hcju/isabelleps huggingface dataset:
ReadMem: fixes a :: "nat"   and E :: "('a \<times> ('b, 'a) obs_event action) llist"   and ad :: "'b"   and al :: "addr_loc"   and v :: "'b val" assumes "a \<in> actions E"   and "action_obs E a = NormalAction (ReadMem ad al v)" shows "a \<in> read_actions E"
cSup_mult: fixes S :: "real set"   and b :: "real"   and c :: "real" assumes "S \<noteq> {}"   and "\<And>x. x \<in> S \<Longrightarrow> x \<le> b"   and "0 \<le> c" shows "c * Sup S = Sup {c * x |x. x \<in> S}"
assms: shows "X \<in> fake bad (synth (analz (extr bad IK CH))) CH"
le_ereal_less: fixes a :: "ereal"   and x :: "real"   and y :: "real" assumes "a \<le> ereal x"   and "x < y" shows "a < ereal y"
eq_iff_9: fixes term1 :: "Syntax.term"   and term2 :: "Syntax.term"   and term1a :: "Syntax.term"   and term2a :: "Syntax.term" shows "(App term1 term2 = App term1a term2a) = (term1 = term1a \<and> term2 = term2a)"
vec_vec_Tensor_left_id: fixes v :: "'a list" shows "vec_vec_Tensor [id] v = v"
release_False_mono: fixes \<R> :: "nat \<Rightarrow> bool option"   and a :: "nat"   and sb :: "'b memref list"   and S :: "nat set" assumes "\<R> a = Some False"   and "outstanding_refs is_volatile_Write\<^sub>s\<^sub>b sb = {}" shows "release sb S \<R> a = Some False"
foundation1: fixes \<tau> :: "'b state \<times> 'b state" shows "\<tau> \<Turnstile> true"
all_before_eq_3: fixes f :: "'c \<Rightarrow> 'd \<Rightarrow> 'b"   and x :: "'a \<Rightarrow> 'c"   and y :: "'a \<Rightarrow> 'd" shows "$f<x, y> \<equiv> f<$x, $y>"
vcons_eq_vcard_eq: fixes x\<^sub>1 :: "V"   and x\<^sub>2 :: "V" assumes "xs\<^sub>1 #\<^sub>\<circ> x\<^sub>1 = xs\<^sub>2 #\<^sub>\<circ> x\<^sub>2" shows "vcard xs\<^sub>1 = vcard xs\<^sub>2"
hyps_4: fixes \<Gamma>\<^sub>2 :: "(name \<times> ty) list"   and u :: "trm"   and \<Gamma>\<^sub>1 :: "(name \<times> ty) list"   and t :: "trm"   and T :: "ty"   and \<theta> :: "(name \<times> trm) list" assumes "\<Gamma>\<^sub>2 \<turnstile> u : U"   and "\<Gamma>\<^sub>1 @ \<Delta>\<^sub>2 @ \<Gamma>\<^sub>2 \<turnstile> t : T"   and "\<turnstile> q \<rhd> u \<Rightarrow> \<theta>"   and "supp q \<sharp>* u" shows "\<Gamma>\<^sub>1 @ \<Gamma>\<^sub>2 \<turnstile> \<theta>\<lparr>t\<rparr> : T"
closed_real_abs_le: fixes r :: "real" shows "closed {w \<in> \<real>. \<bar>Re w\<bar> \<le> r}"
takefill_numeral_Cons: fixes fill :: "'b"   and k :: "num"   and x :: "'b"   and xs :: "'b list" shows "takefill fill (numeral k) (x # xs) = x # takefill fill (pred_numeral k) xs"
partition_between_ref_def: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and h :: "'a \<Rightarrow> 'b"   and lo :: "nat"   and hi :: "nat"   and xs0 :: "'a list" shows "partition_between_ref R h lo hi xs0 = ASSERT (hi < length xs0 \<and> hi < length xs0 \<and> lo \<le> hi) \<bind> (\<lambda>_. choose_pivot3 R h xs0 lo hi \<bind> (\<lambda>k. ASSERT (k < length xs0) \<bind> (\<lambda>_. RETURN (swap xs0 k hi) \<bind> (\<lambda>xs. ASSERT (length xs = length xs0) \<bind> (\<lambda>_. partition_main R h lo hi xs)))))"
