hcju/isabelleps huggingface dataset:
timpls_transformable_to_pred_inv_Var: fixes A :: "'b set set"   and x :: "'e"   and t :: "(('c, 'd, 'b) prot_fun, 'e) Term.term" assumes "timpls_transformable_to_pred A (Var x) t" shows "t = Var x"
T_spaceM_stable_2: fixes x :: "'a"   and n :: "nat" assumes "x \<in> space M" shows "(Tinv ^^ n) x \<in> space M"
lstream'_def: shows "lstream' \<equiv> Abs_lgenerator' lstream"
sumlist_dim: fixes xs :: "'a vec list"   and n :: "nat" assumes "\<And>x. x \<in> set xs \<Longrightarrow> x \<in> carrier_vec n" shows "dim_vec (abelian_monoid.sumlist (module_vec TYPE('a) n) xs) = n"
abs_eq_iff: fixes x :: "('b \<times> 'b) list"   and y :: "('b \<times> 'b) list" shows "(abs_prm x = abs_prm y) = x =p y"
Hadjicostas_nn_integral_def: fixes z :: "real" shows "Hadjicostas_nn_integral z = \<integral>\<^sup>+x\<in>{0<..<1}. ennreal ((- ln x) powr z / (1 - x) * (- ln x + x - 1))\<partial>lborel"
fresh_prod: fixes a :: "'a"   and x :: "'b"   and y :: "'c" shows "a \<sharp> (x, y) = (a \<sharp> x \<and> a \<sharp> y)"
knd_def: fixes a :: "'b LDCFG_node \<times> 'a edge_kind \<times> 'b LDCFG_node" shows "knd a \<equiv> fst (snd a)"
blinfun_inverseD: fixes f :: "'c \<Rightarrow>\<^sub>L 'b"   and f' :: "'b \<Rightarrow>\<^sub>L 'c"   and x :: "'b" assumes "f o\<^sub>L f' = 1\<^sub>L" shows "blinfun_apply f (blinfun_apply f' x) = x"
pcr_cr_eq: shows "pcr_mod_ring = cr_mod_ring"
intros_13: 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 rest :: "'c rule list" assumes "matches \<gamma> m p"   and "\<Gamma> chain = Some rs"   and "\<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>rs, Undecided\<rangle> \<Rightarrow> Undecided" shows "\<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>Rule m (Goto chain) # rest, Undecided\<rangle> \<Rightarrow> Undecided"
add_lt_neutral: fixes a :: "'c DBMEntry"   and b :: "'c DBMEntry" assumes "dbm_add a b \<prec> b" shows "a \<prec> Le (0::'c)"
B_def: shows "B = (\<Union>n. (T ^^ n) --` C)"
m_inst_2: fixes xs :: "'a option list"   and tstp :: "nat + nat" assumes "Mapping.lookup m xs = Some tstp" shows "tstp_lt tstp (cur - (left I - 1)) tp \<and> isl tstp = (0 < left I)"
finite_measure_distr: fixes M :: "'a measure"   and f :: "'a \<Rightarrow> 'b"   and M' :: "'b measure" assumes "finite_measure M"   and "f \<in> M \<rightarrow>\<^sub>M M'" shows "finite_measure (distr M M' f)"
striaghten_topdown_computation: shows "kauff_mat (basic [vert, cup] \<circ> basic [cap, vert]) = kauff_mat (basic [vert] \<circ> basic [vert])"
upd_left_comm: fixes g :: "'c \<Rightarrow> 'b"   and f :: "'a \<Rightarrow> 'c" shows "comp_fun_commute (\<lambda>a c. finfun_update c a (g (f a)))"
sat_prefix_conv: fixes \<pi> :: "(char list \<times> 'a list) prefix"   and \<sigma> :: "(char list \<times> 'a list) trace"   and \<sigma>' :: "(char list \<times> 'a list) trace"   and i :: "nat"   and \<phi> :: "'a MFOTL.formula"   and v :: "'a list" assumes "prefix_of \<pi> \<sigma>"   and "prefix_of \<pi> \<sigma>'"   and "i < Monitor.progress \<sigma> \<phi> (plen \<pi>)" shows "MFOTL.sat \<sigma> v i \<phi> = MFOTL.sat \<sigma>' v i \<phi>"
