hcju/isabelleps huggingface dataset:
trace_trans: fixes c :: "('a, 'b, 'c) configuration"   and t :: "('a, 'b, 'c) event list"   and c' :: "('a, 'b, 'c) configuration"   and t' :: "('a, 'b, 'c) event list"   and c'' :: "('a, 'b, 'c) configuration" assumes "trace c t c'"   and "trace c' t' c''" shows "trace c (t @ t') c''"
p3_1: fixes i :: "nat" shows "local.coeff p3 i \<in> carrier R"
perm_6: fixes pi :: "(name \<times> name) list"   and x3 :: "name"   and x2 :: "name"   and x1 :: "pi" shows "pi \<bullet> [x3\<noteq>x2]x1 = [(pi \<bullet> x3)\<noteq>(pi \<bullet> x2)](pi \<bullet> x1)"
1: shows "irreflexive (x \<sqinter> - (1::'a))"
enumAppend_bound: fixes ls :: "nat list"   and nmax :: "nat"   and lss :: "nat list list"   and x :: "nat" assumes "ls \<in> set (enumAppend nmax lss)"   and "\<forall>y\<in>set lss. \<forall>z\<in>set y. z \<le> nmax"   and "x \<in> set ls" shows "x \<le> nmax"
1: shows "LeftDerives1 v (fst d - length u) (snd d) x'"
DirProds_is_comm_group: fixes I :: "'e set"   and G :: "'e \<Rightarrow> ('f, 'g) monoid_scheme" assumes "\<And>i. i \<in> I \<Longrightarrow> comm_group (G i)" shows "comm_group (DirProds G I)"
that: fixes u :: "real" assumes "0 \<le> u"   and "u \<le> 1"   and "g u \<in> frontier S"   and "path_image (subpath 0 u g) - {g u} \<subseteq> interior S" shows "thesis"
simps: fixes k :: "nat"   and xs :: "'a list" shows "T_mom_select k xs = (if length xs \<le> 20 then T_slow_select k xs else let xss = chop 5 xs; ms = map slow_median xss; idx = ((length xs + 4) div 5 - 1) div 2; x = mom_select idx ms; (ls, es, gs) = partition3 x xs; nl = length ls; ne = length es in (if k < nl then T_mom_select k ls else if k < nl + ne then 0 else T_mom_select (k - nl - ne) gs) + T_mom_select idx ms + T_chop 5 xs + T_map T_slow_median xss + T_partition3 x xs + T_length ls + T_length es + 1)"
a2: shows "1 \<le> a\<^sup>2 * 4 + b\<^sup>2 * 4"
cases: fixes a1 :: "nat"   and a2 :: "'a \<Rightarrow> 'a" assumes "swapidseq a1 a2" obtains "a1 = 0" and "a2 = id"   | n :: "nat" and p :: "'a \<Rightarrow> 'a" and a :: "'a" and b :: "'a" where "a1 = Suc n" and "a2 = Fun.swap a b id \<circ> p" and "swapidseq n p" and "a \<noteq> b"
absorb1: fixes a :: "'d multiset"   and b :: "'d multiset" assumes "a \<subseteq># b" shows "a \<inter># b = a"
tickFree_implies_front_tickFree_single: fixes t :: "'b event list"   and a :: "'b event" assumes "tickFree t" shows "front_tickFree (t @ [a])"
out_arcs_emptyD_dominates: fixes G :: "('d, 'c) pre_digraph"   and x :: "'d"   and y :: "'d" assumes "out_arcs G x = {}" shows "(x, y) \<notin> arcs_ends G"
simps_5: fixes val :: "val" shows "assignsE (Lit val) = {}"
surjective: fixes r :: "('c, 'd) Graph_scheme" shows "r = \<lparr>Graph.verts = V\<^bsub>r\<^esub>, arcs = E\<^bsub>r\<^esub>, \<dots> = Graph.more r\<rparr>"
