hcju/isabelleps huggingface dataset:
post_fst_k_Suc_neg: fixes i :: "nat" assumes "\<not> f i" shows "post_fst_k (Suc i) = post_fst_k i"
hoare_partial_sound: fixes P :: "complex mat"   and S :: "com"   and Q :: "complex mat" assumes "ps_P.hoare_partial P S Q"   and "ps_P.well_com S" shows "ps_P.hoare_partial_correct P S Q"
cy: fixes x :: "real \<times> real" assumes "x \<in> set xs" shows "coll 0 x y"
mult_minus_left: fixes plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and uminus :: "'e \<Rightarrow> 'e"   and times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and a :: "'e"   and b :: "'e" assumes "class.ring plus zero minus uminus times" shows "times (uminus a) b = uminus (times a b)"
c2: shows "card ({(x, y). x < y \<and> swaps ss init ! s = x \<and> swaps ss init ! Suc s = y} \<union> {(x, y). x < y \<and> swaps ss init ! s = y \<and> swaps ss init ! Suc s = x}) = 1"
path_prop: shows "path A (w || r1) p1 \<and> path B (w || r2) p2 \<or> target (w || r1 || r2) (p1, p2) = FAIL \<and> 0 < length w \<and> path A (butlast (w || r1)) p1 \<and> path B (butlast (w || r2)) p2 \<and> succ A (last w) (target (butlast (w || r1)) p1) = {} \<and> succ B (last w) (target (butlast (w || r2)) p2) \<noteq> {}"
Quotient3_circline: shows "Quotient3 circline_eq_clmat abs_circline rep_circline"
compatible_different_no_read: fixes cms\<^sub>1 :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) list"   and mem\<^sub>1 :: "'Var \<Rightarrow> 'Val"   and cms\<^sub>2 :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) list"   and mem\<^sub>2 :: "'Var \<Rightarrow> 'Val"   and mems :: "(('Var \<Rightarrow> 'Val) \<times> ('Var \<Rightarrow> 'Val)) list"   and i :: "nat"   and x :: "'Var" assumes "sound_mode_use (cms\<^sub>1, mem\<^sub>1)"   and "sound_mode_use (cms\<^sub>2, mem\<^sub>2)"   and "makes_compatible (cms\<^sub>1, mem\<^sub>1) (cms\<^sub>2, mem\<^sub>2) mems"   and "map snd cms\<^sub>1 = map snd cms\<^sub>2"   and "i < length cms\<^sub>1"   and "x \<in> differing_vars_lists mem\<^sub>1 mem\<^sub>2 mems i" shows "doesnt_read_or_modify (fst (cms\<^sub>1 ! i)) x \<and> doesnt_read_or_modify (fst (cms\<^sub>2 ! i)) x"
expand_codomain: fixes X :: "'b set set"   and Y :: "'c set set"   and f :: "'b \<Rightarrow> 'c"   and Z :: "'c set set" assumes "ChamberComplexMorphism X Y f"   and "ChamberComplex Z"   and "ChamberComplex.ChamberSubcomplex Z Y" shows "ChamberComplexMorphism X Z f"
ordinal_le_0: fixes x :: "ordinal" shows "(x \<le> 0) = (x = 0)"
some_in_prime_divisor: fixes v :: "'a \<Rightarrow> ant" assumes "valuation K v" shows "(SOME w. w \<in> P\<^bsub> K v\<^esub>) \<in> P\<^bsub> K v\<^esub>"
ntr_valid_preserve_s: fixes c :: "'n list multiset"   and w :: "('p, 'ba) label list"   and c' :: "'n list multiset" assumes "(c, w, c') \<in> ntr fg"   and "valid fg c" shows "valid fg c'"
in_mult: shows "(mset xs, mset ys) \<in> mult {(x, y). gt y x}"
