hcju/isabelleps huggingface dataset:
WT_swap\<^sub>C: fixes \<I>1 :: "('h, 'j) \<I>"   and \<I>2 :: "('i, 'k) \<I>" shows "\<I>1 \<oplus>\<^sub>\<I> \<I>2, \<I>2 \<oplus>\<^sub>\<I> \<I>1 \<turnstile>\<^sub>C swap\<^sub>C \<surd>"
\<tau>rtrancl3p_Nil_eq_\<tau>moves: fixes trsys :: "'g \<Rightarrow> 'h \<Rightarrow> 'g \<Rightarrow> bool"   and \<tau>move :: "'g \<Rightarrow> 'h \<Rightarrow> 'g \<Rightarrow> bool"   and s :: "'g"   and s' :: "'g" shows "\<tau>trsys.\<tau>rtrancl3p trsys \<tau>move s [] s' = (\<tau>trsys.silent_move trsys \<tau>move)\<^sup>*\<^sup>* s s'"
child_lv: fixes d :: "nat"   and p :: "(nat \<times> int) list"   and dir :: "dir" assumes "d < length p" shows "lv (child p dir d) d = lv p d + 1"
is_heap_except_up_def: fixes cmp :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and heap :: "int \<Rightarrow> 'a"   and l :: "int"   and r :: "int"   and i :: "int" shows "is_heap_except_up cmp heap l r i = (\<forall>x. bounded l r x \<longrightarrow> (x \<noteq> i \<longrightarrow> bounded l r (parent l x) \<longrightarrow> cmp (heap (parent l x)) (heap x)) \<and> (parent l x = i \<longrightarrow> bounded l r (parent l (parent l x)) \<longrightarrow> cmp (heap (parent l (parent l x))) (heap x)))"
a_Rec: fixes x :: "Syntax.var"   and \<Gamma> :: "(Syntax.var, ty) fmap"   and y :: "Syntax.var"   and \<tau>\<^sub>1 :: "ty"   and \<tau>\<^sub>2 :: "ty"   and e :: "Syntax.term"   and eP :: "Syntax.term"   and eV :: "Syntax.term" assumes "atom x \<sharp> \<Gamma>"   and "atom y \<sharp> (\<Gamma>, x)"   and "\<Gamma>(x $$:= Fun \<tau>\<^sub>1 \<tau>\<^sub>2) \<turnstile> Lam y e, Lam y eP, Lam y eV : Fun \<tau>\<^sub>1 \<tau>\<^sub>2" shows "\<Gamma> \<turnstile> Rec x (Lam y e), Rec x (Lam y eP), Rec x (Lam y eV) : Fun \<tau>\<^sub>1 \<tau>\<^sub>2"
H_face_cycle_succ_1: shows "H.face_cycle_succ uw = rev_H vw"
is_renaming_def: fixes \<sigma> :: "'s" shows "is_renaming \<sigma> = (\<exists>\<tau>. \<sigma> \<odot> \<tau> = id_subst)"
f1: shows "(r |\<^bsub>rel\<^esub> s) \<oplus>\<^bsub>rec_rng_of_frac\<^esub> (r' |\<^bsub>rel\<^esub> s') \<oplus>\<^bsub>rec_rng_of_frac\<^esub> (r'' |\<^bsub>rel\<^esub> s'') = (s'' \<otimes> (s' \<otimes> r \<oplus> s \<otimes> r') \<oplus> s \<otimes> s' \<otimes> r'' |\<^bsub>rel\<^esub> s \<otimes> s' \<otimes> s'')"
zero_le_one: shows "(0::'d) \<le> (1::'d)"
comb_spec1: fixes X :: "'b"   and a :: "'a"   and b :: "'a" assumes "Q X a" shows "comb a b \<le> SPEC (Q X)"
Gcd_0_iff: fixes one :: "'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and gcd :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and lcm :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and Gcd :: "'b set \<Rightarrow> 'b"   and Lcm :: "'b set \<Rightarrow> 'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and A :: "'b set" assumes "class.semiring_Gcd one zero times gcd lcm Gcd Lcm divide plus minus unit_factor normalize" shows "(Gcd A = zero) = (A \<subseteq> {zero})"
