hcju/isabelleps huggingface dataset:
comp_char: fixes g :: "'a"   and f :: "'a" shows "g \<cdot> f = (if arr f \<and> arr g \<and> C.seq g f then g \<cdot>\<^sub>C f else C.null)"
eval: shows "G\<turnstile>s0 \<midarrow>e-\<succ>b\<rightarrow> s1"
simps_9: fixes bs1 :: "WS1S_Nameful.fo list"   and bs2 :: "WS1S_Nameful.so list"   and X :: "WS1S_Nameful.so"   and \<phi> :: "WS1S_Nameful.ws1s" shows "\<epsilon> bs1 bs2 (All2 X \<phi>) = FAll SO (\<epsilon> bs1 (X # bs2) \<phi>)"
take_subseq: fixes weights :: "'a \<Rightarrow> 'a \<Rightarrow> 'b antichain"   and n :: "nat"   and xs :: "'c list" assumes "graph weights" shows "take n xs \<preceq> xs"
zero_smult: fixes R :: "'a set"   and smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and M :: "'b set"   and m :: "'b" assumes "RModule R smult M"   and "m \<in> M" shows "smult (0::'a) m = (0::'b)"
hextI: fixes h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and h' :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option" assumes "\<forall>a C fs. h a = \<lfloor>(C, fs)\<rfloor> \<longrightarrow> (\<exists>fs'. h' a = \<lfloor>(C, fs')\<rfloor>)" shows "h \<unlhd> h'"
sup_set_def: fixes A :: "'b set"   and B :: "'b set" shows "A \<union> B = p2s ((\<lambda>x. x \<in> A) \<squnion> (\<lambda>x. x \<in> B))"
null_def: shows "C2.null = (THE n. \<forall>f. n \<cdot>\<^sub>2 f = n \<and> f \<cdot>\<^sub>2 n = n)"
ordinal_oLog_base_0: fixes x :: "ordinal" shows "oLog 0 x = 0"
assms: shows "M1 \<preceq>\<lbrakk>T.\<Omega>\<rbrakk> M2"
eq_iff: fixes x :: "'a"   and y :: "'a" assumes "x \<in> U"   and "y \<in> U" shows "(x = y) = (x \<le>\<^sub>o\<^sub>w y \<and> y \<le>\<^sub>o\<^sub>w x)"
ident_relI: fixes G :: "('c, 'b) labeled_graph"   and idt :: "'c" assumes "\<And>x. x \<in> vertices G \<Longrightarrow> (x, x) \<in> getRel idt G"   and "\<And>x y. (x, y) \<in> getRel idt G \<Longrightarrow> x = y"   and "\<And>x y. (x, y) \<in> getRel idt G \<Longrightarrow> x \<in> vertices G" shows "ident_rel idt G"
Iod_carr_segment: fixes a :: "'a" shows "carrier (Iod D (segment D a)) = segment D a"
s2: shows "(\<Sum>a\<in>F. \<langle>f x, f a\<rangle>) = (\<Sum>a\<in>F. 0)"
assms: shows "\<rho> \<in> \<lbrakk>\<lbrakk> \<Psi> \<rbrakk>\<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L"
poly_mod_minus_left: fixes x :: "'a poly"   and y :: "'a poly" shows "- x mod y = - (x mod y)"
comp_fun_commute: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and y :: "'b"   and x :: "'b" assumes "comp_fun_commute f" shows "f y \<circ> f x = f x \<circ> f y"
lookup_None_iff: fixes P :: "'b \<Rightarrow> bool"   and xs :: "('b \<times> 'a) list" shows "(lookup P xs = None) = (\<nexists>x. x \<in> set (map fst xs) \<and> P x)"
measurable_density_eq1: fixes g :: "'c \<Rightarrow> 'd"   and Mg :: "'c measure"   and f :: "'c \<Rightarrow> ennreal"   and Mg' :: "'d measure" shows "(g \<in> density Mg f \<rightarrow>\<^sub>M Mg') = (g \<in> Mg \<rightarrow>\<^sub>M Mg')"
