hcju/isabelleps huggingface dataset:
inject_4: fixes x41 :: "bexp"   and x42 :: "'a"   and x43 :: "'a acom"   and x44 :: "'a"   and x45 :: "'a acom"   and x46 :: "'a"   and y41 :: "bexp"   and y42 :: "'a"   and y43 :: "'a acom"   and y44 :: "'a"   and y45 :: "'a acom"   and y46 :: "'a" shows "(IF x41 THEN {x42} x43 ELSE {x44} x45 {x46} = IF y41 THEN {y42} y43 ELSE {y44} y45 {y46}) = (x41 = y41 \<and> x42 = y42 \<and> x43 = y43 \<and> x44 = y44 \<and> x45 = y45 \<and> x46 = y46)"
GrD1: fixes x :: "'b"   and fx :: "'c"   and A :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "(x, fx) \<in> BNF_Def.Gr A f" shows "x \<in> A"
wivs_least_def: shows "wivs_least \<equiv> - wivs_overflow"
smap_un1: fixes x :: "'a"   and Y :: "'a set" shows "smap (insert x (\<eta> (smap Y))) = smap (\<eta> x \<union> Y)"
snd_Gauss_Jordan_column_k_det_P_is_snd_Gauss_Jordan_column_k_PA: fixes n :: "'b"   and i :: "nat"   and A :: "(('b, 'c) vec, 'd) vec"   and k :: "nat"   and P :: "(('b, 'd) vec, 'd) vec" shows "snd (Gauss_Jordan_column_k_det_P (n, i, A) k) = snd (Gauss_Jordan_column_k_PA (P, i, A) k)"
intro: fixes hom :: "'a \<Rightarrow> 'b" assumes "idom_hom hom" shows "map_poly_idom_hom hom"
Gexp: fixes t :: "'b tree" shows "2 ^ lrh t \<le> size t + 1"
prems: fixes x :: "'a \<times> 'b" assumes "x \<in># add_mset x__ A \<times># {#b#}" shows "P x = Q (fst x)"
head_App: fixes s :: "('c, 'd) tm"   and t :: "('c, 'd) tm" shows "head (App s t) = head s"
t_sift_down_height: fixes l :: "'b tree"   and x :: "'b"   and r :: "'b tree" assumes "braun \<langle>l, x, r\<rangle>" shows "t_sift_down l x r \<le> height \<langle>l, x, r\<rangle>"
flush_all_until_volatile_write_Write\<^sub>s\<^sub>b_non_volatile_commute: fixes i :: "nat"   and ts :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'b memref list \<times> 'c \<times> 'd \<times> 'e) list"   and p :: "'b"   and "is" :: "instr list"   and xs :: "nat \<Rightarrow> nat option"   and a :: "nat"   and sop :: "nat set \<times> ((nat \<Rightarrow> nat option) \<Rightarrow> nat)"   and v :: "nat"   and A :: "nat set"   and L :: "nat set"   and R :: "nat set"   and W :: "nat set"   and sb :: "'b memref list"   and \<D> :: "'c"   and \<O> :: "'d"   and \<R> :: "'e"   and \<D>' :: "'c"   and \<R>' :: "'e"   and m :: "nat \<Rightarrow> nat" assumes "i < length ts"   and "ts ! i = (p, is, xs, Write\<^sub>s\<^sub>b False a sop v A L R W # sb, \<D>, \<O>, \<R>)"   and "\<forall>i<length ts. \<forall>j<length ts. i \<noteq> j \<longrightarrow> (let (uu_, uu_, uu_, sb\<^sub>i, uu_, uu_, uu_) = ts ! i; (uu_, uu_, uu_, sb\<^sub>j, uu_, uu_, uu_) = ts ! j in outstanding_refs is_Write\<^sub>s\<^sub>b sb\<^sub>i \<inter> outstanding_refs is_Write\<^sub>s\<^sub>b (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb\<^sub>j) = {})"   and "\<forall>j<length ts. i \<noteq> j \<longrightarrow> (let (uu_, uu_, uu_, sb\<^sub>j, uu_, uu_, uu_) = ts ! j in a \<notin> outstanding_refs is_Write\<^sub>s\<^sub>b (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb\<^sub>j))" shows "flush_all_until_volatile_write (ts[i := (p, is, xs, sb, \<D>', \<O>, \<R>')]) (m(a := v)) = flush_all_until_volatile_write ts m"
