hcju/isabelleps huggingface dataset:
types_preserved_get_global: fixes s :: "s"   and i :: "nat"   and j :: "nat"   and \<C> :: "t_context"   and \<S> :: "s_context"   and ts :: "t list"   and ts' :: "t list" assumes "typeof (sglob_val s i j) = tg_t (global \<C> ! j)"   and "\<S>\<bullet>\<C> \<turnstile> [$Get_global j] : ts _> ts'" shows "\<S>\<bullet>\<C> \<turnstile> [$C sglob_val s i j] : ts _> ts'"
del_invc_invh: fixes t :: "(('c \<times> 'd) \<times> color) tree"   and x :: "'c" assumes "invh t"   and "invc t" shows "invh (RBT_Map.del x t) \<and> (color t = Red \<and> bheight (RBT_Map.del x t) = bheight t \<and> invc (RBT_Map.del x t) \<or> color t = Black \<and> bheight (RBT_Map.del x t) = bheight t - 1 \<and> invc2 (RBT_Map.del x t))"
ih: fixes xs :: "'a list" shows "zip (xs @ ys) [0..<length xs + length ys] = zip xs [0..<length xs] @ zip ys [length xs..<length xs + length ys]"
F': shows "Rep_Set\<^sub>b\<^sub>a\<^sub>s\<^sub>e (Abs_Set\<^sub>b\<^sub>a\<^sub>s\<^sub>e \<lfloor>\<lfloor>Some ` (H ` ran (heap (snd \<tau>)) - {None})\<rfloor>\<rfloor>) = \<lfloor>\<lfloor>Some ` (H ` ran (heap (snd \<tau>)) - {None})\<rfloor>\<rfloor>"
diff_divide_distrib: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "(a - b) / c = a / c - b / c"
prj_chine_assoc_1: shows "\<nu>\<pi>\<rho>.Prj\<^sub>1 \<cdot> \<nu>\<pi>\<rho>.chine_assoc = \<nu>\<pi>\<rho>.Prj\<^sub>1\<^sub>1"
1: shows "E.Ide (Dom \<mu> \<^bold>\<lfloor>\<^bold>\<star>\<^bold>\<rfloor> Dom \<nu>) \<and> E.Nml (Dom \<mu> \<^bold>\<lfloor>\<^bold>\<star>\<^bold>\<rfloor> Dom \<nu>) \<and> E.Ide (Cod \<mu> \<^bold>\<lfloor>\<^bold>\<star>\<^bold>\<rfloor> Cod \<nu>) \<and> E.Nml (Cod \<mu> \<^bold>\<lfloor>\<^bold>\<star>\<^bold>\<rfloor> Cod \<nu>)"
sorted_job_lower_bound_makespan: fixes T :: "nat \<Rightarrow> nat"   and A :: "nat \<Rightarrow> nat set"   and j :: "nat" assumes "lb T A j"   and "m < j"   and "sorted j" shows "2 * t j \<le> makespan T"
Inf_insert: fixes a :: "'a::type"   and A :: "'a::type set" shows "\<Sqinter> insert (a::'a::type) (A::'a::type set) = a \<sqinter> \<Sqinter> A"
comp_converter_id_right: fixes conv :: "('y, 'z, 'aa, 'ab) converter" shows "conv \<odot> 1\<^sub>C = conv"
res: shows "(case hensel_binary C (factor_node_info l) (factor_node_info r) of (V, W) \<Rightarrow> hensel_main V l @ hensel_main W r) = Gs"
word_eqD: fixes u :: "'b word"   and v :: "'b word"   and x :: "nat" assumes "u = v" shows "u !! x = v !! x"
case_cong: fixes step :: "('c, 'd) step"   and step' :: "('c, 'd) step"   and f1 :: "'e"   and g1 :: "'e"   and f2 :: "'d \<Rightarrow> 'e"   and g2 :: "'d \<Rightarrow> 'e"   and f3 :: "'c \<Rightarrow> 'd \<Rightarrow> 'e"   and g3 :: "'c \<Rightarrow> 'd \<Rightarrow> 'e" assumes "step = step'"   and "step' = Done \<Longrightarrow> f1 = g1"   and "\<And>x2. step' = Skip x2 \<Longrightarrow> f2 x2 = g2 x2"   and "\<And>x31 x32. step' = Yield x31 x32 \<Longrightarrow> f3 x31 x32 = g3 x31 x32" shows "(case step of Done \<Rightarrow> f1 | Skip x2 \<Rightarrow> f2 x2 | Yield x31 x32 \<Rightarrow> f3 x31 x32) = (case step' of Done \<Rightarrow> g1 | Skip x2 \<Rightarrow> g2 x2 | Yield x31 x32 \<Rightarrow> g3 x31 x32)"
