hcju/isabelleps huggingface dataset:
SUP_upper2: fixes Inf :: "'c set \<Rightarrow> 'c"   and Sup :: "'c set \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and i :: "'d"   and A :: "'d set"   and u :: "'c"   and f :: "'d \<Rightarrow> 'c" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "i \<in> A"   and "less_eq u (f i)" shows "less_eq u (Sup (f ` A))"
left_unique_alt_def: fixes R :: "'i \<Rightarrow> 'j \<Rightarrow> bool" shows "left_unique R = (R OO R\<inverse>\<inverse> \<le> (=))"
edge_det: fixes a :: "'edge"   and a' :: "'edge" assumes "valid_edge a"   and "valid_edge a'"   and "sourcenode a = sourcenode a'"   and "targetnode a = targetnode a'" shows "a = a'"
isval_if_inv: fixes e1 :: "exp"   and e2 :: "exp"   and e3 :: "exp"   and P :: "bool" assumes "isval (EIf e1 e2 e3)" shows "P"
Inf_ow_axioms: shows "Inf_ow U\<^sub>2 Inf\<^sub>2"
that: assumes "r = fst (alloc xs h)"   and "h' = snd (alloc xs h)"   and "get h' r = xs"   and "present h' r"   and "\<not> present h r" shows "thesis"
r_2: shows "\<forall>t. t \<noteq> 0 \<and> dist t 0 < r \<longrightarrow> flow0 x t \<noteq> x"
dom_map_ran: fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and al :: "('a \<times> 'b) list" shows "fst ` set (map_ran f al) = fst ` set al"
sifum_example2_def: shows "sifum_example2 \<equiv> sifum_lang_no_dma ev\<^sub>A\<^sub>C ev\<^sub>B\<^sub>C aexp_vars\<^sub>C bexp_vars\<^sub>C \<and> sifum_example2_axioms"
bind_pure_returns_result_I2: fixes f :: "('l, 'm, 'n) prog"   and h :: "'l"   and g :: "'n \<Rightarrow> ('l, 'm, 'o) prog"   and y :: "'o" assumes "pure f h"   and "h \<turnstile> ok f"   and "\<And>x. h \<turnstile> f \<rightarrow>\<^sub>r x \<Longrightarrow> h \<turnstile> g x \<rightarrow>\<^sub>r y" shows "h \<turnstile> Heap_Error_Monad.bind f g \<rightarrow>\<^sub>r y"
exhaust_sel: fixes foo :: "('f, 'g) foo" obtains "foo = make_foo (foo.field_1 foo) (foo.field_2 foo)"
set_update_with_aux: fixes xs :: "('a \<times> 'b) list"   and v :: "'b"   and k :: "'a"   and f :: "'b \<Rightarrow> 'b" assumes "distinct (map fst xs)" shows "list.set (AList.update_with_aux v k f xs) = list.set xs - {k} \<times> UNIV \<union> {(k, f (case map_of xs k of None \<Rightarrow> v | Some v \<Rightarrow> v))}"
Fun_total: fixes F :: "ZF"   and U :: "ZF"   and V :: "ZF"   and a :: "ZF" assumes "Elem F (Fun U V)"   and "Elem a U" shows "\<exists>x. Elem (Opair a x) F"
prems_3: shows "tObsC (t \<leadsto> s) = tObsC (t' \<leadsto> s')"
rexp_DA_axioms: shows "rexp_DA ACI_norm (\<lambda>a r. \<guillemotleft>deriv a r\<guillemotright>) nullable lang"
CompFunCarAudioSystem_the3: shows "the (CompFun CarAudioSystem ''Off'') = {}"
SubstTermP_dbtm: fixes t :: "dbtm"   and i :: "name"   and u :: "dbtm" assumes "wf_dbtm t" shows "{} \<turnstile> SubstTermP \<guillemotleft>Var i\<guillemotright> (quot_dbtm t) (quot_dbtm u) (quot_dbtm (subst_dbtm t i u))"