mtf_eq_mtf2: fixes x :: "'a"   and xs :: "'a list" shows "mtf x xs = mtf2 (length xs - 1) x xs"
p_1: shows "poly (map_poly of_int p) (of_real x) = (0::'a)"
view_determination: fixes x :: "'d \<Longrightarrow> 'e"   and \<sigma> :: "'e"   and u :: "'d"   and \<rho> :: "'e"   and v :: "'d" assumes "weak_lens x"   and "put\<^bsub>x\<^esub> \<sigma> u = put\<^bsub>x\<^esub> \<rho> v" shows "u = v"
exec_rem_consdless: fixes s :: "'a \<Rightarrow> 'b option"   and ops :: "('a, 'b) sas_plus_operator list" shows "execute_serial_plan_sas_plus s (rem_condless_ops s ops) = execute_serial_plan_sas_plus s ops"
strong_release_and_right_distrib: fixes w :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltln"   and \<psi>\<^sub>1 :: "'b ltln"   and \<psi>\<^sub>2 :: "'b ltln" shows "w \<Turnstile>\<^sub>n \<phi> M\<^sub>n (\<psi>\<^sub>1 and\<^sub>n \<psi>\<^sub>2) = w \<Turnstile>\<^sub>n \<phi> M\<^sub>n \<psi>\<^sub>1 and\<^sub>n \<phi> M\<^sub>n \<psi>\<^sub>2"
is_unit_diagonal_U: shows "\<forall>i. is_unit (U $ i $ i)"
card_of_Times_cong2: fixes A :: "'d set"   and B :: "'e set"   and C :: "'f set" assumes "|A| =o |B|" shows "|C \<times> A| =o |C \<times> B|"
emptyE: fixes a :: "'b"   and P :: "bool" assumes "a \<in> {}" shows "P"
rel_\<I>I: fixes C :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and \<I>1 :: "('d, 'f) \<I>"   and \<I>2 :: "('e, 'g) \<I>"   and R :: "'f \<Rightarrow> 'g \<Rightarrow> bool" assumes "rel_set C (outs_\<I> \<I>1) (outs_\<I> \<I>2)"   and "\<And>x y. C x y \<Longrightarrow> rel_set R (responses_\<I> \<I>1 x) (responses_\<I> \<I>2 y)" shows "rel_\<I> C R \<I>1 \<I>2"
content_cball_pos: fixes r :: "real"   and c :: "'b" assumes "0 < r" shows "0 < content (cball c r)"
Span_append: fixes ms :: "'v list"   and ns :: "'v list" shows "RSpan (ms @ ns) = RSpan ms + RSpan ns"
INF_lower2: fixes i :: "'b"   and A :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and u :: "'c" assumes "i \<in> A"   and "f i \<le> u" shows "\<Sqinter>(f ` A) \<le> u"
HC_intros_10: fixes F :: "'b formula"   and G :: "'b formula" shows "AX10 \<turnstile>\<^sub>H (F::'b formula) \<^bold>\<rightarrow> (G::'b formula) \<^bold>\<rightarrow> F \<^bold>\<and> G"
Fubini': fixes f :: "'b \<Rightarrow> 'a \<Rightarrow> ennreal" assumes "(\<lambda>(x, y). f x y) \<in> borel_measurable (N \<Otimes>\<^sub>M M)" shows "\<integral>\<^sup>+ y. \<integral>\<^sup>+ x. f x y \<partial>N \<partial>M = \<integral>\<^sup>+ x. integral\<^sup>N M (f x) \<partial>N"
ptensor_mat_def: fixes m1 :: "'b mat"   and m2 :: "'b mat" shows "ptensor_mat m1 m2 = tensor_mat dims0 vars1' m1 m2"
hd_stack_tree_path_finished: fixes v :: "'v" assumes "stack s \<noteq> []"   and "(hd (stack s), v) \<in> (tree_edges s)\<^sup>+" shows "v \<in> dom (finished s)"
is_operator_applicable_in_def: fixes s :: "'b \<Rightarrow> bool option"   and op :: "'b strips_operator" shows "is_operator_applicable_in s op \<equiv> Let (precondition_of op) (list_all (\<lambda>v. s v = Some True))"