simpeq_def: fixes t :: "tm" shows "simpeq t = (let (c, s) = split0 (simptm t) in if c = 0\<^sub>p then eq s else Eq (CNP 0 c s))"
induct: fixes P :: "pol \<Rightarrow> pol \<Rightarrow> bool"   and a0 :: "pol"   and a1 :: "pol" assumes "\<And>a b. P (Pc a) (Pc b)"   and "\<And>c i Pa. P Pa (Pc c) \<Longrightarrow> P (Pc c) (Pinj i Pa)"   and "\<And>i Pa c. P Pa (Pc c) \<Longrightarrow> P (Pinj i Pa) (Pc c)"   and "\<And>c Pa i Q. P Q (Pc c) \<Longrightarrow> P (Pc c) (PX Pa i Q)"   and "\<And>Pa i Q c. P Q (Pc c) \<Longrightarrow> P (PX Pa i Q) (Pc c)"   and "\<And>x Pa y Q. \<lbrakk>x = y \<Longrightarrow> P Pa Q; \<lbrakk>x \<noteq> y; y < x\<rbrakk> \<Longrightarrow> P (Pinj (x - y) Pa) Q; \<lbrakk>x \<noteq> y; \<not> y < x\<rbrakk> \<Longrightarrow> P (Pinj (y - x) Q) Pa\<rbrakk> \<Longrightarrow> P (Pinj x Pa) (Pinj y Q)"   and "\<And>x Pa Q y R. \<lbrakk>x = 0 \<Longrightarrow> P Pa (PX Q y R); \<lbrakk>x \<noteq> 0; x = 1\<rbrakk> \<Longrightarrow> P R Pa; \<lbrakk>x \<noteq> 0; x \<noteq> 1\<rbrakk> \<Longrightarrow> P R (Pinj (x - 1) Pa)\<rbrakk> \<Longrightarrow> P (Pinj x Pa) (PX Q y R)"   and "\<And>Pa x R y Q. \<lbrakk>y = 0 \<Longrightarrow> P (PX Pa x R) Q; \<lbrakk>y \<noteq> 0; y = 1\<rbrakk> \<Longrightarrow> P R Q; \<lbrakk>y \<noteq> 0; y \<noteq> 1\<rbrakk> \<Longrightarrow> P R (Pinj (y - 1) Q)\<rbrakk> \<Longrightarrow> P (PX Pa x R) (Pinj y Q)"   and "\<And>P1 x P2 Q1 y Q2. \<lbrakk>x = y \<Longrightarrow> P P1 Q1; x = y \<Longrightarrow> P P2 Q2; \<lbrakk>x \<noteq> y; y < x\<rbrakk> \<Longrightarrow> P (PX P1 (x - y) (Pc 0)) Q1; \<lbrakk>x \<noteq> y; y < x\<rbrakk> \<Longrightarrow> P P2 Q2; \<lbrakk>x \<noteq> y; \<not> y < x\<rbrakk> \<Longrightarrow> P (PX Q1 (y - x) (Pc 0)) P1; \<lbrakk>x \<noteq> y; \<not> y < x\<rbrakk> \<Longrightarrow> P P2 Q2\<rbrakk> \<Longrightarrow> P (PX P1 x P2) (PX Q1 y Q2)" shows "P a0 a1"
weight_simp: fixes w :: "'c \<times> nat \<Rightarrow> nat"   and w0 :: "nat"   and scf :: "'c \<times> nat \<Rightarrow> nat \<Rightarrow> nat"   and f :: "'c"   and ts :: "('c, 'd) Term.term list" shows "weight_fun.weight w w0 scf (Fun f ts) = w (f, length ts) + sum_list (map (weight_fun.weight w w0 scf) (scf_list (scf (f, length ts)) ts))"
recvmsg_simps_5: fixes P :: "'a \<Rightarrow> bool"   and m :: "'a" shows "recvmsg P (send m) = True"
rgt0: fixes i :: "'a" assumes "i \<in> K" shows "0 < r i"
replace_coeff_monom: fixes f :: "'c \<Rightarrow> 'b"   and m :: "nat \<Rightarrow>\<^sub>0 nat"   and a :: "'c" assumes "f (0::'c) = (0::'b)" shows "replace_coeff f (monom m a) = monom m (f a)"
is_finite_weak_ranking_def: fixes xs :: "'b set list" shows "is_finite_weak_ranking xs = (is_weak_ranking xs \<and> (\<forall>x\<in>set xs. finite x))"
effect_refE: fixes v :: "'b"   and h :: "heap"   and h' :: "heap"   and r :: "'b ref" assumes "effect (ref v) h h' r" obtains "Ref.get h' r = v" and "Ref.present h' r" and "\<not> Ref.present h r"
