hcju/isabelleps huggingface dataset:
Polys_homomorphismE: fixes X :: "'b set" obtains h :: "(('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c) \<Rightarrow> ('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c" where "\<And>p q. h (p + q) = h p + h q" and "\<And>p q. h (p * q) = h p * h q" and "\<And>p. h (h p) = h p" and "range h = P[X]"
inj_on_Cons1: fixes x :: "'a"   and A :: "'a list set" shows "inj_on ((#) x) A"
power_Suc0_right: fixes a :: "'a" shows "a +^ Suc 0 = a"
TensorDiag_Prim: fixes t :: "'a local.term"   and f :: "'a" assumes "t \<noteq> \<^bold>\<I>" shows "\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<lfloor>\<^bold>\<otimes>\<^bold>\<rfloor> t = \<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<otimes> t"
denot_sound_wrt_ctx_equiv: fixes e1 :: "exp"   and e2 :: "exp" assumes "E e1 = E e2" shows "e1 \<simeq> e2"
\<psi>_2: shows "(domain \<circ> \<psi>) ` N refines X ` A"
l_inv: fixes x :: "'a" assumes "x \<in> carrier R" shows "\<ominus> x \<oplus> x = \<zero>"
card_transfer: fixes A :: "'b \<Rightarrow> 'c \<Rightarrow> bool" assumes "bi_unique A" shows "rel_fun (rel_fset A) (=) fcard fcard"
alpha_qSwap_preserves_qGood1: fixes zs :: "'varSort"   and y :: "'var"   and x :: "'var"   and X :: "('a, 'b, 'varSort, 'var, 'c) qTerm"   and zs' :: "'varSort"   and y' :: "'var"   and x' :: "'var"   and X' :: "('a, 'b, 'varSort, 'var, 'c) qTerm" assumes "X #[[y \<and> x]]_zs #= X' #[[y' \<and> x']]_zs'"   and "qGood X" shows "qGood X'"
wf_trm_subst_range_iff: fixes \<theta> :: "'a \<Rightarrow> ('fun, 'a) Term.term" shows "(\<forall>x. wf\<^sub>t\<^sub>r\<^sub>m (\<theta> x)) = wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (subst_range \<theta>)"
insert: fixes A :: "'b set"   and x :: "'b" assumes "finite A"   and "A \<noteq> {}" shows "Max (insert x A) = max x (Max A)"
n_eq_val_eq_ideal: fixes n :: "nat"   and P :: "nat \<Rightarrow> ('a \<Rightarrow> ant) set"   and x :: "'a"   and y :: "'a" assumes "distinct_pds K n P"   and "x \<in> carrier (O\<^bsub>K P n\<^esub>)"   and "y \<in> carrier (O\<^bsub>K P n\<^esub>)"   and "\<forall>j\<le>n. (\<nu>\<^bsub>K P j\<^esub>) x = (\<nu>\<^bsub>K P j\<^esub>) y" shows "(O\<^bsub>K P n\<^esub>) \<diamondsuit>\<^sub>p x = (O\<^bsub>K P n\<^esub>) \<diamondsuit>\<^sub>p y"
check_no_overlap_def: fixes \<gamma> :: "'b set \<Rightarrow> 'c \<Rightarrow> bool"   and ft :: "('b, 'd) flow_entry_match list" shows "check_no_overlap \<gamma> ft = (\<forall>a\<in>set ft. \<forall>b\<in>set ft. \<forall>p\<in>UNIV. ofe_prio a = ofe_prio b \<and> \<gamma> (ofe_fields a) p \<and> a \<noteq> b \<longrightarrow> \<not> \<gamma> (ofe_fields b) p)"
Ri_indep: fixes N' :: "'a literal multiset set"   and N :: "'a literal multiset set" assumes "N' \<subseteq> Rf N" shows "Ri N \<subseteq> Ri (N - N')"