complex_inner_1_right: fixes x :: "complex" shows "inner x 1 = Re x"
prems_3: fixes d :: "nat" assumes "d < Suc dm" shows "odd (ix p d) \<and> lv b d \<le> lv p d \<and> ix p d < (ix b d + 1) * 2 ^ (lv p d - lv b d) \<and> (ix b d - 1) * 2 ^ (lv p d - lv b d) < ix p d"
subst_id_on_type_scheme_list': fixes A :: "type_scheme list"   and S :: "nat \<Rightarrow> typ" assumes "\<forall>x\<in>free_tv A. S x = TVar x" shows "$ S A = $ id_subst A"
terminates_fewer_guards: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c' :: "('e, 'd, 'f) com"   and s :: "('e, 'f) xstate"   and c :: "('e, 'd, 'f) com" assumes "\<Gamma>\<turnstile>c' \<down> s"   and "c \<subseteq>\<^sub>g c'"   and "\<Gamma>\<turnstile> \<langle>c',s\<rangle> \<Rightarrow>\<notin>range Fault" shows "\<Gamma>\<turnstile>c \<down> s"
unrest_false: fixes x :: "'a \<Longrightarrow> 'b" shows "x \<sharp> false"
cmult_in_iff: fixes c :: "'a"   and f :: "'b \<Rightarrow> 'a"   and F :: "'b filter"   and g :: "'b \<Rightarrow> 'a" assumes "c \<noteq> (0::'a)" shows "((\<lambda>x. c * f x) \<in> \<Theta>[F](g)) = (f \<in> \<Theta>[F](g))"
AbstForm_trans_fm_eq: fixes x :: "hf"   and e :: "(name, hf) finfun"   and A :: "fm"   and x' :: "hf"   and i :: "name" assumes "x = \<lbrakk>\<guillemotleft>A\<guillemotright>\<rbrakk>e"   and "x' = \<lbrakk>quot_dbfm (trans_fm [i] A)\<rbrakk>e" shows "AbstForm (q_Var i) 0 x x'"
has_laplaceD: fixes f :: "real \<Rightarrow> complex"   and L :: "complex"   and s :: "complex" assumes "(f has_laplace L) s" shows "((\<lambda>t. exp (t *\<^sub>R - s) * f t) has_integral L) {0..}"
has_vector_derivative_minus: fixes f :: "real \<Rightarrow> 'a"   and f' :: "'a"   and net :: "real filter" assumes "(f has_vector_derivative f') net" shows "((\<lambda>x. - f x) has_vector_derivative - f') net"
minus_sum: fixes a :: "nat \<Rightarrow> int"   and b :: "nat \<Rightarrow> int" assumes "a \<in> carrier Zp"   and "b \<in> carrier Zp" shows "\<ominus> (a \<oplus> b) = \<ominus> a \<oplus> \<ominus> b"
measurable_emeasure_subprob_algebra: fixes a :: "'a set"   and A :: "'a measure" assumes "a \<in> sets A" shows "(\<lambda>M. emeasure M a) \<in> borel_measurable (subprob_algebra A)"
op_norm_scaleR: fixes c :: "real"   and A :: "(('a, 'b) vec, 'c) vec" shows "\<parallel>c *\<^sub>R A\<parallel>\<^sub>o\<^sub>p = \<bar>c\<bar> * (\<parallel>A\<parallel>\<^sub>o\<^sub>p)"
termFSwSbImorph_def: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'c"   and hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> 'd"   and MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd, 'e) Iteration.model_scheme" shows "termFSwSbImorph h hA MOD \<equiv> termFSwImorph h hA MOD \<and> ipresSubstAll h hA MOD"
