hcju/isabelleps huggingface dataset:
exec_plan_Append: fixes s :: "('c, 'd) fmap"   and as_a :: "(('c, 'd) fmap \<times> ('c, 'd) fmap) list"   and as_b :: "(('c, 'd) fmap \<times> ('c, 'd) fmap) list" shows "exec_plan s (as_a @ as_b) = exec_plan (exec_plan s as_a) as_b"
order_prod_root_unity: fixes ks :: "nat list"   and x :: "complex" assumes "0 \<notin> set ks" shows "order x (prod_root_unity ks) = length (filter (\<lambda>k. x ^ k = 1) ks)"
assms_2: shows "S \<Turnstile>\<^sub>P \<F> \<phi> w \<G>' (i + j)"
heap_top_max: fixes t :: "'b Tree" assumes "t \<noteq> E"   and "is_heap t" shows "val t = Max_mset (Heap.multiset t)"
simps: fixes n :: "nat" shows "A n = gcd (B n) (D n)"
simps_5: fixes va :: "('a, 'b) rbt"   and vb :: "'a"   and vc :: "'b"   and vd :: "('a, 'b) rbt"   and w :: "'a"   and x :: "'b"   and b :: "('a, 'b) rbt"   and s :: "'a"   and t :: "'b"   and c :: "('a, 'b) rbt"   and y :: "'a"   and z :: "'b" shows "balance (Branch R (Branch B va vb vc vd) w x (Branch R b s t c)) y z Empty = Branch R (Branch B (Branch B va vb vc vd) w x b) s t (Branch B c y z Empty)"
shiftr_rev: fixes w :: "'b word"   and n :: "nat" shows "w >> n = word_reverse (word_reverse w << n)"
intros_5: fixes a :: "'b"   and b :: "'b" assumes "a \<le> b" shows "Lt a \<prec> Le b"
mem_eq: shows "\<forall>a\<in>A. m' a = m a"
snd_image_times: fixes A :: "'b set"   and B :: "'a set" shows "snd ` (A \<times> B) = (if A = {} then {} else B)"
Rf_mono: fixes N :: "'a literal multiset set"   and N' :: "'a literal multiset set" assumes "N \<subseteq> N'" shows "sr.Rf N \<subseteq> sr.Rf N'"
arc_simple_path_subpath: fixes g :: "real \<Rightarrow> 'a"   and u :: "real"   and v :: "real" assumes "simple_path g"   and "u \<in> {0..1}"   and "v \<in> {0..1}"   and "g u \<noteq> g v" shows "arc (subpath u v g)"
simps: fixes b :: "'b list"   and G :: "'b list set" shows "( b \<in>B G ) = (b \<in> G \<and> (\<forall>us. us \<in> lists G\<^sub>+ \<and> concat us = b \<longrightarrow> \<^bold>|us\<^bold>| = 1))"
toplevel_summands_ACI_nPlus: fixes r :: "'c rexp"   and s :: "'c rexp" shows "toplevel_summands (ACI_nPlus r s) = toplevel_summands (Plus r s)"
value_of_inv: fixes v :: "'a \<Rightarrow> ant"   and x :: "'a" assumes "valuation K v"   and "x \<in> carrier K"   and "x \<noteq> \<zero>" shows "v (x\<^bsup>\<hyphen> K\<^esup>) = - v x"
eq: assumes "bi = bj" shows "li < nn \<and> lj < nn \<and> bj < length n_as \<and> (nn, la) \<in> set n_as"
lex_pm_plus_monotone: fixes s :: "'a \<Rightarrow>\<^sub>0 'b"   and t :: "'a \<Rightarrow>\<^sub>0 'b"   and u :: "'a \<Rightarrow>\<^sub>0 'b" assumes "lex_pm s t" shows "lex_pm (s + u) (t + u)"
