hcju/isabelleps huggingface dataset:
InvariantGetReasonIsReason_def: fixes GetReason :: "Literal \<Rightarrow> nat option"   and F :: "Literal list list"   and M :: "(Literal \<times> bool) list"   and Q :: "Literal set" shows "InvariantGetReasonIsReason GetReason F M Q \<equiv> \<forall>literal. (literal el elements M \<and> \<not> literal el decisions M \<and> 0 < elementLevel literal M \<longrightarrow> (\<exists>reason. GetReason literal = Some reason \<and> 0 \<le> reason \<and> reason < length F \<and> isReason (F ! reason) literal (elements M))) \<and> (0 < currentLevel M \<and> literal \<in> Q \<longrightarrow> (\<exists>reason. GetReason literal = Some reason \<and> 0 \<le> reason \<and> reason < length F \<and> (isUnitClause (F ! reason) literal (elements M) \<or> clauseFalse (F ! reason) (elements M))))"
chine_composite_in_hom: shows "\<guillemotleft>Chn (\<mu> \<star> \<nu>) : \<mu>.dom.leg0 \<down>\<down> \<nu>.dom.leg1 \<rightarrow>\<^sub>C \<mu>.cod.leg0 \<down>\<down> \<nu>.cod.leg1\<guillemotright>"
inorder_ins: fixes t :: "'b bro"   and h :: "nat"   and a :: "'b" assumes "t \<in> T h"   and "Sorted_Less.sorted (inorder t)" shows "inorder (ins a t) = ins_list a (inorder t)"
init_obj_nonempty: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and C :: "char list"   and Cs :: "char list list"   and fs :: "char list \<Rightarrow> val option" assumes "init_obj P C (Cs, fs)" shows "Cs \<noteq> []"
2: shows "avl \<langle>l, ((a, b), h), r\<rangle>"
List1Red1: fixes uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and e :: "'addr expr1"   and s :: "'heap \<times> 'addr val list"   and ta :: "('addr, 'thread_id, 'heap) external_thread_action"   and e' :: "'addr expr1"   and s' :: "'heap \<times> 'addr val list"   and es :: "'addr expr1 list" assumes "uf,P,t \<turnstile>1 \<langle>e,s\<rangle> -ta\<rightarrow> \<langle>e',s'\<rangle>" shows "uf,P,t \<turnstile>1 \<langle>e # es,s\<rangle> [-ta\<rightarrow>] \<langle>e' # es,s'\<rangle>"
reach_init: fixes T :: "('a, 'b) TS_scheme"   and I :: "'a set" assumes "reach T \<subseteq> I" shows "init T \<subseteq> I"
obtain_min_worklist: fixes a :: "'a \<Rightarrow> 't zmultiset"   and loc' :: "'a" assumes "a loc' \<noteq> {#}\<^sub>z" obtains loc :: "'a" and t :: "'t" where "t \<in>#\<^sub>z a loc" and "\<forall>t' loc'. t' \<in>#\<^sub>z a loc' \<longrightarrow> \<not> t' < t"
upd_def: fixes i :: "nat"   and x :: "'a"   and a :: "'a array" shows "Array.upd i x a = Heap_Monad.guard (\<lambda>h. i < Array.length h a) (\<lambda>h. (a, Array.update a i x h))"
pi_lemma: fixes x :: "nat" shows "rec_exec rec_pi [x] = Pi x"
sup2CI: fixes B :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'a"   and y :: "'b"   and A :: "'a \<Rightarrow> 'b \<Rightarrow> bool" assumes "\<not> B x y \<Longrightarrow> A x y" shows "sup A B x y"
S'_approx: fixes n :: "nat"   and x :: "real" shows "S' n x = ln (real n + x) - ln x + p n x"
cr_mapping_def: shows "cr_mapping \<equiv> \<lambda>x y. x = mapping.rep y"