normalize_gcd_factorial: fixes a :: "'a"   and b :: "'a" shows "normalize (gcd_factorial a b) = gcd_factorial a b"
le_fininf_eq_inf_transp: fixes w\<^sub>1 :: "'action list"   and w\<^sub>2 :: "'action stream"   and w\<^sub>3 :: "'action stream" assumes "w\<^sub>1 \<preceq>\<^sub>F\<^sub>I w\<^sub>2"   and "w\<^sub>2 =\<^sub>I w\<^sub>3" shows "w\<^sub>1 \<preceq>\<^sub>F\<^sub>I w\<^sub>3"
lens_update_def: fixes X :: "'d \<Longrightarrow> 'c"   and f :: "'d \<Rightarrow> 'd"   and \<sigma> :: "'c" shows "update\<^bsub>X\<^esub> f \<sigma> = put\<^bsub>X\<^esub> \<sigma> (f (get\<^bsub>X\<^esub> \<sigma>))"
disj_assoc: fixes x :: "'b"   and y :: "'b"   and z :: "'b" shows "(x OR y) OR z = x OR y OR z"
WL_def: fixes p :: "'d process_name" shows "WL p = (\<lambda>s. W (s p) \<union> ghost_honorary_grey (s p))"
verit_eq_simplify_6: fixes x :: "'a"   and y :: "'a" assumes "x = y" shows "x \<le> y"
assoc: fixes a :: "'c"   and b :: "'c"   and c :: "'c" shows "inf (inf a b) c = inf a (inf b c)"
matrix_vector_mult_diff_distrib: fixes A :: "(('a, 'c) Finite_Cartesian_Product.vec, 'b) Finite_Cartesian_Product.vec"   and x :: "('a, 'c) Finite_Cartesian_Product.vec"   and y :: "('a, 'c) Finite_Cartesian_Product.vec" shows "A *v (x - y) = A *v x - A *v y"
Pbij_inverse_Dom: fixes \<beta> :: "(Location \<times> Location) set" shows "Pbij_Dom \<beta> = Pbij_Rng (Pbij_inverse \<beta>)"
set_trie_empty: shows "set_trie Tries_Binary.empty = {}"
join_comm: fixes x :: "'b"   and y :: "'b" shows "x \<squnion> y = y \<squnion> x"
current_plus_current: fixes \<Gamma> :: "('k, 'l) web_scheme"   and f :: "'k \<times> 'k \<Rightarrow> ennreal"   and g :: "'k \<times> 'k \<Rightarrow> ennreal" assumes "current \<Gamma> f"   and "wave \<Gamma> f"   and "current (quotient_web \<Gamma> f) g" shows "current \<Gamma> (plus_current f g)"
Throw1Throw: fixes uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and a :: "'addr"   and s :: "'heap \<times> 'addr val list" shows "uf,P,t \<turnstile>1 \<langle>throw (Throw a),s\<rangle> -\<lbrace>\<rbrace>\<rightarrow> \<langle>Throw a,s\<rangle>"
bezout_imp_all_fin_gen_ideals_are_principal: fixes I :: "'b set" assumes "finitely_generated_ideal I" shows "principal_ideal I"
reachable_awalk: fixes u :: "'a"   and v :: "'a" shows "(u \<rightarrow>\<^sup>* v) = (\<exists>p. awalk u p v)"
wave_current_of_bipartite: fixes f :: "('v + 'v) \<times> ('v + 'v) \<Rightarrow> ennreal" assumes "current (bipartite_web_of \<Gamma>) f"   and "wave (bipartite_web_of \<Gamma>) f" shows "wave \<Gamma> (current_of_bipartite f)"
ev_inequalities_1: shows "cmod \<alpha> \<le> r (map_vector cmod v)"
G: fixes S :: "'a set" shows "S \<in> \<D> \<longrightarrow> G S \<in> \<F> \<and> S \<subseteq> G S"
A11: shows "[A1]\<^bsub>v,n\<^esub> = [D]\<^bsub>v,n\<^esub> \<inter> {u. u \<turnstile> g}"
