hcju/isabelleps huggingface dataset:
distinct_17: fixes x31 :: "POPLmark.type"   and x32 :: "trm"   and x51 :: "trm"   and x52 :: "POPLmark.type" shows "(\<lambda><:x31. x32) \<noteq> x51 \<bullet>\<^sub>\<tau> x52"
le_iff_add: fixes a :: "'c"   and b :: "'c" shows "(a \<le> b) = (\<exists>c. b = a + c)"
simps_5: fixes x2 :: "nat"   and x1 :: "nat" shows "Intv x2 \<noteq> Const x1"
param_nat1_1: shows "(0, 0) \<in> nat_rel"
gtrp_env_4: shows "c' = add_mset s'_ c'_"
not_OK_eq: fixes x :: "'b err" shows "(\<forall>y. x \<noteq> OK y) = (x = Err)"
splitAt_dist_ram1: fixes vs :: "'c list"   and a :: "'c list"   and ram1 :: "'c"   and b :: "'c list"   and ram2 :: "'c"   and c :: "'c list" assumes "distinct vs"   and "vs = a @ ram1 # b @ ram2 # c" shows "(a, b @ ram2 # c) = splitAt ram1 vs"
isomorphic_group_integer_free_Abelian_group_singleton: fixes x :: "'b" shows "integer_group \<cong> free_Abelian_group {x}"
CardinalityPrognosisApp_axioms: shows "CardinalityPrognosisApp prognosis"
Sup_measure'_def: fixes M :: "'a measure set" shows "local.Sup_measure' M = measure_of (\<Union> (space ` M)) (\<Union> (sets ` M)) (\<lambda>X. SUP P\<in>{P. finite P \<and> P \<subseteq> M}. emeasure (sup_measure.F id P) X)"
cil_commute: fixes w :: "'b list"   and \<alpha> :: "'b \<Rightarrow> 'c set \<times> 'c set"   and w' :: "'b list" shows "w \<otimes>\<^bsub>\<alpha>\<^esub> w' = w' \<otimes>\<^bsub>\<alpha>\<^esub> w"
prems_2: shows "\<forall>xs x xs'. nsx = xs @ x # xs' \<and> obs_intra x S \<noteq> {} \<longrightarrow> (\<exists>x''\<in>set (xs' @ [n]). \<exists>nx. call_of_return_node x'' nx \<and> nx \<notin> S)"
assms_1: shows "pred_stream (\<lambda>u. u \<in> V) xs"
trans: fixes b :: "'c word"   and a :: "'c word"   and c :: "'c word" assumes "b \<le>s a"   and "c \<le>s b" shows "c \<le>s a"
simps_7: fixes x2 :: "'a" shows "Drop \<noteq> ForwardOrEdit x2"
continuous_on_linepath: fixes s :: "real set"   and a :: "'b"   and b :: "'b" shows "continuous_on s (linepath a b)"
is_zero_after_Gauss: fixes a :: "'a"   and k :: "nat"   and A :: "(('b, 'c) vec, 'a) vec"   and m :: "'a"   and ma :: "'a" assumes "is_zero_row_upt_k a k A"   and "\<not> is_zero_row_upt_k m k A"   and "reduced_row_echelon_form_upt_k A k"   and "(GREATEST n. \<not> is_zero_row_upt_k n k A) + (1::'a) \<le> ma"   and "A $ ma $ mod_type_class.from_nat k \<noteq> (0::'b)" shows "is_zero_row_upt_k a k (Gauss_Jordan_in_ij A ((GREATEST m. \<not> is_zero_row_upt_k m k A) + (1::'a)) (mod_type_class.from_nat k))"
card_of_refl: fixes A :: "'a set" shows "|A| =o |A|"
pickBinp_def: fixes binp :: "'e \<Rightarrow> 'f set option" shows "pickBinp binp \<equiv> lift pick binp"
fixp_unfold: fixes b :: "'c"   and f :: "('b \<Rightarrow> 'c) \<Rightarrow> 'b \<Rightarrow> 'c" assumes "flatf_mono b f" shows "flatf_fp b f = f (flatf_fp b f)"
crdt_ops_intro: fixes deps :: "'d \<Rightarrow> 'c set"   and oper :: "'d"   and xs :: "('c \<times> 'd) list"   and oid :: "'c" assumes "\<And>r. r \<in> deps oper \<Longrightarrow> r \<in> fst ` set xs \<and> r < oid"   and "oid \<notin> fst ` set xs"   and "crdt_ops xs deps" shows "crdt_ops (xs @ [(oid, oper)]) deps"