return: fixes vs :: "e list"   and n :: "nat"   and j :: "nat"   and lholed :: "Lholed"   and es :: "e list"   and i :: "nat"   and vls :: "v list" assumes "const_list vs"   and "length vs = n"   and "Lfilled j lholed (vs @ [$Return]) es" shows "\<lparr>[Local n i vls es]\<rparr> \<leadsto> \<lparr>vs\<rparr>"
simps: fixes m :: "nat"   and rs :: "nat"   and l :: "cell list"   and r :: "cell list" shows "wadjust_loop_right_move2 m rs (l, r) = (\<exists>ml mr ln rn. l = Oc # Oc \<up> ml @ Bk # Oc \<up> Suc m \<and> r = Bk \<up> ln @ Bk # Bk # Oc \<up> mr @ Bk \<up> rn \<and> ml + mr = Suc rs \<and> 0 < mr)"
dist_at: fixes vs :: "'c list"   and a :: "'c list"   and r :: "'c"   and b :: "'c list"   and c :: "'c list"   and d :: "'c list" assumes "distinct vs"   and "vs = a @ r # b"   and "vs = c @ r # d" shows "a = c \<and> b = d"
sets_def: fixes M :: "'a measure" shows "sets M = fst (snd (Rep_measure M))"
Limsup_ereal_mult_left: fixes F :: "'a filter"   and c :: "real"   and f :: "'a \<Rightarrow> ereal" assumes "F \<noteq> bot"   and "0 \<le> c" shows "Limsup F (\<lambda>n. ereal c * f n) = ereal c * Limsup F f"
proj_poly_syz_zero: fixes n :: "nat" shows "proj_poly_syz n 0 = 0"
bres_one: fixes x :: "'b \<times> 'c" shows "less_eq_prod one_prod (prod.bres x x)"
option_Max_by_eqI: fixes S :: "'b set"   and S' :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'd" assumes "(S = {}) = (S' = {})"   and "S \<noteq> {} \<and> S' \<noteq> {} \<Longrightarrow> Max_by f S = Max_by g S'" shows "option_Max_by f S = option_Max_by g S'"
suffix_tl: fixes xs :: "'a list" shows "suffix (tl xs) xs"
Cons_4: fixes y :: "'t \<Rightarrow>\<^sub>0 'b" assumes "y \<in> set ys" shows "\<not> is_red (set ((a # xs) @ ys) - {y}) y"
weakStepSim_def: fixes P :: "pi"   and Rel :: "(pi \<times> pi) set"   and Q :: "pi" shows "P \<leadsto><Rel> Q \<equiv> \<forall>Rs. Q \<longmapsto> Rs \<longrightarrow> weakStepSimAct P Rs P Rel"
Norm_nonneg: fixes N :: "'b quasinorm"   and x :: "'b" shows "0 \<le> Norm N x"
com_sifum_secure_def: fixes cmd :: "(var\<^sub>C, aexp\<^sub>C, bexp\<^sub>C) Stmt \<times> (Mode \<Rightarrow> var\<^sub>C set)" shows "conc.com_sifum_secure cmd \<equiv> case cmd of (c, mds\<^sub>s) \<Rightarrow> conc.low_indistinguishable mds\<^sub>s c c"
iT_cut_le_5: fixes r :: "nat"   and m :: "nat"   and c :: "nat"   and t :: "nat" shows "[ r, mod m, c ] \<down>\<le> t = (if t < r then {} else if t < r + m * c then [ r, mod m, (t - r) div m ] else [ r, mod m, c ])"
eventually_def: fixes P :: "'b \<Rightarrow> bool"   and F :: "'b filter" shows "eventually P F = Rep_filter F P"
split: fixes P :: "'a \<Rightarrow> bool"   and f1 :: "'a"   and f2 :: "color \<Rightarrow> ('b, 'c) RBT_Impl.rbt \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> ('b, 'c) RBT_Impl.rbt \<Rightarrow> 'a"   and rbt :: "('b, 'c) RBT_Impl.rbt" shows "P (case rbt of rbt.Empty \<Rightarrow> f1 | Branch x xa xb xc xd \<Rightarrow> f2 x xa xb xc xd) = ((rbt = rbt.Empty \<longrightarrow> P f1) \<and> (\<forall>x21 x22 x23 x24 x25. rbt = Branch x21 x22 x23 x24 x25 \<longrightarrow> P (f2 x21 x22 x23 x24 x25)))"
