hcju/isabelleps huggingface dataset:
cnj_in_Ints_iff: fixes x :: "complex" shows "(cnj x \<in> \<int>) = (x \<in> \<int>)"
minus_eq: fixes x :: "'a"   and y :: "'a" shows "g x (- y) = gm x y"
assms: shows "eval r_phi [i, x\<^sub>0] \<down>"
major: shows "(a, b) \<in> rtrancl_on F r"
sliceable_M: fixes S :: "'a list set"   and v :: "'a option list"   and i :: "nat"   and \<pi> :: "(char list \<times> 'a list) prefix" assumes "mem_restr S v" shows "((i, v) \<in> M (pmap_\<Gamma> (\<lambda>D. D \<inter> relevant_events \<phi> S) \<pi>)) = ((i, v) \<in> M \<pi>)"
hyps: assumes "\<And>k. k < n \<Longrightarrow> f k \<in> carrier_mat d2 d2" shows "matrix_sum d (\<lambda>k. tensor_mat m1 (f k)) n = tensor_mat m1 (matrix_sum d2 f n)"
ext_inject: fixes root :: "'c"   and edges :: "'c edge set"   and more :: "'d"   and root' :: "'c"   and edges' :: "'c edge set"   and more' :: "'d" shows "(\<lparr>root = root, edges = edges, \<dots> = more\<rparr> = \<lparr>root = root', edges = edges', \<dots> = more'\<rparr>) = (root = root' \<and> edges = edges' \<and> more = more')"
vimage_Un: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'d set"   and B :: "'d set" shows "f -` (A \<union> B) = f -` A \<union> f -` B"
assn_aci_9: fixes a :: "assn"   and b :: "assn"   and c :: "assn" shows "a * (b * c) = a * b * c"
map_injective: fixes f :: "'b \<Rightarrow> 'a"   and xs :: "'b list"   and ys :: "'b list" assumes "map f xs = map f ys"   and "inj f" shows "xs = ys"
cont_minus2: fixes below :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a" assumes "class.minus_cpo below" shows "cont ((-) x)"
nres_rel_comp: fixes A :: "('c \<times> 'e) set"   and B :: "('e \<times> 'd) set" shows "\<langle>A\<rangle>nres_rel O \<langle>B\<rangle>nres_rel = \<langle>A O B\<rangle>nres_rel"
simps_1: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'a"   and a :: "'b"   and b :: "'b"   and bs :: "'b list" shows "bisumTail f (a # b # bs) = f a b + bisumTail f (b # bs)"
cr_transaction\<^sub>m_def: shows "transaction_merkle.cr_transaction\<^sub>m \<equiv> \<lambda>x y. x = the_Transaction\<^sub>m y"
assms_3: shows "set (characterize_consistent_signs_at_roots p qs) \<subseteq> set signs"
simps: fixes L1 :: "'at Literal"   and L2 :: "'at Literal" shows "literal_ordering L1 L2 = ((atom L1, atom L2) \<in> atom_ordering)"
lfilled_deterministic: fixes k :: "nat"   and lfilled :: "Lholed"   and es :: "e list"   and les :: "e list"   and les' :: "e list" assumes "Lfilled k lfilled es les"   and "Lfilled k lfilled es les'" shows "les = les'"
gcd_mult: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<in> carrier mk_monoid"   and "b \<in> carrier mk_monoid"   and "c \<in> carrier mk_monoid" shows "c \<otimes>\<^bsub>mk_monoid\<^esub> somegcd mk_monoid a b \<sim>\<^bsub>mk_monoid\<^esub> somegcd mk_monoid (c \<otimes>\<^bsub>mk_monoid\<^esub> a) (c \<otimes>\<^bsub>mk_monoid\<^esub> b)"