cases: fixes s :: "('a, 'b) locks \<times> (('b \<Rightarrow> ('x \<times> 'a \<Rightarrow>f nat) option) \<times> 'c) \<times> ('b \<Rightarrow> 'd wait_set_status option) \<times> 'b set"   and t :: "'b"   and ta :: "('a, 'b, 'e, 'c, 'd, 'f) thread_action" assumes "actions_ok s t ta" obtains "lock_ok_las (locks s) t \<lbrace>ta\<rbrace>\<^bsub>l\<^esub>" and "thread_oks (thr s) \<lbrace>ta\<rbrace>\<^bsub>t\<^esub>" and "cond_action_oks s t \<lbrace>ta\<rbrace>\<^bsub>c\<^esub>" and "wset_actions_ok (wset s) t \<lbrace>ta\<rbrace>\<^bsub>w\<^esub>" and "interrupt_actions_ok (interrupts s) \<lbrace>ta\<rbrace>\<^bsub>i\<^esub>"
funpow_fixpoint: fixes f :: "'b \<Rightarrow> 'b"   and x :: "'b"   and n :: "nat" assumes "f x = x" shows "(f ^^ n) x = x"
R_mult: fixes x :: "'a"   and y :: "'a" assumes "x \<in> R"   and "y \<in> R" shows "x * y \<in> R"
inf_p: fixes x :: "'c" shows "x \<squnion> - x = top"
power_mono: fixes a :: "'a"   and b :: "'a"   and n :: "nat" assumes "a \<le> b"   and "(0::'a) \<le> a" shows "a ^ n \<le> b ^ n"
likelihood_compute_def: fixes s :: "'s"   and os :: "'t list" shows "likelihood_compute s os \<equiv> if s \<in> set state_list then Some (\<Sum>t\<leftarrow>state_list. forward s t os) else None"
distinct_remdups_id: fixes xs :: "'a list" assumes "distinct xs" shows "remdups xs = xs"
Gr_Image_image: fixes A :: "'c set"   and f :: "'c \<Rightarrow> 'b"   and B :: "'c set" shows "BNF_Def.Gr A f `` B = f ` (A \<inter> B)"
assms: shows "b \<in> closed_segment a c \<or> \<not> collinear {a, b, c}"
polyexp_fps: fixes xs :: "('b \<times> nat \<times> 'b) list" shows "Abs_fps (eval_polyexp xs) = polyexp_fps xs"
hrp_comp_def: fixes RR' :: "('d \<Rightarrow> 'b \<Rightarrow> assn) \<times> ('d \<Rightarrow> 'c \<Rightarrow> assn)"   and S :: "('d \<times> 'a) set" shows "hrp_comp RR' S \<equiv> (hr_comp (fst RR') S, hr_comp (snd RR') S)"
inf_def: fixes L :: "('a, 'b) gorder_scheme"   and A :: "'a set" shows "\<Sqinter>\<^bsub>L\<^esub>A = (SOME x. is_glb L x A)"
assoc'_def: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "assoc' a b c \<equiv> tuple (tuple (pr1 a (local.prod b c)) (pr1 b c \<cdot> pr0 a (local.prod b c))) (pr0 b c \<cdot> pr0 a (local.prod b c))"
parts_crypt: fixes K :: "enc_key"   and X :: "msg"   and H :: "msg set" assumes "Crypt K X \<in> parts H" shows "X \<in> parts H"
dns1'': fixes x :: "'a" shows "d x \<cdot> x = x"
linear_simps_1: fixes f :: "'a \<Rightarrow> 'b"   and a :: "'a"   and b :: "'a" assumes "bounded_linear f" shows "f (a + b) = f a + f b"
Enc_keys_clean_Un: fixes G :: "msg set"   and H :: "msg set" shows "Enc_keys_clean (G \<union> H) = (Enc_keys_clean G \<and> Enc_keys_clean H)"
scale_minus_left: fixes a :: "'a"   and x :: "'a" shows "- a * x = - (a * x)"
list_of_oalist_OAlist_tc_empty: shows "list_of_oalist_tc OAlist_tc_empty = []"
N: fixes x :: "'a"   and m :: "nat"   and n :: "nat" assumes "x \<in> X"   and "N \<le> m"   and "N \<le> n" shows "dist (f m x) (f n x) < e / 2"
sgp1: shows "singular_chain (p - Suc 0) (subtopology (powertop_real UNIV) (standard_simplex p)) (chain_boundary p (frag_of (restrict id (standard_simplex p))))"
falsif: shows "falsifies\<^sub>g G C\<^sub>1'"