assms_7: shows "(c\<^sub>0, c\<^sub>1) = combine (p\<^sub>0\<^sub>L, p\<^sub>1\<^sub>L) (p\<^sub>0\<^sub>R, p\<^sub>1\<^sub>R) l ps"
1: shows "d\<^sub>\<kappa> (\<^bold>\<iota>x. \<phi> x) = d\<^sub>\<kappa> (\<^bold>\<iota>x. \<psi> x)"
finsum_empty: fixes f :: "'c \<Rightarrow> 'a vec" shows "finsum V f {} = 0\<^sub>v n"
simps_1: fixes x1 :: "'b"   and x2 :: "'b stream"   and y1 :: "'b"   and y2 :: "'b stream" shows "(x1 ## x2 = y1 ## y2) = (x1 = y1 \<and> x2 = y2)"
simps_25: fixes va :: "\<I>"   and vb :: "'b formula" shows "safe_formula (Neg (Prev va vb)) = False"
assms_2: shows "f1 ++ f2 \<noteq> f2"
rootE: fixes p :: "'a poly" assumes "0 < degree p" obtains z :: "'a" where "poly p z = (0::'a)"
telescope_sums': fixes f :: "nat \<Rightarrow> 'b"   and c :: "'b" assumes "f \<longlonglongrightarrow> c" shows "(\<lambda>n. f n - f (Suc n)) sums (f 0 - c)"
valid_init_mmuaux: fixes L :: "nat set"   and R :: "nat set"   and I :: "\<I>"   and n :: "nat"   and b :: "bool" assumes "L \<subseteq> R" shows "valid_mmuaux (init_args I n L R b) 0 (init_mmuaux (init_args I n L R b)) []"
carrierI: fixes T :: "'b set set"   and t :: "'b set"   and x :: "'b" assumes "t open\<^bsub>T\<^esub>"   and "x \<in> t" shows "x \<in> carrier\<^bsub>T\<^esub>"
new_tree_order: shows "h' \<turnstile> to_tree_order (cast\<^sub>c\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>_\<^sub>d\<^sub>a\<^sub>t\<^sub>a\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r new_character_data_ptr) \<rightarrow>\<^sub>r new_tree_order"
insert_3: shows "sum (rep_nat \<circ> f) A = rep_nat (sum f A)"
Liminf_eq_Limsup: fixes F :: "'b filter"   and f :: "'b \<Rightarrow> 'c"   and f0 :: "'c" assumes "F \<noteq> bot"   and "Liminf F f = f0"   and "Limsup F f = f0" shows "(f \<longlongrightarrow> f0) F"
seqI: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b"   and g :: "'b" assumes "category C"   and "partial_magma.arr C f"   and "partial_magma.arr C g"   and "partial_magma.dom C g = partial_magma.cod C f" shows "partial_magma.arr C (C g f)"
assms: shows "nop_onto A (1\<^sub>\<nat>) f"
vimage_ident: fixes Y :: "'a set" shows "(\<lambda>x. x) -` Y = Y"
continuous_on_s: fixes a :: "real"   and v :: "real"   and T :: "real set"   and s0 :: "real" assumes "braking_movement a v" shows "continuous_on T (movement.s a v s0)"
assms_2: shows "S \<sqsubseteq> (\<lceil>\<not> b\<rceil>\<^sub>< \<and> Q)"
connected_refl: fixes v :: "'a" assumes "v \<in> V" shows "v \<rightarrow>\<^sup>* v"
nat_pow_closed: fixes x :: "nat \<Rightarrow> 'a"   and n :: "nat" assumes "x \<in> carrier P" shows "[n] \<cdot>\<^bsub>P\<^esub> x \<in> carrier P"
fa_step: fixes q :: "'a"   and \<delta> :: "('a, 'b) ta_rule set"   and Q0 :: "'a set"   and l :: "'b"   and qs :: "'a list"   and q' :: "'a" assumes "q \<in> f_accessible_alt \<delta> Q0"   and "q \<rightarrow> l qs \<in> \<delta>"   and "q' \<in> set qs" shows "q' \<in> f_accessible_alt \<delta> Q0"