SetPC_def: fixes f :: "'b \<Rightarrow> bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and p :: "'a \<Rightarrow> 'b \<Rightarrow> real" shows "SetPC f p \<equiv> \<lambda>ab P s. \<Sum>a\<in>supp (p s). p s a * f a ab P s"
raw_ports_compress_src_MatchExpr: fixes \<beta> :: "'b common_primitive \<Rightarrow> ('b, 'c) tagged_packet_scheme \<Rightarrow> ternaryvalue"   and pss :: "ipt_l4_ports list"   and m :: "ipt_l4_ports"   and \<alpha> :: "action \<Rightarrow> ('b, 'c) tagged_packet_scheme \<Rightarrow> bool"   and a :: "action"   and p :: "('b, 'c) tagged_packet_scheme" assumes "primitive_matcher_generic \<beta>"   and "l4_ports_compress pss = MatchExpr m" shows "matches (\<beta>, \<alpha>) (Match (Src_Ports m)) a p = matches (\<beta>, \<alpha>) (alist_and (map (Pos \<circ> Src_Ports) pss)) a p"
ring_simprules_17: fixes x :: "nat \<Rightarrow> 'a"   and y :: "nat \<Rightarrow> 'a" assumes "x \<in> carrier P"   and "y \<in> carrier P" shows "x \<oplus>\<^bsub>P\<^esub> (\<ominus>\<^bsub>P\<^esub> x \<oplus>\<^bsub>P\<^esub> y) = y"
fin_dim: fixes n :: "nat" shows "vectorspace.fin_dim class_ring (module_vec TYPE('b) n)"
uminus_ms_aux_def: fixes xs :: "('a \<times> real) msllist" shows "uminus_ms_aux xs = mslmap (map_prod uminus id) xs"
valid_reqs_def: fixes M :: "'b SecurityInvariant_configured list" shows "valid_reqs M \<equiv> \<forall>m\<in>set M. configured_SecurityInvariant m"
Smith_normal_form_upt_k1_intro_diagonal: fixes A :: "(('a, 'b) vec, 'c) vec"   and k :: "nat" assumes "Smith_normal_form_upt_k A k"   and "isDiagonal A"   and "A $ from_nat (k - 1) $ from_nat (k - 1) dvd A $ from_nat k $ from_nat k" shows "Smith_normal_form_upt_k A (Suc k)"
ranking_stuck: fixes A :: "('a, 'b) nba"   and w :: "'a stream"   and f :: "nat \<times> 'b \<Rightarrow> nat"   and v :: "nat \<times> 'b"   and r :: "(nat \<times> 'b) stream" assumes "ranking A w f"   and "v \<in> gunodes A w"   and "gurun A w r v" obtains n :: "nat" and k :: "nat" where "smap f (gtrace (sdrop n r) (gtarget (stake n r) v)) = sconst k"
segment_bound_1: fixes x :: "'b"   and a :: "'b"   and b :: "'b" assumes "x \<in> closed_segment a b" shows "norm (x - a) \<le> norm (b - a)"
cc_restr_empty: fixes G :: "CoCalls"   and S :: "Vars.var set" assumes "ccField G \<subseteq> - S" shows "G G|` S = \<bottom>"
diff_closed: fixes R :: "'a set"   and smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and M :: "'b set"   and g :: "'b"   and h :: "'b" assumes "RModule R smult M"   and "g \<in> M"   and "h \<in> M" shows "g - h \<in> M"
wp_assign_idx_eq: fixes \<pi> :: "char list \<Rightarrow> com option"   and x :: "char list"   and i :: "aexp"   and a :: "aexp"   and Q :: "(char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> bool"   and s :: "char list \<Rightarrow> int \<Rightarrow> int" shows "wp \<pi> (x[i] ::= a) Q s = Q (s(x := (s x)(aval i s := aval a s)))"
right_total_fun: fixes A :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and B :: "'f \<Rightarrow> 'g \<Rightarrow> bool" assumes "right_unique A"   and "right_total B" shows "right_total (rel_fun A B)"
