hcju/isabelleps huggingface dataset:
homotopic_loops_linear: fixes g :: "real \<Rightarrow> 'b"   and h :: "real \<Rightarrow> 'b"   and S :: "'b set" assumes "path g"   and "path h"   and "pathfinish g = pathstart g"   and "pathfinish h = pathstart h"   and "\<And>t x. t \<in> {0..1} \<Longrightarrow> closed_segment (g t) (h t) \<subseteq> S" shows "homotopic_loops S g h"
wlp_whileI_modset: fixes I :: "(char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> bool"   and s\<^sub>0 :: "char list \<Rightarrow> int \<Rightarrow> int"   and \<pi> :: "char list \<Rightarrow> com option"   and c :: "com"   and b :: "bexp"   and Q :: "(char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> bool" assumes "I s\<^sub>0"   and "\<And>s. \<lbrakk>modifies (ANALYZE (lhsv \<pi> c)) s s\<^sub>0; I s; bval b s\<rbrakk> \<Longrightarrow> wlp \<pi> c I s"   and "\<And>s. \<lbrakk>modifies (ANALYZE (lhsv \<pi> c)) s s\<^sub>0; I s; \<not> bval b s\<rbrakk> \<Longrightarrow> Q s" shows "wlp \<pi> (WHILE b DO c) Q s\<^sub>0"
arr_Set_ArrVal_vdomain: shows "\<D>\<^sub>\<circ> (T\<lparr>ArrVal\<rparr>) = T\<lparr>ArrDom\<rparr>"
S_alt_def: shows "S ({(v, u)} \<union> A) = {r} \<union> fst ` ({(v, u)} \<union> A)"
dg_Hom_vifunion_in_Vset: fixes \<alpha> :: "V"   and \<CC> :: "V"   and A :: "V"   and B :: "V" assumes "digraph \<alpha> \<CC>"   and "A \<subseteq>\<^sub>\<circ> \<CC>\<lparr>Obj\<rparr>"   and "B \<subseteq>\<^sub>\<circ> \<CC>\<lparr>Obj\<rparr>"   and "A \<in>\<^sub>\<circ> Vset \<alpha>"   and "B \<in>\<^sub>\<circ> Vset \<alpha>" shows "\<Union>\<^sub>\<circ> ((\<lambda>a\<in>\<^sub>\<circ>A. \<Union>\<^sub>\<circ> (VLambda B (Hom \<CC> a) `\<^sub>\<circ> B)) `\<^sub>\<circ> A) \<in>\<^sub>\<circ> Vset \<alpha>"
corec1_def: fixes s1 :: "'d \<Rightarrow> ('e, 'e JF1 + 'd, 'e JF2 + 'f) F1"   and s2 :: "'f \<Rightarrow> ('e, 'e JF1 + 'd, 'e JF2 + 'f) F2" shows "corec1 s1 s2 = unfold1 (case_sum (F1map id Inl Inl \<circ> dtor1) s1) (case_sum (F2map id Inl Inl \<circ> dtor2) s2) \<circ> Inr"
f_carrier: fixes i :: "nat" assumes "i < length fs" shows "fs ! i \<in> carrier_vec n"
differing_vars_lists_def: fixes mem\<^sub>1 :: "'Var \<Rightarrow> 'Val"   and mem\<^sub>2 :: "'Var \<Rightarrow> 'Val"   and mems :: "(('Var \<Rightarrow> 'Val) \<times> ('Var \<Rightarrow> 'Val)) list"   and i :: "nat" shows "differing_vars_lists mem\<^sub>1 mem\<^sub>2 mems i \<equiv> differing_vars mem\<^sub>1 (fst (mems ! i)) \<union> differing_vars mem\<^sub>2 (snd (mems ! i))"
primorial'_Suc: fixes n :: "nat" shows "primorial' (Suc n) = nth_prime n * primorial' n"
PrefixSameOnLow: fixes l1 :: "'a list"   and l2 :: "'a list" assumes "prefixList l1 l2" shows "\<forall>index<length l1. l1 ! index = l2 ! index"
