hcju/isabelleps huggingface dataset:
tdghm_dghm_comp_is_tm_tdghm: fixes \<alpha> :: "V"   and \<BB> :: "V"   and \<CC> :: "V"   and \<FF> :: "V"   and \<GG> :: "V"   and \<NN> :: "V"   and \<AA> :: "V"   and \<HH> :: "V" assumes "\<NN> : \<FF> \<mapsto>\<^sub>D\<^sub>G\<^sub>H\<^sub>M\<^sub>.\<^sub>t\<^sub>m \<GG> : \<BB> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>t\<^sub>m\<^bsub>\<alpha>\<^esub> \<CC>"   and "\<HH> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>t\<^sub>m\<^bsub>\<alpha>\<^esub> \<BB>" shows "\<NN> \<circ>\<^sub>T\<^sub>D\<^sub>G\<^sub>H\<^sub>M\<^sub>-\<^sub>D\<^sub>G\<^sub>H\<^sub>M \<HH> : \<FF> \<circ>\<^sub>D\<^sub>G\<^sub>H\<^sub>M \<HH> \<mapsto>\<^sub>D\<^sub>G\<^sub>H\<^sub>M\<^sub>.\<^sub>t\<^sub>m \<GG> \<circ>\<^sub>D\<^sub>G\<^sub>H\<^sub>M \<HH> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>t\<^sub>m\<^bsub>\<alpha>\<^esub> \<CC>"
simps_3: fixes x :: "'c"   and xs :: "'c list"   and y :: "'c"   and ys :: "'c list" shows "list_diff (x # xs) (y # ys) = (if x = y then list_diff xs ys else None)"
simps_23: fixes x41 :: "('a, 'b, 'c) Stmt"   and x42 :: "('a, 'b, 'c) Stmt" shows "Skip \<noteq> x41 ;; x42"
gfp_unfold: fixes f :: "'b::complete_lattice \<Rightarrow> 'b::complete_lattice" assumes "mono (f::'b::complete_lattice \<Rightarrow> 'b::complete_lattice)" shows "gfp (f::'b::complete_lattice \<Rightarrow> 'b::complete_lattice) = f (gfp f)"
inv_closed: fixes x :: "'a" assumes "x \<in> carrier \<G>" shows "inv x \<in> carrier \<G>"
has_vector_derivative_withinD: fixes f :: "real \<Rightarrow> 'b"   and f' :: "'b"   and x0 :: "real"   and S :: "real set" assumes "(f has_vector_derivative f') (at x0 within S)" shows "((\<lambda>x. (f x - f x0) /\<^sub>R (x - x0)) \<longlongrightarrow> f') (at x0 within S)"
wrap_def: shows "wrap \<equiv> \<mu> wrap. \<Lambda> v. Fixrec.run\<cdot>(match_B\<cdot>v\<cdot>(\<Lambda> a. Fixrec.succeed\<cdot>a))"
induct: fixes x :: "event list"   and P :: "event list \<Rightarrow> bool" assumes "x \<in> ns_shared"   and "P []"   and "\<And>evsf X B. \<lbrakk>evsf \<in> ns_shared; P evsf; X \<in> synth (analz (knows Spy evsf))\<rbrakk> \<Longrightarrow> P (Says Spy B X # evsf)"   and "\<And>evs1 NA A B. \<lbrakk>evs1 \<in> ns_shared; P evs1; Nonce NA \<notin> used evs1\<rbrakk> \<Longrightarrow> P (Says A Server \<lbrace>Agent A, Agent B, Nonce NA\<rbrace> # evs1)"   and "\<And>evs2 KAB A' A B NA. \<lbrakk>evs2 \<in> ns_shared; P evs2; Key KAB \<notin> used evs2; KAB \<in> symKeys; Says A' Server \<lbrace>Agent A, Agent B, Nonce NA\<rbrace> \<in> set evs2\<rbrakk> \<Longrightarrow> P (Says Server A (Crypt (shrK A) \<lbrace>Nonce NA, Agent B, Key KAB, Crypt (shrK B) \<lbrace>Key KAB, Agent A\<rbrace>\<rbrace>) # evs2)"   and "\<And>evs3 A S NA B K X. \<lbrakk>evs3 \<in> ns_shared; P evs3; A \<noteq> Server; Says S A (Crypt (shrK A) \<lbrace>Nonce NA, Agent B, Key K, X\<rbrace>) \<in> set evs3; Says A Server \<lbrace>Agent A, Agent B, Nonce NA\<rbrace> \<in> set evs3\<rbrakk> \<Longrightarrow> P (Says A B X # evs3)"   and "\<And>evs4 NB K A' B A. \<lbrakk>evs4 \<in> ns_shared; P evs4; Nonce NB \<notin> used evs4; K \<in> symKeys; Says A' B (Crypt (shrK B) \<lbrace>Key K, Agent A\<rbrace>) \<in> set evs4\<rbrakk> \<Longrightarrow> P (Says B A (Crypt K (Nonce NB)) # evs4)"   and "\<And>evs5 K B' A NB S NA B X. \<lbrakk>evs5 \<in> ns_shared; P evs5; K \<in> symKeys; Says B' A (Crypt K (Nonce NB)) \<in> set evs5; Says S A (Crypt (shrK A) \<lbrace>Nonce NA, Agent B, Key K, X\<rbrace>) \<in> set evs5\<rbrakk> \<Longrightarrow> P (Says A B (Crypt K \<lbrace>Nonce NB, Nonce NB\<rbrace>) # evs5)"   and "\<And>evso B A K NB NA X. \<lbrakk>evso \<in> ns_shared; P evso; Says B A (Crypt K (Nonce NB)) \<in> set evso; Says Server A (Crypt (shrK A) \<lbrace>Nonce NA, Agent B, Key K, X\<rbrace>) \<in> set evso\<rbrakk> \<Longrightarrow> P (Notes Spy \<lbrace>Nonce NA, Nonce NB, Key K\<rbrace> # evso)" shows "P x"
andfold_MatchExp_not_disc_negated_mapMatch: fixes disc :: "'b \<Rightarrow> bool"   and C :: "'c \<Rightarrow> 'b"   and ls :: "'c list" shows "\<not> has_disc_negated disc False (andfold_MatchExp (map (Match \<circ> C) ls))"
nn_integral_stream_space: fixes f :: "('s \<Rightarrow> 's) stream \<Rightarrow> ennreal" assumes "D.random_variable borel f" shows "integral\<^sup>N (stream_space (Pi\<^sub>M UNIV (\<lambda>x. measure_pmf (K x)))) f = \<integral>\<^sup>+ x. \<integral>\<^sup>+ X. f (x ## X) \<partial>stream_space (Pi\<^sub>M UNIV (\<lambda>x. measure_pmf (K x))) \<partial>Pi\<^sub>M UNIV (\<lambda>x. measure_pmf (K x))"
af_G_letter_sat_core: fixes \<G> :: "'a ltl set"   and \<phi> :: "'a ltl"   and \<nu> :: "'a set" assumes "Only_G \<G>"   and "\<G> \<Turnstile>\<^sub>P \<phi>" shows "\<G> \<Turnstile>\<^sub>P af_G_letter \<phi> \<nu>"
in_arcs_imp_in_arcs_ends: fixes e :: "'b" assumes "e \<in> arcs G" shows "tail G e \<rightarrow>\<^bsub>G\<^esub> head G e"
induct: fixes x1 :: "state"   and x2 :: "com"   and x3 :: "state"   and P :: "state \<Rightarrow> com \<Rightarrow> state \<Rightarrow> bool" assumes "x1 -x2\<rightarrow> x3"   and "\<And>t f s. t \<in> f s \<Longrightarrow> P s (Do f) t"   and "\<And>s0 c1 s1 c2 s2. \<lbrakk>s0 -c1\<rightarrow> s1; P s0 c1 s1; s1 -c2\<rightarrow> s2; P s1 c2 s2\<rbrakk> \<Longrightarrow> P s0 (c1; c2) s2"   and "\<And>b s c1 t c2. \<lbrakk>b s; s -c1\<rightarrow> t; P s c1 t\<rbrakk> \<Longrightarrow> P s (IF b THEN c1 ELSE c2) t"   and "\<And>b s c2 t c1. \<lbrakk>\<not> b s; s -c2\<rightarrow> t; P s c2 t\<rbrakk> \<Longrightarrow> P s (IF b THEN c1 ELSE c2) t"   and "\<And>b s c. \<not> b s \<Longrightarrow> P s (WHILE b DO c) s"   and "\<And>b s c t u. \<lbrakk>b s; s -c\<rightarrow> t; P s c t; t -WHILE b DO c\<rightarrow> u; P t (WHILE b DO c) u\<rbrakk> \<Longrightarrow> P s (WHILE b DO c) u"   and "\<And>f s c t g. \<lbrakk>f s -c\<rightarrow> t; P (f s) c t\<rbrakk> \<Longrightarrow> P s (LOCAL f; c; g) (g s t)" shows "P x1 x2 x3"
rep_eq: fixes x :: "'c"   and xa :: "'c set['d]" shows "bmember x xa = (x \<in> set_bset xa)"