sqrt_float_intervalI: fixes x :: "real"   and X :: "float interval"   and p :: "nat" assumes "x \<in>\<^sub>r X" shows "sqrt x \<in>\<^sub>r sqrt_float_interval p X"
simps_1: fixes i :: "'a"   and x :: "int poly" shows "factors_of_factor_tree (Factor_Leaf i x) = {#x#}"
set_tag_name_ok: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and set_tag_name :: "(_) element_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_tag_name_locs :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and h :: "(_) heap"   and element_ptr :: "(_) element_ptr"   and tag :: "char list" assumes "l_set_tag_name\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf set_tag_name set_tag_name_locs"   and "type_wf h"   and "element_ptr |\<in>| element_ptr_kinds h" shows "h \<turnstile> ok set_tag_name element_ptr tag"
gcd_dvd2: fixes one :: "'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and gcd :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and lcm :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize" shows "dvd.dvd times (gcd a b) b"
vs_conf_vs_type_all: fixes P :: "'a prog" shows "vs_conf P (vs_type_all P)"
defined_not_I: fixes \<tau> :: "'d state \<times> 'd state"   and x :: "'d state \<times> 'd state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>" assumes "\<tau> \<Turnstile> \<delta> x" shows "\<tau> \<Turnstile> \<delta> not x"
ccApprox_many_calls: fixes x :: "Vars.var" shows "ccApprox (many_calls x) = {x} G\<times> {x}"
sum_matrix_2_def: fixes f :: "'c \<times> 'c \<Rightarrow> 'd" shows "sum2\<^sub>M f = (\<lambda>e. ab_semigroup_add_0.sum_0 (+) (bot + bot) (\<lambda>k. ab_semigroup_add_0.sum_0 (+) (bot + bot) (\<lambda>l. f (k, l)) {l. True}) {k. True})"
simps: fixes m :: "nat"   and lm :: "nat list"   and l :: "cell list"   and r :: "cell list" shows "wprepare_loop_goon_in_middle m lm (l, r) = (\<exists>rn mr lm1. rev l @ r = Oc \<up> Suc m @ Bk # Bk # <lm> @ Bk \<up> rn \<and> l \<noteq> [] \<and> (if lm1 = [] then r = Oc \<up> mr @ Bk \<up> rn else r = Oc \<up> mr @ Bk # <lm1> @ Bk \<up> rn) \<and> 0 < mr)"
factor_unit: shows "I = IA.additive.Left_Coset zeroA I"
connected_disjoint_Union_open_unique: fixes A :: "'a set set"   and B :: "'a set set" assumes "disjoint A"   and "disjoint B"   and "\<And>S. S \<in> A \<Longrightarrow> open S \<and> connected S \<and> S \<noteq> {}"   and "\<And>S. S \<in> B \<Longrightarrow> open S \<and> connected S \<and> S \<noteq> {}"   and "\<Union> A = \<Union> B" shows "A = B"
mono_D3: fixes P :: "'c process"   and Q :: "'c process"   and S :: "'c process" assumes "P \<sqsubseteq> Q" shows "min_elems (D (P [+] S)) \<subseteq> T (Q [+] S)"
assms_4: shows "inv_dghm \<FF>\<lparr>ArrMap\<rparr>\<lparr>f\<rparr> : inv_dghm \<FF>\<lparr>ObjMap\<rparr>\<lparr>a\<rparr> \<mapsto>\<^bsub>\<AA>\<^esub> inv_dghm \<FF>\<lparr>ObjMap\<rparr>\<lparr>b\<rparr>"
distr: fixes i :: "int" shows "(\<Sum>j<length ws. ws ! j * unity_root k (- i * int j)) * unity_root k (m * i) = (\<Sum>j<length ws. ws ! j * unity_root k (- i * int j) * unity_root k (m * i))"
