hcju/isabelleps huggingface dataset:
uint_distinct: fixes w1 :: "'a Word.word"   and w2 :: "'a Word.word" assumes "w1 \<noteq> w2" shows "uint w1 \<noteq> uint w2"
nth_replicate: fixes i :: "nat"   and n :: "nat"   and x :: "'b" assumes "i < n" shows "replicate n x ! i = x"
assms_1: shows "e1 \<sqsubset>\<^sup>i e2"
1: shows "\<forall>ucl. \<forall>u\<in>GL. out (doo st0 ucl) u = out (doo st0 (purge GH ucl)) u"
induct: fixes x1 :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and x2 :: "com"   and x3 :: "(char list \<Rightarrow> int) \<Rightarrow> nat"   and x4 :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and P :: "((char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool) \<Rightarrow> com \<Rightarrow> ((char list \<Rightarrow> int) \<Rightarrow> nat) \<Rightarrow> ((char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool) \<Rightarrow> bool" assumes "\<turnstile>\<^sub>1 {x1} x2 { x3 \<Down> x4}"   and "\<And>P. P P SKIP (\<lambda>s. Suc 0) P"   and "\<And>P a x. P (\<lambda>l s. P l (s[a/x])) (x ::= a) (\<lambda>s. Suc 0) P"   and "\<And>P b c\<^sub>1 e1 Q c\<^sub>2. \<lbrakk>\<turnstile>\<^sub>1 {\<lambda>l s. P l s \<and> bval b s} c\<^sub>1 { e1 \<Down> Q}; P (\<lambda>l s. P l s \<and> bval b s) c\<^sub>1 e1 Q; \<turnstile>\<^sub>1 {\<lambda>l s. P l s \<and> \<not> bval b s} c\<^sub>2 { e1 \<Down> Q}; P (\<lambda>l s. P l s \<and> \<not> bval b s) c\<^sub>2 e1 Q\<rbrakk> \<Longrightarrow> P P (IF b THEN c\<^sub>1 ELSE c\<^sub>2) (\<lambda>s. e1 s + Suc 0) Q"   and "\<And>P\<^sub>1 x e2' c\<^sub>1 e1 P\<^sub>2 e2 c\<^sub>2 P\<^sub>3 e. \<lbrakk>\<turnstile>\<^sub>1 {\<lambda>l s. P\<^sub>1 l s \<and> l x = e2' s} c\<^sub>1 { e1 \<Down> \<lambda>l s. P\<^sub>2 l s \<and> e2 s \<le> l x}; P (\<lambda>l s. P\<^sub>1 l s \<and> l x = e2' s) c\<^sub>1 e1 (\<lambda>l s. P\<^sub>2 l s \<and> e2 s \<le> l x); \<turnstile>\<^sub>1 {P\<^sub>2} c\<^sub>2 { e2 \<Down> P\<^sub>3}; P P\<^sub>2 c\<^sub>2 e2 P\<^sub>3; x \<notin> support P\<^sub>1; x \<notin> support P\<^sub>2; \<And>l s. P\<^sub>1 l s \<Longrightarrow> e1 s + e2' s \<le> e s\<rbrakk> \<Longrightarrow> P P\<^sub>1 (c\<^sub>1;; c\<^sub>2) e P\<^sub>3"   and "\<And>P b e' y c e'' e. \<lbrakk>\<turnstile>\<^sub>1 {\<lambda>l s. P l s \<and> bval b s \<and> e' s = l y} c { e'' \<Down> \<lambda>l s. P l s \<and> e s \<le> l y}; P (\<lambda>l s. P l s \<and> bval b s \<and> e' s = l y) c e'' (\<lambda>l s. P l s \<and> e s \<le> l y); \<forall>l s. bval b s \<and> P l s \<longrightarrow> 1 + e' s + e'' s \<le> e s; \<forall>l s. \<not> bval b s \<and> P l s \<longrightarrow> 1 \<le> e s; y \<notin> support P\<rbrakk> \<Longrightarrow> P P (WHILE b DO c) e (\<lambda>l s. P l s \<and> \<not> bval b s)"   and "\<And>P' e e' P Q Q' c. \<lbrakk>\<exists>k>0. \<forall>l s. P' l s \<longrightarrow> e s \<le> k * e' s \<and> (\<forall>t. \<exists>l'. P l' s \<and> (Q l' t \<longrightarrow> Q' l t)); \<turnstile>\<^sub>1 {P} c { e \<Down> Q}; P P c e Q\<rbrakk> \<Longrightarrow> P P' c e' Q'" shows "P x1 x2 x3 x4"
