hcju/isabelleps huggingface dataset:
assoc: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<in> D"   and "y \<in> D"   and "z \<in> D" shows "x \<cdot> y \<cdot> z = x \<cdot> (y \<cdot> z)"
tt'': shows "x \<notin> FV (t\<^bsup>[Fvar s,Fvar p]\<^esup>) \<longrightarrow> x \<notin> FV t''"
I: shows "bij_betw I {0..<n} {0..<n}"
ring_simprules_26: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier R"   and "y \<in> carrier R" shows "\<ominus> x \<otimes> y = \<ominus> (x \<otimes> y)"
fresh_2: fixes a :: "W.tvar"   and x1 :: "W.var" shows "a \<sharp> Var x1 = a \<sharp> x1"
inner_diff_left: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and norm :: "'a \<Rightarrow> real"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and sgn :: "'a \<Rightarrow> 'a"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and inner :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "class.real_inner minus dist norm plus zero uminus scaleR sgn uniformity open inner" shows "inner (minus x y) z = inner x z - inner y z"
trancl_id: fixes r :: "('b \<times> 'b) set" assumes "trans r" shows "r\<^sup>+ = r"
rbl_mult_Suc: fixes n :: "nat"   and bina :: "int"   and binb :: "int" shows "rbl_mult (rev (bin_to_bl n bina)) (rev (bin_to_bl (Suc n) binb)) = rbl_mult (rev (bin_to_bl n bina)) (rev (bin_to_bl n binb))"
attach_CNV_RES: fixes callee :: "'r \<Rightarrow> 'n \<Rightarrow> ('o \<times> 'r, 'p, 'q) gpv"   and s :: "'r"   and res :: "'s \<Rightarrow> 'p \<Rightarrow> ('q \<times> 's) spmf"   and s' :: "'s" shows "CNV callee s \<rhd> RES res s' = RES (attach_callee callee res) (s, s')"
mk_disjoint_insert: fixes a :: "'a"   and A :: "'a set" assumes "a \<in> A" shows "\<exists>B. A = a \<triangleright> B \<and> a \<notin> B"
insert_is_Un: fixes a :: "'h"   and A :: "'h set" shows "insert a A = {a} \<union> A"
Q1_def: shows "Q1 = matrix_sum d (\<lambda>l. tensor_P (proj_psi'_l l) (proj_k l)) R"
reds_fresh_fv: fixes \<Gamma> :: "(Vars.var \<times> exp) list"   and e :: "exp"   and L :: "Vars.var list"   and \<Delta> :: "(Vars.var \<times> exp) list"   and z :: "exp"   and x :: "Vars.var" assumes "\<Gamma> : e \<Down>\<^bsub>L\<^esub> \<Delta> : z"   and "x \<in> fv (\<Delta>, z) \<and> (x \<notin> domA \<Delta> \<or> x \<in> set L)" shows "x \<in> fv (\<Gamma>, e)"
that: fixes a :: "'a"   and b :: "'a" assumes "a \<noteq> b"   and "a \<in> Q"   and "b \<in> Q"   and "\<forall>c\<in>Q. a \<noteq> c \<and> b \<noteq> c \<longrightarrow> [[a c b]]" shows "thesis"
app_LB: fixes n :: "nat"   and vv :: "nat \<Rightarrow> 'a \<Rightarrow> ant"   and x :: "nat \<Rightarrow> 'a"   and j :: "nat"   and m :: "nat" assumes "vals_nonequiv K (Suc n) vv"   and "\<forall>l\<le>Suc n. x l \<in> carrier K"   and "j \<le> Suc n" shows "\<forall>N>(\<Psi>\<^bsub>K Suc n vv x m\<^esub>) j. an m \<le> vv j (\<Sigma>\<^sub>e K (\<lambda>j\<in>{h. h \<le> Suc n}. x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>)) Suc n \<plusminus> -\<^sub>a x j)"
language_state_prefix: fixes w1 :: "('e \<times> 'f) list"   and w2 :: "('e \<times> 'f) list"   and M :: "('e, 'f, 'g) FSM"   and q :: "'g" assumes "w1 @ w2 \<in> LS M q" shows "w1 \<in> LS M q"
