hcju/isabelleps huggingface dataset:
dbl_inc_def: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and one :: "'a"   and x :: "'a" assumes "class.neg_numeral minus plus zero uminus" shows "neg_numeral.dbl_inc plus one x = plus (plus x x) one"
compose: fixes f :: "'b \<Rightarrow> 'c"   and F :: "'b filter"   and g :: "'b \<Rightarrow> 'c"   and h' :: "'d \<Rightarrow> 'b"   and G :: "'d filter" assumes "f \<in> \<Theta>[F](g)"   and "filterlim h' F G" shows "(\<lambda>x. f (h' x)) \<in> \<Theta>[G](\<lambda>x. g (h' x))"
assms: shows "intersect_segment_xline' prec (p0, p1) x = Some (m, M)"
stable_def: shows "stable \<equiv> \<lambda>f g. \<forall>x y. x \<in> f \<longrightarrow> (x, y) \<in> g \<longrightarrow> y \<in> f"
removeProcs_subset: fixes ps :: "'a pState_scheme list"   and dead :: "bool"   and sd :: "bool"   and ps' :: "'a pState_scheme list"   and dcs :: "integer list" assumes "removeProcs ps = (dead, sd, ps', dcs)" shows "set ps' \<subseteq> set ps"
"3_1": assumes "x = y" shows "set (merge_wrt rel xs ys) = set xs \<union> set ys"
guarantees_Int_right: fixes Z :: "'b program set"   and X :: "'b program set"   and Y :: "'b program set" shows "Z guarantees X \<inter> Y = (Z guarantees X) \<inter> (Z guarantees Y)"
INF_empty: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a"   and f :: "'b \<Rightarrow> 'a" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "Inf (f ` {}) = top"
simps_3: fixes m :: "nat"   and b :: "'b"   and ts :: "'a FOL_Fitting.term list" shows "closed m (Pred b ts) = closedts m ts"
cfs_a_inv: fixes p :: "nat \<Rightarrow> 'a"   and n :: "nat" assumes "p \<in> carrier P" shows "(\<ominus>\<^bsub>P\<^esub> p) n = \<ominus> p n"
tp_correct': fixes ly :: "nat list"   and ap :: "abc_inst list"   and tp :: "(action \<times> nat) list"   and lm :: "nat list"   and l :: "cell list"   and r :: "cell list"   and ires :: "cell list"   and stp :: "nat"   and am :: "nat list" assumes "ly = layout_of ap"   and "tp = tm_of ap"   and "crsp ly (0, lm) (Suc 0, l, r) ires"   and "abc_steps_l (0, lm) ap stp = (length ap, am)" shows "\<exists>stp k. steps0 (Suc 0, l, r) tp stp = (start_of ly (length ap), Bk # Bk # ires, <am> @ Bk \<up> k)"
norm_EM_remainder_le_strong_nat': fixes Y :: "real set"   and n :: "nat"   and D :: "real"   and a :: "real"   and f :: "real \<Rightarrow> 'b"   and g :: "real \<Rightarrow> real"   and g' :: "real \<Rightarrow> real" assumes "finite Y"   and "\<forall>x. \<bar>pbernpoly n x\<bar> \<le> D"   and "continuous_on {a..} f"   and "continuous_on {a..} g"   and "continuous_on {a..} g'"   and "(g \<longlongrightarrow> 0) at_top"   and "\<And>x. a \<le> x \<Longrightarrow> norm (f x) \<le> g' x"   and "\<And>x. x \<in> {a..} - Y \<Longrightarrow> (g has_real_derivative - g' x) (at x)" shows "\<forall>x. a \<le> real x \<longrightarrow> norm (EM_remainder n f (int x)) \<le> D / fact n * g (real x)"
right_inverse_matrix_dim: fixes A :: "'b matrix"   and X :: "'b matrix" assumes "right_inverse_matrix A X" shows "nrows A = ncols X"