is_bal_insrt: fixes t :: "'a tree\<^sub>0"   and x :: "'a" assumes "is_bal t" shows "is_bal (insrt\<^sub>0 x t)"
simps_4: fixes uu :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and uv :: "'b" shows "Semantics.matches uu MatchAny uv = True"
neg_closed: fixes R :: "'a set"   and g :: "'a" assumes "Ring1 R"   and "g \<in> R" shows "- g \<in> R"
nat_filtration_borel_measurable_singleton: fixes f :: "bool stream \<Rightarrow> 'a"   and n :: "nat"   and x :: "bool stream" assumes "f \<in> borel_measurable (nat_filtration n)" shows "f -` {f x} \<in> sets (nat_filtration n)"
that: fixes D :: "'c set"   and E :: "'c set" assumes "f ` A = g ` D"   and "f ` B = g ` E"   and "D \<subseteq> C"   and "E \<subseteq> C" shows "thesis"
approx_trans3: fixes x :: "'a star"   and r :: "'a star"   and s :: "'a star" assumes "x \<approx> r"   and "x \<approx> s" shows "r \<approx> s"
has_vderiv_on_eq_rhs: fixes f :: "real \<Rightarrow> 'a"   and g' :: "real \<Rightarrow> 'a"   and T :: "real set"   and f' :: "real \<Rightarrow> 'a" assumes "(f has_vderiv_on g') T"   and "\<And>x. x \<in> T \<Longrightarrow> g' x = f' x" shows "(f has_vderiv_on f') T"
sum_is_subspace: fixes W1 :: "'c set"   and W2 :: "'c set" assumes "subspace K W1 V"   and "subspace K W2 V" shows "subspace K (subspace_sum W1 W2) V"
c: shows "nat_of_ipv4addr ((ipv4addr_of_nat (d + 256 * c + 65536 * b + 16777216 * a) >> 8) && mask 8) = c"
triangle_right_implies_left: assumes "(g \<star> \<epsilon>) \<cdot> (\<eta> \<star> g) = g" shows "(\<epsilon> \<star> f) \<cdot> (f \<star> \<eta>) = f"
preserves_ide: fixes a :: "'d" assumes "D.ide a" shows "D.ide (trg\<^sub>D a)"
frrestriction_fconverse: fixes r :: "V"   and A :: "V" shows "r\<inverse>\<^sub>\<bullet> \<restriction>\<^sup>r\<^sub>\<bullet> A = (r \<restriction>\<^sup>l\<^sub>\<bullet> A)\<inverse>\<^sub>\<bullet>"
ntr_add_context_s: fixes c :: "'n list multiset"   and e :: "('p, 'ba) label list"   and c' :: "'n list multiset"   and cn :: "'n list multiset" assumes "(c, e, c') \<in> ntr fg"   and "mon_w fg e \<inter> mon_c fg cn = {}" shows "(c + cn, e, c' + cn) \<in> ntr fg"
two_composable_arrows_of_spans_axioms: shows "two_composable_arrows_of_spans (\<cdot>) prj0 prj1 \<nu> (\<pi> \<star> \<rho>)"
rep_eq: fixes x :: "'a fset" shows "fMax x = Max (fset x)"
Bp_if_B: fixes t :: "'c bro"   and h :: "nat" assumes "t \<in> B h" shows "t \<in> Bp h"
blinding_respects_hashes_def: fixes h :: "'c \<Rightarrow> 'd"   and bo :: "'c \<Rightarrow> 'c \<Rightarrow> bool" shows "blinding_respects_hashes h bo \<equiv> bo \<le> vimage2p h h (=)"
Inj: fixes X :: "msg"   and H :: "msg set" assumes "X \<in> H" shows "X \<in> items H"
rcos_const_imp_mem: fixes I :: "'a set"   and R :: "('a, 'b) ring_scheme"   and i :: "'a" assumes "ideal I R"   and "i \<in> carrier R"   and "I +>\<^bsub>R\<^esub> i = I" shows "i \<in> I"
neg_divide_less_eq: fixes c :: "'a"   and b :: "'a"   and a :: "'a" assumes "c < (0::'a)" shows "(b / c < a) = (a * c < b)"