bisim_inv_red_other: fixes t' :: "'t"   and x :: "'x1"   and m1 :: "'m1"   and xx :: "'x2"   and m2 :: "'m2"   and t :: "'t"   and x1 :: "'x1"   and x2 :: "'x2"   and x1' :: "'x1"   and ta1 :: "('l, 't, 'x1, 'm1, 'w, 'o) thread_action"   and x1'' :: "'x1"   and m1' :: "'m1"   and x2' :: "'x2"   and ta2 :: "('l, 't, 'x2, 'm2, 'w, 'o) thread_action"   and x2'' :: "'x2"   and m2' :: "'m2" assumes "t' \<turnstile> (x, m1) \<approx> (xx, m2)"   and "t \<turnstile> (x1, m1) \<approx> (x2, m2)"   and "r1.silent_moves t (x1, m1) (x1', m1)"   and "r1.r_syntax t x1' m1 ta1 x1'' m1'"   and "\<not> False"   and "r2.silent_moves t (x2, m2) (x2', m2)"   and "r2.r_syntax t x2' m2 ta2 x2'' m2'"   and "\<not> False"   and "t \<turnstile> (x1'', m1') \<approx> (x2'', m2')"   and "ta1 \<sim>m ta2" shows "t' \<turnstile> (x, m1') \<approx> (xx, m2')"
Inf_subdistl_iso: fixes f :: "'a \<Rightarrow> 'c" assumes "f \<circ> Inf \<le> Inf \<circ> (`) f" shows "mono f"
cmult_S: fixes n :: "nat"   and sb :: "nat" assumes "0 < n"   and "sb \<in> S" shows "n * sb \<in> S"
cnt1: shows "count (hmsetmset (local.encode (Suc e) (n div base))) (local.encode 0 e) = 0"
assms: shows "(X, Y) \<in> mult2_alt_s ns s"
not_prime_elem_zero: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a" assumes "class.comm_semiring_1 times one plus zero" shows "\<not> comm_semiring_1.prime_elem times one zero zero"
iso_is_arr: fixes f :: "('c \<Rightarrow> 's, 'c \<Rightarrow> 's) concrete_category.arr" assumes "Cop_S.iso f" shows "Cop_S.arr f"
lunit_eqI: fixes f :: "'a"   and a :: "'a" assumes "\<guillemotleft>f : C.tensor C.unity a \<rightarrow> a\<guillemotright>"   and "C.tensor C.unity f = C.tensor \<iota> a \<cdot> C.assoc' C.unity C.unity a" shows "f = C.lunit a"
gbinomial_int_n_n: fixes n :: "nat" shows "int n gchoose n = 1"
retract_of_space_singleton: fixes a :: "'b"   and X :: "'b topology" shows "({a} retract_of_space X) = (a \<in> topspace X)"
extremum_unique: fixes a :: "'a" shows "(top \<le> a) = (a = top)"
lookup_subtensor: fixes "is" :: "nat list"   and A :: "'b tensor" assumes "is \<lhd> dims A" shows "lookup A is = hd (vec (fold (\<lambda>i A. subtensor A i) is A))"
i_aggregate_i_take: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'b"   and k :: "nat"   and ag :: "'b list \<Rightarrow> 'b" shows "f_aggregate (f \<Down> n) k ag = i_aggregate f k ag \<Down> (n div k)"
finite_chamber: fixes x :: "'b set" assumes "codomain.chamber x" shows "finite x"
simps_1: fixes s :: "'c list"   and uu :: "'c list"   and uv :: "(nat \<times> nat list) list" shows "steps' s uu uv 0 = s"
Bit_B0_2t: fixes k :: "int" shows "k BIT False = 2 * k"
list_less_iff_prefix: fixes a :: "'a list"   and b :: "'a list" shows "(a < b) = strict_prefix a b"
bet_cong_eq: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "Bet A B C"   and "Bet A C D"   and "Cong B C A D" shows "C = D \<and> A = B"
zcount_inject: fixes M :: "'b zmultiset"   and N :: "'b zmultiset" shows "(zcount M = zcount N) = (M = N)"
