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The lemma named "operative_approximableI" is defined with a fixed function 'f' that maps from a Euclidean space 'b' to a Banach space 'a'. It assumes that 'e' is greater than or equal to zero. The lemma shows that the operation 'conj' with the value 'True' is operative on 'i' if and only if there exists a function 'g' ... |
If a natural number 'a' is less than another natural number 'b', and the modulus of 'a' with respect to 'm' is greater than or equal to the modulus of 'b' with respect to 'm', then the quotient of 'a' divided by 'm' is less than the quotient of 'b' divided by 'm'. |
A real number 'x' is in the set of solutions to the depressed quartic equation (an equation of the form x^4 + px^2 + qx + r = 0) if and only if it satisfies the equation. |
The lemma named "col_conga_col" assumes that points A, B, and C are collinear and the angle ABC is congruent to the angle DEF. It shows that points D, E, and F are also collinear. |
The ideal of B equals the universal set if and only if the number 1 is in the ideal of B. |
If x is in HFinite and y is in HFinite, then the sum of x and y is also in HFinite. |
The lemma named "find_fas_init_correct" assumes that "G" is a graph and the set obtained by applying the reflexive transitive closure of the edge relation of "G" to the initial vertices of "G" is finite. It shows that the function "find_fas_init" applied to "G" and "FI" is less than or equal to the specification functi... |
The theorem named "lehmers_theorem'" is defined with a fixed natural number 'p'. It assumes that all elements in the list 'ps' are prime numbers, 'a' is equivalent to 'a', and 'n' is equivalent to 'n'. It also assumes that for all elements 'p' in the list 'ps', the modular exponentiation of 'a' and the quotient of 'n -... |
In the context of an infinite coin toss space, the lemma named "pseudo_proj_True_shift" shows that if the length of list 'l' is equal to 'n', then the pseudo projection of 'n' on the shifted list 'l' with constant 'True' is equal to the shifted list 'l' with constant 'True'. |
The category is small cocomplete if for any two morphisms from an object to another object, there exists a coequalizer in the category, and for any discrete diagram in the category, there exists a coproduct in the category. |
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Assuming the following conditions:
1. The administrative agreement sets of the free variables (FV) excluding 'n' and the set 'S' excluding 'n' with respect to 'X', 'σ', and 'τ' are in agreement.
2. The set 'S' is a subset of the free variables (FV).
3. The set 'S' is finite.
4. The image of the function 'τ' applied t... |
The lemma named "inputFreeTrans" is defined with fixed variables: "a" and "x" of type name, "P" and "P'" of type pi, and "α" of type freeRes. It assumes that there is a transition from the input prefix process "a<x>.P" to the process "P'" with the action "α". The lemma then shows that this assumption leads to a contrad... |
The function 'can_select' applied to a predicate 'P' and the set of elements in the list 'xs' is equivalent to checking if there is an element in the list that satisfies the predicate 'P'. If there is no comparison function for the elements of the list, the code will abort with an error message. If there is a compariso... |
The theorem I_qe_FM1:
Assuming that for every 'a' in the set 'as', 'a' is less, and for every 'a' in the set 'as', 'a' depends on 'R'. It shows that 'R.I' of 'qe_FM1 as' and 'xs' is equal to there exists an 'x' such that for every 'a' in the set 'as', 'I_R' of 'a' and the list consisting of 'x' and 'xs' holds true. Thi... |
Applying the function "of_nat" to any number "n" and any element "x" is equivalent to simply applying "of_nat" to the number "n". |
The least common multiple of two polynomial ordered association lists, "xs" and "ys", is equal to a polynomial ordered association list that is the result of mapping a function that returns the maximum of two values over "xs" and "ys". This is applicable for "xs" and "ys" that are of the type "oalist_tc", where the key... |
If a property P is a subset of the infinite power of A, then the safety of A and P is equivalent to P being closed. |
The lemma named "part_insert" assumes two conditions: first, that "P" is a partition of "A", and second, that the intersection of "X" and "A" is empty. Under these assumptions, it shows that "insert X P" is a partition of the union of "A" and "X". |
The lemma named "compare_2" assumes that the variables "x" and "y" both satisfy the condition "invariant_2". It shows that the result of the function "compare_2" with "x" and "y" as arguments is equal to the result of the function "compare" with the real values of "x" and "y" as arguments. |
The lemma "no_vs_change_valid_in_snapshot" assumes that "as" is in the valid plans of "PROB", "as" satisfies the precondition in state "s", and the variables that change in "as" with respect to "vs" and "s" is an empty list. Under these assumptions, it shows that "as" is in the valid plans of the snapshot of "PROB" res... |
If the word representation of a natural number 'r' is equal to 'a', and 'r' is less than 2 raised to the length of 'a', then the unsigned natural number representation of 'a' is 'r'. This is applicable for 'a' which is a word of any length. |
: given two sets A and B that are small, a function π such that for any two elements x and y in A, if (x,y) is in relation r, then the pair (πx, πy) is in relation s, and given that r is a well-founded relation and total on A, s is a well-founded and transitive relation, and the image of A under π is a subset of B, it ... |
If a list 'xs' contains distinct elements and the length of 'xs' is equal to the length of another list 'bs', then the cardinality (or size) of the set obtained by filtering the list 'xs' based on the boolean values in 'bs' is equal to the count of 'True' values in the list 'bs'. |
The lemma named "Fun_canonical" assumes that an empty list leads "v" to a function from "T1" to "T2". It shows that if "v" is a valid value, then there exist "x", "t", and "S" such that "v" is equal to a function that maps "x" to "t" in the context of "S". |
The amalgamation of i and j is in the set P. |
:
- If xi' is in the packet xi, then the identifier of xi' is equal to the identifier of xi.
