hcju/isabelleps huggingface dataset:
IH2: fixes T :: "secLevel" assumes "\<Gamma> \<turnstile> e2 : T"   and "x \<in> fv e2"   and "\<Gamma> x = Some High" shows "\<Gamma> \<turnstile> e2 : High"
cases: fixes x :: "'b list \<times> 'b list" obtains ys :: "'b list" where "x = ([], ys__)"   | xa :: "'b" and xs :: "'b list" and ys :: "'b list" where "x = (xa # xs, ys)"
inf_top_right: fixes inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and top :: "'b"   and x :: "'b" assumes "class.bounded_semilattice_inf_top inf less_eq less top" shows "inf x top = x"
cong_l_def: fixes F :: "'a set"   and a :: "'a"   and b :: "'a" shows "cong_l F a b \<equiv> d2 a b \<in> F"
esqrt_square2: fixes x :: "ereal" shows "esqrt (x * x) = \<bar>x\<bar>"
out_degree_def: fixes G :: "('a, 'b) pre_digraph"   and v :: "'a" shows "out_degree G v \<equiv> card (out_arcs G v)"
deduct_num_if_deduct: fixes arity :: "'a \<Rightarrow> nat"   and Ana :: "('a, 'b) Term.term \<Rightarrow> ('a, 'b) Term.term list \<times> ('a, 'b) Term.term list"   and public :: "'a \<Rightarrow> bool"   and M :: "('a, 'b) Term.term set"   and t :: "('a, 'b) Term.term" assumes "intruder_model arity Ana"   and "intruder_model.intruder_deduct arity public Ana M t" shows "\<exists>k. intruder_model.intruder_deduct_num arity public Ana M k t"
ge_game_leftright_refl: fixes y :: "game"   and x :: "game" shows "(zin y (right_options x) \<longrightarrow> \<not> ge_game (x, y)) \<and> (zin y (left_options x) \<longrightarrow> \<not> ge_game (y, x)) \<and> ge_game (x, x)"
chine_composite_1: shows "Chn ((cod \<mu> \<star> cod \<nu>) \<star> cod \<pi>) = cods.\<nu>.leg0 \<cdot> cods.\<mu>\<nu>.prj\<^sub>0 \<down>\<down> cods.\<pi>.leg1"
prime_elem_multiplicity_mult_distrib: 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 unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and p :: "'a"   and x :: "'a"   and y :: "'a" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize"   and "comm_semiring_1.prime_elem times one zero p"   and "x \<noteq> zero"   and "y \<noteq> zero" shows "factorial_semiring.multiplicity times one p (times x y) = factorial_semiring.multiplicity times one p x + factorial_semiring.multiplicity times one p y"
subst_var_inv_wf_trms: fixes \<delta> :: "'a \<Rightarrow> ('b, 'a) Term.term"   and X :: "'a set" shows "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (subst_range (subst_var_inv \<delta> X))"
intros: fixes A :: "fmla"   and i :: "name"   and x :: "trm" shows "A(i::=x) IMP exi i A \<in> special_axioms"
simps_3: fixes x :: "'g"   and xs :: "'g list"   and xss :: "'g list list" shows "transpose ((x # xs) # xss) = (x # concat (map (case_list [] (\<lambda>h t. [h])) xss)) # transpose (xs # concat (map (case_list [] (\<lambda>h t. [t])) xss))"
nsI': fixes u :: "'b list"   and v :: "'b list" assumes "u \<noteq> \<epsilon>"   and "\<exists>z. z \<cdot> u = v" shows "u \<le>ns v"