HPhase1or2ReadThen_HInv5_p2: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and d :: "Disk"   and r :: "Proc" assumes "HPhase1or2ReadThen s s' p d r"   and "HInv4c s p"   and "Inv2c_inner s p"   and "phase s p = 2"   and "\<exists>D\<in>MajoritySet. \<exists>q. \<forall>d\<in>D. bal (dblock s p) < mbal (disk s d q) \<and> \<not> hasRead s p d q" shows "\<exists>D\<in>MajoritySet. \<exists>q. \<forall>d\<in>D. bal (dblock s' p) < mbal (disk s' d q) \<and> \<not> hasRead s' p d q"
funcset_mem: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and B :: "'b set"   and x :: "'a" assumes "f \<in> A \<rightarrow> B"   and "x \<in> A" shows "f x \<in> B"
a: shows "((\<lambda>x. D x * (F (\<gamma>2 (\<phi> x)) \<bullet> b * (vector_derivative \<gamma>2 (at (\<phi> x) within {0..1}) \<bullet> b))) has_integral integral {\<phi> 0..\<phi> 1} (\<lambda>x. F (\<gamma>2 x) \<bullet> b * (vector_derivative \<gamma>2 (at x within {0..1}) \<bullet> b))) {0..1}"
mconn_refl': fixes E :: "('b \<times> 'b) set" shows "refl (mconn E)"
borel_integrable_compact: fixes S :: "'c set"   and f :: "'c \<Rightarrow> 'd" assumes "compact S"   and "continuous_on S f" shows "integrable lborel (\<lambda>x. indicat_real S x *\<^sub>R f x)"
Quo4: fixes x :: "nat"   and y :: "nat" assumes "0 < x" shows "y < x + x * Quo x y"
Return_changed: fixes ch :: "state \<Rightarrow> PrIds \<Rightarrow> ('a, 'b) chan"   and p :: "PrIds"   and v :: "state \<Rightarrow> 'b" shows "\<turnstile> Return ch p v \<longrightarrow> <Return ch p v>_(rtrner ch!p)"
sb: shows "\<Union> (range A) \<subseteq> \<Omega>"
assms_3: shows "\<Gamma>,\<gamma>,p\<turnstile> \<langle>rs, Undecided\<rangle> \<Rightarrow> Undecided"
gamma1_is_compos_of_scal_and_g2: shows "gamma1 = (\<lambda>x. (x, g2 x)) \<circ> (\<lambda>x. a + (b - a) * x)"
hom_poly_of_vec: fixes v :: "int vec" shows "of_int_poly (poly_of_vec v) = poly_of_vec (of_int_vec v)"
inject_1: fixes x2 :: "('c \<Rightarrow> 'd) \<Rightarrow> bool"   and y2 :: "('c \<Rightarrow> 'd) \<Rightarrow> bool" shows "(Assume x2 = Assume y2) = (x2 = y2)"
one_Measurement_System_ext_def: shows "1 = \<lparr>unit_sys = unit, \<dots> = 1::'d\<rparr>"
Suc_Suc_induct: fixes i :: "nat"   and ys :: "recf list"   and n :: "nat" assumes "i < Suc (Suc 0)"   and "primerec (ys ! 0) n"   and "primerec (ys ! 1) n" shows "primerec (ys ! i) n"
eq: fixes x :: "nat" assumes "0 < x" shows "{j. p (2 ^ j * oddpart x) = 1} = index ` {i. p i = 1 \<and> oddpart x = oddpart i}"
sel_1: fixes x1 :: "'a"   and x2 :: "'b" shows "cc (x1, x2) = x1"
ext_inject: fixes es :: "'c"   and ps :: "'a \<Rightarrow> 'e"   and pubActs :: "'b \<times> ('a \<Rightarrow> 'd)"   and more :: "'f"   and es' :: "'c"   and ps' :: "'a \<Rightarrow> 'e"   and pubActs' :: "'b \<times> ('a \<Rightarrow> 'd)"   and more' :: "'f" shows "(\<lparr>es = es, ps = ps, pubActs = pubActs, \<dots> = more\<rparr> = \<lparr>es = es', ps = ps', pubActs = pubActs', \<dots> = more'\<rparr>) = (es = es' \<and> ps = ps' \<and> pubActs = pubActs' \<and> more = more')"
assms_2: shows "(f \<circ> g) ` A \<subseteq> carrier G"