rawpsubstT_pls: fixes r1 :: "'trm"   and r2 :: "'trm"   and txs :: "('trm \<times> 'var) list" assumes "r1 \<in> trm"   and "r2 \<in> trm"   and "snd ` set txs \<subseteq> var"   and "fst ` set txs \<subseteq> trm" shows "rawpsubstT (pls r1 r2) txs = pls (rawpsubstT r1 txs) (rawpsubstT r2 txs)"
Rep_Assertion_inverse: fixes x :: "'c Assertion" shows "Abs_Assertion (Rep_Assertion x) = x"
tendsto_uniform_limitI: fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and S :: "'b set"   and l :: "'b \<Rightarrow> 'c"   and F :: "'a filter"   and x :: "'b" assumes "uniform_limit S f l F"   and "x \<in> S" shows "((\<lambda>y. f y x) \<longlongrightarrow> l x) F"
countable_imp_null_set_lborel: fixes A :: "'b set" assumes "countable A" shows "A \<in> null_sets lborel"
rel_F1_F2: shows "rel_fundefs (Fstd_get F1) (Finca_get F2)"
simTrans_simAbs_cong: fixes t :: "'s Trace"   and ec :: "'rep"   and a :: "'a"   and ec' :: "'rep" assumes "t \<in> jkbpC"   and "simAbs ec = sim_equiv_class a t"   and "simAbs ec = simAbs ec'" shows "simAbs ` set (simTrans a ec) = simAbs ` set (simTrans a ec')"
additionTr2: fixes n :: "nat" shows "\<forall>f h. f \<in> {j. j \<le> Suc n} \<rightarrow> carrier A \<and> h \<in> {j. j \<le> Suc n} \<rightarrow> {j. j \<le> Suc n} \<and> inj_on h {j. j \<le> Suc n} \<longrightarrow> \<Sigma>\<^sub>e A cmp f h Suc n = \<Sigma>\<^sub>e A f Suc n"
dirichlet_prod'_def: fixes f :: "nat \<Rightarrow> 'a"   and g :: "real \<Rightarrow> 'a"   and x :: "real" shows "dirichlet_prod' f g x = sum_upto (\<lambda>m. f m * g (x / real m)) x"
join_closed: fixes L :: "('a, 'b) gorder_scheme"   and x :: "'a"   and y :: "'a" assumes "weak_upper_semilattice L"   and "x \<in> carrier L"   and "y \<in> carrier L" shows "x \<squnion>\<^bsub>L\<^esub> y \<in> carrier L"
fbspec: fixes A :: "'c fset"   and P :: "'c \<Rightarrow> bool"   and x :: "'c" assumes "fBall A P"   and "x |\<in>| A" shows "P x"
intros: fixes tval :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and aval :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and cf :: "('a, 'c) com \<times> 'b"   and cf' :: "('a, 'c) com \<times> 'b"   and s'' :: "'b" assumes "PL.MtransC tval aval cf cf'"   and "PL.transT tval aval cf' s''" shows "PL.MtransT tval aval cf s''"
stop_type: fixes \<Gamma> :: "'Var \<Rightarrow> Sec option" shows "\<turnstile> \<Gamma> {Stop} \<Gamma>"
subset_dfs_def: fixes A :: "bool list bdd list \<times> bool list"   and x :: "bool list" shows "subset_dfs A x = gen_dfs (subset_succs A) subset_ins subset_memb subset_empt [x]"
simps_4: fixes \<Gamma> :: "char list set"   and s\<^sub>1 :: "stmt"   and s\<^sub>2 :: "stmt" shows "wfs \<Gamma> (Seq s\<^sub>1 s\<^sub>2) = (wfs \<Gamma> s\<^sub>1 \<and> wfs \<Gamma> s\<^sub>2)"
remove_nth_def: fixes n :: "nat"   and xs :: "'a list" shows "remove_nth n xs \<equiv> take n xs @ drop (Suc n) xs"