object_ptr_kinds_M_eq2_h: shows "|h \<turnstile> object_ptr_kinds_M|\<^sub>r = |h2 \<turnstile> object_ptr_kinds_M|\<^sub>r"
extend_indets_zero_iff: fixes p :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd" shows "(extend_indets p = 0) = (p = 0)"
homotopy_eqv_contractibility: fixes S :: "'b set"   and T :: "'c set" assumes "S homotopy_eqv T" shows "contractible S = contractible T"
ext_inject: fixes round :: "'g \<Rightarrow> nat"   and state :: "'g \<Rightarrow> 'f"   and sent :: "'g \<Rightarrow> 'g set"   and rcvd :: "'g \<Rightarrow> 'g \<Rightarrow> 'h option"   and network :: "('g \<times> nat \<times> 'g \<times> 'h) set"   and more :: "'i"   and round' :: "'g \<Rightarrow> nat"   and state' :: "'g \<Rightarrow> 'f"   and sent' :: "'g \<Rightarrow> 'g set"   and rcvd' :: "'g \<Rightarrow> 'g \<Rightarrow> 'h option"   and network' :: "('g \<times> nat \<times> 'g \<times> 'h) set"   and more' :: "'i" shows "(\<lparr>round = round, state = state, sent = sent, rcvd = rcvd, network = network, \<dots> = more\<rparr> = \<lparr>round = round', state = state', sent = sent', rcvd = rcvd', network = network', \<dots> = more'\<rparr>) = (round = round' \<and> state = state' \<and> sent = sent' \<and> rcvd = rcvd' \<and> network = network' \<and> more = more')"
defNode_eq: fixes \<alpha>e :: "'a \<Rightarrow> ('b \<times> 'c \<times> 'b) set"   and \<alpha>n :: "'a \<Rightarrow> 'b list"   and invar :: "'a \<Rightarrow> bool"   and inEdges' :: "'a \<Rightarrow> 'b \<Rightarrow> ('b \<times> 'c) list"   and Entry :: "'a \<Rightarrow> 'b"   and defs :: "'a \<Rightarrow> 'b \<Rightarrow> 'd set"   and "uses" :: "'a \<Rightarrow> 'b \<Rightarrow> 'd set"   and phis :: "'a \<Rightarrow> 'b \<times> 'd \<Rightarrow> 'd list option"   and n :: "'b"   and g :: "'a"   and v :: "'d" assumes "CFG_SSA_wf \<alpha>e \<alpha>n invar inEdges' Entry defs uses phis"   and "n \<in> set (\<alpha>n g)"   and "v \<in> CFG_SSA_base.allDefs defs phis g n" shows "CFG_SSA_wf_base.defNode \<alpha>n defs phis g v = n"
Input: fixes a :: "name"   and x :: "name"   and P :: "pi"   and u :: "name" shows "a<x>.P \<longmapsto>\<^sub>e a<u> \<prec>\<^sub>e P[x::=u]"
fps_add_comm: fixes f :: "('c, 'd) fps"   and g :: "('c, 'd) fps" shows "f + g = g + f"
cases: fixes \<Gamma> :: "('a, 'b) web_scheme" assumes "loose \<Gamma>" obtains "\<And>f. \<lbrakk>current \<Gamma> f; wave \<Gamma> f\<rbrakk> \<Longrightarrow> f = (\<lambda>_. 0)" and "\<not> hindrance \<Gamma> (\<lambda>_. 0)"
nth_default_replicate_eq: fixes dflt :: "'a"   and n :: "nat"   and x :: "'a"   and i :: "nat" shows "nth_default dflt (replicate n x) i = (if i < n then x else dflt)"
inj_on_diff: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and B :: "'a set" assumes "inj_on f A" shows "inj_on f (A - B)"
B: shows "B \<subseteq> A \<and> finite B \<and> card B = card C + 1"
poly_rat_def: fixes x :: "rat" shows "poly_rat x = (case quotient_of x of (n, d) \<Rightarrow> [:- n, d:])"
