hcju/isabelleps huggingface dataset:
rbt_lookup_rbt_bulkload: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and xs :: "('a \<times> 'b) list" assumes "class.linorder less_eq less" shows "ord.rbt_lookup less (ord.rbt_bulkload less xs) = map_of xs"
height_r_bal: fixes r :: "'a tree\<^sub>0"   and l :: "'a tree\<^sub>0"   and n :: "'a" assumes "height r = height l + 2" shows "height (r_bal\<^sub>0 n l r) = height l + 2 \<or> height (r_bal\<^sub>0 n l r) = height l + 3"
dom_rhs: shows "\<D>\<^sub>\<circ> [\<NN>\<lparr>NTMap\<rparr>, \<NN>\<lparr>NTDGDom\<rparr>, \<NN>\<lparr>NTDGCod\<rparr>, \<NN>\<lparr>NTDom\<rparr>, \<NN>\<lparr>NTCod\<rparr>]\<^sub>\<circ> = 5\<^sub>\<nat>"
rel_is_equivalence: fixes S :: "'c set"   and is_open :: "'c set \<Rightarrow> bool"   and \<FF> :: "'c set \<Rightarrow> 'd set"   and \<rho> :: "'c set \<Rightarrow> 'c set \<Rightarrow> 'd \<Rightarrow> 'd"   and b :: "'d"   and add_str :: "'c set \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and mult_str :: "'c set \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and zero_str :: "'c set \<Rightarrow> 'd"   and one_str :: "'c set \<Rightarrow> 'd"   and I :: "'c set set" assumes "direct_lim S is_open \<FF> \<rho> b add_str mult_str zero_str one_str I" shows "equivalence (Sigma I \<FF>) {(x, y). direct_lim.rel \<FF> \<rho> I x y}"
lookup_lower: fixes pair_of_term :: "'c \<Rightarrow> 'd \<times> 'e"   and term_of_pair :: "'d \<times> 'e \<Rightarrow> 'c"   and ord :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ord_strict :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ord_term :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and ord_term_strict :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and p :: "'c \<Rightarrow>\<^sub>0 'f"   and u :: "'c"   and v :: "'c" assumes "ordered_term pair_of_term term_of_pair ord ord_strict ord_term ord_term_strict" shows "lookup (ordered_term.lower ord_term p u) v = (if ord_term_strict v u then lookup p v else (0::'f))"
single_valuedI: fixes r :: "('d \<times> 'e) set" assumes "\<And>x y z. \<lbrakk>(x, y) \<in> r; (x, z) \<in> r\<rbrakk> \<Longrightarrow> y = z" shows "single_valued r"
r_def: shows "r = fst (solve_factored_ratfps' p fctrs)"
path_path_shift: fixes \<pi> :: "nat \<Rightarrow> 'n"   and m :: "nat" assumes "is_path \<pi>" shows "is_path (\<pi> \<guillemotleft> m)"
simps: fixes k :: "nat"   and xs :: "'b list" shows "brauns k xs = (if xs = [] then [] else let ys = take (2 ^ k) xs; zs = drop (2 ^ k) xs; ts = brauns (k + 1) zs in nodes ts ys (drop (2 ^ k) ts))"
update_by_fun_gr_pair_def: fixes k :: "'a"   and f :: "'c \<Rightarrow> 'c"   and xs :: "('a \<times> 'c) list" shows "update_by_fun_gr_pair k f xs = (if xs = [] then let v = f (0::'c) in if v = (0::'c) then [] else [(k, v)] else if comp k (fst (last xs)) = Gt then let v = f (0::'c) in if v = (0::'c) then xs else xs @ [(k, v)] else update_by_fun_pair k f xs)"