assms: shows "s' = snd (fst (write_state_reg_instr instr s)) \<and> get_S (cpu_reg_val PSR s) = 0"
assoc'_eq_inv_assoc: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i> :: "'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and f :: "'b"   and g :: "'b"   and h :: "'b" assumes "bicategory V H \<a> \<i> src trg"   and "partial_magma.ide V f"   and "partial_magma.ide V g"   and "partial_magma.ide V h"   and "src f = trg g"   and "src g = trg h" shows "bicategory.\<a>' V H \<a> src trg f g h = category.inv V (\<a> f g h)"
case_distrib: fixes h :: "'e \<Rightarrow> 'd"   and f1 :: "'f \<Rightarrow> 'e"   and f2 :: "'g \<Rightarrow> 'h \<Rightarrow> 'e"   and generat :: "('f, 'g, 'h) generat" shows "h (case generat of Pure x \<Rightarrow> f1 x | IO x xa \<Rightarrow> f2 x xa) = (case generat of Pure x \<Rightarrow> h (f1 x) | IO x1 x2 \<Rightarrow> h (f2 x1 x2))"
representation_of_cones_functor_def: fixes J :: "'ag \<Rightarrow> 'ag \<Rightarrow> 'ag"   and C :: "'ah \<Rightarrow> 'ah \<Rightarrow> 'ah"   and D :: "'ag \<Rightarrow> 'ah"   and S :: "'ai \<Rightarrow> 'ai \<Rightarrow> 'ai"   and \<phi> :: "'ah \<times> 'ah \<Rightarrow> 'ah \<Rightarrow> 'ai"   and \<iota> :: "('ag \<Rightarrow> 'ah) \<Rightarrow> 'ai"   and a :: "'ah"   and \<Phi> :: "'ah \<Rightarrow> 'ai" shows "representation_of_cones_functor J C D S \<phi> \<iota> a \<Phi> \<equiv> ((category C \<and> dual_category C) \<and> category J \<and> diagram J C D) \<and> (replete_concrete_set_category S UNIV \<iota> \<and> cones_functor J C D S \<iota>) \<and> hom_functor C S \<phi> \<and> representation_of_functor C S \<phi> (cones_functor.map J C D S \<iota>) a \<Phi>"
case_cong: fixes obs_event :: "('a, 'b) obs_event"   and obs_event' :: "('a, 'b) obs_event"   and f1 :: "'a \<Rightarrow> String.literal \<Rightarrow> 'a val list \<Rightarrow> 'a val \<Rightarrow> 'c"   and g1 :: "'a \<Rightarrow> String.literal \<Rightarrow> 'a val list \<Rightarrow> 'a val \<Rightarrow> 'c"   and f2 :: "'a \<Rightarrow> addr_loc \<Rightarrow> 'a val \<Rightarrow> 'c"   and g2 :: "'a \<Rightarrow> addr_loc \<Rightarrow> 'a val \<Rightarrow> 'c"   and f3 :: "'a \<Rightarrow> addr_loc \<Rightarrow> 'a val \<Rightarrow> 'c"   and g3 :: "'a \<Rightarrow> addr_loc \<Rightarrow> 'a val \<Rightarrow> 'c"   and f4 :: "'a \<Rightarrow> htype \<Rightarrow> 'c"   and g4 :: "'a \<Rightarrow> htype \<Rightarrow> 'c"   and f5 :: "'b \<Rightarrow> 'c"   and g5 :: "'b \<Rightarrow> 'c"   and f6 :: "'b \<Rightarrow> 'c"   and g6 :: "'b \<Rightarrow> 'c"   and f7 :: "'a \<Rightarrow> 'c"   and g7 :: "'a \<Rightarrow> 'c"   and f8 :: "'a \<Rightarrow> 'c"   and g8 :: "'a \<Rightarrow> 'c"   and f9 :: "'b \<Rightarrow> 'c"   and g9 :: "'b \<Rightarrow> 'c"   and f10 :: "'b \<Rightarrow> 'c"   and g10 :: "'b \<Rightarrow> 'c" assumes "obs_event = obs_event'"   and "\<And>x11 x12 x13 x14. obs_event' = ExternalCall x11 x12 x13 x14 \<Longrightarrow> f1 x11 x12 x13 x14 = g1 x11 x12 x13 x14"   and "\<And>x21 x22 x23. obs_event' = ReadMem x21 x22 