hcju/isabelleps huggingface dataset:
scc_root_of_node_exists: fixes v :: "'a" assumes "v \<in> dom (discovered s)" shows "\<exists>r. scc_root s r (scc_of E v)"
testN_post2: fixes k :: "nat" assumes "\<not> f k" shows "adjoint (testN k) * post * testN k = 0\<^sub>m N N"
1: shows "\<Y> (\<Z> k u) (\<P> k u) k \<subseteq> \<W> (\<P> k u) k"
is_span: shows "C.span \<nu>\<mu>.cod.leg0 \<nu>\<mu>.cod.leg1"
\<phi>_uniqueness_2: shows "local.iso r\<^sub>0s\<^sub>1.\<phi>"
cblinfun_plus_image_distr: fixes A :: "'b \<Rightarrow>\<^sub>C\<^sub>L 'a"   and B :: "'b \<Rightarrow>\<^sub>C\<^sub>L 'a"   and S :: "'b ccsubspace" shows "(A + B) *\<^sub>S S \<le> A *\<^sub>S S \<squnion> B *\<^sub>S S"
assms_3: shows "rel_interior T \<inter> rel_interior U \<noteq> {}"
eq1: shows "\<b> ps (Suc (card X)) = \<a> ps"
disjoint_family_on_def: fixes A :: "'c \<Rightarrow> 'd set"   and S :: "'c set" shows "disjoint_family_on A S = (\<forall>m\<in>S. \<forall>n\<in>S. m \<noteq> n \<longrightarrow> A m \<inter> A n = {})"
lcm_unique: fixes a :: "'a"   and d :: "'a"   and b :: "'a" shows "(a dvd d \<and> b dvd d \<and> normalize d = d \<and> (\<forall>e. a dvd e \<and> b dvd e \<longrightarrow> d dvd e)) = (d = lcm a b)"
const_res_subseq_prop_1: fixes s :: "nat \<Rightarrow> nat \<Rightarrow> int"   and k :: "nat" assumes "s \<in> carrier Zp\<^bsup>\<omega>\<^esup>" shows "\<forall>m. const_res_subseq k s m k = const_res k s"
snd_apfst: fixes f :: "'f \<Rightarrow> 'e"   and x :: "'f \<times> 'd" shows "snd (apfst f x) = snd x"
bexI: fixes P :: "'g \<Rightarrow> bool"   and x :: "'g"   and A :: "'g set" assumes "P x"   and "x \<in> A" shows "\<exists>x\<in>A. P x"
simps_13: fixes f :: "'d \<Rightarrow> 'c"   and x21 :: "'d"   and x22 :: "'d llist" shows "lmap f (LCons x21 x22) = LCons (f x21) (lmap f x22)"
risk_free_asset_martingale: fixes N :: "bool stream measure"   and q :: "real" assumes "N = bernoulli_stream q"   and "0 < q"   and "q < 1" shows "martingale N G (discounted_value r (prices Mkt risk_free_asset))"
lunit_agreement: fixes a :: "'c local.term set" assumes "ide a" shows "lunit a = \<l>[a]"
pq: shows "map_spmf fst pq = pair_spmf p q"
matching_srcs_pull_out_accu: fixes d :: "'a word"   and rs :: "'a simple_rule list"   and a1 :: "'a wordinterval"   and a2 :: "'a wordinterval" shows "wordinterval_to_set (matching_srcs d rs (wordinterval_union a1 a2)) = wordinterval_to_set (matching_srcs d rs a2) - wordinterval_to_set a1"
safe_max_def: fixes X :: "'a set" shows "safe_max X = (if X = {} then None else Some (Max X))"
SHORun_def: fixes A :: "('a, 'b, 'c, 'd) CHOAlgorithm_scheme"   and rho :: "nat \<Rightarrow> 'a \<Rightarrow> 'b"   and HOs :: "nat \<Rightarrow> 'a \<Rightarrow> 'a set"   and SHOs :: "nat \<Rightarrow> 'a \<Rightarrow> 'a set" shows "SHORun A rho HOs SHOs \<equiv> CSHORun A rho HOs SHOs (\<lambda>r q. undefined)"
simps: fixes l :: "'b sort_key extended"   and u :: "'b sort_key extended" shows "derive_left (l, u) = (derive_pos (\<gamma> l sort_dir.Right) l, derive_pos (\<gamma> l sort_dir.Right) u)"
