hcju/isabelleps huggingface dataset:
PHILs\<^sub>n\<^sub>o\<^sub>r\<^sub>m_unfold: fixes n :: "nat"   and ps :: "nat list" assumes "0 < n"   and "length ps = Suc n" shows "PHILs\<^sub>n\<^sub>o\<^sub>r\<^sub>m (Suc n) ps = (PHILs\<^sub>n\<^sub>o\<^sub>r\<^sub>m n (butlast ps)|||RPHIL\<^sub>n\<^sub>o\<^sub>r\<^sub>m n (ps ! n))"
poly_zero: fixes p :: "'a poly"   and x :: "'a" assumes "poly p x = (0::'a)" shows "(p = 0) = (degree p = 0)"
fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s_inI_2: fixes f :: "('c, 'd) Term.term \<times> ('e, 'd) Term.term"   and F :: "(('c, 'd) Term.term \<times> ('e, 'd) Term.term) list"   and x :: "'d" assumes "f \<in> set F"   and "x \<in> fv (fst f)" shows "x \<in> fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s F"
naturality: fixes f :: "'a \<times> 'a" assumes "C.VV.arr f" shows "(fst (C.VV.cod f) \<star>\<^sub>C snd (C.VV.cod f)) \<cdot>\<^sub>C (fst f \<star>\<^sub>C snd f) = (fst f \<star>\<^sub>C snd f) \<cdot>\<^sub>C (fst (C.VV.dom f) \<star>\<^sub>C snd (C.VV.dom f))"
less_eq_option_None_code: fixes x :: "'a option" shows "(None \<le> x) = True"
m_defs_1: fixes cval :: "'a \<Rightarrow> 'b \<Rightarrow> real"   and indis :: "('b \<times> 'b) set"   and aval :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and tval :: "'d \<Rightarrow> 'b \<Rightarrow> bool"   and theta :: "(('d, 'c, 'a) cmd \<times> ('d, 'c, 'a) cmd) set"   and c :: "('d, 'c, 'a) cmd"   and d :: "('d, 'c, 'a) cmd"   and s :: "'b"   and t :: "'b"   and P :: "nat set set"   and F :: "nat set \<Rightarrow> nat set" assumes "PL_Indis cval indis" shows "PL_Indis.mC_C aval tval cval indis theta c d s t P F \<equiv> PL_Indis.mC_C_part c d P F \<and> inj_on F P \<and> PL_Indis.mC_C_wt aval tval cval c d s t P F \<and> PL_Indis.mC_C_eff_cont aval tval cval indis theta c d s t P F"
simps_3: fixes v :: "'c"   and va :: "'c"   and vb :: "'c Tree"   and vc :: "'c Tree" shows "siftDown (T v (T va vb vc) E) = (if val (T va vb vc) \<le> v then T v (T va vb vc) E else T (val (T va vb vc)) (siftDown (set_val (T va vb vc) v)) E)"
assms: shows "h \<turnstile> get_parent child \<rightarrow>\<^sub>r Some ptr"
axioms_1: fixes f :: "'a \<Rightarrow> 'b" assumes "bounded_clinear f" shows "clinear f"
maxsimpchain_def: fixes xs :: "'a set list" shows "gallery xs \<equiv> Ball (set xs) chamber \<and> adjacentchain xs"
generate_incl: fixes H :: "'a set" assumes "H \<subseteq> carrier G" shows "generate G H \<subseteq> carrier G"
smallstepsV_ps_append: fixes \<pi> :: "Syntax.term list"   and eV :: "Syntax.term"   and i :: "nat"   and \<pi>' :: "Syntax.term list"   and eV' :: "Syntax.term"   and X :: "Syntax.term list" shows "\<lless>\<pi>, eV\<ggreater> V\<rightarrow>i \<lless>\<pi>', eV'\<ggreater> = \<lless>\<pi> @ X, eV\<ggreater> V\<rightarrow>i \<lless>\<pi>' @ X, eV'\<ggreater>"
