hcju/isabelleps huggingface dataset:
strong_cases: fixes a1 :: "lam"   and a2 :: "lam"   and x :: "name"   and xa :: "name" assumes " a1 \<longrightarrow>\<^sub>1 a2" obtains x :: "name" where "a1 = Var x" and "a2 = Var x"   | t1 :: "lam" and t2 :: "lam" and s1 :: "lam" and s2 :: "lam" where "a1 = App t1a__ s1__" and "a2 = App t2a__ s2__" and " t1a__ \<longrightarrow>\<^sub>1 t2a__" and " s1__ \<longrightarrow>\<^sub>1 s2__"   | t1 :: "lam" and t2 :: "lam" where "\<lbrakk>x__ \<sharp> a1; x__ \<sharp> a2\<rbrakk> \<Longrightarrow> a1 = Lam [x__].t1b__ \<and> a2 = Lam [x__].t2b__ \<and>  t1b__ \<longrightarrow>\<^sub>1 t2b__"   | s1 :: "lam" and s2 :: "lam" and t1 :: "lam" and t2 :: "lam" where "\<lbrakk>xa \<sharp> a1; xa \<sharp> a2\<rbrakk> \<Longrightarrow> a1 = App (Lam [xa].t1) s1 \<and> a2 = t2[xa::=s2] \<and> xa \<sharp> (s1, s2) \<and>  t1 \<longrightarrow>\<^sub>1 t2 \<and>  s1 \<longrightarrow>\<^sub>1 s2"
retract_of_def: fixes T :: "'b set"   and S :: "'b set" shows "(T retract_of S) = (\<exists>r. retraction S T r)"
add: fixes f :: "'m \<Rightarrow> 'n"   and x :: "'m"   and y :: "'m" assumes "Modules.additive f" shows "f (x + y) = f x + f y"
scale_zero_right: fixes scale :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'a" assumes "module scale" shows "scale a (0::'b) = (0::'b)"
a_group: fixes G :: "('a, 'b) ring_scheme" assumes "abelian_group G" shows "Group.group (add_monoid G)"
proper_ParT_properL: fixes cval :: "'a \<Rightarrow> 'b \<Rightarrow> real"   and cl :: "('c, 'd, 'e) cmd list" assumes "PL cval" shows "proper (ParT cl) = properL cl"
multiple_blocks_sub: fixes n :: "nat" assumes "0 < n" shows "\<B> \<subseteq># multiple_blocks n"
wlsBinp_imp_goodBinp: fixes delta :: "'opSym"   and binp :: "'bindex \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set option" assumes "wlsBinp delta binp" shows "goodBinp binp"
card_sccs_da: shows "card H.sccs = Suc (card G.sccs)"
appendPath_path_via: fixes wf_md :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list \<Rightarrow> char list \<Rightarrow> char list \<times> ty list \<times> ty \<times> char list list \<times> expr \<Rightarrow> bool"   and P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and Cs :: "char list list"   and C :: "char list"   and Cs' :: "char list list"   and Cs'' :: "char list list" assumes "wf_prog wf_md P"   and "P \<turnstile> Path last Cs to C via Cs' "   and "P \<turnstile> Path last Cs to C via Cs'' "   and "Cs = Cs @\<^sub>p Cs'" shows "Cs = Cs @\<^sub>p Cs''"
two_comparisons_into_case_order_9: fixes comp :: "'c \<Rightarrow> 'c \<Rightarrow> order"   and x :: "'c"   and y :: "'c"   and P :: "'d"   and R :: "'d"   and Q :: "'d" assumes "comparator comp" shows "(if x = y then P else if lt_of_comp comp y x then R else Q) = (case comp x y of Eq \<Rightarrow> P | Lt \<Rightarrow> Q | Gt \<Rightarrow> R)"
normalize_power: fixes a :: "'a"   and n :: "nat" shows "normalize (a ^ n) = normalize a ^ n"
