hcju/isabelleps huggingface dataset:
invariance_of_domain_affine_sets: fixes U :: "'a set"   and S :: "'a set"   and V :: "'b set"   and f :: "'a \<Rightarrow> 'b" assumes "openin (top_of_set U) S"   and "affine U"   and "affine V"   and "aff_dim V \<le> aff_dim U"   and "continuous_on S f"   and "f ` S \<subseteq> V"   and "inj_on f S" shows "openin (top_of_set V) (f ` S)"
g_add_def: fixes m1 :: "'s"   and m2 :: "'s" shows "g_add m1 m2 \<equiv> iterate m2 (\<lambda>(x, y). update x y) m1"
malt_SE_def: fixes S :: "('a \<Rightarrow> ('b \<times> 'a) option) list" shows "\<Sqinter>\<^sub>S\<^sub>E S = foldr (\<sqinter>\<^sub>S\<^sub>E) S fail\<^sub>S\<^sub>E"
resolution_class_union: fixes S :: "'a set set" assumes "resolution_class S" shows "\<Union> S = \<V>"
j: shows "fst (prod_dfs A B x) ! j1 ! j2 = Some k"
elims_2: fixes x :: "('d, 'e, 'f) FSM"   and xa :: "'d list set" assumes "is_det_state_cover x xa" obtains M :: "('d, 'e, 'f) FSM" and V :: "'d 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)"
B_prism_def: shows "B\<^sub>\<triangle> = \<lparr>prism_match = fromB, prism_build = B\<rparr>"
pair_is_static_simp1: fixes p :: "'a \<times> 'b" shows "is_static p = is_static (snd p)"
fg: fixes x :: "'b" assumes "x \<in> topspace Y" shows "f (g x) = x"
src_inv: fixes \<mu> :: "'a" assumes "B.iso \<mu>" shows "src\<^sub>B (B.inv \<mu>) = src\<^sub>B \<mu>"
conj_disj_distr: fixes P :: "'a upred"   and Q :: "'a upred"   and R :: "'a upred" shows "(P \<and> (Q \<or> R)) = (P \<and> Q \<or> P \<and> R)"
Lemma_4: fixes js :: "nat list"   and k :: "nat" assumes "\<And>f h. associative f \<Longrightarrow> foldl1 f (map h js) = foldl1 f (map h [0..<k + 1])"   and "js \<noteq> []" shows "js = [0..<k + 1]"
count_image_mset_le_imp_lt_raw: fixes A :: "'b set"   and M :: "'b multiset"   and N :: "'b multiset"   and f :: "'b \<Rightarrow> 'c"   and a :: "'b" assumes "finite A"   and "A = set_mset M \<union> set_mset N"   and "count (image_mset f M) (f a) + count N a < count (image_mset f N) (f a) + count M a" shows "\<exists>b. f b = f a \<and> count M b < count N b"
entails_trans2: fixes A :: "assn"   and D :: "assn"   and B :: "assn"   and C :: "assn" assumes "A \<Longrightarrow>\<^sub>A D * B"   and "B \<Longrightarrow>\<^sub>A C" shows "A \<Longrightarrow>\<^sub>A D * C"
box_below: fixes below :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b"   and c :: "'b"   and d :: "'b" assumes "class.po below"   and "below a b"   and "below c a"   and "below b d" shows "below c d"
simps_1: fixes t :: "'trm" shows "rawpsubstT t [] = t"
check_TunerMode_CTRL: shows "(TunerMode_CTRL_States, TunerMode_CTRL_Init, TunerMode_CTRL_Labels, TunerMode_CTRL_Delta) \<in> seqauto"