window_def: fixes as :: "'b list" shows "window as = (\<lambda>(l, r). 0 < l \<and> l \<le> r \<and> r \<le> length as)"
nxs1: shows "\<langle>vcard xs\<^sub>1, x\<^sub>1\<rangle> \<notin>\<^sub>\<circ> xs\<^sub>1"
obj_fields_marked_invL_def: shows "obj_fields_marked_invL = (\<lambda>s. (atS gc obj_fields_marked_locs s \<longrightarrow> obj_fields_marked s\<down> \<and> gc_tmp_ref s\<down> \<in> gc_W s\<down>) \<and> (atS gc obj_fields_marked_good_ref_locs s \<longrightarrow> obj_at_field_on_heap (\<lambda>r. gc_ref s\<down> = Some r \<or> marked r s\<down>) (gc_tmp_ref s\<down>) (gc_field s\<down>) s\<down>) \<and> (atS gc mark_loop_mo_locs s \<longrightarrow> (\<forall>x. gc_ref s\<down> \<noteq> None \<and> (gc_the_ref s\<down> reaches x) s\<down> \<longrightarrow> obj_at (\<lambda>s. True) x s\<down>)) \<and> (at gc mark_loop_fields s \<longrightarrow> gc_tmp_ref s\<down> \<in> gc_W s\<down>) \<and> (at gc mark_loop_mark_field_done s \<longrightarrow> gc_ref s\<down> \<noteq> None \<longrightarrow> marked (gc_the_ref s\<down>) s\<down>) \<and> (at gc mark_loop_blacken s \<longrightarrow> gc_field_set s\<down> = {}) \<and> (atS gc ghost_honorary_grey_empty_locs s \<longrightarrow> gc_ghost_honorary_grey s\<down> = {}))"
alphaAbs_sym: fixes A :: "('c, 'd, 'varSort, 'var, 'e) qAbs"   and A' :: "('c, 'd, 'varSort, 'var, 'e) qAbs" assumes "A $= A'" shows "A' $= A"
fds_const_uminus: fixes b :: "'b" shows "fds_const (- b) = - fds_const b"
alw_ev_shift: fixes \<phi> :: "'a stream \<Rightarrow> bool"   and xs1 :: "'a stream"   and xl :: "'a list" assumes "alw \<phi> xs1" shows "ev (alw \<phi>) (xl @- xs1)"
utility_less: fixes x :: "'a"   and y :: "'a" assumes "x \<prec>[le] y" shows "u x < u y"
typeof_NoSuchFieldError: fixes h :: "heap" assumes "preallocated h" shows "typeof\<^bsub>h\<^esub> (Addr (addr_of_sys_xcpt NoSuchFieldError)) = \<lfloor>Class NoSuchFieldError\<rfloor>"
that: fixes T1 :: "'a ugraph"   and T2 :: "'a ugraph" assumes "tree T1"   and "tree T2"   and "nodes T1 \<inter> nodes T2 = {}"   and "nodes T = nodes T1 \<union> nodes T2"   and "edges T1 \<union> edges T2 = E'"   and "nodes T1 = {u. (x, u) \<in> E'\<^sup>*}"   and "nodes T2 = {u. (y, u) \<in> E'\<^sup>*}"   and "x \<in> nodes T1"   and "y \<in> nodes T2" shows "thesis"
val_preservation: fixes b :: "'a bexp"   and s :: "'a \<Rightarrow> bool" shows "val_bool_expr (bool_expr_of_bexp b) s = bval b s"
"case": fixes f :: "'e \<Rightarrow> 'd"   and x :: "'e" shows "(case NondetT x of NondetT x \<Rightarrow> f x) = f x"