standard_decomp_geE: fixes X :: "'x set"   and ps :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list"   and T :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set"   and k :: "nat"   and d :: "nat" assumes "finite X"   and "valid_decomp X ps"   and "cone_decomp T ps"   and "standard_decomp k ps"   and "k \<le> d" obtains qs :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list" where "valid_decomp X qs" and "cone_decomp T qs" and "standard_decomp d qs" and "monomial_decomp ps \<Longrightarrow> monomial_decomp qs" and "hom_decomp ps \<Longrightarrow> hom_decomp qs"
forall_coeffs_conv: fixes P :: "'a \<Rightarrow> bool"   and p :: "'a poly" assumes "P (0::'a)" shows "(\<forall>n. P (coeff p n)) = (\<forall>c\<in>set (coeffs p). P c)"
bind_conv_join_option: fixes x :: "'d option"   and f :: "'d \<Rightarrow> 'c option" shows "x \<bind> f = join_option (map_option f x)"
dvd_imp_poly_roots_subset: fixes q :: "'a poly"   and p :: "'a poly" assumes "q \<noteq> 0"   and "p dvd q" shows "poly_roots p \<subseteq># poly_roots q"
H: shows "s @ [tick] \<in> {d. d \<noteq> [] \<and> hd d \<in> ev ` A \<and> (\<exists>a. ev a = hd d \<and> tl d \<in> D (P a))}"
enabledE: fixes F :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool"   and s :: "nat \<Rightarrow> 'a" assumes "(Enabled F) s" obtains u :: "nat \<Rightarrow> 'a" where "F (first s ## u)"
dist_decreases_open_segment: fixes x :: "'b"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "x \<in> open_segment a b" shows "dist c x < dist c a \<or> dist c x < dist c b"
len_of_finite_2_def: fixes x :: "Enum.finite_2 itself" shows "len_of x \<equiv> 2"
sig_red_singleD2: fixes sing_reg :: "'t \<Rightarrow> 't \<Rightarrow> bool"   and top_tail :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and p :: "'t \<Rightarrow>\<^sub>0 'b"   and q :: "'t \<Rightarrow>\<^sub>0 'b"   and f :: "'t \<Rightarrow>\<^sub>0 'b"   and t :: "'a" assumes "sig_red_single sing_reg top_tail p q f t" shows "t + punit.lt (rep_list f) \<in> keys (rep_list p)"
livenessI: fixes A :: "'b set"   and P :: "'b llist set" assumes "\<And>s. s \<in> A\<^sup>\<star> \<Longrightarrow> \<exists>t\<in>A\<^sup>\<infinity>. s @@ t \<in> P" shows "liveness A P"
vec_nth_inject: fixes x :: "('b, 'a) vec"   and y :: "('b, 'a) vec" shows "(($) x = ($) y) = (x = y)"
sum_UNIV_set_def: fixes f :: "nat \<Rightarrow> 'c"   and n :: "nat" shows "sum_UNIV_set f n = sum f {..<n}"
iso_finf: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "mono f"   and "mono g" shows "mono (f \<sqinter> g)"
VPow_vsingleton_iff: fixes X :: "V"   and Y :: "V" shows "(VPow X = set {Y}) = (X = 0 \<and> Y = 0)"
Blacken_Roots: shows "\<turnstile> Blacken_Roots  \<lbrace>\<acute>Proper \<and> Roots \<subseteq> Blacks \<acute>M\<rbrace>"
E'_to_E: fixes u :: "'v"   and v :: "'v" assumes "(u, v) \<in> g_E G'" shows "(fi u, fi v) \<in> E"
reindex_nontrivial: fixes A :: "'b set"   and h :: "'b \<Rightarrow> 'c"   and g :: "'c \<Rightarrow> 'a" assumes "finite A"   and "\<And>x y. \<lbrakk>x \<in> A; y \<in> A; x \<noteq> y; h x = h y\<rbrakk> \<Longrightarrow> g (h x) = (0::'a)" shows "sum g (h ` A) = sum (g \<circ> h) A"
S_le: fixes p :: "nat" assumes "prime p" shows "S n p \<le> \<lfloor>log (real p) (real (2 * n))\<rfloor>"