process_ret_sound': fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and chain :: "char list"   and rs :: "'c rule list"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and m :: "'c match_expr"   and t :: "state" assumes "\<Gamma>(chain \<mapsto> rs),\<gamma>,p\<turnstile> \<langle>process_ret (add_match m rs), Undecided\<rangle> \<Rightarrow> t" shows "\<Gamma>(chain \<mapsto> rs),\<gamma>,p\<turnstile> \<langle>[Rule m (Call chain)], Undecided\<rangle> \<Rightarrow> t"
map_add_eq_empty_iff: fixes f :: "'c \<Rightarrow> 'd option"   and g :: "'c \<Rightarrow> 'd option" shows "(f ++ g = Map.empty) = (f = Map.empty \<and> g = Map.empty)"
cont_def: fixes f :: "'c \<Rightarrow> 'd" shows "cont f \<equiv> \<forall>C. C \<noteq> {} \<longrightarrow> f (Sup C) = Sup (f ` C)"
normalize_prop: shows "\<forall>no\<in>Nodes n ll. var (repb no) \<le> var no \<and> (\<exists>not nort. Dag (repb no) (repb \<propto> low) (repb \<propto> high) nort \<and> Dag no low high not \<and> reduced nort \<and> ordered nort var \<and> set_of nort \<subseteq> repb ` Nodes n ll \<and> (\<forall>no\<in>set_of nort. repb no = no) \<and> (\<exists>nobdt. bdt not var = Some nobdt \<and> (\<exists>norbdt. bdt nort var = Some norbdt \<and> nobdt \<sim> norbdt)))"
Input: fixes P :: "pi"   and s :: "(name \<times> name) list"   and Q :: "pi"   and x :: "name"   and a :: "name" assumes "P[
] \<simeq>\<^sup>s Q[
]"   and "x \<sharp> s" shows "a<x>.P[
] \<simeq> a<x>.Q[
]"
prod_ord_raw_alt: fixes xs :: "('a \<times> 'e) list \<times> 'o"   and ys :: "('a \<times> 'f) list \<times> 'o"   and P :: "'a \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool" assumes "oalist_inv xs"   and "oalist_inv ys" shows "prod_ord_raw P xs ys = (\<forall>k\<in>fst ` set (fst xs) \<union> fst ` set (fst ys). P k (lookup_raw xs k) (lookup_raw ys k))"
elims_3: fixes x :: "('e, 'f, 'varSort, 'var, 'g) qAbs" assumes "\<not> qGoodAbs x" obtains xs :: "'varSort" and xa :: "'var" and X :: "('e, 'f, 'varSort, 'var, 'g) qTerm" where "x = qAbs xs xa X" and "\<not> qGood X"
spEdges_def: shows "spEdges \<equiv> {e. \<exists>p. e \<in> set p \<and> isShortestPath s p t}"
arrow_of_spans_axioms: shows "arrow_of_spans (\<cdot>) (trg f)"
simps: fixes f :: "heap \<Rightarrow> ('a \<times> heap) option" shows "execute (Heap f) = f"
disjuncts: fixes p :: "fm"   and bs :: "real list" shows "(\<exists>q\<in>set (disjuncts p). Ifm bs q) = Ifm bs p"
leI: fixes less_eq :: "'q \<Rightarrow> 'q \<Rightarrow> bool"   and less :: "'q \<Rightarrow> 'q \<Rightarrow> bool"   and x :: "'q"   and y :: "'q" assumes "class.linorder less_eq less"   and "\<not> less x y" shows "less_eq y x"
compose_extensional: fixes A :: "'b set"   and f :: "'d \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'd" shows "compose A f g \<in> extensional A"
R_pre_neq_Nil: shows "R_pre \<noteq> []"
pt_bij4: fixes pi :: "('b \<times> 'b) list"   and x :: "'a"   and y :: "'a" assumes "pt TYPE('a) TYPE('b)"   and "at TYPE('b)"   and "pi \<bullet> x = pi \<bullet> y" shows "x = y"
nz: shows "\<lbrace>Y \<^bold>\<in> HPow (Vset (rank X)). Transset Y \<and> X \<le> Y\<rbrace> \<noteq> 0"