induct: fixes x1 :: "nat"   and x2 :: "nat"   and P :: "nat \<Rightarrow> nat \<Rightarrow> bool" assumes "less_nat x1 x2"   and "\<And>y. P 0 (Suc y)"   and "\<And>x y. \<lbrakk>less_nat x y; P x y\<rbrakk> \<Longrightarrow> P (Suc x) (Suc y)" shows "P x1 x2"
x_pow_n: fixes n :: "nat" shows "monom (1::'c) 1 ^ n = monom (1::'c) n"
mask_Suc_double: fixes n :: "nat" shows "mask (Suc n) = (1::'a) OR (2::'a) * mask n"
mult_eq_self_implies_10: fixes m :: "nat"   and n :: "nat" assumes "m = m * n" shows "n = 1 \<or> m = 0"
vars_of_monom_in_alt_def2: fixes xs :: "'a list"   and \<V> :: "'a set" shows "vars_of_monom_in xs \<V> = fold (\<lambda>x b. b \<and> x \<in> \<V>) xs True"
swapidseq_swap: fixes a :: "'a"   and b :: "'a" shows "swapidseq (if a = b then 0 else 1) (Fun.swap a b id)"
trms_irreducible_lemma: fixes C :: "'a eclause"   and \<sigma> :: "('a \<times> 'a trm) list"   and S :: "'a eclause set"   and t :: "'a trm" assumes "all_trms_irreducible (subst_set (trms_ecl C) \<sigma>) (\<lambda>t. trm_rep t S)" shows "trms_irreducible C \<sigma> S t"
support_wpt: fixes c :: "com"   and Q :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool" shows "support (wp\<^sub>t c Q) \<subseteq> support Q"
assms_1: shows "\<exists>t. local.prefixes t j \<up>"
sgnx_at_left_at_right_IVT: fixes p :: "real poly"   and a :: "real"   and b :: "real" assumes "sgnx (poly p) (at_right a) \<noteq> sgnx (poly p) (at_left b)"   and "a < b" shows "\<exists>x>a. x < b \<and> poly p x = 0"
grid_mono: fixes j :: "nat"   and n :: "nat" assumes "j \<le> n" shows "t j \<le> t n"
normalize_quot_proj_2: fixes b :: "'a"   and a :: "'a" assumes "b \<noteq> (0::'a)" shows "snd (normalize_quot (a, b)) = normalize b div gcd a b"
addition_set_inc_z: fixes z :: "'e"   and f :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and i :: "'e \<Rightarrow> 'e"   and A :: "'e set" shows "z \<in> addition_set f (aug_pm_set z i A)"
TRACES_def: fixes P :: "('c event list \<times> 'c event set) set \<times> 'c event list set" shows "TRACES P = {tr. \<exists>a. a \<in> FAILURES P \<and> tr = fst a}"
e: shows "(x * e1 + (q - s1)) mod q = (x * y + x * q - x * b + q * q + q * q + q - x * y - x * (q - b) - (q - r)) mod q"
dghm_cn_cov_comp_ArrMap_vsv: fixes \<alpha> :: "V"   and \<BB> :: "V"   and \<CC> :: "V"   and \<GG> :: "V"   and \<AA> :: "V"   and \<FF> :: "V" assumes "\<GG> : op_dg \<BB> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<CC>"   and "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<BB>" shows "vsv ((\<GG> \<^sub>D\<^sub>G\<^sub>H\<^sub>M\<circ> \<FF>)\<lparr>ArrMap\<rparr>)"
nexts_def: fixes F :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" shows "\<circle>F \<equiv> %s. F (tail s)"