stable_rank_bounded: fixes x :: "nat"   and j :: "nat"   and i :: "nat" assumes "stable_rank x j"   and "infinite (succeed i)"   and "q\<^sub>0 \<notin> F" shows "j < i"
\<epsilon>_uh_nonneg: shows "0 \<le> local.hash.\<epsilon>_uh"
ps_to_int_def: fixes G :: "diagram"   and \<sigma> :: "node \<rightharpoonup>\<^sub>f topbot" shows "ps_to_int G \<sigma> \<equiv> \<Otimes>v|\<in>|fmdom \<sigma>. (case lookup \<sigma> v of Top \<Rightarrow> top_ass | Bot \<Rightarrow> bot_ass) (G^\<Lambda> v)"
finite_simplices: shows "\<forall>x\<in>X. finite x"
assms_2: shows "set \<phi>2s \<subseteq> fmla"
emeasure_positive: fixes M :: "'b measure" shows "positive (sets M) (emeasure M)"
assms: shows "hn_refine \<Gamma> c \<Gamma>'' R a"
simps_2: fixes f :: "'a"   and Tl :: "'a trm list" shows "nvT (Fn f Tl) = {}"
is_extensional: fixes f :: "'a \<times> 'a \<times> 'a" assumes "\<not> VVV.arr f" shows "\<alpha>.map f = null"
closure_complement: fixes S :: "'a set" shows "closure (- S) = - interior S"
of_class_code2: fixes thy :: "exetheory"   and c :: "String.literal" assumes "exe_wf_theory' thy" shows "const_type (sig (translate_theory thy)) (const_of_class c) = lookup (\<lambda>k. k = const_of_class c) (execonst_type_of (exesig thy))"
weak_unit_char_2: fixes a :: "'d" shows "weak_composition.weak_unit (\<cdot>\<^sub>D) (\<star>\<^sub>D) a = D.isomorphic a (trg\<^sub>D a)"
PO_hoare_defs_2: fixes r :: "('b \<times> 'a) set"   and s :: "'b set" shows "r `` s = {y. \<exists>x\<in>s. (x, y) \<in> r}"
szip_sset: fixes u :: "'c stream"   and v :: "'d stream" shows "sset (u ||| v) \<subseteq> sset u \<times> sset v"
tab_simps_2: shows "src \<tau>.r.tab = src (tab\<^sub>0 (local.dom \<tau>))"
insert_Nil: fixes x :: "'a" shows "List.insert x [] = [x]"
leg1_composite_2: shows "HHfgh.dom.leg1 = fgh.\<mu>.leg1 \<cdot> fgh.Prj\<^sub>1\<^sub>1"
nc_c: shows "NC \<cdot> 1\<^sub>\<pi> = 1\<^sub>\<pi>"
zeta_even_numeral: fixes n :: "num" shows "zeta (numeral (num.Bit0 n)) = complex_of_real ((- 1) ^ Suc (numeral n) * bernoulli (numeral (num.Bit0 n)) * (2 * pi) ^ numeral (num.Bit0 n) / (2 * fact (numeral (num.Bit0 n))))"
DirProds_group_imp_groups: fixes G :: "'e \<Rightarrow> ('f, 'g) monoid_scheme"   and I :: "'e set"   and i :: "'e" assumes "Group.group (DirProds G I)"   and "i \<in> I" shows "Group.group (G i)"
cring_lcf_of_sub_in_ltrm: fixes g :: "nat \<Rightarrow> 'a"   and f :: "nat \<Rightarrow> 'a"   and n :: "nat" assumes "g \<in> carrier P"   and "f \<in> carrier P"   and "deg R f = n"   and "0 < deg R g"   and "f (deg R f) \<otimes> g (deg R g) [^] n \<noteq> \<zero>" shows "Cring_Poly.compose R (monom P (f (deg R f)) (deg R f)) g (deg R (Cring_Poly.compose R (monom P (f (deg R f)) (deg R f)) g)) = f (deg R f) \<otimes> g (deg R g) [^] n"
integral_norm_bound: fixes M :: "'c measure"   and f :: "'c \<Rightarrow> 'b" shows "norm (integral\<^sup>L M f) \<le> LINT x|M. norm (f x)"