finter_commute: fixes A :: "'e fset"   and B :: "'e fset" shows "A |\<inter>| B = B |\<inter>| A"
llist_all2_trans: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and xs :: "'a llist"   and ys :: "'a llist"   and zs :: "'a llist" assumes "llist_all2 P xs ys"   and "llist_all2 P ys zs"   and "transp P" shows "llist_all2 P xs zs"
simps_2: fixes a :: "'a"   and n :: "nat" shows "a ^ Suc n = a * a ^ n"
rel_eq: shows "rel_nondetT' (=) (=) = (=)"
nontr: shows "at 1 within {0..<1} \<noteq> bot"
netglobalsimp: fixes P :: "(nat \<Rightarrow> state) \<Rightarrow> bool"   and s :: "((state \<times> (state, msg, pseqp, pseqp label) seqp) \<times> msg list \<times> (msg list, msg, unit, unit label) seqp) net_state" shows "netglobal P s = P (fst (initmissing (netgmap (\<lambda>(p, q). (fst (Fun.id p), snd (Fun.id p), q)) s)))"
iff: shows "\<forall>i'\<in>{0..<Suc n}. f i' = f i \<longrightarrow> i' = i"
get_ancestors_def: shows "get_ancestors \<equiv> l_get_root_node\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_ancestors get_parent"
ptensor_mat_mult: fixes m1 :: "'a mat"   and m2 :: "'a mat"   and m3 :: "'a mat"   and m4 :: "'a mat" assumes "m1 \<in> carrier_mat ps2_P.d1 ps2_P.d1"   and "m2 \<in> carrier_mat ps2_P.d1 ps2_P.d1"   and "m3 \<in> carrier_mat ps2_P.d2 ps2_P.d2"   and "m4 \<in> carrier_mat ps2_P.d2 ps2_P.d2" shows "tensor_P (m1 * m2) (m3 * m4) = tensor_P m1 m3 * tensor_P m2 m4"
simps: fixes x :: "'b"   and lx :: "'b hp list"   and y :: "'b"   and ly :: "'b hp list" shows "link (Hp x lx) (Hp y ly) = (if x < y then Hp x (Hp y ly # lx) else Hp y (Hp x lx # ly))"
preserves_cod: fixes f :: "'a" assumes "B.arr f" shows "cod (UP f) = UP (B.cod f)"
conjug_rq: fixes z :: "'b list"   and x :: "'b list"   and y :: "'b list" assumes "z \<cdot> x = y \<cdot> z" shows "y = (z \<cdot> x)\<^sup><\<inverse>z "
CastRedsFail: fixes e :: "char list exp"   and s :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and a :: "nat"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and D :: "char list"   and fs :: "char list \<times> char list \<Rightarrow> val option"   and C :: "char list" assumes "P \<turnstile> \<langle>e,s\<rangle> \<rightarrow>* \<langle>addr a,s'\<rangle>"   and "hp s' a = \<lfloor>(D, fs)\<rfloor>"   and "(D, C) \<notin> (subcls1 P)\<^sup>*" shows "P \<turnstile> \<langle>Cast C e,s\<rangle> \<rightarrow>* \<langle>THROW ClassCast,s'\<rangle>"
grouphom: shows "group_hom G (BijGroup M) \<phi>"
ex_maximal_wave: fixes \<Gamma> :: "('a, 'b) web_scheme" assumes "countable \<^bold>E\<^bsub>\<Gamma>\<^esub>" shows "\<exists>f. current \<Gamma> f \<and> wave \<Gamma> f \<and> (\<forall>w. current \<Gamma> w \<and> wave \<Gamma> w \<and> f \<le> w \<longrightarrow> f = w)"
IH_2: assumes "vars p2 \<subseteq> {..<n}" shows "(\<lambda>f. insertion f p2) \<in> borel_measurable (lborel_f n)"
cms''_def: shows "cms'' = cms[n := fst lcn'']"