x23 \<Longrightarrow> f2 x21 x22 x23 = g2 x21 x22 x23"   and "\<And>x31 x32 x33. obs_event' = WriteMem x31 x32 x33 \<Longrightarrow> f3 x31 x32 x33 = g3 x31 x32 x33"   and "\<And>x41 x42. obs_event' = NewHeapElem x41 x42 \<Longrightarrow> f4 x41 x42 = g4 x41 x42"   and "\<And>x5. obs_event' = ThreadStart x5 \<Longrightarrow> f5 x5 = g5 x5"   and "\<And>x6. obs_event' = ThreadJoin x6 \<Longrightarrow> f6 x6 = g6 x6"   and "\<And>x7. obs_event' = SyncLock x7 \<Longrightarrow> f7 x7 = g7 x7"   and "\<And>x8. obs_event' = SyncUnlock x8 \<Longrightarrow> f8 x8 = g8 x8"   and "\<And>x9. obs_event' = ObsInterrupt x9 \<Longrightarrow> f9 x9 = g9 x9"   and "\<And>x10. obs_event' = ObsInterrupted x10 \<Longrightarrow> f10 x10 = g10 x10" shows "(case obs_event of ExternalCall x11 x12 x13 x14 \<Rightarrow> f1 x11 x12 x13 x14 | ReadMem x21 x22 x23 \<Rightarrow> f2 x21 x22 x23 | WriteMem x31 x32 x33 \<Rightarrow> f3 x31 x32 x33 | NewHeapElem x41 x42 \<Rightarrow> f4 x41 x42 | ThreadStart x5 \<Rightarrow> f5 x5 | ThreadJoin x6 \<Rightarrow> f6 x6 | SyncLock x7 \<Rightarrow> f7 x7 | SyncUnlock x8 \<Rightarrow> f8 x8 | ObsInterrupt x9 \<Rightarrow> f9 x9 | ObsInterrupted x10 \<Rightarrow> f10 x10) = (case obs_event' of ExternalCall x11 x12 x13 x14 \<Rightarrow> g1 x11 x12 x13 x14 | ReadMem x21 x22 x23 \<Rightarrow> g2 x21 x22 x23 | WriteMem x31 x32 x33 \<Rightarrow> g3 x31 x32 x33 | NewHeapElem x41 x42 \<Rightarrow> g4 x41 x42 | ThreadStart x5 \<Rightarrow> g5 x5 | ThreadJoin x6 \<Rightarrow> g6 x6 | SyncLock x7 \<Rightarrow> g7 x7 | SyncUnlock x8 \<Rightarrow> g8 x8 | ObsInterrupt x9 \<Rightarrow> g9 x9 | ObsInterrupted x10 \<Rightarrow> g10 x10)"
assms_1: shows "AE x in M. f x \<le> g x"
dghm_ObjMap_vsubset_Vset: shows "\<FF>\<lparr>ObjMap\<rparr> \<subseteq>\<^sub>\<circ> Vset \<alpha>"
recurrent_class: fixes x :: "'s" assumes "recurrent x" shows "acc `` {x} = communicating `` {x}"
inj_on_Un_image_eq_iff: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and B :: "'c set" assumes "inj_on f (A \<union> B)" shows "(f ` A = f ` B) = (A = B)"
init_item_\<beta>: fixes r :: "symbol \<times> symbol list"   and k :: "nat" shows "item_\<beta> (init_item r k) = snd r"
arrow_of_spans_axioms: shows "arrow_of_spans (\<cdot>) (\<mu> \<star> \<nu>)"
is_sentence_cons: fixes \<NN> :: "symbol set"   and \<TT> :: "symbol set"   and \<RR> :: "(symbol \<times> symbol list) set"   and \<SS> :: "symbol"   and x :: "symbol"   and xs :: "symbol list" assumes "CFG \<NN> \<TT> \<RR> \<SS>" shows "CFG.is_sentence \<NN> \<TT> (x # xs) = (CFG.is_symbol \<NN> \<TT> x \<and> CFG.is_sentence \<NN> \<TT> xs)"
cases: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a stream" assumes "PTA_Reachability.pairwise P a" obtains aa :: "'a" and b :: "'a" and xs :: "'a stream" where "a = aa ## b ## xs" and "P aa b" and "PTA_Reachability.pairwise P (b ## xs)"
assms_4: shows "\<Gamma> \<turnstile>\<^sub>c e2 : BOOL"
TUA: shows "NTUadmit \<sigma> \<theta> U"
