hcju/isabelleps huggingface dataset:
self_finance_updated: fixes Mkt :: "('b, 'c) discrete_market"   and asset :: "'c"   and n :: "nat"   and w :: "'b"   and pf :: "'c \<Rightarrow> nat \<Rightarrow> 'b \<Rightarrow> real"   and v :: "real" assumes "prices Mkt asset (Suc n) w \<noteq> 0"   and "portfolio pf" shows "cls_val_process Mkt (self_finance Mkt v pf asset) (Suc n) w = cls_val_process Mkt pf (Suc n) w + prices Mkt asset (Suc n) w * remaining_qty Mkt v pf asset (Suc n) w"
ifmi_les_hlp: fixes s :: "bdd"   and v :: "nat"   and ni1 :: "nat"   and ni2 :: "nat"   and x1 :: "nat"   and s' :: "bdd"   and nia :: "nat"   and n :: "nat ifex" assumes "pointermap_sane (dpm s)"   and "pointermap_getmk (v, ni1, ni2) (dpm s) = (x1, dpm s')"   and "Rmi_g nia n s" shows "Rmi_g nia n s'"
subst_respects_ltl_prop_entailment_generalized: fixes S :: "'c ltl set"   and y :: "'c ltl"   and \<A> :: "'c ltl set"   and m :: "'c ltl \<Rightarrow> 'c ltl option" assumes "\<And>\<A>. (\<And>x. x \<in> S \<Longrightarrow> \<A> \<Turnstile>\<^sub>P x) \<Longrightarrow> \<A> \<Turnstile>\<^sub>P y"   and "\<And>x. x \<in> S \<Longrightarrow> \<A> \<Turnstile>\<^sub>P subst x m" shows "\<A> \<Turnstile>\<^sub>P subst y m"
nhop_addpreRT: fixes dip :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option"   and pre :: "nat set"   and ip :: "nat" assumes "dip \<in> kD rt" shows "nhop (the (addpreRT rt dip pre)) ip = nhop rt ip"
InvariantQCharacterizationAfterApplyUnitPropagate: fixes state :: "State" assumes "InvariantConsistent (getM state)"   and "InvariantWatchListsContainOnlyClausesFromF (getWatchList state) (getF state)"   and "InvariantWatchListsUniq (getWatchList state)"   and "InvariantWatchListsCharacterization (getWatchList state) (getWatch1 state) (getWatch2 state)"   and "InvariantWatchesEl (getF state) (getWatch1 state) (getWatch2 state)"   and "InvariantWatchesDiffer (getF state) (getWatch1 state) (getWatch2 state)"   and "InvariantWatchCharacterization (getF state) (getWatch1 state) (getWatch2 state) (getM state)"   and "InvariantConflictFlagCharacterization (getConflictFlag state) (getF state) (getM state)"   and "InvariantQCharacterization (getConflictFlag state) (getQ state) (getF state) (getM state)"   and "InvariantUniqQ (getQ state)"   and "getQ state \<noteq> []"   and "\<not> getConflictFlag state" shows "let state'' = applyUnitPropagate state in InvariantQCharacterization (getConflictFlag state'') (getQ state'') (getF state'') (getM state'')"
unit_prod: fixes a :: "'a"   and b :: "'a" assumes "a dvd (1::'a)"   and "b dvd (1::'a)" shows "a * b dvd (1::'a)"
remove: fixes A :: "'a set"   and x :: "'a"   and g :: "'a \<Rightarrow> 'b" assumes "finite A"   and "x \<in> A" shows "prod g A = g x * prod g (A - {x})"
defined_offending: fixes sinvar :: "'b graph \<Rightarrow> ('b \<Rightarrow> 'c) \<Rightarrow> bool"   and G :: "'b graph"   and nP :: "'b \<Rightarrow> 'c" assumes "SecurityInvariant_preliminaries sinvar"   and "wf_graph G"   and "\<not> sinvar G nP" shows "SecurityInvariant_withOffendingFlows.set_offending_flows sinvar G nP \<noteq> {}"
