hcju/isabelleps huggingface dataset:
le_trans: fixes L :: "('d, 'e) gorder_scheme"   and x :: "'d"   and y :: "'d"   and z :: "'d" assumes "weak_partial_order L"   and "x \<sqsubseteq>\<^bsub>L\<^esub> y"   and "y \<sqsubseteq>\<^bsub>L\<^esub> z"   and "x \<in> carrier L"   and "y \<in> carrier L"   and "z \<in> carrier L" shows "x \<sqsubseteq>\<^bsub>L\<^esub> z"
Loss_Delay_length_y: fixes a :: "nat \<Rightarrow> aType list"   and x :: "nat \<Rightarrow> aType list"   and i1 :: "nat \<Rightarrow> ECall_Info list"   and d :: "nat"   and y :: "nat \<Rightarrow> aType list"   and i2 :: "nat \<Rightarrow> ECall_Info list"   and lose :: "nat \<Rightarrow> bool list"   and i :: "nat \<Rightarrow> ECall_Info list"   and t :: "nat" assumes "\<forall>t. length (a t) \<le> Suc 0"   and "Delay x i1 d y i2"   and "Loss lose a i2 x i" shows "length (y t) \<le> Suc 0"
eq_iff: fixes a :: "'x \<Rightarrow>\<^sub>0 nat"   and b :: "'x \<Rightarrow>\<^sub>0 nat" shows "(a = b) = (lex_pm b a \<and> lex_pm a b)"
infinite: fixes A :: "'b set" assumes "infinite A" shows "Max A = the None"
cscalar_prod_vec_of_basis_enum: fixes \<phi> :: "'b"   and \<psi> :: "'b" shows "vec_of_basis_enum \<phi> \<bullet>c vec_of_basis_enum \<psi> = \<psi> \<bullet>\<^sub>C \<phi>"
assms_4: shows "NT \<eta> : YFtor C @@ X \<Longrightarrow> YFtor C @@ Y"
bdd_below_Int2: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and B :: "'b set"   and A :: "'b set" assumes "class.preorder less_eq less"   and "preorder.bdd_below less_eq B" shows "preorder.bdd_below less_eq (A \<inter> B)"
is_map: shows "is_left_adjoint TfTgh_TfHgh.chine"
APP_def: fixes f :: "'j \<Rightarrow> 'i"   and a :: "'j" shows "f $ a \<equiv> f a"
atCs_empty: fixes cs :: "('h, 'g, 'i, 'j) com list" shows "(atCs cs = {}) = (cs = [])"
induct: fixes x1 :: "(('sf, 'sc, 'sz) formula list \<times> ('sf, 'sc, 'sz) formula list) list"   and x2 :: "('sf, 'sc, 'sz) formula list \<times> ('sf, 'sc, 'sz) formula list"   and x3 :: "nat"   and x4 :: "nat"   and x5 :: "rrule"   and P :: "(('sf, 'sc, 'sz) formula list \<times> ('sf, 'sc, 'sz) formula list) list \<Rightarrow> ('sf, 'sc, 'sz) formula list \<times> ('sf, 'sc, 'sz) formula list \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> rrule \<Rightarrow> bool" assumes "rrule_ok x1 x2 x3 x4 x5"   and "\<And>SG i j C p q. snd (SG ! i) ! j = (p && q) \<Longrightarrow> P SG C i j AndR"   and "\<And>SG i j C p q. snd (SG ! i) ! j = (p \<rightarrow> q) \<Longrightarrow> P SG C i j ImplyR"   and "\<And>SG i j C p q. snd (SG ! i) ! j = (p \<leftrightarrow> q) \<Longrightarrow> P SG C i j EquivR"   and "\<And>SG i j C. \<lbrakk>j < length (snd (SG ! i)); \<And>\<Gamma> q. SG ! i \<noteq> (\<Gamma>, [q])\<rbrakk> \<Longrightarrow> P SG C i j CohideR"   and "\<And>SG i j C. \<lbrakk>j < length (snd (SG ! i)); \<And>q. SG ! i \<noteq> ([], [q])\<rbrakk> \<Longrightarrow> P SG C i j CohideRR"   and "\<And>SG i j C. snd (SG ! i) ! j = TT \<Longrightarrow> P SG C i j TrueR" shows "P x1 x2 x3 x4 x5"
