hcju/isabelleps huggingface dataset:
meas_single_meas: fixes f :: "'c \<Rightarrow> 'd"   and M :: "'c measure"   and N :: "'d measure" assumes "f \<in> M \<rightarrow>\<^sub>M N"   and "\<forall>r\<in>range f \<inter> space N. \<exists>A\<in>sets N. range f \<inter> A = {r}" shows "point_measurable M (space N) f"
assms_2: shows "list_all (\<lambda>seg. 0 \<le> det3 (fst seg) (snd seg) p) (segments_of_aform X)"
Def_bool1: fixes x :: "bool" shows "(Def x \<noteq> FF) = x"
one_zeroI: assumes "carrier R = {\<zero>}" shows "\<one> = \<zero>"
step: fixes e :: "((nat \<times> nat) \<times> transition) fset"   and s :: "nat"   and r :: "nat \<Rightarrow>f value option"   and l :: "String.literal"   and i :: "value list"   and p :: "value list"   and t :: "(String.literal \<times> value list \<times> value list) list" assumes "\<exists>(s', T)|\<in>|possible_steps e s r l i. evaluate_outputs T i r = map Some p \<and> accepts_trace e s' (evaluate_updates T i r) t" shows "accepts_trace e s r ((l, i, p) # t)"
greaterThanAtMost_empty_iff2: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and k :: "'a"   and l :: "'a" assumes "class.preorder less_eq less" shows "({} = ord.greaterThanAtMost less_eq less k l) = (\<not> less k l)"
HPhase1or2ReadElse_def: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and d :: "Disk"   and q :: "Proc" shows "HPhase1or2ReadElse s s' p d q = (Phase1or2ReadElse s s' p d q \<and> HNextPart s s')"
ivl_cong: fixes a :: "'a"   and c :: "'a"   and b :: "'a"   and d :: "'a"   and g :: "'a \<Rightarrow> 'b"   and h :: "'a \<Rightarrow> 'b" assumes "a = c"   and "b = d"   and "\<And>x. \<lbrakk>c \<le> x; x < d\<rbrakk> \<Longrightarrow> g x = h x" shows "prod g {a..<b} = prod h {c..<d}"
Lxy_set_filter: fixes xs :: "'b list"   and S :: "'b set" shows "set (Lxy xs S) = S \<inter> set xs"
valid_plan_then_is_serial_sol: fixes \<pi>s :: "char list list" assumes "valid_plan \<pi>s" shows "SAS_Plus_Semantics.is_serial_solution_for_problem abs_prob (abs_ast_plan \<pi>s)"
obj_simps_4: fixes a :: "'a" assumes "H.obj a" shows "V.dom a = a"
set_zip_leftD: fixes x :: "'d"   and y :: "'e"   and xs :: "'d list"   and ys :: "'e list" assumes "(x, y) \<in> set (zip xs ys)" shows "x \<in> set xs"
simps_2: fixes v :: "'v"   and v1 :: "'v"   and w :: "'w"   and v2 :: "'v"   and ps :: "('v \<times> 'w \<times> 'v) list"   and v' :: "'v" shows "is_trail v ((v1, w, v2) # ps) v' = (v = v1 \<and> (v1, w, v2) \<in> E \<and> (v1, w, v2) \<notin> set ps \<and> (v2, w, v1) \<notin> set ps \<and> is_trail v2 ps v')"
Fix_lemma: fixes \<phi> :: "(((Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list) \<times> ((Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list) \<times> (Location \<times> Location) set \<Rightarrow> bool) \<Rightarrow> ((Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list) \<times> ((Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list) \<times> (Location \<times> Location) set \<Rightarrow> bool" assumes "Monotone \<phi>" shows "\<phi> (FIX \<phi>) = FIX \<phi>"
diff_def: fixes x :: "'c bit0"   and y :: "'c bit0" shows "x - y = Abs_bit0 ((Rep_bit0 x - Rep_bit0 y) mod int CARD('c bit0))"