get_root_node_locs_def: shows "Shadow_DOM.get_root_node_locs = Shadow_DOM.get_ancestors_locs"
freshBoundOutputAction: fixes P :: "pi"   and a :: "name"   and x :: "name"   and P' :: "pi"   and c :: "name" assumes "P \<longmapsto>\<^sub>e a<\<nu>x> \<prec>\<^sub>e P'"   and "c \<sharp> P" shows "c \<noteq> a"
arr_cod: fixes f :: "('a, 'a set) Maps.arr" assumes "Maps.arr f" shows "Maps.arr (Maps.cod f)"
simps: fixes hb_weak :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and hb :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c list" assumes "happens_before hb_weak hb" shows "happens_before.hb_consistent hb a = (a = [] \<or> (\<exists>xs y. a = xs @ [y] \<and> happens_before.hb_consistent hb xs \<and> (\<forall>x\<in>set xs. \<not> hb y x)))"
dim: fixes xs :: "'a vec list"   and n :: "nat" assumes "\<And>x. x \<in> set xs \<Longrightarrow> x \<in> carrier_vec n" shows "dim_vec (abelian_monoid.sumlist (module_vec TYPE('a) n) xs) = n"
Fvars_ground_aux: fixes t :: "tm"   and B :: "name set" assumes "Fvars t \<subseteq> B" shows "ground_aux t (atom ` B)"
no_outstanding_vol_write_takeWhile_append: fixes sb :: "'b memref list"   and xs :: "'b memref list" assumes "outstanding_refs is_volatile_Write\<^sub>s\<^sub>b sb = {}" shows "takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) (sb @ xs) = sb @ takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) xs"
app\<^sub>i_Getfield: fixes F :: "char list"   and C :: "char list"   and P :: "'a prog"   and pc :: "nat"   and mxs :: "nat"   and T\<^sub>r :: "ty"   and T :: "ty"   and ST :: "ty list"   and LT :: "ty err list" shows "app\<^sub>i (Getfield F C, P, pc, mxs, T\<^sub>r, T # ST, LT) = (\<exists>T\<^sub>f. P \<turnstile> C sees F,NonStatic:T\<^sub>f in C \<and> subtype P T (Class C))"
relcompEpair: fixes a :: "'a"   and c :: "'b"   and r :: "('a \<times> 'c) set"   and s :: "('c \<times> 'b) set" assumes "(a, c) \<in> r O s" obtains b :: "'c" where "(a, b) \<in> r" and "(b, c) \<in> s"
5: shows "set (stake (length as) xs) \<inter> a \<noteq> {}"
assms_3: shows "(card E choose s) * (r - 1) ^ k * fact k < card F"
full_pmdl_iff: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and K :: "'k set" shows "(p \<in> full_pmdl K) = (component_of_term ` keys p \<subseteq> K)"
elims: fixes x :: "'a list \<times> 'a list"   and y :: "'a list \<times> 'a list" assumes "tl_queue_t x = y" obtains "x = ([], [])" and "y = ([], [])"   | l :: "'a" where "x = ([], [l__])" and "y = ([], [])"   | l :: "'a" and v :: "'a" and va :: "'a list" where "x = ([], l # v # va)" and "y = (tl (rev (v # va)), [l])"   | a :: "'a" and as :: "'a list" and fs :: "'a list" where "x = (a # as, fs)" and "y = (as, fs)"
dual\<^sub>s\<^sub>t_fv: fixes A :: "('b, 'a) strand_step list" shows "fv\<^sub>s\<^sub>t (dual\<^sub>s\<^sub>t A) = fv\<^sub>s\<^sub>t A"
prems: shows "upd_cs cs (a' # as') = c' # cs'"
poly_2i_def: shows "poly_2i \<equiv> [:4, 0, 1:]"
False: shows "\<not> (q = 0 \<or> p = 0)"