WTrt_hext_mono: 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 T :: "ty"   and h' :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option" assumes "P,E,h \<turnstile> e : T"   and "h \<unlhd> h'" shows "P,E,h' \<turnstile> e : T"
prob_space: fixes s :: "'s" shows "T.prob s (space (T s)) = 1"
equalityE: fixes A :: "'b set"   and B :: "'b set" assumes "A = B" obtains "order.greater_eq B A" and "order.greater_eq A B"
1: shows "[\<^bold>\<not>\<lparr>K,x\<^sup>P\<rparr> in dw]"
PO_a0n_inv1_niagree_init: shows "init a0n \<subseteq> a0n_inv1_niagree"
ln_approx_aux1_def: fixes x' :: "real"   and e :: "real"   and n :: "nat"   and x :: "real" shows "ln_approx_aux1 x' e n x = ln_approx_aux2 x' e n ((x - 1) / (x + 1))"
outer_invar_from_cscc_invarI: fixes it :: "'v set"   and L :: "'v set list"   and D :: "'v set" assumes "cscc_outer_invar it (L, D)" shows "outer_invar it D"
tuple_ABC_in_hom: shows "Maps.in_hom tuple_ABC (Maps.MkIde (src fg\<^sub>0h\<^sub>1.p\<^sub>0)) (Maps.MkIde (src f\<^sub>0gh\<^sub>1.p\<^sub>0))"
sumlist_snoc: fixes xs :: "'a list"   and x :: "'a" shows "R.sumlist (xs @ [x]) = R.sumlist xs + x"
intros_2: fixes i :: "'a"   and p :: "'a pfm"   and q :: "'a pfm" shows "\<turnstile>\<^sub>! (K\<^sub>! i p \<^bold>\<and>\<^sub>! K\<^sub>! i (p \<^bold>\<longrightarrow>\<^sub>! q) \<^bold>\<longrightarrow>\<^sub>! K\<^sub>! i q)"
int1: shows "integrable M (\<lambda>x. f (T x) * max (g x) 0)"
step_1_def: fixes A :: "'b mat" shows "??.Jordan_Normal_Form_Existence.step_1 A = ??.Jordan_Normal_Form_Existence.step_1_main (dim_row A) 0 0 A"
div_int_poly: fixes p :: "int poly"   and q :: "int poly"   and r :: "int poly" shows "(div_int_poly p q = Some r) = (q \<noteq> 0 \<and> p = r * q)"
pth_a: fixes x :: "'b"   and y :: "'b" shows "0 *\<^sub>R x + y \<equiv> y"
finite_cone_empty: fixes n :: "nat" shows "gram_schmidt.finite_cone n {} = {0\<^sub>v n}"
iMOD_mult: fixes r :: "nat"   and m :: "nat"   and k :: "nat" shows "[ r, mod m ] \<otimes> k = [ r * k, mod m * k ]"
cmp_def: fixes \<mu>\<nu> :: "'b \<times> 'b" shows "HoF.cmp \<mu>\<nu> = (if B.VV.arr \<mu>\<nu> then H (F (fst \<mu>\<nu> \<star>\<^sub>B snd \<mu>\<nu>)) \<cdot>\<^sub>D H (\<Phi>\<^sub>F (B.VV.dom \<mu>\<nu>)) \<cdot>\<^sub>D \<Phi>\<^sub>H (F (B.dom (fst \<mu>\<nu>)), F (B.dom (snd \<mu>\<nu>))) else D.null)"
finite_same_card_bij_on_ext_funcset: fixes A :: "'c set"   and B :: "'d set" assumes "finite A"   and "finite B"   and "card A = card B" shows "\<exists>f. f \<in> A \<rightarrow>\<^sub>E B \<and> bij_betw f A B"
that: fixes B :: "'a set" assumes "card B = n"   and "B \<subseteq> A" shows "thesis"
P2_game_alt_def: fixes m1 :: "'msg1"   and m2 :: "'msg2"   and D :: "'view2 \<Rightarrow> bool spmf" shows "P2_game_alt m1 m2 D = coin_spmf \<bind> (\<lambda>b. funct m1 m2 \<bind> (\<lambda>(out1, out2). R2 m1 m2 \<bind> (\<lambda>rview. S2 m2 out2 \<bind> (\<lambda>sview. D (if b then rview else sview) \<bind> (\<lambda>b'. return_spmf (b = b'))))))"