redT_elims: fixes s :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and t :: "'t"   and ta :: "('l, 't, 'x, 'm, 'w, 'o) thread_action"   and s' :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set" assumes "s -t\<triangleright>ta\<rightarrow> s'" obtains x :: "'x" and x' :: "'x" and m' :: "'m" and ws' :: "'t \<Rightarrow> 'w wait_set_status option" where "t \<turnstile> \<langle>xa__, shr s\<rangle> -ta\<rightarrow> \<langle>x', m'\<rangle>" and "thr s t = \<lfloor>(xa__, no_wait_locks)\<rfloor>" and "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>" and "redT_updWs t (wset s) \<lbrace>ta\<rbrace>\<^bsub>w\<^esub> ws'" and "s' = (redT_updLs (locks s) t \<lbrace>ta\<rbrace>\<^bsub>l\<^esub>, (redT_updTs (thr s) \<lbrace>ta\<rbrace>\<^bsub>t\<^esub>(t \<mapsto> (x', redT_updLns (locks s) t no_wait_locks \<lbrace>ta\<rbrace>\<^bsub>l\<^esub>)), m'), ws', redT_updIs (interrupts s) \<lbrace>ta\<rbrace>\<^bsub>i\<^esub>)"   | x :: "'x" and ln :: "'l \<Rightarrow>f nat" and n :: "'l" where "thr s t = \<lfloor>(x, ln)\<rfloor>" and "ta = (K$ [], [], [], [], [], convert_RA ln)" and "\<not> waiting (wset s t)" and "may_acquire_all (locks s) t ln" and "0 < ln $ n" and "s' = (acquire_all (locks s) t ln, (thr s(t \<mapsto> (x, no_wait_locks)), shr s), wset s, interrupts s)"
simps_2: fixes b :: "'c" shows "local.plus_oracle_stop (Inr b) = map_spmf (apfst (map_option Inr)) (right s b)"
infinite_finite_induct: fixes P :: "'b set \<Rightarrow> bool"   and A :: "'b set" assumes "\<And>A. infinite A \<Longrightarrow> P A"   and "P {}"   and "\<And>x F. \<lbrakk>finite F; x \<notin> F; P F\<rbrakk> \<Longrightarrow> P (insert x F)" shows "P A"
down_chain_imp_eq: fixes f :: "nat \<Rightarrow> nat" assumes "\<forall>i. f (Suc i) \<le> f i" shows "\<exists>N. \<forall>i>N. f i = f (Suc i)"
rk_AA'C: fixes A :: "Points"   and C :: "Points"   and P :: "Points"   and A' :: "Points" assumes "rk {A, C, P} = 3"   and "rk {A, A'} = 2"   and "rk {A, A', P} = 2" shows "3 \<le> rk {A, A', C}"
norm_add_rule_thm: fixes x1 :: "'a"   and b1 :: "real"   and x2 :: "'a"   and b2 :: "real" assumes "norm x1 \<le> b1"   and "norm x2 \<le> b2" shows "norm (x1 + x2) \<le> b1 + b2"
inject: fixes a :: "'e"   and b :: "'f"   and a' :: "'e"   and b' :: "'f" shows "((a, b) = (a', b')) = (a = a' \<and> b = b')"
h2: shows "t mod cycleLength (nC k) mem schedule (nC k)"
mat_addrow_def: fixes ad :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and mul :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and k :: "nat"   and l :: "nat"   and A :: "'a mat" shows "mat_addrow_gen ad mul a k l A = Matrix.mat (dim_row A) (dim_col A) (\<lambda>(i, j). if k = i then ad (mul a (A $$ (l, j))) (A $$ (i, j)) else A $$ (i, j))"
finite_Un: fixes F :: "'g set"   and G :: "'g set" shows "finite (F \<union> G) = (finite F \<and> finite G)"
vector_derivative_works: fixes f :: "real \<Rightarrow> 'b"   and net :: "real filter" shows "(f differentiable net) = (f has_vector_derivative vector_derivative f net) net"
update_convs_7: fixes label' :: "t list list \<Rightarrow> t list list"   and types_t :: "tf list"   and func_t :: "tf list"   and global :: "tg list"   and table :: "nat option"   and memory :: "nat option"   and local :: "t list"   and label :: "t list list"   and return :: "t list option"   and more :: "'a" shows "label_update label' \<lparr>types_t = types_t, func_t = func_t, global = global, table = table, memory = memory, local = local, label = label, return = return, \<dots> = more\<rparr> = \<lparr>types_t = types_t, func_t = func_t, global = global, table = table, memory = memory, local = local, label = label' label, return = return, \<dots> = more\<rparr>"