zone_set_mono: fixes A :: "('d \<Rightarrow> 'e) set"   and B :: "('d \<Rightarrow> 'e) set"   and r :: "'d list" assumes "A \<subseteq> B" shows "A\<^bsub>r \<rightarrow> 0\<^esub> \<subseteq> B\<^bsub>r \<rightarrow> 0\<^esub>"
ThrowNull: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and e :: "char list exp"   and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and s\<^sub>1 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)" assumes "P \<turnstile> \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>null,s\<^sub>1\<rangle>" shows "P \<turnstile> \<langle>throw e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>THROW NullPointer,s\<^sub>1\<rangle>"
ln_2_ge: shows "5677 / 8192 \<le> ln 2"
emb_step_under_args_head: fixes p :: "dir list"   and d :: "dir"   and t :: "('c, 'd) tm" assumes "\<not> list_all (\<lambda>x. x = dir.Left) p" shows "head (emb_step_at p d t) = head t"
size_fset_overloaded_simps: fixes x :: "'a fset" shows "size x = (\<Sum>x\<in>fset x. Suc 0)"
shiftr_shiftl1: fixes c :: "nat"   and b :: "nat"   and a :: "'a word" assumes "c \<le> b" shows "a >> b << c = (a >> b - c) AND NOT (mask c)"
idom_potent_nonzero: fixes R :: "('e, 'f) Ring_scheme"   and x :: "'e"   and n :: "nat" assumes "Idomain R"   and "x \<in> carrier R"   and "x \<noteq> \<zero>\<^bsub>R\<^esub>" shows "x^\<^bsup>R n\<^esup> \<noteq> \<zero>\<^bsub>R\<^esub>"
i_5: shows "f \<in> insert x F \<rightarrow> carrier G"
freshBottoChain_1: fixes xvec :: "name list" shows "xvec \<sharp>* (\<bottom>)"
remove_1_mset_id_iff_notin: fixes M :: "'b multiset"   and a :: "'b" shows "(remove1_mset a M = M) = (a \<notin># M)"
\<eta>_in_hhom: shows "\<guillemotleft>\<eta> : src f \<rightarrow> src f\<guillemotright>"
wf_update: fixes i :: "'agent"   and Ri' :: "'alt \<Rightarrow> 'alt \<Rightarrow> bool" assumes "i \<in> agents"   and "total_preorder_on alts Ri'" shows "is_pref_profile (R(i := Ri'))"
assms: fixes x :: "'a" assumes "x \<in> space M" shows "monotone ord (\<le>) (\<lambda>f. F f x)"
ratfps_eqI: fixes x :: "'a ratfps"   and y :: "'a ratfps" assumes "fps_of_ratfps x = fps_of_ratfps y" shows "x = y"
le_imp_diff_is_add: fixes a :: "'a multiset"   and b :: "'a multiset"   and c :: "'a multiset" assumes "a \<subseteq># b"   and "a \<subseteq># b" shows "(b - a = c) = (b = c + a)"
WT_calleeD: fixes \<I> :: "('c, 'd) \<I>"   and callee :: "'c \<Rightarrow> ('d \<times> 'e) spmf"   and ret :: "'d"   and s :: "'e"   and out :: "'c" assumes "\<I> \<turnstile>c callee \<surd>"   and "(ret, s) \<in> set_spmf (callee out)"   and "out \<in> outs_\<I> \<I>" shows "ret \<in> responses_\<I> \<I> out"