slice_kind_Pred_empty_obs_nearer_not_SOME: fixes a :: "'edge"   and S :: "'node SDG_node set"   and Q :: "('var \<Rightarrow> 'val option) \<Rightarrow> bool"   and mex :: "'node"   and x :: "nat" assumes "sourcenode a \<notin> \<lfloor>HRB_slice S\<rfloor>\<^bsub>CFG\<^esub>"   and "kind a = (Q)\<^sub>\<surd>"   and "obs_intra (sourcenode a) \<lfloor>HRB_slice S\<rfloor>\<^bsub>CFG\<^esub> = {}"   and "method_exit mex"   and "get_proc (sourcenode a) = get_proc mex"   and "distance (targetnode a) mex x"   and "distance (sourcenode a) mex (x + 1)"   and "targetnode a \<noteq> (SOME n'. \<exists>a'. sourcenode a = sourcenode a' \<and> distance (targetnode a') mex x \<and> valid_edge a' \<and> intra_kind (kind a') \<and> targetnode a' = n')" shows "slice_kind S a = (\<lambda>s. False)\<^sub>\<surd>"
length_product: fixes xs :: "'a list"   and ys :: "'b list" shows "length (List.product xs ys) = length xs * length ys"
finfun_apply: fixes x :: "'a \<Rightarrow>f 'b" shows "($) x \<in> finfun"
Cons_lift: fixes P :: "'b com"   and Q :: "'b com"   and s :: "'b"   and xs :: "('b com option \<times> 'b) list" shows "(Some (Seq P Q), s) # map (lift Q) xs = map (lift Q) ((Some P, s) # xs)"
strictly_preserves_assoc: fixes a :: "'c local.term set"   and b :: "'c local.term set"   and c :: "'c local.term set" assumes "\<F>C.arr a"   and "\<F>C.arr b"   and "\<F>C.arr c" shows "F (\<F>C.assoc a b c) = \<a>\<^sub>D[F a, F b, F c]"
hball_cong: fixes A :: "hf"   and A' :: "hf"   and P :: "hf \<Rightarrow> bool"   and P' :: "hf \<Rightarrow> bool" assumes "A = A'"   and "\<And>x. x \<^bold>\<in> A' \<Longrightarrow> P x = P' x" shows "HBall A P = HBall A' P'"
arr_cod_iff_arr: fixes f :: "J.Arr" shows "J.arr (J.cod f) = J.arr f"
concat_bit_nonnegative_iff: fixes n :: "nat"   and k :: "int"   and l :: "int" shows "(0 \<le> concat_bit n k l) = (0 \<le> l)"
subfield_iff_1: fixes K :: "'c set" assumes "field (S\<lparr>carrier := K\<rparr>)"   and "K \<subseteq> carrier S" shows "subfield K S"
list_wset_rel_def: fixes R :: "('a \<times> 'c) set" shows "\<langle>R\<rangle>list_wset_rel = br set top O \<langle>R\<rangle>set_rel"
B': shows "x # xs \<simeq> {ys', zs, P}"
bisimilarity_implies_equivalence: fixes P :: "'state"   and Q :: "'state" assumes "P \<sim>\<cdot> Q" shows "P =\<cdot> Q"
left_in_interval: fixes a :: "'a"   and b :: "'a" assumes "a \<in> A"   and "b \<in> A"   and "interval r a b \<noteq> {}" shows "a \<in> interval r a b"
strictly_precedes_e: fixes \<Gamma> :: "'b constr list"   and n :: "nat"   and K\<^sub>1 :: "clock"   and K\<^sub>2 :: "clock"   and \<Psi> :: "'b TESL_atomic list"   and \<Phi> :: "'b TESL_atomic list" shows "\<Gamma>, n \<turnstile> (K\<^sub>1 strictly precedes K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^sub>e \<lceil>#\<^sup>\<le> K\<^sub>2 n, #\<^sup>< K\<^sub>1 n\<rceil> \<in> \<lambda>(x, y). x \<le> y # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 strictly precedes K\<^sub>2) # \<Phi>"
