hcju/isabelleps huggingface dataset:
Struct_axioms_def: fixes wtFsym :: "'b \<Rightarrow> bool"   and arOf :: "'b \<Rightarrow> 'c list"   and resOf :: "'b \<Rightarrow> 'c"   and intT :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and intF :: "'b \<Rightarrow> 'd list \<Rightarrow> 'd"   and intP :: "'e \<Rightarrow> 'd list \<Rightarrow> bool" shows "Struct_axioms wtFsym arOf resOf intT intF intP \<equiv> (\<forall>f al. wtFsym f \<longrightarrow> list_all2 intT (arOf f) al \<longrightarrow> intT (resOf f) (intF f al)) \<and> intP = intP"
uwalk_length_def: fixes p :: "nat list" shows "uwalk_length p \<equiv> length (uwalk_edges p)"
fmdom'I: fixes m :: "('b, 'a) fmap"   and x :: "'b"   and y :: "'a" assumes "fmlookup m x = Some y" shows "x \<in> fmdom' m"
csp_defs_1: fixes P :: "('c, 'd) alpha_rp_scheme \<times> ('c, 'd) alpha_rp_scheme \<Rightarrow> bool" shows "CSP1 P \<equiv> P \<or> (\<lambda>(A, A'). \<not> ok A \<and> tr A \<le> tr A')"
remove_child_locs_def: shows "Shadow_DOM.remove_child_locs \<equiv> l_remove_child\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_remove_child_locs Shadow_DOM.set_child_nodes_locs set_disconnected_nodes_locs"
finite_alt_consistency: fixes C :: "('a, 'b) form set set" assumes "alt_consistency C"   and "subset_closed C" shows "alt_consistency (mk_finite_char C)"
conversep_conversep: fixes r :: "'a \<Rightarrow> 'b \<Rightarrow> bool" shows "r\<inverse>\<inverse>\<inverse>\<inverse> = r"
get_exists_matching_src_ips_def: fixes iface :: "iface"   and m :: "'b common_primitive match_expr" shows "get_exists_matching_src_ips iface m \<equiv> let (i_matches, uu_) = primitive_extractor (is_Iiface, iiface_sel) m in if \<forall>is\<in>set i_matches. case is of Pos i \<Rightarrow> match_iface i (iface_sel iface) | Neg i \<Rightarrow> \<not> match_iface i (iface_sel iface) then let (ip_matches, uu_) = primitive_extractor (is_Src, src_sel) m in if ip_matches = [] then UNIV else \<Inter>ips\<in>set ip_matches. case ips of Pos ip \<Rightarrow> ipt_iprange_to_set ip | Neg ip \<Rightarrow> - ipt_iprange_to_set ip else {}"
unowned_shared': fixes \<S> :: "nat \<Rightarrow> bool option"   and ts :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'c \<times> 'd \<times> nat set \<times> 'e) list"   and a :: "nat" assumes "unowned_shared \<S> ts"   and "\<forall>i<length ts. a \<notin> owned (ts ! i)" shows "a \<in> dom \<S>"
deltafun_apply_eq: fixes b :: "'c"   and a :: "'d" shows "(b \<delta> a) a = b"
assms: shows "low_equal s1 s2 \<and> memory_write 10 addr (st_byte_mask instr addr) (st_data0 instr curr_win rd addr s2) s1 = Some y \<and> memory_write 10 addr (st_byte_mask instr addr) (st_data0 instr curr_win rd addr s2) s2 = Some ya \<and> memory_write 10 (addr + 4) 15 (user_reg_val curr_win (rd OR 1) y) y = Some yb \<and> memory_write 10 (addr + 4) 15 (user_reg_val curr_win (rd OR 1) ya) ya = Some yc"