- If xi' is in the new packet xi, then the identifier of xi' is equal to the identifier of xi. |
If the value of the function "three_step" applied to "r" is greater than 0, then the value of the function "three_step" applied to "r" must be either 1 or 2. |
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Assuming the system is consistent, and given two conditions HBL4 and WHBL2:
HBL4: For any two formulas φ1 and φ2, if both φ1 and φ2 are subsets of the set of all formulas and neither φ1 nor φ2 have any free variables, then if the conjunction of the provability predicates of φ1 and φ2 is provable, then the provabili... |
The star of x times x is less than or equal to the star of x. |
The hypercomplex number derived from a complex numeral is equal to the hypercomplex number derived from a hyperreal numeral. |
The lemma named "Psi o in hom" assumes that "yx" is an identity in the category "DopxC". It shows that the mapping of "yx" under "Psi o" is a morphism from the mapping of "yx" under "Hom_DopxG" to the mapping of "yx" under "Hom_FopxC" in the category "S". |
If a relation R is symmetric, then the relation on options of R is also symmetric. |
The step 1 of protocol m1a refines the step 1 of protocol m1x under the relation R1x1a. |
An empty multiset is valid in the flow graph. |
The lemma named "is_density_exprD" makes the following assumptions and shows the following results:
Assumption:
An expression 'e' is a density expression with respect to the variables 'vs', the shifted variables 'vs'', the context 'Γ', and the density 'δ'.
Results:
1. The expression 'e' under the context 'Γ' shifted... |
The lemma named "swap_registers" assumes that registers R and S are compatible. It shows that the product of register R with a and register S with b is equal to the product of register S with b and register R with a. |
If x is in the result set of the generative probabilistic value (gpv) restricted by \<I>', then x is also in the result set of the original generative probabilistic value (gpv). |
If a property P is in a set S and every property in set S is non-negative, then the infimum (greatest lower bound) of the expected values of set S is less than or equal to property P. |
If the length of the list "xs" is equal to "vl", then the recursive execution of the function "rec_strt" with "vl" as an argument on the list "xs" is equal to the result of the function "strt" on the list "xs". |
Applying the function "hensel_fpxs2" to 0 results in "h". |
The length of the list obtained by taking elements from the start of list 'xs' while condition 'P' is true, is less than or equal to the length of the original list 'xs'. |
The relation "mod_ring_rel" holds true for the pair (1, 1). |
The property of being founded for a subset P of the increment level image of set A is equivalent to the property of being founded for the subset P of set A itself. |
Taking the first 'n' elements from a list 'a' results in a prefix of the original list 'a'. |
The truncation of x rounded up to p decimal places is greater than or equal to zero if and only if x is greater than or equal to zero. |
If the cast shadow root pointer is not equal to the document pointer, then getting the shadow root from the shadow root pointer after putting the document into the document pointer and the document heap is the same as getting the shadow root from the shadow root pointer and the heap. |
The diamond operation applied to the join (or sup, supremum) of x and y, and then directed towards z, is equal to the join of the diamond operation applied separately to x and z, and y and z. |
Subtracting an empty list from any list "xs" results in the original list "xs". |
The lemma named "zmult_zless_mono2" establishes that for any three integers i, j, and k, if i is less than j and k is greater than 0, then the product of k and i is less than the product of k and j. |
An empty list remains empty even after applying a substitution operation on it. |
The lemma assumes that the composition of the functions 'wrap' and 'unwrap' equals the identity function. It then shows that the composition of 'wrap' and the strictified version of 'unwrap' also equals the identity function. |
If 'x' is co-total, then the dual of 'x' is total. |
The lemma named "tensor_mat_zero1" shows that the tensor product of a zero matrix of dimensions d1 by d1 and another matrix m1 equals a zero matrix of dimensions d by d. |
The lemma named "Borsukian_separation_compact" is about a set S of complex numbers. It assumes that the set S is compact. It shows that the set S is Borsukian if and only if the complement of S is connected. |
If "n" is not equal to zero, then the length of the result of the function "bin_rsplit" with parameters "n" and a pair consisting of "nw" and "w", is equal to the integer division of "(nw + n - 1)" by "n". |