pred_inject_2: fixes P :: "'a \<Rightarrow> bool"   and a :: "'a bdd"   and aa :: "'a bdd" shows "bdd_all P (Branch a aa) = (bdd_all P a \<and> bdd_all P aa)"
List_unique: fixes h :: "'c \<Rightarrow> 'c"   and x :: "'c"   and as :: "'c list"   and bs :: "'c list" assumes "List h x as"   and "List h x bs" shows "as = bs"
less_eq_sqmatrix_def: fixes A :: "('c, 'd) sqmatrix"   and B :: "('c, 'd) sqmatrix" shows "A \<le> B \<equiv> A + B = B"
algfree_cond_def: fixes R :: "('d, 'e) Ring_scheme"   and S :: "('d, 'f) Ring_scheme"   and X :: "'d" shows "algfree_cond R S X = (\<forall>c. pol_coeff S c \<and> (\<forall>k\<le>fst c. \<Sigma>\<^sub>e R (\<lambda>j. snd c j \<cdot>\<^sub>r\<^bsub>R\<^esub> X^\<^bsup>R j\<^esup>) k = \<zero>\<^bsub>R\<^esub> \<longrightarrow> (\<forall>j\<le>k. snd c j = \<zero>\<^bsub>S\<^esub>)))"
to_complex_of_complex: fixes z :: "complex" shows "to_complex (of_complex z) = z"
rel_option_unfold: fixes R :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and x :: "'d option"   and y :: "'e option" shows "rel_option R x y = (Option.is_none x = Option.is_none y \<and> (\<not> Option.is_none x \<longrightarrow> \<not> Option.is_none y \<longrightarrow> R (the x) (the y)))"
rbl_add_app2: fixes bla :: "bool list"   and blb :: "bool list"   and blc :: "bool list" assumes "length bla \<le> length blb" shows "rbl_add bla (blb @ blc) = rbl_add bla blb"
prv_imp_implies_isTrue: fixes \<phi> :: "'fmla"   and \<chi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "\<chi> \<in> fmla"   and "Fvars \<phi> = {}"   and "Fvars \<chi> = {}"   and "prv (imp \<phi> \<chi>)"   and "isTrue \<phi>" shows "isTrue \<chi>"
hypnat_diff_split: fixes P :: "nat star \<Rightarrow> bool"   and a :: "nat star"   and b :: "nat star" shows "P (a - b) = ((a < b \<longrightarrow> P 0) \<and> (\<forall>d. a = b + d \<longrightarrow> P d))"
reduced_word_singleton: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and a :: "'b"   and A :: "'b set" assumes "class.monoid_add plus zero"   and "a \<in> A"   and "a \<noteq> zero" shows "monoid_add.reduced_word_for plus zero A (monoid_add.sum_list plus zero [a]) [a]"
iT_Plus_Plus_neg_assoc2: fixes a :: "nat"   and b :: "nat"   and I :: "nat set" assumes "a \<le> b" shows "I \<oplus> a \<oplus>- b = I \<oplus>- b - a"
def_hashmap_size_gState_ext_def: fixes uu :: "'a gState_scheme itself" shows "def_hashmap_size uu \<equiv> 10"
Interp_as_UNION: fixes S :: "'b literal multiset \<Rightarrow> 'b literal multiset"   and N :: "'b literal multiset set"   and C :: "'b literal multiset" assumes "ground_resolution_with_selection S" shows "ground_resolution_with_selection.Interp S N C = \<Union> (ground_resolution_with_selection.production S N ` {D. D \<le> C})"