blocks_1: shows "set ((i_begin, i_end) # bs) \<subseteq> set (identify_blocks A k)"
image_mset_union: fixes f :: "'l \<Rightarrow> 'k"   and M :: "'l multiset"   and N :: "'l multiset" shows "image_mset f (M + N) = image_mset f M + image_mset f N"
supp_EF: fixes x :: "'b fs_set \<times> 'a" shows "supp (EF x) = supp x"
product_in_factor_by_sum: fixes x :: "i"   and y :: "i"   and z :: "i" assumes "O x y" shows "P (x \<otimes> y) (x \<otimes> (y \<oplus> z))"
intro: fixes f :: "'c \<Rightarrow> 'c" assumes "bij f" shows "bijection f"
IDirProds_finite: fixes I :: "'d set"   and S :: "'d \<Rightarrow> 'a set" assumes "finite I"   and "\<forall>i\<in>I. subgroup (S i) G"   and "\<forall>i\<in>I. finite (S i)" shows "finite (IDirProds G S I)"
rep_in_span: fixes scale :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and r :: "'d \<Rightarrow>\<^sub>0 'c" assumes "module scale" shows "module.rep scale r \<in> module.span scale (keys r)"
urep: shows "r \<le> ub \<and> (snd ([\<theta>]<>\<nu>) \<equiv>\<^sub>E ub)"
disjnt_Un2: fixes C :: "'b set"   and A :: "'b set"   and B :: "'b set" shows "disjnt C (A \<union> B) = (disjnt C A \<and> disjnt C B)"
indRelRTPO_impl_TRel_is_strong_reduction_simulation_rev: fixes TRel :: "('procT \<times> 'procT) set" assumes "strong_reduction_simulation ((indRelRTPO TRel)\<inverse>) (STCal Source Target)" shows "strong_reduction_simulation ((TRel\<^sup>+)\<inverse>) Target"
pref_nemp: fixes u :: "'b list"   and v :: "'b list" assumes "u \<noteq> \<epsilon>" shows "u \<cdot> v \<noteq> \<epsilon>"
convex_hull_list_def: fixes Vs :: "'a vec list" shows "m.convex_hull_list Vs = {x. \<exists>c. m.convex_lincomb_list c Vs x}"
map_poly_smult: fixes f :: "'d \<Rightarrow> 'c"   and c :: "'d"   and p :: "'d poly" assumes "f (0::'d) = (0::'c)"   and "\<And>c x. f (c * x) = f c * f x" shows "map_poly f (Polynomial.smult c p) = Polynomial.smult (f c) (map_poly f p)"
conjug_lq: fixes x :: "'b list"   and z :: "'b list"   and y :: "'b list" assumes "x \<cdot> z = z \<cdot> y" shows "y = z\<inverse>\<^sup>>(x \<cdot> z)"
ServiceCenter_a_msg: fixes i :: "nat \<Rightarrow> ECall_Info list"   and a :: "nat \<Rightarrow> aType list" assumes "ServiceCenter i a" shows "msg (Suc 0) a"
diff_less_conv: fixes i :: "nat"   and j :: "nat"   and k :: "nat" assumes "0 < i" shows "(j - k < i) = (j < i + k)"
zero_order_5: fixes n :: "'k multiset" shows "(\<not> {#} \<subset># n) = (n = {#})"
thif: fixes a :: "bool"   and b :: "bool"   and c :: "'b"   and d :: "'b" shows "(if a \<or> b then c else d) = (if a then c else if b then c else d)"
poly_roots_of_int_conv_sum_prime_factors: fixes q :: "int poly" shows "poly_roots (Ring_Hom_Poly.of_int_poly q) = (\<Sum>p\<in>#prime_factorization q. poly_roots (Ring_Hom_Poly.of_int_poly p))"