greatest_def: fixes r :: "('b \<times> 'b) set"   and X :: "'b set" shows "greatest r X = {x \<in> X. \<forall>y\<in>X. (y, x) \<in> r}"
real_poly_roots_induct: fixes P :: "real poly \<Rightarrow> bool"   and p :: "real poly" assumes "\<And>p x. P p \<Longrightarrow> P (p * [:- x, 1:])"   and "\<And>p a b. \<lbrakk>b \<noteq> 0; P p\<rbrakk> \<Longrightarrow> P (p * [:a * a + b * b, - 2 * a, 1:])"   and "\<And>a. P [:a:]" shows "P p"
ts_ord_trans: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "n x \<sqsubseteq> n y"   and "n y \<sqsubseteq> n z" shows "n x \<sqsubseteq> n z"
intro: fixes prb :: "('f, 'd, 'e) pre_RedBlack" assumes "finite (pred (init_conf prb))"   and "\<forall>e\<in>pred (init_conf prb). finite (Bexp.vars e)"   and "finite_lts (black prb)" shows "finite_RedBlack prb"
assms_1: shows "valid ((Pr n f g, xs, []) # rest)"
open_lessThan: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and "open" :: "'c set \<Rightarrow> bool"   and a :: "'c" assumes "class.order_topology less_eq less open" shows "open (ord.lessThan less a)"
remove_vertex_walk: fixes xs :: "'a list"   and x :: "'a" assumes "H.walk xs"   and "x \<notin> set xs" shows "Digraph.walk (H.remove_vertex x) xs"
T_opt_def: fixes s :: "'b list"   and rs :: "'b list" shows "T\<^sub>p_opt s rs = \<Sqinter> {T\<^sub>p s rs as |as. length as = length rs}"
hm_delete_rule: fixes m :: "'a \<Rightarrow> 'b option"   and ht :: "('a, 'b) hashtable"   and k :: "'a" shows "<is_hashmap m ht> hm_delete k ht <is_hashmap (m |` (- {k}))>"
valid_pre: fixes ds :: "'b set"   and P' :: "'b set \<Rightarrow> 'a set \<Rightarrow> bool"   and Q :: "'b set \<Rightarrow> 'a set \<Rightarrow> bool"   and P :: "'b set \<Rightarrow> 'a set \<Rightarrow> bool" assumes "valid ds P' Q"   and "\<And>fp. P ds fp \<Longrightarrow> P' ds fp" shows "valid ds P Q"
while_option_finite_decreasing_Some: fixes f :: "'b \<Rightarrow> 'b"   and s :: "'b" assumes "mono f"   and "finite UNIV"   and "f s \<le> s" shows "\<exists>P. while_option (\<lambda>A. f A \<noteq> A) f s = Some P"
plus_log_le_2log_plus: fixes x :: "real"   and y :: "real"   and b :: "real" assumes "0 < x"   and "0 < y"   and "1 < b" shows "log b x + log b y \<le> 2 * log b (x + y)"
assms_1: shows "\<NN> : \<FF> \<mapsto>\<^sub>D\<^sub>G\<^sub>H\<^sub>M \<GG> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<BB>"
hom_prod_list: fixes xs :: "real poly list" shows "map_poly complex_of_real (prod_list xs) = prod_list (map (map_poly complex_of_real) xs)"
ent_false: fixes P :: "assn" shows "false \<Longrightarrow>\<^sub>A P"
finite_pointwise: fixes U :: "'b set"   and G :: "'c triple set"   and P' :: "'b \<Rightarrow> 'c assn"   and c0 :: "'b \<Rightarrow> com"   and Q' :: "'b \<Rightarrow> 'c assn"   and P :: "'b \<Rightarrow> 'c assn"   and Q :: "'b \<Rightarrow> 'c assn" assumes "finite U"   and "\<forall>p. G|-{P' p}. c0 p .{Q' p} \<longrightarrow> G|-{P p}. c0 p .{Q p}"   and "G||-(\<lambda>p. {P' p}. c0 p .{Q' p}) ` U" shows "G||-(\<lambda>p. {P p}. c0 p .{Q p}) ` U"