HaddP_SUCC_Ex2: fixes H :: "fm set"   and x :: "tm"   and y :: "tm"   and z :: "tm"   and z' :: "name" assumes "H \<turnstile> HaddP x (SUCC y) z"   and "atom z' \<sharp> (x, y, z)" shows "H \<turnstile> SyntaxN.Ex z' (HaddP x y (Var z') AND z EQ SUCC (Var z'))"
start_point_no_cycle: fixes x :: "'a" assumes "has_start_points_path x" shows "\<not> cycle x"
assms_2: assumes "set p \<subseteq> verts G"   and "set (vwalk_arcs p) \<subseteq> arcs_ends G \<and> p \<noteq> []" shows "P"
list_of_eucl_nth: fixes n :: "nat"   and x :: "'a" assumes "n < DIM('a)" shows "list_of_eucl x ! n = x \<bullet> Basis_list ! n"
AtGuardExpr: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "'d \<Rightarrow> ('e, 'd, 'f) ann list option"   and p :: "('e, 'd, 'f) ann"   and e :: "('e, 'd, 'f) com"   and a :: "'e set \<times> ('e, 'd, 'f) com"   and r :: "'e set"   and f :: "'f"   and b :: "'e set" assumes "atomicsR \<Gamma> \<Theta> p e a" shows "atomicsR \<Gamma> \<Theta> (AnnRec r p) (Guard f b e) a"
mset_eq_perm: fixes xs :: "'c list"   and ys :: "'c list" shows "(mset xs = mset ys) = (xs <~~> ys)"
cutoff_r: fixes k :: "'b"   and v :: "real"   and l :: "'b kdt"   and r :: "'b kdt"   and p :: "(real, 'b) vec"   and c :: "(real, 'b) vec" assumes "invar (Node k v l r)"   and "p $ k \<le> v"   and "dist p c \<le> dist (p $ k) v" shows "\<forall>q\<in>set_kdt r. dist p c \<le> dist p q"
merge_guards_simps_3: fixes r :: "('d \<times> 'd) set" shows "merge_guards (Spec r) = Spec r"
path_CondFalse: fixes prog :: "cmd"   and l :: "nat"   and as :: "(w_node \<times> (char list \<Rightarrow> val option) edge_kind \<times> w_node) list"   and n' :: "w_node"   and b :: "expr"   and c\<^sub>1 :: "cmd" assumes "While_CFG.path prog (_ l _) as n'" shows "While_CFG.path (if (b) c\<^sub>1 else prog) ((_ l _) \<oplus> #:c\<^sub>1 + 1) (as \<oplus>s #:c\<^sub>1 + 1) (n' \<oplus> #:c\<^sub>1 + 1)"
hyps_2: assumes "max_reg a2 = None" shows "aval a2 (join_ir i r) = aval a2 (join_ir i r')"
fatom_of_hatom_bij: shows "bij_betw fatom_of_hatom UNIV ground_fatoms"
closure_greaterThanAtMost: fixes a :: "'b"   and b :: "'b" assumes "a < b" shows "closure {a<..b} = {a..b}"
empty_not_nhd: fixes x :: "'a" shows "{} \<notin> nhds x"
cscalar_prod_adjoint: fixes M :: "complex mat"   and nB :: "nat"   and nA :: "nat"   and v :: "complex vec"   and u :: "complex vec" assumes "M \<in> carrier_mat nB nA"   and "dim_vec v = nA"   and "dim_vec u = nB" shows "v \<bullet>c (mat_adjoint M *\<^sub>v u) = (M *\<^sub>v v) \<bullet>c u"
zip_add_closed: fixes as :: "'r list"   and bs :: "'r list" assumes "set as \<subseteq> R"   and "set bs \<subseteq> R" shows "set (map2 (+) as bs) \<subseteq> R"
canonic_int_ordered_def: fixes S :: "'at Literal set set" shows "canonic_int_ordered S = {A. canonic_int_fun_ordered A S}"
Domainp_pcr_real: shows "Domainp pcr_real = Real.cauchy"
ring_axioms: shows "ring {0} ring0.subtraction ring0.subtraction 0 0"
sum: shows "(\<Sum>i\<in>UNIV. f i *s X $ i) = 0"
Plus_3: shows "s \<in> \<gamma>\<^sub>o S"