dom_assoc: fixes a :: "'d"   and b :: "'d"   and c :: "'d" assumes "ide a"   and "ide b"   and "ide c" shows "local.dom \<a>[a, b, c] = (a \<otimes> b) \<otimes> c"
WTrtFAss: 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 E :: "char list \<Rightarrow> ty option"   and h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and e\<^sub>1 :: "expr"   and C :: "char list"   and Cs :: "char list list"   and F :: "char list"   and T :: "ty"   and e\<^sub>2 :: "expr"   and T' :: "ty" assumes "P,E,h \<turnstile> e\<^sub>1 : Class C"   and "Cs \<noteq> []"   and "P \<turnstile> C has least F:T via Cs"   and "P,E,h \<turnstile> e\<^sub>2 : T'"   and "P \<turnstile> T' \<le> T" shows "P,E,h \<turnstile> e\<^sub>1\<bullet>F{Cs} := e\<^sub>2 : T"
star_n_less: fixes X :: "nat \<Rightarrow> 'a"   and Y :: "nat \<Rightarrow> 'a" shows "(star_n X < star_n Y) = (\<forall>\<^sub>F n in \<U>. X n < Y n)"
graph_lm_3: fixes x :: "nat"   and y :: "nat"   and f :: "nat \<Rightarrow> nat" shows "((x, y) \<in> graph f) = (y = f x)"
hyps_5: fixes Cf' :: "varDef list" assumes "fields(CT,D) = Cf'"   and "CT OK" shows "Dg = Cf'"
zero_fds_def: shows "0 = fds (\<lambda>_. 0::'a)"
star_rtc_least: fixes x :: "'a"   and y :: "'a" assumes "(1::'a) + x + y \<cdot> y \<le> y" shows "x\<^sup>\<star> \<le> y"
Ex_enabled: fixes x :: "'s" shows "\<exists>\<omega>. enabled x \<omega>"
N_def: shows "N \<equiv> SOME N. N \<noteq> 0"
fs_deriv: fixes k :: "nat"   and x :: "real" assumes "k < N"   and "x \<in> {real a..} - Y" shows "(fs k has_vector_derivative fs (Suc k) x) (at x)"
less: shows "\<forall>j<b. (x # xs) ! j < a"
eval_prod_lam: fixes a :: "'a"   and b :: "'a"   and c :: "'a"   and g :: "'a" assumes "ide a"   and "ide b"   and "ide c"   and "\<guillemotleft>g : local.prod a b \<rightarrow> c\<guillemotright>" shows "\<guillemotleft>\<Lambda> a b c g : a \<rightarrow> exp b c\<guillemotright> \<and> g = C (eval b c) (local.prod (\<Lambda> a b c g) b)"
atLeast_iff: fixes i :: "'b"   and k :: "'b" shows "(i \<in> {k..}) = (k \<le> i)"
reflects_iso: fixes f :: "'a"   and a' :: "'a"   and a :: "'a" assumes "Right_a.in_hom f a' a"   and "Right_a.iso (H\<^sub>R a f)" shows "Right_a.iso f"
r_def: shows "r = Abs_poly_mapping (\<lambda>k. q k when k \<in> A)"
uInf: fixes P :: "'s \<Rightarrow> real" assumes "unitary P" shows "unitary (Inf_utrans (snd ` {x. P (fst x) (snd x) \<and> feasible (fst x) \<and> (\<forall>Q. unitary Q \<longrightarrow> unitary (snd x Q)) \<and> le_trans (fst x) (lfp_trans (\<lambda>a. wp (body ;; Embed a \<^bsub>\<guillemotleft> G \<guillemotright>\<^esub>\<oplus> Skip))) \<and> le_utrans (gfp_trans (\<lambda>a. wlp (body ;; Embed a \<^bsub>\<guillemotleft> G \<guillemotright>\<^esub>\<oplus> Skip))) (snd x)}) P)"
assms_5: shows "($$) \<eta>2 \<in> extensional (obj\<^bsub>NTCatDom \<eta>2\<^esub>)"