pCons_cases: fixes p :: "'a poly" obtains a :: "'a" and q :: "'a poly" where "p = pCons a q"
iIN_I: fixes n :: "nat"   and x :: "nat"   and d :: "nat" assumes "n \<le> x"   and "x \<le> n + d" shows "x \<in> [n\<dots>,d]"
mertens_bound_real_strong: fixes x :: "real" assumes "1 \<le> x" shows "\<MM> x - ln x \<in> {- 1 - 9 / pi\<^sup>2 - ln (1 + frac x / real (nat \<lfloor>x\<rfloor>))<..ln 4}"
tensor_P_dim: fixes A :: "'a mat"   and B :: "'a mat" shows "tensor_P A B \<in> carrier_mat d d"
finite_dom_iff_fsubst: fixes \<theta> :: "'a \<Rightarrow> ('b, 'a) Term.term" shows "finite (subst_domain \<theta>) = (\<theta> \<in> fsubst)"
times_prod_def: fixes x :: "'c \<times> 'd"   and y :: "'c \<times> 'd" shows "x \<oplus> y = (fst x, snd x \<oplus> snd y)"
real_differentiableE: fixes f :: "real \<Rightarrow> real"   and x :: "real"   and s :: "real set" assumes "f differentiable at x within s" obtains df :: "real" where "(f has_real_derivative df) (at x within s)"
negate_embed: fixes P :: "'b \<Rightarrow> bool"   and s :: "'b" shows "\<guillemotleft> \<N> P \<guillemotright> s = 1 - \<guillemotleft> P \<guillemotright> s"
conjugate_isotone_pp: fixes f :: "'a \<Rightarrow> 'a"   and g :: "'a \<Rightarrow> 'a" assumes "conjugate f g" shows "isotone (uminus \<circ> uminus \<circ> f)"
no_self_loop: fixes c :: "nat \<times> nat \<Rightarrow> 'a"   and s :: "nat"   and t :: "nat" assumes "Network c s t" shows "\<forall>u. (u, u) \<notin> Graph.E c"
UN_extend_simps_3: fixes A :: "'a set"   and B :: "'b \<Rightarrow> 'a set"   and C :: "'b set" shows "A \<union> \<Union> (B ` C) = (if C = {} then A else \<Union>x\<in>C. A \<union> B x)"
prime_factorization_eqI: 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 multiset"   and n :: "'a" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize"   and "\<And>p. p \<in># P \<Longrightarrow> normalization_semidom.prime zero times one normalize p"   and "normalize (comm_monoid_mult.prod_mset times one P) = normalize n" shows "factorial_semiring.prime_factorization zero times one normalize n = P"
distinct_Ex1: fixes xs :: "'e list"   and x :: "'e" assumes "distinct xs"   and "x \<in> set xs" shows "\<exists>!i. i < length xs \<and> xs ! i = x"
entails_concl_compact: fixes Bot :: "'a set"   and entails :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool"   and EE :: "'a set"   and CC :: "'a set" assumes "concl_compact_consequence_relation Bot entails"   and "finite EE"   and "entails CC EE" shows "\<exists>CC'\<subseteq>CC. finite CC' \<and> entails CC' EE"
dfa_accepting_def: fixes A :: "nat bdd list \<times> bool list"   and q :: "nat" shows "dfa_accepting A q = snd A ! q"
not_less_bot: fixes a :: "'a" shows "\<not> a < 0"
odd_add: fixes a :: "'a"   and b :: "'a" shows "odd (a + b) = (odd a \<noteq> odd b)"