f: fixes x :: "real" assumes "x \<in> S" shows "0 \<le> f x \<and> f x \<le> 1"
sel_1: fixes x21 :: "'c"   and x22 :: "'c llist" shows "lhd (LCons x21 x22) = x21"
case_two_tau_not_quite_total_1: fixes i :: "nat"   and t :: "nat" assumes "\<forall>T. gap i T \<le> gap i t" shows "\<tau> i (gap i t) \<up>"
perm_boolE: fixes pi :: "('a \<times> 'a) list"   and P :: "bool" assumes "pi \<bullet> P" shows "P"
strong_postdominate_path_targetnode: fixes n' :: "'node"   and n :: "'node"   and as :: "'edge list"   and n'' :: "'node" assumes "n' strongly-postdominates n"   and "n -as\<rightarrow>* n''"   and "n' \<notin> set (sourcenodes as)" shows "n' strongly-postdominates n''"
bind_simps_1: fixes f :: "'d \<Rightarrow> 'c list" shows "List.bind [] f = []"
induct: fixes x :: "nat list \<Rightarrow> nat"   and P :: "(nat list \<Rightarrow> nat) \<Rightarrow> bool" assumes "PRIMREC x"   and "P SC"   and "\<And>k. P (CONSTANT k)"   and "\<And>i. P (PROJ i)"   and "\<And>g fs. \<lbrakk>PRIMREC g; P g; \<forall>f\<in>set fs. PRIMREC f \<and> P f\<rbrakk> \<Longrightarrow> P (COMP g fs)"   and "\<And>f g. \<lbrakk>PRIMREC f; P f; PRIMREC g; P g\<rbrakk> \<Longrightarrow> P (PREC f g)" shows "P x"
ref_union_closed_def: fixes P :: "'e process" shows "ref_union_closed P \<equiv> \<forall>xs A. (\<exists>X. X \<in> A) \<longrightarrow> (\<forall>X\<in>A. (xs, X) \<in> failures P) \<longrightarrow> (xs, \<Union>X\<in>A. X) \<in> failures P"
l: shows "lo < lo' - 1 \<or> lo' - 1 \<le> lo"
fds_converges_Re_le: fixes f :: "'a fds"   and s0 :: "'a"   and s :: "'a" assumes "fds_converges f s0"   and "s0 \<bullet> (1::'a) < s \<bullet> (1::'a)" shows "fds_converges f s"
inverse_arrowsI: fixes g :: "'a"   and f :: "'a" assumes "A.ide (A g f)"   and "A.ide (A f g)" shows "A.inverse_arrows f g"
hom_component_zero_iff: fixes p :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c"   and n :: "nat" shows "(hom_component p n = 0) = (\<forall>t\<in>keys p. deg_pm t \<noteq> n)"
val_Zp_0_imp_unit: fixes x :: "nat \<Rightarrow> int" assumes "val_Zp x = 0"   and "x \<in> carrier Zp" shows "x \<in> Units Zp"
llength_lappend: fixes xs :: "'b llist"   and ys :: "'b llist" shows "llength (xs $ ys) = llength xs + llength ys"
pno_equiv: shows "uninst_code.phi_equiv_mapping (\<lambda>_. s1_phis) g s1_phi_nodes_of (uninst_code.ssa.phiNodes_of (\<lambda>g. s1_phis) g)"
collinear_aff_dim: fixes S :: "'b set" shows "collinear S = (aff_dim S \<le> 1)"
complement_vsubset: fixes I :: "V"   and J :: "V" shows "I -\<^sub>\<circ> J \<subseteq>\<^sub>\<circ> I"
charts_submanifold_Int_carrier: shows "charts_submanifold (S \<inter> carrier) = charts_submanifold S"
single_numeral: fixes n :: "num" shows "monomial (numeral n) (0::'b) = numeral n"
SDom_def: fixes \<sigma> :: "('f, 'g, 'h) subst" shows "SDom \<sigma> = {Inl x |x. x \<in> dom (SFunctions \<sigma>)} \<union> {Inr (Inl x) |x. x \<in> dom (SContexts \<sigma>)} \<union> {Inr (Inr x) |x. x \<in> dom (SPredicates \<sigma>)} \<union> {Inr (Inr x) |x. x \<in> dom (SPrograms \<sigma>)}"