star_conv_concat: fixes A :: "'b list set" shows "star A = {concat ws |ws. set ws \<subseteq> A}"
PP_PPf: 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 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 fls :: "'c"   and dsj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and "proof" :: "'d set"   and prfOf :: "'d \<Rightarrow> 'c \<Rightarrow> bool"   and encPf :: "'d \<Rightarrow> 'b"   and Pf :: "'c"   and \<phi> :: "'c" assumes "CleanRepr_Proofs var trm fmla Var FvarsT substT Fvars subst num eql cnj imp all exi prv bprv enc fls dsj proof prfOf encPf Pf"   and "\<phi> \<in> fmla" shows "HBL1.PP var subst (CleanRepr_Proofs.P var Var FvarsT Fvars subst exi Pf) (enc \<phi>) = exi (Generic_Syntax.Variable var 11) (CleanRepr_Proofs.PPf var Var FvarsT Fvars subst Pf (Var (Generic_Syntax.Variable var 11)) (enc \<phi>))"
ballE: fixes A :: "'a set"   and P :: "'a \<Rightarrow> bool"   and x :: "'a" assumes "\<forall>x\<in>A. P x" obtains "P x"   | "x \<notin> A"
dims_matricize_1: fixes rmodes :: "nat set"   and T :: "'m tensor" shows "dim_row (matricize rmodes T) = prod_list (nths (dims T) rmodes)"
wf_trancl: fixes r :: "('b \<times> 'b) set" assumes "wf r" shows "wf (r\<^sup>+)"
assms: shows "mod_type_class.to_nat a + 1 \<noteq> CARD('a)"
in_set_insert: fixes x :: "'a"   and xs :: "'a list" assumes "x \<in> list.set xs" shows "List.insert x xs = xs"
comm_monoidI: fixes G :: "('c, 'd) monoid_scheme" assumes "\<And>x y. \<lbrakk>x \<in> carrier G; y \<in> carrier G\<rbrakk> \<Longrightarrow> x \<otimes>\<^bsub>G\<^esub> y \<in> carrier G"   and "\<one>\<^bsub>G\<^esub> \<in> carrier G"   and "\<And>x y z. \<lbrakk>x \<in> carrier G; y \<in> carrier G; z \<in> carrier G\<rbrakk> \<Longrightarrow> x \<otimes>\<^bsub>G\<^esub> y \<otimes>\<^bsub>G\<^esub> z = x \<otimes>\<^bsub>G\<^esub> (y \<otimes>\<^bsub>G\<^esub> z)"   and "\<And>x. x \<in> carrier G \<Longrightarrow> \<one>\<^bsub>G\<^esub> \<otimes>\<^bsub>G\<^esub> x = x"   and "\<And>x y. \<lbrakk>x \<in> carrier G; y \<in> carrier G\<rbrakk> \<Longrightarrow> x \<otimes>\<^bsub>G\<^esub> y = y \<otimes>\<^bsub>G\<^esub> x" shows "Group.comm_monoid G"
subdivFace'_incr_faces: fixes g :: "graph"   and f :: "face"   and u :: "nat"   and v :: "nat"   and n :: "nat"   and ovs :: "nat option list" assumes "pre_subdivFace' g f u v n ovs"   and "minGraphProps g"   and "f \<in> \<F> g" shows "|finals (subdivFace' g f v n ovs)| = Suc |finals g| \<and> |nonFinals g| - Suc 0 \<le> |nonFinals (subdivFace' g f v n ovs)|"
assms_2: shows "\<guillemotleft>g : c \<rightarrow> d\<guillemotright>"
funpow_swap1: fixes f :: "'g \<Rightarrow> 'g"   and n :: "nat"   and x :: "'g" shows "f ((f ^^ n) x) = (f ^^ n) (f x)"