abs_eq_iff: fixes x :: "('b, 'c, 'd) Tree"   and y :: "('b, 'c, 'd) Tree" shows "(abs_Tree\<^sub>\<alpha> x = abs_Tree\<^sub>\<alpha> y) = (x =\<^sub>\<alpha> y)"
ces_Un: fixes v1 :: "'c \<times> nat"   and es :: "('c \<times> nat) edge list"   and v2 :: "'c \<times> nat"   and subs1 :: "(('c \<times> nat) \<times> 'c \<times> nat) set"   and subs2 :: "(('c \<times> nat) \<times> 'c \<times> nat) set" assumes "SubRel.ces v1 es v2 subs1" shows "SubRel.ces v1 es v2 (subs1 \<union> subs2)"
delete_def: fixes x :: "'b"   and t :: "('b \<times> color) tree" shows "RBT_Set.delete x t = paint Black (RBT_Set.del x t)"
Gt_lt_conv: fixes xa :: "'o"   and x :: "'a"   and y :: "'a" shows "(key_compare (rep_key_order xa) x y = Gt) = lt xa y x"
thetaZOCh_def: fixes ch :: "'c"   and c1 :: "('a, 'b, 'c) cmd"   and c2 :: "('a, 'b, 'c) cmd"   and d1 :: "('d, 'e, 'c) cmd"   and d2 :: "('d, 'e, 'c) cmd" shows "thetaZOCh ch c1 c2 d1 d2 \<equiv> {(Ch ch c1 c2, Ch ch d1 d2)}"
Assign: fixes P :: "(char list \<Rightarrow> int) \<Rightarrow> enat"   and x :: "char list"   and a :: "aexp" shows "\<turnstile>\<^sub>2\<^sub>' {\<lambda>s. eSuc (P (s[a/x]))} x ::= a {P}"
decomp_closure_subterms_composable: fixes M :: "('fun, 'var) Term.term set"   and M' :: "('fun, 'var) Term.term set"   and t' :: "('fun, 'var) Term.term"   and t :: "('fun, 'var) Term.term" assumes "decomp_closure M M'"   and "M' \<turnstile>\<^sub>c t'"   and "M' \<turnstile> t"   and "t \<sqsubseteq> t'" shows "M' \<turnstile>\<^sub>c t"
dghm_comp_ArrMap_app: fixes \<alpha> :: "V"   and \<BB> :: "V"   and \<CC> :: "V"   and \<GG> :: "V"   and \<AA> :: "V"   and \<FF> :: "V"   and f :: "V" assumes "\<GG> : \<BB> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<CC>"   and "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<BB>"   and "f \<in>\<^sub>\<circ> \<AA>\<lparr>Arr\<rparr>" shows "(\<GG> \<circ>\<^sub>D\<^sub>G\<^sub>H\<^sub>M \<FF>)\<lparr>ArrMap\<rparr>\<lparr>f\<rparr> = \<GG>\<lparr>ArrMap\<rparr>\<lparr>\<FF>\<lparr>ArrMap\<rparr>\<lparr>f\<rparr>\<rparr>"
sorted_iff_nth_mono: fixes xs :: "'a list list" shows "linorder.sorted (\<lambda>x y. y \<le>lex x) xs = (\<forall>i j. i \<le> j \<longrightarrow> j < \<^bold>|xs\<^bold>| \<longrightarrow> xs ! j \<le>lex xs ! i)"
eint_minus_ineq': fixes c :: "eint"   and d :: "eint"   and b :: "eint"   and e :: "eint" assumes "c \<noteq> \<infinity>"   and "d \<le> b"   and "0 < e"   and "e \<noteq> \<infinity>" shows "e * (d - c) \<le> e * (b - c)"
bounded_subset_cboxE: fixes X :: "'b set" assumes "\<And>i. i \<in> Basis \<Longrightarrow> bounded ((\<lambda>x. x \<bullet> i) ` X)" obtains a :: "'b" and b :: "'b" where "X \<subseteq> cbox a b"
exhaust_sel: fixes tm :: "'a tm" obtains "tm = ??.Time_Monad.tm.TM (tm.val tm) (un_TM2 tm)"