encR: fixes S :: "'procS"   and TRel :: "('procT \<times> 'procT) set" shows "SourceTerm S \<lesssim>\<lbrakk>\<cdot>\<rbrakk>RT<TRel> TargetTerm (\<lbrakk>S\<rbrakk>)"
\<omega>_limit_set_in_compact_tendsto: fixes K :: "'a set"   and x :: "'a" assumes "compact K"   and "K \<subseteq> X"   and "x \<in> X"   and "rev.trapped_forward x K" shows "((\<lambda>t. infdist (rev.flow0 x t) (rev.\<omega>_limit_set x)) \<longlongrightarrow> 0) at_top"
has_vector_derivative_def: fixes f :: "real \<Rightarrow> 'b"   and f' :: "'b"   and net :: "real filter" shows "(f has_vector_derivative f') net = D f \<mapsto> (\<lambda>x. x *\<^sub>R f') net"
convex_imp_starlike: fixes S :: "'b set" assumes "convex S"   and "S \<noteq> {}" shows "starlike S"
servTicket_form: fixes authK :: "nat"   and servK :: "nat"   and B :: "agent"   and Ts :: "msg"   and servTicket :: "msg"   and evs :: "event list" assumes "Crypt authK \<lbrace>Key servK, Agent B, Ts, servTicket\<rbrace> \<in> parts (knows Spy evs)"   and "Key authK \<notin> analz (knows Spy evs)"   and "evs \<in> kerbIV_gets" shows "servK \<notin> range shrK \<and> servK \<in> symKeys \<and> (\<exists>A. servTicket = Crypt (shrK B) \<lbrace>Agent A, Agent B, Key servK, Ts\<rbrace>)"
case_cong: fixes option :: "'e option"   and option' :: "'e option"   and f1 :: "'f"   and g1 :: "'f"   and f2 :: "'e \<Rightarrow> 'f"   and g2 :: "'e \<Rightarrow> 'f" assumes "option = option'"   and "option' = None \<Longrightarrow> f1 = g1"   and "\<And>x2. option' = Some x2 \<Longrightarrow> f2 x2 = g2 x2" shows "(case option of None \<Rightarrow> f1 | Some x2 \<Rightarrow> f2 x2) = (case option' of None \<Rightarrow> g1 | Some x2 \<Rightarrow> g2 x2)"
truncate_up_lt: fixes x :: "real"   and y :: "real"   and prec :: "nat" assumes "x < y" shows "x < truncate_up prec y"
LIMSEQ_iff_nz: fixes X :: "nat \<Rightarrow> 'b"   and L :: "'b" shows "X \<longlonglongrightarrow> L = (\<forall>r>0. \<exists>no>0. \<forall>n\<ge>no. dist (X n) L < r)"
weak_ranking_total_preorder_2: shows "of_weak_ranking (weak_ranking le) = le"
nb\<^sub>3: fixes opa :: "('a, 'b) sas_plus_operator" assumes "opa \<in> set (map (strips_op_to_sasp \<Psi>) ops')" shows "opa \<in> set (\<Psi>\<^sub>\<O>\<^sub>+)"
IH: fixes F :: "'fmla set" assumes "F \<subseteq> fmla"   and "finite F"   and "snd ` set txs \<subseteq> var"   and "fst ` set txs \<subseteq> trm" shows "prv (eqv (rawpsubst (scnj F) txs) (scnj ((\<lambda>\<phi>. rawpsubst \<phi> txs) ` F)))"
ct_eq_common_tsome: fixes x :: "ct"   and y :: "ct" shows "ct_eq x y = (\<exists>t. ct_eq x (TSome t) \<and> ct_eq (TSome t) y)"
assms: fixes x :: "'a" shows "f\<cdot>x = g\<cdot>x \<union> \<Union> (Rep_cfun f ` R\<cdot>x)"
eq_iff: fixes a :: "nat list"   and b :: "nat list" shows "(a = b) = (b \<le>\<^sub>v a \<and> a \<le>\<^sub>v b)"
emeasure_ge_1_iff: fixes M :: "'c measure"   and A :: "'c set" assumes "prob_space M" shows "(1 \<le> emeasure M A) = (emeasure M A = 1)"