v: fixes x :: "'a" shows "(\<lambda>n. cblinfun_apply (X n) x) \<longlonglongrightarrow> v x"
rabin_def: shows "rabin \<equiv> \<lambda>(I, F) w. infs I w \<and> fins F w"
outside: fixes x :: "'a" assumes "x \<notin> S" shows "p' x = x"
vcpower_index_vempty: fixes A :: "V" shows "A ^\<^sub>\<times> []\<^sub>\<circ> = ZFC_in_HOL.set {[]\<^sub>\<circ>}"
simps_1: shows "\<lbrakk>\<lbrakk> [] \<rbrakk>\<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L = {uu_. True}"
measure_measure_spmf_def: fixes p :: "'h spmf" shows "measure_measure_spmf p = Sigma_Algebra.measure (measure_spmf p)"
simps_1: fixes x :: "'c"   and d :: "'b" shows "assoc x d [] = d"
labeled_list_insert_eq_cases_1: fixes d :: "'a"   and D :: "('b \<times> 'a) list"   and i :: "'b" assumes "d \<notin> set (unlabel D)" shows "List.insert d (unlabel D) = unlabel (List.insert (i, d) D)"
gval_fold_rev_invalid: fixes G :: "'a gexp list"   and s :: "'a \<Rightarrow> value option" assumes "gval (fold gAnd (rev G) (Bc True)) s = invalid" shows "gval (fold gAnd G (Bc True)) s = invalid"
homeomorphic_map_openness_eq: fixes X :: "'c topology"   and Y :: "'d topology"   and f :: "'c \<Rightarrow> 'd"   and U :: "'c set" assumes "homeomorphic_map X Y f" shows "openin X U = (U \<subseteq> topspace X \<and> openin Y (f ` U))"
hom0: shows "mat\<^sub>h (0\<^sub>m (dim_row A) 0) = 0\<^sub>m (dim_row A) 0"
Suc_3: shows "v0_v1_Digraph G v0 v1"
orderI: fixes a :: "'b::semilattice_inf"   and b :: "'b::semilattice_inf" assumes "(a::'b::semilattice_inf) = a \<sqinter> (b::'b::semilattice_inf)" shows "(a::'b::semilattice_inf) \<le> (b::'b::semilattice_inf)"
iterate_add_less_iff2: fixes i :: "nat"   and j :: "nat"   and u :: "'a"   and m :: "'a"   and n :: "'a" assumes "i \<le> j" shows "(iterate_add i u + m < iterate_add j u + n) = (m < iterate_add (j - i) u + n)"
betaupright_upright_vut: fixes va :: "'c zmultiset"   and vb :: "'c zmultiset"   and t :: "'c" assumes "beta_upright va vb"   and "upright vb"   and "vacant_upto (va + vb) t" shows "vacant_upto va t"
cases: fixes freshenLC :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and renameLCs :: "('a \<Rightarrow> 'a) \<Rightarrow> 'b \<Rightarrow> 'b"   and lconsts :: "'b \<Rightarrow> 'a set"   and closed :: "'b \<Rightarrow> bool"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and subst_lconsts :: "'c \<Rightarrow> 'a set"   and subst_renameLCs :: "('a \<Rightarrow> 'a) \<Rightarrow> 'c \<Rightarrow> 'c"   and anyP :: "'b"   and antecedent :: "'d \<Rightarrow> ('b, 'a) antecedent list"   and consequent :: "'d \<Rightarrow> 'b list"   and rules :: "'d stream"   and assumptions :: "'b list"   and conclusions :: "'b list"   and a1 :: "('b, 'd) graph_node"   and a2 :: "nat"   and a3 :: "'c"   and a4 :: "'b fset \<times> 'b" assumes "Abstract_Task freshenLC renameLCs lconsts closed subst subst_lconsts subst_renameLCs anyP antecedent consequent rules assumptions conclusions"   and "Abstract_Task.local_fresh_check freshenLC lconsts subst_lconsts anyP antecedent a1 a2 a3 a4" obtains \<Gamma> :: "'b fset" and i :: "nat" and n :: "('b, 'd) graph_node" and s :: "'c" and c :: "'b" where "a1 = n" and "a2 = i" and "a3 = s" and "a4 = (\<Gamma> \<turnstile> c)" and "\<And>f. f |\<in>| \<Gamma> \<Longrightarrow> freshenLC i ` Abstract_Task.all_local_vars anyP antecedent n \<inter> lconsts f = {}" and "freshenLC i ` Abstract_Task.all_local_vars anyP antecedent n \<inter> subst_lconsts s = {}"