eval_monom_nonzero: fixes basis :: "(real \<Rightarrow> real) list"   and monom :: "real \<times> real list" assumes "basis_wf basis"   and "fst monom \<noteq> 0" shows "\<forall>\<^sub>F x in at_top. eval_monom monom basis x \<noteq> 0"
divisors_nat_2: fixes n :: "nat" shows "distinct (divisors_nat n)"
Vv_inter_Ei_is_Vvi: shows "V\<^bsub>\<V>\<^esub> \<inter> E\<^bsub>ESi\<^esub> = V\<^bsub>\<V>i\<^esub>"
timpl_closure'_mono: fixes TI :: "('a set \<times> 'a set) set"   and TI' :: "('a set \<times> 'a set) set" assumes "TI \<subseteq> TI'" shows "timpl_closure' TI \<subseteq> timpl_closure' TI'"
v_S2: fixes cfg :: "'s cfg" assumes "cfg \<in> valid_cfg"   and "state cfg \<in> S2" shows "v cfg = 1"
finite_imp_inj_to_nat_seg: fixes A :: "'b set" assumes "finite A" shows "\<exists>f n. f ` A = {i. order.greater n i} \<and> inj_on f A"
A: fixes x :: "'a" assumes "cont g" shows "cont (\<lambda>y. y `;` g x)"
simps_8: fixes \<phi> :: "atom fm" shows "clearQuantifiers (AllQ \<phi>) = (let \<phi>' = clearQuantifiers \<phi> in if freeIn 0 \<phi>' then lowerFm 0 1 \<phi>' else AllQ \<phi>')"
axioms_1: fixes \<V> :: "'c set"   and \<B> :: "'c set multiset"   and \<V>' :: "'d set"   and \<B>' :: "'d set multiset"   and bij_map :: "'c \<Rightarrow> 'd" assumes "design_isomorphism \<V> \<B> \<V>' \<B>' bij_map" shows "incidence_system_isomorphism \<V> \<B> \<V>' \<B>' bij_map"
U_par_st: fixes a :: "'a"   and A :: "'a set"   and R :: "('a \<times> 'a set) set" shows "((a, A) \<in> R \<parallel> Multirelations.U) = (\<exists>B\<subseteq>A. (a, B) \<in> R)"
lower_exp_float_interval: fixes p :: "nat"   and x :: "float interval" shows "lower (exp_float_interval p x) = lb_exp p (lower x)"
sum_set_cond_linear: fixes P :: "'b set \<Rightarrow> bool"   and f :: "'b set \<Rightarrow> 'c set"   and I :: "'d set"   and S :: "'d \<Rightarrow> 'b set" assumes "\<And>A B. \<lbrakk>P A; P B\<rbrakk> \<Longrightarrow> P (A + B)"   and "P {0::'b}"   and "\<And>A B. \<lbrakk>P A; P B\<rbrakk> \<Longrightarrow> f (A + B) = f A + f B"   and "f {0::'b} = {0::'c}"   and "\<And>i. i \<in> I \<Longrightarrow> P (S i)" shows "f (sum S I) = sum (f \<circ> S) I"
problem_plan_bound_s_ge_min_pls: fixes PROB :: "(('c, bool) fmap \<times> ('c, bool) fmap) set"   and s :: "('c, bool) fmap"   and as :: "(('c, bool) fmap \<times> ('c, bool) fmap) list"   and k :: "nat" assumes "finite PROB"   and "s \<in> valid_states PROB"   and "as \<in> valid_plans PROB"   and "problem_plan_bound_s PROB s \<le> k" shows "Inf (PLS s as) \<le> problem_plan_bound_s PROB s"
diameter_lower_bounded: fixes S :: "'b set"   and d :: "real" assumes "bounded S"   and "0 < d"   and "d < diameter S" shows "\<exists>x\<in>S. \<exists>y\<in>S. d < dist x y"
card_seteq: fixes B :: "'b set"   and A :: "'b set" assumes "finite B"   and "A \<subseteq> B"   and "card B \<le> card A" shows "A = B"
Rep_matrix_cases: fixes y :: "nat \<Rightarrow> nat \<Rightarrow> 'b" assumes "y \<in> matrix" obtains x :: "'b matrix" where "y = Rep_matrix x"