cn_weak: shows "\<forall>k\<le>n. 0 < k \<longrightarrow> (\<exists>c. v c = k)"
basis_iff_basis_in: fixes B :: "'a set" shows "\<E>.basis B = \<E>.basis_in \<E> B"
cases: fixes a1 :: "('s, 'v) tm"   and a2 :: "('s, 'v) tm" assumes "gt_chop a1 a2" obtains t :: "('s, 'v) tm" and s :: "('s, 'v) tm" where "a1 = t" and "a2 = s" and "is_App t" and "chop t \<ge>\<^sub>t s"
E: shows "\<forall>xs X. xs \<in> snd (Rep_process P) \<longrightarrow> (xs, X) \<in> fst (Rep_process P)"
lmap_eq_LNil: fixes f :: "'c \<Rightarrow> 'b"   and xs :: "'c llist" shows "(lmap f xs = LNil) = (xs = LNil)"
ExcessTab_vertex: fixes p :: "nat \<times> nat"   and g :: "graph" assumes "p \<in> set (ExcessTable g (vertices g))" shows "fst p \<in> \<V> g"
gb_schema_aux_term1_wf_on: fixes d :: "'a \<Rightarrow> nat"   and K :: "'k set"   and gs :: "(('t \<Rightarrow>\<^sub>0 'd) \<times> nat \<times> 'e) list"   and m :: "nat" assumes "dickson_grading d"   and "finite K" shows "wfp_on (\<lambda>x y. (x, y) \<in> gb_schema_aux_term1) {x. args_to_set (gs, x) \<subseteq> dgrad_p_set d m \<and> component_of_term ` Keys (args_to_set (gs, x)) \<subseteq> K}"
set_rw: shows "{column i (fst (QR_decomposition A)) |i. i \<le> k} = insert (column k (fst (QR_decomposition A))) {column i (fst (QR_decomposition A)) |i. i < k}"
list_interval2: fixes a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" shows "list_interval [a, b] [c, d] = {[x, y] |x y. x \<in> {a..c} \<and> y \<in> {b..d}}"
M_5: fixes vs :: "nat \<Rightarrow> nat"   and x :: "nat" assumes "vs 0 = x"   and "vs 1 = 0"   and "x \<in> A \<union> {}" shows "\<exists>vs'. ((2, vs), 0, vs') \<in> (step M)\<^sup>* \<or> ((2, vs), Suc 0, vs') \<in> (step M)\<^sup>*"
game\<^sub>0_def: shows "game\<^sub>0 = TRY lift_spmf key_gen \<bind> (\<lambda>(pk, uu_). inline interceptor (\<A>\<^sub>1 pk) {} \<bind> (\<lambda>(((m\<^sub>1, m\<^sub>2), \<sigma>), s). assert_gpv (valid_plains m\<^sub>1 m\<^sub>2) \<bind> (\<lambda>_. lift_spmf coin_spmf \<bind> (\<lambda>b. encrypt pk (if b then m\<^sub>1 else m\<^sub>2) \<bind> (\<lambda>c. inline interceptor (\<A>\<^sub>2 c \<sigma>) s \<bind> (\<lambda>(b', s'). Done (b' = b))))))) ELSE lift_spmf coin_spmf"
aas_not_empt: shows "\<forall>i<n. AAs ! i \<noteq> {#}"
relevant_vars_subl: fixes l :: "fterm literal"   and \<sigma>\<^sub>1 :: "char list \<Rightarrow> fterm"   and \<sigma>\<^sub>2 :: "char list \<Rightarrow> fterm" assumes "\<forall>x\<in>vars\<^sub>l l. \<sigma>\<^sub>1 x = \<sigma>\<^sub>2 x" shows "l \<cdot>\<^sub>l \<sigma>\<^sub>1 = l \<cdot>\<^sub>l \<sigma>\<^sub>2"
effective_matrix_kauff_mat_1: fixes ws :: "wall" assumes "is_tangle_diagram ws" shows "rat_poly.row_length (kauff_mat ws) = 2 ^ nat (domain_wall ws)"
fv_list\<^sub>s\<^sub>s\<^sub>t\<^sub>p_is_fv\<^sub>s\<^sub>s\<^sub>t\<^sub>p: fixes a :: "('d, 'c) stateful_strand_step" shows "fv\<^sub>s\<^sub>s\<^sub>t\<^sub>p a = set (fv_list\<^sub>s\<^sub>s\<^sub>t\<^sub>p a)"