SLC_commute: fixes s\<^sub>3 :: "'r \<Rightarrow> (('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('r, 'l, 'v) expr) option"   and s\<^sub>3' :: "'r \<Rightarrow> (('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('r, 'l, 'v) expr) option"   and r' :: "'r"   and s\<^sub>2 :: "'r \<Rightarrow> (('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('r, 'l, 'v) expr) option"   and r :: "'r"   and s\<^sub>2' :: "'r \<Rightarrow> (('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('r, 'l, 'v) expr) option" assumes "s\<^sub>3 = s\<^sub>3'"   and "revision_step r' s\<^sub>2 s\<^sub>3"   and "revision_step r s\<^sub>2' s\<^sub>3'" shows "s\<^sub>3 \<approx> s\<^sub>3' \<and> (revision_step r' s\<^sub>2 s\<^sub>3 \<or> s\<^sub>2 = s\<^sub>3) \<and> (revision_step r s\<^sub>2' s\<^sub>3' \<or> s\<^sub>2' = s\<^sub>3')"
IH1: fixes s :: "'a net_state" assumes "s \<in> init (pnet np p1)"   and "netmap s i = Some v" shows "v \<in> init (np i)"
induct: fixes x1 :: "Var \<Rightarrow> L"   and x2 :: "BExpr"   and x3 :: "L"   and P :: "(Var \<Rightarrow> L) \<Rightarrow> BExpr \<Rightarrow> L \<Rightarrow> bool" assumes "(x1, x2, x3) \<in> HS_B"   and "\<And>G e1 p1 e2 p2 p f. \<lbrakk>(G, e1, p1) \<in> HS_E; (G, e2, p2) \<in> HS_E; p = LUB p1 p2\<rbrakk> \<Longrightarrow> P G (compB f e1 e2) p"   and "\<And>G b p q. \<lbrakk>(G, b, p) \<in> HS_B; P G b p; LEQ p q\<rbrakk> \<Longrightarrow> P G b q" shows "P x1 x2 x3"
simps_5: fixes r :: "'b mrexp3"   and s :: "'b mrexp3"   and uw :: "'b set"   and ux :: "bool" shows "strip3 (Times3 r s uw ux) = Times (strip3 r) (strip3 s)"
hcomplex_i_mult_eq: shows "iii * iii = - 1"
init_fin_\<tau>diverge_RunningI: fixes t :: "'t"   and x :: "'x"   and m :: "'m" assumes "\<tau>diverge t (x, m)" shows "\<tau>trsys.\<tau>diverge (init_fin t) init_fin_\<tau>move ((Running, x), m)"
meet_isor: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<le> y" shows "z \<cdot> x \<le> z \<cdot> y"
upOne_def: fixes n :: "int" shows "upOne n = n + 1"
next_inI: fixes A :: "'c set"   and a :: "'c"   and b :: "'c"   and u :: "'c"   and v :: "'c" assumes "finite A"   and "A \<subseteq> {a..b}"   and "a \<in> A"   and "b \<in> A"   and "a < b"   and "u \<in> {a..<b}"   and "v \<in> A"   and "u < v"   and "{u<..<v} \<inter> A = {}" shows "next_in A u = v"
simps: fixes ant :: "('a, 'b) form multiset"   and suc :: "('a, 'b) form multiset" shows "succ ( ant \<Rightarrow>* suc) = suc"
sim_while_wlp_aux: fixes b :: "bexp"   and b' :: "bexp"   and \<pi> :: "char list \<Rightarrow> com option"   and c :: "com"   and c' :: "com"   and s :: "char list \<Rightarrow> int \<Rightarrow> int"   and t :: "char list \<Rightarrow> int \<Rightarrow> int" assumes "bval b = bval b'"   and "wlp \<pi> c = wlp \<pi> c'"   and "\<pi>: (WHILE b DO c, s) \<Rightarrow> t" shows "\<pi>: (WHILE b' DO c', s) \<Rightarrow> t"