assms: shows "\<forall>M w. M, w \<Turnstile>\<^sub>! p"
continuous_reward_until: fixes \<iota> :: "'c \<Rightarrow> 'c \<Rightarrow> ennreal"   and \<rho> :: "'c \<Rightarrow> ennreal"   and X :: "'c set" assumes "MC_with_rewards \<iota> \<rho>" shows "sup_continuous (\<lambda>F s \<omega>. if s \<in> X then 0 else \<rho> s + \<iota> s (shd \<omega>) + F (shd \<omega>) (stl \<omega>))"
less_imp_not_eq2: fixes x :: "'c word"   and y :: "'c word" assumes "x <s y" shows "(y = x) = False"
interval_integral_iexp: fixes a :: "real"   and b :: "real" shows "complex_interval_lebesgue_integral lborel (ereal a) (ereal b) iexp = \<i> * iexp a - \<i> * iexp b"
weak_reduction_contrasimulation_and_closures_1: fixes Rel :: "('b \<times> 'b) set"   and Cal :: "'b processCalculus" assumes "weak_reduction_contrasimulation Rel Cal" shows "weak_reduction_contrasimulation (Rel\<^sup>=) Cal"
lunit'_naturality: fixes \<mu> :: "'a" assumes "B.arr \<mu>" shows "(trg\<^sub>B \<mu> \<star>\<^sub>B \<mu>) \<cdot>\<^sub>B B.lunit' (B.dom \<mu>) = B.lunit' (B.cod \<mu>) \<cdot>\<^sub>B \<mu>"
H_def: shows "H = geodesic_subsegment G p D (dist p x)"
lfp_fixpoint: fixes f :: "'b::complete_lattice \<Rightarrow> 'b::complete_lattice" assumes "mono (f::'b::complete_lattice \<Rightarrow> 'b::complete_lattice)" shows "(f::'b::complete_lattice \<Rightarrow> 'b::complete_lattice) (lfp f) = lfp f"
f: fixes i :: "'b"   and j :: "'b"   and x :: "'a" assumes "i \<in> I"   and "j \<in> I"   and "x \<in> topspace X \<inter> T i \<inter> T j" shows "f i x = f j x"
generate_ring_incl: fixes H :: "'a set" assumes "H \<subseteq> carrier R" shows "generate_ring R H \<subseteq> carrier R"
simps_2: fixes f :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and x :: "'c"   and xs :: "'c list" shows "foldr f (x # xs) = f x \<circ> foldr f xs"
pmat_extension_def: fixes m :: "'a mat" shows "st1.pmat_extension m = st1.ptensor_mat m (1\<^sub>m st4.d2)"
csrp_fn_mem: fixes I :: "'a set"   and x :: "'a set" assumes "ideal R I"   and "x \<in> carrier (R /\<^sub>r I)" shows "csrp_fn R I x \<in> carrier R"
try_SE_def: fixes ioprog :: "'b \<Rightarrow> ('c \<times> 'b) option" shows "try_SE ioprog = (\<lambda>\<sigma>. case ioprog \<sigma> of None \<Rightarrow> Some (None, \<sigma>) | Some (outs, \<sigma>') \<Rightarrow> Some (Some outs, \<sigma>'))"
that: fixes a :: "'a" assumes "a \<in># A"   and "p dvd a" shows "thesis"
bigo_mult7: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "\<forall>x. f x \<noteq> (0::'d)" shows "O(f * g) \<subseteq> O(f) * O(g)"
assms_3: fixes x :: "'a" assumes "x \<in> agents" shows "R1 x = R2 x"
coeff_a_inv: fixes p :: "nat \<Rightarrow> 'a"   and n :: "nat" assumes "p \<in> carrier P" shows "coeff P (\<ominus>\<^bsub>P\<^esub> p) n = \<ominus> coeff P p n"