simps_3: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and Exit :: "'b"   and Def :: "'b \<Rightarrow> 'c set"   and Use :: "'b \<Rightarrow> 'c set"   and ParamDefs :: "'b \<Rightarrow> 'c list"   and ParamUses :: "'b \<Rightarrow> 'c set list"   and m :: "'b"   and x :: "nat" assumes "SDG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main Exit Def Use ParamDefs ParamUses" shows "SDG.valid_SDG_node sourcenode targetnode kind valid_edge procs (Formal_out (m, x)) = (\<exists>a Q p f ins outs. valid_edge a \<and> kind a = Q\<hookleftarrow>\<^bsub>p\<^esub>f \<and> sourcenode a = m \<and> (p, ins, outs) \<in> set procs \<and> x < length outs)"
pc: shows "path_connected {r. 0 \<le> r \<and> P r}"
comp_null_1: fixes f :: "'a" shows "C.null \<cdot> f = C.null"
arrI: fixes f :: "'a" assumes "\<guillemotleft>f : trg (src f) \<rightarrow> trg e\<guillemotright>" shows "hom'.arr f"
r_pdec122_def: shows "r_pdec122 \<equiv> Cn 1 r_pdec1 [r_pdec22]"
lcm_proj2_iff_int: fixes m :: "int"   and n :: "int" shows "(lcm m n = \<bar>n\<bar>) = (m dvd n)"
min_elt_def: fixes B :: "'a set" shows "min_elt B = (SOME x. x \<in> B \<and> (\<forall>y\<in>A. P y x \<longrightarrow> y \<notin> B))"
ImpI: fixes F :: "'b formula"   and \<Gamma> :: "'b formula set"   and G :: "'b formula" assumes "F \<triangleright> \<Gamma> \<turnstile> G" shows "\<Gamma> \<turnstile> F \<^bold>\<rightarrow> G"
pr0: shows "\<guillemotleft>pr0 a b : local.prod a b \<rightarrow> b\<guillemotright>"
exhaust: fixes y :: "'a option" obtains "y = None"   | x2 :: "'a" where "y = \<lfloor>x2\<rfloor>"
join_restrict: fixes X :: "'b option list set"   and n :: "nat"   and A :: "nat set"   and Y :: "'b option list set"   and B :: "nat set"   and b :: "bool"   and v :: "'b option list" assumes "\<And>v. v \<in> X \<Longrightarrow> wf_tuple n A v"   and "\<And>v. v \<in> Y \<Longrightarrow> wf_tuple n B v"   and "\<not> b \<Longrightarrow> B \<subseteq> A" shows "(v \<in> join X b Y) = (wf_tuple n (A \<union> B) v \<and> restrict A v \<in> X \<and> join_cond b Y (restrict B v))"
cases: fixes x :: "'a cstate \<times> 'a insec_query" obtains m' :: "'a" where "x = (Void, Edit m'__)"   | m :: "'a" and m' :: "'a" where "x = (Store m__, Edit m')"   | m :: "'a" where "x = (Store ma__, Forward)"   | m :: "'a" where "x = (Store mb__, Drop)"   | m :: "'a" where "x = (Store m, Look)"   | uu_ :: "'a insec_query" where "x = (cstate.Fail, uu___)"   | v :: "'a" and uu_ :: "'a insec_query" where "x = (cstate.Collect v__, uu_)"   | "x = (Void, Look)"   | "x = (cstate.Fail, Look)"   | v :: "'a" where "x = (cstate.Collect va__, Look)"   | "x = (Void, Forward)"   | "x = (cstate.Fail, Forward)"   | v :: "'a" where "x = (cstate.Collect vb__, Forward)"   | "x = (Void, Drop)"   | "x = (cstate.Fail, Drop)"   | v :: "'a" where "x = (cstate.Collect v, Drop)"