finite_chain3_betw: fixes a :: "'a"   and b :: "'a"   and c :: "'a"   and X :: "'a set" assumes "[[a .. b .. c]X]" shows "[[a b c]]"
that: fixes N :: "nat set" assumes "N \<subseteq> M"   and "infinite N"   and "\<And>S n. \<lbrakk>S \<subseteq> N; finite S; strongly_accepts \<F> S N; n \<in> N; S \<lless> {n}\<rbrakk> \<Longrightarrow> strongly_accepts \<F> (insert n S) N" shows "thesis"
integrable_componentwise_iff: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set" shows "(f integrable_on A) = (\<forall>b\<in>Basis. (\<lambda>x. f x \<bullet> b) integrable_on A)"
Girard_Newton: fixes X :: "nat set"   and k :: "nat" assumes "finite X" shows "(- 1) ^ k * of_nat k * sym_mpoly X k + (\<Sum>i<k. (- 1) ^ i * sym_mpoly X i * powsum_mpoly X (k - i)) = 0"
a: shows "vec_vec_Tensor (z # zs) y = local.times z y @ vec_vec_Tensor zs y"
transp_rtranclp: fixes R :: "'c \<Rightarrow> 'c \<Rightarrow> bool" shows "transp R\<^sup>*\<^sup>*"
eval_ground\<^sub>t\<^sub>s: fixes ts :: "fterm list"   and E :: "char list \<Rightarrow> hterm" assumes "ground\<^sub>t\<^sub>s ts" shows "eval\<^sub>t\<^sub>s E HFun ts = hterms_of_fterms ts"
Union_disjoint: fixes C :: "'d set set"   and A :: "'d set" shows "(\<Union> C \<inter> A = {}) = (\<forall>B\<in>C. B \<inter> A = {})"
face': fixes S :: "'a set"   and T :: "'a set" assumes "S \<in> \<F>"   and "T \<in> \<F>" shows "S \<inter> T face_of S \<and> S \<inter> T face_of T"
cpI4: fixes f :: "('a state \<times> 'a state \<Rightarrow> 'b) \<Rightarrow> ('a state \<times> 'a state \<Rightarrow> 'c) \<Rightarrow> ('a state \<times> 'a state \<Rightarrow> 'd) \<Rightarrow> ('a state \<times> 'a state \<Rightarrow> 'e) \<Rightarrow> 'a state \<times> 'a state \<Rightarrow> 'f"   and P :: "('a state \<times> 'a state \<Rightarrow> 'g) \<Rightarrow> 'a state \<times> 'a state \<Rightarrow> 'b"   and Q :: "('a state \<times> 'a state \<Rightarrow> 'g) \<Rightarrow> 'a state \<times> 'a state \<Rightarrow> 'c"   and R :: "('a state \<times> 'a state \<Rightarrow> 'g) \<Rightarrow> 'a state \<times> 'a state \<Rightarrow> 'd"   and S :: "('a state \<times> 'a state \<Rightarrow> 'g) \<Rightarrow> 'a state \<times> 'a state \<Rightarrow> 'e" assumes "\<forall>W X Y Z \<tau>. f W X Y Z \<tau> = f (\<lambda>_. W \<tau>) (\<lambda>_. X \<tau>) (\<lambda>_. Y \<tau>) (\<lambda>_. Z \<tau>) \<tau>"   and "cp P"   and "cp Q"   and "cp R"   and "cp S" shows "cp (\<lambda>X. f (P X) (Q X) (R X) (S X))"
get_ancestors_di_pure: fixes ptr :: "(_) object_ptr"   and h :: "(_) heap" shows "pure (get_ancestors_di ptr) h"
Quotient_alt_def4: fixes R :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and Abs :: "'d \<Rightarrow> 'e"   and Rep :: "'e \<Rightarrow> 'd"   and T :: "'d \<Rightarrow> 'e \<Rightarrow> bool" shows "Quotient R Abs Rep T = ((\<forall>a b. T a b \<longrightarrow> Abs a = b) \<and> (\<forall>b. T (Rep b) b) \<and> R = T OO T\<inverse>\<inverse>)"