Hash_Aenc: fixes X :: "msg"   and X' :: "msg"   and Y' :: "msg" shows "Hash X \<noteq> Aenc X' Y'"
P_def: shows "P = wP_of_wstate (llast Sts)"
llast_linfinite: fixes xs :: "'b llist" assumes "\<not> lfinite xs" shows "llast xs = undefined"
fml_sound: fixes \<phi> :: "formula"   and \<nu> :: "fin_string + fin_string \<Rightarrow> 32 Word.word"   and b :: "bool"   and \<nu>' :: "fin_string \<Rightarrow> real" assumes "[[\<phi>]]\<nu> \<down> b"   and "\<nu> REP \<nu>'" shows "[\<phi>]\<nu>' \<downharpoonright> b"
that: shows "\<forall>i\<in>A. \<forall>j\<in>B. i < j"
qe_inv: shows "\<forall>bs p. qfree p \<longrightarrow> qfree (qe p) \<and> Ifm bs (qe p) = Ifm bs (E p)"
ctermsl_cterms: fixes q :: "('a, 'b, 'c, 'd) seqp"   and p :: "('a, 'b, 'c, 'd) seqp"   and \<Gamma> :: "'c \<Rightarrow> ('a, 'b, 'c, 'd) seqp" assumes "q \<in> ctermsl p"   and "not_call q"   and "sterms \<Gamma> p \<subseteq> cterms \<Gamma>"   and "wellformed \<Gamma>" shows "q \<in> cterms \<Gamma>"
that: fixes zs :: "'a list" assumes "xs = ys @ zs"   and "\<And>z. z \<in> set zs \<Longrightarrow> P z"   and "no_trailing P ys" shows "thesis"
coreflexive_comp_inf_comp: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<le> top"   and "y \<le> top" shows "x \<sqinter> z \<sqinter> (y \<sqinter> z) = x \<sqinter> y \<sqinter> z"
supp_finite_set_at_base: fixes S :: "'a set" assumes "finite S" shows "supp S = atom ` S"
opp_Point: fixes x :: "'b"   and y :: "'b" shows "opp (Point x y) = Point x (- y)"
OrdP_ORD_OF: fixes H :: "fm set"   and n :: "nat" shows "H \<turnstile> OrdP (ORD_OF n)"
intro_elim_2_a: fixes v :: "i"   and \<phi> :: "\<o>"   and \<psi> :: "\<o>" assumes "[\<phi> \<^bold>& \<psi> in v]" shows "[\<phi> in v]"
assms: shows "g \<in> M \<rightarrow>\<^sub>M M"
strict_implies_order: fixes a :: "'b list"   and b :: "'b list" assumes "strict_prefix a b" shows "??.Sublist.prefix a b"
InitQ: fixes cfg1 :: "('p, 'v, 's) configuration"   and Q :: "'p set" shows "qReachable cfg1 Q cfg1"
perm_below_to_right: fixes permute :: "perm \<Rightarrow> 'a \<Rightarrow> 'a"   and below :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and \<pi> :: "perm"   and x :: "'a"   and y :: "'a" assumes "class.cont_pt permute below" shows "(\<pi> \<bullet> x \<sqsubseteq> y) = (x \<sqsubseteq> - \<pi> \<bullet> y)"
term_variants_pred_subst': fixes P :: "'a \<Rightarrow> 'a list"   and t :: "('a, 'b) Term.term"   and \<delta> :: "'b \<Rightarrow> ('a, 'b) Term.term"   and s :: "('a, 'b) Term.term" assumes "term_variants_pred P (t \<cdot> \<delta>) s"   and "\<forall>x\<in>fv t \<union> fv s. (\<exists>y. \<delta> x = Var y) \<or> (\<exists>f. \<delta> x = Fun f [] \<and> P f = [])" shows "\<exists>u. term_variants_pred P t u \<and> s = u \<cdot> \<delta>"