lebesgue_integral_count_space_finite: fixes A :: "'d set"   and f :: "'d \<Rightarrow> 'e" assumes "finite A" shows "integral\<^sup>L (count_space A) f = sum f A"
convex_le_Inf_differential: fixes I :: "real set"   and f :: "real \<Rightarrow> real"   and x :: "real"   and y :: "real" assumes "convex_on I f"   and "x \<in> interior I"   and "y \<in> I" shows "f x + (INF t\<in>{x<..} \<inter> I. (f x - f t) / (x - t)) * (y - x) \<le> f y"
cases: fixes a1 :: "('d \<times> ('c, real) cconstraint \<times> 'e \<times> 'c list \<times> 'd) set \<times> ('d \<Rightarrow> ('c, real) cconstraint)"   and a2 :: "'d"   and a3 :: "('c \<Rightarrow> real) set"   and a4 :: "'d"   and a5 :: "('c \<Rightarrow> real) set" assumes "a1 \<turnstile> \<langle>a2, a3\<rangle> \<leadsto>\<^sub>\<alpha> \<langle>a4, a5\<rangle>" obtains A :: "('d \<times> ('c, real) cconstraint \<times> 'e \<times> 'c list \<times> 'd) set \<times> ('d \<Rightarrow> ('c, real) cconstraint)" and l :: "'d" and Z :: "('c \<Rightarrow> real) set" and l' :: "'d" and Z' :: "('c \<Rightarrow> real) set" where "a1 = A" and "a2 = l" and "a3 = Z" and "a4 = l'" and "a5 = Closure\<^sub>\<alpha> Z'" and "A \<turnstile> \<langle>l, Z\<rangle> \<leadsto> \<langle>l', Z'\<rangle>"
seq_distrib_left_axioms: shows "seq_distrib_left (;) nil"
B_def: shows "B \<equiv> \<Union>s\<in>A\<^sup>\<star>. {suff A s}"
permute_CValue_def: fixes p :: "perm"   and v :: "CValue" shows "p \<bullet> v = v"
adds_lcs: fixes s :: "'c"   and t :: "'c" shows "s adds lcs s t"
index_uminus_mat_2: fixes A :: "'a mat" shows "dim_row (- A) = dim_row A"
leD: fixes y :: "('x, nat) pp"   and x :: "('x, nat) pp" assumes "ord y x" shows "\<not> ord_strict x y"
fun_relD: fixes f :: "'e \<Rightarrow> 'f"   and f' :: "'g \<Rightarrow> 'h"   and A :: "('e \<times> 'g) set"   and B :: "('f \<times> 'h) set"   and x :: "'e"   and x' :: "'g" assumes "(f, f') \<in> A \<rightarrow> B"   and "(x, x') \<in> A" shows "(f x, f' x') \<in> B"
RS_simps_19: shows "Maps.Dom S\<^sub>1 = src (tab\<^sub>0 s)"
step_Normal_elim_cases_2: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and css :: "(('e, 'd, 'f) com list \<times> ('e, 'd, 'f) com list) list"   and f :: "'f"   and u :: "('e, 'd, 'f) com list \<times> (('e, 'd, 'f) com list \<times> ('e, 'd, 'f) com list) list \<times> ('e, 'f) xstate" assumes "\<Gamma>\<turnstile> ([], css, Fault f) \<rightarrow> u" obtains nrms :: "('e, 'd, 'f) com list" and abrs :: "('e, 'd, 'f) com list" and cssa :: "(('e, 'd, 'f) com list \<times> ('e, 'd, 'f) com list) list" where "css = (nrms, abrs) # cssa" and "u = (nrms, cssa, Fault f)"
Lambert_W'_ex1: fixes x :: "real" assumes "x \<in> {- exp (- 1)..<0}" shows "\<exists>!w. w \<le> - 1 \<and> w * exp w = x"