static_portfolio_self_financing: fixes p :: "'b \<Rightarrow> nat \<Rightarrow> 'c \<Rightarrow> real"   and Mkt :: "('c, 'b) discrete_market" assumes "\<forall>x\<in>support_set p. \<forall>w i. p x i w = p x (Suc i) w" shows "self_financing Mkt p"
fundamental_property: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and sem :: "'g \<Rightarrow> ('c \<Rightarrow> 'd option) list \<Rightarrow> 'g \<Rightarrow> ('c \<Rightarrow> 'd option) list \<Rightarrow> bool"   and identifies :: "'b \<Rightarrow> 'g \<Rightarrow> bool"   and n :: "'b"   and c :: "'g"   and cf :: "'c \<Rightarrow> 'd option"   and c' :: "'g"   and s' :: "('c \<Rightarrow> 'd option) list"   and cfs' :: "(('c \<Rightarrow> 'd option) \<times> 'e) list" assumes "CFG_semantics_wf sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main sem identifies"   and "identifies n c"   and "sem c [cf] c' s'" shows "\<exists>n' as. CFG.valid_path' sourcenode targetnode kind valid_edge get_return_edges n as n' \<and> identifies n' c' \<and> CFG.preds procs (CFG.kinds kind as) [(cf, undefined)] \<and> CFG.transfers procs (CFG.kinds kind as) [(cf, undefined)] = cfs' \<and> map fst cfs' = s'"
pell_valuation_solution_pos_nat: fixes z :: "nat \<times> nat" assumes "solution z" shows "0 < pell_valuation (case z of (x, y) \<Rightarrow> (int x, int y))"
not_less: fixes x :: "'a"   and y :: "'a" shows "(\<not> x \<prec> y) = (y \<preceq> x)"
queue_correct_enq: fixes q :: "'b queue"   and x :: "'b" assumes "invar q" shows "list (enq x q) = list q @ [x]"
asm: shows "(as \<inter> supp x, x) \<approx>set (=) supp p (as' \<inter> supp x', x')"
elims: fixes x :: "int \<times> nat"   and y :: "int \<times> nat" assumes "incr2 x = y" obtains xa :: "int" and n :: "nat" where "x = (xa, n)" and "y = (xa, n + 1)"
union_eq_empty: fixes M :: "'c multiset"   and N :: "'c multiset" shows "(M + N = {#}) = (M = {#} \<and> N = {#})"
invertible_def: fixes M :: "'c set"   and composition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit :: "'c"   and u :: "'c" assumes "monoid M composition unit"   and "u \<in> M" shows "monoid.invertible M composition unit u = (\<exists>v\<in>M. composition u v = unit \<and> composition v u = unit)"
DDH_advantage_def: fixes n :: "nat"   and \<A> :: "'grp \<Rightarrow> 'grp \<Rightarrow> 'grp \<Rightarrow> bool spmf" shows "local.ddh.DDH_advantage n \<A> = \<bar>spmf (local.ddh.DDH0 n \<A>) True - spmf (local.ddh.DDH1 n \<A>) True\<bar>"
inEdges_def: fixes g :: "'g"   and n :: "'node" shows "inEdges g n \<equiv> map (\<lambda>(f, d). (f, d, n)) (inEdges' g n)"
simps_4: fixes V :: "'c set"   and s :: "('d, 'c) Term.term"   and t :: "('d, 'c) Term.term"   and S :: "('d, 'c) strand_step list" shows "wf\<^sub>s\<^sub>t V (\<langle>assign: s \<doteq> t\<rangle>\<^sub>s\<^sub>t # S) = (fv t \<subseteq> V \<and> wf\<^sub>s\<^sub>t (V \<union> fv s) S)"
rep_eq: fixes x :: "uint16" shows "lsb x = lsb (Rep_uint16 x)"
fps_add_idem: fixes f :: "('c, 'd) fps" shows "f + f = f"
unit_factor_field: fixes x :: "'a" shows "unit_factor x = x"
