hcju/isabelleps huggingface dataset:
is_rewrite_ord_canon_rewriterD2: fixes rword :: "'t \<times> ('a \<Rightarrow>\<^sub>0 'b) \<Rightarrow> 't \<times> ('a \<Rightarrow>\<^sub>0 'b) \<Rightarrow> bool"   and A :: "('t \<Rightarrow>\<^sub>0 'b) set"   and u :: "'t"   and p :: "'t \<Rightarrow>\<^sub>0 'b"   and q :: "'t \<Rightarrow>\<^sub>0 'b" assumes "is_rewrite_ord rword"   and "is_canon_rewriter rword A u p"   and "is_canon_rewriter rword A u q" shows "lt p = lt q"
assms_5: shows "Esigma a * Esigma m = a * (m + n)"
dim_def: shows "Ker.dim = (LEAST n. \<exists>A. finite A \<and> card A = n \<and> A \<subseteq> carrier (V.vs kerT) \<and> Ker.gen_set A)"
positions_in_row_def: fixes s :: "('c \<times> bool list) stream"   and i :: "nat" shows "positions_in_row s i = Option.these (sset (smap2 (\<lambda>p (uu_, bs). if bs ! i then Some p else None) nats s))"
add: fixes b1 :: "'a"   and b2 :: "'a" shows "f' (b1 + b2) = f' b1 + f' b2"
mediant_def: shows "mediant \<equiv> \<lambda>((a, c), b, d). (a + b, c + d)"
tendsto_norm_zero_cancel: fixes f :: "'b \<Rightarrow> 'c"   and F :: "'b filter" assumes "((\<lambda>x. norm (f x)) \<longlongrightarrow> 0) F" shows "(f \<longlongrightarrow> (0::'c)) F"
cl8_var: fixes R :: "('a \<times> 'a set) set"   and S :: "('a \<times> 'a set) set" shows "Multirelations.d R \<cdot> S = R \<cdot> 1\<^sub>\<pi> \<parallel> S"
cases: fixes x :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<times> 'a list" obtains P :: "'a \<Rightarrow> 'a \<Rightarrow> bool" where "xa__ = (P__, [])"   | P :: "'a \<Rightarrow> 'a \<Rightarrow> bool" and x :: "'a" and ys :: "'a list" where "xa__ = (P, x # ys)"
bernoulli_denom_def: fixes n :: "nat" shows "bernoulli_denom n = (if n = 1 then 2 else if n = 0 \<or> odd n then 1 else \<Prod>{p. prime p \<and> p - 1 dvd n})"
trivial_in_allVars: fixes g :: "'g"   and v :: "'val" assumes "uninst_code.ssa.trivial uses phis g v" shows "v \<in> uninst_code.ssa.allVars uses phis g"
reals_Archimedean3: fixes x :: "real" assumes "0 < x" shows "\<forall>y. \<exists>n. y < real n * x"
simps_54: fixes x6 :: "msg"   and x5 :: "nat" shows "Hash x6 \<noteq> Key x5"
cong_refl: fixes divide :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and modulo :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and euclidean_size :: "'d \<Rightarrow> nat"   and division_segment :: "'d \<Rightarrow> 'd"   and b :: "'d"   and a :: "'d" assumes "class.unique_euclidean_semiring divide plus minus zero times one modulo euclidean_size division_segment" shows "unique_euclidean_semiring.cong modulo b b a"
"2_3": shows "A \<turnstile> l \<longrightarrow>\<^bsup>g,a,r\<^esup> l'"
simps_1: fixes f :: "'c" shows "\<^bold>\<lfloor>\<^bold>\<langle>f\<^bold>\<rangle>\<^bold>\<rfloor> = \<^bold>\<langle>f\<^bold>\<rangle>"