AlphaNegImpl: fixes A :: "('a, 'b) form"   and B :: "('a, 'b) form"   and G :: "('a, 'b) form list" assumes "\<stileturn> A # Neg B # G" shows "\<stileturn> Neg (Impl A B) # G"
br_def: fixes \<alpha> :: "'a \<Rightarrow> 'b"   and I :: "'a \<Rightarrow> bool" shows "br \<alpha> I \<equiv> {(c, a). a = \<alpha> c \<and> I c}"
conj_one_right: fixes x :: "'b" shows "x AND - (1::'b) = x"
eval_primfuns_nonzero: fixes fs :: "(primfun \<times> real) list" shows "\<forall>\<^sub>F x in at_top. eval_primfuns fs x \<noteq> 0"
v_def: shows "v = (\<lambda>n. inv_into X f (u n))"
map_2: fixes f1 :: "'d \<Rightarrow> 'b"   and f2 :: "'e \<Rightarrow> 'c"   and x2 :: "'e error_result" shows "map_result f1 f2 (Rerr x2) = Rerr (map_error_result f2 x2)"
awalk_Cons_iff: fixes a :: "'a \<times> nat \<times> 'a"   and u :: "'a"   and e :: "'a \<times> nat \<times> 'a"   and es :: "('a \<times> nat \<times> 'a) list"   and w :: "'a" shows "H.awalk a u (e # es) w = (e \<in> arcs (add_arc a) \<and> u = tail G e \<and> H.awalk a (head G e) es w)"
nu_star2: fixes x :: "'a" assumes "\<And>x. x\<^sup>\<star> \<cdot> x\<^sup>\<star> \<le> x\<^sup>\<star>" shows "\<nu> x\<^sup>\<star> \<cdot> (1\<^sub>\<sigma> + \<tau> x) \<le> x\<^sup>\<star>"
dghm_ObjMap_app_in_HomCod_Obj: fixes \<alpha> :: "V"   and \<AA> :: "V"   and \<BB> :: "V"   and \<FF> :: "V"   and a :: "V" assumes "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<BB>"   and "a \<in>\<^sub>\<circ> \<AA>\<lparr>Obj\<rparr>" shows "\<FF>\<lparr>ObjMap\<rparr>\<lparr>a\<rparr> \<in>\<^sub>\<circ> \<BB>\<lparr>Obj\<rparr>"
fmfilter_subset: fixes P :: "'c \<Rightarrow> bool"   and m :: "('c, 'd) fmap" shows "fmfilter P m \<subseteq>\<^sub>f m"
reachable_step: fixes a :: "'a"   and b :: "'a" assumes "reachable a"   and "E a b" shows "reachable b"
trancl_supteq: shows "{\<unrhd>}\<^sup>+ = {\<unrhd>}"
Det_def: shows "Det \<equiv> \<lambda>((a, c), b, d). a * d - b * c"
dist_less_imp_nth_equal: fixes f :: "'b fps"   and g :: "'b fps"   and i :: "nat"   and j :: "nat" assumes "dist f g < inverse (2 ^ i)"   and "j \<le> i" shows "f $ j = g $ j"
ucl_iff: fixes R :: "('a \<times> 'a set) set" shows "(\<forall>a A B. (a, A) \<in> R \<and> A \<subseteq> B \<longrightarrow> (a, B) \<in> R) = (R \<parallel> Multirelations.U = R)"
"3_1": fixes t :: "('d, 'c) Term.term"   and \<theta> :: "'c \<Rightarrow> ('d, 'c) Term.term" shows "fv\<^sub>s\<^sub>s\<^sub>t\<^sub>p (send\<langle>t\<rangle> \<cdot>\<^sub>s\<^sub>s\<^sub>t\<^sub>p \<theta>) = fv (t \<cdot> \<theta>)"
hm_isEmpty_rule': fixes m :: "'a \<Rightarrow> 'b option"   and l :: "('a \<times> 'b) list list"   and ht :: "('a, 'b) hashtable" shows "<is_hashmap' m l ht> hm_isEmpty ht <\<lambda>r. is_hashmap' m l ht * \<up> (r = (m = Map.empty))>"
fls_shift_eq_1_iff: fixes n :: "int"   and f :: "'b fls" shows "(fls_shift n f = 1) = (f = fls_X_intpow n)"