ln7: shows "ln 7 \<in> {3 / 2..15 / 7}"
usubstappf_exists: fixes \<sigma> :: "(char \<Rightarrow> trm option) \<times> (char \<Rightarrow> trm option) \<times> (char \<Rightarrow> fml option) \<times> (char \<Rightarrow> game option)"   and U :: "variable set"   and x :: "variable"   and \<phi> :: "fml" assumes "usubstappf \<sigma> U (Exists x \<phi>) \<noteq> undeff" shows "usubstappf \<sigma> U (Exists x \<phi>) = Afml (Exists x (the (usubstappf \<sigma> (U \<union> {x}) \<phi>)))"
Or_def: fixes P :: "('a, 'b, 'c) formula"   and Q :: "('a, 'b, 'c) formula" shows "(P || Q) = ! (! P && ! Q)"
nontrivial: fixes \<Delta> :: "('a, 'b) network_scheme"   and g :: "'a \<times> 'a \<Rightarrow> ennreal" assumes "flow_network \<Delta> g" shows "\<^bold>V\<^bsub>\<Delta>\<^esub> - {source \<Delta>, sink \<Delta>} \<noteq> {}"
AbstTerm_imp_abst_dbtm: fixes v :: "hf"   and i :: "hf"   and x :: "hf"   and x' :: "hf"   and e :: "(name, hf) finfun" assumes "AbstTerm v i x x'" shows "\<exists>t. x = \<lbrakk>quot_dbtm t\<rbrakk>e \<and> x' = \<lbrakk>quot_dbtm (abst_dbtm (decode_Var v) (nat_of_ord i) t)\<rbrakk>e"
that: fixes V :: "'a set" assumes "T \<subseteq> V"   and "openin (top_of_set S) V"   and "homotopic_with_canon (\<lambda>x. True) V U f g" shows "thesis"
1: shows "u @ [a] @ v\<^sub>1 =\<^sub>F u @ [a] @ v\<^sub>2"
cblinfun_inv_right: fixes A :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'b" assumes "iso_cblinfun A" shows "A o\<^sub>C\<^sub>L cblinfun_inv A = id_cblinfun"
integral_component_eq_cart: fixes f :: "'a \<Rightarrow> (real, 'b) vec"   and s :: "'a set"   and k :: "'b" assumes "f integrable_on s" shows "integral s (\<lambda>x. f x $ k) = integral s f $ k"
restrict_in_dom: fixes x :: "'d"   and A :: "'d set"   and m :: "'d \<Rightarrow> 'e option" assumes "x \<in> A" shows "(m|\<^bsub>A\<^esub>) x = m x"
\<delta>_le_i_plus_ix: shows "length \<delta> \<le> i + ix"
fun_diff_alt: fixes f :: "'b \<Rightarrow> 'a"   and g :: "'b \<Rightarrow> 'a" shows "{a \<in> Field s. f a \<noteq> g a} = (SUPP f \<union> SUPP g) \<inter> {a. f a \<noteq> g a}"
gcd_dvd1: fixes a :: "'b"   and b :: "'b" shows "gcd a b dvd a"
simps_1: fixes xs :: "'a list"   and c :: "int" shows "feval xs (FCnst c) = \<guillemotleft>c\<guillemotright>"
trunc_simps_1: fixes f :: "nat \<Rightarrow> 'a" assumes "f \<in> carrier P" shows "f = Cring_Poly.truncate R f \<oplus>\<^bsub>P\<^esub> monom P (f (deg R f)) (deg R f)"
gsfw_validI: fixes fw :: "'b simple_rule list" assumes "simple_fw_valid fw" shows "gsfw_valid (map simple_rule_dtor fw)"
eval_wp_Skip: fixes P :: "'a \<Rightarrow> real" shows "wp Skip P = P"
assms_1: shows "A o\<^sub>C\<^sub>L B = B"
WTrtWhile: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and E :: "char list \<Rightarrow> ty option"   and h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and e :: "expr"   and c :: "expr"   and T :: "ty" assumes "P,E,h \<turnstile> e : Boolean"   and "P,E,h \<turnstile> c : T" shows "P,E,h \<turnstile> while (e) c : Void"