cvalid_def: fixes \<Gamma> :: "'e \<Rightarrow> ('f, 'e, 'g) com option"   and \<Theta> :: "('f set \<times> 'e \<times> 'f set \<times> 'f set) set"   and F :: "'g set"   and P :: "'f set"   and c :: "('f, 'e, 'g) com"   and Q :: "'f set"   and A :: "'f set" shows "\<Gamma>,\<Theta>\<Turnstile>\<^bsub>/F\<^esub> P c Q,A \<equiv> (\<forall>(P, p, Q, A)\<in>\<Theta>. \<Gamma>\<Turnstile>\<^bsub>/F\<^esub> P Call p Q,A) \<longrightarrow> \<Gamma>\<Turnstile>\<^bsub>/F\<^esub> P c Q,A"
uv: shows "Uvariation (F 0) \<omega> (U \<union> {RVar x, DVar x})"
plus_uexpr_def: fixes u :: "('e, 'f) uexpr"   and v :: "('e, 'f) uexpr" shows "u + v = bop (+) u v"
push_forward_def: fixes X :: "('a \<Rightarrow> real) \<Rightarrow> real" shows "d12.push_forward X = restrict0 d12.dest.diff_fun_space (\<lambda>g. X (restrict0 d12.src.carrier (g \<circ> f)))"
head_def: fixes t :: "('c, 'd) FingerTree" shows "head t = FingerTreeStruc.head (Rep_FingerTree t)"
PdXXX_linear: fixes d :: "'b" shows "Linear_order (Pd' d)"
mr_not_DA: fixes s :: "((int \<times> int) set set, int) Combinators list"   and x :: "int \<times> (int \<times> int) \<times> (int \<times> int) \<times> DummyContent"   and p :: "((int \<times> int) set set, int) Combinators list"   and a :: "(int \<times> int) set set"   and ab :: "(int \<times> int) set set" assumes "wellformed_policy1_strong s"   and "applied_rule_rev C x p = \<lfloor>DenyAllFromTo a ab\<rfloor>"   and "set p = set s" shows "applied_rule_rev C x s \<noteq> \<lfloor>DenyAll\<rfloor>"
assms: shows "\<parallel>nid\<parallel>\<^bsub>t n\<^esub>"
backward_finite_path_root_path: fixes r :: "'a"   and x :: "'a" assumes "backward_finite_path_root r x" shows "path x"
ahm_iteratei_def: shows "ArrayHashMap_Impl.ahm_iteratei \<equiv> rec_hashmap (\<lambda>a n. ahm_iteratei_aux a)"
ext: fixes g :: "'a"   and f :: "'a" assumes "g \<cdot> f \<noteq> S.null" shows "S.seq g f"
P\<delta>i: fixes i :: "'a" shows "0 < (SOME \<delta>. 0 < \<delta> \<and> (\<forall>x2. x2 \<noteq> x1 \<and> dist x2 x1 < \<delta> \<longrightarrow> dist (f i x2) (f i x1) < \<epsilon> / real CARD('a))) \<and> (\<forall>x2. x2 \<noteq> x1 \<and> dist x2 x1 < (SOME \<delta>. 0 < \<delta> \<and> (\<forall>x2. x2 \<noteq> x1 \<and> dist x2 x1 < \<delta> \<longrightarrow> dist (f i x2) (f i x1) < \<epsilon> / real CARD('a))) \<longrightarrow> dist (f i x2) (f i x1) < \<epsilon> / real CARD('a))"
intuitionistic_sep_impl_sep_true_P: fixes P :: "'a \<Rightarrow> bool"   and h :: "'a" assumes "P h"   and "intuitionistic P" shows "((\<lambda>s. True) \<longrightarrow>* P) h"
CEMU_def: shows "CEMU \<equiv> CEM \<and> CMU"
ex: shows "\<forall>ip\<in>kD rt. 1 \<le> the (dhops rt ip)"
FTI: assumes "n = Trueif" shows "ospec (fti s) (\<lambda>(r, s'). (r, ft) \<in> Q s \<and> I' s')"
r_into_rtranclp: fixes r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" assumes "r x y" shows "r\<^sup>*\<^sup>* x y"