intro: fixes set_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and get_shadow_root_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" assumes "\<And>ptr h h' ptr'. \<forall>w\<in>set_disconnected_nodes_locs ptr. h \<turnstile> w \<rightarrow>\<^sub>h h' \<longrightarrow> (\<forall>r\<in>get_shadow_root_locs ptr'. r h h')" shows "l_set_disconnected_nodes_get_shadow_root set_disconnected_nodes_locs get_shadow_root_locs"
assms_4: shows "prv (eqv \<chi> \<phi>)"
not_less_bot: fixes bot :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a" assumes "class.order_bot bot less_eq less" shows "\<not> less a bot"
extremally_disconnected_not_part_not_ed_ou_card: fixes X :: "'a set" assumes "extremally_disconnected X"   and "\<not> part X"   and "\<not> ed_ou X" shows "card {f. CK_nf_pos f} = 5"
parOutputCases: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi"   and M :: "'a"   and xvec :: "name list"   and N :: "'a"   and R :: "('a, 'b, 'c) psi"   and C :: "'e"   and Prop :: "('a, 'b, 'c) psi \<Rightarrow> bool" assumes "\<Psi> \<rhd> P \<parallel> Q \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> R"   and "xvec \<sharp>* \<Psi>"   and "xvec \<sharp>* P"   and "xvec \<sharp>* Q"   and "xvec \<sharp>* M"   and "\<And>P' A\<^sub>Q \<Psi>\<^sub>Q. \<lbrakk>\<Psi> \<otimes> \<Psi>\<^sub>Q \<rhd> P \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> P'; extractFrame Q = \<langle>A\<^sub>Q, \<Psi>\<^sub>Q\<rangle>; distinct A\<^sub>Q; A\<^sub>Q \<sharp>* \<Psi>; A\<^sub>Q \<sharp>* P; A\<^sub>Q \<sharp>* Q; A\<^sub>Q \<sharp>* M; A\<^sub>Q \<sharp>* xvec; A\<^sub>Q \<sharp>* N; A\<^sub>Q \<sharp>* C; A\<^sub>Q \<sharp>* xvec; distinct xvec\<rbrakk> \<Longrightarrow> Prop (P' \<parallel> Q)"   and "\<And>Q' A\<^sub>P \<Psi>\<^sub>P. \<lbrakk>\<Psi> \<otimes> \<Psi>\<^sub>P \<rhd> Q \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> Q'; extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>; distinct A\<^sub>P; A\<^sub>P \<sharp>* \<Psi>; A\<^sub>P \<sharp>* P; A\<^sub>P \<sharp>* Q; A\<^sub>P \<sharp>* M; A\<^sub>P \<sharp>* xvec; A\<^sub>P \<sharp>* N; A\<^sub>P \<sharp>* C; A\<^sub>P \<sharp>* xvec; distinct xvec\<rbrakk> \<Longrightarrow> Prop (P \<parallel> Q')" shows "Prop R"
Nil: fixes r :: "('b \<times> 'c) set" shows "([], []) \<in> listrel r"
execn_normalize_to_execn: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "('e, 'd, 'f) com"   and s :: "('e, 'f) xstate"   and n :: "nat"   and t :: "('e, 'f) xstate" assumes "\<Gamma>\<turnstile> \<langle>Language.normalize c,s\<rangle> =n\<Rightarrow> t" shows "\<Gamma>\<turnstile> \<langle>c,s\<rangle> =n\<Rightarrow> t"
cases: fixes a1 :: "x \<Rightarrow> x option"   and a2 :: "s"   and a3 :: "s" assumes "tr_s a1 a2 a3" obtains T :: "x \<Rightarrow> x option" and s_s'_list :: "(s \<times> s) list" where "a1 = Ta__" and "a2 = s_block (map (\<lambda>(s_XXX, s_'). s_XXX) s_s'_list)" and "a3 = s_block (map (\<lambda>(s_XXX, s_'). s_') s_s'_list)" and "list_all (\<lambda>f. f) (map (\<lambda>(s_XXX, s_'). tr_s Ta__ s_XXX s_') s_s'_list)"   | T :: "x \<Rightarrow> x option" and var :: "char list" and var' :: "char