Hilbert_poly_def: fixes b :: "nat \<Rightarrow> nat" shows "Hilbert_poly b = (\<lambda>z. let n = card X in z - int (b (Suc n)) + int n gchoose n - 1 - (\<Sum>i = 1..n. z - int (b i) + int i - 1 gchoose i))"
distinct_53: fixes x6 :: "poly"   and x81 :: "poly"   and x82 :: "nat"   and x83 :: "poly" shows "Neg x6 \<noteq> CN x81 x82 x83"
preserves_equivalence_maps: fixes f :: "'a" assumes "equivalence_map f" shows "S.equivalence_map (S.UP f)"
independent_explicit_module: fixes s :: "'a set" shows "cindependent s = (\<forall>t u v. finite t \<longrightarrow> t \<subseteq> s \<longrightarrow> (\<Sum>v\<in>t. u v *\<^sub>C v) = (0::'a) \<longrightarrow> v \<in> t \<longrightarrow> u v = 0)"
one_le_inverse_iff: fixes x :: "'a" shows "((1::'a) \<le> inverse x) = ((0::'a) < x \<and> x \<le> (1::'a))"
invariant_s_def: fixes t :: "(nat \<times> linear_poly) list"   and as :: "('i \<times> 'a atom) set"   and s :: "('i, 'a) state" shows "invariant_s t as s = (\<not> \<U> s \<and> \<Turnstile>\<^sub>n\<^sub>o\<^sub>l\<^sub>h\<^sub>s s \<and> \<triangle> (\<T> s) \<and> \<nabla> s \<and> \<diamond> s \<and> (\<forall>v. v \<Turnstile>\<^sub>t \<T> s = v \<Turnstile>\<^sub>t t) \<and> index_valid as s \<and> (snd ` as) \<doteq> \<B> s \<and> as \<Turnstile>\<^sub>i \<B>\<I> s)"
bdd_above_Iio: fixes b :: "'a" shows "bdd_above {..<b}"
SetMarkInv_def: fixes i :: "I" shows "SetMarkInv i = (case i of init \<Rightarrow> Init | loop \<Rightarrow> Loop | final \<Rightarrow> Final)"
is_reduced_GB_def: fixes B :: "('a \<Rightarrow>\<^sub>0 'c) set" shows "punit.is_reduced_GB B \<equiv> punit.is_Groebner_basis B \<and> punit.is_auto_reduced B \<and> punit.is_monic_set B \<and> 0 \<notin> B"
interval_lowerbound_def: fixes s :: "'a set" shows "interval_lowerbound s = (\<Sum>i\<in>Basis. (INF x\<in>s. x \<bullet> i) *\<^sub>R i)"
Pareto_def: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" shows "Pareto R x y = ((\<exists>j. R j x x) \<and> (\<forall>i. R i x x \<longrightarrow> R i x y))"
resultant_impl_main: fixes G2 :: "'a poly"   and G1 :: "'a poly" assumes "length (coeffs G2) \<le> length (coeffs G1)" shows "resultant_impl_main div_exp G1 G2 = resultant G1 G2"
is_Prim0_Trg: fixes t :: "'a E.term" shows "E.is_Prim\<^sub>0 (E.Trg t)"
all_le_eq: fixes d :: "nat"   and c :: "nat" shows "(\<forall>a. (a < d) = (a < c)) = (d = c)"
obsE: fixes n' :: "'node"   and n :: "'node"   and S :: "'node set" assumes "n' \<in> obs n S" obtains as :: "'edge list" where "n -as\<rightarrow>* n'" and "\<forall>nx\<in>set (sourcenodes as). nx \<notin> S" and "n' \<in> S"
keys_def: fixes t :: "('a, 'b) RBT_Impl.rbt" shows "RBT_Impl.keys t = list.map fst (RBT_Impl.entries t)"
lossless_sample_uniform_units: fixes p :: "nat" assumes "1 < p" shows "lossless_spmf (sample_uniform_units p)"
prv_eqv_imp: fixes \<phi> :: "'fmla"   and \<chi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "\<chi> \<in> fmla" shows "prv (imp (eqv \<phi> \<chi>) (eqv \<chi> \<phi>))"