zero_def: shows "0 \<equiv> Abs_G (0::'b)"
termi_impl_not_inf: fixes c :: "com"   and s :: "state" assumes "c \<down> s" shows "\<not> PsHoareTotal.inf [c] s"
SNF_1x2_works: shows "\<forall>A\<in>carrier_mat 1 2. is_SNF A (1\<^sub>m 1, Smith_1x2_JNF A)"
lincomb_singles: fixes a :: "'f"   and m :: "'v" shows "lincomb' [a] [m] = a \<cdot> m"
defNode_ex1: fixes v :: "'val"   and g :: "'g" assumes "v \<in> allVars g" shows "\<exists>!n. n \<in> set (\<alpha>n g) \<and> v \<in> allDefs g n"
in_kD_case: fixes dip :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option"   and en :: "'a"   and es :: "nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set \<Rightarrow> 'a" assumes "dip \<in> kD rt" shows "(case rt dip of None \<Rightarrow> en | Some r \<Rightarrow> es r) = es (the (rt dip))"
apply_rsp': fixes R1 :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and R2 :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'd"   and x :: "'a"   and y :: "'b" assumes "rel_fun R1 R2 f g"   and "R1 x y" shows "R2 (f x) (g y)"
take_MkI_Cons: fixes n :: "int"   and x :: "'b"   and xs :: "['b]" assumes "0 < n" shows "take\<cdot>(MkI\<cdot>n)\<cdot>(x : xs) = x : take\<cdot>(MkI\<cdot>(n - 1))\<cdot>xs"
assms: shows "i - Suc (length x) < Suc (length y)"
linear_eq_0_on_span: fixes f :: "'a \<Rightarrow> 'b"   and b :: "'a set"   and x :: "'a" assumes "clinear f"   and "\<And>x. x \<in> b \<Longrightarrow> f x = (0::'b)"   and "x \<in> cspan b" shows "f x = (0::'b)"
wlp_while_unfold: fixes \<pi> :: "char list \<Rightarrow> com option"   and b :: "bexp"   and c :: "com"   and Q :: "(char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> bool"   and s :: "char list \<Rightarrow> int \<Rightarrow> int" shows "wlp \<pi> (WHILE b DO c) Q s = (if bval b s then wlp \<pi> c (wlp \<pi> (WHILE b DO c) Q) s else Q s)"
to_fun_def: fixes f :: "nat \<Rightarrow> nat \<Rightarrow> int" shows "to_fun f \<equiv> to_function Zp f"
power_le_one: 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 zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and a :: "'c"   and n :: "nat" assumes "class.linordered_semidom plus minus less_eq less zero times one"   and "less_eq zero a"   and "less_eq a one" shows "less_eq (power.power one times a n) one"
PhiWhileOp_def: fixes b :: "BExpr"   and \<Phi> :: "(Var \<Rightarrow> Val) \<times> (Var \<Rightarrow> Val) \<Rightarrow> bool" shows "PhiWhileOp b \<Phi> = (\<lambda>\<Psi> (s, t). (evalB b t \<longrightarrow> (\<exists>r. \<Phi> (r, t) \<and> (\<forall>w.  r \<approx> w  \<longrightarrow> \<Psi> (s, w)))) \<and> (\<not> evalB b t \<longrightarrow>  s \<approx> t ))"
v11_vrange_ex1_eq: fixes b :: "V" shows "(b \<in>\<^sub>\<circ> \<R>\<^sub>\<circ> r) = (\<exists>!a. a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> r \<and> r\<lparr>a\<rparr> = b)"