bin_lcp_all_lcp: fixes ws :: "'a list list" assumes "ws \<in> lists {x, y}" shows "i.\<alpha> \<le>p concat ws \<cdot> i.\<alpha>"
assms_6: shows "vars_of_lit (subst_lit L' \<sigma>) = {}"
rcsTr10: fixes H :: "'a set"   and a :: "'a"   and x :: "'a"   and y :: "'a" assumes "G \<guillemotright> H "   and "a \<in> carrier G"   and "x \<in> H \<bullet> a"   and "y \<in> H \<bullet> a" shows "x \<cdot> \<rho> y \<in> H"
\<W>_def: fixes P :: "(symbol \<times> character list) list set"   and k :: "nat" shows "\<W> P k = {u \<in> \<X> k. \<exists>p\<in>by_length k P. admissible (p @ [u])}"
simps_2: fixes \<nu> :: "'a set" shows "af_letter false \<nu> = false"
enn2real_positive_iff: fixes x :: "ennreal" shows "(0 < enn2real x) = (0 < x \<and> x < \<top>)"
homotopy_equivalence_homology_group_isomorphism: fixes X :: "'b topology"   and Y :: "'c topology"   and f :: "'b \<Rightarrow> 'c"   and g :: "'c \<Rightarrow> 'b"   and p :: "int" assumes "continuous_map X Y f"   and "continuous_map Y X g"   and "homotopic_with (\<lambda>h. True) X X (g \<circ> f) id"   and "homotopic_with (\<lambda>k. True) Y Y (f \<circ> g) id" shows "hom_induced p X {} Y {} f \<in> Group.iso (homology_group p X) (homology_group p Y)"
simps_1: fixes t :: "('b \<times> real) tree"   and A :: "'b set" shows "rinss [] t A = return (tree_sigma (count_space A \<Otimes>\<^sub>M lborel)) t"
assms: shows "x \<in> lset (ltake (enat (lfind P lst)) lst)"
prod_ord_pairI: fixes xs :: "('a \<times> 'd) list"   and ys :: "('a \<times> 'e) list"   and P :: "'a \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> bool" assumes "oalist_inv_raw xs"   and "oalist_inv_raw ys"   and "\<And>k. k \<in> fst ` set xs \<union> fst ` set ys \<Longrightarrow> P k (lookup_pair xs k) (lookup_pair ys k)" shows "prod_ord_pair P xs ys"
aff_dim_geq: fixes V :: "'c set" shows "- 1 \<le> aff_dim V"
emeasure_pair_measure_alt: fixes X :: "('c \<times> 'd) set"   and N :: "'c measure" assumes "X \<in> sets (N \<Otimes>\<^sub>M lborel)" shows "emeasure (N \<Otimes>\<^sub>M lborel) X = \<integral>\<^sup>+ x. emeasure lborel (Pair x -` X) \<partial>N"
fw_canonical_subs: fixes n :: "nat"   and k :: "nat"   and m :: "nat \<Rightarrow> nat \<Rightarrow> 'a" assumes "\<forall>i xs. i \<le> n \<and> set xs \<subseteq> {0..k} \<longrightarrow> (0::'a) \<le> len m i i xs"   and "k \<le> n" shows "canonical_subs n {0..k} (fw m n k)"
len: fixes L :: "fin_string" assumes "List.member (fin_string_upto n) L" shows "length (Rep_fin_string L) \<le> n"
simps_2: fixes x :: "char list"   and a :: "'a"   and ps :: "(char list \<times> 'a) list" shows "fun_rep ((x, a) # ps) = (fun_rep ps)(x := a)"