Gamma_asymp_equiv_aux: shows "Gamma \<sim>[at_top] (\<lambda>x. exp c * x powr (x - 1 / 2) / exp x)"
select_convs_15: fixes getSATFlag :: "ExtendedBool"   and getF :: "Literal list list"   and getM :: "(Literal \<times> bool) list"   and getConflictFlag :: "bool"   and getConflictClause :: "nat"   and getQ :: "Literal list"   and getReason :: "Literal \<Rightarrow> nat option"   and getWatch1 :: "nat \<Rightarrow> Literal option"   and getWatch2 :: "nat \<Rightarrow> Literal option"   and getWatchList :: "Literal \<Rightarrow> nat list"   and getC :: "Literal list"   and getCl :: "Literal"   and getCll :: "Literal"   and getCn :: "nat"   and more :: "'a" shows "more \<lparr>getSATFlag = getSATFlag, getF = getF, getM = getM, getConflictFlag = getConflictFlag, getConflictClause = getConflictClause, getQ = getQ, getReason = getReason, getWatch1 = getWatch1, getWatch2 = getWatch2, getWatchList = getWatchList, getC = getC, getCl = getCl, getCll = getCll, getCn = getCn, \<dots> = more\<rparr> = more"
intros_2: fixes rule :: "'rule"   and c :: "'form"   and ants :: "('form, 'var) antecedent fset"   and \<Gamma> :: "'form fset"   and a :: "nat"   and s :: "'subst" assumes "nat_rule rule c ants"   and "\<And>ant f. \<lbrakk>ant |\<in>| ants; f |\<in>| \<Gamma>\<rbrakk> \<Longrightarrow> freshenLC a ` a_fresh ant \<inter> lconsts f = {}"   and "\<And>ant. ant |\<in>| ants \<Longrightarrow> freshenLC a ` a_fresh ant \<inter> subst_lconsts s = {}" shows "eff (NatRule rule) (\<Gamma> \<turnstile> subst s (freshen a c)) ((\<lambda>ant. (\<lambda>p. subst s (freshen a p)) |`| a_hyps ant |\<union>| \<Gamma> \<turnstile> subst s (freshen a (a_conc ant))) |`| ants)"
arg_complex_of_real_positive: fixes k :: "real" assumes "0 < k" shows "arg (cor k) = 0"
comp_id: fixes f :: "'g \<Rightarrow> 'h" shows "f \<circ> id = f"
a: shows "s to x in t = s to x' in ([(x, x')] \<bullet> t)"
FnA: shows "{f^n \<bottom> |. n} \<subseteq> A"
star_sub_one: fixes x :: "'a" assumes "x \<le> top" shows "top = top"
memb_imp_not_empty: fixes x :: "'e"   and S :: "'e set" assumes "x \<in> S" shows "S \<noteq> {}"
first_vec_two_block_non_inter_is_first_vec: fixes A :: "'a mat"   and B :: "'a mat"   and v :: "'a vec"   and n :: "nat" assumes "dim_col A + dim_col B = dim_vec v"   and "dim_row A = n" shows "vec_first (two_block_non_interfering A B *\<^sub>v v) n = A *\<^sub>v vec_first v (dim_col A)"
Aexp_def: fixes cccExp :: "exp \<Rightarrow> Arity \<rightarrow> (Vars.var \<Rightarrow> Arity\<^sub>\<bottom>) \<times> CoCalls"   and e :: "exp" shows "CoCallArityAnalysis.Aexp cccExp e = (\<Lambda> a. fst (cccExp e\<cdot>a))"
nolb_Or: fixes \<phi>\<^sub>1 :: "atom fm"   and \<phi>\<^sub>2 :: "atom fm"   and xs :: "'a list"   and l :: "'a"   and x :: "'a" shows "nolb (Or \<phi>\<^sub>1 \<phi>\<^sub>2) xs l x = (nolb \<phi>\<^sub>1 xs l x \<and> nolb \<phi>\<^sub>2 xs l x)"