Cons_1: fixes state :: "State"   and newWl :: "nat list" assumes "InvariantWatchesEl (getF state) (getWatch1 state) (getWatch2 state)"   and "\<forall>c. c \<in> set Wl' \<longrightarrow> 0 \<le> c \<and> c < length (getF state)" shows "let state' = notifyWatches_loop literal Wl' newWl state in getM state' = getM state \<and> getF state' = getF state \<and> getSATFlag state' = getSATFlag state \<and> isPrefix (getQ state) (getQ state')"
slg_rel_def: fixes R :: "('b \<times> 'c) set" shows "\<langle>R\<rangle>slg_rel \<equiv> {(succs, G). \<forall>v. (succs v, G `` {v}) \<in> \<langle>R\<rangle>list_set_rel}"
push_bit_minus: fixes n :: "nat"   and a :: "'a" shows "push_bit n (- a) = - push_bit n a"
wf_firstRecursiveCall: fixes n :: "nat"   and V :: "nat set"   and Qp :: "(nat set \<times> 'a option list set) set"   and Qn :: "(nat set \<times> 'a option list set) set"   and I :: "nat set"   and J :: "nat set"   and Q_I_pos :: "(nat set \<times> 'a option list set) set"   and Q_I_neg :: "(nat set \<times> 'a option list set) set" assumes "rwf_query n V Qp Qn"   and "2 \<le> card V"   and "(I, J) = getIJ Qp Qn V"   and "Q_I_pos = projectQuery I (filterQuery I Qp)"   and "Q_I_neg = filterQueryNeg I Qn" shows "rwf_query n I Q_I_pos Q_I_neg"
disc_nodes: shows "h \<turnstile> get_disconnected_nodes owner_document \<rightarrow>\<^sub>r prev_disc_nodes"
x_in_set: shows "x \<in> set (reduction_signs p qs signs subsets (M_mat signs subsets))"
reflect_poly_prod_list: fixes xs :: "'a poly list" shows "reflect_poly (prod_list xs) = prod_list (map reflect_poly xs)"
simps_2: fixes typeof_addr :: "'b \<Rightarrow> 'c \<Rightarrow> htype option"   and h :: "'b" shows "heap_base.typeof_h typeof_addr h Unit = typeof Unit"
\<G>_suffix: fixes \<phi> :: "'b ltln"   and w :: "nat \<Rightarrow> 'b set"   and i :: "nat" shows "\<G> \<phi> w \<subseteq> \<G> \<phi> (suffix i w)"
trancl_induct: fixes a :: "'k"   and b :: "'k"   and r :: "('k \<times> 'k) set"   and P :: "'k \<Rightarrow> bool" assumes "(a, b) \<in> r\<^sup>+"   and "\<And>y. (a, y) \<in> r \<Longrightarrow> P y"   and "\<And>y z. \<lbrakk>(a, y) \<in> r\<^sup>+; (y, z) \<in> r; P y\<rbrakk> \<Longrightarrow> P z" shows "P b"
symcl_converse: fixes A :: "('b \<times> 'b) set" shows "(A\<^sup>\<leftrightarrow>)\<inverse> = A\<^sup>\<leftrightarrow>"
induct: fixes P :: "Core.term \<Rightarrow> nat \<Rightarrow> Core.term list \<Rightarrow> bool"   and a0 :: "Core.term"   and a1 :: "nat"   and a2 :: "Core.term list" assumes "\<And>i lev args. P (Bv i) lev args"   and "\<And>T body lev args. P body (lev + 1) (map (\<lambda>t. lift t 0) args) \<Longrightarrow> P (Abs T body) lev args"   and "\<And>f t lev u. \<lbrakk>P f lev u; P t lev u\<rbrakk> \<Longrightarrow> P (f $ t) lev u"   and "\<And>v va uu_ uv_. P (Ct v va) uu_ uv_"   and "\<And>v va uu_ uv_. P (Fv v va) uu_ uv_" shows "P a0 a1 a2"