induct: fixes x1 :: "char list \<Rightarrow> Types.ty"   and x2 :: "bexp"   and P :: "(char list \<Rightarrow> Types.ty) \<Rightarrow> bexp \<Rightarrow> bool" assumes "x1 \<turnstile> x2"   and "\<And>\<Gamma> v. P \<Gamma> (Bc v)"   and "\<And>\<Gamma> b. \<lbrakk>\<Gamma> \<turnstile> b; P \<Gamma> b\<rbrakk> \<Longrightarrow> P \<Gamma> (bexp.Not b)"   and "\<And>\<Gamma> b1 b2. \<lbrakk>\<Gamma> \<turnstile> b1; P \<Gamma> b1; \<Gamma> \<turnstile> b2; P \<Gamma> b2\<rbrakk> \<Longrightarrow> P \<Gamma> (And b1 b2)"   and "\<And>\<Gamma> a1 \<tau> a2. \<lbrakk>\<Gamma> \<turnstile> a1 : \<tau>; \<Gamma> \<turnstile> a2 : \<tau>\<rbrakk> \<Longrightarrow> P \<Gamma> (Less a1 a2)" shows "P x1 x2"
map_comp_def: fixes f :: "'f \<Rightarrow> 'e option"   and g :: "'d \<Rightarrow> 'f option" shows "f \<circ>\<^sub>m g = (\<lambda>k. case g k of None \<Rightarrow> None | Some v \<Rightarrow> f v)"
phl_skip: fixes x :: "'a" shows "x ; 1' \<le> 1' ; x"
quot_one_subset: shows "UNIV // \<approx>{[]} \<subseteq> {{[]}, UNIV - {[]}}"
monoid_axioms: shows "Group_Theory.monoid stfx.carrier_direct_lim stfx.mult_rel (stfx.class_of V (one_str\<^sub>Y V))"
comp_eq_dest_lhs: fixes a :: "'o \<Rightarrow> 'n"   and b :: "'m \<Rightarrow> 'o"   and c :: "'m \<Rightarrow> 'n"   and v :: "'m" assumes "a \<circ> b = c" shows "a (b v) = c v"
minus_perm_def: fixes p1 :: "'a perm"   and p2 :: "'a perm" shows "p1 - p2 = p1 + - p2"
llength_llist_of_tllist: fixes xs :: "('a, 'b) tllist" shows "llength (llist_of_tllist xs) = tlength xs"
nullspace_inter_rowspace: fixes A :: "((real, 'c) vec, 'b) vec"   and x :: "(real, 'c) vec" shows "(A *v x = 0 \<and> x \<in> span (rows A)) = (x = 0)"
update_until_eq: shows "update_until args rel1 rel2 nt auxlist = map (\<lambda>x. case x of (t, a1, a2) \<Rightarrow> (t, if pos then join a1 True rel1 else a1 \<union> rel1, if mem (nt - t) I then a2 \<union> join rel2 pos a1 else a2)) auxlist @ [(nt, rel1, if left I = 0 then rel2 else empty_table)]"
abs_pred_trm_def: fixes P :: "'a trm \<Rightarrow> bool"   and t :: "'a trm" shows "abs_pred P t = True"
sep_impl_sep_true_P: fixes P :: "'a \<Rightarrow> bool"   and h :: "'a" assumes "((\<lambda>s. True) \<longrightarrow>* P) h" shows "P h"
delta_def: shows "delta = \<bar>(u - real_of_rat x) / 2\<bar>"
subgroupI: fixes H :: "'a set" assumes "H \<subseteq> carrier G"   and "H \<noteq> {}"   and "\<And>a. a \<in> H \<Longrightarrow> inv a \<in> H"   and "\<And>a b. \<lbrakk>a \<in> H; b \<in> H\<rbrakk> \<Longrightarrow> a \<otimes> b \<in> H" shows "H \<le> G"
IH: fixes x :: "nat" shows "eval (r_contains xs) [x] \<down>= (if x \<in> set xs then 1 else 0)"
det_identity: fixes t :: "real \<times> real"   and p :: "real \<times> real"   and q :: "real \<times> real"   and s :: "real \<times> real"   and r :: "real \<times> real" shows "det3 t p q * det3 t s r + det3 t q r * det3 t s p + det3 t r p * det3 t s q = 0"
eq_invI: fixes f :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "bijection f"   and "inv f a = inv f b" shows "a = b"
integrable2_aux: fixes x :: "real"   and z :: "complex"   and y :: "real" assumes "- 2 < x"   and "Re z \<in> {x..y}" shows "complex_interval_lebesgue_integrable lborel (ereal (exp (- 1))) 1 (f z)"