rGamma_neg_1: shows "rGamma (- (1::'a)) = (0::'a)"
invariance_of_domain: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'a" assumes "continuous_on S f"   and "open S"   and "inj_on f S" shows "open (f ` S)"
setdist_neq_0_sing_1: fixes x :: "'b"   and S :: "'b set"   and a :: "real" assumes "setdist {x} S = a"   and "a \<noteq> 0" shows "S \<noteq> {} \<and> x \<notin> closure S"
abs_eq: fixes x :: "'b" shows "single x = Abs_ttree {[], [x]}"
simps_4: fixes c :: "int"   and P :: "pol"   and i :: "nat"   and Q :: "pol" shows "Pc c \<langle>*\<rangle> PX P i Q = (if c = 0 then Pc 0 else mkPX (P \<langle>*\<rangle> Pc c) i (Q \<langle>*\<rangle> Pc c))"
uniform_distrI_borel_atLeastAtMost: fixes X :: "'a \<Rightarrow> real"   and a :: "real"   and b :: "real" assumes "random_variable borel X"   and "a < b"   and "\<And>t. \<lbrakk>a \<le> t; t \<le> b\<rbrakk> \<Longrightarrow> prob {x \<in> space M. X x \<le> t} = (t - a) / (b - a)" shows "distributed M lborel X (\<lambda>x. ennreal (indicat_real {a..b} x / content {a..b}))"
elements_matI: fixes A :: "'b mat"   and nr :: "nat"   and nc :: "nat"   and i :: "nat"   and j :: "nat"   and a :: "'b" assumes "A \<in> carrier_mat nr nc"   and "i < nr"   and "j < nc"   and "a = A $$ (i, j)" shows "a \<in> elements_mat A"
of_nat_in_nonpos_Ints_iff: fixes n :: "nat" shows "(of_nat n \<in> \<int>\<^sub>\<le>\<^sub>0) = (n = 0)"
poly_poly_poly_x_minus_y: fixes p :: "'a poly"   and q :: "'a poly"   and x :: "'a" shows "poly (poly (poly_x_minus_y p) q) x = poly p (x - poly q x)"
pm_of_poly_one: fixes x :: "'a" shows "pm_of_poly x 1 = 1"
Pi_cong: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b"   and B :: "'a \<Rightarrow> 'b set" assumes "\<And>w. w \<in> A \<Longrightarrow> f w = g w" shows "(f \<in> Pi A B) = (g \<in> Pi A B)"
manifold_eucl_carrier: shows "manifold_eucl.carrier = UNIV"
nop_onto_vdomain: shows "\<D>\<^sub>\<circ> vreal_plus = \<real>\<^sub>\<circ> ^\<^sub>\<times> 2\<^sub>\<nat>"
wf_atable_subset: fixes getIJ :: "(nat set \<times> 'b option list set) set \<Rightarrow> (nat set \<times> 'b option list set) set \<Rightarrow> nat set \<Rightarrow> nat set \<times> nat set"   and n :: "nat"   and V :: "nat set"   and X :: "'c option list set"   and Y :: "'c option list set" assumes "getIJ getIJ"   and "table n V X"   and "Y \<subseteq> X" shows "table n V Y"
rev_existence_ivl_eq0: fixes y :: "'a" shows "existence_ivl0 y = uminus ` rev.existence_ivl0 y"
successors_def: fixes g :: "'a"   and m :: "'node" shows "gp2.successors g m \<equiv> concat (map (\<lambda>n. if m \<in> set (gp2.predecessors g n) then [n] else []) (\<alpha>n2 g))"
rel_gpv''_conversep: fixes A :: "'d \<Rightarrow> 'a \<Rightarrow> bool"   and C :: "'e \<Rightarrow> 'b \<Rightarrow> bool"   and R :: "'f \<Rightarrow> 'c \<Rightarrow> bool" shows "rel_gpv'' A\<inverse>\<inverse> C\<inverse>\<inverse> R\<inverse>\<inverse> = (rel_gpv'' A C R)\<inverse>\<inverse>"