remove_def: fixes x :: "'d"   and A :: "'d set" shows "Set.remove x A = A - {x}"
update_def: fixes x :: "'c"   and y :: "'d"   and t :: "('c \<times> 'd) tree234" shows "update x y t = tree\<^sub>i (upd x y t)"
add_cancel_right_right: fixes plus :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and minus :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and zero :: "'g"   and a :: "'g"   and b :: "'g" assumes "class.cancel_comm_monoid_add plus minus zero" shows "(a = plus a b) = (b = zero)"
simps_1: fixes \<phi>\<^sub>1 :: "atom fm"   and \<phi>\<^sub>2 :: "atom fm" shows "inf\<^sub>+ (And \<phi>\<^sub>1 \<phi>\<^sub>2) = and (inf\<^sub>+ \<phi>\<^sub>1) (inf\<^sub>+ \<phi>\<^sub>2)"
ppf: shows "prime_product_factor (nat (\<bar>a\<bar> * b)) = (sq, fact)"
dnso6: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and zero :: "'b"   and domain_op :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "class.domain_near_semiring_one_zerol plus times one zero domain_op less_eq less" shows "domain_op zero = zero"
lm031: fixes f :: "'a \<Rightarrow> 'b"   and x :: "'a" shows "argmax f {x} = {x}"
f: shows "f \<in> M \<rightarrow>\<^sub>M S"
push_down_older_seniors_senior: fixes \<Sigma> :: "'c set"   and \<delta> :: "'d \<Rightarrow> 'c \<Rightarrow> 'd"   and q\<^sub>0 :: "'d"   and w :: "nat \<Rightarrow> 'c"   and x :: "nat"   and n :: "nat"   and y :: "nat" assumes "semi_mojmir \<Sigma> \<delta> q\<^sub>0 w"   and "\<not> semi_mojmir_def.sink \<Sigma> \<delta> q\<^sub>0 (semi_mojmir_def.token_run \<delta> q\<^sub>0 w x n)"   and "\<not> semi_mojmir_def.sink \<Sigma> \<delta> q\<^sub>0 (semi_mojmir_def.token_run \<delta> q\<^sub>0 w y n)"   and "semi_mojmir_def.older_seniors \<Sigma> \<delta> q\<^sub>0 w x n = semi_mojmir_def.older_seniors \<Sigma> \<delta> q\<^sub>0 w y n" shows "semi_mojmir_def.senior \<delta> q\<^sub>0 w x n = semi_mojmir_def.senior \<delta> q\<^sub>0 w y n"
assms_1: shows "x\<^sup>\<omega> = (0::'a)"
ell2_norm_finite_support: fixes S :: "'c set"   and x :: "'c ell2" assumes "finite S"   and "\<And>i. i \<notin> S \<Longrightarrow> Rep_ell2 x i = 0" shows "norm x = sqrt (\<Sum>i\<in>S. (cmod (Rep_ell2 x i))\<^sup>2)"
pps_to_list_sorted_wrt: fixes pair_of_term :: "'c \<Rightarrow> 'd \<times> 'e"   and term_of_pair :: "'d \<times> 'e \<Rightarrow> 'c"   and ord :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ord_strict :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ord_term :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and ord_term_strict :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and S :: "'c set" assumes "ordered_term pair_of_term term_of_pair ord ord_strict ord_term ord_term_strict" shows "sorted_wrt ord_term_strict\<inverse>\<inverse> (ordered_term.pps_to_list ord_term S)"
iT_defs_2: fixes n :: "nat" shows "[\<dots>n] \<equiv> {..n}"
partition_by_biflows_def: fixes E :: "('a \<times> 'a) list" shows "partition_by_biflows E \<equiv> (rembiflowdups (filter_for_biflows E), remdups (filter_for_uniflows E))"