list" and x :: "x" and x' :: "x" where "a1 = Tb__" and "a2 = s_ass var__ x__" and "a3 = s_ass var'__ x'__" and "(case Tb__ (x_var var__) of None \<Rightarrow> var__ | Some (x_var var') \<Rightarrow> var' | Some x_this \<Rightarrow> var__) = var'__" and "(case Tb__ x__ of None \<Rightarrow> x__ | Some x' \<Rightarrow> x') = x'__"   | T :: "x \<Rightarrow> x option" and var :: "char list" and var' :: "char list" and x :: "x" and x' :: "x" and f :: "char list" where "a1 = Tc__" and "a2 = s_read vara__ xa__ f__" and "a3 = s_read var'a__ x'a__ f__" and "(case Tc__ (x_var vara__) of None \<Rightarrow> vara__ | Some (x_var var') \<Rightarrow> var' | Some x_this \<Rightarrow> vara__) = var'a__" and "(case Tc__ xa__ of None \<Rightarrow> xa__ | Some x' \<Rightarrow> x') = x'a__"   | T :: "x \<Rightarrow> x option" and x :: "x" and x' :: "x" and y :: "x" and y' :: "x" and f :: "char list" where "a1 = Td__" and "a2 = s_write xb__ f y__" and "a3 = s_write x'b__ f y'__" and "(case Td__ xb__ of None \<Rightarrow> xb__ | Some x' \<Rightarrow> x') = x'b__" and "(case Td__ y__ of None \<Rightarrow> y__ | Some x' \<Rightarrow> x') = y'__"   | T :: "x \<Rightarrow> x option" and x :: "x" and x' :: "x" and y :: "x" and y' :: "x" and s1 :: "s" and s1' :: "s" and s2 :: "s" and s2' :: "s" where "a1 = Te__" and "a2 = s_if xc__ y s1 s2" and "a3 = s_if x'c__ y' s1' s2'" and "(case Te__ xc__ of None \<Rightarrow> xc__ | Some x' \<Rightarrow> x') = x'c__" and "(case Te__ y of None \<Rightarrow> y | Some x' \<Rightarrow> x') = y'" and "tr_s Te__ s1 s1'" and "tr_s Te__ s2 s2'"   | T :: "x \<Rightarrow> x option" and var :: "char list" and var' :: "char list" and ctx :: "ctx" and cl :: "cl" where "a1 = Tf__" and "a2 = s_new varb__ ctx cl" and "a3 = s_new var'b__ ctx cl" and "(case Tf__ (x_var varb__) of None \<Rightarrow> varb__ | Some (x_var var') \<Rightarrow> var' | Some x_this \<Rightarrow> varb__) = var'b__"   | T :: "x \<Rightarrow> x option" and var :: "char list" and var' :: "char list" and x :: "x" and x' :: "x" and y_y'_list :: "(x \<times> x) list" and meth :: "char list" where "a1 = T" and "a2 = s_call var x meth (map (\<lambda>(y_XXX, y_'). y_XXX) y_y'_list)" and "a3 = s_call var' x' meth (map (\<lambda>(y_XXX, y_'). y_') y_y'_list)" and "(case T (x_var var) of None \<Rightarrow> var | Some (x_var var') \<Rightarrow> var' | Some x_this \<Rightarrow> var) = var'" and "(case T x of None \<Rightarrow> x | Some x' \<Rightarrow> x') = x'" and "list_all (\<lambda>f. f) (map (\<lambda>(y_XXX, y_'). (case T y_XXX of None \<Rightarrow> y_XXX | Some x' \<Rightarrow> x') = y_') y_y'_list)"
simps_18: fixes f1 :: "'b"   and f2 :: "nat \<Rightarrow> 'b"   and f3 :: "PreSimplyTyped.type \<Rightarrow> PreSimplyTyped.type \<Rightarrow> 'b"   and f4 :: "PreSimplyTyped.type \<Rightarrow> PreSimplyTyped.type \<Rightarrow> 'b"   and x31 :: "PreSimplyTyped.type"   and x32 :: "PreSimplyTyped.type" shows "(case TArr x31 x32 of TUnit \<Rightarrow> f1 | TVar x \<Rightarrow> f2 x | TArr x xa \<Rightarrow> f3 x xa | TPair x xa \<Rightarrow> f4 x xa) = f3 x31 x32"