select_append_1: fixes ys :: "'b list"   and zs :: "'b list"   and k :: "nat" assumes "\<And>y z. \<lbrakk>y \<in> set ys; z \<in> set zs\<rbrakk> \<Longrightarrow> y \<le> z"   and "k < length ys" shows "select k (ys @ zs) = select k ys"
closed_Times: fixes S :: "'a set"   and T :: "'b set" assumes "closed S"   and "closed T" shows "closed (S \<times> T)"
hid_is_CSP: fixes P :: "('c, 'd) action"   and cs :: "'c set" shows "is_CSP_process (hid P cs)"
eint_ord_number_2: fixes m :: "num"   and n :: "num" shows "(eint (numeral m) < eint (numeral n)) = (numeral m < numeral n)"
idiff_enat_enat: fixes a :: "nat"   and b :: "nat" shows "enat a - enat b = enat (a - b)"
cycle_enum_cycle: fixes n :: "nat" assumes "cycles \<Delta> \<noteq> {}" shows "cycle \<Delta> (enum_cycle n)"
s1: shows "0 \<le> real_of_int (root_int_floor_pos p x)"
obeta: fixes x :: "name"   and t :: "lam"   and t' :: "lam" shows "App (Lam [x].t) t' \<longrightarrow>o t[x::=t']"
inv_comm: fixes x :: "nat \<Rightarrow> int"   and y :: "nat \<Rightarrow> int" assumes "x \<oplus>\<^bsub>UP INTEG\<^esub> y = \<zero>\<^bsub>UP INTEG\<^esub>"   and "x \<in> carrier (UP INTEG)"   and "y \<in> carrier (UP INTEG)" shows "y \<oplus>\<^bsub>UP INTEG\<^esub> x = \<zero>\<^bsub>UP INTEG\<^esub>"
valid_K_cfg: fixes K :: "'a \<Rightarrow> 'a pmf set"   and S :: "'a set"   and cfg :: "'a cfg"   and cfg' :: "'a cfg" assumes "Finite_Markov_Decision_Process K S"   and "cfg \<in> Finite_Markov_Decision_Process.valid_cfg K S"   and "cfg' \<in> set_pmf (K_cfg cfg)" shows "cfg' \<in> Finite_Markov_Decision_Process.valid_cfg K S"
exp_ge_one_plus_x_over_n_power_n: fixes n :: "nat"   and x :: "real" assumes "- real n \<le> x"   and "0 < n" shows "(1 + x / real n) ^ n \<le> exp x"
times_divide_eq_right: fixes R :: "('b, 'c) ring_scheme"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "field R"   and "a \<in> carrier R"   and "b \<in> carrier R"   and "c \<in> carrier R"   and "c \<noteq> \<zero>\<^bsub>R\<^esub>" shows "a \<otimes>\<^bsub>R\<^esub> (b \<oslash>\<^bsub>R\<^esub> c) = a \<otimes>\<^bsub>R\<^esub> b \<oslash>\<^bsub>R\<^esub> c"
override_on_insert: fixes f :: "'d \<Rightarrow> 'e"   and g :: "'d \<Rightarrow> 'e"   and x :: "'d"   and X :: "'d set" shows "override_on f g (insert x X) = (override_on f g X)(x := g x)"
g_is_pr: shows "(\<lambda>x. a) \<in> PrimRec1"
inject_1: fixes x11 :: "'a"   and x12 :: "'b"   and y11 :: "'a"   and y12 :: "'b" shows "(Tip x11 x12 = Tip y11 y12) = (x11 = y11 \<and> x12 = y12)"
hyps: fixes i :: "nat"   and m :: "nat \<Rightarrow> nat" assumes "i < length ts"   and "ts ! i = (p, is, \<theta>, sb, \<D>, \<O>, \<R>)" shows "flush_all_until_volatile_write (ts[i := (p', is', \<theta>, sb @ [Write\<^sub>s\<^sub>b True a sop v A L R W], \<D>', \<O>, \<R>')]) m = flush_all_until_volatile_write ts m"