prWlsAbs_def: fixes gA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> ('varSort \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> ('varSort, 'c) sAbs"   and SEM :: "('d, 'e, 'f, 'sort, 'g, 'c, 'h) semDom_scheme" shows "prWlsAbs gA SEM \<equiv> \<forall>us s A val. wlsAbs (us, s) A \<and> sWlsVal SEM val \<longrightarrow> sWlsAbs SEM (us, s) (gA A val)"
mulex_on_add_right: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set"   and M :: "'b multiset"   and N :: "'b multiset"   and a :: "'b" assumes "mulex_on P A M N"   and "a \<in> A" shows "mulex_on P A M (add_mset a N)"
k_adju_var: fixes y :: "'b" shows "\<exists>F. \<forall>x f. (F x \<le> f) = (x \<le> f y)"
CHOinitConfig_def: fixes A :: "('f, 'g, 'h, 'i) CHOAlgorithm_scheme"   and cfg :: "'f \<Rightarrow> 'g"   and coord :: "'f \<Rightarrow> 'f" shows "CHOinitConfig A cfg coord \<equiv> \<forall>p. CinitState A p (cfg p) (coord p)"
OUTfromVCorrect1_def: fixes x :: "chanID" shows "OUTfromVCorrect1 x \<equiv> OUTfromV x \<noteq> {} \<longrightarrow> (\<exists>Z. x \<in> OUT Z \<and> (\<forall>v\<in>OUTfromV x. v \<in> VAR Z))"
n_st_mono: fixes S1 :: "ivl Abs_State.st"   and X :: "char list set"   and S2 :: "ivl Abs_State.st" assumes "set (Abs_State.dom S1) \<subseteq> X"   and "set (Abs_State.dom S2) \<subseteq> X"   and "S1 \<sqsubseteq> S2" shows "n_st n_ivl X S1 \<le> n_st n_ivl X S2"
int_gr_induct: fixes k :: "int"   and i :: "int"   and P :: "int \<Rightarrow> bool" assumes "k < i"   and "P (k + 1)"   and "\<And>i. \<lbrakk>k < i; P i\<rbrakk> \<Longrightarrow> P (i + 1)" shows "P i"
arity_rewrite_fundef_body: fixes fd :: "'a fundef"   and n :: "nat"   and x :: "'a" shows "arity (rewrite_fundef_body fd n x) = arity fd"
Let_def: fixes s :: "'h"   and f :: "'h \<Rightarrow> 'g" shows "Let s f \<equiv> f s"
kleisli_assoc: fixes z :: "'c \<Rightarrow> 'c option"   and x :: "'c \<Rightarrow> 'c option"   and y :: "'c \<Rightarrow> 'c option" shows "(z \<rhd> x) \<rhd> y = z \<rhd> x \<rhd> y"
fls_def: shows "fls \<equiv> neg (zer EQ zer)"
comp_red_basis_pmdl: fixes xs :: "('t \<Rightarrow>\<^sub>0 'c) list" assumes "is_Groebner_basis (set xs)" shows "pmdl (set (comp_red_basis xs)) = pmdl (set xs)"
rm_vars_apply': fixes \<delta> :: "'d \<Rightarrow> ('e, 'd) Term.term"   and vs :: "'d set" assumes "subst_domain \<delta> \<inter> vs = {}" shows "rm_vars vs \<delta> = \<delta>"
pp_point_linorder_matrix_unique: fixes f :: "'c \<times> 'c \<Rightarrow> 'd" shows "matrix_stone_relation_algebra.pp_point f = (\<exists>!i. \<forall>j. f (i, j) \<noteq> bot \<and> (\<forall>k. f (i, j) = f (i, k)) \<and> (\<forall>k. i \<noteq> k \<longrightarrow> f (k, j) = bot))"
gamma_differentiable: shows "\<forall>x\<in>s. \<gamma> differentiable at x"
utp_top: fixes hcond :: "'a hrel \<Rightarrow> 'a hrel"   and P :: "'a hrel" assumes "utp_theory_lattice hcond"   and "P is hcond" shows "P \<sqsubseteq> \<top>\<^bsub>utp_order hcond\<^esub>"
uminus_Assertion_def: fixes p :: "'a Assertion" shows "- p = abs_wpt (neg_assert {\<cdot> p })"
