hcju/isabelleps huggingface dataset:
moebius_comp_inv_left: fixes M :: "moebius" shows "moebius_comp (moebius_inv M) M = id_moebius"
rep_eq: fixes x :: "'a regular"   and xa :: "'a regular" shows "Rep_regular (x * xa) = Rep_regular x * Rep_regular xa"
simps: fixes a1 :: "'a list"   and a2 :: "('b, 'c) sequent" shows "((a1, a2) \<in> Ax) = ((\<exists>i. a1 = [] \<and> a2 = ( \<LM> At i  \<RM> \<Rightarrow>* \<LM> At i  \<RM>)) \<or> a1 = [] \<and> a2 = ( \<LM> ff  \<RM> \<Rightarrow>* \<Empt>))"
rel_setI: fixes A :: "'a set"   and B :: "'b set"   and R :: "'a \<Rightarrow> 'b \<Rightarrow> bool" assumes "\<And>x. x \<in> A \<Longrightarrow> \<exists>y\<in>B. R x y"   and "\<And>y. y \<in> B \<Longrightarrow> \<exists>x\<in>A. R x y" shows "rel_set R A B"
ennreal_SUP: fixes f :: "'a \<Rightarrow> real"   and A :: "'a set" assumes "(\<Squnion>a\<in>A. ennreal (f a)) \<noteq> \<top>"   and "A \<noteq> {}" shows "ennreal (\<Squnion> (f ` A)) = (\<Squnion>a\<in>A. ennreal (f a))"
some_limit_def: shows "some_limit = (SOME a. has_as_limit a)"
simps_5: fixes fv :: "'a \<Rightarrow> 'b set"   and safe :: "safety \<Rightarrow> 'a \<Rightarrow> bool"   and g :: "safety"   and r :: "'a Regex.regex"   and s :: "'a Regex.regex" shows "Regex.safe_regex fv safe Past g (regex.Times r s) = ((g = Lax \<or> Regex.fv_regex fv s \<subseteq> Regex.fv_regex fv r) \<and> Regex.safe_regex fv safe Past g r \<and> Regex.safe_regex fv safe Past Lax s)"
content_times_primitive_part: fixes p :: "'b poly" shows "smult (content p) (primitive_part p) = p"
y: fixes ps :: "char list \<Rightarrow> int option"   and n :: "nat" assumes "P (ps, n)" shows "pre C (\<lambda>(ps, n). (Q \<and>* (\<lambda>s. True)) (ps, n div k)) (ps, k * n)"
elims: fixes x :: "nat"   and xa :: "'a fps"   and y :: "'a fps" assumes "fps_nth_deriv x xa = y" obtains f :: "'a fps" where "x = 0" and "xa = f__" and "y = f__"   | n :: "nat" and f :: "'a fps" where "x = Suc n" and "xa = f" and "y = fps_nth_deriv n (fps_deriv f)"
direct_decompE: fixes A :: "'b set"   and ss :: "'b set list"   and a :: "'b" assumes "direct_decomp A ss"   and "a \<in> A" obtains qs :: "'b list" where "qs \<in> listset ss" and "a = sum_list qs"
emsapss_decode_help6_def: fixes mHash :: "bit list"   and DB :: "bit list"   and H :: "bit list"   and emBits :: "nat" shows "emsapss_decode_help6 mHash DB H emBits = emsapss_decode_help7 mHash (maskedDB_zero DB emBits) H emBits"
WT_gpv_inline_invar: fixes gpv :: "('b, 'call, 'ret) gpv"   and s :: "'s" assumes "\<I> \<turnstile>g gpv \<surd>"   and "I s" shows "\<I>' \<turnstile>g inline callee gpv s \<surd>"
shared_weight_net_Input: fixes M :: "nat" shows "shared_weight_net (Input M)"
preserves_dom: fixes f :: "'c" assumes "C.arr f" shows "C.dom (C.map f) = C.map (C.dom f)"