path_is_trancl: fixes E :: "('a \<times> 'a) set"   and u :: "'a"   and l :: "'a list"   and v :: "'a" assumes "path E u l v"   and "l \<noteq> []" shows "(u, v) \<in> E\<^sup>+"
permutes_subset: fixes p :: "'b \<Rightarrow> 'b"   and S :: "'b set"   and T :: "'b set" assumes "p permutes S"   and "S \<subseteq> T" shows "p permutes T"
length_list_of_vec: fixes v :: "'a vec" shows "length (list_of_vec v) = dim_vec v"
induct: fixes S :: "'c set set"   and x :: "'c set"   and P :: "'c set \<Rightarrow> bool" assumes "generate_topology_on S x"   and "P {}"   and "\<And>a b. \<lbrakk>generate_topology_on S a; P a; generate_topology_on S b; P b\<rbrakk> \<Longrightarrow> P (a \<inter> b)"   and "\<And>K. \<lbrakk>\<And>k. k \<in> K \<Longrightarrow> generate_topology_on S k; \<And>k. k \<in> K \<Longrightarrow> P k\<rbrakk> \<Longrightarrow> P (\<Union> K)"   and "\<And>s. s \<in> S \<Longrightarrow> P s" shows "P x"
SC_discr_imp_SC_WbisT: fixes c :: "('test, 'atom) com" assumes "noWhile c"   and "SC_discr c" shows "SC_WbisT c"
ssubst: fixes t :: "'e"   and s :: "'e"   and P :: "'e \<Rightarrow> bool" assumes "t = s"   and "P s" shows "P t"
div_result: fixes instr :: "sparc_operation \<times> inst_operand list"   and rs1 :: "word5"   and rd :: "word5"   and operand2 :: "word32"   and s :: "'b sparc_state" shows "snd (div_comp instr rs1 rd operand2 s) = False"
psubset_eq: fixes A :: "'e set"   and B :: "'e set" shows "(A \<subset> B) = (A \<subseteq> B \<and> A \<noteq> B)"
map_def: fixes gf :: "'d \<times> 'd" shows "HomD.map gf = (if DopxD.arr gf then S.mkArr (HomD.set (DopxD.dom gf)) (HomD.set (DopxD.cod gf)) (\<phi>D (DopxD.cod gf) \<circ> (\<lambda>h. snd gf \<cdot>\<^sub>D h \<cdot>\<^sub>D fst gf) \<circ> \<psi>D (DopxD.dom gf)) else S.null)"
strict_implies_not_eq: fixes a :: "'b list"   and b :: "'b list" assumes "strict_suffix a b" shows "a \<noteq> b"
is_Zp_continuousI: fixes f :: "(nat \<Rightarrow> int) \<Rightarrow> nat \<Rightarrow> int" assumes "f \<in> carrier (function_ring (carrier Zp) Zp)"   and "\<And>s. is_Zp_cauchy s \<Longrightarrow> is_Zp_cauchy (f \<circ> s)" shows "is_Zp_continuous f"
taut_test_rec: fixes t :: "'d ifex"   and env :: "('d, bool) mapping" shows "taut_test_rec t env = (\<forall>s. agree s env \<longrightarrow> val_ifex t s)"
Lin2: assumes "Lin t t' \<and> s = s'"   and "a = Linearize id1" shows "thesis"
\<beta>s: shows "\<Theta>,\<Gamma> \<turnstile> mk_eq (Abs_fv x \<tau>' s $ Fv x \<tau>') (subst_bv (Fv x \<tau>') (bind_fv (x, \<tau>') s))"
ff_list_pairs: fixes X :: "'b fract list" shows "\<exists>xs. X = map (\<lambda>(x, y). Fraction_Field.Fract x y) xs \<and> (0::'b) \<notin> snd ` set xs"
hd_exp_times: fixes xs :: "('a \<times> real) msllist"   and ys :: "('a \<times> real) msllist" shows "ms_aux_hd_exp (times_ms_aux xs ys) = (case (ms_aux_hd_exp xs, ms_aux_hd_exp ys) of (None, b) \<Rightarrow> None | (Some x, None) \<Rightarrow> None | (Some x, Some y) \<Rightarrow> Some (x + y))"