The head of the trace, given parameters n, f, g, xs, and x, is equal to the evaluation of the primitive recursive function (Pr) with parameters n, f, g, and the list formed by prepending x to xs. |
The inverse of e1 is -e1, the inverse of e2 is -e2, the inverse of e3 is -e3, the inverse of e4 is -e4, the inverse of e5 is -e5, the inverse of e6 is -e6, and the inverse of e7 is -e7. |
Taking "info" from the result of taking "info" from "l" at the "nxt" position and then extracting "info" from "l" at the "nxt" position is equivalent to just taking "info" from "l" at the "nxt" position. |
The power of an assignment "f" to the power of "n" is equal to the assignment of "f" to the power of "n". |
Being confluent is equivalent to being Church-Rosser. |
If the variable "nu" is within the set of numbers greater than 0 and less than or equal to "lambda + mu", and if "tortoise" is equal to the sequence at "mu", and "m" is equal to "mu", then after executing "find_lambda", "nu" will still be within the set of numbers greater than 0 and less than or equal to "lambda + mu",... |
The imaginary unit 'i' is not equal to zero. |
If for all states 's' and outputs 'x' such that 'x' is in the set of outputs of interface 'I' and 'I' holds for state 's', the sub-probability monad function (SPMF) 'callee' is lossless, and if 'I' holds for state 's', then the resource derived from the oracle 'callee' with state 's' is a lossless resource for interfac... |
The lemma "lt_sqn_imp_update_strictly_fresher" assumes that "dip" is in the set of valid destinations of the routing table of "nhip", the old sequence number "osn" is less than the sequence number of "dip" in the routing table of "nhip", and the routing table "rt" is not equal to the updated routing table with "dip", "... |
The lemma named "s_mul_ext_singleton" assumes that the pair (a, b) is in the set s. It then shows that the pair of multisets containing a and b respectively is in the multiset extension of s with respect to ns. This lemma is marked as a simplification rule and can be used to introduce new facts. |
If the well-formed discriminant selector pair (disc, sel) is compatible with a set C, and if a match m is in normalized negation normal form, and if the primitive extractor function applied to the pair (disc, sel) and m results in a pair (as, ms), then the following holds:
If ms is in normalized negation normal form, ... |
In the context of a ring, if 'a' and 'b' are elements of the carrier of the ring, then the product of 'a' and 'b' is equal to the product of the additive inverse of 'a' and the additive inverse of 'b'. |
The lemma named "Fun_ide" assumes that "a" is an identity. It shows that the function of "a" is equal to the restriction of the function that maps "x" to "x" on the set of "a". |
For a transfer rule, the function "put_option" preserves the relation between two option monads when the relation between their states and their monadic values is preserved. This is expressed in terms of a relation between functions and states, and a relation between option monads. |
Applying the "dflt_None_set" function to an empty set results in "None". |
For any set X, if a list 'l' is conformant to 'L' under the context of a program 'G' and a state 's', then for all lists of values 'vs' and types 'Ts', if the variable names 'vns' are distinct and the length of 'Ts' is equal to the length of 'vns', and if each value in 'vs' is conformant to the corresponding type in 'T... |
When looking up a key 'k'' in a list 'xs' that has been updated by a function 'f' at key 'k', the result is as follows: if 'k' is the same as 'k'', then the function 'f' is applied to the lookup of 'k'' in 'xs'; otherwise, the identity function is applied to the lookup of 'k'' in 'xs'. |
If a pair of address and location (ad, al) is in the action location of a program P and execution environment E for an action a, and another pair of address and location (ad', al') is also in the action location of the same program P and execution environment E for the same action a, then the two addresses ad and ad' m... |
If "CNV x y" holds, then x is equal to y. |
If there is a transfer relation 'R' between 'x' and 'y', then there is a transfer relation 'crel_vs R' between 'Wrap x' and 'return y'. |
If x is approximately equal to r, and x is also approximately equal to s, then r is approximately equal to s. |
If "i" is less than "j + 1", then substituting "bv2" in "subst_bvs1'" of "t" with "Suc j" and a map that lifts "v" by "i" in "vs", and then substituting "i" with "subst_bvs1'" of "u" and "j" in "vs", is equal to substituting "bv2" in "t" with "i" and "u", and then substituting "j" with "vs" in "subst_bvs1'". |