bdd_all_make_bdd: fixes P :: "'a \<Rightarrow> bool"   and f :: "nat list \<Rightarrow> 'a"   and n :: "nat"   and xs :: "nat list" shows "bdd_all P (make_bdd f n xs) = (\<forall>ys\<in>set (mk_nat_vecs n). P (f (xs @ ys)))"
eval_final: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and e :: "char list exp"   and s :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and e' :: "char list exp"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)" assumes "P \<turnstile> \<langle>e,s\<rangle> \<Rightarrow> \<langle>e',s'\<rangle>" shows "final e'"
dgrad_p_set_le_red_rtrancl: fixes d :: "'a \<Rightarrow> nat"   and F :: "('t \<Rightarrow>\<^sub>0 'd) set"   and p :: "'t \<Rightarrow>\<^sub>0 'd"   and q :: "'t \<Rightarrow>\<^sub>0 'd" assumes "dickson_grading d"   and "(red F)\<^sup>*\<^sup>* p q" shows "dgrad_p_set_le d {q} (insert p F)"
pseudo_divmod_main: fixes d :: "'b poly"   and lc :: "'b"   and r :: "'b poly"   and dr :: "nat"   and q :: "'b poly"   and n :: "nat"   and q' :: "'b poly"   and r' :: "'b poly" assumes "d \<noteq> 0"   and "lc = lead_coeff d"   and "degree r \<le> dr"   and "pseudo_divmod_main lc q r d dr n = (q', r')"   and "n = 1 + dr - degree d \<or> dr = 0 \<and> n = 0 \<and> r = 0" shows "(r' = 0 \<or> degree r' < degree d) \<and> smult (lc ^ n) (d * q + r) = d * q' + r'"
Astack_transform_alts: fixes as :: "Arity list"   and S :: "stack_elem list" shows "Astack (transform_alts as S) = Astack S"
Pareto_strictI: fixes x :: "'alt pmf"   and y :: "'alt pmf"   and i :: "'agent" assumes "\<And>i. i \<in> agents \<Longrightarrow> (SD \<circ> R) i x y"   and "i \<in> agents"   and "x \<prec>[(SD \<circ> R) i] y" shows "x \<prec>[Pareto (SD \<circ> R)] y"
is_class_def: 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" shows "is_class P C \<equiv> class P C \<noteq> None"
abs_pred_sterm_def: fixes P :: "sterm \<Rightarrow> bool"   and t :: "sterm" shows "abs_pred P t = (\<forall>cs. t = Sabs cs \<longrightarrow> (\<forall>pat t. (pat, t) \<in> set cs \<longrightarrow> P t) \<longrightarrow> P t)"
prv_\<phi>L_eqv: fixes var :: "'a set"   and trm :: "'b set"   and fmla :: "'c set"   and Var :: "'a \<Rightarrow> 'b"   and num :: "'b set"   and FvarsT :: "'b \<Rightarrow> 'a set"   and substT :: "'b \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b"   and Fvars :: "'c \<Rightarrow> 'a set"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'c"   and eql :: "'b \<Rightarrow> 'b \<Rightarrow> 'c"   and cnj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and imp :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and all :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and exi :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and prv :: "'c \<Rightarrow> bool"   and bprv :: "'c \<Rightarrow> bool"   and enc :: "'c \<Rightarrow> 'b"   and S :: "'c"   and P :: "'c"   and \<phi> :: "'c" assumes "Loeb_Form var trm fmla Var num FvarsT substT Fvars subst eql cnj imp all exi prv bprv enc S P"   and "\<phi> \<in> fmla"   and "Fvars \<phi> = {}" shows "prv (Syntax_with_Connectives.eqv cnj imp (Loeb_Form.\<phi>L var fmla Fvars subst cnj imp bprv enc P \<phi>) (imp (HBL1.PP var subst P (enc (Loeb_Form.\<phi>L var fmla Fvars subst cnj imp bprv enc P \<phi>))) \<phi>))"