complete_def: fixes A :: "'b set"   and r :: "('b \<times> 'b) set" shows "complete A r \<equiv> \<forall>x\<in>A. \<forall>y\<in>A - {x}. x \<^bsub>r\<^esub>\<preceq> y \<or> y \<^bsub>r\<^esub>\<preceq> x"
Rev_PriK_def: shows "Rev_PriK \<equiv> SOME Rev_PriK. \<exists>Sec_PriK Tok_PriK. inj Sec_PriK \<and> inj Tok_PriK \<and> Rev_PriK \<notin> range Sec_PriK \<and> Rev_PriK \<notin> range Tok_PriK \<and> range Sec_PriK \<inter> range Tok_PriK = {}"
in'_2: shows "real_of_rat y \<le> real_of_rat r"
substdbasis_expansion_unique: fixes d :: "'a set"   and f :: "'a \<Rightarrow> real"   and x :: "'a" assumes "d \<subseteq> Basis" shows "((\<Sum>i\<in>d. f i *\<^sub>R i) = x) = (\<forall>i\<in>Basis. (i \<in> d \<longrightarrow> f i = x \<bullet> i) \<and> (i \<notin> d \<longrightarrow> x \<bullet> i = 0))"
modular_1_aux: fixes f :: "'a \<Rightarrow> 'a"   and g :: "'a \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a" assumes "is_conjugation f g" shows "f (x \<cdot> - g y) \<cdot> y = 0"
is_nth_power_upto_unit_def: fixes n :: "nat"   and x :: "'a" shows "is_nth_power_upto_unit n x = (\<exists>u. is_unit u \<and> is_nth_power n (u * x))"
get_scdom_component_contains_get_dom_component: fixes h :: "(_) heap"   and ptr' :: "(_) object_ptr"   and sc :: "(_) object_ptr list"   and thesis :: "bool"   and ptr :: "(_) object_ptr" assumes "\<And>c. \<lbrakk>h \<turnstile> get_dom_component ptr' \<rightarrow>\<^sub>r c; set c \<subseteq> set sc\<rbrakk> \<Longrightarrow> thesis"   and "heap_is_wellformed h"   and "type_wf h"   and "known_ptrs h"   and "h \<turnstile> get_scdom_component ptr \<rightarrow>\<^sub>r sc"   and "ptr' \<in> set sc" shows "thesis"
hyps: assumes "P []"   and "\<And>x. P [x]"   and "\<And>y xs x. P (y # xs) \<Longrightarrow> P (x # y # xs)" shows "P xs"
simps_2: fixes f :: "'f \<Rightarrow> 'g \<Rightarrow> 'e"   and x1 :: "'f"   and x2 :: "'g" shows "(case (x1, x2) of (x, xa) \<Rightarrow> f x xa) = f x1 x2"
axioms_2: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i> :: "'b \<Rightarrow> 'b" assumes "prebicategory_with_units V H \<a> \<i>" shows "weak_composition V H"
lifting: fixes C\<^sub>1 :: "fterm literal set"   and C\<^sub>2 :: "fterm literal set"   and C\<^sub>1' :: "fterm literal set"   and C\<^sub>2' :: "fterm literal set"   and L\<^sub>1' :: "fterm literal set"   and L\<^sub>2' :: "fterm literal set"   and \<sigma> :: "char list \<Rightarrow> fterm" assumes "finite C\<^sub>1 \<and> finite C\<^sub>2"   and "vars\<^sub>l\<^sub>s C\<^sub>1 \<inter> vars\<^sub>l\<^sub>s C\<^sub>2 = {}"   and "instance_of\<^sub>l\<^sub>s C\<^sub>1' C\<^sub>1 \<and> instance_of\<^sub>l\<^sub>s C\<^sub>2' C\<^sub>2"   and "applicable C\<^sub>1' C\<^sub>2' L\<^sub>1' L\<^sub>2' \<sigma>" shows "\<exists>L\<^sub>1 L\<^sub>2 \<tau>. applicable C\<^sub>1 C\<^sub>2 L\<^sub>1 L\<^sub>2 \<tau> \<and> instance_of\<^sub>l\<^sub>s (resolution C\<^sub>1' C\<^sub>2' L\<^sub>1' L\<^sub>2' \<sigma>) (resolution C\<^sub>1 C\<^sub>2 L\<^sub>1 L\<^sub>2 \<tau>)"