The lemma named "Var_Idem" assumes that variable 'v' is not a subterm of 't'. It shows that the pair (v, t) is idempotent. |
If a function 'f' is injective on a set 'A' and 'x' is a subset of the image of 'A' under 'f', then the image of the inverse of 'f' on 'x' under 'f' is equal to 'x'. |
The lemma named "subject_reduction2" assumes two conditions: first, that expression 'e' reduces to expression 'e'', and second, that under context 'Γ', expression 'e' has type 'T'. Given these assumptions, it shows that under the same context 'Γ', the reduced expression 'e'' also has type 'T'. |
The lemma named "weakCongI" in the context of Weak Congruence theory, fixes two psi-calculi expressions P and Q, both of type psi. It assumes that for any psi and substitution sigma, if sigma is a well-formed substitution, then the application of sigma to P is weakly equivalent to the application of sigma to Q. Under t... |
The sum of the real representations of two numbers, r1 and r2, is equal to the real representation of the sum of r1 and r2 if r1 and r2 are compatible according to the "mau_compatible" function. If they are not compatible, the program will abort with the error message "different base", and the sum of the real represent... |
The function 'tsub' that takes two arguments from the set of integers (ZN) and returns an integer (ZN), corresponds to the subtraction operation ("-"). |
If a function 'f' is 'k-smooth' on a set 'S' and 'k' is not equal to zero, then 'f' is differentiable on 'S'. |
Mapping the bit function over a list of integers from 0 to the size of x is equal to the reverse of the binary representation of x. |
The lemma demonstrates respect for the quotient in three instances:
1. It shows that if two pairs are equal, then their corresponding alpha abstraction sets are also equal.
2. It shows that if two pairs are equal, then their corresponding alpha abstraction results are also equal.
3. It shows that if two pairs are equa... |
The error function complement (erfc) of a real number x is less than or equal to 1 if and only if x is greater than or equal to 0. |
The lemma "Can_implies_coherent_iff_coherent_Inv" assumes that "t" is canonical. It shows that "t" is coherent if and only if the inverse of "t" is coherent. |
The mapping of the Bernoulli probability mass function of a probability 'p' is defined as follows:
- If 'p' is less than or equal to 0, then the mapping is updated such that the value for 'False' is 1 and the mapping is otherwise empty.
- If 'p' is greater than or equal to 1, then the mapping is updated such that th... |
If "n" and "d" are even integers, and "d" is greater than or equal to zero, then the remainder of "n + 1" divided by "d" is equal to the remainder of "n" divided by "d" plus one. |
If there does not exist a list "us" such that "us" is a prefix of "xs" and "P us" holds, then "ps" is the maximum prefix of "xs" satisfying "P" if and only if "ps" is an empty list. |
The lemma named "concat_empty_fset" shows that the concatenation of an empty finite set equals an empty finite set. |
The range of the combination of map x and map y is a subset of the union of the range of map x and the range of map y. |
A relation R is bijective if and only if R is single-valued and the inverse of R is also single-valued. |
The frame satisfies the first, third, and fourth frame conditions, and the counterexample condition, and it does not satisfy the McKinsey's condition for negation. |
The multiset obtained by adding an element "s" to a multiset "c" is equal to the multiset obtained by adding an element "r'" to a multiset "c'". The second assumption "CASES" states that if "s" is equal to "r'" and "c" is equal to "c'", then the proposition "P" holds. Alternatively, if "c'" is the sum of the multiset o... |
In the context of an integral domain, if we assume that a monoid is a factorial monoid, then it can be shown that this monoid is a unique factorization domain. This is defined in terms of the multiplication operation, the number 1, the addition operation, the number 0, the subtraction operation, and the unary minus ope... |
"x is equal to y if the undual of x is equal to the undual of y". |
:
1. If the intersection of sets I and J is empty and an element 'i' belongs to set I, then the merge of sets I and J with respect to the pair (x, y) at 'i' equals to x at 'i'.
2. If the intersection of sets I and J is empty and an element 'i' belongs to set J, then the merge of sets I and J with respect to the pair (... |
Adding an empty list to any list "xs" of elements from an additive monoid results in the original list "xs". |
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