ospecD2: fixes x :: "'a option"   and P :: "'a \<Rightarrow> bool" assumes "ospec x P" shows "\<exists>y. x = Some y \<and> P y"
act_def: shows "act \<equiv> x$ = Suc<Suc<$x>>"
subsetCE: fixes A :: "'b set"   and B :: "'b set"   and c :: "'b" assumes "A \<subseteq> B" obtains "c \<notin> A"   | "c \<in> B"
lookup_pm_of_idx_pm: fixes xs :: "'c list"   and f :: "nat \<Rightarrow>\<^sub>0 'd" shows "lookup (pm_of_idx_pm xs f) = (\<lambda>x. lookup f (Min {i. i < length xs \<and> xs ! i = x}) when x \<in> set xs)"
merge_commute: fixes I :: "'d set"   and J :: "'d set"   and x :: "'d \<Rightarrow> 'e"   and y :: "'d \<Rightarrow> 'e" assumes "I \<inter> J = {}" shows "merge I J (x, y) = merge J I (y, x)"
Abs_filter_inverse: fixes y :: "'b set" assumes "y \<in> filters" shows "Filters.filter.Rep_filter (Filters.filter.Abs_filter y) = y"
lm039: fixes P :: "('c \<times> 'd) set"   and X :: "'c set"   and Q :: "('c \<times> 'd) set" shows "P outside X \<inter> Q || X = {}"
Abs_seqauto_inverse: fixes y :: "'d set \<times> 'd \<times> (('d, 'e) expr \<times> ('f data \<Rightarrow> bool) \<times> 'e set \<times> 'f pupdate) set \<times> ('d \<times> (('d, 'e) expr \<times> ('f data \<Rightarrow> bool) \<times> 'e set \<times> 'f pupdate) \<times> 'd) set" assumes "y \<in> seqauto" shows "Rep_seqauto (Abs_seqauto y) = y"
height_balL2: fixes l :: "('c \<times> nat) tree"   and r :: "('c \<times> nat) tree"   and a :: "'c" assumes "avl l"   and "avl r"   and "height l \<noteq> height r + 2" shows "height (balL l a r) = 1 + max (height l) (height r)"
ind_sheaf_def: fixes V :: "'a set" shows "local.ind_sheaf V \<equiv> \<FF> (U \<inter> V)"
type_const_return: fixes i :: "nat"   and lholed :: "Lholed"   and vs :: "e list"   and LI :: "e list"   and \<C> :: "t_context"   and tcs :: "t list"   and \<S> :: "s_context"   and ts :: "t list"   and ts' :: "t list"   and \<C>' :: "t_context" assumes "Lfilled i lholed (vs @ [$Return]) LI"   and "return \<C> = Some tcs"   and "length tcs = length vs"   and "\<S>\<bullet>\<C> \<turnstile> LI : ts _> ts'"   and "const_list vs" shows "\<S>\<bullet>\<C>' \<turnstile> vs : [] _> tcs"
extended_Gromov_distance_def: fixes x :: "'a Gromov_completion"   and y :: "'a Gromov_completion" shows "extended_Gromov_distance x y = (if x \<in> Gromov_boundary \<or> y \<in> Gromov_boundary then \<infinity> else ereal (dist (inv to_Gromov_completion x) (inv to_Gromov_completion y)))"
llbound_not_bot: assumes "I \<noteq> {}" shows "llbound \<noteq> bot"