big_step_t_determ2: fixes c :: "com"   and s :: "char list \<Rightarrow> int"   and p :: "nat"   and t :: "char list \<Rightarrow> int"   and q :: "nat"   and u :: "char list \<Rightarrow> int" assumes "(c, s) \<Rightarrow> p \<Down> t"   and "(c, s) \<Rightarrow> q \<Down> u" shows "u = t \<and> p = q"
assms: shows "wffp (sl1 @ [State st1 u1 c1] @ [Idle st2] @ sl2)"
subset_\<P>k: fixes u :: "nat"   and v :: "nat"   and k :: "nat" assumes "u \<le> v" shows "\<P> k u \<subseteq> \<P> k v"
setLyx: shows "set (Lxy init {y, x}) = {x, y}"
integral_indicator: fixes S :: "'c set"   and T :: "'c set" assumes "S \<inter> T \<in> lmeasurable" shows "integral T (indicat_real S) = Sigma_Algebra.measure lebesgue (S \<inter> T)"
I: shows "\<rho> = - (- \<rho> \<squnion> \<bottom>\<bottom>)"
bounded_linear_sub: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "bounded_linear f"   and "bounded_linear g" shows "bounded_linear (\<lambda>x. f x - g x)"
Keys_def: fixes F :: "('d \<Rightarrow>\<^sub>0 'e) set" shows "Keys F = \<Union> (keys ` F)"
vP_def: fixes k :: "'k2"   and v :: "'v2" shows "vP k v = (\<exists>k' v'. m1.\<alpha> m k' = Some v' \<and> f (k', v') = Some (k, v))"
h1: shows "\<forall>I \<omega> X. (\<omega> \<in> game_sem I \<alpha> X) = (\<omega> \<in> game_sem I \<alpha> (selectlike X \<omega> M))"
has_white_path_to_reaches: fixes x :: "'a"   and y :: "'a"   and s :: "'b process_name \<Rightarrow> ('c, 'b, 'd, 'a) Model.local_state" assumes "(x has_white_path_to y) s" shows "(x reaches y) s"
measure_negligible_finite_Union_image: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'b set" assumes "finite S"   and "\<And>x. x \<in> S \<Longrightarrow> f x \<in> lmeasurable"   and "pairwise (\<lambda>x y. negligible (f x \<inter> f y)) S" shows "Sigma_Algebra.measure lebesgue (\<Union> (f ` S)) = (\<Sum>x\<in>S. Sigma_Algebra.measure lebesgue (f x))"
nf_RRR_9: shows "test (K \<circ> C \<circ> K \<circ> C) = [True, True, False, False, False]"
Neg: fixes a :: "'c"   and ps :: "('b, 'c) fm list"   and branch :: "(('b, 'c) fm list \<times> 'c) list"   and p :: "('b, 'c) fm"   and A :: "'c set"   and n :: "nat" assumes "(\<^bold>\<not> \<^bold>\<not> p) at a in' set ((ps, a) # branch)"   and "new p a ((ps, a) # branch)"   and "A, Suc n \<turnstile> (p # ps, a) # branch" shows "A, n \<turnstile> (ps, a) # branch"
round_robin_reschedule_Cons_2: fixes t :: "'t"   and t0 :: "'t"   and queue :: "'t list"   and n0 :: "nat"   and s :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set" assumes "t \<noteq> t0" shows "round_robin_reschedule t0 (t # queue) n0 s = (case round_robin_step n0 (t # queue, n0) s t of None \<Rightarrow> round_robin_reschedule t0 (queue @ [t]) n0 s | \<lfloor>ttaxm\<sigma>\<rfloor> \<Rightarrow> \<lfloor>ttaxm\<sigma>\<rfloor>)"