intro: fixes U\<^sub>1 :: "'a set"   and bot\<^sub>1 :: "'a"   and le\<^sub>1 :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and ls\<^sub>1 :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and U\<^sub>2 :: "'b set"   and bot\<^sub>2 :: "'b"   and le\<^sub>2 :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and ls\<^sub>2 :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "order_bot_ow U\<^sub>1 bot\<^sub>1 le\<^sub>1 ls\<^sub>1"   and "order_bot_ow U\<^sub>2 bot\<^sub>2 le\<^sub>2 ls\<^sub>2" shows "order_bot_pair_ow U\<^sub>1 bot\<^sub>1 le\<^sub>1 ls\<^sub>1 U\<^sub>2 bot\<^sub>2 le\<^sub>2 ls\<^sub>2"
setdist_attains_inf: fixes B :: "'b set"   and A :: "'b set" assumes "compact B"   and "B \<noteq> {}" obtains y :: "'b" where "y \<in> B" and "setdist A B = infdist y A"
odd_square_cong_4_int: fixes x :: "int" assumes "odd x" shows "[x\<^sup>2 = 1] (mod 4)"
comm_monoid_rng_of_frac: shows "Group.comm_monoid rec_monoid_rng_of_frac"
cond_precise: shows "\<forall>s. s \<in> wa_invar WC \<and> \<alpha>1 s \<in> wa_cond WB \<longrightarrow> s \<in> wa_cond WC"
Sup_supdistl_iso: fixes f :: "'b \<times> 'c \<Rightarrow> 'd" assumes "Sup \<circ> (`) f \<le> f \<circ> Sup_prod" shows "prod.mono f"
impl_of_ahm_empty: shows "impl_of ahm_empty_const = ArrayHashMap_Impl.ahm_empty ()"
comp_arr_mkPoint: fixes S :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<AA> :: "('c \<times> 'c) set"   and f :: "'b"   and t :: "'b" assumes "set_category S \<AA>"   and "partial_magma.arr S f"   and "t \<in> set_category_data.set S (set_category.some_img S \<AA>) (partial_magma.dom S f)" shows "S f (set_category.mkPoint S \<AA> (partial_magma.dom S f) t) = set_category.mkPoint S \<AA> (partial_magma.cod S f) (set_category_given_img.Fun S (set_category.some_img S \<AA>) f t)"
Diff_mono: fixes A :: "'b set"   and C :: "'b set"   and D :: "'b set"   and B :: "'b set" assumes "A \<subseteq> C"   and "D \<subseteq> B" shows "A - B \<subseteq> C - D"
wf_list_graph_iff_wf_graph: fixes G :: "'c list_graph" shows "wf_graph (list_graph_to_graph G) = wf_list_graph_axioms G"
zero_le_even_power: fixes 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 times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and n :: "nat"   and a :: "'b" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one"   and "even n" shows "less_eq zero (power.power one times a n)"
elims_3: fixes x :: "nat"   and xa :: "cell list \<times> cell list" assumes "\<not> inv_loop1_exit x xa" obtains n :: "nat" and l :: "cell list" and r :: "cell list" where "x = n" and "xa = (l, r)" and "\<not> (0 < n \<and> (l, r) = ([], Bk # Oc # Bk \<up> n @ Oc \<up> n))"
startsby_zero_power_prefix: fixes a :: "'a fps"   and k :: "nat" assumes "fps_nth a 0 = (0::'a)" shows "\<forall>n<k. fps_nth (a ^ k) n = (0::'a)"
eqset: shows "D = (\<Union>b\<in>{y \<in> Y ` space M \<inter> space N. P (img_dce M Y X y)}. Y -` {b}) \<inter> space M"
tpOfV_getVars_nth: fixes i :: "nat"   and \<sigma>l :: "'tp list" assumes "i < length \<sigma>l" shows "tpOfV (getVars \<sigma>l ! i) = \<sigma>l ! i"