assms_2: shows "H \<subseteq> dgrad_sig_set' (length fs) d"
init_rev_wrap_1: shows "[] \<equiv> rev_Nil_wrap"
valid_refs_imp_valid_refs_inv: fixes s :: "'e process_name \<Rightarrow> ('f, 'e, 'g, 'h) Model.local_state" assumes "valid_refs s"   and "no_grey_refs s"   and "\<forall>p. sys_mem_store_buffers p s = []"   and "\<forall>m. ghost_honorary_root (s (mutator m)) = {}" shows "valid_refs_inv s"
inf_top_left: fixes x :: "'a" shows "\<top> \<sqinter> x = x"
ennreal_divide_eq_top_iff: fixes a :: "ennreal"   and b :: "ennreal" shows "(a / b = top) = (a \<noteq> 0 \<and> b = 0 \<or> a = top \<and> b \<noteq> top)"
abs_of_pos: fixes abs :: "'b \<Rightarrow> '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 a :: "'b" assumes "class.ordered_ab_group_add_abs abs plus zero minus uminus less_eq less"   and "less zero a" shows "abs a = a"
plossless_swap_lassocr: fixes \<I>1 :: "('i, 'l) \<I>"   and \<I>2 :: "('j, 'm) \<I>"   and \<I>3 :: "('k, 'n) \<I>" shows "plossless_converter (\<I>1 \<oplus>\<^sub>\<I> (\<I>2 \<oplus>\<^sub>\<I> \<I>3)) (\<I>2 \<oplus>\<^sub>\<I> (\<I>1 \<oplus>\<^sub>\<I> \<I>3)) swap_lassocr"
word_minus_one_le: fixes x :: "'b word" shows "(- 1 \<le> x) = (x = - 1)"
eqv_class_eqD: fixes a :: "'b"   and b :: "'b" assumes "\<lfloor>a\<rfloor> = \<lfloor>b\<rfloor>" shows "a \<sim> b"
padic_val_def: fixes p :: "int"   and f :: "nat \<Rightarrow> int" shows "padic_val p f \<equiv> if f = padic_zero p then - 1 else int (LEAST k. f (Suc k) \<noteq> \<zero>\<^bsub>residue_ring (p ^ Suc k)\<^esub>)"
hyps_4: shows "Q \<lesssim>\<lbrakk>\<cdot>\<rbrakk>R<SRel,TRel> R"
isomorphic_implies_ide_1: fixes f :: "'b"   and g :: "'b" assumes "D.isomorphic f g" shows "D.ide f"
replicate_add: fixes n :: "nat"   and m :: "nat"   and x :: "'a" shows "x \<up> (n + m) = x \<up> n @ x \<up> m"
abc_dec_2_LE_def: shows "abc_dec_2_LE \<equiv> inv_image lex_square abc_dec_2_measure"
perp2__col: fixes X :: "'a"   and Y :: "'a"   and A :: "'a"   and B :: "'a"   and Z :: "'a" assumes "X Y Perp A B"   and "X Z Perp A B" shows "Col X Y Z"
isomorphic_def: fixes a :: "('a, 'a set) Maps.arr"   and a' :: "('a, 'a set) Maps.arr" shows "Maps.isomorphic a a' = (\<exists>f. Maps.in_hom f a a' \<and> Maps.iso f)"
pseudo_proj_Suc_preimage: fixes n :: "nat"   and x :: "bool stream" shows "range (pseudo_proj_True (Suc n)) \<inter> pseudo_proj_True n -` {pseudo_proj_True n x} = {pseudo_proj_True n x, pseudo_proj_False n x}"
eval_hom': fixes R :: "('c, 'd) ring_scheme"   and S :: "('e, 'f) ring_scheme"   and h :: "'c \<Rightarrow> 'e"   and a :: "'c"   and p :: "'c list" assumes "ring_hom_ring R S h"   and "a \<in> carrier R"   and "set p \<subseteq> carrier R" shows "h (ring.eval R p a) = ring.eval S (map h p) (h a)"