elims_3: fixes x :: "'b mat \<times> 'b mat \<times> 'b mat"   and xa :: "(('b, 'c) Finite_Cartesian_Product.vec, 'c) Finite_Cartesian_Product.vec \<times> (('b, 'd) Finite_Cartesian_Product.vec, 'c) Finite_Cartesian_Product.vec \<times> (('b, 'd) Finite_Cartesian_Product.vec, 'd) Finite_Cartesian_Product.vec" assumes "\<not> HMA_M3 x xa" obtains P :: "'b mat" and A :: "'b mat" and Q :: "'b mat" and P' :: "(('b, 'c) Finite_Cartesian_Product.vec, 'c) Finite_Cartesian_Product.vec" and A' :: "(('b, 'd) Finite_Cartesian_Product.vec, 'c) Finite_Cartesian_Product.vec" and Q' :: "(('b, 'd) Finite_Cartesian_Product.vec, 'd) Finite_Cartesian_Product.vec" where "x = (P, A, Q)" and "xa = (P', A', Q')" and "\<not> (Mod_Type_Connect.HMA_M P P' \<and> Mod_Type_Connect.HMA_M A A' \<and> Mod_Type_Connect.HMA_M Q Q')"
sinh_conv_sin: fixes z :: "complex" shows "sinh z = - \<i> * sin (\<i> * z)"
H: shows "H \<equiv> vec_of_list (coeffs h @ replicate (degree u - length (coeffs h)) 0)"
r_step_Pr_def: shows "r_step_Pr \<equiv> Cn 1 r_ifz [r_e2ls, Cn 1 r_ifz [r_e2rv, Cn 1 r_prod_encode [Cn 1 r_cons [r_triple_encode (Cn 1 r_pdec122 [r_e2i]) (Cn 1 r_tl [r_e2xs]) Z, r_e2stack], Z], Cn 1 r_prod_encode [Cn 1 r_cons [r_triple_encode r_e2i r_e2xs (Cn 1 r_singleton_encode [Cn 1 r_dec [r_e2rv]]), r_e2tail], Z]], Cn 1 r_ifeq [r_e2lenls, Cn 1 S [Cn 1 r_hd [r_e2xs]], Cn 1 r_prod_encode [r_e2tail, Cn 1 S [Cn 1 r_hd [r_e2ls]]], Cn 1 r_ifz [r_e2rv, Cn 1 r_prod_encode [Cn 1 r_cons [r_triple_encode (Cn 1 r_pdec222 [r_e2i]) (Cn 1 r_cons [Cn 1 r_dec [r_e2lenls], Cn 1 r_cons [Cn 1 r_hd [r_e2ls], Cn 1 r_tl [r_e2xs]]]) Z, r_e2stack], Z], Cn 1 r_prod_encode [Cn 1 r_cons [r_triple_encode r_e2i r_e2xs (Cn 1 r_cons [Cn 1 r_dec [r_e2rv], r_e2ls]), r_e2tail], Z]]]]"
sup_hmultiset_def: fixes A :: "hmultiset"   and B :: "hmultiset" shows "sup A B = (if A < B then B else A)"
intros_23: fixes s :: "s"   and vs :: "v list"   and es :: "e list"   and i :: "nat"   and s' :: "s"   and vs' :: "v list"   and es' :: "e list"   and k :: "nat"   and lholed :: "Lholed"   and les :: "e list"   and les' :: "e list" assumes "\<lparr>s;vs;es\<rparr> \<leadsto>_ i \<lparr>s';vs';es'\<rparr>"   and "Lfilled k lholed es les"   and "Lfilled k lholed es' les'" shows "\<lparr>s;vs;les\<rparr> \<leadsto>_ i \<lparr>s';vs';les'\<rparr>"
simps: fixes \<pi> :: "nat \<Rightarrow> 'n"   and n :: "nat" shows "cs\<^bsup>\<pi>\<^esup> n = (if Ex (is_icdi n \<pi>) then cs\<^bsup>\<pi>\<^esup> The (is_icdi n \<pi>) @ [\<pi> n] else [\<pi> n])"
sig_inv_set_closed_uminus: fixes r :: "'t \<Rightarrow>\<^sub>0 'b"   and j :: "nat" assumes "r \<in> sig_inv_set' j" shows "- r \<in> sig_inv_set' j"
is_extensional: fixes f :: "'a1" assumes "\<not> A1.arr f" shows "A1.map f = A1.null"
ide_eval_Ide: fixes t :: "'a E.term" assumes "E.Ide t" shows "B.ide (EVAL t)"
insertions_last_ins: fixes xs :: "('c \<times> ('c, 'd) list_op) list"   and oid :: "'c"   and ref :: "'c option"   and val :: "'d" shows "insertions (xs @ [(oid, Insert ref val)]) = insertions xs @ [(oid, ref)]"