u_gt_s_if_ui: assumes "is_App u"   and "chop u \<ge>\<^sub>t t" shows "u >\<^sub>t s"
method_exit_unique: fixes n :: "'node"   and n' :: "'node" assumes "method_exit n"   and "method_exit n'"   and "get_proc n = get_proc n'" shows "n = n'"
leg0_composite: shows "HH\<mu>\<nu>\<pi>.leg0 = \<pi>.leg0 \<cdot> \<mu>\<nu>_\<pi>.prj\<^sub>0"
list_of_map_def: fixes f :: "nat \<Rightarrow> 'c option" shows "list_of_map f = (if f = Map.empty then [] else map (the \<circ> f) [0..<Suc (GREATEST x. x \<in> dom f)])"
list_assn_len: fixes A :: "'b \<Rightarrow> 'c \<Rightarrow> assn"   and xs :: "'b list"   and ys :: "'c list"   and h :: "heap \<times> nat set" assumes "h \<Turnstile> list_assn A xs ys" shows "length xs = length ys"
cong: fixes succs :: "'c \<Rightarrow> 'c list" shows "DFS.reachable succs \<equiv> DFS.reachable succs"
unlift_equiv: fixes x :: "'a itrm"   and y :: "'a itrm" assumes "x \<simeq> y" shows "unlift x \<leftrightarrow> unlift y"
entries_rbtreeify_g: fixes n :: "nat"   and kvs :: "('a \<times> 'b) list" assumes "n \<le> Suc (length kvs)" shows "entries (fst (rbtreeify_g n kvs)) = take (n - 1) kvs"
src_trg: fixes \<mu> :: "'d" assumes "D.arr \<mu>" shows "src\<^sub>D (trg\<^sub>D \<mu>) = trg\<^sub>D \<mu>"
continuous_on_Icc_at_leftD: fixes a :: "'c"   and b :: "'c"   and f :: "'c \<Rightarrow> 'd" assumes "continuous_on {a..b} f"   and "a < b" shows "(f \<longlongrightarrow> f b) (at_left b)"
\<xi>: shows "[\<lparr>A!,b\<^sup>P\<rparr> \<^bold>& (\<^bold>\<lambda>z. \<lparr>R,a\<^sup>P\<rparr>) \<^bold>= (\<^bold>\<lambda>z. \<lparr>R,b\<^sup>P\<rparr>) \<^bold>& \<^bold>\<not>\<lbrace>b\<^sup>P,\<^bold>\<lambda>z. \<lparr>R,a\<^sup>P\<rparr>\<rbrace> in v]"
coalg_def: fixes B1 :: "'d set"   and B2 :: "'e set"   and s1 :: "'d \<Rightarrow> ('f, 'd, 'e) F1"   and s2 :: "'e \<Rightarrow> ('f, 'd, 'e) F2" shows "coalg B1 B2 s1 s2 = ((\<forall>a\<in>B1. s1 a \<in> F1in UNIV B1 B2) \<and> (\<forall>a\<in>B2. s2 a \<in> F2in UNIV B1 B2))"
Pi_mem: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and B :: "'c \<Rightarrow> 'd set"   and x :: "'c" assumes "f \<in> Pi A B"   and "x \<in> A" shows "f x \<in> B x"
map\<^sub>0_simp: fixes a :: "'c" assumes "C.obj a" shows "FG.map\<^sub>0 a = F.map\<^sub>0 (G.map\<^sub>0 a)"
lens_equiv'_def: fixes X :: "'a \<Longrightarrow> 'b"   and Y :: "'c \<Longrightarrow> 'b" shows "X \<approx>\<^sub>L' Y = (\<forall>s\<^sub>1 s\<^sub>2. s\<^sub>1 \<triangleleft>\<^bsub>X\<^esub> s\<^sub>2 = s\<^sub>1 \<triangleleft>\<^bsub>Y\<^esub> s\<^sub>2)"
hide_rules_2: fixes c :: "'b \<Rightarrow> 'c"   and B :: "'c set"   and a :: "'b"   and P :: "'c process" assumes "\<forall>y. c y \<notin> B" shows "(c`!`a \<rightarrow> P) \ B = (c`!`a \<rightarrow> P \ B)"
change_on_either_hyp_change: fixes i :: "nat"   and t :: "nat" assumes "\<not> change_on_neither (block i t) (gap i t)" shows "\<sigma> i (\<tau> i \<triangleright> (t + 1)) \<noteq> \<sigma> i (\<tau> i \<triangleright> (gap i t - 1))"