cod_pr1: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and pr0 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and pr1 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "elementary_category_with_binary_products C pr0 pr1"   and "partial_magma.ide C a"   and "partial_magma.ide C b" shows "partial_magma.cod C (pr1 a b) = a"
wf_prog_def: fixes wf_md :: "'a prog \<Rightarrow> char list \<Rightarrow> 'a mdecl \<Rightarrow> bool"   and P :: "'a prog" shows "wf_prog wf_md P \<equiv> wf_syscls P \<and> (\<forall>c\<in>set P. wf_cdecl wf_md P c) \<and> distinct_fst P"
seqstar_inductr: fixes z :: "'a"   and y :: "'a"   and x :: "'a" assumes "z + y ; x \<le> y" shows "z ; x\<^sup>; \<le> y"
sets_lebesgue_almost_borel: fixes S :: "'a set" assumes "S \<in> sets lebesgue" obtains B :: "'a set" and N :: "'a set" where "B \<in> sets borel" and "negligible N" and "B \<union> N = S"
Lambert_W'_strict_antimono: fixes x :: "real"   and y :: "real" assumes "- exp (- 1) \<le> x"   and "x < y"   and "y < 0" shows "Lambert_W' y < Lambert_W' x"
rep_eq: fixes x :: "'e \<Rightarrow> ('d, 'c) uexpr"   and xa :: "('e, 'c) uexpr" shows "\<lbrakk>msubst x xa\<rbrakk>\<^sub>e = (\<lambda>b. \<lbrakk>x (\<lbrakk>xa\<rbrakk>\<^sub>e b)\<rbrakk>\<^sub>e b)"
induct: fixes P :: "'a ifex list \<Rightarrow> bool"   and a0 :: "'a ifex list" assumes "P []"   and "\<And>v uu_ uv_ r. P r \<Longrightarrow> P (IF v uu_ uv_ # r)"   and "\<And>r. P r \<Longrightarrow> P (Trueif # r)"   and "\<And>r. P r \<Longrightarrow> P (Falseif # r)" shows "P a0"
\<tau>inv: fixes s1 :: "'s1"   and s2 :: "'s2"   and tl1 :: "'tl1"   and s1' :: "'s1"   and tl2 :: "'tl2"   and s2' :: "'s2" assumes "s1 \<approx> s2"   and "s1 -1-tl1\<rightarrow> s1'"   and "s2 -2-tl2\<rightarrow> s2'"   and "s1' \<approx> s2'"   and "tl1 \<sim> tl2" shows "\<tau>move1 s1 tl1 s1' = \<tau>move2 s2 tl2 s2'"
assms_5: shows "Q_I_pos = projectQuery I (filterQuery I Q)"
prm_compose_push: fixes \<pi> :: "'b prm"   and a :: "'b"   and b :: "'b" shows "\<pi> \<diamondop> [a \<leftrightarrow> b] = [\<pi> $ a \<leftrightarrow> \<pi> $ b] \<diamondop> \<pi>"
unique_CryptKey: fixes B :: "agent"   and A :: "agent"   and SesKey :: "nat"   and T :: "msg"   and evs :: "event list"   and B' :: "agent"   and A' :: "agent"   and T' :: "msg" assumes "Crypt (shrK B) \<lbrace>Agent A, Agent B, Key SesKey, T\<rbrace> \<in> parts (knows Spy evs)"   and "Crypt (shrK B') \<lbrace>Agent A', Agent B', Key SesKey, T'\<rbrace> \<in> parts (knows Spy evs)"   and "Key SesKey \<notin> analz (knows Spy evs)"   and "evs \<in> kerbIV_gets" shows "A = A' \<and> B = B' \<and> T = T'"
pdevs_val_pdevs_of_list_append: fixes f :: "nat \<Rightarrow> real"   and I :: "real set"   and g :: "nat \<Rightarrow> real"   and xs :: "'b list"   and ys :: "'b list" assumes "f \<in> UNIV \<rightarrow> I"   and "g \<in> UNIV \<rightarrow> I" obtains e :: "nat \<Rightarrow> real" where "pdevs_val f (pdevs_of_list xs) + pdevs_val g (pdevs_of_list ys) = pdevs_val e (pdevs_of_list (xs @ ys))" and "e \<in> UNIV \<rightarrow> I"