domain_eq: shows "Domainp pcr_circline_mat = (\<lambda>H. hermitean H \<and> H \<noteq> mat_zero)"
word_le_sub1: fixes x :: "'b word" assumes "x \<noteq> 0" shows "(1 \<le> x) = (0 \<le> x - 1)"
assms_2: shows "- z \<le> - - x \<squnion> - y"
sum_diff: fixes A :: "'a set"   and B :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "finite A"   and "B \<subseteq> A" shows "sum f (A - B) = sum f A - sum f B"
minus_permutation_abs_eq: fixes f :: "'a \<Rightarrow> 'a"   and g :: "'a \<Rightarrow> 'a" assumes "bij f"   and "bij g" shows "Abs_permutation f - Abs_permutation g = Abs_permutation (f \<circ> the_inv g)"
subspace_finrank: fixes smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and V :: "'b set"   and M :: "'b set"   and N :: "'b set" assumes "VectorSpace smult V"   and "R_scalar_mult.R_finrank UNIV smult M"   and "N \<subseteq> M" shows "R_scalar_mult.R_finrank UNIV smult N"
steps_r_sound: fixes A :: "('d \<times> ('e, real) cconstraint \<times> 'f \<times> 'e list \<times> 'd) set \<times> ('d \<Rightarrow> ('e, real) cconstraint)"   and \<R> :: "('e \<Rightarrow> real) set set"   and l :: "'d"   and R :: "('e \<Rightarrow> real) set"   and l' :: "'d"   and R' :: "('e \<Rightarrow> real) set"   and X :: "'e set"   and k :: "'e \<Rightarrow> nat"   and u :: "'e \<Rightarrow> real" assumes "A,\<R> \<turnstile> \<langle>l, R\<rangle> \<leadsto>* \<langle>l', R'\<rangle>"   and "\<R> = {region X I r |I r. valid_region X k I r}"   and "R' \<noteq> {}"   and "u \<in> R" shows "\<exists>u'\<in>R'. A \<turnstile> \<langle>l, u\<rangle> \<rightarrow>* \<langle>l', u'\<rangle>"
BlockRedsFinal: fixes e\<^sub>0 :: "char list exp"   and s\<^sub>0 :: "heap \<times> sfields \<times> sheap"   and b\<^sub>0 :: "bool"   and e\<^sub>2 :: "char list exp"   and h\<^sub>2 :: "heap"   and l\<^sub>2 :: "sfields"   and sh\<^sub>2 :: "sheap"   and b\<^sub>2 :: "bool"   and P :: "(char list list \<times> char list exp) prog"   and h\<^sub>0 :: "heap"   and l\<^sub>0 :: "sfields"   and V :: "char list"   and sh\<^sub>0 :: "sheap"   and T :: "ty" assumes "P \<turnstile> \<langle>e\<^sub>0,s\<^sub>0,b\<^sub>0\<rangle> \<rightarrow>* \<langle>e\<^sub>2,(h\<^sub>2, l\<^sub>2, sh\<^sub>2),b\<^sub>2\<rangle>"   and "final e\<^sub>2"   and "s\<^sub>0 = (h\<^sub>0, l\<^sub>0(V := None), sh\<^sub>0)" shows "P \<turnstile> \<langle>{V:T; e\<^sub>0},(h\<^sub>0, l\<^sub>0, sh\<^sub>0),b\<^sub>0\<rangle> \<rightarrow>* \<langle>e\<^sub>2,(h\<^sub>2, l\<^sub>2(V := l\<^sub>0 V), sh\<^sub>2),b\<^sub>2\<rangle>"
hmem_not_refl: fixes x :: "hf" shows "x \<^bold>\<notin> x"
simps_1: fixes \<nu> :: "'a"   and u :: "'a local.term" shows "\<^bold>\<langle>\<nu>\<^bold>\<rangle>\<^sub>0 \<^bold>\<lfloor>\<^bold>\<cdot>\<^bold>\<rfloor> u = u"
bdd_above_norm: fixes X :: "'b set" shows "bdd_above (norm ` X) = bounded X"
UP_def: shows "replete_setcat.UP \<equiv> setcat.UP"
length_take_le: fixes n :: "nat"   and xs :: "'b list" shows "length (xs \<down> n) \<le> length xs"