iwlsFSbSw_def: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'c, 'd, 'e, 'f) Iteration.model_scheme" shows "iwlsFSbSw MOD \<equiv> iwlsFSb MOD \<and> igSwapAllIPresIGWlsAll MOD \<and> igSwapCls MOD"
leq_refl: fixes leq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c" assumes "class.quasi_order leq" shows "leq x x"
smap_id: shows "smap \<circ> \<eta> = id"
Dom_def: fixes L :: "('a \<times> 'b) list" shows "Dom L = {l. \<exists>a. lookup L l = Some a}"
norm_list: fixes f :: "dB \<Rightarrow> dB"   and u :: "dB"   and e :: "nat \<Rightarrow> LambdaType.type"   and T :: "LambdaType.type"   and i :: "nat"   and as :: "dB list"   and Us :: "LambdaType.type list" assumes "\<And>t t'. t \<rightarrow>\<^sub>\<beta>\<^sup>* t' \<Longrightarrow> f t \<rightarrow>\<^sub>\<beta>\<^sup>* f t'"   and "\<And>t. NF t \<Longrightarrow> NF (f t)"   and "NF u"   and "e \<turnstile> u : T"   and "e\<langle>i:T\<rangle> \<tturnstile> as : Us"   and "listall (\<lambda>t. \<forall>e T' u i. e\<langle>i:T\<rangle> \<turnstile> t : T' \<longrightarrow> NF u \<longrightarrow> e \<turnstile> u : T \<longrightarrow> (\<exists>t'. t[u/i] \<rightarrow>\<^sub>\<beta>\<^sup>* t' \<and> NF t')) as" shows "\<exists>as'. \<forall>j. dB.Var j \<degree>\<degree> map (\<lambda>t. f (t[u/i])) as \<rightarrow>\<^sub>\<beta>\<^sup>* dB.Var j \<degree>\<degree> map f as' \<and> NF (dB.Var j \<degree>\<degree> map f as')"
tpairs_id_cons: fixes xs :: "abc_inst list"   and x :: "abc_inst" shows "tpairs_of (xs @ [x]) = tpairs_of xs @ [(start_of (layout_of (xs @ [x])) (length xs), x)]"
Basis_list_vec_def: shows "Basis_list = concat (map (\<lambda>n. map (axis n) Basis_list) enum_class.enum)"
partition_permI: fixes p :: "'b \<Rightarrow> bool"   and xs :: "'b list" shows "filter p xs @ filter (Not \<circ> p) xs <~~> xs"
vector_derivative_linepath_within: fixes x :: "real"   and a :: "'b"   and b :: "'b" assumes "x \<in> {0..1}" shows "vector_derivative (linepath a b) (at x within {0..1}) = b - a"
Seq: fixes P :: "(char list \<Rightarrow> int) \<Rightarrow> bool"   and c\<^sub>1 :: "com"   and Q :: "(char list \<Rightarrow> int) \<Rightarrow> bool"   and c\<^sub>2 :: "com"   and R :: "(char list \<Rightarrow> int) \<Rightarrow> bool" assumes "\<turnstile> {P} c\<^sub>1 {Q}"   and "\<turnstile> {Q} c\<^sub>2 {R}" shows "\<turnstile> {P} c\<^sub>1;; c\<^sub>2 {R}"
in_set_independent_pdevs_invariant_nlex: fixes x :: "real \<times> real"   and xs :: "(real \<times> real) list" assumes "x \<in> set (independent_pdevs xs)"   and "\<And>x. x \<in> set xs \<Longrightarrow> Counterclockwise_2D_Arbitrary.lex x 0"   and "\<And>x. x \<in> set xs \<Longrightarrow> x \<noteq> 0" shows "Counterclockwise_2D_Arbitrary.lex x 0"
P: shows "prime (a\<^sup>2 + N * b\<^sup>2)"
bddh_make_bdd: fixes n :: "nat"   and f :: "nat list \<Rightarrow> 'a"   and xs :: "nat list" shows "bddh n (make_bdd f n xs)"
vint_of_int_def: shows "vint_of_int = (\<lambda>x. (real_of_int x)\<^sub>\<real>)"