gb_schema_aux_simps: fixes data :: "nat \<times> nat \<times> 'd"   and bs :: "(('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) list"   and ps :: "((('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) \<times> ('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) list" assumes "struct_spec sel ap ab compl" shows "gb_schema_aux data bs ps = (if ps = [] then gs @ bs else let sps = sel gs bs ps (snd data); ps0 = ps -- sps; aux = compl gs bs ps0 sps (snd data); remcomps = fst data - count_const_lt_components (fst aux) in if remcomps = 0 then full_gb (gs @ bs) else let (hs, data') = add_indices aux (snd data) in gb_schema_aux (remcomps, data') (ab gs bs hs data') (ap gs bs ps0 hs data'))"
is_closed_def: fixes T :: "'b set set"   and s :: "'b set" shows "(s closed\<^bsub>T\<^esub>) = (carrier\<^bsub>T\<^esub> - s open\<^bsub>T\<^esub>)"
ex1: shows "\<exists>!s. vs = fst s @ r2 # snd s"
length_sorted_list_of_set: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and A :: "'c set" assumes "class.linorder less_eq less" shows "length (linorder.sorted_list_of_set less_eq A) = card A"
member_seq_def: fixes xp :: "'a Predicate.seq" shows "local.member_seq xp = local.member_pred (Predicate.pred_of_seq xp)"
sol: shows "(sol solves_ode (\<lambda>a. ODEs I vid1)) {0..t} {x. Predicates I vid1 (\<chi>i. dterm_sem I (local.empty i) (mk_v I (OVar vid1) (sol 0, b) x))}"
Podd: shows "odd (p\<^sup>2 + N * q\<^sup>2)"
enum_step: shows "\<forall>i. \<Gamma>\<turnstile> f i \<rightarrow> f (Suc i)"
keys_to_list_punit_def: fixes cmp_term :: "'a nat_term_order" shows "keys_to_list_punit cmp_term \<equiv> ordered_term.keys_to_list (le_of_nat_term_order cmp_term)"
bound: fixes z :: "'d"   and n :: "nat"   and f :: "nat \<Rightarrow> 'd"   and m :: "nat" assumes "bound z n f"   and "n < m" shows "f m = z"
removing_offending_flows_makes_invariant_hold: fixes G :: "'a graph"   and nP :: "'a \<Rightarrow> 'b" shows "\<forall>F\<in>set_offending_flows G nP. sinvar (delete_edges G F) nP"
co_out1c: shows "connected (outside (path_image c1 \<union> path_image c))"
fmlookup_of_list: fixes m :: "('a \<times> 'b) list" shows "fmlookup (fmap_of_list m) = map_of m"
lens_plus_sub_comm: fixes X :: "'d \<Longrightarrow> 'e"   and Y :: "'f \<Longrightarrow> 'e" assumes "X \<bowtie> Y" shows "X +\<^sub>L Y \<subseteq>\<^sub>L Y +\<^sub>L X"
ideal_points_different_4: fixes u :: "complex_homo"   and v :: "complex_homo"   and i1 :: "complex_homo"   and i2 :: "complex_homo" assumes "u \<in> unit_disc"   and "v \<in> unit_disc"   and "u \<noteq> v"   and "ideal_points (poincare_line u v) = {i1, i2}" shows "v \<noteq> i1"
X_1: fixes x :: "bool" assumes "Aa__ x" shows "B"
continuous_map_eq: fixes X :: "'b topology"   and X' :: "'c topology"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "continuous_map X X' f"   and "\<And>x. x \<in> topspace X \<Longrightarrow> f x = g x" shows "continuous_map X X' g"
paint_rbt_greater: fixes less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and v :: "'a"   and c :: "color"   and t :: "('a, 'b) rbt" shows "ord.rbt_greater less v (paint c t) = ord.rbt_greater less v t"
fg_phi: shows "\<forall>x\<in>{0..<n}. f x = g (\<phi> x)"