cases: fixes a1 :: "nat option \<times> (nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (val list \<times> val list \<times> char list \<times> char list \<times> nat) list"   and a2 :: "nat option \<times> (nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (val list \<times> val list \<times> char list \<times> char list \<times> nat) list"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> nat \<times> nat \<times> instr list \<times> (nat \<times> nat \<times> char list \<times> nat \<times> nat) list) list) list" assumes "P \<turnstile> a1 -jvm\<rightarrow>\<^sub>1 a2" obtains \<sigma> :: "nat option \<times> (nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (val list \<times> val list \<times> char list \<times> char list \<times> nat) list" and \<sigma>' :: "nat option \<times> (nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (val list \<times> val list \<times> char list \<times> char list \<times> nat) list" where "a1 = \<sigma>" and "a2 = \<sigma>'" and "exec (P, \<sigma>) = \<lfloor>\<sigma>'\<rfloor>"
ph1: fixes t :: "real" assumes "t \<in> {0..1}" shows "p (h1 t) = g1 t"
cont_on_bot: shows "continuous_on (path_image (\<lambda>x. twoC (0, x))) F"
contour_integral_split: fixes a :: "complex"   and b :: "complex"   and f :: "complex \<Rightarrow> complex"   and k :: "real"   and c :: "complex" assumes "continuous_on (closed_segment a b) f"   and "0 \<le> k"   and "k \<le> 1"   and "c - a = k *\<^sub>R (b - a)" shows "contour_integral (linepath a b) f = contour_integral (linepath a c) f + contour_integral (linepath c b) f"
map2_val_compat'_def: fixes f :: "('a \<times> 'd) list \<times> 'o \<Rightarrow> ('a \<times> 'e) list \<times> 'o" shows "map2_val_compat' f = (\<forall>zs. oalist_inv zs \<longrightarrow> oalist_inv (f zs) \<and> snd (f zs) = snd zs \<and> fst ` set (fst (f zs)) \<subseteq> fst ` set (fst zs))"
ln_gt_zero: fixes x :: "real" assumes "1 < x" shows "0 < real_ln x"
guha_umstaendlich: fixes a :: "'c"   and fe :: "('d, 'c) flow_entry_match"   and ft :: "('d, 'c) flow_entry_match list"   and \<gamma> :: "'d set \<Rightarrow> 'e \<Rightarrow> bool"   and p :: "'e" assumes "a = ofe_action fe"   and "fe \<in> set ft"   and "\<gamma> (ofe_fields fe) p"   and "\<forall>fe'\<in>set ft. ofe_prio fe < ofe_prio fe' \<longrightarrow> \<not> \<gamma> (ofe_fields fe') p" shows "guha_table_semantics \<gamma> ft p (Some a)"
inorder_delete: fixes t :: "('c \<times> 'd) tree23"   and x :: "'c" assumes "complete t"   and "sorted1 (inorder t)" shows "inorder (Tree23_Map.delete x t) = AList_Upd_Del.del_list x (inorder t)"
a1: shows "wf_graph \<lparr>nodes = V, edges = E\<rparr>"
simps_1: fixes r :: "real" shows "ennexp (ereal r) = ennreal (exp r)"
simps: fixes vid1 :: "'a"   and vid2 :: "'a"   and vid3 :: "'a"   and fid1 :: "'b"   and fid2 :: "'b"   and fid3 :: "'b"   and pid1 :: "'c"   and pid2 :: "'c"   and pid3 :: "'c"   and pid4 :: "'c"   and D :: "('b, 'c, 'a) formula list \<times> ('b, 'c, 'a) formula list"   and S :: "(nat \<times> ('b, 'c, 'a) step) list" assumes "ids vid1 vid2 vid3 fid1 fid2 fid3 pid1 pid2 pid3 pid4" shows "ids.proof_result (D, S) = ids.deriv_result (ids.start_proof D) S"
Sum2: fixes Q :: "ccs"   and \<alpha> :: "act"   and Q' :: "ccs"   and P :: "ccs" assumes "Q \<longmapsto>\<alpha> \<prec> Q'" shows "P \<oplus> Q \<longmapsto>\<alpha> \<prec> Q'"