partition_main_inv_def: fixes R :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and h :: "'d \<Rightarrow> 'c"   and lo :: "nat"   and hi :: "nat"   and xs0 :: "'d list"   and p :: "nat \<times> nat \<times> 'd list" shows "partition_main_inv R h lo hi xs0 p \<equiv> case p of (i, j, xs) \<Rightarrow> j < length xs \<and> j \<le> hi \<and> i < length xs \<and> lo \<le> i \<and> i \<le> j \<and> mset xs = mset xs0 \<and> (\<forall>k. lo \<le> k \<and> k < i \<longrightarrow> R (h (xs ! k)) (h (xs ! hi))) \<and> (\<forall>k. i \<le> k \<and> k < j \<longrightarrow> R (h (xs ! hi)) (h (xs ! k))) \<and> (\<forall>k<lo. xs ! k = xs0 ! k) \<and> (\<forall>k. j \<le> k \<and> k < length xs \<longrightarrow> xs ! k = xs0 ! k)"
noops_complete: fixes \<Psi> :: "('a, 'b) sas_plus_problem"   and \<pi> :: "('a, 'b) sas_plus_operator list"   and n :: "nat" assumes "SAS_Plus_Semantics.is_serial_solution_for_problem \<Psi> \<pi>" shows "SAS_Plus_Semantics.is_serial_solution_for_problem (prob_with_noop \<Psi>) (replicate n empty_sasp_action @ \<pi>)"
apath_def: fixes G :: "('c, 'd) pre_digraph"   and u :: "'c"   and p :: "'d list"   and v :: "'c" shows "pre_digraph.apath G u p v \<equiv> pre_digraph.awalk G u p v \<and> distinct (pre_digraph.awalk_verts G u p)"
widen_trans2: fixes G :: "prog"   and U :: "ty"   and T :: "ty"   and S :: "ty" assumes "G\<turnstile>U\<preceq>T"   and "G\<turnstile>S\<preceq>U"   and "wf_prog G" shows "G\<turnstile>S\<preceq>T"
semiring_simprules_9: fixes R :: "('c, 'd) ring_scheme"   and x :: "'c" assumes "semiring R"   and "x \<in> carrier R" shows "\<one>\<^bsub>R\<^esub> \<otimes>\<^bsub>R\<^esub> x = x"
con_rews_2: shows "ValTT \<noteq> \<bottom>"
minus_minus: 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" assumes "class.group_add minus plus zero uminus" shows "uminus (uminus a) = a"
strict_coboundedI2: fixes b :: "'c DBMEntry"   and c :: "'c DBMEntry"   and a :: "'c DBMEntry" assumes "b \<prec> c" shows "linordered_monoid.min a b \<prec> c"
list_induct_first2: fixes P :: "'a list \<Rightarrow> bool"   and xs :: "'a list" assumes "P []"   and "\<And>x. P [x]"   and "\<And>x1 x2 xs. P xs \<Longrightarrow> P (x1 # x2 # xs)" shows "P xs"
of_bool_def: fixes p :: "bool" shows "of_bool p = (if p then 1::'a else (0::'a))"
simps_7: fixes v :: "'a" shows "nnf (\<^bold>\<not> (Atom v)) = \<^bold>\<not> (Atom v)"
main: assumes "real_of_rat l \<le> x"   and "x \<le> real_of_rat r"   and "poly (real_of_int_poly p) x = 0"   and "root_cond plr x"   and "\<And>y. (x = y) = root_cond plr y"   and "\<And>y. (y = x) = root_cond plr y" shows "thesis"
finsum_cong': fixes A :: "'b set"   and B :: "'b set"   and f :: "'b \<Rightarrow> 'a"   and g :: "'b \<Rightarrow> 'a" assumes "A = B"   and "True"   and "\<And>i. i \<in> B \<Longrightarrow> f i = g i" shows "sum f A = sum g B"
antisym_conv2: fixes x :: "'b list"   and y :: "'b list" assumes "x \<le>s y" shows "(\<not> x <s y) = (x = y)"
dist_self: fixes dist :: "'d \<Rightarrow> 'd \<Rightarrow> real"   and uniformity :: "('d \<times> 'd) filter"   and "open" :: "'d set \<Rightarrow> bool"   and x :: "'d" assumes "class.metric_space dist uniformity open" shows "dist x x = 0"