indecomposable_imp_Ex_less_sets: fixes \<alpha> :: "V"   and A :: "V set"   and x :: "V"   and A1 :: "V set" assumes "indecomposable \<alpha>"   and "1 < \<alpha>"   and "tp A = \<alpha>"   and "small A"   and "A \<subseteq> ON"   and "x \<in> A"   and "tp A1 = \<alpha>"   and "A1 \<subseteq> A" obtains A2 :: "V set" where "tp A2 = \<alpha>" and "A2 \<subseteq> A1" and "{x} \<lless> A2"
of_nat_eq_1_iff: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and n :: "nat" assumes "class.semiring_char_0 one times plus zero" shows "(semiring_1.of_nat one plus zero n = one) = (n = 1)"
isCont_Polygamma: fixes z :: "'a"   and f :: "'a \<Rightarrow> 'b"   and n :: "nat" assumes "isCont f z"   and "f z \<notin> \<int>\<^sub>\<le>\<^sub>0" shows "isCont (\<lambda>x. Polygamma n (f x)) z"
mojmir_axioms: shows "mojmir \<Sigma> \<up>af\<^sub>G (Abs \<phi>) w {q. \<G> \<Turnstile>\<^sub>P Rep q}"
csubspace_raw_def: shows "csubspace \<equiv> module.subspace (*\<^sub>C)"
span_eq_iff: fixes s :: "'b set" shows "(vs1.span s = s) = vs1.subspace s"
in_image_Neg_is_neg: fixes L :: "'a literal"   and \<sigma> :: "'s"   and AA :: "'a set" assumes "L \<cdot>l \<sigma> \<in> Neg ` AA" shows "is_neg L"
cases: fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and B :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and C :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and R :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and a1 :: "('a, 'c, 'e, 'g) converter"   and a2 :: "('b, 'd, 'f, 'h) converter" assumes "rel_converter A B C R a1 a2" obtains conv1 :: "('a, 'c, 'e, 'g) converter" and conv2 :: "('b, 'd, 'f, 'h) converter" where "a1 = conv1" and "a2 = conv2" and "rel_fun A (rel_gpv'' (rel_prod B (rel_converter A B C R)) C R) (run_converter conv1) (run_converter conv2)"
power2_eq_square: fixes one :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and a :: "'d" assumes "class.monoid_mult one times" shows "power.power one times a 2 = times a a"
star_one: shows "mone\<^sup>\<odot> = mone"
the_unique_part_alternative_def: fixes A :: "'c set"   and P :: "'c set set"   and x :: "'c" assumes "partition_on A P"   and "x \<in> A" shows "(THE X. x \<in> X \<and> X \<in> P) = {y. \<exists>X\<in>P. x \<in> X \<and> y \<in> X}"
d_sub'_c: shows "Cl \<in> FL.Red_F_\<G> {Dl} \<or> Dl \<sqsubset> Cl"
Ioc_subset_iff: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" assumes "class.linorder less_eq less" shows "(ord.greaterThanAtMost less_eq less a b \<subseteq> ord.greaterThanAtMost less_eq less c d) = (less_eq b a \<or> less_eq c a \<and> less_eq b d)"
in_set_member: fixes x :: "'c"   and xs :: "'c list" shows "(x \<in> set xs) = List.member xs x"
UP_triangle_5: assumes "(g \<star> \<epsilon>) \<cdot> \<a>[g, f, g] \<cdot> (\<eta> \<star> g) = \<r>\<^sup>-\<^sup>1[g] \<cdot> \<l>[g]" shows "S.cmp\<^sub>U\<^sub>P (trg f, f) \<cdot>\<^sub>S (S.UP \<epsilon> \<cdot>\<^sub>S S.cmp\<^sub>U\<^sub>P (f, g) \<star>\<^sub>S S.UP f) \<cdot>\<^sub>S (S.UP f \<star>\<^sub>S S.inv (S.cmp\<^sub>U\<^sub>P (g, f)) \<cdot>\<^sub>S S.UP \<eta>) \<cdot>\<^sub>S S.inv (S.cmp\<^sub>U\<^sub>P (f, src f)) = (S.cmp\<^sub>U\<^sub>P (trg f, f) \<cdot>\<^sub>S (UP.unit (trg f) \<star>\<^sub>S S.UP f)) \<cdot>\<^sub>S (S.UP f \<star>\<^sub>S S.inv (UP.unit (src f))) \<cdot>\<^sub>S S.inv (S.cmp\<^sub>U\<^sub>P (f, src f))"