check_tableau_normalized: fixes s :: "('i, 'a) state" assumes "\<not> \<U> s"   and "\<Turnstile>\<^sub>n\<^sub>o\<^sub>l\<^sub>h\<^sub>s s"   and "\<diamond> s"   and "\<triangle> (\<T> s)"   and "\<nabla> s" shows "\<triangle> (\<T> (check s))"
det_four_block_mat_lower_right_id: fixes A :: "'a mat"   and m :: "nat"   and B :: "'a mat"   and n :: "nat"   and C :: "'a mat"   and D :: "'a mat" assumes "A \<in> carrier_mat m m"   and "B = 0\<^sub>m m (n - m)"   and "C = 0\<^sub>m (n - m) m"   and "D = 1\<^sub>m (n - m)"   and "m < n" shows "Determinant.det (four_block_mat A B C D) = Determinant.det A"
toFunction_def: fixes R :: "('b \<times> 'c) set" shows "toFunction R = (,,) R"
assms_2: shows "Bisimulation_Invariants B C sim2 PB QB PC QC"
SepSubset: fixes A :: "ZF"   and p :: "ZF \<Rightarrow> bool" shows "HOLZF.subset (Sep A p) A"
l_one: fixes G :: "('a, 'b) monoid_scheme"   and x :: "'a" assumes "Group.monoid G"   and "x \<in> carrier G" shows "\<one>\<^bsub>G\<^esub> \<otimes>\<^bsub>G\<^esub> x = x"
ntrp_mon_increasing_s: fixes s :: "'n list"   and c :: "'n list multiset"   and e :: "('p, 'ba) label list el_step"   and s' :: "'n list"   and c' :: "'n list multiset" assumes "((s, c), e, s', c') \<in> ntrp fg" shows "mon_s fg s \<subseteq> mon_s fg s' \<and> mon_c fg c \<subseteq> mon_c fg c'"
weakTransitionAlpha: fixes P :: "pi"   and a :: "name"   and x :: "name"   and P' :: "pi"   and y :: "name" assumes "P \<Longrightarrow>a<\<nu>x> \<prec> P'"   and "y \<sharp> P" shows "P \<Longrightarrow>a<\<nu>y> \<prec> [(x, y)] \<bullet> P'"
simps_3: fixes x2 :: "'x" shows "Some x2 \<noteq> None"
Trie_Mapping_def: fixes t :: "('b, 'c) trie" shows "Trie_Mapping t = Mapping (lookup t)"
isomorphic_unit_left: fixes f :: "'a" assumes "C.ide f" shows "trg\<^sub>C f \<star>\<^sub>C f \<cong>\<^sub>C f"
sub_dist: fixes s :: "'s" shows "sum (p s) (supp (p s)) \<le> 1"
list_slice_Nil: fixes k :: "nat" shows "list_slice [] k = []"
interchange: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<nu> :: "'b"   and \<mu> :: "'b"   and \<tau> :: "'b"   and \<sigma> :: "'b" assumes "weak_composition V H"   and "partial_magma.arr V (V \<nu> \<mu>)"   and "partial_magma.arr V (V \<tau> \<sigma>)" shows "H (V \<nu> \<mu>) (V \<tau> \<sigma>) = V (H \<nu> \<tau>) (H \<mu> \<sigma>)"
disc_inter_disc_compl: fixes H :: "ocircline" shows "disc H \<inter> disc_compl H = {}"
positions_in_row_unique: fixes w :: "('b \<times> bool list) list"   and i :: "nat" assumes "\<exists>!p. p < length w \<and> snd (w ! p) ! i" shows "the_elem (positions_in_row w i) = (THE p. p < length w \<and> snd (w ! p) ! i)"
finite_symvars_imp_ex_greatest_symvar: fixes SV :: "('c \<times> nat) set"   and v :: "'c" assumes "finite SV"   and "\<exists>sv\<in>SV. fst sv = v" shows "\<exists>sv\<in>{sv \<in> SV. fst sv = v}. \<forall>sv'\<in>{sv \<in> SV. fst sv = v}. snd sv' \<le> snd sv"
absorb_iff1: fixes a :: "'c multiset"   and b :: "'c multiset" shows "(a \<subseteq># b) = (a \<inter># b = a)"