intros_10: fixes P :: "(char list list \<times> char list exp) prog"   and v :: "val"   and s :: "heap \<times> sfields \<times> sheap" shows "P \<turnstile> \<langle>Val v,s\<rangle> \<Rightarrow> \<langle>Val v,s\<rangle>"
verit_le_mono_div_int: fixes A :: "int"   and B :: "int"   and n :: "int" assumes "A < B"   and "0 < n" shows "A div n + (if B mod n = 0 then 1 else 0) \<le> B div n"
ncols_iarray_def: fixes A :: "'c iarray iarray" shows "ncols_iarray A = IArray.length (A !! 0)"
flow_initial_time: fixes t0 :: "real"   and x0 :: "'a" assumes "t0 \<in> T"   and "x0 \<in> X" shows "flow t0 x0 t0 = x0"
list_set_rel_def: fixes R :: "('b \<times> 'c) set" shows "\<langle>R\<rangle>list_set_rel \<equiv> \<langle>R\<rangle>list_rel O br set distinct"
basic_trans_rules_17: fixes a :: "'a"   and b :: "'a" assumes "a \<noteq> b"   and "a \<le> b" shows "a < b"
suma_right_comm: 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 D :: "'a"   and E :: "'a"   and F :: "'a"   and G :: "'a"   and H :: "'a"   and I :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.SumA Bet Cong A B C D E F G H I" shows "Tarski_neutral_dimensionless.SumA Bet Cong A B C D E F I H G"
geodesic_segment_topology_4: fixes G :: "'b set" assumes "geodesic_segment G" shows "bounded G"
quadratic_hensel_loop: fixes q :: "int"   and S :: "int poly"   and T :: "int poly"   and D :: "int poly"   and H :: "int poly" assumes "local.quadratic_hensel_loop j = (q, S, T, D, H)" shows "poly_mod.Mp q C = poly_mod.Mp q (D * H) \<and> monic D \<and> poly_mod.Mp p D1 = poly_mod.Mp p D \<and> poly_mod.Mp p H1 = poly_mod.Mp p H \<and> poly_mod.Mp q (D * S + H * T) = poly_mod.Mp q 1 \<and> poly_mod.Mp q D = D \<and> poly_mod.Mp q H = H \<and> poly_mod.Mp q S = S \<and> poly_mod.Mp q T = T \<and> q = p ^ j"
orientable_6_7: fixes x :: "'a" shows "(local.comp_inf.symmetric x \<longrightarrow> (\<exists>y. y \<squnion> id y = x \<and> y \<sqinter> id y \<le> x \<sqinter> top)) = (local.comp_inf.symmetric x \<longrightarrow> (\<exists>y. y \<squnion> id y = x \<and> y \<sqinter> id y = x \<sqinter> top))"
elims: fixes x :: "int poly \<times> rat \<times> rat"   and y :: "int poly" assumes "poly_real_alg_1 x = y" obtains p :: "int poly" and uu_ :: "rat" and uv_ :: "rat" where "x = (p, uu_, uv_)" and "y = p"
surjective: shows "e1.Class ` Sigma I \<FF>1 = e1.Partition"
fps_divide_1: fixes a :: "'b fps" shows "a / 1 = a"
odd_Suc_div_two: fixes n :: "nat" assumes "odd n" shows "Suc n div 2 = Suc (n div 2)"
evaln_elim_cases_15: fixes G :: "prog"   and s :: "st"   and e :: "expr"   and n :: "nat"   and x :: "vals"   and s' :: "state" assumes "G\<turnstile>Norm s \<midarrow>In1r (Expr e)\<succ>\<midarrow>n\<rightarrow> (x, s')" obtains v :: "val" where "x = \<diamondsuit>" and "G\<turnstile>Norm s \<midarrow>e-\<succ>v\<midarrow>n\<rightarrow> s'"