map_cong: fixes x :: "'d stream"   and ya :: "'d stream"   and f :: "'d \<Rightarrow> 'e"   and g :: "'d \<Rightarrow> 'e" assumes "x = ya"   and "\<And>z. z \<in> sset ya \<Longrightarrow> f z = g z" shows "smap f x = smap g ya"
eprops: shows "\<forall>a b. a \<noteq> b \<longrightarrow> fmul a e \<noteq> fmul b e"
typed_secure: fixes m :: "Mode \<Rightarrow> 'Var 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" assumes "\<turnstile> \<Gamma>_of_mds m,\<S>_of_mds m,{} {c} \<Gamma>',\<S>',P'"   and "mds_yields_stable_types m" shows "com_sifum_secure (c, m)"
sent_before_recv: fixes i :: "'a"   and k :: "nat"   and s :: "'a \<times> nat"   and m :: "('a \<times> nat, 'b) message"   and j :: "nat" assumes "event_at (i, k) (Receive s m)"   and "j < length (events i)"   and "k < j" shows "event_at s (Send m) \<and> happened_immediately_before\<^sup>+\<^sup>+ s (i, j)"
Chains_def: fixes r :: "('c \<times> 'c) set" shows "Chains r = {C. \<forall>a\<in>C. \<forall>b\<in>C. (a, b) \<in> r \<or> (b, a) \<in> r}"
Cons: fixes x :: "'a"   and y :: "'a"   and xs :: "'a list" assumes "x \<rightarrow> y"   and "local.steps (y # xs)" shows "local.steps (x # y # xs)"
coeffs_pCons_eq_cCons: fixes a :: "'b"   and p :: "'b poly" shows "coeffs (pCons a p) = a ## coeffs p"
apply_rsp': fixes R1 :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and R2 :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'e"   and x :: "'b"   and y :: "'c" assumes "(R1 ===> R2) f g"   and "R1 x y" shows "R2 (f x) (g y)"
mtype_functional: fixes CT :: "nat \<Rightarrow> classDef option"   and m :: "nat"   and C :: "nat"   and Cs :: "nat list"   and C0 :: "nat"   and Ds :: "nat list"   and D0 :: "nat" assumes "mtype(CT,m,C) = Cs \<rightarrow> C0"   and "mtype(CT,m,C) = Ds \<rightarrow> D0" shows "Ds = Cs \<and> D0 = C0"
hom_graph_unique': fixes f :: "'a \<Rightarrow> 'b"   and S :: "'a set"   and x :: "'a formula"   and a :: "'b"   and T :: "'a set"   and a' :: "'b" assumes "hom_graph f S x a"   and "hom_graph f T x a'" shows "a = a'"
star_inductl_var_eq: fixes x :: "'a"   and y :: "'a" assumes "x ; y = y" shows "x\<^sup>\<star> ; y \<le> y"
assms_3: shows "X \<in>\<^sub>\<circ> Vfrom A i"
sorted_sorted_wrt: shows "sorted = sorted_wrt (\<le>)"
dickson_leD3: fixes ord :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and ord_strict :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and d :: "'a \<Rightarrow> nat"   and m :: "nat"   and s :: "'a"   and t :: "'a" assumes "gd_powerprod ord ord_strict"   and "gd_powerprod.dickson_le ord d m s t" shows "ord s t"
simps_1: fixes a :: "symbol list"   and b :: "symbol list" shows "Derivation a [] b = (a = b)"
simps_153: fixes x101 :: "Term.var"   and x102 :: "expr"   and x61 :: "unop"   and x62 :: "expr" shows "x101:=x102 \<noteq> UnOp x61 x62"
Encode_def: fixes var :: "'a set"   and trm :: "'b set"   and fmla :: "'c set"   and Var :: "'a \<Rightarrow> 'b"   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 num :: "'b set"   and enc :: "'c \<Rightarrow> 'b" shows "Encode var trm fmla Var FvarsT substT Fvars subst num enc \<equiv> Syntax_with_Numerals var trm fmla Var FvarsT substT Fvars subst num \<and> Encode_axioms fmla num enc"