MWLfSteps_det_cases_3: fixes c1 :: "('exp, 'id) MWLfCom"   and c2 :: "('exp, 'id) MWLfCom"   and m :: "'id \<Rightarrow> 'val"   and W :: "('exp, 'id) MWLfCom list"   and m' :: "'id \<Rightarrow> 'val" assumes "SRabbr (c1;c2) m W m'" obtains "W = [c2]" and "SRabbr c1 m [] m'"   | c1' :: "('exp, 'id) MWLfCom" and V :: "('exp, 'id) MWLfCom list" where "W = (c1';c2) # V" and "SRabbr c1 m (c1' # V) m'"
remove_term_coeff: fixes m0 :: "nat \<Rightarrow>\<^sub>0 nat"   and p :: "'b mpoly"   and m :: "nat \<Rightarrow>\<^sub>0 nat" shows "coeff (remove_term m0 p) m = (coeff p m when m \<noteq> m0)"
elims_3: fixes x :: "('c, 'd) tm"   and xa :: "dir list" assumes "\<not> position_of x xa" obtains uv_ :: "('c, 'd) hd" and uw_ :: "dir" and ux_ :: "dir list" where "x = Hd uv_" and "xa = uw_ # ux_"   | t :: "('c, 'd) tm" and s :: "('c, 'd) tm" and ds :: "dir list" where "x = App t__ sa__" and "xa = dir.Left # ds__" and "\<not> position_of t__ ds__"   | t :: "('c, 'd) tm" and s :: "('c, 'd) tm" and ds :: "dir list" where "x = App t s" and "xa = dir.Right # ds" and "\<not> position_of s ds"
that: fixes ws :: "('a, 'b) tm list" assumes "ws \<noteq> []"   and "hd ws = subst \<rho> w"   and "last ws = u"   and "\<forall>i. Suc i < length ws \<longrightarrow> (\<exists>p d. emb_step_at p d (ws ! i) = ws ! Suc i \<and> \<not> list_all (\<lambda>x. x = dir.Left) p)"   and "\<forall>i<length ws. head (ws ! i) = head w \<and> num_args (ws ! i) = num_args w"   and "\<forall>i<length ws. \<forall>k<num_args w. args (subst \<rho> w) ! k \<unrhd>\<^sub>e\<^sub>m\<^sub>b args (ws ! i) ! k" shows "thesis"
SD_efficient_auxdef: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> bool"   and p :: "'c pmf" shows "SD_efficient R p = (\<not> (\<exists>q\<in>lotteries_on {x. \<exists>i. R i x x}. p \<prec>[Pareto (SD \<circ> R)] q))"
neg_minus_divide_le_eq: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and 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 c :: "'b"   and b :: "'b"   and a :: "'b" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less c zero" shows "less_eq (uminus (divide b c)) a = less_eq (times a c) (uminus b)"
t_var: fixes \<Gamma> :: "nat \<Rightarrow> nat option"   and x :: "nat"   and C :: "nat"   and CT :: "nat \<Rightarrow> classDef option" assumes "\<Gamma> x = Some C" shows "CT;\<Gamma> \<turnstile> Var x : C"
l_span_gen_mono: fixes K :: "'a set"   and H :: "'a set"   and A :: "'b set" assumes "K \<subseteq> H"   and "H \<subseteq> carrier M"   and "ideal R A" shows "linear_span R M A K \<subseteq> linear_span R M A H"
neg_eq_iff_add_eq_0: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.group_add minus plus zero uminus" shows "(uminus a = b) = (plus a b = zero)"
integral_finite_singleton: fixes f :: "nat set set \<Rightarrow> real" shows "expectation f = (\<Sum>x\<in>Pow S_edges. f x * prob {x})"
fps_lr_inverse_one_minus_fps_X_1: shows "fps_left_inverse (1 - fps_X) (1::'b) = Abs_fps (\<lambda>n. 1::'b)"