nat_abs_dvd_iff: fixes k :: "int"   and n :: "nat" shows "(nat \<bar>k\<bar> dvd n) = (k dvd int n)"
repeat_stabilize_funpow: fixes n :: "nat"   and f :: "'b \<Rightarrow> 'b"   and v :: "'b" shows "repeat_stabilize n f v = (f ^^ n) v"
assms_1: shows "p = {l}broadcast(fmsg) . qq"
ccpo_Sup_least_inv: fixes Sup :: "'b set \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and C :: "'b set"   and x :: "'b" assumes "class.ccpo Sup less_eq less"   and "Complete_Partial_Order.chain less_eq C"   and "less x (Sup C)" obtains y :: "'b" where "y \<in> C" and "\<not> less_eq y x"
min_set_def: fixes X :: "'b set" shows "min_set X = {y \<in> X. \<forall>x\<in>X. x \<le> y \<longrightarrow> x = y}"
execn_Normal_elim_cases_3: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "('e, 'd, 'f) com"   and s :: "'e"   and n :: "nat"   and t :: "('e, 'f) xstate" assumes "\<Gamma>\<turnstile> \<langle>c,Abrupt s\<rangle> =n\<Rightarrow> t" obtains "t = Abrupt s"
int_less: fixes a :: "'a interval"   and b :: "'a interval" shows "(a < b) = (low a < low b \<or> low a = low b \<and> high a < high b)"
assms_3: fixes i :: "nat" assumes "p i \<noteq> 0" shows "odd i"
map_pmf_def: fixes f :: "'b \<Rightarrow> 'a"   and M :: "'b pmf" shows "map_pmf f M = bind_pmf M (\<lambda>x. return_pmf (f x))"
fps_sin_def: fixes c :: "'a" shows "fps_sin c = Abs_fps (\<lambda>n. if even n then 0::'a else (- (1::'a)) ^ ((n - 1) div 2) * c ^ n / of_nat (fact n))"
list_all2_all_nthI: fixes a :: "'e list"   and b :: "'f list"   and P :: "'e \<Rightarrow> 'f \<Rightarrow> bool" assumes "length a = length b"   and "\<And>n. n < length a \<Longrightarrow> P (a ! n) (b ! n)" shows "list_all2 P a b"
float_add: fixes a1 :: "int"   and e1 :: "int"   and a2 :: "int"   and e2 :: "int" shows "float (a1, e1) + float (a2, e2) = (if e1 \<le> e2 then float (a1 + a2 * 2 ^ nat (e2 - e1), e1) else float (a1 * 2 ^ nat (e1 - e2) + a2, e2))"
power2_le_imp_le: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and x :: "'b"   and y :: "'b" assumes "class.linordered_semidom plus minus less_eq less zero times one"   and "less_eq (power.power one times x 2) (power.power one times y 2)"   and "less_eq zero y" shows "less_eq x y"
favorite_def: fixes R :: "'d \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> bool"   and i :: "'d" shows "favorite R i = the_elem (favorites R i)"
intro: fixes H :: "'c set"   and G :: "('c, 'd) monoid_scheme" assumes "H \<subseteq> carrier G"   and "\<And>x y. \<lbrakk>x \<in> H; y \<in> H\<rbrakk> \<Longrightarrow> x \<otimes>\<^bsub>G\<^esub> y \<in> H"   and "\<one>\<^bsub>G\<^esub> \<in> H"   and "\<And>x. x \<in> H \<Longrightarrow> inv\<^bsub>G\<^esub> x \<in> H" shows "subgroup H G"
add: fixes b1 :: "(real, 'n) vec"   and b2 :: "(real, 'n) vec" shows "f (b1 + b2) = f b1 + f b2"