return_None_bind_spmf: fixes f :: "'d \<Rightarrow> 'c spmf" shows "return_pmf None \<bind> f = return_pmf None"
simps: fixes s :: "nat set"   and tab :: "'b option list set"   and st :: "nat set"   and tup :: "'b option list" shows "New_max.semiJoin (s, tab) (st, tup) = (s, Set.filter (New_max.isSameIntersection tup (s \<inter> st)) tab)"
hypothesis: shows "\<forall>n<m. Partition n 2 = n div 2"
Sup_ereal_mult_left': fixes Y :: "'a set"   and x :: "real"   and f :: "'a \<Rightarrow> ereal" assumes "Y \<noteq> {}"   and "0 \<le> x" shows "ereal x * \<Squnion> (f ` Y) = (SUP i\<in>Y. ereal x * f i)"
simps_1: fixes s :: "'a stream" shows "[] @- s = s"
cbl_unit_def: shows "cbl = Some (\<lambda>_. [], \<lambda>_. ())"
clock_sim_def: shows "clock_sim \<equiv> \<lambda>t. (clock_commonAbs t, tLast t)"
prems_1: shows "\<forall>a\<in>set (H # Hs). AbGroup a"
eval_const: fixes s :: "'a"   and r :: "'a" assumes "s \<in> carrier R"   and "r \<in> carrier R" shows "eval R R id s (monom P r 0) = id r"
distinct_54: fixes x81 :: "nat"   and x82 :: "msg"   and x6 :: "msg" shows "Crypt x81 x82 \<noteq> Hash x6"
less_log_of_power: fixes b :: "real"   and n :: "nat"   and m :: "real" assumes "b ^ n < m"   and "1 < b" shows "real n < log b m"
simps_3: fixes uu :: "char"   and v :: "char"   and va :: "char list" shows "iface_name_is_wildcard (uu # v # va) = iface_name_is_wildcard (v # va)"
fps_deriv_eq_iff: fixes f :: "'a fps"   and g :: "'a fps" shows "(fps_deriv f = fps_deriv g) = (f = fps_const (fps_nth f 0 - fps_nth g 0) + g)"
complex_cnj_power: fixes x :: "complex"   and n :: "nat" shows "cnj (x ^ n) = cnj x ^ n"
truth_provable: fixes H :: "fmla set" shows "H \<turnstile> Robinson_Arithmetic.neg fls"
rem_effectless_works_9: fixes as :: "('d \<times> ('e, 'f) fmap) list" shows "subseq (rem_effectless_act as) as"
"2_4": shows "list_all (typ_ok_sig \<Sigma>) (map snd inst)"
mult_ac_3: fixes b :: "'a"   and a :: "'a"   and c :: "'a" shows "b * (a * c) = a * (b * c)"
step_Some: fixes e :: "((nat \<times> nat) \<times> transition) fset"   and s :: "nat"   and r :: "nat \<Rightarrow>f value option"   and l :: "String.literal"   and i :: "value list"   and t :: "transition"   and s' :: "nat"   and p :: "value option list"   and r' :: "nat \<Rightarrow>f value option" shows "(step e s r l i = Some (t, s', p, r')) = (random_member (possible_steps e s r l i) = Some (s', t) \<and> evaluate_outputs t i r = p \<and> evaluate_updates t i r = r')"
oalist_inv_raw_filter: fixes xs :: "('a \<times> 'b) list"   and P :: "'a \<times> 'b \<Rightarrow> bool" assumes "tc.oalist_inv_raw xs" shows "tc.oalist_inv_raw (filter P xs)"
pairs_sup_closed: fixes x :: "'b regular"   and y :: "'b dense Filters.filter"   and z :: "'b regular"   and w :: "'b dense Filters.filter" assumes "(x, y) \<in> stone_phi.pairs"   and "(z, w) \<in> stone_phi.pairs" shows "stone_phi.pairs_sup (x, y) (z, w) \<in> stone_phi.pairs"