assms_1: fixes X :: "'a"   and Y :: "'a" assumes "f X < f Y" shows "R X Y"
smult_minus_1_back: fixes v :: "'c" assumes "v \<in> carrier V" shows "\<ominus>\<^bsub>V\<^esub> v = \<ominus>\<^bsub>K\<^esub> \<one>\<^bsub>K\<^esub> \<odot>\<^bsub>V\<^esub> v"
hR: shows "reciprocal_poly p P \<circ>\<^sub>p [:1, 1:] = R * [:- (1 / x - 1), 1:]"
bits_minus_1_mod_2_eq: shows "- (1::'a) mod (2::'a) = (1::'a)"
is_ground_clss_def: fixes CC :: "'a literal multiset set" shows "is_ground_clss CC = Ball CC is_ground_cls"
add_closed_converse_left: fixes g :: "'v"   and x :: "'v" assumes "g + x \<in> V"   and "x \<in> V" shows "g \<in> V"
scale_right_diff_distrib: fixes a :: "'c"   and x :: "'c"   and y :: "'c" shows "a * (x - y) = a * x - a * y"
integrable_fst: fixes f :: "'b \<Rightarrow> 'a \<Rightarrow> 'c" assumes "integrable (M2 \<Otimes>\<^sub>M M1) (\<lambda>(x, y). f x y)" shows "integrable M2 (\<lambda>x. integral\<^sup>L M1 (f x))"
val_zero: fixes a :: "('a, 'b) IEEE.float" assumes "IEEE.is_zero a" shows "valof a = 0"
fbox_g_orbital_inv: fixes P :: "'b \<Rightarrow> bool"   and I :: "'b \<Rightarrow> bool"   and f :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and G :: "'b \<Rightarrow> bool"   and U :: "'b \<Rightarrow> real set"   and S :: "'b set"   and t\<^sub>0 :: "real"   and Q :: "'b \<Rightarrow> bool" assumes "P \<le> I"   and "I \<le> |x\<acute>=f & G on U S @ t\<^sub>0] I"   and "I \<le> Q" shows "P \<le> |x\<acute>=f & G on U S @ t\<^sub>0] Q"
union_single_eq_diff: fixes x :: "'b"   and M :: "'b multiset"   and N :: "'b multiset" assumes "add_mset x M = N" shows "M = N - {#x#}"
separating_RF: fixes \<Gamma> :: "('a, 'b) web_scheme"   and S :: "'a set" shows "separating \<Gamma> (RF\<^bsub>\<Gamma>\<^esub> S) = separating \<Gamma> S"
onorm_compose: fixes f :: "'d \<Rightarrow> 'e"   and g :: "'f \<Rightarrow> 'd" assumes "bounded_linear f"   and "bounded_linear g" shows "onorm (f \<circ> g) \<le> onorm f * onorm g"
set_mset_single: fixes b :: "'c" shows "set_mset {#b#} = {b}"
permutes_surj: fixes p :: "'b \<Rightarrow> 'b"   and s :: "'b set" assumes "p permutes s" shows "surj p"
OrdP_cases_disj: fixes p :: "name"   and x :: "tm"   and H :: "fm set" assumes "atom p \<sharp> x" shows "insert (OrdP x) H \<turnstile> x EQ Zero OR SyntaxN.Ex p (OrdP (Var p) AND x EQ SUCC (Var p))"
UINF_as_Sup_collect: fixes A :: "'c set"   and f :: "'c \<Rightarrow> ('a, 'b) uexpr" shows "(\<Sqinter> P \<in> A \<bullet> f P) = \<Sqinter>(f ` A)"
simps_15: fixes x21 :: "'a"   and x22 :: "'a list" shows "set (x21 # x22) = insert x21 (set x22)"
prob_space_imp_subprob_space: fixes M :: "'b measure" assumes "prob_space M" shows "subprob_space M"
d_2: shows "snd x = snd (normalize_quot x) * d"
counit_simps_4: shows "src \<psi> = trg g'"
has_prod_one: shows "(\<lambda>n. 1::'e) has_prod (1::'e)"
A2_DAcc_level0: shows "DAcc level0 sA2 = {sA3}"
set_treeE: fixes a :: "'b"   and t :: "'b tree" assumes "a \<in> set_tree t" shows "\<exists>l r. \<langle>l, a, r\<rangle> \<in> subtrees t"