set_2: fixes x21 :: "'b"   and x22 :: "'b list" shows "ran\<^sub>l (x21 # x22) = insert x21 (ran\<^sub>l x22)"
contc: shows "continuous_map (nsphere p) (nsphere p) (\<lambda>x. a)"
is_ground_subst_def: fixes subst_atm :: "'b \<Rightarrow> 'c \<Rightarrow> 'b"   and \<sigma> :: "'c" shows "substitution_ops.is_ground_subst subst_atm \<sigma> = (\<forall>A. substitution_ops.is_ground_atm subst_atm (subst_atm A \<sigma>))"
cut_same_sconst: fixes x :: "'b"   and xs :: "'b list" shows "cut_same x (xs @- sconst x) = fin_cut_same x xs"
max_x_impl'_conv: fixes i :: "nat"   and a :: "nat list"   and y :: "nat list"   and b :: "nat list" assumes "i < length a"   and "length y = length b" shows "max_x_impl' a b y i = max_x_impl a b y i"
finite_cartesian_product: fixes A :: "'g set"   and B :: "'h set" assumes "finite A"   and "finite B" shows "finite (A \<times> B)"
assertion_fun_disj_less_one: shows "assertion_fun = Apply.disjunctive \<inter> {x. x \<le> id}"
GoTo: fixes i :: "'c"   and branch :: "(('d, 'c) fm list \<times> 'c) list"   and A :: "'c set"   and n :: "nat" assumes "i \<in> branch_nominals branch"   and "A, n \<turnstile> ([], i) # branch" shows "A, Suc n \<turnstile> branch"
field_intro2: fixes R :: "('a, 'b) ring_scheme" assumes "cring R"   and "\<zero>\<^bsub>R\<^esub> \<noteq> \<one>\<^bsub>R\<^esub>"   and "\<And>x. x \<in> carrier R - {\<zero>\<^bsub>R\<^esub>} \<Longrightarrow> x \<in> Units R" shows "field R"
integrable_Gamma_integral_bound: fixes a :: "real"   and c :: "real" assumes "- 1 < a"   and "0 \<le> c" shows "(\<lambda>x. if x \<in> {0..c} then x powr a else exp (- x / 2)) integrable_on {0..}"
simps: fixes pi :: "('c \<times> 'c) list"   and x :: "'a"   and y :: "'b" shows "pi \<bullet> (x, y) = (pi \<bullet> x, pi \<bullet> y)"
solves: shows "(ll.flow 0 (\<chi>i. x) solves_ode (\<lambda>t x. \<chi>i. 0)) (ll.existence_ivl 0 (\<chi>i. x)) UNIV"
mult_cancel_right: fixes a :: "'a"   and c :: "'a"   and b :: "'a" shows "(a * c = b * c) = (c = (0::'a) \<or> a = b)"
simps: fixes ly :: "nat list"   and n :: "nat"   and as :: "nat"   and lm :: "nat list"   and s :: "nat"   and l :: "cell list"   and r :: "cell list"   and ires :: "cell list" shows "findnth_inv ly n (as, lm) (s, l, r) ires = (if s = 0 then False else if s \<le> Suc (2 * n) then if s mod 2 = 1 then inv_locate_a (as, lm) ((s - 1) div 2, l, r) ires else inv_locate_b (as, lm) ((s - 1) div 2, l, r) ires else False)"
nonneg: fixes x :: "'a" shows "0 \<le> f x"
aform_isFDERIV: fixes optns :: "nat" shows "(\<lambda>N xs fas vs. nres_of (aform_isFDERIV optns N xs fas vs), isFDERIV_spec) \<in> nat_rel \<rightarrow> \<langle>nat_rel\<rangle>list_rel \<rightarrow> fas_rel \<rightarrow> aforms_rel \<rightarrow> \<langle>bool_rel\<rangle>nres_rel"