arg_cong2: fixes a :: "'g"   and b :: "'g"   and c :: "'h"   and d :: "'h"   and f :: "'g \<Rightarrow> 'h \<Rightarrow> 'i" assumes "a = b"   and "c = d" shows "f a c = f b d"
local_init_comp: fixes u' :: "'a \<Rightarrow> 'b"   and init' :: "('a \<Rightarrow> 'b) set"   and S :: "('a \<Rightarrow> 'c) \<Rightarrow> 'd \<Rightarrow> 'e"   and init :: "('a \<Rightarrow> 'c) set"   and S' :: "('a \<Rightarrow> 'b) \<Rightarrow> 'f \<Rightarrow> 'd" assumes "u' \<in> init'"   and "\<forall>u. sconjunctive (S u)" shows "local_init init S \<circ> local_init init' S' = local_init (zip_set init init') (\<lambda>u. S (fst \<circ> u) \<circ> S' (snd \<circ> u))"
executableI_words_fin: fixes p :: "'state"   and w :: "'action list" assumes "p \<in> nodes"   and "path w p" shows "set w \<subseteq> executable"
valid_reads: fixes i :: "nat"   and p :: "'a"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and sb :: "'a memref list"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option" assumes "i < length ts\<^sub>s\<^sub>b"   and "ts\<^sub>s\<^sub>b ! i = (p, is, \<theta>, sb, \<D>, \<O>, \<R>)" shows "reads_consistent False \<O> m\<^sub>s\<^sub>b sb"
LSM_mem: fixes M :: "('e, 'a, 'f) Module_scheme"   and T :: "'e set"   and t :: "'e" assumes "R module M"   and "T \<subseteq> carrier M"   and "t \<in> T" shows "t \<in> LSM\<^bsub>R\<^esub> M T"
comp_arr_inv: fixes f :: "'a \<times> 'a"   and g :: "'a \<times> 'a" assumes "VoV.inverse_arrows f g" shows "VoV.comp f g = VoV.dom g"
replace_comp_support_set: fixes pf1 :: "'c \<Rightarrow> nat \<Rightarrow> 'd \<Rightarrow> real"   and x :: "'c"   and pf2 :: "'c \<Rightarrow> nat \<Rightarrow> 'd \<Rightarrow> real" shows "support_set (qty_replace_comp pf1 x pf2) \<subseteq> support_set pf1 - {x} \<union> support_set pf2"
vec_neg: fixes x :: "'a vec" assumes "x \<in> carrier_vec n" shows "\<ominus>\<^bsub>V\<^esub> x = - x"
envconf_def: 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" shows "P \<turnstile> E \<surd> \<equiv> \<forall>V T. E V = \<lfloor>T\<rfloor> \<longrightarrow> is_type P T"
wf_list_graph_iff_wf_graph: fixes G :: "'a list_graph" shows "wf_graph (list_graph_to_graph G) = wf_list_graph_axioms G"
igFreshIGAbs1_fromMOD: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) Recursion.model" assumes "gFreshGAbs1 MOD" shows "igFreshIGAbs1 (fromMOD MOD)"
lgridE: fixes p :: "(nat \<times> int) list"   and b :: "(nat \<times> int) list"   and ds :: "nat set"   and lm :: "nat" assumes "p \<in> lgrid b ds lm" obtains "p \<in> grid b ds" and "level p < lm"
lim: shows "LIM n sequentially. real (nth_prime (n - 1)) :> at_top"
suffix_to_prefix: fixes xs :: "'c list"   and ys :: "'c list" shows "suffix xs ys = prefix (rev xs) (rev ys)"
monad_altc_optionT: fixes return :: "'b option \<Rightarrow> 'c"   and bind :: "'c \<Rightarrow> ('b option \<Rightarrow> 'c) \<Rightarrow> 'c"   and altc :: "'d cset \<Rightarrow> ('d \<Rightarrow> 'c) \<Rightarrow> 'c" assumes "monad_altc return bind altc" shows "monad_altc (return_option return) (bind_option return bind) (altc_option altc)"