l_insert_before_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms: shows "l_insert_before_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_parent get_parent_locs get_child_nodes get_child_nodes_locs set_child_nodes set_child_nodes_locs get_ancestors adopt_node adopt_node_locs set_disconnected_nodes set_disconnected_nodes_locs get_disconnected_nodes get_disconnected_nodes_locs get_owner_document insert_before insert_before_locs DocumentClass.type_wf DocumentClass.known_ptr DocumentClass.known_ptrs heap_is_wellformed parent_child_rel"
length_remove1: fixes x :: "'g"   and xs :: "'g list" shows "length (remove1 x xs) = (if x \<in> set xs then length xs - 1 else length xs)"
compact_segment: fixes a :: "'b"   and b :: "'b" shows "compact (closed_segment a b)"
is_extensional: fixes f :: "'j" assumes "\<not> J.arr f" shows "E f = C.null"
ring_hom_ring_def: fixes R :: "('c, 'd) ring_scheme"   and S :: "('e, 'f) ring_scheme"   and h :: "'c \<Rightarrow> 'e" shows "ring_hom_ring R S h \<equiv> ring R \<and> ring S \<and> ring_hom_ring_axioms R S h"
length_dense: shows "\<forall>ts v. (0 < \<parallel>ext v\<parallel>) = (\<exists>va u. v=va\<parallel>u \<and> 0 < \<parallel>ext va\<parallel> \<and> 0 < \<parallel>ext u\<parallel>)"
zip_eq_conv: fixes xs :: "'a list"   and ys :: "'b list"   and zs :: "('a \<times> 'b) list" assumes "length xs = length ys" shows "(zip xs ys = zs) = (map fst zs = xs \<and> map snd zs = ys)"
add_right_imp_eq: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and b :: "'b"   and a :: "'b"   and c :: "'b" assumes "class.cancel_semigroup_add plus"   and "plus b a = plus c a" shows "b = c"
bot_action: shows "bot \<equiv> action_of (R (false \<turnstile> true))"
IH: fixes E' :: "(('f, 'v) Term.term \<times> ('f, 'v) Term.term) multiset" assumes "(E', E) \<in> unifless"   and "unifiable (set_mset E')" shows "(E', {#}) \<in> unif\<^sup>*"
downset_inj: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "class.order less_eq less" shows "inj (ord.downset less_eq)"
simps: fixes xs :: "int list"   and i :: "nat" shows "der_stack xs i = (if has_left xs i then the_left xs i # der_stack xs (the_left xs i) else [])"
simps: fixes xs :: "real list"   and n :: "nat"   and l :: "real list" shows "insert_into xs n l = take n xs @ l @ drop n xs"
assms_1: fixes c :: "'a" shows "\<forall>\<^sub>F x in F. c \<le> f x"
unit_factor: fixes a :: "'a list"   and b :: "'a list" assumes "a \<otimes>\<^bsub>K [X]\<^esub> b \<in> Units (K [X])"   and "a \<in> carrier (K [X])"   and "b \<in> carrier (K [X])" shows "a \<in> Units (K [X])"
sigma_finite: fixes i :: "'i" obtains A :: "nat \<Rightarrow> 'a set" where "range A \<subseteq> sets (M i)" and "\<Union> (range A) = space (M i)" and "\<And>i. emeasure (M i) (A i) \<noteq> \<infinity>"
in_support_on: fixes x :: "'c"   and S :: "'c set"   and f :: "'c \<Rightarrow> 'd" shows "(x \<in> support_on S f) = (x \<in> S \<and> f x \<noteq> (0::'d))"
pid_system_automaton_simps_3: shows "pid.V0 = {pid_init}"