const_name_simps_2: fixes size :: "'c \<Rightarrow> nat"   and frees :: "'c \<Rightarrow> name fset"   and subst :: "'c \<Rightarrow> (name, 'c) fmap \<Rightarrow> 'c"   and "consts" :: "'c \<Rightarrow> name fset"   and app :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unapp :: "'c \<Rightarrow> ('c \<times> 'c) option"   and const :: "name \<Rightarrow> 'c"   and unconst :: "'c \<Rightarrow> name option"   and free :: "name \<Rightarrow> 'c"   and unfree :: "'c \<Rightarrow> name option"   and t :: "'c" assumes "class.pre_term size frees subst consts app unapp const unconst free unfree"   and "pre_term.is_const unconst t" shows "const (pre_term.const_name unconst t) = t"
subspace_span: fixes scale :: "'d \<Rightarrow> 'e \<Rightarrow> 'e"   and S :: "'e set" assumes "module scale" shows "module.subspace scale (module.span scale S)"
igFreshIGAbs1STR_def: fixes MOD :: "('h, 'i, 'j, 'k, 'l, 'm, 'n, 'o, 'p) model_scheme" shows "igFreshIGAbs1STR MOD \<equiv> \<forall>ys y X. igFreshAbs MOD ys y (igAbs MOD ys y X)"
lens_id_unique: fixes Y :: "'d \<Longrightarrow> 'e"   and X :: "'d \<Longrightarrow> 'd" assumes "weak_lens Y"   and "Y = X ;\<^sub>L Y" shows "X = 1\<^sub>L"
F: shows "offs_next ns ub (x # xs) index key mi ma j = ns ! i"
not_par_inter_uniqueness: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p"   and X :: "'p"   and Y :: "'p" assumes "A \<noteq> B"   and "C \<noteq> D"   and "\<not> A B Par C D"   and "Col A B X"   and "Col C D X"   and "Col A B Y"   and "Col C D Y" shows "X = Y"
picard_lindeloef_sq_mtx_affine: fixes T :: "real set"   and A :: "real \<Rightarrow> 'b sq_mtx"   and B :: "real \<Rightarrow> (real, 'b) vec"   and t\<^sub>0 :: "real"   and S :: "(real, 'b) vec set" assumes "continuous_on T A"   and "continuous_on T B"   and "t\<^sub>0 \<in> T"   and "is_interval T"   and "open T"   and "open S" shows "picard_lindeloef (\<lambda>t s. A t *\<^sub>V s + B t) T S t\<^sub>0"
powr_zero_eq_one: fixes x :: "'c" shows "x powr (0::'c) = (if x = (0::'c) then 0::'c else (1::'c))"
add_zmset_remove_trivial: fixes x :: "'b"   and M :: "'b zmultiset" shows "add_zmset x M - {#x#}\<^sub>z = M"
tendsto_power_strong: fixes f :: "'a \<Rightarrow> 'b"   and a :: "'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> nat"   and b :: "nat" assumes "(f \<longlongrightarrow> a) F"   and "(g \<longlongrightarrow> b) F" shows "((\<lambda>x. f x ^ g x) \<longlongrightarrow> a ^ b) F"
invD_3: fixes w :: "nat \<Rightarrow> real"   and m :: "nat"   and S :: "nat \<Rightarrow> 'b set"   and C :: "nat set"   and R :: "'b set" assumes "Set_Cover w m S"   and "Set_Cover.inv w m S C R" shows "\<exists>c. sum w C = sum c (Set_Cover.U m S - R) \<and> (\<forall>i. 0 \<le> c i) \<and> (\<forall>k\<in>{1..m}. sum c (S k \<inter> (Set_Cover.U m S - R)) \<le> (\<Sum>j = card (S k \<inter> R) + 1..card (S k). inverse (real j)) * w k)"