in_span_adjust: fixes w :: "'a vec"   and us :: "'a vec list" assumes "w \<in> carrier_vec n"   and "set us \<subseteq> carrier_vec n"   and "distinct us" shows "w \<in> local.span (insert (adjuster n w us + w) (set us))"
GB_replace_red_stable_GB_dgrad_p_set: fixes d :: "'a \<Rightarrow> nat"   and G :: "('t \<Rightarrow>\<^sub>0 'c) set"   and m :: "nat"   and p :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c" assumes "dickson_grading d"   and "G \<subseteq> dgrad_p_set d m"   and "is_Groebner_basis G"   and "p \<in> G"   and "red (G - {p}) p q" shows "is_Groebner_basis (insert q (G - {p}))"
simps_2: fixes m :: "int"   and d :: "int"   and i :: "int"   and k :: "int"   and ks :: "int list" shows "hd_coeff1 m (Dvd d i (k # ks)) = (if k = 0 then Dvd d i (k # ks) else let m' = m div k in Dvd (m' * d) (m' * i) (1 # m' *\<^sub>s ks))"
embed_measure_eq_distr': fixes f :: "'c \<Rightarrow> 'd"   and M :: "'c measure" assumes "inj_on f (space M)" shows "embed_measure M f = distr M (embed_measure M f) f"
Reals_1: shows "(1::'a) \<in> \<real>"
vsv_appI: fixes a :: "V"   and b :: "V" assumes "\<langle>a, b\<rangle> \<in>\<^sub>\<circ> \<FF>\<lparr>ArrMap\<rparr>" shows "\<FF>\<lparr>ArrMap\<rparr>\<lparr>a\<rparr> = b"
strict_coboundedI2: fixes b :: "'g"   and c :: "'g"   and a :: "'g" assumes "b < c" shows "min a b < c"
m_assoc: fixes G :: "('a, 'b) monoid_scheme"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "Group.monoid G"   and "x \<in> carrier G"   and "y \<in> carrier G"   and "z \<in> carrier G" shows "x \<otimes>\<^bsub>G\<^esub> y \<otimes>\<^bsub>G\<^esub> z = x \<otimes>\<^bsub>G\<^esub> (y \<otimes>\<^bsub>G\<^esub> z)"
nsets_def: fixes A :: "'a set"   and n :: "nat" shows "[A]\<^bsup>n\<^esup> \<equiv> {N. N \<subseteq> A \<and> finite N \<and> card N = n}"
conseqPrePost: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "('e set \<times> 'd \<times> 'e set \<times> 'e set) set"   and F :: "'f set"   and P' :: "'e set"   and c :: "('e, 'd, 'f) com"   and Q' :: "'e set"   and A' :: "'e set"   and P :: "'e set"   and Q :: "'e set"   and A :: "'e set" assumes "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P' c Q',A'"   and "P \<subseteq> P'"   and "Q' \<subseteq> Q"   and "A' \<subseteq> A" shows "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P c Q,A"
Suc: fixes w :: "int" shows "bin_sign (bin_sc n b w) = bin_sign w"
q1_def: shows "q1 = except q {u. deg_pm u < poly_deg q}"
igSubstIGOp_def: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'd, 'e, 'f, 'g) model_scheme" shows "igSubstIGOp MOD \<equiv> \<forall>ys y Y delta inp binp. igWls MOD (asSort ys) Y \<and> igWlsInp MOD delta inp \<and> igWlsBinp MOD delta binp \<longrightarrow> igSubst MOD ys Y y (igOp MOD delta inp binp) = igOp MOD delta (igSubstInp MOD ys Y y inp) (igSubstBinp MOD ys Y y binp)"
rep_eq: fixes x :: "'a ACI_rexp" shows "ACI_nullable x = nullable (rep_ACI_rexp x)"