visit_subset_dfs: fixes S :: "'b"   and xs :: "'a list"   and y :: "'a" assumes "invariant S"   and "list_all is_node xs"   and "is_node y"   and "memb y S" shows "memb y (dfs S xs)"
bind_ask: fixes f :: "'r \<Rightarrow> 'm"   and g :: "'a \<times> 's \<Rightarrow> 'm" shows "bind (ask f) g = ask (\<lambda>r. bind (f r) g)"
weakBisimWeakCoinduct: fixes P :: "pi"   and Q :: "pi"   and X :: "(pi \<times> pi) set" assumes "(P, Q) \<in> X"   and "\<And>P Q. (P, Q) \<in> X \<Longrightarrow> P \<leadsto><X> Q"   and "\<And>P Q. (P, Q) \<in> X \<Longrightarrow> (Q, P) \<in> X" shows "P \<approx> Q"
reachable_def: fixes y :: "'c" shows "reachable y = (\<lambda>s. \<exists>x. x \<in> mut_roots s \<union> mut_ghost_honorary_root s \<union> tso_store_refs s \<and> (x reaches y) s)"
ag_eq_diffzero1: fixes A :: "('c, 'd) aGroup_scheme"   and a :: "'c"   and b :: "'c" assumes "aGroup A"   and "a \<in> carrier A"   and "b \<in> carrier A" shows "(a = b) = (-\<^sub>a\<^bsub>A\<^esub> a \<plusminus>\<^bsub>A\<^esub> b = \<zero>\<^bsub>A\<^esub>)"
hyps: fixes st' :: "nat"   and l' :: "cell list"   and r' :: "cell list" assumes "steps0 (st, l, r) ap stp = (st', l', r')"   and "0 < st'" shows "steps0 (st, l, r) (adjust0 ap) stp = (st', l', r')"
DFT_def: fixes n :: "nat"   and a :: "nat \<Rightarrow> complex" shows "DFT n a \<equiv> \<lambda>i. \<Sum>j = 0..<n. FFT.root n ^ (i * j) * a j"
sisoL_siso: fixes cval :: "'a \<Rightarrow> 'b \<Rightarrow> real"   and indis :: "('b \<times> 'b) set"   and aval :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and tval :: "'d \<Rightarrow> 'b \<Rightarrow> bool"   and cl :: "('d, 'c, 'a) cmd list"   and c :: "('d, 'c, 'a) cmd" assumes "PL_Indis cval indis"   and "PL_Indis.sisoL aval tval cval indis cl"   and "c \<in> set cl" shows "PL_Indis.siso aval tval cval indis c"
dim: fixes i :: "nat" assumes "i < length vs" shows "vs ! i \<in> carrier_vec n"
step: fixes i :: "nat" assumes "(rel_chain_fun i a b r, rel_chain_fun (Suc i) a b r) \<in> r" shows "(rel_chain_fun (Suc i) a b r, rel_chain_fun (Suc (Suc i)) a b r) \<in> r"
assms: shows "sgn (poly p a) \<noteq> sgn (poly p b)"
substInp_empInp: fixes ys :: "'varSort"   and Y :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set"   and y :: "'var" shows "empInp %[Y / y]_ys = empInp"
alg_dvd_uminus_right: fixes x :: "'b"   and y :: "'b" assumes "x alg_dvd y" shows "x alg_dvd - y"
times_X_not_zero: fixes p :: "nat \<Rightarrow> 'a" assumes "p \<in> carrier P"   and "p \<noteq> \<zero>\<^bsub>P\<^esub>" shows "X_poly R \<otimes>\<^bsub>P\<^esub> p \<noteq> \<zero>\<^bsub>P\<^esub>"
list_repr_def: fixes i :: "nat"   and ba :: "'b list \<times> 'b list"   and xs :: "'b list" shows "list_repr i ba xs = (i \<le> length xs \<and> fst ba = rev (take i xs) \<and> snd ba = drop i xs)"
assms_3: fixes k :: "nat"   and lholed :: "Lholed"   and i :: "nat" assumes "Lfilled k lholed [$Br i] cs_es" shows "i < k"