diff_union_cancelL: fixes N :: "'b multiset"   and M :: "'b multiset" shows "N + M - N = M"
dghm_comp_ArrMap_vsv: fixes \<alpha> :: "V"   and \<BB> :: "V"   and \<CC> :: "V"   and \<GG> :: "V"   and \<AA> :: "V"   and \<FF> :: "V" assumes "\<GG> : \<BB> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<CC>"   and "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<BB>" shows "vsv ((\<GG> \<circ>\<^sub>D\<^sub>G\<^sub>H\<^sub>M \<FF>)\<lparr>ArrMap\<rparr>)"
lmeasurable_iff_has_integral: fixes S :: "'b set" shows "(S \<in> lmeasurable) = (indicat_real S has_integral Sigma_Algebra.measure lebesgue S) UNIV"
alt_matrix_char_R: fixes signs :: "rat list list"   and subsets :: "(nat list \<times> nat list) list" shows "alt_matrix_A_R signs subsets = M_mat_R signs subsets"
isCont_Re: fixes a :: "'a"   and g :: "'a \<Rightarrow> complex" assumes "isCont g a" shows "isCont (\<lambda>x. Re (g x)) a"
Idempotent_def: fixes H :: "'a upred \<Rightarrow> 'a upred" shows "Idempotent H = (\<forall>P. H (H P) = H P)"
nth_enumerate_eq: fixes m :: "nat"   and xs :: "'a list"   and n :: "nat" assumes "m < length xs" shows "List.enumerate n xs ! m = (n + m, xs ! m)"
behaviourally_equiv_implements: fixes jp :: "'a \<Rightarrow> ('obs, 'aAct, 'b) Protocol"   and jp' :: "'a \<Rightarrow> ('obs, 'aAct, 'c) Protocol" assumes "behaviourally_equiv jp jp'" shows "implements jp = implements jp'"
comm: fixes a :: "'c"   and b :: "'c" shows "a + b = b + a"
less_eqE: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a" assumes "class.canonically_ordered_monoid_add plus zero less_eq less"   and "less_eq a b" obtains c :: "'a" where "b = plus a c"
"1_1": shows "\<forall>ys. length ys < length xs \<longrightarrow> ys \<noteq> [] \<longrightarrow> toplevel_summands (lderivs ys (Star r)) \<subseteq> {Times (lderivs ysa r) (Star r) |ysa. \<exists>zs. ysa \<noteq> [] \<and> zs @ ysa = ys}"
univ_poly_units_incl: fixes K :: "'a list set" assumes "subring K (K [X])" shows "Units (K [X]\<^bsub>K [X]\<^esub>) \<subseteq> {[k] |k. k \<in> K - {\<zero>\<^bsub>K [X]\<^esub>}}"
sel_1: fixes x1 :: "'g" shows "generat.result (Pure x1) = x1"
strict_implies_not_eq: fixes a :: "'a"   and b :: "'a" assumes "a \<in> U"   and "b \<in> U"   and "a <\<^sub>o\<^sub>w b" shows "a \<noteq> b"
var_mset_subst_db_subseteq: fixes s :: "('s, 'a) tm"   and t :: "('s, 'a) tm"   and i :: "nat"   and x :: "'a" assumes "vars_mset s \<subseteq># vars_mset t" shows "vars_mset (subst_db i x s) \<subseteq># vars_mset (subst_db i x t)"
tab_in_vhom': shows "\<guillemotleft>tab : tab\<^sub>1 r \<Rightarrow> r \<star> tab\<^sub>0 r\<guillemotright>"
simps_6: fixes \<phi> :: "'a ltln"   and \<psi> :: "'a ltln" shows "is_\<mu>LTL (\<phi> or\<^sub>n \<psi>) = (is_\<mu>LTL \<phi> \<and> is_\<mu>LTL \<psi>)"
nat_div: fixes b :: "nat"   and a :: "nat" assumes "0 < b"   and "b < a" shows "a div b = (a - b) div b + 1"