extendEmpty: fixes C :: "'b sequent" shows "extend ( \<Empt> \<Rightarrow>* \<Empt>) C = C"
x_or: shows "x = u\<^sub>0 \<or> x = u\<^sub>1"
assms: shows "ciso\<^bsub>CatDom F\<^esub> k"
get_nondetT_def: shows "get = nondetM_base.get_nondet get"
isotone_def: fixes A :: "('d, 'e) gorder_scheme"   and B :: "('f, 'g) gorder_scheme"   and f :: "'d \<Rightarrow> 'f" shows "isotone A B f \<equiv> weak_partial_order A \<and> weak_partial_order B \<and> (\<forall>x\<in>carrier A. \<forall>y\<in>carrier A. x \<sqsubseteq>\<^bsub>A\<^esub> y \<longrightarrow> f x \<sqsubseteq>\<^bsub>B\<^esub> f y)"
l_distr: fixes x :: "nat \<Rightarrow> 'a"   and y :: "nat \<Rightarrow> 'a"   and z :: "nat \<Rightarrow> 'a" assumes "x \<in> carrier P"   and "y \<in> carrier P"   and "z \<in> carrier P" shows "(x \<oplus>\<^bsub>P\<^esub> y) \<otimes>\<^bsub>P\<^esub> z = x \<otimes>\<^bsub>P\<^esub> z \<oplus>\<^bsub>P\<^esub> y \<otimes>\<^bsub>P\<^esub> z"
call_result: fixes \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and m :: "'c match_expr"   and p :: "'d"   and \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and chain :: "char list"   and rs :: "'c rule list"   and X :: "final_decision"   and rrs :: "'c rule list" assumes "matches \<gamma> m p"   and "\<Gamma> chain = Some rs"   and "\<Gamma>,\<gamma>,p\<turnstile> \<langle>rs, Undecided\<rangle> \<Rightarrow>\<^sub>s Decision X" shows "\<Gamma>,\<gamma>,p\<turnstile> \<langle>Rule m (Call chain) # rrs, Undecided\<rangle> \<Rightarrow>\<^sub>s Decision X"
mem_Units: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and y :: "'b" assumes "class.comm_semiring_1 times one plus zero" shows "(y \<in> Units \<lparr>carrier = UNIV - {zero}, monoid.mult = times, one = one\<rparr>) = dvd.dvd times y one"
inconsistentAppend: fixes valuation1 :: "Literal list"   and valuation2 :: "Literal list" assumes "inconsistent (valuation1 @ valuation2)" shows "inconsistent valuation1 \<or> inconsistent valuation2 \<or> (\<exists>literal. literalTrue literal valuation1 \<and> literalFalse literal valuation2)"
irreducible_cong: fixes G :: "('a, 'b) monoid_scheme"   and a :: "'a"   and a' :: "'a" assumes "monoid_cancel G"   and "Divisibility.irreducible G a"   and "a \<sim>\<^bsub>G\<^esub> a'"   and "a \<in> carrier G"   and "a' \<in> carrier G" shows "Divisibility.irreducible G a'"
mono_def: fixes less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and f :: "'g \<Rightarrow> 'h" assumes "class.order less_eq less" shows "order.mono less_eq f = (\<forall>x y. less_eq x y \<longrightarrow> f x \<le> f y)"
lfp_induct': fixes f :: "'g \<Rightarrow> 'g"   and P :: "'g" assumes "mono f"   and "\<And>m. \<lbrakk>m \<le> lfp f; m \<le> P\<rbrakk> \<Longrightarrow> f m \<le> P" shows "lfp f \<le> P"
simps_2: fixes z :: "complex" shows "Im (cnj z) = - Im z"