assms: fixes x :: "nat" assumes "\<And>y. y < x \<Longrightarrow> P y" shows "P x"
ideal_eq_zero_iff: fixes F :: "'b set" shows "(ideal F = {0::'b}) = (F \<subseteq> {0::'b})"
covering_space_lift_homotopic_function: fixes C :: "'c set"   and p :: "'c \<Rightarrow> 'd"   and S :: "'d set"   and U :: "'e set"   and g :: "'e \<Rightarrow> 'c"   and f :: "'e \<Rightarrow> 'd"   and f' :: "'e \<Rightarrow> 'd" assumes "covering_space C p S"   and "continuous_on U g"   and "g ` U \<subseteq> C"   and "\<And>y. y \<in> U \<Longrightarrow> p (g y) = f y"   and "homotopic_with_canon (\<lambda>x. True) U S f f'" obtains g' :: "'e \<Rightarrow> 'c" where "continuous_on U g'" and "g' ` U \<subseteq> C" and "\<And>y. y \<in> U \<Longrightarrow> p (g' y) = f' y"
mat_gtI: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a mat"   and B :: "'a mat"   and i :: "nat"   and sd :: "nat"   and j :: "nat"   and gt :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "ord.mat_ge less_eq A B"   and "i < sd"   and "j < sd"   and "gt (A $$ (i, j)) (B $$ (i, j))" shows "ord.mat_gt less_eq gt sd A B"
sum_upto_moebius_mu_integral: fixes x :: "real" assumes "1 < x" shows "((\<lambda>t. M t / t) has_integral M x * ln x - H x) {1..x}"
closedin_Union: fixes S :: "'c set set"   and U :: "'c topology" assumes "finite S"   and "\<And>T. T \<in> S \<Longrightarrow> closedin U T" shows "closedin U (\<Union> S)"
alg_dvd_imp_algebraic_int: fixes x :: "'a"   and y :: "'a" assumes "x alg_dvd y"   and "algebraic_int x" shows "algebraic_int y"
multiple_is_wellformed: fixes n :: "nat" shows "incidence_system \<V> (multiple_blocks n)"
lin_dense: fixes p :: "fm"   and l :: "'b"   and u :: "'b"   and vs :: "'b list"   and x :: "'b"   and bs :: "'b list"   and y :: "'b" assumes "islin p"   and "\<forall>t. l < t \<and> t < u \<longrightarrow> t \<notin> (\<lambda>(c, t). - Itm vs (x # bs) t / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup>) ` set (uset p)"   and "l < x"   and "x < u"   and "Ifm vs (x # bs) p"   and "l < y"   and "y < u" shows "Ifm vs (y # bs) p"
in_mk_uedge_img: fixes a :: "nat"   and b :: "nat"   and A :: "(nat \<times> nat) set" assumes "(a, b) \<in> A \<or> (b, a) \<in> A" shows "{a, b} \<in> mk_uedge ` A"
lookup_delete: fixes t :: "('c, 'd) trie"   and ks :: "'c list"   and ks' :: "'c list" assumes "invar_trie t" shows "lookup_trie (delete_trie ks t) ks' = (if ks = ks' then None else lookup_trie t ks')"
quot_to_ratfps_quot_of_ratfps: fixes x :: "'b ratfps" shows "quot_to_ratfps (quot_of_ratfps x) = x"
rbt_delete: fixes t :: "(('c \<times> 'd) \<times> tcolor \<times> 'c \<times> 'd) tree"   and k :: "'c" assumes "rbt t" shows "rbt (delete k t)"
int_pow_mult: fixes x :: "'a"   and i :: "int"   and j :: "int" assumes "x \<in> carrier G" shows "x [^] (i + j) = x [^] i \<otimes> x [^] j"
integrate_delete_result: shows "is_certified_associated_string (M \<union> {Delete m}) (integrate_delete m s)"