lm014: fixes X :: "'c set"   and x :: "'d"   and Y :: "'c set"   and y :: "'d" shows "runiq (X \<times> {x} +< Y \<times> {y})"
tag_Star_non_empty_injI: fixes A :: "'a list set"   and x :: "'a list"   and y :: "'a list"   and z :: "'a list" assumes "tag_Star A x = tag_Star A y"   and "x @ z \<in> A\<star>"   and "x \<noteq> []" shows "y @ z \<in> A\<star>"
or_id: fixes v :: "i"   and x :: "i"   and y :: "i"   and w :: "i" assumes "(v = x \<or> v = y) \<and> O w v" shows "O w x \<or> O w y"
distribF: shows "dist_over_inf (F::'a::refinement_lattice \<Rightarrow> 'a::refinement_lattice)"
intro: fixes sourcenode :: "'f \<Rightarrow> 'g"   and targetnode :: "'f \<Rightarrow> 'g"   and kind :: "'f \<Rightarrow> 'h edge_kind"   and valid_edge :: "'f \<Rightarrow> bool"   and Entry :: "'g"   and Exit :: "'g"   and Def :: "'g \<Rightarrow> 'i set"   and Use :: "'g \<Rightarrow> 'i set"   and state_val :: "'h \<Rightarrow> 'i \<Rightarrow> 'j" assumes "StrongPostdomination sourcenode targetnode kind valid_edge Entry Exit"   and "CFGExit_wf sourcenode targetnode kind valid_edge Entry Def Use state_val Exit" shows "WeakControlDependencePDG sourcenode targetnode kind valid_edge Entry Def Use state_val Exit"
transfer_return: fixes R :: "'e \<Rightarrow> 'f \<Rightarrow> bool" shows "(R ===> local.rel_state R) State_Monad.return return"
make_affine_def: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and p :: "'b \<times> 'b \<times> 'b" assumes "class.ell_field divide inverse times one plus zero minus uminus" shows "ell_field.make_affine divide zero p = (let (x, y, z) = p in if z = zero then Infinity else Point (divide x z) (divide y z))"
sorted_list_of_multiset_def: fixes M :: "'a multiset" shows "sorted_list_of_multiset M = fold_mset insort [] M"
c: shows "c = Le_Constraint Leq_Rel (poly_of_vec (row A i)) (b $ i)"
uconnectedV_tracl: fixes F :: "nat uprod set" shows "(Restr (uconnected F) V)\<^sup>* = Restr (uconnected F) V \<union> nat_rel"
eWls_nchotomy: fixes eX :: "'a withERR"   and MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'a, 'b) model"   and s :: "'sort" shows "(\<exists>X. eX = OK X \<and> igWls MOD s X) \<or> \<not> eWls MOD s eX"
simps_1: fixes x1 :: "bool"   and x2 :: "nat"   and y1 :: "bool"   and y2 :: "nat" shows "(Lit x1 x2 = Lit y1 y2) = (x1 = y1 \<and> x2 = y2)"
SeqHRP_graph_aux_def: shows "SeqHRP_graph_aux \<equiv> lfp (\<lambda>p x1 x2. \<exists>SeqHRP_aux l s k sl sl' m n sm sm' sn sn' x x'. x1 = (x, x', s, k) \<and> x2 = LstSeqP s k (HPair x x') AND All2 l (SUCC k) (SyntaxN.Ex sl (SyntaxN.Ex sl' (HPair (Var l) (HPair (Var sl) (Var sl')) IN s AND (OrdP (Var sl) AND WRP (Var sl) (Var sl') OR SyntaxN.Ex m (SyntaxN.Ex n (SyntaxN.Ex sm (SyntaxN.Ex sm' (SyntaxN.Ex sn (SyntaxN.Ex sn' (Var m IN Var l AND Var n IN Var l AND HPair (Var m) (HPair (Var