ideE: fixes V\<^sub>B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H\<^sub>B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a>\<^sub>B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i>\<^sub>B :: "'b \<Rightarrow> 'b"   and src\<^sub>B :: "'b \<Rightarrow> 'b"   and trg\<^sub>B :: "'b \<Rightarrow> 'b"   and F :: "('b bicategorical_language.term, 'b) concrete_category.arr" assumes "strictified_bicategory V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B"   and "partial_magma.ide (concrete_category.COMP {A. bicategorical_language.Nml V\<^sub>B src\<^sub>B trg\<^sub>B A \<and> bicategorical_language.Ide V\<^sub>B src\<^sub>B trg\<^sub>B A} (\<lambda>A B. {\<mu>. bicategorical_language.Src src\<^sub>B A = bicategorical_language.Src src\<^sub>B B \<and> bicategorical_language.Trg trg\<^sub>B A = bicategorical_language.Trg trg\<^sub>B B \<and> partial_magma.in_hom V\<^sub>B \<mu> (Coherence.evaluation_map.eval V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B (\<lambda>\<mu>. if partial_magma.arr V\<^sub>B \<mu> then \<mu> else partial_magma.null V\<^sub>B) A) (Coherence.evaluation_map.eval V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B (\<lambda>\<mu>. if partial_magma.arr V\<^sub>B \<mu> then \<mu> else partial_magma.null V\<^sub>B) B)}) (\<lambda>_ _ _. V\<^sub>B)) F" obtains "partial_magma.arr (concrete_category.COMP {A. bicategorical_language.Nml V\<^sub>B src\<^sub>B trg\<^sub>B A \<and> bicategorical_language.Ide V\<^sub>B src\<^sub>B trg\<^sub>B A} (\<lambda>A B. {\<mu>. bicategorical_language.Src src\<^sub>B A = bicategorical_language.Src src\<^sub>B B \<and> bicategorical_language.Trg trg\<^sub>B A = bicategorical_language.Trg trg\<^sub>B B \<and> partial_magma.in_hom V\<^sub>B \<mu> (Coherence.evaluation_map.eval V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B (\<lambda>\<mu>. if partial_magma.arr V\<^sub>B \<mu> then \<mu> else partial_magma.null V\<^sub>B) A) (Coherence.evaluation_map.eval V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B (\<lambda>\<mu>. if partial_magma.arr V\<^sub>B \<mu> then \<mu> else partial_magma.null V\<^sub>B) B)}) (\<lambda>_ _ _. V\<^sub>B)) F" and "concrete_category.Dom F = concrete_category.Cod F" and "partial_magma.ide V\<^sub>B (concrete_category.Map F)" and "concrete_category.Map F = Coherence.evaluation_map.eval V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B (\<lambda>\<mu>. if partial_magma.arr V\<^sub>B \<mu> then \<mu> else partial_magma.null V\<^sub>B) (concrete_category.Dom F)" and "concrete_category.Map F = Coherence.evaluation_map.eval V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B (\<lambda>\<mu>. if partial_magma.arr V\<^sub>B \<mu> then \<mu> else partial_magma.null V\<^sub>B) (concrete_category.Cod F)"
powsum_mpoly_empty: fixes k :: "nat" shows "powsum_mpoly {} k = 0"
vector_mul_lcancel: fixes a :: "'a"   and x :: "('a, 'b) Finite_Cartesian_Product.vec"   and y :: "('a, 'b) Finite_Cartesian_Product.vec" shows "(a *s x = a *s y) = (a = (0::'a) \<or> x = y)"