g2_def: shows "g2 \<equiv> \<lambda>u. (- ln u) powr x * (u - ln u - 1) / (1 - u)"
nth_last_index: fixes x :: "'b"   and xs :: "'b list" assumes "x \<in> set xs" shows "xs ! last_index xs x = x"
Nset_pre_mem: fixes f :: "nat \<Rightarrow> nat"   and n :: "nat"   and k :: "nat" assumes "f \<in> {i. i \<le> Suc n} \<rightarrow> {i. i \<le> Suc n}"   and "inj_on f {i. i \<le> Suc n}"   and "f (Suc n) = Suc n"   and "k \<le> n" shows "f k \<in> {i. i \<le> n}"
integrable_restrict_UNIV: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "S \<in> sets lebesgue" shows "integrable lebesgue (\<lambda>x. if x \<in> S then f x else (0::'b)) = integrable (lebesgue_on S) f"
KbaseTr1: fixes n :: "nat"   and P :: "nat \<Rightarrow> ('a \<Rightarrow> ant) set" assumes "distinct_pds K n P" shows "KbaseP K P n (Kb\<^bsub>K n P\<^esub>)"
RHS_def: shows "RHS \<equiv> pCons (0::'a) (poly_shift (Suc (nat (- m))) (fls_prpart f))"
Zp_converges_toI: fixes s :: "nat \<Rightarrow> nat \<Rightarrow> int"   and a :: "nat \<Rightarrow> int" assumes "s \<in> carrier Zp\<^bsup>\<omega>\<^esup>"   and "a \<in> carrier Zp"   and "\<And>n. \<exists>N. \<forall>k>N. s k n = a n" shows "Zp_converges_to s a"
avg_correct: fixes s :: "(nat, unit) HashMap.hashmap" assumes "my_invar s" shows "avg s = average (my_\<alpha> s)"
eq_iff: fixes x :: "('d, 'e) Term.term"   and y :: "('d, 'e) Term.term" shows "(x = y) = (y \<unrhd> x \<and> x \<unrhd> y)"
eq_fun_eq_val: fixes f :: "'d \<Rightarrow> 'e"   and g :: "'d \<Rightarrow> 'e"   and x :: "'d" assumes "f = g" shows "f x = g x"
app5: fixes ck :: "bool"   and env :: "v sem_env"   and s1 :: "'b state"   and es :: "exp0 list"   and s2 :: "'b state"   and vs :: "v list"   and op0 :: "op0" assumes "evaluate_list ck env s1 (rev es) (s2, Rval vs) \<and> do_app (refs s2, ffi s2) op0 (rev vs) = None \<and> op0 \<noteq> Opapp" shows "evaluate ck env s1 (App op0 es) (s2, Rerr (Rabort Rtype_error))"
order_divides: fixes a :: "'a"   and n :: "nat"   and p :: "'a poly" shows "([:- a, 1::'a:] ^ n dvd p) = (p = 0 \<or> n \<le> Polynomial.order a p)"
IH: fixes S :: "'a set"   and p :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "swapidseq_ext S n p"   and "a \<noteq> b" shows "swapidseq_ext (insert a (insert b S)) (Suc n) (p \<circ> Fun.swap a b id)"
some_gcd_connect: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.idom times one plus zero minus uminus"   and "a \<noteq> zero"   and "b \<noteq> zero" shows "somegcd \<lparr>carrier = UNIV - {zero}, monoid.mult = times, one = one\<rparr> a b = dvd.some_gcd times a b"
comm_neutral: fixes a :: "'e" shows "a \<cdot> (1::'e) = a"
cod_simp: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and Arr :: "'b \<Rightarrow> bool"   and f :: "'b" assumes "subcategory C Arr"   and "partial_magma.arr (subcategory.comp C Arr) f" shows "partial_magma.cod (subcategory.comp C Arr) f = partial_magma.cod C f"