Smith_1x2_eucl_JNF_def: fixes A :: "'b mat" shows "Smith_1x2_eucl_JNF A = (if A $$ (0, 0) = (0::'b) \<and> A $$ (0, 1) \<noteq> (0::'b) then let Q = swaprows_mat 2 0 1; A' = swapcols 0 1 A in (A', Q) else if A $$ (0, 0) \<noteq> (0::'b) \<and> A $$ (0, 1) \<noteq> (0::'b) then let bezout_matrix_right = (bezout_matrix_JNF A\<^sup>T 0 1 0 euclid_ext2)\<^sup>T in (A * bezout_matrix_right, bezout_matrix_right) else (A, 1\<^sub>m 2))"
prism_diff_def: fixes X :: "'c \<Longrightarrow>\<^sub>\<triangle> 'd"   and Y :: "'e \<Longrightarrow>\<^sub>\<triangle> 'd" shows "(X \<nabla> Y) = (range build\<^bsub>X\<^esub> \<inter> range build\<^bsub>Y\<^esub> = {})"
initial_no_ret: fixes p :: "'b" assumes "initialproc fg p" shows "entry fg p \<noteq> return fg p"
word_or_not: fixes x :: "'b word" shows "x OR NOT x = - 1"
60: fixes x :: "'a" shows "- x \<le> - - - x"
vsubst_psubst_idEnv: fixes ys :: "'varSort"   and z :: "'var"   and y :: "'var"   and X :: "('a, 'b, 'varSort, 'var, 'c) qTerm set" shows "X #[z // y]_ys = X #[idEnv [y \<leftarrow> Var ys z]_ys]"
measure_spmf_zero_iff: fixes p :: "'c spmf"   and A :: "'c set" shows "(Sigma_Algebra.measure (measure_spmf p) A = 0) = (set_spmf p \<inter> A = {})"
hom_one: shows "act \<one>\<^bsub>G\<^esub> = \<one>\<^bsub>BijGroup X\<^esub>"
lessThan_eq_iff: fixes x :: "'c"   and y :: "'c" shows "({..<x} = {..<y}) = (x = y)"
lex_pm_iff: fixes s :: "'c \<Rightarrow>\<^sub>0 'd"   and t :: "'c \<Rightarrow>\<^sub>0 'd" shows "lex_pm s t = (\<forall>x. lookup s x \<le> lookup t x \<or> (\<exists>y<x. lookup s y \<noteq> lookup t y))"
Rep_uexpr_inverse: fixes x :: "('c, 'd) uexpr" shows "mk\<^sub>e \<lbrakk>x\<rbrakk>\<^sub>e = x"
interleave_append_sym: fixes s :: "'a event list"   and t :: "'a event list"   and C :: "'a event set"   and u1 :: "'a event list"   and u2 :: "'a event list" assumes "s setinterleaves ((t, u1 @ u2), C)" shows "\<exists>t1 t2 s1 s2. t = t1 @ t2 \<and> s = s1 @ s2 \<and> s1 setinterleaves ((t1, u1), C) \<and> s2 setinterleaves ((t2, u2), C)"
that: shows "(R0 ===> state_dp_consistency.crel_vs R1 OO local.rel_state (=)) x y"
dom_def: fixes m :: "'c \<Rightarrow> 'd option" shows "dom m = _Collect a (m a \<noteq> None)"
cmp_interpretation: shows "identity_arrow_of_tabulations_in_maps (\<cdot>) (\<star>) \<a> \<i> src trg (f \<star> g \<star> h) TfHgh.\<rho>\<sigma>.tab (tab\<^sub>0 (g \<star> h) \<star> TfHgh.p\<^sub>0) (tab\<^sub>1 f \<star> TfHgh.p\<^sub>1) (f \<star> g \<star> h) HfHgh.tab (tab\<^sub>0 (f \<star> g \<star> h)) (tab\<^sub>1 (f \<star> g \<star> h)) (f \<star> g \<star> h)"
ex_W_cmpser: fixes m :: "nat"   and f :: "nat \<Rightarrow> 'a set" assumes "w_cmpser G m f" shows "\<exists>g. W_cmpser G (card (f ` {i. i \<le> m}) - 1) g \<and> g ` {i. i \<le> card (f ` {i. i \<le> m}) - 1} = f ` {i. i \<le> m}"