set_iterator_genord_axioms: shows "set_iterator_genord iti S0 R"
Oops_range_spies1: fixes A :: "agent"   and KeyA :: "nat"   and authK :: "nat"   and Peer :: "msg"   and Ta :: "msg"   and authTicket :: "msg"   and evs :: "event list" assumes "Says Kas A \<lbrace>Crypt KeyA \<lbrace>Key authK, Peer, Ta\<rbrace>, authTicket\<rbrace> \<in> set evs"   and "evs \<in> kerbV" shows "authK \<notin> range shrK \<and> authK \<in> symKeys"
transpose_one: fixes n :: "nat" shows "1\<^sub>m n\<^sup>t = 1\<^sub>m n"
distinct_concat: fixes xs :: "'b list list" assumes "distinct xs"   and "\<And>ys. ys \<in> set xs \<Longrightarrow> distinct ys"   and "\<And>ys zs. \<lbrakk>ys \<in> set xs; zs \<in> set xs; ys \<noteq> zs\<rbrakk> \<Longrightarrow> set ys \<inter> set zs = {}" shows "distinct (concat xs)"
Step1_2_3: fixes rmhist :: "state \<Rightarrow> PrIds \<Rightarrow> histState"   and p :: "PrIds" shows "\<turnstile> [HNext rmhist p]_(c p, r p, m p, rmhist!p) \<and> ImpNext p \<and> \<not> unchanged (e p, c p, r p, m p, rmhist!p) \<and> $S3 rmhist p \<longrightarrow> S4 rmhist p$ \<and> RPCFwd crCh rmCh rst p \<and> unchanged (e p, c p, m p, rmhist!p) \<or> S6 rmhist p$ \<and> RPCFail crCh rmCh rst p \<and> unchanged (e p, c p, m p)"
rotate_downneg_inv: fixes w1 :: "wall"   and w2 :: "wall" assumes "rotate_downneg w1 w2" shows "kauff_mat w1 = kauff_mat w2"
weakBisimE_3: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi"   and \<Psi>' :: "'b" assumes "\<Psi> \<rhd> P \<approx> Q" shows "\<Psi> \<otimes> \<Psi>' \<rhd> P \<approx> Q"
veval'_wellformed: fixes \<Gamma> :: "(name, value) fmap"   and t :: "sterm"   and v :: "value" assumes "\<Gamma> \<turnstile>\<^sub>v t \<down> v"   and "pre_strong_term_class.wellformed t"   and "wellformed_venv \<Gamma>" shows "vwellformed v"
12371: fixes x :: "'a"   and y :: "'a" shows "- x \<sqinter> - (- - x \<sqinter> y) = - x"
subst_graph_aux_def: shows "subst_graph_aux \<equiv> lfp (\<lambda>p x1 x2. (\<exists>subst_aux i x. x1 = (i, x, zer) \<and> x2 = zer) \<or> (\<exists>subst_aux i x k. x1 = (i, x, Var k) \<and> x2 = (if i = k then x else Var k)) \<or> (\<exists>subst_aux i x t. x1 = (i, x, suc t) \<and> x2 = suc (((\<lambda>uu_. uu_) \<circ> subst_aux) (i, x, t)) \<and> p (i, x, t) (((\<lambda>uu_. uu_) \<circ> subst_aux) (i, x, t))) \<or> (\<exists>subst_aux i x t u. x1 = (i, x, pls t u) \<and> x2 = pls (((\<lambda>uu_. uu_) \<circ> subst_aux) (i, x, t)) (((\<lambda>uu_. uu_) \<circ> subst_aux) (i, x, u)) \<and> p (i, x, t) (((\<lambda>uu_. uu_) \<circ> subst_aux) (i, x, t)) \<and> p (i, x, u) (((\<lambda>uu_. uu_) \<circ> subst_aux) (i, x, u))) \<or> (\<exists>subst_aux i x t u. x1 = (i, x, tms t u) \<and> x2 = tms (((\<lambda>uu_. uu_) \<circ> subst_aux) (i, x, t)) (((\<lambda>uu_. uu_) \<circ> subst_aux) (i, x, u)) \<and> p (i, x, t) (((\<lambda>uu_. uu_) \<circ> subst_aux) (i, x, t)) \<and> p (i, x, u) (((\<lambda>uu_. uu_) \<circ> subst_aux) (i, x, u))))"