multiset_order_inclusion: fixes E :: "'b multiset"   and F :: "'b multiset"   and r :: "('b \<times> 'b) set" assumes "E \<subset># F"   and "trans r" shows "(E, F) \<in> mult r"
mat_O_mult_alpha: shows "mat_O *\<^sub>v \<alpha> = \<alpha>"
mprod_expR_memTr: fixes R :: "('c, 'd) Ring_scheme"   and e :: "nat \<Rightarrow> nat"   and n :: "nat"   and f :: "nat \<Rightarrow> 'c" assumes "Ring R" shows "e \<in> {j. j \<le> n} \<rightarrow> {j. 0 \<le> j} \<and> f \<in> {j. j \<le> n} \<rightarrow> carrier R \<longrightarrow> mprod_expR R e f n \<in> carrier R"
IH: shows "\<chi> (a + m * n) = \<chi> a"
Complete_def: fixes atom_ordering :: "('a \<times> 'a) set"   and Rule :: "'a Literal set \<Rightarrow> 'a Literal set \<Rightarrow> 'a Literal set \<Rightarrow> bool" assumes "propositional_atoms atom_ordering" shows "propositional_atoms.Complete Rule = (\<forall>S. saturated_binary_rule Rule S \<longrightarrow> propositional_atoms.all_fulfill finite S \<longrightarrow> {} \<notin> S \<longrightarrow> propositional_atoms.satisfiable S)"
final2_simulation: fixes t :: "'t"   and s1 :: "'x1 \<times> 'm1"   and s2 :: "'x2 \<times> 'm2" assumes "t \<turnstile> s1 \<approx> s2"   and "case s2 of (x2, m) \<Rightarrow> final2 x2" shows "\<exists>s1'. r1.silent_moves t s1 s1' \<and> t \<turnstile> s1' \<approx> s2 \<and> (case s1' of (x1, m) \<Rightarrow> final1 x1)"
conv2: shows "\<I>2 \<turnstile>g run_converter conv1 a \<surd>"
no_trivial_inverse: fixes x :: "'a" assumes "x \<noteq> (zero_class.zero::'a)" shows "\<nexists>y. x + y = (zero_class.zero::'a)"
all_same_arities_generalize_fundef: fixes F :: "'fenv_ubx"   and f :: "'fun"   and fd :: "('dyn, 'var, 'fun, 'op, 'opinl, 'opubx, 'ubx1, 'ubx2) Ubx.instr fundef" assumes "Fubx_get F f = Some fd" shows "all_same_arities (Fubx_get F) (Fubx_get (Fubx_add F f (Subx.generalize_fundef fd)))"
comp_swap: fixes f :: "'d \<Rightarrow> 'c"   and a :: "'b"   and b :: "'b"   and g :: "'b \<Rightarrow> 'd" shows "f \<circ> Fun.swap a b g = Fun.swap a b (f \<circ> g)"
K_def: shows "K = {Pi\<^sub>E UNIV X |X. (\<forall>i. X i \<in> B2) \<and> finite {i. X i \<noteq> UNIV}}"
f: shows "integral\<^sup>N M f \<noteq> \<infinity>"
diff_def: fixes n :: "nat" shows "diff n = (if n = 0 then x else if n = 1 then \<lambda>t. f (t, x t) else if n = 2 then \<lambda>t. f' (t, x t) $ (1, f (t, x t)) else (\<lambda>s. f'' (s, x s) $ (1, f (s, x s)) $ (1, f (s, x s)) + f' (s, x s) $ (0, f' (s, x s) $ (1, f (s, x s)))))"
liveVal_use_path: fixes g :: "'g"   and v :: "'val" assumes "liveVal g v" obtains ns :: "'node list" and m :: "'node" where "g \<turnstile> defNode g v-ns\<rightarrow>m" and "var g v \<in> oldUses g m" and "\<And>x. x \<in> set (tl ns) \<Longrightarrow> var g v \<notin> oldDefs g x"