get_child_nodes_pure: fixes ptr :: "(_) object_ptr"   and h :: "(_) heap" shows "pure (get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M ptr) h"
gcd_dvd1: fixes a :: "'b"   and b :: "'b" shows "normalization_euclidean_semiring_class.gcd a b dvd a"
can_in_hom: fixes t :: "'a E.term"   and u :: "'a E.term" assumes "E.Ide t"   and "E.Ide u"   and "\<^bold>\<lfloor>t\<^bold>\<rfloor> = \<^bold>\<lfloor>u\<^bold>\<rfloor>" shows "\<guillemotleft>B.can u t : EVAL t \<Rightarrow>\<^sub>B EVAL u\<guillemotright>"
monad_state_stateT: fixes return :: "'b \<times> 'c \<Rightarrow> 'd"   and bind :: "'d \<Rightarrow> ('b \<times> 'c \<Rightarrow> 'd) \<Rightarrow> 'd" assumes "monad return bind" shows "monad_state (return_state return) (bind_state bind) get_state put_state"
N1C: shows "E' `` N1 \<subseteq> N1"
split_list_first_unique: fixes u\<^sub>1 :: "'a list"   and a :: "'a"   and u\<^sub>2 :: "'a list"   and v\<^sub>1 :: "'a list"   and v\<^sub>2 :: "'a list" assumes "u\<^sub>1 @ [a] @ u\<^sub>2 = v\<^sub>1 @ [a] @ v\<^sub>2"   and "a \<notin> set u\<^sub>1"   and "a \<notin> set v\<^sub>1" shows "u\<^sub>1 = v\<^sub>1"
Taylor: shows "((\<lambda>xa. (t + h - xa) *\<^sub>R f' (xa, x xa) (1, f xa (x xa))) has_integral x (t + h) - (x t + h *\<^sub>R f t (x t))) {t..t + h}"
map_1: shows "G\<^sub>2 = map_ugraph f\<^sub>1 G\<^sub>1"
Image_mono: fixes r' :: "('e \<times> 'f) set"   and r :: "('e \<times> 'f) set"   and A' :: "'e set"   and A :: "'e set" assumes "r' \<subseteq> r"   and "A' \<subseteq> A" shows "r' `` A' \<subseteq> r `` A"
ordinal_numberTr2: fixes D :: "'c Order"   and x :: "'c Order set" assumes "Worder D"   and "x = ordinal_number D" shows "D \<in> x"
simps_3: fixes f1 :: "'a"   and f2 :: "'a" shows "(case SEND of SEND \<Rightarrow> f1 | RECV \<Rightarrow> f2) = f1"
cong: fixes w :: "int \<Rightarrow> int \<Rightarrow> nat" shows "Optimal_BST.min_wpl w \<equiv> Optimal_BST.min_wpl w"
boundedByInsert: fixes N :: "nat"   and n :: "nat"   and delta :: "'a"   and B :: "(nat \<times> 'a) set" shows "boundedBy N (insert (n, delta) B) = (n < N \<and> boundedBy N B)"
ipoly_root_delta_def: fixes p :: "int poly" shows "ipoly_root_delta p = Min (insert 1 {\<bar>x - y\<bar> |x y. poly (real_of_int_poly p) x = 0 \<and> poly (real_of_int_poly p) y = 0 \<and> x \<noteq> y}) / 4"
sim_first2: fixes s :: "nat \<Rightarrow> 'b"   and t :: "nat \<Rightarrow> 'b" assumes "s \<approx> t" shows "s 0 = t 0"
composition_closed: fixes a :: "nat"   and b :: "nat" assumes "a \<in> {0}"   and "b \<in> {0}" shows "0 \<in> {0}"
nsg_def: fixes G :: "('c, 'd) Group_scheme"   and H :: "'c set" shows "G \<triangleright> H = (G \<guillemotright> H  \<and> (\<forall>x\<in>carrier G. H \<bullet>\<^bsub>G\<^esub> x = x \<diamondsuit>\<^bsub>G\<^esub> H))"
traverse_B_B: fixes f :: "'c \<Rightarrow> ('d, 'e) state" shows "traverse_rev_rev (B \<circ> B \<circ> f) = B \<circ> B \<circ> traverse f"
natoms: fixes k :: "'a" assumes "k \<in> atoms F" shows "k \<notin> fst ` set (tseytin_assmt F)"