prerecording_event: fixes channel :: "nat \<Rightarrow> ('d \<times> 'd) option"   and trans :: "'d \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> bool"   and send :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and recv :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and init :: "('d, 'e, 'f) configuration"   and final :: "('d, 'e, 'f) configuration"   and t :: "('d, 'e, 'f) event list"   and i :: "nat" assumes "computation channel trans send recv init final" shows "computation.prerecording_event channel trans send recv init t i \<equiv> i < length t \<and> regular_event (t ! i) \<and> \<not> ps (distributed_system.s channel trans send recv init t i) (occurs_on (t ! i)) \<noteq> None"
add_cancel_right_left: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and a :: "'c"   and b :: "'c" assumes "class.cancel_comm_monoid_add plus minus zero" shows "(a = plus b a) = (b = zero)"
len_start_is_len_verts: shows "length start = length (g_V G)"
r_next_def: shows "r_next \<equiv> Cn 1 r_append [recf.Id 1 0, Cn 1 r_badblock [Cn 1 r_sub [r_search, r_length], Cn 1 r_phi [r_s, r_search]]]"
complex_of_real_mono_iff: fixes x :: "real"   and y :: "real" shows "(complex_of_real x \<le> complex_of_real y) = (x \<le> y)"
x0_hb_bound8: fixes x :: "real"   and i :: "nat" assumes "x\<^sub>1 \<le> x"   and "i < k" shows "x\<^sub>0 < bs ! i * x - hb * x / ln x powr (1 + e)"
assms_2: shows "set (coeffs p) \<subseteq> \<real>"
that_1: assumes "c"   and "run t h h' r" shows "thesis"
elem_ccNeighbors: fixes y :: "Vars.var"   and x :: "Vars.var"   and G :: "CoCalls" shows "(y \<in> ccNeighbors x G) = y--x\<in>G"
less_imp_not_eq2: fixes x :: "'a word"   and y :: "'a word" assumes "x <s y" shows "(y = x) = False"
rep_eq: fixes x :: "'a fset"   and xa :: "'a fset" shows "(x |\<subseteq>| xa) = (fset x \<subseteq> fset xa)"
frechet_derivative_plus: fixes x :: "'b \<Rightarrow> 'c"   and a :: "'b"   and y :: "'b \<Rightarrow> 'c" assumes "x differentiable at a"   and "y differentiable at a" shows "frechet_derivative (\<lambda>xa. x xa + y xa) (at a) = (\<lambda>xa. frechet_derivative x (at a) xa + frechet_derivative y (at a) xa)"
m: shows "v' # tl (verticesFrom f v') = fst (splitAt v (verticesFrom f v')) @ v # snd (splitAt v (verticesFrom f v'))"
fds_truncate_truncate': fixes m :: "nat"   and f :: "'a fds" shows "fds_truncate m (fds_truncate m f) = fds_truncate m f"
div_left_ineq: fixes id :: "'b"   and f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and x :: "nat"   and y :: "nat"   and z :: "nat" assumes "Matrix_Tensor.mult id f"   and "x < y * z" shows "x div z < y"
obt_u_fnt_carrier: fixes carrier :: "'b set"   and relation :: "('b \<times> 'b) set" assumes "finite carrier"   and "rational_preference carrier relation" obtains u :: "'b \<Rightarrow> real" where "ordinal_utility carrier relation u"
iFROM_inext_if: fixes t :: "nat"   and n :: "nat" shows "inext t [n\<dots>] = (if t \<in> [n\<dots>] then Suc t else t)"