d_def: fixes fs :: "int vec list"   and k :: "nat" shows "fs'.d fs k = gs.Gramian_determinant fs k"
h3b: shows "length (take_indices signs (find_nonzeros_from_input_vec (solve_for_lhs_R p qs subsets (M_mat_R signs subsets)))) \<le> length subsets"
map_key_mapping_of_PP: fixes p :: "('a, 'b) pp \<Rightarrow>\<^sub>0 'c" shows "Poly_Mapping.map_key pp.mapping_of (Poly_Mapping.map_key PP p) = p"
Since: fixes \<sigma> :: "(char list \<times> 'a list) trace"   and j :: "nat"   and n :: "nat"   and R :: "'a list set"   and \<phi> :: "'a mformula"   and \<phi>' :: "'a MFOTL.formula"   and \<psi> :: "'a mformula"   and \<psi>' :: "'a MFOTL.formula"   and pos :: "bool"   and \<phi>'' :: "'a MFOTL.formula"   and buf :: "'a option list set list \<times> 'a option list set list"   and nts :: "nat list"   and I :: "\<I>"   and aux :: "(nat \<times> 'a option list set) list" assumes "wf_mformula \<sigma> j n R \<phi> \<phi>'"   and "wf_mformula \<sigma> j n R \<psi> \<psi>'"   and "if pos then \<phi>'' = \<phi>' else \<phi>'' = formula.Neg \<phi>'"   and "safe_formula \<phi>'' = pos"   and "fv \<phi>' \<subseteq> fv \<psi>'"   and "wf_mbuf2' \<sigma> j n R \<phi>' \<psi>' buf"   and "wf_ts \<sigma> j \<phi>' \<psi>' nts"   and "wf_since_aux \<sigma> n R pos \<phi>' I \<psi>' aux (Monitor.progress \<sigma> (formula.Since \<phi>'' I \<psi>') j)" shows "wf_mformula \<sigma> j n R (MSince pos \<phi> I \<psi> buf nts aux) (formula.Since \<phi>'' I \<psi>')"
IA: shows "A' * IA' = 1\<^sub>m n"
match_rews_4: fixes a :: "'a"   and slist :: "[:'a:]"   and k :: "'b match" assumes "a \<noteq> \<bottom>"   and "slist \<noteq> \<bottom>" shows "match_snil\<cdot>(a :# slist)\<cdot>k = Fixrec.fail"
invariantD: fixes A :: "('c, 'd) automaton"   and I :: "'d \<Rightarrow> bool"   and P :: "'c \<Rightarrow> bool"   and s :: "'c" assumes "A \<TTurnstile> (I \<rightarrow>) P"   and "s \<in> reachable A I" shows "P s"
Seq: shows "S = (\<Union>n. cbox (vec (- real n)) (vec (real n)) \<inter> S)"
finfun_update_twist: fixes a :: "'a"   and a' :: "'a"   and f :: "('a, 'b) finfun"   and b :: "'b"   and b' :: "'b" assumes "a \<noteq> a'" shows "finfun_update (finfun_update f a b) a' b' = finfun_update (finfun_update f a' b') a b"
distinct_17: fixes x1 :: "agent"   and x101 :: "nat"   and x102 :: "msg" shows "Agent x1 \<noteq> Crypt x101 x102"
intro: fixes C :: "('d, 'e, 'f) Category_scheme" assumes "ExtCategory C"   and "Category_axioms C" shows "Category C"
refl: fixes D :: "('c, 'd) Order_scheme"   and a :: "'c" assumes "Order D"   and "a \<in> carrier D" shows "(a, a) \<in> rel D"
add_strict_right_mono: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c"   and b :: "'c"   and c :: "'c" assumes "class.ordered_cancel_ab_semigroup_add plus minus less_eq less"   and "less a b" shows "less (plus a c) (plus b c)"