mset_quicksort: fixes R :: "('a \<times> 'a) set"   and xs :: "'a list" shows "mset (quicksort R xs) = mset xs"
1: shows "t\<^sup>+ ; e \<le> - 1'"
in_lots_2: shows "mix_pmf \<alpha> q r \<in> \<P> outcomes"
imirror_cut_le: fixes I :: "nat set"   and t :: "nat" assumes "finite I"   and "t \<le> iMin I + Max I" shows "imirror I \<down>\<le> mirror_elem t I = imirror_bounds (I \<down>\<ge> t) (iMin I) (Max I)"
differentiable_difg: fixes m :: "nat"   and x :: "real" assumes "m < n"   and "0 \<le> x"   and "x \<le> h" shows "difg m differentiable at x"
rel_mono_strong: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'a list"   and y :: "'b list"   and Ra :: "'a \<Rightarrow> 'b \<Rightarrow> bool" assumes "list_all2 R x y"   and "\<And>z yb. \<lbrakk>z \<in> set x; yb \<in> set y; R z yb\<rbrakk> \<Longrightarrow> Ra z yb" shows "list_all2 Ra x y"
strict_implies_not_eq: fixes b :: "'d"   and a :: "'d" assumes "b < a" shows "a \<noteq> b"
eq_fin_alt_def: fixes w\<^sub>1 :: "'action list"   and w\<^sub>2 :: "'action list" shows "(w\<^sub>1 =\<^sub>F w\<^sub>2) = (w\<^sub>1 \<preceq>\<^sub>F w\<^sub>2 \<and> w\<^sub>2 \<preceq>\<^sub>F w\<^sub>1)"
db\<^sub>s\<^sub>s\<^sub>t_def: fixes S :: "('c, 'd) stateful_strand_step list"   and I :: "'d \<Rightarrow> ('c, 'd) Term.term" shows "db\<^sub>s\<^sub>s\<^sub>t S I \<equiv> db'\<^sub>s\<^sub>s\<^sub>t S I []"
index_one_mat_div_mod: fixes i :: "nat"   and m :: "nat"   and n :: "nat"   and j :: "nat" assumes "i < 2 ^ (m + n)"   and "j < 2 ^ (m + n)" shows "1\<^sub>m (2 ^ m) $$ (i div 2 ^ n, j div 2 ^ n) * 1\<^sub>m (2 ^ n) $$ (i mod 2 ^ n, j mod 2 ^ n) = 1\<^sub>m (2 ^ (m + n)) $$ (i, j)"
foldseq_zerostart2: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and n :: "nat"   and s :: "nat \<Rightarrow> 'b" assumes "\<forall>x. f (0::'b) x = x"   and "\<forall>i<n. s i = (0::'b)" shows "foldseq f s n = s n"
xs_2: assumes "n \<noteq> 0 \<or> f \<noteq> 0" shows "\<not> monom 1 1 dvd g"
intros_1: fixes a :: "'b set"   and A :: "'b set set"   and sp :: "'b set" assumes "a \<in> A" shows "a \<in> sigma_sets sp A"
simps_15: fixes x21 :: "'g"   and x22 :: "'g list" shows "set (x21 # x22) = insert x21 (set x22)"
simps: fixes binop :: "'b ltln \<Rightarrow> 'b ltln \<Rightarrow> 'b ltln"   and \<phi> :: "'b ltln"   and i :: "enat"   and \<psi> :: "'b ltln"   and j :: "enat" shows "combine binop (\<phi>, i) (\<psi>, j) = (let \<chi> = binop (X\<^sub>n' (the_enat_0 (i - j)) \<phi>) (X\<^sub>n' (the_enat_0 (j - i)) \<psi>) in (\<chi>, if is_constant \<chi> then \<infinity> else min i j))"