3: fixes xa :: "'a"   and ya :: "'a" assumes "(xa, ya) = bezout_coefficients y1 (prod_list (y2 # ys))" shows "decompose (xa * x) (y2 # ys) = decompose' (xa * x) (y2 # ys) (tl (rev (decompose_aux (1::'a) (rev (y2 # ys)))))"
rel_restrict_empty: fixes R :: "('a \<times> 'a) set" shows "rel_restrict R {} = R"
oassert_def: fixes P :: "bool" shows "oassert P \<equiv> if P then oreturn () else ofail"
basic_identity\<^sub>\<kappa>_def: shows "(\<^bold>=\<^sub>\<kappa>) \<equiv> \<lambda>x y. x \<^bold>=\<^sub>E y \<^bold>\<or> \<lparr>A!,x\<rparr> \<^bold>& \<lparr>A!,y\<rparr> \<^bold>& \<^bold>\<box>(\<^bold>\<forall>F. \<lbrace>x,F\<rbrace> \<^bold>\<equiv> \<lbrace>y,F\<rbrace>)"
Ints_cases: fixes q :: "'a" assumes "q \<in> \<int>" obtains z :: "int" where "q = of_int z"
assms_4: fixes x1 :: "'a acom"   and x2 :: "'a acom" assumes "P x1"   and "P x2" shows "P (x1 \<nabla> x2)"
refine: fixes mrel :: "('a \<times> 'b) set"   and node_rel :: "('c \<times> 'd) set"   and node_eq_impl :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and node_hash_impl :: "nat \<Rightarrow> 'c \<Rightarrow> nat"   and node_def_hash_size :: "nat"   and G_impl :: "('c \<Rightarrow> bool, 'c \<Rightarrow> 'c list, 'c list, 'a) gen_g_impl_scheme"   and G :: "('d, 'b) graph_rec_scheme" assumes "fr_graph_impl_loc mrel node_rel node_eq_impl node_hash_impl node_def_hash_size G_impl G" shows "(compute_SCC_code node_eq_impl node_hash_impl node_def_hash_size G_impl, fr_graph.compute_SCC_impl G) \<in> \<langle>\<langle>\<langle>node_rel\<rangle>list_set_rel\<rangle>list_rel\<rangle>nres_rel"
indexed_var_def: fixes i :: "'d" shows "\<X>\<^bsub>i\<^esub> = indexed_const \<one> \<Otimes> i"
div_self: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and a :: "'b" assumes "class.semidom_divide divide plus minus zero times one"   and "a \<noteq> zero" shows "divide a a = one"
sets_dens_ctxt_measure_eq: fixes V :: "nat set"   and V' :: "nat set"   and \<Gamma> :: "nat \<Rightarrow> pdf_type"   and \<delta> :: "(nat \<Rightarrow> val) \<Rightarrow> ennreal"   and \<rho> :: "nat \<Rightarrow> val" shows "sets (dens_ctxt_measure (V, V', \<Gamma>, \<delta>) \<rho>) = sets (state_measure (V \<union> V') \<Gamma>)"
assoc_in_vhom\<^sub>A\<^sub>W\<^sub>C: fixes f :: "'a"   and g :: "'a"   and h :: "'a" assumes "ide f"   and "ide g"   and "ide h"   and "f \<star> g \<noteq> null"   and "g \<star> h \<noteq> null" shows "\<guillemotleft>\<a>[f, g, h] : (f \<star> g) \<star> h \<Rightarrow> f \<star> g \<star> h\<guillemotright>"
to_omega_to_stream: fixes w :: "nat \<Rightarrow> 'b" shows "to_omega (to_stream w) = w"
ds_ord_def: fixes x :: "'a"   and y :: "'a" shows "domain_semigroup.ds_ord ans_d (\<cdot>) x y = (x = ans_d x \<cdot> y)"
mult_1s_ring_1_2: fixes b :: "'a" shows "b * - Numeral1 = - b"
prems_1: shows "\<exists>a. trace c (t'' @ [ev]) a"
wset_thread_ok_upd: fixes ls :: "'a \<Rightarrow> 'b wait_set_status option"   and ts :: "'a \<Rightarrow> ('c \<times> 'd \<Rightarrow>f nat) option"   and t :: "'a"   and xw :: "'c \<times> 'd \<Rightarrow>f nat" assumes "wset_thread_ok ls ts" shows "wset_thread_ok ls (ts(t \<mapsto> xw))"
insertion_diff: fixes f :: "nat \<Rightarrow> 'a"   and p :: "'a mpoly"   and q :: "'a mpoly" shows "insertion f (p - q) = insertion f p - insertion f q"