row_upd_row: fixes M :: "('c, 'd) sq_matrix"   and i :: "'d"   and x :: "('c, 'd) vec" shows "Square_Matrix.row (upd_row M i x) i = x"
fixing_arr_gives_natural_transformation_2: fixes f2 :: "'c" assumes "C.arr f2" shows "natural_transformation (\<cdot>\<^sup>o\<^sup>p) (\<cdot>\<^sub>S) (\<lambda>f1. Hom.map (f1, C.dom f2)) (\<lambda>f1. Hom.map (f1, C.cod f2)) (\<lambda>f1. Hom.map (f1, f2))"
intro: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and prj0 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and prj1 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<mu> :: "'b arrow_of_spans_data"   and \<nu> :: "'b arrow_of_spans_data" assumes "span_bicategory C prj0 prj1"   and "arrow_of_spans C \<mu>"   and "arrow_of_spans C \<nu>"   and "two_composable_arrows_of_spans_axioms C \<mu> \<nu>" shows "two_composable_arrows_of_spans C prj0 prj1 \<mu> \<nu>"
has_derivative_bounded_linear: fixes f :: "'a \<Rightarrow> 'b"   and f' :: "'a \<Rightarrow> 'b"   and F :: "'a filter" assumes "(f has_derivative f') F" shows "bounded_linear f'"
lsteps_f_add': fixes xs :: "'a list"   and f :: "'a \<Rightarrow> nat"   and l :: "nat"   and u :: "nat"   and v :: "nat"   and m :: "nat" shows "lsteps xs f l u v = lsteps xs (\<lambda>x. f x + m) (l + m) u v"
Inf_closed_clop_var: fixes f :: "'b \<Rightarrow> 'b"   and X :: "'b set" assumes "clop f"   and "\<forall>x\<in>X. x \<in> range f" shows "\<Sqinter> X \<in> range f"
norm_blinfun: fixes f :: "'d \<Rightarrow>\<^sub>L 'c"   and x :: "'d" shows "norm (f *\<^sub>v x) \<le> norm f * norm x"
red_diff_in_pmdl: fixes F :: "('t \<Rightarrow>\<^sub>0 'c) set"   and p :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c" assumes "red F p q" shows "p - q \<in> pmdl F"
order_le_less: fixes x :: "'a"   and y :: "'a" shows "(x \<sqsubseteq> y) = (x \<sqsubset> y \<or> x = y)"
simps_2: fixes om :: "'f \<Rightarrow> 'g"   and am :: "'d \<Rightarrow> 'e"   and more :: "'h" shows "\<lparr>om = om, am = am, \<dots> = more\<rparr> \<^bsub>\<a>\<^esub> = am"
symmetric: fixes P :: "pi"   and Q :: "pi" assumes "P \<sim>\<^sub>l Q" shows "Q \<sim>\<^sub>l P"
counit_simps_1: shows "seq \<tau> (h \<star> \<xi> \<star> k)"
assms_1: shows "\<^bold>- y \<sqsubseteq> x"
raw_match_term_map_le: fixes t :: "Core.term"   and u :: "Core.term"   and subs :: "variable \<times> String.literal set \<Rightarrow> typ option"   and subs' :: "variable \<times> String.literal set \<Rightarrow> typ option" assumes "raw_match_term t u subs = Some subs'" shows "subs \<subseteq>\<^sub>m subs'"
arr_mkArr: fixes A :: "'s set"   and B :: "'s set"   and F :: "'s \<Rightarrow> 's" shows "arr (mkArr A B F) = (A \<subseteq> Univ \<and> B \<subseteq> Univ \<and> F \<in> A \<rightarrow> B)"
plus_le_conv: fixes A :: "'c set"   and r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and x :: "'c"   and y :: "'c"   and z :: "'c" assumes "Semilat A r f"   and "x \<in> A"   and "y \<in> A"   and "z \<in> A" shows "(x \<squnion>\<^bsub>f\<^esub> y \<sqsubseteq>\<^bsub>r\<^esub> z) = (x \<sqsubseteq>\<^bsub>r\<^esub> z \<and> y \<sqsubseteq>\<^bsub>r\<^esub> z)"