char_one: shows "principal_char (G\<lparr>carrier := adjoin G H a\<rparr>) \<one>\<^bsub>G\<lparr>carrier := adjoin G H a\<rparr>\<^esub> = 1"
coclop_Sup_closed: fixes f :: "'b \<Rightarrow> 'b" assumes "coclop f" shows "Sup_closed_set (Fix f)"
subprob_space_measure_pmf: fixes x :: "'b pmf" shows "subprob_space (measure_pmf x)"
sel_2: fixes x1 :: "'b match_expr"   and x2 :: "action" shows "get_action (Rule x1 x2) = x2"
wellformed_items_\<pi>: fixes I :: "item set"   and T :: "(symbol \<times> character list) set"   and k :: "nat" assumes "wellformed_items I"   and "T \<subseteq> \<X> k" shows "wellformed_items (\<pi> k T I)"
summable_mono_reindex: fixes g :: "nat \<Rightarrow> nat"   and f :: "nat \<Rightarrow> 'a" assumes "strict_mono g"   and "\<And>n. n \<notin> range g \<Longrightarrow> f n = (0::'a)" shows "summable (\<lambda>n. f (g n)) = summable f"
DirProds_order: fixes I :: "'c set"   and G :: "'c \<Rightarrow> ('d, 'e) monoid_scheme" assumes "finite I" shows "order (DirProds G I) = prod (order \<circ> G) I"
z_neq_minf: fixes z :: "int" shows "ant z \<noteq> - \<infinity>"
outer_measure_agrees: fixes f :: "'a set \<Rightarrow> ennreal"   and s :: "'a set" assumes "positive M f"   and "countably_additive M f"   and "s \<in> M" shows "outer_measure M f s = f s"
newleft_lemma: fixes p :: "nat"   and r :: "nat"   and a :: "nat" shows "rec_exec rec_newleft [p, r, a] = newleft p r a"
beta_into_beta_eta_reds: fixes s :: "dB"   and t :: "dB" assumes "s \<rightarrow>\<^sub>\<beta> t" shows "s \<rightarrow>\<^sub>\<beta>\<^sub>\<eta>\<^sup>* t"
bspec: fixes A :: "'a set"   and P :: "'a \<Rightarrow> bool"   and x :: "'a" assumes "\<forall>x::'a\<in>A::'a set. (P::'a \<Rightarrow> bool) x"   and "(x::'a) \<in> (A::'a set)" shows "(P::'a \<Rightarrow> bool) (x::'a)"
d_\<kappa>_Ints: fixes i :: "nat"   and l :: "nat"   and t :: "nat" assumes "i < length fs"   and "l \<le> i"   and "t < l" shows "d l * \<kappa> i l t \<in> \<int>"
Smith_2x2_eucl_def: fixes A :: "(('b, 2) Finite_Cartesian_Product.vec, 2) Finite_Cartesian_Product.vec" shows "Smith_2x2_eucl A = (let (P, D, Q) = diagonalize_2x2 A; (P', S, Q') = diagonal_to_Smith_PQ D euclid_ext2 in (P' ** P, S, Q ** Q'))"
reachable_def: fixes v :: "nat" shows "reachable v \<equiv> v \<in> (g_E G)\<^sup>* `` {0}"
isom_gch_units_transpTr7: fixes E :: "('e, 'f) Group_scheme"   and i :: "'g"   and n :: "'g"   and j :: "'g"   and g :: "'g \<Rightarrow> ('h, 'i) Group_scheme"   and h :: "'g \<Rightarrow> ('j, 'k) Group_scheme" assumes "Ugp E"   and "i \<le> n"   and "j \<le> n"   and "g j \<cong> h i"   and "Group (h i)"   and "Group (g j)"   and "\<not> g j \<cong> E" shows "\<not> h i \<cong> E"
read_buffer_typed: fixes \<Gamma> :: "Eg1.var \<Rightarrow> bexp set option"   and \<S> :: "Eg1.var set \<times> Eg1.var set"   and P :: "bexp set" assumes "sifum_example"   and "\<Gamma> = Map.empty"   and "\<S> = ({}, {})"   and "P = {}" shows "\<exists>\<Gamma>' \<S>' P'. has_type \<Gamma> \<S> P read_buffer \<Gamma>' \<S>' P'"