le_fin_le_fininf_transp: fixes w\<^sub>1 :: "'action list"   and w\<^sub>2 :: "'action list"   and w\<^sub>3 :: "'action stream" assumes "w\<^sub>1 \<preceq>\<^sub>F w\<^sub>2"   and "w\<^sub>2 \<preceq>\<^sub>F\<^sub>I w\<^sub>3" shows "w\<^sub>1 \<preceq>\<^sub>F\<^sub>I w\<^sub>3"
ccw'_sortedP_scaled_inl: fixes X :: "'a \<times> (real \<times> real) pdevs" shows "ccw'.sortedP 0 (map ((*\<^sub>R) 2) (ccw.selsort 0 (inl (snd X))))"
corthogonal_distinct: fixes us :: "'a vec list" assumes "corthogonal us" shows "distinct us"
bounded_iff: fixes b :: "'a"   and c :: "'a"   and a :: "'a" shows "(b \<squnion> c \<le> a) = (b \<le> a \<and> c \<le> a)"
Sup_closed_coclop_var: fixes f :: "'b \<Rightarrow> 'b"   and X :: "'b set" assumes "coclop f"   and "\<forall>x\<in>X. x \<in> range f" shows "\<Squnion> X \<in> range f"
B1_def: shows "B1 = {GREATEST x. x \<in> U |U. U \<in> A}"
representation_of_functor_def: fixes C :: "'ag \<Rightarrow> 'ag \<Rightarrow> 'ag"   and S :: "'ah \<Rightarrow> 'ah \<Rightarrow> 'ah"   and \<phi> :: "'ag \<times> 'ag \<Rightarrow> 'ag \<Rightarrow> 'ah"   and F :: "'ag \<Rightarrow> 'ah"   and a :: "'ag"   and \<Phi> :: "'ag \<Rightarrow> 'ah" shows "representation_of_functor C S \<phi> F a \<Phi> \<equiv> (category C \<and> dual_category C \<and> replete_set_category S) \<and> (functor (dual_category.comp C) S F \<and> hom_functor C S \<phi>) \<and> yoneda_functor_fixed_object C S \<phi> a \<and> natural_isomorphism (dual_category.comp C) S (yoneda_functor.Y C S \<phi> a) F \<Phi>"
removeKeyList_eq: fixes as :: "'c list"   and ps :: "('c \<times> 'd) list" shows "removeKeyList as ps = filter (\<lambda>p. \<forall>a\<in>set as. a \<noteq> fst p) ps"
relation_mono: fixes A :: "'b set"   and C :: "'b set"   and B :: "'c set"   and D :: "'c set" assumes "A \<subseteq> C"   and "B \<subseteq> D" shows "A \<times> B \<subseteq> C \<times> D"
Macaulay_list_def: fixes ps :: "('t \<Rightarrow>\<^sub>0 'c) list" shows "Macaulay_list ps = filter (\<lambda>p. p \<noteq> 0) (mat_to_polys (Keys_to_list ps) (row_echelon (Macaulay_mat ps)))"
progression_def: fixes R :: "'s \<Rightarrow> 's \<Rightarrow> bool"   and S :: "'s \<Rightarrow> 's \<Rightarrow> bool" shows "R \<rightarrow> S = (\<forall>s1 s2. R s1 s2 \<longrightarrow> wf_state s1 \<and> wf_state s2 \<and> final s1 = final s2 \<and> (\<forall>x\<in>set (\<sigma> n). BNF_Greatest_Fixpoint.image2p post post S (post (delta x s1)) (post (delta x s2))))"
cts_wp_SetDC_const: fixes S :: "'a set"   and a :: "'a \<Rightarrow> bool \<Rightarrow> ('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real" assumes "\<And>x. x \<in> S \<Longrightarrow> bd_cts (wp (a x))"   and "\<And>x. x \<in> S \<Longrightarrow> healthy (wp (a x))"   and "finite S"   and "S \<noteq> {}" shows "bd_cts (wp (SetDC a (\<lambda>_. S)))"
comm_group_abelian_groupI: fixes G :: "('e, 'f) ring_scheme" assumes "comm_group (add_monoid G)" shows "abelian_group G"
let4: fixes l :: "nat"   and binds :: "((nat \<times> char list) \<times> lambda) list"   and c' :: "call"   and x :: "lambda" assumes "call.Let l binds c' \<in> calls x" shows "snd ` set binds \<subseteq> lambdas x"