sum_slice1: fixes n'' :: "'node SDG_node"   and p :: "'pname"   and n' :: "'node SDG_node"   and n :: "'node SDG_node" assumes "n'' s-p\<rightarrow>\<^bsub>sum\<^esub> n'"   and "n' \<in> sum_SDG_slice1 n" shows "n'' \<in> sum_SDG_slice1 n"
prj_in_hom_5: shows "\<guillemotleft>fgf.Prj\<^sub>1\<^sub>0 : Chn (f \<star> g \<star> f) \<rightarrow>\<^sub>C g.apex\<guillemotright>"
cons: fixes c :: "'a"   and a :: "'b"   and c' :: "'a"   and t :: "('a \<times> 'b \<times> 'a) set"   and w :: "'b list"   and c'' :: "'a" assumes "(c, a, c') \<in> t"   and "(c', w, c'') \<in> trcl t" shows "(c, a # w, c'') \<in> trcl t"
S\<^sub>e_E2: fixes s :: "'s" assumes "s \<in> S1"   and "\<And>t. (s, t) \<in> E \<Longrightarrow> t \<in> S\<^sub>e" shows "s \<in> S\<^sub>e"
set_iterator_abs_def: fixes \<alpha> :: "'e \<Rightarrow> 'f"   and invar :: "'e \<Rightarrow> bool"   and it :: "('g \<Rightarrow> bool) \<Rightarrow> ('e \<Rightarrow> 'g \<Rightarrow> 'g) \<Rightarrow> 'g \<Rightarrow> 'g"   and S0 :: "'f set" shows "set_iterator_abs \<alpha> invar it S0 \<equiv> set_iterator_abs_genord \<alpha> invar it S0 (\<lambda>_ _. True)"
simps: fixes A :: "'a"   and B :: "'b"   and C :: "'c"   and D :: "'d" shows "mat_diagonal (A, B, C, D) = (B = (0::'b) \<and> C = (0::'c))"
PiM_def: fixes I :: "'d set"   and M :: "'d \<Rightarrow> 'e measure" shows "Pi\<^sub>M I M = extend_measure (\<Pi>\<^sub>E i\<in>I. space (M i)) {(J, X). (J \<noteq> {} \<or> I = {}) \<and> finite J \<and> J \<subseteq> I \<and> X \<in> (\<Pi> j\<in>J. sets (M j))} (\<lambda>(J, X). prod_emb I M J (Pi\<^sub>E J X)) (\<lambda>(J, X). \<Prod>j\<in>J \<union> {i \<in> I. emeasure (M i) (space (M i)) \<noteq> 1}. if j \<in> J then emeasure (M j) (X j) else emeasure (M j) (space (M j)))"
MPair_synth_analz: fixes X :: "msg"   and Y :: "msg"   and H :: "msg set" shows "(\<lbrace>X, Y\<rbrace> \<in> synth (analz H)) = (X \<in> synth (analz H) \<and> Y \<in> synth (analz H))"
distinct_mset_singleton: fixes a :: "'b" shows "distinct_mset {#a#}"
simps_2: fixes uv :: "('b, 'c) model"   and g :: "'d \<Rightarrow> 'b"   and w :: "'b"   and i :: "'d" shows "(uv, g, w \<Turnstile> Nom i) = (w = g i)"
scf_term_subst: fixes scf :: "'c \<times> nat \<Rightarrow> nat \<Rightarrow> nat"   and t :: "('c, 'e) Term.term"   and \<sigma> :: "'e \<Rightarrow> ('c, 'd) Term.term" shows "scf_term scf (t \<cdot> \<sigma>) = scf_term scf t \<cdot> (\<lambda>x. scf_term scf (\<sigma> x))"
suc_inField: fixes B :: "'b set" assumes "B \<subseteq> Field s"   and "s.AboveS B \<noteq> {}" shows "s.suc B \<in> Field s"
disc_7: fixes x11 :: "name"   and x12 :: "value list" shows "\<not> is_Vrecabs (Vconstr x11 x12)"
simps_2: fixes a :: "'a"   and n :: "nat" shows "power.power 1' (;) a (Suc n) = a ; power.power 1' (;) a n"
suminf_dirichlet_prod: fixes f :: "nat \<Rightarrow> 'a"   and g :: "nat \<Rightarrow> 'a"   and s :: "'a" assumes "f 0 = (0::'a)"   and "g 0 = (0::'a)"   and "summable (\<lambda>n. norm (f n / nat_power n s))"   and "summable (\<lambda>n. norm (g n / nat_power n s))" shows "(\<Sum>n. dirichlet_prod f g n / nat_power n s) = (\<Sum>n. f n / nat_power n s) * (\<Sum>n. g n / nat_power n s)"