soconf_upd_obj: fixes P :: "'b prog"   and h :: "heap"   and C :: "char list"   and obj :: "sfields"   and a :: "nat"   and Ca :: "char list"   and fs :: "Objects.fields"   and fs' :: "Objects.fields" assumes "P,h,C \<turnstile>\<^sub>s obj \<surd>"   and "h a = \<lfloor>(Ca, fs)\<rfloor>" shows "P,h(a \<mapsto> (Ca, fs')),C \<turnstile>\<^sub>s obj \<surd>"
SeqStTermP_Eats: fixes s :: "name"   and k :: "name"   and s1 :: "tm"   and s2 :: "tm"   and k1 :: "tm"   and k2 :: "tm"   and t1 :: "tm"   and t2 :: "tm"   and u1 :: "tm"   and u2 :: "tm"   and v :: "tm"   and i :: "tm" assumes "atom s \<sharp> (k, s1, s2, k1, k2, t1, t2, u1, u2, v, i)"   and "atom k \<sharp> (t1, t2, u1, u2, v, i)" shows "{SeqStTermP v i t1 u1 s1 k1, SeqStTermP v i t2 u2 s2 k2} \<turnstile> SyntaxN.Ex s (SyntaxN.Ex k (SeqStTermP v i (Q_Eats t1 t2) (Q_Eats u1 u2) (Var s) (Var k)))"
expr_sem_pair_vars: fixes \<sigma> :: "nat \<Rightarrow> val"   and x :: "nat"   and y :: "nat" shows "expr_sem \<sigma> <Var x ,  Var y> = return_val <|\<sigma> x, \<sigma> y|>"
graph_def: fixes G :: "('c, 'd) graph_rec_scheme" shows "graph G \<equiv> g_V0 G \<subseteq> g_V G \<and> g_E G \<subseteq> g_V G \<times> g_V G"
left_neutral: fixes a :: "'f" shows "(0::'f) + a = a"
map_converter_id_move_right: fixes f :: "'a \<Rightarrow> 'e"   and g :: "'f \<Rightarrow> 'b"   and f' :: "'e \<Rightarrow> 'c"   and g' :: "'d \<Rightarrow> 'f" shows "map_converter f g f' g' 1\<^sub>C = map_converter id id (f' \<circ> f) (g \<circ> g') 1\<^sub>C"
HaddP_Mem_cancel_right_Mem: fixes H :: "fm set"   and x' :: "tm"   and y :: "tm"   and z' :: "tm"   and x :: "tm"   and z :: "tm" assumes "H \<turnstile> HaddP x' y z'"   and "H \<turnstile> HaddP x y z"   and "H \<turnstile> x' IN x"   and "H \<turnstile> OrdP x" shows "H \<turnstile> z' IN z"
less_eq_hf_def: fixes A :: "hf"   and B :: "hf" shows "(A \<le> B) = (\<forall>x. x \<^bold>\<in> A \<longrightarrow> x \<^bold>\<in> B)"
prc_compat: fixes f :: "'f \<times> nat"   and g :: "'f \<times> nat"   and s1 :: "bool"   and ns1 :: "bool"   and h :: "'f \<times> nat"   and s2 :: "bool"   and ns2 :: "bool"   and s :: "bool"   and ns :: "bool" assumes "prc f g = (s1, ns1)"   and "prc g h = (s2, ns2)"   and "prc f h = (s, ns)" shows "(ns1 \<and> ns2 \<longrightarrow> ns) \<and> (ns1 \<and> s2 \<longrightarrow> s) \<and> (s1 \<and> ns2 \<longrightarrow> s)"
lemma_order_pderiv1: fixes a :: "'a"   and n :: "nat"   and q :: "'a poly" shows "pderiv ([:- a, 1::'a:] ^ Suc n * q) = [:- a, 1::'a:] ^ Suc n * pderiv q + smult (of_nat (Suc n)) (q * [:- a, 1::'a:] ^ n)"
cod_comp: fixes g :: "'a2"   and f :: "'a2" assumes "C2.seq g f" shows "C2.cod (g \<cdot>\<^sub>2 f) = C2.cod g"