carrier_substitute1: fixes t :: "'b ttree"   and f :: "'b \<Rightarrow> 'b ttree"   and T :: "'b set" shows "carrier t \<subseteq> carrier (substitute f T t)"
exhaust: fixes y :: "p" obtains "y = noreq"   | "y = req"
complete_ht_new: shows "<emp> hm_new <is_hashtable (replicate (def_hashmap_size TYPE('a)) [])>"
ccompatible_def: fixes \<R> :: "('b \<Rightarrow> 'c) set set"   and cc :: "('b, 'c) cconstraint" shows "ccompatible \<R> cc \<equiv> \<forall>R\<in>\<R>. R \<subseteq> \<lbrace>cc\<rbrace> \<or> \<lbrace>cc\<rbrace> \<inter> R = {}"
one_is_fermat_liar: fixes n :: "nat" assumes "1 < n" shows "fermat_liar 1 n"
add_mset_eq_single: fixes b :: "'a"   and M :: "'a multiset"   and a :: "'a" shows "(add_mset b M = {#a#}) = (b = a \<and> M = {#})"
reduce_basis_1: fixes fs :: "int vec list" assumes "local.reduce_basis = fs" shows "lattice_of fs = L"
gpv_2: shows "\<I> \<turnstile>g gpv \<surd>"
less_oFixI: fixes a :: "ordinal"   and F :: "ordinal \<Rightarrow> ordinal" assumes "a < F a" shows "a < oFix F a"
a: shows "[x \<^bold>=\<^sub>E y \<^bold>& y \<^bold>=\<^sub>E z in v]"
sorted_wrt_Cons: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and ys :: "'a list" shows "sorted_wrt P (x # ys) = (Ball (set ys) (P x) \<and> sorted_wrt P ys)"
mod_add_mod_if: fixes a :: "nat"   and m :: "nat"   and b :: "nat" shows "(a mod m + b mod m) mod m = (if a mod m + b mod m < m then a mod m + b mod m else a mod m + b mod m - m)"
mono_Span_subset: fixes R :: "('a, 'b) ring_scheme"   and K :: "'a set"   and Us :: "'a list"   and Vs :: "'a list" assumes "ring R"   and "subfield K R"   and "set Us \<subseteq> ring.Span R K Vs"   and "set Vs \<subseteq> carrier R" shows "ring.Span R K Us \<subseteq> ring.Span R K Vs"
inject: fixes x :: "hmultiset multiset"   and ya :: "hmultiset multiset" shows "(HMSet x = HMSet ya) = (x = ya)"
is_dghmD_7: fixes \<alpha> :: "V"   and \<AA> :: "V"   and \<BB> :: "V"   and \<FF> :: "V" assumes "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<BB>" shows "\<FF>\<lparr>HomCod\<rparr> = \<BB>"
latestAct_prop_2: fixes n :: "nat"   and nid :: "'nid"   and t :: "nat \<Rightarrow> cnf" assumes "\<exists>n'. latestAct_cond nid t n n'" shows "\<langle>nid \<leftarrow> t\<rangle>\<^bsub>n\<^esub> < n"
fib_rec_odd: fixes n :: "nat" shows "fib (Suc (2 * n)) = (fib n)\<^sup>2 + (fib (Suc n))\<^sup>2"
mat_leqb_eqc_split_correct2: fixes c :: "rat vec"   and A :: "rat mat"   and b :: "rat vec"   and X :: "(nat, rat) mapping"   and x :: "rat vec"   and y :: "rat vec" assumes "dim_vec c = dim_row A\<^sup>T"   and "dim_vec b = dim_col A\<^sup>T"   and "\<langle>X\<rangle> \<Turnstile>\<^sub>c\<^sub>s set (mat_leqb_eqc A b c)"   and "(x, y) = split_n_m_x (dim_row A\<^sup>T) (dim_col A\<^sup>T) X" shows "[y\<^sub>v*A]=c"
simps_2: fixes k :: "'a"   and v :: "'c"   and k' :: "'a"   and v' :: "'c"   and xs :: "('a \<times> 'c) list" shows "update_by_pair (k, v) ((k', v') # xs) = (case comp k k' of Eq \<Rightarrow> if v = (0::'c) then xs else (k, v) # xs | Lt \<Rightarrow> if v = (0::'c) then (k', v') # xs else (k, v) # (k', v') # xs | Gt \<Rightarrow> (k', v') # update_by_pair (k, v) xs)"