behp_beh_eq: fixes T :: "('a, 'b) TS_scheme" shows "behp T = (\<lambda>x. x \<in> beh T)"
dX_Int_Field_Pd: fixes X :: "'a set"   and d :: "'b" assumes "PdXXX.dX X d \<subseteq> Field (Pd' d)" shows "X \<inter> Field (Pd' d) = PdXXX.dX X d"
finite_pow2_eq: fixes n :: "nat" shows "finite {i. 2 ^ i = n}"
simps_2: fixes x :: "name"   and K :: "stack"   and t :: "trm"   and s :: "trm" assumes "x \<sharp> (K, t)" shows "t \<star> [x]s\<ggreater>K = (t to x in s) \<star> K"
connected_Un: fixes s :: "'c set"   and t :: "'c set" assumes "connected s"   and "connected t"   and "s \<inter> t \<noteq> {}" shows "connected (s \<union> t)"
hindrance_conv_hindrance_by: fixes \<Gamma> :: "('a, 'b) web_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal" shows "hindrance \<Gamma> f = (\<exists>\<epsilon>>0. hindrance_by \<Gamma> f \<epsilon>)"
Times_insert_left: fixes A :: "'a multiset"   and x :: "'b"   and B :: "'b multiset" shows "A \<times># add_mset x B = A \<times># B + {#(a, x). a \<in># A#}"
R_2: fixes R' :: "'a list" assumes "new_last \<leadsto>R'\<leadsto>\<^bsub>H\<^esub> v1" shows "length R \<le> length R'"
well_order_on: fixes A :: "'c set" shows "\<exists>r. well_order_on A r"
wf_digraph_axioms: shows "wf_digraph (with_proj G)"
zero_space\<^sub>N_def: fixes N :: "'c quasinorm" shows "zero_space\<^sub>N N = {f. eNorm N f = 0}"
inorder_pairs_sorted: fixes t :: "('c, 'd) tree" shows "tree_sorted t = Lists_Ex.strict_sorted (map fst (in_traverse_pairs t))"
recvmsg_simps_3: fixes P :: "'a \<Rightarrow> bool"   and ip :: "nat"   and m :: "'a" shows "recvmsg P (unicast ip m) = True"
MkElem_IMG: fixes p :: "('e set, 'e \<Rightarrow> 'e) concrete_category.arr"   and a :: "('e set, 'e \<Rightarrow> 'e) concrete_category.arr" assumes "p \<in> hom unity a" shows "MkArr {U} (Dom a) (\<lambda>_\<in>{U}. DOWN (IMG p)) = p"
Goal: fixes Pa :: "pi"   and Qa :: "pi"   and xa :: "name"   and \<sigma>' :: "(name \<times> name) list"   and aa :: "name" assumes "Pa \<sim>\<^sup>s Qa"   and "xa \<sharp> \<sigma>'" shows "aa<xa>.Pa[<\<sigma>'>] \<sim> aa<xa>.Qa[<\<sigma>'>]"
D_def: shows "D = C \<inter> {..(int p - 1) div 2}"
univ: fixes Rep :: "'a \<Rightarrow> 'b"   and Abs :: "'b \<Rightarrow> 'a"   and A :: "'b set" assumes "type_definition Rep Abs A" shows "UNIV = Abs ` A"
S_verts_def: shows "ES.S_verts \<equiv> {1..n}"
datarefin_def: fixes S :: "'b"   and S1 :: "'b"   and D :: "'b"   and D1 :: "'b" shows "datarefin S S1 D D1 = (D * S \<le> S1 * D1)"
swapVal_ident: fixes xs :: "'varSort"   and x :: "'var"   and val :: "'varSort \<Rightarrow> 'var \<Rightarrow> 'b" shows "val ^[x \<and> x]_xs = val"
induct: fixes P :: "atom fm \<Rightarrow> bool"   and a0 :: "atom fm" assumes "\<And>\<phi>\<^sub>1 \<phi>\<^sub>2. \<lbrakk>P \<phi>\<^sub>1; P \<phi>\<^sub>2\<rbrakk> \<Longrightarrow> P (And \<phi>\<^sub>1 \<phi>\<^sub>2)"   and "\<And>\<phi>\<^sub>1 \<phi>\<^sub>2. \<lbrakk>P \<phi>\<^sub>1; P \<phi>\<^sub>2\<rbrakk> \<Longrightarrow> P (Or \<phi>\<^sub>1 \<phi>\<^sub>2)"   and "\<And>i k ks. P (Atom (Le i (k # ks)))"   and "P TrueF"   and "P FalseF"   and "\<And>va. P (Atom (Le va []))"   and "\<And>va vb vc. P (Atom (Dvd va vb vc))"   and "\<And>va vb vc. P (Atom (NDvd va vb vc))"   and "\<And>v. P (Neg v)"   and "\<And>v. P (ExQ v)" shows "P a0"