x_def: fixes xs :: "'a list"   and i :: "nat" shows "??.Newton_Interpolation.x xs i = xs ! i"
exhaust: fixes y :: "'c option" obtains "y = None"   | x2 :: "'c" where "y = Some x2"
wf_ll_Nodes_pret: fixes pret :: "dag"   and levellista :: "ref list list"   and var :: "ref \<Rightarrow> nat"   and nb :: "nat"   and x :: "ref" assumes "wf_ll pret levellista var"   and "nb < length levellista"   and "x \<in> Nodes nb levellista" shows "x \<in> set_of pret \<and> var x < nb"
intro: fixes \<eta> :: "'d \<Rightarrow> 'e"   and R :: "'d set"   and addition :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and multiplication :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and unit :: "'d"   and R' :: "'e set"   and addition' :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and multiplication' :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero' :: "'e"   and unit' :: "'e" assumes "ring_homomorphism \<eta> R addition multiplication zero unit R' addition' multiplication' zero' unit'"   and "bijective_map \<eta> R R'" shows "ring_isomorphism \<eta> R addition multiplication zero unit R' addition' multiplication' zero' unit'"
composite: assumes "\<not> prime n" shows "pmf (fermat_test n) True < 1 / 2"
lift_def: fixes Q :: "'b com" shows "lift Q \<equiv> \<lambda>(P, s). if P = None then (Some Q, s) else (Some (Seq (the P) Q), s)"
fcs_x_eq: fixes x :: "'b" assumes "x \<in> segment G.face_cycle_succ a' a"   and "G.face_cycle_succ x \<noteq> a" shows "H.face_cycle_succ x = G.face_cycle_succ x"
map_family_fun: fixes I :: "'c set"   and A :: "'c \<Rightarrow> ('d, 'e) aGroup_scheme"   and S :: "('f, 'g) aGroup_scheme"   and g :: "'c \<Rightarrow> 'f \<Rightarrow> 'd"   and x :: "'f" assumes "\<forall>j\<in>I. aGroup (A j)"   and "aGroup S"   and "\<forall>j\<in>I. g j \<in> aHom S (A j)"   and "x \<in> carrier S" shows "(\<lambda>y\<in>carrier S. \<lambda>j\<in>I. g j y) x \<in> carrier (a\<Pi>\<^bsub>I\<^esub> A)"
star_of_approx_zero_iff: fixes k :: "'a" shows "(star_of k \<approx> 0) = (k = (0::'a))"
Ax_canonical': fixes k :: "'b"   and \<Gamma> :: "'b formula multiset"   and \<Delta> :: "'b formula multiset"   and n :: "nat" shows "Atom k, \<Gamma> \<Rightarrow> Atom k, \<Delta> \<down> Suc n"
vector_derivative_def: fixes f :: "real \<Rightarrow> 'b"   and net :: "real filter" shows "vector_derivative f net = (SOME f'. (f has_vector_derivative f') net)"
len2: shows "length (takeWhile ((\<noteq>) n1) (rev (x # xs))) < length (rev xs)"