run_altc_sample_writer: fixes altc_sample :: "'d \<Rightarrow> ('e \<Rightarrow> 'a) \<Rightarrow> 'a"   and p :: "'d"   and f :: "'e \<Rightarrow> ('b, 'c, 'a) writerT" shows "run_writer (altc_sample_writer altc_sample p f) = altc_sample p (\<lambda>p. run_writer (f p))"
pi: shows "contour_integral p (\<lambda>x. 1 / (x - z)) = complex_of_real (2 * pi) * \<i> * winding_number \<gamma> z"
t_def: shows "approx_tm prec ord I a f env = Some t"
quat_eqI: fixes x :: "quat"   and y :: "quat" assumes "quat.Re x = quat.Re y"   and "Im1 x = Im1 y"   and "Im2 x = Im2 y"   and "Im3 x = Im3 y" shows "x = y"
intros: fixes M :: "'c measure"   and f :: "'c \<Rightarrow> 'd"   and x :: "'d" assumes "has_bochner_integral M f x" shows "integrable M f"
satisfies_eq_def: fixes v :: "nat \<Rightarrow> 'a"   and eq :: "nat \<times> linear_poly" shows "v \<Turnstile>\<^sub>e eq \<equiv> v (lhs eq) = rhs eq \<lbrace> v \<rbrace>"
induct: fixes P :: "'a list \<Rightarrow> bool"   and a0 :: "'a list" assumes "P []"   and "\<And>x ys. P ys \<Longrightarrow> P (x # ys)" shows "P a0"
simps_5: fixes v :: "'b rexp"   and va :: "'b rexp" shows "le_rexp (Times v va) Zero = False"
intros_2: fixes x :: "'d" shows "ignore_Inl (Inr x) (Inr x)"
deg_pm_superset: fixes s :: "'a \<Rightarrow>\<^sub>0 'b"   and A :: "'a set" assumes "keys s \<subseteq> A"   and "finite A" shows "deg_pm s = sum (lookup s) A"
bin_rest_ge_0: fixes i :: "int" shows "(0 \<le> i div 2) = (0 \<le> i)"
tranclpD: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "R\<^sup>+\<^sup>+ x y" shows "\<exists>z. R x z \<and> R\<^sup>*\<^sup>* z y"
size_Un_Int: fixes A :: "'b multiset"   and B :: "'b multiset" shows "size A + size B = size (A \<union># B) + size (A \<inter># B)"
L_def: fixes n :: "nat"   and fs_init :: "int vec list" shows "LLL.L n fs_init = vec_module.lattice_of n fs_init"
case_1: fixes f1 :: "'a"   and f2 :: "'a"   and f3 :: "'a" shows "(case Acute of Acute \<Rightarrow> f1 | Grave \<Rightarrow> f2 | Macron \<Rightarrow> f3) = f1"
multiset_heapify: fixes t :: "'c Tree" shows "Heap.multiset (heapify t) = Heap.multiset t"
digraph_isomorphism_invI: fixes hom :: "('c, 'd, 'e, 'f) digraph_isomorphism" assumes "H.digraph_isomorphism hom" shows "pre_digraph.digraph_isomorphism (app_iso hom (app_iso hom G)) (H.inv_iso hom)"
bilinear_ladd: fixes h :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and x :: "'a"   and y :: "'a"   and z :: "'b" assumes "bilinear h" shows "h (x + y) z = h x z + h y z"
cont_times2: fixes below :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a" assumes "class.times_cpo below" shows "cont ((*) x)"
eq_upto_def: fixes B :: "(nat \<Rightarrow> 'b) set"   and f :: "nat \<Rightarrow> 'b"   and i :: "nat" shows "eq_upto B f i = {g \<in> B. \<forall>j<i. f j = g j}"
simps_6: fixes v :: "vname aexp"   and va :: "vname aexp" shows "enumerate_regs (Times v va) = enumerate_regs v \<union> enumerate_regs va"