fixp_induct: fixes P :: "(nat \<Rightarrow> nat \<Rightarrow> nat spmf) \<Rightarrow> bool" assumes "spmf.admissible (\<lambda>fdr_alt. P (curry fdr_alt))"   and "P (\<lambda>fdr_alt v. return_pmf None)"   and "\<And>c. P c \<Longrightarrow> P (\<lambda>a aa. local.fdr_step a aa \<bind> (\<lambda>a. case a of (v', c') \<Rightarrow> if c' < n then return_spmf c' else c (v' - n) (c' - n)))" shows "P local.fdr_alt"
add_mono: fixes plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and a :: "'e"   and b :: "'e"   and c :: "'e"   and d :: "'e" assumes "class.ordered_ab_semigroup_add plus less_eq less"   and "less_eq a b"   and "less_eq c d" shows "less_eq (plus a c) (plus b d)"
continuous_on_linepath: fixes s :: "real set"   and a :: "'c"   and b :: "'c" shows "continuous_on s (linepath a b)"
Red_dom_lcl: fixes E :: "char list \<Rightarrow> ty option"   and e :: "expr"   and h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and l :: "char list \<Rightarrow> val option"   and e' :: "expr"   and h' :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and l' :: "char list \<Rightarrow> val option"   and 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" assumes "P,E \<turnstile> \<langle>e,(h, l)\<rangle> \<rightarrow> \<langle>e',(h', l')\<rangle>" shows "dom l' \<subseteq> dom l \<union> fv e"
assms_2: shows "\<forall>i\<le>n. Le (0::'b) \<le> M' i i"
isometries_subspaces: fixes S :: "'b set"   and T :: "'c set" assumes "subspace S"   and "subspace T"   and "dim S = dim T" obtains f :: "'b \<Rightarrow> 'c" and g :: "'c \<Rightarrow> 'b" where "linear f" and "linear g" and "f ` S = T" and "g ` T = S" and "\<And>x. x \<in> S \<Longrightarrow> norm (f x) = norm x" and "\<And>x. x \<in> T \<Longrightarrow> norm (g x) = norm x" and "\<And>x. x \<in> S \<Longrightarrow> g (f x) = x" and "\<And>x. x \<in> T \<Longrightarrow> f (g x) = x"
jordan_matrix_pow: fixes n_as :: "(nat \<times> 'b) list"   and r :: "nat" shows "jordan_matrix n_as ^\<^sub>m r = diag_block_mat (map (\<lambda>(n, a). jordan_block n a ^\<^sub>m r) n_as)"
isolated_ret_def: fixes fg :: "('c, 'd, 'e, 'f, 'g) flowgraph_rec_scheme"   and p :: "'d" shows "isolated_ret fg p \<equiv> (\<forall>u l. (u, l, return fg p) \<notin> edges fg) \<and> entry fg p \<noteq> return fg p"
ivl_integral_norm_bound_ivl_integral: fixes f :: "real \<Rightarrow> 'a"   and a :: "real"   and b :: "real"   and g :: "real \<Rightarrow> real" assumes "f integrable_on closed_segment a b"   and "g integrable_on closed_segment a b"   and "\<And>x. x \<in> closed_segment a b \<Longrightarrow> norm (f x) \<le> g x" shows "norm (ivl_integral a b f) \<le> \<bar>ivl_integral a b g\<bar>"
gram_schmidt_sub0_result: fixes us :: "'a vec list"   and ws :: "'a vec list"   and us' :: "'a vec list" assumes "gram_schmidt_sub0 n us ws = us'"   and "set ws \<subseteq> carrier_vec n"   and "set us \<subseteq> carrier_vec n"   and "distinct us"   and "lin_indpt (set us)"   and "corthogonal us" shows "set us' \<subseteq> carrier_vec n \<and> distinct us' \<and> corthogonal us' \<and> local.span (set (us @ ws)) = local.span (set us')"