lossless_def: fixes \<A> :: "'grp \<Rightarrow> 'grp \<Rightarrow> 'grp set spmf" shows "local.lcdh.lossless \<A> = (\<forall>\<alpha> \<beta>. lossless_spmf (\<A> \<alpha> \<beta>))"
Chains_wo: fixes R :: "('a \<times> 'a) set set"   and r :: "('a \<times> 'a) set" assumes "R \<in> Chains I"   and "r \<in> R" shows "Well_order r \<and> downset_on (Field r) p \<and> extension_on (Field r) r p"
extension_lemmaE: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'c"   and U :: "'a set" assumes "closedin (top_of_set carrier) A"   and "diff_fun_on A f"   and "A \<subseteq> U"   and "U \<subseteq> carrier"   and "open U" obtains f' :: "'a \<Rightarrow> 'c" where "diff_fun k charts f'" and "\<And>x. x \<in> A \<Longrightarrow> f' x = f x" and "csupport_on carrier f' \<inter> carrier \<subseteq> U"
c2: fixes i :: "nat" shows "e \<le> infdist (flow0 x ((s \<circ> r) i)) (\<omega>_limit_set x)"
propa: shows "propa f (step (hd b) (a ! hd b)) a (tl b) = (ss, w)"
bi_total_alt_def: fixes A :: "'i \<Rightarrow> 'j \<Rightarrow> bool" shows "bi_total A = (left_total A \<and> right_total A)"
cosh_plus_sinh: fixes x :: "'a" shows "cosh x + sinh x = exp x"
closeSubstI: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi"   and Rel :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set" assumes "\<And>\<sigma>. wellFormedSubst \<sigma> \<Longrightarrow> (\<Psi>, P[<\<sigma>>], Q[<\<sigma>>]) \<in> Rel" shows "(\<Psi>, P, Q) \<in> closeSubst Rel"
simps_14: fixes f1 :: "'a"   and f2 :: "'a"   and f3 :: "'a"   and f4 :: "'a" shows "(case Package of Private \<Rightarrow> f1 | Package \<Rightarrow> f2 | Protected \<Rightarrow> f3 | Public \<Rightarrow> f4) = f2"
none_MT_rulessubset: fixes a :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators list"   and b :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators list" assumes "none_MT_rules Cp a"   and "set b \<subseteq> set a" shows "none_MT_rules Cp b"
len_nonFinals_makeFaceFinal: fixes f :: "face"   and g :: "graph" assumes "\<not> final f"   and "f \<in> \<F> g" shows "|nonFinals (makeFaceFinal f g)| = |nonFinals g| - 1"
F_basis: fixes a :: "'a"   and b :: "'b" assumes "a \<in> basisA'"   and "b \<in> basisB" shows "F a b \<in> basis"
is_ff_dghmI: fixes \<alpha> :: "V"   and \<AA> :: "V"   and \<BB> :: "V"   and \<FF> :: "V" assumes "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>f\<^sub>a\<^sub>i\<^sub>t\<^sub>h\<^sub>f\<^sub>u\<^sub>l\<^bsub>\<alpha>\<^esub> \<BB>"   and "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>f\<^sub>u\<^sub>l\<^sub>l\<^bsub>\<alpha>\<^esub> \<BB>" shows "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>f\<^sub>f\<^bsub>\<alpha>\<^esub> \<BB>"
closure_minimal: fixes S :: "'c set"   and T :: "'c set" assumes "S \<subseteq> T"   and "closed T" shows "closure S \<subseteq> T"
sym: fixes A :: "wall"   and B :: "wall" assumes "A ~f B" shows "B ~f A"