relH_sym: fixes as :: "nat set"   and h :: "heap"   and h' :: "heap" assumes "relH as h h'" shows "relH as h' h"
eq_0_size: fixes t :: "'b tree" shows "(0 = size t) = (t = \<langle>\<rangle>)"
e_if_cong: fixes e1 :: "exp"   and e1' :: "exp"   and e2 :: "exp"   and e2' :: "exp"   and e3 :: "exp"   and e3' :: "exp" assumes "E e1 = E e1'"   and "E e2 = E e2'"   and "E e3 = E e3'" shows "E (EIf e1 e2 e3) = E (EIf e1' e2' e3')"
ptensor_mat_carrier: fixes m1 :: "'a mat"   and m2 :: "'a mat" shows "st2.ptensor_mat m1 m2 \<in> carrier_mat st2.d0 st2.d0"
between_imp_mod_gr0: fixes m :: "nat"   and k :: "nat"   and n :: "nat" assumes "m * k < n"   and "n < m * k + m" shows "0 < n mod m"
int_and_extra_simps_2: fixes w :: "int" shows "w AND - 1 = w"
onorm_open_ball: fixes f :: "'b \<Rightarrow>\<^sub>L 'c" shows "\<parallel>f\<parallel> = Sup {\<parallel>f *\<^sub>v x\<parallel> |x. \<parallel>x\<parallel> < 1}"
hyps_2: shows "\<forall>t. \<Gamma>\<turnstile> \<langle>c,s\<rangle> \<Rightarrow> t \<longrightarrow> \<Gamma>\<turnstile>cs,css \<Down> t \<and> \<not> AlternativeSmallStep.inf \<Gamma> cs css t"
set_permutations_of_list_impl: fixes xs :: "'a list" shows "set (permutations_of_list_impl xs) = permutations_of_multiset (mset xs)"
pdiv_def: fixes x :: "'a mpoly"   and y :: "'a mpoly" shows "x pdiv y = (THE (a, q). \<exists>r. pseudo_divmod_rel a x y q r)"
first_coeff_le_tail: fixes g :: "complex poly" shows "(cmod (lead_coeff g))\<^sup>2 \<le> norm2 g"
H2: shows "geodesic_segment_param G p E \<in> H"
add_edge_def: shows "add_edge \<equiv> \<lambda>(u, v) d g. M_update u ((d, v) # the_default [] (M_lookup g u)) g"
invalidate_clone: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> assn"   and x :: "'a"   and y :: "'b" shows "R x y \<Longrightarrow>\<^sub>A invalid_assn R x y * R x y"
Inf_lower: fixes Inf :: "'d set \<Rightarrow> 'd"   and Sup :: "'d set \<Rightarrow> 'd"   and inf :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sup :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and bot :: "'d"   and top :: "'d"   and x :: "'d"   and A :: "'d set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "x \<in> A" shows "less_eq (Inf A) x"
list_ex_simps_2: fixes P :: "'b \<Rightarrow> bool" shows "list_ex P [] = False"
b: shows "\<forall>x\<in>S. \<bar>x - l\<bar> \<le> e"
push_values: fixes k :: "'c"   and p :: "'d"   and q :: "('c, 'd) pq" shows "set |push k p q| = set |q| \<union> {k}"
EM_remainder_limit_1: fixes b :: "int" assumes "a \<le> b" shows "EM_remainder N (fs N) b = C' - EM_remainder' N (fs N) (real_of_int a) (real_of_int b)"
weight_def: fixes w :: "'b set \<Rightarrow> nat"   and g :: "'b ugraph" shows "weight w g \<equiv> (\<Sum>e\<in>edges g. w (uedge e)) div 2"
image_UN: fixes f :: "'d \<Rightarrow> 'c"   and B :: "'e \<Rightarrow> 'd set"   and A :: "'e set" shows "f ` \<Union> (B ` A) = (\<Union>x\<in>A. f ` B x)"