elims_2: fixes x :: "nat \<times> nat list"   and xa :: "nat \<times> cell list \<times> cell list"   and xb :: "cell list" assumes "inv_check_left_moving_on_leftmost x xa xb" obtains as :: "nat" and lm :: "nat list" and s :: "nat" and l :: "cell list" and r :: "cell list" and ires :: "cell list" where "x = (as, lm)" and "xa = (s, l, r)" and "xb = ires" and "\<exists>rn. l = ires \<and> r = [Bk, Bk] @ <lm> @ Bk \<up> rn"
plugged_redex_not_val: fixes r :: "('a, 'b, 'c) expr"   and \<E> :: "('a, 'b, 'c) cntxt"   and t :: "('a, 'b, 'c) val" assumes "redex r" shows "\<E> [r] \<noteq> VE t"
lt_slength_0_2: fixes xs :: "[:'a:]" assumes "xs \<noteq> \<bottom>" shows "lt\<cdot>(slength\<cdot>xs)\<cdot>(slength\<cdot>xs + 1) = TT"
phi_wf: fixes g :: "'g"   and v :: "'val"   and vs :: "'val list" assumes "phi g v = Some vs" shows "length (old.predecessors g (defNode g v)) = length vs"
composite_simps_7: shows "cod (cod \<mu> \<star> cod \<nu> \<star> cod \<pi>) = cod (cod \<mu>) \<star> cod (cod \<nu>) \<star> cod (cod \<pi>)"
c_def: shows "c = max 0 (Max (set (map c' n_as)))"
span_alt: fixes B :: "'a set" shows "cspan B = {\<Sum>x | f x \<noteq> 0. f x *\<^sub>C x |f. {x. f x \<noteq> 0} \<subseteq> B \<and> finite {x. f x \<noteq> 0}}"
hyps_3: assumes "\<forall>a\<in>A. a dvd c"   and "\<forall>a1 a2. a1 \<in> A \<and> a2 \<in> A \<and> a1 \<noteq> a2 \<longrightarrow> comm_monoid_mult_class.coprime a1 a2" shows "\<Prod>A dvd c"
dg_prod_Cod_app: fixes x :: "V"   and I :: "V"   and \<AA> :: "V \<Rightarrow> V" assumes "x \<in>\<^sub>\<circ> dg_prod I \<AA>\<lparr>Arr\<rparr>" shows "dg_prod I \<AA>\<lparr>Cod\<rparr>\<lparr>x\<rparr> = (\<lambda>i\<in>\<^sub>\<circ>I. \<AA> i\<lparr>Cod\<rparr>\<lparr>x\<lparr>i\<rparr>\<rparr>)"
simps_2: fixes k :: "nat"   and x :: "'a"   and ts :: "('a btree \<times> 'a) list"   and t :: "'a btree" shows "del k x (Node ts t) = (case split ts x of (ls, []) \<Rightarrow> rebalance_last_tree k ls (del k x t) | (ls, (sub, sep) # rs) \<Rightarrow> if sep \<noteq> x then rebalance_middle_tree k ls (del k x sub) sep rs t else if sub = Leaf then Node (ls @ rs) t else let (sub_s, max_s) = split_max k sub in rebalance_middle_tree k ls sub_s max_s rs t)"
cod_char: fixes f :: "'a \<times> 'a \<times> 'a" shows "B.VxVxV.cod f = (if B.arr (fst f) \<and> B.VxV.arr (snd f) then (B.cod (fst f), B.VxV.cod (snd f)) else (B.null, B.VxV.null))"
preserves_hseq: fixes \<mu> :: "'c"   and \<nu> :: "'c" assumes "C.arr \<mu> \<and> C.arr \<nu> \<and> src\<^sub>C \<mu> = trg\<^sub>C \<nu>" shows "D.arr (F \<mu>) \<and> D.arr (F \<nu>) \<and> src\<^sub>D (F \<mu>) = trg\<^sub>D (F \<nu>)"
sterms_choice_disj: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'c, 'a, 'd) seqp"   and p :: "('b, 'c, 'a, 'd) seqp"   and p1 :: "('b, 'c, 'a, 'd) seqp"   and p2 :: "('b, 'c, 'a, 'd) seqp" assumes "wellformed \<Gamma>" shows "(p \<in> sterms \<Gamma> (p1 \<oplus> p2)) = (p \<in> sterms \<Gamma> p1 \<or> p \<in> sterms \<Gamma> p2)"
surjective: fixes r :: "('e, 'f) partial_object_scheme" shows "r = \<lparr>carrier = carrier r, \<dots> = partial_object.more r\<rparr>"
connected_Times: fixes S :: "'b set"   and T :: "'c set" assumes "connected S"   and "connected T" shows "connected (S \<times> T)"