induct: fixes P :: "(nat \<times> 'a) list \<Rightarrow> (nat \<times> 'a) list \<Rightarrow> bool"   and a0 :: "(nat \<times> 'a) list"   and a1 :: "(nat \<times> 'a) list" assumes "\<And>arr. P arr []"   and "\<And>v va. P [] (v # va)"   and "\<And>i a arr j b brr. \<lbrakk>i < j \<Longrightarrow> P arr ((j, b) # brr); \<lbrakk>\<not> i < j; j < i\<rbrakk> \<Longrightarrow> P ((i, a) # arr) brr; \<lbrakk>\<not> i < j; \<not> j < i\<rbrakk> \<Longrightarrow> P arr brr\<rbrakk> \<Longrightarrow> P ((i, a) # arr) ((j, b) # brr)" shows "P a0 a1"
lprefix_lappend_same: fixes xs :: "'c llist"   and ys :: "'c llist"   and zs :: "'c llist" shows "lappend xs ys \<sqsubseteq> lappend xs zs = (lfinite xs \<longrightarrow> ys \<sqsubseteq> zs)"
elims: fixes x :: "'a list"   and y :: "'a list set" assumes "state_set x = y" obtains "x = []" and "y = {}"   | s :: "'a" and ss :: "'a list" where "x = s # ss" and "y = insert [s] ((#) s ` state_set ss)"
Hyp: shows "subgroup (N <#>\<^bsub>G\<lparr>carrier := K\<rparr>\<^esub> H) (G\<lparr>carrier := K\<rparr>)"
assms: shows "(\<Gamma> @ VarB x Q # \<Delta>)\<turnstile>S<:T"
Liminf_llist_imp_exists_index: fixes x :: "'b"   and Xs :: "'b set llist" assumes "x \<in> Liminf_llist Xs" shows "\<exists>i. enat i < llength Xs \<and> x \<in> lnth Xs i"
wrap_def: shows "wrap \<equiv> \<Lambda> f xs. f\<cdot>(c2a\<cdot>xs)"
to_fract_mult: fixes x :: "'a"   and y :: "'a" shows "to_fract (x * y) = to_fract x * to_fract y"
pinf_3: fixes t :: "'a" shows "\<exists>z. \<forall>x>z. (x = t) = False"
FVDiff_sub: fixes f :: "('e, 'd) trm" shows "FVT f \<subseteq> FVDiff f"
restrict_shared_fuse: fixes S :: "nat \<Rightarrow> bool option"   and A :: "nat set"   and L :: "nat set"   and B :: "nat set"   and M :: "nat set" shows "S \<ominus>\<^bsub>A\<^esub> L \<ominus>\<^bsub>B\<^esub> M = S \<ominus>\<^bsub>(A \<union> B)\<^esub> L \<union> M"
simps_2: fixes i :: "nat"   and xs :: "nat list"   and bezout :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<times> 'b \<times> 'b \<times> 'b \<times> 'b"   and P :: "'b mat"   and A :: "'b mat"   and Q :: "'b mat" shows "diagonal_to_Smith_aux_PQ_JNF (i # xs) bezout (P, A, Q) = diagonal_to_Smith_aux_PQ_JNF xs bezout (Diagonal_to_Smith_row_i_PQ_JNF i bezout (P, A, Q))"
EF_def: fixes p :: "'b set" shows "\<^bold>E\<^bold>F p = lfp (\<lambda>s. p \<union> \<^bold>E\<^bold>X s)"
intros_2: fixes xs :: "'a list" shows "interleavings xs [] xs"
check_eqv_def: fixes n :: "nat"   and \<phi> :: "'a formula"   and \<psi> :: "'a formula" shows "fast.check_eqv n \<phi> \<psi> = (wf_formula n (FOr \<phi> \<psi>) \<and> fast.check_eqvRE enum_class.enum n (Plus (rexp_of'' n (norm \<phi>)) One) (Plus (rexp_of'' n (norm \<psi>)) One))"
Id_refine: fixes S :: "'c nres" shows "S \<le> \<Down> Id S"