simps_2: fixes \<tau> :: "ty"   and \<sigma> :: "ty" shows "RED (\<tau> \<rightarrow> \<sigma>) = _Collect t (\<forall>u\<in>RED \<tau>. App t u \<in> RED \<sigma>)"
update_preserves_le: fixes ev\<^sub>A :: "('a \<Rightarrow> 'b) \<Rightarrow> 'c \<Rightarrow> 'b"   and ev\<^sub>B :: "('a \<Rightarrow> 'b) \<Rightarrow> 'd \<Rightarrow> bool"   and aexp_vars :: "'c \<Rightarrow> 'a set"   and bexp_vars :: "'d \<Rightarrow> 'a set"   and dma :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> Sec"   and \<C>_vars :: "'a \<Rightarrow> 'a set"   and mds\<^sub>1 :: "Mode \<Rightarrow> 'a set"   and mds\<^sub>2 :: "Mode \<Rightarrow> 'a set"   and annos :: "'a ModeUpd list" assumes "sifum_modes ev\<^sub>A ev\<^sub>B aexp_vars bexp_vars dma \<C>_vars"   and "mds\<^sub>1 \<le> mds\<^sub>2" shows "sifum_modes.update_annos mds\<^sub>1 annos \<le> sifum_modes.update_annos mds\<^sub>2 annos"
assms_5: shows "h \<turnstile> get_elements_by_class_name ptr' name \<rightarrow>\<^sub>r results"
mbisim_wset_thread_ok1: fixes s1 :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x1 \<times> 'l \<Rightarrow>f nat) option) \<times> 'm1) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and s2 :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x2 \<times> 'l \<Rightarrow>f nat) option) \<times> 'm2) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set" assumes "s1 \<approx>m s2" shows "wset_thread_ok (wset s1) (thr s1)"
assms_3: assumes "a \<in>\<^sub>\<circ> x" shows "R"
Example2_lemma2_Suc0: fixes j :: "nat"   and n :: "nat"   and b :: "nat \<Rightarrow> nat" assumes "j < n"   and "b j = 0" shows "Suc (sum b {0..<n}) = sum (b(j := Suc 0)) {0..<n}"
Cons_1: assumes "set Us \<subseteq> carrier R"   and "\<And>Ks. \<lbrakk>set Ks \<subseteq> K; combine Ks Us = \<zero>\<rbrakk> \<Longrightarrow> set (take (length Us) Ks) \<subseteq> {\<zero>}" shows "local.independent K Us"
many_strongly_connected_iff_13: fixes x :: "'a" assumes "is_inj x" shows "many_strongly_connected x = (x\<^sup>T ; x ; x\<^sup>\<star> \<le> x\<^sup>+)"
finite_imp_ANR: fixes S :: "'a set" assumes "finite S" shows "ANR S"
cases: fixes a :: "('fenv_ubx, 'a, ('fun, 'b) frame) state" assumes "Subx.final a" obtains F :: "'fenv_ubx" and f :: "'fun" and fd :: "('dyn, 'var, 'fun, 'op, 'opinl, 'opubx, 'ubx1, 'ubx2) Ubx.instr fundef" and pc :: "nat" and H :: "'a" and \<Sigma> :: "'b list" where "a = State F H [Frame f pc \<Sigma>]" and "Fubx_get F f = Some fd" and "pc = length (body fd)"
index_minus_mat_1: fixes i :: "nat"   and B :: "'c mat"   and j :: "nat"   and A :: "'c mat" assumes "i < dim_row B"   and "j < dim_col B" shows "(A - B) $$ (i, j) = A $$ (i, j) - B $$ (i, j)"
right_uniqueD: fixes A :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and x :: "'d"   and y :: "'e"   and z :: "'e" assumes "right_unique A"   and "A x y"   and "A x z" shows "y = z"
case_2: fixes f1 :: "'c"   and f2 :: "'d \<Rightarrow> 'd list \<Rightarrow> 'c"   and x21 :: "'d"   and x22 :: "'d list" shows "(case x21 # x22 of [] \<Rightarrow> f1 | x # xa \<Rightarrow> f2 x xa) = f2 x21 x22"