induct: fixes x1 :: "com \<times> (char list \<Rightarrow> int)"   and x2 :: "nat"   and x3 :: "char list \<Rightarrow> int"   and P :: "com \<times> (char list \<Rightarrow> int) \<Rightarrow> nat \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool" assumes "x1 \<Rightarrow> x2 \<Down> x3"   and "\<And>s. P (SKIP, s) (Suc 0) s"   and "\<And>x a s. P (x ::= a, s) (Suc 0) (s(x := aval a s))"   and "\<And>c1 s1 x s2 c2 y s3 z. \<lbrakk>(c1, s1) \<Rightarrow> x \<Down> s2; P (c1, s1) x s2; (c2, s2) \<Rightarrow> y \<Down> s3; P (c2, s2) y s3; z = x + y\<rbrakk> \<Longrightarrow> P (c1;; c2, s1) z s3"   and "\<And>b s c1 x t y c2. \<lbrakk>bval b s; (c1, s) \<Rightarrow> x \<Down> t; P (c1, s) x t; y = x + 1\<rbrakk> \<Longrightarrow> P (IF b THEN c1 ELSE c2, s) y t"   and "\<And>b s c2 x t y c1. \<lbrakk>\<not> bval b s; (c2, s) \<Rightarrow> x \<Down> t; P (c2, s) x t; y = x + 1\<rbrakk> \<Longrightarrow> P (IF b THEN c1 ELSE c2, s) y t"   and "\<And>b s c. \<not> bval b s \<Longrightarrow> P (WHILE b DO c, s) (Suc 0) s"   and "\<And>b s1 c x s2 y s3 z. \<lbrakk>bval b s1; (c, s1) \<Rightarrow> x \<Down> s2; P (c, s1) x s2; (WHILE b DO c, s2) \<Rightarrow> y \<Down> s3; P (WHILE b DO c, s2) y s3; 1 + x + y = z\<rbrakk> \<Longrightarrow> P (WHILE b DO c, s1) z s3" shows "P x1 x2 x3"
step: shows "\<theta>\<turnstile> (s, t) \<rightarrow>\<^sub>s ((s', t'), is)"
mult_left_mono: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<le> b" shows "c * a \<le> c * b"
assms_4: shows "VS\<^sub>2.dim U\<^sub>M\<^sub>_\<^sub>2 = VS\<^sub>1.dim U\<^sub>M\<^sub>_\<^sub>1"
power_commuting_commutes: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b"   and n :: "nat" assumes "class.monoid_mult one times"   and "times x y = times y x" shows "times (power.power one times x n) y = times y (power.power one times x n)"
vector_mult_component: fixes x :: "('c, 'd) vec"   and y :: "('c, 'd) vec"   and i :: "'d" shows "(x * y) $ i = x $ i * y $ i"
elims_2: fixes x :: "('e, 'f, 'g) FSM"   and xa :: "'e list set" assumes "is_det_state_cover x xa" obtains M :: "('e, 'f, 'g) FSM" and V :: "'e list set" where "x = M" and "xa = V" and "\<exists>f. is_det_state_cover_ass M f \<and> V = f ` d_reachable M (initial M)"
powers_2: fixes a :: "'a"   and b :: "'a" shows "a + b = b + a"
intro: fixes G :: "('a, 'b) pre_digraph"   and rev_G :: "'b \<Rightarrow> 'b"   and H :: "('a, 'b) pre_digraph"   and rev_H :: "'b \<Rightarrow> 'b"   and u :: "'a"   and v :: "'a"   and w :: "'a"   and uv :: "'b"   and uw :: "'b"   and vw :: "'b" assumes "subdivision_step G rev_G H rev_H (u, v, w) (uv, uw, vw)" shows "subdiv_step G rev_G H rev_H u v w uv uw vw"
zeta_neg_numeral: fixes n :: "num" shows "zeta (- numeral n) = - complex_of_real (bernoulli' (numeral (Num.inc n)) / numeral (Num.inc n))"
d\<mu>: shows "rat_of_int (d\<mu> i j) = rat_of_int (d fs (Suc j)) * gs.\<mu> i j"
simps_1: fixes n :: "String.literal" shows "name (Free n) = n"
less_1: fixes w :: "'a stream"   and thesis :: "bool" assumes "w !! 0 < wa__ !! 0"   and "\<And>k. sdrop k w = sconst (w !! k) \<Longrightarrow> thesis"   and "sdescending w" shows "thesis"