sum: fixes a :: "complex"   and g :: "'e \<Rightarrow> 'd"   and A :: "'e set" shows "a *\<^sub>C sum g A = (\<Sum>x\<in>A. a *\<^sub>C g x)"
LeastI2: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and P :: "'b \<Rightarrow> bool"   and a :: "'b"   and Q :: "'b \<Rightarrow> bool" assumes "class.wellorder less_eq less"   and "P a"   and "\<And>x. P x \<Longrightarrow> Q x" shows "Q (ord.Least less_eq P)"
cnj_idem: fixes a :: "'a" shows "cnj_class.cnj (cnj_class.cnj a) = a"
c_iop_welldefTr1: fixes N :: "'a set"   and a :: "'a" assumes "G \<triangleright> N"   and "a \<in> carrier G" shows "c_iop G N (N \<bullet> a) \<subseteq> N \<bullet> \<rho> a"
ptautology: fixes p :: "'a pfm"   and M :: "('a, 'b) kripke"   and w :: "'b" assumes "ptautology p" shows "M, w \<Turnstile>\<^sub>! p"
polyhedron_Int_affine_parallel_minimal: fixes S :: "'b set" shows "polyhedron S = (\<exists>F. finite F \<and> S = affine hull S \<inter> \<Inter> F \<and> (\<forall>h\<in>F. \<exists>a b. a \<noteq> (0::'b) \<and> h = {x. a \<bullet> x \<le> b} \<and> (\<forall>x\<in>affine hull S. x + a \<in> affine hull S)) \<and> (\<forall>F'\<subset>F. S \<subset> affine hull S \<inter> \<Inter> F'))"
int_simps_21: fixes P :: "'b \<Rightarrow> bool" shows "\<turnstile> (P \<and> \<not> P) = #False"
eq_refl: fixes x :: "'c list"   and y :: "'c list" assumes "x = y" shows "sublist x y"
prime_factorization_Lcm_factorial: fixes A :: "'a set" assumes "(0::'a) \<notin> A"   and "subset_mset.bdd_above (prime_factorization ` A)" shows "prime_factorization (Lcm_factorial A) = Sup (prime_factorization ` A)"
inv: fixes R :: "'a set"   and smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and M :: "'b set"   and smult' :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and T :: "'b \<Rightarrow> 'c"   and N :: "'c set" assumes "RModuleIso R smult M smult' T N" shows "RModuleIso R smult' N smult (the_inv_into M T \<down> N) M"
foundation25': fixes \<tau> :: "'a state \<times> 'a state"   and Q :: "'a state \<times> 'a state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>"   and P :: "'a state \<times> 'a state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>" assumes "\<tau> \<Turnstile> Q" shows "\<tau> \<Turnstile> P or Q"
intro: fixes sort :: "'a list \<Rightarrow> 'a list" assumes "\<And>l. sorted (sort l)"   and "\<And>l. sort l <~~> l" shows "Sort sort"
tl_P_new_1: shows "tl P_new \<noteq> []"
less_Assertion_def: fixes p :: "'a Assertion"   and q :: "'a Assertion" shows "(p < q) = ({\<cdot> p } < {\<cdot> q })"
sorted_list_of_set_bij_betw: fixes A :: "'a set" assumes "finite A" shows "bij_betw ((!) (sorted_list_of_set A)) {..<card A} A"
zero_in_nonpos_Ints: shows "(0::'a) \<in> \<int>\<^sub>\<le>\<^sub>0"
eq_swap_sym: fixes v :: "'item list"   and w :: "'item list" assumes "v =\<^sub>S w" shows "w =\<^sub>S v"
lastListInList: fixes list :: "'a list" assumes "list \<noteq> []" shows "last list \<in> set list"