less_sets_Suc_Max: fixes A :: "nat set" assumes "finite A" shows "A \<lless> {Suc (Max A)..}"
exec_Call_body': fixes p :: "'d"   and \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and s :: "('e, 'f) xstate"   and t :: "('e, 'f) xstate" assumes "p \<in> dom \<Gamma>" shows "\<Gamma>\<turnstile> \<langle>Call p,s\<rangle> \<Rightarrow> t = \<Gamma>\<turnstile> \<langle>the (\<Gamma> p),s\<rangle> \<Rightarrow> t"
antisym: fixes b :: "'k multiset"   and a :: "'k multiset" assumes "b \<subseteq># a"   and "a \<subseteq># b" shows "a = b"
lift_spmf_None: shows "lift_spmf (return_pmf None) = Fail"
is_execution_fragment_def: fixes A :: "('e set \<times> 'e set \<times> 'e set) \<times> 'f set \<times> ('f \<times> 'e \<times> 'f) set"   and ex :: "(nat \<Rightarrow> 'e option) \<times> (nat \<Rightarrow> 'f)" shows "is_execution_fragment A ex \<equiv> let act = fst ex; state = snd ex in \<forall>n a. (act n = None \<longrightarrow> state (Suc n) = state n) \<and> (act n = Some a \<longrightarrow> (state n, a, state (Suc n)) \<in> trans_of A)"
indep_var_eq: fixes S :: "'c measure"   and X :: "'b \<Rightarrow> 'c"   and T :: "'c measure"   and Y :: "'b \<Rightarrow> 'c" shows "X.indep_var S X T Y = ((X.random_variable S X \<and> X.random_variable T Y) \<and> X.indep_set (sigma_sets (space (distr M S X)) {X -` A \<inter> space (distr M S X) |A. A \<in> sets S}) (sigma_sets (space (distr M S X)) {Y -` A \<inter> space (distr M S X) |A. A \<in> sets T}))"
record_simps_36: fixes f :: "char list list set \<Rightarrow> char list list set"   and x :: "'b state" shows "clock (update_defined_mods f x) = clock x"
literal_of_asciis_of_literal: fixes s :: "literal" shows "literal_of_asciis (asciis_of_literal s) = s"
rel_def: shows "rel = inv_image finite_psubset (\<lambda>S. nodeAbs ` {n. isNode n \<and> \<not> memb n S})"
lim2: shows "(((\<lambda>x. - 1 / ln x) \<circ> real_of_ereal) \<longlongrightarrow> 0) (at_left \<infinity>)"
r_kind_def: shows "r_kind \<equiv> Cn 1 r_ifz [recf.Id 1 0, Z, Cn 1 r_ifeq [recf.Id 1 0, r_const 1, r_const 1, r_pdec1]]"
list_all_length: fixes P :: "'a \<Rightarrow> bool"   and xs :: "'a list" shows "list_all P xs = (\<forall>n<length xs. P (xs ! n))"
poly_pcompose: fixes p :: "'a poly"   and q :: "'a poly"   and x :: "'a" shows "poly (p \<circ>\<^sub>p q) x = poly p (poly q x)"
taut2emptyrem: fixes \<phi> :: "'a"   and A :: "'a set" assumes "\<tturnstile> \<phi>" shows "A .\<bottom>. \<phi> = {}"
left_commute: fixes b :: "'b"   and a :: "'b"   and c :: "'b" shows "sup b (sup a c) = sup a (sup b c)"
ide_char: fixes f :: "('e set, 'e \<Rightarrow> 'e) concrete_category.arr" shows "ide f = (f \<noteq> Null \<and> Dom f \<in> {A. |A| <o \<AA>} \<and> Cod f = Dom f \<and> Map f = (\<lambda>x\<in>Dom f. x))"