type_soundness: fixes \<Gamma> :: "'Var \<Rightarrow> 'BExp set option"   and \<S> :: "'Var set \<times> 'Var set"   and P :: "'BExp set"   and c :: "('Var, 'AExp, 'BExp) Stmt"   and \<Gamma>' :: "'Var \<Rightarrow> 'BExp set option"   and \<S>' :: "'Var set \<times> 'Var set"   and P' :: "'BExp set"   and mds :: "Mode \<Rightarrow> 'Var set"   and mem\<^sub>1 :: "'Var \<Rightarrow> 'Val"   and mem\<^sub>2 :: "'Var \<Rightarrow> 'Val" assumes "\<turnstile> \<Gamma>,\<S>,P {c} \<Gamma>',\<S>',P'"   and "tyenv_wellformed mds \<Gamma> \<S> P"   and "\<forall>x. type_max (to_total \<Gamma> x) mem\<^sub>1 = Low \<longrightarrow> mem\<^sub>1 x = mem\<^sub>2 x"   and "pred P mem\<^sub>1"   and "pred P mem\<^sub>2"   and "\<forall>x\<in>dom \<Gamma>. x \<notin> mds AsmNoReadOrWrite \<longrightarrow> type_max (the (\<Gamma> x)) mem\<^sub>1 \<le> dma mem\<^sub>1 x" shows "\<langle>c, mds, mem\<^sub>1\<rangle> \<approx> \<langle>c, mds, mem\<^sub>2\<rangle>"
domain_1'': fixes x :: "'a"   and y :: "'a" shows "d (x \<cdot> y) \<le> d x"
ltl_implies_satisfiable_prefix: fixes w :: "nat \<Rightarrow> 'a set"   and \<phi> :: "'a ltln"   and i :: "nat" assumes "w \<Turnstile>\<^sub>n \<phi>" shows "\<not> af \<phi> (prefix i w) \<sim> false\<^sub>n"
map_G_parametric: fixes L1 :: "'b \<Rightarrow> 'e \<Rightarrow> bool"   and L1' :: "'d \<Rightarrow> 'f \<Rightarrow> bool"   and L2 :: "'g \<Rightarrow> 'r \<Rightarrow> bool"   and L2' :: "'h \<Rightarrow> 's \<Rightarrow> bool"   and Co1 :: "'i \<Rightarrow> 't \<Rightarrow> bool"   and Co1' :: "'j \<Rightarrow> 'u \<Rightarrow> bool"   and Co2 :: "'k \<Rightarrow> 'v \<Rightarrow> bool"   and Co2' :: "'l \<Rightarrow> 'w \<Rightarrow> bool"   and Contra1' :: "'m \<Rightarrow> 'x \<Rightarrow> bool"   and Contra1 :: "'n \<Rightarrow> 'y \<Rightarrow> bool"   and Contra2' :: "'o \<Rightarrow> 'z \<Rightarrow> bool"   and Contra2 :: "'p \<Rightarrow> 'aa \<Rightarrow> bool" shows "rel_fun (rel_fun L1 L1') (rel_fun (rel_fun L2 L2') (rel_fun (rel_fun Co1 Co1') (rel_fun (rel_fun Co2 Co2') (rel_fun (rel_fun Contra1' Contra1) (rel_fun (rel_fun Contra2' Contra2) (rel_fun (rel_G L1 L2 Co1 Co2 Contra1 Contra2) (rel_G L1' L2' Co1' Co2' Contra1' Contra2'))))))) map_G map_G"
inf_aci_1: fixes x :: "'a"   and y :: "'a" shows "x \<sqinter> y = y \<sqinter> x"
emeasure_completion: fixes S :: "'a set"   and M :: "'a measure" assumes "S \<in> sets (completion M)" shows "emeasure (completion M) S = emeasure M (main_part M S)"
spEdges_ss_E: shows "spEdges \<subseteq> E"
bird_unfold: shows "bird = Node (1, 1) (pure recip \<diamondop> (pure succ \<diamondop> bird)) (pure succ \<diamondop> (pure recip \<diamondop> bird))"
quotients_length: shows "length quotients + 1 = length \<GG>"
countable_web_quotient_web: fixes f :: "'v \<times> 'v \<Rightarrow> ennreal" shows "countable_web (quotient_web (quotient_web \<Gamma> f__) f)"
simps_1: fixes nodes :: "'c set"   and edges :: "('c \<times> 'c) set"   and more :: "'d" shows "nodes \<lparr>nodes = nodes, edges = edges, \<dots> = more\<rparr> = nodes"
simps_1: fixes a :: "nat"   and h :: "nat \<Rightarrow> nat \<Rightarrow> nat" shows "PrimRecOp1 a h 0 = a"
other_steps_def: fixes U :: "('e \<Rightarrow> 'f) \<Rightarrow> ('e \<Rightarrow> 'f) \<Rightarrow> bool"   and J :: "'e set" shows "other_steps U J \<equiv> \<forall>\<sigma> \<sigma>'. U \<sigma> \<sigma>' \<longrightarrow> (\<forall>j\<in>J. \<sigma>' j = \<sigma> j)"