gso_cong: fixes x :: "nat"   and f1 :: "'a vec list"   and f2 :: "'a vec list"   and n :: "nat" assumes "\<And>i. i \<le> x \<Longrightarrow> f1 ! i = f2 ! i" shows "gram_schmidt_fs.gso n f1 x = gram_schmidt_fs.gso n f2 x"
usolves_odeD_4: fixes y :: "real \<Rightarrow> 'b"   and f :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and t0 :: "real"   and T :: "real set"   and X :: "'b set"   and T' :: "real set"   and z :: "real \<Rightarrow> 'b"   and t :: "real" assumes "(y usolves_ode f from t0) T X"   and "t0 \<in> T'"   and "is_interval T'"   and "T' \<subseteq> T"   and "(z solves_ode f) T' X"   and "z t0 = y t0"   and "t \<in> T'" shows "z t = y t"
eq1: shows "(2 \<cdot>\<^sub>m proj_psi - 1\<^sub>m N) *\<^sub>v psi'_l l = 2 \<cdot>\<^sub>m proj_psi *\<^sub>v psi'_l l - psi'_l l"
of_real_eq_0_iff: fixes x :: "real" shows "(of_real x = (0::'a)) = (x = 0)"
qty_single_updated: fixes Mkt :: "('b, 'c) discrete_market"   and asset :: "'c"   and qty :: "nat \<Rightarrow> 'b \<Rightarrow> real"   and n :: "nat"   and w :: "'b" shows "cls_val_process Mkt (qty_single asset qty) (Suc n) w = prices Mkt asset (Suc n) w * qty (Suc n) w"
merge_respects_hashes: fixes h :: "'a \<Rightarrow> 'b"   and bo :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and m :: "'a \<Rightarrow> 'a \<Rightarrow> 'a option"   and a :: "'a"   and b :: "'a" assumes "merkle_interface h bo m" shows "(h a = h b) = (\<exists>ab. m a b = Some ab)"
alg_mult_eq_count_roots: fixes p :: "'a list list" assumes "p \<in> carrier (poly_ring (poly_ring R))" shows "UP.alg_mult p = count (UP.roots p)"
trcl_rev_pair_induct: fixes xc1 :: "'a"   and xc2 :: "'b"   and xb :: "'c list"   and xa1 :: "'a"   and xa2 :: "'b"   and S :: "(('a \<times> 'b) \<times> 'c \<times> 'a \<times> 'b) set"   and P :: "'a \<Rightarrow> 'b \<Rightarrow> 'c list \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> bool" assumes "((xc1, xc2), xb, xa1, xa2) \<in> trcl S"   and "\<And>a b. P a b [] a b"   and "\<And>a b w aa ba e ab bb. \<lbrakk>((a, b), w, aa, ba) \<in> trcl S; ((aa, ba), e, ab, bb) \<in> S; P a b w aa ba\<rbrakk> \<Longrightarrow> P a b (w @ [e]) ab bb" shows "P xc1 xc2 xb xa1 xa2"
length_Cons: fixes x :: "'b"   and xs :: "'b list" shows "length (x # xs) = Suc (length xs)"
S_SN: shows "SN {(x, y). fst (kbo x y)}"
nodes_alt_def: fixes A :: "('e, 'f, 'd, 'g) FSM_scheme" shows "nodes A = \<Union> (reachable A ` {p. p = initial A})"
conjugate_sprod_vec: fixes v :: "'a vec"   and n :: "nat"   and w :: "'a vec" assumes "v \<in> carrier_vec n"   and "w \<in> carrier_vec n" shows "conjugate (v \<bullet> w) = conjugate v \<bullet>c w"
adjunction_data_in_bicategory_def: fixes V :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and H :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<a> :: "'c \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<i> :: "'c \<Rightarrow> 'c"   and src :: "'c \<Rightarrow> 'c"   and trg :: "'c \<Rightarrow> 'c"   and f :: "'c"   and g :: "'c"   and \<eta> :: "'c"   and \<epsilon> :: "'c" shows "adjunction_data_in_bicategory V H \<a> \<i> src trg f g \<eta> \<epsilon> \<equiv> bicategory V H \<a> \<i> src trg \<and> adjunction_data_in_bicategory_axioms V H src f g \<eta> \<epsilon>"