bind_foldl_spmf_pair_append: fixes f :: "'e \<Rightarrow> 'f \<Rightarrow> ('d list \<times> 'e) spmf"   and a :: "'d list"   and c :: "'d list"   and b :: "'e"   and es :: "'f list"   and g :: "'d list \<Rightarrow> 'e \<Rightarrow> 'c spmf" shows "foldl_spmf (\<lambda>(x, y) e. map_spmf (apfst ((@) x)) (f y e)) (return_spmf (a @ c, b)) es \<bind> (\<lambda>(x, y). g x y) = foldl_spmf (\<lambda>(x, y) e. map_spmf (apfst ((@) x)) (f y e)) (return_spmf (c, b)) es \<bind> (\<lambda>(x, y). g (a @ x) y)"
obj_fields_marked_good_ref_subseteq_hp_IdleMarkSweep_locs: shows "obj_fields_marked_good_ref_locs \<subseteq> hp_IdleMarkSweep_locs"
rep_eq: fixes x :: "'a dual"   and xa :: "'a dual" shows "undual (x - xa) = - (undual xa - undual x)"
wf_trm_subtermeq: fixes t :: "('fun, 'a) Term.term"   and s :: "('fun, 'a) Term.term" assumes "wf\<^sub>t\<^sub>r\<^sub>m t"   and "s \<sqsubseteq> t" shows "wf\<^sub>t\<^sub>r\<^sub>m s"
Diff_insert2: fixes A :: "'b set"   and a :: "'b"   and B :: "'b set" shows "A - (a \<triangleright> B) = A - {a} - B"
init_segment_Un: fixes S :: "nat set"   and S' :: "nat set" assumes "S \<lless> S'" shows "init_segment S (S \<union> S')"
Rep_BinomialHeap_invar: fixes x :: "('a, 'b) BinomialHeap" shows "BinomialHeapStruc.invar (Rep_BinomialHeap x)"
inj_prop: fixes g :: "'a" assumes "g \<in> carrier G" shows "inj_on (\<phi> g) E"
preserve_order_def: fixes x :: "'b"   and y :: "'b"   and u :: "'c"   and v :: "'c" shows "preserve_order x y u v \<equiv> (x < y \<longrightarrow> u < v) \<and> (y < x \<longrightarrow> v < u)"
Gp_smult_nsg1: fixes H :: "'a set"   and N :: "'a set" assumes "G \<guillemotright> H "   and "G \<triangleright> N" shows "\<natural>N \<diamondop> H \<triangleright> N"
finite_insert: fixes a :: "'c"   and A :: "'c set" shows "finite ({a} \<union> A) = finite A"
absorb_SomeD: fixes j :: "jump"   and a :: "abopt"   and x :: "abrupt" assumes "absorb j a = Some x" shows "a = Some x"
commute: fixes a :: "'a"   and b :: "'a" shows "max a b = max b a"
assoc_in_hom_1: fixes f :: "'a"   and g :: "'a"   and h :: "'a" assumes "ide f"   and "ide g"   and "ide h"   and "src f = trg g"   and "src g = trg h" shows "\<guillemotleft>\<a>[f, g, h] : src h \<rightarrow> trg f\<guillemotright>"
induced_hom_Abs_freelist_conv_sum_list: fixes S :: "'b set"   and P :: "('b \<times> bool) list set"   and f :: "'b \<Rightarrow> 'c"   and ss :: "'b list" assumes "GroupByPresentationInducedFun S P f"   and "ss \<in> lists S" shows "GroupByPresentationInducedFun.induced_hom S P f (BinOpSetGroup.Abs_G_perm (FreeGroup S // Restr (lcoset_rel \<langle>\<Union>g\<in>FreeGroup S. lconjby g ` Abs_freeword ` P\<rangle>) (FreeGroup S)) (+) (Restr (lcoset_rel \<langle>\<Union>g\<in>FreeGroup S. lconjby g ` Abs_freeword ` P\<rangle>) (FreeGroup S) `` {Abs_freelist ss})) = sum_list (map f ss)"
convK: shows "\<forall>i\<in>I. convex (K i)"