ptensor_mat_dim_row: fixes m1 :: "'b mat"   and m2 :: "'b mat" shows "dim_row (ptensor_mat m1 m2) = d0"
continuous_map_euclidean_top_of_set: fixes f :: "'c \<Rightarrow> 'd"   and S :: "'d set" assumes "f -` S = UNIV"   and "continuous_on UNIV f" shows "continuous_map euclidean (top_of_set S) f"
t: shows "Arr t \<and> Ide (Dom t) \<and> Ide (Cod t) \<and> Ide \<^bold>\<lfloor>Dom t\<^bold>\<rfloor> \<and> Ide \<^bold>\<lfloor>Cod t\<^bold>\<rfloor> \<and> arr \<lbrace>t\<rbrace> \<and> arr \<lbrace>Dom t\<rbrace> \<and> ide \<lbrace>Dom t\<rbrace> \<and> arr \<lbrace>Cod t\<rbrace> \<and> ide \<lbrace>Cod t\<rbrace>"
one_fsmult: fixes m :: "'v" assumes "m \<in> V" shows "(1::'f) \<sharp>\<cdot> m = m"
valid_disj_PL: fixes Cn :: "'b set \<Rightarrow> 'b set"   and true_PL :: "'b"   and false_PL :: "'b"   and imp_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and not_PL :: "'b \<Rightarrow> 'b"   and conj_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and disj_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and equiv_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and p :: "'b"   and q :: "'b" assumes "Supraclassical_logic Cn true_PL false_PL imp_PL not_PL conj_PL disj_PL equiv_PL" shows "(disj_PL p q \<in> Cn {}) = (q \<in> Cn {not_PL p})"
similar_mat_wit_trans: fixes A :: "'c mat"   and B :: "'c mat"   and P :: "'c mat"   and Q :: "'c mat"   and C :: "'c mat"   and P' :: "'c mat"   and Q' :: "'c mat" assumes "similar_mat_wit A B P Q"   and "similar_mat_wit B C P' Q'" shows "similar_mat_wit A C (P * P') (Q' * Q)"
is_class_Throwable: fixes wf_md :: "'a prog \<Rightarrow> String.literal \<Rightarrow> 'a mdecl \<Rightarrow> bool"   and P :: "'a prog" assumes "wf_prog wf_md P" shows "is_class P Throwable"
rell_U_alt_def: fixes L1 :: "'n \<Rightarrow> 't \<Rightarrow> bool" shows "rell_U L1 = rel_U L1 (=) (=) (=) (=)"
sorted_hd_b_cons: shows "Sorted_Less.sorted (rev (hd b # a ! hd b))"
vardict_inv_variable_inv: fixes ss :: "(integer \<times> edge list) iarray"   and proc :: "nat \<times> edgeIndex \<times> procArg list \<times> varDecl list"   and v :: "(String.literal, variable) assoc_list"   and k :: "String.literal"   and x :: "variable" assumes "vardict_inv ss proc v"   and "lm.lookup k v = Some x" shows "variable_inv x"
os_it_has_next_def: fixes it :: "'b node ref option" shows "os_it_has_next it \<equiv> return (it \<noteq> None)"
basis_criterion: fixes A :: "'a vec set" assumes "finite A"   and "A \<subseteq> carrier_vec n" shows "basis A = (\<forall>v. v \<in> carrier_vec n \<longrightarrow> (\<exists>!a. a \<in> A \<rightarrow>\<^sub>E UNIV \<and> lincomb a A = v))"
v2_def: shows "V2 = \<lparr>ext = l2, lan = lan v, own = own v\<rparr>"
distinct_disc_3: fixes event :: "('a, 'b, 'c) event" assumes "isTrans event" shows "\<not> isRecv event"