r_g: shows "\<turnstile> \<rho>,\<eta> : \<Gamma>"
space_subprob_algebra_empty_iff: fixes N :: "'c measure" shows "(space (subprob_algebra N) = {}) = (space N = {})"
c_failures_failures: fixes step :: "'g \<Rightarrow> 'e \<Rightarrow> 'g"   and out :: "'g \<Rightarrow> 'e \<Rightarrow> 'f"   and s\<^sub>0 :: "'g" shows "failures (c_process step out s\<^sub>0) = c_failures step out s\<^sub>0"
invfun_l: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and B :: "'d set"   and a :: "'c" assumes "f \<in> A \<rightarrow> B"   and "inj_on f A"   and "surj_to f A B"   and "a \<in> A" shows "(f\<inverse>\<^bsub>B,A\<^esub>) (f a) = a"
le_pred_imp_less: fixes n :: "nat"   and m :: "nat" assumes "0 < n"   and "m \<le> n - Suc 0" shows "m < n"
YMapInvApp: fixes X :: "'d"   and C :: "('d, 'e, 'f) LSCategory_scheme"   and B :: "'d"   and F :: "('d, ZF, 'e, ZF, \<lparr>mor2ZF :: 'e \<Rightarrow> ZF, \<dots> :: 'f\<rparr>, unit) Functor"   and x :: "ZF" assumes "X \<in> obj\<^bsub>C\<^esub>"   and "B \<in> obj\<^bsub>C\<^esub>"   and "LSCategory C" shows "YMapInv C X F x $$ B = ZFfun (Hom\<^bsub>C\<^esub> B X) (F @@ B) (\<lambda>f. (F ## z2m\<^bsub>C\<^esub>f) |@| x)"
divides_cong_l: fixes G :: "('c, 'd) monoid_scheme"   and x :: "'c"   and x' :: "'c"   and y :: "'c" assumes "Group.monoid G"   and "x \<sim>\<^bsub>G\<^esub> x'"   and "x' divides\<^bsub>G\<^esub> y"   and "x \<in> carrier G" shows "x divides\<^bsub>G\<^esub> y"
v_def: shows "v = {w. w \<notin> path_image \<gamma> \<and> winding_number \<gamma> w = 0}"
multi_self_add_other_not_self: fixes M :: "'b multiset"   and x :: "'b" shows "(M = add_mset x M) = False"
funpow_swap1: fixes f :: "'b \<Rightarrow> 'b"   and n :: "nat"   and x :: "'b" shows "f (f^n x) = f^n (f x)"
ell2_pointwise_ortho: fixes x :: "'c ell2"   and y :: "'c ell2" assumes "\<And>i. Rep_ell2 x i = 0 \<or> Rep_ell2 y i = 0" shows "is_orthogonal x y"
unif: shows "unifier\<^sub>l\<^sub>s \<sigma> L"
f21: fixes out :: "'c"   and c :: "('a, 'c, 'b) rpv"   and gpv :: "('a, 'c, 'b) gpv"   and ba :: "('d, 'g, ('d, 'g, 'e) rpv) generat" assumes "IO out c \<in> set_spmf (the_gpv gpv)"   and "rel_generat A C (rel_fun R (rel_gpv'' A C R)) (IO out c) ba" shows "Domainp C out"
shunting: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(- x \<sqinter> - y \<le> - z) = (- y \<le> - - x \<squnion> - z)"
insertCI: fixes a :: "'b"   and B :: "'b set"   and b :: "'b" assumes "(a::'b) \<notin> (B::'b set) \<Longrightarrow> a = (b::'b)" shows "(a::'b) \<in> (b::'b) \<triangleright> (B::'b set)"
conforms_set_locals: fixes x :: "abopt"   and s :: "st"   and G :: "prog"   and L' :: "lenv"   and l :: "lname \<Rightarrow> val option"   and L :: "lenv" assumes "(x, s)\<Colon>\<preceq>(G, L')"   and "G,s\<turnstile>l[\<sim>\<Colon>\<preceq>]L"   and "x = Some (Jump Ret) \<longrightarrow> l Result \<noteq> None" shows "(x, set_locals l s)\<Colon>\<preceq>(G, L)"