succsr_isNode: fixes x :: "'a"   and y :: "'a" assumes "(x, y) \<in> succsr\<^sup>*"   and "isNode x" shows "isNode y"
sel_2: fixes f :: "'b \<Rightarrow> 'b"   and x :: "'b" shows "ltl (iterates f x) = iterates f (f x)"
that: assumes "r = fst (the (execute c h))"   and "h' = snd (the (execute c h))"   and "success c h" shows "thesis"
singleActionChain_2: fixes P :: "pi"   and \<alpha> :: "freeRes"   and P' :: "pi" assumes "P \<longmapsto> \<alpha> \<prec> P'" shows "P \<Longrightarrow>\<alpha> \<prec> P'"
basisD_2: fixes X :: "(nat \<times> int \<times> nat) set"   and x :: "nat \<times> int \<times> nat" assumes "s.SpanningForest X"   and "x \<in> E - X" shows "\<not> (forest \<lparr>nodes = fst ` E \<union> (snd \<circ> snd) ` E, edges = insert x X\<rparr> \<and> subgraph \<lparr>nodes = fst ` E \<union> (snd \<circ> snd) ` E, edges = insert x X\<rparr> \<lparr>nodes = fst ` E \<union> (snd \<circ> snd) ` E, edges = E\<rparr>)"
ub_def: shows "ub \<equiv> max ub1 ub2"
measure_lebesgue_affine: fixes m :: "real"   and \<delta> :: "'b"   and S :: "'b set" shows "Sigma_Algebra.measure lebesgue ((\<lambda>x. m *\<^sub>R x + \<delta>) ` S) = \<bar>m\<bar> ^ DIM('b) * Sigma_Algebra.measure lebesgue S"
csol_4: fixes t :: "real"   and x0 :: "'a" assumes "t \<in> existence_ivl t0 x0" shows "(csol t0 x0 t solves_ode f) {t0--t} X"
AE_stream_space: fixes P :: "'a stream \<Rightarrow> bool" assumes "Measurable.pred (stream_space M) P" shows "almost_everywhere (stream_space M) P = (AE x in M. AE X in stream_space M. P (x ## X))"
nat_power2_diff: fixes b :: "nat"   and a :: "nat" assumes "b \<le> a" shows "(a - b)\<^sup>2 = a\<^sup>2 + b\<^sup>2 - 2 * a * b"
filterlim_inverse_at_top_right: shows "filterlim inverse at_top (at_right 0)"
P_def: shows "P \<equiv> {Pi' I F |F. \<forall>i\<in>I. F i \<in> E i}"
is_wt_instance_of_cond_def: fixes \<Gamma> :: "('a, 'b) Term.term \<Rightarrow> 'c"   and t :: "('a, 'b) Term.term"   and s :: "('a, 'b) Term.term" shows "is_wt_instance_of_cond \<Gamma> t s \<equiv> \<Gamma> t = \<Gamma> s \<and> (case mgu t s of None \<Rightarrow> False | Some \<delta> \<Rightarrow> inj_on \<delta> (fv t) \<and> (\<forall>x\<in>fv t. is_Var (\<delta> x)))"
HasFooInt_def: shows "HasFooInt = \<lparr>access = Public, imethods = [(foo_sig, foo_mhead)], isuperIfs = []\<rparr>"
diff_conv_add_uminus: fixes a :: "'a"   and b :: "'a" shows "a - b = a + - b"
Mf: shows "(\<lambda>x. f (ln x) * ennreal (inverse x)) \<in> borel_measurable borel"
zero_order_1: fixes x :: "'f multiset" shows "{#} \<subseteq># x"