affine_translation: fixes S :: "'a set"   and a :: "'a" shows "affine S = affine ((+) a ` S)"
simps_3: fixes ts :: "nat"   and v :: "nat" shows "max_tstp (Inl ts) (Inr v) = Inl ts"
psi_at_0: fixes s :: "nat \<Rightarrow> nat option"   and j :: "nat" assumes "r1_bc s" shows "r1_bc.psi s j 0 \<down>= j"
sup_ext: fixes H :: "'b set"   and h :: "'b \<Rightarrow> real"   and c :: "('b \<times> real) set set"   and M :: "('b \<times> real) set set"   and E :: "'b set"   and p :: "'b \<Rightarrow> real"   and F :: "'b set"   and f :: "'b \<Rightarrow> real" assumes "graph H h = \<Union> c"   and "M = norm_pres_extensions E p F f"   and "c \<in> chains M"   and "\<exists>x. x \<in> c" shows "graph F f \<subseteq> graph H h"
assms_3: shows "distributed M S X (\<lambda>x. ennreal (Px x))"
m_o2: fixes X :: "char list set"   and o1 :: "'a st option"   and o2 :: "'a st option" assumes "finite X"   and "top_on\<^sub>o o1 (- X)"   and "top_on\<^sub>o o2 (- X)"   and "o1 < o2" shows "m\<^sub>o o2 X < m\<^sub>o o1 X"
scan_take: fixes k :: "nat"   and f :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and xs :: "'c list"   and a :: "'b" shows "take k (scan f xs a) = scan f (take k xs) a"
propRuleCharacterise: fixes Ps :: "'a list"   and C :: "sequent" assumes "(Ps, C) \<in> upRules" shows "\<exists>F Fs. C = ( \<Empt> \<Rightarrow>* \<LM> Cpd0 F Fs  \<RM>) \<or> C = ( \<LM> Cpd0 F Fs  \<RM> \<Rightarrow>* \<Empt>)"
right_invariant_def: fixes r :: "('b list \<times> 'b list) set" shows "right_invariant r \<equiv> \<forall>u v w. (u, v) \<in> r \<longrightarrow> (u @ w, v @ w) \<in> r"
an_npn: fixes n :: "nat"   and m :: "nat" shows "an (n + m) = an n + an m"
Units_def: shows "Units = {u \<in> S \<rightarrow>\<^sub>E S. invertible u}"
lcs_adds_plus: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and lcs :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and s :: "'b"   and t :: "'b" assumes "class.lcs_powerprod plus minus zero lcs" shows "plus.adds plus (lcs s t) (plus s t)"
UN_upper: fixes a :: "'c"   and A :: "'c set"   and B :: "'c \<Rightarrow> 'd set" assumes "a \<in> A" shows "B a \<subseteq> \<Union> (B ` A)"
linear_diff: fixes f :: "'b \<Rightarrow> 'c"   and x :: "'b"   and y :: "'b" assumes "linear (*a) (*b) f" shows "f (x - y) = f x - f y"
FeqG: shows "insertAssertion F \<Psi> \<hookrightarrow>\<^sub>F insertAssertion G \<Psi>"
closure_finite_cspan: fixes T :: "'d set" assumes "finite T" shows "closure (cspan T) = cspan T"
proper_interval_Compl_set_aux: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and proper_interval :: "'a option \<Rightarrow> 'a option \<Rightarrow> bool"   and xs :: "'a list"   and ys :: "'a list" assumes "class.linorder_proper_interval less_eq less proper_interval"   and "finite UNIV"   and "linorder.sorted less_eq xs"   and "distinct xs"   and "linorder.sorted less_eq ys"   and "distinct ys" shows "proper_intrvl.proper_interval_Compl_set_aux less proper_interval None xs ys = (\<exists>A. ord.set_less_aux less_eq (- set xs) A \<and> ord.set_less_aux less_eq A (set ys))"