map_of_ins_list: fixes x :: "'c"   and y :: "'d"   and ps :: "('c \<times> 'd) list" shows "map_of (upd_list x y ps) = map_of ps(x \<mapsto> y)"
simps_7: fixes bs :: "nat \<Rightarrow> 'a"   and t :: "'a poly"   and n :: "nat" shows "insertion bs (Pw t n) = insertion bs t ^ n"
path_weightp_def: fixes l1 :: "'vtx"   and l2 :: "'vtx"   and s :: "'lbl" shows "path_weightp l1 l2 s \<equiv> \<exists>xs. path l1 l2 xs \<and> s = sum_weights (map (\<lambda>(s, l, t). l) xs)"
is_extensional: fixes f :: "'b \<times> 'b" assumes "\<not> B.VV.arr f" shows "fst f \<star>\<^sub>B snd f = B.null"
UN_extend_simps_10: fixes B :: "'f \<Rightarrow> 'c set"   and f :: "'d \<Rightarrow> 'f"   and A :: "'d set" shows "(\<Union>a\<in>A. B (f a)) = \<Union> (B ` f ` A)"
disj_cancel_right: fixes x :: "'a" shows "x \<^bold>\<squnion> \<sim> x = \<one>"
mbufnt_take_add'_2: fixes f :: "event_data option list set list \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> 'a"   and z :: "'a"   and xss :: "event_data option list set list list"   and buf :: "event_data option list set list list"   and nts :: "nat list"   and z' :: "'a"   and buf' :: "event_data option list set list list"   and nts' :: "nat list"   and j :: "nat"   and P :: "String.literal \<Rightarrow> nat option"   and j' :: "nat"   and P' :: "String.literal \<Rightarrow> nat option"   and m :: "modality"   and g :: "safety"   and r :: "Formula.formula Regex.regex"   and mr :: "mregex"   and \<phi>s :: "Formula.formula list"   and \<sigma> :: "(String.literal \<times> event_data list) trace"   and V :: "String.literal \<Rightarrow> (nat \<Rightarrow> event_data list set) option"   and n :: "nat"   and R :: "event_data list set" assumes "mbufnt_take f z (mbufn_add xss buf) nts = (z', buf', nts')"   and "pred_mapping (\<lambda>x. x \<le> j) P"   and "pred_mapping (\<lambda>x. x \<le> j') P'"   and "rel_mapping (\<le>) P P'"   and "safe_regex m g r"   and "to_mregex r = (mr, \<phi>s)"   and "wf_mbufn' \<sigma> P V j n R r buf"   and "list_all2 (\<lambda>i t. t = \<tau> \<sigma> i) [progress_regex \<sigma> P r j..<j'] nts"   and "list_all3 list_all2 (map (\<lambda>\<phi> i. qtable n (fv \<phi>) (mem_restr R) (\<lambda>v. Formula.sat \<sigma> V (map the v) i \<phi>)) \<phi>s) (map2 upt (map (\<lambda>\<phi>. Monitor.progress \<sigma> P \<phi> j) \<phi>s) (map (\<lambda>\<phi>. Monitor.progress \<sigma> P' \<phi> j') \<phi>s)) xss"   and "j \<le> j'" shows "wf_ts_regex \<sigma> P' j' r nts'"
main: shows "(\<Sum>u'\<in>set us. (- (w \<bullet>c u') / (u' \<bullet>c u') \<cdot>\<^sub>v u') \<bullet>c u) = - (w \<bullet>c u)"
prefix_Nil: fixes xs :: "'b list" shows "prefix xs [] = (xs = [])"
d_i_1: shows "real_of_int (d fs' i) < real_of_rat reduction * real_of_int (d fs i)"
not_zero_i_A: shows "\<not> is_zero_row i A"
fun_isol: fixes f :: "'b \<Rightarrow> 'c" assumes "mono f" shows "mono ((\<circ>) f)"
zero_leastkA: shows "is_zero_row_upt_k (LEAST n. A $ n $ mod_type_class.from_nat k \<noteq> (0::'a) \<and> mod_type_class.from_nat i \<le> n) k A"