induces_adjunction: shows "adjunction (\<cdot>\<^sub>D) (\<cdot>\<^sub>C) replete_setcat.comp Adj.\<phi>C Adj.\<phi>D Adj.G.F G Adj.G.\<phi> Adj.G.\<psi> (meta_adjunction.\<eta> (\<cdot>\<^sub>C) Adj.G.F Adj.G.\<phi>) (meta_adjunction.\<epsilon> (\<cdot>\<^sub>D) Adj.G.F G Adj.G.\<psi>) (transformation_by_components.map Adj.DopxC.comp replete_setcat.comp (Adj.HomC.map \<circ> product_functor.map (\<cdot>\<^sup>o\<^sup>p) (\<cdot>\<^sub>D) Adj.Cop.comp (\<cdot>\<^sub>D) (dual_functor.map Adj.G.F) D.map) (meta_adjunction.\<Phi>o (\<cdot>\<^sub>D) (\<cdot>\<^sub>C) Adj.G.F G Adj.G.\<phi>)) (inverse_transformation.map Adj.DopxC.comp replete_setcat.comp Adj.Hom_DopxG.map (transformation_by_components.map Adj.DopxC.comp replete_setcat.comp (Adj.HomC.map \<circ> product_functor.map (\<cdot>\<^sup>o\<^sup>p) (\<cdot>\<^sub>D) Adj.Cop.comp (\<cdot>\<^sub>D) (dual_functor.map Adj.G.F) D.map) (meta_adjunction.\<Phi>o (\<cdot>\<^sub>D) (\<cdot>\<^sub>C) Adj.G.F G Adj.G.\<phi>)))"
cUnion_cinsert: fixes x :: "'c cset"   and A :: "'c cset cset" shows "cUnion (cinsert x A) = cUn x (cUnion A)"
trimmed_ms_aux_def: fixes xs :: "('a \<times> real) msllist" shows "trimmed_ms_aux xs = (case xs of MSLNil \<Rightarrow> False | MSLCons (C, x) xa \<Rightarrow> trimmed C)"
distinct_fst_half_segments: fixes X :: "(real \<times> real) \<times> (real \<times> real) pdevs" shows "distinct (map fst (half_segments_of_aform X))"
push_bit_eq_mult: fixes n :: "nat"   and a :: "'a" shows "push_bit n a = a * (2::'a) ^ n"
FORWEAK_mono_rule'': fixes S :: "'a set"   and d :: "'b nres"   and P :: "'b \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'b nres"   and I :: "'a set \<Rightarrow> 'b \<Rightarrow> bool"   and c :: "'b \<Rightarrow> 'b \<Rightarrow> 'b nres" assumes "S = {} \<Longrightarrow> d \<le> SPEC P"   and "\<And>s. s \<in> S \<Longrightarrow> f s \<le> SPEC (I {s})"   and "\<And>it it' \<sigma>. \<lbrakk>I it \<sigma>; it' \<subseteq> it; it \<subseteq> S\<rbrakk> \<Longrightarrow> I it' \<sigma>"   and "\<And>x it \<sigma>. \<lbrakk>x \<in> S; x \<notin> it; it \<subseteq> S; I it \<sigma>\<rbrakk> \<Longrightarrow> f x \<le> SPEC (\<lambda>f'. c \<sigma> f' \<le> SPEC (I (insert x it)))"   and "\<And>\<sigma>. I S \<sigma> \<Longrightarrow> P \<sigma>" shows "FORWEAK S d f c \<le> SPEC P"
l11_29_b: fixes C :: "'p"   and A :: "'p"   and B :: "'p"   and D :: "'p"   and E :: "'p"   and F :: "'p" assumes "\<exists>Q. C InAngle A B Q \<and> A B Q CongA D E F" shows "A B C LeA D E F"
m1_defs_1: shows "m1 \<equiv> \<lparr>init = m1_init, trans = m1_trans, obs = id\<rparr>"
comp_substs_length: fixes subst_atm :: "'b \<Rightarrow> 'c \<Rightarrow> 'b"   and id_subst :: "'c"   and comp_subst :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and renamings_apart :: "'b literal multiset list \<Rightarrow> 'c list"   and atm_of_atms :: "'b list \<Rightarrow> 'b"   and \<tau>s :: "'c list"   and \<sigma>s :: "'c list" assumes "substitution subst_atm id_subst comp_subst renamings_apart atm_of_atms" shows "length (substitution_ops.comp_substs comp_subst \<tau>s \<sigma>s) = min (length \<tau>s) (length \<sigma>s)"