UN_sets_chop: fixes n :: "nat"   and xs :: "'b list" assumes "0 < n" shows "\<Union> (set ` set (chop n xs)) = set xs"
abs_eq: fixes x :: "'c set" shows "csets x = acset ` {B. B \<subseteq> x \<and> countable B}"
select_convs_2: fixes lasso_reach :: "'e list"   and lasso_va :: "'e"   and lasso_cysfx :: "'e list"   and more :: "'f" shows "lasso_va \<lparr>lasso_reach = lasso_reach, lasso_va = lasso_va, lasso_cysfx = lasso_cysfx, \<dots> = more\<rparr> = lasso_va"
mem_Collect_eq: fixes a :: "'m"   and P :: "'m \<Rightarrow> bool" shows "(a \<in> Collect P) = P a"
semiring_0_def: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" shows "class.semiring_0 plus zero times \<equiv> class.comm_monoid_add plus zero \<and> class.mult_zero times zero \<and> class.semiring plus times"
distinct_right: fixes l :: "'c tree"   and x :: "'c"   and r :: "'c tree"   and y :: "'c" assumes "all_distinct (Node l x False r)"   and "y \<in> set_of r" shows "x \<noteq> y"
MethdI: fixes G :: "prog"   and s :: "state"   and C :: "qtname"   and sig :: "sig"   and v :: "val"   and s' :: "state" assumes "G\<turnstile>s \<midarrow>body G C sig-\<succ>v\<rightarrow> s'" shows "G\<turnstile>s \<midarrow>Methd C sig-\<succ>v\<rightarrow> s'"
select_convs_1: fixes access :: "acc_modi"   and static :: "bool"   and pars :: "vname list"   and resT :: "ty"   and more :: "'b" shows "pars \<lparr>access = access, static = static, pars = pars, resT = resT, \<dots> = more\<rparr> = pars"
star_absorb: fixes x :: "'a"   and y :: "'a" assumes "x \<sqinter> y = bot" shows "x \<sqinter> top = x"
bar2: fixes ws :: "letter list list" assumes "\<And>w. bar (w # ws)" shows "bar ws"
map_project_def: fixes f :: "'h \<Rightarrow> 'g option"   and A :: "'h set" shows "List.map_project f A = {b. \<exists>a\<in>A. f a = Some b}"
d4: shows "H \<turnstile> SyntaxN.Ex x (SyntaxN.Ex y (SyntaxN.Ex x' (SyntaxN.Ex y' (HPair a b EQ HPair (Var x) (Var y) AND HPair a b' EQ HPair (Var x') (Var y') AND OrdP (Var x) AND OrdP (Var x') AND (Var x EQ Var x' IMP Var y EQ Var y')))))"
cong: fixes g :: "'e \<Rightarrow> 'a"   and h :: "'e \<Rightarrow> 'a" assumes "\<And>a. g a = h a" shows "G g = G h"
set_remdups: fixes xs :: "'b list" shows "set (remdups xs) = set xs"
delete\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t_def: fixes ptr :: "(_) object_ptr"   and h :: "(_) heap" shows "delete\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t ptr h = (if ptr |\<in>| object_ptr_kinds h then Some (Heap (fmdrop ptr (the_heap h))) else None)"
DSourcesA3_L0: shows "DSources level0 sA3 = {sA2}"
QTrans: shows "Q \<Longrightarrow>\<^sub>l\<^sup>^\<alpha> \<prec> Q'"
grdLit_def: fixes x :: "Preliminaries.var" shows "grdLit x \<equiv> Neg (Pr (Guard (GE.tpOfV x)) [Var x])"
simps_12: fixes \<phi> :: "atom fm" shows "nnf (Neg (Neg \<phi>)) = nnf \<phi>"
comp_arr_inv: fixes f :: "'a \<times> 'a"   and g :: "'a \<times> 'a" assumes "CC.inverse_arrows f g" shows "CC.comp f g = CC.dom g"