simps_1: fixes N :: "'a"   and P :: "('a, 'b, 'c) psi" shows "inputChain [] N P = \<rparr>N.P"
if_SE_def: fixes c :: "'c \<Rightarrow> bool"   and E :: "'c \<Rightarrow> ('d \<times> 'c) option"   and F :: "'c \<Rightarrow> ('d \<times> 'c) option" shows "_if_SE c E F = (\<lambda>\<sigma>. if c \<sigma> then E \<sigma> else F \<sigma>)"
connected_singleton: fixes u :: "'b" shows "Undirected_Graph.connected (ins_node u graph_empty)"
degree_sum_smaller: fixes n :: "nat"   and A :: "'a set"   and f :: "'a \<Rightarrow> 'b poly" assumes "0 < n"   and "finite A"   and "\<And>x. x \<in> A \<Longrightarrow> degree (f x) < n" shows "degree (sum f A) < n"
fmdrop_set_fmdom: fixes A :: "('d, 'e) fmap" shows "fmdrop_set (fmdom' A) A = fmempty"
mono_Inf: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and f :: "'b \<Rightarrow> 'c"   and A :: "'b set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "order.mono less_eq f" shows "f (Inf A) \<le> \<Sqinter> (f ` A)"
add_diff_cancel_right: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and c :: "'a"   and b :: "'a" assumes "class.cancel_ab_semigroup_add plus minus" shows "minus (plus a c) (plus b c) = minus a b"
Fail_HInv3: fixes s :: "state"   and s' :: "state"   and t :: "Proc"   and p :: "Proc"   and q :: "Proc"   and d :: "Disk" assumes "Fail s s' t"   and "HInv3_inner s p q d" shows "HInv3_inner s' p q d"
Ps_sym: fixes i :: "nat" assumes "i < length cms" shows "sym (P_rel (cms ! i))"
atom_map_Sup_pres: fixes X :: "'a set" shows "atom_map (\<Squnion> X) = \<Union> (atom_map ` X)"
sum_list_cons_def: fixes s :: "'s" shows "local.sum_list_cons s = sum_list (unstream g s)"
arr_eqI': fixes f :: "('e set, 'e \<Rightarrow> 'e) concrete_category.arr"   and f' :: "('e set, 'e \<Rightarrow> 'e) concrete_category.arr" assumes "par f f'"   and "\<And>t. \<guillemotleft>t : unity \<rightarrow> local.dom f\<guillemotright> \<Longrightarrow> f \<cdot> t = f' \<cdot> t" shows "f = f'"
nonzero_inverse_mult_distrib: fixes a :: "'a"   and b :: "'a" assumes "a \<noteq> (0::'a)"   and "b \<noteq> (0::'a)" shows "inverse (a * b) = inverse b * inverse a"
simps_5: fixes k :: "name"   and e :: "name list"   and A :: "fm" assumes "atom k \<sharp> e" shows "trans_fm e (SyntaxN.Ex k A) = DBEx (trans_fm (k # e) A)"
proper_it_loc_def: fixes it :: "'a \<Rightarrow> ('b list \<Rightarrow> bool) \<Rightarrow> ('b \<Rightarrow> 'b list \<Rightarrow> 'b list) \<Rightarrow> 'b list \<Rightarrow> 'b list"   and it' :: "'a \<Rightarrow> ('c \<Rightarrow> bool) \<Rightarrow> ('b \<Rightarrow> 'c \<Rightarrow> 'c) \<Rightarrow> 'c \<Rightarrow> 'c" shows "proper_it_loc it it' \<equiv> proper_it' it it'"
cases: fixes r :: "gState" obtains vars :: "(String.literal, variable) assoc_list" and channels :: "channel list" and timeout :: "bool" and procs :: "pState list" where "r = \<lparr>gState.vars = vars, channels = channels, timeout = timeout, procs = procs\<rparr>"
