hcju/isabelleps huggingface dataset:
callee: fixes s :: "'s"   and x :: "'call" assumes "I s"   and "x \<in> outs_\<I> \<I>" shows "plossless_gpv \<I>' (callee s x)"
comp_null_2: fixes f :: "('a \<Rightarrow> 'b, 'a \<Rightarrow> 'b) concrete_category.arr" shows "f \<cdot>\<^sub>[\<^sub>A\<^sub>,\<^sub>B\<^sub>] A_B.null = A_B.null"
one_le_numeral: fixes n :: "num" shows "(1::'a) \<le> numeral n"
of_nat_eq_0_iff: fixes one :: "'e"   and times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and m :: "nat" assumes "class.semiring_char_0 one times plus zero" shows "(semiring_1.of_nat one plus zero m = zero) = (m = 0)"
circuit_dep: fixes X :: "'a set" assumes "\<E>.circuit X" shows "\<not> indep_in \<E> X"
ring_ideal_imp_quot_ideal: fixes I :: "'a set"   and J :: "'a set" assumes "ideal I R"   and "ideal J R" shows "ideal ((+>) I ` J) (R Quot I)"
ntrs_c_no_mon_s: fixes s :: "'n list"   and c :: "'n list multiset"   and w :: "('p, 'ba) label list"   and s' :: "'n list"   and c' :: "'n list multiset" assumes "((s, c), w, s', c') \<in> ntrs fg" shows "mon_c fg c' = mon_c fg c"
simps_2: fixes c :: "color"   and uu :: "('b, 'c) RBT_Impl.rbt"   and uv :: "'b"   and uw :: "'c"   and ux :: "('b, 'c) RBT_Impl.rbt" shows "color_of (Branch c uu uv uw ux) = c"
VVp_to_V: fixes v :: "'a"   and p :: "Player" assumes "v \<in> G'.VV p" shows "v \<in> V\<^bsub>subgame V'\<^esub>"
card_carrier_vec: fixes n :: "nat" shows "card (carrier_vec n) = CARD('c) ^ n"
axioms_1: fixes \<alpha> :: "V"   and \<AA> :: "V"   and \<BB> :: "V"   and \<FF> :: "V" assumes "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<BB>" shows "\<Z> \<alpha>"
collapse: fixes complex_interval :: "complex_interval" shows "Complex_Interval (Re_interval complex_interval) (Im_interval complex_interval) = complex_interval"
Suc: shows "\<exists>M. \<forall>z. M \<le> norm z \<longrightarrow> norm (\<Sum>i\<le>n. c i * z ^ i) \<le> e * norm z ^ Suc n"
ereal_Limsup_uminus: fixes net :: "'a filter"   and f :: "'a \<Rightarrow> ereal" shows "Limsup net (\<lambda>x. - f x) = - Liminf net f"
simps: fixes a :: "'a stream"   and A :: "'a set" shows "(a \<in> streams A) = (\<exists>aa s. a = aa ## s \<and> aa \<in> A \<and> s \<in> streams A)"
Quotient_ltl_prop_equiv_quotient: shows "Quotient (\<equiv>\<^sub>P) Abs Rep cr_ltl_prop_equiv_quotient"
map_simp: fixes f2 :: "'c" assumes "C.arr f2" shows "Ya.map f2 = Ya.Cop_S.MkArr (\<lambda>f1. Hom.map (f1, C.dom f2)) (\<lambda>f1. Hom.map (f1, C.cod f2)) (\<lambda>f1. Hom.map (f1, f2))"
Seqs_def: fixes ts :: "expr list"   and t :: "expr" shows "ts ?? t = fold (?) (rev ts) t"
first_time_image: fixes f :: "nat \<Rightarrow> nat"   and sub :: "'b run"   and r :: "'b run"   and c :: "clock"   and n :: "nat"   and t :: "'b tag_const" assumes "dilating f sub r" shows "first_time sub c n t = first_time r c (f n) t"
gb_schema_aux_Nil: fixes data :: "nat \<times> nat \<times> 'd"   and bs :: "(('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) list" shows "gb_schema_aux data bs [] = gs @ bs"