CFG_SSA_axioms_def: fixes \<alpha>n :: "'d \<Rightarrow> 'e list"   and inEdges' :: "'d \<Rightarrow> 'e \<Rightarrow> ('e \<times> 'f) list"   and defs :: "'d \<Rightarrow> 'e \<Rightarrow> 'g set"   and phis :: "'d \<Rightarrow> 'e \<times> 'g \<Rightarrow> 'g list option" shows "CFG_SSA_axioms \<alpha>n inEdges' defs phis \<equiv> ((\<forall>g. finite (dom (phis g))) \<and> (\<forall>g n v vs. phis g (n, v) = Some vs \<longrightarrow> n \<in> set (\<alpha>n g))) \<and> (\<forall>g n v args. phis g (n, v) = Some args \<longrightarrow> length (graph_path_base.predecessors inEdges' g n) = length args) \<and> (\<forall>n g. n \<in> set (\<alpha>n g) \<longrightarrow> defs g n \<inter> CFG_SSA_base.phiDefs phis g n = {}) \<and> (\<forall>n g m. n \<in> set (\<alpha>n g) \<longrightarrow> m \<in> set (\<alpha>n g) \<longrightarrow> n \<noteq> m \<longrightarrow> CFG_SSA_base.allDefs defs phis g n \<inter> CFG_SSA_base.allDefs defs phis g m = {})"
assms_1: shows "A \<Turnstile> (S, U \<rightarrow>) onl \<Gamma> P"
A: shows "inverse c * (ln x - (\<Sum>k<n. 2 * y ^ (2 * k + 1) / real (2 * k + 1))) \<in> {0..1 / (1 - y\<^sup>2) / real (2 * n + 1)}"
hyps_2: shows "xs @ [x] = cs\<^bsup>\<pi>\<^esup> n"
spaceN_diff: fixes x :: "'b"   and N :: "'b quasinorm"   and y :: "'b" assumes "x \<in> space\<^sub>N N"   and "y \<in> space\<^sub>N N" shows "x - y \<in> space\<^sub>N N"
irreflexive_inf_arc_asymmetric: fixes x :: "'a" assumes "x \<sqinter> top = bot"   and "(x \<sqinter> top \<sqinter> id (x \<sqinter> top) \<le> top \<and> top \<sqinter> (x \<sqinter> top) = top) \<and> id x \<sqinter> top \<sqinter> id (id x \<sqinter> top) \<le> top \<and> top \<sqinter> (id x \<sqinter> top) = top" shows "x \<sqinter> id x = bot"
induct: fixes S :: "Vars.var set"   and G :: "CoCalls"   and x1 :: "Vars.var set"   and x2 :: "Vars.var list"   and P :: "Vars.var set \<Rightarrow> Vars.var list \<Rightarrow> bool" assumes "valid_lists' S G x1 x2"   and "\<And>prefix. P prefix []"   and "\<And>prefix x xs. \<lbrakk>prefix \<subseteq> ccNeighbors x G; valid_lists' S G (insert x prefix) xs; P (insert x prefix) xs; x \<in> S\<rbrakk> \<Longrightarrow> P prefix (x # xs)" shows "P x1 x2"
less_linear: fixes x :: "complex"   and y :: "complex" shows "x <\<^sub>\<complex> y \<or> x = y \<or> y <\<^sub>\<complex> x"
pcr_cr_eq: shows "pcr_ell2 (=) = cr_ell2"
parallel_reduction_redexE: fixes x :: "'a"   and T :: "PreSimplyTyped.type"   and A :: "'a trm"   and B :: "'a trm"   and X :: "'a trm" assumes "App (Fn x T A) B >> X" shows "X = App (Fn x T A) B \<or> (\<exists>A' B'. A >> A' \<and> B >> B' \<and> X = A'[x ::= B']) \<or> (\<exists>A' B'. Fn x T A >> Fn x T A' \<and> B >> B' \<and> X = App (Fn x T A') B')"
expand_combine_infmatrix: fixes f :: "'c \<Rightarrow> 'd \<Rightarrow> 'b"   and A :: "nat \<Rightarrow> nat \<Rightarrow> 'c"   and B :: "nat \<Rightarrow> nat \<Rightarrow> 'd"   and j :: "nat"   and i :: "nat" shows "combine_infmatrix f A B j i = f (A j i) (B j i)"
insert_subset: fixes x :: "'b"   and A :: "'b set"   and B :: "'b set" shows "(x \<triangleright> A \<subseteq> B) = (x \<in> B \<and> A \<subseteq> B)"
degree_m_smult_le: fixes c :: "int"   and f :: "int poly" shows "degree_m (Polynomial.smult c f) \<le> degree_m f"
cases: fixes x :: "'a \<times> ('b \<times> 'c) list \<times> 'a" obtains ko :: "'a" and xs :: "('b \<times> 'c) list" and ox :: "'a" where "x = (ko, xs, ox)"