supp_2: fixes exp :: "exp"   and var :: "Vars.var" shows "supp (App exp var) = supp exp \<union> supp var"
X: shows "[:x (Suc (l + j)), - (1::'a):] = - local.X (Suc l + j)"
subst0_bound0: fixes p :: "fm"   and t :: "Cooper.num" assumes "qfree p"   and "numbound0 t" shows "bound0 (subst0 t p)"
0: shows "\<forall>f\<in>injections X Y. x \<notin> Domain f"
rel_spmf_inf: fixes ord :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d spmf"   and q :: "'d spmf" assumes "ord_spmf ord p q"   and "ord_spmf ord q p"   and "reflp ord"   and "transp ord" shows "rel_spmf (ord \<sqinter> ord\<inverse>\<inverse>) p q"
eq_box: fixes a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" shows "(box a b = box c d) = (box a b = {} \<and> box c d = {} \<or> a = c \<and> b = d)"
assms: fixes e :: "'a" assumes "e \<in> set_pmf p" shows "pmf p e = pmf q e"
cos1_sin0: fixes \<phi> :: "real" assumes "cos \<phi> = 1"   and "sin \<phi> = 0" shows "\<exists>x. \<phi> = 2 * real_of_int x * pi"
pickFT_theFT: fixes cl :: "('g, 'h, 'i) cmd list" assumes "WtFT cl = 1" shows "pickFT cl \<in> theFT cl"
rev_transfer: fixes A :: "'c \<Rightarrow> 'd \<Rightarrow> bool" shows "(list_all2 A ===> list_all2 A) rev rev"
alt_seq_Zp_cauchy: fixes s :: "nat \<Rightarrow> nat \<Rightarrow> int" assumes "is_Zp_cauchy s" shows "is_Zp_cauchy (alt_seq s)"
vpeq_subj_subj_def: fixes u :: "partition_id_t"   and s :: "state_t"   and t :: "state_t" shows "vpeq_subj_subj u s t \<equiv> \<forall>v. (Policy.sp_spec_subj_subj u v \<longrightarrow> sp_impl_subj_subj s u v = sp_impl_subj_subj t u v) \<and> (Policy.sp_spec_subj_subj v u \<longrightarrow> sp_impl_subj_subj s v u = sp_impl_subj_subj t v u)"
last_rev: fixes xs :: "'b list" assumes "xs \<noteq> []" shows "last (rev xs) = hd xs"
estep: fixes c :: "(recf \<times> nat list \<times> nat list) list \<times> nat option" assumes "valid (fst c)" shows "estep (encode_config c) = encode_config (step c)"
left_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 "plus (uminus a) a = zero"
comp_wiring_def: shows "(\<circ>\<^sub>w) = (\<lambda>(f, g) (f', g'). (f' \<circ> f, g \<circ> g'))"
rel_conversep: fixes R1 :: "'j \<Rightarrow> 'h \<Rightarrow> bool"   and R2 :: "'k \<Rightarrow> 'i \<Rightarrow> bool" shows "rel_sum R1\<inverse>\<inverse> R2\<inverse>\<inverse> = (rel_sum R1 R2)\<inverse>\<inverse>"
syzygy_module_listE: fixes s :: "'t \<Rightarrow>\<^sub>0 'c"   and bs :: "('t \<Rightarrow>\<^sub>0 'c) list" assumes "s \<in> syzygy_module_list bs" obtains s' :: "('t \<Rightarrow>\<^sub>0 'c) \<Rightarrow>\<^sub>0 'a \<Rightarrow>\<^sub>0 'c" where "s' \<in> pmdl.syzygy_module (set bs)" and "s = atomize_poly (idx_pm_of_pm bs s')"
S_transition_eqvt: fixes P :: "'state"   and \<alpha>\<^sub>SP' :: "(('act, 'pred) S_action, 'state) residual"   and p :: "perm" assumes "P \<rightarrow>\<^sub>S \<alpha>\<^sub>SP'" shows "p \<bullet> P \<rightarrow>\<^sub>S p \<bullet> \<alpha>\<^sub>SP'"