unit_simps_2: fixes V\<^sub>C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H\<^sub>C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a>\<^sub>C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i>\<^sub>C :: "'b \<Rightarrow> 'b"   and src\<^sub>C :: "'b \<Rightarrow> 'b"   and trg\<^sub>C :: "'b \<Rightarrow> 'b"   and V\<^sub>D :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and H\<^sub>D :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<a>\<^sub>D :: "'c \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<i>\<^sub>D :: "'c \<Rightarrow> 'c"   and src\<^sub>D :: "'c \<Rightarrow> 'c"   and trg\<^sub>D :: "'c \<Rightarrow> 'c"   and F :: "'b \<Rightarrow> 'c"   and \<Phi> :: "'b \<times> 'b \<Rightarrow> 'c"   and a :: "'b" assumes "pseudofunctor V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D F \<Phi>"   and "horizontal_homs.obj V\<^sub>C src\<^sub>C a" shows "src\<^sub>D (pseudofunctor.unit \<i>\<^sub>C V\<^sub>D H\<^sub>D \<i>\<^sub>D src\<^sub>D F \<Phi> a) = weak_arrow_of_homs.map\<^sub>0 src\<^sub>D F a"
set_category_data_def: fixes S :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" shows "set_category_data S \<equiv> category S"
nat_pow_pow: fixes x :: "int \<times> int"   and n :: "nat"   and m :: "nat" assumes "x \<in> carrier R" shows "(x [^]\<^bsub>R\<^esub> n) [^]\<^bsub>R\<^esub> m = x [^]\<^bsub>R\<^esub> (n * m)"
safe_neg_def: fixes \<phi> :: "formula" shows "safe_neg \<phi> = (\<not> safe_formula \<phi> \<longrightarrow> safe_formula (remove_neg \<phi>))"
iso_def: fixes f :: "'a1 \<times> 'a2" shows "local.iso f \<equiv> \<exists>g. inverse_arrows f g"
ERETURN_rule_iff: fixes x :: "'d"   and \<Phi> :: "'c \<Rightarrow> bool"   and \<Psi> :: "'d \<Rightarrow> bool" shows "(ERETURN x \<le> ESPEC \<Phi> \<Psi>) = \<Psi> x"
3: shows "\<^bold>\<lfloor>Dom (fst fg) \<^bold>\<star> Dom (snd fg)\<^bold>\<rfloor> = \<^bold>\<lfloor>Dom (fst fg) \<^bold>\<lfloor>\<^bold>\<star>\<^bold>\<rfloor> Dom (snd fg)\<^bold>\<rfloor>"
conjunction: fixes p :: "fm"   and q :: "fm" shows "valid (Con' p q) = (valid p \<and> valid q)"
assms: shows "a \<in>\<^sub>\<circ> \<F>\<^sub>\<circ> r"
heq'': shows "frechet I \<theta> (fst (mk_v I ODE (sol 0, b) (sol s))) \<circ> (\<lambda>t'. t' *\<^sub>R (\<chi>i. if i \<in> ODE_vars I ODE then ODE_sem I ODE (sol s) $ i else 0)) = (\<lambda>t'. t' *\<^sub>R frechet I \<theta> (fst (mk_v I ODE (sol 0, b) (sol s))) (snd (mk_v I ODE (sol 0, b) (sol s))))"
prod_coprime_left: fixes one :: "'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and A :: "'b set"   and f :: "'b \<Rightarrow> 'a"   and a :: "'a" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize"   and "\<And>i. i \<in> A \<Longrightarrow> algebraic_semidom.coprime times one (f i) a" shows "algebraic_semidom.coprime times one (comm_monoid_mult.prod times one f A) a"
lm054: fixes bids :: "'d \<times> 'e set \<Rightarrow> 'f"   and N :: "'d set"   and G :: "'e set" shows "runiq (summedBidVectorRel bids N G)"