weight_spmf'_def: fixes p :: "'a spmf" shows "weight_spmf' p = weight_spmf p"
VA': shows "Vagree \<nu> \<omega> (\<Union>i\<in>{i. Inr i \<in> SIGT \<theta>}. FVT (\<sigma> i))"
g_mc: shows "(GREATEST n. \<not> is_zero_row_upt_k n k A) + (1::'rows) \<le> mc"
subst_cls_mono_mset: fixes C :: "'a literal multiset"   and D :: "'a literal multiset"   and \<sigma> :: "'s" assumes "C \<subseteq># D" shows "C \<cdot> \<sigma> \<subseteq># D \<cdot> \<sigma>"
restrict_map_insert: fixes f :: "'b \<Rightarrow> 'c option"   and a :: "'b"   and A :: "'b set" shows "f |` insert a A = (f |` A)(a := f a)"
of_nat_id: fixes n :: "nat" shows "semiring_1_class.of_nat n = n"
disjoint_iff_not_equal: fixes A :: "'a set"   and B :: "'a set" shows "(A \<inter> B = {}) = (\<forall>x\<in>A. \<forall>y\<in>B. x \<noteq> y)"
ass: fixes f1 :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and f2 :: "'a \<Rightarrow> 'b \<Rightarrow> 'd" assumes "\<forall>x1 x2. P x1 x2 (f1 x1 x2) (f2 x1 x2)" shows "False"
P_lemma: fixes x :: "int" assumes "x \<in> local.S1" shows "\<exists>!y. local.P x y"
ppp: shows "p = int_of_uint32 (uint32_of_int p)"
non_empty_cycle_root_move_root_msc: fixes r :: "'a"   and x :: "'a"   and q :: "'a" assumes "non_empty_cycle_root r x" shows "x\<^sup>T\<^sup>\<star> ; q = x\<^sup>\<star> ; q"
assoc'_simps_5: fixes f :: "'d"   and g :: "'d"   and h :: "'d" assumes "D.ide f"   and "D.ide g"   and "D.ide h"   and "src\<^sub>D f = trg\<^sub>D g"   and "src\<^sub>D g = trg\<^sub>D h" shows "D.cod \<a>\<^sub>D\<^sup>-\<^sup>1[f, g, h] = (D.cod f \<star>\<^sub>D D.cod g) \<star>\<^sub>D D.cod h"
affine_hull_subset_span: fixes s :: "'a set" shows "affine hull s \<subseteq> span s"
hensel_1_def: fixes ty :: "'b itself"   and u :: "'a mod_ring poly"   and v :: "'c mod_ring poly"   and w :: "'c mod_ring poly" shows "hensel_1 ty u v w \<equiv> if v = 1 then (1, u) else let (s, t) = bezout_coefficients (#v) (#w); (a, b) = Hensel_Lifting_Type_Based.dupe_monic (#v) (#w) s t 1 in (#v + Polynomial.smult (of_nat (Knuth_ex_4_6_2_22_base.q TYPE('c))) (#Knuth_ex_4_6_2_22_main.v' TYPE('c) b u v w), #w + Polynomial.smult (of_nat (Knuth_ex_4_6_2_22_base.q TYPE('c))) (#Knuth_ex_4_6_2_22_main.w' TYPE('c) a b u v w))"
actions_ok1_ex_actions_ok2: fixes final1 :: "'a \<Rightarrow> bool"   and r1 :: "('d, 'b, 'a, 'c, 'e, 'f) semantics"   and final2 :: "'g \<Rightarrow> bool"   and r2 :: "('d, 'b, 'g, 'h, 'e, 'f) semantics"   and bisim :: "'b \<Rightarrow> 'a \<times> 'c \<Rightarrow> 'g \<times> 'h \<Rightarrow> bool"   and bisim_wait :: "'a \<Rightarrow> 'g \<Rightarrow> bool"   and \<tau>move1 :: "('d, 'b, 'a, 'c, 'e, 'f) \<tau>moves"   and \<tau>move2 :: "('d, 'b, 'g, 'h, 'e, 'f) \<tau>moves"   and s1 :: "('i, 'b) locks \<times> (('b \<Rightarrow> ('a \<times> 'i \<Rightarrow>f nat) option) \<times> 'c) \<times> ('b \<Rightarrow> 'j wait_set_status option) \<times> 'b set"   and t :: "'b"   and ta1 :: "('i, 'b, 'a, 'c, 