elims_3: fixes x :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xa :: "'b list" assumes "\<not> sorted_wrt x xa" obtains P :: "'b \<Rightarrow> 'b \<Rightarrow> bool" and x :: "'b" and ys :: "'b list" where "xa__ = P" and "xa = x # ys" and "\<not> (Ball (set ys) (P x) \<and> sorted_wrt P ys)"
leD: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and y :: "'c"   and x :: "'c" assumes "class.order less_eq less"   and "less_eq y x" shows "\<not> less x y"
compact_cbox: fixes a :: "'b"   and b :: "'b" shows "compact (cbox a b)"
trans_resolve1: fixes n1 :: "nat"   and x :: "nat"   and y :: "nat"   and n2 :: "nat" assumes "0 < n1"   and "x + n1 \<le> y"   and "y + n2 \<le> x" shows "False"
normalize_content: fixes p :: "'b poly" shows "normalize (content p) = content p"
apex_def: shows "src.apex \<equiv> C.dom src.leg0"
powser_uniformly_convergent: fixes r :: "real"   and a :: "nat \<Rightarrow> 'b"   and \<xi> :: "'b" assumes "ereal r < conv_radius a" shows "uniformly_convergent_on (cball \<xi> r) (\<lambda>n x. \<Sum>i<n. a i * (x - \<xi>) ^ i)"
pa_step_correct_2: fixes H1 :: "('a, 'b) hashedTa"   and H2 :: "('c, 'b) hashedTa"   and Q :: "('a \<times> 'c, unit) HashMap.hashmap"   and W :: "('a \<times> 'c) list"   and \<delta>d :: "('a \<times> 'c, 'b) ta_rule dlist" assumes "hashedTa H1"   and "hashedTa H2"   and "hta_has_idx_s H1"   and "hta_has_idx_sf H2"   and "(Q, W, \<delta>d) \<in> pa_invar H1 H2"   and "pa_cond (Q, W, \<delta>d)" shows "(pa_\<alpha> (Q, W, \<delta>d), pa_\<alpha> (pa_step H1 H2 (Q, W, \<delta>d))) \<in> frp_step (ls.\<alpha> (hta_\<delta> H1)) (ls.\<alpha> (hta_\<delta> H2))"
wary_AppE\<^sub>h: fixes s :: "('s, 'v) tm"   and t :: "('s, 'v) tm" assumes "wary (App s t)" obtains "wary s" and "wary t" and "of_nat (num_args s) < arity_hd\<^sub>h (head s)"
ResSet_def: fixes m :: "int"   and X :: "int set" shows "ResSet m X = (\<forall>y1 y2. y1 \<in> X \<and> y2 \<in> X \<and> [y1 = y2] (mod m) \<longrightarrow> y1 = y2)"
ran_is_image: shows "ran (idx xs) = (the \<circ>\<circ> idx) xs ` dom (idx xs)"
hcomp_ide_src: fixes f :: "('a E.term, 'a) arr" assumes "ide f" shows "f \<star> src f = f"
pfun_minus_unit: fixes f :: "('a, 'b) pfun" shows "f - {}\<^sub>p = f"
p_def: shows "p = real (Suc n) / (1 - d)"
NSDERIV_Id: fixes x :: "'a" shows "NSDERIV (\<lambda>x. x) x :> (1::'a)"
abs_int_poly_dvd: fixes p :: "int poly"   and q :: "int poly" shows "(abs_int_poly p dvd q) = (p dvd q)"
IH_2: fixes ds :: "bool list" shows "treesize (cutoff red ds T2) \<le> treesize T2"
trad_foundation: fixes z :: "V" assumes "z \<noteq> 0" shows "\<exists>w. w \<in>\<^sub>\<circ> z \<and> w \<inter>\<^sub>\<circ> z = 0"