semiconnex: fixes A :: "'c set"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" assumes "semiconnex A less"   and "x \<in> A"   and "y \<in> A" shows "less x y \<or> x = y \<or> less y x"
\<rho>_in_terms_of_rep: shows "\<rho> = (T.T0.trnr\<^sub>\<epsilon> r \<rho> \<star>\<^sub>C f) \<cdot>\<^sub>C T.T0.trnr\<^sub>\<eta> g (g \<star>\<^sub>C T.T0.some_right_adjoint f)"
connected_imp_connected_closure: fixes S :: "'c set" assumes "connected S" shows "connected (closure S)"
invertible_left_inverse2: fixes u :: "'a"   and v :: "'a" assumes "multiplicative.invertible u"   and "u \<in> R"   and "v \<in> R" shows "multiplicative.inverse u \<cdot> (u \<cdot> v) = v"
finite_ordLess_infinite2: fixes A :: "'b set"   and B :: "'c set" assumes "finite A"   and "infinite B" shows "|A| <o |B|"
Invariants_func_charac: fixes f :: "'a \<Rightarrow> 'c"   and N :: "'c measure" assumes "f \<in> M \<rightarrow>\<^sub>M N"   and "\<And>x. x \<in> space M \<Longrightarrow> f (T x) = f x" shows "f \<in> Invariants \<rightarrow>\<^sub>M N"
arith_simps_15: fixes m :: "num"   and n :: "num" shows "num.Bit1 m * num.Bit1 n = num.Bit1 (m + n + num.Bit0 (m * n))"
apx_norm_eq: fixes M :: "nat \<Rightarrow> nat \<Rightarrow> real DBMEntry" assumes "canonical M n"   and "[M]\<^bsub>v,n\<^esub> \<subseteq> V"   and "dbm_int M n" shows "Approx\<^sub>\<beta> ([M]\<^bsub>v,n\<^esub>) = [DBM_Normalization.norm M (\<lambda>x. real ((k \<circ> v') x)) n]\<^bsub>v,n\<^esub>"
scale_ms_real: fixes c :: "real"   and c' :: "real" shows "scale_ms c c' = c * c'"
to_poly_UP_pre_univ_prop: shows "UP_pre_univ_prop R P (to_polynomial R)"
UN_Ioc_eq_UNIV: shows "(\<Union>n. {- real n<..real n}) = UNIV"
False: shows "\<not> snd next \<noteq> snd \<phi>"
rlim: shows "r \<longlonglongrightarrow>\<^bsub>\<^esub> \<infinity>"
expand_eq_expand\<^sub>T: fixes \<phi> :: "'b ltlr"   and n_ns :: "'b node \<times> 'b node set" assumes "expand_inv \<phi> n_ns" shows "expand\<^sub>T n_ns = expand n_ns"
PrimeDmd: fixes P :: "state \<Rightarrow> bool"   and sigma :: "behavior" assumes "sigma \<Turnstile> Init P$" shows "sigma \<Turnstile> \<diamond>P"
dma_type_wellformed: fixes x :: "'Var" shows "type_wellformed (dma_type x)"
normalize_primitive_extract_normalizes_n_primitive: fixes m :: "'a match_expr"   and disc :: "'a \<Rightarrow> bool"   and sel :: "'a \<Rightarrow> 'b"   and C :: "'b \<Rightarrow> 'a"   and f :: "'b negation_type list \<Rightarrow> 'b list"   and P :: "'b \<Rightarrow> bool" assumes "normalized_nnf_match m"   and "wf_disc_sel (disc, sel) C"   and "\<forall>as. \<forall>a'\<in>set (f as). P a'" shows "\<forall>m'\<in>set (normalize_primitive_extract (disc, sel) C f m). normalized_n_primitive (disc, sel) P m'"
SUP_eq_and_INF_eq: fixes A :: "'a set"   and B :: "'b set"   and f :: "'a \<Rightarrow> 'd"   and g :: "'b \<Rightarrow> 'd" assumes "\<And>i. i \<in> A \<Longrightarrow> \<exists>j\<in>B. f i = g j"   and "\<And>j. j \<in> B \<Longrightarrow> \<exists>i\<in>A. g j = f i" shows "\<Squnion> (f ` A) = \<Squnion> (g ` B) \<and> \<Sqinter> (f ` A) = \<Sqinter> (g ` B)"