ListRedsVal: fixes e :: "char list exp"   and s\<^sub>0 :: "heap \<times> sfields \<times> sheap"   and b\<^sub>0 :: "bool"   and v :: "val"   and s\<^sub>1 :: "heap \<times> sfields \<times> sheap"   and b\<^sub>1 :: "bool"   and P :: "(char list list \<times> char list exp) prog"   and es :: "char list exp list"   and es' :: "char list exp list"   and s\<^sub>2 :: "heap \<times> sfields \<times> sheap"   and b\<^sub>2 :: "bool" assumes "P \<turnstile> \<langle>e,s\<^sub>0,b\<^sub>0\<rangle> \<rightarrow>* \<langle>Val v,s\<^sub>1,b\<^sub>1\<rangle>"   and "P \<turnstile> \<langle>es,s\<^sub>1,b\<^sub>1\<rangle> [\<rightarrow>]* \<langle>es',s\<^sub>2,b\<^sub>2\<rangle>" shows "P \<turnstile> \<langle>e # es,s\<^sub>0,b\<^sub>0\<rangle> [\<rightarrow>]* \<langle>Val v # es',s\<^sub>2,b\<^sub>2\<rangle>"
index_row_2: fixes A :: "'d mat"   and i :: "nat" shows "dim_vec (Matrix.row A i) = dim_col A"
funion_upper2: fixes B :: "'b fset"   and A :: "'b fset" shows "B |\<subseteq>| A |\<union>| B"
parallel_reduction_prm: fixes A :: "'a trm"   and A' :: "'a trm"   and \<pi> :: "'a prm" assumes "A >> A'" shows "\<pi> \<cdot> A >> \<pi> \<cdot> A'"
LSC_weaken: fixes \<Gamma> :: "'b formula multiset"   and F :: "'b formula" assumes "\<Gamma> \<Rightarrow>\<^sub>n" shows "F, \<Gamma> \<Rightarrow>\<^sub>n"
prj_simps_10: shows "C.arr \<mu>_\<nu>_\<pi>\<rho>.Prj\<^sub>1"
path_edge: fixes a :: "'b LDCFG_node \<times> ('c, 'd, 'e, 'f) edge_kind \<times> 'b LDCFG_node" assumes "lift_valid_edge valid_edge sourcenode targetnode kind Entry Exit a" shows "CFGExit.path (src a) [a] (trg a)"
split_at_conv_split_at_it: fixes n :: "nat"   and xs :: "'a list" shows "split_at n xs = split_at_it n xs"
times_binomial_minus1_eq: fixes k :: "nat"   and n :: "nat" assumes "0 < k" shows "k * (n choose k) = n * (n - 1 choose (k - 1))"
Node_1: assumes "wbt l"   and "l \<noteq> \<langle>\<rangle>" shows "2 \<le> (1 - \<alpha>) ^ (height l - 1) * real (size1 l)"
ucast_mono: fixes x :: "'b word"   and y :: "'b word" assumes "x < y"   and "y < 2 ^ LENGTH('c)" shows "ucast x < ucast y"
coeff_higher_pderiv: fixes m :: "nat"   and f :: "'b poly"   and n :: "nat" shows "coeff ((pderiv ^^ m) f) n = pochhammer (of_nat (Suc n)) m * coeff f (n + m)"
graph_restrict_def: fixes G :: "('a, 'b) graph_rec_scheme"   and R :: "'a set" shows "graph_restrict G R \<equiv> \<lparr>g_V = g_V G, g_E = rel_restrict (g_E G) R, g_V0 = g_V0 G - R, \<dots> = graph_rec.more G\<rparr>"
take_equal_bv_prepend: fixes emBits :: "nat"   and M :: "bit list" assumes "176 + sLen \<le> emBits"   and "roundup emBits 8 * 8 - emBits \<le> 7" shows "take (roundup emBits 8 * 8 - length (sha1 M) - sLen - 16) (maskedDB_zero (generate_DB (generate_PS emBits 160)) emBits) = bv_prepend (roundup emBits 8 * 8 - length (sha1 M) - sLen - 16) \<zero> []"
heq: shows "h ` span ((+) (- a) ` S) = T"
Suc_innermost_id2: fixes i :: "nat"   and n :: "nat"   and j :: "nat"   and j' :: "nat"   and i' :: "nat"   and m :: "nat \<Rightarrow> nat \<Rightarrow> 'a"   and k :: "nat" assumes "i \<le> n"   and "j \<le> n"   and "j' < j"   and "i' \<le> i" shows "fw m n (Suc k) i' j' i j = fw m n k i j i j"
cblinfun_of_mat_uminus: fixes M :: "complex mat" assumes "M \<in> carrier_mat (length canonical_basis) (length canonical_basis)" shows "cblinfun_of_mat (- M) = - cblinfun_of_mat M"
bool_list_fin: fixes m :: "nat" shows "finite {bl. length bl = m}"