WTrtFAcc: 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 h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and E :: "char list \<Rightarrow> ty option"   and e :: "char list exp"   and C :: "char list"   and F :: "char list"   and T :: "ty"   and D :: "char list" assumes "P,E,h \<turnstile> e : Class C"   and "P \<turnstile> C has F:T in D" shows "P,E,h \<turnstile> e\<bullet>F{D} : T"
1091: fixes x :: "'a"   and y :: "'a" shows "- (x \<squnion> y) \<sqinter> x = bot"
open_in_topspace_empty: fixes X :: "'c topology"   and S :: "'c set" assumes "topspace X = {}" shows "openin X S = (S = {})"
env_single: fixes a :: "'c el_step" shows "env [a] = (case a of LOC e \<Rightarrow> [] | ENV e \<Rightarrow> [a])"
partial_density_operator_def: fixes A :: "complex mat" shows "partial_density_operator A = (positive A \<and> trace A \<le> 1)"
option_bind_alt: fixes x :: "'e option"   and f :: "'e \<Rightarrow> 'd option" shows "x \<bind> f = (case x of None \<Rightarrow> None | Some v \<Rightarrow> f v)"
l1_inv3E: fixes x :: "l1_state" assumes "x \<in> l1_inv3" obtains "\<And>A B Rb Ra gny. \<lbrakk>guessed_runs Rb = \<lparr>role = Resp, owner = B, partner = A\<rparr>; in_progressS (progress x Rb) {xny, xgnx, xgny, xsk}; guessed_frame Rb xgny = Some gny; guessed_frame Rb xgnx = Some (Exp Gen (NonceF (Ra $ nx))); guessed_runs Ra = \<lparr>role = Init, owner = A, partner = B\<rparr>; progress x Ra = Some {xnx, xgnx}; can_signal x A B\<rbrakk> \<Longrightarrow> signalsInit x (Commit A B (Exp gny (NonceF (Ra $ nx)))) < signalsInit x (Running A B (Exp gny (NonceF (Ra $ nx))))"
progress_simps_14: fixes \<sigma> :: "(String.literal \<times> event_data list) trace"   and P :: "String.literal \<Rightarrow> nat option"   and \<phi> :: "Formula.formula"   and I :: "\<I>"   and \<psi> :: "Formula.formula"   and j :: "nat" shows "Monitor.progress \<sigma> P (formula.Since \<phi> I \<psi>) j = min (Monitor.progress \<sigma> P \<phi> j) (Monitor.progress \<sigma> P \<psi> j)"
wf_lex: fixes r :: "('a \<times> 'a) set" assumes "wf r" shows "wf (lex r)"
emeasure_lborel_box_finite: fixes a :: "'a"   and b :: "'a" shows "emeasure lborel (box a b) < \<infinity>"
F: fixes a :: "'a"   and b :: "'a"   and c :: "'a"   and z :: "'a" shows "a * b * c l\<rightarrow> z = a * (b * c) l\<rightarrow> z"
Quotient3_ltln\<^sub>Q: shows "Quotient3 (\<sim>\<^sub>Q) abs_ltln\<^sub>Q rep_ltln\<^sub>Q"
llist_all2_flip: fixes P :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and xs :: "'a llist"   and ys :: "'b llist" shows "llist_all2 (flip P) xs ys = llist_all2 P ys xs"
sets_restrict_space_iff: fixes \<Omega> :: "'a set"   and M :: "'a measure"   and A :: "'a set" assumes "\<Omega> \<inter> space M \<in> sets M" shows "(A \<in> sets (restrict_space M \<Omega>)) = (A \<subseteq> \<Omega> \<and> A \<in> sets M)"
cont: fixes x :: "'a"   and d :: "'b \<Rightarrow>\<^sub>L 'a" assumes "(x, d) \<in> X0" shows "continuous (at x within S) (return_time p2)"
f1''_less_f1': fixes f1 :: "real"   and f2 :: "real"   and f1' :: "real"   and f2' :: "real"   and e :: "real"   and c :: "real"   and l0 :: "nat"   and f1'' :: "real"   and f2'' :: "real" assumes "Table2 f1 f2 f1' f2' e c l0 f1'' f2''" shows "f1'' < f1'"