list_induct2_snoc_Cons: fixes P :: "'b list \<Rightarrow> 'a list \<Rightarrow> bool"   and xs :: "'b list"   and ys :: "'a list" assumes "\<And>ys. P [] ys"   and "\<And>xs. P xs []"   and "\<And>xs x y ys. P xs ys \<Longrightarrow> P (xs @ [x]) (y # ys)" shows "P xs ys"
x_readable: shows "\<forall>i<length cms\<^sub>1. x \<notin> snd (cms\<^sub>1 ! i) AsmNoRead"
simps_7: fixes \<sigma> :: "(char list \<times> 'a list) trace"   and v :: "'a list"   and i :: "nat"   and I :: "\<I>"   and \<phi> :: "'a MFOTL.formula" shows "MFOTL.sat \<sigma> v i (formula.Next I \<phi>) = (mem (\<tau> \<sigma> (Suc i) - \<tau> \<sigma> i) I \<and> MFOTL.sat \<sigma> v (Suc i) \<phi>)"
assms_7: shows "InvariantEquivalentZL (getF state) (getM state) F0"
fcard_fminus1_less: fixes x :: "'b"   and A :: "'b fset" assumes "x |\<in>| A" shows "fcard (A |-| {|x|}) < fcard A"
assms_2: shows "U \<inter> X frontier_of S \<noteq> {}"
inject: fixes x1 :: "tag"   and x2 :: "agent"   and x3 :: "agent"   and x4 :: "msg"   and y1 :: "tag"   and y2 :: "agent"   and y3 :: "agent"   and y4 :: "msg" shows "(Chan x1 x2 x3 x4 = Chan y1 y2 y3 y4) = (x1 = y1 \<and> x2 = y2 \<and> x3 = y3 \<and> x4 = y4)"
vseq_implies_hpar_2: fixes \<nu> :: "'a"   and \<mu> :: "'a" assumes "seq \<nu> \<mu>" shows "trg \<nu> = trg \<mu>"
st: shows "(Inum (x # bs) t / real_of_int n + Inum (x # bs) s / real_of_int m) / 2 = Inum (x # bs) (Add (Mul m t) (Mul n s)) / real_of_int (2 * n * m)"
nres_monad2: fixes M :: "'b nres" shows "M \<bind> RETURN = M"
smap_union_def: fixes smap1 :: "'b \<Rightarrow> 'a set"   and smap2 :: "'b \<Rightarrow> 'a set"   and k :: "'b" shows "smap1 \<union>. smap2 k = smap1 k \<union> smap2 k"
dense_repr_replicate_zero: fixes R :: "('c, 'd) ring_scheme"   and n :: "nat"   and p :: "'c list" assumes "ring R" shows "ring.dense_repr R (replicate n \<zero>\<^bsub>R\<^esub> @ p) = ring.dense_repr R p"
take_Suc_last: fixes as :: "nat"   and xs :: "'a list" assumes "Suc as \<le> length xs" shows "take (Suc as) xs = take as xs @ [xs ! as]"
case1: fixes y :: "game" assumes "zin y (right_options x)" shows "\<not> ge_game (zero_game, plus_game y (neg_game x))"
mk_linear_orders_distinct: fixes r :: "'a list"   and C :: "'a set"   and B :: "'a set" assumes "r \<in> local.mk_linear_orders C B"   and "B \<subseteq> A" shows "distinct r"
vderiv_on_composeI: fixes f :: "real \<Rightarrow> 'b"   and f' :: "real \<Rightarrow> 'b"   and g :: "real \<Rightarrow> real"   and T :: "real set"   and g' :: "real \<Rightarrow> real"   and h :: "real \<Rightarrow> 'b" assumes " D f = f' on g ` T"   and " D g = g' on T"   and "h = (\<lambda>t. g' t *\<^sub>R f' (g t))" shows " D (\<lambda>t. f (g t)) = h on T"
int_induct2: fixes P :: "nat \<Rightarrow> nat \<Rightarrow> bool"   and n :: "nat"   and m :: "nat" assumes "P 0 0"   and "\<And>n m. P n m \<Longrightarrow> P (Suc n) m"   and "\<And>n m. P n m \<Longrightarrow> P n (Suc m)" shows "P n m"