natom: fixes n :: "nat"   and m :: "nat"   and p :: "nat"   and vs :: "nat list"   and xs :: "(nat \<times> form) list"   and nfs :: "(nat \<times> form) list" assumes "(n, (m, NAtom p vs) # xs) \<in> deriv nfs"   and "\<not> is_axiom (s_of_ns ((m, NAtom p vs) # xs))" shows "(Suc n, xs @ [(0, NAtom p vs)]) \<in> deriv nfs"
A00D: shows "four_block_mat A (0\<^sub>m n m) (0\<^sub>m m n) D \<in> carrier_mat (n + m) (n + m)"
homotopic_loops_imp_homotopic_circlemaps: fixes S :: "'a set"   and p :: "real \<Rightarrow> 'a"   and q :: "real \<Rightarrow> 'a" assumes "homotopic_loops S p q" shows "homotopic_with_canon (\<lambda>h. True) (sphere 0 1) S (p \<circ> (\<lambda>z. Arg2pi z / (2 * pi))) (q \<circ> (\<lambda>z. Arg2pi z / (2 * pi)))"
stirling_integral_bound_aux_integral2: fixes a :: "real"   and b :: "real"   and c :: "real"   and n :: "nat"   and l :: "real"   and r :: "real" assumes "0 \<le> a"   and "0 < b"   and "0 \<le> c"   and "1 < n" obtains I :: "real" where "((\<lambda>x. c / max b \<bar>x - a\<bar> ^ n) has_integral I) {l..r}" and "I \<le> 2 * c * (real n / real (n - 1)) / b ^ (n - 1)"
simps_1: fixes x :: "'v" shows "arity_hd (Var x) = \<infinity>"
dur_Suc: fixes xs :: "('c \<Rightarrow> real) stream"   and i :: "nat" shows "dur xs (Suc i) - dur xs i = elapsed (xs !! i) (xs !! Suc i)"
one_discr_def: shows "1 \<equiv> Discr (1::'a)"
Hausdorff_space_product_topology: fixes X :: "'a \<Rightarrow> 'b topology"   and I :: "'a set" shows "Hausdorff_space (product_topology X I) = ((\<Pi>\<^sub>E i\<in>I. topspace (X i)) = {} \<or> (\<forall>i\<in>I. Hausdorff_space (X i)))"
hequiv_trans: fixes A :: "'c set"   and H :: "(('d, 'c) fm list \<times> 'c) set"   and i :: "'c"   and k :: "'c"   and j :: "'c" assumes "Hintikka A H"   and "i \<in> A"   and "k \<in> A"   and "hequiv H i j"   and "hequiv H j k" shows "hequiv H i k"
assms_3: shows "\<not> (\<exists>w\<ge>x. w \<le> y \<and> a * w\<^sup>2 + b * w + c = 0)"
simps_1: fixes E :: "'a \<Rightarrow> 'm"   and x :: "'a" shows "eval E (Var x) = E x"
comp_null_1: fixes f :: "'b" shows "C' C'.null f = C'.null"
unit_in_set_ar_cos: fixes A :: "('c, 'd) aGroup_scheme"   and H :: "'c set" assumes "aGroup A"   and "A +> H" shows "H \<in> set_ar_cos A H"
continuous_on_vec_lambda: fixes S :: "'b set"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c" assumes "\<And>i. continuous_on S (f i)" shows "continuous_on S (\<lambda>x. \<chi>i. f i x)"
congruent_triangle_sss_18: fixes a1 :: "'a"   and b1 :: "'a"   and a2 :: "'b"   and b2 :: "'b"   and c1 :: "'a"   and c2 :: "'b" assumes "dist a1 b1 = dist a2 b2"   and "dist b1 c1 = dist b2 c2"   and "dist a1 c1 = dist a2 c2" shows "angle b1 c1 a1 = angle a2 c2 b2"
fs_i_fs_j_sum_\<kappa>: fixes i :: "nat"   and l :: "nat"   and j :: "nat" assumes "i < length fs"   and "l \<le> i"   and "j < l" shows "- (fs ! i \<bullet> fs ! j) = (\<Sum>t = 0..<l. fs ! t \<bullet> fs ! j * \<kappa> i l t)"