ltl_def: fixes llist :: "'b llist" shows "ltl llist = (case llist of LNil \<Rightarrow> LNil | LCons x21 x22 \<Rightarrow> x22)"
onorm_pos_le: fixes f :: "'d \<Rightarrow> 'e" assumes "bounded_linear f" shows "0 \<le> onorm f"
inf_pres_def: fixes X :: "('a, 'b) gorder_scheme"   and Y :: "('c, 'd) gorder_scheme"   and f :: "'a \<Rightarrow> 'c" shows "inf_pres X Y f \<equiv> complete_lattice X \<and> complete_lattice Y \<and> (\<forall>A\<subseteq>carrier X. f (\<Sqinter>\<^bsub>X\<^esub>A) = \<Sqinter>\<^bsub>Y\<^esub>f ` A)"
l9_10: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and P :: "'a"   and Q :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "\<not> Tarski_neutral_dimensionless.Col Bet A P Q" shows "\<exists>C. Tarski_neutral_dimensionless.TS Bet P Q A C"
paste_def: fixes P :: "('c \<times> 'd) set"   and Q :: "('c \<times> 'd) set" shows "P +< Q = Outside' (Domain Q) P \<union> Q"
simps_2: fixes s :: "nat \<Rightarrow> 'a"   and k :: "nat" shows "inf_truncate s (Suc k) = inf_truncate s k @ [s (Suc k)]"
that: fixes ys :: "'a list"   and y :: "'a"   and zs :: "'a list" assumes "distinct ys"   and "y \<in> set ys"   and "xs = ys @ [y] @ zs" shows "thesis"
Bex_rev_wrap_def: fixes A :: "'b list set"   and P :: "'b list \<Rightarrow> bool" shows "Bex_rev_wrap A P = (\<exists>x\<in>rev ` A. P (rev x))"
gen_fib_recurrence: fixes a :: "nat"   and b :: "nat"   and n :: "nat" shows "gen_fib a b (Suc (Suc n)) = gen_fib a b n + gen_fib a b (Suc n)"
ground_fterm_of_hterm: fixes t :: "hterm" shows "ground\<^sub>t (fterm_of_hterm t)"
flow_initial_time: fixes t0 :: "real"   and x0 :: "('b, 'sz) vec" assumes "t0 \<in> UNIV"   and "x0 \<in> UNIV" shows "ll.flow t0 x0 t0 = x0"
real_poly_add: fixes p :: "'a poly"   and q :: "'a poly" assumes "set (coeffs p) \<subseteq> \<real>"   and "set (coeffs q) \<subseteq> \<real>" shows "set (coeffs (p + q)) \<subseteq> \<real>"
continuous_on_IccI: fixes f :: "'a \<Rightarrow> 'b"   and a :: "'a"   and b :: "'a" assumes "(f \<longlongrightarrow> f a) (at_right a)"   and "(f \<longlongrightarrow> f b) (at_left b)"   and "\<And>x. \<lbrakk>a < x; x < b\<rbrakk> \<Longrightarrow> f \<midarrow>x\<rightarrow> f x"   and "a < b" shows "continuous_on {a..b} f"
C: shows "finite C \<and> C \<subseteq> carrier V \<and> C \<subseteq> V.module.span B \<and> C \<inter> A = {} \<and> int (card C) \<le> int (card B) - int (card A) \<and> V.module.span (A \<union> C) = V.module.span B"
Int_space_eq2: fixes x :: "'b set"   and M :: "'b measure" assumes "x \<in> null_sets M" shows "x \<inter> space M = x"
IH_1: fixes ps :: "(real, 'a) vec list" assumes "invar l"   and "distinct ps"   and "set ps \<inter> set_kdt l = {}" shows "distinct (nearest_nbors n ps p l)"
IH: fixes k :: "nat" shows "n choose k \<le> 2 ^ n"
m_c_widen: fixes C1 :: "ivl st option acom"   and C2 :: "ivl st option acom" assumes "strip C1 = strip C2"   and "top_on\<^sub>c C1 (- vars C1)"   and "top_on\<^sub>c C2 (- vars C2)"   and "\<not> C2 \<le> C1" shows "m_c (C1 \<nabla> C2) < m_c C1"