aOps: shows "MapOps simAbs (\<Union>a. sim_equiv_class a ` jkbpC) aOps"
graphE: fixes x :: "'b"   and y :: "real"   and F :: "'b set"   and f :: "'b \<Rightarrow> real" assumes "(x, y) \<in> graph F f" obtains "x \<in> F" and "y = f x"
neg_bounds_compare_1: fixes c :: "'a"   and b :: "'a option" assumes "\<not> c \<ge>\<^sub>u\<^sub>b b" shows "c <\<^sub>u\<^sub>b b"
space_eq: fixes v :: "view"   and v' :: "view"   and ts :: "traffic"   and c :: "cars" shows "own v = own v' \<longrightarrow> space ts v c = space ts v' c"
partition_filter1: fixes P :: "'a \<Rightarrow> bool"   and xs :: "'a list" shows "fst (List.partition P xs) = filter P xs"
split_list_last_prop: fixes xs :: "'b list"   and P :: "'b \<Rightarrow> bool" assumes "\<exists>x\<in>set xs. P x" shows "\<exists>ys x zs. xs = ys @ x # zs \<and> P x \<and> (\<forall>z\<in>set zs. \<not> P z)"
assms_1: shows "eq \<phi> \<psi> \<pi>l"
dgrad_p_set_closed_tail: fixes p :: "'a \<Rightarrow>\<^sub>0 'c"   and d :: "'a \<Rightarrow> nat"   and m :: "nat" assumes "p \<in> punit.dgrad_p_set d m" shows "punit.tail p \<in> punit.dgrad_p_set d m"
R_cring: fixes m :: "nat" assumes "0 < m" shows "cring (residue_ring (p ^ m))"
iso_fSup: fixes F :: "('b \<Rightarrow> 'c) set" assumes "\<forall>f\<in>F. mono f" shows "mono (\<Squnion> F)"
elims: fixes x :: "('a \<times> 'b) set"   and y :: "('a \<times> 'b) tree" assumes "treap_of x = y" obtains KP :: "('a \<times> 'b) set" where "x = KP" and "y = (if infinite KP \<or> KP = {} then \<langle>\<rangle> else let m = arg_min_on snd KP; L = {p \<in> KP. fst p < fst m}; R = {p \<in> KP. fst m < fst p} in \<langle>treap_of L, m, treap_of R\<rangle>)"
ltl_def: fixes llist :: "'a llist" shows "ltl llist = (case llist of LNil \<Rightarrow> LNil | LCons x21 x22 \<Rightarrow> x22)"
row_four_block_mat_2: fixes A :: "'b mat"   and nr1 :: "nat"   and nc1 :: "nat"   and B :: "'b mat"   and nc2 :: "nat"   and C :: "'b mat"   and nr2 :: "nat"   and D :: "'b mat"   and i :: "nat" assumes "A \<in> carrier_mat nr1 nc1"   and "B \<in> carrier_mat nr1 nc2"   and "C \<in> carrier_mat nr2 nc1"   and "D \<in> carrier_mat nr2 nc2"   and "\<not> i < nr1"   and "i < nr1 + nr2" shows "row (four_block_mat A B C D) i = row C (i - nr1) @\<^sub>v row D (i - nr1)"
singleton_lst_proj_2: fixes l :: "'a"   and l' :: "'a"   and a :: "'b" assumes "l \<noteq> l'" shows "proj_unl l' [(ln l, a)] = []"
closure_open_Int_superset: fixes S :: "'a set"   and T :: "'a set" assumes "open S"   and "S \<subseteq> closure T" shows "closure (S \<inter> T) = closure S"
RCOSETS_def: fixes G :: "('b, 'c) monoid_scheme"   and H :: "'b set" shows "rcosets\<^bsub>G\<^esub> H = (\<Union>a\<in>carrier G. {H #>\<^bsub>G\<^esub> a})"