intersection_ofE: fixes P :: "'d set set \<Rightarrow> bool"   and Q :: "'d set \<Rightarrow> bool"   and S :: "'d set"   and R :: "'d set \<Rightarrow> bool" assumes "(P intersection_of Q) S"   and "\<And>T. \<lbrakk>P T; T \<subseteq> Collect Q\<rbrakk> \<Longrightarrow> R (\<Inter> T)" shows "R S"
DJ_qe: fixes qe :: "fm \<Rightarrow> fm"   and vs :: "'b list" assumes "\<forall>bs p. qfree p \<longrightarrow> qfree (qe p) \<and> Ifm vs bs (qe p) = Ifm vs bs (E p)" shows "\<forall>bs p. qfree p \<longrightarrow> qfree (DJ qe p) \<and> Ifm vs bs (DJ qe p) = Ifm vs bs (E p)"
simps_7: fixes r :: "'c rexp" shows "\<guillemotleft>Star r\<guillemotright> = Star \<guillemotleft>r\<guillemotright>"
zle: fixes z :: "int"   and w :: "int" shows "(z \<le> w) = (\<not> w < z)"
simps_2: fixes R :: "('c, 'd) ring_scheme"   and v :: "'c"   and va :: "'c list"   and p2 :: "'c list" assumes "ring R" shows "ring.poly_mult R (v # va) p2 = ring.poly_add R (map ((\<otimes>\<^bsub>R\<^esub>) (lead_coeff (v # va))) p2 @ replicate (degree (v # va)) \<zero>\<^bsub>R\<^esub>) (ring.poly_mult R (tl (v # va)) p2)"
assms_4: fixes z :: "'a" assumes "z \<in> s - cball a r" shows "f a \<le> f z"
execute_bind_success: fixes f :: "'b Heap"   and h :: "heap"   and g :: "'b \<Rightarrow> 'c Heap" assumes "success f h" shows "execute (f \<bind> g) h = execute (g (fst (the (execute f h)))) (snd (the (execute f h)))"
aform_sup_inner_refine: fixes optns :: "nat" shows "(RETURN o2 aform_sup_inner optns, Sup_inners) \<in> aforms_rel \<rightarrow> rl_rel \<rightarrow> \<langle>rnv_rel\<rangle>nres_rel"
contract': fixes F :: "'b formula"   and \<Gamma> :: "'b formula multiset"   and \<Delta> :: "'b formula multiset"   and n :: "nat" shows "(F, F, \<Gamma> \<Rightarrow> \<Delta> \<down> n \<longrightarrow> F, \<Gamma> \<Rightarrow> \<Delta> \<down> n) \<and> (\<Gamma> \<Rightarrow> F, F, \<Delta> \<down> n \<longrightarrow> \<Gamma> \<Rightarrow> F, \<Delta> \<down> n)"
pmf_of_mapping_def: fixes m :: "('b, real) mapping" shows "pmf_of_mapping m = embed_pmf (Mapping.lookup_default 0 m)"
ddh_1_def: fixes \<A> :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool spmf" shows "local.ddh.ddh_1 \<A> = sample_uniform (order \<G>) \<bind> (\<lambda>x. sample_uniform (order \<G>) \<bind> (\<lambda>y. sample_uniform (order \<G>) \<bind> (\<lambda>z. \<A> (\<^bold>g [^] x) (\<^bold>g [^] y) (\<^bold>g [^] z))))"
simps_120: fixes f1 :: "'a"   and f2 :: "'a"   and f3 :: "'b \<Rightarrow> 'a"   and f4 :: "'b \<Rightarrow> 'a"   and f5 :: "'b ltln \<Rightarrow> 'b ltln \<Rightarrow> 'a"   and f6 :: "'b ltln \<Rightarrow> 'b ltln \<Rightarrow> 'a"   and f7 :: "'b ltln \<Rightarrow> 'a"   and f8 :: "'b ltln \<Rightarrow> 'b ltln \<Rightarrow> 'a"   and f9 :: "'b ltln \<Rightarrow> 'b ltln \<Rightarrow> 'a"   and f10 :: "'b ltln \<Rightarrow> 'b ltln \<Rightarrow> 'a"   and f11 :: "'b ltln \<Rightarrow> 'b ltln \<Rightarrow> 'a" shows "(case true\<^sub>n of true\<^sub>n \<Rightarrow> f1 | false\<^sub>n \<Rightarrow> f2 | prop\<^sub>n(x) \<Rightarrow> f3 x | nprop\<^sub>n(x) \<Rightarrow> f4 x | x and\<^sub>n xa \<Rightarrow> f5 x xa | x or\<^sub>n xa \<Rightarrow> f6 x xa | X\<^sub>n x \<Rightarrow> f7 x | x U\<^sub>n xa \<Rightarrow> f8 x xa | x R\<^sub>n xa \<Rightarrow> f9 x xa | x W\<^sub>n xa \<Rightarrow> f10 x xa | x M\<^sub>n xa \<Rightarrow> f11 x xa) = f1"