int_left_shift_def: fixes x :: "int \<Rightarrow> 'b" shows "int_left_shift x = (\<lambda>i. x (i + 1))"
exhaustive_above_iff: fixes xs :: "'a list"   and x :: "'a" assumes "sorted xs"   and "distinct xs"   and "\<forall>x'\<in>set xs. x < x'" shows "exhaustive_above x xs = (set xs = {z. x < z})"
num_def: fixes P :: "'b \<Rightarrow> bool"   and xs :: "'b stream" shows "num P xs \<equiv> LEAST n. list_all (Not \<circ> P) (stake n xs) \<and> P (xs !! n)"
strict_bicategory_axioms: shows "strict_bicategory (\<cdot>\<^sub>S) (\<star>\<^sub>S) S.\<a> S.\<i> S.src S.trg"
Rep_game_inject: fixes x :: "game"   and y :: "game" shows "(Rep_game x = Rep_game y) = (x = y)"
Units_m_closed: fixes x :: "'a"   and y :: "'a" assumes "x \<in> Units R"   and "y \<in> Units R" shows "x \<otimes> y \<in> Units R"
snd_proj_mem: fixes B :: "'c"   and M :: "('d \<times> 'c) set" shows "(B \<in> snd_proj M) = (\<exists>A. (A, B) \<in> M)"
esize_iff_infinite: fixes A :: "'b set" shows "(esize A = \<infinity>) = infinite A"
BinOp_\<tau>red1t_xt1: fixes uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and h :: "'heap"   and e1 :: "'addr expr1"   and xs :: "'addr val list"   and e1' :: "'addr expr1"   and xs' :: "'addr val list"   and bop :: "bop"   and e2 :: "'addr expr1" assumes "\<tau>red1gt uf P t h (e1, xs) (e1', xs')" shows "\<tau>red1gt uf P t h (e1 \<guillemotleft>bop\<guillemotright> e2, xs) (e1' \<guillemotleft>bop\<guillemotright> e2, xs')"
disc_14: fixes x4 :: "real" shows "is_RealVal (RealVal x4)"
assms_5: shows "h \<turnstile> get_child_nodes ptr \<rightarrow>\<^sub>h h'"
strip_while_append: fixes P :: "'b \<Rightarrow> bool"   and x :: "'b"   and xs :: "'b list" assumes "\<not> P x" shows "strip_while P (xs @ [x]) = xs @ [x]"
scene_top_greatest: fixes X :: "'a scene" shows "X \<subseteq>\<^sub>S \<top>\<^sub>S"
nonzero_fun_is_valid_fun: fixes f :: "'a \<Rightarrow> real" assumes "nonzero_prob_fun f" shows "valid_prob_fun f"
order_iff: fixes a :: "'a"   and b :: "'a" shows "(a \<le> b) = (a = min a b)"
continuous_on_sequentially: fixes s :: "'a set"   and f :: "'a \<Rightarrow> 'b" shows "continuous_on s f = (\<forall>x. \<forall>a\<in>s. (\<forall>n. x n \<in> s) \<and> x \<longlonglongrightarrow> a \<longrightarrow> (f \<circ> x) \<longlonglongrightarrow> f a)"
list_all2_cr_fset: fixes xs :: "'b list list"   and ys :: "'b fset list" shows "list_all2 cr_fset xs ys = (map abs_fset xs = ys)"
simps_13: fixes v :: "'a rexp"   and va :: "'a rexp" shows "local.le_rexp (Plus v va) One = False"
rCase: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and M :: "'a"   and B :: "('a, 'b, 'c) boundOutput"   and A\<^sub>P :: "name list"   and \<Psi>\<^sub>P :: "'b"   and \<phi> :: "'c"   and Cs :: "('c \<times> ('a, 'b, 'c) psi) list"   and C :: "'d" assumes "\<Psi> \<rhd> P \<longmapsto> ROut M B"   and "extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>"   and "distinct A\<^sub>P"   and "\<And>C. Prop C \<Psi> P M B A\<^sub>P \<Psi>\<^sub>P"   and "(\<phi>, P) mem Cs"   and "\<Psi> \<turnstile> \<phi>"   and "guarded P"   and "\<Psi>\<^sub>P \<simeq> \<one>"   and "supp \<Psi>\<^sub>P = {}"   and "A\<^sub>P \<sharp>* \<Psi>"   and "A\<^sub>P \<sharp>* P"   and "A\<^sub>P \<sharp>* M"   and "A\<^sub>P \<sharp>* B"   and "A\<^sub>P \<sharp>* C" shows "Prop C \<Psi> (Cases Cs) M B [] (\<one>)"