arr_fromArr: fixes x :: "'a"   and Arr :: "'a set" assumes "x \<in> Arr" shows "partial_magma.arr (concrete_category.COMP {False, True} (\<lambda>x y. Collect (graph.path {False, True} Arr (\<lambda>_. False) (\<lambda>_. True) x y)) (\<lambda>_ _ _ g f. f @ g)) (quiver.fromArr Arr x)"
g_inj: fixes i :: "nat"   and j :: "nat" assumes "g i = g j" shows "i = j"
map_option_idI: fixes x :: "'a option"   and f :: "'a \<Rightarrow> 'a" assumes "\<And>y. y \<in> set_option x \<Longrightarrow> f y = y" shows "map_option f x = x"
downshift_def: fixes P :: "nat fset" shows "downshift P = (\<lambda>x. x - Suc 0) |`| (P |-| {|0|})"
Red1NewFail: fixes h :: "'heap"   and C :: "String.literal"   and uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and l :: "'addr val list" assumes "allocate h (Class_type C) = {}" shows "uf,P,t \<turnstile>1 \<langle>new C,(h, l)\<rangle> -\<lbrace>\<rbrace>\<rightarrow> \<langle>Throw (addr_of_sys_xcpt OutOfMemory),(h, l)\<rangle>"
simps_5: fixes b :: "nat"   and t1 :: "Formula.trm"   and t2 :: "Formula.trm" shows "Formula.fvi b (formula.LessEq t1 t2) = Formula.fvi_trm b t1 \<union> Formula.fvi_trm b t2"
trace_minus: fixes A :: "('c, 'd) sq_matrix"   and B :: "('c, 'd) sq_matrix" shows "Square_Matrix.trace (A - B) = Square_Matrix.trace A - Square_Matrix.trace B"
insert_subset: fixes x :: "'h"   and A :: "'h set"   and B :: "'h set" shows "(insert x A \<subseteq> B) = (x \<in> B \<and> A \<subseteq> B)"
prems_1: shows "\<forall>xs\<in>set (xs # xss). length xs = d"
subdegreeI: fixes f :: "'a fps"   and d :: "nat" assumes "fps_nth f d \<noteq> (0::'a)"   and "\<And>i. i < d \<Longrightarrow> fps_nth f i = (0::'a)" shows "subdegree f = d"
S_def: shows "S = (\<sharp>\<cdot>) ((1::'f) / ordG) \<circ> sum CP G"
E: shows "(t1, X - {tick}) \<in> F P \<and> (u1, X - {tick}) \<in> F Q"
assms_7: shows "\<not> ([[b a c d]] \<or> [[b a d c]] \<or> [[b c a d]] \<or> [[b d a c]] \<or> [[c a b d]] \<or> [[c b a d]])"
induct: fixes x :: "'a list"   and P :: "'a list \<Rightarrow> bool" assumes "walk x"   and "P []"   and "\<And>v. v \<in> V \<Longrightarrow> P [v]"   and "\<And>v w vs. \<lbrakk>v \<rightarrow> w; walk (w # vs); P (w # vs)\<rbrakk> \<Longrightarrow> P (v # w # vs)" shows "P x"
sorted_Cons_iff: fixes x :: "'c"   and xs :: "'c list" shows "sorted (x # xs) = ((\<forall>y\<in>set xs. x < y) \<and> sorted xs)"
less_nrows: fixes m :: "nat"   and A :: "'b matrix" shows "(m < nrows A) = (\<exists>j i. m \<le> j \<and> Rep_matrix A j i \<noteq> (0::'b))"
Ints_power: fixes one :: "'b"   and times :: "'b \<Rightarrow> '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 a :: "'b"   and n :: "nat" assumes "class.ring_1 one times plus zero minus uminus"   and "a \<in> ring_1.Ints one plus zero minus" shows "power.power one times a n \<in> ring_1.Ints one plus zero minus"