set_remisodups: fixes Us :: "('a, 'g) aezfun set list" shows "set (remisodups Us) \<subseteq> set Us"
inverse_fds_def: fixes f :: "'b fds" shows "inverse f = fds (dirichlet_inverse (fds_nth f) (inverse (fds_nth f 1)))"
not_subseteq: fixes A :: "'e set"   and B :: "'e set" assumes "\<not> A \<subseteq> B" shows "\<exists>a\<in>A. a \<notin> B"
L2_set_eq_0_iff: fixes A :: "'a set"   and f :: "'a \<Rightarrow> real" assumes "finite A" shows "(L2_set f A = 0) = (\<forall>x\<in>A. f x = 0)"
simps_18: fixes V :: "'g"   and a :: "'h"   and e :: "('f, 'g, 'h) exp"   and A :: "'f set option" shows "\<D> (insync\<^bsub>V\<^esub> (a) e) A = \<D> e A"
edges_sym': fixes u :: "'a"   and v :: "'a"   and g :: "'a ugraph" assumes "(u, v) \<in> edges g" shows "(v, u) \<in> edges g"
measurable_pmf_measure2: fixes N :: "'b measure"   and M :: "'c pmf" shows "N \<rightarrow>\<^sub>M measure_pmf M = N \<rightarrow>\<^sub>M count_space UNIV"
atLeastLessThan_iff: fixes i :: "'a word"   and l :: "'a word"   and u :: "'a word" shows "(i \<in> signed.atLeastLessThan l u) = (l \<le>s i \<and> i <s u)"
EL: fixes l :: "'b"   and r :: "'b" assumes "l \<in> S_Min_Le"   and "r \<in> S_Max_Lt" shows "l < r"
col_space_eq_range: fixes f :: "('b, 'c) vec \<Rightarrow> ('b, 'd) vec" assumes "linear (*s) (*s) f" shows "col_space (matrix f) = range f"
factorize_complex_main_def: fixes p :: "complex poly" shows "factorize_complex_main p \<equiv> let (c, pis) = yun_factorization gcd p in if \<forall>(p, i)\<in>set pis. degree p \<le> 2 \<or> (\<forall>x\<in>set (coeffs p). x \<in> \<rat>) then Some (c, concat (map (\<lambda>(p, i). map (\<lambda>r. (r, i)) (roots_of_complex_main p)) pis)) else None"
ordinal_mod_less: fixes y :: "ordinal"   and x :: "ordinal" assumes "0 < y" shows "x mod y < y"
hyps: shows "(a + b) ^ n = (\<Sum>k\<le>n. of_nat (n choose k) * a ^ k * b ^ (n - k))"
Pw_pref: shows "\<forall>w. Preorder (Pw w) \<and> Total (Pw w)"
right_unique_rel: assumes "right_unique (in_rel A)" shows "right_unique (rel_sctn (in_rel A))"
get_msgs_K: fixes snd_f :: "'d \<Rightarrow> 'd \<Rightarrow> 'f \<Rightarrow> 'e"   and cfg :: "'d \<Rightarrow> 'f"   and HOs :: "'g \<Rightarrow> 'd \<Rightarrow> 'd set"   and r :: "'g"   and p :: "'d"   and SHOs :: "'g \<Rightarrow> 'd \<Rightarrow> 'd set" shows "get_msgs snd_f cfg (K (HOs r p)) (K (SHOs r p)) p = get_msgs snd_f cfg (HOs r) (SHOs r) p"
plusinf_nb: fixes p :: "fm" assumes "islin p" shows "bound0 (plusinf p)"
absorb_iff2: fixes a :: "'c \<times> 'c \<Rightarrow> 'd"   and b :: "'c \<times> 'c \<Rightarrow> 'd" shows "(a \<preceq> b) = (a \<oplus> b = b)"