set_decode_Suc: fixes n :: "nat"   and x :: "nat" shows "(Suc n \<in> set_decode x) = (n \<in> set_decode (x div 2))"
arr_dom_iff_arr: fixes f :: "'b" shows "D.arr (D.dom f) = D.arr f"
wt_subst_rm_vars: fixes \<delta> :: "'var \<Rightarrow> ('fun, 'var) Term.term"   and X :: "'var set" assumes "wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<delta>" shows "wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t (rm_vars X \<delta>)"
minus_minus: fixes x :: "'c" assumes "x \<in> carrier V" shows "\<ominus>\<^bsub>V\<^esub> (\<ominus>\<^bsub>V\<^esub> x) = x"
reach_foldl_def: fixes \<Sigma> :: "'b set"   and \<delta> :: "'c \<Rightarrow> 'b \<Rightarrow> 'c"   and q\<^sub>0 :: "'c" assumes "\<Sigma> \<noteq> {}" shows "reach \<Sigma> \<delta> q\<^sub>0 = {foldl \<delta> q\<^sub>0 w |w. set w \<subseteq> \<Sigma>}"
bd_inv_ins: fixes t :: "('a, 'b) rbt"   and x :: "'a"   and v :: "'b" assumes "bd_inv t" shows "bd_inv (ins x v t) \<and> black_depth t = black_depth (ins x v t)"
inputChainFresh: fixes x :: "name"   and xvec :: "name list"   and N :: "'a"   and P :: "('a, 'b, 'c) psi" shows "x \<sharp> inputChain xvec N P = (x mem xvec \<or> x \<sharp> N \<and> x \<sharp> P)"
elims: fixes x :: "'e list"   and xa :: "'e list set"   and y :: "'e list" assumes "mcp' x xa = y" obtains z :: "'e list" and W :: "'e list set" where "x = z" and "xa = W" and "y = The (mcp z W)"
lens_put_irr2: fixes X :: "'a \<Longrightarrow> 'b"   and Y :: "'c \<Longrightarrow> 'b"   and \<sigma> :: "'b"   and u :: "'a" assumes "X \<bowtie> Y" shows "get\<^bsub>Y\<^esub> (put\<^bsub>X\<^esub> \<sigma> u) = get\<^bsub>Y\<^esub> \<sigma>"
bex_cong: fixes A :: "'b set"   and B :: "'b set"   and P :: "'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool" assumes "A = B"   and "\<And>x. x \<in> B \<Longrightarrow> P x = Q x" shows "(\<exists>x\<in>A. P x) = (\<exists>x\<in>B. Q x)"
pr0_simps_2: fixes a :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and b :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "ide a"   and "ide b" shows "local.dom (pr0 a b) = local.prod a b"
fps_X_pow_eq_fps_X_pow_iff: fixes m :: "nat"   and n :: "nat" shows "(fps_X ^ m = fps_X ^ n) = (m = n)"
eq: shows "AE x in M. RN_deriv M (density M (\<lambda>x. ennreal (f x))) x = ennreal (f x)"
terminates_Normal_elim_cases_8: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and p :: "'d"   and s :: "'e" assumes "\<Gamma>\<turnstile>Call p \<down> Normal s" obtains bdy :: "('e, 'd, 'f) com" where "\<Gamma> p = Some bdy" and "\<Gamma>\<turnstile>bdy \<down> Normal s"   | "\<Gamma> p = None"
ennreal_less_zero_iff: fixes x :: "real" shows "(0 < ennreal x) = (0 < x)"
simps: fixes ae :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and a :: "Arity"   and as :: "Arity list"   and \<Gamma> :: "(Vars.var \<times> exp) list"   and e :: "exp"   and S :: "stack_elem list" shows "a_transform (ae, a, as) (\<Gamma>, e, S) = (map_transform Aeta_expand ae (map_transform ccTransform ae \<Gamma>), ccTransform a e, transform_alts as S)"