self_finance_risk_free_update_ind: fixes pf :: "'a \<Rightarrow> nat \<Rightarrow> bool stream \<Rightarrow> real"   and v :: "real"   and n :: "nat"   and w :: "bool stream" assumes "portfolio pf" shows "cls_val_process Mkt (self_finance Mkt v pf risk_free_asset) (Suc n) w = cls_val_process Mkt pf (Suc n) w + (1 + r) * (val_process Mkt (self_finance Mkt v pf risk_free_asset) n w - val_process Mkt pf n w)"
ipd_def: fixes x :: "'n" shows "ipd x = (THE y. y ipd\<rightarrow> x)"
STL3: fixes F :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool" shows "\<turnstile> \<box>\<box>F = \<box>F"
arr_dom_iff_arr: fixes f :: "('o, 'a) arr" shows "arr (local.dom f) = arr f"
ivls_of_aforms: fixes xs :: "real list"   and XS :: "(real \<times> real pdevs) list"   and prec :: "nat" assumes "xs \<in> Joints XS" shows "bounded_by xs (ivls_of_aforms prec XS)"
id_subst: fixes v :: "('b, 'c) uexpr" shows "id \<dagger> v = v"
ennreal_suminf_SUP_eq_directed: fixes I :: "'b set"   and f :: "'b \<Rightarrow> nat \<Rightarrow> ennreal" assumes "\<And>N i j. \<lbrakk>i \<in> I; j \<in> I; finite N\<rbrakk> \<Longrightarrow> \<exists>k\<in>I. \<forall>n\<in>N. f i n \<le> f k n \<and> f j n \<le> f k n" shows "(\<Sum>n. SUP i\<in>I. f i n) = (SUP i\<in>I. \<Sum>n. f i n)"
convex_Int: fixes s :: "'c set"   and t :: "'c set" assumes "convex s"   and "convex t" shows "convex (s \<inter> t)"
commute: fixes a :: "'b upred"   and b :: "'b upred" shows "(a \<or> b) = (b \<or> a)"
p: shows "\<forall>b\<in>f ` A. bij_betw (p b) {a \<in> A. f a = b} {a \<in> A. f' a = b}"
lin: fixes x :: "'x"   and y :: "'x" shows "R x y \<or> R y x"
simps_2: shows "VARto stA2 = {data4, data12}"
extend_in_C: fixes C :: "(nat, 'b) form set set"   and S :: "(nat, 'b) form set"   and f :: "nat \<Rightarrow> (nat, 'b) form"   and n :: "nat" assumes "alt_consistency C"   and "S \<in> C"   and "infinite (- \<Union> (params ` S))" shows "extend S C f n \<in> C"
mem_next: fixes A :: "ordinal set"   and x :: "ordinal" assumes "normal_set A" shows "(LEAST z. z \<in> A \<and> x < z) \<in> A"
simps: fixes f :: "'a \<Rightarrow> 'b"   and x :: "'a"   and xs :: "'a list" shows "argmin f (x # xs) = (if xs = [] then x else let m = argmin f xs in if f x < f m then x else m)"
Diagonalize_rep_preserves_seq: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and g :: "'b local.term set"   and f :: "'b local.term set" assumes "free_monoidal_category C"   and "partial_magma.arr (free_monoidal_category.comp C) (free_monoidal_category.comp C g f)" shows "monoidal_language.Arr C (monoidal_language.Diagonalize C (free_monoidal_category.rep C g)) \<and> monoidal_language.Arr C (monoidal_language.Diagonalize C (free_monoidal_category.rep C f)) \<and> monoidal_language.Dom C (monoidal_language.Diagonalize C (free_monoidal_category.rep C g)) = monoidal_language.Cod C (monoidal_language.Diagonalize C (free_monoidal_category.rep C f))"