preserves_retraction: fixes e :: "'a" assumes "retraction e" shows "retraction (L e)"
b: shows "\<Sqinter> {e::'a::refinement_lattice. (c::'a::refinement_lattice) \<sqsubseteq> e \<parallel> (i::'a::refinement_lattice)} \<sqsubseteq> \<Sqinter> {f::'a::refinement_lattice. (d::'a::refinement_lattice) \<sqsubseteq> f \<parallel> i}"
zerosign_def: fixes s :: "nat"   and a :: "('a, 'b) IEEE.float" shows "zerosign s a = (if is_zero a then if s = 0 then 0 else - 0 else a)"
inv_is_inverse: fixes f :: "'a" assumes "R.iso f" shows "R.inverse_arrows f (R.inv f)"
permutation_ghc_sort: fixes xs :: "int list" shows "mset (ghc_sort xs) = mset xs"
preserves_iso: fixes f :: "'b" assumes "local.iso f" shows "local.iso (\<rr>'.map f)"
flip_signed_order2: fixes x :: "'a \<times> bool" shows "flip_signed (flip_signed x) = x"
"3_2": assumes "x \<noteq> y"   and "rel x y"   and "\<And>ya. \<lbrakk>ya \<in> set (y # ys); P ya\<rbrakk> \<Longrightarrow> False" shows "filter P (merge_wrt rel xs (y # ys)) = filter P xs"
hyps_5: shows "h\<^sub>2 a = \<lfloor>(C, fs)\<rfloor>"
zero_jkA: shows "is_zero_row_upt_k j k A"
simps_2: fixes P :: "(('a, 'b) vec, 'b) vec"   and A :: "(('a, 'c) vec, 'b) vec"   and n :: "nat"   and i :: "'b"   and j :: "'c" shows "row_add_iterate_PA (P, A) (Suc n) i j = (if Suc n = mod_type_class.to_nat i then row_add_iterate_PA (P, A) n i j else row_add_iterate_PA (row_add P (mod_type_class.from_nat (Suc n)) i (- A $ mod_type_class.from_nat (Suc n) $ j), row_add A (mod_type_class.from_nat (Suc n)) i (- A $ mod_type_class.from_nat (Suc n) $ j)) n i j)"
P2: shows "Col A B P' \<and> A B Perp P P'"
Smith_1xn_aux_invertible_Q: fixes Smith_1x2 :: "'a mat \<Rightarrow> 'a mat \<times> 'a mat"   and Smith_2x2 :: "'a mat \<Rightarrow> 'a mat \<times> 'a mat \<times> 'a mat"   and div_op :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and S' :: "'a mat"   and Q' :: "'a mat"   and i :: "nat"   and A :: "'a mat"   and S :: "'a mat"   and Q :: "'a mat"   and n :: "nat" assumes "Smith_Impl Smith_1x2 Smith_2x2 div_op"   and "(S', Q') = Smith_Impl.Smith_1xn_aux Smith_1x2 i A (S, Q)"   and "A \<in> carrier_mat 1 n"   and "Q \<in> carrier_mat n n"   and "i < n"   and "invertible_mat Q" shows "invertible_mat Q'"
assms_3: shows "(S, T) \<in> \<langle>A\<rangle>set_rel"
trcl_uncons: fixes c :: "'a"   and a :: "'b"   and w :: "'b list"   and c' :: "'a"   and t :: "('a \<times> 'b \<times> 'a) set" assumes "(c, a # w, c') \<in> trcl t" shows "\<exists>ch. (c, a, ch) \<in> t \<and> (ch, w, c') \<in> trcl t"
Well_order_isMinim_exists: fixes B :: "'a set" assumes "B \<subseteq> Field r"   and "B \<noteq> {}" shows "\<exists>b. local.isMinim B b"
toArr_fromArr: fixes x :: "'a"   and Arr :: "'a set" assumes "x \<in> Arr" shows "toArr (quiver.fromArr Arr x) = x"
domI: fixes m :: "'e \<Rightarrow> 'd option"   and a :: "'e"   and b :: "'d" assumes "m a = Some b" shows "a \<in> dom m"