ins_is_def: fixes C :: "Class"   and m :: "Method"   and l :: "int"   and ins :: "Instr" shows "ins_is C m l ins = (\<exists>M. mbody_is C m M \<and> get_ins M l = Some ins)"
mset_single_cases2: fixes s :: "'a"   and c :: "'a multiset"   and r' :: "'a"   and c' :: "'a multiset" assumes "add_mset s c = add_mset r' c'" obtains "s = r'" and "c = c'"   | "c' = c' - {#s#} + {#s#}" and "c = c - {#r'#} + {#r'#}" and "c - {#r'#} = c' - {#s#}"
ennreal_tendsto_cmult: fixes c :: "ennreal"   and f :: "'c \<Rightarrow> ennreal"   and x :: "ennreal"   and F :: "'c filter" assumes "c < \<top>"   and "(f \<longlongrightarrow> x) F" shows "((\<lambda>x. c * f x) \<longlongrightarrow> c * x) F"
mono_rtranclp: fixes x :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and y :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a" assumes "\<And>a b. x a b \<longrightarrow> y a b" shows "x\<^sup>*\<^sup>* a b \<longrightarrow> y\<^sup>*\<^sup>* a b"
sparsegrid'_parents: fixes b :: "(nat \<times> int) list"   and dm :: "nat"   and p' :: "(nat \<times> int) list"   and d :: "nat"   and p :: "(nat \<times> int) list" assumes "b \<in> sparsegrid' dm"   and "p' \<in> parents d b p" shows "p' \<in> sparsegrid' dm"
univ_ce_def: shows "univ_ce = {c_pair n x |n x. x \<in> nat_to_ce_set n}"
starfun2_star_n: fixes f :: "'c \<Rightarrow> 'd \<Rightarrow> 'b"   and X :: "nat \<Rightarrow> 'c"   and Y :: "nat \<Rightarrow> 'd" shows "(*f2* f) (star_n X) (star_n Y) = star_n (\<lambda>n. f (X n) (Y n))"
evnoddD: fixes x :: "nat \<times> nat"   and A :: "(nat \<times> nat) set"   and b :: "nat" assumes "x \<in> evnodd A b" shows "x \<in> A"
H1: fixes x1 :: "char list"   and x2 :: "char list multiset"   and p :: "int mpoly" assumes "x1 \<in># x2" shows "(Var (\<phi> x1) * poly_of_vars x2 - p \<in> More_Modules.ideal polynomial_bool) = (poly_of_vars x2 - p \<in> More_Modules.ideal polynomial_bool)"
vE: shows "valid_graph \<lparr>nodes = fst ` E \<union> (snd \<circ> snd) ` E, edges = E\<rparr>"
k_in: shows "k \<in> {0..<length \<pi>}"
vertical_composite_axioms: shows "vertical_composite A B F G F \<sigma> \<sigma>'"
rPar1: fixes \<Psi>\<^sub>Q :: "'b"   and P' :: "('a, 'b, 'c) psi"   and A\<^sub>Q :: "name list" assumes "\<Psi> \<otimes> \<Psi>\<^sub>Q \<rhd> P \<longmapsto> x\<alpha> \<prec> P'"   and "extractFrame Q = \<langle>A\<^sub>Q, \<Psi>\<^sub>Q\<rangle>"   and "distinct A\<^sub>Q"   and "A\<^sub>Q \<sharp>* \<Psi>"   and "A\<^sub>Q \<sharp>* P"   and "A\<^sub>Q \<sharp>* Q"   and "A\<^sub>Q \<sharp>* x\<alpha>"   and "A\<^sub>Q \<sharp>* P'"   and "A\<^sub>Q \<sharp>* C"   and "xT = P' \<parallel> Q" shows "Prop"
P1: shows "y = [p] \<cdot> \<one> [^] nat (ord_Zp y) \<otimes> ac_Zp y"
satisfies_sumlist_le_constraints: fixes cs :: "'a le_constraint list"   and v :: "nat \<Rightarrow> 'a" assumes "\<And>c. c \<in> set cs \<Longrightarrow> v \<Turnstile>\<^sub>l\<^sub>e c" shows "v \<Turnstile>\<^sub>l\<^sub>e sum_list cs"