SIA_implies_BSIA_for_modified_view: fixes ES :: "'b ES_rec"   and \<V> :: "'b V_rec"   and \<rho> :: "'b V_rec \<Rightarrow> 'b set"   and \<V>' :: "'b V_rec"   and \<rho>' :: "'b V_rec \<Rightarrow> 'b set" assumes "BSPTaxonomyDifferentCorrections ES \<V>"   and "SIA \<rho> \<V> Tr\<^bsub>ES\<^esub>"   and "\<V>' = \<lparr>V = V\<^bsub>\<V>\<^esub> \<union> N\<^bsub>\<V>\<^esub>, N = {}, C = C\<^bsub>\<V>\<^esub>\<rparr>"   and "\<rho> \<V> = \<rho>' \<V>'" shows "BSIA \<rho>' \<V>' Tr\<^bsub>ES\<^esub>"
f0n: fixes n :: "nat" shows "f 0 \<le> n"
analz_Un_implSecureSet_aux_3: fixes G :: "msg set"   and H :: "msg set"   and Ag :: "(agent \<times> agent) set" assumes "Enc_keys_clean (G \<union> H)" shows "analz (implSecureSet_aux Ag G \<union> H) \<subseteq> synth (analz (G \<union> H)) \<union> - payload"
coprime_def: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and a :: "'c"   and b :: "'c" assumes "class.algebraic_semidom divide plus minus zero times one" shows "algebraic_semidom.coprime times one a b = (\<forall>c. dvd.dvd times c a \<longrightarrow> dvd.dvd times c b \<longrightarrow> dvd.dvd times c one)"
assms: shows "P \<turnstile>\<^sub>S \<phi>"
PiM_empty: fixes M :: "'b \<Rightarrow> 'c measure" shows "Pi\<^sub>M {} M = count_space {\<lambda>_. undefined}"
lhd_lappend: fixes xs :: "'c llist"   and ys :: "'c llist" shows "lhd (lappend xs ys) = (if lnull xs then lhd ys else lhd xs)"
U0_def: shows "U0 = {p x |x. x \<in> A}"
in_lex_prod: fixes a :: "'b"   and b :: "'c"   and a' :: "'b"   and b' :: "'c"   and r :: "('b \<times> 'b) set"   and s :: "('c \<times> 'c) set" shows "(((a, b), a', b') \<in> r <*lex*> s) = ((a, a') \<in> r \<or> a = a' \<and> (b, b') \<in> s)"
execute_ref: fixes v :: "'b"   and h :: "heap" shows "execute (ref v) h = Some (Ref.alloc v h)"
associatedD1: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.normalization_semidom divide plus minus zero times one unit_factor normalize"   and "normalize a = normalize b" shows "dvd.dvd times a b"
nested_prop_atoms_subfrmlsn: fixes \<phi> :: "'a ltln" shows "nested_prop_atoms \<phi> \<subseteq> subfrmlsn \<phi>"
measurable_empty_iff: fixes N :: "'c measure"   and f :: "'d \<Rightarrow> 'c"   and M :: "'d measure" assumes "space N = {}" shows "(f \<in> M \<rightarrow>\<^sub>M N) = (space M = {})"
assms_3: shows "pmdl B \<subseteq> pmdl A"
numVals_def: fixes x :: "nat" shows "numVals x \<equiv> length (snd (snd (astDom ! x)))"
bounded_linear_mult_right: fixes x :: "'a" shows "bounded_linear ((*) x)"
S8: shows "c_sum (c_pair x y) \<le> x + y"
replace_primitive_matchexpr_replaces_disc: fixes m :: "'b match_expr"   and disc :: "'b \<Rightarrow> bool"   and sel :: "'b \<Rightarrow> 'c"   and C :: "'c \<Rightarrow> 'b"   and replace_f :: "'c negation_type \<Rightarrow> 'b match_expr" assumes "normalized_nnf_match m"   and "wf_disc_sel (disc, sel) C"   and "\<forall>a. \<not> has_disc disc (replace_f a)" shows "\<not> has_disc disc (replace_primitive_matchexpr (disc, sel) replace_f m)"
csk: shows "cs\<^bsup>\<pi>\<^esup> k = map \<pi> (sorted_list_of_set ({i. k cd\<^bsup>\<pi>\<^esup>\<rightarrow> i} \<union> {k}))"