simps_3: fixes ve :: "nat"   and vf :: "nat"   and vg :: "'b option"   and vh :: "'b tree"   and vi :: "'b tree"   and v :: "nat"   and va :: "nat"   and vb :: "'b option"   and vc :: "'b tree"   and vd :: "'b tree" shows "combine (Node ve vf vg vh vi) (Node v va vb vc vd) = Node (l (Node ve vf vg vh vi)) (r (Node v va vb vc vd)) (val (Node ve vf vg vh vi) + val (Node v va vb vc vd)) (discharge (Node ve vf vg vh vi)) (Node v va vb vc vd)"
fbd_ffd: shows "bd\<^sub>\<F> = fd\<^sub>\<F> \<circ> op\<^sub>K"
comp_assoc: fixes h :: "'a"   and g :: "'a"   and f :: "'a" shows "(h \<cdot>\<^sub>C g) \<cdot>\<^sub>C f = h \<cdot>\<^sub>C g \<cdot>\<^sub>C f"
intersect_list'_language: fixes AA :: "('c, 'd) dba list" shows "DGBA.language (intersect_list' AA) = \<Inter> (DBA.language ` set AA)"
pref_compIff: fixes u :: "'b list"   and v :: "'b list" shows "(u \<bowtie> v) = (u \<le>p v \<or> v \<le>p u)"
assms_11: fixes \<Gamma> :: "(Vars.var \<times> exp) list"   and scrut :: "exp"   and e1 :: "exp"   and e2 :: "exp"   and S :: "stack_elem list" assumes "I (\<Gamma>, scrut ? e1 : e2, S)" shows "P (\<Gamma>, scrut ? e1 : e2, S) (\<Gamma>, scrut, Alts e1 e2 # S)"
dim_vec_conjugate: fixes v :: "'a vec" shows "dim_vec (conjugate v) = dim_vec v"
assms: shows "\<Turnstile> F \<^bold>\<rightarrow> H"
insert: fixes A :: "'c set"   and x :: "'c"   and g :: "'c \<Rightarrow> 'a" assumes "finite A"   and "x \<notin> A" shows "F.F g (insert x A) = g x \<^bold>* F.F g A"
ipurge_ref_def: fixes I :: "('f \<times> 'f) set"   and D :: "'e \<Rightarrow> 'f"   and u :: "'f"   and xs :: "'e list"   and X :: "'e set" shows "ipurge_ref I D u xs X \<equiv> {x \<in> X. (u, D x) \<notin> I \<and> (\<forall>v\<in>sinks I D u xs. (v, D x) \<notin> I)}"
rel_funE: fixes A :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and B :: "'h \<Rightarrow> 'i \<Rightarrow> bool"   and f :: "'f \<Rightarrow> 'h"   and g :: "'g \<Rightarrow> 'i"   and x :: "'f"   and y :: "'g" assumes "(A ===> B) f g"   and "A x y" obtains "B (f x) (g y)"
has_integral_erfc: fixes x :: "real" assumes "0 \<le> x" shows "((\<lambda>t. exp (- t\<^sup>2)) has_integral sqrt pi / 2 * erfc x) {x..}"
mem_val_alt_def: fixes asi :: "word8"   and add :: "word36"   and state :: "'b sparc_state" shows "mem_val_alt asi add state \<equiv> let r1 = mem state asi add; asi8 = word_of_int 8; asi9 = word_of_int 9; asi10 = word_of_int 10; asi11 = word_of_int 11 in if r1 = None \<and> uint asi = 8 then let r2 = mem state asi9 add in r2 else if r1 = None \<and> uint asi = 9 then let r2 = mem state asi8 add in r2 else if r1 = None \<and> uint asi = 10 then let r2 = mem state asi11 add in if r2 = None then let r3 = mem state asi8 add in if r3 = None then mem state asi9 add else r3 else r2 else if r1 = None \<and> uint asi = 11 then let r2 = mem state asi10 add in if r2 = None then let r3 = mem state asi8 add in if r3 = None then mem state asi9 add else r3 else r2 else r1"