f_range_onD': fixes c :: "'d"   and f :: "'d set \<Rightarrow> 'd set"   and B :: "'d set"   and A :: "'d set" assumes "c \<in> f B"   and "f_range_on A f"   and "B \<subseteq> A" shows "c \<in> B"
extr_payload: fixes X :: "msg"   and Bad :: "agent set"   and NI :: "msg set"   and I :: "msg set" assumes "X \<in> extr Bad NI (local.abs I)"   and "NI \<subseteq> payload" shows "X \<in> payload"
cod_cod: fixes f :: "'a1" assumes "C1.arr f" shows "C1.cod (C1.cod f) = C1.cod f"
ev1: shows "\<forall>\<^sub>F n in sequentially. 1 < t n"
externals_def: fixes asig :: "'e set \<times> 'e set \<times> 'e set" shows "externals asig \<equiv> inputs asig \<union> outputs asig"
derived_in_carrier: fixes H :: "(nat \<times> nat) set" assumes "H \<subseteq> carrier S'" shows "derived S' H \<subseteq> carrier S'"
cnf_of_encode_goal_state_set_i: fixes \<Pi> :: "'b strips_problem"   and t :: "nat" shows "cnf ((\<Phi>\<^sub>G \<Pi>) t) = \<Union> {cnf (encode_state_variable t (index (\<Pi>\<^sub>\<V>) v) ((\<Pi>\<^sub>G) v)) |v. v \<in> set (\<Pi>\<^sub>\<V>) \<and> (\<Pi>\<^sub>G) v \<noteq> None}"
transp: shows "transp (\<lambda>a b. cmp a b = Equiv)"
ln_ge_zero: fixes x :: "real" assumes "1 \<le> x" shows "0 \<le> real_ln x"
GB_adds_lt: fixes G :: "('t \<Rightarrow>\<^sub>0 'c) set"   and f :: "'t \<Rightarrow>\<^sub>0 'c" assumes "is_Groebner_basis G"   and "f \<in> pmdl G"   and "f \<noteq> 0" obtains g :: "'t \<Rightarrow>\<^sub>0 'c" where "g \<in> G" and "g \<noteq> 0" and "lt g adds\<^sub>t lt f"
pred_less_eq_le: fixes m :: "nat"   and n :: "nat" assumes "0 < m" shows "(m - Suc 0 < n) = (m \<le> n)"
flrestriction_vsubset_frel: fixes r :: "V"   and A :: "V" shows "r \<restriction>\<^sup>l\<^sub>\<bullet> A \<subseteq>\<^sub>\<circ> r"
perm_weak_filter: fixes P :: "'b list \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool"   and xs :: "'b list" assumes "\<forall>xs ys. xs <~~> ys \<longrightarrow> P xs = P ys"   and "\<forall>x xs. P xs \<longrightarrow> P (x # xs)"   and "P (filter Q xs)" shows "P xs"
pfa_get_def: shows "pfa_get \<equiv> arl_get"
dehomo_subst_def: fixes x :: "'b" shows "dehomo_subst x = (\<lambda>y. if y = x then 1 else monomial (1::'c) (monomial 1 y))"
quasi_inverse_antipar_1: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i> :: "'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and f :: "'b" assumes "bicategory V H \<a> \<i> src trg"   and "bicategory.equivalence_map V H \<a> \<i> src trg f" shows "src (bicategory.some_quasi_inverse V H \<a> \<i> src trg f) = trg f"
1: shows "h \<turnstile> filter_M P xs \<rightarrow>\<^sub>r ys"
F2_add_f: shows "Finca_add F2 f (rewrite_fundef_body fd2 pc instr) = F2'"
sel_rews_7: fixes a :: "'r"   and tree1 :: "'r tree"   and tree2 :: "'r tree" shows "label\<cdot>(Node\<cdot>a\<cdot>tree1\<cdot>tree2) = a"
Rep_range: shows "range Rep_bit1 = {0..<1 + 2 * int CARD('a)}"
s_1: shows "s \<in> trms\<^sub>l\<^sub>s\<^sub>s\<^sub>t (S \<cdot>\<^sub>l\<^sub>s\<^sub>s\<^sub>t \<delta>)"
nxt_Collapse_resp: fixes x :: "'a" shows "(\<lambda>q. Collapse.eq_right_lang `` {nxt Collapse_dfa q x}) respects Collapse.eq_right_lang"