res_top: fixes f :: "'a \<Rightarrow> 'a" assumes "residuated f" shows "residual f \<top> = \<top>"
pax1: fixes F :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" shows "|~ \<circle>\<not> F = \<not> \<circle>F"
geod_dist2: fixes a :: "nat"   and n :: "nat"   and b :: "nat"   and p :: "nat" assumes "a \<le> 2 ^ n"   and "b \<le> 2 ^ p"   and "real a / 2 ^ n \<le> real b / 2 ^ p" shows "dist (geod n a) (geod p b) = (real b / 2 ^ p - real a / 2 ^ n) * dist x0 y0"
WT_sim: shows "\<I>_middle, \<I>_inner \<turnstile>\<^sub>C sim_inner \<surd>"
simps_1: fixes x :: "'a option" shows "None \<triangle> x = None"
comp_refl: fixes u :: "'b list" shows "u \<bowtie> u"
i1: fixes x :: "('a, 'n) vec"   and y :: "('a, 'n) vec" shows "inner x y = cnj_class.cnj (inner y x)"
atom_sup_iff: fixes x :: "'a" shows "atom x = (x \<noteq> \<bottom> \<and> (\<forall>y z. (x \<le> y \<or> x \<le> z) = (x \<le> y \<squnion> z)))"
abs_summable_on_Sigma_iff: fixes A :: "'a set"   and B :: "'a \<Rightarrow> 'b set"   and f :: "'a \<times> 'b \<Rightarrow> 'c" assumes "countable A"   and "\<And>x. x \<in> A \<Longrightarrow> countable (B x)" shows "(f abs_summable_on Sigma A B) = ((\<forall>x\<in>A. (\<lambda>y. f (x, y)) abs_summable_on B x) \<and> (\<lambda>x. \<Sum>\<^sub>ay\<in>B x. norm (f (x, y))) abs_summable_on A)"
card_1_singletonE: fixes A :: "'e set" assumes "card A = 1" obtains x :: "'e" where "A = {x}"
substT_liftT_1: fixes k :: "nat"   and k' :: "nat"   and n :: "nat"   and T :: "POPLmarkRecord.type"   and U :: "POPLmarkRecord.type" assumes "k \<le> k'"   and "k' < k + n" shows "\<up>\<^sub>\<tau> n k T[k' \<mapsto>\<^sub>\<tau> U]\<^sub>\<tau> = \<up>\<^sub>\<tau> (n - 1) k T"
bia_empty: shows "before q s \<inter> ({q} \<union> after q s) = {}"
dickson_less_p_zero: fixes m :: "nat" shows "dickson_less_p (\<lambda>_. 0) m = (\<prec>\<^sub>p)"
s1_1_inj1: fixes x1 :: "nat"   and y1 :: "nat"   and x2 :: "nat"   and y2 :: "nat" assumes "s1_1 x1 y1 = s1_1 x2 y2" shows "x1 = x2"
that: fixes m :: "'c multiset" assumes "length Qs \<le> count m i"   and "indexed_eval (P # Qs) i m \<noteq> \<zero>" shows "thesis"
ans4: fixes antidomain_op :: "'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and zero :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "class.antidomain_near_semiring antidomain_op plus times one zero less_eq less" shows "antidomain_op (plus x y) = times (antidomain_op x) (antidomain_op y)"
textbook_true: fixes \<tau> :: "'a state \<times> 'a state" shows "I\<lbrakk>true\<rbrakk> \<tau> = \<lfloor>\<lfloor>True\<rfloor>\<rfloor>"
measurable_If: fixes f :: "'b \<Rightarrow> 'c"   and M :: "'b measure"   and M' :: "'c measure"   and g :: "'b \<Rightarrow> 'c"   and P :: "'b \<Rightarrow> bool" assumes "f \<in> M \<rightarrow>\<^sub>M M'"   and "g \<in> M \<rightarrow>\<^sub>M M'"   and "{x \<in> space M. P x} \<in> sets M" shows "(\<lambda>x. if P x then f x else g x) \<in> M \<rightarrow>\<^sub>M M'"
less_le: fixes x :: "'e"   and y :: "'e" shows "(x < y) = (x \<le> y \<and> x \<noteq> y)"
def_C': shows "C' = cl_ecl P1 - {L1}"