ereal_mult_left_mono: fixes a :: "ereal"   and b :: "ereal"   and c :: "ereal" assumes "a \<le> b"   and "0 \<le> c" shows "c * a \<le> c * b"
p1: fixes l :: "nat" assumes "l < R" shows "tensor_P (1\<^sub>m N) (mat_incr K) * tensor_P (proj_psi_l (l + 1)) (proj_k l) * tensor_P (1\<^sub>m N) (adjoint (mat_incr K)) = tensor_P (proj_psi_l (l + 1)) (proj_k (l + 1))"
card_of_Un_infinite_simps_2: fixes A :: "'a set"   and B :: "'a set" assumes "infinite A"   and "|B| \<le>o |A|" shows "|B \<union> A| =o |A|"
pl: assumes "w1 = POS_INF \<and> w2 \<noteq> NEG_INF \<and> w2 <s 0" shows "P w1 w2"
norm_column_le_norm: fixes A :: "'b sq_mtx"   and i :: "'b" shows "\<parallel>A $$ i\<parallel> \<le> \<parallel>A\<parallel>"
assms: shows "w\<^sub>1 @ v\<^sub>1 =\<^sub>F w\<^sub>2"
assms: shows "m(a \<mapsto> x) = n(a \<mapsto> y)"
0: shows "Poly_Mapping.single v ` {x. lookup (mapping_of p) (Poly_Mapping.single v x) \<noteq> (0::'a)} \<subseteq> {k. lookup (mapping_of p) k \<noteq> (0::'a)}"
elimmeets: fixes meets :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and \<I> :: "'b \<Rightarrow> bool"   and p :: "'b"   and s :: "'b"   and r :: "'b"   and q :: "'b" assumes "class.interval meets \<I>" shows "(meets p s \<and> (\<exists>t. meets p t \<and> meets t s) \<and> (\<exists>t. meets r t \<and> meets t q)) = False"
step: fixes a :: "'a"   and x :: "'a" assumes "iterates_abovep a x" shows "iterates_abovep a (f x)"
subset_finite_imp_set_less_eq_aux: fixes A :: "'a set"   and B :: "'a set" assumes "A \<subseteq> B"   and "finite B" shows "A \<sqsubseteq>' B"
f_closed: fixes a :: "'ar"   and b :: "'br" shows "f (Rep_a a) (Rep_b b) \<in> \<UU>C"
B: shows "\<forall>n. (\<exists>t1 t2. x = t1 @ t2 \<and> t1 @ [tick] \<in> T S \<and> (t2, X) \<in> F (Y n)) \<or> (\<exists>t1 t2. x = t1 @ t2 \<and> t1 @ [tick] \<in> T S \<and> t2 \<in> D (Y n))"
all_zero_k: shows "\<forall>m. A $ m $ mod_type_class.from_nat k = (0::'a)"
b': shows "\<forall>r'\<in>R. rightPrincipal r' (F \<nabla> [x].A) \<longrightarrow> ( \<Gamma>' \<Rightarrow>* \<Delta>') \<in> set (fst r')"
fdia_d_simp: fixes x :: "'a"   and y :: "'a" shows "\<langle>x| ar ar y = \<langle>x| y"
simps_2: fixes r :: "'b llist \<rightarrow> 'b llist \<rightarrow> 'b llist"   and x :: "'b"   and xs :: "'b llist" shows "lrev_body2\<cdot>r\<cdot>(x :@ xs) = list2H\<cdot>(wrapH\<cdot>r\<cdot>xs) oo list2H\<cdot>(x :@ lnil)"
frameIntComposition: fixes \<Psi> :: "'b"   and \<Psi>' :: "'b"   and A\<^sub>F :: "name list"   and \<Psi>\<^sub>F :: "'b" assumes "\<Psi> \<simeq> \<Psi>'" shows "\<langle>A\<^sub>F, \<Psi> \<otimes> \<Psi>\<^sub>F\<rangle> \<simeq>\<^sub>F \<langle>A\<^sub>F, \<Psi>' \<otimes> \<Psi>\<^sub>F\<rangle>"