assms_1: fixes x :: "'a" assumes "x \<in># A" shows "prime x"
per_cong: fixes A :: "'p"   and B :: "'p"   and P :: "'p"   and R :: "'p"   and X :: "'p" assumes "A \<noteq> B"   and "A \<noteq> P"   and "Per B A P"   and "Per A B R"   and "Cong A P B R"   and "Col A B X"   and "Bet P X R" shows "Cong A R P B"
iso_set_trans: fixes h :: "'c \<Rightarrow> 'd"   and G :: "('c, 'e) monoid_scheme"   and H :: "('d, 'f) monoid_scheme"   and i :: "'d \<Rightarrow> 'g"   and I :: "('g, 'h) monoid_scheme" assumes "h \<in> Group.iso G H"   and "i \<in> Group.iso H I" shows "i \<circ> h \<in> Group.iso G I"
sylvester_mat_sub_index: fixes i :: "nat"   and m :: "nat"   and n :: "nat"   and j :: "nat"   and p :: "'a poly"   and q :: "'a poly" assumes "i < m + n"   and "j < m + n" shows "sylvester_mat_sub m n p q $$ (i, j) = (if i < n then if i \<le> j \<and> j - i \<le> m then coeff p (m + i - j) else (0::'a) else if i - n \<le> j \<and> j \<le> i then coeff q (i - j) else (0::'a))"
node_not_in_ancestors_h2: shows "cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r node \<notin> set ancestors_h2"
cases: fixes a1 :: "pi"   and a2 :: "pi" assumes "a1 \<sim> a2" obtains P :: "pi" and Q :: "pi" where "a1 = P" and "a2 = Q" and "P \<leadsto>[bisim] Q" and "Q \<sim> P"
pirreducibleI: fixes K :: "'a list set"   and p :: "'a list list" assumes "subring K (K [X])"   and "p \<in> carrier (K [X]\<^bsub>K [X]\<^esub>)"   and "p \<noteq> []"   and "p \<notin> Units (K [X]\<^bsub>K [X]\<^esub>)"   and "\<And>q r. \<lbrakk>q \<in> carrier (K [X]\<^bsub>K [X]\<^esub>); r \<in> carrier (K [X]\<^bsub>K [X]\<^esub>); p = q \<otimes>\<^bsub>K [X]\<^bsub>K [X]\<^esub>\<^esub> r\<rbrakk> \<Longrightarrow> q \<in> Units (K [X]\<^bsub>K [X]\<^esub>) \<or> r \<in> Units (K [X]\<^bsub>K [X]\<^esub>)" shows "pirreducible\<^bsub>K [X]\<^esub> K p"
solution_pell_power: fixes D :: "nat"   and z :: "'a \<times> 'a"   and n :: "nat" assumes "pell D"   and "pell.solution D z" shows "pell.solution D (pell.pell_power D z n)"
am_get_def: fixes am :: "nat \<Rightarrow> nat list"   and u :: "nat" shows "am_get am u \<equiv> assert (u \<in> V) \<bind> (\<lambda>_. return (am u))"
hom_prod: fixes f :: "'c \<Rightarrow> 'b"   and X :: "'c set" shows "[:prod f X:] = (\<Prod>x\<in>X. [:f x:])"
cases: fixes a1 :: "'a"   and a2 :: "'c"   and r :: "('c \<times> 'a) set" assumes "(a1, a2) \<in> r\<inverse>" obtains a :: "'c" and b :: "'a" where "a1 = b" and "a2 = a" and "(a, b) \<in> r"
InfinitesimalI2: fixes x :: "'c star" assumes "\<And>r. 0 < r \<Longrightarrow> hnorm x < hypreal_of_real r" shows "x \<in> Infinitesimal"