case_3: fixes f1 :: "'b"   and f2 :: "'c \<Rightarrow> 'c Predicate.pred \<Rightarrow> 'b"   and f3 :: "'c Predicate.pred \<Rightarrow> 'c Predicate.seq \<Rightarrow> 'b"   and x31 :: "'c Predicate.pred"   and x32 :: "'c Predicate.seq" shows "(case seq.Join x31 x32 of seq.Empty \<Rightarrow> f1 | seq.Insert x xa \<Rightarrow> f2 x xa | seq.Join x xa \<Rightarrow> f3 x xa) = f3 x31 x32"
strategyproofness_1: shows "(pmf (sds R38) a < pmf (sds R28) a \<or> pmf (sds R38) a + pmf (sds R38) c < pmf (sds R28) a + pmf (sds R28) b) \<or> pmf (sds R38) a = pmf (sds R28) a \<and> pmf (sds R38) a + pmf (sds R38) c = pmf (sds R28) a + pmf (sds R28) b"
ftv_tvars: fixes Tvs :: "W.tvar list" shows "supp Tvs = set Tvs"
distinct_24: fixes x4 :: "nat"   and x3 :: "nat" shows "Key x4 \<noteq> Nonce x3"
norm_pos2: fixes j :: "nat" assumes "j < m" shows "0 < \<parallel>gs2.gso j\<parallel>\<^sup>2"
angle: fixes a :: "'a"   and b :: "'a" assumes "(a, b) \<in> R\<^sub>d" shows "(b, a\<^sup>\<bullet>) \<in> R\<^sub>d"
le_iff_add: fixes plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and a :: "'e"   and b :: "'e" assumes "class.canonically_ordered_monoid_add plus zero less_eq less" shows "less_eq a b = (\<exists>c. b = plus a c)"
observable_move_equal_length_1: fixes S :: "'node SDG_node set"   and f :: "'edge \<Rightarrow> ('var, 'val, 'ret, 'pname) edge_kind"   and ms :: "'node list"   and s :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list"   and a :: "'edge"   and ms' :: "'node list"   and s' :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" assumes "S,f \<turnstile> (ms,s) -a\<rightarrow> (ms',s')" shows "length ms = length s"
bintrunc_bintrunc_l: fixes n :: "nat"   and m :: "nat"   and w :: "int" assumes "n \<le> m" shows "take_bit m (take_bit n w) = take_bit n w"
smallest_accepting_rank_properties_6: fixes \<Sigma> :: "'b set set"   and \<G> :: "'b ltl set"   and w :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltl"   and i :: "nat" assumes "ltl_FG_to_rabin \<Sigma> \<G> w"   and "mojmir_def.smallest_accepting_rank \<Sigma> \<up>af\<^sub>G (Abs \<phi>) w {q. \<G> \<Turnstile>\<^sub>P Rep q} = Some i" shows "i < semi_mojmir_def.max_rank \<Sigma> \<up>af\<^sub>G (Abs \<phi>)"
mult_right_isotone: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<le> y" shows "z \<sqinter> x \<le> z \<sqinter> y"
add_new_ts_vmsaux_def: shows "add_new_ts_vmsaux = (\<lambda>args nt (t, auxlist). (nt, filter (\<lambda>(t, rel). enat (nt - t) \<le> right (args_ivl args)) auxlist))"
permits_acc_inheritance: fixes G :: "prog"   and m :: "qtname \<times> memberdecl"   and statC :: "qtname"   and accC :: "qtname"   and dynC :: "qtname" assumes "G \<turnstile> m in statC permits_acc_from accC"   and "G\<turnstile>dynC\<preceq>\<^sub>C statC" shows "G \<turnstile> m in dynC permits_acc_from accC"
eint2_cases: fixes y :: "eint"   and ya :: "eint" obtains int :: "int" and inta :: "int" where "y = eint int__" and "ya = eint inta"   | int :: "int" where "y = eint intb__" and "ya = \<infinity>"   | int :: "int" where "y = \<infinity>" and "ya = eint int"   | "y = \<infinity>" and "ya = \<infinity>"