step: fixes p :: "'s1 spmf"   and q :: "'s2 spmf"   and a :: "'a" assumes "S p q"   and "a \<in> A" shows "p \<bind> (\<lambda>s. map_spmf fst (callee1 s a)) = q \<bind> (\<lambda>s. map_spmf fst (callee2 s a))"
generate_fieldE_2: fixes H :: "'a set"   and E :: "'a set" assumes "H \<subseteq> carrier R"   and "E = generate_field R H" shows "H \<subseteq> E"
SINK_def: shows "SINK \<equiv> \<lambda>f. {x. SINKp f x}"
upper: fixes x :: "'c Gromov_completion"   and y :: "'c Gromov_completion" shows "turn_into_distance extended_predist x y \<le> extended_predist x y"
pruned_def: fixes g :: "'g" shows "pruned g = (\<forall>n\<in>set (\<alpha>n g). \<forall>val. val \<in> phiDefs g n \<longrightarrow> liveVal g val)"
subst_all_insert: fixes \<delta> :: "'e \<Rightarrow> ('c, 'd) Term.term"   and t :: "('c, 'e) Term.term"   and M :: "('c, 'e) Term.term set" shows "insert t M \<cdot>\<^sub>s\<^sub>e\<^sub>t \<delta> = insert (t \<cdot> \<delta>) (M \<cdot>\<^sub>s\<^sub>e\<^sub>t \<delta>)"
insertion_lowerPoly1: fixes i :: "nat"   and p :: "'b mpoly"   and prfx :: "'b list"   and x :: "'b"   and L :: "'b list" assumes "i \<notin> vars p"   and "length prfx = i" shows "insertion (nth_default (0::'b) (prfx @ x # L)) p = insertion (nth_default (0::'b) (prfx @ L)) (lowerPoly i 1 p)"
times2_conv_add: fixes j :: "nat" shows "j + j = 2 * j"
ls_update_map_of_eq: fixes k :: "'b"   and v :: "'a"   and ls :: "('b \<times> 'a) list" shows "map_of (fst (ls_update k v ls)) k = Some v"
as_needed_asses_submap_exec_v: fixes v :: "'d"   and f :: "('d, 'e) fmap"   and g :: "('d, 'e) fmap" assumes "v \<in> fmdom' f"   and "f \<subseteq>\<^sub>f g" shows "v \<in> fmdom' g"
subspace_0: fixes scale :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and S :: "'c set" assumes "module scale"   and "module.subspace scale S" shows "(0::'c) \<in> S"
finite_imp_compact: fixes s :: "'b set" assumes "finite s" shows "compact s"
h4: shows "aEval (Leq (A + B * Var var)) (L[var := sqrt Cv]) = eval (Or (And (fm.Atom (Leq A)) (fm.Atom (Leq (B\<^sup>2 * c - A\<^sup>2)))) (And (fm.Atom (Leq B)) (fm.Atom (Leq (A\<^sup>2 - B\<^sup>2 * c))))) (L[var := sqrt Cv])"
pth_2: fixes x :: "'a"   and y :: "'a" shows "x - y \<equiv> x + - y"
rel_fundef_rewrite_body: fixes fd1 :: "('dyn, 'a, 'b, 'c, 'opinl) Inca.instr fundef"   and fd2 :: "('dyn, 'a, 'b, 'c, 'opinl, 'opubx, 'ubx1, 'ubx2) Ubx.instr fundef"   and pc :: "nat"   and instr :: "('dyn, 'a, 'b, 'c, 'opinl, 'opubx, 'ubx1, 'ubx2) Ubx.instr" assumes "rel_fundef (\<lambda>x y. x = norm_instr y) fd1 fd2"   and "norm_instr (body fd2 ! pc) = norm_instr instr" shows "rel_fundef (\<lambda>x y. x = norm_instr y) fd1 (rewrite_fundef_body fd2 pc instr)"
uniformity_octo_def: shows "uniformity = (INF e\<in>{0<..}. principal {(x, y). dist x y < e})"
eqpoll_def: fixes A :: "'c set"   and B :: "'d set" shows "A \<approx> B \<equiv> \<exists>f. bij_betw f A B"
cod_def: fixes f :: "'a" shows "C.cod f = (if C.codomains f \<noteq> {} then SOME b. b \<in> C.codomains f else C.null)"