cp\<^sub>P\<^sub>O\<^sub>S\<^sub>T: fixes self :: "'\<AA> state \<times> '\<AA> state \<Rightarrow> '\<alpha>0"   and res :: "'\<AA> state \<times> '\<AA> state \<Rightarrow> 'res"   and \<tau> :: "'\<AA> state \<times> '\<AA> state" shows "POST self res \<tau> = POST (\<lambda>_. self \<tau>) (\<lambda>_. res \<tau>) \<tau>"
i: shows "i \<in> set (\<sigma> (f, length ss))"
insert_nth_take_drop: fixes n :: "nat"   and x :: "'a"   and xs :: "'a list" shows "insert_nth n x xs = take n xs @ [x] @ drop n xs"
cobounded2: fixes a :: "'b DBMEntry"   and b :: "'b DBMEntry" shows "linordered_monoid.min a b \<preceq> b"
cmult_succ: fixes m :: "V"   and n :: "V" shows "ZFC_in_HOL.succ m \<otimes> n = n \<oplus> m \<otimes> n"
tu_lemma: fixes u\<^sub>1 :: "32 Word.word"   and r1 :: "real"   and u\<^sub>2 :: "32 Word.word"   and r2 :: "real"   and l\<^sub>1 :: "32 Word.word"   and l\<^sub>2 :: "32 Word.word" assumes "u\<^sub>1 \<equiv>\<^sub>U r1"   and "u\<^sub>2 \<equiv>\<^sub>U r2"   and "l\<^sub>1 \<equiv>\<^sub>L r1"   and "l\<^sub>2 \<equiv>\<^sub>L r2" shows "tu l\<^sub>1 u\<^sub>1 l\<^sub>2 u\<^sub>2 \<equiv>\<^sub>U r1 * r2"
assms_2: shows "b\<^sup>2 - 4 * a * c = 0"
sigma_sets_image: fixes S :: "'b set"   and \<Omega> :: "'b set"   and M :: "'b set set"   and f :: "'b \<Rightarrow> 'b" assumes "S \<in> sigma_sets \<Omega> M"   and "M \<subseteq> Pow \<Omega>"   and "f ` \<Omega> = \<Omega>"   and "inj_on f \<Omega>"   and "\<And>y. y \<in> M \<Longrightarrow> f ` y \<in> M" shows "f ` S \<in> sigma_sets \<Omega> M"
induct: fixes x :: "event list"   and P :: "event list \<Rightarrow> bool" assumes "x \<in> recur"   and "P []"   and "\<And>evsf X B. \<lbrakk>evsf \<in> recur; P evsf; X \<in> synth (analz (knows Spy evsf))\<rbrakk> \<Longrightarrow> P (Says Spy B X # evsf)"   and "\<And>evs1 NA A B. \<lbrakk>evs1 \<in> recur; P evs1; Nonce NA \<notin> used evs1\<rbrakk> \<Longrightarrow> P (Says A B Hash[Key (shrK A)] \<lbrace>Agent A, Agent B, Nonce NA, END\<rbrace> # evs1)"   and "\<And>evs2 NB A' B PA C. \<lbrakk>evs2 \<in> recur; P evs2; Nonce NB \<notin> used evs2; Says A' B PA \<in> set evs2\<rbrakk> \<Longrightarrow> P (Says B C Hash[Key (shrK B)] \<lbrace>Agent B, Agent C, Nonce NB, PA\<rbrace> # evs2)"   and "\<And>evs3 B' PB RB K B. \<lbrakk>evs3 \<in> recur; P evs3; Says B' Server PB \<in> set evs3; (PB, RB, K) \<in> respond evs3\<rbrakk> \<Longrightarrow> P (Says Server B RB # evs3)"   and "\<And>evs4 B C XH NB XA A NA Pa C' KBC KAB RA. \<lbrakk>evs4 \<in> recur; P evs4; Says B C \<lbrace>XH, Agent B, Agent C, Nonce NB, XA, Agent A, Agent B, Nonce NA, Pa\<rbrace> \<in> set evs4; Says C' B \<lbrace>Crypt (shrK B) \<lbrace>Key KBC, Agent C, Nonce NB\<rbrace>, Crypt (shrK B) \<lbrace>Key KAB, Agent A, Nonce NB\<rbrace>, RA\<rbrace> \<in> set evs4\<rbrakk> \<Longrightarrow> P (Says B A RA # evs4)" shows "P x"
e_in_Rats: fixes k :: "nat" assumes "\<And>k. k \<in> {1..n} \<Longrightarrow> p k \<in> \<rat>" shows "e k \<in> \<rat>"