edge_det: fixes a :: "'b LDCFG_node \<times> ('c, 'd, 'e, 'f) edge_kind \<times> 'b LDCFG_node"   and a' :: "'b LDCFG_node \<times> ('c, 'd, 'e, 'f) edge_kind \<times> 'b LDCFG_node" assumes "lift_valid_edge valid_edge sourcenode targetnode kind Entry Exit a"   and "lift_valid_edge valid_edge sourcenode targetnode kind Entry Exit a'"   and "src a = src a'"   and "trg a = trg a'" shows "a = a'"
vinsert_in_VsetI: fixes \<alpha> :: "V"   and a :: "V"   and b :: "V" assumes "Limit \<alpha>"   and "a \<in>\<^sub>\<circ> Vset \<alpha>"   and "b \<in>\<^sub>\<circ> Vset \<alpha>" shows "vinsert a b \<in>\<^sub>\<circ> Vset \<alpha>"
simps_12: fixes f1 :: "'b"   and f2 :: "BitList \<Rightarrow> 'b \<Rightarrow> 'b"   and f3 :: "BitList \<Rightarrow> 'b \<Rightarrow> 'b" shows "rec_BitList f1 f2 f3 BitListNil = f1"
geodesic_segment_split_1: fixes y :: "'a"   and x :: "'a"   and z :: "'a" assumes "y \<in> {x--z}" shows "{x--z} = {x--y} \<union> {y--z}"
awalk_verts_induce: fixes G :: "('c, 'd) pre_digraph"   and S :: "'c set" assumes "wf_digraph G" shows "pre_digraph.awalk_verts (G \<restriction> S) = pre_digraph.awalk_verts G"
kop_epi: fixes f :: "('e \<Rightarrow> 'd set) \<Rightarrow> 'f"   and g :: "('e \<Rightarrow> 'd set) \<Rightarrow> 'f" assumes "f \<circ> op\<^sub>K = g \<circ> op\<^sub>K" shows "f = g"
rm_vars_img_fv_subset: fixes V :: "'a set"   and s :: "'a \<Rightarrow> ('b, 'a) Term.term" shows "range_vars (rm_vars V s) \<subseteq> range_vars s"
vector_minus_component: fixes x :: "('b, 'c) Finite_Cartesian_Product.vec"   and y :: "('b, 'c) Finite_Cartesian_Product.vec"   and i :: "'c" shows "(x - y) $h i = x $h i - y $h i"
lookup_make_fmap1: fixes k :: "'b"   and v :: "'a" shows "lookup [ {|k|} |=> v ] k = v"
Plus_assoc: fixes r :: "'a language"   and s :: "'a language"   and t :: "'a language" shows "Plus (Plus r s) t = Plus r (Plus s t)"
p\<^sub>0_simps: shows "trg fg\<^sub>0h\<^sub>1.p\<^sub>0 = src (tab\<^sub>1 h)"
times_square_lepoll: fixes A :: "'a set" shows "A \<lesssim> A \<times> A"
degree_prod_eq_sum_degree: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b poly" assumes "\<forall>i\<in>A. f i \<noteq> 0" shows "degree (prod f A) = (\<Sum>i\<in>A. degree (f i))"
continuous_at_imp_continuous_on: fixes s :: "'d set"   and f :: "'d \<Rightarrow> 'e" assumes "\<forall>x\<in>s. isCont f x" shows "continuous_on s f"
list_of_list_to_matrix_def: fixes xs :: "'d list list" shows "list_of_list_to_matrix xs = (\<chi>i. list_to_vec (xs ! mod_type_class.to_nat i))"
runit_char_3: fixes C :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and T :: "'d \<times> 'd \<Rightarrow> 'd"   and \<alpha> :: "'d \<times> 'd \<times> 'd \<Rightarrow> 'd"   and \<iota> :: "'d"   and a :: "'d" assumes "monoidal_category C T \<alpha> \<iota>"   and "partial_magma.ide C a" shows "\<exists>!f. partial_magma.in_hom C f (T (a, monoidal_category.unity C \<iota>)) a \<and> T (f, monoidal_category.unity C \<iota>) = C (T (a, \<iota>)) (\<alpha> (a, monoidal_category.unity C \<iota>, monoidal_category.unity C \<iota>))"