nprv_strong_cut2: fixes F :: "'fmla set"   and \<phi>1 :: "'fmla"   and \<phi>2 :: "'fmla"   and \<psi> :: "'fmla" assumes "nprv F \<phi>1"   and "nprv (insert \<phi>1 F) \<phi>2"   and "nprv (insert \<phi>2 (insert \<phi>1 F)) \<psi>"   and "F \<subseteq> fmla"   and "finite F"   and "\<phi>1 \<in> fmla"   and "\<phi>2 \<in> fmla"   and "\<psi> \<in> fmla" shows "nprv F \<psi>"
filterlim_uminus_at_bot: fixes f :: "'c \<Rightarrow> real"   and F :: "'c filter" shows "filterlim f at_bot F = (LIM x F. - f x :> at_top)"
subst_lift_transform: fixes t :: "'b \<Rightarrow> exp \<Rightarrow> exp"   and e :: "exp"   and x :: "Vars.var"   and y :: "Vars.var"   and a :: "'b\<^sub>\<bottom>" assumes "\<And>a. (t a e)[x::=y] = t a e[x::=y]" shows "(lift_transform t a e)[x::=y] = lift_transform t a e[x::=y]"
plus_enat_simps_3: fixes q :: "enat" shows "q + \<infinity> = \<infinity>"
ksup_comm: fixes f :: "'a \<Rightarrow> 'b set"   and g :: "'a \<Rightarrow> 'b set" shows "f \<squnion> g = g \<squnion> f"
simply_connected_eq_unbounded_complement_components: fixes S :: "complex set" assumes "open S" shows "simply_connected S = (connected S \<and> (\<forall>C\<in>components (- S). \<not> bounded C))"
zeroi_func: fixes R :: "('e, 'f) Ring_scheme"   and A :: "'e set" assumes "Ring R"   and "ideal R A" shows "zeroi R \<in> {j. j \<le> 0} \<rightarrow> A"
element_ord_generates_cyclic: fixes a :: "nat \<Rightarrow> int" assumes "a \<in> carrier (DirProds (\<lambda>p. Z (p ^ multiplicity p n)) (prime_factors n))"   and "DP.ord a = order (DirProds (\<lambda>p. Z (p ^ multiplicity p n)) (prime_factors n))" shows "Finite_And_Cyclic_Groups.cyclic_group (DirProds (\<lambda>p. Z (p ^ multiplicity p n)) (prime_factors n)) a"
hn_rel_def: fixes P :: "'e \<Rightarrow> 'd \<Rightarrow> assn"   and m :: "'e nres" shows "hn_rel P m \<equiv> \<lambda>r. \<exists>\<^sub>Ax. P x r * \<up> (RETURN x \<le> m)"
gen_model_kripke: fixes M :: "('e, 'f, 'g) KripkeStructure"   and w :: "'g" shows "kripke (gen_model M w)"
anno_type: fixes \<Gamma>' :: "'Var \<Rightarrow> Sec option"   and \<Gamma> :: "'Var \<Rightarrow> Sec option"   and upd :: "'Var ModeUpd"   and c :: "('Var, 'AExp, 'BExp) Stmt"   and \<Gamma>'' :: "'Var \<Rightarrow> Sec option" assumes "\<Gamma>' = \<Gamma> \<oplus> upd"   and "\<turnstile> \<Gamma>' {c} \<Gamma>''"   and "c \<noteq> Stop"   and "\<forall>x. to_total \<Gamma> x \<sqsubseteq> to_total \<Gamma>' x" shows "\<turnstile> \<Gamma> {c@[upd]} \<Gamma>''"
spoly_def: fixes p :: "'a \<Rightarrow>\<^sub>0 'c"   and q :: "'a \<Rightarrow>\<^sub>0 'c" shows "punit.spoly p q = (let v1 = punit.lt p; v2 = punit.lt q in if local.punit.component_of_term v1 = local.punit.component_of_term v2 then let t1 = local.punit.pp_of_term v1; t2 = local.punit.pp_of_term v2; l = lcs t1 t2 in local.punit.monom_mult ((1::'c) / lookup p v1) (l - t1) p - local.punit.monom_mult ((1::'c) / lookup q v2) (l - t2) q else 0)"