inv_unique': fixes x :: "'a vec"   and y :: "'a vec" assumes "x \<in> carrier_vec n"   and "y \<in> carrier_vec n"   and "x + y = 0\<^sub>v n"   and "y + x = 0\<^sub>v n" shows "y = inv\<^bsub>\<lparr>carrier = carrier_vec n, mult = (+), one = 0\<^sub>v n, \<dots> = undefined\<rparr>\<^esub> x"
collision_d: fixes a\<^sub>e :: "real"   and v\<^sub>e :: "real"   and s\<^sub>e :: "real"   and a\<^sub>o :: "real"   and v\<^sub>o :: "real"   and s\<^sub>o :: "real"   and S :: "real set" assumes "safe_distance a\<^sub>e v\<^sub>e s\<^sub>e a\<^sub>o v\<^sub>o s\<^sub>o" shows "safe_distance.collision a\<^sub>e v\<^sub>e s\<^sub>e a\<^sub>o v\<^sub>o s\<^sub>o S = (\<exists>t\<in>S. safe_distance.d a\<^sub>e v\<^sub>e s\<^sub>e a\<^sub>o v\<^sub>o s\<^sub>o t = 0)"
simps_1: fixes x :: "'b"   and y :: "'b" shows "(Some x \<le> Some y) = (x \<le> y)"
intpow_uminus: fixes x :: "'b"   and n :: "int" shows "x ^\<^sub>Z - n = inverse (x ^\<^sub>Z n)"
exu_def: fixes x :: "'var"   and \<phi> :: "'fmla" shows "exu x \<phi> \<equiv> let y = getFr [x] [] [\<phi>] in cnj (exi x \<phi>) (exi y (all x (imp \<phi> (eql (Var x) (Var y)))))"
h2: fixes i :: "nat"   and j :: "nat" assumes "length (pivot_positions A) \<le> dim_row A"   and "i < dim_row A"   and "j < dim_col (1\<^sub>m (length (pivot_positions A)) @\<^sub>r 0\<^sub>m (dim_row A - length (pivot_positions A)) (length (pivot_positions A)))" shows "take_cols A (map snd (pivot_positions A)) $$ (i, j) = (1\<^sub>m (length (pivot_positions A)) @\<^sub>r 0\<^sub>m (dim_row A - length (pivot_positions A)) (length (pivot_positions A))) $$ (i, j)"
simps_14: fixes s :: "'b"   and t :: "'c" shows "balance rbt.Empty s t rbt.Empty = Branch color.B rbt.Empty s t rbt.Empty"
pred_equals_eq: fixes R :: "'e set"   and S :: "'e set" shows "((\<lambda>x. x \<in> R) = (\<lambda>x. x \<in> S)) = (R = S)"
det_upperdiagonal: fixes A :: "(('e, 'd) vec, 'd) vec" assumes "\<And>i j. j < i \<Longrightarrow> A $ i $ j = (0::'e)" shows "det A = (\<Prod>i\<in>UNIV. A $ i $ i)"
igWlsDisj_fromMOD: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) Recursion.model" assumes "gWlsDisj MOD" shows "igWlsDisj (fromMOD MOD)"
simps_5: fixes uw :: "'b heap list" shows "cost Merge uw = 1"
proj_Cons_4: fixes l :: "'a strand_label"   and n :: "'a"   and a :: "'b"   and A :: "('a strand_label \<times> 'b) list" assumes "l = ln n" shows "proj n ((l, a) # A) = (l, a) # proj n A"
c_6: fixes stateful_strand_step :: "('e, 'f) stateful_strand_step" assumes "is_InSet stateful_strand_step" shows "\<langle>the_check stateful_strand_step: the_elem_term stateful_strand_step \<in> the_set_term stateful_strand_step\<rangle> = stateful_strand_step"
deriv_pderiv: fixes p :: "'a poly" shows "deriv (poly p) = poly (pderiv p)"
as_rel_def: fixes R :: "('c \<times> 'd) set" shows "\<langle>R\<rangle>as_rel \<equiv> br as_raw_\<alpha> as_raw_invar O \<langle>R\<rangle>list_rel"
sums_zeta_nat: fixes n :: "nat" assumes "1 < n" shows "(\<lambda>k. 1 / of_nat (Suc k) ^ n) sums zeta (of_nat n)"
assms_1: shows "\<forall>x. x \<in> s \<longrightarrow> (\<forall>y. y \<in> s \<longrightarrow> f1 x \<noteq> f2 y)"