assms_1: shows "rt1 \<sqsubseteq>\<^bsub>i\<^esub> rt2"
asym: fixes b :: "'b word"   and a :: "'b word" assumes "b <s a"   and "a <s b" shows "False"
bs_eq_def: fixes s1 :: "integer"   and s2 :: "integer" shows "bs_eq s1 s2 \<equiv> s1 = s2"
fls_right_inverse_eq_inverse: fixes f :: "'a fls" shows "fls_right_inverse f (inverse (fls_nth f (fls_subdegree f))) = inverse f"
match_subst_some: fixes t :: "term"   and u :: "'b"   and env :: "(name, 'b) fmap" assumes "match t u = Some env"   and "linear t" shows "subst (convert_term t) env = u"
sum_alt: fixes p' :: "real" shows "(\<Sum>i<k. as ! i * bs ! i powr p') = (\<Sum>i<k. as ! i * exp (p' * ln (bs ! i)))"
rel_fun_iff_leq_vimage2p: fixes R :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and S :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'f"   and g :: "'e \<Rightarrow> 'g" shows "rel_fun R S f g = (R \<le> vimage2p f g S)"
l_get_M\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_lemmas_axioms: shows "l_get_M\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_lemmas ShadowRootClass.type_wf"
degree_one_associatedI: fixes K :: "'a list set"   and p :: "'a list list" assumes "subfield K (poly_ring R)"   and "p \<in> carrier (K [X]\<^bsub>poly_ring R\<^esub>)"   and "degree p = 1" shows "p \<sim>\<^bsub>K [X]\<^bsub>poly_ring R\<^esub>\<^esub> [\<one>\<^bsub>poly_ring R\<^esub>, inv\<^bsub>poly_ring R\<^esub> lead_coeff p \<otimes>\<^bsub>poly_ring R\<^esub> UP.const_term p]"
key_compare: fixes x :: "'a key_order" shows "key_compare x \<in> {compare. comparator compare}"
simps_2: fixes var :: "nat"   and a :: "'a" shows "atom_const (Geq var a) = a"
sequence_unique_limpt: fixes f :: "nat \<Rightarrow> 'b"   and l :: "'b"   and l' :: "'b" assumes "f \<longlonglongrightarrow> l"   and "l' islimpt range f" shows "l' = l"
min_range_mono_def: fixes p :: "nat"   and F :: "floatarith \<Rightarrow> floatarith"   and DF :: "floatarith \<Rightarrow> floatarith"   and l :: "float"   and u :: "float"   and X :: "(real \<times> real pdevs) \<times> real" shows "min_range_mono p F DF l u X = (let L = floatarith.Num l; U = floatarith.Num u in approx p (floatarith.Min (DF L) (DF U)) [] \<bind> (\<lambda>aivl. let a = lower aivl; A = floatarith.Num a in approx p (Half (F L + F U - A * (L + U))) [] \<bind> (\<lambda>bivl. let (b, be) = mid_err bivl; (B, Be) = (floatarith.Num (float_of b), floatarith.Num (float_of be)) in approx p (Half (F U - F L - A * (U - L)) + Be) [] \<bind> (\<lambda>divl. Some (affine_unop p (real_of_float a) b (real_of_float (upper divl)) X)))))"
polynomial_function_bounded_linear: fixes f :: "'c \<Rightarrow> 'd" assumes "bounded_linear f" shows "polynomial_function f"
timpl_closure_set_finite: fixes M :: "(('fun, 'atom, 'sets) prot_fun, 'b) Term.term set"   and TI :: "('sets set \<times> 'sets set) list" assumes "finite M"   and "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s M" shows "finite (timpl_closure_set M (set TI))"