assms_4: shows "NTCod \<eta>1 = NTCod \<eta>2"
chain_mono: fixes A :: "'b set"   and P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and C :: "'b set" assumes "\<And>x y. \<lbrakk>x \<in> A; y \<in> A; P x y\<rbrakk> \<Longrightarrow> Q x y"   and "chain_on P C A" shows "chain_on Q C A"
i_shrink_eq_Msg_iEx_iAll_conv: fixes k :: "nat"   and m :: "'b message_af"   and s :: "nat \<Rightarrow> 'b message_af"   and t :: "nat" assumes "0 < k"   and "m \<noteq> NoMsg" shows "((s \<div> k) t = m) = (\<diamond> t1 [t * k\<dots>,k - Suc 0]. s t1 = m \<and> (\<box> t2 [Suc t1\<dots>]. t2 \<le> t * k + k - Suc 0 \<longrightarrow> s t2 = NoMsg))"
Image_Int_subset: fixes R :: "('f \<times> 'e) set"   and A :: "'f set"   and B :: "'f set" shows "R `` (A \<inter> B) \<subseteq> R `` A \<inter> R `` B"
h: shows "- V = \<Union> {{x} |x. x \<notin> V}"
lnth_0_conv_lhd: fixes xs :: "'c llist" assumes "\<not> lnull xs" shows "xs $ 0 = lhd xs"
assigns_trel: fixes \<sigma> :: "'b \<Rightarrow> 'b"   and \<rho> :: "'b \<Rightarrow> 'b" shows "(\<sigma>, \<langle>\<rho>\<rangle>\<^sub>a) \<rightarrow>\<^sub>u (\<rho> \<circ> \<sigma>, II)"
adds_term_antisym: fixes u :: "'t"   and v :: "'t" assumes "u adds\<^sub>t v"   and "v adds\<^sub>t u" shows "u = v"
dense: fixes x :: "'b"   and y :: "'b" assumes "x < y" shows "\<exists>z>x. z < y"
add_0: fixes a :: "'h" shows "(0::'h) + a = a"
\<pi>_no_tokens_item_end: fixes x :: "item"   and k :: "nat"   and I :: "item set" assumes "x \<in> \<pi> k {} I" shows "item_end x = k \<or> x \<in> I"
drop_upt: fixes m :: "nat"   and i :: "nat"   and j :: "nat" shows "drop m [i..<j] = [i + m..<j]"
state_rank_step: fixes q :: "'b"   and n :: "nat" shows "state_rank q (Suc n) = step (\<lambda>q. state_rank q n) (w n) q"
weak_reduction_bisimulations_impl_inverse_is_simulation: fixes Rel :: "('a \<times> 'a) set"   and Cal :: "'a processCalculus" assumes "weak_reduction_bisimulation Rel Cal" shows "weak_reduction_simulation (Rel\<inverse>) Cal"
abs_eq: fixes x :: "('c, 'd) phi \<Rightarrow> 'c \<times> 'd Filters.filter" assumes "eq_onp (\<lambda>pf. \<forall>f. pf f \<in> triple.pairs (Rep_phi f)) x x" shows "- Abs_lifted_pair x = Abs_lifted_pair (\<lambda>f. triple.pairs_uminus (Rep_phi f) (x f))"
uminus_uint16_def: shows "uminus \<equiv> map_fun Rep_uint16 Abs_uint16 uminus"
f0: shows "\<exists>l. v = l \<cdot>\<^sub>v w \<or> w = l \<cdot>\<^sub>v v"
lat_dist2: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<sqinter> y + z = (x + z) \<sqinter> (y + z)"
simps_1: shows "perm2 [] = (\<lambda>n. [])"