that: fixes X' :: "(real \<times> real pdevs) \<times> real" assumes "X = Some X'" shows "thesis"
integral_add: fixes M :: "'d measure"   and f :: "'d \<Rightarrow> 'e"   and g :: "'d \<Rightarrow> 'e" assumes "integrable M f"   and "integrable M g" shows "LINT x|M. f x + g x = integral\<^sup>L M f + integral\<^sup>L M g"
in_arcs_app_iso_cases: fixes a :: "'e"   and hom :: "('g, 'h, 'f, 'e) digraph_isomorphism"   and G :: "('g, 'h) pre_digraph" assumes "a \<in> arcs (app_iso hom G)" obtains a0 :: "'h" where "a = iso_arcs hom a0" and "a0 \<in> arcs G"
Y_seq'_def: fixes n :: "nat"   and \<omega> :: "real" shows "Y_seq' n \<omega> = (if \<omega> \<in> {\<omega> \<in> {0<..<1}. \<not> isCont M.I \<omega>} then 0 else \<mu>.I n \<omega>)"
homeomorphic_spheres: fixes d :: "real"   and e :: "real"   and a :: "'c"   and b :: "'c" assumes "0 < d"   and "0 < e" shows "sphere a d homeomorphic sphere b e"
CHECK_simps_2: fixes e :: "'c" shows "CHECK False e = THROW e"
elims_1: fixes x :: "'b match_expr"   and y :: "bool" assumes "has_primitive x = y" obtains "x = MatchAny" and "\<not> y"   | a :: "'b" where "x = Match a" and "y"   | m :: "'b match_expr" where "x = MatchNot m" and "y = has_primitive m"   | m1 :: "'b match_expr" and m2 :: "'b match_expr" where "x = MatchAnd m1 m2" and "y = (has_primitive m1 \<or> has_primitive m2)"
cod_eqI: fixes b :: "'a"   and f :: "'a" assumes "C.ide b"   and "C.seq b f" shows "C.cod f = b"
upt_last_append: fixes a :: "nat"   and b :: "nat" assumes "a \<le> b" shows "[a..<b] @ [b] = [a..<Suc b]"
sub_tl: shows "sub2_wit ((snd (mus_adjuster f n_gs [] (0\<^sub>v n)) + f) # gs') fs = (tl mus, tl gs)"
lincomb_list_def: fixes c :: "nat \<Rightarrow> rat"   and vs :: "rat vec list" shows "gs.lincomb_list c vs = gs.M.sumlist (map (\<lambda>i. c i \<cdot>\<^sub>v vs ! i) [0..<length vs])"
IH_1: fixes \<Gamma> :: "'a formula multiset"   and \<Delta> :: "'a formula multiset" shows "(F1, F1, \<Gamma> \<Rightarrow> \<Delta> \<longrightarrow> F1, \<Gamma> \<Rightarrow> \<Delta>) \<and> (\<Gamma> \<Rightarrow> F1, F1, \<Delta> \<longrightarrow> \<Gamma> \<Rightarrow> F1, \<Delta>)"
is_full_pmdl_alt: fixes B :: "('t \<Rightarrow>\<^sub>0 'c) set" shows "is_full_pmdl B = (pmdl B = full_pmdl (component_of_term ` Keys B))"
induct: fixes x1 :: "(any_unop + boolean_unop + numeric_unop + string_unop + collection_unop) + (super_binop + boolean_binop + numeric_binop + string_binop + collection_binop) + (string_ternop + collection_ternop) + String.literal"   and x2 :: "call_kind"   and x3 :: "'a OCL_Types.type"   and x4 :: "'a OCL_Types.type list"   and x5 :: "'a OCL_Types.type"   and P :: "(any_unop + boolean_unop + numeric_unop + string_unop + collection_unop) + (super_binop + boolean_binop + numeric_binop + string_binop + collection_binop) + (string_ternop + collection_ternop) + String.literal \<Rightarrow> call_kind \<Rightarrow> 'a OCL_Types.type \<Rightarrow> 'a OCL_Types.type