seq_assoc: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x ; (y ; z) = x ; y ; z"
antisymmetric_subset: fixes B :: "'a set" assumes "B \<subseteq> X" shows "antisymmetric B (\<sqsubseteq>)"
elims_3: fixes x :: "'b trm \<Rightarrow> 'b trm \<Rightarrow> bool"   and xa :: "'b literal" assumes "\<not> validate_ground_lit x xa" obtains I :: "'b trm \<Rightarrow> 'b trm \<Rightarrow> bool" and E :: "'b equation" where "x = I__" and "xa = Pos E__" and "\<not> validate_ground_eq I__ E__"   | I :: "'b trm \<Rightarrow> 'b trm \<Rightarrow> bool" and E :: "'b equation" where "x = I" and "xa = Neg E" and "\<not> \<not> validate_ground_eq I E"
SD_efficient_same_support: fixes p :: "'alt pmf"   and p' :: "'alt pmf" assumes "set_pmf p = set_pmf p'"   and "p \<in> lotteries_on alts" shows "SD_efficient R p = SD_efficient R p'"
r_step_prim: shows "prim_recfn 1 r_step"
hom_simps_3: fixes x :: "'a list"   and y :: "'a list" assumes "x \<in> carrier (K [X])"   and "y \<in> carrier (K [X])" shows "local.eval (x \<oplus>\<^bsub>K [X]\<^esub> y) x = local.eval x x \<oplus> local.eval y x"
delta_nonneg: fixes dist :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and uniformity :: "('c \<times> 'c) filter"   and "open" :: "'c set \<Rightarrow> bool"   and deltaG :: "'c itself \<Rightarrow> real" assumes "class.Gromov_hyperbolic_space dist uniformity open deltaG" shows "0 \<le> deltaG TYPE('c)"
simps_26: fixes f1 :: "bool \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'b"   and f2 :: "bool \<Rightarrow> nat \<Rightarrow> nat set \<times> ((nat \<Rightarrow> nat option) \<Rightarrow> nat) \<Rightarrow> nat set \<Rightarrow> nat set \<Rightarrow> nat set \<Rightarrow> nat set \<Rightarrow> 'b"   and f3 :: "nat \<Rightarrow> nat \<Rightarrow> nat set \<times> ((nat \<Rightarrow> nat option) \<Rightarrow> nat) \<Rightarrow> ((nat \<Rightarrow> nat option) \<Rightarrow> bool) \<Rightarrow> (nat \<Rightarrow> nat \<Rightarrow> nat) \<Rightarrow> nat set \<Rightarrow> nat set \<Rightarrow> nat set \<Rightarrow> nat set \<Rightarrow> 'b"   and f4 :: "'b"   and f5 :: "nat set \<Rightarrow> nat set \<Rightarrow> nat set \<Rightarrow> nat set \<Rightarrow> 'b"   and x21 :: "bool"   and x22 :: "nat"   and x23 :: "nat set \<times> ((nat \<Rightarrow> nat option) \<Rightarrow> nat)"   and x24 :: "nat set"   and x25 :: "nat set"   and x26 :: "nat set"   and x27 :: "nat set" shows "(case Write x21 x22 x23 x24 x25 x26 x27 of Read x xa xb \<Rightarrow> f1 x xa xb | Write x xa xb xc xd xe xf \<Rightarrow> f2 x xa xb xc xd xe xf | RMW x xa xb xc xd xe xf xg xh \<Rightarrow> f3 x xa xb xc xd xe xf xg xh | Fence \<Rightarrow> f4 | Ghost x xa xb xc \<Rightarrow> f5 x xa xb xc) = f2 x21 x22 x23 x24 x25 x26 x27"
intrinsic_borderI: fixes k :: "'a list"   and ys :: "'a list" assumes "strict_border k ys" shows "strict_border (intrinsic_border ys) ys"
le_or_fail_def: fixes m :: "'a nres"   and m' :: "'a nres" shows "m \<le>\<^sub>n m' \<equiv> nofail m \<longrightarrow> m \<le> m'"