intro: fixes block_collection :: "'a set multiset"   and point_set :: "'a set" assumes "int (size block_collection) = int (card point_set)" shows "symmetric_bibd_axioms point_set block_collection"
map_transfer: fixes Rb :: "'a \<Rightarrow> 'c \<Rightarrow> bool"   and Sd :: "'b \<Rightarrow> 'd \<Rightarrow> bool" shows "((Rb ===> Sd) ===> list_all2 Rb ===> list_all2 Sd) map map"
col_transitivity_1: fixes P :: "'p"   and Q :: "'p"   and A :: "'p"   and B :: "'p" assumes "P \<noteq> Q"   and "Col P Q A"   and "Col P Q B" shows "Col P A B"
vnm_utility_axioms_def: fixes carrier :: "'b set"   and le :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and u :: "'b \<Rightarrow> real" shows "vnm_utility_axioms carrier le u \<equiv> \<forall>x y. x \<in> carrier \<longrightarrow> y \<in> carrier \<longrightarrow> (u x \<le> u y) = le x y"
mset_permute_list: fixes f :: "nat \<Rightarrow> nat"   and xs :: "'b list" assumes "f permutes {..<length xs}" shows "mset (permute_list f xs) = mset xs"
homotopic_with_def: fixes P :: "('b \<Rightarrow> 'c) \<Rightarrow> bool"   and X :: "'b topology"   and Y :: "'c topology"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" shows "homotopic_with P X Y f g \<equiv> \<exists>h. continuous_map (prod_topology (top_of_set {0..1}) X) Y h \<and> (\<forall>x. h (0, x) = f x) \<and> (\<forall>x. h (1, x) = g x) \<and> (\<forall>t\<in>{0..1}. P (\<lambda>x. h (t, x)))"
dghm_ObjMap_app_in_HomCod_Obj: fixes a :: "V" assumes "a \<in>\<^sub>\<circ> \<AA>\<lparr>Obj\<rparr>" shows "\<KK>\<lparr>ObjMap\<rparr>\<lparr>a\<rparr> \<in>\<^sub>\<circ> \<BB>\<lparr>Obj\<rparr>"
map_member_minus: fixes x :: "'a"   and y :: "'b"   and f :: "'a \<Rightarrow> 'b option"   and g :: "'a \<Rightarrow> 'b option" shows "((x, y) \<in>\<^sub>m f -- g) = ((x, y) \<in>\<^sub>m f \<and> \<not> (x, y) \<in>\<^sub>m g)"
rtrancl_trancl_reflcl: fixes r :: "('b \<times> 'b) set" shows "r\<^sup>* = (r\<^sup>+)\<^sup>="
insert_compr: fixes a :: "'c"   and B :: "'c set" shows "insert a B = {x. x = a \<or> s2p B x}"
a_subid_aux2: fixes x :: "'a"   and y :: "'a" shows "x \<cdot> ad y \<le> x"
except_zero: fixes S :: "'a set" shows "except 0 S = 0"
d: shows "distinct (map fst (map (\<lambda>(x, y). ((x, y), \<rho> x y)) l))"
simps_2: fixes hs :: "'b tree"   and uu :: "'b"   and uv :: "'b tree" shows "T_del_min \<langle>hs, uu, uv\<rangle> = T_pass\<^sub>2 (pass\<^sub>1 hs) + T_pass\<^sub>1 hs + 1"
update_since_1: fixes \<sigma> :: "(char list \<times> 'a list) trace"   and n :: "nat"   and R :: "'a list set"   and pos :: "bool"   and \<phi> :: "'a MFOTL.formula"   and I :: "\<I>"   and \<psi> :: "'a MFOTL.formula"   and aux :: "(nat \<times> 'a option list set) list"   and ne :: "nat"   and rel1 :: "'a option list set"   and rel2 :: "'a option list set"   and rel :: "'a option list set"   and aux' :: "(nat \<times> 'a option list set) list" assumes "wf_since_aux \<sigma> n R pos \<phi> I \<psi> aux ne"   and "qtable n (fv \<phi>) (mem_restr R) (\<lambda>v. MFOTL.sat \<sigma> (map the v) ne \<phi>) rel1"   and "qtable n (fv \<psi>) (mem_restr R) (\<lambda>v. MFOTL.sat \<sigma> (map the v) ne \<psi>) rel2"   and "(rel, aux') = update_since I pos rel1 rel2 (\<tau> \<sigma> ne) aux"   and "fv \<phi> \<subseteq> fv \<psi>" shows "wf_since_aux \<sigma> n R pos \<phi> I \<psi> aux' (Suc ne)"