lookup_row_to_poly: fixes ts :: "'c list"   and r :: "'d vec"   and i :: "nat" assumes "distinct ts"   and "dim_vec r = length ts"   and "i < length ts" shows "lookup (row_to_poly ts r) (ts ! i) = r $ i"
coeff_Var: fixes p :: "nat"   and m :: "nat \<Rightarrow>\<^sub>0 nat" shows "MPoly_Type.coeff (Var p) m = ((1::'a) when m = monomial 1 p)"
between_imp_mod_le: fixes b :: "nat"   and m :: "nat"   and k :: "nat"   and n :: "nat" assumes "b < m"   and "m * k \<le> n"   and "n \<le> m * k + b" shows "n mod m \<le> b"
smult_minus_right: fixes a :: "'a"   and p :: "'a poly" shows "smult a (- p) = - smult a p"
span_breakdown_eq: fixes x :: "'d"   and a :: "'d"   and S :: "'d set" shows "(x \<in> cspan (insert a S)) = (\<exists>k. x - k *\<^sub>C a \<in> cspan S)"
add_mset_remove_trivial_eq: fixes N :: "'b multiset"   and a :: "'b" shows "(N = add_mset a (N - {#a#})) = (a \<in># N)"
Red_preserves_type: fixes P :: "(char list list \<times> char list exp) prog"   and e :: "char list exp"   and s :: "heap \<times> sfields \<times> sheap"   and b :: "bool"   and e' :: "char list exp"   and s' :: "heap \<times> sfields \<times> sheap"   and b' :: "bool"   and E :: "char list \<Rightarrow> ty option"   and T :: "ty" assumes "wf_J_prog P"   and "P \<turnstile> \<langle>e,s,b\<rangle> \<rightarrow>* \<langle>e',s',b'\<rangle>"   and "P,E \<turnstile> s \<surd>"   and "iconf (shp s) e"   and "P,E,hp s,shp s \<turnstile> e : T" shows "\<exists>T'. P \<turnstile> T' \<le> T \<and> P,E,hp s',shp s' \<turnstile> e' : T'"
const: fixes c :: "real" shows "real_polynomial_function (\<lambda>x. c)"
simps: fixes I :: "'at set"   and C :: "'at Literal set" shows "I \<Turnstile> C = (\<exists>L. L \<in> C \<and> I \<Turnstile> L)"
scale_cancel_right: fixes a :: "'c"   and x :: "'c"   and b :: "'c" shows "(a * x = b * x) = (a = b \<or> x = (0::'c))"
beta_alpha_orth: shows "inner_prod \<beta> \<alpha> = 0"
piecewise_continuous_on_imp_absolutely_integrable: fixes a :: "real"   and b :: "real"   and I :: "real set"   and f :: "real \<Rightarrow> 'c" assumes "piecewise_continuous_on a b I f" shows "f absolutely_integrable_on {a..b}"
left_neutral: fixes a :: "'c" shows "(0::'c) + a = a"
\<phi>adj\<phi>: shows "\<phi>* o\<^sub>C\<^sub>L \<phi> = id_cblinfun"
glbound_in_ofI: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and b :: "'a"   and P :: "'a set"   and x :: "'a"   and y :: "'a" assumes "ordering less_eq less"   and "b \<in> P"   and "ordering.lbound_of less_eq x y b"   and "\<And>a. \<lbrakk>a \<in> P; ordering.lbound_of less_eq x y a\<rbrakk> \<Longrightarrow> less_eq a b" shows "ordering.glbound_in_of less_eq P x y b"
trace_eq_simcl_bindI: fixes p :: "'a spmf"   and S :: "'b spmf \<Rightarrow> 'c spmf \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'b spmf"   and g :: "'a \<Rightarrow> 'c spmf" assumes "\<And>x. x \<in> set_spmf p \<Longrightarrow> S (f x) (g x)" shows "trace_eq_simcl S (p \<bind> f) (p \<bind> g)"