'j, 'k) thread_action"   and ta2 :: "('i, 'b, 'g, 'h, 'j, 'k) thread_action" assumes "FWdelay_bisimulation_obs final1 r1 final2 r2 bisim bisim_wait \<tau>move1 \<tau>move2"   and "final_thread.actions_ok final1 s1 t ta1"   and "ta_bisim bisim ta1 ta2" obtains s2 :: "('i, 'b) locks \<times> (('b \<Rightarrow> ('g \<times> 'i \<Rightarrow>f nat) option) \<times> 'h) \<times> ('b \<Rightarrow> 'j wait_set_status option) \<times> 'b set" where "final_thread.actions_ok final2 s2 t ta2"
composition_def: fixes \<sigma>\<^sub>1 :: "char list \<Rightarrow> fterm"   and \<sigma>\<^sub>2 :: "char list \<Rightarrow> fterm"   and x :: "char list" shows "(\<sigma>\<^sub>1 \<cdot> \<sigma>\<^sub>2) x = \<sigma>\<^sub>1 x \<cdot>\<^sub>t \<sigma>\<^sub>2"
ed_ou_witness_Compl: fixes A :: "ed_ou_witness set" shows "- A = {ed_ou_witness.a, ed_ou_witness.b, ed_ou_witness.c, d, e} - A"
if_True: fixes x :: "'c"   and y :: "'c" shows "(if True then x else y) = x"
Retr_defs_1: fixes theta :: "(('test, 'atom, 'choice) cmd \<times> ('test, 'atom, 'choice) cmd) set" shows "Sretr theta \<equiv> {(c, d). matchC_C theta c d}"
append_eq_append_conv2: fixes xs :: "'c list"   and ys :: "'c list"   and zs :: "'c list"   and ts :: "'c list" shows "(xs \<cdot> ys = zs \<cdot> ts) = (\<exists>us. xs = zs \<cdot> us \<and> us \<cdot> ys = ts \<or> xs \<cdot> us = zs \<and> ys = us \<cdot> ts)"
simps_9: fixes v :: "nat" shows "pratt2_max \<top>\<^sub>2 [v]\<^sub>2 = \<top>\<^sub>2"
norm_eq_L2: fixes z :: "quat" shows "norm z = L2_set (quat_proj z) {..3}"
f_at_Suc: shows "\<forall>y. f (Suc y) = h y (f y)"
simps_21: fixes f1 :: "'a"   and f2 :: "'a"   and f3 :: "'a"   and f4 :: "'a"   and f5 :: "'a" shows "(case a\<^sub>1 of a\<^sub>1 \<Rightarrow> f1 | a\<^sub>2 \<Rightarrow> f2 | a\<^sub>3 \<Rightarrow> f3 | a\<^sub>4 \<Rightarrow> f4 | a\<^sub>5 \<Rightarrow> f5) = f1"
wf_interp_for_formula_FEXISTS: fixes I :: "(nat + nat set) list"   and \<phi> :: "'a formula"   and w :: "'a list" assumes "wf_formula (length I) (FEXISTS \<phi>)"   and "w \<noteq> []" shows "wf_interp_for_formula (w, I) (FEXISTS \<phi>) = (\<forall>P\<subseteq>{0..length w - 1}. wf_interp_for_formula (w, Inr P # I) \<phi>)"
con_rews_2: fixes a :: "'c" shows "Just\<cdot>a \<noteq> \<bottom>"
sub_positive: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and n :: "num"   and m :: "num" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less zero (neg_numeral.sub minus plus one n m) = (m < n)"
Mempty_parametric: fixes A :: "'d \<Rightarrow> 'e \<Rightarrow> bool" shows "rel_mset A {#} {#}"
alphaAbs_all_qFresh_def: fixes xs :: "'f"   and x :: "'g"   and X :: "('h, 'i, 'f, 'g, 'j) qTerm"   and xs' :: "'f"   and x' :: "'g"   and X' :: "('h, 'i, 'f, 'g, 'j) qTerm" assumes "FixVars TYPE('d) TYPE('e)" shows "alphaAbs_all_qFresh xs x X xs' x' X' \<equiv> xs = xs' \<and> (\<forall>y. qFresh xs y X \<and> qFresh xs y X' \<longrightarrow> X #[[y \<and> x]]_xs #= X' #[[y \<and> x']]_xs)"