P2: shows "Bet C' B' D' \<and> Cong B' D' B' C'"
p_def: shows "p \<equiv> \<lambda>i. if i = k then 1 else 0"
weakTransition_def: fixes \<Psi> :: "'b"   and Q :: "('a, 'b, 'c) psi"   and P :: "('a, 'b, 'c) psi"   and \<alpha> :: "'a action"   and P' :: "('a, 'b, 'c) psi" shows "\<Psi> : Q \<rhd> P \<Longrightarrow>\<alpha> \<prec> P' \<equiv> \<exists>P''. \<Psi> \<rhd> P \<Longrightarrow>\<^sup>^\<^sub>\<tau> P'' \<and> insertAssertion (extractFrame Q) \<Psi> \<hookrightarrow>\<^sub>F insertAssertion (extractFrame P'') \<Psi> \<and> \<Psi> \<rhd> P'' \<longmapsto> \<alpha> \<prec> P'"
resr_comp1: fixes y :: "'a"   and x :: "'a" shows "y \<cdot> (y \<rightarrow> y \<cdot> x) = y \<cdot> x"
Eq_def: shows "(EQ) \<equiv> (id ---> id ---> abs_fm) Eq_raw"
crel_vs_def: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and v :: "'b"   and s :: "(heap, 'c) state" shows "state_dp_consistency.crel_vs R v s \<equiv> \<forall>M. state_dp_consistency.cmem M \<and> P M \<longrightarrow> (case run_state s M of (v', M') \<Rightarrow> R v v' \<and> state_dp_consistency.cmem M' \<and> P M')"
15: fixes x :: "'a" shows "x \<squnion> - x = top"
simps_9: fixes F :: "'b formula"   and G :: "'b formula" shows "nnf (F \<^bold>\<and> G) = nnf F \<^bold>\<and> nnf G"
no_multi: fixes G :: "('a, 'b) graph"   and v :: "'a"   and w :: "'b"   and u :: "'a"   and w' :: "'b" assumes "valid_unSimpGraph G"   and "(v, w, u) \<in> edges G"   and "(v, w', u) \<in> edges G" shows "w = w'"
dropWhile_eq_Cons_conv: fixes P :: "'g \<Rightarrow> bool"   and xs :: "'g list"   and y :: "'g"   and ys :: "'g list" shows "(dropWhile P xs = y # ys) = (xs = takeWhile P xs @ y # ys \<and> \<not> P y)"
exhaust_sel: fixes pred :: "'a pred" obtains "pred = Pred (eval pred)"
good_ruleset_append: fixes rs\<^sub>1 :: "'c rule list"   and rs\<^sub>2 :: "'c rule list" shows "good_ruleset (rs\<^sub>1 @ rs\<^sub>2) = (good_ruleset rs\<^sub>1 \<and> good_ruleset rs\<^sub>2)"
generate_field_in_carrier: fixes H :: "'a set"   and h :: "'a" assumes "H \<subseteq> carrier R"   and "h \<in> generate_field R H" shows "h \<in> carrier R"
rep_eq: fixes x :: "'a Filters.filter"   and xa :: "'a Filters.filter" shows "Filters.filter.Rep_filter (x \<squnion> xa) = filter_sup (Filters.filter.Rep_filter x) (Filters.filter.Rep_filter xa)"
shift1_invI: fixes B :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) set"   and qs :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list" assumes "B = {q \<in> set qs. poly_deg (fst q) = d \<and> m < card (snd q)}"   and "shift2_inv qs" shows "shift1_inv (qs, B)"
gcd_divides_l: fixes a :: "'a"   and b :: "'a" assumes "a \<in> carrier G"   and "b \<in> carrier G" shows "somegcd G a b divides a"
qd: shows "discrim (p \<bullet> (M *v p)) (2 * (p \<bullet> (M *v q))) (q \<bullet> (M *v q)) = 4 * quarter_discrim p q"