coeff_max_zeroTr: fixes c :: "nat \<times> (nat \<Rightarrow> 'a)" assumes "pol_coeff S c" shows "\<forall>j. j \<le> fst c \<and> c_max S c < j \<longrightarrow> snd c j = \<zero>\<^bsub>S\<^esub>"
\<epsilon>_def: shows "\<epsilon> = Min (insert 1 {pmf p x / (pmf p' x - pmf q' x) |x. pmf q' x < pmf p' x})"
map_scale_zero_right: fixes k :: "'d" shows "k \<cdot> 0 = 0"
def_coinduct: fixes A :: "'a"   and f :: "'a \<Rightarrow> 'a"   and X :: "'a" assumes "A \<equiv> gfp f"   and "mono f"   and "X \<le> f (X \<squnion> A)" shows "X \<le> A"
m: shows "1 / (q - 1) * 1 / q ^ m < e"
add_rel_def: fixes X :: "('a \<times> 'a) set"   and Y :: "('a \<times> 'a) set" shows "add_rel X Y \<equiv> let x = SOME x. x \<in> X; y = SOME y. y \<in> Y in add_rel_aux (fst x) (snd x) (fst y) (snd y)"
legal_rulesI: fixes r :: "('a, 'L) ta_rule"   and \<delta> :: "('a, 'L) ta_rule set"   and Q :: "'a set" assumes "r \<in> \<delta>"   and "rule_states r \<subseteq> Q"   and "A (rhsl r) = Some (length (rhsq r))" shows "r \<in> legal_rules Q"
RedT_preserves_wset_thread_ok: fixes final :: "'a \<Rightarrow> bool"   and r :: "('d, 'b, 'a, 'c, 'e, 'f) semantics"   and convert_RA :: "'d \<Rightarrow>f nat \<Rightarrow> 'f list"   and s :: "('d, 'b) locks \<times> (('b \<Rightarrow> ('a \<times> 'd \<Rightarrow>f nat) option) \<times> 'c) \<times> ('b \<Rightarrow> 'e wait_set_status option) \<times> 'b set"   and ttas :: "('b \<times> ('d, 'b, 'a, 'c, 'e, 'f) thread_action) list"   and s' :: "('d, 'b) locks \<times> (('b \<Rightarrow> ('a \<times> 'd \<Rightarrow>f nat) option) \<times> 'c) \<times> ('b \<Rightarrow> 'e wait_set_status option) \<times> 'b set" assumes "multithreaded_base.RedT final r convert_RA s ttas s'"   and "wset_thread_ok (wset s) (thr s)" shows "wset_thread_ok (wset s') (thr s')"
assms_2: shows "\<forall>M\<in>Chains leq. finite M"
indaut_halfchsys_pgal_map_f: fixes X :: "'b set set"   and f :: "'b \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'b"   and C0 :: "'b set"   and Cs :: "'b set list" assumes "OpposedThinChamberComplexFoldings X f g C0"   and "order.greater_eq (f \<turnstile> ChamberComplex.chamber_system X) (set Cs)"   and "SimplicialComplex.pmaxsimpchain X Cs" shows "SimplicialComplex.pmaxsimpchain X (OpposedThinChamberComplexFoldings.induced_automorphism X f g \<Turnstile> Cs)"
extendAssoc: fixes S :: "'a sequent"   and R :: "'a sequent"   and c :: "'a sequent" shows "extend S (extend R c) = extend (extend S R) c"
simps_1: fixes c :: "poly"   and p :: "poly" shows "head (CN c 0 p) = head p"
simps_6: fixes \<tau> :: "'a OCL_Types.type"   and \<sigma> :: "'a OCL_Types.type" shows "OrderedSet \<tau> \<squnion> \<sigma> = (case \<sigma> of Collection \<rho> \<Rightarrow> Collection (\<tau> \<squnion> \<rho>) | Set \<rho> \<Rightarrow> Collection (\<tau> \<squnion> \<rho>) | OrderedSet \<rho> \<Rightarrow> OrderedSet (\<tau> \<squnion> \<rho>) | Bag \<rho> \<Rightarrow> Collection (\<tau> \<squnion> \<rho>) | Sequence \<rho> \<Rightarrow> Collection (\<tau> \<squnion> \<rho>) | _ \<Rightarrow> OclSuper)"