modelExpand: fixes valuation :: "Literal list"   and formula :: "Literal list list"   and literal :: "Literal" assumes "model valuation formula"   and "var literal \<notin> vars valuation" shows "model (valuation @ [literal]) formula"
finite_field_ops_int_def: fixes p :: "int" shows "finite_field_ops_int p \<equiv> Arith_Ops_Record 0 1 (plus_p p) (mult_p p) (minus_p p) (uminus_p p) (divide_p p) (inverse_p p) (\<lambda>x y. if y = 0 then x else 0) (\<lambda>x. if x = 0 then 0 else 1) (\<lambda>x. x) (\<lambda>x. x) (\<lambda>x. x) (\<lambda>x. 0 \<le> x \<and> x < p)"
subst_sst_cons: fixes a :: "('c, 'd) stateful_strand_step"   and A :: "('c, 'd) stateful_strand_step list"   and \<delta> :: "'d \<Rightarrow> ('c, 'd) Term.term" shows "a # A \<cdot>\<^sub>s\<^sub>s\<^sub>t \<delta> = (a \<cdot>\<^sub>s\<^sub>s\<^sub>t\<^sub>p \<delta>) # (A \<cdot>\<^sub>s\<^sub>s\<^sub>t \<delta>)"
linearD: shows "\<exists>es. char_poly D = (\<Prod>a\<leftarrow>es. [:- a, 1::'a:])"
not_inf: shows "u \<noteq> \<infinity>\<^sub>h"
sort_of_atom_eq: fixes permute :: "perm \<Rightarrow> 'b \<Rightarrow> 'b"   and atom :: "'b \<Rightarrow> atom"   and a :: "'b"   and b :: "'b" assumes "class.at permute atom" shows "sort_of (atom a) = sort_of (atom b)"
prime_elem_multD: fixes a :: "'a"   and b :: "'a" assumes "prime_elem (a * b)" shows "is_unit a \<or> is_unit b"
intro: fixes carrier :: "'b set"   and relation :: "('b \<times> 'b) set" assumes "total_on carrier relation" shows "rational_preference_axioms carrier relation"
IH_2: shows "(\<Sum>m\<in>monomials p2. MPoly_Type.coeff p2 m * (\<Prod>k\<in>keys m. f k ^ lookup m k)) = (\<Sum>m\<in>monomials (lowerPoly i j (liftPoly i j p2)). MPoly_Type.coeff (lowerPoly i j (liftPoly i j p2)) m * (\<Prod>k\<in>keys m. f k ^ lookup m k))"
chk_\<rho>t_\<rho>s_if: assumes "chksubs (>\<^sub>t) t s" shows "chksubs (>\<^sub>t) (subst \<rho> t) (subst \<rho> s)"
body_def: shows "body \<equiv> \<lambda>(b, x). map_spmf (\<lambda>b'. (\<not> b', x + (if b' then 0 else 1))) (bernoulli p)"
uniq_zero: fixes a :: "'a"   and b :: "'a"   and p\<^sub>1 :: "'a point"   and p\<^sub>2 :: "'a point" assumes "a \<in> carrier R"   and "b \<in> carrier R"   and "nonsingular a b"   and "on_curve a b p\<^sub>1"   and "on_curve a b p\<^sub>2"   and "local.add a p\<^sub>1 p\<^sub>2 = p\<^sub>2" shows "p\<^sub>1 = Infinity"
hyps: obtains n :: "nat" where "\<And>l. n \<le> l \<Longrightarrow> card (level A w (2 * k) l) \<le> card (nodes A) - k"
abs_summable_on_product: fixes A :: "'b set"   and B :: "'c set"   and f :: "'b \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "countable A"   and "countable B"   and "f abs_summable_on A"   and "g abs_summable_on B" shows "(\<lambda>(x, y). f x * g y) abs_summable_on A \<times> B"
tensor_mat_minus1: fixes m1 :: "'a mat"   and dims :: "nat list"   and vars :: "nat set"   and m2 :: "'a mat"   and m3 :: "'a mat" assumes "m1 \<in> carrier_mat (partial_state.d1 dims vars) (partial_state.d1 dims vars)"   and "m2 \<in> carrier_mat (partial_state.d1 dims vars) (partial_state.d1 dims vars)"   and "m3 \<in> carrier_mat (partial_state.d2 dims vars) (partial_state.d2 dims vars)" shows "tensor_mat dims vars (m1 - m2) m3 = tensor_mat dims vars m1 m3 - tensor_mat dims vars m2 m3"