frechet_derivative_works: fixes f :: "'a \<Rightarrow> 'b"   and net :: "'a filter" shows "(f differentiable net) = (f has_derivative frechet_derivative f net) net"
worse_in_no_better: fixes carrier :: "'a set"   and relation :: "('a \<times> 'a) set"   and x :: "'a"   and y :: "'a" assumes "rational_preference carrier relation"   and "x \<succeq>[relation] y" shows "y \<in> no_better_than y carrier relation"
walk_to_path: fixes xs :: "'a list"   and v :: "'a"   and w :: "'a" assumes "H.walk xs"   and "xs \<noteq> []"   and "hd xs = v"   and "last xs = w" shows "\<exists>ys. v \<leadsto>ys\<leadsto>\<^bsub>H\<^esub> w \<and> set ys \<subseteq> set xs"
length_nth_simps_3: fixes x :: "'a"   and xs :: "'a list" shows "(x # xs) ! 0 = x"
alpha_tNot: fixes t1 :: "('a, 'b, 'c, 'd) Tree"   and t2 :: "('a, 'b, 'c, 'd) Tree" shows "(tNot t1 =\<^sub>\<alpha> tNot t2) = (t1 =\<^sub>\<alpha> t2)"
distinct_91: shows "LessOrEqual \<noteq> GreaterThan"
is_path_f': assumes "infinite (deriv s)" shows "\<forall>n. f n \<in> deriv s \<and> fst (f n) = n \<and> infinite (deriv (snd (f n))) \<and> \<not> is_axiom (s_of_ns (snd (f n)))"
omega_sum_less_\<omega>\<omega>: fixes ms :: "nat list" shows "omega_sum ms < \<omega> \<up> \<omega>"
wf_dual_rexp_dual_of: fixes \<Sigma> :: "'a list"   and n :: "nat"   and r :: "'a atom rexp" assumes "M2L_Equivalence_Checking.wf \<Sigma> n r" shows "wf_dual \<Sigma> n (rexp_dual_of r)"
sets_expr_sem: fixes \<Gamma> :: "nat \<Rightarrow> pdf_type"   and w :: "expr"   and t :: "pdf_type"   and \<sigma> :: "nat \<Rightarrow> val" assumes "\<Gamma> \<turnstile> w : t"   and "\<forall>x\<in>free_vars w. val_type (\<sigma> x) = \<Gamma> x" shows "sets (expr_sem \<sigma> w) = sets (stock_measure t)"
gen_model_subset: fixes M :: "('b, 'c, 'd) KripkeStructure"   and M' :: "('b, 'c, 'd) KripkeStructure"   and T :: "'d set"   and t :: "'d" assumes "kripke M"   and "kripke M'"   and "\<And>a. Restr (relations M a) T = Restr (relations M' a) T"   and "(\<Union> (range (relations M)))\<^sup>* `` {t} \<subseteq> T"   and "(\<Union> (range (relations M')))\<^sup>* `` {t} \<subseteq> T"   and "t \<in> worlds M"   and "t \<in> worlds M'"   and "valuation M = valuation M'" shows "gen_model M t = gen_model M' t"
1: shows "(\<lceil>c\<rceil>\<^sub>< \<Rightarrow> \<lceil>d\<^sub>1 \<and> d\<^sub>2\<rceil>\<^sub>>) \<sqsubseteq> (\<lceil>c\<rceil>\<^sub>< \<Rightarrow> \<lceil>d\<^sub>1\<rceil>\<^sub>>) |a|b|\<^sub>\<sigma> (\<lceil>c\<rceil>\<^sub>< \<Rightarrow> \<lceil>d\<^sub>2\<rceil>\<^sub>>)"
assms: shows "insertion f (liftPoly 0 z (c - Const C)) = 0"
uint_nonnegative: fixes w :: "'c word" shows "0 \<le> uint w"
stake_add: fixes i :: "nat"   and \<sigma> :: "nat \<Rightarrow> 'b"   and j :: "nat" shows "stake i \<sigma> @ \<sigma>(i \<rightarrow> j) = stake (i + j) \<sigma>"
least_equality: fixes x :: "hf"   and a :: "hf" assumes "x \<^bold>\<in> a"   and "\<And>y. y \<^bold>\<in> a \<Longrightarrow> Rep_hf x \<le> Rep_hf y" shows "least a = x"