R: fixes a :: "'a"   and b :: "'b" assumes "R a b" shows "R a y = R x b"
assms_2: shows " s1 \<longrightarrow>\<^sub>\<beta>\<^sup>* s2"
powser_ms_aux'_def: fixes cs :: "real msstream"   and xs :: "('a \<times> real) msllist" shows "powser_ms_aux' cs xs = powser_ms_aux (msllist_of_msstream cs) xs"
step: fixes x :: "'a node \<times> 'a node set" shows "f x \<le> SPEC (expand_rslt_exist_eq x)"
echelon_echelon_form_of_upt_k: fixes k :: "nat"   and A :: "(('d, 'e) vec, 'f) vec"   and bezout :: "'d \<Rightarrow> 'd \<Rightarrow> 'd \<times> 'd \<times> 'd \<times> 'd \<times> 'd" assumes "k < ncols A"   and "is_bezout_ext bezout" shows "echelon_form_upt_k (echelon_form_of_upt_k A k bezout) (Suc k)"
assms_2: shows "strictly_maximal_literal P1 (Pos A)"
mod_div_trivial: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and a :: "'a"   and b :: "'a" assumes "class.euclidean_semiring_cancel divide plus minus zero times one modulo euclidean_size" shows "divide (modulo a b) b = zero"
HEndPhase1_Inv2a: fixes s :: "state"   and s' :: "state"   and p :: "Proc" assumes "HEndPhase1 s s' p"   and "HInv1 s"   and "Inv2a s"   and "Inv2c_inner s p" shows "Inv2a s'"
st1: shows "inv (CONJ f) = inv (inv ni_bij \<circ> f \<circ> ni_bij)"
chamber_retraction1: fixes X :: "'b set set"   and f :: "'b \<Rightarrow> 'b"   and C :: "'b set" assumes "ChamberComplexFolding X f"   and "C \<in> f \<turnstile> ChamberComplex.chamber_system X" shows "fixespointwise f C"
lfinite_code_2: fixes x :: "'c"   and xs :: "'c llist" shows "lfinite (LCons x xs) = lfinite xs"
assertion_inf_comp_eq: fixes x :: "'a"   and y :: "'a" assumes "x \<in> assertion"   and "y \<in> assertion" shows "x \<sqinter> y = x * y"
assms: shows "\<forall>\<^sub>F x in at_top. a x \<le> b x"
distinct_fst_def: shows "distinct_fst \<equiv> distinct \<circ> map fst"
inf_absorb2: fixes inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and y :: "'c"   and x :: "'c" assumes "class.semilattice_inf inf less_eq less"   and "less_eq y x" shows "inf x y = y"
circline_point_0_cmat_def: shows "circline_point_0_cmat = (1, 0, 0, 0)"
arccos_eq_Re_Arccos: fixes x :: "real" assumes "\<bar>x\<bar> \<le> 1" shows "arccos x = Re (Arccos (complex_of_real x))"
unwind_def: fixes \<Delta> :: "'state \<Rightarrow> 'value list \<Rightarrow> 'state \<Rightarrow> 'value list \<Rightarrow> bool" shows "unwind \<Delta> \<equiv> \<forall>s vl s1 vl1. reachNT s \<and> reach s1 \<and> \<Delta> s vl s1 vl1 \<longrightarrow> vl \<noteq> [] \<and> exit s (hd vl) \<or> iaction \<Delta> s vl s1 vl1 \<or> (vl \<noteq> [] \<or> vl1 = []) \<and> reaction \<Delta> s vl s1 vl1"
two_comparisons_into_case_order_3: fixes comp :: "'a \<Rightarrow> 'a \<Rightarrow> order"   and x :: "'a"   and y :: "'a"   and P :: "'b"   and Q :: "'b"   and R :: "'b" assumes "comparator comp" shows "(if le_of_comp comp x y then if le_of_comp comp y x then P else Q else R) = (case comp x y of Eq \<Rightarrow> P | Lt \<Rightarrow> Q | Gt \<Rightarrow> R)"