select_convs_6: fixes id :: "nat"   and no :: "nat"   and nhid :: "nat"   and msg :: "msg"   and num :: "nat"   and sid :: "nat"   and more :: "'a" shows "sid \<lparr>state.id = id, no = no, nhid = nhid, msg = msg, num = num, sid = sid, \<dots> = more\<rparr> = sid"
rExt: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi"   and \<Psi>' :: "'b" assumes "(\<Psi>, P, Q) \<in> Rel" shows "(\<Psi> \<otimes> \<Psi>', P, Q) \<in> Rel"
wt_int: fixes \<xi> :: "Preliminaries.var \<Rightarrow> univ"   and T :: "'fsym trm" assumes "Ik.wtE \<xi>"   and "wt T" shows "Ik_intT (tpOf T) (Ik.int \<xi> T)"
wf_liftB: fixes \<Gamma> :: "binding list"   and i :: "nat"   and T :: "POPLmarkRecord.type" assumes "\<Gamma> \<turnstile>\<^sub>w\<^sub>f"   and "\<Gamma>\<langle>i\<rangle> = \<lfloor>VarB T\<rfloor>" shows "\<Gamma> \<turnstile>\<^sub>w\<^sub>f \<up>\<^sub>\<tau> (Suc i) 0 T"
norm_eq: fixes x :: "'a"   and y :: "'b" shows "(norm x = norm y) = (x \<bullet> x = y \<bullet> y)"
elims_3: fixes x :: "char list"   and xa :: "char list" assumes "\<not> lleq_charlist x xa" obtains v :: "char" and va :: "char list" where "xa__ = v # va" and "xa = []"   | x :: "char" and xs :: "char list" and y :: "char" and ys :: "char list" where "xa__ = x # xs" and "xa = y # ys" and "\<not> (if x = y then lleq_charlist xs ys else x < y)"
scale_zero_right: fixes a :: "'a" shows "scale a (0::'b) = (0::'b)"
sum_case_split: fixes P :: "'b \<Rightarrow> bool"   and f :: "hf \<Rightarrow> 'b"   and g :: "hf \<Rightarrow> 'b"   and a :: "hf" shows "P (sum_case f g a) = ((\<forall>x. a = HF.Inl x \<longrightarrow> P (f x)) \<and> (\<forall>y. a = HF.Inr y \<longrightarrow> P (g y)) \<and> (\<not> is_hsum a \<longrightarrow> P undefined))"
of_drop_to_bl: fixes n :: "nat"   and x :: "'a word" shows "of_bl (drop n (to_bl x)) = x && mask (size x - n)"
in_lo_2: shows "mix_pmf \<alpha> x r \<in> lotteries_on outcomes"
chine_char: shows "\<tau>\<mu>.chine \<cong> \<tau>.chine \<star> \<mu>.chine"
cnj_cong: fixes H :: "fmla set"   and A :: "fmla"   and A' :: "fmla"   and B :: "fmla"   and B' :: "fmla" assumes "H \<turnstile> A IFF A'"   and "H \<turnstile> B IFF B'" shows "H \<turnstile> A AND B IFF A' AND B'"
Always_eq_UN_invariant: fixes A :: "'b set" shows "Always A = \<Union> (invariant ` Pow A)"
coeff_sdiv_poly: fixes f :: "'b poly"   and a :: "'b"   and n :: "nat" shows "coeff (sdiv_poly f a) n = coeff f n div a"
mult_imp_le_div_pos: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and y :: "'b"   and z :: "'b"   and x :: "'b" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less zero y"   and "less_eq (times z y) x" shows "less_eq z (divide x y)"
bmod_comp_def: fixes R :: "'d \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and f :: "'e \<Rightarrow> 'c"   and g :: "'f \<Rightarrow> 'c"   and x :: "'d" shows "\<otimes> R f g x = \<Squnion>{f y \<cdot> g z |y z. R x y z}"