ideal_inc_elem0val_whole: fixes v :: "'a \<Rightarrow> ant"   and x :: "'a"   and I :: "'a set" assumes "valuation K v"   and "x \<in> carrier K"   and "v x = 0"   and "ideal (Vr K v) I"   and "x \<in> I" shows "I = carrier (Vr K v)"
bvars\<^sub>e\<^sub>s\<^sub>t_cons: fixes x :: "('fun, 'var) extstrand_step"   and X :: "('fun, 'var) extstrand_step list" shows "bvars\<^sub>e\<^sub>s\<^sub>t (x # X) = bvars\<^sub>e\<^sub>s\<^sub>t [x] \<union> bvars\<^sub>e\<^sub>s\<^sub>t X"
normal_kernel: fixes G :: "('a, 'b) monoid_scheme"   and H :: "('c, 'd) monoid_scheme"   and h :: "'a \<Rightarrow> 'c" assumes "group_hom G H h" shows "kernel G H h \<lhd> G"
P: assumes "\<forall>s. e l1 s = e l2 s" shows "e l1 = e l2"
starfun_ln_ge_zero: fixes x :: "real star" assumes "1 \<le> x" shows "0 \<le> (*f* real_ln) x"
a: shows "S.ide a \<and> (\<exists>\<phi>. bij_betw \<phi> (S.hom S.unity a) (cones S.unity))"
open_seq_step_invariant: fixes A :: "('a \<times> ('a, 'b, 'c, 'd) seqp, 'b seq_action) automaton"   and I :: "'b seq_action \<Rightarrow> bool"   and P :: "('a \<times> ('a, 'b, 'c, 'd) seqp) \<times> 'b seq_action \<times> 'a \<times> ('a, 'b, 'c, 'd) seqp \<Rightarrow> bool"   and i :: "nat"   and OA :: "((nat \<Rightarrow> 'a) \<times> ('a, 'b, 'c, 'd) seqp, 'b seq_action) automaton"   and \<Gamma> :: "'c \<Rightarrow> ('a, 'b, 'c, 'd) seqp"   and ANY :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "A \<TTurnstile>\<^sub>A (I \<rightarrow>) P"   and "initiali i (init OA) (init A)"   and "automaton.trans OA = oseqp_sos \<Gamma> i"   and "automaton.trans A = seqp_sos \<Gamma>" shows "OA \<Turnstile>\<^sub>A (act I, other ANY {i} \<rightarrow>) seqll i P"
Bernstein_Poly_nonneg: fixes c :: "real"   and x :: "real"   and d :: "real"   and j :: "nat"   and p :: "nat" assumes "c \<le> x"   and "x \<le> d" shows "0 \<le> poly (Bernstein_Poly j p c d) x"
f17: shows "snd x \<otimes> snd x' \<otimes> (r \<otimes> r') \<in> carrier R"
nonneg_bounded: fixes f :: "'b \<Rightarrow> 'c" assumes "bounded_linear f" shows "\<exists>K\<ge>0. \<forall>x. \<parallel>f x\<parallel> \<le> \<parallel>x\<parallel> * K"
exhaust: fixes y :: "protocol" obtains "y = tcp"   | "y = udp"
setmult_closed: fixes H :: "'c set"   and G :: "('c, 'd) monoid_scheme"   and K1 :: "'c set"   and K2 :: "'c set" assumes "H \<lhd> G"   and "K1 \<in> rcosets\<^bsub>G\<^esub> H"   and "K2 \<in> rcosets\<^bsub>G\<^esub> H" shows "K1 <#>\<^bsub>G\<^esub> K2 \<in> rcosets\<^bsub>G\<^esub> H"
child_not_in_any_children: fixes p :: "(_) object_ptr"   and children :: "(_) node_ptr list" assumes "h2 \<turnstile> get_child_nodes p \<rightarrow>\<^sub>r children" shows "node \<notin> set children"
le_funE: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c"   and x :: "'b" assumes "f \<le> g" obtains "f x \<le> g x"
closed_halfspace_Re_ge: fixes b :: "real" shows "closed {z. b \<le> Re z}"
simps_3: fixes uu :: "('e \<Rightarrow> 'd) option" shows "uu <*> None = None"
simps_1: fixes X :: "tyvrs"   and Y :: "tyvrs"   and T :: "ty" shows "Tvar X[Y \<mapsto> T]\<^sub>\<tau> = (if X = Y then T else Tvar X)"