equality: fixes r :: "('f, 'g) monoid_scheme"   and r' :: "('f, 'g) monoid_scheme" assumes "carrier r = carrier r'"   and "(\<otimes>\<^bsub>r\<^esub>) = (\<otimes>\<^bsub>r'\<^esub>)"   and "\<one>\<^bsub>r\<^esub> = \<one>\<^bsub>r'\<^esub>"   and "monoid.more r = monoid.more r'" shows "r = r'"
sdrop_simps_2: fixes n :: "nat"   and s :: "'b stream" shows "stl (sdrop n s) = sdrop (Suc n) s"
left_adjoints_compose: fixes f :: "'b"   and f' :: "'b" assumes "D.is_left_adjoint f"   and "D.is_left_adjoint f'"   and "src\<^sub>D f = trg\<^sub>D f'" shows "D.is_left_adjoint (f \<star>\<^sub>D f')"
span_eq: fixes scale :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and S :: "'b set"   and T :: "'b set" assumes "module scale" shows "(module.span scale S = module.span scale T) = (S \<subseteq> module.span scale T \<and> T \<subseteq> module.span scale S)"
word_cat_eq': fixes a :: "'c word"   and b :: "'d word" shows "word_cat a b = word_of_int (concat_bit LENGTH('d) (uint b) (uint a))"
maxne0_impl: fixes x :: "nat list"   and a :: "nat list" assumes "length x = length a" shows "maxne0_impl x a = maxne0 x a"
supp_eqvt: fixes p :: "perm"   and x :: "'b" shows "p \<bullet> supp x = supp (p \<bullet> x)"
is_ground_cls_as_atms: fixes C :: "'a literal multiset" shows "is_ground_cls C = Ball (atms_of C) is_ground_atm"
radical_sqrt_simultaneous_linear_quadratic: fixes a :: "real"   and b :: "real"   and c :: "real"   and d :: "real"   and e :: "real"   and f :: "real"   and x :: "real"   and y :: "real" assumes "a \<in> radical_sqrt"   and "b \<in> radical_sqrt"   and "c \<in> radical_sqrt"   and "d \<in> radical_sqrt"   and "e \<in> radical_sqrt"   and "f \<in> radical_sqrt"   and "\<not> (d = 0 \<and> e = 0 \<and> f = 0)"   and "(x - a)\<^sup>2 + (y - b)\<^sup>2 = c"   and "d * x + e * y = f" shows "x \<in> radical_sqrt \<and> y \<in> radical_sqrt"
qesimps_1: shows "qe_FM = R\<^sub>e.lift_dnfeq_qe qe_FM\<^sub>1"
disj_Un_runiq: fixes P :: "('a \<times> 'b) set"   and Q :: "('a \<times> 'b) set" assumes "runiq P"   and "runiq Q"   and "Domain P \<inter> Domain Q = {}" shows "runiq (P \<union> Q)"
hyps: fixes qs :: "real poly list" assumes "length qs < length qsa__" shows "p \<noteq> 0 \<and> 0 < length qs \<and> (\<forall>q. List.member qs q \<longrightarrow> coprime p q) \<longrightarrow> satisfies_properties p qs (get_subsets (calculate_data p qs)) (get_signs (calculate_data p qs)) (get_matrix (calculate_data p qs))"
hoare_diagram3: fixes D :: "I \<times> I \<Rightarrow> 'a \<Rightarrow> 'a"   and X :: "nat \<Rightarrow> I \<Rightarrow> 'a"   and P :: "I \<Rightarrow> 'a"   and Q :: "I \<Rightarrow> 'a" assumes "dmono D"   and "\<forall>u i j. \<Turnstile> X u i{| D (i, j) |}SUP_L_P X (i, u) j"   and "P \<sqsubseteq> \<Squnion>range X"   and "\<Squnion>range X \<sqinter> - grd (step D) \<sqsubseteq> Q" shows "\<Turnstile> P{| pt D |}Q"
restrict_read_only_mono: fixes \<S> :: "nat \<Rightarrow> bool option"   and \<S>' :: "nat \<Rightarrow> bool option"   and A :: "nat set"   and L :: "nat set" assumes "read_only \<S> \<subseteq> read_only \<S>'" shows "read_only (\<S> \<ominus>\<^bsub>A\<^esub> L) \<subseteq> read_only (\<S>' \<ominus>\<^bsub>A\<^esub> L)"