assms_3: fixes x :: "'a" assumes "x \<in> A" shows "x \<sqsubseteq> z"
ltl_prop_equiv_equivp: shows "equivp (\<equiv>\<^sub>P)"
induct: fixes P :: "fm \<Rightarrow> bool"   and a0 :: "fm" assumes "\<And>a. P (Lt a)"   and "\<And>a. P (Le a)"   and "\<And>a. P (Gt a)"   and "\<And>a. P (Ge a)"   and "\<And>a. P (Eq a)"   and "\<And>a. P (NEq a)"   and "\<And>i a. P (Dvd i a)"   and "\<And>i a. P (NDvd i a)"   and "\<And>p. P p \<Longrightarrow> P (NOT p)"   and "\<And>p q. \<lbrakk>P p; P q\<rbrakk> \<Longrightarrow> P (And p q)"   and "\<And>p q. \<lbrakk>P p; P q\<rbrakk> \<Longrightarrow> P (Or p q)"   and "\<And>p q. \<lbrakk>P p; P q\<rbrakk> \<Longrightarrow> P (Imp p q)"   and "\<And>p q. \<lbrakk>P p; P q\<rbrakk> \<Longrightarrow> P (Iff p q)"   and "P T"   and "P F"   and "\<And>v. P (E v)"   and "\<And>v. P (A v)" shows "P a0"
Integral_zero: fixes a :: "real"   and f :: "real \<Rightarrow> real" shows "Integral (a, a) f 0"
W1a: fixes a :: "'a" shows "(1::'a) l\<rightarrow> a = a"
meas_M': shows "M \<in> dens_ctxt_measure \<Y> \<rho> \<rightarrow>\<^sub>M subprob_algebra (stock_measure t)"
ln: assumes "length lm \<le> Suc n" shows "l = Oc # Oc \<up> lm ! n @ Bk \<up> ln @ Bk # Bk # ires"
61156: fixes x :: "'a"   and y :: "'a" shows "- - (- x \<sqinter> y) = - x \<sqinter> - - y"
tendsto_iff_Liminf_eq_Limsup: fixes F :: "'b filter"   and f :: "'b \<Rightarrow> 'c"   and f0 :: "'c" assumes "F \<noteq> \<bottom>" shows "(f \<longlongrightarrow> f0) F = (Liminf F f = f0 \<and> Limsup F f = f0)"
assms_2: fixes n :: "('a, 'b) node_scheme"   and v :: "nat set" assumes "n \<in> ns"   and "P n" shows "P (n\<lparr>incoming := v\<rparr>)"
abcd_dcba_only_1: fixes a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" assumes "[[a b c d]]" shows "\<not> [[a b d c]]"
setc_def_to_intro: fixes A :: "'a set"   and P :: "'a \<Rightarrow> bool"   and x :: "'a" assumes "A \<equiv> {x. P x}"   and "P x" shows "x \<in> A"
incseq_bounded: fixes X :: "nat \<Rightarrow> real"   and B :: "real" assumes "incseq X"   and "\<forall>i. X i \<le> B" shows "Bseq X"
ev: shows "eigen_vector A (\<chi>i. complex_of_real (v $h i)) 0"
bind_return_spmf: fixes x :: "'d spmf" shows "x \<bind> return_spmf = x"
elims_2: fixes x :: "'v"   and xa :: "('v \<times> 'w \<times> 'v) list"   and xb :: "'v" assumes "is_trail x xa xb" obtains v :: "'v" and v' :: "'v" where "x = vb__" and "xa = []" and "xb = v'a__" and "vb__ = v'a__ \<and> v'a__ \<in> V"   | v :: "'v" and v1 :: "'v" and w :: "'w" and v2 :: "'v" and ps :: "('v \<times> 'w \<times> 'v) list" and v' :: "'v" where "x = v" and "xa = (v1, w, v2) # ps" and "xb = v'" and "v = v1 \<and> (v1, w, v2) \<in> E \<and> (v1, w, v2) \<notin> set ps \<and> (v2, w, v1) \<notin> set ps \<and> is_trail v2 ps v'"