bin_rsplit_def: fixes n :: "nat"   and w :: "nat \<times> int" shows "bin_rsplit n w = bin_rsplit_aux n (fst w) (snd w) []"
collinear_affine_hull: fixes S :: "'b set" shows "collinear S = (\<exists>u v. S \<subseteq> affine hull {u, v})"
assms: shows "\<forall>v. freedb e v \<longrightarrow> \<rho>\<cdot>v = \<rho>'\<cdot>v"
eq2: shows "weak_ranking_aux (\<Union> (set As)) = As"
dd_lt_d: shows "\<forall>C\<in>DD. C < D"
vconverse_vsingleton: fixes a :: "V"   and b :: "V" shows "(set {\<langle>a, b\<rangle>})\<inverse>\<^sub>\<circ> = set {\<langle>b, a\<rangle>}"
cases: fixes x :: "nat \<times> nat" obtains xa :: "nat" and n :: "nat" where "x = (xa, n)"
test_bit_bin: fixes w :: "'b word"   and n :: "nat" shows "w !! n = (n < LENGTH('b) \<and> bit (uint w) n)"
has_integral_Un: fixes f :: "'a \<Rightarrow> 'b"   and i :: "'b"   and S :: "'a set"   and j :: "'b"   and T :: "'a set" assumes "(f has_integral i) S"   and "(f has_integral j) T"   and "negligible (S \<inter> T)" shows "(f has_integral i + j) (S \<union> T)"
state_ext_def: fixes id :: "nat"   and no :: "nat"   and nhid :: "nat"   and msg :: "msg"   and num :: "nat"   and sid :: "nat"   and more :: "'a" shows "\<lparr>state.id = id, no = no, nhid = nhid, msg = msg, num = num, sid = sid, \<dots> = more\<rparr> \<equiv> Record.iso_tuple_cons state_ext_Tuple_Iso (Record.iso_tuple_cons Record.tuple_iso_tuple (Record.iso_tuple_cons Record.tuple_iso_tuple id (Record.iso_tuple_cons Record.tuple_iso_tuple no nhid)) (Record.iso_tuple_cons Record.tuple_iso_tuple msg (Record.iso_tuple_cons Record.tuple_iso_tuple num sid))) more"
set_pmf_of_multiset: fixes M :: "'c multiset" assumes "M \<noteq> {#}" shows "set_pmf (pmf_of_multiset M) = set_mset M"
of_int_less_iff: fixes w :: "int"   and z :: "int" shows "(of_int w < of_int z) = (w < z)"
H': fixes aa :: "char list list"   and ab :: "char list list"   and ba :: "int" assumes "(q, q') \<in> sorted_poly_rel"   and "distinct aa"   and "sorted_wrt var_order aa"   and "(ab, ba) \<in> set q" shows "remdups_mset (mset aa + mset ab) = mset (mult_monoms aa ab)"
5: shows "case s of (t, n, dl) \<Rightarrow> n = size t \<and> bal_i (size t + dl) (height t) \<and> bal_d (size t) dl"
distinct_8: fixes x51 :: "expr"   and x52 :: "com" shows "while (x51) x52 \<noteq> Skip"
coeff_div_poly: fixes a :: "'a"   and f :: "'a poly"   and n :: "nat" shows "coeff (div_poly a f) n = coeff f n div a"
div_eq_0_conv: fixes n :: "nat"   and m :: "nat" shows "(n div m = 0) = (m = 0 \<or> n < m)"
subset_inter_diff_empty: fixes s :: "'a set"   and t :: "'a set"   and u :: "'a set" assumes "s \<subseteq> t" shows "s \<inter> (u - t) = {}"
fpxs_val_of_fls: fixes f :: "'b fls" shows "fpxs_val (fpxs_of_fls f) = rat_of_int (fls_subdegree f)"
lookup_single_not_eq: fixes k :: "'c"   and k' :: "'c"   and v :: "'d" assumes "k \<noteq> k'" shows "lookup (monomial v k) k' = (0::'d)"
fps_catalan_nth: fixes n :: "nat" shows "fps_nth fps_catalan n = of_nat (catalan n)"
c: shows "(\<Sum>x\<in>A. real_of_int (f x) / real_of_int (g x)) = real_of_int c / real_of_int d'"