mat1I_def: fixes ze :: "'a"   and on :: "'a"   and n :: "nat" shows "mat1I ze on n \<equiv> map (vec1I ze on n) [0..<n]"
induct: fixes P :: "'b btree \<Rightarrow> bool"   and a0 :: "'b btree" assumes "P Leaf"   and "\<And>ts t. P t \<Longrightarrow> P (Node ts t)" shows "P a0"
induct: fixes P :: "'b tree \<Rightarrow> 'b tree \<Rightarrow> bool"   and a0 :: "'b tree"   and a1 :: "'b tree" assumes "\<And>t2. P \<langle>\<rangle> t2"   and "\<And>v va vb. P \<langle>v, va, vb\<rangle> \<langle>\<rangle>"   and "\<And>l1 a1 r1 l2 a2 r2. \<lbrakk>a1 \<le> a2 \<Longrightarrow> P l1 r1; \<not> a1 \<le> a2 \<Longrightarrow> P l2 r2\<rbrakk> \<Longrightarrow> P \<langle>l1, a1, r1\<rangle> \<langle>l2, a2, r2\<rangle>" shows "P a0 a1"
rotate_toppos_inv: fixes w1 :: "wall"   and w2 :: "wall" assumes "rotate_toppos w1 w2" shows "kauff_mat w1 = kauff_mat w2"
simps: fixes xs :: "('c \<times> 'd) list"   and k :: "'c"   and v :: "'d" shows "add (T xs) k v = T ((k, v) # xs)"
simps_4: fixes typeof_addr :: "'a \<Rightarrow> 'b \<Rightarrow> htype option"   and h :: "'a"   and va :: "bool" shows "heap_base.typeof_h typeof_addr h (Bool va) = typeof (Bool va)"
obj_def: fixes a :: "'c" shows "D.obj a \<equiv> D.arr a \<and> src\<^sub>D a = a"
Equality: fixes M\<^sub>0 :: "('fun, 'var) Term.term set"   and \<I> :: "'var \<Rightarrow> ('fun, 'var) Term.term"   and S :: "('fun, 'var) extstrand_step list"   and t :: "('fun, 'var) Term.term"   and t' :: "('fun, 'var) Term.term"   and a :: "poscheckvariant" assumes "sem\<^sub>e\<^sub>s\<^sub>t_d M\<^sub>0 \<I> S"   and "Unifier \<I> t t'" shows "sem\<^sub>e\<^sub>s\<^sub>t_d M\<^sub>0 \<I> (S @ [Step \<langle>a: t \<doteq> t'\<rangle>\<^sub>s\<^sub>t])"
univalent_comp_left_dist_inf: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "univalent x" shows "x * (y \<sqinter> z) = x * y \<sqinter> x * z"
lookup_update': fixes k :: "'c"   and v :: "'b"   and m :: "('c, 'b) mapping"   and k' :: "'c" shows "lookup (update k v m) k' = (if k = k' then Some v else lookup m k')"
MapsTo_def: fixes CC :: "('g, 'h, 'i) Category_scheme"   and f :: "'h"   and X :: "'g"   and Y :: "'g" shows "f maps\<^bsub>CC\<^esub> X to Y \<equiv> f \<in> mor\<^bsub>CC\<^esub> \<and> dom\<^bsub>CC\<^esub> f = X \<and> cod\<^bsub>CC\<^esub> f = Y"
delete_shadow_root_get_M\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t: fixes h :: "(_) heap"   and shadow_root_ptr :: "(_) shadow_root_ptr"   and h' :: "(_) heap"   and shadow_root_ptr' :: "(_) shadow_root_ptr"   and getter :: "('c, 'a, 'b, 'k option) RShadowRoot_scheme \<Rightarrow> 'm" assumes "h \<turnstile> delete\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_M shadow_root_ptr \<rightarrow>\<^sub>h h'"   and "shadow_root_ptr \<noteq> shadow_root_ptr'" shows "preserved (get_M\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t shadow_root_ptr' getter) h h'"