rtrancl_into_trancl1: fixes a :: "'a"   and b :: "'a"   and r :: "('a \<times> 'a) set"   and c :: "'a" assumes "(a, b) \<in> r\<^sup>*"   and "(b, c) \<in> r" shows "(a, c) \<in> r\<^sup>+"
sub_commutative: fixes x :: "'b"   and y :: "'b" shows "! (d x) \<squnion> ! (d y) = ! (d y) \<squnion> ! (d x)"
prod_ge_1: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and A :: "'c set"   and f :: "'c \<Rightarrow> 'b" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero"   and "\<And>x. x \<in> A \<Longrightarrow> less_eq one (f x)" shows "less_eq one (comm_monoid_mult.prod times one f A)"
False: shows "A $$ (k, l) \<noteq> (0::'a)"
Rep_formula_simps_1: fixes x :: "'b formula"   and y :: "'b formula" shows "(x \<le> y) = (Rep_formula x \<subseteq> Rep_formula y)"
freshChainSimps_6: fixes x :: "name"   and p :: "(name \<times> name) list" assumes "x \<sharp> p" shows "p \<bullet> x = x"
finite_subset: fixes A :: "'q set"   and B :: "'q set" assumes "A \<subseteq> B"   and "finite B" shows "finite A"
simps: fixes a :: "'a llist" shows "valid_path a = (a = LNil \<or> (\<exists>v. a = LCons v LNil \<and> v \<in> V) \<or> (\<exists>v w Ps. a = LCons v Ps \<and> v \<in> V \<and> w \<in> V \<and> v \<rightarrow> w \<and> valid_path Ps \<and> \<not> lnull Ps \<and> lhd Ps = w))"
set: fixes x1 :: "'b"   and x2 :: "'b stream" shows "sset (x1 ## x2) = insert x1 (sset x2)"
chain_def: fixes A :: "'a set set"   and C :: "'a set set" shows "subset.chain A C = (C \<subseteq> A \<and> (\<forall>x\<in>C. \<forall>y\<in>C. (\<subset>)\<^sup>=\<^sup>= x y \<or> (\<subset>)\<^sup>=\<^sup>= y x))"
blocking_on_imp_not_stable: fixes ds :: "'b set"   and X :: "'a set"   and h :: "'c"   and X'' :: "'a set" assumes "TruePref_tax.blocking_on ds X h X''" shows "\<not> TruePref_tax.stable_on ds X"
simps_11: fixes f1 :: "'b \<Rightarrow> 'a"   and f2 :: "'c \<Rightarrow> 'a"   and f3 :: "'d \<Rightarrow> 'a"   and x2 :: "'c" shows "(case In2 x2 of In1 x \<Rightarrow> f1 x | In2 x \<Rightarrow> f2 x | In3 x \<Rightarrow> f3 x) = f2 x2"
Some_le: fixes x :: "'b"   and u :: "'b option" shows "(Some x \<le> u) = (\<exists>y. u = Some y \<and> x \<le> y)"
power_similiar_sq_mtx_diag_eq: fixes P :: "'a sq_mtx"   and A :: "'a sq_mtx"   and f :: "'a \<Rightarrow> real"   and n :: "nat" assumes "mtx_invertible P"   and "A = P\<^sup>-\<^sup>1 * sq_mtx_diag f * P" shows "A ^ n = P\<^sup>-\<^sup>1 * (\<d>\<i>\<a>\<g> i. f i ^ n) * P"
above_aboveS_trans: fixes r :: "('a \<times> 'a) set"   and b :: "'a"   and a :: "'a"   and c :: "'a" assumes "trans r"   and "antisym r"   and "b \<in> above r a"   and "c \<in> aboveS r b" shows "c \<in> aboveS r a"
strong_del_point_design: fixes p :: "'a" shows "design (des.del_point p) (des.str_del_point_blocks p)"
ord_le_eq_trans: fixes less_eq :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and a :: "'b::type"   and b :: "'b::type"   and c :: "'b::type" assumes "(less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (a::'b::type) (b::'b::type)"   and "(b::'b::type) = (c::'b::type)" shows "(less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (a::'b::type) (c::'b::type)"