list \<Rightarrow> 'a OCL_Types.type \<Rightarrow> bool" assumes "op_type x1 x2 x3 x4 x5"   and "\<And>op k \<tau> \<upsilon>. unop_type op k \<tau> \<upsilon> \<Longrightarrow> P (Inl op) k \<tau> [] \<upsilon>"   and "\<And>op k \<tau> \<sigma> \<upsilon>. binop_type op k \<tau> \<sigma> \<upsilon> \<Longrightarrow> P (Inr (Inl op)) k \<tau> [\<sigma>] \<upsilon>"   and "\<And>op k \<tau> \<sigma> \<rho> \<upsilon>. ternop_type op k \<tau> \<sigma> \<rho> \<upsilon> \<Longrightarrow> P (Inr (Inr (Inl op))) k \<tau> [\<sigma>, \<rho>] \<upsilon>"   and "\<And>\<tau> op \<pi> oper. operation \<tau> op \<pi> oper \<Longrightarrow> P (Inr (Inr (Inr op))) DotCall \<tau> \<pi> (oper_type oper)" shows "P x1 x2 x3 x4 x5"
Crypt_valid_pubSK: fixes C :: "agent"   and SKi :: "nat"   and evs :: "event list" assumes "Crypt (priSK RCA) \<lbrace>Agent C, Key SKi, onlySig\<rbrace> \<in> parts (knows Spy evs)"   and "evs \<in> set_pur" shows "SKi = pubSK C"
simps: fixes x :: "int"   and xs :: "int list" shows "even_case x xs = collatz (x div 2) xs"
is_Groebner_basis_empty: shows "punit.is_Groebner_basis {}"
cases: fixes x :: "'a tm" obtains v :: "'a" and n :: "nat" where "x = TM v n"
get_document_put_shadow_root: fixes document_ptr :: "(_) document_ptr"   and shadow_root_ptr :: "(_) shadow_root_ptr"   and shadow_root :: "('o, 'p, 'q, 'r option) RShadowRoot_scheme"   and h :: "(_) heap" assumes "document_ptr \<noteq> cast shadow_root_ptr" shows "get document_ptr (put shadow_root_ptr shadow_root h) = get document_ptr h"
abs_conv_abscissa_add_le: shows "abs_conv_abscissa (f + - g) \<le> max (abs_conv_abscissa f) (abs_conv_abscissa (- g))"
not_le_imp_less: fixes y :: "'b"   and x :: "'b" assumes "\<not> y \<le> x" shows "x < y"
induct: fixes x1 :: "('e \<times> ('f, 'g) cconstraint \<times> 'h \<times> 'f list \<times> 'e) set \<times> ('e \<Rightarrow> ('f, 'g) cconstraint)"   and x2 :: "'e"   and x3 :: "nat \<Rightarrow> nat \<Rightarrow> 'g DBMEntry"   and x4 :: "'f \<Rightarrow> nat"   and x5 :: "nat"   and x6 :: "'e"   and x7 :: "nat \<Rightarrow> nat \<Rightarrow> 'g DBMEntry"   and P :: "('e \<times> ('f, 'g) cconstraint \<times> 'h \<times> 'f list \<times> 'e) set \<times> ('e \<Rightarrow> ('f, 'g) cconstraint) \<Rightarrow> 'e \<Rightarrow> (nat \<Rightarrow> nat \<Rightarrow> 'g DBMEntry) \<Rightarrow> ('f \<Rightarrow> nat) \<Rightarrow> nat \<Rightarrow> 'e \<Rightarrow> (nat \<Rightarrow> nat \<Rightarrow> 'g DBMEntry) \<Rightarrow> bool" assumes "x1 \<turnstile> \<langle>x2, x3\<rangle> \<leadsto>*\<^bsub>x4,x5\<^esub> \<langle>x6, x7\<rangle>"   and "\<And>A l D v n. P A l D v n l D"   and "\<And>A l D v n l' D' l'' D''. \<lbrakk>A \<turnstile> \<langle>l, D\<rangle> \<leadsto>\<^bsub>v,n\<^esub> \<langle>l', D'\<rangle>; A \<turnstile> \<langle>l', D'\<rangle> \<leadsto>*\<^bsub>v,n\<^esub> \<langle>l'', D''\<rangle>; P A l' D' v n l'' D''\<rbrakk> \<Longrightarrow> P A l D v n l'' D''" shows "P x1 x2 x3 x4 x5 x6 x7"
