes "A1.seq (fst g) (fst f)"   and "A2.seq (snd g) (snd f)" shows "g \<cdot>\<^sub>A\<^sub>1\<^sub>x\<^sub>A\<^sub>2 f = (fst g \<cdot>\<^sub>A\<^sub>1 fst f, snd g \<cdot>\<^sub>A\<^sub>2 snd f)"
restrict_map_dom: fixes S :: "'c set"   and m :: "'c \<Rightarrow> 'd option" assumes "S = dom m" shows "m |` S = m"
empty_sets: fixes M :: "'b measure" shows "{} \<in> null_sets M"
swap_prems_eq: fixes A :: "prop"   and B :: "prop"   and C :: "prop" shows "(\<lbrakk>PROP A; PROP B\<rbrakk> \<Longrightarrow> PROP C) \<equiv> (\<lbrakk>PROP B; PROP A\<rbrakk> \<Longrightarrow> PROP C)"
find_nu: shows "\<lbrace>\<lambda>s. True\<rbrace> find_nu \<lbrace>\<lambda>s. nu s \<in> {0<..lambda + mu} \<and> (seq \<circ> (\<lambda>s. nu s + nu s)) s = (seq \<circ> nu) s \<and> hare s = (seq \<circ> nu) s\<rbrace>"
sel_1: fixes x11 :: "'a"   and x12 :: "'a"   and x13 :: "'a"   and x14 :: "'b" shows "pac_src1 (Add x11 x12 x13 x14) = x11"
open_preimage_identity: fixes B :: "'a set" assumes "is_open B" shows "identity S \<^sup>\<inverse> S B = B"
assms: shows "\<Gamma> \<turnstile> M \<turnstile> \<Delta>"
gval_gAnd: fixes g1 :: "'a gexp"   and g2 :: "'a gexp"   and s :: "'a \<Rightarrow> value option" shows "gval (gAnd g1 g2) s = gval g1 s \<and>? gval g2 s"
ennreal_enn2real: fixes r :: "ennreal" assumes "r < top" shows "ennreal (enn2real r) = r"
irreducible_prodE: fixes a :: "'a"   and b :: "'a" assumes "irreducible G (a \<otimes> b)"   and "a \<in> carrier G"   and "b \<in> carrier G" obtains "irreducible G a" and "b \<in> Units G"   | "a \<in> Units G" and "irreducible G b"
update_by_fun_gr_raw_eq_update_by_fun_raw: fixes xs :: "('a \<times> 'c) list \<times> 'o"   and k :: "'a"   and f :: "'c \<Rightarrow> 'c" assumes "oalist_inv xs" shows "update_by_fun_gr_raw k f xs = update_by_fun_raw k f xs"
Q_decompose: shows "of_int_poly Q = Polynomial.smult (of_nat c) (\<Prod>X\<in>Roots'. [:- root' X, 1:])"
rep_eq: fixes x :: "uint64" shows "Rep_uint (uint_of_uint64 x) = ucast (Rep_uint64 x)"
abs_0_eq: fixes abs :: "'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 less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b" assumes "class.ordered_ab_group_add_abs abs plus zero minus uminus less_eq less" shows "(zero = abs a) = (a = zero)"
access_type_safe: fixes s :: "Store"   and l :: "Location" shows "\<tau> s@@l \<le> ltype l"
atLeastLessThan_concat: fixes m :: "nat"   and n :: "nat"   and p :: "nat"   and g :: "nat \<Rightarrow> 'a" assumes "m \<le> n"   and "n \<le> p" shows "sum g {m..<n} + sum g {n..<p} = sum g {m..<p}"
nth_insert_nth_back: fixes j :: "nat"   and i :: "nat"   and xs :: "'b list"   and x :: "'b" assumes "j < i"   and "i \<le> length xs" shows "insert_nth j x xs ! i = xs ! (i - 1)"
SUP_enat_add_left: fixes I :: "'d set"   and f :: "'d \<Rightarrow> enat"   and c :: "enat" assumes "I \<noteq> {}" shows "(\<Squnion>i\<in>I. f i + c) = \<Squnion> (f ` I) + c"
set_cond_spmf: fixes p :: "'d spmf"   and A :: "'d set" shows "set_spmf (cond_spmf p A) = set_spmf p \<inter> A"
split_list_prop: fixes xs :: "'a list"   and P :: "'a \<Rightarrow> bool" assumes "\<exists>x\<in>set xs. P x" shows "\<exists>ys x zs. xs = ys @ x # zs \<and> P x"
inj_on_Un: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set"   and B :: "'b set" shows "inj_on f (A \<union> B) = (inj_on f A \<and> inj_on f B \<and> f ` (A - B) \<inter> f ` (B - A) = {})"
append_replicate_commute: fixes n :: "nat"   and x :: "'g"   and k :: "nat" shows "replicate n x @ replicate k x = replicate k x @ replicate n x"
preservation_of_barbs_and_closures_1: fixes Rel :: "('b \<times> 'b) set"   and CWB :: "('b, 'c) calculusWithBarbs" assumes "rel_preserves_binary_pred Rel (\<lambda>a. hasBarb a CWB)" shows "rel_preserves_binary_pred (Rel\<^sup>=) (\<lambda>a. hasBarb a CWB)"
mod_self: fixes divide :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and modulo :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and a :: "'d" assumes "class.semidom_modulo divide plus minus zero times one modulo" shows "modulo a a = zero"
f_minus: fixes R :: "('e, 'f) ring_scheme"   and M :: "('e, 'g, 'h) module_scheme"   and N :: "('e, 'i, 'j) module_scheme"   and f :: "'g \<Rightarrow> 'i"   and v :: "'g"   and w :: "'g" assumes "mod_hom R M N f"   and "v \<in> carrier M"   and "w \<in> carrier M" shows "f (v \<ominus>\<^bsub>M\<^esub> w) = f v \<ominus>\<^bsub>N\<^esub> f w"
sup_loc_refl: fixes P :: "'a prog"   and LT :: "ty err list" shows "P \<turnstile> LT [\<le>\<^sub>\<top>] LT"
seqSubstCases: fixes Cs :: "('c \<times> ('a, 'b, 'c) psi) list"   and \<sigma> :: "(name list \<times> 'a list) list" shows "Cases Cs[<\<sigma>>] = Cases caseListSeqSubst Cs \<sigma>"
R2: fixes p :: "'a"   and x :: "'a"   and q :: "'a" assumes "H p x q" shows "x \<le> R p q"
simps_2: fixes x :: "'b"   and n :: "'b"   and l :: "'b tree"   and r :: "'b tree"   and h :: "nat" shows "delete x (MKT n l r h) = (if x = n then delete_root (MKT n l r h) else if x < n then let l' = delete x l in mkt_bal_r n l' r else Let (delete x r) (mkt_bal_l n l))"
Sup_ow_axioms: shows "Sup_ow U\<^sub>2 Sup\<^sub>2"
map_id: fixes t :: "('d, 'e, 'f) F1" shows "F1map id id id t = t"
sem_5b1: fixes X :: "w \<Rightarrow> bool"   and Y :: "w \<Rightarrow> bool" shows "ob X Y \<longrightarrow> ob X (Y \<sqinter> X)"