cases: fixes \<I> :: "('a, 'b) \<I>"   and \<I>' :: "('c, 'd) \<I>"   and a :: "('a, 'b, 'c, 'd) converter" assumes "\<I>, \<I>' \<turnstile>\<^sub>C a \<surd>" obtains conv :: "('a, 'b, 'c, 'd) converter" where "a = conv" and "\<And>q. q \<in> outs_\<I> \<I> \<Longrightarrow> \<I>' \<turnstile>g run_converter conv q \<surd>" and "\<And>q r conv'. \<lbrakk>q \<in> outs_\<I> \<I>; (r, conv') \<in> results_gpv \<I>' (run_converter conv q)\<rbrakk> \<Longrightarrow> r \<in> responses_\<I> \<I> q \<and> \<I>, \<I>' \<turnstile>\<^sub>C conv' \<surd>"
assms_1: shows "Inr (out, callee', rpv') \<in> set_spmf (inline1 callee gpv s)"
simps: fixes pat :: "[:'r:]" shows "endswith\<cdot>pat = selem\<cdot>pat oo stails"
listsum_cong: fixes xs :: "'a list"   and ys :: "'a list"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "xs = ys"   and "\<And>y. y \<in> set ys \<Longrightarrow> f y = g y" shows "ListSum xs f = ListSum ys g"
FG_subring_FH: shows "Ring1 FG \<and> FG \<subseteq> FH"
nxt_repeatable: fixes t :: "'b ttree"   and x :: "'b" assumes "repeatable t"   and "possible t x" shows "nxt t x = t"
is_zero_null: fixes p :: "'b poly" shows "is_zero p = (p = 0)"
assms_6: shows "finite (E\<^sup>* `` {src})"
pdouble_correct: fixes a :: "'a"   and p :: "'a \<times> 'a \<times> 'a" assumes "a \<in> carrier R"   and "in_carrierp p" shows "make_affine (pdouble a p) = local.add a (make_affine p) (make_affine p)"
is_Zp_continuous_def: fixes f :: "(nat \<Rightarrow> int) \<Rightarrow> nat \<Rightarrow> int" shows "is_Zp_continuous f = (f \<in> carrier (function_ring (carrier Zp) Zp) \<and> (\<forall>s. is_Zp_cauchy s \<longrightarrow> is_Zp_cauchy (f \<circ> s)))"
inject: fixes x :: "'a \<Rightarrow> bool"   and ya :: "'a \<Rightarrow> bool" shows "(Pred x = Pred ya) = (x = ya)"
linord_of_list_linord_of_listP: fixes xy :: "'a \<times> 'a"   and xs :: "'a list" shows "(xy \<in> linord_of_list xs) = linord_of_listP (fst xy) (snd xy) xs"
simps_2: fixes x :: "'b"   and xs :: "['b]" shows "length\<cdot>(x : xs) = length\<cdot>xs + 1"
simps_1: fixes var :: "nat"   and xs :: "real list" shows "convert_atom_list var [] xs = Some []"
scd: fixes m :: "nat" assumes "m \<le> Suc (Suc n)" shows "geom_proc m x = geom_proc m y"
inj_onI_pair: fixes f :: "'c \<Rightarrow> 'd"   and S :: "'c set"   and T :: "('c \<times> 'c) set" assumes "inj_on f S"   and "T \<subseteq> S \<times> S" shows "inj_on (\<lambda>(u, v). (f u, f v)) T"
blv_in: shows "blv \<in># filter_mset ((\<subseteq>) p) \<B>"
in_set_right: fixes y :: "'c"   and r :: "'c tree"   and l :: "'c tree"   and x :: "'c" assumes "y \<in> set_of r" shows "y \<in> set_of (Node l x False r)"
perm_equiv: shows "(Abs ^^ n) (var_dist vs s) \<leftrightarrow> (Abs ^^ n) (var_dist vs t)"
even_even_mod_4_iff: fixes n :: "nat" shows "even n = even (n mod 4)"