mset_heap_merge: fixes ts\<^sub>1 :: "'b tree list"   and ts\<^sub>2 :: "'b tree list" shows "mset_heap (merge ts\<^sub>1 ts\<^sub>2) = mset_heap ts\<^sub>1 + mset_heap ts\<^sub>2"
num_eq_iff: fixes x :: "num"   and y :: "num" shows "(x = y) = (nat_of_num x = nat_of_num y)"
weak_equality_l_def: fixes p :: "(char list list \<times> int) list"   and q :: "(char list list \<times> int) list" shows "weak_equality_l p q = RETURN (p = q)"
simps_1: fixes n :: "nat"   and m :: "nat"   and uu :: "'b" shows "cut_bound n m (Atom uu) = m + n"
col_def: fixes m :: "'c list list"   and i :: "nat" shows "col m i \<equiv> m ! i"
simps_2: shows "ternary_to_bool TernaryFalse = Some False"
fst_snd_snd_echelon_form_of_column_k_det_le_nrows: fixes i :: "nat"   and A :: "(('c, 'd) vec, 'e) vec"   and bezout :: "'c \<Rightarrow> 'c \<Rightarrow> 'c \<times> 'c \<times> 'c \<times> 'c \<times> 'c"   and n :: "'c"   and k :: "nat" assumes "i \<le> nrows A" shows "snd (snd (echelon_form_of_column_k_det bezout (n, A, i) k)) \<le> nrows A"
brI: fixes a :: "'e"   and \<alpha> :: "'f \<Rightarrow> 'e"   and c :: "'f"   and I :: "'f \<Rightarrow> bool" assumes "a = \<alpha> c"   and "I c" shows "(c, a) \<in> br \<alpha> I"
ereal_of_enat_mult: fixes m :: "enat"   and n :: "enat" shows "ereal_of_enat (m * n) = ereal_of_enat m * ereal_of_enat n"
k_in: shows "k \<in> {0..<length (trace_parallel_plan_strips (\<Pi>\<^sub>I) \<pi>) - 1}"
covering_space_locally_eq: fixes C :: "'c set"   and p :: "'c \<Rightarrow> 'd"   and S :: "'d set"   and \<phi> :: "'c set \<Rightarrow> bool"   and \<psi> :: "'d set \<Rightarrow> bool" assumes "covering_space C p S"   and "\<And>T. \<lbrakk>T \<subseteq> C; \<phi> T\<rbrakk> \<Longrightarrow> \<psi> (p ` T)"   and "\<And>q U. \<lbrakk>U \<subseteq> S; continuous_on U q; \<psi> U\<rbrakk> \<Longrightarrow> \<phi> (q ` U)" shows "locally \<psi> S = locally \<phi> C"
tape_of_nl_append_one: fixes lm :: "nat list"   and a :: "nat" assumes "lm \<noteq> []" shows "<lm @ [a]> = <lm> @ Bk # Oc \<up> Suc a"
embed_measure_eq_distr': fixes f :: "'a \<Rightarrow> 'b"   and M :: "'a measure" assumes "inj_on f (space M)" shows "embed_measure M f = distr M (embed_measure M f) f"
generatorE: fixes x :: "'a" assumes "x \<in> carrier \<G>" obtains n :: "nat" where "x = \<^bold>g [^] n"
dy_fake_msg_subset_synth_analz: fixes bad :: "agent set"   and IK :: "msg set"   and chan :: "chan set"   and T :: "msg set" assumes "extr bad IK chan \<subseteq> T" shows "dy_fake_msg bad IK chan \<subseteq> synth (analz T)"
inputChainEqvt: fixes p :: "(name \<times> name) list"   and xvec :: "name list"   and N :: "'a"   and P :: "('a, 'b, 'c) psi" shows "p \<bullet> inputChain xvec N P = inputChain (p \<bullet> xvec) (p \<bullet> N) (p \<bullet> P)"
ideal_zero_forall: shows "\<forall>I. ideal R I \<longrightarrow> \<zero> \<in> I"
left_ideal_generated_def: fixes S :: "'a set" shows "left_ideal_generated S = \<Inter> {I. left_ideal I \<and> S \<subseteq> I}"