simps_3: fixes xs_ys :: "(int \<times> int) list" shows "interpolation_poly_int Neville_Aitken xs_ys = (let rxs_ys = map (\<lambda>(x, y). (rat_of_int x, rat_of_int y)) xs_ys; rp = interpolation_poly Neville_Aitken rxs_ys in if Ball (set (coeffs rp)) is_int_rat then Some (map_poly int_of_rat rp) else None)"
exhaust_disc: fixes status :: "status" obtains "is_failed status"   | "is_success status"   | "is_found status"
rb0': shows "r k (b $ 0) \<noteq> (0::'a)"
poincare_between_sandwich: fixes u :: "complex_homo"   and v :: "complex_homo" assumes "u \<in> unit_disc"   and "v \<in> unit_disc"   and "poincare_between u v u" shows "u = v"
trd_aux_irred: fixes fs :: "('t \<Rightarrow>\<^sub>0 'c) list"   and r :: "'t \<Rightarrow>\<^sub>0 'c"   and p :: "'t \<Rightarrow>\<^sub>0 'c" assumes "\<not> is_red (set fs) r" shows "\<not> is_red (set fs) (trd_aux fs p r)"
sel_2: fixes x1 :: "'a list"   and x2 :: "nat" shows "arity (Fundef x1 x2) = x2"
\<tau>move_heap: fixes t :: "'t"   and x :: "'x"   and m :: "'m"   and ta :: "('l, 't, 'x, 'm, 'w, 'o) thread_action"   and x' :: "'x"   and m' :: "'m" assumes "t \<turnstile> \<langle>x, m\<rangle> -ta\<rightarrow> \<langle>x', m'\<rangle>"   and "\<tau>move (x, m) ta (x', m')" shows "m = m'"
SubstTermP_trivial_dbtm: fixes i :: "name"   and u :: "dbtm" assumes "atom i \<sharp> u" shows "{} \<turnstile> SubstTermP \<guillemotleft>Var i\<guillemotright> Zero (quot_dbtm u) (quot_dbtm u)"
lprodp_lprod_eq: fixes R :: "('b \<times> 'b) set" shows "lprodp (\<lambda>x xa. (x, xa) \<in> R) = (\<lambda>x xa. (x, xa) \<in> lprod R)"
test_add_comp: fixes p :: "'a" assumes "test p" shows "p + !p = (1::'a)"
sat_preserving_redundancy_criterion_axioms: shows "sat_preserving_redundancy_criterion \<Gamma> Rf Ri"
dense: fixes l :: "'a"   and u :: "'a"   and y :: "'a" assumes "\<forall>y\<in>{l<..<x}. y \<notin> L"   and "\<forall>y\<in>{x<..<u}. y \<notin> U"   and "l < x"   and "x < u"   and "l < y"   and "y < u" shows "P y"
red_diff_rtrancl': fixes F :: "('a \<Rightarrow>\<^sub>0 'c) set"   and p :: "'a \<Rightarrow>\<^sub>0 'c"   and q :: "'a \<Rightarrow>\<^sub>0 'c"   and r :: "'a \<Rightarrow>\<^sub>0 'c" assumes "(punit.red F)\<^sup>*\<^sup>* (p - q) r" obtains p' :: "'a \<Rightarrow>\<^sub>0 'c" and q' :: "'a \<Rightarrow>\<^sub>0 'c" where "(punit.red F)\<^sup>*\<^sup>* p p'" and "(punit.red F)\<^sup>*\<^sup>* q q'" and "r = p' - q'"
dghm_dghm_comp_dghm_id_left: shows "dghm_id \<BB> \<circ>\<^sub>D\<^sub>G\<^sub>H\<^sub>M \<GG> = \<GG>"
focus_sum: fixes X :: "'a set"   and f :: "'c \<Rightarrow> ('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and I :: "'c set" shows "focus X (sum f I) = (\<Sum>i\<in>I. focus X (f i))"
simps_8: fixes x31 :: "'c"   and x32 :: "('c \<Rightarrow> 'd) \<Rightarrow> 'd"   and x2 :: "('c \<Rightarrow> 'd) \<Rightarrow> bool" shows "Assign x31 x32 \<noteq> Assume x2"