spec_is_ringed_space: shows "ringed_space Spec is_zariski_open sheaf_spec sheaf_spec_morphisms \<O>b add_sheaf_spec mult_sheaf_spec zero_sheaf_spec one_sheaf_spec"
lookup_lift_poly_syz: fixes n :: "nat"   and b :: "'t \<Rightarrow>\<^sub>0 'c"   and i :: "nat" shows "lookup (lift_poly_syz n b i) = (\<lambda>x. if pair_of_term x = (0::'a, i) then 1::'c else if n \<le> component_of_term x then lookup b (map_component (\<lambda>k. k - n) x) else (0::'c))"
type_definition_perm: shows "type_definition Rep_perm Abs_perm perm"
bbM: fixes i :: "nat"   and s :: "'s" shows "wp b (M i) s \<le> c"
substitute_either: fixes f :: "'b \<Rightarrow> 'b ttree"   and T :: "'b set"   and t :: "'b ttree"   and t' :: "'b ttree" shows "substitute f T (t \<oplus>\<oplus> t') = substitute f T t \<oplus>\<oplus> substitute f T t'"
case_distrib: fixes h :: "'c \<Rightarrow> 'b"   and f1 :: "'c"   and f2 :: "'d \<Rightarrow> 'd list \<Rightarrow> 'c"   and list :: "'d list" shows "h (case list of [] \<Rightarrow> f1 | x # xa \<Rightarrow> f2 x xa) = (case list of [] \<Rightarrow> h f1 | x1 # x2 \<Rightarrow> h (f2 x1 x2))"
assms_2: shows "P \<Longrightarrow>\<^sub>A Q"
FactGroup_iso: fixes G :: "('b, 'c) monoid_scheme"   and H :: "('d, 'e) monoid_scheme"   and h :: "'b \<Rightarrow> 'd" assumes "group_hom G H h"   and "h ` carrier G = carrier H" shows "G Mod kernel G H h \<cong> H"
Tree\<^sub>\<alpha>_free_5: fixes t1\<^sub>\<alpha> :: "('e, 'f, 'g) Tree\<^sub>\<alpha>"   and \<alpha> :: "'g"   and t2\<^sub>\<alpha> :: "('e, 'f, 'g) Tree\<^sub>\<alpha>" shows "Not\<^sub>\<alpha> t1\<^sub>\<alpha> \<noteq> Act\<^sub>\<alpha> \<alpha> t2\<^sub>\<alpha>"
inv_is_inverse: fixes f :: "'c" assumes "C.iso f" shows "C.inverse_arrows f (C.inv f)"
WT_fv: 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 e :: "expr"   and T :: "ty" assumes "P,E \<turnstile> e :: T" shows "fv e \<subseteq> dom E"
Infinitesimal_mult: fixes x :: "'a star"   and y :: "'a star" assumes "x \<in> Infinitesimal"   and "y \<in> Infinitesimal" shows "x * y \<in> Infinitesimal"
\<epsilon>: fixes z :: "complex" assumes "cmod z = \<epsilon>" shows "cmod (f z) \<le> B"
diagonal_step_row_def: fixes A :: "(('d, 'e) vec, 'f) vec"   and i :: "nat"   and j :: "nat"   and c :: "'d"   and v :: "'d"   and a :: "'f" shows "diagonal_step_row A i j c v a = (\<chi>b. if a = from_nat i \<and> b = from_nat i then c else if a = from_nat j \<and> b = from_nat j then v * A $ from_nat j $ from_nat j else A $ a $ b)"
SubstForm_imp_subst_dbfm: fixes v :: "hf"   and u :: "hf"   and x :: "hf"   and x' :: "hf"   and e :: "(name, hf) finfun" assumes "SubstForm v u x x'" obtains t :: "dbtm" and A :: "dbfm" where "u = \<lbrakk>quot_dbtm t\<rbrakk>e" and "x = \<lbrakk>quot_dbfm A\<rbrakk>e" and "x' = \<lbrakk>quot_dbfm (subst_dbfm t (decode_Var v) A)\<rbrakk>e"
comp_eq_dest_lhs: fixes a :: "'i \<Rightarrow> 'h"   and b :: "'g \<Rightarrow> 'i"   and c :: "'g \<Rightarrow> 'h"   and v :: "'g" assumes "a \<circ> b = c" shows "a (b v) = c v"