ClassI: fixes a :: "'a set \<times> 'b"   and b :: "'a set \<times> 'b" assumes "(a, b) \<in> {(x, y). x \<sim> y}" shows "a \<in> rel.Class b"
dep3_event_permute_5: fixes \<E> :: "'b set"   and \<P> :: "'b set set"   and Q :: "'b set"   and R :: "'b set"   and S :: "'b set"   and x :: "'b" assumes "MinkowskiPrimitive \<E> \<P>"   and "MinkowskiPrimitive.dep3_event \<P> Q R S x" shows "MinkowskiPrimitive.dep3_event \<P> S R Q x"
eval_cases_12: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and E :: "char list \<Rightarrow> ty option"   and V :: "char list"   and T :: "ty"   and e\<^sub>1 :: "expr"   and s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and e' :: "expr"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)" assumes "P,E \<turnstile> \<langle>{V:T; e\<^sub>1},s\<rangle> \<Rightarrow> \<langle>e',s'\<rangle>" obtains h\<^sub>0 :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option" and l\<^sub>0 :: "char list \<Rightarrow> val option" and h\<^sub>1 :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option" and l\<^sub>1 :: "char list \<Rightarrow> val option" where "s = (h\<^sub>0, l\<^sub>0)" and "s' = (h\<^sub>1, l\<^sub>1(V := l\<^sub>0 V))" and "P,E(V \<mapsto> T) \<turnstile> \<langle>e\<^sub>1,(h\<^sub>0, l\<^sub>0(V := None))\<rangle> \<Rightarrow> \<langle>e',(h\<^sub>1, l\<^sub>1)\<rangle>"
mat_mult_row_code: fixes m :: "(('c, 'd) Finite_Cartesian_Product.vec, 'e) Finite_Cartesian_Product.vec"   and m' :: "(('c, 'b) Finite_Cartesian_Product.vec, 'd) Finite_Cartesian_Product.vec"   and f :: "'e" shows "($h) (mat_mult_row m m' f) = (\<lambda>c. \<Sum>k\<in>UNIV. m $h f $h k * m' $h k $h c)"
preserves_arr: fixes f :: "'c \<times> 'c" assumes "C.VV.arr f" shows "C.hseq (fst f) (snd f)"
cancel_eq: fixes F :: "ordinal \<Rightarrow> ordinal"   and x :: "ordinal"   and y :: "ordinal" assumes "normal F" shows "(F x = F y) = (x = y)"
init_u: fixes a\<^sub>e :: "real"   and v\<^sub>e :: "real"   and s\<^sub>e :: "real"   and a\<^sub>o :: "real"   and v\<^sub>o :: "real"   and s\<^sub>o :: "real"   and \<delta> :: "real"   and t :: "real" assumes "safe_distance_normal a\<^sub>e v\<^sub>e s\<^sub>e a\<^sub>o v\<^sub>o s\<^sub>o \<delta>"   and "t \<le> 0" shows "safe_distance_normal.u a\<^sub>e v\<^sub>e s\<^sub>e \<delta> t = s\<^sub>e"
T_deter_rand: fixes s0 :: "'a list"   and qs :: "'a list"   and A :: "('a list \<Rightarrow> 'b) \<times> ('a list \<times> 'b \<Rightarrow> 'a \<Rightarrow> (nat \<times> nat list) \<times> 'b)"   and x :: "'b" shows "real (T\<^sub>p s0 qs (Partial_Cost_Model.off2 A (s0, x) qs)) = T\<^sub>p_on_rand' (Partial_Cost_Model.embed A) (return_pmf (s0, x)) qs"
fmrel_restrict_fset: fixes P :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and m :: "'f \<rightharpoonup>\<^sub>f 'd"   and n :: "'f \<rightharpoonup>\<^sub>f 'e"   and A :: "'f fset" assumes "fmrel P m n" shows "fmrel P (fmrestrict_fset A m) (fmrestrict_fset A n)"