hyps: fixes u :: "nat" shows "wellformed_items (\<J> k u) \<and> wellformed_items (\<I> k)"
on_triple_dest: fixes R :: "('g \<times> 'f) set"   and E :: "('e \<times> 'g \<times> 'g) set"   and G :: "('e \<times> 'f \<times> 'f) set"   and l :: "'e"   and x :: "'g"   and y :: "'g"   and xx :: "'f"   and yy :: "'f" assumes "on_triple R `` E \<subseteq> G"   and "(l, x, y) \<in> E"   and "(x, xx) \<in> R"   and "(y, yy) \<in> R" shows "(l, xx, yy) \<in> G"
elementary_monoidal_category_def: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and tensor :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unity :: "'b"   and lunit :: "'b \<Rightarrow> 'b"   and runit :: "'b \<Rightarrow> 'b"   and assoc :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b" shows "elementary_monoidal_category C tensor unity lunit runit assoc \<equiv> category C \<and> elementary_monoidal_category_axioms C tensor unity lunit runit assoc"
infinite: fixes P :: "'a \<Rightarrow> bool" assumes "eventually P F" shows "infinite {x. P x}"
servK_authentic: fixes authK :: "nat"   and servK :: "nat"   and B :: "agent"   and Ts :: "nat"   and servTicket :: "msg"   and evs :: "event list" assumes "Crypt authK \<lbrace>Key servK, Agent B, Number Ts, servTicket\<rbrace> \<in> parts (knows Spy evs)"   and "Key authK \<notin> analz (knows Spy evs)"   and "authK \<notin> range shrK"   and "evs \<in> kerbIV_gets" shows "\<exists>A. Says Tgs A (Crypt authK \<lbrace>Key servK, Agent B, Number Ts, servTicket\<rbrace>) \<in> set evs"
rel_spmf_mono_strong: fixes A :: "'b \<Rightarrow> 'd \<Rightarrow> bool"   and f :: "'b spmf"   and g :: "'d spmf"   and B :: "'b \<Rightarrow> 'd \<Rightarrow> bool" assumes "rel_spmf A f g"   and "\<And>x y. \<lbrakk>A x y; x \<in> set_spmf f; y \<in> set_spmf g\<rbrakk> \<Longrightarrow> B x y" shows "rel_spmf B f g"
elims_1: fixes x :: "('b, 'c) Combinators list"   and y :: "bool" assumes "wellformed_policy1_strong x = y" obtains "x = []" and "\<not> y"   | xa :: "('b, 'c) Combinators" and xs :: "('b, 'c) Combinators list" where "x = xa # xs" and "y = (xa = DenyAll \<and> DenyAll \<notin> set xs)"
absolutely_integrable_imp_integrable: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "f absolutely_integrable_on S"   and "S \<in> sets lebesgue" shows "integrable (lebesgue_on S) f"
full_subcategory_def: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and Ide :: "'b \<Rightarrow> bool" shows "full_subcategory C Ide \<equiv> category C \<and> full_subcategory_axioms C Ide"
zero_smallest: fixes j :: "'a" assumes "j \<in> Field r" shows "(r.zero, j) \<in> r"
encode_all_operator_preconditions_def: fixes \<Pi> :: "'b strips_problem"   and ops :: "'b strips_operator list"   and t :: "nat" shows "encode_all_operator_preconditions \<Pi> ops t \<equiv> let l = List.product [0..<t] ops in foldr (\<^bold>\<and>) (map (\<lambda>(t, op). encode_operator_precondition \<Pi> t op) l) (\<^bold>\<not> \<bottom>)"
zero_less_oSuc: fixes x :: "ordinal" shows "0 < oSuc x"