determination_of_hom: fixes K :: "'a set"   and A :: "('c, 'd) ring_scheme"   and h :: "'a list \<Rightarrow> 'c"   and g :: "'a list \<Rightarrow> 'c"   and p :: "'a list" assumes "subring K R"   and "ring_hom_ring (K [X]) A h"   and "ring_hom_ring (K [X]) A g"   and "\<And>k. k \<in> K \<Longrightarrow> h [k] = g [k]"   and "h X = g X"   and "p \<in> carrier (K [X])" shows "h p = g p"
add_left_mono: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c"   and b :: "'c"   and c :: "'c" assumes "class.ordered_ab_semigroup_add plus less_eq less"   and "less_eq a b" shows "less_eq (plus c a) (plus c b)"
NC_def: shows "NC = map (\<lambda>(r, a). (s (r, a), ns (r, a))) C"
return_bind: fixes x :: "'c"   and f :: "'c \<Rightarrow> 'a + 'b" shows "Inr x \<bind> f = f x"
intro: fixes type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) heap \<Rightarrow> bool"   and set_disconnected_nodes :: "(_) document_ptr \<Rightarrow> (_) node_ptr list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and type_wf :: "(_) heap \<Rightarrow> bool"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and known_ptr\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) object_ptr \<Rightarrow> bool"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_child_nodes_locs\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" assumes "l_set_disconnected_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M set_disconnected_nodes set_disconnected_nodes_locs"   and "l_get_child_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf known_ptr type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_child_nodes get_child_nodes_locs get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_child_nodes_locs\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M" shows "l_set_disconnected_nodes_get_child_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf set_disconnected_nodes set_disconnected_nodes_locs known_ptr type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_child_nodes get_child_nodes_locs get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_child_nodes_locs\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M"
cases: fixes x :: "('a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b) \<times> ('a \<times> 'b) list \<times> ('a \<times> 'b) list" obtains f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b" and k :: "'a" and v :: "'b" and as :: "('a \<times> 'b) list" and k' :: "'a" and v' :: "'b" and bs :: "('a \<times> 'b) list" where "x = (f__, (k, v) # as__, (k', v') # bs__)"   | f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b" and bs :: "('a \<times> 'b) list" where "x = (fa__, [], bs)"   | f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b" and as :: "('a \<times> 'b) list" where "x = (f, as, [])"
ag_single_inc_a: fixes a :: "'a" assumes "ag_single_gen A a" shows "a \<in> carrier A"
multi_drop_mem_not_eq: fixes c :: "'a"   and B :: "'a multiset" assumes "c \<in># B" shows "B - {#c#} \<noteq> B"
fps_idem: fixes f :: "'a \<Rightarrow> 'a" assumes "f \<in> carrier L \<rightarrow> carrier L"   and "Idem\<^bsub>L\<^esub> f" shows "fps L f {.=}\<^bsub>L\<^esub> f ` carrier L"