gt_iff_wt_unary_diff_same: fixes ground_heads_var :: "'a \<Rightarrow> 'b set"   and gt_sym :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and \<epsilon> :: "nat"   and \<delta> :: "nat"   and extf :: "'b \<Rightarrow> (('b, 'a) tm \<Rightarrow> ('b, 'a) tm \<Rightarrow> bool) \<Rightarrow> ('b, 'a) tm list \<Rightarrow> ('b, 'a) tm list \<Rightarrow> bool"   and arity_sym :: "'b \<Rightarrow> enat"   and arity_var :: "'a \<Rightarrow> enat"   and wt_sym :: "'b \<Rightarrow> nat"   and t :: "('b, 'a) tm"   and s :: "('b, 'a) tm" assumes "kbo_std ground_heads_var gt_sym \<epsilon> \<delta> extf arity_sym arity_var wt_sym" shows "kbo_std.gt ground_heads_var gt_sym \<delta> extf arity_sym wt_sym t s = (kbo_std.gt_wt ground_heads_var \<delta> arity_sym wt_sym t s \<or> kbo_std.gt_unary ground_heads_var gt_sym \<delta> extf arity_sym wt_sym t s \<or> kbo_std.gt_diff ground_heads_var gt_sym \<delta> arity_sym wt_sym t s \<or> kbo_std.gt_same ground_heads_var gt_sym \<delta> extf arity_sym wt_sym t s)"
mlex_prod_def: fixes f :: "'a \<Rightarrow> nat"   and R :: "('a \<times> 'a) set" shows "f <*mlex*> R = inv_image (less_than <*lex*> R) (\<lambda>x. (f x, x))"
rep_eq: fixes x :: "ivl" shows "is_empty_ivl x = is_empty_rep (rep_ivl x)"
neg_less_0_iff_less: 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 a :: "'a" assumes "class.ordered_ab_group_add plus zero minus uminus less_eq less" shows "less (uminus a) zero = less zero a"
neq_Red: fixes c :: "color" shows "(c \<noteq> Red) = (c = Black)"
Blinfun_cases: fixes x :: "'a \<Rightarrow>\<^sub>L 'b" obtains y :: "'a \<Rightarrow> 'b" where "x = Blinfun y" and "y \<in> {f. bounded_linear f}"
I: shows "no_smallest_element_skipped (E \<union> {e}) F"
substNext_def: fixes g :: "'g" shows "step.substNext g \<equiv> \<lambda>v. if v = step.chooseNext g then step.substitution g else v"
rrep_sqn_greater_dsn: shows "paodv i \<TTurnstile> (recvmsg rreq_rrep_sn \<rightarrow>) onl \<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V (\<lambda>(\<xi>, l). l \<in> {PRrep-:1..PRrep-:6} \<longrightarrow> 1 \<le> sqn (rt \<xi>) (dip \<xi>))"
Poly_berlekamp_mat: fixes k :: "nat"   and u :: "'a mod_ring poly" assumes "k < degree u" shows "Poly (list_of_vec (row (local.berlekamp_mat u) k)) = [:0, 1:] ^ (CARD('a) * k) mod u"
contractible_imp_holomorphic_sqrt: fixes f :: "complex \<Rightarrow> complex"   and S :: "complex set" assumes "f holomorphic_on S"   and "contractible S"   and "\<And>z. z \<in> S \<Longrightarrow> f z \<noteq> 0" obtains g :: "complex \<Rightarrow> complex" where "g holomorphic_on S" and "\<And>z. z \<in> S \<Longrightarrow> f z = (g z)\<^sup>2"
degree_greater_iff: fixes k :: "nat"   and p :: "'a mpoly"   and x :: "nat" shows "(k < MPoly_Type.degree p x) = (\<exists>m\<in>monomials p. k < lookup m x)"
real_root_power: fixes n :: "nat"   and x :: "real"   and k :: "nat" assumes "0 < n" shows "root n (x ^ k) = root n x ^ k"
validEqvt: fixes p :: "(name \<times> name) list"   and P :: "pi" shows "valid (p \<bullet> P) = valid P"
step_next: fixes ls :: "nat"   and key :: "nat" shows "g_step ls key = ls \<or> c_tl (g_step ls key) = ls"