greaterThanLessThan_empty: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and l :: "'a"   and k :: "'a" assumes "class.order less_eq less"   and "less_eq l k" shows "ord.greaterThanLessThan less k l = {}"
encode_problem_parallel_complete_vi_d: fixes \<Pi> :: "'b strips_problem"   and \<pi> :: "'b strips_operator list list"   and k :: "nat"   and op :: "'b strips_operator" assumes "is_parallel_solution_for_problem \<Pi> \<pi>"   and "k < length \<pi>"   and "op \<notin> set (\<pi> ! k)" shows "\<not> valuation_for_plan \<Pi> \<pi> (Operator k (index (\<Pi>\<^sub>\<O>) op))"
ordermap_eq_self: fixes \<alpha> :: "V"   and x :: "V" assumes "Ord \<alpha>"   and "x \<in> elts \<alpha>" shows "ordermap (elts \<alpha>) VWF x = x"
cancel: shows "(A \<inter> B \<subseteq> {\<one>}) = (\<forall>x\<in>A. \<forall>y\<in>B. \<forall>x'\<in>A. \<forall>y'\<in>B. x \<otimes> y = x' \<otimes> y' \<longrightarrow> x = x' \<and> y = y')"
right_neutral: fixes a :: "'c" shows "a \<cdot> (1::'c) = a"
that_1: shows "min 1 (max 0 p) = min 1 (max 0 r) * min 1 (max 0 s)"
flow_in_domain: fixes t :: "real"   and x0a :: "'a"   and t0 :: "real"   and x0 :: "'a \<Rightarrow>\<^sub>L 'a" assumes "t \<in> Y.mvar.existence_ivl x0a t0 x0" shows "Y.mvar.flow x0a t0 x0 t \<in> UNIV"
CHOICE: fixes p1 :: "('b, 'c, 'a, 'd) seqp"   and p2 :: "('b, 'c, 'a, 'd) seqp" assumes "wellformed \<Gamma>"   and "P p1"   and "P p2" shows "P (p1 \<oplus> p2)"
nnfis_nn: fixes a :: "real"   and f :: "'b \<Rightarrow> real"   and M :: "'b set set \<times> ('b set \<Rightarrow> real)" assumes "a \<in> nnfis f M" shows "nonnegative f"
fold_nop: fixes \<Delta>' :: "('c \<times> 't) zmultiset"   and c :: "('c, 't, 'd) Propagate.configuration_scheme"   and \<Delta> :: "('c \<times> 't) zmultiset"   and loc :: "'c"   and t :: "'t" shows "zcount (c_pts (Finite_Set.fold (\<lambda>(loc', t') c. apply_cm c loc' t' (zcount \<Delta>' (loc', t'))) c (set_zmset \<Delta> - {(loc, t)})) loc) t = zcount (c_pts c loc) t"
pref_ext_nemp: fixes r :: "'b list"   and u :: "'b list"   and v :: "'b list" assumes "r \<le>p u"   and "v \<noteq> \<epsilon>" shows "r <p u \<cdot> v"
mset_subset_insertD: fixes x :: "'k"   and A :: "'k multiset"   and B :: "'k multiset" assumes "add_mset x A \<subset># B" shows "x \<in># B \<and> A \<subset># B"
bprv_diag_eqv: fixes \<phi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "Fvars \<phi> = {xx}" shows "bprv (eqv (diag \<phi>) (subst \<phi> \<langle>diag \<phi>\<rangle> xx))"
simTauChain: fixes Q :: "ccs"   and Q' :: "ccs"   and P :: "ccs"   and Rel :: "(ccs \<times> ccs) set" assumes "Q \<Longrightarrow>\<^sub>\<tau> Q'"   and "(P, Q) \<in> Rel"   and "\<And>R S. (R, S) \<in> Rel \<Longrightarrow> R \<leadsto>\<^sup>^<Rel> S" obtains P' :: "ccs" where "P \<Longrightarrow>\<^sub>\<tau> P'" and "(P', Q') \<in> Rel"
leg0_uniquely_isomorphic_1: shows "t\<^sub>0 \<star> \<omega>.chine \<cong> r\<^sub>0"
sublist_entire: fixes xs :: "'c list" shows "WB_Sort.sublist xs 0 (length xs - 1) = xs"
A: shows "\<forall>x. xa x \<longrightarrow> Ex (xb[x ..])"