nth_swap_id: fixes i :: "nat"   and xs :: "'b list" assumes "Suc i < length xs" shows "swap i xs ! i = xs ! (i + 1)"
HOnextConfig_def: fixes A :: "('a, 'b, 'c, 'd) CHOAlgorithm_scheme"   and r :: "nat"   and cfg :: "'a \<Rightarrow> 'b"   and HO :: "'a \<Rightarrow> 'a set"   and cfg' :: "'a \<Rightarrow> 'b" shows "HOnextConfig A r cfg HO cfg' \<equiv> SHOnextConfig A r cfg HO HO cfg'"
sets_pair_countable: fixes S1 :: "'c set"   and S2 :: "'d set"   and M :: "'c measure"   and N :: "'d measure" assumes "countable S1"   and "countable S2"   and "sets M = Pow S1"   and "sets N = Pow S2" shows "sets (M \<Otimes>\<^sub>M N) = Pow (S1 \<times> S2)"
i_Exec_Comp_Stream_Acc_Output_def: fixes k :: "nat"   and output_fun :: "'e \<Rightarrow> 'd message_af"   and trans_fun :: "'f message_af \<Rightarrow> 'e \<Rightarrow> 'e"   and input :: "nat \<Rightarrow> 'f message_af"   and c :: "'e" shows "i_Exec_Comp_Stream_Acc_Output k output_fun trans_fun input c \<equiv> (output_fun \<circ> i_Exec_Comp_Stream trans_fun (input \<odot> k) c) \<div> k"
downclosed_downset: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b" assumes "class.preorder less_eq less" shows "ord.downclosed_set less_eq (ord.downset less_eq x)"
simps_33: fixes x31 :: "('a, 'b, 'c) Stmt"   and x32 :: "'a ModeUpd"   and x41 :: "('a, 'b, 'c) Stmt"   and x42 :: "('a, 'b, 'c) Stmt" shows "x31@[x32] \<noteq> x41 ;; x42"
is_failed_def: fixes status :: "status" shows "is_failed status = (status = FAILED)"
assms: shows "(Fun f S, Fun g T) \<in> timpl_closure' TI"
convex_real_interval_8: fixes a :: "real"   and b :: "real" shows "convex {a<..<b}"
Refinement_using_invariants: fixes R :: "('e \<times> 'f) set"   and I :: "'e set"   and J :: "'f set"   and Sa :: "('e, 'g) spec"   and Sc :: "('f, 'h) spec"   and pi :: "'h \<Rightarrow> 'g" assumes "{R \<inter> I \<times> J} TS.trans Sa, TS.trans Sc {> R}"   and "{I \<inter> Domain (R \<inter> UNIV \<times> J)} TS.trans Sa {> I}"   and "{J \<inter> Range (R \<inter> I \<times> UNIV)} TS.trans Sc {> J}"   and "init Sc \<subseteq> R `` init Sa"   and "init Sa \<subseteq> I"   and "init Sc \<subseteq> J"   and "obs_consistent (R \<inter> I \<times> J) pi Sa Sc" shows "refines (R \<inter> I \<times> J) pi Sa Sc"
sgn_div_eq_sgn_mult: fixes a :: "int"   and b :: "int" assumes "a div b \<noteq> 0" shows "sgn (a div b) = sgn (a * b)"
erH_multTr: fixes A :: "('g, 'h) Ring_scheme"   and B :: "('g, 'i) Ring_scheme"   and Y :: "'g"   and h :: "'a \<Rightarrow> 'g"   and c :: "nat \<times> (nat \<Rightarrow> 'a)"   and m :: "nat" assumes "PolynRg A B Y"   and "h \<in> rHom S B"   and "pol_coeff S c" shows "\<forall>f g. pol_coeff S (m, f) \<and> pol_coeff S (fst c + m, g) \<and> polyn_expr R X (fst c) c \<cdot>\<^sub>r polyn_expr R X m (m, f) = polyn_expr R X (fst c + m) (fst c + m, g) \<longrightarrow> polyn_expr A Y (fst c) (cf_h h c) \<cdot>\<^sub>r\<^bsub>A\<^esub> polyn_expr A Y m (cf_h h (m, f)) = polyn_expr A Y (fst c + m) (cf_h h (fst c + m, g))"