of_nat_eq_of_nat_power_cancel_iff: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and b :: "nat"   and w :: "nat"   and x :: "nat" assumes "class.semiring_char_0 one times plus zero" shows "(power.power one times (semiring_1.of_nat one plus zero b) w = semiring_1.of_nat one plus zero x) = (b ^ w = x)"
eventually_inside: shows "\<forall>\<^sub>F x in at (poincare_map {x \<in> R. x \<bullet> n = c} x). x \<bullet> n = c \<longrightarrow> x \<in> R"
eq_for_lvar: fixes x :: "nat"   and t :: "(nat \<times> linear_poly) list" assumes "x \<in> lvars t" shows "eq_for_lvar_code t x \<in> set t \<and> lhs (eq_for_lvar_code t x) = x"
is_starext_starfun: fixes f :: "'c \<Rightarrow> 'c" shows "is_starext (*f* f) f"
HomFtorCompDef': fixes C :: "('d, 'e, 'f) LSCategory_scheme"   and X :: "'d"   and f :: "'e"   and g :: "'e" assumes "LSCategory C"   and "X \<in> obj\<^bsub>C\<^esub>"   and "f \<approx>>\<^bsub>C\<^esub> g" shows "Hom\<^bsub>C\<^esub>[X,f] \<approx>>\<^bsub>SET'\<^esub> (Hom\<^bsub>C\<^esub>[X,g])"
ltl_const_equiv_def: fixes \<phi> :: "'b ltln"   and \<psi> :: "'b ltln" shows "\<phi> \<sim>\<^sub>C \<psi> \<equiv> \<phi> = \<psi> \<or> eval \<phi> = eval \<psi> \<and> eval \<psi> \<noteq> Maybe"
fract_expand_2: shows "1 = Fraction_Field.Fract (1::'c) (1::'c)"
has_binary_products_if: assumes "has_products {0, 1}" shows "has_binary_products"
le: fixes i :: "'b" assumes "i \<in> K" shows "g i \<le> f i"
that: assumes "(f ^^ funpow_dist1 f x y) x = y"   and "funpow_dist1 f x y < funpow_dist1 f x z" shows "thesis"
h: assumes "\<exists>x\<in>{r<..k}. 0 < a * x\<^sup>2 + b * x + c" shows "\<exists>x\<in>{r<..k}. 0 \<le> a * x\<^sup>2 + b * x + c"
div_free_CHAOS\<^sub>S\<^sub>K\<^sub>I\<^sub>P: fixes A :: "'a set" shows "D (CHAOS\<^sub>S\<^sub>K\<^sub>I\<^sub>P A) = {}"
erlang_density_nonneg: fixes l :: "real"   and k :: "nat"   and x :: "real" assumes "0 \<le> l" shows "0 \<le> erlang_density k l x"
prv_dsj_cnj_distrib1: fixes \<phi> :: "'fmla"   and \<chi>1 :: "'fmla"   and \<chi>2 :: "'fmla" assumes "\<phi> \<in> fmla"   and "\<chi>1 \<in> fmla"   and "\<chi>2 \<in> fmla" shows "prv (imp (dsj \<phi> (cnj \<chi>1 \<chi>2)) (cnj (dsj \<phi> \<chi>1) (dsj \<phi> \<chi>2)))"
split_max_Leaf: fixes t :: "('b \<times> bal) tree"   and x :: "'b" assumes "t \<noteq> \<langle>\<rangle>"   and "avl t" shows "(split_max t = (\<langle>\<rangle>, x)) = (t = \<langle>\<langle>\<rangle>, (x, Bal), \<langle>\<rangle>\<rangle>)"