1: assumes "arr F"   and "Dom F = Cod F"   and "B.ide (Map F)"   and "Map F = EVAL (Dom F)"   and "Map F = EVAL (Cod F)" shows "T"
af_decompose_1: fixes \<phi> :: "'a ltl"   and \<psi> :: "'a ltl"   and w :: "'a set list" shows "af (\<phi> and \<psi>) w = af \<phi> w and af \<psi> w"
full_Suc_elim: fixes n :: "nat"   and t :: "'a tree23" assumes "full (Suc n) t" obtains l :: "'a tree23" and r :: "'a tree23" and p :: "int \<times> 'a" where "t = Branch2 l__ p__ r__" and "full n l__" and "full n r__"   | l :: "'a tree23" and m :: "'a tree23" and r :: "'a tree23" and p :: "int \<times> 'a" and q :: "int \<times> 'a" where "t = Branch3 l p m q r" and "full n l" and "full n m" and "full n r"
sep_conj_def: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and sep_disj :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool" assumes "class.sep_algebra plus zero sep_disj" shows "sep_algebra.sep_conj plus sep_disj P Q \<equiv> \<lambda>h. \<exists>x y. sep_disj x y \<and> h = plus x y \<and> P x \<and> Q y"
downset_set_upset_set_dual: shows "(`) \<partial> \<circ> \<Down> = \<Up> \<circ> (`) \<partial>"
Aab0: fixes a :: "'rows"   and b :: "nat" assumes "a \<noteq> from_nat b"   and "b < ncols A" shows "A $ a $ from_nat b = (0::'a)"
simps: fixes p1 :: "int poly"   and p2 :: "int poly"   and l1 :: "rat"   and r1 :: "rat"   and sr1 :: "rat"   and l2 :: "rat"   and r2 :: "rat"   and sr2 :: "rat" shows "compare_1 p1 p2 l1 r1 sr1 l2 r2 sr2 = (if r1 < l2 then Lt else if r2 < l1 then Gt else let (l1', r1', sr1') = tighten_poly_bounds p1 l1 r1 sr1; (l2', r2', sr2') = tighten_poly_bounds p2 l2 r2 sr2 in compare_1 p1 p2 l1' r1' sr1' l2' r2' sr2')"
special_coset_subset_rev_mono: fixes a :: "'w set"   and b :: "'w set"   and w :: "'w" assumes "a \<in> Pow S"   and "b \<in> Pow S"   and "order.greater_eq (((+o) w \<circ> genby) b) (((+o) w \<circ> genby) a)" shows "order.greater_eq b a"
H1: fixes s :: "'a set" assumes "s open" shows "\<exists>t. t open\<^bsub>T\<^esub> \<and> s = t \<inter> carrier"
f_def: fixes i :: "nat" shows "f i = insertion (nth_default 0 (L[var := C])) (isolate_variable_sparse p var i) * (A + B * C) ^ i"
elem_mset_set: fixes A :: "'b set"   and x :: "'b" assumes "finite A" shows "(x \<in># mset_set A) = (x \<in> A)"
g_def: fixes b :: "'c set" shows "g b = ((with (0::'a) zero' plus' scale' : \<guillemotleft>independent\<guillemotright> b) \<and> (with zero' plus' scale' : \<guillemotleft>span\<guillemotright> b) = (with zero' plus' scale' : \<guillemotleft>span\<guillemotright> V'))"
SLN_left: fixes P :: "assn" shows "Automation.SLN * P = P"