simps: fixes a1 :: "'a list"   and a2 :: "'a list"   and r :: "('a \<times> 'a) set" shows "((a1, a2) \<in> genPrefix r) = (a1 = [] \<and> a2 = [] \<or> (\<exists>xs ys x y. a1 = x # xs \<and> a2 = y # ys \<and> (xs, ys) \<in> genPrefix r \<and> (x, y) \<in> r) \<or> (\<exists>xs ys zs. a1 = xs \<and> a2 = ys @ zs \<and> (xs, ys) \<in> genPrefix r))"
OrderingSetIso_axioms: shows "OrderingSetIso (\<^bold>\<le>) (\<^bold><) (\<^bold>\<le>*) (\<^bold><*) P f"
comp_mwb_lens: fixes x :: "'a \<Longrightarrow> 'b"   and y :: "'b \<Longrightarrow> 'c" assumes "mwb_lens x"   and "mwb_lens y" shows "mwb_lens (x ;\<^sub>L y)"
Extend_def: fixes S :: "('a, 'b) fm set"   and f :: "nat \<Rightarrow> ('a, 'b) fm" shows "Extend S f \<equiv> \<Union> (range (extend S f))"
relin_Ex: fixes x :: "'a" assumes "x \<in> rel_interior S" shows "\<exists>e>0. \<forall>x'\<in>affine hull S. dist x' x < e \<longrightarrow> x' \<in> rel_interior S"
assms_2: shows "s' = s\<lparr>s.mem := (s.mem s)[i := mem']\<rparr>"
arrI: fixes f :: "'a" assumes "left b f" shows "S.arr f"
is_strict_rewrite_ord_def: fixes rel :: "'t \<times> ('a \<Rightarrow>\<^sub>0 'b) \<Rightarrow> 't \<times> ('a \<Rightarrow>\<^sub>0 'b) \<Rightarrow> bool" shows "is_strict_rewrite_ord rel = is_rewrite_ord (\<lambda>x y. \<not> rel y x)"
is_type_void: fixes P :: "'b prog" shows "is_type P Void = True"
ty_cmd_pres_aux: fixes \<Gamma>l :: "String.literal \<Rightarrow> unit option"   and \<Gamma>g :: "String.literal \<Rightarrow> unit option"   and loop :: "bool"   and c :: "cmd"   and a :: "action + brk_ctd"   and c' :: "cmd" assumes "ty_cmd ((\<Gamma>l, \<Gamma>g), loop) c"   and "cfg c a c'" shows "ty_cmd ((\<Gamma>l, \<Gamma>g), loop) c' \<or> a = Inr AIBreak \<or> a = Inr AIContinue"
clear_signs: shows "signs_smash (get_signs_R (calculate_data_R p qs1)) (get_signs_R (calculate_data_R p qs2)) = get_signs_R (snd (combine_systems_R p (qs1, calculate_data_R p qs1) (qs2, calculate_data_R p qs2)))"
disj_matrices_zero2: fixes A :: "'a matrix" shows "disj_matrices A 0"
inverse_functor_unique': fixes C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and D :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and F :: "'d \<Rightarrow> 'c"   and G :: "'c \<Rightarrow> 'd"   and F' :: "'d \<Rightarrow> 'c" assumes "inverse_functors C D F G"   and "inverse_functors C D F' G" shows "F = F'"
zero_neq_one_ow_axioms_def: fixes one :: "'b"   and zero :: "'b" shows "zero_neq_one_ow_axioms one zero \<equiv> zero \<noteq> one"
rep_eq: shows "poly.coeff 0 = (\<lambda>_. 0::'a)"
zth: fixes f :: "nat \<Rightarrow> nat" assumes "f \<in> {f. (\<forall>i\<in>{0..<n}. f i \<in> {0..<n}) \<and> (\<forall>i. i \<notin> {0..<n} \<longrightarrow> f i = i)} - {p. p permutes {0..<n}}" shows "det (mat\<^sub>r n n (\<lambda>i. A $$ (i, f i) \<cdot>\<^sub>v row B (f i))) = (0::'a)"
has_disc_negated_primitive_extractor: fixes m :: "'b match_expr"   and disc :: "'b \<Rightarrow> bool"   and sel :: "'b \<Rightarrow> 'c" assumes "normalized_nnf_match m" shows "has_disc_negated disc False m = (\<exists>a. Neg a \<in> set (fst (primitive_extractor (disc, sel) m)))"