RedsRInit: fixes e\<^sub>0 :: "char list exp"   and s\<^sub>0 :: "heap \<times> sfields \<times> sheap"   and b\<^sub>0 :: "bool"   and v :: "val"   and h\<^sub>1 :: "heap"   and l\<^sub>1 :: "sfields"   and sh\<^sub>1 :: "sheap"   and b\<^sub>1 :: "bool"   and P :: "(char list list \<times> char list exp) prog"   and C :: "char list"   and sfs :: "sfields"   and i :: "init_state"   and sh\<^sub>2 :: "sheap"   and C' :: "char list"   and Cs :: "char list list"   and e :: "char list exp"   and e' :: "char list exp"   and s' :: "heap \<times> sfields \<times> sheap"   and b' :: "bool" assumes "P \<turnstile> \<langle>e\<^sub>0,s\<^sub>0,b\<^sub>0\<rangle> \<rightarrow>* \<langle>Val v,(h\<^sub>1, l\<^sub>1, sh\<^sub>1),b\<^sub>1\<rangle>"   and "sh\<^sub>1 C = \<lfloor>(sfs, i)\<rfloor>"   and "sh\<^sub>2 = sh\<^sub>1(C \<mapsto> (sfs, Done))"   and "C' = last (C # Cs)"   and "P \<turnstile> \<langle>INIT C' (Cs,True) \<leftarrow> e,(h\<^sub>1, l\<^sub>1, sh\<^sub>2),b\<^sub>1\<rangle> \<rightarrow>* \<langle>e',s',b'\<rangle>" shows "P \<turnstile> \<langle>RI (C,e\<^sub>0) ; Cs \<leftarrow> e,s\<^sub>0,b\<^sub>0\<rangle> \<rightarrow>* \<langle>e',s',b'\<rangle>"
Norm_Distr2: fixes r :: "'d \<times> 'e \<Rightarrow> 'c"   and P :: "'f \<Rightarrow> 'd decision option"   and Q :: "('g \<Rightarrow> 'e decision option) list"   and d :: "'b \<Rightarrow> 'f \<times> 'g" shows "r o\<^sub>f (P \<Otimes>\<^sub>2 list2policy Q \<circ> d) = list2policy ((P \<Otimes>\<^sub>L Q) (\<Otimes>\<^sub>2) r d)"
disj_not2: fixes P :: "bool"   and Q :: "bool" shows "(P \<or> \<not> Q) = (Q \<longrightarrow> P)"
vdomain_vrange_is_vempty: shows "(\<D>\<^sub>\<circ> (\<CC>\<lparr>Cod\<rparr>) = []\<^sub>\<circ>) = (\<R>\<^sub>\<circ> (\<CC>\<lparr>Cod\<rparr>) = []\<^sub>\<circ>)"
o_def: fixes f :: "'d \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'd" shows "f \<circ> g = (\<lambda>x. f (g x))"
simps_2: fixes v :: "char list" shows "coste (Var v) = 0"
Bang_2: fixes P :: "pi"   and \<alpha> :: "freeRes"   and P' :: "pi" assumes "P \<parallel> !P \<Longrightarrow>\<alpha> \<prec> P'" shows "!P \<Longrightarrow>\<alpha> \<prec> P'"
TQ_def: fixes x :: "('a, 'n) vec" shows "TQ x = Q *v x"
lipschitz_on_compose: fixes C :: "real"   and U :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and D :: "real"   and g :: "'c \<Rightarrow> 'd" assumes "C-lipschitz_on U f"   and "D-lipschitz_on (f ` U) g" shows "(D * C)-lipschitz_on U (g \<circ> f)"
atoms_list_conjE: fixes a :: "'a"   and fs :: "'a fm list" assumes "a \<in> atoms (list_conj fs)" shows "a \<in> \<Union> (atoms ` set fs)"
simps_1: shows "eval_ternary_Or TernaryTrue TernaryTrue = TernaryTrue"
F_def: shows "F = (\<lambda>x. x powr (e + 1) / (e + 1))"
IH: fixes \<phi> :: "'fmla"   and s :: "'trm"   and y :: "'var" assumes "\<phi> \<in> fmla"   and "snd ` set txs \<subseteq> var"   and "fst ` set txs \<subseteq> trm"   and "\<forall>x\<in>snd ` set txs. x \<notin> FvarsT s"   and "\<forall>t\<in>fst ` set txs. y \<notin> FvarsT t"   and "distinct (map snd txs)"   and "s \<in> trm"   and "y \<in> var"   and "y \<notin> snd ` set txs" shows "subst (rawpsubst \<phi> txs) s y = rawpsubst \<phi> ((s, y) # txs)"
simps_11: fixes x3 :: "'b"   and x2 :: "'b" shows "mid x3 \<noteq> right x2"
1: shows "qGood (X #[[x' \<and> x]]_xs)"
Vvar: shows "|V| <o |UNIV| \<or> finite V"