fds_converges_altdef2: fixes f :: "'a fds"   and s :: "'a" shows "fds_converges f s = convergent (\<lambda>N. eval_fds (fds_truncate N f) s)"
roots4_generic_def: fixes depressed_solver :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a list"   and p :: "'a poly" shows "roots4_generic depressed_solver p = (let cs = coeffs p; cs = coeffs p; a4 = cs ! 4; a3 = cs ! 3; a2 = cs ! 2; a1 = cs ! 1; a0 = cs ! 0; b = a3 / a4; c = a2 / a4; d = a1 / a4; e = a0 / a4; b2 = b * b; b3 = b2 * b; b4 = b3 * b; b4' = b / (4::'a); p = c - (3::'a) / (8::'a) * b2; q = (b3 - (4::'a) * b * c + (8::'a) * d) / (8::'a); r = (- (3::'a) * b4 + (256::'a) * e - (64::'a) * b * d + (16::'a) * b2 * c) / (256::'a) in Let (depressed_solver p q r) (map (\<lambda>y. y - b4')))"
concat_bit_def: fixes n :: "nat"   and k :: "int"   and l :: "int" shows "concat_bit n k l = take_bit n k OR push_bit n l"
strip_while_eq_obtain_trailing: fixes P :: "'f \<Rightarrow> bool"   and xs :: "'f list"   and ys :: "'f list" assumes "strip_while P xs = ys" obtains zs :: "'f list" where "xs = ys @ zs" and "\<And>z. z \<in> set zs \<Longrightarrow> P z" and "no_trailing P ys"
subset_step: fixes p :: "real poly"   and qs1 :: "real poly list"   and signs1 :: "rat list list"   and qs2 :: "real poly list"   and signs2 :: "rat list list" assumes "set (characterize_consistent_signs_at_roots_copr p qs1) \<subseteq> set signs1"   and "set (characterize_consistent_signs_at_roots_copr p qs2) \<subseteq> set signs2" shows "set (characterize_consistent_signs_at_roots_copr p (qs1 @ qs2)) \<subseteq> set (signs_smash signs1 signs2)"
ba_less_eq: fixes x :: "'a"   and y :: "'a" shows "(x \<le> y) = (x \<squnion> y = y)"
lb: shows "insertion (nth_default 0 (xs' @ x # xs)) (liftPoly 0 0 b) = B"
fair_chain_def: fixes Rs :: "(('e, 'f) labeled_graph \<times> ('e, 'f) labeled_graph) set"   and S :: "nat \<Rightarrow> ('e, 'g) labeled_graph" shows "fair_chain Rs S \<equiv> chain S \<and> (\<forall>R f i. R \<in> Rs \<and> graph_homomorphism (fst R) (S i) f \<longrightarrow> (\<exists>j. extensible R (S j) f))"
acyclic_star_inf_conv: fixes w :: "'a" assumes "pd_kleene_allegory_class.acyclic w" shows "w\<^sup>\<star> \<sqinter> w\<^sup>T\<^sup>\<star> = (1::'a)"
freshAbs_psubstAbs_I1: fixes A :: "('a, 'b, 'varSort, 'var, 'c) qAbs set"   and rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qTerm set option"   and zs :: "'varSort"   and z :: "'var" assumes "goodAbs A"   and "goodEnv rho"   and "freshAbs zs z A"   and "freshEnv zs z rho" shows "freshAbs zs z A $[rho]"
pmf_equiv_intro1: fixes p :: "'b pmf"   and q :: "'b pmf" assumes "\<And>e. e \<in> set_pmf p \<Longrightarrow> pmf p e = pmf q e" shows "p = q"
longest_common_prefix_prefix2: fixes xs :: "'c list"   and ys :: "'c list" shows "prefix (longest_common_prefix xs ys) ys"
minus_of_int: fixes x :: "'a"   and n :: "int" shows "frac (x - of_int n) = frac x"