sm) (Var sm')) IN s AND HPair (Var n) (HPair (Var sn) (Var sn')) IN s AND Var sl EQ HPair (Var sm) (Var sn) AND Var sl' EQ Q_HPair (Var sm') (Var sn'))))))))))) \<and> atom l \<sharp> (s, k, sl, sl', m, n, sm, sm', sn, sn') \<and> atom sl \<sharp> (s, sl', m, n, sm, sm', sn, sn') \<and> atom sl' \<sharp> (s, m, n, sm, sm', sn, sn') \<and> atom m \<sharp> (s, n, sm, sm', sn, sn') \<and> atom n \<sharp> (s, sm, sm', sn, sn') \<and> atom sm \<sharp> (s, sm', sn, sn') \<and> atom sm' \<sharp> (s, sn, sn') \<and> atom sn \<sharp> (s, sn') \<and> atom sn' \<sharp> s)"
product_integrable_prod: fixes I :: "'a set"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> 'd" assumes "finite I"   and "\<And>i. i \<in> I \<Longrightarrow> integrable ((count_space \<circ> B') i) (f i)" shows "integrable (Pi\<^sub>M I (count_space \<circ> B')) (\<lambda>x. \<Prod>i\<in>I. f i (x i))"
dgrad_p_set_le_red: fixes d :: "'a \<Rightarrow> nat"   and F :: "('a \<Rightarrow>\<^sub>0 'c) set"   and p :: "'a \<Rightarrow>\<^sub>0 'c"   and q :: "'a \<Rightarrow>\<^sub>0 'c" assumes "dickson_grading d"   and "punit.red F p q" shows "punit.dgrad_p_set_le d {q} (insert p F)"
ij_le1_1: shows "i < card {i. i < dim_row A \<and> i \<in> I}"
Collect_disj_eq: fixes P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool" shows "{x. P x \<or> Q x} = {x. P x} \<union> {x. Q x}"
list_of_fset_def: fixes s :: "'b fset" shows "list_of_fset s = (SOME l. fset_from_list l = s \<and> distinct l)"
adjoint_transpose_right_4: fixes u :: "'a"   and v :: "'a"   and \<nu> :: "'a" assumes "ide u"   and "ide v"   and "src v = trg f\<^sup>*"   and "src u = trg f"   and "\<guillemotleft>\<nu> : v \<Rightarrow> u \<star> f\<guillemotright>" shows "T0.trnr\<^sub>\<eta> v (T0.trnr\<^sub>\<epsilon> u \<nu>) = \<nu>"
is_Groebner_basis_empty: shows "is_Groebner_basis {}"
compl_unique: fixes x :: "'a"   and y :: "'a" assumes "x \<cdot> y = 0"   and "x + y = 1" shows "- x = y"
U0_V0_not_in_BC: shows "U\<^sub>0 \<union> V\<^sub>0 \<notin> BC"
hom_imp_img_subring: fixes h :: "'a \<Rightarrow> 'e"   and S :: "('e, 'f) ring_scheme"   and K :: "'a set" assumes "h \<in> ring_hom R S"   and "subring K R" shows "ring (S\<lparr>carrier := h ` K, one := h \<one>, zero := h \<zero>\<rparr>)"
f4_red_aux_def: fixes bs :: "(('t \<Rightarrow>\<^sub>0 'd) \<times> nat \<times> 'e) list"   and ps :: "((('t \<Rightarrow>\<^sub>0 'd) \<times> nat \<times> 'e) \<times> ('t \<Rightarrow>\<^sub>0 'd) \<times> nat \<times> 'e) list" shows "f4_red_aux bs ps = (let aux = sym_preproc (map fst bs) (pdata_pairs_to_list ps) in Macaulay_red (fst aux) (snd aux))"
demand_Suc_Least: fixes f :: "C \<rightarrow> 'a" assumes "f\<cdot>\<bottom> = \<bottom>"   and "demand f \<noteq> C\<^sup>\<infinity>" shows "demand f = C\<^bsup>Suc (LEAST n. f\<cdot>C\<^bsup>Suc n\<^esup> \<noteq> \<bottom>)\<^esup>"