ProcRec1: fixes \<Gamma> :: "'f \<Rightarrow> ('d, 'f, 'g) com option"   and \<Theta> :: "('d set \<times> 'f \<times> 'd set \<times> 'd set) set"   and P :: "'e \<Rightarrow> 'd set"   and p :: "'f"   and r :: "(('d \<times> 'f) \<times> 'd \<times> 'f) set"   and Q :: "'e \<Rightarrow> 'd set"   and A :: "'e \<Rightarrow> 'd set"   and F :: "'g set" assumes "\<forall>\<sigma> Z. \<Gamma>,\<Theta> \<union> (\<Union>Z. {(P Z \<inter> {s. ((s, p), \<sigma>, p) \<in> r}, p, Q Z, A Z)})\<turnstile>\<^sub>t\<^bsub>/F\<^esub> ({\<sigma>} \<inter> P Z) the (\<Gamma> p) (Q Z),(A Z)"   and "wf r"   and "p \<in> dom \<Gamma>" shows "\<forall>Z. \<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> (P Z) Call p (Q Z),(A Z)"
indep_vars_def2: fixes M' :: "'b \<Rightarrow> 'c measure"   and X :: "'b \<Rightarrow> real \<Rightarrow> 'c"   and I :: "'b set" shows "\<mu>.indep_vars M' X I = ((\<forall>i\<in>I. \<mu>.random_variable (M' i) (X i)) \<and> \<mu>.indep_sets (\<lambda>i. {X i -` A \<inter> space \<mu> |A. A \<in> sets (M' i)}) I)"
K_in_space: fixes x :: "real \<times> 'a" shows "K x \<in> space (prob_algebra S)"
r_Linear_order: fixes r :: "('a \<times> 'a) set" assumes "MaxR r" shows "Linear_order r"
quadroot_in_def: fixes m :: "real"   and M :: "real"   and a :: "real"   and b :: "real"   and c :: "real" shows "quadroot_in m M a b c = root_in m M (\<lambda>x. a * x\<^sup>2 + b * x + c)"
sort_eqv_sym: fixes x :: "String.literal set"   and y :: "String.literal set" assumes "sort_eqv cs x y" shows "sort_eqv cs y x"
type_definition_pp: shows "type_definition pp.mapping_of PP UNIV"
add_le_cancel_left: fixes c :: "'c multiset"   and a :: "'c multiset"   and b :: "'c multiset" shows "(c + a \<subseteq># c + b) = (a \<subseteq># b)"
get_fst_lens: fixes x :: "'b"   and y :: "'c" shows "get\<^bsub>fst\<^sub>L\<^esub> (x, y) = x"
fst_snd_flip: fixes xy :: "'g \<times> 'h" shows "fst xy = (snd \<circ> (\<lambda>(x, y). (y, x))) xy"
ok_sym: fixes F :: "'c program"   and G :: "'c program" assumes "F ok G" shows "G ok F"
N2C: shows "E' `` N2 \<subseteq> N2"
select_convs_7: fixes iiface :: "iface"   and oiface :: "iface"   and src :: "'a word \<times> nat"   and dst :: "'a word \<times> nat"   and proto :: "protocol"   and sports :: "16 word \<times> 16 word"   and dports :: "16 word \<times> 16 word"   and more :: "'b" shows "dports \<lparr>iiface = iiface, oiface = oiface, src = src, dst = dst, proto = proto, sports = sports, dports = dports, \<dots> = more\<rparr> = dports"
1: shows "\<tau> \<Turnstile> \<delta> x \<or> \<tau> |\<noteq> \<delta> x"
A: fixes pa :: "'a poly fract"   and qa :: "'a poly fract" assumes "fractrel (quot_of_fract pa) (quot_of_fract qa)" shows "pa = qa"
quot_Var: fixes x :: "name" shows "\<guillemotleft>Var x\<guillemotright> = SUCC (ORD_OF (nat_of_name x))"
extremum_uniqueI: fixes a :: "'b word" assumes "- 1 \<le> a" shows "a = - 1"
24: shows "irreflexive (e * (w \<sqinter> - prim_EP w v e)\<^sup>\<star> * e * prim_P w v e\<^sup>T\<^sup>\<star> * (w \<sqinter> - prim_EP w v e)\<^sup>\<star>)"