zero_spaceN_cmult: fixes x :: "'c"   and N :: "'c quasinorm"   and c :: "real" assumes "x \<in> zero_space\<^sub>N N" shows "c *\<^sub>R x \<in> zero_space\<^sub>N N"
dg_def: shows "dg \<equiv> fls_subdegree g"
random_permutation_def: fixes \<sigma> :: "'d \<Rightarrow> 'b option"   and x :: "'d" shows "random_permutation \<sigma> x = (case \<sigma> x of None \<Rightarrow> spmf_of_set (A - ran \<sigma>) \<bind> (\<lambda>y. return_spmf (y, \<sigma>(x \<mapsto> y))) | Some y \<Rightarrow> return_spmf (y, \<sigma>))"
simps_1: fixes var :: "nat"   and a :: "'a" shows "atom_const (Leq var a) = a"
assms_9: shows "InvariantCllCharacterization (getCl state) (getCll state) (getC state) (getM state)"
ConjI: fixes D :: "(('a \<Rightarrow> state \<Rightarrow> bool) \<times> com \<times> ('a \<Rightarrow> state \<Rightarrow> bool)) set"   and C :: "(('a \<Rightarrow> state \<Rightarrow> bool) \<times> com \<times> ('a \<Rightarrow> state \<Rightarrow> bool)) set" assumes "\<forall>(P, c, Q)\<in>D. C \<turnstile> {P} c {Q}" shows "C \<tturnstile> D"
5: shows "top * e * w\<^sup>T\<^sup>\<star> \<le> top * e * (w\<^sup>T \<sqinter> - v\<^sup>T)\<^sup>\<star> \<squnion> - - v\<^sup>T"
divides_as_fmsubset: fixes as :: "'a list"   and a :: "'a"   and bs :: "'a list"   and b :: "'a" assumes "wfactors G as a"   and "wfactors G bs b"   and "a \<in> carrier G"   and "b \<in> carrier G"   and "set as \<subseteq> carrier G"   and "set bs \<subseteq> carrier G" shows "a divides b = (fmset G as \<subseteq># fmset G bs)"
simps: fixes a1 :: "('procS, 'procT) Proc"   and a2 :: "('procS, 'procT) Proc"   and SRel :: "('procS \<times> 'procS) set"   and TRel :: "('procT \<times> 'procT) set" shows "a1 \<sim>\<lbrakk>\<cdot>\<rbrakk><SRel,TRel> a2 = ((\<exists>S. S \<in>S a1 \<and> \<lbrakk>S\<rbrakk> \<in>T a2) \<or> (\<exists>S. \<lbrakk>S\<rbrakk> \<in>T a1 \<and> S \<in>S a2) \<or> (\<exists>S1 S2. S1 \<in>S a1 \<and> S2 \<in>S a2 \<and> (S1, S2) \<in> SRel) \<or> (\<exists>T1 T2. T1 \<in>T a1 \<and> T2 \<in>T a2 \<and> (T1, T2) \<in> TRel) \<or> (\<exists>P Q R. a1 = P \<and> a2 = R \<and> P \<sim>\<lbrakk>\<cdot>\<rbrakk><SRel,TRel> Q \<and> Q \<sim>\<lbrakk>\<cdot>\<rbrakk><SRel,TRel> R))"
isCont_mult: fixes a :: "'a"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "isCont f a"   and "isCont g a" shows "isCont (\<lambda>x. f x * g x) a"
a_very_costrict'': fixes x :: "'a" shows "(ad x = (1::'a)) = (x = (0::'a))"
map_idx_eq_map2: fixes f :: "'c \<Rightarrow> nat \<Rightarrow> 'b"   and xs :: "'c list"   and n :: "nat" shows "map_idx f xs n = map2 f xs [n..<n + length xs]"
ide_trg: fixes \<mu> :: "'b" assumes "B.arr \<mu>" shows "B.ide (trg\<^sub>B \<mu>)"
ordIso_imp_ordLeq: fixes r :: "('a \<times> 'a) set"   and r' :: "('b \<times> 'b) set" assumes "r =o r'" shows "r \<le>o r'"
insert_closed: fixes g :: "'e"   and gens :: "'e set" assumes "g \<in> gens" shows "\<iota> g \<in> carrier \<F>\<^bsub>gens\<^esub>"