odd: fixes n :: "nat" shows "P (Suc (2 * n))"
is_extensional: fixes f :: "'a" assumes "\<not> S.arr f" shows "\<rr>.map f = S.null"
allocationsUniverseOutside: fixes a :: "('a \<times> 'b set) set"   and X :: "'a set" assumes "a \<in> allocationsUniverse" shows "Outside' X a \<in> allocationsUniverse"
nth_fls_subdegree_zero_iff: fixes f :: "'b fls" shows "(fls_nth f (fls_subdegree f) = (0::'b)) = (f = 0)"
ldrop_simps_2: fixes xs :: "'c llist" shows "ldrop 0 xs = xs"
assms_5: fixes dir :: "('i, 'a) Direction"   and x :: "nat"   and c :: "'a" assumes "dir = Positive \<or> dir = Negative"   and "a = LE dir x c"   and "\<not> \<unrhd>\<^sub>u\<^sub>b (lt dir) c (UB dir s x)"   and "\<not> \<lhd>\<^sub>l\<^sub>b (lt dir) c (LB dir s x)"   and "\<not> lt dir c (\<langle>\<V> s\<rangle> x)" shows "P' (lt dir) (UBI dir) (LBI dir) (UB dir) (LB dir) (UBI_upd dir) (UI dir) (LI dir) (LE dir) (GE dir) (update\<B>\<I> (UBI_upd dir) i x c s)"
simps_2: fixes m :: "nat"   and y :: "nat"   and x :: "nat" shows "mod_exp_nat.efficient_power m y x (Suc 0) = x * y mod m"
him_mult_left: fixes a :: "real"   and q :: "octo" shows "HIm (a *\<^sub>R q) = a *\<^sub>R HIm q"
l_inv: fixes x :: "'a vec" assumes "x \<in> carrier_vec n" shows "\<ominus>\<^bsub>V\<^esub> x + x = 0\<^sub>v n"
continuous_on_add: fixes s :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "continuous_on s f"   and "continuous_on s g" shows "continuous_on s (\<lambda>x. f x + g x)"
dom_restrict: fixes m :: "'c \<Rightarrow> 'd option"   and A :: "'c set" shows "dom (m |` A) = dom m \<inter> A"
connected_contains_Icc: fixes A :: "'a set"   and a :: "'a"   and b :: "'a" assumes "connected A"   and "a \<in> A"   and "b \<in> A" shows "{a..b} \<subseteq> A"
cong: fixes \<O>\<^sub>s :: "'d set"   and \<K> :: "'c \<Rightarrow> 'c pmf"   and \<O> :: "'c \<Rightarrow> 'd pmf"   and \<S> :: "'c set" shows "HMM2.is_decoding \<O>\<^sub>s \<K> \<O> \<S> \<equiv> HMM2.is_decoding \<O>\<^sub>s \<K> \<O> \<S>"
simps_9: fixes \<C> :: "t_context"   and t1 :: "t"   and t2 :: "t"   and sx :: "sx option"   and ts :: "checker_type" shows "check_single \<C> (Cvtop t1 Convert t2 sx) ts = (if convert_cond t1 t2 sx then type_update ts [TSome t2] (Type [t1]) else Bot)"
add_nonneg_eq_0_iff: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "class.ordered_comm_monoid_add plus zero less_eq less"   and "less_eq zero x"   and "less_eq zero y" shows "(plus x y = zero) = (x = zero \<and> y = zero)"
invc_baliL: fixes l :: "('b \<times> color) tree"   and r :: "('b \<times> color) tree"   and a :: "'b" assumes "invc2 l"   and "invc r" shows "invc (baliL l a r)"