table_fvi_tl: fixes n :: "nat"   and b :: "nat"   and \<phi> :: "'b MFOTL.formula"   and X :: "'a option list set" assumes "table (Suc n) (MFOTL.fvi b \<phi>) X" shows "table n (MFOTL.fvi (Suc b) \<phi>) (tl ` X)"
tl_def: fixes list :: "'g list" shows "tl list = (case list of [] \<Rightarrow> [] | x21 # x22 \<Rightarrow> x22)"
sep_substate_disj_add': fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and sep_disj :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "class.sep_algebra plus zero sep_disj"   and "sep_disj x y" shows "sep_algebra.sep_substate plus sep_disj x (plus y x)"
sel_1: fixes x1 :: "'c match_expr"   and x2 :: "action" shows "get_match (Rule x1 x2) = x1"
matched_is_SDG_path: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and Exit :: "'b"   and Def :: "'b \<Rightarrow> 'c set"   and Use :: "'b \<Rightarrow> 'c set"   and ParamDefs :: "'b \<Rightarrow> 'c list"   and ParamUses :: "'b \<Rightarrow> 'c set list"   and n :: "'b SDG_node"   and ns :: "'b SDG_node list"   and n' :: "'b SDG_node" assumes "SDG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main Exit Def Use ParamDefs ParamUses"   and "SDG.matched sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses n ns n'" obtains ns' :: "'b SDG_node list" where "SDG.intra_sum_SDG_path sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses n ns' n'"
abs_eq: fixes xa :: "'a list"   and x :: "'a \<Rightarrow> 'b" shows "Mapping.tabulate xa x = Mapping (map_of (map (\<lambda>k. (k, x k)) xa))"
dom_locals_init_lvars: fixes m :: "methd"   and G :: "prog"   and C :: "qtname"   and sig :: "sig"   and pvs :: "val list"   and e :: "expr"   and a :: "val"   and s :: "state" assumes "m = mthd (the (methd G C sig))"   and "length (pars m) = length pvs" shows "dom (locals (snd (init_lvars G C sig (invmode m e) a pvs s))) = parameters m"
freq_0_left: fixes t\<^sub>1 :: "'a tree"   and t\<^sub>2 :: "'a tree"   and a :: "'a" assumes "alphabet t\<^sub>1 \<inter> alphabet t\<^sub>2 = {}"   and "a \<in> alphabet t\<^sub>2" shows "freq t\<^sub>1 a = 0"
domA_not_fresh: fixes x :: "'a"   and \<Gamma> :: "('a \<times> 'b) list" assumes "x \<in> domA \<Gamma>" shows "\<not> atom x \<sharp> \<Gamma>"
assms_7: shows "C2 \<in> carrier_mat n2 nc1"
subst_ext: fixes \<sigma> :: "'s"   and \<tau> :: "'s" assumes "\<And>A. A \<cdot>a \<sigma> = A \<cdot>a \<tau>" shows "\<sigma> = \<tau>"
oid_preserve: fixes x :: "'\<AA>" assumes "x \<in> ran (heap (fst \<tau>)) \<or> x \<in> ran (heap (snd \<tau>))"   and "H x \<noteq> \<bottom>" shows "oid_of (H x) = oid_of x"
pptop_subtop_ttop: fixes A :: "'b set" shows "subtopology (\<Union>t\<in>topo (\<Union>s\<in>A\<^sup>\<star>. {suff A s}). {t - {LNil}}) (topo (\<Union>s\<in>A\<^sup>\<star>. {suff A s}))"
simps_2: fixes a :: "'a set"   and b :: "'a set" shows "(RES a \<le> RES b) = (a \<subseteq> b)"
compose_assoc: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and B :: "'d set"   and h :: "'f \<Rightarrow> 'e"   and g :: "'d \<Rightarrow> 'f" assumes "f \<in> A \<rightarrow> B" shows "compose A h (compose A g f) = compose A (compose B h g) f"