change_on_0_init_tau: fixes i :: "nat"   and t :: "nat" assumes "change_on_0 (block i t) (gap i t)" shows "\<tau> i \<triangleright> (t + 1) = block i t"
set_eq: shows "set (remdups (map (consistent_sign_vec_copr qs) (characterize_root_list_p p))) = set signs \<inter> consistent_sign_vec_copr qs ` set (characterize_root_list_p p)"
bf_ite_def: fixes i :: "'b \<Rightarrow> bool"   and t :: "'b \<Rightarrow> 'c"   and e :: "'b \<Rightarrow> 'c" shows "bf_ite i t e \<equiv> \<lambda>l. if i l then t l else e l"
ex_in_conv: fixes A :: "'b set" shows "(\<exists>x. x \<in> A) = (A \<noteq> {})"
int_pow_consistent: fixes H :: "'c set"   and x :: "'c"   and n :: "int" assumes "subgroup H H"   and "x \<in> H" shows "x [^]\<^bsub>H\<^esub> n = x [^]\<^bsub>H\<lparr>carrier := H\<rparr>\<^esub> n"
Rep_seqauto_select: fixes SA :: "('d, 'e, 'f) seqauto" shows "(States SA, InitState SA, Labels SA, Delta SA) \<in> seqauto"
mon0_closed: shows "monom P a 0 \<in> carrier P"
plus_le_conv: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "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)"
simps: fixes p :: "'a Point"   and cone :: "'a Cone" shows "outsideCone p cone = (slope cone * slope cone * time2 (vertex cone) p < space2 (vertex cone) p)"
is_partition_of_def: fixes P :: "'a set set"   and A :: "'a set" shows "P partitions A = is_partition_of' P A"
simps_51: fixes f1 :: "bool \<Rightarrow> 'a"   and f2 :: "int \<Rightarrow> 'a"   and f3 :: "int \<Rightarrow> 'a"   and f4 :: "int \<Rightarrow> 'a"   and f5 :: "CTypeId \<Rightarrow> nat \<Rightarrow> 'a"   and f6 :: "Arraytype \<Rightarrow> nat \<Rightarrow> 'a"   and f7 :: "'a"   and x3 :: "int" shows "(case shortV x3 of boolV x \<Rightarrow> f1 x | intgV x \<Rightarrow> f2 x | shortV x \<Rightarrow> f3 x | byteV x \<Rightarrow> f4 x | objV x xa \<Rightarrow> f5 x xa | arrV x xa \<Rightarrow> f6 x xa | nullV \<Rightarrow> f7) = f3 x3"
a: shows "\<nexists>l. D \<in> P_of_state (lnth Sts l) \<and> D \<notin> P_of_state (lnth Sts (Suc l)) \<and> i \<le> l \<and> enat (Suc l) < llength Sts"
min_0_1_5: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and x :: "num" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "ord.min less_eq one (numeral.numeral one plus x) = one"
single_eq_single_zmset: fixes a :: "'a"   and b :: "'a" shows "({#a#}\<^sub>z = {#b#}\<^sub>z) = (a = b)"
pos_assign_monom: fixes \<alpha> :: "'c \<Rightarrow> 'd"   and m :: "'c monom" assumes "pos_assign \<alpha>" shows "(0::'d) \<le> eval_monom \<alpha> m"
inter_filter: fixes A :: "'d set"   and g :: "'d \<Rightarrow> 'a"   and P :: "'d \<Rightarrow> bool" assumes "finite A" shows "F g {x \<in> A. P x} = F (\<lambda>x. if P x then g x else \<^bold>1) A"