is_aligned_mask_out_add_eq_sub: fixes p :: "'b word"   and n :: "nat"   and a :: "'b word"   and b :: "'b word" assumes "is_aligned p n" shows "p + a - b AND NOT (mask n) = p + (a - b AND NOT (mask n))"
of_nat_Real: fixes x :: "nat" shows "real x = Real.Real (\<lambda>n. rat_of_nat x)"
minus_1_mod_2_eq: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and division_segment :: "'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a" assumes "class.unique_euclidean_ring_with_nat divide plus minus zero times one modulo euclidean_size division_segment uminus" shows "modulo (uminus one) (numeral.numeral one plus (num.Bit0 num.One)) = one"
rel_setI: fixes A :: "'e set"   and B :: "'f set"   and R :: "'e \<Rightarrow> 'f \<Rightarrow> bool" assumes "\<And>x. x \<in> A \<Longrightarrow> \<exists>y\<in>B. R x y"   and "\<And>y. y \<in> B \<Longrightarrow> \<exists>x\<in>A. R x y" shows "rel_set R A B"
f: shows "\<forall>(x, K)\<in>p. f (x, K) \<notin> s \<and> f (x, K) \<in> interior K"
eq: fixes s :: "'a" assumes "s \<in> orbit f t" shows "g s = f s"
simps_1: fixes x :: "'b" shows "num_funs (Var x) = 0"
simps_29: fixes f1 :: "'e \<Rightarrow> 'f \<Rightarrow> 'f \<Rightarrow> 'd"   and f2 :: "nat \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> 'f \<Rightarrow> 'g \<Rightarrow> 'd"   and f3 :: "nat \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> 'f \<Rightarrow> 'g \<Rightarrow> 'd"   and f4 :: "'e \<Rightarrow> 'd"   and f5 :: "nat \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'd"   and x4 :: "'e" shows "(case Snapshot x4 of Trans x xa xb \<Rightarrow> f1 x xa xb | Send x xa xb xc xd xe \<Rightarrow> f2 x xa xb xc xd xe | Recv x xa xb xc xd xe \<Rightarrow> f3 x xa xb xc xd xe | Snapshot x \<Rightarrow> f4 x | RecvMarker x xa xb \<Rightarrow> f5 x xa xb) = f4 x4"
fBall_cong: fixes A :: "'b fset"   and B :: "'b fset"   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 "fBall A P = fBall B Q"
of_int_code_1: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and k :: "num" assumes "class.ring_1 one times plus zero minus uminus" shows "ring_1.of_int one plus zero minus (Int.Neg k) = uminus (numeral.numeral one plus k)"
ts_inv_ok_inv_ext_upd_invs: fixes ts :: "'g \<Rightarrow> ('h \<times> 'i \<Rightarrow>f nat) option"   and I :: "'g \<Rightarrow> 'j option"   and tas :: "('g, 'k, 'l) new_thread_action list"   and P :: "'j \<Rightarrow> 'g \<Rightarrow> 'k \<Rightarrow> 'l \<Rightarrow> bool" assumes "ts_inv_ok ts I"   and "thread_oks ts tas" shows "I \<subseteq>\<^sub>m upd_invs I P tas"
vint_distributive_law: fixes x :: "V"   and y :: "V"   and z :: "V" assumes "x \<in>\<^sub>\<circ> \<int>\<^sub>\<circ>"   and "y \<in>\<^sub>\<circ> \<int>\<^sub>\<circ>"   and "z \<in>\<^sub>\<circ> \<int>\<^sub>\<circ>" shows "x *\<^sub>\<int> (y +\<^sub>\<int> z) = x *\<^sub>\<int> y +\<^sub>\<int> (x *\<^sub>\<int> z)"
cong: fixes a :: "'b"   and b :: "'b"   and c :: "'b"   and d :: "'b" assumes "a = b"   and "c = d" shows "a * c = b * d"
exists_minimal: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and P :: "'a \<Rightarrow> bool"   and x :: "'a" assumes "class.wellorder less_eq less"   and "P x" shows "\<exists>x. P x \<and> (\<forall>y. P y \<longrightarrow> less_eq x y)"