SDG_call_or_param_in_edge_unique_CFG_call_edge: fixes n :: "'node SDG_node"   and Vopt :: "'var option"   and p :: "'pname"   and n' :: "'node SDG_node" assumes "SDG_edge n Vopt (Some (p, True)) n'" shows "\<exists>!a. valid_edge a \<and> sourcenode a = parent_node n \<and> targetnode a = parent_node n' \<and> (\<exists>Q r fs. kind a = Q:r\<hookrightarrow>\<^bsub>p\<^esub>fs)"
seqd: fixes \<rho> :: "complex mat" assumes "\<rho> \<in> carrier_mat d d"   and "partial_density_operator \<rho>" shows "matrix_seq d (matrix_sum d (\<lambda>n. denote_while_n (M 0) (M 1) (denote S) n \<rho>))"
inject_1: fixes x1 :: "'m"   and y1 :: "'m" shows "(Inl x1 = Inl y1) = (x1 = y1)"
foldl_spmf_pair_left: fixes f :: "'r \<Rightarrow> 't \<Rightarrow> 'r spmf"   and l :: "'r"   and r :: "'s"   and es :: "'t list" shows "foldl_spmf (\<lambda>(l, r) e. map_spmf (\<lambda>l'. (l', r)) (f l e)) (return_spmf (l, r)) es = map_spmf (\<lambda>l'. (l', r)) (foldl_spmf f (return_spmf l) es)"
lass_val_of_spec: fixes e :: "'a exp"   and a :: "'a \<times> val" assumes "lass_val_of e = \<lfloor>a\<rfloor>" shows "e = fst a:=Val (snd a)"
extends_to_adjoint_equivalence: shows "\<exists>G \<eta> \<epsilon>. adjoint_equivalence (\<cdot>\<^sub>C) (\<cdot>\<^sub>D) G F \<eta> \<epsilon>"
ProblemIkPol_axioms: shows "ProblemIkPol wtFsym wtPsym arOf resOf parOf \<Phi> infTp"
has_integral_subset_le: fixes s :: "'a set"   and t :: "'a set"   and f :: "'a \<Rightarrow> real"   and i :: "real"   and j :: "real" assumes "s \<subseteq> t"   and "(f has_integral i) s"   and "(f has_integral j) t"   and "\<forall>x\<in>t. 0 \<le> f x" shows "i \<le> j"
red_no_deadlock: fixes s :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and t :: "'t"   and ta :: "('l, 't, 'x, 'm, 'w, 'o) thread_action"   and s' :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set" assumes "s -t\<triangleright>ta\<rightarrow> s'"   and "t \<in> deadlocked s" shows "False"
ord_eq_less_trans: fixes a :: "'c list"   and b :: "'c list"   and c :: "'c list" assumes "a = b"   and "strict_suffix b c" shows "strict_suffix a c"
t_dcast: fixes CT :: "nat \<Rightarrow> classDef option"   and \<Gamma> :: "nat \<Rightarrow> nat option"   and e0 :: "exp"   and D :: "nat"   and C :: "nat" assumes "CT;\<Gamma> \<turnstile> e0 : D"   and "CT \<turnstile> C <: D"   and "C \<noteq> D" shows "CT;\<Gamma> \<turnstile> Cast C e0 : C"
power_suc_below_plus: fixes x :: "'a"   and n :: "nat" shows "power.power 1' (;) x (Suc n) \<le> x\<^sup>+"
nonzero: shows "0 < measure_pmf.prob (p \<bind> f) A"
subst_apply_term_empty: fixes t :: "('a, 'b) Term.term" shows "t \<cdot> term.Var = t"
mult_dvd_mono: fixes times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and a :: "'c"   and b :: "'c"   and c :: "'c"   and d :: "'c" assumes "class.comm_monoid_mult times one"   and "dvd.dvd times a b"   and "dvd.dvd times c d" shows "dvd.dvd times (times a c) (times b d)"
subseteq_mset_def: fixes A :: "'c multiset"   and B :: "'c multiset" shows "(A \<subseteq># B) = (\<forall>a. count A a \<le> count B a)"