is_extensional: fixes f :: "J.Arr" assumes "\<not> J.arr f" shows "\<chi> f = C.null"
strict_subtuple_trans''': fixes zm :: "'d \<rightharpoonup>\<^sub>f 'c"   and P :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and ym :: "'d \<rightharpoonup>\<^sub>f 'c"   and xm :: "'d \<rightharpoonup>\<^sub>f 'c" assumes "acyclicP_on (fmran' zm) P"   and "strict_subtuple (\<lambda>x y. P x y \<or> x = y) xm ym"   and "strict_subtuple (\<lambda>x y. P x y \<or> x = y)\<^sup>*\<^sup>* ym zm" shows "strict_subtuple (\<lambda>x y. P x y \<or> x = y)\<^sup>*\<^sup>* xm zm"
order_on_defs_1: fixes A :: "'a set"   and r :: "('a \<times> 'a) set" shows "preorder_on A r \<equiv> refl_on A r \<and> trans r"
quasi_measure_preserving_comp: fixes f :: "'b \<Rightarrow> 'c"   and M :: "'b measure"   and N :: "'c measure"   and g :: "'c \<Rightarrow> 'd"   and P :: "'d measure" assumes "f \<in> quasi_measure_preserving M N"   and "g \<in> quasi_measure_preserving N P" shows "g \<circ> f \<in> quasi_measure_preserving M P"
DirProd_iso_trans: fixes G :: "('a, 'b) monoid_scheme"   and G2 :: "('c, 'd) monoid_scheme"   and H :: "('e, 'f) monoid_scheme"   and I :: "('g, 'h) monoid_scheme" assumes "Group.group G"   and "G \<cong> G2"   and "H \<cong> I" shows "G \<times>\<times> H \<cong> G2 \<times>\<times> I"
setc: shows "(\<Sum>s\<in>set (map (consistent_sign_vec qs) (characterize_root_list_p p)) \<inter> {s. z_R I s = 1} \<union> set (map (consistent_sign_vec qs) (characterize_root_list_p p)) \<inter> {s. z_R I s = - 1}. z_R I s * rat_of_int (int (card {x. poly p x = 0 \<and> consistent_sign_vec qs x = s}))) + (\<Sum>s\<in>set (map (consistent_sign_vec qs) (characterize_root_list_p p)) \<inter> {s. z_R I s = 0}. z_R I s * rat_of_int (int (card {x. poly p x = 0 \<and> consistent_sign_vec qs x = s}))) = (\<Sum>s\<in>set (map (consistent_sign_vec qs) (characterize_root_list_p p)) \<inter> {s. z_R I s = 1}. z_R I s * rat_of_int (int (card {x. poly p x = 0 \<and> consistent_sign_vec qs x = s}))) + (\<Sum>s\<in>set (map (consistent_sign_vec qs) (characterize_root_list_p p)) \<inter> {s. z_R I s = - 1}. z_R I s * rat_of_int (int (card {x. poly p x = 0 \<and> consistent_sign_vec qs x = s}))) + (\<Sum>s\<in>set (map (consistent_sign_vec qs) (characterize_root_list_p p)) \<inter> {s. z_R I s = 0}. z_R I s * rat_of_int (int (card {x. poly p x = 0 \<and> consistent_sign_vec qs x = s})))"
iIN_eq_conv: fixes n :: "nat"   and d :: "nat"   and n' :: "nat"   and d' :: "nat" shows "([n\<dots>,d] = [n'\<dots>,d']) = (n = n' \<and> d = d')"
Di_def: fixes a :: "nat list"   and b :: "nat list"   and i :: "nat"   and y :: "nat list" shows "hlde_ops.Di a b i y = {dij a b i j - 1 |j. j < length y \<and> eij a b i j \<le> y ! j}"
cInf_singleton: fixes Inf :: "'c set \<Rightarrow> 'c"   and Sup :: "'c set \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and x :: "'c" assumes "class.conditionally_complete_lattice Inf Sup inf less_eq less sup" shows "Inf {x} = x"