take_induct: fixes P :: "'a\<cdot>('b, 'c) stateT \<Rightarrow> bool"   and x :: "'a\<cdot>('b, 'c) stateT" assumes "\<And>n. P (stateT_take n\<cdot>x)" shows "P x"
leftd: shows "c \<cdot>\<^sub>m (M * adjoint M) \<in> carrier_mat n n"
id: shows "separable_i ff_ops (of_int_poly_i ff_ops (Mp f)) = separable f''"
pth_1: fixes x :: "'b" shows "x \<equiv> 1 *\<^sub>R x"
chamber_vertex_conv_special_coset: fixes w :: "'w"   and X :: "'w set" assumes "w \<in> W"   and "X \<in> dual_order.poset_simplex_map \<P> {w}" shows "\<exists>s\<in>S. X = w +o \<langle>S - {s}\<rangle>"
nonintSfmla_def: shows "nonintSfmla \<equiv> fall2 (\<lambda>\<pi>' \<pi> \<pi>l. imp (alw (eqButGH (length \<pi>l) (Suc (length \<pi>l)))) (alw (eqOnGL (length \<pi>l) (Suc (length \<pi>l)))) (\<pi>l @ [\<pi>, \<pi>']))"
cptn_append_is_cptn: fixes b :: "'b com option \<times> 'b"   and c1 :: "('b com option \<times> 'b) list"   and a :: "'b com option \<times> 'b"   and c2 :: "('b com option \<times> 'b) list" assumes "b # c1 \<in> cptn"   and "a # c2 \<in> cptn"   and "(b # c1) ! length c1 = a" shows "b # c1 @ c2 \<in> cptn"
fact_not_neg: fixes n :: "nat" shows "\<not> fact n < (0::'a)"
prime_dvd_multD: fixes p :: "'a"   and a :: "'a"   and b :: "'a" assumes "prime p"   and "p dvd a * b" shows "p dvd a \<or> p dvd b"
member_fold_insert: fixes xs :: "'b list"   and dxs :: "'b set_dlist"   and x :: "'b" assumes "ID CEQ('b) \<noteq> None" shows "DList_Set.member (fold DList_Set.insert xs dxs) x = (DList_Set.member dxs x \<or> x \<in> set xs)"
simps_7: fixes ac :: "poscheckvariant"   and t :: "('fun, 'var) Term.term"   and s :: "('fun, 'var) Term.term"   and A :: "('fun, 'var) stateful_strand_step list"   and D :: "(('fun, 'var) Term.term \<times> ('fun, 'var) Term.term) list" shows "tr (\<langle>ac: t \<in> s\<rangle> # A) D = concat (map (\<lambda>B. map (\<lambda>d. \<langle>ac: pair (t, s) \<doteq> pair d\<rangle>\<^sub>s\<^sub>t # B) D) (tr A D))"
Ra_def: fixes P :: "'b process"   and s :: "'b event list" shows "Ra P s = {X. (s, X) \<in> F P}"
map_eq_imp_length_eq: fixes f :: "'b \<Rightarrow> 'a"   and xs :: "'b list"   and g :: "'c \<Rightarrow> 'a"   and ys :: "'c list" assumes "map f xs = map g ys" shows "length xs = length ys"
simps: fixes todo :: "'a list"   and fin :: "'a set" shows "mk_rtrancl_main todo fin = (if set todo = {} then fin else let (a, tod) = case todo of h # t \<Rightarrow> (h, t) in if \<exists>b\<in>fin. subsumes b a then mk_rtrancl_main tod fin else mk_rtrancl_main (r a @ tod) (insert a fin))"
matrix_vector_mult_def: fixes m :: "(('c, 'e) Finite_Cartesian_Product.vec, 'd) Finite_Cartesian_Product.vec"   and x :: "('c, 'e) Finite_Cartesian_Product.vec" shows "m *v x \<equiv> \<chi>i. \<Sum>j\<in>UNIV. m $h i $h j * x $h j"
from_nat_into_inj_infinite: fixes A :: "'c set"   and m :: "nat"   and n :: "nat" assumes "countable A"   and "infinite A" shows "(from_nat_into A m = from_nat_into A n) = (m = n)"
in_store_sops_conv: fixes sop :: "nat set \<times> ((nat \<Rightarrow> nat option) \<Rightarrow> nat)"   and xs :: "instr list" shows "(sop \<in> store_sops xs) = ((\<exists>volatile a A L R W. Write volatile a sop A L R W \<in> set xs) \<or> (\<exists>a t cond ret A L R W. RMW a t sop cond ret A L R W \<in> set xs))"