Y_def: shows "Y \<equiv> \<lambda>i \<omega>. if i \<in> I then X i \<omega> else (0::'b)"
simps_7: fixes a :: "MIR.num"   and b :: "MIR.num" shows "zsplit0 (Sub a b) = (let (ia, a') = zsplit0 a; (ib, b') = zsplit0 b in (ia - ib, Sub a' b'))"
intros_5: fixes \<Gamma> :: "'b formula set"   and F :: "'b formula"   and G :: "'b formula" assumes "\<Gamma> \<turnstile> F \<^bold>\<and> G" shows "\<Gamma> \<turnstile> F"
toTree_def: fixes t :: "('a \<times> 'b) list" shows "FingerTree.toTree t \<equiv> Abs_FingerTree (FingerTreeStruc.toTree t)"
add_commute: fixes x :: "'a"   and y :: "'a" assumes "x \<in> E"   and "y \<in> E" shows "x + y = y + x"
yoneda_functor_fixed_object_axioms: shows "yoneda_functor_fixed_object (\<cdot>) (\<cdot>\<^sub>S) \<phi> a"
assms_1: fixes x :: "'a" shows "x \<in> \<gamma> x"
right_total_OO: fixes A :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and B :: "'d \<Rightarrow> 'e \<Rightarrow> bool" assumes "right_total A"   and "right_total B" shows "right_total (A OO B)"
right_total_conversep: fixes A :: "'d \<Rightarrow> 'c \<Rightarrow> bool" shows "right_total A\<inverse>\<inverse> = left_total A"
compress_normalize_output_interfaces_def: fixes m :: "'b common_primitive match_expr" shows "compress_normalize_output_interfaces m \<equiv> compress_normalize_primitive (is_Oiface, oiface_sel) OIface compress_interfaces m"
fds_converges_add: fixes f :: "'b fds"   and s :: "'b"   and g :: "'b fds" assumes "fds_converges f s"   and "fds_converges g s" shows "fds_converges (f + g) s"
weakOpen: fixes \<Psi> :: "'b"   and Q :: "('a, 'b, 'c) psi"   and P :: "('a, 'b, 'c) psi"   and M :: "'a"   and xvec :: "name list"   and yvec :: "name list"   and N :: "'a"   and P' :: "('a, 'b, 'c) psi"   and x :: "name" assumes "\<Psi> : Q \<rhd> P \<Longrightarrow>M\<lparr>\<nu>*(xvec @ yvec)\<rparr>\<langle>N\<rangle> \<prec> P'"   and "x \<in> supp N"   and "x \<sharp> \<Psi>"   and "x \<sharp> M"   and "x \<sharp> xvec"   and "x \<sharp> yvec" shows "\<Psi> : \<lparr>\<nu>x\<rparr>Q \<rhd> \<lparr>\<nu>x\<rparr>P \<Longrightarrow>M\<lparr>\<nu>*(xvec @ x # yvec)\<rparr>\<langle>N\<rangle> \<prec> P'"
Red_I_subs_Red_I_diff_Red_F: fixes N :: "'f set" shows "Red_I N \<subseteq> Red_I (N - Red_F N)"
clearjunk_delete: fixes x :: "'a"   and al :: "('a \<times> 'b) list" shows "AList.clearjunk (AList.delete x al) = AList.delete x (AList.clearjunk al)"
below_meet2: fixes z :: "'c"   and x :: "'c" assumes "z \<sqsubseteq> x" shows "x \<sqinter> z = z"
R: fixes x :: "nat" assumes "\<And>y. y < x \<Longrightarrow> P y" shows "P x"
simps_2: fixes f :: "('fsym, 'tp) efsym"   and Tl :: "('fsym, 'tp) efsym trm list" shows "TE.tpOf (Fn f Tl) = TE_resOf f"
pfun_app_add: fixes x :: "'a"   and g :: "('a, 'b) pfun"   and f :: "('a, 'b) pfun" assumes "x \<in> pdom g" shows "(f + g)(x)\<^sub>p = g(x)\<^sub>p"