SortedByIteratorTE: fixes \<Gamma> :: "String.literal \<rightharpoonup>\<^sub>f 'a OCL_Types.type"   and src :: "'a expr"   and k :: "call_kind"   and its :: "String.literal list"   and its_ty :: "'a OCL_Types.type option"   and body :: "'a expr"   and \<tau> :: "'a OCL_Types.type" assumes "\<Gamma> \<turnstile>\<^sub>E SortedByIteratorCall src k its its_ty body : \<tau>" obtains \<tau>' :: "'a OCL_Types.type" and \<sigma> :: "'a OCL_Types.type" and \<rho> :: "'a OCL_Types.type" where "k = ArrowCall" and "\<Gamma> \<turnstile>\<^sub>I (src, its, its_ty, body) : (\<tau>', \<sigma>, \<rho>)" and "length its \<le> Suc 0" and "to_ordered_collection \<tau>' \<tau>"
isSuccOrd_def: shows "isSuccOrd \<equiv> \<exists>j\<in>Field r. \<forall>i\<in>Field r. (i, j) \<in> r"
insert_DiffM2: fixes x :: "'a"   and M :: "'a multiset" assumes "x \<in># M" shows "M - {#x#} + {#x#} = M"
App_base_hom: fixes n :: "nat"   and vv :: "nat \<Rightarrow> 'a \<Rightarrow> ant" assumes "vals_nonequiv K (Suc n) vv"   and "\<forall>j\<le>Suc n. n_val K (vv j) = vv j" shows "\<forall>j\<le>Suc n. App_base K vv (Suc n) j \<in> carrier K"
cmult_assoc: fixes i :: "V"   and j :: "V"   and k :: "V" shows "i \<otimes> j \<otimes> k = i \<otimes> (j \<otimes> k)"
FORKs\<^sub>n\<^sub>o\<^sub>r\<^sub>m_1_dir2: fixes N :: "nat"   and fs :: "nat list" assumes "DiningPhilosophers N"   and "0 < length fs" shows "DiningPhilosophers.FORK\<^sub>n\<^sub>o\<^sub>r\<^sub>m N 0 (fs ! 0) \<sqsubseteq> DiningPhilosophers.FORKs\<^sub>n\<^sub>o\<^sub>r\<^sub>m N 1 fs"
get_owner_document_disconnected_nodes: fixes h :: "(_) heap"   and document_ptr :: "(_) document_ptr"   and disc_nodes :: "(_) node_ptr list"   and node_ptr :: "(_) node_ptr" assumes "Shadow_DOM.heap_is_wellformed h"   and "h \<turnstile> get_disconnected_nodes document_ptr \<rightarrow>\<^sub>r disc_nodes"   and "node_ptr \<in> set disc_nodes"   and "ShadowRootClass.known_ptrs h"   and "ShadowRootClass.type_wf h" shows "h \<turnstile> Shadow_DOM.get_owner_document (cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r node_ptr) \<rightarrow>\<^sub>r document_ptr"
return_pmf_parametric: fixes A :: "'c \<Rightarrow> 'd \<Rightarrow> bool" shows "rel_fun A (rel_pmf A) return_pmf return_pmf"
prems_2: shows "w \<Turnstile>\<^sub>n (\<phi>1 M\<^sub>n \<phi>2)[X]\<^sub>\<nu>"
Equal2: shows "sat_models ` {Abs \<phi> |\<phi>. nested_propos \<phi> \<subseteq> P} = {sat_models (Abs \<phi>) |\<phi>. nested_propos \<phi> \<subseteq> P}"
essentialE: fixes G :: "('k, 'l) graph_scheme"   and B :: "'k set"   and S :: "'k set"   and x :: "'k" assumes "essential G B S x" obtains p :: "'k list" and y :: "'k" where "path G x p y" and "y \<in> B" and "\<And>z. \<lbrakk>x \<noteq> y; z \<in> set p\<rbrakk> \<Longrightarrow> z = x \<or> z \<notin> S"