natUnion_upperbound: fixes f :: "nat \<Rightarrow> 'a set"   and G :: "'a set" assumes "\<And>n. f n \<subseteq> G" shows "natUnion f \<subseteq> G"
add_assoc: fixes a :: "'c DBMEntry"   and b :: "'c DBMEntry"   and c :: "'c DBMEntry" shows "dbm_add (dbm_add a b) c = dbm_add a (dbm_add b c)"
assms_2: shows "(g \<longlongrightarrow> (0::'c)) net"
projection_intersection_neutral: fixes l :: "'b list"   and X :: "'b set"   and Y :: "'b set" assumes "set l \<subseteq> X" shows "l \<upharpoonleft> (X \<inter> Y) = l \<upharpoonleft> Y"
measurable_eP: fixes n :: "nat" shows "eP n \<in> Pi\<^sub>M {0..<n} M \<rightarrow>\<^sub>M subprob_algebra (Pi\<^sub>M {0..<Suc n} M)"
8: shows "sset (gtrace r u) \<subseteq> clean A w V"
config'_rand_snoc: fixes A :: "('a list \<Rightarrow> 'b pmf) \<times> ('a list \<times> 'b \<Rightarrow> 'a \<Rightarrow> ((nat \<times> nat list) \<times> 'b) pmf)"   and s :: "('a list \<times> 'b) pmf"   and rs :: "'a list"   and r :: "'a" shows "Partial_Cost_Model.config'_rand A s (rs @ [r]) = Partial_Cost_Model.config'_rand A s rs \<bind> (\<lambda>s. snd A s r \<bind> (\<lambda>(a, is'). return_pmf (step (fst s) r a, is')))"
term_is_le_rel_canc_left: fixes rel :: "'t \<Rightarrow> 't \<Rightarrow> bool"   and t :: "'a"   and u :: "'t"   and v :: "'t" assumes "ord_term_lin.is_le_rel rel" shows "rel (t \<oplus> u) (t \<oplus> v) = rel u v"
sumlist_as_summset: fixes xs :: "'a list" assumes "set xs \<subseteq> carrier G" shows "sumlist xs = summset (mset xs)"
simps_6: fixes v :: "'b"   and va :: "'b"   and vb :: "'b"   and vc :: "'b"   and vd :: "'c"   and l :: "'b set list" shows "smaller (DenyAllFromTo v va) (AllowPortFromTo vb vc vd) l = (DenyAllFromTo v va = AllowPortFromTo vb vc vd \<or> (if bothNet (DenyAllFromTo v va) = bothNet (AllowPortFromTo vb vc vd) then case AllowPortFromTo vb vc vd of DenyAllFromTo a b \<Rightarrow> DenyAllFromTo v va = DenyAllFromTo b a | _ \<Rightarrow> True else position (bothNet (DenyAllFromTo v va)) l \<le> position (bothNet (AllowPortFromTo vb vc vd)) l))"
power_down_simp: fixes p :: "nat"   and x :: "real"   and n :: "nat" shows "power_down p x (Suc n) = (if odd n then truncate_down (Suc p) ((power_down p x (Suc n div 2))\<^sup>2) else truncate_down (Suc p) (x * power_down p x n))"
inject: fixes x :: "'e"   and ya :: "'e" shows "(object_ptr.Ext x = object_ptr.Ext ya) = (x = ya)"
diff_empty: fixes M :: "'a multiset" shows "M -s {} = M"
nt_match_list_matches: fixes \<gamma> :: "('c \<Rightarrow> 'd \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'd \<Rightarrow> bool)"   and a :: "action"   and p :: "'d"   and l :: "'c negation_type list" shows "nt_match_list \<gamma> a p l = matches \<gamma> (alist_and l) a p"
cas_append_iff: fixes G :: "('c, 'd) pre_digraph"   and u :: "'c"   and p :: "'d list"   and q :: "'d list"   and v :: "'c" shows "pre_digraph.cas G u (p @ q) v = (pre_digraph.cas G u p (last (pre_digraph.awalk_verts G u p)) \<and> pre_digraph.cas G (last (pre_digraph.awalk_verts G u p)) q v)"
Sup_downset_id: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "Sup \<circ> ord.downset less_eq = id"