L_prepare: shows "L = (ladder_n L 0, ladder_j L 0, ladder_i L 0) # (ladder_n L 1, ladder_j L 1, ladder_ix L 1) # drop 2 L"
5: fixes am :: "of_match_field set"   and bm :: "of_match_field set"   and ab :: "32 simple_match"   and bb :: "32 simple_match" assumes "OF_match_fields_unsafe am p"   and "OF_match_fields_unsafe bm p"   and "am \<noteq> bm"   and "am \<in> set (simple_match_to_of_match ab ifs)"   and "bm \<in> set (simple_match_to_of_match bb ifs)"   and "\<not> ab \<noteq> bb" shows "False"
vors: shows "i < |ls| \<and> j < |ls| \<and> ls ! i = x \<and> ls ! j = y \<and> Suc i = j"
bounded_iff: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "semilattice_order f less_eq less" shows "less_eq a (f b c) = (less_eq a b \<and> less_eq a c)"
Range_iff: fixes a :: "'g"   and r :: "('h \<times> 'g) set" shows "(a \<in> Range r) = (\<exists>y. (y, a) \<in> r)"
lowEquivReflexive: fixes \<Gamma> :: "char list \<Rightarrow> secLevel option"   and s1 :: "char list \<Rightarrow> val option" shows "\<Gamma> \<turnstile> s1 \<approx>\<^sub>L s1"
mayDiverge_Par_R: fixes c2 :: "('test, 'atom) com"   and s :: "'state"   and c1 :: "('test, 'atom) com" assumes "mayDiverge c2 s" shows "mayDiverge (c1 | c2) s"
simps_3: fixes ParamUses :: "'h \<Rightarrow> 'g set list" shows "lift_ParamUses ParamUses NewExit = []"
remdups_sort_def: fixes xs :: "'a list" shows "remdups_sort xs = remdups_adj (sort xs)"
symmetric_mk_symmetric: fixes G :: "('c, 'd) pre_digraph" shows "symmetric (with_proj (mk_symmetric G))"
assms_3: fixes t :: "'a"   and queue :: "'a list" assumes "t \<noteq> t0"   and "t0 \<in> set queue"   and "P (queue @ [t])" shows "P (t # queue)"
lr_fps_denominator_def: fixes cs :: "'b list" shows "lr_fps_denominator cs = Poly (rev cs)"
invert_opposite_sides_of_square: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b"   and g :: "'b"   and h :: "'b"   and k :: "'b" assumes "category C"   and "partial_magma.arr C (C f g)"   and "C f g = C h k"   and "category.iso C f"   and "category.iso C k" shows "partial_magma.arr C (C g (category.inv C k)) \<and> partial_magma.arr C (C (category.inv C f) h) \<and> C g (category.inv C k) = C (category.inv C f) h"
tendsto_zero_mult_left_iff: fixes c :: "'b"   and a :: "nat \<Rightarrow> 'b" assumes "c \<noteq> (0::'b)" shows "(\<lambda>n. c * a n) \<longlonglongrightarrow> (0::'b) = a \<longlonglongrightarrow> (0::'b)"
eSum_single: fixes n :: "nat"   and f :: "nat \<Rightarrow> ant"   and j :: "nat" assumes "\<forall>j\<le>n. f j \<in> Z\<^sub>\<infinity>"   and "j \<le> n"   and "\<forall>l\<in>{h. h \<le> n} - {j}. f l = 0" shows "ASum f n = f j"
vrestr_meas: fixes M :: "'b measure"   and T :: "'b \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'b"   and A :: "'b set" assumes "qmpt M T"   and "g \<in> M \<rightarrow>\<^sub>M M"   and "A \<in> sets M" shows "qmpt.vimage_restr M g A \<in> sets M"