use_iso1: fixes A :: "('a, 'b) gorder_scheme"   and f :: "'a \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a" assumes "Mono\<^bsub>A\<^esub> f"   and "x \<in> carrier A"   and "y \<in> carrier A"   and "x \<sqsubseteq>\<^bsub>A\<^esub> y" shows "f x \<sqsubseteq>\<^bsub>A\<^esub> f y"
cl_ord_def: shows "cl_ord = {(x, y). (mset_cl x, mset_cl y) \<in> mult (mult trm_ord)}"
bit_iff_and_push_bit_not_eq_0: fixes a :: "'a"   and n :: "nat" shows "bit a n = (a AND push_bit n (1::'a) \<noteq> (0::'a))"
some_gcd_ff_list_smult: fixes a :: "'b fract"   and xs :: "'b fract list" assumes "a \<noteq> 0" shows "some_gcd_ff_list (map ((*) a) xs) =dff a * some_gcd_ff_list xs"
VC_cfn: fixes p :: "nat"   and i :: "nat" assumes "correct p (te p (i + 1))"   and "te p (i + 1) < te p (i + 2)" shows "VC p (te p (i + 1)) = cfn p (\<theta> p (i + 1))"
finite_Assoc_List_set_image: fixes A :: "('b, 'c) assoc_list set" assumes "finite (Assoc_List.set ` A)" shows "finite A"
Cons_nth_drop_Suc: fixes i :: "nat"   and xs :: "'d list" assumes "i < length xs" shows "xs ! i # xs \<up> Suc i = xs \<up> i"
mat_of_cblinfun_scaleC: fixes a :: "complex"   and F :: "'b \<Rightarrow>\<^sub>C\<^sub>L 'c" shows "mat_of_cblinfun (a *\<^sub>C F) = a \<cdot>\<^sub>m mat_of_cblinfun F"
assms_3: fixes x :: "'a"   and y :: "'a" assumes "xs = [x, y]" shows "P"
distinct_disc_9: fixes event :: "('a, 'b, 'c) event" assumes "isSend event" shows "\<not> isRecv event"
run_altc_sample_writer: fixes altc_sample :: "'b \<Rightarrow> ('d \<Rightarrow> 'm) \<Rightarrow> 'm"   and p :: "'b"   and f :: "'d \<Rightarrow> ('w, 'a, 'm) writerT" shows "run_writer (local.altc_sample_writer altc_sample p f) = altc_sample p (\<lambda>p. run_writer (f p))"
preprocess_part_2_def: fixes as :: "('b \<times> 'c atom) list"   and t :: "(nat \<times> linear_poly) list" shows "preprocess_part_2 as t = preprocess_opt (atom_var ` snd ` set as) [] t"
ffold_fun_left_comm: fixes x :: "'e \<times> nat"   and z :: "nat"   and A :: "('e \<times> nat) fset" shows "tcf x (ffold tcf z A) = ffold tcf (tcf x z) A"
star_unfold_part: fixes x :: "'a" shows "x \<cdot> x\<^sup>\<star> \<le> x\<^sup>\<star>"
cancel_same: fixes xs :: "'d list" shows "cancel (xs @ xs) xs"
swap1: fixes m :: "'n"   and n :: "'n" assumes "m \<noteq> n" shows "P (\<chi>i j. mat 1 $ i $ Fun.swap m n id j)"
A_1: shows "S = [M]\<^bsub>v,n\<^esub> \<and> dbm_int M n"
component_stable: fixes F :: "'b program"   and G :: "'b program"   and A :: "'b set" assumes "F \<le> G"   and "G \<in> stable A" shows "F \<in> stable A"
zeta_def: shows "zeta = (\<Sum>k. 1 / of_nat (Suc k) ^ n)"
to_nat_0: shows "mod_type_class.to_nat (0::'a) = 0"
terminal_of_token_simp: fixes a :: "symbol"   and b :: "character list" shows "terminal_of_token (a, b) = a"
polyhedra_are_convex: fixes A :: "'a mat"   and nr :: "nat"   and b :: "'a vec"   and P :: "'a vec set" assumes "A \<in> carrier_mat nr n"   and "b \<in> carrier_vec nr"   and "P = polyhedron A b" shows "convex P"
fmdom'_restrict_fset_precise: fixes A :: "'c fset"   and m :: "('c, 'd) fmap" shows "fmdom (fmrestrict_fset A m) = fmdom m |\<inter>| A"