GetElim1: fixes M :: "Var list \<times> (int \<times> Instr) list \<times> int"   and l :: "int"   and os :: "Val list"   and S :: "(Var \<times> Val) list"   and h :: "(Addr \<times> Class \<times> (Field \<times> Val) list) list"   and n :: "nat"   and ll :: "int"   and t :: "Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list"   and d :: "Class"   and F :: "Field" assumes "(M, l, (os, S, h), n, ll, t) \<in> Step"   and "get_ins M l = Some (getfield d F)" shows "\<exists>a Flds v ops. h\<down>a = Some (d, Flds) \<and> Flds\<down>F = Some v \<and> os = RVal (Loc a) # ops \<and> n = Suc 0 \<and> t = (v # ops, S, h) \<and> ll = l + 1"
cases: fixes a1 :: "('d, 'e) ta_rule set"   and a2 :: "'e tree"   and a3 :: "'d" assumes "accs a1 a2 a3" obtains q :: "'d" and f :: "'e" and qs :: "'d list" and \<delta> :: "('d, 'e) ta_rule set" and ts :: "'e tree list" where "a1 = \<delta>" and "a2 = NODE f ts" and "a3 = q" and "q \<rightarrow> f qs \<in> \<delta>" and "length ts = length qs" and "\<And>i. i < length qs \<Longrightarrow> accs \<delta> (ts ! i) (qs ! i)"
elims: fixes x :: "('b \<times> nat) tree"   and y :: "('b \<times> nat) tree" assumes "del_min x = y" obtains "x = \<langle>\<rangle>" and "y = \<langle>\<rangle>"   | l :: "('b \<times> nat) tree" and uu_ :: "'b \<times> nat" and r :: "('b \<times> nat) tree" where "x = \<langle>l, uu_, r\<rangle>" and "y = merge l r"
qq3: shows "supp r \<subseteq> q \<bullet> set as \<union> set bs"
separation: fixes x :: "'a"   and y :: "'a" assumes "x \<in> F"   and "y \<in> F"   and "x \<noteq> y" shows "D \<le> dist x y"
E_def: shows "E \<equiv> {A \<times> B |A B. A \<in> sets M1 \<and> B \<in> sets M2}"
ustar_alt_def: fixes P :: "'b hrel" shows "P\<^sup>\<star> = (\<Sqinter> i \<bullet> (power.power II (;;) P) i)"
Seq_t_assoc: fixes c1 :: "com"   and c2 :: "com"   and c3 :: "com"   and s :: "char list \<Rightarrow> int"   and p :: "nat"   and s' :: "char list \<Rightarrow> int" shows "(c1;; c2;; c3, s) \<Rightarrow> p \<Down> s' = (c1;; (c2;; c3), s) \<Rightarrow> p \<Down> s'"
simps_2: fixes e :: "String.literal \<Rightarrow> 'a"   and f :: "String.literal \<Rightarrow> 'a list \<Rightarrow> 'a"   and t :: "tm"   and l :: "tm list" shows "semantics_list e f (t # l) = semantics_term e f t # semantics_list e f l"
elims: fixes x :: "'b run"   and xa :: "clock"   and xb :: "nat"   and y :: "nat" assumes "#\<^sub>< x xa xb = y" obtains \<rho> :: "'b run" and K :: "clock" where "x = \<rho>__" and "xa = K__" and "xb = 0" and "y = 0"   | \<rho> :: "'b run" and K :: "clock" and n :: "nat" where "x = \<rho>" and "xa = K" and "xb = Suc n" and "y = #\<^sub>\<le> \<rho> K n"
subst_1: fixes y :: "name"   and t' :: "lam"   and x :: "name" shows "Var x[y::=t'] = (if x = y then t' else Var x)"
bind_lzero: fixes f :: "'d \<Rightarrow> 'c option" shows "None \<bind> f = None"