distinct_length_2_or_more: fixes a :: "'b"   and b :: "'b"   and xs :: "'b list" shows "cycle (a # b # xs) = (a \<noteq> b \<and> cycle (a # xs) \<and> cycle (b # xs))"
vars1: shows "free_vars e \<subseteq> V \<union> V'"
that: assumes "r = fst (alloc (map f [0..<n]) h)"   and "h' = snd (alloc (map f [0..<n]) h)"   and "get h' r = map f [0..<n]"   and "present h' r"   and "\<not> present h r" shows "thesis"
Suc: fixes new :: "'a list"   and "have" :: "'b" shows "set_of (relpow_i new have n) = {uu_. \<exists>a b m. uu_ = b \<and> a \<in> set new \<and> m \<le> n \<and> (a, b) \<in> (rel \<inter> {a. case a of (a, b) \<Rightarrow> b \<notin> set_of have}) ^^ m} \<union> set_of have"
States_Int_not_mem: fixes F :: "('a, 'b, 'c) seqauto set"   and SA :: "('a, 'b, 'c) seqauto" assumes "\<Union> (States ` F) \<inter> States SA = {}" shows "SA \<notin> F"
in_progressS_trans: fixes s :: "l2_state"   and R :: "fid_t"   and S :: "Runs.var set"   and s' :: "l2_state" assumes "in_progressS (progress s R) S"   and "(s, s') \<in> TS.trans l2" shows "in_progressS (progress s' R) S"
alw_Next'_alw_invs: fixes s :: "('b, 't, 'loc) Combined.configuration stream" assumes "holds all_invs s"   and "alw (holds (\<lambda>c. cri.InvMsgInGlob (exchange_config c))) s"   and "alw Next s" shows "alw (holds all_invs) s"
accepts_def: fixes fsm :: "'b fsm"   and xs :: "'b list" shows "accepts fsm xs \<equiv> \<exists>st st'. reaches fsm st xs st' \<and> st \<^bold>\<in> init fsm \<and> st' \<^bold>\<in> final fsm"
left_idem: fixes a :: "'a upred"   and b :: "'a upred" shows "(a \<and> a \<and> b) = (a \<and> b)"
inject_6: fixes x6 :: "Formula.formula"   and y6 :: "Formula.formula" shows "(formula.Neg x6 = formula.Neg y6) = (x6 = y6)"
snth_siterate: fixes f :: "'d \<Rightarrow> 'd"   and x :: "'d"   and n :: "nat" shows "siterate f x !! n = (f ^^ n) x"
igConsIPresIGWls_def: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'd, 'e, 'f, 'g) model_scheme" shows "igConsIPresIGWls MOD \<equiv> igVarIPresIGWls MOD \<and> igAbsIPresIGWls MOD \<and> igOpIPresIGWls MOD"
int_of_integer_code: fixes k :: "integer" shows "int_of_integer k = (if k < 0 then - int_of_integer (- k) else if k = 0 then 0 else let (l, j) = divmod_integer k 2; l' = 2 * int_of_integer l in if j = 0 then l' else l' + 1)"
leg1_in_hom_1: shows "\<guillemotleft>p\<^sub>1 : src p\<^sub>0 \<rightarrow> trg (k\<^sup>* \<star> h)\<guillemotright>"
wpc_simpl: fixes t :: "'d itself"   and Rs :: "(('e, 'f) labeled_graph \<times> ('e, 'f) labeled_graph) set"   and S :: "nat \<Rightarrow> ('e, 'g) labeled_graph" assumes "Simple_WPC t Rs S" shows "WPC t Rs S"
s_ns_mul_ext: fixes A :: "'b multiset"   and B :: "'b multiset"   and ns :: "('b \<times> 'b) set"   and s :: "('b \<times> 'b) set" assumes "(A, B) \<in> s_mul_ext ns s" shows "(A, B) \<in> ns_mul_ext ns s"
gtr_find_thread: fixes c :: "'a multiset"   and e :: "'b"   and c' :: "'c multiset"   and gtrs :: "(('a \<times> 'a multiset) \<times> 'b \<times> 'c \<times> 'c multiset) set" assumes "(c, e, c') \<in> gtr gtrs" obtains s :: "'a" and ce :: "'a multiset" and s' :: "'c" and ce' :: "'c multiset" where "c = add_mset s ce" and "c' = add_mset s' ce'" and "((s, ce), e, s', ce') \<in> gtrs"