induct: fixes P :: "'b \<Rightarrow> ('b \<times> color) tree \<Rightarrow> bool"   and a0 :: "'b"   and a1 :: "('b \<times> color) tree" assumes "\<And>x. P x \<langle>\<rangle>"   and "\<And>x l a r. \<lbrakk>cmp x a = LT \<Longrightarrow> P x l; cmp x a = GT \<Longrightarrow> P x r\<rbrakk> \<Longrightarrow> P x (B l a r)"   and "\<And>x l a r. \<lbrakk>cmp x a = LT \<Longrightarrow> P x l; cmp x a = GT \<Longrightarrow> P x r\<rbrakk> \<Longrightarrow> P x (R l a r)" shows "P a0 a1"
simps_2: fixes CatDom :: "('d, 'e, 'f) Category_scheme"   and CatCod :: "('a, 'b, 'c) Category_scheme"   and MapM :: "'e \<Rightarrow> 'b"   and more :: "'g" shows "CatCod \<lparr>CatDom = CatDom, CatCod = CatCod, MapM = MapM, \<dots> = more\<rparr> = CatCod"
WT_sim: fixes \<eta> :: "nat" shows "\<I>_real2 \<eta>, \<I>_inner2 \<eta> \<turnstile>\<^sub>C sim2 \<eta> \<surd>"
subst_apply_labeled_stateful_strand_def: fixes S :: "('d strand_label \<times> ('e, 'f) stateful_strand_step) list"   and \<theta> :: "'f \<Rightarrow> ('e, 'f) Term.term" shows "S \<cdot>\<^sub>l\<^sub>s\<^sub>s\<^sub>t \<theta> \<equiv> map (\<lambda>x. x \<cdot>\<^sub>l\<^sub>s\<^sub>s\<^sub>t\<^sub>p \<theta>) S"
elims_2: fixes x :: "('a list \<times> 's \<times> 's) list \<times> ('s \<times> 's) list \<times> ('s \<times> 's) set" assumes "test_invariant x" obtains ws :: "('a list \<times> 's \<times> 's) list" and uu_ :: "('s \<times> 's) list" and uv_ :: "('s \<times> 's) set" where "x = (ws, uu_, uv_)" and "case ws of [] \<Rightarrow> False | (w, p, q) # x \<Rightarrow> final p = final q"
IA: fixes p :: "('a, 'b) fm"   and t :: "'a tm" shows "\<turnstile> \<^bold>\<forall> p \<^bold>\<longrightarrow> p\<langle>t/0\<rangle>"
fps_monom_poly_def: fixes c :: "'b"   and k :: "nat" shows "fps_monom_poly c k = (if k = 0 then 1 else pCons (0::'b) (eulerian_poly k) \<circ>\<^sub>p [:0::'b, c:])"
idNMT: fixes p :: "('a, 'b) Combinators list" assumes "p \<noteq> []" shows "insertDenies p \<noteq> []"
that: fixes B :: "'a fset" assumes "A = finsert x B"   and "x |\<notin>| B" shows "thesis"
preserves_dom: fixes f :: "'a \<times> 'a" assumes "VoV.arr f" shows "VoV.dom (VoV.map f) = VoV.map (VoV.dom f)"
continuous_on_subset: fixes s :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and t :: "'b set" assumes "continuous_on s f"   and "t \<subseteq> s" shows "continuous_on t f"
assms: shows "t \<in> {- pi / 2<..3 / 2 * pi}"
finite_FV: fixes \<phi> :: "('a, 'k) aformula"   and k :: "'k" shows "finite (FV \<phi> k)"
insert_remove: fixes A :: "'e set"   and g :: "'e \<Rightarrow> 'f"   and x :: "'e" assumes "finite A" shows "prod g (insert x A) = g x * prod g (A - {x})"
s': shows "s' \<in>\<^sub>A path_summary loc loc2"
simps_3: fixes orderA :: "('d, 'f) gorder_scheme"   and orderB :: "('e, 'g) gorder_scheme"   and lower :: "'d \<Rightarrow> 'e"   and upper :: "'e \<Rightarrow> 'd"   and more :: "'h" shows "\<pi>\<^sup>*\<^bsub>\<lparr>orderA = orderA, orderB = orderB, lower = lower, upper = upper, \<dots> = more\<rparr>\<^esub> = lower"