walk_cheaper_path_imp_neg_cyc: fixes u :: "'a"   and p :: "'b list"   and v :: "'a"   and f :: "'b \<Rightarrow> real" assumes "awalk u p v"   and "ereal (awalk_cost f p) < (INF p\<in>{p. apath u p v}. ereal (awalk_cost f p))" shows "\<exists>w c. awalk w c w \<and> w \<in> set (awalk_verts u p) \<and> awalk_cost f c < 0"
rel_FGl_def: fixes L1 :: "'y \<Rightarrow> 'al \<Rightarrow> bool"   and L2 :: "'z \<Rightarrow> 'am \<Rightarrow> bool"   and L3 :: "'af \<Rightarrow> 'ar \<Rightarrow> bool"   and Co1 :: "'aa \<Rightarrow> 'an \<Rightarrow> bool"   and Co2 :: "'ab \<Rightarrow> 'ao \<Rightarrow> bool"   and Co3 :: "'ag \<Rightarrow> 'as \<Rightarrow> bool"   and Co4 :: "'ah \<Rightarrow> 'at \<Rightarrow> bool"   and Contra1 :: "'ac \<Rightarrow> 'ap \<Rightarrow> bool"   and Contra2 :: "'ad \<Rightarrow> 'aq \<Rightarrow> bool"   and Contra3 :: "'ai \<Rightarrow> 'au \<Rightarrow> bool"   and Contra4 :: "'aj \<Rightarrow> 'av \<Rightarrow> bool" shows "rel_FGl L1 L2 L3 Co1 Co2 Co3 Co4 Contra1 Contra2 Contra3 Contra4 = rel_F (rel_G L1 L2 Co1 Co2 Contra1 Contra2) L1 L3 Co1 Co3 Co4 Contra1 Contra3 Contra4"
Join_absorb: fixes F :: "'c program" shows "F \<squnion> F = F"
bis_cong: fixes B1 :: "'d set"   and B2 :: "'e set"   and s1 :: "'d \<Rightarrow> ('f, 'd, 'e) F1"   and s2 :: "'e \<Rightarrow> ('g, 'd, 'e) F2"   and B1' :: "'h set"   and B2' :: "'i set"   and s1' :: "'h \<Rightarrow> ('f, 'h, 'i) F1"   and s2' :: "'i \<Rightarrow> ('g, 'h, 'i) F2"   and R1 :: "('d \<times> 'h) set"   and R2 :: "('e \<times> 'i) set"   and R1' :: "('d \<times> 'h) set"   and R2' :: "('e \<times> 'i) set" assumes "bis B1 B2 s1 s2 B1' B2' s1' s2' R1 R2"   and "R1' = R1"   and "R2' = R2" shows "bis B1 B2 s1 s2 B1' B2' s1' s2' R1' R2'"
weakenTransition_def: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and \<alpha> :: "'a action"   and P' :: "('a, 'b, 'c) psi" shows "\<Psi> \<rhd> P \<Longrightarrow>\<alpha> \<prec> P' \<equiv> (\<exists>P''' P''. \<Psi> \<rhd> P \<Longrightarrow>\<^sup>^\<^sub>\<tau> P''' \<and> \<Psi> \<rhd> P''' \<longmapsto> \<alpha> \<prec> P'' \<and> \<Psi> \<rhd> P'' \<Longrightarrow>\<^sup>^\<^sub>\<tau> P') \<or> P = P' \<and> \<alpha> = \<tau>"
InfImpl: fixes F :: "'a \<Rightarrow> bool"   and sigma :: "behavior"   and G :: "'a \<Rightarrow> bool"   and H :: "'a \<Rightarrow> bool" assumes "sigma \<Turnstile> \<box>\<diamond>F"   and "sigma \<Turnstile> \<box>G"   and "\<turnstile> F \<and> G \<longrightarrow> H" shows "sigma \<Turnstile> \<box>\<diamond>H"
sum_simp_setup_4: shows "\<forall>v\<in>cf.V. sum f' (cf.outgoing v) = (\<Sum>u\<in>cf.V. f' (v, u))"
double_adj: fixes A :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'b" shows "A** = A"
unat_arith_simps_6: fixes x :: "'b word"   and y :: "'b word" shows "unat (x div y) = unat x div unat y"
invertible_left_cancel: fixes x :: "'a set"   and y :: "'a set"   and z :: "'a set" assumes "quotient.invertible x"   and "x \<in> Partition"   and "y \<in> Partition"   and "z \<in> Partition" shows "(x [\<cdot>] y = x [\<cdot>] z) = (y = z)"