Lm_follow: fixes m :: "bool"   and r :: "(bool \<times> 'b) rexp" shows "Lm (follow m r) = tl ` Lm r \<union> (if m then lang (strip r) else {}) - {[]}"
exhaust: fixes y :: "'f option" obtains "y = None"   | x2 :: "'f" where "y = \<lfloor>x2\<rfloor>"
jkbpCn_init_inv: fixes s :: "'s"   and n :: "nat" assumes "tInit s \<in> jkbpC\<^bsub>n\<^esub>" shows "s \<in> set envInit"
assms_3: fixes h :: "'a set" assumes "h \<in> F" shows "a h \<noteq> (0::'a) \<and> h = {x. a h \<bullet> x \<le> b h}"
initial_d_sb: fixes ts :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'c \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list"   and \<S> :: "nat \<Rightarrow> bool option"   and valid :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'd list \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list \<Rightarrow> bool" assumes "initial\<^sub>d ts \<S> valid" shows "initial\<^sub>s\<^sub>b (map (\<lambda>(p, is, \<theta>, sb, \<D>, \<O>, \<R>). (p, is, \<theta>, [], \<D>, \<O>, \<R>)) ts) \<S>"
tt_less: fixes u :: "'t"   and p :: "'t \<Rightarrow>\<^sub>0 'c"   and v :: "'t" assumes "u \<in> keys p"   and "u \<prec>\<^sub>t v" shows "tt p \<prec>\<^sub>t v"
Step2: fixes l :: "Locs"   and rmhist :: "state \<Rightarrow> PrIds \<Rightarrow> histState" shows "\<turnstile> #l \<in> #MemLoc \<and> (\<forall>p. IPImp p \<and> HistP rmhist p) \<longrightarrow> MSpec memCh mm (resbar rmhist) l"
UP_sum_is_Zp_continuous: fixes a :: "nat \<Rightarrow> nat \<Rightarrow> int"   and b :: "nat \<Rightarrow> nat \<Rightarrow> int" assumes "a \<in> carrier (UP Zp)"   and "b \<in> carrier (UP Zp)"   and "is_Zp_continuous (to_fun a)"   and "is_Zp_continuous (to_fun b)" shows "is_Zp_continuous (to_fun (a \<oplus>\<^bsub>UP Zp\<^esub> b))"
lasso_run_rel_def_internal: fixes R :: "('e \<times> 'f) set" shows "lasso_run_rel R \<equiv> br run_of_lasso (\<lambda>_. True) O (nat_rel \<rightarrow> R)"
not_\<omega>\<^sub>z_le_of_nat: fixes n :: "nat" shows "\<not> \<omega>\<^sub>z \<le> of_nat n"
resultant_sub_def: fixes m :: "nat"   and n :: "nat"   and p :: "'b poly"   and q :: "'b poly" shows "resultant_sub m n p q = det (sylvester_mat_sub m n p q)"
crename_OrR2_aux: fixes R :: "trm"   and a :: "coname"   and b :: "coname"   and c :: "coname"   and M :: "trm"   and e :: "coname" assumes "R[a\<turnstile>c>b] = OrR2 <c>.M e" shows "a = e \<and> a = b \<or> a \<noteq> e"
Abs_bit1'_code: fixes x :: "int" shows "Rep_bit1 (Abs_bit1' x) = x mod int CARD('b bit1)"
more_permissive_def: fixes p :: "'d \<Rightarrow> 'e decision option"   and q :: "'d \<Rightarrow> 'e decision option" shows "p \<sqsubseteq>\<^sub>A q = (\<forall>x. case q x of \<bottom> \<Rightarrow> True | \<lfloor>allow y\<rfloor> \<Rightarrow> \<exists>z. p x = \<lfloor>allow z\<rfloor> | \<lfloor>deny y\<rfloor> \<Rightarrow> True)"